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REACHABLE SET CONTROL
FOR
PREFERRED AXIS HOMING MISSILES

















By

DONALD J. CAUGHLIN, JR.


A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN
PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY



UNIVERSITY OF FLORIDA


1988

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Copyright 1988

By

DONALD J. CAUGHLIN JR.

..
















To Barbara

Amy

Jon

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ACKNOWLEDGMENTS

The author wishes to express his gratitude to his committee chairman, Dr. T.E Bullock, for his instruction, helpful suggestions, and encouragement. Appreciation is also expressed for the support and many helpful comments from the other committee members, Dr. Basile, Dr. Couch, Dr. Smith, and Dr. Svoronos.

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TABLE OF CONTENTS

ACK NOW LEDG M ENTS iv

LIST OF FIG U RES vii

K EY TO SYM BOLS ix

ABSTRACT xiv

CHAPTER

I INTRODU CTION I

11 BACK G ROU ND 4
M issile Dynam ics 5
Linear Accelerations 6
M oment Equations 6
Linear Quadratic G aussian Control Law 7

III CONSTRAINED CONTROL 13

IV CONSTRAINED CONTROL WITH UNMODELED SETPOINT AN D PLANT VARIATIONS 25
Linear Optim al Control with U ncertainty and Constraints 31
Control Technique 32 Discussion 36 Procedure 37

V REACH ABLE SET CONTROL EX AM PLE 41
Performance Comparison Reachable Set and LQG Control 41
Sum m ary 54

VI REACHABLE SET CONTROL FOR PREFERRED AXIS HOM ING M ISSILES 55
Acceleration Control 56
System M odel 56
Disturbance M odel 58
Reference M odel 60
Roll Control 62
Definition 62 Controller 66
K alm an Filter 67 Reachable Set Controller 68
Structure 68 Application 72

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VII RESU LTS AND DISCU SSION 76
Simulation 77
Trajectory Param eters 78
Results 78
Determ inistic Results 78
Stochastic Results 81
Conclusions 87
Reachable Set Control 87 Singer M odel 87

APPENDIX

A SIM U LATION RESU LTS 88

B SAM PLED-DATA CONVERSION 94
System M odel 94
Sampled Data Equations 96
System 96
Target Disturbance 98
M inim um Control Reference 99
Summ ary 100

C SAM PLED DATA COST FU NCTIONS 101

D LQG CONTROLLER DECOM POSITION 107

E CONTROLLER PARAM ETERS III
Control Law III
Filter 112

LIST OF REFERENCES 113

BIOGRAPHICAL SK ETCH 117

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LIST OF FIGURES


Figure Page

2.1 Missile Reference System 4

4.1 Feedback System and Notation 28

4.2 Reachable Set Control Objective 33

4.3 Intersection of Missile Reachable Sets Based on
Uncertain Target Motion and Symmetric Constraints 38

4.4 Intersection of Missile Reachable Sets Based on
Uncertain Target Motion and
Unsymmetric Constraints 38 5.1 Terminal Performance of Linear Optimal Control 43

5.2 Initial Acceleration of Linear Optimal Control 43

5.3 Linear Optimal Acceleration vs Time 45

5.4 Linear Optimal Velocity vs Time A5

5.5 Linear Optimal Position vs Time 46

5.6 Unconstrained and Constrained Acceleration 47

5.7 Unconstrained and Constrained Velocity vs Time 48

5.8 Unconstrained and Constrained Position vs Time 48

5.9 Acceleration Profile
With and Without Target Set Uncertainty 50

5.10 Velocity vs Time
With and Without Target Set Uncertainty 50

5.11 Position vs Time
With and Without Target Set Uncertainty 51

5.12 Acceleration vs Time
LQG and Reachable Set Control 52

..







5.13 Velocity vs Time
LQG and Reachable Set Control 53

5.14 Position vs Time
LQG and Reachable Set Control 53

6.1 Reachable Set Control D isturbance processes 60

6.2. Roll A ngle Error Definition from Seeker A ngles 63

6.3. Roll Control Zones 65

6.4 Target M issile System 74

6.5 Com m and G enerator/Tracker 75

7.1 RM S M issile Acceleration 76

7.2 Engagem ent G eom etry 77

7.3 Determ inistic Results 80

7.4 Stochastic Results 81

7.5 M easured vs Actual Z A xis Velocity 84

7.6 Performance Using Position Estimates
and Actual Velocities 86

A. I XY Missile & Target Positions
Reachable Set Control 89 A.2 XY Missile & Target Positions
Baseline Control Law 89

A.3 XZ Missile & Target Positions
Reachable Set Control 90 A.4 XZ Missile & Target Positions
Baseline Control Law 90

A .5 M issile Acceleration Reachable Set Control 91

A .6 M issile Acceleration Baseline Control Law 91

A.7 Missile Roll Commands & Rate
Reachable Set Control 92

A.8 Missile Roll Commands & Rate
Baseline Control Law 92

A .9 M issile Roll A ngle Error 93

viii

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KEY TO SYMBOLS

a(.) Reference control input vector.

aMx Missile inertial x axis acceleration.

aTx Target inertial x axis acceleration.

Ax Specific force (drag) along X body axis.

Azb,Ayb Desired linear acceleration about Z and Y body axes.

B(.) Reference control input matrix.

C(.) Reference state output matrix.

D(.) Feedforward state output matrix.

Do Stability parameter Equilibrium drag coefficient.

DOwt Stability parameter Change in drag due to weight.

Du Stability parameter Change in drag due to velocity.

Doc Stability parameter Change in drag due to angle of attack.

Doc Stability parameter Change in drag due to angle of attack rate.

Dq Stability parameter Change in drag due to pitch rate.

DO Stability parameter Change in drag due to pitch angle.

D6e Stability parameter Change in drag due to pitch canard deflection
angle.

E(.) Feedforward reference output matrix.

eo Roll angle error.

F(.) System matrix describing the dynamic interaction between state
variables.

G(.) System control input matrix.

G*(ti) Optimal control feedback gain matrix.

ix

..








Gl(ti) G2(ti) G3(ti)

g

H(.) IXII IZ
Ixx,IyyIzz

J

L(.) LOt LOwt Lu


Loc Loc


Lq LO L~e



LB LB Lp

Lr L~a L~r


Optimal system state feedback gain matrix. Optimal target state feedback gain matrix. Optimal reference state feedback gain matrix. Acceleration due to gravity. System state output matrix.

Moment of inertial with respect to the given axis. Cost to go function for the mathematical optimization. System noise input matrix.

Stability parameter Equilibrium change in Z axis velocity. Stability parameter Change in Z axis velocity due to weight. Stability parameter Change in Z axis velocity due to forward velocity.

Stability parameter Change in Z axis velocity due to angle of attack.

Stability parameter Change in Z axis velocity due to angle of attack rate.

Stability parameter Change in Z axis velocity due to pitch rate. Stability parameter Change in Z axis velocity due to pitch angle. Stability parameter Change in Z axis velocity due to pitch canard deflection angle.

Stability parameter Equilibrium change in roll rate. Stability parameter Change in roll rate due to sideslip angle. Stability parameter Change in roll rate due to sideslip angle rate. Stability parameter Change in roll rate due to roll rate. Stability parameter Change in roll rate due to yaw rate. Stability parameter Change in roll rate due to roll canard deflection angle.

Stability parameter Change in roll rate due to yaw canard deflection angle.

..








M Mass of the missile.

M0 Stability parameter Equilibrium pitch rate.

Mu Stability parameter Change in pitch rate due to forward velocity.

MOc Stability parameter Change in pitch rate due to angle of attack.

MOc Stability parameter Change in pitch rate due to angle of attack
rate.

Mq Stability parameter Change in pitch rate due to pitch rate.
MSe Stability parameter Change in pitch rate due to pitch canard

deflection angle.

No Stability parameter Equilibrium yaw rate.

NB3 Stability parameter Change in yaw rate due to sideslip angle.

NB3 Stability parameter Change in yaw rate due to sideslip angle rate.

Np Stability parameter Change in yaw rate due to roll rate.

Nr Stability parameter Change in yaw rate due to yaw rate.

NSa Stability parameter Change in yaw rate due to roll canard
deflection angle.

N6r Stability parameter Change in yaw rate due to yaw canard
deflection angle.

Nx,Ny,Nz Components of applied acceleration on respective missile body axis. P Solution to the Riccati equation.

P,Q,R Angular rates about the X,Y, and Z body axis respectively.

Q(.) State weighting matrix.

R(.) Control weighting matrix.

R(.) Reference state vector.

S(.) State-Control cross weighting matrix.

T(.) Target disturbance state vector.

Tgo Time-to-go.

U System input vector.

..








U,V,W Vx,VyVz, Vs Ws


Wt


IVtot X(.) X,Y,Z YO Yowt YB

YB


Yp Yr YO

Y6a


Linear velocities with respect to the X,Y, and Z body axis respectively.

State velocity.

System noise process. Zero mean white Gaussian noise modeling uncorrelated state disturbances.

Zero mean white Gaussian noise driving first order Markov process modeling correlated state disturbances. Total missile velocity. System state vector. Body stabilized axis. Stability parameter Equilibrium change in Y axis velocity. Stability parameter Change in Y axis velocity due to weight. Stability parameter Change in Y axis rate due to sideslip angle. Stability parameter Change in Y axis velocity due to sideslip
angle rate.

Stability parameter Change in Y axis velocity due to roll rate. Stability parameter Change in Y axis velocity due to yaw rate. Stability parameter Change in Y axis velocity due to roll angle. Stability parameter Change in Y axis velocity due to roll canard deflection angle. Stability parameter Change in Y axis velocity due to yaw canard deflection angle. Angle of attack.

Angle of Sideslip. System noise transition matrix. Reference state transition matrix. Target disturbance state transition matrix. System state transition matrix.

xii

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AX Target model correlation time.

Oa Target elevation aspect angle.

09 Seeker elevation gimbal angle.

Oa Target azimuth aspect angle.

Og Seeker azimuth gimbal angle.

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Abstract of Dissertation Presented to the Graduate School of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy







REACHABLE SET CONTROL FOR
PREFERRED AXIS HOMING MISSILES By

Donald J. Caughlin, Jr.

April 1988



Chairman: T.E. Bullock
Major Department: Electrical Engineering

The application of modern control methods to the guidance and control of preferred axis terminal homing missiles is non-trivial in that it requires controlling a coupled, non-linear plant with severe control variable constraints, to intercept an evading target. In addition, the range of initial conditions is quite large and is limited only by the seeker geometry and aerodynamic performance of the missile. This is the problem: Linearization will cause plant parameter errors that modify the linear trajectory. In non-trivial trajectories, both Ny and Nz acceleration commands will, at some time, exceed the maximum value. The two point boundary problem is too complex to complete in real time and other formulations are not capable of handling plant parameter variations and control variable constraints.

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Reachable Set Control directly adapts Linear Quadratic Gaussian (LQG) synthesis to the Preferred Axis missile, as well as a large class of nonlinear problems where plant uncertainty and control constraints prohibit effective fixed- final -time linear control. It is a robust control technique that controls a continuous system with sampled data and minimizes the effects of modeling errors. As a stochastic command generator/ tracker, it specifies and maintains a minimum control trajectory to minimize the terminal impact of errors generated by plant parameter (transfer function) or target set uncertainty while rejecting system noise and target set disturbances. Also, Reachable Set Control satisfies the Optimality Principle by insuring that saturated control, if required, will occur during the initial portion of the trajectory. With large scale dynamics determined by a dual reference in the command generator, the tracker gains can be optimized to the response time of the system. This separation results in an "adaptable" controller because gains are based on plant dynamics and cost while the overall system is smoothly driven from some large displacement to a region where the relatively high gain controller remains linear.

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CHAPTER I
INTRODUCTION

The application of modern control methods to the guidance and control of preferred axis terminal homing missiles has had only limited success [1,2,3]. This guidance problem is non-trivial in that it requires controlling a coupled, non-linear plant with severe control variable constraints, to intercept an evading target. In addition, the range of initial conditions is quite large and limited only by the seeker geometry and aerodynamic performance of the missile.

There are three major control issues that must be addressed: the coupled non-linear plant of the Preferred Axis Missile; the severe control variable constraints; and implementation in the missile where the solution is required to control trajectories lasting one (1) to two (2) seconds real time.

There have been a number of recent advances in non-linear control but these techniques have not reached the point where real time implementation in an autonomous missile controller is practical [4,5,6]. Investigation of non-linear techniques during this research did not improve the situation. Consequently, primarily due to limitations imposed by real time implementation, linear suboptimal control schemes were emphasized.

Bryson & Ho introduced a number of techniques for optimal control with inequality constraints on the control variables [7]. Each of these use variational techniques to generate constrained and unconstrained arcs that must be pieced together to construct the optimal trajectory.

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2

In general, real time solution of optimal control problems with bounded control is not possible [8]. In fact, with the exception of space applications, the optimal control solution has not been applied [9,10]. When Linear Quadratic Gaussian (LQG) techniques are used, the problem is normally handled via saturated linear control, where the control is calculated as if no constraints existed and then simply limited. This technique has been shown to be seriously deficient. In this case, neither stability nor controllability can be assured. Also, this technique can cause an otherwise initially controllable trajectory to become uncontrollable [I I].

Consequently, a considerable amount of time is spent adjusting the gains of the controller so that control input will remain below its maximum value. This adjustment, however, will force the controller to operate below its maximum capability [12]. Also, in the case of the terminal homing missile, the application of LQG controllers that do not violate an input constraint lead to an increasing acceleration profile and (terminally) low gain systems [131. As a result, the performance of these controllers is not desirable.

While it is always possible to tune a regulator to control the system to a given trajectory, the variance of the initial conditions, the time to intercept the target (normally a few seconds for a short range high performance missile), and the lack of a globally optimal trajectory due to the nonlinear nature, the best policy is to develop a suboptimal real time controller.

The problem of designing a globally stable and controllable high performance guidance system for the preferred axis terminal homing missile is treated in this dissertation. Chapter 2 provides adequate background information on the missile guidance problem. Chapter 3 covers recent work on constrained

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3

control techniques. Chapters 4 and 5 discuss Robust Control and introduce "Reachable Set" Control, while Chapter 6 applies the technique to control of a preferred axis homing missile. The performance of "Reachable Set" control is presented in Chapter 7.

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CHAPTER 11
BACKGROUND

The preferred axis orientation missile has significant control input constraints and complicated coupled angular dynamics associated with the maneuvering. In the generic missile considered, the Z axis acceleration (see

Figure 2.1) was structurally limited to 100 "g" with further limits on "g" resulting from a maximum angle of attack as a function of dynamic pressure. Even though the Z axis was capable of 100 "g", the "skid- to- turn" capability of the Y axis was constrained to 5 "g" or less because of aerodynamic limitations a 20:1 difference. In addition to pitch (Nz) and yaw (Ny) accelerations, the missile can roll up to 500 degrees per second to align the primary maneuver plane with the plane of intercept. Hence, bank-to-turn.






IY














z

Figure 2.1 Missile Reference System.

4

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5

The classical technique for homing missile guidance is proportional navigation (pro nav). This technique controls the seeker gimbal angle rate to zero which (given constant velocity) causes the missile to fly a straight line trajectory toward the target [14,15]. In the late 70's an effort was made to use modern control theory to improve guidance laws for air-to-air missiles. For recent research on this problem see, for example, [11]. As stated in the introduction, these efforts have not significantly improved the performance of the preferred axis homing missile.

Of the modern techniques, two basic methodologies have emerged: one was a body-axis oriented control law that used singular perturbation techniques to uncouple the pitch & roll axis [16,17]. This technique assumed that roll rate is the fast variable, an assumption that may not be true during the terminal phase of an intercept. The second technique was an inertial point mass formulation that controls inertial accelerations [18]. The acceleration commands are fixed with respect to the missile body; but, since these commands can be related to the inertial reference via the Euler Angles, the solution is straight forward. Both of these methods have usually assumed unlimited control available and the inertial technique has relied on the autopilot to control the missile roll angle, and therefore attitude, to derotate from the inertial to body axis.



Missile Dynamics



The actual missile dynamics are a coupled set of nonlinear forces and moments resolved along the (rotating) body axes of the missile [19]. Linearization of the equations about a "steady state" or trim condition,

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6

neglecting higher order terms, results in the following set of equations (using standard notation, see symbol key in the preface):


oc = Q PB + Azb /IVtot B = R Poc + Ayb / IVtotI Linear Accelerations


1
U= RV- QW - D0+ DOwt)
M

DuU- Doccx- DocoC- DqQ- DOO Dse6e + Ax/M



V= PW- RU + Y0 + YOwt}
M

+ YBB+ Y3B+ YpP+ YrR+ Y0o + Ysa6a + Y6r6r

1
W= QU- PV + (L+ Lwt)
M

LuU- Lcc- Lococ- LqQ- L0O L6e6e



Moment Equations




Q = Mo/Iyy + MuU + Mcxoc + Moc + MqQ + M6e6e



+ PR- ( p2 R2)
Iyy Iyy

..










R N0/Izz + NBB+ NB + NpP+ NrR+ N~a6a + N6r6r (7)

(I yy-Ixx) Ixz
PQ- (QR- P)Izz Izz




P = LO/Ixx + LBB + Lf3B + LPP + LrR + L6a6a + L~r6r (8)

(Ixx-I yy) Ixz
+ QR- ( PQ- R)Izz Ixx



Linear Quadratic Gaussian Control Law


For all of the modern development models, a variation of a fixed-final-time LQG controller was used to shape the trajectory. Also, it was expected that the autopilot would realize the commanded acceleration. First, consider the effect of the unequal body axis constraints. Assume that 100 "g" was commanded in each axis resulting in an acceleration vector 45 degrees from Nz. If Ny is only capable of 5 "g", the resultant vector will be 42 degrees in error, an error that will have to be corrected by succeeding guidance commands. Even if the missile has the time or capability to complete a

successful intercept, the trajectory can not be considered optimal.

Now consider the nonlinear nature of the dynamics. The inertial linear system is accurately modeled as a double integrator of the acceleration to determine position. However, the acceleration command is a function of the missile state, equation (1), and therefore, it is not possible to arbitrarily assign the input acceleration. And, given a body axis linear acceleration, the inertial

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8

component will be severely modified by the rotation (especially roll) of the reference frame. All of these effects are neglected in the linearization.

This then is the problem: In the intercept trajectories worth discussing, Ny, Nz, and roll acceleration commands will, at some time, saturate. High order linear approximations do not adequately model the effects of nonlinear dynamics, and the complete two point boundary value problem with control input dynamics and constraints is too difficult to complete in real time.

Although stochastic models are discussed in Bryson and Ho [71, and a specific technique is introduced by Fiske [18], the general procedure has been to use filtered estimates and a dynamic -programming- like definition of optimality (using the Principle of Optimality) with Assumed Certainty Equivalence to find control policies [20,21,22]. Therefore, all of the controllers actually designed for the preferred axis missiles are deterministic laws cascaded with a Kalman Filter. The baseline for our analysis is an advanced control law proposed by Fiske [181. Given the finite dimensional linear system:



x(t) =Fx(t) + GuMt (9)


where


( Ax
x= u Ay
Vx AzJ



and


F 0 1 G= []

..









with the cost functional:
ftf
J=xfPfxf + o J UTRudr (10)
1Jt0





Application of the Maximum principle results in a linear optimal control law: 3(Tgo) 3(Tgo)2 (11)
ui(t) = xi(t) + Vi(t)
3- +(Tgo)3 3-y +(Tgo)3



Coordinates used for this system are "relative inertial." The orientation of the inertial system is established at the launch point. The distances and velocities are the relative measures between the missile and the target. Consequently, the set point is zero, with the reference frame moving with the missile similar to a "moving earth" reference used in navigation.

Since Fisk's control law was based on a point mass model, the control law did not explicitly control the roll angle PHI (o). The roll angle was controlled by a bank-to-turn autopilot [23]. Therefore, the guidance problem was decomposed into two components, trajectory formation and control. The autopilot attempted to control the roll so that the preferred axis (the -Z axis) was directed toward the plane of intercept. The autopilot used to control the missile was designed to use proportional navigation and is a classical combination of single loop systems.

Recently, Williams and Friedland have developed a new bank-to-turn autopilot based on modern state space methods [24]. In order to accurately control the banking maneuver, the missile dynamics are augmented to include the kinematic relations describing the change in the commanded specific force

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10

vector with bank angle. To determine the actual angle through which the

vehicle must roll, define the roll angle error: eo = tan- Ayb (12)

Azb


Using the standard relations for the derivative of a vector in a rotating reference frame, the following relationships follow from the assumption that A I << A IB:

Azb = P(Ayb) (13)

Ayb = + P(Azb) (14)

The angle eo represents the error between the actual and desired roll angle of the missile. Differentiating eo yields:


(Azb)(Ayb) (Ayb)(Azb)
(Azb)2 + (Ayb)2



which, after substituting components of A x w, shows that eo = P (16)

Simplifying the nonlinear dynamics of (1) (8), the following model was used:


= Q PB + Azb /IVtot (17)



B = R Poc + Ayb / lMtot (18)


(IZZ Ixx)
Q = Moc + MqQ + MseSe + PR (19)
Iyy

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11
(Iyy Ixx)
R= NBB + NrR + N6r~r + PQ(20)
Izz


P = LpP + L6a6a (21)


where


Az =Zco + Z6qq (22)


Ay= Z13B + Zsq6q (23)



Using this model directly, the autopilot would be designed as an eighth-order system with time-varying coefficients. However, even though these equations contain bilinear terms involving the roll rate P as well as pitch/yaw cross-coupling terms, the roll dynamics alone, represent a second order system that is independent of pitch and yaw. Therefore, using an "Adiabatic Approximation" where the optimal solution of the time-varying system is approximated by a sequence of solutions of the time-invariant algebraic Riccati equation for the optimum control law at each instant of time, the model was separated into roll and pitch/yaw subsystems [25]. Now, similar to a singular perturbations technique, the function of the roll channel is to provide the necessary orientation of the missile so that the specific force acceleration lies on the Z (preferred) axis of the missile. Using this approximation, the system is assumed to be in steady state, and all coefficients- -including roll rate--are assumed to be constant. Linear Quadratic Gaussian (LQG) synthesis is used, with an algebraic Riccati equation, on a second and sixth order system. And, when necessary, the gains are scheduled as a function of the flight condition.

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12

While still simplified, this formulation differs significantly from previous controllers in two respects. First, the autopilot explicitly controls the roll angle; and second, the pitch and yaw dynamics are coupled.

Even though preliminary work with this controller demonstrated improved tracking performance by the autopilot, overall missile performance, measured by miss distance and time to intercept, did not improve. However, the autopilot still relies on a trajectory generated by the baseline controller ( e.g. Azb in 17). Consequently, the missile performance problem is not in the autopilot, the error source is in the linear optimal control law which forms the trajectory. "Reachable Set Control" is a LQG formulation that can minimize these errors.

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CHAPTER III
CONSTRAINED CONTROL

In Chapters I and 11, we covered the non-linear plant, the dynamics neglected in the linearization, the impact of control variable constraints, and the inability of improved autopilots to reduce the terminal error. To solve this problem, we must consider the optimal control of systems subject to input constraints. Although a search of the constrained control literature did not provide any suitable technique for real time implementation, some of the underlying concepts were used in the formulation of "Reachable Set Control." This Chapter reviews some of these results to focus on the constrained control problem and illustrate the concepts.

Much of the early work was based on research reported by Tufts and Shnidman [261 which justified the use of saturated linear control. However, as stated in the introduction, with saturated linear control, controllability is not assured. If the system, boundary values and final time are such that there is no solution with any allowable control (If the trajectory is not controllable), then the boundary condition will not be met by either a zero terminal error or penalty function controller. While constrained control can be studied in a classical way by searching for the effect of the constraint on the value of the performance function, this procedure is not suitable for real time control of a system with a wide range of initial conditions [27]. Some of the techniques that could be implemented in real time are outlined below.

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14

Lim used a linearized gain to reduce the problem to a parameter optimization [8]. Given the system model: x = Fx + Gu + Lw (1)

with state x, constant F, G, and L, scaler control u, and Lw representing zero mean Gaussian white noise with covariance LLT. Consider the problem of

choosing a feedback law such that in steady state, assuming it exists, the expected quadratic cost

tf
J = E{ lim [ (x(t)TQx(t) + Au(t)2) dt + x(tf)Tp(tf)x(tf)] } (2) tf 00


is minimized. The weighting matrix Q is assumed to be positive semidefinite and A > 0. Dynamic programming leads to Bellman's equation:

min ( I tr[LTVxx(x)L] + (Fx + Gu)TVx(x) + xTQx + Au2 } cx* (3)

Jul
and, assuming a V(x) satisfying (3), the optimal solution u(x) = SAT { (1/2A)GTVx(x) } A 0 (4)

= SGN GTVx(x)) A= 0

However, (3) cannot be solved analytically, and Vx in general is a nonlinear function of x. Consider a modified problem by assuming a control of the form: u(x) = SAT( gTx } A0 0 (5a)

= SGN gTx A = 0 (5b)

where g is a constant (free) vector.

Assume further that x is Gaussian with known covariance W (positive definite). Using statistical linearization, a linearized gain k can be obtained by minimizing


E(u(x) kTx)2

..









which results in


for (5a): k (gTwg)- } g, (7a)



where
fz
4D(z) (2/7r), exp { -1y2} dy for (5b): k = (2/r) (gTwg)- g (7b)


From (1), with u = kTx, the stable covariance matrix W and steady state P are determined by

(F + GkT)W + W(F + GkT)T + LLT = 0 (8)
and
(F + GkT)Tp + P(F + GkT) + P + ),kkT = 0 (9)


The solution to (3), without the minimum, is V(x) = xTpx (10)
and
oc= tr ( LTPL (11)

The problem is to choose g such that the expected cost oc by statistical

linearization is a minimum. However, a minimum may not exist. In fact, from [8], a minimum does not exist when the noise disturbance is large. Since we are considering robust control problems with plant uncertainty or significant modeling errors, the noise will be large and the minimum will be replaced by a greatest lower bound. As oc approached the greatest lower bound, the control approached bang-bang operation. A combination of plant errors and the rapid dynamics of some systems (such as the preferred axis missile) would preclude acceptable performance with bang-bang control.

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16

Frankena and Sivan suggested a criterion that reduce the two-point boundary problem to an initial value problem [12]. They suggest controlling the plant while minimizing this performance index:

It1
J = /2)llx(t)ll2Q(t)+Ilx(t)lls(t)}dt (12)
to

+ (1/2)llx(t l)112p(t)


With the constraint Ilu(t)IIR(t) <- I


Applying the maximum principle to the Hamiltonian developed from


x(t) = F(t)x(t) + G(t)u(t) to < t < tl (13)
with
x(t0) = x0


provides the adjoint differential equation S(t)x(t)
),(t) = Q(t)x(t) + FT(t)A(t) A(tl) = -plx(tl) (14)
IIx(t)Ils(t)


With u(t) = R-I(t)GT(t)A found by maximizing the Hamiltonian, the constraint in

(12) can be expressed as


R- l(t)GT(t)A(t)
u(t) =(15) U II JR- I (t)GT(t)A,(t)IIR(t) (5



The desired control exists if a matrix P(t) can be defined such that A(t) = P(t)x(t) (16)
and from (14)


P(tl) = -PI < 0


(17)

..










For GTpx* 0 and lixils 0 0, P will be the solution of PGR-1GTp S
P + PF + Q FTp (18)
11R- 1GTpxlIR IIxIIs


Now choosing S = PGR-1GTP results in a Lyapunov equation and will insure negative definite P(t) if F is a stability matrix. Therefore, with this choice of weighting functions to transform the problem to a single boundary condition, a stable F matrix is required. This is a significant restriction and not applicable to the system under consideration.

Gutman and Hagander developed a design for saturated linear controllers for systems with control constraints [9,281. The design begins with a low-gain stabilizing control, solves a Lyapunov equation to find a region of stability and associated stability matrix, and then sums the controls in a saturation function to form the constrained control. Begin with the stabilizable continuous linear time invariant system


x = Fx+ Gu x(0) = x0 (19)


with admissible control inputs ui, such that


gi -5 ui -- hi i = 1 .m where gi and hi are the control constraints. Consider an n x m matrix L- [ 11 12 .1 lm] (20)
such that

Fc a Fc(L) -(F + GLT) (21)


is a stability matrix.

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18

Associated with each of the controls are sets that define allowable conditions. The set D is the set of initial conditions from which it is desired to stabilize the system to the origin. The low gain stabilizing control L defines the set E:

E- E(L)-{ z I z E R (22)

and gi < liTz < hi ) i-1.m which is the set of states at which the stabilizing linear feedback does not initially exceed the constraints. Another set is F: F a-F(L) n { (eFct)-1 E} (23)

tE[O,oo)

F is a subset of E such that along all trajectories emanating from F, the stabilizing linear state feedback does not exceed the constraint. The region of stability for the solution of the Lyapunov equation is defined by fl I(L,P,c) (24)

{ x [ xTpx< c}

where V(x) = xTpx is the Lyapunov function candidate for the stability matrix Fc, and c is to be determined.

The control technique follows: Step 1: Determine D.

Step 2: Find L by solving a LQG problem. The control penalty is increased until the control LTx satisfies the constraint in (19) for x in D. If D is such that the control constraint can not be satisfied, then this design is not appropriate. Step 3: Find P and c. First find a P = pT > 0 such that the Lyapunov equation PFc + FcTP > 0. Now determine n) by choosing c in (24) such that D C 0 C E: sup xTpx _< c _< min xTpx (25)
xED xE6E

..







19

where 6E designates the boundary of E. If this fails, choose another P, or select a "lower gain" in order to enlarge E, or finally, a reduction in the size of D might be considered.

Step 4: Set up the control according to

u = SAT[ (LT -KGTP)x 1(26) where K is defined

ki 0

K= [ kn k10, i 1,2,.,m (27)



and tune the parameters ki by simulations.

A sufficient condition for the algorithm to work is

DC c) 0cE. (28)

Unfortunately, determining the stability region was trial and error; and, once found, further tuning of a diagonal gain matrix is required. In essence, this was a technique for determining a switching surface between a saturated and linear control. Also, when the technique was applied to an actual problem, inadequacies in the linear model were not compensated for. Given the nonlinear nature of the preferred axis missile, range of initial conditions, and the trial and error tuning required for each of these conditions, the procedure would not be adequate for preferred axis terminal homing missile control. A notable feature of the control scheme, however, was the ability to maintain a stable system with a saturated control during much of the initial portion of the trajectory.

Another technique for control with bounded input was proposed by Spong et al. [29]. This procedure used an optimal decision strategy to develop a pointwise optimal control that minimized the deviation between the actual and

..







20

desired vector of joint accelerations, subject to input computation of the control law is reduced to the solution quadratic programming problem. Key to this solution is the desired trajectory in state space. Suppose that a dynamical described by


with


constraints. The of a weighted availability of a system can be


x(t) = f(x(t)) + G(x(t))u(t) luil ui,max


which can be written as


(29)


Nu< c


Fix time t > 0, let s(t,x0,t0,u(t)) (or s(t) for short), denote the solution to (29) corresponding to the given input function u(t). At time t, ds/dt is the velocity vector of the system, and is given explicitly by the right hand side of (29). Define the set Ct = C(s(t))


C(s(t)) = ( c(t,o) G RN Ioc


(30)


= f(s(t)) + G(s(t))w, w E fl )
with
fl= wI Nw< c}


Therefore, for each t and any allowable u(t), ds/dt lies in the set Ct. In other words, the set Ct contains the allowable velocities of the solution s(t). Assume that there exists a desired trajectory yd, and an associated vector field v(t) = v(s(t),yd(t),t)), which is the desired (state) velocity of s(t) to attain yd.

Consider the following "optimal decision strategy" for a given positive definite matrix Q: Choose the input u(t) so that the corresponding solution s(t) satisfies (d/dt)s(t,u(t)) = s*(t), where s*(t) is chosen at each t to minimize


min ( (oc v(s(t),yd(t),t))TQ( c v(s(t),yd(t),t)) } OcECt


(31)

..









This is equivalent to the minimization


min { luTGTQGu (GTQ(v-f))Tu } subject to, Nu(t) _< c (32)
u


We may now solve the quadratic programming problem to yield a pointwise

optimal control law for (29).

At each time t, the optimal decision strategy attempts to "align" the closed loop system with the desired velocity v(t) as nearly as possible in a least squares sense. In this way the authors retain the desirable properties of v(t) within the constraints imposed by the control. Reachable Set Control builds on this technique: it will determine the desired trajectory and optimally track it.

Finally, minimum-time control to the origin using a constrained acceleration has also been solved by a transformation to a two-dimensional unconstrained control problem [30]. By using a trigonometric transformation, the control is defined by an angular variable, u(t) = f{cos(13),sin(13)}, and the control problem was modified to the control of this angle. The constrained linear

problem is converted to an unconstrained nonlinear problem that forces a numerical solution. This approach removes the effect of the constraints at the expense of the continuous application of the maximum control. Given the aerodynamic performance (range and velocity) penalty of maximum control and the impact on attainable roll rates due to reduced stability at high angle of attack, this concept did not fit preferred axis homing missiles.

An important assumption in the previous techniques was that the constrained system was controllable. In fact, unlike (unconstrained) linear systems, controllability becomes a function of the set admissible controls, the initial state, the time-to-go, and the target state. To illustrate this, some of the relevant points from [31,32] will be presented. An admissible control is one

..







22

that satisfies the condition u(.) : [O,oo) 11 E Rm where 0 is the control restraint set. The collection of all admissible controls will be denoted by M(f). The target set X is a specified subset in Rn. A system is defined to be fl-controllable from an initial state x(t0) = x0 to the target set X at T if there exists U(.) E M(fl) such that x(T,u(.),x0) E X. A system would be globally f-controllable to X if it is fl-controllable to X from every x(t0) E Rn.

In order to present the necessary and sufficient conditions for fl-controllability, consider the following system: x(t) = F(t)x(t) + G(t,u(t)) x(t0) = x0 (33)

and the adjoint defined by:

z(t) = F(t)Tz(t) z(t0) = z0 t e [O,oo) (34)

with the state transition matrix 4(t,r) and solution z(t) = 4(tOt)TzO (35)

The interior B and surface S of the unit ball in Rn are defined as B=(zoeRn :IIzoll_5 1 ) (36)

S =(z0ERn : IIzo1= I ) (37)

The scaler function J(.): Rn x R x Rn x Rn R is defined by


J(x0,t,x,z0) = x0z0 + max [ GT(r,w)z(r) ]dr x(t)Tz(t) (38)
0 eWfl

Given the relatively mild assumptions of [32], a necessary condition for

(33) to be fl-controllable to X from x(t0) is max min J(x0,T,x,z0) = 0 (39)
xeX z0EB

while a sufficient condition is

sup min J(x0,T,x,z0) > 0 (40)
xCX z0ES

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23

The principle behind the conditions arises from the definition of the

adjoint system -- Z(t). Using reciprocity, the adjoint is formed by reversing the role of the input and output, and running the system in reverse time [33]. Consider

x(t) = F(t)x(t) + G(t)u(t) x(t0) = x0 (41)

y(t) = H(t)x(t)
and:
z(t) = F(t)Tz(t) + HT(t)p1(t) z(t0) = z0 (42)

o(t) = GT(t)z(t)

Therefore

zT(t)x(t) = zT(F(t)x(t) + G(t)u(t)) (43)
and
(d/dt)(zT(t)x(t)) = zT(t)x(t) + zT(t)x(t) (44)
= PT(t)H(t)x(t) + zT(t)G(t)u(t)


Integrating both sides from to to tf yields the adjoint lemma:


zT(tf)x(tf) zT(t0)x(t0) --J (IjT(t)H(t)x(t) + zT(t)G(t)u(t)) dt (45)



The adjoint defined in (31) does not have an input. Consequently, the integral in (35) is a measure of the effect of the control applied to the original system. By searching for the maximum GT(r,w)z(r), it provides the boundary of the effect of allowable control on the system (33). Restricting the search over the target set to the min ( J(x0,t,x,z0) : t c [0,T], z0 c S ) or min { J(x0,t,x,z0) : t e [0,T], z0 e B ) minimizes the effect of the specific selection of z0 on the reachable set and insures that the search is over a function that is jointly continuous in (t,x). Consequently, (35) compares the autonomous

growth of the system, the reachable boundary of the allowable input, and the desired target set and time. Therefore, if J = 0, the adjoint lemma is be

..







24

identically satisfied at the boundary of the control constraint set (necessary); J > 0 guarantees that a control can be found to satisfy the lemma. If the lemma is satisfied, then the initial and final conditions are connected by an allowable trajectory. The authors 1321 go on to develop a zero terminal error steering control for conditions where the target set is closed and max min J(x0,T,x,z0) >: 0 (46)
xCX z0ES

But their control technique has two shortcomings: First; it requires the selection of z0. The initial condition zo is not specified but limited to lizoll = 1. A particular zo must be selected to meet the prescribed conditions and the equality in (43) for a given boundary condition, and is therefore not suitable for real time applications. And second; the steering control searched M(il) for the supremum of J, making the control laws bang-bang in nature, again not suitable for homing missile control.

While a direct search of Ox~ is not appropriate for a preferred axis missile steering control, a "dual" system, similar to the adjoint system used in the formulation of the controllability function J, can be used to determine the amount of control required to maintain controllability. Once controllability is assured, then a cost function that penalizes the state deviation (as opposed to a zero terminal error controller) can be used to control the system to an arbitrarily small distance from the reference.

..














CHAPTER IV
CONSTRAINED CONTROL
WITH
UNMODELED SETPOINT AND PLANT VARIATIONS

Chapter III reviewed a number of techniques to control systems subject to control variable constraints. While none of the techniques were judged adequate for real time implementation of a preferred-axis homing missile controller, some of the underlying concepts can be used to develop a technique that can function in the presence of control constraints: (1) Use of a "dual system" that can be used to maintain a controllable system (trajectory); (2) an "optimal decision strategy" to minimize the deviation between the actual and desired trajectory generated by the "dual systemn" and (3) initially saturated control and optimal (real time) selection of the switching surface to linear control with zero terminal error.

However, in addition to, and compounding the limitations imposed by control constraints, we must also consider the sensitivity of the control to unmodeled disturbances and robustness under plant variations. In the stochastic problem, there are three major sources of plant variations. First, there will be modeling errors (linearization/reductions) that will cause the dynamics of the system to evolve in a different or "perturbed" fashion. Second, there may be the unmodeled uncertainty in the system state due to Gaussian assumptions. And finally, in the fixed final time problem, there may be errors in the final time, especially if it is a function of the uncertain state or impacted by the modeling reductions. Since the primary objective of this research is the zero error control of a dynamical system in fixed time, most of the more recent 25

..







26

optimization techniques (eg. LQG/LTR,H') did not apply. At this time, these techniques seemed to be more attuned to loop shaping or robust stabilization questions.

A fundamental proposition that forms the basis of Reachable Set Control is that excessive terminal errors encountered when using an optimal feedback control for an initially controllable trajectory (a controllable system that can meet the boundary conditions with allowable control values) are caused by the combination of control constraints and uncertainty (errors) in the target set stemming from unmodeled plant perturbations (modeling errors) or set point dynamics.

First, a distinction must be made between a feedback and closed-loop controller. Feedback control is defined as a control system with real-time measurement data fed back from the actual system but no knowledge of the form, precision, or even the existence of future measurements. Closed-loop control exploits the knowledge that the loop will remain closed throughout the future interval to the final time. It adds to the information provided to a feedback controller, anticipates that measurements will be taken in the future, and allows prior assessment of the impact of future measurements. If Certainty Equivalence applies, the feedback law is a closed-loop law. Under the Linear Quadratic Gaussian (LQG) assumptions, there is nothing to be gained by anticipating future measurements. In the mathematical optimization, external disturbances can be rejected, and the mean value of the terminal error can be made arbitrarily close to zero by a suitable choice of control cost.

For the following discussion, the "system" consists of a controllable plant and an uncontrollable reference or target. The system state is the relative difference between the plant state and reference. Since changes in the system

..







27

boundary condition can be caused by either a change in the reference point or plant output perturbations similar to those discussed in Chapter II, some definitions are necessary. The set of boundary conditions for the combined plant and target system, allowing for unmodeled plant and reference perturbations, will be referred to as the target set. Changes, or potential for change, in the target set caused only by target (reference) dynamics will be referred to as variations in the set point. The magnitude of these changes is assumed to be bounded. Admissible plant controls are restricted to a control restraint set that limits the input vector. Since there are bounds on the input control, the system becomes non-linear in nature, and each trajectory must be evaluated for controllability. Assume that the system (trajectory) is pointwise controllable from the initial to the boundary condition.

Before characterizing the effects of plant and set point variations, we must consider the form of the plant and it's perturbations. If we assume that the plant is nonlinear and time-varying, there is not much that can be deduced about the target set perturbations. However, if have a reduced order linear model of a combined linear and nonlinear process, or a reasonable linearization

of a nonlinear model, then the plant can be considered as linear and time-varying. For example, in the case of a Euclidean trajectory, the system model (a double integrator) is exact and linear. Usually, neglected higher order or nonlinear dynamics or constraints modify the accelerations and lead to trajectory (plant) perturbations. Consequently, in this case, the plant can be accurately represented as a Linear Time Invariant System with (possibly) time varying perturbations.

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28

Consider the feedback interconnection of the systems K and P where K is a sampled-data dynamic controller and P the (continuous) controlled system:


Figure 4.1 Feedback System and Notation



Assuming that the feedback system is well defined and Bounded Input Bounded Output (BIBO) stable, at any sample time ti, the system can be defined in terms of the following functions:


e(ti) u(ti) Y(ti)


= r(ti) Y(ti) = Ke(ti) = Pu(ti)

..







29

with the operator G = G[K,P] as the operator that maps the input e(ti) to the output y(ti) [341.

At any time, the effect of a plant perturbation AP can also be characterized as a perturbation in the target set.


If P = P0 + AP (4a)
or
P = P(I+AP) (4b)
then
y(ti) = YO(ti) + Ay(ti) (5)


where Ay(ti) represents the deviation from the "nominal" output caused by either the additive or multiplicative plant perturbation. Therefore, e(ti) = r(ti) (y0(ti) + Ay(ti)) (6)

= (r(ti) + Ay(ti)) y0(ti) (7)

= Ar(ti) y0(ti) (8)

with Ar(ti) representing a change in the target set that was unknown to the controller. These changes are then fed back to the controller but could be handled a priori in a closed loop controller design as target set uncertainty.

Now consider the effect of constraints. If the control is not constrained, and target set errors are generated by plant variations or target maneuvers, the feedback controller can recover from these intermediate target set errors by using large (impulsive) terminal controls. The modeled problem remains linear. While the trajectory is not the optimal closed-loop trajectory, the trajectory is optimal based on the model and information set available.

Even with unmodeled control variable constraints, and a significant displacement of the initial condition, an exact plant model allows the linear stochastic optimal controller to generate an optimal trajectory. The switching time from saturated to linear control is properly (automatically) determined and,

..







30

as in the linear case, the resulting linear control will drive the state to within an arbitrarily small distance from the estimate of the boundary condition.

If the control constraint set covers the range of inputs required by the control law, the law will always be able to accommodate target set errors in the remaining time-to-go. This is, in effect, the unconstrained case. If, however, the cost-to-go is higher and/or the deviation from the boundary condition is of sufficient magnitude relative to the time remaining to require inputs outside the boundary of the control constraint set, the system will not follow the trajectory assumed by the system model. If this is the case as time-to-go approaches zero, the boundary condition will not be met, the system is not controllable (to the boundary condition). As time-to-go decreases, the effects of the constraints become more important.

With control input constraints, and intermediate target set errors caused by unmodeled target maneuvers or plant variations, it may not be possible for the linear control law to recover from the midcourse errors by relying on large terminal control. In this case, an optimal trajectory is not generated by the feedback controller, and, at the final time, the system is left with large terminal errors.

Consequently, if external disturbances are adequately modeled, terminal errors that are orders of magnitude larger than predicted by the open loop optimal control are caused by the combination of control constraints and target set uncertainty.

..







31

Linear Ovtimal Control with Uncertainty and Constraints



An optimal solution must meet the boundary conditions. To accomplish this, plant perturbations and constraints must be considered a priori. They should be included as a priori information in the system model, they must be physically realizable, and they must be deterministic functions of a priori information, past controls, current measurements, and the accuracy of future measurements.

From the control point of view, we have seen that the effect of plant parameter errors and set point dynamics can be grouped as target set uncertainty. This uncertainty can cause a terminally increasing acceleration profile even when an optimal feedback control calls for a decreasing input (see Chapter 5). With the increasing acceleration caused by midcourse target set uncertainty, the most significant terminal limitation becomes the control input constraints. (These constraints not only affect controllability, they also limit how quickly the system can recover from errors.) If the initial control is saturated while the terminal portion linear, the control is still optimal. If the final control is going to be saturated, however, the controller must account for this saturation.

The controller could anticipate the saturation and correct the linear portion of the trajectory to meet the final boundary condition. This control, however, requires a closed form solution for x(t), carries an increased cost for an unrealized constraint, and is known to be valid for monotonic ( single switching time) trajectories only [I I].

Another technique available is LQG synthesis. However, LQG assumes

controllability in minimizing a quadratic cost to balance the control error and

..







32

input magnitudes. As we have seen, the effects of plant parameter and reference variations, combined with control variable constraints, can adversely impact controllability. The challenge of LQG is the proper formulation of the problem to function with control variable constraints while compensating for unmodeled set point and plant variations. Reachable Set Control uses LQG synthesis and overcomes the limitations of an anticipative control to insure a controllable trajectory.



Control Technioue



Reachable Set Control can be thought of as a fundamentally different robust control technique based on the concepts outlined above. The usual discussion of robust feedback control (stabilization) centers on the development of controllers that function even in the presence of plant variations. Using either a frequency domain or state space approach, and modeling the uncertainty, bounds on the allowable plant or perturbations are developed that guarantee stability [35]. These bounds are determined for the specific plant under consideration and a controller is designed so that expected plant variations are contained within the stability bounds. Building on ideas presented above, however, this same problem can be approached in an entirely different way.

This new approach begins with the same assumptions as standard techniques, specifically a controllable system and trajectory. But, with Reachable Set Control, we will not attempt to model the plant or parameter uncertainty, nor the set point variation. We will, instead, reformulate the problem so that the system remains controllable, and thus stable, throughout the trajectory even in the presence of plant perturbations and severe control input constraints.

..







33

Before we develop an implementable technique, consider the desired result of Reachable Set Control (and the origin of the name) by using a twodimensional missile intercept problem as an example. At time t = tl, not any specific time during the intercept, the target is at some location TI and the missile is at MI as shown in Figure 4.2. Consider these locations as origins of two independent, target and missile centered, reference systems. From these initial locations, given the control inputs available, reachable sets for each system can be defined as a function of time (not shown explicitly). The target set is circular because is maneuver direction is unknown but its capability bounded, and the missile reachable set exponential because the x axis control is constant and uncontrollable while the z axis acceleration is symmetric and bounded. The objective of Reachable Set Control is to maintain the reachable target set in the interior of the missile reachable set. Hence, Reachable Set Control.

x




Target Reachable







1Ti



z
Missile Reachable

Set
Figure 4.2 Reachable Set Control Objective

..







34

As stated, Reachable Set Control would be difficult to implement as a control strategy. Fortunately, however, further analysis leads to a simple, direct, and optimal technique that is void of complicated algorithms or ad-hoc procedures.

First, consider the process. The problem addressed is the control of fixed-terminal-time systems. The true cost is the displacement of the state at the final time and only at the final time. In the terminal homing missile problem, this is the closest approach, or miss distance. In another problem, it may be fuel remaining at the final time, or possibly a combination of the two. In essence, with respect to the direct application of this technique, there is no preference for one trajectory over another or no intermediate cost based on the displacement of the state from the boundary condition. The term "direct application" was used because constrained path trajectories, such as those required by robotics, or the infinite horizon problem, like the control of the depth of a submarine can be addressed by separating the problem into several distinct intervals--each with a fixed terminal time--or a switching surface when the initial objective is met [36].

Given a plant with dynamics

x(t) = f(x,t) + g(u(w),t) x(t0) = x0 (9)

y(t) = h(x(t),t)

modeled by

x(t) = F(t)x(t) + G(t)u(t) (10)

yx(t) = H(t)x(t)

..










with final condition x(tf) and a compact control restraint set flx. Let Ox denote the set of controls u(t) for which u(t) E fOx for t ( [0,oo). The reachable set

X(tO,tf,xO,fnx) x: x(tf) = solution to (10) (11)
with xo for some u(.) E M(nx) }

is the set of all states reachable from x0 in time tf.

In addition to the plant and model in (9 & 10), we define the reference


r(t) = a(x,t) + b(a(w),t) r(t0) = r0 (12)

y(t) = c(x(t),t)

modeled by

r(t) = A(t)r(t) + B(t)a(t) (13)

Yr(t) = C(t)r(t)


and similarly defined set R(to,tf,ro,flr),

R(to,tf,ro,fOr) r: r(tf) = solution to (13) (14)
with r0 for some a(.) E M(fOr) )

as the set of all reference states reachable from r0 in time tf.

Associated with the plant and reference, at every time t, is the following system:

e(t) = yx(t) Yr(t) (15)

.'m (10 & 13), we see that yx(t) and Yr(t) are output functions that incorporate the significant characteristics of the plant and reference that will

be controlled.

The design objective is

e(tf) = 0 (16)

and we want to maximize the probability of success and minimize the effect of errors generated by the deviation of the reference and plant from their associated models. To accomplish this with a sampled-data feedback control law,

..







36

we will select the control u(ti) such that, at the next sample time (ti+i), the target reachable set will be covered by the plant reachable set and, in steady state, if e(tf) = 0, the control will not change.



Discussion



Recalling that the performance objective at the final time is the real measure of effectiveness, and assuming that the terminal performance is directly related to target set uncertainty, this uncertainty should be reduced with time-to-go. Now consider the trajectory remembering that the plant model is approximate (linearized or reduced order), and that the reference has the capability to change and possibly counter the control input. (This maneuverability does not have to be taken in the context of a differential game. It is only intended to allow for unknown set point dynamics.) During the initial portion

of the trajectory, the target set uncertainty is the highest. First, at this point, the unknown (future) reference changes have the capability of the largest displacement. Second, the plant distance from the uncertain set point is the greatest and errors in the plant model will generate the largest target set errors because of the autonomous response and the magnitude of the control inputs required to move the plant state to the set point.

Along the trajectory, the contribution of the target (reference) maneuverability to set point uncertainty will diminish with time. This statement assumes that the target (reference) capability to change does not increase faster than the appropriate integral of its' input variable. Regardless of the initial maneuverability of the target, the time remaining is decreasing, and consequently, the

..









ability to move the set point decreases. Target motion is smaller and it's position is more and more certain.

Selection of the control inputs in the initial stages of the trajectory that will result in a steady state control (that contains the target reachable set within the plant reachable set) reduces target set uncertainty by establishing the plant operating point and defining the effective plant transfer function.

At this point, we do not have a control procedure, only the motivation to keep the target set within the reachable set of the plant along with a desire to attain steady state performance during the initial stages of the trajectory. The specific objectives are to minimize target set uncertainty, and most importantly, to maintain a controllable trajectory. The overall objective is better performance in terms of terminal errors.



Procedure



A workable control law that meets the objectives can be deduced from Figure 4.3. Here we have the same reachable set for the uncertain target, but this time, several missile origins are placed at the extremes of target motion. From these origins, the system is run backward from the final time to the current time using control values from the boundary of the control constraint set to provide a unique set of states that are controllable to the specific origin. If the intersection of these sets is non-empty, any potential target location is reachable from this intersection. Figure 4.4 is similar, but this time the missile control restraint set is not symmetric. Figure 4.4 shows a case where the missile acceleration control is constrained to the set


A= [AminAmax] where 0 :5 Amin:5 Amax


(17)

..











Target Reachable


z / "'- All target positions
Reachable




Figure 4.3 Intersection of Missile Reachable Sets Based on Uncertain Target Motion and Symmetric Constraints


Target Reachable Set


-All Target Locations Reachable
Figure 4.4 Intersection of Missile Reachable Sets Based on Uncertain Target Motion and Unsymmetric Constraints

..










Since controllability is assumed, which for constrained control includes the control bounds and the time interval, the extreme left and right (near and far) points of the set point are included in the set drawn from the origin.

To implement the technique, construct a dual system that incorporates functional constraints, uncontrollable modes, and uses a suitable control value from the control constraint set as the input. From the highest probability target position at the final time, run the dual system backward in time from the final boundary condition. Regulate the plant (system) to the trajectory defined by

the dual system. In this way, the fixed- final- time zero terminal error control is accomplished by re-formulating the problem as optimal regulation to the dual

trajectory.

In general, potential structures of the constraint set preclude a specific point (origin, center, etc.) from always being the proper input to the dual system.

Regulation to a "dual" trajectory from the current target position will insure that the origin of the target reachable set remains within the reachable set of the plant. Selection of a suitable interior point from the control restraint set as input to the dual system will insure that the plant has sufficient control power to prevent the target reachable set from escaping from the interior of the plant reachable set.

Based on unmodeled set point uncertainty, symmetric control constraints, and a double integrator for the plant, a locus exists that will keep the target in the center of the missile reachable set. If the set point is not changed, this trajectory can be maintained without additional inputs. For a symmetric control restraint set, especially as the time-to-go approaches zero, Reachable Set Control is control to a "coasting" (null control) trajectory.

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40

If the control constraints are not symmetric, such as Figure 4.4, a locus of points that maintains the target in the center of the reachable set is the trajectory based on the system run backward from the final time target location with the acceleration command equal to the midpoint of the set A. Pictured in Figures 4.2 to 4.4 were trajectories that are representative of the double integrator. Other plant models would have different trajectories.

Reachable Set Control is a simple technique for minimizing the effects of target set uncertainty and improving terminal the performance of a large class of systems. We can minimize the effects of modeling errors (or target set uncertainty) by a linear optimal regulator that controls the system to a steady state control. Given the well known and desirable characteristics of LQG synthesis, this technique can be used as the basis for control to the desired "steady state control" trajectory. The technique handles constraints by insuring an initially constrained trajectory. Also, since the large scale dynamics are controlled by the "dual" reference trajectory, the tracking problem be optimized to the response time of the system under consideration. This results in an "adaptable" controller because gains are based on plant dynamics and cost while the overall system is smoothly driven from some large displacement to a region where the relatively high gain LQG controller will remain linear.

..













CHAPTER V REACHABLE SET CONTROL EXAMPLE Performance Comparison Reachable Set and LQG Control

In order to demonstrate the performance of "Reachable Set Control" we will contrast its performance with the performance of a linear optimal controller when there is target set uncertainty combined with input constraints.

Consider, for example, the finite dimensional linear system:


d2x
= u x(t0) = x0 (1)
dt2

with the quadratic cost

1 C tf
J = --XfTPfxf + J u(r)Tu(r)dr (2)
22



where
tf E [0,oo) and
'1 _0


Application of maximum principle yields the following linear optimal control law:

I
u = + x(tf)(t-tf) (3)
'


x0 + x0 tf where x(tf) = (4)
(t f)3
1+
3,1

..












Appropriately defining t, to, and tf, the control law can be equivalently expressed in an open loop or feedback form with the latter incorporating the usual disturbance rejection properties. The optimal control will tradeoff the cost of the integrated square input with the final error penalty. Consequently, even in the absence of constraints, the terminal performance of the control is a function of the initial displacement, time allowed to drive the state to zero, and the weighting factor -1. To illustrate this, Figure 5.1 presents the terminal states (miss distance and velocity) of the linear optimal controller. This plot is a composite of trajectories with different run times ranging from 0 to 3.0 seconds. The figure presents the values of position and velocity at the final time t = tf that result f rom an initial position of 1000 feet and with velocity of 1000 feet/sec with -y = 10-4. Figure 5.2 depicts, as a function of the run time, the initial acceleration (at t = 0.) associated with each of the trajectories shown in Figure 5.1. From these two plots, the impact of short run times is evident: the miss distance will be higher, and the initial acceleration command will be greater. Since future set point (target) motion is unknown, the suboptimal feedback controller is reset at each sample time to accommodate this motion. The word reset is significant. The optimal control is a function of the initial condition at time t = to, time, and the final time. A feedback realization becomes a function of the initial condition and time to go only. In this case, set point motion (target set uncertainty) can place the controller in a position where the time-to-go is small but the state deviation is large.

..












Velocity
2000
0
-2000
-4000
-6000
-8000
-10000
-12000
-14000


800 1000 1200


Figure 5.1 Terminal Performance of Linear Optimal Control


Acceleration


0


-100000


-200000


-300000


-400000


0.5 1 1.5 2
Final Time

Figure 5.2 Initial Acceleration of Linear Optimal Control


0 200 400 600
Position

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44

While short control times will result in poorer performance and higher accelerations, it does not take a long run time to drive the terminal error to near zero. Also, from (4) we see that the terminal error can be driven to an arbitrarily small value by selection of the control weighting. Figure 5.1 presented the final values of trajectories running from 0 to 3 seconds. Figures 5.3 through 5.5 are plots of the trajectory parameters for the two second trajectory (with the same initial conditions) along with the zero control trajectory values. These values are determined by starting at the boundary conditions of the optimal control trajectory and running the system backward with zero acceleration. For example, if we start at the final velocity and run backwards in time along the optimal trajectory, for each point in time, there is a velocity (the null control velocity) that will take the corresponding position of the optimal control trajectory to the boundary without additional input. The null control position begins at the origin at the final time, and moving backward in time, is the position that will take the system to the boundary condition at the current velocity. Therefore, these are the positions and velocities (respectively) that will result in the boundary condition without additional input. As t => tf the optimal trajectory acceleration approaches zero. Therefore, the zero control trajectory converges to the linear optimal trajectory. If the system has a symmetric control constraint set, Reachable Set Control will control the system position to the zero control (constant velocity) trajectory.

..










Acceleration
0

-500

-1000 -1500 -2000 -2500


0.5 1 1.5
Time


Figure 5.3 Linear Optimal Acceleration vs Time


Velocity
1500 1000

500

0

-500

-1000

-1500


0 0.5 1 1.5 2
Time


Figure 5.4 Linear Optimal Velocity vs Time

..











Position
2000 1000


0


-1000


-2000


-3000


0.5 1 1.5


Time
Figure 5.5 Linear Optimal Position vs Time Consider now the same problem with input constraints. Since U(t) is a linear function of time and the final state, it is monotonic and the constrained optimal control is

1
u = SAT(- x(tf)(t-tf)) (5)
,y
In this case, controllability is in question, and is a function of the initial conditions and the time-to-go. Assuming controllability, the final state will be given by:


x0 + x0tf a(tl)SGN(x(tf)[tf-(tl/2)] x(tf) (tf-tl)3

1+
3-1

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47
where t1 is the switching time from saturated to linear control. The open loop switch time can be shown to be

t1 = tf + { 3(tf)2 6(xO+xtf)/a )1 (7)

or the closed loop control can be used directly. In either case, the optimal control will correctly control the system to a final state X(tf) near zero. Figures 5.6 through 5.8 illustrate the impact of the constraint on the closed loop optimal control. In each plot, the optimal constrained and unconstrained trajectory is shown.









Acceleration
1000

0

-1000 -2000 -3000

-4000 Constrained Acceleration
-5000 1 1 1
0 0.5 1 1.5 2
Time


Figure 5.6 Unconstrained and Constrained Acceleration

..












Velocity

2000


0


-2000


-4000


0.5 1 1.5
Tim e


Figure 5.7 Unconstrained and Constrained Velocity vs Time


Position



2000 1 500
Constrained Position
1 000

500

0 0.5 1 1.5
Time Figure 5.8 Unconstrained and Constrained Position vs Time

..










Now consider the effects of target set uncertainty on the deterministic optimal control by using the same control law for a 2.0 second trajectory where the boundary condition is not constant but changes. The reason for the target uncertainty and selection of the boundary condition can be seen by analyzing the components of the modeled system. Assume that system actually consists of an uncontrollable reference (target) plant as well as controlled (missile) plant with the geometry modeled by the difference in their states. Therefore, the final set point (relative distance) is zero, but the boundary condition along the controlled (missile) trajectory is the predicted target position at the final time. This predicted position at the final time is the boundary condition for the controlled plant.

Figures 5.9 through 5.11 are plots of linear optimal trajectories using the control law in (5,6). There are two trajectories in each plot. The boundary condition for one trajectory is fixed at zero, the set point for the other trajectory is the pointwise zero control value (predicted target state at the final time). Figures 5.9 through 5.11 demonstrate the impact of this uncertainty on the linear optimal control law by comparing the uncertain constrained control with the constrained control that has a constant boundary condition.

..










Acceleration
1 000

0

-1000

-2000 -3000

-4000 -5000


Velocity

2000


0


-2000


-4000


0.5 1 1.5 2
Time

Figure 5.9 Acceleration Profile
With and Without Target Set Uncertainty


0.5


1
Time


1 .5


Figure 5.10 Velocity vs Time
With and Without Target Set Uncertainty


Uncertain Target Set

..















Position

1400

1 2 0 0 . .

1000

800 600
Uncertain Target Set
400 200

0
0 0.5 1 1.5 2
Time

Figure 5.11 Position vs Time With and Without Target Set Uncertainty When there is target set uncertainty, simulated by the varying set point, the initial acceleration is insufficient to prevent saturation during the terminal phase. Consequently, the boundary condition is not met.

The final set of plots, Figures 5.12 through 5.14, contrast the performance of the optimal LQG closed loop controller that we have been discussing and the Reachable Set Control technique. In these trajectories, the final set point is zero but there is target set uncertainty again simulated by a time varying boundary condition (predicted target position) that converges to zero. Although properly shown as a fixed final time controller, the Reachable Set Control results in Figures 5.12 through 5.14 are from a simple steady state (fixed gain) optimal tracker referenced to the zero control trajectory r.

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52
The system model for each technique is

d2x
= u (8)
dt2
with x(t0) = (x-r)0

The linear optimal controller has a quadratic cost of
J (x-r)fTPf(x-r)f + -Y (9)



The reachable set controller minimizes

J = J [(x-r)TQ(x-r)+u(r)Tu(r)]dr (10)
b

And, in either case, the value for r(t) is the position that will meet the
boundary condition at the final time without further input.


Acceleration
1000
Reachable Set Control
0 . .

-1000 -2000 -3000 -4000
-5000 '
0 0.5 1 1.5 2
Tim e

Figure 5.12 Acceleration vs Time LQG and Reachable Set Control

..











Velocity 2000


0
. . . . . . . .



-2000 Reachable Set Control


-4000


0.5


1.5


Time


Figure 5.13 Velocity vs Time LQG and Reachable Set Control


Position


0.5 1 1.5 2
Time


Figure 5.14 Position vs Time LQG and Reachable Set Control


1400 1200 1000 800 600 400 200
0

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54

Summary


The improved performance of Reachable Set Control is obvious from Figure 5-14. While demonstrated for a specific plant, and symmetric control constraint set, Reachable Set Control is capable of improving the terminal performance of a large class of systems. It minimized the effects of modeling errors (or target set uncertainty) by regulating the system to the zero control state. The technique handled constraints and insured an initially constrained trajectory. The tracking problem could be optimized to the response time of the system under consideration by smoothly driving the system from some large displacement to a region where the relatively high gain LQG controller remained linear.

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CHAPTER VI
REACHABLE SET CONTROL FOR PREFERRED AXIS HOMING MISSILES

As stated in Chapter II, the most promising techniques that can extend the inertial point mass formulation are based on singular perturbations [37,38,39]. When applied to the preferred axis missile, each of these techniques leads to a controller that is optimal in some sense. However, a discussion of "optimality" notwithstanding, the best homing missile intercept trajectory is the one that arrives at the final "control point" with the highest probability of hitting the target. This probability can be broken down into autonomous and forced

events. If nothing is changed, what is the probability of a hit or what is the miss distance? If the target does not maneuver, can additional control inputs result in a hit? And, in the worse case, if the target maneuvers (or an estimation error is corrected) will the missile have adequate maneuverability to correct the trajectory? None of the nonlinear techniques based on singular perturbations attempt to control uncertainty or address the terminally constrained trajectories caused by increasing acceleration profile.

Unfortunately, an increasing acceleration profile has been observed in all of the preferred axis homing missile controllers. In many cases, the generic bank-to-turn missile of [11,18] was on all three constraints (NyNzP) during the latter portion of the trajectory. If the evading target is able to put the missile in this position without approaching it's own maneuver limits, it will not be possible for the missile to counter the final evasive maneuver. The missile is no longer controllable to the target set. The "standard" solution to the increasing acceleration profile is a varying control cost. However, without additional 55

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56

additional modifications, this type of solution results in a trajectory dependent control. As we have seen, Reachable Set Control is an LQG control implementation that moves the system to the point where further inputs are not required. A Reachable Set Controller that will reject target and system disturbances, can satisfy y both the mathematical and heuristic optimality requirements by minimizing the cost yet maintaining a controllable system.

Since the roll control has different characteristics, the discussion of the preferred axis homing missile controller using the Reachable Set Control technique will be separated into translational and roll subsystems. The translational subsystem has a suitable null control trajectory defined by the initial velocity and uncontrollable acceleration provided by the rocket motor. The roll subsystem, however, is significantly different. In order for the

preferred axis missile to function, the preferred axis must be properly aligned. Consequently, both roll angle error and roll rate should be zero at all times. In this case, the null control trajectory collapses to the origin.



Acceleration Control

System Model

Since we want to control the relative target-missile inertial system to the zero state, the controller will be defined in this reference frame. Each of the individual system states are defined (in relative coordinates) as target state minus missile state.

Begin with the deterministic system:


x(t) = Fx(t) + Gu(t)

..









where
X
Y

x= Vxu= Ny
Vy Nz
Vz

and

F- G-0 0 -I

Since the autopilot model is a linear approximation and the inertial model assumes instantaneous response, modeling errors will randomly affect the trajectory. Atmospheric and other external influences will disturb the system. Also, the determination of the state will require the use of noisy measurements. Consequently, the missile intercept problem should be approached via a stochastic optimal control law. Because the Reachable Set Control technique will minimize the effect of plant parameter variations (modeling errors) and unmodeled target maneuvers to maintain controllability, we can use an LQG controller. Assuming Certainty Equivalence, this controller consists of an optimal linear (Kalman) filter cascaded with the optimal feedback gain matrix of the corresponding deterministic optimal control problem. Disturbances and modeling errors can be accounted for by suitably extending the system description [40]:


x(t) = F(t)x(t) + G(t)u(t) + Vs(t) (2)


by adding a noise process Vs(-,.) to the dynamics equations with


Vs(t,w) E Rn

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58

Therefore, let the continuous time state description be formally given by the linear stochastic differential equation


dx(t) = F(t)x(t)dt + G(t)u(t)dt + L(t)dB(t) (4)


(with 13(.,.) a Wiener process) that has the solution:


x(t) = 4(t,t0)x(t0) + 1 (t,r)G(r)u(r)dr (5)

+ '1(t,r)L(r)dB(r)


characterized by a covariance and mean whose trajectory can be adequately represented as:


x(t) = F(t)x(t) + G(t)u(t) + Lws(t) (6)


where ws(.,.) is a zero mean white Gaussian noise of strength Ws(t) for all t.


Efws(t)ws(t)T) Ws(t) (7)


Disturbance Model

In the process of the intercept, it is expected that the target will attempt to counter the missile threat. While it is theoretically possible to have an adequate truth model and sufficiently sophisticated algorithms to adapt system parameters or detect the maneuvers, the short time of flight and maneuver

detection delays make this approach unrealistic at this time. Even though the actual evasive maneuvers will be discretely initiated and carried out in finite time, the effect of these maneuvers, combined with unmodeled missile states, appear as continuous, correlated and uninterrupted disturbances on the system. Therefore, even though a minimum square error, unbiased estimate can be made of the system state it would be very unusual for the estimates of the target

..







59

state to converge with zero error. Since the optimal solution to the linear stochastic differential equation is a Gauss-Markov process, time correlated processes can be included by augmenting the system state to include the disturbance process.

Let the time-correlated target (position) disturbance be modeled by the following:


T(t) = N(t)T(t) + wt(t) (8)

with

Tx(t)

T(t) = Ty(t)

Tz(t)

and
E(wt(t)wt(t)T) = Wt(t)


While the target disturbance resulting from an unknown acceleration is localized to a single plane with respect to the body axis of the target, the target orientation is unknown to the inertial model. Consequently, following the methodology of the Singer Model, each axis will be treated equally [41]. Since the disturbance is first order Markov, it's components will be:


N(t) = (1/Tc)[II (9)
and
Wt(t) = (2at2/Tc)[II (10)


where Tc is the correlation time, and orr is the RMS value of the disturbance process. The Power Spectral Density of the disturbance is: 2at2/Tc
'tt(w) = w2 + (I/Tc)2


Figure 6.1 summarizes the noise interactions with the system.

..














Atmospheric disturbances Actuator Errors Autopilot Errors


Linear System Physical Model


Target Accelerations


Figure 6.1 Reachable Set Control Disturbance processes.

With appropriate dimensions, the nine state (linear) augmented system model becomes:


x(t) F(t)I + r x~t) Ju(t) + 01 [ws(t) 1

T(t) O N(t) T(t) 0 JO M lwt(t)

Reference Model

Reachable Set Control requires a supervisory steering control (reference) that includes the environmental impact on the controlled dynamic system. Recalling the characteristics of the dual system, one was developed that explicitly ran (1) backward in time after determining the terminal conditions. However, in developing this control for this preferred axis missile a number of factors actually simplify the computation of the reference trajectory:

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61

(1) The control constraint set for this preferred axis missile is symmetric. Consequently, the reference trajectory for an intercept condition, is a null control (coasting) trajectory.

(2) The body axis X acceleration is provided by the missile motor, and is not controllable but known. This uncontrollable acceleration will contribute to the total inertial acceleration vector, must be considered by the controller, and is the only acceleration present on an intercept (coasting) trajectory.

(3) The termination of the intercept is the closest approach, which now becomes the fixed-final-time (tf). The time-to-go (tgo) is defined with respect to the current time (t) by:

t = tf tgo (13)

(4) The final boundary condition for the system state (target minus missile) is zero.

In summary, the intercept positions are zero, the initial velocity is given, and the average acceleration is a constant. Therefore, it is sufficient to reverse the direction of the initial velocity and average acceleration then run the system forward in time for tgo seconds from the origin to determine the current position of the coasting trajectory. Let: r(t) = A(t)r(t) + B(t)a(t) (14)
with
r(O) = r0 = 0
and
A(t) = F(t) and B(t) = G(t)


then r(t) is the point from which the autonomous system dynamics will take the system to desired boundary condition.

Because of the disturbances, target motion, and modeling errors, future control inputs are random vectors. Therefore, the best policy is not to

..







62

determine the input over the control period [tQ,tf] a priori but to reconsider the situation at each instant t on the basis of all available information. At each update, if the system is controllable, the reference (and system state) will approach zero as tgo approaches zero.

Since the objective of the controller is to drive the system state to zero, we do not require a tracker that will maintain the control variable at a desired non-zero value with zero steady state error in the presence of unmodeled constant disturbances. There are disturbances, but the final set point is zero, and therefore, a PI controller is not required.



Roll Control

Definition

The roll mode is most significant source of modeling errors in the preferred axis homing missile. While non-linear and high order dynamics associated with the equations of motion, autopilot, and control actuators are

neglected, the double integrator is an exact model for determining inertial position from inertial accelerations. The linear system, however, is referenced with respect to the body axis. Consequently, to analyze the complete dynamics, the angular relationship between the body axis and inertial references must be considered. Recall Friedland's linearized (simplified) equations. The angular relationships determine the orientation of the body axis reference and the roll rate appeared in the dynamics of all angular relationships. Yet, to solve the system using linear techniques, the system must be uncoupled via a steady state (Adiabatic) assumption. Also, the roll angle is inertially defined and the effect of the linear accelerations on the error is totally neglected.

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63

From a geometric point of view, this mode controls the range of the orthogonal linear acceleration commands and the constrained controllability of the trajectory. With a 20:1 ratio in the pitch and yaw accelerations, the ability to point the preferred axis in the "proper" direction is absolutely critical. Consequently, effective roll control is essential to the performance of the preferred axis homing missile.

The first problem in defining the roll controller, is the determination of the "proper" direction. There are two choices. The preferred axis could be aligned with the target position or the direction of the commanded acceleration. The first selection is the easiest to implement. The seeker gimbal angles provide a direct measure of intercept geometry (Figure 6.2), and the roll angle error is defined directly:


O= Tan- (sin(obg)/tan(Og)) (15)













g l g



Y

z

Figure 6.2. Roll Angle Error Definition from Seeker Angles.

This selection, however, is not the most robust. Depending on the initial geometry, the intercept point may not be in the plane defined by the current

..







64

line of sight (LOS) and longitudinal axis of the missile. In this case, the missile must continually adjust its orientation (roll) to maintain the target in the preferred plane. As range decreases the angular rates increase, with the very real possibility of saturation and poor terminal performance.

Consequently, the second definition of roll angle error should be used. Considering the dynamics of the intercept, however, aligning the preferred axis with the commanded acceleration vector is not as straightforward as it seems. Defining the roll angle error as

Oe = Tan-I(Ay/Az) (16)
leads to significant difficulties. From the previous discussion, it is obvious that roll angle errors must be minimized so that the preferred axis acceleration can be used to control the intercept. The roll controller must have a high gain. Assume, for example, the missile is on the intercept trajectory. Therefore, both Ay and Az will be zero. Now, if the target moves slightly in the Yb direction and the missile maneuvers to correct the deviation, the roll angle error instantaneously becomes 90 degrees. High gain roll control inputs to correct this situation are counter productive. The small Ay may be adequate to completely correct the situation before the roll mode can respond. Now, the combination of linear and roll control leads to instability as the unnecessary roll rate generates errors in future linear accelerations.

The problems resulting from the definition of equation 16 can be overcome be re-examining the roll angle error. First, Ay and Az combine to generate a resultant vector at an angle from the preferred axis. In the process of applying constraints, the acceleration angle that results from the linear accelerations can be increased or decreased by the presence of the constraint. If the angle is decreased, the additional roll is needed to line up the preferred axis and the

..







65

desired acceleration vector. If the constrained (actual) acceleration angle is increased beyond the (unconstrained) desired value by the unsymmetric action of the constraints, then the roll controller must allow for this "over control" caused by the constraints.

Define the roll angle error as the difference between the actual and

desired acceleration vectors after the control constraints are considered. This definition allows for the full skid to turn capability of the missile in accelerating toward the intercept point and limits rolling to correct large deviations in acceleration angle from the preferred axis that are generated by small accelerations.

Referring to Figure 6.3, three zones can be associated with the following definitions:

Oec = Tan-(Ay/Az) (17)

Oea = Tan-1(Ny/Nz) (18)

Oer = Oec Oea (19)
A Zone I

Zone 11


Zone III


/b







Zb


Figure 6.3. Roll Control Zones.

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66

Here NY and Nz are the constrained acceleration values. In Zone 1, Oer = 0The linear acceleration can complete the intercept without further roll angle change. This is the desired locus for the roll controller. Both Ay and Az are limited in Zone II. This is the typical situation for the initial position of a demanding intercept. The objective of the roll controller is to keep the intercept acceleration out of Zone III where only Ay is limited. In this case, the Ay acceleration is insufficient to complete the intercept yet significant roll angle change may be required to make the trajectory controllable. Controller

A dual mode roll controller was developed to accommodate the range of situations and minimize roll angle error. Zone I requires a lower gain controller that will stabilize the roll rate and maintain Oer small. Zones II and III require high gain controllers. To keep Zone II trajectories from entering Zone III, the Oec will be controlled to zero rather than the roll angle error. Since the linear control value is also a function of the roll angle error, roll angle errors are determined by comparing the desired and actual angles of a fixed high gain reachable set controller. If the actual linear commands are used and a linear acceleration is small because of large roll angle errors, the actual amount of roll needed to line up the preferred axis and the intercept point, beyond the capability of the linear accelerations, will not be available because they have been limited by the existing roll angle error that must be corrected.

Unlike the inertial motions, the linear model for the roll controller accounts for the (roll) damping and recognizes that the input is a roll rate change command:


0 = -WO + WP (20)

..







67

Therefore, the roll mode elements (that will be incorporated into the model are) are:

Fr -- Gr= (21)


Also, the dual mode controller will require an output function and weighting matrix that includes both roll angle and roll rate.



Kalman Filter



The augmented system model (13) is not block diagonal. Consequently, the augmented system filter will not decouple into two independent system and

reference filters. Rather, a single, higher order filter was required to generate the state and disturbance estimates.

A target model (the Singer model) was selected and modified to track maneuvering targets from a Bank-To-Turn missile [41,42]. Using this model, a continuous-discrete Extended Kalman Filter was developed. The filter used a 9 state target model for the relative motion (target missile):



I F I + [ u(t) + ( 0 I (22)

aT(t) O N aT(t) 0 -N wt(t)


with u(t) the known missile acceleration, N the correlation coefficient, and wt(t) an assumed Gaussian white noise input with zero mean.

Azimuth, elevation, range, and range rate measurements were available from passive IR, semi-active, analog radar, and digitally processed radar sensors. The four measurements are seeker azimuth (0), seeker elevation (0), range (r), and range rate (dr/dt):

..







68

0 = -Tan- 1(z(x2+y2)-1/2) (23)

= +Tan-l(y/x) x > 0

= r + Tan-l{y/x) x< 0

r = (x2+y2+z2-1/2

r = {xx+yy+zz}{x2+y2+z2}-1/2


Noise statistics for the measurements are a function of range, and are designed to simulate glint and scintillation in a relatively inexpensive missile seeker. In contrast to the linear optimal filter, the order of the measurements for the extended filter is important. In this simulation, the elevation angle (0) was processed first, followed by azimuth (0t), range (r), and range rate (dr/dt). In addition, optimal estimates were available from the fusion of the detailed (digital) radar model and IR seeker.



Reachable Set Controller



Structure

The Target-Missile System is shown in Figure 6.4. The combination of the augmented system state and the dual reference that generates the minimum control trajectory for the reachable set concept is best described as a Command Generator/Tracker and is shown in Figure 6.5. In a single system of equations the controller models the system response, including time correlated position disturbances, and provides the reference trajectory. Since only noise-corrupted measurements of the controlled system are available, optimal estimates of the actual states were used.

Because of the processing time required for the filter and delays in the autopilot response, a continuous-discrete Extended Kalman Filter, and a sampled

..







69

data (discrete) controller was used. This controller incorporated discrete cross-coupling terms to control the deviations between the sampling times as well the capability to handle non-coincident sample and control intervals (Appendices B and C).

Combining the linear and roll subsystems with a first order roll mode for the roll angle state, the model for the preferred axis homing missile becomes:



x(t) F(t) I x(t +[ G(t) u(t) + L 0 w)(t)
T(t) N(t) T(t) 0 O wt(t)


the reference:


r(t) = A(t)r(t) + B(t)a(t) (25)


with the tracking error:

rx(t)
e(t) --[ Yx(t) Yr(t) ] =[ H(t) 1 0 1 -C(t) ] T(t) (26)
Ir(t)J


The initial state is modeled as an n-dimensional Gaussian random variable with mean x0 and covariance P0. E(ws(t)ws(t)T) = Ws(t) is the strength of the system (white noise) disturbances to be rejected, and E{wt(t)wt(t)T} = Wt(t) is an input to a stationary first order Gauss-Markov process that models target acceleration. The positions are the primary variables of interest, and the output matrices will select these terms. Along with the roll rate, these are the variables that will be penalized by the control cost and the states where disturbances will directly impact the performance of the system.

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70

The components of the controlled system are:


x(t) y(t) z(t) o(t)

x(t) y(t) z(t) o(t) Nx(t) Ny(t) Nz(t) P(t)


'Tx(t) T(t) Ty(t)

Tz(t)


















e(t) -


cos(6)cos(4)
A
a(t) cos(O)sin(O) A

-sin(O)


(27)


r(t) =


x(t) y(t) z(t) 0(t)

0

0

0

O(t)


rx(t) ry(t) rz(t) ro(t) rx(t) ry(t) rz(t) ro(t) rx(t) ry(t) rz(t) ro(t)

0 0 0

ro(t)


A
where A is the average acceleration from the rocket motor. In block form, with appropriate dimensions, the system matrices are:


F(t) = A(t) = F= G(t) = B(t) = G= 0J


(28)


H(t) = C(t) = [ I hw I


N(t) = (l/Tc)[I]

L(t) = I


where the Ow, Iw, and hw terms are required to specify the roll axis system and control terms:


x(t) =


u(t) =


M(t) = I

..









i=j=8 1 i=j=8
{ 0 otherwise 1 0 otherwise


{ +w i=8,j=4
(Iw)ij I otherwise


The performance objective for the LQG synthesis is to minimize an
appropriate continuous-time quadratic cost: Js(t) = E(Jd(t)II(t)} (29)

where JS is the stochastic cost, 1(t) is the information set available at time t, and Jd a deterministic cost function:


Jd = efTPfef + J {e(r)TQ(r)e(r)+u(r)TR(r)u(r))dr (30)


Dividing the interval of interest into N+1 intervals for discrete time control, and summing the integral cost generates the following (see Appendix C):

Jd = e(tN+ I )Tp(tN+ 1 )e(tN+ 1) (31)
N [ e(ti)T ] wxx(ti) Wxu(ti) ] e(ti)]
+ E
i-0 u(ti) Wxu(ti)T Wuu(ti) u(ti)


which can be related to the augmented state X = [ x T r ]T by:

Jd = X(tN+ 1)Tp(tN+l )X(tN+ i) (32)
N [ X(ti)T Q(t1) S(t1) X(ti)
+ EH
i=0 u(ti) S(ti)T R(ti) u(ti)

..







72

In general, with the cost terms defined for the augmented state (Appendix C), the optimal (discrete) solution to the LQG tracker can be expressed as: [x(ti)
u*(ti) = -[G*(ti)] T(ti) (33)
1 r(ti)
where

G*(ti) = [ R(ti) + GT(ti)P(ti+l)G(ti) ] -1 (34)

GT(ti)P(ti+l)4(ti+l,ti) + sT(ti) I
and

P(ti) = Q(ti) + 4T(ti+l,)P(ti+l)4(ti+l,) (35)

[ GT(ti)P(ti+l)4D(ti+l,) + sT(ti) ]TG*(ti)


Since only the positions (and roll rate) are penalized, the Riccati recursion is quite sparse. Consequently, by partitioning the gain and Riccati equations, and explicitly carrying out the matrix operations, considerable computational improvements are possible over the straightforward implementation of a 19 by 19 tracker (Appendix D).

Application

The tracking error and control costs were determined from the steady

state tracker used in the example in Chapter 5. First, missile seeker and

aerodynamic limitations were analyzed to determine the most demanding intercept attainable by the simulated hardware. Then, autopilot delays were incorporated to estimate that amount of time that a saturated control would require to turn the missile after correcting a 90 degree (limit case) roll angle error. The steady state regulator was used to interactively place the closed loop poles and select a control cost combination that generated non zero control for the desired length of time. These same values were used in the time varying Reachable Set Controller with the full up autopilot simulation to determine the

..







73

terminal error cost and control delay time. To maintain a basis of comparison, the Kalman Filter parameters were not modified for this controller. Appendix E contains initial conditions for the controller and estimator dynamics.

During the initialization sequence (safety delay) for a given run, time varying fixed- final-time LQG regulator gains are calculated (via 36) based on the initial estimate of the time to go. Both high and low roll control gains were computed. These solutions used the complete Riccati recursion and cost based on the sampled data system, included a penalty on the final state (to control transient behavior as tgo approaches zero), and allowed for non-coincident sample and control.

Given an estimated tgo, at each time t, the Command Generator / Tracker computed the reference position and required roll angle that leads to an intercept without additional control input. The high or low roll control gain was selected based on the mode. Then the precomputed gains (that are a function of tgo) are used with the state and correlated disturbance estimate from the filter, roll control zone, and the reference r to generate the control (which is applied only to the missile system). Because of symmetry, the tracker gain for the state term equaled the reference gain, so that, in effect, except for the correlated noise, the current difference between the state x and the reference r determined the control value.

During the intercept, between sample times when the state is extrapolated by the filter dynamics, tgo was calculated based on this new extrapolation and appropriate gains used. This technique demonstrated better performance than using a constant control value over the duration of the sample interval and justified the computational penalty of the continuous discrete implementation of the controller and filter.

..










Measuremrenit Noise D


Reachable Set Controller


Conti nuous- Discrete Kalman Filter


Measurement Noise


Dynamic Disturbances
Figure 6.4 Target Missile System


Missile

..










Reference Variable Dynamics


Yr(t) e(t)
. . . C . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . -. 0 . . . . . . . o


/namic Disturbances I Measurement Noise Yx(t)


Z(t)


Continuous
Discrete


Kalman
Filter


Figure 6.5 Command Generator / Tracker


Missile Target


System

..













CHAPTER VII
RESULTS AND DISCUSSION

As an additional reference, before comparing the results of Reachable Set Control to the baseline control, consider an air-to-air missile problem from [13]. In this example, the launch direction is along the line of sight, the missile velocity is constant, and the autopilot response to commands is instantaneous. The controller has noisy measurements of target angular location, a priori knowledge of the time to go, and stochastically models the target maneuver. Even with this relatively simple problem, the acceleration profile increases sharply near the final time. Unfortunately, this acceleration profile is typical, and has been observed in all previous optimal control laws. Reachable Set Control fixes this problem.




300
0


200



too

E
ad 01

12 6 0
Time to go (sec)



Figure 7.1 RMS Missile Acceleration 76

..












Simulation



The performance of Reachable Set Control was determined via a high fidelity Bank-To-Turn simulation developed at the University of Florida and used for a number of previous evaluations. The simulation is based on the coupled non-linear missile dynamics of chapter 11 equations (1) to (8) and is a continuous -discrete system that has the capability of comparing control laws and estimators at any sample time. In addition to the non-linear aerodynamic parameters, the simulation models the Rockwell Bank-To-Turn autopilot, sensor (seeker and accelerometer) dynamics, has a non-standard atmosphere, and mass model of the missile to calculate time-varying moments of inertia and the missile specific acceleration from the time varying rocket motor.

Figure 7.2 presents the engagement geometry and some of the variables used to define the initial conditions.








LOS



Missile


Figure 7.2 Engagement Geometry

..







78

The simulated target is a three (3) dimensional, nine (9) "g" maneuvering target. Initially, the target trajectory is a straight line. Once the range from the missile to target is less than 6000 feet, the target initiates an instantaneous 9 "g" evasive maneuver in a plane determined by the target roll angle, an input parameter. If the launch range is within 6000 feet, the evasive maneuver begins immediately. There is a .4 second "safety" delay between missile launch and autopilot control authority.

Trajectory Parameters

The performance of the control laws was measured with and without sensor noise using continuous and sampled data measurements. The integration step was .005 seconds and the measurement step for the Extended Kalman Filter was .05 seconds. The trajectory presented for comparison has an initial offset angle of 40 degrees (0g) and 180 degree aspect (0a), and a target roll of 90 degrees away from the missile. This angle off and target maneuver is one of the most demanding intercept for a preferred axis missile since it must roll through 90 degrees before the preferred axis is aligned with the target. Other intercepts were run with different conditions and target maneuvers to verify the robustness of Reachable Set Control and the miss distances were similar or less that this trajectory.



Results

Deterministic Results

These results are the best comparison of control concepts since both Linear Optimal Control and Reachable Set Control are based on assumed Certainty Equivalence.

..







79

Representative deterministic results are presented in Table 7.1 and Figure 7.3. Figures A.1 through A.9 present relevant parameters for the 4000 foot deterministic trajectories.



Table 7.1 Deterministic Control Law Performance Initial Control Time Miss
Range Distance
(feet) (sec) (feet)


5500 Baseline 2.34 8
Reachable 2.34 6

5000 Baseline 2.21 13
Reachable 2.21 10

4800 Baseline 2.17 15
Reachable 2.17 4

4600 Baseline 2.13 29
Reachable 2.13 6

4400 Baseline 2.06 38
Reachable 2.08 7

4200 Baseline 2.02 35
Reachable 2.05 5

4000 Baseline 1.98 54
Reachable 2.00 13

3900 Baseline 1.98 43
Reachable 1.98 8

3800 Baseline 1.98 40
Reachable 1.98 8

3700 Baseline 2.02 44
Reachable 1.99 10

3600 Baseline 1.99 136
Reachable 1.99 65

..











Miss Distance (feet)

140 1 20 1 00
80 60
4 0 .
20

B 500 Z


0OGO


4500
initial Rainge


Figure 7.3 Deterministic Results


An analysis of trajectory parameters revealed that one of the major performance limitations was the Rockwell autopilot. Designed for proportional navigation with noisy (analog) seeker angle rates, the self adaptive loops in the autopilot penalized a high gain control law such as Reachable Set Control. This penalty prevented Reachable Set Control from demonstrating quicker intercepts and periodic control that were seen with a perfect autopilot on a similar simulation used during the research. However, even with the autopilot penalty, Reachable Set Control was able to significantly improve missile performance near the inner launch boundary. This verifies the theoretical analysis, since this is the region where the target set errors, control constraints, and short run times affect the linear law most significantly.


Baseline Guidance Law


Reachable Set Control


5000


5500

..










Stochastic Results

Stochastic performance was determined by 100 runs at each initial condition. At the termination of the run, the miss distance and Time of Flight (TOF) was recorded. During each of these runs, the estimator and seeker

(noise) error sequences were tracked. Both sequences were analyzed to insure gaussian seeker noise, and an unbiased estimator (with respect to each axis). From the final performance data, the mean and variance of the miss distance was calculated. Also, from the estimator and seeker sequences, the root mean square (RMS) error and variance for each run was determined to identify some general characteristics of the process. The average of these numbers is presented. Care must be taken in interpreting these numbers. Since the measurement error is a function of the trajectory as well as instantaneous trajectory parameters, a single number is not adequate to completely describe the total process. Table 7.2 and Figure 7.4 present average results using the guidance laws with noisy measurements and the Kalman filter. Miss Distance (f eet)


2 5 0 ----------Baseline Guidance Law
200 150

1 00.

50RecalSe Coto


S5 500 40 00 45 00 50 00 5-5 00
Initial Rainge

Figure 7.4 Stochastic Results

..










Table 7.2 Stochastic Control Law Performance


Initial Control Range (feet)


5500 Baseline
Reachable

5000 Baseline
Reachable

4800 Baseline
Reachable

4600 Baseline
Reachable

4400 Baseline
Reachable

4200 Baseline
Reachable

4000 Baseline
Reachable

3900 Baseline
Reachable

3800 Baseline
Reachable

3700 Baseline
Reachable

3600 Baseline
Reachable


Time (sec)


2.38
2.41 2.23 2.25

2.18 2.19 2.13 2.13

2.08 2.07

2.04 2.03

2.01 2.01 2.00 2.00


2.00 1.99

1.98 1.98

1.95 1.95


Miss Distance Mean Variance (feet)


RMS Error EKF Seeker (feet)(deg)


273 11470 11
83 2677 11


240 7874 89 3937

193 7540 114 3708

172 5699 107 1632

129 4324 85 1421

123 3375 62 673

105 2745 66 1401

105 3637 79 4356

124 5252 105 10217


159 5240 10
176 13078 9


230 6182 239 14808

..







83

The first runs made with Reachable Set Control were not as good as the results presented. Reachable Set Control was only slightly (10 to 20 feet) superior to the baseline guidance law and was well below expectations. Yet, the performance of the filter with respect to position error was reasonable, many of the individual runs had miss distances near 20 feet, and most of the errors were in the Z axis. Analyzing several trajectories from various initial conditions led to two main conclusions. First, the initial and terminal seeker errors were quite large, especially compared to the constant 5 mrad tracking accuracy assumed by many studies [2,3,18]. Second, the non-linear coupled nature of the preferred axis missile, combined with range dependent seeker errors, and the system (target) model, makes the terminal performance a strong function of the particular sequence of seeker errors. For example, Figure 7.5 compares the actual and estimated Z axis velocity (Target Missile) from a single 4000 foot run. The very first elevation measurement generated a 14 foot Z axis position error. A reasonable number considering the range. The Z axis velocity error, however, was quite large, 409 feet per second, and never completely eliminated by the filter. Recalling that the target velocity is 969 feet per second, is approximately co-altitude with the missile and maneuvers primarily in the XY plane, this error is significant when compared to the actual Z axis velocity (2 feet per second). Also, this is the axis that defines the roll angle error and, consequently, roll rate of the missile. Errors of this magnitude cause the primary maneuver plane of the missile to roll away from the target limiting (via the constraints) the ability of the missile to maneuver.

Further investigation confirmed that the filter was working properly. Although the time varying noise prevents a direct comparison for an entire trajectory, these large velocity errors are consistent with the covariance ratios

..







84

in [41]. The filter model was developed to track maneuvering targets. The penalty for tracking maneuvering targets is the inability to precisely define all of the trajectory parameters (ie. velocity). More accurate (certain) models track better, but risk losing track (diverging) when the target maneuvers unexpectedly. The problem with the control then, was the excessive deviations in the velocity. To verify this, the simulation was modified to use estimates of position, but to use actual velocities. Figure 7.6 and Table 7.3 has these results. As seen from the table, the control performance is quite good considering the noise statistics and autopilot.











Z Axis Velocity (feet/sec)


600 400
200 Actudl
2 0 0 Ilk .
u ."".

. . -"
-200
-400 Estimate

-600

0 0.5 1 1.5 2
Time


Figure 7.5 Measured vs Actual Z Axis Velocity

..












Table 7.5 Stochastic Control Law Performance
Using Actual Velocities

Initial Control Time Miss Distance RMS Error
Range Mean Variance EKF Seeker
(feet) (sec) (feet) (feet) (deg)


5500 Baseline 2.35 34 318 10 1.5
Reachable 2.37 38 504 11 1.5

5000 Baseline 2.21 50 506 10 1.7
Reachable 2.23 38 623 10 1.6

4800 Baseline 2.17 55 367 10 1.6
Reachable 2.19 48 526 10 1.5

4600 Baseline 2.12 61 366 10 1.6
Reachable 2.14 52 326 10 1.6

4400 Baseline 2.06 62 333 10 1.7
Reachable 2.10 45 415 10 1.7

4200 Baseline 2.02 60 277 10 1.7
Reachable 2.06 44 463 10 1.8

4000 Baseline 1.98 62 330 10 1.9
Reachable 2.02 53 1011 9 1.9

3900 Baseline 1.98 55 436 10 1.9
Reachable 2.00 64 2052 10 1.9

3800 Baseline 1.98 53 400 9 2.0
Reachable 1.99 91 2427 10 1.8

3700 Baseline 1.99 63 235 10 2.0
Reachable 1.99 140 3110 10 1.7

3600 Baseline 1.98 138 354 10 1.8
Reachable 1.96 213 4700 10 1.6

..

PAGE 1

REACHABLE SET CONTROLFORPREFERREDAXIS HOMING MISSILESByDONALDJ.CAUGHLIN,JR. A DISSERTATION PRESENTED TOTHEGRADUATESCHOOL OFTHEUNIVERSITYOFFLORIDA IN PARTIALFULFILLMENTOFTHEREQUIREMENTS FORTHEDEGREEOFDOCTOR OF PHILOSOPHY UNIVERSITYOFFLORIDA1988

PAGE 2

Copyright 1988ByDONALD J.CAUGHLINJR.

PAGE 3

To BarbaraAmyJon

PAGE 4

ACKNOWLEDGMENTS The author wishestoexpress his gratitude to his committee chairman, Dr. T.E Bullock, for his instruction, helpful suggestions, and encouragement. Appreciationisalso expressed for the support and many helpful commentsfromthe other committee members, Dr. Basile, Dr. Couch, Dr. Smith, and Dr. Svoronos. iv

PAGE 5

TABLE OF CONTENTS ACKNOWLEDGMENTSivLIST OF FIGURES viiKEYTO SyMBOLS ix ABSTRACTxivCHAPTER I INTRODUCTION 1 IIBACKGROUND4 Missile Dynamics 5 Linear Accelerations 6 Moment Equations (i Linear Quadratic Gaussian Control Law 7 III CONSTRAINED CONTROL13IV CONSTRAINED CONTROL WITH UNMODELED SETPOINT ANDPLANTVARIA TIONS25Linear Optimal Control with Uncertainty and Constraints31Control Technique32Discussion36Procedure.37V REACHABLE SET CONTROL EXAMPLE.41Performance Comparison Reachable SetandLQGControl....41Summary.54VI REACHABLE SET CONTROL FORPREFERREDAXIS HOMING MISSILES55Acceleration Control.56System Model.56Disturbance Model58Reference Model.60Roll Control. 62 Definition 62 Controller 66 Kalman FiIter67Reachable Set Controller68Structure 68 Application 72 v

PAGE 6

VII RESULTS AND DISCUSSION76Simulation77Trajectory Parameters78Results78Deterministic Results78Stochastic Results81Conclusions87Reachable Set Control. 87 Singer Model. 87APPENDIXA SIMULATION RESUL TS88BSAMPLED-DATA CONVERSION 94 System Model 94 Sampled Data Equations 96 System 96 Target Disturbance 98 Minimum Control Reference 99 Summary 1 00 C SAMPLED DATA COST FUNCTIONS101DLQGCONTROLLERDECOMPOSITION.107 ECONTROLLERPARAMETERS111Control Law111Filter 112 LISTOFREFERENCES 113 BIOGRAPHICAL SKETCH 117 vi

PAGE 7

LISTOFFIGURES Figure Page2.1Missile Reference System 44.1Feedback System and Notation284.2 Reachable Set Control Objective334.3 IntersectionofMissile Reachable Sets Based on Uncertain Target Motion and Symmetric Constraints.384.4 IntersectionofMissile Reachable Sets Based on Uncertain Target Motion and U nsymmetric Constraints385.1Terminal PerformanceofLinear Optimal Control...435.2 Initial AccelerationofLinear Optimal Control.435.3 Linear Optimal AccelerationvsTime455.4 Linear Optimal VelocityvsTime455.5 Linear Optimal PositionvsTime lt6 5.6 Unconstrained and Constrained Acceleration .47 5.7 Unconstrained and Constrained Velocity vs Time.485.8 Unconstrained and Constrained PositionvsTime.485.9 Acceleration Profile With and Without Target Set Uncertainty 50 5.10 VelocityvsTime With and Without Target Set Uncertainty 505.11PositionvsTime With and Without Target Set Uncertainty515.12 AccelerationvsTime LQG and Reachable Set Control. 52 vii

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5.13 VelocityvsTime LQG and Reachable Set Control.535.14 PositionvsTime LQG and Reachable Set Control.536.1Reachable Set Control Disturbance processes606.2. Roll Angle Error Definition from Seeker Angles636.3. Roll Control Zones 65 6.4 Target Missile System746.5 Command Generator/Tracker757.1RMS Missile Acceleration767.2 Engagement Geometry777.3 Deterministic Results807.4 Stochastic Results817.5 MeasuredvsActual Z Axis Velocity 84 7.6 Performance Using Position Estimates and Actual Velocities86A.I XY Missile&Target Positions Reachable Set Control.89A.2 XY Missile&Target Positions Baseline Control Law89A.3XZMissile&Target Positions Reachable Set Control.90AAXZMissile&Target Positions Baseline Control Law 90 A.5 Missile Acceleration Reachable Set Control...91A.6 Missile Acceleration Baseline Control Law91A.7 Missile Roll Commands&Rate Reachable Set Control.92A.8 Missile Roll Commands&Rate Baseline Control Law92A.9 Missile Roll Angle Error93viii

PAGE 9

a()B()q.)D()DODOwtDoc DocDq DO E() F() G()KEYTO SYMBOLS Reference controlinputvector. Missile inertial x axis acceleration. Target inertial x axis acceleration. Specific force (drag) along X body axis. Desired linear acceleration about Z and Y body axes. Reference controlinputmatrix. Reference stateoutputmatrix. Feedforward stateoutputmatrix. Stability parameter Equilibrium drag coefficient. Stability parameter Change in drag due to weight. Stability parameter Changeindrag due to velocity. Stability parameter Change in drag due to angleofattack. Stability parameter Change in drag due to angleofattack rate. Stability parameter Change in drag due to pitch rate. Stability parameter Change in drag due to pitch angle. Stability parameter Changeindrag due to pitch canard deflection angle. Feedforward referenceoutputmatrix. Roll angle error. System matrix describing the dynamic interaction between state variables. System controlinputmatrix. Optimal control feedback gain matrix. ix

PAGE 10

G l(ti) Optimal system state feedback gain matrix. G2(ti) Optimal target state feedback gain matrix. G3(ti) Optimal reference state feedback gain matrix. g Acceleration duetogravity. H(.) System state output matrix. Ixx,Iyy,Izz Momentofinertial withrespect to the given axis.JCosttogofunction for the mathematical optimization. L() System noise input matrix.LOStability parameter Equilibrium change in Z axis velocity.LOwt Stability parameter Change in Z axis velocity duetoweight. L u Stability parameter Change in Z axis velocity duetoforward velocity.LexLexStability parameter Change in Z axis velocity duetoangleofattack. Stability parameter Change in Z axis velocity due to angleofattack rate. Stability parameter Change in Z axis velocity duetopitch rate. Stability parameter Change in Z axis velocity duetopitch angle. Stability parameter Change in Z axis velocity duetopitch canard deflection angle. Stability parameter Equilibrium change in roll rate. Stability parameter Change in roll rate duetosideslip angle. Stability parameter Change in roll rate due to sideslip angle rate. Stability parameter Change in roll rate due to roll rate. Stability parameter Change in roll rate due to yaw rate. Stability parameter Change in roll rate due to roll canard deflection angle. Stability parameter Change in roll rate due to yaw canard deflection angle.x

PAGE 11

M Massofthemissile.MOStabilityparameter Equilibriumpitchrate. M u StabilityparameterChangeinpitchrateduetoforwardvelocity.McxStabilityparameterChange inpitchratedueto angleofattack.McxStabilityparameterChange inpitchratedueto angleofattackrate. M q StabilityparameterChange inpitchrateduetopitchrate. MOe StabilityparameterChangeinpitchrateduetopitchcanarddeflectionangle.NOStabilityparameter-Equilibriumyaw rate. N B Stabilityparameter Change in yaw ratedueto sideslip angle. N B StabilityparameterChange in yaw ratedueto sideslip angle rate. N p Stabilityparameter Change in yawratedueto roll rate. N r StabilityparameterChange in yawrateduetoyawrate. NOa StabilityparameterChangeinyawratedueto rollcanarddeflectionangle. NOr StabilityparameterChangeinyawratedueto yawcanarddeflectionangle.Nx,Ny,NzComponentsofappliedaccelerationonrespective missilebodyaxis. P Solution totheRiccatiequation.P,Q,RAngularratesabouttheX,Y,andZbodyaxis respectively.Q(.)State weightingmatrix.R(.)Controlweightingmatrix.R(.)Referencestate vector. S()State-Controlcross weighting matrix. T(.)Targetdisturbancestate vector. TgoTime-to-go.U Systeminputvector. xi

PAGE 12

V,V,W Linear velocities with respect to the X,Y,andZbodyaxis respectively.vs System noise process . Zero mean white Gaussian noise modeling uncorrelated state disturbances. Zero mean white Gaussian noise driving firstorderMarkov process modeling correlated state disturbances. IVtotlX(.)X,Y,ZYoYOwtex B Total missile velocity. System state vector. Body stabilized axis. Stability parameter Equilibrium change inYaxis velocity. Stability parameter Change inYaxis velocity due to weight. Stability parameter Change in Y axis ratedueto sideslip angle. Stability parameter Change in Y axis velocitydueto sideslip angle rate. Stability parameter ChangeinY axis velocitydueto roll rate. Stability parameter ChangeinY axis velocity due to yaw rate. Stability parameter ChangeinY axis velocity due to roll angle. Stability parameter Change in Y axis velocity due to rollcanarddeflection angle. Stability parameter Change in Y axis velocity due to yawcanarddeflection angle. Angleofattack. AngleofSideslip. System noise transition matrix. Reference state transition matrix. Target disturbance state transition matrix. System state transition matrix. xii

PAGE 13

Target model correlation time. Target elevation aspect angle. Seeker elevation gimbal angle. Target azimuth aspect angle. Seeker azimuth gimbal angle. xiii

PAGE 14

AbstractofDissertation Presented to the Graduate Schoolofthe UniversityofFlorida in Partial Fulfillmentofthe Requirements for the DegreeofDoctorofPhilosophy REACHABLE SET CONTROLFORPREFERRED AXIS HOMING MISSILESByDonald J. Caughlin, Jr. April1988Chairman: T.E. Bullock Major Department: Electrical Engineering The applicationofmodern control methodstothe guidance and controlofpreferred axis terminal homing missilesisnon-trivial in that it requires controlling a coupled, non-linear plant with severe control variable constraints, to intercept an evading target. In addition, the rangeofinitial conditionsisquite large andislimited only by the seeker geometry and aerodynamic performanceofthe missile. Thisisthe problem: Linearization will cause plant parameter errors that modify the linear trajectory. In non-trivial trajectories, both Ny andNzacceleration commands will, at some time, exceed the maximum value. The two point boundary problem is too complextocomplete in real time and other formulations are not capableofhandling plant parameter variations and control variable constraints. xiv

PAGE 15

Reachable Set Control directly adapts Linear Quadratic Gaussian (LQG) synthesis to the Preferred Axis missile,aswellasa large classofnonlinear problems where plant uncertainty and control constraints prohibit effectivefixed-final-timelinear control.Itisa robust control technique that controls a continuous system with sampled data and minimizes the effectsofmodeling errors. As a stochastic command generator/tracker, it specifies and maintains a minimum control trajectory to minimize the terminal impactoferrors generated by plant parameter (transfer function)ortarget set uncertainty while rejecting system noise and target set disturbances. Also, Reachable Set Control satisfies the Optimality Principle by insuring that saturated control,ifrequired, will occur during the initial portionofthe trajectory. With large scale dynamics determined by a dual reference in the command generator, the tracker gains can be optimized to the response timeofthe system. This separation results in an "adaptable" controller because gains are based on plant dynamics and cost while the overall systemissmoothly driven from some large displacement to a region where the relatively high gain controller remains linear. xv

PAGE 16

CHAPTER I INTRODUCTION The applicationofmodern control methods to the guidance and controlofpreferred axis terminal homing missiles has had only limited success [l,2,3]. This guidance problemisnon-trivial in thatitrequires controlling a coupled, non-linear plant with severe control variable constraints, to intercept an evading target. In addition, the rangeofinitial conditionsisquite largeandlimited only by the seeker geometry and aerodynamic performanceofthe missile. There are three major control issues that must be addressed: the couplednon-linearplantofthe Preferred Axis Missile; the severe control variable constraints; and implementation in the missile where the solutionisrequired to control trajectories lasting one(I)to two (2) seconds real time. There have been a numberofrecent advances in non-linear controlbutthese techniques have not reached the point where real time implementation in an autonomous missile controllerispractical [4,5,6]. Investigationofnon-linear techniques during this research did not improve the situation. Consequently, primarilydueto limitations imposed by real time implementation, linear suboptimal control schemes were emphasized. Bryson&Ho introduced a numberoftechniques for optimal control with inequality constraints on the control variables [7]. Eachofthese use variational techniques to generate constrained and unconstrained arcs that must be pieced together to construct the optimal trajectory.

PAGE 17

2Ingeneral, real time solutionofoptimal control problems with bounded controlisnot possible [8]. In fact, with the exceptionofspace applications, the optimal control solution has not been applied [9,10]. When Linear Quadratic Gaussian (LQG) techniques are used, the problemisnormally handled via saturated linear control, where the control is calculatedasifno constraints existed and then simply limited. This technique has been shown to be seriously deficient. In this case, neither stability nor controllability can be assured. Also, this technique can cause an otherwise initially controllable trajectory to become uncontrollable [11]. Consequently, a considerable amountoftime is spent adjusting the gainsofthe controller so that control input will remain below its maximum value. This adjustment, however, will force the controller to operate below its maximum capability [12]. Also, in the caseofthe terminal homing missile, the applicationofLQG controllers that do not violate an input constraint lead to an increasing acceleration profile and (terminally) low gain systems [13].Asa result, the performanceofthese controllersisnot desirable. Whileitisalwayspossible to tune a regulator to control the system to a given trajectory, the varianceofthe initial conditions, the time to intercept the target (normally a few seconds for a short range high performance missile), and the lackofa globally optimal trajectory due to the nonlinear nature, the best policyisto develop a suboptimal real time controller. The problemofdesigning a globally stable and controllable high performance guidance system for the preferred axis terminal homing missile is treated in this dissertation. Chapter 2 provides adequate background information on the missile guidance problem. Chapter 3 covers recent work on constrained

PAGE 18

3 control techniques. Chapters 4 and 5 discuss Robust Control and introduce "Reachable Set" Control, while Chapter 6 applies the technique to controlofapreferredaxis homing missile.Theperformanceof"Reachable Set" controlispresented in Chapter7.

PAGE 19

CHAPTER II BACKGROUND The preferred axis orientation missile has significant control input constraints and complicated coupled angular dynamics associated with the maneuvering. In the generic missile considered, the Z axis acceleration (see Figure 2.1) was structurally limited to100"g"with further limits on"g"resulting from a maximum angleofattackasa functionofdynamic pressure. Even though the Z axis was capableof100"g",the "skid-to-turn" capabilityofthe Y axis was constrained to 5"g"or less becauseofaerodynamic limitations a20:1difference. In addition to pitch (Nz) and yaw (Ny) accelerations, the missile can roll upto500 degrees per second to align the primary maneuver plane with the planeofintercept. Hence,bank-to-turn. x zFigure2.1Missile Reference System.4

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5The classical technique for homing missile guidanceisproportional navigation (pro nav). This technique controls the seeker gimbal angle ratetozero which (given constant velocity) causes the missiletofly a straight line trajectory toward the target [14,15]. In the late 70's aneffortwas made to use modern control theory to improve guidance laws forair-to-airmissiles.Forrecent research on this problem see, for example, [11].Asstated in the introduction, these efforts have not significantly improved the performanceofthe preferred axis homing missile.Ofthe modern techniques, two basic methodologies have emerged: one was a body-axis oriented control law that used singular perturbation techniques to uncouple the pitch&roll axis [16,17]. This technique assumed that roll rateisthe fast variable, an assumption that may not be true during the terminal phaseofan intercept. The second techniquewasan inertial point mass formulationthat controls inertial accelerations [18]. The acceleration commands are fixed with respecttothe missile body; but, since these commands can be related to the inertial reference via the Euler Angles, the solution is straight forward. Bothofthese methods have usually assumed unlimited control available and the inertial technique has relied on the autopilottocontrol the missile roll angle, and therefore attitude,tode rotate from the inertial to body axis. Missile Dynamics The actual missile dynamics are a coupled setofnonlinear forces and moments resolved along the (rotating) body axesofthe missile [19]. Linearizationofthe equations about a "steady state"ortrim condition,

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6neglectinghigherorderterms, resultsinthefollowingsetofequations (usingstandardnotation,see symbol keyinthe preface):ex=Q-PB+ Azb / IV totl B =-R-Pex+Ayb/ IVtotlLinearAccelerationsIU=RV-QW-{DO+DOwt }MIV=PW -RU+-{yo+ YOwtl M+ Yf3B+YaB+ YpP+YrR+ Y0 0 + Yoaoa + Yoror IW=QU-PV +-{LO+LOwt} MMomentEquations Q=MO/I yy +M u U+Mexex+Mexex+MqQ+ Moeoe (1)(2)(3) (4)(5) (6)+-----IyyIyy

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7 (7)+(8)+Linear Quadratic Gaussian Control Law For allofthe modern development models, a variationofa fixed-final-time LQG controller was usedtoshape the trajectory. Also, it was expected that the autopilot would realize the commanded acceleration. First, consider the effectofthe unequal body axis constraints. Assume that100"g"was commanded in each axis resulting in an acceleration vector45degrees from Nz.IfNyisonly capableof5"g",the resultant vector will be 42 degrees in error, an error that will havetobe corrected by succeeding guidance commands. Evenifthe missile has the timeorcapability to complete a successful intercept, the trajectory can not be considered optimal. Now consider the nonlinear natureofthe dynamics. The inertial linear systemisaccurately modeledasa double integratorofthe acceleration to determine position. However, the acceleration commandisa functionofthe missile state, equation(1),and therefore, itisnot possible to arbitrarily assign the input acceleration. And, given a body axis linear acceleration, the inertial

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8component will be severely modified by the rotation (especially roll)ofthe reference frame. Allofthese effects are neglectedinthe linearization. This thenisthe problem: In the intercept trajectories worth discussing, Ny, Nz, and roll acceleration commands will,atsome time, saturate. Highorder,linear approximations do not adequately model the effectsofnonlinear dynamics, and the complete two point boundary value problem with control input dynamics and constraintsistoo difficult to complete in real time. Although stochastic models are discussed in Bryson and Ho [7], and a specific techniqueisintroduced by Fiske [18], the general procedure has been to use filtered estimates and adynamic-programming-likedefinitionofoptimality (using the PrincipleofOptimality) with Assumed Certainty Equivalence to find control policies [20,21,22]. Therefore, allofthe controllers actually designed for the preferred axis missiles are deterministic laws cascaded with a Kalman Filter. The baseline forouranalysisisan advanced control law proposed by Fiske [18]. Given the finite dimensional linear system: where andxx(t)=Fx(t)+Gu(t)xyz VxVyVz(9)

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9with the cost functional:Jtfl=xfPfxf+ 1 uTRudrtoR=I(10)Applicationofthe Maximum principle results in a linear optimal control law:=-----3(Tgo) 31 +(Tgo)3+3(Tgo)2 31 +(Tgo)3(11)Coordinates usedforthis system are "relative inertial."Theorientationofthe inertial systemisestablished at the launch point.Thedistances and velocities are the relative measures between the missileandthe target. Consequently, the set pointiszero, with the reference frame moving with the missile similartoa "moving earth" reference used in navigation. Since Fisk's control law was based on a point mass model, the control law did not explicitly control the roll angle PHI (0). Theroll angle was controlledbyabank-to-turnautopilot [23]. Therefore, the guidance problemwasdecomposed into two components, trajectory formationandcontrol. The autopilot attempted to control the rollsothat thepreferredaxis (the-Zaxis) was directed toward the planeofintercept.Theautopilot used to control the missile was designed to use proportional navigationandisa classical combinationofsingle loop systems. Recently, WilliamsandFriedland have developed a newbank-to-turnautopilot based on modern state space methods [24]. Inorderto accurately control the banking maneuver, the missile dynamics are augmented to include the kinematic relations describing the change in the commanded specific force

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10vectorwithbankangle.Todeterminethe actual anglethroughwhichthe vehicle must roll,definethe roll angle error:Ayb e", =tan-l{--} Azb (12) Using thestandardrelationsforthe derivativeofavectorina rotatingreferenceframe,the following relationships followfromthe assumptionthatA11AlB:Azb=-P(Ayb)Ayb=+P(Azb) (13) (14)Theangle e", represents theerrorbetween the actualanddesiredroll angleofthe missile.Differentiating e", yields: (Azb)(Ayb) (Ayb)(Azb) e", =(Azb)2+(AYb)2which,aftersubstituting componentsofAxw, showsthat(15) (16) Simplifying the nonlinear dynamicsof(1) (8), the following model was used:oe=Q-PJ3+Azb / IVtotl 13 =-R-Poe+Ayb /IVtotl(Izz -Ixx ) Q=Moeoe+MqQ+ MSeSe +-----IyyPR(17) (18) (19)

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wherePQ(20)(21)(22)(23)U sing this model directly, the autopilot would be designedasan eighth-order system with time-varying coefficients. However, even though these equations contain bilinear terms involving the roll rate Paswellaspitch/yaw cross-coupling terms, the roll dynamics alone, represent a second order system thatisindependentofpitch and yaw. Therefore, using an "Adiabatic Approximation" where the optimal solutionofthe time-varying systemisapproximated by a sequenceofsolutionsofthe time-invariant algebraic Riccati equation for the optimum control law at each instantoftime, the model was separated into roll and pitch/yaw subsystems [25]. Now, similartoa singular perturbations technique, the functionofthe roll channelisto provide the necessary orientationofthe missilesothat the specific force acceleration lies on the Z (preferred) axisofthe missile. Using this approximation, the systemisassumed to be in steady state, and all coefficients--including rollrate--areassumed to be constant. Linear Quadratic Gaussian (LQG) synthesisisused, with an algebraic Riccati equation, on a second and sixth order system. And, when necessary, the gains are scheduledasa functionofthe flight condition.

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12While still simplified, this formulation differs significantly from previous controllers in two respects. First, the autopilot explicitly controls the roll angle; and second, the pitch and yaw dynamics are coupled. Even though preliminary work with this controller demonstrated improved tracking performance by the autopilot, overall missile performance, measured by miss distance and time to intercept, did not improve. However, the autopilot still relies on a trajectory generated by the baseline controller ( e.g. Azb in 17). Consequently, the missile performance problemisnot in the autopilot, the error sourceisin the linear optimal control law whichforms the trajectory. "Reachable Set Control"isa LQG formulation that can minimize these errors.

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CHAPTER III CONSTRAINED CONTROL In Chapters I and II,wecovered the non-linear plant, the dynamics neglected in the linearization, the impactofcontrol variable constraints, and the inabilityofimproved autopilots to reduce the terminal error. To solve this problem,wemust consider the optimal controlofsystems subject to input constraints. Although a searchofthe constrained control literature did not provide any suitable technique for real time implementation, someofthe underlying concepts were used in the formulationof"Reachable Set Control." This Chapter reviews someofthese results to focus on the constrained control problem and illustrate the concepts. Muchofthe early work was based on research reported by Tufts and Shnidman [26] which justified the useofsaturated linear control. However,asstated in the introduction, with saturated linear control, controllabilityisnot assured.Ifthe system, boundary values and final time are such that thereisno solution with any allowable control(Ifthe trajectoryisnot controllable), then the boundary condition will not be met by either azero terminal errororpenalty function controller. While constrained control can be studied in a classical way by searching for the effectofthe constraint on the valueofthe performance function, this procedureisnot suitable for real time controlofa system with a wide rangeofinitial conditions [27]. Someofthe techniques that could be implemented in real time are outlined below.13

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14Lim used a linearized gain to reduce the problemtoa parameter optimization [8]. Given the system model: x=Fx+Gu+Lw (1) with state x, constant F,G,and L, scaler control u, and Lw representing zero mean Gaussian white noise with covariance LLT. Consider the problemofchoosing a feedback law such that in steady state, assuming it exists, the expected quadratic costItfJ=E{lim [ (x(t)TQx(t)+ >.u(t)2) dt+X(tf)Tp(tf)X(tf)] }tf-00 to (2)isminimized. The weighting matrix Qisassumed to be positive semidefinite and >. O.Dynamic programming leadstoBellman's equation: min { t tr[LTYxx(x)L)+(Fx+Gu)Tyx(x)+xTQx+ >.u2 }= 0:.* and, assuming a Y(x) satisfying(3),the optimal solution u(x)=SAT {(1/2>')G T y x (x) }=SGN {GTyx(x)}(3)(4)However,(3)cannot be solved analytically, andYx in generalisa nonlinear functionofx. Consider a modified problem by assuming a controlofthe form: u(x)=SAT { gT x }=SGN{gTx } where gisa constant (free) vector. >'=0 (5a)(5b)Assume further that xisGaussian with known covarianceW(positive definite). Using statistical linearization, a linearized gain k can be obtained by minimizing E{u(x) -kTx}2(6)

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15which results in for (Sa): for (Sb): where 4>(z) = (2MtJ;xp (-ty 2)dyk= (2/7l")t .(gTWg)-t.g(7a)(7b)From (1), with u determined by kT x, the stable covariance matrixWandsteady statePareand(F+GkT)W+W(F+GkT)T+LLT=0 (F+GkT)Tp+P(F+GkT )+P+ >.kkT =0(8)(9)andThe solution to (3), without the minimum, isv(x)=xTpxex=tr{LT PL } (10)(11) The problemisto choose g suchthatthe expected costexby statistical linearizationisa minimum. However, a minimum may not exist. In fact, from [8], a minimum does not exist when the noise disturbance is large. Since we are considering robust control problems with plant uncertainty or significant modeling errors, the noise will be large and the minimum will be replaced by a greatest lower bound.Asexapproached the greatest lower bound, the control approached bang-bang operation. A combinationofplant errorsandthe rapid dynamicsofsome systems (such as the preferred axis missile) would preclude acceptable performance withbang-bangcontrol.

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16Frankenaand Sivan suggested a criterionthatreduce thetwo-pointboundary problem to an initial value problem [12]. They suggest controlling the plant while minimizing this performance index: With the constraint lIu(t)IIR(t) 1 Applying the maximum principle to the Hamiltonian developed from(12)x(t)=F(t)x(t)+G(t)u(t) with x(to)=xOprovides the adjoint differential equationto
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17ForGT Px:;: 0 andIIxliS :;: 0, P will be the solutionofPGR-IGTpS P+PF+=Q+---FTPIIR I GTpxIlRIIxliS(18)Now choosing S=PGR-IGTpresults in a Lyapunovequationandwill insure negativedefiniteP(t)ifFisa stability matrix.Therefore,withthis choiceofweighting functions totransformthe problem to a singleboundarycondition, a stable F matrixisrequired. Thisisa significant restrictionandnot applicable to the systemunderconsideration.Gutmanand Hagander developed a designforsaturatedlinear controllersforsystems with controlconstraints[9,28].Thedesign beginswitha low-gain stabilizing control, solves a Lyapunov equation tofinda regionofstability and associated stability matrix, andthensums the controlsina saturation function toformthe constrained control. Begin with the stabilizable continuous linear time invariant system x=Fx+Guwith admissible controlinputsui,suchthatx(O)=xO=l,...,m(19)where gi and hi are the control constraints. ConsiderannxmmatrixL == [11I12I...I1 m ] suchthatisa stability matrix.(20)(21)

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18Associated with eachofthe controlsaresetsthatdefineallowable conditions.Theset Disthe setofinitial conditionsfromwhichitisdesired to stabilize the system to the origin.Thelow gain stabilizing control LdefinesthesetE:E == E(L) == (zIzER(22)i=I,...,mwhichis the setofstatesatwhichthestabilizing linearfeedbackdoesnotinitiallyexceedthe constraints.Anotherset is F: F == F(L) ==n {(eFct)-1E}tE[O,oo)(23)(24) Fisa subsetofE suchthatalong all trajectories emanatingfromF,thestabilizing linear state feedback doesnotexceed the constraint.Theregionofstabilityforthe solutionofthe Lyapunovequationisdefinedby 0== O(L,P,c) == {xIxTPx c}whereV(x)=xTpxis the Lyapunovfunctioncandidateforthe stabilitymatrixFc,andc is to be determined.Thecontrol technique follows: Step1:DetermineD. Step2:FindLbysolving aLQGproblem.Thecontrol penalty is increaseduntilthecontrolLTx satisfies theconstraintin(19)forxinD.IfDissuchthatthecontrolconstraintcan notbesatisfied,thenthis designisnotappropriate.Step3:FindPandc.FirstfindaP=PT>0suchthattheLyapunovequationPFc+F cTp>O.Now determine 0 bychoosing cin(24) suchthatD 0 E:supxT Px c minxT PxxEDxE&E(25)

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19where 6E designates the boundaryofE.Ifthis fails, choose another P, or select a "lower gain" in order to enlarge E,orfinally, a reduction in the sizeofD might be considered. Step4:Set up the control according to u= SAT[ (L T -KGTp)x]where Kisdefined(26)K=[ki0]ki 0,i= 1,2, ... ,m (27)okmand tune the parameters ki by simulations. A sufficient condition for the algorithm to workisD E.(28)Unfortunately, determining the stability region was trial and error; and, once found, further tuningofa diagonal gain matrixisrequired. In essence, thiswasa technique for determining a switching surface between a saturated and linear control. Also, when the techniquewasapplied to an actual problem, inadequacies in the linear model were not compensated for. Given the nonlinear natureofthe preferred axis missile, rangeofinitial conditions, and the trial and error tuning required for eachofthese conditions, the procedure would not be adequate for preferred axis terminal homing missile control. A notable featureofthe control scheme, however, was the ability to maintain a stable system with a saturated control during muchofthe initial portionofthe trajectory. Another technique for control with bounded input was proposed by Spong et al. [29]. This procedure used an optimal decision strategytodevelop a pointwise optimal control that minimized the deviation between the actual and

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20desired vectorofjointaccelerations, subject toinputconstraints. The computationofthe control law is reduced to the solutionofa weighted quadratic programming problem. Key to this solution is the availabilityofa desired trajectory in state space. Suppose that a dynamical system can be described by with which can be writtenasx(t)=f(x(t+G(x(tu(t) c (29) Fix time t 0, let s(t,xo,to,u(t (or s(t) for short), denote the solution to (29) corresponding to the giveninputfunction u(t). At time t,ds/dtisthe velocity vectorofthe system, andisgiven explicitly by the right hand sideof(29). Define the set C t=C(s(t with C(s(t=(ex(t,w)ERNIex=f(s(t+G(s(tw,wE {} } {} ={wI NW:5 C}(30)Therefore, for each t andany allowable u(t),ds/dtliesinthe set Ct. In other words, the set C t contains the allowable velocitiesofthe solution s(t). Assume that there exists a desired trajectory yd, and an associated vector field v( t)=v(s(t),yd(t),t, which is the desired (state) velocityofs(t) to attain yd. Consider the following "optimal decision strategy" for a given positive definite matrixQ:Choose theinputu(t)sothat the corresponding solution s(t) satisfies(d/dt)s(t,u(t=s*(t), where s*(t)ischosenateach t to minimize min {(ex v(s(t),yd(t),tTQ(exv(s(t),yd(t),t }exECt(31)

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21Thisisequivalenttothe minimization min { !UTGTQGu -(GTQ(v-fTu } subject to, Nu(t) cu(32)Wemay now solve the quadratic programming problemtoyield a pointwise optimal control law for (29). At each time t, the optimal decision strategy attemptsto"align" the closed loop system with the desired velocity v(t)asnearlyaspossible in a least squares sense. In this way the authors retain the desirable propertiesofv(t) within the constraints imposed by the control. Reachable Set Control builds on this technique:itwill determine the desired trajectory and optimally track it. Finally, minimum-time control to the origin using a constrained acceleration has also been solved by a transformationtoa two-dimensionalun-constrained control problem[30].Byusing a trigonometric transformation, the controlisdefined by an angular variable, u(t) f{cos(I3),sin(I3)}, and the control problem was modifiedtothe controlofthis angle. The constrained linear problemisconvertedtoan unconstrained nonlinear problem that forces a numerical solution. This approach removes the effectofthe constraintsatthe expenseofthe continuous applicationofthe maximum control. Given the aerodynamic performance (range and velocity) penaltyofmaximumcontrol and the impact on attainable roll rates due to reduced stability at high angleofattack, this concept did not fit preferred axis homing missiles.Animportant assumption in the previous techniques was that the constrained systemwascontrollable. In fact, unlike (unconstrained) linear systems, controllability becomes a functionofthe set admissible controls, the initial state, the time-to-go, and the target state. To illustrate this, someofthe relevant points from [31,32] will be presented. An admissible controlisone

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22 that satisfies the condition u() :[0,00) 0 ERm where 0 is the control restraint set. The collectionofall admissible controls will be denoted by M(O). The target set Xisa specified subset in Rn.Asystem is defined to be O-controllable from an initial statex(tO)= xo to the target set X at Tifthere existsU()EM(O)such that x(T,u(),xO) EX.Asystem would be globally O-controllable to XifitisO-controllable toXfromeveryx(tO)ERn.Inordertopresent the necessary and sufficient conditions for O-controllability, consider the following system: x(t)=F(t)x(t)+G(t,u(tand the adjoint defined by: =F(t)T z(t) with the state transition matrix q>(t,r) and solution x(to) =xOZ(to) =zO(33)t f [0,00)(34)(35)The interior B and surface Softhe unit ball in Rn are defined asB={ZO f RnS={zOfRn llzoll I}IIzoll=I}(36) (37)The scaler functionJ(.):Rn xRxRnxRn-R is defined by J(xo,t,x,zo) =XOTZO+Jtmax [GT(r,w)z(r)]dr-x(t)Tz(t)o WfO (38)Given the relatively mild assumptionsof[32], a necessary conditionfor(33) to be O-controllable to X fromx(tO)ismax min J(xO,T,x,ZO) =0 XfXzOfB while a sufficient conditionissup min J(xO,T,x,zO)>0 XfXZOfS (39) (40)

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23The principle behind the conditions arises from the definitionofthe adjoint system--Z(t). Using reciprocity, the adjointisformed by reversing the roleofthe input and output, and running the system in reverse time [33]. Consider x(t)=F(t)x(t)+G(t)u(t) y(t)=H(t)x(t) and: z(t)=-F(t)Tz(t)+ o(t)=GT(t)z(t) Therefore x(to)=xOz(to)=zO(41) (42)and zT(t)x(t)=zT(F(t)x(t)+G(t)u(t(d/dt)(zT(t)x(t= T(t)x(t)+z T(t)x'(t)= +zT(t)G(t)u(t)(43) (44)Integrating both sides from tototfyields the adjoint lemma:(45)The adjoint defined in (31) does not have an input. Consequently, the integral in (35)isa measureofthe effectofthe control applied to the original system.Bysearching for the maximumGT(r,w)z(r),it provides the boundaryoftheeffectofallowable control on the system (33). Restricting the search over the target settothe min ( J(xo,t,x,zO) :t f [O,T],zo f S}ormin ( J(xo,t,x,zo) :t f [O,T],zo f B} minimizes the effectofthe specific selectionof Z() on the reachable set and insures that the search is over a function thatisjointly continuous in (t,x). Consequently, (35) compares the autonomous growthofthe system, the reachable boundaryofthe allowable input, and the desired target set and time. Therefore,ifJ=0, the adjoint lemmaisbe

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24identically satisfied at the boundaryofthe control constraint set (necessary); J>0 guarantees that a control can be found to satisfy the lemma.Ifthe lemmaissatisfied, then the initial and final conditions are connected by an allowable trajectory. The authors [32]goon to develop a zero terminal error steering control for conditions where the target setisclosed and max min J(xO,T,x,zO) 0 xXZOS (46)But their control technique has two shortcomings: First; it requires the selectionofz00The initial conditionzoisnot specified but limited toIIz01l=1.A particularzomust be selected to meet the prescribed conditions and the equality in (43 ) for a given boundary condition, andistherefore not suitable for real time applications. And second; the steering control searched M(O) for the supremumofJ, making the control laws bang-bang in nature, again not suitable for homing missile control. While a direct searchof Ox isnot appropriate for a preferred axis missile steering control, a "dual" system, similar to the adjoint system used in the formulationofthe controllability function J, can be usedtodetermine the amountofcontrol requiredtomaintain controllability. Once controllabilityisassured, then a cost function that penalizes the state deviation (as opposed to a zero terminal error controller) can be used to control the systemtoan arbitrarily small distance from the reference.

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CHAPTERIV CONSTRAINEDCONTROLWITHUNMODELEDSETPOINTANDPLANTVARIAnONSChapterIII reviewed anumberoftechniques tocontrolsystemssubjectto control variable constraints. While noneofthe techniques werejudgedadequateforreal time implementationofapreferred-axishoming missile controller, someofthe underlying conceptscanbe used to develop atechniquethatcanfunctionin the presenceofcontrol constraints:(l)Useofa "dual system"thatcanbe used to maintain a controllable system (trajectory); (2) an "optimal decision strategy" to minimize the deviation betweentheactualanddesiredtrajectorygenerated by the "dual system;"and(3) initiallysaturatedcontrolandoptimal (realtime) selectionofthe switchingsurfacetolinearcontrolwithzero terminalerror.However, inadditionto,andcompounding the limitations imposedbycontrol constraints, wemustalso consider the sensitivityofthecontrol to unmodeled disturbancesandrobustnessunderplant variations.Inthe stochastic problem, there are threemajorsourcesofplantvariations.First,therewill be modeling errors (linearization/reductions)thatwill causethedynamicsofthe system to evolve in adifferentor"perturbed" fashion. Second,theremay betheunmodeleduncertaintyinthe system statedueto Gaussian assumptions.Andfinally,inthe fixed final time problem, there may beerrorsinthe final time, especiallyifitisafunctionoftheuncertain stateorimpactedbythe modeling reductions. Since theprimaryobjectiveofthis researchisthe zeroerrorcontrolofa dynamical systeminfixed time, mostofthemore recent 25

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26optimization techniques (eg. LQG/LTR,WXl) did not apply. At this time, these techniques seemedtobe more attuned to loop shapingorrobust stabilization questions. A fundamental proposition that forms the basisofReachable Set Controlisthat excessive terminal errors encountered when using an optimal feedback control for an initially controllable trajectory (a controllable system that can meet the boundary conditions with allowable control values) are caused by the combinationofcontrol constraints and uncertainty (errors) in the target set stemming from unmodeled plant perturbations (modeling errors)orset point dynamics. First, a distinction must be made between a feedback and closed-loop controller. Feedback controlisdefinedasa control system with real-time measurement data fed back from the actual system but no knowledgeofthe form, precision, or even the existenceoffuture measurements. Closed-loop control exploits the knowledge that the loop will remain closed throughout the future intervaltothe final time.Itaddstothe information provided to a feedback controller, anticipates that measurements will be taken in the future, and allows prior assessmentofthe impactoffuture measurements.IfCertainty Equivalence applies, the feedback lawisa closed-loop law.Underthe Linear Quadratic Gaussian (LQG) assumptions, thereisnothing to be gained by anticipating future measurements. In the mathematical optimization, external disturbances can be rejected, and the mean valueofthe terminal error can be made arbitrarily close to zero by a suitable choiceofcontrol cost. For the following discussion, the "system" consistsofa controllable plant and an uncontrollable referenceortarget. The system stateisthe relative difference between the plant state and reference. Since changes in the system

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27boundary condition can be caused by either a change in the reference point or plant output perturbations similartothose discussed in Chapter II, some definitions are necessary. The setofboundary conditions for the combined plant and target system, allowing for unmodeled plant and reference perturbations, will be referredtoasthe target set. Changes, or potential for change, in the target set caused only by target (reference) dynamics will be referredtoasvariations in the set point. The magnitudeofthese changesisassumed to be bounded. Admissible plant controls are restrictedtoa control restraint set that limits the input vector. Since there are bounds on the input control, the system becomes non-linear in nature, and each trajectory must be evaluated for controllability. Assume that the system (trajectory)ispointwise controllable from the initialtothe boundary condition. Before characterizing the effectsofplant and set point variations, we must consider the formofthe plant and it's perturbations.Ifweassume that the plantisnonlinear and time-varying, thereisnot much that can be deduced about the target set perturbations. However,ifhave a reduced order linear modelofa combined linear and nonlinear process,ora reasonable linearizationofa nonlinear model, then the plant can be consideredaslinear and time-varying. For example, in the caseofa Euclidean trajectory. the system model (a double integrator)isexact and linear. Usually, neglected higher order or nonlinear dynamics or constraints modify the accelerations and lead to trajectory (plant) perturbations. Consequently, in this case, the plant can be accurately representedasa Linear Time Invariant System with (possibly) time varying perturbations.

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28Consider the feedback interconnectionofthe systems K and P where Kisa sampled-data dynamic controller and P the (continuous) controlled system:r-K uG--p_ .: Figure4.1Feedback System and Notation Assuming that the feedback systemiswell defined and Bounded Input Bounded Output (BIBO) stable, at any sample time ti, the system can be defined in termsofthe following functions: e(ti)=r(ti) y(ti) u( ti)=Ke( ti)y(ti)=Pu( ti)(1) (2)(3)

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with the operator G29G[K,P]asthe operator that maps the input e(ti)tothe output y(ti) [34]. At any time, the effectofa plant perturbation DoP can alsobecharacterizedasa perturbation in the target set. or thenIfP=Po + DoP P= P(I+DoP) y(ti)= YO(tj) + Doy(ti) (4a)(4b)(5)where Doy(ti) represents the deviation from the "nominal" output caused by either the additive or multiplicative plant perturbation. Therefore, e(ti)=r(ti) (YO(ti)+ Doy(ti = (r(tj) + Doy(ti -YO(ti)= Dor(ti) -YO(ti)(6)(7)(8)with Dor(ti) representing a change in the target set that was unknown to the controller. These changes are then fed backtothe controller but could be handled a priori in a closed loop controller designastarget set uncertainty. Now consider theeffectofconstraints.Ifthe controlisnot constrained, and target set errors are generated by plant variationsortarget maneuvers, the feedback controller can recover from these intermediate target set errors by using large (impulsive) terminal controls. The modeled problem remains linear. While the trajectoryisnot the optimal closed-loop trajectory, the trajectoryisoptimal based on the model and information set available. Even with unmodeled control variable constraints, and a significant dis placementofthe initial condition, an exact plant model allows the linear stochastic optimal controllertogenerate an optimal trajectory. The switching time from saturated to linear controlisproperly (automatically) determined and,

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30asin the linear case, the resulting linear control will drive the statetowithin an arbitrarily small distance from the estimateofthe boundary condition.Ifthecontrol constraint set covers the rangeofinputs required by the control law, the law will always be able to accommodate target set errors in the remaining time-to-go. This is, in effect, the unconstrained case. If, however, the cost-to-goishigherand/orthe deviation from the boundary conditionisofsufficient magnitude relative to the time remainingtorequire inputs outside the boundaryofthe control constraint set, the system will not follow the trajectory assumed by the system model.Ifthisisthe caseastime-to-goapproaches zero, the boundary condition will not be met, the systemisnot controllable (to the boundary condition).Astime-to-go decreases, the effectsofthe constraints become more important. With control input constraints, and intermediate target set errors caused by unmodeled target maneuvers or plant variations, it may not be possible for the linear control lawtorecover from the midcourse errors by relying on large terminal control. In this case, an optimal trajectoryisnot generated by the feedback controller, and, at the final time, the systemisleft with large terminal errors. Consequently,ifexternal disturbances are adequately modeled, terminal errors that are ordersofmagnitude larger than predicted by the open loop optimal control are caused by the combinationofcontrol constraints and target set uncertainty.

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31Linear Optimal Control with Uncertainty and Constraints An optimal solution must meet the boundary conditions. To accomplish this, plant perturbations and constraints must be considered a priori. They should be includedasa priori information in the system model, they must be physically realizable, and they must be deterministic functionsofa priori information, past controls, current measurements, and the accuracyoffuture measurements. From the control pointofview,wehave seen that the effectofplant parameter errors and set point dynamics can be groupedastarget set uncertainty. This uncertainty can cause a terminally increasing acceleration profile even when an optimal feedback control calls for a decreasing input (see Chapter 5). With the increasing acceleration caused by midcourse target set uncertainty, the most significant terminal limitation becomes the control input constraints. (These constraints not only affect controllability, they also limit how quickly the system can recover from errors.)Ifthe initial controlissaturated while the terminal portion linear, the controlisstill optimal.Ifthe final controlisgoingtobesaturated, however, the controller must account for this saturation. The controller could anticipate the saturation and correct the linear portionofthe trajectorytomeet the final boundary condition. This control, however, requires a closed formsolution for x(t), carries an increased cost for an unrealized constraint, and is known to be valid for monotonic ( single switching time) trajectories only [11]. Another technique availableisLQG synthesis. However, LQG assumes controllability in minimizing a quadratic cost to balance thecontrol error and

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32input magnitudes.Aswehave seen, the effectsofplant parameter and reference variations, combined with control variable constraints, can adversely impact controllability. The challengeofLQGisthe proper formulationofthe problem to function with control variable constraints while compensating for unmodeled set point and plant variations. Reachable Set Control uses LQG synthesis and overcomes the limitationsofan anticipative controltoinsure a controllable trajectory. Control Technique Reachable Set Control can be thoughtofasa fundamentally different robust control technique based on the concepts outlined above. The usual discussionofrobust feedback control (stabilization) centers on the developmentofcontrollers that function even in the presenceofplant variations. Using either a frequency domain or state space approach, and modeling the uncertain ty, bounds on the allowable plant or perturbations are developed that guarantee stability [35]. These bounds are determined for the specific plant under consideration and a controller is designedsothat expected plant variations are contained within the stability bounds. Building on ideas presented above, however, this same problem can be approached in an entirely different way. This new approach begins with the same assumptionsasstandard techniques, specifically a controllable system and trajectory. But, with Reachable Set Control,wewill not attempt to model the plant or parameter uncertainty, nor the set point variation.Wewill, instead, reformulate the problemsothat the system remains controllable, and thus stable, throughout the trajectory even in the presenceofplant perturbations and severe control input constraints.

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33Before we developanimplementable technique, considerthedesired resultofReachable Set Control (and the originofthe name) by using atwo-dimensional missile intercept problemasan example.Attime t= t}. not any specific time during the intercept, the targetisatsome location T1andthe missileisatMIasshown in Figure 4.2. Consider these locationsasoriginsoftwo independent, target and missile centered, reference systems.Fromthese initial locations, given the control inputs available, reachable sets for each system can be definedasafunctionoftime (not shown explicitly).Thetarget setiscircular because is maneuver directionisunknownbutits capabilitybounded,and the missile reachable set exponential because the x axis controlisconstant and uncontrollable while the z axis accelerationissymmetricandbounded.TheobjectiveofReachable Set Controlisto maintain the reachable targetsetin the interiorofthe missile reachable set. Hence, Reachable Set Control.xTargetReachableM1zMissileReachableSetFigure 4.2 Reachable Set Control Objective

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34As stated, Reachable Set Control would bedifficultto implementasa control strategy. Fortunately, however,furtheranalysis leads to a simple, direct, and optimal technique thatisvoidofcomplicated algorithms orad-hocprocedures. First, consider the process. The problem addressed is the controloffixed-terminal-timesystems. The true cost is the displacementofthe state at the final time and only at the final time. In the terminal homing missile problem, thisisthe closest approach, or miss distance. In another problem, it may be fuel remaining at the final time, or possibly a combinationofthe two. In essence, with respect to the direct applicationofthis technique, thereisno preferenceforone trajectory over another or no intermediate cost based on the displacementofthe state from the boundary condition. The term "direct application" was used because constrained path trajectories, suchasthose required by robotics, or the infinite horizon problem, like the controlofthe depthofa submarine can be addressed by separating the problem into several distinctintervals--eachwith a fixed terminaltime--ora switching surface when the initial objective is met [36]. Given a plant with dynamics x(t)=f(x,t)+g(u(w),t) y(t)=h(x(t),t) modeled by x(t)=F(t)x(t)+G(t)u(t) yx(t)=H(t)x(t) x(to)=xo(9) (10)

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35with final condition x(tf) and a compact control restraint setOx.Let Ox denote the setofcontrols u(t) for which u(t)E Ox for tE [0,00).The reachable set X(tO,tf,xO'Ox) == (x:x(tf)=solutionto(10) withxofor some u()EM(Ox) }isthe setofall states reachable fromxoin time tf. In addition to the plant and model in(9&10),wedefine the reference(11)r(t)=a(x,t)+b(a(w),t) y(t)=c(x(t),t) modeled by r(t)=A(t)r(t)+B(t)a(t)Yr(t)=C(t)r(t) and similarly defined set R(to,tf,ro,Or), r(to)=rO(12) (13) R(to,tf,ro,Or) == (r:r(tf)=solution to (13) withrOfor some a()EM(Or) }asthe setofall reference states reachable fromrOin time tf. (14) Associated with the plant and reference, at every time t,isthe following system: e(t)=yx(t) -Yr(t)(15) ;;m (10&13), we see that yx(t) andYr(t)are output functions that incorporate the significant characteristicsofthe plant and reference that will be controlled. The design objectiveise(tf)=0(16)and we want to maximize the probabilityofsuccess and minimize the effectoferrors generated by the deviationofthe reference and plant from their associated models. To accomplish this with a sampled-data feedback control law,

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36wewill select the control u(ti) such that, at the next sample time (ti+l), the target reachable set will be covered by the plant reachable set and, in steady state,ife(tf)=0, the control will not change. Discussion Recalling that the performance objective at the final timeisthe real measureofeffectiveness, and assuming that the terminal performanceisdirectly related to target set uncertainty, this uncertainty should be reduced with time-to-go. Now consider the trajectory remembering that the plant model is approximate (linearized or reduced order), and that the reference has the capability to change and possibly counter the control input. (This maneuverabili ty does not havetobe taken in the contextofa differential game.Itis only intended to allow for unknown set point dynamics.) During the initial portionofthe trajectory, the target set uncertaintyisthe highest. First, at this point, the unknown (future) reference changes have the capabilityofthe largest displacement. Second, the plant distance from the uncertain set pointisthe greatest and errors in the plant model will generate the largest target set errors becauseofthe autonomous response and the magnitudeofthe control inputs required to move the plant state to the set point. Along the trajectory, the contributionofthe target (reference) maneuvera bility to set point uncertainty will diminish with time. This statement assumes that the target (reference) capability to change does not increase faster than the appropriate integralofits' input variable. Regardlessofthe initial maneu verabilityofthe target, the time remainingisdecreasing, and consequently, the

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37ability to move the set point decreases. Target motionissmaller and it's positionismore and more certain. Selectionofthe control inputs in the initial stagesofthe trajectory that will result in a steady state control (that contains the target reachable set within the plant reachable set) reduces target set uncertainty by establishing the plant operating point and defining the effective plant transfer function. At this point,wedo not have a control procedure, only the motivation to keep the target set within the reachable setofthe plant along with a desire to attain steady state performance during the initial stagesofthe trajectory. The specific objectives are to minimize target set uncertainty, and most importantly, to maintain a controllable trajectory. The overall objectiveisbetter performance in termsofterminal errors. Procedure A workable control law that meets the objectives can be deduced from Figure 4.3. Herewehave the same reachable setforthe uncertain target, but this time, several missile origins are placed at the extremesoftarget motion. From these origins, the systemisrun backward from the final timetothe current time using control values from the boundaryofthe control constraint set to provide a unique setofstates that are controllable to the specific origin.Ifthe intersectionofthese setsisnon-empty, any potential target locationisreachable from this intersection. Figure 4.4 is similar, but this time the missile control restraint setisnot symmetric. Figure 4.4 shows a case where the missile acceleration controlisconstrainedtothe set A=[Amin,Amax] where 0 Amax(17)

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z38xTargetReachable AlltargetpositionsReachableFigure 4.3 IntersectionofMissile Reachable Sets BasedonUncertain Target Motion and Symmetric ConstraintsxTargetReachableSetAllTargetLocationsReachableFigure 4.4 IntersectionofMissile Reachable Sets Based on Uncertain Target Motion and Unsymmetric Constraints

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39Since controllabilityisassumed, which for constrained control includes the control bounds and the time interval, the extreme left and right (near and far) pointsofthe set point are included in the set drawn from the origin. To implement the technique, construct a dual system that incorporates functional constraints, uncontrollable modes, and uses a suitable control value from the control constraint setasthe input. From the highest probability target positionatthe final time, run the dual system backward in time from the final boundary condition. Regulate the plant (system) to the trajectory defined by the dual system. In this way, the fixed-final-time zero terminal error controlisaccomplished by re-formulating the problemasoptimal regulationtothe dual trajectory. In general, potential structuresofthe constraint set preclude a specific point (origin, center, etc.) from always being the proper inputtothe dual system. Regulation to a "dual" trajectory from the current target position will insure that the originofthe target reachable set remains within the reachable setofthe plant. Selectionofa suitable interior point from the control restraint setasinputto the dual system will insure that the plant has sufficient control power to prevent the target reachable set from escaping from the interiorofthe plant reachable set. Based on unmodeled set point uncertainty, symmetric control constraints, and a double integrator for the plant, a locus exists that will keep the target in the centerofthe missile reachable set.Ifthe set pointisnot changed, this trajectory can be maintained without additional inputs. For a symmetric control restraint set, especiallyasthe time-to-go approaches zero, Reachable Set Controliscontrol to a "coasting" (null control) trajectory.

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40Ifthe control constraints are not symmetric, such asFigure4.4, a locusofpoints that maintains the target in thecenterofthe reachable set is the trajectory based on the systemrunbackward from the final time target location with the acceleration command equal to the midpointofthe set A. Pictured in Figures 4.2 to 4.4 were trajectories that are representativeofthe double integrator. Other plant models would havedifferenttrajectories. Reachable Set Control is a simple techniqueforminimizing the effectsoftarget set uncertainty and improving terminal the performanceofa large classofsystems.Wecan minimize the effectsofmodeling errors (or target setuncertainty) by a linear optimal regulator that controls the system to a steady state control. Given the well known and desirable characteristicsofLQGsynthesis, this technique canbeusedasthe basisforcontrol to the desired "steady state control" trajectory.Thetechnique handles constraints by insuringaninitially constrained trajectory. Also, since the large scale dynamics are controlled by the "dual" reference trajectory, the tracking problem be optimized to the response timeofthe systemunderconsideration. This results inan"adaptable" controller because gains are basedonplant dynamicsandcost while the overall system is smoothlydrivenfrom some large displacement to a region where the relatively high gainLQGcontroller will remain linear.

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CHAPTERV REACHABLE SETCONTROLEXAMPLEPerformance Comparison Reachable SetandLOGControlInorderto demonstrate theperformanceof"Reachable Set Control" we will contrast its performance with theperformanceofa linear optimal controllerwhenthereistarget setuncertaintycombinedwithinputconstraints. Consider,forexample, thefinitedimensional linear system:withthequadratic cost x(to)=xo(1)whereJ1=-XfTpfXf 2(2)andtfE [0,00) '1 0Applicationofmaximum principle yields the following linear optimal control law:where1u=+-x(tf)(t-tf) '1 xo+xo tfx(tf)=1+41(3) (4)

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42Appropriately defining t, to, andtf,the control law can be equivalently expressed in an open loop or feedback form with the latter incorporating the usual disturbance rejection properties. The optimal control will tradeoff the costofthe integrated square input with the final error penalty. Consequently, even in the absenceofconstraints, the terminal performanceofthecontrolisa functionofthe initial displacement, time allowed to drive the statetozero, and the weighting factor ,. To illustrate this, Figure5.1presents the terminal states (miss distance and velocity)ofthe linear optimal controller. This plotisa compositeoftrajectories with different run times ranging from 0to3.0 seconds. The figure presents the valuesofposition and velocity at the final time t=tfthat result from an initial positionof1000 feet and with velocityof1000feet/sec with =104 Figure 5.2 depicts,asa functionofthe run time, the initial acceleration (at t=0.) associated with eachofthe trajectories shown in Figure 5.1. From these two plots, the impactofshort run timesisevident: the miss distance will be higher, and the initial acceleration command will be greater. Since future set point (target) motionisunknown, the suboptimal feedback controllerisreset at each sample time to accommodate this motion. The word resetissignificant. The optimal controlisa functionofthe initial condition at time t=to, time, and the final time. A feedback realization becomes a functionofthe initial condition and time to go only. In this case, set point motion (target set uncertainty) can place the controller in a position where the time-to-goissmall but the state deviationislarge.

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Velocity2000o-2000-4000 -6000 -8000-10000 -12000-14000o20043400600800Position10001200Figure5.1Terminal PerformanceofLinear Optimal ControlAccelerationo-100000--200000-300000v-400000o0.51Final Time1.52Figure 5.2 Initial AccelerationofLinear Optimal Control

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44While short control times will result in poorer performance and higher accelerations, it does not take a long run time to drive the terminal error to near zero. Also, from (4)wesee that the terminal error can be driventoan arbitrarily small value by selectionofthe control weighting. Figure5.1presented the final valuesoftrajectories running from 0 to 3 seconds. Figures 5.3 through 5.5 are plotsofthe trajectory parameters for the two second trajectory (with the same initial conditions) along with the zero control trajectory values. These values are determined by starting at the boundary conditionsofthe optimal control trajectory and running the system backward with zero acceleration. For example,ifwestart at the final velocity and run backwards in time along the optimal trajectory, for each point in time, thereisa velocity (the null control velocity) that will take the corresponding positionofthe optimal control trajectorytothe boundary without additional input. The null control position begins at the origin at the final time, and moving backward in time,isthe position that will take the system to the boundary condition at the current velocity. Therefore, these are the positions and velocities (respectively) that will result in the boundary condition without additional input.Ast=>tfthe optimal trajectory acceleration approaches zero. Therefore, the zero control trajectory converges to the linear optimal trajectory.Ifthe system has a symmetric control constraint set, Reachable Set Control will control the system position to the zero control (constant velocity) trajectory.

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Accelerationo-500-1000-1500 -2000-250000.5451Time1 .52Figure 5.3 Linear Optimal Acceleration vs Time r--Null Control VelocityVelocity1500 10005000-500........-1000 -15000.50..............1Time1.52Figure 5.4 Linear Optimal Velocity vs Time

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46Position20001000......o-1000-2000-3000o NullControlPosition0.511.5TimeFigure 5.5 Linear Optimal PositionvsTime2Consider now the same problemwithinputconstraints. Since Vet)is a linear functionoftime and the final state,itismonotonicandthe constrained optimal controlisIu=SAT(-x(tf)(t-tf(5) "I In this case, controllability is in question,andisa functionofthe initial conditionsandthetime-to-go.Assuming controllability, the final state will be given by: x(tf)=xQ+ xQtf a(tl)SGN(x(tf)[tf-(tl/2)](tf-t1)31+---(6)

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47wheretlisthe switching time from saturated to linear control. The open loop switch time can be shown to be(7)or the closed loop control can be used directly. In either case, the optimal control will correctly control the system to a final state X(tf) near zero. Figures 5.6 through 5.8 illustrate the impactofthe constraint on the closed loop optimal control. In each plot, the optimal constrained and unconstrained trajectoryisshown.Acceleration1000o-1000-2000-3000 -4000.............-500000.51Time1 .52Figure 5.6 Unconstrained and Constrained Acceleration

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48Velocity2000o-2000-4000Constrained o0.51Time1 .52Figure 5.7 Unconstrained and Constrained VelocityvsTime ...... ...........21 .51Time0.5Constrained Position------l Position250020001500 10005000 0Figure 5.8 Unconstrained and Constrained Position vs Time

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49Now consider the effectsoftarget set uncertainty on the deterministic optimal control by using the same control law for a 2.0 second trajectory where the boundary conditionisnot constantbutchanges. The reason for the target uncertainty and selectionofthe boundary condition can be seen by analyzing the componentsofthe modeled system. Assume that system actually consistsofan uncontrollable reference (target) plant as wellascontrolled (missile) plant with the geometry modeled by the difference in their states. Therefore, the final set point (relative distance)iszero,butthe boundary condition along the controlled (missile) trajectoryisthe predicted target position at the final time. This predicted position at the final timeisthe boundary condition for the controlled plant. Figures 5.9 through5.11are plotsoflinear optimal trajectories using the control law in (5,6). There are two trajectoriesineach plot. The boundary condition for one trajectoryisfixed at zero, the set point for the other trajectory is the pointwise zero control value (predicted target state at the final time). Figures 5.9 through5.11demonstrate the impactofthis uncertainty on the linear optimal control law by comparing the uncertain constrained control with the constrained control that has a constant boundary condition.

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Acceleration1000o-1000 -2000 -3000 -400050......UncertainTarget Set -500000.51Time1 .52Velocity2000o-2000 -4000Figure 5.9 Acceleration ProfileWithand Without Target Set Uncertainty TargetSet \ .......................o0.51Time1 .52Figure 5.10 Velocity vs TimeWithand Without Target Set Uncertainty

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51PositionUncertainTargetSet ---------1400 1200 1000800600400200o00.51Time1 .52Figure5.11PositionvsTime With and Without Target Set Uncertainty When thereistarget set uncertainty, simulated by the varying set point, the initial accelerationisinsufficient to prevent saturation during the terminal phase. Consequently, the boundary conditionisnot met. The final setofplots, Figures 5.12 through 5.14, contrast the performanceofthe optimal LQG closed loop controller that we have been discussing and the Reachable Set Control technique.Inthese trajectories, the final set pointiszerobutthereistarget set uncertainty again simulatedbya time varying boundary condition (predicted target position) that convergestozero. Although properly shownasa fixed final time controller, the Reachable Set Control results in Figures 5.12 through 5.14 are from a simple steady state (fixed gain) optimal tracker referencedtothe zero control trajectory r.

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52Thesystem modelforeach techniqueis(8)withx(to)=(x-r)oThelinear optimal controller has a quadratic costofJ=2Thereachable set controller minimizes(9)J= J : 1)[(x-r)TQ(x-r)+u(r)Tu(r)]dr(10)And,ineithercase, the value for r(t)isthe position that will meet theboundaryconditionatthe final time withoutfurtherinput.Acceleration1000o-1000 -2000 -3000-4000ReachableSetControl -500000.51Time1 .52Figure 5.12 Acceleration vsTimeLQGand Reachable Set Control

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Velocity2000o-2000 -400053.......................................................-__-_. "'-ReachableSetControlo0.51 Time1 .5 2".'.ReachableSetControl Position140012001000800600400200o0Figure 5.13 Velocity vs Time LQG and Reachable Set Control"..............0.51TimeFigure 5.14 Position vs Time LQG and Reachable Set Control1 .5'"2

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54Summary The improved performanceofReachable Set Controlisobvious from Figure 5.14. While demonstrated for a specific plant, and symmetric control constraint set, Reachable Set Controliscapableofimproving the terminal performanceofa large classofsystems.Itminimized the effectsofmodeling errors (or target set uncertainty) by regulating the system to the zero control state. The technique handled constraints and insured an initially constrained trajectory. The tracking problem could be optimized to the response timeofthe system under considerationbysmoothly driving the system from some large displacementtoa region where the relatively high gain LQG controller remained linear.

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CHAPTER VI REACHABLE SET CONTROL FOR PREFERRED AXIS HOMING MISSILES As statedinChapter II, the most promising techniques that can extend the inertial point mass formulation are based on singular perturbations [37,38,39]. When applied to the preferred axis missile, eachofthese techniques leads to a controller that is optimal in some sense. However, a discussionof"optimality" notwithstanding, the best homing missile intercept trajectory is the one that arrives at the final "control point" with the highest probabilityofhitting the target. This probability can be broken down into autonomous and forced events.Ifnothing is changed, what is the probabilityofa hitorwhat is the miss distance?Ifthe target does not maneuver, can additional control inputs result in a hit? And, in the worse case,ifthe target maneuvers (or an estimation error iscorrected) will the missile have adequate maneuverability to correct the trajectory? Noneofthe nonlinear techniques based on singular perturbations attempt to control uncertaintyoraddress the terminally constrained trajectories caused by increasing acceleration profile. Unfortunately, an increasing acceleration profile has been observed in allofthe preferred axis homing missile controllers. In many cases, the genericbank-to-turnmissileof[11,18] was on all three constraints (Ny,Nz,P) during the latter portionofthe trajectory.Ifthe evading targetisable toputthe missileinthis position without approaching it's own maneuver limits,itwill not be possible for the missile to counter the final evasive maneuver. The missileisno longer controllable to the target set. The "standard" solution to the increasing acceleration profileisa varyingcontrol cost.55However, without additional

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56 additional modifications, this typeofsolution results in a trajectorydependentcontrol. As we have seen, Reachable Set Control is anLQG controlimplementationthatmoves the system to the point wherefurtherinputs are not required. A Reachable Set Controller that willrejecttarget and system disturbances,cansatisfy both the mathematicalandheuristic optimality requirementsbyminimizing the cost yet maintaining a controllable system. Since the roll control hasdifferentcharacteristics, the discussionofthepreferredaxis homing missile controller using the Reachable Set Control technique will be separated into translationalandroll subsystems.Thetranslational subsystem has a suitable null control trajectorydefinedbythe initial velocity and uncontrollable acceleration provided by the rocket motor.Theroll subsystem, however, is significantlydifferent.Inorderforthepreferredaxis missile to function, thepreferredaxis must be properly aligned. Consequently, both roll angleerrorandroll rate should be zero at all times. In this case, the null control trajectory collapses to the origin. Acceleration Control System Model Since we want to control the relative target-missile inertial system to the zero state, the controller will bedefinedinthis reference frame. Eachofthe individual system states aredefined(in relative coordinates) as target state minus missile state. Begin with the deterministic system: x(t)=Fx(t)+Gu(t)(I)

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57where andxyzx=Vx Vy Vzu= Since the autopilot modelisa linear approximation and the inertial model assumes instantaneous response, modeling errors will randomly affect the trajectory. Atmospheric and other external influences will disturb the system. Also, the determinationofthe state will requirethe useofnoisy measurements. Consequently, the missile intercept problem should be approached via a stochastic optimal control law. Because the Reachable Set Control technique will minimize the effectofplant parameter variations (modeling errors) and un modeled target maneuverstomaintain controllability, we can use an LQG controller. Assuming Certainty Equivalence, this controller consistsofan optimal linear (Kalman) filter cascaded with the optimal feedback gain matrixofthe corresponding deterministic optimal control problem. Disturbances and modeling errors can be accounted for by suitably extending the system description[40]:x(t)=F(t)x(t)+G(t)u(t)+Vs(t) by adding a noise process V s (") to the dynamics equations with(2) (3)

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58Therefore, let the continuous time state description be formally given by the linear stochastic differential equationx(t)= ct>( t, to)x( to) dx(t)=F(t)x(t)dt+G(t)u(t)dt+L(t)df3(t) (with13(,.)a Wiener process) that has the solution: ct>( t,r)G(r)u(r)dr+
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state to converge with zero error.59Since the optimal solution to the linear stochasticdifferentialequation is a Gauss-Markov process, time correlated processescanbe includedbyaugmenting the system state to include the disturbance process.Letthe time-correlated target (position) disturbance be modeled by the following: T(t)=N(t)T(t)+Wt(t)(8)with[Tx(t)IT(t)=Ty(t) Tz(t) and =Wt(t) While the target disturbance resulting from anunknownacceleration is localized to a single plane with respect to the body axisofthe target, the target orientation is unknown to the inertial model. Consequently, following the methodologyofthe Singer Model, each axis will be treated equally [41]. Since the disturbance is firstorderMarkov, it's components will be:andN(t)=-(I/Tc)[I] Wt(t)=(20't2 / T c)[I](9) (10)where Tcisthe correlation time, and O'r isthe RMS valueofthe disturbance process.ThePower Spectral Densityofthe disturbanceis: Wtt(W) =20't2 / T cw2+O/Tc)2(11) Figure 6.1 summarizes the noise interactions with the system.

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60WhiteGaussianNoiseJ-------'AtmosphericdisturbancesActuatorErrorsAutopilotErrors"DeterministicControls LinearSystem OutputsPhysicalModelSystemResponse Jl TargetAccelerationsShapingFilterWhite Gaussian NoiseFigure6.1Reachable Set Control Disturbance processes.Withappropriate dimensions, the nine state (linear) augmented system model becomes:[ ]T(t)=Reference Model Reachable Set Control requires a supervisory steering control (reference) that includes the environmental impact on the controlled dynamic system. Recalling the characteristicsofthe dual system, one was developed that explicitlyran(I) backward in time after determining the terminal conditions. However, in developing this control for this preferred axis missile a numberoffactors actually simplify the computationofthe reference trajectory:

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61(I)Thecontrol constraint setforthispreferredaxis missile is symmetric.Consequently.thereferencetrajectoryforaninterceptcondition.is a null control (coasting)trajectory.(2)Thebodyaxis X acceleration isprovidedbythemissile motor,andisnotcontrollablebutknown. This uncontrollableaccelerationwillcontributetothetotal inertial acceleration vector.mustbeconsideredbythe controller.andistheonlyacceleration present onanintercept(coasting)trajectory.(3)Theterminationoftheinterceptistheclosestapproach.whichnowbecomes thefixed-final-time(tf).Thetime-to-go(tgo) isdefinedwith respect tothecurrenttime (t) by: t=tftgo(13)(4)Thefinalboundaryconditionforthesystem state (target minus missile) is zero.Insummary,theinterceptpositions are zero.theinitial velocity is given.andthe average acceleration is a constant.Therefore.itissufficientto reversethedirectionoftheinitial velocityandaverageaccelerationthenrunthesystemforwardintimefortgo secondsfromtheorigintodeterminethecurrentpositionofthecoasting trajectory. Let:withandr(t)=A(t)r(t)+B(t)a(t)r(O)=rO=0A(t)=F(t)andB(t)=G(t)(14)thenr(t) isthepointfromwhich theautonomoussystem dynamics will takethesystem todesiredboundarycondition. Becauseofthe disturbances. targetmotion,andmodelingerrors,futurecontrolinputsarerandomvectors.Therefore,thebest policy isnotto

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62determine the input over the control period [to,tfl a priori buttoreconsider the situation at each instant t on the basisofall available information. At each update,ifthe systemiscontrollable, the reference (and system state) will approach zeroastgo approaches zero. Since the objectiveofthe controlleristo drive the system state to zero,wedo not require a tracker that will maintain the control variable at a desired non-zero value with zero steady state error in the presenceofunmodeled constant disturbances. There are disturbances, but the final set pointiszero, and therefore, aPIcontrollerisnot required. Roll Control Definition The roll modeismost significant sourceofmodeling errors in the preferred axis homing missile. While non-linear and high order dynamics associated with the equationsofmotion, autopilot, and control actuators are neglected, the double integratorisan exact modelfordetermining inertial position from inertial accelerations. The linear system, however,isreferenced with respect to the body axis. Consequently, to analyze the complete dynamics, the angular relationship between the body axis and inertial references must be considered. Recall Friedland's linearized (simplified) equations. The angular relationships determine the orientationofthe body axis reference and the roll rate appeared in the dynamicsofall angular relationships. Yet,tosolve the system using linear techniques, the system must be uncoupled via a steady state (Adiabatic) assumption. Also, the roll angleisinertially defined and the effectofthe linear accelerations on the erroristotally neglected.

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63Froma geometric pointofview, this mode controls the rangeofthe orthogonal linear acceleration commandsandthe constrained controllabilityofthe trajectory. With a20:I ratio in thepitchandyaw accelerations, the ability topointthepreferredaxis in the "proper" direction is absolutely critical. Consequently, effective roll control is essential to the performanceofthepreferredaxis homing missile.Thefirst problem in defining the roll controller, is the determinationofthe "proper" direction. There are two choices.Thepreferredaxis could be aligned with the target positionorthedirectionofthe commanded acceleration.Thefirstselectionisthe easiest to implement.Theseeker gimbal angles provide adirectmeasureofintercept geometry (Figure 6.2), and the roll angleerrorisdefineddirectly:(15)TargetLOS eyzxFigure 6.2. Roll AngleErrorDefinitionfromSeeker Angles. This selection, however,isnot the most robust. Depending on the initial geometry, the intercept point may not beinthe planedefinedbythecurrent

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64lineofsight (LOS) and longitudinal axisofthe missile. In this case, the missile must continually adjust its orientation (roll) to maintain the target in the preferred plane.Asrange decreases the angular rates increase, with the very real possibilityofsaturation and poor terminal performance. Consequently, the second definitionofroll angle error should be used. Considering the dynamicsofthe intercept, however, aligning the preferred axis with the commanded acceleration vectorisnotasstraightforwardasit seems. Defining the roll angle erroras 0e =Tan-I{A y /A z } (16) leadstosignificant difficulties. From the previous discussion, itisobvious that roll angle errors must be minimizedsothat the preferred axis acceleration canbeusedtocontrol the intercept. The roll controller must have a high gain. Assume, for example, the missileison the intercept trajectory. Therefore, both A y and A z will be zero. Now,ifthe target moves slightly in theYbdirection and the missile maneuvers to correct the deviation, the roll angle error instan taneously becomes 90 degrees. High gain roll control inputs to correct this situation are counter productive. The small A y may be adequate to completely correct the situation before the roll mode can respond. Now, the combinationoflinear and roll control leads to instabilityasthe unnecessary roll rate generates errors in future linear accelerations. The problems resulting from the definitionofequation16can be overcomebere-examining the roll angle error. First, A y and A z combine to generate a resultant vector at an angle from the preferred axis. In the processofapplying constraints, the acceleration angle that results from the linear accelerations canbeincreased or decreased by the presenceofthe constraint.Ifthe angleisdecreased, the additional rollisneeded to line up the preferred axis and the

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65desired acceleration vector.Ifthe constrained (actual) acceleration angleisincreased beyond the (unconstrained) desired value by the unsymmetric actionofthe constraints, then the roll controller must allow for this "over control" caused by the constraints. Define the roll angle errorasthe difference between the actual and desired acceleration vectors after the control constraints are considered. This definition allows for the full skid toturncapabilityofthe missile in accelerat ing toward the intercept point and limits rolling to correct large deviations in acceleration angle from the preferred axis that are generated by small accelera tions. ReferringtoFigure 6.3, three zones can be associated with the following definitions: "ec =Tan-I(Ay/Az } 'lea =Tan-I(Ny/Nz } "er = "ec 'lea;ZoneIZoneIIZoneIIIFigure 6.3. Roll Control Zones.(17) (18) (19)

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66Here Ny and N z are the constrained acceleration values. In Zone I, "er =O.The linear acceleration can complete the intercept without further roll angle change. Thisisthe desired locus for the roll controller. Both A y and A z are limited in Zone II. Thisisthe typical situation for the initial positionofa demanding intercept. The objectiveofthe roll controlleristokeep the intercept acceleration outofZone III where only A yislimited. In this case, the A y accelerationisinsufficient to complete the intercept yet significant roll angle change may be requiredtomake the trajectory controllable. Controller A dual mode roll controllerwasdeveloped to accommodate the rangeofsituations and minimize roll angle error. Zone I requires a lower gain controller that will stabilize the roll rate and maintain "er small. Zones II and III require high gain controllers. To keep Zone II trajectories from entering Zone III, the "ec will be controlled to zero rather than the roll angle error. Since the linear control valueisalso a functionofthe roll angle error, roll angle errors are determined by comparing the desired and actual anglesofa fixed high gain reachable set controller.Ifthe actual linear commands are used and a linear accelerationissmall becauseoflarge roll angle errors, the actual amountofroll needed to line up the preferred axis and the intercept point, beyond the capabilityofthe linear accelerations, will not be available because they have been limited by the existing roll angle error that must be corrected. Unlike the inertial motions, the linear model for the roll controller accounts for the (roll) damping and recognizesthatthe inputisa roll rate change command: = -w" +wP(20)

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67Therefore, the roll mode elements (that will be incorporated into the model are) are: (21) Also, the dual mode controller will require an output function and weighting matrix that includes both roll angle and roll rate. Kalman Filter The augmented system model (13)isnot block diagonal. Consequently, the augmented system filter will not decouple into two independent system and reference filters. Rather, a single, higher order filter was required to generate the state and disturbance estimates. A target model (the Singer model)wasselected and modified to trackma-neuvering targets from a Bank-To-Turnmissile [41,42]. Using this model, a continuous-discrete Extended Kalman Filter was developed. The filter used a9 state target model for the relative motion (target missile): (22) with u(t) the known missile acceleration, N the correlation coefficient, and Wt(t) an assumed Gaussian white noise input with zero mean. Azimuth, elevation, range, and range rate measurements were available from passive JR, semi-active, analog radar, and digitally processed radar sensors. The four measurements are seeker azimuth (,p), seeker elevation(0),range (r), and range rate (dr/dt):

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68 6 =-Tan-l{z(x2 +y2)-1/2} t/J =+Tan-l{y/x}= 1r +Tan-l{y/x}r={x2+y2+z 2}-1/2 r= x 0x 0(23)Noise statistics for the measurements are a functionofrange, and are designed to simulate glintandscintillation in a relatively inexpensive missile seeker. In contrast to the linear optimal filter, theorderofthe measurements for the extended filterisimportant. In this simulation, the elevation angle (6) was processed first, followed by azimuth (t/J), range (r), and range rate(dr/dt).In addition, optimal estimates were available from the fusionofthe detailed (digital) radar model and IR seeker. Reachable Set Controller StructureTheTarget-Missile System is showninFigure 6.4. The combinationofthe augmented system state and the dual referencethatgenerates the minimum control trajectory for the reachable set concept is best describedasa CommandGenerator/Trackerandisshown in Figure 6.5.Ina single systemofequations the controller models the system response, including time correlated position disturbances, and provides the reference trajectory. Since only noise-corrupted measurementsofthe controlled system are available, optimal estimatesofthe actual states were used. Becauseofthe processing time requiredforthe filter and delays in the autopilot response, a continuous-discrete Extended Kalman Filter,anda sampled

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69data(discrete) controller was used. This controller incorporated discrete cross-coupling terms to control the deviations between the sampling timesaswell the capability to handle non-coincident sampleandcontrol intervals (Appendices B andC).Combining the linear and roll subsystems with a firstorderroll mode for the roll angle state, the model for the preferred axis homing missile becomes: the reference: r(t)=A(t)r(t)+B(t)a(t) with the tracking error: e(t)=[yx(t) Yr(t) ]=[H(t)I0I-C(t)] r(t)(25)(26)The initial stateismodeled asann-dimensional Gaussian random variable with meanxOand covariancePO.E{ws(t)ws(t)T}=Ws(t) is the strengthofthe system (white noise) disturbances to be rejected,andE{wt(t)Wt(t)T}=Wt(t)isaninput to a stationary firstorderGauss-Markov processthatmodels target acceleration. The positions are the primary variablesofinterest,andthe output matrices will select these terms. Along with the roll rate, these are the variables that will be penalized by the control costandthe states where disturbances will directly impact the performanceofthe system.

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In block form, with appropriate dimensions, the system matrices are:F(t)=A(t)=F=[0I]OOwG(t)=B(t)=G=[0]Iw(28)H(t)=C(t)=[Ihw ]N(t)=-(ljTc)[I]L(t)=IM(t)=Iwhere the Ow, I w and h w terms are required to specify the roll axis system and control terms:

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71{-wi=j=8{1i=j=8 (Ow>ij = (hw>ij =0otherwise0otherwise{+wi=8,j=4 (Iw)ij = I otherwise The performance objective for the LQG synthesisisto minimize an appropriate continuous-time quadratic cost: Js(t) = E{Jd(t)II(t)}(29)where J sisthe stochastic cost, I(t)isthe information set available at time t, and Jd a deterministic cost function:(30)Dividing the intervalofinterest into N+I intervals for discrete time control, and summing the integral cost generates the following (see AppendixC):(31)which can be related to the augmented state X=[xTr ]T by:][ X(tj) u(ti)](32)

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72In general, with the cost terms defined for the augmented state (AppendixC),the optimal (discrete) solution to the LQG tracker can be expressedas:where*=-[G(ti)] [ x(ti) T(ti) r(ti)](33)and G*(ti)=[R(ti) + GT(ti)P(ti+l)G(ti) ] -1 [ + ST(tj) ] P(ti)=Q(ti) + -[+ ST(ti) ]TG*(ti)(34) (35)Since only the positions (and roll rate) are penalized, the Riccati recursionisquite sparse. Consequently, by partitioning the gain and Riccati equations, and explicitly carrying out the matrix operations, considerable computationalim-provements are possible over the straightforward implementationofa19by19tracker (Appendix D). Application The tracking error and control costs were determined from the steady state tracker used in the example in Chapter5.First, missile seeker and aerodynamic limitations were analyzed to determine the most demanding intercept attainable by the simulated hardware. Then, autopilot delays were incorporated to estimate that amountoftime that a saturated control would require to turn the missile after correcting a 90 degree (limit case) roll angle error. The steady state regulator was used to interactively place the closed loop polesandselect a control cost combination that generated non zero control for the desired lengthoftime. These same values were used in the time varying Reachable Set Controller with the full up autopilot simulation to determine the

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73terminal error cost and control delay time. To maintain a basisofcomparison, the Kalman Filter parameters were not modified for this controller. Appendix E contains initial conditions for the controller and estimator dynamics. During the initialization sequence (safety delay) for a given run, time varying fixed-final-time LQG regulator gains are calculated (via 36) based on the initial estimateofthe time togo.Both high and low roll control gains were computed. These solutions used the complete Riccati recursion and cost based on the sampled data system, included a penalty on the final state (to control transient behaviorastgo approaches zero), and allowed for non-coincident sample and control. Given an estimated tgo, at each time t, the Command GeneratorITracker computed the reference position and required roll angle that leads to an intercept without additional control input. The highorlow roll control gain was selected based on the mode. Then the precomputed gains (that are a functionoftgo) are used with the state and correlated disturbance estimate from the filter, roll control zone, and the reference r to generate the control (whichisapplied only to the missile system). Becauseofsymmetry, the tracker gain for the state term equaled the reference gain,sothat, in effect, except for the correlated noise, the current difference between the state x and the reference r determined the control value. During the intercept, between sample times when the stateisextrapolated by the filter dynamics, tgo was calculated based on this new extrapolation and appropriate gains used. This technique demonstrated better performance than using a constant control value over the durationofthe sample interval and justified the computational penalty of the continuous discrete implementationofthe controller and filter.

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MeasurementNoise InterceptGeometry--Sensors--TargetControl--Disturbances TargetDynamics--OutputFunction+ l' Command -ZeroOrder GeneratorI .. Hold -DIATrackerStateEstimationSample&Hold AIDSensorsReachableSetControllerContinuous-DiscreteKalmanFilterMeasurementNoise Autopilot--Actuators Missile MissileDynamics-OutputFunctionDynamicDisturbancesFigure 6.4 Target Missile System

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r-Reference ModelReferenceVariableDynamicsa....Missile Model Yr(t)e(t) ,----,LJ-DynamicDisturbancesMeasurementNoise ", G1G3u(ti)I---{ "')-------1 G2Missile -TargetSystem X(t)Q!LJi Z(t)Yx(t)X(ti)TargetManeuverModelA(ti)ContinuousDiscreteKalmanFilter-Figure 6.5 Command Generator / Tracker

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CHAPTER VII RESULTS AND DISCUSSIONAsan additional reference, before comparing the resultsofReachable Set Control to the baseline control, consider anair-to-airmissile problem from [13]. In this example,the launch directionisalong the lineofsight, the missile velocityisconstant, and the autopilot response to commandsisinstantaneous. The controller has noisy measurementsoftarget angular location, a priori knowledgeofthe time to go, and stochastically models the target maneuver. Even with this relatively simple problem, the acceleration profile increases sharply near the final time. Unfortunately, this acceleration profileistypical, and has been observed in all previous optimal control laws. Reachable Set Control fixes this problem. c.9-...oa:)_u'"u u0oen--....t= .EenE0=: 300 200100o6Time to go (sec) O'--....L------l-.---l 12Figure7.1RMS Missile Acceleration76

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77Simulation The performanceofReachable Set Control was determined via a highfi-delity Bank-To-Turn simulation developed at the UniversityofFlorida and used for a numberofprevious evaluations. The simulationisbased on the coupled non-linear missile dynamicsofchapter II equations (1)to(8) andisa continuous-discrete system that has the capabilityofcomparing control laws and estimators at any sample time. In addition to the non-linear aerodynamic parameters, the simulation models the Rockwell Bank-To-Turn autopilot, sensor (seeker and accelerometer) dynamics, has a non-standard atmosphere, and mass modelofthe missile to calculate time-varying momentsofinertia and the missile specific acceleration from the time varying rocket motor. Figure 7.2 presents the engagement geometry and someofthe variables used to define the initial conditions.Missiley,ZTarget G ax D ..... U9 ;./\j! 9Figure 7.2 Engagement Geometry

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78The simulated targetisa three (3) dimensional, nine (9)"g"maneuvering target. Initially, the target trajectoryisa straight line. Once the range from the missile to targetisless than 6000 feet, the target initiates an instantaneous 9"g"evasive maneuver in a plane determined by the target roll angle, an input parameter.Ifthe launch rangeiswithin 6000 feet,the evasive maneuver begins immediately. Thereisa.4second "safety" delay between missile launch and autopilot control authority. Trajectory Parameters The performanceofthe control laws was measured with and without sensor noise using continuous and sampled data measurements. The integration step was .005 seconds and the measurement step for the Extended Kalman Filter was.05seconds. The trajectory presented for comparison has an initial offset angleof40 degrees (tPg) and180degree aspect (tPa), and a target rollof90 degrees away from the missile. This angleoffand target maneuverisoneofthe most demanding intercept for a preferred axis missile since it must roll through 90 degrees before the preferred axisisaligned with the target. Other intercepts were run with different conditions and target maneuvers to verify the robustnessofReachable Set Control and the miss distances were similar or less that this trajectory. Results Deterministic Results These results are the best comparisonofcontrol concepts since both Linear Optimal Control and Reachable Set Control are based on assumedCertainty Equivalence.

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79Representative deterministic results are presented in Table7.1and Figure 7.3. Figures A.l throughA.9present relevant parameters for the 4000 foot deterministic trajectories. Table7.1Deterministic Control Law Performance Initial Control Time Miss Range Distance (feet) (sec) (feet) 5500 Baseline 2.34 8 Reachable 2.34 6 5000 Baseline2.2113Reachable2.21104800 Baseline 2.1715Reachable 2.17 4 4600 Baseline 2.13 29 Reachable 2.13 6 4400 Baseline 2.0638Reachable 2.08 7 4200 Baseline 2.0235Reachable 2.05 5 4000 Baseline1.9854Reachable 2.00133900 Baseline1.9843Reachable1.988 3800 Baseline1.9840 Reachable1.988 3700 Baseline 2.02 44 Reachable 1.99103600 Baseline 1.99136Reachable 1.9965

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80MissDistance(feet) BaselineGuidanceLaw ............./ SetControl140 f-120 100 80 f-60-40 20 e-j500 40004500Initial Range50005500Figure7.3Deterministic Results An analysisoftrajectory parameters revealed that oneofthe major performance limitations was the Rockwell autopilot. Designed for proportional navigation with noisy (analog) seeker angle rates, the self adaptive loops in the autopilot penalized a high gain control law suchasReachable Set Control. This penalty prevented Reachable Set Control from demonstrating quicker intercepts and periodic control that were seen with a perfect autopilot on a similar simulation used during the research. However, even with the autopilot penalty, Reachable Set Controlwasabletosignificantly improve missile performance near the inner launchboundary. This verifies the theoretical analysis, since thisisthe region where the target set errors, control constraints, and short run times affect the linear law most significantly.

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81Stochastic Results Stochastic performance was determined by 100 runsateach initial condition.Atthe terminationoftherun,the miss distanceandTimeofFlight (TOF) was recorded. During eachofthese runs, the estimatorandseeker (noise)errorsequences were tracked. Both sequences were analyzed to insure gaussian seeker noise, and an unbiased estimator (with respect to each axis).Fromthe final performance data, the meanandvarianceofthe miss distance was calculated. Also, from the estimator and seeker sequences, the root mean square (RMS)errorand varianceforeach run was determined toidentifysome general characteristicsofthe process. The averageofthese numbersispresented. Care must be taken in interpreting these numbers. Since the measurement errorisafunctionofthe trajectoryaswell as instantaneoustrajectoryparameters, a singlenumberisnot adequate to completely describe the total process. Table 7.2andFigure 7.4 present average results using the guidance laws with noisy measurementsandthe Kalman filter.MissDistance(feet)250 200150 100BaselineGuidanceLaw..,..' ., .....50 ReachableSetControl S500 40004500Initial Range50005500Figure 7.4 Stochastic Results

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82 Table 7.2 Stochastic Control Law Performance Initial Control Time Miss Distance RMS Error Range Mean VarianceEKFSeeker (feet) (sec) (feet) (feet)(deg) 5500 Baseline 2.38 273 11470111.3Reachable2.41832677111.55000 Baseline 2.23 240 7874111.4Reachable 2.25 89 3937101.64800 Baseline 2.181937540101.4Reachable 2.191143708101.64600 Baseline 2.13 172 5699101.4Reachable 2.131071632101.54400 Baseline 2.081294324101.5Reachable 2.07 851421101.64200 Baseline 2.041233375101.6Reachable 2.03 62 673101.74000 Baseline2.011052745101.7Reachable2.01661401101.83900 Baseline 2.001053637101.8Reachable 2.00794356101.83800 Baseline 2.001245252101.8Reachable 1.9910510217102.0 3700 Baseline1.981595240101.8Reachable1.9817613078 91.83600 Baseline1.95230 6182101.7Reachable1.95239 14808101.7

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83The first runs made with Reachable Set Control were notasgoodasthe results presented. Reachable Set Control was only slightly (10to20feet) superiortothe baseline guidance law and was well below expectations. Yet, the performanceofthe filter with respect to position error was reasonable, manyofthe individual runs had miss distances near20feet,andmostofthe errors were in the Z axis. Analyzing several trajectories from various initial conditions ledtotwo main conclusions. First, the initial and terminal seeker errors were quite large, especially comparedtothe constant 5 mrad tracking accuracy assumed by many studies [2,3,18]. Second, the non-linear coupled natureofthe preferred axis missile, combined with range dependent seeker errors, and the system (target) model, makes the terminal performance a strong functionofthe particular sequenceofseeker errors. For example, Figure 7.5 compares the actual and estimated Z axis velocity (Target Missile) from a single 4000 foot run. The very first elevation measurement generated a14foot Z axis position error. A reasonable number considering the range. The Z axis velocity error, however, was quite large, 409 feet per second, and never completely eliminated by the filter. Recalling that the target velocityis969 feet per second,isapproximately co-altitude with the missile and maneuvers primarily in the XY plane, this errorissignificant when compared to the actual Z axis velocity(2feet per second). Also, thisisthe axis that defines the rollangle error and, consequently, roll rateofthe missile. Errorsofthis magnitude cause the primary maneuver planeofthe missile to roll away from the target limiting (via the constraints) the abilityofthe missile to maneuver. Further investigation confirmed that the filter was working properly. Although the time varying noise prevents a direct comparison for an entire trajectory, these large velocity errors are consistent with the covariance ratios

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84 in [41]. The filter model was developed to track maneuvering targets. Thepen-alty for tracking maneuvering targetsisthe inability to precisely define allofthe trajectory parameters (ie. velocity). More accurate (certain) models track better,butrisk losing track (diverging) when the target maneuvers unexpectedly. The problem with the control then, was the excessive deviations in the velocity. To verify this, the simulation was modifiedtouse estimatesofposition,buttouseactual velocities. Figure 7.6 and Table7.3has these results.Asseen from the table, the control performanceisquite good considering the noise statistics and autopilot.ZAxis Velocity(feet/sec)600400200 Actual, 0-200 -400 -6000 0.5 11 .5 2TimeFigure7.5MeasuredvsActual Z Axis Velocity

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85 Table 7.5 Stochastic ControlLawPerformance Using Actual Velocities Initial Control Time Miss Distance RMSErrorRange Mean VarianceEKFSeeker (feet) (sec) (feet) (feet) (deg) 5500 Baseline 2.35 34 318101.5 Reachable 2.3738504II1.5 5000 Baseline 2.21 50 506101.7Reachable 2.2338623101.6 4800 Baseline 2.1755367101.6 Reachable 2.19 48 526101.54600 Baseline 2.1261366101.6 Reachable 2.1452326101.6 4400 Baseline 2.06 62 333101.7 Reachable 2.1045415101.7 4200 Baseline 2.02 60 277101.7 Reachable 2.06 44 463101.84000 Baseline 1.98 62 330101.9 Reachable 2.0253lOll9 1.9 3900 Baseline 1.9855436101.9 Reachable 2.00 64 2052101.9 3800 Baseline 1.98 53 400 9 2.0 Reachable 1.99912427101.8 3700 Baseline 1.99 63 235102.0 Reachable 1.991403110101.7 3600 Baseline 1.98138354101.8 Reachable 1.96 213 4700101.6

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86MissDistance(feet)25020015010050 ,. ReachableSetControlBaseline GUidance/ ................................................... -------_...------9500 40004500InitialRange5000 5500Figure 7.6 Performance Using Position Estimates and Actual Velocities While the long term solution to the problemisa better target model that will accommodate both tracking and control requirements, the same model was used in order to provide a better comparison with previous research. For the same reason, the Kalman filter was not tuned to function better with the higher gain reachable set controller. However, target velocity changes used for the generationofthe roll angle error were limited to the equivalentofa 20 degree per second target turn rate. This limited the performance on a single run, but precluded the 300 foot miss that followed a 20 foot hit.

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87Conclusions Reachable Set ControlAsseen from the Tables7.1and 7.2, Reachable Set Controlwasinherently more accurate than the baseline linear law, especiallyin the more realistic case where noiseisadded and sampled data measurements are used. In addition, Reachable set control did insure an initially constrained trajectory for controllable trajectories, and required minimal accelerations duringthe terminal phaseofthe intercept. Unlike previous constrained control schemes, Reachable Set Control was better able to accommodate tinmodeled non-linearities and provide adequate performance with a suboptimal sampled data controller. While demonstrated with a Preferred Axis Missile, Reachable Set Controlisa general technique that could be used on most trajectory control problems. Singer Model Unmodified, the Singer model provides an excellent basis for a maneuver ing target tracker, but itisnot a good model to use for linear control. Neither control law penalized velocity errors. In fact, the baseline control law did not define a velocity error. Yet the requirementsoflinearity, and the integration from velocity to position, require better velocities estimates than are provided by this model (for this quality seeker).

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APPENDIXASIMULAnONRESULTS

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YPosition(feet)40003000 20001000o0 13 footmiss 100020003000XPosition(feet)FigureA.lXYMissile&Target Positions Reachable Set Control4000YPosition(feet)40003000 200054 feet 1000Behind Targeto0100020003000XPosition(feet)Figure A.2XYMissile&Target Positions Baseline Control Law894000

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90

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91Acceleration(g)10050 -o-50 f-t >1009CommandedAcceleration-100o0.51Time1 .52FigureA.SMissile Acceleration Reachable Set ControlAcceleration(g)10050o-50 I>1009CommandedAcceleration-100o0.51Time1 .52Figure A.6 Missile Acceleration Baseline Control Law

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92Roll Rate (deg/sec)6004002000-200-400 -60000.51Time...1.52Figure A.7 Missile Roll Commands&Rate Reachable Set ControlActualRoIIRate(deg/sec)600400200o-200 -400Commanded '.:-600o0.51Time1 .52Figure A.8 Missile Roll Commands&Rate Baseline Control Law

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93Roll AngleError(deg)50o-50o0.51Time1 .52Figure A.9 Missile Roll Angle Error

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APPENDIX B SAMPLED-DA TA CONVERSION System Model The system model for the formulationofthe sampled-data control law consistsofthe plant dynamics with time correlated and white disturbances: [ ]=[F(t) ][X(t)]+[GO(t)] u(t)+T(t) 0 N(t) T(t) the reference: r(t)=A(t)r(t)+B(t)a(t) with the tracking error:[ L0][WS(t) ](1)OMWt(t)(2)e(t)=[yx(t) -Yr(t)]=[H(t)I0I-C(t) ] )(3)r(t)andquadratic cost Js(t)=E{Jd(t)II(t)}, where(4)E{ws(t)ws(t)T}=Ws(t)isthe strengthofthe system (white noise) disturbances to be rejected. E{wt(t)Wt(t)T}=Wt(t) is an input to a stationary first order Gauss-Markov process that models target acceleration (see CH VI). The following assumes a constant cycle time that defines the sampling interval and a possible delay between sampling and controlof =ti' ti94

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9SThecomponentsofthe controlled system arex(t)=x(t)y(t)Z(t)0(t)x(t)T(t)= [::::: ]Tz(t) r(t)(5)o0u(t)y(t) z(t)0(t)Nx(t)Ny(t)Nz(t)P(t)e(t)x(t)y(t) z(t)0(t)ry(t) rz(t)r0(t)rx(t) ry(t) rz(t)r0(t)o0o0In block form, withappropriatedimensions, the system matrices are F(t) = A(t) =F=[0I]OwG(t)=B(t)=G= (6)H(t)=C(t)=[Ihw ]i=j=8 otherwise{+wi=8,j=4(Iw)ij=I otherwise L(t)=I(hw)jj={ M(t)=Ii=j=8 otherwise

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96Sampled-dataEquationsSystemFromthecontinuoussystem,thediscrete-timesampled-datasystemmodelcanbesummarizedinthefollowingsetofequations:y(ti')=H x(ti)+D u(ti)ForatimeinvariantFwithaconstantsampleinterval: (t .t)-eF(t..-1.)eF(.lt) with .lt -ttxHI'1-HI1-HI-1I000 .lt 00 0 0I00 0 .ltO 0 00I0 0 0 .ltO 0 00I0 0 0 x 48x(.lt) =0 000I000 0 0000I00 0 000 00I0 0 000 000 x 88 with x 48(.lt) = (l!w(.lt)})x 88(.lt) =exp{ -w(.lt)} (7) (8)

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97Also, since G and H are time invariantandthe u(ti) are piecewise continuous, theinputandoutputmatrices (allowing fornon-coincidentsamplingandcontrol) arefi+lG= cP(ti+l,r)G(r)dr (9)t.1H=H cP(ti',ti) (10) r D=H (11)t.1Therefore, At 2j2 00 0 0 At2j2 0 0 00 At 2j2 0 0 00 At-cPx4S(At) G= At 0 0 0 (12) 0 At 0 0 0 0 At 0 0 0 0 l-cPxSS(At) I00 0 At' 000 0I00 0 At' 00H=00I00 0 At' 0(13)000I0 0o cPX4S(At') 00 000 000 0 0 0 00 000 0 0 0 00 000 00 0 00 0o cPXSS(At')At,2j2 00 0 0 At,2 j 2 0 0 00 At,2j2 0 0 00 At' -cPX4S(At') D= At' 00 0 (14) 0 At' 0 0 0 0 At' 0 0 0 0 l-cPxSS(At')

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98Target DisturbanceFortime correlated disturbances, Wt(t)=Wt=2 at2 ITc where at is the RMS valueofthe noise process T(,.), and T c is the correlation time.N(t)=-(1/Tc)[I]The sampled data disturbanceis(15) (16)Wt(ti)isa sequenceofzero mean mutually uncorrelated random variables, and is givenby:with =T c(l the sampled data impact on the system.(17)

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100 C=C
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APPENDIX C SAMPLED DATA COST FUNCTIONS Assume that the performance objectiveisto minimize an appropriate continuous-time quadratic cost.(1)where J sisthe stochastic cost, Jd a deterministic cost, and I(t)isthe information set available at time t. Let Jd(t)=erT Prer +rr(e(t)TQ(t)e(')+u(t)TR(t)u(tdT to (2)Dividing the intervalofinterest into N+l control intervals for discrete time control, Jd(t)=e(tn+l)TPfe(tn+l)N Jti+1 + [t{e(t)TQ(t)e(t) + u(t)TR(t)u(t)}dr]1whereforall tE [tj,ti+l)' u(t)=u(ti). Substituting the following for e(t):(3)e(t)= + t.1+Jt ti101(4)

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102thedeterministiccost becomes: (5)+N E [i=OJti+1 [ [4>( t,ti)e(ti)tI+{Jt 4>(t,r)G(r)dr}u(ti) t.1+Jt {4>(t,r)L(r)dr}w(ti)]T t.1+{It 4>(t,r)G(r)dr}u(ti) t.I+{It 4>(t,r)L(r)dr}w(ti)] t.1Makingthefollowingsubstitutions:(6){It 4>(t,r)G(r)dr}T t.1Q(t) {Jr 4>(r,ti)G(r)dr}]dt .t1(7)J'i+l r'l;(t,T)L(T)dT} TQ(t) Www(ti)=[{t.t.I 1.{r 'l;(t,T)L(T)dT}]dt t.1 f+l4>(t,ti) T 1: Wxu(ti)=[Q(t){ 4>(t,r)G(r)dr}]dt t.1 1(8) (9)

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103fi+1J: Wxw(ti)=[ t.1iWWX(ti)=[f+'( f T t.t.1 1WUW(ti)=[fi+1( f T Q(t)t.t.1 1( f t.1WWU(ti)=[fi+'( f T Q(t)t.t.1 1( f t.1And,allowingforthefactthatforall tE[ti,ti+l):u(t)=u(ti) e(t)=e(ti) w(t)=w(ti)thedeterministiccostcanbe expressed as:Jd=e(tN+ 1)1r Pfe(tN+l)+N E (e(ti) 1r Wxx(ti)e(ti) + u(ti) 1r Wuu(ti)u(ti)i=O+ W(ti)1rWWW(ti)W(ti) + 2e(ti)1rWXU(ti)U(ti) + 2e(ti)1rWXW(ti)W(ti) +2u(ti)1rWUW(ti)W(ti)}(10) (11) (12) (13) (14) (15) (16)

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104RecallthatJs(t)=E{Jd(t)II(t)}andthatw(.,.) is zeromeananduncorrelatedwitheithereoru. Sinceiftwo variables x&yareuncorrelatedthenE{x,y}=E{x}E{y}.Therefore,E{2e(ti)T Wxw (ti)W(ti)}=2E{e(ti)T w (ti)}E{w(ti)}=0 E{2u(ti)TWxw(ti)W(ti)}=2E{u(ti)T w (ti)}E{w(ti)}=0and(17)Jt t.1(18)Jti+1=tr[Q(t){t.1Jt s(r)L(r) T Tdr}dt)t.1(19)Consequently,forconsiderationinJ sJd=e(tN+l)TPf e(tN+l)N+ E (e(ti)TWxx(ti)e(ti) + U(ti)TWuu(ti)U(ti) +Jw(ti)1=0(20)Wxu(ti) Wuu(ti) (21)

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105Forsmall sample times, the weighting functionscanbeapproximatedby:WxxQ(t).M Wxu(I/2)Q(t)G(t)(.6.t)2 Wuu-[R(t)+( I13)G( t) T Q(t)G(t)(.6.t)2](.6.t) (22)(23)(24)Inorderto relate the valuesofthe sampleddataweighting terms on the systemerrorto the state variables (using discrete variable notation), let: P(ti)=[H 0 -C]Tpr[ H0-C](25)Q(ti)=[H 0-C]T Wxx(ti)[ H0-C](26)R(ti)=Wuu (27) S(ti)=[H 0-C]T Wxu (28)Therefore,withthe augmented state variableX=[xTr ]T:ExpandingQandST](29)[ -CTWXX(ti)Hoo o ] CTWXX(ti)C(30)

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106With non-coincident sampling and control, the penalty terms are modifiedasfollows: e(ti')=[H0-C] +rti' This modification adds additional terms:[DE][U]ati'(31 ) (32){[HOC] +[DE][U]}T.r ti' a ti'C] [DE][U]}r ti' a ti' AgainwithWxx(ti)=Wxx(ti') and Wxu(ti')=Wxu(ti), the additional terms can be grouped withRandS to generate:R(ti')=R(ti)+DTWxx(ti)D+DTWxu(ti) S(ti')=S(ti)+DTWxx(ti)[ H0-C](33) (34)

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APPENDIXDLQGCONTROLLERDECOMPOSITIONThecombinationofthesystem, targetdisturbance,andthereference, result in a19state controller.However,thedecoupledstructure,symmetry,andthe zerosinthe controlinputandcost matricescanbeexploitedto streamline the calculations. In general, theoptimalsolution to theLQGtrackercanbe expressed as: whereand G*(ti) =[R(ti) +GT(ti)P(ti+l)G(ti)rl [GT(ti)P(ti+l)cI>(ti+l,ti) + ST(ti)] P(ti)=Q(ti) + cI>T(ti+}.ti)P(ti+l)cI>(ti+l,ti) [GT(ti)P(ti+l)cI>(ti+}.ti) + ST(ti)]TG*(ti)(I)(2)(3)Toreduce thenumberofcalculations,partitionthe gainandRiccatiequationsuchthat: Evaluating terms:(4)GT (ti)P( ti+ 1 )G(til=[ =[=[GT10I0][PIIPl2P13 ][G]P21P22 P23 0P31P32 P33 0GTPIIIGTPI2IGTP13J [g] GTPIIG]107(5)

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108Therefore,Now consider, GT (ti)P(ti+ 1 I ,ti) 0=[GT1010][PIIP12P13j 00P21P22 P230 0P31P32 P3300 0=[GTPIIIGTPl21GTP13 ] 00 0 0 00 (6) (7)[ I + I oFromAppendix C, S=[H0-C]TWxu Substituting, the required terms for the gain computation become:,ti) + ST(ti))= + WxuTHI + o(8)Consequently, with(9)the optimal control can be expressedas:G1(ti)=GI + WxuTH] G2(tj)=GI + 0G3(ti)=GI WxuTC](10)(11)(12)

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109Partitioningequation3:[PHPI2Pl3] [ 0-HTWXX('i)C](13) P21 P22 P23=0 (ti) P31 P32 P33(ti)-CTWXX(ti)H 0 4>n 0 T 4>n 0 4>x [ PI IP12Pl31 4>x 0 0 0 0 + 0 4>T 0 P21 P22P23 0 4>T 0 0 0 4>r P31 P32P33 (ti+ 1) 0 0 4>r4>n 0 -([ GTI0101[PIIPI2Pl3] 4>x 0 0P2IP22 P23 0 4>T 0P3IP32 P33ti+I)0 0 4>r +[WxuTHI01-WxuTC]}T [GIlG21G3]=[ -CTWxXCti)Ho oo ] CTWXX(ti)C (14)+ {4>xTPll}{[4>n TOJPll+ 4>xTP2t>{4>rTP3t>{4>x TPI2} {4>x TPI3} {[4>n T OJPI2+4>TTP22}{[4>n T 0]P13+4> T T P23 }{4>rTP32}{4>rTP33}[;X4>n )('i+l)0 4>T 0 GTPI2I GTPl31 [;x4>nil -{[GTPIII0 4>T 0 + [WxuTHI0I-WxuTCnT [GIIG21G3]

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= -cT Wxx(ti)H 110 g oCTWXX(ti)C (15) o + o o TH}T [GIIG21G3]o (ti+l)ForthepropagationoftheRiccatiequationonly these termsarerequiredtogeneratethecontrol gain:Pll(ti)=HTWxx(tj)H + TH}T][GIlPI2(ti)= + o-[[G2] P13(ti)=-HTWxx(ti)C + -(16)(17)(18)

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APPENDIX E CONTROLLER ANDFILTERPARAMETERS Controller Control Delay=.21seconds Target Maneuver Correlation time=.21seconds Riccati initialization (Pf) Linear controller::Pf=IE+2 for sample times<.01secondPf=lEIfor sampletimes>.01second Roll rate controller:Pf=1.0Quadratic Cost Terms (Continuous) Linear Accelerations Q=320.R=l.Roll Control (Angle) (Rate)QIlQ22RHI GAIN IE+lIE-4lE-3LOGAINlE-4lE-3Sample Time=integration (not sample) stepforthe continuous-discrete filter System Disturbance Input=(AT AM) (Tf-TO)2/ 2.III

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112Filter Target Correlation Time = 2.0 seconds Riccati initialization (Pf)Positions:PII= 2500 Velocities: P44 = 2.0E6 Covariance:Pl4= 5.0E4 Maneuver Excitation Matrix (2a2) = 5120000 Seeker Measurement Noise Azimuth&Elevation R= (SITH*SITH/(RNG HAT*RNGHA T)+SOTH*SOTH +SITHI*SITHI*RNG HAT**4)/MEAS ST Range R= (SOR*SOR+SIR*SIR*RNG HAT**4)/MEAS ST Range Rate R= (SODR*SODR+SIDR*SIDR*(RNG HAT**4) )/MEAS ST With SITH = SIPH =1.5SOTH = SOPH = .225E-4 SITHI= SIPHI= 0.0 SOR = SODR = 3.0 SIR =IE-8SIDR = .2E-IO

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LISTOFREFERENCES [1] A. Arrow, "StatusandConcerns for Bank-To-TurnControlofTactical Missiles," GACIACPR-85-01,"Proceedingsofthe Workshop onBank-To-TurnControlled Terminal Homing Missiles," Vol1,Joint ServiceGuidanceandControl Committee, January 1985. [2] J.R. McClendon and P.L. Verges, "ApplicationsofModern Control and Estimation Theory to the GuidanceandControlofTacticalAir-to-AirMissiles," Technical ReportRG-81-20,"Research onFurureArmyModular Missile," USArmyMissile Command, Redstone Arsenal, Alabama, March 1981. [3] N.K. Gupta, J.W. Fuller,andT.L. Riggs, "Modern ControlTheoryMethodsforAdvanced Missile Guidance," Technical ReportRG-81-20,"Research onFurureArmy Modular Missile," USArmyMissile Command, Redstone Arsenal, Alabama, March 1981. [4] N.B. Nedeljkovic, "New Algorithms for Unconstrained Nonlinear Optimal Control Problems," IEEE Transactions on Automatic Control, Vol.AC-26,No.4,pp868-884, August 1981. [5] W.T. Baumann and W.J.Rugh,"Feedback ControlofNonlinear Systems byExtendedLinearization," IEEE Transactions on Automatic Control, Vol.AC-31,No.1,pp. 40-46, January 1986. [6] E.D. Sontag, "Controllability and Linearized Regulation,"DepartmentofMathematics, Rutgers University, New Brunswick, NJ (unpublished),14February 1097.[7]A.E. Bryson and Y.C. Ho, Applied Optimal Control, Blaisdell Publishing Company, Waltham, Massachusetts, 1969. [8] Y.-S. Lim, "LinearizationandOptimizationofStochastic Systems with Bounded Control," IEEE Transactions on Automatic Control, Vol.AC-15,No.1,pp49-52, February 1970. [9] P.-O.GutmanandP.Hagander,"ANew DesignofConstrained ControllersforLinearSystems," IEEE Transactions on Automatic Control, Vol.AC-30,No. 1,pp22-33, January 1985. [10] R.L. Kousut, "Suboptimal ControlofLinearTime-InvariantSystems Subject to Control Structure Constraints," IEEE TransactionsonAutomatic Control, Vol.AC-15,No.5,pp 557-562, October 1970.113

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114[11] D.J. Caughlin, "Bank-To-TurnControl," Master's Thesis, UniversityofFlorida, 1983. [12] J.F. Frankena and R. Sivan,"Anon-linear optimal control law for linear systems," INT. J. CONTROL, Vol 30, NoI,pp159-178, 1979. [13] P.S. Maybeck, Stochastic Models. Estimation. and Control, Volume 3, Academic Press, New York, 1982. [14] S.A. Murtaugh and H.E. Criel, "FundamentalsofProportional Navigation," IEEESpectrum, pp.75-85, December 1966. [15] L.A. Stockum, and I.C. Weimer, "Optimal and Suboptimal Guidance for a Short Range Homing Missile," IEEE Trans. on Aerospace and Electronic Systems, Vol. AES-12,No.3,pp 355-361, May 1976. [16]B.Stridhar, and N.K. Gupta, "Missile Guidance Laws Based on Singular Perturbation Methodology," AIAA JournalofGuidance and Control, Vol 3,No.2,1980. [17] R.K. Aggarwal and C.R. Moore, "Near-Optimal Guidance LawforaBank-To-TurnMissile," Proceedings 1984 American Control Conference, Volume 3, pp. 1408-1415, June 1984. [18] P.H. Fiske, "Advanced Digital Guidance and Control Concepts forAir-To-AirTactical Missiles,"AFATL-TR-77-130,Air Force Armament Laboratory, United States Air Force, Eglin Air Force Base, Florida, January 1980. [19] USAF Test Pilot School, "Stability and Control Flight Test Theory,"AFFTC-77-I,revised February 1977. [20] L.C. Kramer andM.Athans, "On the ApplicationofDeterministic Optimization MethodstoStochastic Control Problems," IEEE Transactions on Automatic Control, Vol. AC-19,No.I,pp22-30, February 1974. [21]Y.Bar-Shalom and E. Tse, "DualEffect,Certainty Equivalence, and Separation in Stochastic Control," IEEE Transactions on Automatic Control, Vol. AC-19,No.5,pp 494-500, October 1974. [22] H. Van DE Water and J.C. Willems, "The Certainty Equivalence Property in Stochastic Control Theory," IEEE Transactions on Automatic Control, Vol. AC-26,No.5,pp1080-1086, October 1981. [23] "Bank-To-TurnConfiguration Aerodynamic Analysis Report" Rockwell International Report No. C77-142Ij034C, date unknown.

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115[24] D.E Williams andB.Friedland, "DesignofAn AutopilotforBank-To-TurnHoming Missile Using Modern Control and Estimation Theory," ProcFifthMeetingofthe Coordinating Group on Modern Control Theory, 15-27 October 1983, (Picatinny) Dover, New Jersey, October 1983, pp. 397-419. [25]B.Friedland,etaI., "On the "Adiabatic ApproximationforDesignofControl Laws for Linear, Time-Varying Systems,"IEEETransactions on Automatic Control, Vol.AC-32,No.1,pp. 62-63, January 1987. [26] D.W. Tufts and D.A. Shnidman, "Optimum Waveforms Subject to Both Energy and Peak-Value Constraints," Proceedingsofthe IEEE, September 1964. [27] M. Pontier and J. Szpirglas, "Linear Stochastic Control with Constraints, IEEE Transactions on Automatic Control, Vol.AC-29,No. 12, pp 1100-1103, December 1984. [28] P.-O.GutmanandS.Gutman,"ANote on the ControlofUncertain Linear Dynamical Systems with Constrained Control Input," IEEE Transactions on Automatic Control, Vol. AC-30,No.5,pp 484-486, May 1985. [29]M.W.Spong, J.S. Thorp, and J.M. Kleinwaks, "The ControlofRobot Manipulators withBounded Input," IEEE TransactionsonAutomatic Control, Vol. AC-31,No.6,pp 483-489, June 1986. [30]D.Feng and B.H. Krogh, "Acceleration-Constrained Time Optimal Control in n Dimensions," IEEE Transactions on Automatic Control, Vol. AC-31, No. 10, pp 955-958, October 1986. [31] B.R. BarmishandW.E. Schmitendorf, "New Results on ControllabilityofSystemsofthe Form x(t)=A(t)x(t)+F(t,u(t,"IEEETransactions on Automatic Control, Vol. AC-25,No.3,pp 540-547, June 1980. [32] W.-G. HwangandW.E. Schmitendorf, "Controllability Results for Systems with a Nonconvex Target," IEEE TransactionsonAutomatic Control, Vol. AC-29,No.9,pp 794-802, September 1984. [33] T. Kaliath, Linear Systems, Prentice-Hall, Inc., Englewood Cliffs, New Jersey, 1980. [34] M. Vidyasager, Nonlinear Systems Analysis, Prentice-Hall, Englewood Cliffs, New Jersey, 1978. [35] K. Zhou, and P. Khargonekar, "Stability Robustness Bounds for Linear State-Space Models with Structured Uncertainity," Transactions on Automatic Control, Vol.AC-32,No.7,pp 621-623, July 1987. [36] K.G. Shin and N.D. McKay, "Minimum Time ControlofRobotic Manipulators with Geometric Path Constraints," IEEE TransactionsonAutomatic Control, Vol. AC-30,No.6,pp 531-541, June 1985.

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116[37] A. Sabari and H. Khalil. "Stabilization and RegulationofNonlinear Singularly Perturbed Systems Composite Control," IEEE Transactions on Automatic Control, Vol.AC-30,No.8,pp. 739-747, August 1985. [38] M. Sampei and K.Furuta,"On Time Scaling for Nonlinear Systems: Application to Linearization." IEEE Transactions on Automatic Control, Vol.AC-31.No.5,pp 459-462, May 1986. [39] I.J. Ha and E.G. Gilbert."AComplete CharacterizationofDecoupling Control Laws for a General ClassofNonlinear Systems," IEEE Transactions on Automatic Control, Vol.AC-31,No.9,pp. 823-830. September 1986. [40] H. K wakernaak and R. Sivan, Linear Optimal Control Systems, Wiley-Interscience. New York, 1972. [41] R.A. Singer. "Estimating Optimal Tracking Filter Performance for Manned Maneuvering Targets," IEEE Transactions on Aerospace and Electronic Systems, Vol. AES6,No.4,pp 473-483, July 1970. [42] M.E Warren and T.E. Bullock, "DevelopmentandComparisonofOptimal Filter Techniques with Application toAir-to-AirMissiles," Electrical Engineering Department. UniversityofFlorida, Prepared for the Air Force Armament Laboratory, Eglin Air Force Base, Florida, March 1980.

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BIOGRAPHICAL SKETCH Donald J. Caughlin, Jr., was born in San Pedro, California on17Dec. 1946. He graduated from the United States Air Force Academy, earning aB.S.in physics, and chose pilot training insteadofan Atomic Energy Commission Fellowship. Since then he has flown over 3100 hours in over 60differentaircraft and completed one tour in Southeast Asia flying theA-ISkyraider. A Distinguished Graduateofthe United StatesAirForce Test Pilot School, Don has spent muchofhis career in research, development,andtest at both major test facilities--Eglin AFB in Florida, and Edwards AFBinCalifornia. Don Caughlinisa Lieutenant Colonel in theUnitedStatesAirForcecurrently assigned as the Assistant for SeniorOfficerManagement at HeadquartersAirForce Systems Command. In addition to theB.S.in physics,Lt.Colonel Caughlin has an M.B.A. from the UniversityofUtah, and a masters degreeinelectrical engineering from the UniversityofFlorida. Heisa memberofthe SocietyofExperimental Test Pilots and IEEE. Lt. Colonel Caughlinismarried to theformerBarbara SchultzofMontgomery, Alabama. They have two children, a daughter, Amy Marie, age eight, and Jon Andrew, age four.117

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AUTHOR: TITLE:Internet Distribution Consent AgreementIn reference to the following dissertation: Caughlin, Donald Reachable Set Control for Preferred Axis Homing Missles (record number: 1102747) PUBLICATION DATE: 1988I, fiJoJ as copyright holder for theaforementioned dissertation, hereby grant specific and limited archive and distribution rights to the BoardofTrusteesofthe UniversityofFlorida and its agents. I authorize the UniversityofFlorida to digitize and distribute the dissertation describedabove for nonprofit, educational purposes via the Internet or successive technologies. This is a non-exclusive grantofpermissions for specific off-line and on-line uses for an indefinite term. Off-line uses shall be limited to those specificallyallowedby"Fair Use" as prescribedbythe termsofUnited States copyright legislation (cf, Title 17, U.S. Code) as well as to the maintenance and preservationofa digital archive copy. DigbiMliM allows the UniversityofFlorida to generate imageand text-based versions as appropriate.and.to,provide and enhance access using search software. rohibits useofthe digitim4 versivns use or profit.personal information blurredDateofSignaturePleaseprint,signandreturnto: CathleenMartyniakUF Dissertation Project PreservationDepartmentUniversityofFloridaLibrariesP.O. Box117007 Gainesville,FL32611-7007512812008



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REACHABLE SET CONTROLFORPREFERREDAXIS HOMING MISSILESByDONALDJ.CAUGHLIN,JR. A DISSERTATION PRESENTED TOTHEGRADUATESCHOOL OFTHEUNIVERSITYOFFLORIDA IN PARTIALFULFILLMENTOFTHEREQUIREMENTS FORTHEDEGREEOFDOCTOR OF PHILOSOPHY UNIVERSITYOFFLORIDA1988

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Copyright 1988ByDONALD J.CAUGHLINJR.

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To BarbaraAmyJon

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ACKNOWLEDGMENTS The author wishestoexpress his gratitude to his committee chairman, Dr. T.E Bullock, for his instruction, helpful suggestions, and encouragement. Appreciationisalso expressed for the support and many helpful commentsfromthe other committee members, Dr. Basile, Dr. Couch, Dr. Smith, and Dr. Svoronos. iv

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TABLE OF CONTENTS ACKNOWLEDGMENTSivLIST OF FIGURES viiKEYTO SyMBOLS ix ABSTRACTxivCHAPTER I INTRODUCTION 1 IIBACKGROUND4 Missile Dynamics 5 Linear Accelerations 6 Moment Equations (i Linear Quadratic Gaussian Control Law 7 III CONSTRAINED CONTROL13IV CONSTRAINED CONTROL WITH UNMODELED SETPOINT ANDPLANTVARIA TIONS25Linear Optimal Control with Uncertainty and Constraints31Control Technique32Discussion36Procedure.37V REACHABLE SET CONTROL EXAMPLE.41Performance Comparison Reachable SetandLQGControl....41Summary.54VI REACHABLE SET CONTROL FORPREFERREDAXIS HOMING MISSILES55Acceleration Control.56System Model.56Disturbance Model58Reference Model.60Roll Control. 62 Definition 62 Controller 66 Kalman FiIter67Reachable Set Controller68Structure 68 Application 72 v

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VII RESULTS AND DISCUSSION76Simulation77Trajectory Parameters78Results78Deterministic Results78Stochastic Results81Conclusions87Reachable Set Control. 87 Singer Model. 87APPENDIXA SIMULATION RESUL TS88BSAMPLED-DATA CONVERSION 94 System Model 94 Sampled Data Equations 96 System 96 Target Disturbance 98 Minimum Control Reference 99 Summary 1 00 C SAMPLED DATA COST FUNCTIONS101DLQGCONTROLLERDECOMPOSITION.107 ECONTROLLERPARAMETERS111Control Law111Filter 112 LISTOFREFERENCES 113 BIOGRAPHICAL SKETCH 117 vi

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LISTOFFIGURES Figure Page2.1Missile Reference System 44.1Feedback System and Notation284.2 Reachable Set Control Objective334.3 IntersectionofMissile Reachable Sets Based on Uncertain Target Motion and Symmetric Constraints.384.4 IntersectionofMissile Reachable Sets Based on Uncertain Target Motion and U nsymmetric Constraints385.1Terminal PerformanceofLinear Optimal Control...435.2 Initial AccelerationofLinear Optimal Control.435.3 Linear Optimal AccelerationvsTime455.4 Linear Optimal VelocityvsTime455.5 Linear Optimal PositionvsTime lt6 5.6 Unconstrained and Constrained Acceleration .47 5.7 Unconstrained and Constrained Velocity vs Time.485.8 Unconstrained and Constrained PositionvsTime.485.9 Acceleration Profile With and Without Target Set Uncertainty 50 5.10 VelocityvsTime With and Without Target Set Uncertainty 505.11PositionvsTime With and Without Target Set Uncertainty515.12 AccelerationvsTime LQG and Reachable Set Control. 52 vii

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5.13 VelocityvsTime LQG and Reachable Set Control.535.14 PositionvsTime LQG and Reachable Set Control.536.1Reachable Set Control Disturbance processes606.2. Roll Angle Error Definition from Seeker Angles636.3. Roll Control Zones 65 6.4 Target Missile System746.5 Command Generator/Tracker757.1RMS Missile Acceleration767.2 Engagement Geometry777.3 Deterministic Results807.4 Stochastic Results817.5 MeasuredvsActual Z Axis Velocity 84 7.6 Performance Using Position Estimates and Actual Velocities86A.I XY Missile&Target Positions Reachable Set Control.89A.2 XY Missile&Target Positions Baseline Control Law89A.3XZMissile&Target Positions Reachable Set Control.90AAXZMissile&Target Positions Baseline Control Law 90 A.5 Missile Acceleration Reachable Set Control...91A.6 Missile Acceleration Baseline Control Law91A.7 Missile Roll Commands&Rate Reachable Set Control.92A.8 Missile Roll Commands&Rate Baseline Control Law92A.9 Missile Roll Angle Error93viii

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a()B()q.)D()DODOwtDoc DocDq DO E() F() G()KEYTO SYMBOLS Reference controlinputvector. Missile inertial x axis acceleration. Target inertial x axis acceleration. Specific force (drag) along X body axis. Desired linear acceleration about Z and Y body axes. Reference controlinputmatrix. Reference stateoutputmatrix. Feedforward stateoutputmatrix. Stability parameter Equilibrium drag coefficient. Stability parameter Change in drag due to weight. Stability parameter Changeindrag due to velocity. Stability parameter Change in drag due to angleofattack. Stability parameter Change in drag due to angleofattack rate. Stability parameter Change in drag due to pitch rate. Stability parameter Change in drag due to pitch angle. Stability parameter Changeindrag due to pitch canard deflection angle. Feedforward referenceoutputmatrix. Roll angle error. System matrix describing the dynamic interaction between state variables. System controlinputmatrix. Optimal control feedback gain matrix. ix

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G l(ti) Optimal system state feedback gain matrix. G2(ti) Optimal target state feedback gain matrix. G3(ti) Optimal reference state feedback gain matrix. g Acceleration duetogravity. H(.) System state output matrix. Ixx,Iyy,Izz Momentofinertial withrespect to the given axis.JCosttogofunction for the mathematical optimization. L() System noise input matrix.LOStability parameter Equilibrium change in Z axis velocity.LOwt Stability parameter Change in Z axis velocity duetoweight. L u Stability parameter Change in Z axis velocity duetoforward velocity.LexLexStability parameter Change in Z axis velocity duetoangleofattack. Stability parameter Change in Z axis velocity due to angleofattack rate. Stability parameter Change in Z axis velocity duetopitch rate. Stability parameter Change in Z axis velocity duetopitch angle. Stability parameter Change in Z axis velocity duetopitch canard deflection angle. Stability parameter Equilibrium change in roll rate. Stability parameter Change in roll rate duetosideslip angle. Stability parameter Change in roll rate due to sideslip angle rate. Stability parameter Change in roll rate due to roll rate. Stability parameter Change in roll rate due to yaw rate. Stability parameter Change in roll rate due to roll canard deflection angle. Stability parameter Change in roll rate due to yaw canard deflection angle.x

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M Massofthemissile.MOStabilityparameter Equilibriumpitchrate. M u StabilityparameterChangeinpitchrateduetoforwardvelocity.McxStabilityparameterChange inpitchratedueto angleofattack.McxStabilityparameterChange inpitchratedueto angleofattackrate. M q StabilityparameterChange inpitchrateduetopitchrate. MOe StabilityparameterChangeinpitchrateduetopitchcanarddeflectionangle.NOStabilityparameter-Equilibriumyaw rate. N B Stabilityparameter Change in yaw ratedueto sideslip angle. N B StabilityparameterChange in yaw ratedueto sideslip angle rate. N p Stabilityparameter Change in yawratedueto roll rate. N r StabilityparameterChange in yawrateduetoyawrate. NOa StabilityparameterChangeinyawratedueto rollcanarddeflectionangle. NOr StabilityparameterChangeinyawratedueto yawcanarddeflectionangle.Nx,Ny,NzComponentsofappliedaccelerationonrespective missilebodyaxis. P Solution totheRiccatiequation.P,Q,RAngularratesabouttheX,Y,andZbodyaxis respectively.Q(.)State weightingmatrix.R(.)Controlweightingmatrix.R(.)Referencestate vector. S()State-Controlcross weighting matrix. T(.)Targetdisturbancestate vector. TgoTime-to-go.U Systeminputvector. xi

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V,V,W Linear velocities with respect to the X,Y,andZbodyaxis respectively.vs System noise process . Zero mean white Gaussian noise modeling uncorrelated state disturbances. Zero mean white Gaussian noise driving firstorderMarkov process modeling correlated state disturbances. IVtotlX(.)X,Y,ZYoYOwtex B Total missile velocity. System state vector. Body stabilized axis. Stability parameter Equilibrium change inYaxis velocity. Stability parameter Change inYaxis velocity due to weight. Stability parameter Change in Y axis ratedueto sideslip angle. Stability parameter Change in Y axis velocitydueto sideslip angle rate. Stability parameter ChangeinY axis velocitydueto roll rate. Stability parameter ChangeinY axis velocity due to yaw rate. Stability parameter ChangeinY axis velocity due to roll angle. Stability parameter Change in Y axis velocity due to rollcanarddeflection angle. Stability parameter Change in Y axis velocity due to yawcanarddeflection angle. Angleofattack. AngleofSideslip. System noise transition matrix. Reference state transition matrix. Target disturbance state transition matrix. System state transition matrix. xii

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Target model correlation time. Target elevation aspect angle. Seeker elevation gimbal angle. Target azimuth aspect angle. Seeker azimuth gimbal angle. xiii

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AbstractofDissertation Presented to the Graduate Schoolofthe UniversityofFlorida in Partial Fulfillmentofthe Requirements for the DegreeofDoctorofPhilosophy REACHABLE SET CONTROLFORPREFERRED AXIS HOMING MISSILESByDonald J. Caughlin, Jr. April1988Chairman: T.E. Bullock Major Department: Electrical Engineering The applicationofmodern control methodstothe guidance and controlofpreferred axis terminal homing missilesisnon-trivial in that it requires controlling a coupled, non-linear plant with severe control variable constraints, to intercept an evading target. In addition, the rangeofinitial conditionsisquite large andislimited only by the seeker geometry and aerodynamic performanceofthe missile. Thisisthe problem: Linearization will cause plant parameter errors that modify the linear trajectory. In non-trivial trajectories, both Ny andNzacceleration commands will, at some time, exceed the maximum value. The two point boundary problem is too complextocomplete in real time and other formulations are not capableofhandling plant parameter variations and control variable constraints. xiv

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Reachable Set Control directly adapts Linear Quadratic Gaussian (LQG) synthesis to the Preferred Axis missile,aswellasa large classofnonlinear problems where plant uncertainty and control constraints prohibit effectivefixed-final-timelinear control.Itisa robust control technique that controls a continuous system with sampled data and minimizes the effectsofmodeling errors. As a stochastic command generator/tracker, it specifies and maintains a minimum control trajectory to minimize the terminal impactoferrors generated by plant parameter (transfer function)ortarget set uncertainty while rejecting system noise and target set disturbances. Also, Reachable Set Control satisfies the Optimality Principle by insuring that saturated control,ifrequired, will occur during the initial portionofthe trajectory. With large scale dynamics determined by a dual reference in the command generator, the tracker gains can be optimized to the response timeofthe system. This separation results in an "adaptable" controller because gains are based on plant dynamics and cost while the overall systemissmoothly driven from some large displacement to a region where the relatively high gain controller remains linear. xv

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CHAPTER I INTRODUCTION The applicationofmodern control methods to the guidance and controlofpreferred axis terminal homing missiles has had only limited success [l,2,3]. This guidance problemisnon-trivial in thatitrequires controlling a coupled, non-linear plant with severe control variable constraints, to intercept an evading target. In addition, the rangeofinitial conditionsisquite largeandlimited only by the seeker geometry and aerodynamic performanceofthe missile. There are three major control issues that must be addressed: the couplednon-linearplantofthe Preferred Axis Missile; the severe control variable constraints; and implementation in the missile where the solutionisrequired to control trajectories lasting one(I)to two (2) seconds real time. There have been a numberofrecent advances in non-linear controlbutthese techniques have not reached the point where real time implementation in an autonomous missile controllerispractical [4,5,6]. Investigationofnon-linear techniques during this research did not improve the situation. Consequently, primarilydueto limitations imposed by real time implementation, linear suboptimal control schemes were emphasized. Bryson&Ho introduced a numberoftechniques for optimal control with inequality constraints on the control variables [7]. Eachofthese use variational techniques to generate constrained and unconstrained arcs that must be pieced together to construct the optimal trajectory.

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2Ingeneral, real time solutionofoptimal control problems with bounded controlisnot possible [8]. In fact, with the exceptionofspace applications, the optimal control solution has not been applied [9,10]. When Linear Quadratic Gaussian (LQG) techniques are used, the problemisnormally handled via saturated linear control, where the control is calculatedasifno constraints existed and then simply limited. This technique has been shown to be seriously deficient. In this case, neither stability nor controllability can be assured. Also, this technique can cause an otherwise initially controllable trajectory to become uncontrollable [11]. Consequently, a considerable amountoftime is spent adjusting the gainsofthe controller so that control input will remain below its maximum value. This adjustment, however, will force the controller to operate below its maximum capability [12]. Also, in the caseofthe terminal homing missile, the applicationofLQG controllers that do not violate an input constraint lead to an increasing acceleration profile and (terminally) low gain systems [13].Asa result, the performanceofthese controllersisnot desirable. Whileitisalwayspossible to tune a regulator to control the system to a given trajectory, the varianceofthe initial conditions, the time to intercept the target (normally a few seconds for a short range high performance missile), and the lackofa globally optimal trajectory due to the nonlinear nature, the best policyisto develop a suboptimal real time controller. The problemofdesigning a globally stable and controllable high performance guidance system for the preferred axis terminal homing missile is treated in this dissertation. Chapter 2 provides adequate background information on the missile guidance problem. Chapter 3 covers recent work on constrained

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3 control techniques. Chapters 4 and 5 discuss Robust Control and introduce "Reachable Set" Control, while Chapter 6 applies the technique to controlofapreferredaxis homing missile.Theperformanceof"Reachable Set" controlispresented in Chapter7.

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CHAPTER II BACKGROUND The preferred axis orientation missile has significant control input constraints and complicated coupled angular dynamics associated with the maneuvering. In the generic missile considered, the Z axis acceleration (see Figure 2.1) was structurally limited to100"g"with further limits on"g"resulting from a maximum angleofattackasa functionofdynamic pressure. Even though the Z axis was capableof100"g",the "skid-to-turn" capabilityofthe Y axis was constrained to 5"g"or less becauseofaerodynamic limitations a20:1difference. In addition to pitch (Nz) and yaw (Ny) accelerations, the missile can roll upto500 degrees per second to align the primary maneuver plane with the planeofintercept. Hence,bank-to-turn. x zFigure2.1Missile Reference System.4

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5The classical technique for homing missile guidanceisproportional navigation (pro nav). This technique controls the seeker gimbal angle ratetozero which (given constant velocity) causes the missiletofly a straight line trajectory toward the target [14,15]. In the late 70's aneffortwas made to use modern control theory to improve guidance laws forair-to-airmissiles.Forrecent research on this problem see, for example, [11].Asstated in the introduction, these efforts have not significantly improved the performanceofthe preferred axis homing missile.Ofthe modern techniques, two basic methodologies have emerged: one was a body-axis oriented control law that used singular perturbation techniques to uncouple the pitch&roll axis [16,17]. This technique assumed that roll rateisthe fast variable, an assumption that may not be true during the terminal phaseofan intercept. The second techniquewasan inertial point mass formulationthat controls inertial accelerations [18]. The acceleration commands are fixed with respecttothe missile body; but, since these commands can be related to the inertial reference via the Euler Angles, the solution is straight forward. Bothofthese methods have usually assumed unlimited control available and the inertial technique has relied on the autopilottocontrol the missile roll angle, and therefore attitude,tode rotate from the inertial to body axis. Missile Dynamics The actual missile dynamics are a coupled setofnonlinear forces and moments resolved along the (rotating) body axesofthe missile [19]. Linearizationofthe equations about a "steady state"ortrim condition,

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6neglectinghigherorderterms, resultsinthefollowingsetofequations (usingstandardnotation,see symbol keyinthe preface):ex=Q-PB+ Azb / IV totl B =-R-Pex+Ayb/ IVtotlLinearAccelerationsIU=RV-QW-{DO+DOwt }MIV=PW -RU+-{yo+ YOwtl M+ Yf3B+YaB+ YpP+YrR+ Y0 0 + Yoaoa + Yoror IW=QU-PV +-{LO+LOwt} MMomentEquations Q=MO/I yy +M u U+Mexex+Mexex+MqQ+ Moeoe (1)(2)(3) (4)(5) (6)+-----IyyIyy

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7 (7)+(8)+Linear Quadratic Gaussian Control Law For allofthe modern development models, a variationofa fixed-final-time LQG controller was usedtoshape the trajectory. Also, it was expected that the autopilot would realize the commanded acceleration. First, consider the effectofthe unequal body axis constraints. Assume that100"g"was commanded in each axis resulting in an acceleration vector45degrees from Nz.IfNyisonly capableof5"g",the resultant vector will be 42 degrees in error, an error that will havetobe corrected by succeeding guidance commands. Evenifthe missile has the timeorcapability to complete a successful intercept, the trajectory can not be considered optimal. Now consider the nonlinear natureofthe dynamics. The inertial linear systemisaccurately modeledasa double integratorofthe acceleration to determine position. However, the acceleration commandisa functionofthe missile state, equation(1),and therefore, itisnot possible to arbitrarily assign the input acceleration. And, given a body axis linear acceleration, the inertial

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8component will be severely modified by the rotation (especially roll)ofthe reference frame. Allofthese effects are neglectedinthe linearization. This thenisthe problem: In the intercept trajectories worth discussing, Ny, Nz, and roll acceleration commands will,atsome time, saturate. Highorder,linear approximations do not adequately model the effectsofnonlinear dynamics, and the complete two point boundary value problem with control input dynamics and constraintsistoo difficult to complete in real time. Although stochastic models are discussed in Bryson and Ho [7], and a specific techniqueisintroduced by Fiske [18], the general procedure has been to use filtered estimates and adynamic-programming-likedefinitionofoptimality (using the PrincipleofOptimality) with Assumed Certainty Equivalence to find control policies [20,21,22]. Therefore, allofthe controllers actually designed for the preferred axis missiles are deterministic laws cascaded with a Kalman Filter. The baseline forouranalysisisan advanced control law proposed by Fiske [18]. Given the finite dimensional linear system: where andxx(t)=Fx(t)+Gu(t)xyz VxVyVz(9)

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9with the cost functional:Jtfl=xfPfxf+ 1 uTRudrtoR=I(10)Applicationofthe Maximum principle results in a linear optimal control law:=-----3(Tgo) 31 +(Tgo)3+3(Tgo)2 31 +(Tgo)3(11)Coordinates usedforthis system are "relative inertial."Theorientationofthe inertial systemisestablished at the launch point.Thedistances and velocities are the relative measures between the missileandthe target. Consequently, the set pointiszero, with the reference frame moving with the missile similartoa "moving earth" reference used in navigation. Since Fisk's control law was based on a point mass model, the control law did not explicitly control the roll angle PHI (0). Theroll angle was controlledbyabank-to-turnautopilot [23]. Therefore, the guidance problemwasdecomposed into two components, trajectory formationandcontrol. The autopilot attempted to control the rollsothat thepreferredaxis (the-Zaxis) was directed toward the planeofintercept.Theautopilot used to control the missile was designed to use proportional navigationandisa classical combinationofsingle loop systems. Recently, WilliamsandFriedland have developed a newbank-to-turnautopilot based on modern state space methods [24]. Inorderto accurately control the banking maneuver, the missile dynamics are augmented to include the kinematic relations describing the change in the commanded specific force

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10vectorwithbankangle.Todeterminethe actual anglethroughwhichthe vehicle must roll,definethe roll angle error:Ayb e", =tan-l{--} Azb (12) Using thestandardrelationsforthe derivativeofavectorina rotatingreferenceframe,the following relationships followfromthe assumptionthatA11AlB:Azb=-P(Ayb)Ayb=+P(Azb) (13) (14)Theangle e", represents theerrorbetween the actualanddesiredroll angleofthe missile.Differentiating e", yields: (Azb)(Ayb) (Ayb)(Azb) e", =(Azb)2+(AYb)2which,aftersubstituting componentsofAxw, showsthat(15) (16) Simplifying the nonlinear dynamicsof(1) (8), the following model was used:oe=Q-PJ3+Azb / IVtotl 13 =-R-Poe+Ayb /IVtotl(Izz -Ixx ) Q=Moeoe+MqQ+ MSeSe +-----IyyPR(17) (18) (19)

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wherePQ(20)(21)(22)(23)U sing this model directly, the autopilot would be designedasan eighth-order system with time-varying coefficients. However, even though these equations contain bilinear terms involving the roll rate Paswellaspitch/yaw cross-coupling terms, the roll dynamics alone, represent a second order system thatisindependentofpitch and yaw. Therefore, using an "Adiabatic Approximation" where the optimal solutionofthe time-varying systemisapproximated by a sequenceofsolutionsofthe time-invariant algebraic Riccati equation for the optimum control law at each instantoftime, the model was separated into roll and pitch/yaw subsystems [25]. Now, similartoa singular perturbations technique, the functionofthe roll channelisto provide the necessary orientationofthe missilesothat the specific force acceleration lies on the Z (preferred) axisofthe missile. Using this approximation, the systemisassumed to be in steady state, and all coefficients--including rollrate--areassumed to be constant. Linear Quadratic Gaussian (LQG) synthesisisused, with an algebraic Riccati equation, on a second and sixth order system. And, when necessary, the gains are scheduledasa functionofthe flight condition.

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12While still simplified, this formulation differs significantly from previous controllers in two respects. First, the autopilot explicitly controls the roll angle; and second, the pitch and yaw dynamics are coupled. Even though preliminary work with this controller demonstrated improved tracking performance by the autopilot, overall missile performance, measured by miss distance and time to intercept, did not improve. However, the autopilot still relies on a trajectory generated by the baseline controller ( e.g. Azb in 17). Consequently, the missile performance problemisnot in the autopilot, the error sourceisin the linear optimal control law whichforms the trajectory. "Reachable Set Control"isa LQG formulation that can minimize these errors.

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CHAPTER III CONSTRAINED CONTROL In Chapters I and II,wecovered the non-linear plant, the dynamics neglected in the linearization, the impactofcontrol variable constraints, and the inabilityofimproved autopilots to reduce the terminal error. To solve this problem,wemust consider the optimal controlofsystems subject to input constraints. Although a searchofthe constrained control literature did not provide any suitable technique for real time implementation, someofthe underlying concepts were used in the formulationof"Reachable Set Control." This Chapter reviews someofthese results to focus on the constrained control problem and illustrate the concepts. Muchofthe early work was based on research reported by Tufts and Shnidman [26] which justified the useofsaturated linear control. However,asstated in the introduction, with saturated linear control, controllabilityisnot assured.Ifthe system, boundary values and final time are such that thereisno solution with any allowable control(Ifthe trajectoryisnot controllable), then the boundary condition will not be met by either azero terminal errororpenalty function controller. While constrained control can be studied in a classical way by searching for the effectofthe constraint on the valueofthe performance function, this procedureisnot suitable for real time controlofa system with a wide rangeofinitial conditions [27]. Someofthe techniques that could be implemented in real time are outlined below.13

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14Lim used a linearized gain to reduce the problemtoa parameter optimization [8]. Given the system model: x=Fx+Gu+Lw (1) with state x, constant F,G,and L, scaler control u, and Lw representing zero mean Gaussian white noise with covariance LLT. Consider the problemofchoosing a feedback law such that in steady state, assuming it exists, the expected quadratic costItfJ=E{lim [ (x(t)TQx(t)+ >.u(t)2) dt+X(tf)Tp(tf)X(tf)] }tf-00 to (2)isminimized. The weighting matrix Qisassumed to be positive semidefinite and >. O.Dynamic programming leadstoBellman's equation: min { t tr[LTYxx(x)L)+(Fx+Gu)Tyx(x)+xTQx+ >.u2 }= 0:.* and, assuming a Y(x) satisfying(3),the optimal solution u(x)=SAT {(1/2>')G T y x (x) }=SGN {GTyx(x)}(3)(4)However,(3)cannot be solved analytically, andYx in generalisa nonlinear functionofx. Consider a modified problem by assuming a controlofthe form: u(x)=SAT { gT x }=SGN{gTx } where gisa constant (free) vector. >'=0 (5a)(5b)Assume further that xisGaussian with known covarianceW(positive definite). Using statistical linearization, a linearized gain k can be obtained by minimizing E{u(x) -kTx}2(6)

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15which results in for (Sa): for (Sb): where 4>(z) = (2MtJ;xp (-ty 2)dyk= (2/7l")t .(gTWg)-t.g(7a)(7b)From (1), with u determined by kT x, the stable covariance matrixWandsteady statePareand(F+GkT)W+W(F+GkT)T+LLT=0 (F+GkT)Tp+P(F+GkT )+P+ >.kkT =0(8)(9)andThe solution to (3), without the minimum, isv(x)=xTpxex=tr{LT PL } (10)(11) The problemisto choose g suchthatthe expected costexby statistical linearizationisa minimum. However, a minimum may not exist. In fact, from [8], a minimum does not exist when the noise disturbance is large. Since we are considering robust control problems with plant uncertainty or significant modeling errors, the noise will be large and the minimum will be replaced by a greatest lower bound.Asexapproached the greatest lower bound, the control approached bang-bang operation. A combinationofplant errorsandthe rapid dynamicsofsome systems (such as the preferred axis missile) would preclude acceptable performance withbang-bangcontrol.

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16Frankenaand Sivan suggested a criterionthatreduce thetwo-pointboundary problem to an initial value problem [12]. They suggest controlling the plant while minimizing this performance index: With the constraint lIu(t)IIR(t) 1 Applying the maximum principle to the Hamiltonian developed from(12)x(t)=F(t)x(t)+G(t)u(t) with x(to)=xOprovides the adjoint differential equationto
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17ForGT Px:;: 0 andIIxliS :;: 0, P will be the solutionofPGR-IGTpS P+PF+=Q+---FTPIIR I GTpxIlRIIxliS(18)Now choosing S=PGR-IGTpresults in a Lyapunovequationandwill insure negativedefiniteP(t)ifFisa stability matrix.Therefore,withthis choiceofweighting functions totransformthe problem to a singleboundarycondition, a stable F matrixisrequired. Thisisa significant restrictionandnot applicable to the systemunderconsideration.Gutmanand Hagander developed a designforsaturatedlinear controllersforsystems with controlconstraints[9,28].Thedesign beginswitha low-gain stabilizing control, solves a Lyapunov equation tofinda regionofstability and associated stability matrix, andthensums the controlsina saturation function toformthe constrained control. Begin with the stabilizable continuous linear time invariant system x=Fx+Guwith admissible controlinputsui,suchthatx(O)=xO=l,...,m(19)where gi and hi are the control constraints. ConsiderannxmmatrixL == [11I12I...I1 m ] suchthatisa stability matrix.(20)(21)

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18Associated with eachofthe controlsaresetsthatdefineallowable conditions.Theset Disthe setofinitial conditionsfromwhichitisdesired to stabilize the system to the origin.Thelow gain stabilizing control LdefinesthesetE:E == E(L) == (zIzER(22)i=I,...,mwhichis the setofstatesatwhichthestabilizing linearfeedbackdoesnotinitiallyexceedthe constraints.Anotherset is F: F == F(L) ==n {(eFct)-1E}tE[O,oo)(23)(24) Fisa subsetofE suchthatalong all trajectories emanatingfromF,thestabilizing linear state feedback doesnotexceed the constraint.Theregionofstabilityforthe solutionofthe Lyapunovequationisdefinedby 0== O(L,P,c) == {xIxTPx c}whereV(x)=xTpxis the Lyapunovfunctioncandidateforthe stabilitymatrixFc,andc is to be determined.Thecontrol technique follows: Step1:DetermineD. Step2:FindLbysolving aLQGproblem.Thecontrol penalty is increaseduntilthecontrolLTx satisfies theconstraintin(19)forxinD.IfDissuchthatthecontrolconstraintcan notbesatisfied,thenthis designisnotappropriate.Step3:FindPandc.FirstfindaP=PT>0suchthattheLyapunovequationPFc+F cTp>O.Now determine 0 bychoosing cin(24) suchthatD 0 E:supxT Px c minxT PxxEDxE&E(25)

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19where 6E designates the boundaryofE.Ifthis fails, choose another P, or select a "lower gain" in order to enlarge E,orfinally, a reduction in the sizeofD might be considered. Step4:Set up the control according to u= SAT[ (L T -KGTp)x]where Kisdefined(26)K=[ki0]ki 0,i= 1,2, ... ,m (27)okmand tune the parameters ki by simulations. A sufficient condition for the algorithm to workisD E.(28)Unfortunately, determining the stability region was trial and error; and, once found, further tuningofa diagonal gain matrixisrequired. In essence, thiswasa technique for determining a switching surface between a saturated and linear control. Also, when the techniquewasapplied to an actual problem, inadequacies in the linear model were not compensated for. Given the nonlinear natureofthe preferred axis missile, rangeofinitial conditions, and the trial and error tuning required for eachofthese conditions, the procedure would not be adequate for preferred axis terminal homing missile control. A notable featureofthe control scheme, however, was the ability to maintain a stable system with a saturated control during muchofthe initial portionofthe trajectory. Another technique for control with bounded input was proposed by Spong et al. [29]. This procedure used an optimal decision strategytodevelop a pointwise optimal control that minimized the deviation between the actual and

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20desired vectorofjointaccelerations, subject toinputconstraints. The computationofthe control law is reduced to the solutionofa weighted quadratic programming problem. Key to this solution is the availabilityofa desired trajectory in state space. Suppose that a dynamical system can be described by with which can be writtenasx(t)=f(x(t+G(x(tu(t) c (29) Fix time t 0, let s(t,xo,to,u(t (or s(t) for short), denote the solution to (29) corresponding to the giveninputfunction u(t). At time t,ds/dtisthe velocity vectorofthe system, andisgiven explicitly by the right hand sideof(29). Define the set C t=C(s(t with C(s(t=(ex(t,w)ERNIex=f(s(t+G(s(tw,wE {} } {} ={wI NW:5 C}(30)Therefore, for each t andany allowable u(t),ds/dtliesinthe set Ct. In other words, the set C t contains the allowable velocitiesofthe solution s(t). Assume that there exists a desired trajectory yd, and an associated vector field v( t)=v(s(t),yd(t),t, which is the desired (state) velocityofs(t) to attain yd. Consider the following "optimal decision strategy" for a given positive definite matrixQ:Choose theinputu(t)sothat the corresponding solution s(t) satisfies(d/dt)s(t,u(t=s*(t), where s*(t)ischosenateach t to minimize min {(ex v(s(t),yd(t),tTQ(exv(s(t),yd(t),t }exECt(31)

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21Thisisequivalenttothe minimization min { !UTGTQGu -(GTQ(v-fTu } subject to, Nu(t) cu(32)Wemay now solve the quadratic programming problemtoyield a pointwise optimal control law for (29). At each time t, the optimal decision strategy attemptsto"align" the closed loop system with the desired velocity v(t)asnearlyaspossible in a least squares sense. In this way the authors retain the desirable propertiesofv(t) within the constraints imposed by the control. Reachable Set Control builds on this technique:itwill determine the desired trajectory and optimally track it. Finally, minimum-time control to the origin using a constrained acceleration has also been solved by a transformationtoa two-dimensionalun-constrained control problem[30].Byusing a trigonometric transformation, the controlisdefined by an angular variable, u(t) f{cos(I3),sin(I3)}, and the control problem was modifiedtothe controlofthis angle. The constrained linear problemisconvertedtoan unconstrained nonlinear problem that forces a numerical solution. This approach removes the effectofthe constraintsatthe expenseofthe continuous applicationofthe maximum control. Given the aerodynamic performance (range and velocity) penaltyofmaximumcontrol and the impact on attainable roll rates due to reduced stability at high angleofattack, this concept did not fit preferred axis homing missiles.Animportant assumption in the previous techniques was that the constrained systemwascontrollable. In fact, unlike (unconstrained) linear systems, controllability becomes a functionofthe set admissible controls, the initial state, the time-to-go, and the target state. To illustrate this, someofthe relevant points from [31,32] will be presented. An admissible controlisone

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22 that satisfies the condition u() :[0,00) 0 ERm where 0 is the control restraint set. The collectionofall admissible controls will be denoted by M(O). The target set Xisa specified subset in Rn.Asystem is defined to be O-controllable from an initial statex(tO)= xo to the target set X at Tifthere existsU()EM(O)such that x(T,u(),xO) EX.Asystem would be globally O-controllable to XifitisO-controllable toXfromeveryx(tO)ERn.Inordertopresent the necessary and sufficient conditions for O-controllability, consider the following system: x(t)=F(t)x(t)+G(t,u(tand the adjoint defined by: =F(t)T z(t) with the state transition matrix q>(t,r) and solution x(to) =xOZ(to) =zO(33)t f [0,00)(34)(35)The interior B and surface Softhe unit ball in Rn are defined asB={ZO f RnS={zOfRn llzoll I}IIzoll=I}(36) (37)The scaler functionJ(.):Rn xRxRnxRn-R is defined by J(xo,t,x,zo) =XOTZO+Jtmax [GT(r,w)z(r)]dr-x(t)Tz(t)o WfO (38)Given the relatively mild assumptionsof[32], a necessary conditionfor(33) to be O-controllable to X fromx(tO)ismax min J(xO,T,x,ZO) =0 XfXzOfB while a sufficient conditionissup min J(xO,T,x,zO)>0 XfXZOfS (39) (40)

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23The principle behind the conditions arises from the definitionofthe adjoint system--Z(t). Using reciprocity, the adjointisformed by reversing the roleofthe input and output, and running the system in reverse time [33]. Consider x(t)=F(t)x(t)+G(t)u(t) y(t)=H(t)x(t) and: z(t)=-F(t)Tz(t)+ o(t)=GT(t)z(t) Therefore x(to)=xOz(to)=zO(41) (42)and zT(t)x(t)=zT(F(t)x(t)+G(t)u(t(d/dt)(zT(t)x(t= T(t)x(t)+z T(t)x'(t)= +zT(t)G(t)u(t)(43) (44)Integrating both sides from tototfyields the adjoint lemma:(45)The adjoint defined in (31) does not have an input. Consequently, the integral in (35)isa measureofthe effectofthe control applied to the original system.Bysearching for the maximumGT(r,w)z(r),it provides the boundaryoftheeffectofallowable control on the system (33). Restricting the search over the target settothe min ( J(xo,t,x,zO) :t f [O,T],zo f S}ormin ( J(xo,t,x,zo) :t f [O,T],zo f B} minimizes the effectofthe specific selectionof Z() on the reachable set and insures that the search is over a function thatisjointly continuous in (t,x). Consequently, (35) compares the autonomous growthofthe system, the reachable boundaryofthe allowable input, and the desired target set and time. Therefore,ifJ=0, the adjoint lemmaisbe

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24identically satisfied at the boundaryofthe control constraint set (necessary); J>0 guarantees that a control can be found to satisfy the lemma.Ifthe lemmaissatisfied, then the initial and final conditions are connected by an allowable trajectory. The authors [32]goon to develop a zero terminal error steering control for conditions where the target setisclosed and max min J(xO,T,x,zO) 0 xXZOS (46)But their control technique has two shortcomings: First; it requires the selectionofz00The initial conditionzoisnot specified but limited toIIz01l=1.A particularzomust be selected to meet the prescribed conditions and the equality in (43 ) for a given boundary condition, andistherefore not suitable for real time applications. And second; the steering control searched M(O) for the supremumofJ, making the control laws bang-bang in nature, again not suitable for homing missile control. While a direct searchof Ox isnot appropriate for a preferred axis missile steering control, a "dual" system, similar to the adjoint system used in the formulationofthe controllability function J, can be usedtodetermine the amountofcontrol requiredtomaintain controllability. Once controllabilityisassured, then a cost function that penalizes the state deviation (as opposed to a zero terminal error controller) can be used to control the systemtoan arbitrarily small distance from the reference.

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CHAPTERIV CONSTRAINEDCONTROLWITHUNMODELEDSETPOINTANDPLANTVARIAnONSChapterIII reviewed anumberoftechniques tocontrolsystemssubjectto control variable constraints. While noneofthe techniques werejudgedadequateforreal time implementationofapreferred-axishoming missile controller, someofthe underlying conceptscanbe used to develop atechniquethatcanfunctionin the presenceofcontrol constraints:(l)Useofa "dual system"thatcanbe used to maintain a controllable system (trajectory); (2) an "optimal decision strategy" to minimize the deviation betweentheactualanddesiredtrajectorygenerated by the "dual system;"and(3) initiallysaturatedcontrolandoptimal (realtime) selectionofthe switchingsurfacetolinearcontrolwithzero terminalerror.However, inadditionto,andcompounding the limitations imposedbycontrol constraints, wemustalso consider the sensitivityofthecontrol to unmodeled disturbancesandrobustnessunderplant variations.Inthe stochastic problem, there are threemajorsourcesofplantvariations.First,therewill be modeling errors (linearization/reductions)thatwill causethedynamicsofthe system to evolve in adifferentor"perturbed" fashion. Second,theremay betheunmodeleduncertaintyinthe system statedueto Gaussian assumptions.Andfinally,inthe fixed final time problem, there may beerrorsinthe final time, especiallyifitisafunctionoftheuncertain stateorimpactedbythe modeling reductions. Since theprimaryobjectiveofthis researchisthe zeroerrorcontrolofa dynamical systeminfixed time, mostofthemore recent 25

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26optimization techniques (eg. LQG/LTR,WXl) did not apply. At this time, these techniques seemedtobe more attuned to loop shapingorrobust stabilization questions. A fundamental proposition that forms the basisofReachable Set Controlisthat excessive terminal errors encountered when using an optimal feedback control for an initially controllable trajectory (a controllable system that can meet the boundary conditions with allowable control values) are caused by the combinationofcontrol constraints and uncertainty (errors) in the target set stemming from unmodeled plant perturbations (modeling errors)orset point dynamics. First, a distinction must be made between a feedback and closed-loop controller. Feedback controlisdefinedasa control system with real-time measurement data fed back from the actual system but no knowledgeofthe form, precision, or even the existenceoffuture measurements. Closed-loop control exploits the knowledge that the loop will remain closed throughout the future intervaltothe final time.Itaddstothe information provided to a feedback controller, anticipates that measurements will be taken in the future, and allows prior assessmentofthe impactoffuture measurements.IfCertainty Equivalence applies, the feedback lawisa closed-loop law.Underthe Linear Quadratic Gaussian (LQG) assumptions, thereisnothing to be gained by anticipating future measurements. In the mathematical optimization, external disturbances can be rejected, and the mean valueofthe terminal error can be made arbitrarily close to zero by a suitable choiceofcontrol cost. For the following discussion, the "system" consistsofa controllable plant and an uncontrollable referenceortarget. The system stateisthe relative difference between the plant state and reference. Since changes in the system

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27boundary condition can be caused by either a change in the reference point or plant output perturbations similartothose discussed in Chapter II, some definitions are necessary. The setofboundary conditions for the combined plant and target system, allowing for unmodeled plant and reference perturbations, will be referredtoasthe target set. Changes, or potential for change, in the target set caused only by target (reference) dynamics will be referredtoasvariations in the set point. The magnitudeofthese changesisassumed to be bounded. Admissible plant controls are restrictedtoa control restraint set that limits the input vector. Since there are bounds on the input control, the system becomes non-linear in nature, and each trajectory must be evaluated for controllability. Assume that the system (trajectory)ispointwise controllable from the initialtothe boundary condition. Before characterizing the effectsofplant and set point variations, we must consider the formofthe plant and it's perturbations.Ifweassume that the plantisnonlinear and time-varying, thereisnot much that can be deduced about the target set perturbations. However,ifhave a reduced order linear modelofa combined linear and nonlinear process,ora reasonable linearizationofa nonlinear model, then the plant can be consideredaslinear and time-varying. For example, in the caseofa Euclidean trajectory. the system model (a double integrator)isexact and linear. Usually, neglected higher order or nonlinear dynamics or constraints modify the accelerations and lead to trajectory (plant) perturbations. Consequently, in this case, the plant can be accurately representedasa Linear Time Invariant System with (possibly) time varying perturbations.

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28Consider the feedback interconnectionofthe systems K and P where Kisa sampled-data dynamic controller and P the (continuous) controlled system:r-K uG--p_ .: Figure4.1Feedback System and Notation Assuming that the feedback systemiswell defined and Bounded Input Bounded Output (BIBO) stable, at any sample time ti, the system can be defined in termsofthe following functions: e(ti)=r(ti) y(ti) u( ti)=Ke( ti)y(ti)=Pu( ti)(1) (2)(3)

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with the operator G29G[K,P]asthe operator that maps the input e(ti)tothe output y(ti) [34]. At any time, the effectofa plant perturbation DoP can alsobecharacterizedasa perturbation in the target set. or thenIfP=Po + DoP P= P(I+DoP) y(ti)= YO(tj) + Doy(ti) (4a)(4b)(5)where Doy(ti) represents the deviation from the "nominal" output caused by either the additive or multiplicative plant perturbation. Therefore, e(ti)=r(ti) (YO(ti)+ Doy(ti = (r(tj) + Doy(ti -YO(ti)= Dor(ti) -YO(ti)(6)(7)(8)with Dor(ti) representing a change in the target set that was unknown to the controller. These changes are then fed backtothe controller but could be handled a priori in a closed loop controller designastarget set uncertainty. Now consider theeffectofconstraints.Ifthe controlisnot constrained, and target set errors are generated by plant variationsortarget maneuvers, the feedback controller can recover from these intermediate target set errors by using large (impulsive) terminal controls. The modeled problem remains linear. While the trajectoryisnot the optimal closed-loop trajectory, the trajectoryisoptimal based on the model and information set available. Even with unmodeled control variable constraints, and a significant dis placementofthe initial condition, an exact plant model allows the linear stochastic optimal controllertogenerate an optimal trajectory. The switching time from saturated to linear controlisproperly (automatically) determined and,

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30asin the linear case, the resulting linear control will drive the statetowithin an arbitrarily small distance from the estimateofthe boundary condition.Ifthecontrol constraint set covers the rangeofinputs required by the control law, the law will always be able to accommodate target set errors in the remaining time-to-go. This is, in effect, the unconstrained case. If, however, the cost-to-goishigherand/orthe deviation from the boundary conditionisofsufficient magnitude relative to the time remainingtorequire inputs outside the boundaryofthe control constraint set, the system will not follow the trajectory assumed by the system model.Ifthisisthe caseastime-to-goapproaches zero, the boundary condition will not be met, the systemisnot controllable (to the boundary condition).Astime-to-go decreases, the effectsofthe constraints become more important. With control input constraints, and intermediate target set errors caused by unmodeled target maneuvers or plant variations, it may not be possible for the linear control lawtorecover from the midcourse errors by relying on large terminal control. In this case, an optimal trajectoryisnot generated by the feedback controller, and, at the final time, the systemisleft with large terminal errors. Consequently,ifexternal disturbances are adequately modeled, terminal errors that are ordersofmagnitude larger than predicted by the open loop optimal control are caused by the combinationofcontrol constraints and target set uncertainty.

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31Linear Optimal Control with Uncertainty and Constraints An optimal solution must meet the boundary conditions. To accomplish this, plant perturbations and constraints must be considered a priori. They should be includedasa priori information in the system model, they must be physically realizable, and they must be deterministic functionsofa priori information, past controls, current measurements, and the accuracyoffuture measurements. From the control pointofview,wehave seen that the effectofplant parameter errors and set point dynamics can be groupedastarget set uncertainty. This uncertainty can cause a terminally increasing acceleration profile even when an optimal feedback control calls for a decreasing input (see Chapter 5). With the increasing acceleration caused by midcourse target set uncertainty, the most significant terminal limitation becomes the control input constraints. (These constraints not only affect controllability, they also limit how quickly the system can recover from errors.)Ifthe initial controlissaturated while the terminal portion linear, the controlisstill optimal.Ifthe final controlisgoingtobesaturated, however, the controller must account for this saturation. The controller could anticipate the saturation and correct the linear portionofthe trajectorytomeet the final boundary condition. This control, however, requires a closed formsolution for x(t), carries an increased cost for an unrealized constraint, and is known to be valid for monotonic ( single switching time) trajectories only [11]. Another technique availableisLQG synthesis. However, LQG assumes controllability in minimizing a quadratic cost to balance thecontrol error and

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32input magnitudes.Aswehave seen, the effectsofplant parameter and reference variations, combined with control variable constraints, can adversely impact controllability. The challengeofLQGisthe proper formulationofthe problem to function with control variable constraints while compensating for unmodeled set point and plant variations. Reachable Set Control uses LQG synthesis and overcomes the limitationsofan anticipative controltoinsure a controllable trajectory. Control Technique Reachable Set Control can be thoughtofasa fundamentally different robust control technique based on the concepts outlined above. The usual discussionofrobust feedback control (stabilization) centers on the developmentofcontrollers that function even in the presenceofplant variations. Using either a frequency domain or state space approach, and modeling the uncertain ty, bounds on the allowable plant or perturbations are developed that guarantee stability [35]. These bounds are determined for the specific plant under consideration and a controller is designedsothat expected plant variations are contained within the stability bounds. Building on ideas presented above, however, this same problem can be approached in an entirely different way. This new approach begins with the same assumptionsasstandard techniques, specifically a controllable system and trajectory. But, with Reachable Set Control,wewill not attempt to model the plant or parameter uncertainty, nor the set point variation.Wewill, instead, reformulate the problemsothat the system remains controllable, and thus stable, throughout the trajectory even in the presenceofplant perturbations and severe control input constraints.

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33Before we developanimplementable technique, considerthedesired resultofReachable Set Control (and the originofthe name) by using atwo-dimensional missile intercept problemasan example.Attime t= t}. not any specific time during the intercept, the targetisatsome location T1andthe missileisatMIasshown in Figure 4.2. Consider these locationsasoriginsoftwo independent, target and missile centered, reference systems.Fromthese initial locations, given the control inputs available, reachable sets for each system can be definedasafunctionoftime (not shown explicitly).Thetarget setiscircular because is maneuver directionisunknownbutits capabilitybounded,and the missile reachable set exponential because the x axis controlisconstant and uncontrollable while the z axis accelerationissymmetricandbounded.TheobjectiveofReachable Set Controlisto maintain the reachable targetsetin the interiorofthe missile reachable set. Hence, Reachable Set Control.xTargetReachableM1zMissileReachableSetFigure 4.2 Reachable Set Control Objective

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34As stated, Reachable Set Control would bedifficultto implementasa control strategy. Fortunately, however,furtheranalysis leads to a simple, direct, and optimal technique thatisvoidofcomplicated algorithms orad-hocprocedures. First, consider the process. The problem addressed is the controloffixed-terminal-timesystems. The true cost is the displacementofthe state at the final time and only at the final time. In the terminal homing missile problem, thisisthe closest approach, or miss distance. In another problem, it may be fuel remaining at the final time, or possibly a combinationofthe two. In essence, with respect to the direct applicationofthis technique, thereisno preferenceforone trajectory over another or no intermediate cost based on the displacementofthe state from the boundary condition. The term "direct application" was used because constrained path trajectories, suchasthose required by robotics, or the infinite horizon problem, like the controlofthe depthofa submarine can be addressed by separating the problem into several distinctintervals--eachwith a fixed terminaltime--ora switching surface when the initial objective is met [36]. Given a plant with dynamics x(t)=f(x,t)+g(u(w),t) y(t)=h(x(t),t) modeled by x(t)=F(t)x(t)+G(t)u(t) yx(t)=H(t)x(t) x(to)=xo(9) (10)

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35with final condition x(tf) and a compact control restraint setOx.Let Ox denote the setofcontrols u(t) for which u(t)E Ox for tE [0,00).The reachable set X(tO,tf,xO'Ox) == (x:x(tf)=solutionto(10) withxofor some u()EM(Ox) }isthe setofall states reachable fromxoin time tf. In addition to the plant and model in(9&10),wedefine the reference(11)r(t)=a(x,t)+b(a(w),t) y(t)=c(x(t),t) modeled by r(t)=A(t)r(t)+B(t)a(t)Yr(t)=C(t)r(t) and similarly defined set R(to,tf,ro,Or), r(to)=rO(12) (13) R(to,tf,ro,Or) == (r:r(tf)=solution to (13) withrOfor some a()EM(Or) }asthe setofall reference states reachable fromrOin time tf. (14) Associated with the plant and reference, at every time t,isthe following system: e(t)=yx(t) -Yr(t)(15) ;;m (10&13), we see that yx(t) andYr(t)are output functions that incorporate the significant characteristicsofthe plant and reference that will be controlled. The design objectiveise(tf)=0(16)and we want to maximize the probabilityofsuccess and minimize the effectoferrors generated by the deviationofthe reference and plant from their associated models. To accomplish this with a sampled-data feedback control law,

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36wewill select the control u(ti) such that, at the next sample time (ti+l), the target reachable set will be covered by the plant reachable set and, in steady state,ife(tf)=0, the control will not change. Discussion Recalling that the performance objective at the final timeisthe real measureofeffectiveness, and assuming that the terminal performanceisdirectly related to target set uncertainty, this uncertainty should be reduced with time-to-go. Now consider the trajectory remembering that the plant model is approximate (linearized or reduced order), and that the reference has the capability to change and possibly counter the control input. (This maneuverabili ty does not havetobe taken in the contextofa differential game.Itis only intended to allow for unknown set point dynamics.) During the initial portionofthe trajectory, the target set uncertaintyisthe highest. First, at this point, the unknown (future) reference changes have the capabilityofthe largest displacement. Second, the plant distance from the uncertain set pointisthe greatest and errors in the plant model will generate the largest target set errors becauseofthe autonomous response and the magnitudeofthe control inputs required to move the plant state to the set point. Along the trajectory, the contributionofthe target (reference) maneuvera bility to set point uncertainty will diminish with time. This statement assumes that the target (reference) capability to change does not increase faster than the appropriate integralofits' input variable. Regardlessofthe initial maneu verabilityofthe target, the time remainingisdecreasing, and consequently, the

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37ability to move the set point decreases. Target motionissmaller and it's positionismore and more certain. Selectionofthe control inputs in the initial stagesofthe trajectory that will result in a steady state control (that contains the target reachable set within the plant reachable set) reduces target set uncertainty by establishing the plant operating point and defining the effective plant transfer function. At this point,wedo not have a control procedure, only the motivation to keep the target set within the reachable setofthe plant along with a desire to attain steady state performance during the initial stagesofthe trajectory. The specific objectives are to minimize target set uncertainty, and most importantly, to maintain a controllable trajectory. The overall objectiveisbetter performance in termsofterminal errors. Procedure A workable control law that meets the objectives can be deduced from Figure 4.3. Herewehave the same reachable setforthe uncertain target, but this time, several missile origins are placed at the extremesoftarget motion. From these origins, the systemisrun backward from the final timetothe current time using control values from the boundaryofthe control constraint set to provide a unique setofstates that are controllable to the specific origin.Ifthe intersectionofthese setsisnon-empty, any potential target locationisreachable from this intersection. Figure 4.4 is similar, but this time the missile control restraint setisnot symmetric. Figure 4.4 shows a case where the missile acceleration controlisconstrainedtothe set A=[Amin,Amax] where 0 Amax(17)

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z38xTargetReachable AlltargetpositionsReachableFigure 4.3 IntersectionofMissile Reachable Sets BasedonUncertain Target Motion and Symmetric ConstraintsxTargetReachableSetAllTargetLocationsReachableFigure 4.4 IntersectionofMissile Reachable Sets Based on Uncertain Target Motion and Unsymmetric Constraints

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39Since controllabilityisassumed, which for constrained control includes the control bounds and the time interval, the extreme left and right (near and far) pointsofthe set point are included in the set drawn from the origin. To implement the technique, construct a dual system that incorporates functional constraints, uncontrollable modes, and uses a suitable control value from the control constraint setasthe input. From the highest probability target positionatthe final time, run the dual system backward in time from the final boundary condition. Regulate the plant (system) to the trajectory defined by the dual system. In this way, the fixed-final-time zero terminal error controlisaccomplished by re-formulating the problemasoptimal regulationtothe dual trajectory. In general, potential structuresofthe constraint set preclude a specific point (origin, center, etc.) from always being the proper inputtothe dual system. Regulation to a "dual" trajectory from the current target position will insure that the originofthe target reachable set remains within the reachable setofthe plant. Selectionofa suitable interior point from the control restraint setasinputto the dual system will insure that the plant has sufficient control power to prevent the target reachable set from escaping from the interiorofthe plant reachable set. Based on unmodeled set point uncertainty, symmetric control constraints, and a double integrator for the plant, a locus exists that will keep the target in the centerofthe missile reachable set.Ifthe set pointisnot changed, this trajectory can be maintained without additional inputs. For a symmetric control restraint set, especiallyasthe time-to-go approaches zero, Reachable Set Controliscontrol to a "coasting" (null control) trajectory.

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40Ifthe control constraints are not symmetric, such asFigure4.4, a locusofpoints that maintains the target in thecenterofthe reachable set is the trajectory based on the systemrunbackward from the final time target location with the acceleration command equal to the midpointofthe set A. Pictured in Figures 4.2 to 4.4 were trajectories that are representativeofthe double integrator. Other plant models would havedifferenttrajectories. Reachable Set Control is a simple techniqueforminimizing the effectsoftarget set uncertainty and improving terminal the performanceofa large classofsystems.Wecan minimize the effectsofmodeling errors (or target setuncertainty) by a linear optimal regulator that controls the system to a steady state control. Given the well known and desirable characteristicsofLQGsynthesis, this technique canbeusedasthe basisforcontrol to the desired "steady state control" trajectory.Thetechnique handles constraints by insuringaninitially constrained trajectory. Also, since the large scale dynamics are controlled by the "dual" reference trajectory, the tracking problem be optimized to the response timeofthe systemunderconsideration. This results inan"adaptable" controller because gains are basedonplant dynamicsandcost while the overall system is smoothlydrivenfrom some large displacement to a region where the relatively high gainLQGcontroller will remain linear.

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CHAPTERV REACHABLE SETCONTROLEXAMPLEPerformance Comparison Reachable SetandLOGControlInorderto demonstrate theperformanceof"Reachable Set Control" we will contrast its performance with theperformanceofa linear optimal controllerwhenthereistarget setuncertaintycombinedwithinputconstraints. Consider,forexample, thefinitedimensional linear system:withthequadratic cost x(to)=xo(1)whereJ1=-XfTpfXf 2(2)andtfE [0,00) '1 0Applicationofmaximum principle yields the following linear optimal control law:where1u=+-x(tf)(t-tf) '1 xo+xo tfx(tf)=1+41(3) (4)

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42Appropriately defining t, to, andtf,the control law can be equivalently expressed in an open loop or feedback form with the latter incorporating the usual disturbance rejection properties. The optimal control will tradeoff the costofthe integrated square input with the final error penalty. Consequently, even in the absenceofconstraints, the terminal performanceofthecontrolisa functionofthe initial displacement, time allowed to drive the statetozero, and the weighting factor ,. To illustrate this, Figure5.1presents the terminal states (miss distance and velocity)ofthe linear optimal controller. This plotisa compositeoftrajectories with different run times ranging from 0to3.0 seconds. The figure presents the valuesofposition and velocity at the final time t=tfthat result from an initial positionof1000 feet and with velocityof1000feet/sec with =104 Figure 5.2 depicts,asa functionofthe run time, the initial acceleration (at t=0.) associated with eachofthe trajectories shown in Figure 5.1. From these two plots, the impactofshort run timesisevident: the miss distance will be higher, and the initial acceleration command will be greater. Since future set point (target) motionisunknown, the suboptimal feedback controllerisreset at each sample time to accommodate this motion. The word resetissignificant. The optimal controlisa functionofthe initial condition at time t=to, time, and the final time. A feedback realization becomes a functionofthe initial condition and time to go only. In this case, set point motion (target set uncertainty) can place the controller in a position where the time-to-goissmall but the state deviationislarge.

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Velocity2000o-2000-4000 -6000 -8000-10000 -12000-14000o20043400600800Position10001200Figure5.1Terminal PerformanceofLinear Optimal ControlAccelerationo-100000--200000-300000v-400000o0.51Final Time1.52Figure 5.2 Initial AccelerationofLinear Optimal Control

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44While short control times will result in poorer performance and higher accelerations, it does not take a long run time to drive the terminal error to near zero. Also, from (4)wesee that the terminal error can be driventoan arbitrarily small value by selectionofthe control weighting. Figure5.1presented the final valuesoftrajectories running from 0 to 3 seconds. Figures 5.3 through 5.5 are plotsofthe trajectory parameters for the two second trajectory (with the same initial conditions) along with the zero control trajectory values. These values are determined by starting at the boundary conditionsofthe optimal control trajectory and running the system backward with zero acceleration. For example,ifwestart at the final velocity and run backwards in time along the optimal trajectory, for each point in time, thereisa velocity (the null control velocity) that will take the corresponding positionofthe optimal control trajectorytothe boundary without additional input. The null control position begins at the origin at the final time, and moving backward in time,isthe position that will take the system to the boundary condition at the current velocity. Therefore, these are the positions and velocities (respectively) that will result in the boundary condition without additional input.Ast=>tfthe optimal trajectory acceleration approaches zero. Therefore, the zero control trajectory converges to the linear optimal trajectory.Ifthe system has a symmetric control constraint set, Reachable Set Control will control the system position to the zero control (constant velocity) trajectory.

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Accelerationo-500-1000-1500 -2000-250000.5451Time1 .52Figure 5.3 Linear Optimal Acceleration vs Time r--Null Control VelocityVelocity1500 10005000-500........-1000 -15000.50..............1Time1.52Figure 5.4 Linear Optimal Velocity vs Time

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46Position20001000......o-1000-2000-3000o NullControlPosition0.511.5TimeFigure 5.5 Linear Optimal PositionvsTime2Consider now the same problemwithinputconstraints. Since Vet)is a linear functionoftime and the final state,itismonotonicandthe constrained optimal controlisIu=SAT(-x(tf)(t-tf(5) "I In this case, controllability is in question,andisa functionofthe initial conditionsandthetime-to-go.Assuming controllability, the final state will be given by: x(tf)=xQ+ xQtf a(tl)SGN(x(tf)[tf-(tl/2)](tf-t1)31+---(6)

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47wheretlisthe switching time from saturated to linear control. The open loop switch time can be shown to be(7)or the closed loop control can be used directly. In either case, the optimal control will correctly control the system to a final state X(tf) near zero. Figures 5.6 through 5.8 illustrate the impactofthe constraint on the closed loop optimal control. In each plot, the optimal constrained and unconstrained trajectoryisshown.Acceleration1000o-1000-2000-3000 -4000.............-500000.51Time1 .52Figure 5.6 Unconstrained and Constrained Acceleration

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48Velocity2000o-2000-4000Constrained o0.51Time1 .52Figure 5.7 Unconstrained and Constrained VelocityvsTime ...... ...........21 .51Time0.5Constrained Position------l Position250020001500 10005000 0Figure 5.8 Unconstrained and Constrained Position vs Time

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49Now consider the effectsoftarget set uncertainty on the deterministic optimal control by using the same control law for a 2.0 second trajectory where the boundary conditionisnot constantbutchanges. The reason for the target uncertainty and selectionofthe boundary condition can be seen by analyzing the componentsofthe modeled system. Assume that system actually consistsofan uncontrollable reference (target) plant as wellascontrolled (missile) plant with the geometry modeled by the difference in their states. Therefore, the final set point (relative distance)iszero,butthe boundary condition along the controlled (missile) trajectoryisthe predicted target position at the final time. This predicted position at the final timeisthe boundary condition for the controlled plant. Figures 5.9 through5.11are plotsoflinear optimal trajectories using the control law in (5,6). There are two trajectoriesineach plot. The boundary condition for one trajectoryisfixed at zero, the set point for the other trajectory is the pointwise zero control value (predicted target state at the final time). Figures 5.9 through5.11demonstrate the impactofthis uncertainty on the linear optimal control law by comparing the uncertain constrained control with the constrained control that has a constant boundary condition.

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Acceleration1000o-1000 -2000 -3000 -400050......UncertainTarget Set -500000.51Time1 .52Velocity2000o-2000 -4000Figure 5.9 Acceleration ProfileWithand Without Target Set Uncertainty TargetSet \ .......................o0.51Time1 .52Figure 5.10 Velocity vs TimeWithand Without Target Set Uncertainty

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51PositionUncertainTargetSet ---------1400 1200 1000800600400200o00.51Time1 .52Figure5.11PositionvsTime With and Without Target Set Uncertainty When thereistarget set uncertainty, simulated by the varying set point, the initial accelerationisinsufficient to prevent saturation during the terminal phase. Consequently, the boundary conditionisnot met. The final setofplots, Figures 5.12 through 5.14, contrast the performanceofthe optimal LQG closed loop controller that we have been discussing and the Reachable Set Control technique.Inthese trajectories, the final set pointiszerobutthereistarget set uncertainty again simulatedbya time varying boundary condition (predicted target position) that convergestozero. Although properly shownasa fixed final time controller, the Reachable Set Control results in Figures 5.12 through 5.14 are from a simple steady state (fixed gain) optimal tracker referencedtothe zero control trajectory r.

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52Thesystem modelforeach techniqueis(8)withx(to)=(x-r)oThelinear optimal controller has a quadratic costofJ=2Thereachable set controller minimizes(9)J= J : 1)[(x-r)TQ(x-r)+u(r)Tu(r)]dr(10)And,ineithercase, the value for r(t)isthe position that will meet theboundaryconditionatthe final time withoutfurtherinput.Acceleration1000o-1000 -2000 -3000-4000ReachableSetControl -500000.51Time1 .52Figure 5.12 Acceleration vsTimeLQGand Reachable Set Control

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Velocity2000o-2000 -400053.......................................................-__-_. "'-ReachableSetControlo0.51 Time1 .5 2".'.ReachableSetControl Position140012001000800600400200o0Figure 5.13 Velocity vs Time LQG and Reachable Set Control"..............0.51TimeFigure 5.14 Position vs Time LQG and Reachable Set Control1 .5'"2

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54Summary The improved performanceofReachable Set Controlisobvious from Figure 5.14. While demonstrated for a specific plant, and symmetric control constraint set, Reachable Set Controliscapableofimproving the terminal performanceofa large classofsystems.Itminimized the effectsofmodeling errors (or target set uncertainty) by regulating the system to the zero control state. The technique handled constraints and insured an initially constrained trajectory. The tracking problem could be optimized to the response timeofthe system under considerationbysmoothly driving the system from some large displacementtoa region where the relatively high gain LQG controller remained linear.

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CHAPTER VI REACHABLE SET CONTROL FOR PREFERRED AXIS HOMING MISSILES As statedinChapter II, the most promising techniques that can extend the inertial point mass formulation are based on singular perturbations [37,38,39]. When applied to the preferred axis missile, eachofthese techniques leads to a controller that is optimal in some sense. However, a discussionof"optimality" notwithstanding, the best homing missile intercept trajectory is the one that arrives at the final "control point" with the highest probabilityofhitting the target. This probability can be broken down into autonomous and forced events.Ifnothing is changed, what is the probabilityofa hitorwhat is the miss distance?Ifthe target does not maneuver, can additional control inputs result in a hit? And, in the worse case,ifthe target maneuvers (or an estimation error iscorrected) will the missile have adequate maneuverability to correct the trajectory? Noneofthe nonlinear techniques based on singular perturbations attempt to control uncertaintyoraddress the terminally constrained trajectories caused by increasing acceleration profile. Unfortunately, an increasing acceleration profile has been observed in allofthe preferred axis homing missile controllers. In many cases, the genericbank-to-turnmissileof[11,18] was on all three constraints (Ny,Nz,P) during the latter portionofthe trajectory.Ifthe evading targetisable toputthe missileinthis position without approaching it's own maneuver limits,itwill not be possible for the missile to counter the final evasive maneuver. The missileisno longer controllable to the target set. The "standard" solution to the increasing acceleration profileisa varyingcontrol cost.55However, without additional

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56 additional modifications, this typeofsolution results in a trajectorydependentcontrol. As we have seen, Reachable Set Control is anLQG controlimplementationthatmoves the system to the point wherefurtherinputs are not required. A Reachable Set Controller that willrejecttarget and system disturbances,cansatisfy both the mathematicalandheuristic optimality requirementsbyminimizing the cost yet maintaining a controllable system. Since the roll control hasdifferentcharacteristics, the discussionofthepreferredaxis homing missile controller using the Reachable Set Control technique will be separated into translationalandroll subsystems.Thetranslational subsystem has a suitable null control trajectorydefinedbythe initial velocity and uncontrollable acceleration provided by the rocket motor.Theroll subsystem, however, is significantlydifferent.Inorderforthepreferredaxis missile to function, thepreferredaxis must be properly aligned. Consequently, both roll angleerrorandroll rate should be zero at all times. In this case, the null control trajectory collapses to the origin. Acceleration Control System Model Since we want to control the relative target-missile inertial system to the zero state, the controller will bedefinedinthis reference frame. Eachofthe individual system states aredefined(in relative coordinates) as target state minus missile state. Begin with the deterministic system: x(t)=Fx(t)+Gu(t)(I)

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57where andxyzx=Vx Vy Vzu= Since the autopilot modelisa linear approximation and the inertial model assumes instantaneous response, modeling errors will randomly affect the trajectory. Atmospheric and other external influences will disturb the system. Also, the determinationofthe state will requirethe useofnoisy measurements. Consequently, the missile intercept problem should be approached via a stochastic optimal control law. Because the Reachable Set Control technique will minimize the effectofplant parameter variations (modeling errors) and un modeled target maneuverstomaintain controllability, we can use an LQG controller. Assuming Certainty Equivalence, this controller consistsofan optimal linear (Kalman) filter cascaded with the optimal feedback gain matrixofthe corresponding deterministic optimal control problem. Disturbances and modeling errors can be accounted for by suitably extending the system description[40]:x(t)=F(t)x(t)+G(t)u(t)+Vs(t) by adding a noise process V s (") to the dynamics equations with(2) (3)

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58Therefore, let the continuous time state description be formally given by the linear stochastic differential equationx(t)= ct>( t, to)x( to) dx(t)=F(t)x(t)dt+G(t)u(t)dt+L(t)df3(t) (with13(,.)a Wiener process) that has the solution: ct>( t,r)G(r)u(r)dr+
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state to converge with zero error.59Since the optimal solution to the linear stochasticdifferentialequation is a Gauss-Markov process, time correlated processescanbe includedbyaugmenting the system state to include the disturbance process.Letthe time-correlated target (position) disturbance be modeled by the following: T(t)=N(t)T(t)+Wt(t)(8)with[Tx(t)IT(t)=Ty(t) Tz(t) and =Wt(t) While the target disturbance resulting from anunknownacceleration is localized to a single plane with respect to the body axisofthe target, the target orientation is unknown to the inertial model. Consequently, following the methodologyofthe Singer Model, each axis will be treated equally [41]. Since the disturbance is firstorderMarkov, it's components will be:andN(t)=-(I/Tc)[I] Wt(t)=(20't2 / T c)[I](9) (10)where Tcisthe correlation time, and O'r isthe RMS valueofthe disturbance process.ThePower Spectral Densityofthe disturbanceis: Wtt(W) =20't2 / T cw2+O/Tc)2(11) Figure 6.1 summarizes the noise interactions with the system.

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60WhiteGaussianNoiseJ-------'AtmosphericdisturbancesActuatorErrorsAutopilotErrors"DeterministicControls LinearSystem OutputsPhysicalModelSystemResponse Jl TargetAccelerationsShapingFilterWhite Gaussian NoiseFigure6.1Reachable Set Control Disturbance processes.Withappropriate dimensions, the nine state (linear) augmented system model becomes:[ ]T(t)=Reference Model Reachable Set Control requires a supervisory steering control (reference) that includes the environmental impact on the controlled dynamic system. Recalling the characteristicsofthe dual system, one was developed that explicitlyran(I) backward in time after determining the terminal conditions. However, in developing this control for this preferred axis missile a numberoffactors actually simplify the computationofthe reference trajectory:

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61(I)Thecontrol constraint setforthispreferredaxis missile is symmetric.Consequently.thereferencetrajectoryforaninterceptcondition.is a null control (coasting)trajectory.(2)Thebodyaxis X acceleration isprovidedbythemissile motor,andisnotcontrollablebutknown. This uncontrollableaccelerationwillcontributetothetotal inertial acceleration vector.mustbeconsideredbythe controller.andistheonlyacceleration present onanintercept(coasting)trajectory.(3)Theterminationoftheinterceptistheclosestapproach.whichnowbecomes thefixed-final-time(tf).Thetime-to-go(tgo) isdefinedwith respect tothecurrenttime (t) by: t=tftgo(13)(4)Thefinalboundaryconditionforthesystem state (target minus missile) is zero.Insummary,theinterceptpositions are zero.theinitial velocity is given.andthe average acceleration is a constant.Therefore.itissufficientto reversethedirectionoftheinitial velocityandaverageaccelerationthenrunthesystemforwardintimefortgo secondsfromtheorigintodeterminethecurrentpositionofthecoasting trajectory. Let:withandr(t)=A(t)r(t)+B(t)a(t)r(O)=rO=0A(t)=F(t)andB(t)=G(t)(14)thenr(t) isthepointfromwhich theautonomoussystem dynamics will takethesystem todesiredboundarycondition. Becauseofthe disturbances. targetmotion,andmodelingerrors,futurecontrolinputsarerandomvectors.Therefore,thebest policy isnotto

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62determine the input over the control period [to,tfl a priori buttoreconsider the situation at each instant t on the basisofall available information. At each update,ifthe systemiscontrollable, the reference (and system state) will approach zeroastgo approaches zero. Since the objectiveofthe controlleristo drive the system state to zero,wedo not require a tracker that will maintain the control variable at a desired non-zero value with zero steady state error in the presenceofunmodeled constant disturbances. There are disturbances, but the final set pointiszero, and therefore, aPIcontrollerisnot required. Roll Control Definition The roll modeismost significant sourceofmodeling errors in the preferred axis homing missile. While non-linear and high order dynamics associated with the equationsofmotion, autopilot, and control actuators are neglected, the double integratorisan exact modelfordetermining inertial position from inertial accelerations. The linear system, however,isreferenced with respect to the body axis. Consequently, to analyze the complete dynamics, the angular relationship between the body axis and inertial references must be considered. Recall Friedland's linearized (simplified) equations. The angular relationships determine the orientationofthe body axis reference and the roll rate appeared in the dynamicsofall angular relationships. Yet,tosolve the system using linear techniques, the system must be uncoupled via a steady state (Adiabatic) assumption. Also, the roll angleisinertially defined and the effectofthe linear accelerations on the erroristotally neglected.

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63Froma geometric pointofview, this mode controls the rangeofthe orthogonal linear acceleration commandsandthe constrained controllabilityofthe trajectory. With a20:I ratio in thepitchandyaw accelerations, the ability topointthepreferredaxis in the "proper" direction is absolutely critical. Consequently, effective roll control is essential to the performanceofthepreferredaxis homing missile.Thefirst problem in defining the roll controller, is the determinationofthe "proper" direction. There are two choices.Thepreferredaxis could be aligned with the target positionorthedirectionofthe commanded acceleration.Thefirstselectionisthe easiest to implement.Theseeker gimbal angles provide adirectmeasureofintercept geometry (Figure 6.2), and the roll angleerrorisdefineddirectly:(15)TargetLOS eyzxFigure 6.2. Roll AngleErrorDefinitionfromSeeker Angles. This selection, however,isnot the most robust. Depending on the initial geometry, the intercept point may not beinthe planedefinedbythecurrent

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64lineofsight (LOS) and longitudinal axisofthe missile. In this case, the missile must continually adjust its orientation (roll) to maintain the target in the preferred plane.Asrange decreases the angular rates increase, with the very real possibilityofsaturation and poor terminal performance. Consequently, the second definitionofroll angle error should be used. Considering the dynamicsofthe intercept, however, aligning the preferred axis with the commanded acceleration vectorisnotasstraightforwardasit seems. Defining the roll angle erroras 0e =Tan-I{A y /A z } (16) leadstosignificant difficulties. From the previous discussion, itisobvious that roll angle errors must be minimizedsothat the preferred axis acceleration canbeusedtocontrol the intercept. The roll controller must have a high gain. Assume, for example, the missileison the intercept trajectory. Therefore, both A y and A z will be zero. Now,ifthe target moves slightly in theYbdirection and the missile maneuvers to correct the deviation, the roll angle error instan taneously becomes 90 degrees. High gain roll control inputs to correct this situation are counter productive. The small A y may be adequate to completely correct the situation before the roll mode can respond. Now, the combinationoflinear and roll control leads to instabilityasthe unnecessary roll rate generates errors in future linear accelerations. The problems resulting from the definitionofequation16can be overcomebere-examining the roll angle error. First, A y and A z combine to generate a resultant vector at an angle from the preferred axis. In the processofapplying constraints, the acceleration angle that results from the linear accelerations canbeincreased or decreased by the presenceofthe constraint.Ifthe angleisdecreased, the additional rollisneeded to line up the preferred axis and the

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65desired acceleration vector.Ifthe constrained (actual) acceleration angleisincreased beyond the (unconstrained) desired value by the unsymmetric actionofthe constraints, then the roll controller must allow for this "over control" caused by the constraints. Define the roll angle errorasthe difference between the actual and desired acceleration vectors after the control constraints are considered. This definition allows for the full skid toturncapabilityofthe missile in accelerat ing toward the intercept point and limits rolling to correct large deviations in acceleration angle from the preferred axis that are generated by small accelera tions. ReferringtoFigure 6.3, three zones can be associated with the following definitions: "ec =Tan-I(Ay/Az } 'lea =Tan-I(Ny/Nz } "er = "ec 'lea;ZoneIZoneIIZoneIIIFigure 6.3. Roll Control Zones.(17) (18) (19)

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66Here Ny and N z are the constrained acceleration values. In Zone I, "er =O.The linear acceleration can complete the intercept without further roll angle change. Thisisthe desired locus for the roll controller. Both A y and A z are limited in Zone II. Thisisthe typical situation for the initial positionofa demanding intercept. The objectiveofthe roll controlleristokeep the intercept acceleration outofZone III where only A yislimited. In this case, the A y accelerationisinsufficient to complete the intercept yet significant roll angle change may be requiredtomake the trajectory controllable. Controller A dual mode roll controllerwasdeveloped to accommodate the rangeofsituations and minimize roll angle error. Zone I requires a lower gain controller that will stabilize the roll rate and maintain "er small. Zones II and III require high gain controllers. To keep Zone II trajectories from entering Zone III, the "ec will be controlled to zero rather than the roll angle error. Since the linear control valueisalso a functionofthe roll angle error, roll angle errors are determined by comparing the desired and actual anglesofa fixed high gain reachable set controller.Ifthe actual linear commands are used and a linear accelerationissmall becauseoflarge roll angle errors, the actual amountofroll needed to line up the preferred axis and the intercept point, beyond the capabilityofthe linear accelerations, will not be available because they have been limited by the existing roll angle error that must be corrected. Unlike the inertial motions, the linear model for the roll controller accounts for the (roll) damping and recognizesthatthe inputisa roll rate change command: = -w" +wP(20)

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67Therefore, the roll mode elements (that will be incorporated into the model are) are: (21) Also, the dual mode controller will require an output function and weighting matrix that includes both roll angle and roll rate. Kalman Filter The augmented system model (13)isnot block diagonal. Consequently, the augmented system filter will not decouple into two independent system and reference filters. Rather, a single, higher order filter was required to generate the state and disturbance estimates. A target model (the Singer model)wasselected and modified to trackma-neuvering targets from a Bank-To-Turnmissile [41,42]. Using this model, a continuous-discrete Extended Kalman Filter was developed. The filter used a9 state target model for the relative motion (target missile): (22) with u(t) the known missile acceleration, N the correlation coefficient, and Wt(t) an assumed Gaussian white noise input with zero mean. Azimuth, elevation, range, and range rate measurements were available from passive JR, semi-active, analog radar, and digitally processed radar sensors. The four measurements are seeker azimuth (,p), seeker elevation(0),range (r), and range rate (dr/dt):

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68 6 =-Tan-l{z(x2 +y2)-1/2} t/J =+Tan-l{y/x}= 1r +Tan-l{y/x}r={x2+y2+z 2}-1/2 r= x 0x 0(23)Noise statistics for the measurements are a functionofrange, and are designed to simulate glintandscintillation in a relatively inexpensive missile seeker. In contrast to the linear optimal filter, theorderofthe measurements for the extended filterisimportant. In this simulation, the elevation angle (6) was processed first, followed by azimuth (t/J), range (r), and range rate(dr/dt).In addition, optimal estimates were available from the fusionofthe detailed (digital) radar model and IR seeker. Reachable Set Controller StructureTheTarget-Missile System is showninFigure 6.4. The combinationofthe augmented system state and the dual referencethatgenerates the minimum control trajectory for the reachable set concept is best describedasa CommandGenerator/Trackerandisshown in Figure 6.5.Ina single systemofequations the controller models the system response, including time correlated position disturbances, and provides the reference trajectory. Since only noise-corrupted measurementsofthe controlled system are available, optimal estimatesofthe actual states were used. Becauseofthe processing time requiredforthe filter and delays in the autopilot response, a continuous-discrete Extended Kalman Filter,anda sampled

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69data(discrete) controller was used. This controller incorporated discrete cross-coupling terms to control the deviations between the sampling timesaswell the capability to handle non-coincident sampleandcontrol intervals (Appendices B andC).Combining the linear and roll subsystems with a firstorderroll mode for the roll angle state, the model for the preferred axis homing missile becomes: the reference: r(t)=A(t)r(t)+B(t)a(t) with the tracking error: e(t)=[yx(t) Yr(t) ]=[H(t)I0I-C(t)] r(t)(25)(26)The initial stateismodeled asann-dimensional Gaussian random variable with meanxOand covariancePO.E{ws(t)ws(t)T}=Ws(t) is the strengthofthe system (white noise) disturbances to be rejected,andE{wt(t)Wt(t)T}=Wt(t)isaninput to a stationary firstorderGauss-Markov processthatmodels target acceleration. The positions are the primary variablesofinterest,andthe output matrices will select these terms. Along with the roll rate, these are the variables that will be penalized by the control costandthe states where disturbances will directly impact the performanceofthe system.

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In block form, with appropriate dimensions, the system matrices are:F(t)=A(t)=F=[0I]OOwG(t)=B(t)=G=[0]Iw(28)H(t)=C(t)=[Ihw ]N(t)=-(ljTc)[I]L(t)=IM(t)=Iwhere the Ow, I w and h w terms are required to specify the roll axis system and control terms:

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71{-wi=j=8{1i=j=8 (Ow>ij = (hw>ij =0otherwise0otherwise{+wi=8,j=4 (Iw)ij = I otherwise The performance objective for the LQG synthesisisto minimize an appropriate continuous-time quadratic cost: Js(t) = E{Jd(t)II(t)}(29)where J sisthe stochastic cost, I(t)isthe information set available at time t, and Jd a deterministic cost function:(30)Dividing the intervalofinterest into N+I intervals for discrete time control, and summing the integral cost generates the following (see AppendixC):(31)which can be related to the augmented state X=[xTr ]T by:][ X(tj) u(ti)](32)

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72In general, with the cost terms defined for the augmented state (AppendixC),the optimal (discrete) solution to the LQG tracker can be expressedas:where*=-[G(ti)] [ x(ti) T(ti) r(ti)](33)and G*(ti)=[R(ti) + GT(ti)P(ti+l)G(ti) ] -1 [ + ST(tj) ] P(ti)=Q(ti) + -[+ ST(ti) ]TG*(ti)(34) (35)Since only the positions (and roll rate) are penalized, the Riccati recursionisquite sparse. Consequently, by partitioning the gain and Riccati equations, and explicitly carrying out the matrix operations, considerable computationalim-provements are possible over the straightforward implementationofa19by19tracker (Appendix D). Application The tracking error and control costs were determined from the steady state tracker used in the example in Chapter5.First, missile seeker and aerodynamic limitations were analyzed to determine the most demanding intercept attainable by the simulated hardware. Then, autopilot delays were incorporated to estimate that amountoftime that a saturated control would require to turn the missile after correcting a 90 degree (limit case) roll angle error. The steady state regulator was used to interactively place the closed loop polesandselect a control cost combination that generated non zero control for the desired lengthoftime. These same values were used in the time varying Reachable Set Controller with the full up autopilot simulation to determine the

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73terminal error cost and control delay time. To maintain a basisofcomparison, the Kalman Filter parameters were not modified for this controller. Appendix E contains initial conditions for the controller and estimator dynamics. During the initialization sequence (safety delay) for a given run, time varying fixed-final-time LQG regulator gains are calculated (via 36) based on the initial estimateofthe time togo.Both high and low roll control gains were computed. These solutions used the complete Riccati recursion and cost based on the sampled data system, included a penalty on the final state (to control transient behaviorastgo approaches zero), and allowed for non-coincident sample and control. Given an estimated tgo, at each time t, the Command GeneratorITracker computed the reference position and required roll angle that leads to an intercept without additional control input. The highorlow roll control gain was selected based on the mode. Then the precomputed gains (that are a functionoftgo) are used with the state and correlated disturbance estimate from the filter, roll control zone, and the reference r to generate the control (whichisapplied only to the missile system). Becauseofsymmetry, the tracker gain for the state term equaled the reference gain,sothat, in effect, except for the correlated noise, the current difference between the state x and the reference r determined the control value. During the intercept, between sample times when the stateisextrapolated by the filter dynamics, tgo was calculated based on this new extrapolation and appropriate gains used. This technique demonstrated better performance than using a constant control value over the durationofthe sample interval and justified the computational penalty of the continuous discrete implementationofthe controller and filter.

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MeasurementNoise InterceptGeometry--Sensors--TargetControl--Disturbances TargetDynamics--OutputFunction+ l' Command -ZeroOrder GeneratorI .. Hold -DIATrackerStateEstimationSample&Hold AIDSensorsReachableSetControllerContinuous-DiscreteKalmanFilterMeasurementNoise Autopilot--Actuators Missile MissileDynamics-OutputFunctionDynamicDisturbancesFigure 6.4 Target Missile System

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r-Reference ModelReferenceVariableDynamicsa....Missile Model Yr(t)e(t) ,----,LJ-DynamicDisturbancesMeasurementNoise ", G1G3u(ti)I---{ "')-------1 G2Missile -TargetSystem X(t)Q!LJi Z(t)Yx(t)X(ti)TargetManeuverModelA(ti)ContinuousDiscreteKalmanFilter-Figure 6.5 Command Generator / Tracker

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CHAPTER VII RESULTS AND DISCUSSIONAsan additional reference, before comparing the resultsofReachable Set Control to the baseline control, consider anair-to-airmissile problem from [13]. In this example,the launch directionisalong the lineofsight, the missile velocityisconstant, and the autopilot response to commandsisinstantaneous. The controller has noisy measurementsoftarget angular location, a priori knowledgeofthe time to go, and stochastically models the target maneuver. Even with this relatively simple problem, the acceleration profile increases sharply near the final time. Unfortunately, this acceleration profileistypical, and has been observed in all previous optimal control laws. Reachable Set Control fixes this problem. c.9-...oa:)_u'"u u0oen--....t= .EenE0=: 300 200100o6Time to go (sec) O'--....L------l-.---l 12Figure7.1RMS Missile Acceleration76

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77Simulation The performanceofReachable Set Control was determined via a highfi-delity Bank-To-Turn simulation developed at the UniversityofFlorida and used for a numberofprevious evaluations. The simulationisbased on the coupled non-linear missile dynamicsofchapter II equations (1)to(8) andisa continuous-discrete system that has the capabilityofcomparing control laws and estimators at any sample time. In addition to the non-linear aerodynamic parameters, the simulation models the Rockwell Bank-To-Turn autopilot, sensor (seeker and accelerometer) dynamics, has a non-standard atmosphere, and mass modelofthe missile to calculate time-varying momentsofinertia and the missile specific acceleration from the time varying rocket motor. Figure 7.2 presents the engagement geometry and someofthe variables used to define the initial conditions.Missiley,ZTarget G ax D ..... U9 ;./\j! 9Figure 7.2 Engagement Geometry

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78The simulated targetisa three (3) dimensional, nine (9)"g"maneuvering target. Initially, the target trajectoryisa straight line. Once the range from the missile to targetisless than 6000 feet, the target initiates an instantaneous 9"g"evasive maneuver in a plane determined by the target roll angle, an input parameter.Ifthe launch rangeiswithin 6000 feet,the evasive maneuver begins immediately. Thereisa.4second "safety" delay between missile launch and autopilot control authority. Trajectory Parameters The performanceofthe control laws was measured with and without sensor noise using continuous and sampled data measurements. The integration step was .005 seconds and the measurement step for the Extended Kalman Filter was.05seconds. The trajectory presented for comparison has an initial offset angleof40 degrees (tPg) and180degree aspect (tPa), and a target rollof90 degrees away from the missile. This angleoffand target maneuverisoneofthe most demanding intercept for a preferred axis missile since it must roll through 90 degrees before the preferred axisisaligned with the target. Other intercepts were run with different conditions and target maneuvers to verify the robustnessofReachable Set Control and the miss distances were similar or less that this trajectory. Results Deterministic Results These results are the best comparisonofcontrol concepts since both Linear Optimal Control and Reachable Set Control are based on assumedCertainty Equivalence.

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79Representative deterministic results are presented in Table7.1and Figure 7.3. Figures A.l throughA.9present relevant parameters for the 4000 foot deterministic trajectories. Table7.1Deterministic Control Law Performance Initial Control Time Miss Range Distance (feet) (sec) (feet) 5500 Baseline 2.34 8 Reachable 2.34 6 5000 Baseline2.2113Reachable2.21104800 Baseline 2.1715Reachable 2.17 4 4600 Baseline 2.13 29 Reachable 2.13 6 4400 Baseline 2.0638Reachable 2.08 7 4200 Baseline 2.0235Reachable 2.05 5 4000 Baseline1.9854Reachable 2.00133900 Baseline1.9843Reachable1.988 3800 Baseline1.9840 Reachable1.988 3700 Baseline 2.02 44 Reachable 1.99103600 Baseline 1.99136Reachable 1.9965

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80MissDistance(feet) BaselineGuidanceLaw ............./ SetControl140 f-120 100 80 f-60-40 20 e-j500 40004500Initial Range50005500Figure7.3Deterministic Results An analysisoftrajectory parameters revealed that oneofthe major performance limitations was the Rockwell autopilot. Designed for proportional navigation with noisy (analog) seeker angle rates, the self adaptive loops in the autopilot penalized a high gain control law suchasReachable Set Control. This penalty prevented Reachable Set Control from demonstrating quicker intercepts and periodic control that were seen with a perfect autopilot on a similar simulation used during the research. However, even with the autopilot penalty, Reachable Set Controlwasabletosignificantly improve missile performance near the inner launchboundary. This verifies the theoretical analysis, since thisisthe region where the target set errors, control constraints, and short run times affect the linear law most significantly.

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81Stochastic Results Stochastic performance was determined by 100 runsateach initial condition.Atthe terminationoftherun,the miss distanceandTimeofFlight (TOF) was recorded. During eachofthese runs, the estimatorandseeker (noise)errorsequences were tracked. Both sequences were analyzed to insure gaussian seeker noise, and an unbiased estimator (with respect to each axis).Fromthe final performance data, the meanandvarianceofthe miss distance was calculated. Also, from the estimator and seeker sequences, the root mean square (RMS)errorand varianceforeach run was determined toidentifysome general characteristicsofthe process. The averageofthese numbersispresented. Care must be taken in interpreting these numbers. Since the measurement errorisafunctionofthe trajectoryaswell as instantaneoustrajectoryparameters, a singlenumberisnot adequate to completely describe the total process. Table 7.2andFigure 7.4 present average results using the guidance laws with noisy measurementsandthe Kalman filter.MissDistance(feet)250 200150 100BaselineGuidanceLaw..,..' ., .....50 ReachableSetControl S500 40004500Initial Range50005500Figure 7.4 Stochastic Results

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82 Table 7.2 Stochastic Control Law Performance Initial Control Time Miss Distance RMS Error Range Mean VarianceEKFSeeker (feet) (sec) (feet) (feet)(deg) 5500 Baseline 2.38 273 11470111.3Reachable2.41832677111.55000 Baseline 2.23 240 7874111.4Reachable 2.25 89 3937101.64800 Baseline 2.181937540101.4Reachable 2.191143708101.64600 Baseline 2.13 172 5699101.4Reachable 2.131071632101.54400 Baseline 2.081294324101.5Reachable 2.07 851421101.64200 Baseline 2.041233375101.6Reachable 2.03 62 673101.74000 Baseline2.011052745101.7Reachable2.01661401101.83900 Baseline 2.001053637101.8Reachable 2.00794356101.83800 Baseline 2.001245252101.8Reachable 1.9910510217102.0 3700 Baseline1.981595240101.8Reachable1.9817613078 91.83600 Baseline1.95230 6182101.7Reachable1.95239 14808101.7

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83The first runs made with Reachable Set Control were notasgoodasthe results presented. Reachable Set Control was only slightly (10to20feet) superiortothe baseline guidance law and was well below expectations. Yet, the performanceofthe filter with respect to position error was reasonable, manyofthe individual runs had miss distances near20feet,andmostofthe errors were in the Z axis. Analyzing several trajectories from various initial conditions ledtotwo main conclusions. First, the initial and terminal seeker errors were quite large, especially comparedtothe constant 5 mrad tracking accuracy assumed by many studies [2,3,18]. Second, the non-linear coupled natureofthe preferred axis missile, combined with range dependent seeker errors, and the system (target) model, makes the terminal performance a strong functionofthe particular sequenceofseeker errors. For example, Figure 7.5 compares the actual and estimated Z axis velocity (Target Missile) from a single 4000 foot run. The very first elevation measurement generated a14foot Z axis position error. A reasonable number considering the range. The Z axis velocity error, however, was quite large, 409 feet per second, and never completely eliminated by the filter. Recalling that the target velocityis969 feet per second,isapproximately co-altitude with the missile and maneuvers primarily in the XY plane, this errorissignificant when compared to the actual Z axis velocity(2feet per second). Also, thisisthe axis that defines the rollangle error and, consequently, roll rateofthe missile. Errorsofthis magnitude cause the primary maneuver planeofthe missile to roll away from the target limiting (via the constraints) the abilityofthe missile to maneuver. Further investigation confirmed that the filter was working properly. Although the time varying noise prevents a direct comparison for an entire trajectory, these large velocity errors are consistent with the covariance ratios

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84 in [41]. The filter model was developed to track maneuvering targets. Thepen-alty for tracking maneuvering targetsisthe inability to precisely define allofthe trajectory parameters (ie. velocity). More accurate (certain) models track better,butrisk losing track (diverging) when the target maneuvers unexpectedly. The problem with the control then, was the excessive deviations in the velocity. To verify this, the simulation was modifiedtouse estimatesofposition,buttouseactual velocities. Figure 7.6 and Table7.3has these results.Asseen from the table, the control performanceisquite good considering the noise statistics and autopilot.ZAxis Velocity(feet/sec)600400200 Actual, 0-200 -400 -6000 0.5 11 .5 2TimeFigure7.5MeasuredvsActual Z Axis Velocity

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85 Table 7.5 Stochastic ControlLawPerformance Using Actual Velocities Initial Control Time Miss Distance RMSErrorRange Mean VarianceEKFSeeker (feet) (sec) (feet) (feet) (deg) 5500 Baseline 2.35 34 318101.5 Reachable 2.3738504II1.5 5000 Baseline 2.21 50 506101.7Reachable 2.2338623101.6 4800 Baseline 2.1755367101.6 Reachable 2.19 48 526101.54600 Baseline 2.1261366101.6 Reachable 2.1452326101.6 4400 Baseline 2.06 62 333101.7 Reachable 2.1045415101.7 4200 Baseline 2.02 60 277101.7 Reachable 2.06 44 463101.84000 Baseline 1.98 62 330101.9 Reachable 2.0253lOll9 1.9 3900 Baseline 1.9855436101.9 Reachable 2.00 64 2052101.9 3800 Baseline 1.98 53 400 9 2.0 Reachable 1.99912427101.8 3700 Baseline 1.99 63 235102.0 Reachable 1.991403110101.7 3600 Baseline 1.98138354101.8 Reachable 1.96 213 4700101.6

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86MissDistance(feet)25020015010050 ,. ReachableSetControlBaseline GUidance/ ................................................... -------_...------9500 40004500InitialRange5000 5500Figure 7.6 Performance Using Position Estimates and Actual Velocities While the long term solution to the problemisa better target model that will accommodate both tracking and control requirements, the same model was used in order to provide a better comparison with previous research. For the same reason, the Kalman filter was not tuned to function better with the higher gain reachable set controller. However, target velocity changes used for the generationofthe roll angle error were limited to the equivalentofa 20 degree per second target turn rate. This limited the performance on a single run, but precluded the 300 foot miss that followed a 20 foot hit.

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87Conclusions Reachable Set ControlAsseen from the Tables7.1and 7.2, Reachable Set Controlwasinherently more accurate than the baseline linear law, especiallyin the more realistic case where noiseisadded and sampled data measurements are used. In addition, Reachable set control did insure an initially constrained trajectory for controllable trajectories, and required minimal accelerations duringthe terminal phaseofthe intercept. Unlike previous constrained control schemes, Reachable Set Control was better able to accommodate tinmodeled non-linearities and provide adequate performance with a suboptimal sampled data controller. While demonstrated with a Preferred Axis Missile, Reachable Set Controlisa general technique that could be used on most trajectory control problems. Singer Model Unmodified, the Singer model provides an excellent basis for a maneuver ing target tracker, but itisnot a good model to use for linear control. Neither control law penalized velocity errors. In fact, the baseline control law did not define a velocity error. Yet the requirementsoflinearity, and the integration from velocity to position, require better velocities estimates than are provided by this model (for this quality seeker).

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APPENDIXASIMULAnONRESULTS

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YPosition(feet)40003000 20001000o0 13 footmiss 100020003000XPosition(feet)FigureA.lXYMissile&Target Positions Reachable Set Control4000YPosition(feet)40003000 200054 feet 1000Behind Targeto0100020003000XPosition(feet)Figure A.2XYMissile&Target Positions Baseline Control Law894000

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90

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91Acceleration(g)10050 -o-50 f-t >1009CommandedAcceleration-100o0.51Time1 .52FigureA.SMissile Acceleration Reachable Set ControlAcceleration(g)10050o-50 I>1009CommandedAcceleration-100o0.51Time1 .52Figure A.6 Missile Acceleration Baseline Control Law

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92Roll Rate (deg/sec)6004002000-200-400 -60000.51Time...1.52Figure A.7 Missile Roll Commands&Rate Reachable Set ControlActualRoIIRate(deg/sec)600400200o-200 -400Commanded '.:-600o0.51Time1 .52Figure A.8 Missile Roll Commands&Rate Baseline Control Law

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93Roll AngleError(deg)50o-50o0.51Time1 .52Figure A.9 Missile Roll Angle Error

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APPENDIX B SAMPLED-DA TA CONVERSION System Model The system model for the formulationofthe sampled-data control law consistsofthe plant dynamics with time correlated and white disturbances: [ ]=[F(t) ][X(t)]+[GO(t)] u(t)+T(t) 0 N(t) T(t) the reference: r(t)=A(t)r(t)+B(t)a(t) with the tracking error:[ L0][WS(t) ](1)OMWt(t)(2)e(t)=[yx(t) -Yr(t)]=[H(t)I0I-C(t) ] )(3)r(t)andquadratic cost Js(t)=E{Jd(t)II(t)}, where(4)E{ws(t)ws(t)T}=Ws(t)isthe strengthofthe system (white noise) disturbances to be rejected. E{wt(t)Wt(t)T}=Wt(t) is an input to a stationary first order Gauss-Markov process that models target acceleration (see CH VI). The following assumes a constant cycle time that defines the sampling interval and a possible delay between sampling and controlof =ti' ti94

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9SThecomponentsofthe controlled system arex(t)=x(t)y(t)Z(t)0(t)x(t)T(t)= [::::: ]Tz(t) r(t)(5)o0u(t)y(t) z(t)0(t)Nx(t)Ny(t)Nz(t)P(t)e(t)x(t)y(t) z(t)0(t)ry(t) rz(t)r0(t)rx(t) ry(t) rz(t)r0(t)o0o0In block form, withappropriatedimensions, the system matrices are F(t) = A(t) =F=[0I]OwG(t)=B(t)=G= (6)H(t)=C(t)=[Ihw ]i=j=8 otherwise{+wi=8,j=4(Iw)ij=I otherwise L(t)=I(hw)jj={ M(t)=Ii=j=8 otherwise

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96Sampled-dataEquationsSystemFromthecontinuoussystem,thediscrete-timesampled-datasystemmodelcanbesummarizedinthefollowingsetofequations:y(ti')=H x(ti)+D u(ti)ForatimeinvariantFwithaconstantsampleinterval: (t .t)-eF(t..-1.)eF(.lt) with .lt -ttxHI'1-HI1-HI-1I000 .lt 00 0 0I00 0 .ltO 0 00I0 0 0 .ltO 0 00I0 0 0 x 48x(.lt) =0 000I000 0 0000I00 0 000 00I0 0 000 000 x 88 with x 48(.lt) = (l!w(.lt)})x 88(.lt) =exp{ -w(.lt)} (7) (8)

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97Also, since G and H are time invariantandthe u(ti) are piecewise continuous, theinputandoutputmatrices (allowing fornon-coincidentsamplingandcontrol) arefi+lG= cP(ti+l,r)G(r)dr (9)t.1H=H cP(ti',ti) (10) r D=H (11)t.1Therefore, At 2j2 00 0 0 At2j2 0 0 00 At 2j2 0 0 00 At-cPx4S(At) G= At 0 0 0 (12) 0 At 0 0 0 0 At 0 0 0 0 l-cPxSS(At) I00 0 At' 000 0I00 0 At' 00H=00I00 0 At' 0(13)000I0 0o cPX4S(At') 00 000 000 0 0 0 00 000 0 0 0 00 000 00 0 00 0o cPXSS(At')At,2j2 00 0 0 At,2 j 2 0 0 00 At,2j2 0 0 00 At' -cPX4S(At') D= At' 00 0 (14) 0 At' 0 0 0 0 At' 0 0 0 0 l-cPxSS(At')

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98Target DisturbanceFortime correlated disturbances, Wt(t)=Wt=2 at2 ITc where at is the RMS valueofthe noise process T(,.), and T c is the correlation time.N(t)=-(1/Tc)[I]The sampled data disturbanceis(15) (16)Wt(ti)isa sequenceofzero mean mutually uncorrelated random variables, and is givenby:with =T c(l the sampled data impact on the system.(17)

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100 C=C
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APPENDIX C SAMPLED DATA COST FUNCTIONS Assume that the performance objectiveisto minimize an appropriate continuous-time quadratic cost.(1)where J sisthe stochastic cost, Jd a deterministic cost, and I(t)isthe information set available at time t. Let Jd(t)=erT Prer +rr(e(t)TQ(t)e(')+u(t)TR(t)u(tdT to (2)Dividing the intervalofinterest into N+l control intervals for discrete time control, Jd(t)=e(tn+l)TPfe(tn+l)N Jti+1 + [t{e(t)TQ(t)e(t) + u(t)TR(t)u(t)}dr]1whereforall tE [tj,ti+l)' u(t)=u(ti). Substituting the following for e(t):(3)e(t)= + t.1+Jt ti101(4)

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102thedeterministiccost becomes: (5)+N E [i=OJti+1 [ [4>( t,ti)e(ti)tI+{Jt 4>(t,r)G(r)dr}u(ti) t.1+Jt {4>(t,r)L(r)dr}w(ti)]T t.1+{It 4>(t,r)G(r)dr}u(ti) t.I+{It 4>(t,r)L(r)dr}w(ti)] t.1Makingthefollowingsubstitutions:(6){It 4>(t,r)G(r)dr}T t.1Q(t) {Jr 4>(r,ti)G(r)dr}]dt .t1(7)J'i+l r'l;(t,T)L(T)dT} TQ(t) Www(ti)=[{t.t.I 1.{r 'l;(t,T)L(T)dT}]dt t.1 f+l4>(t,ti) T 1: Wxu(ti)=[Q(t){ 4>(t,r)G(r)dr}]dt t.1 1(8) (9)

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103fi+1J: Wxw(ti)=[ t.1iWWX(ti)=[f+'( f T t.t.1 1WUW(ti)=[fi+1( f T Q(t)t.t.1 1( f t.1WWU(ti)=[fi+'( f T Q(t)t.t.1 1( f t.1And,allowingforthefactthatforall tE[ti,ti+l):u(t)=u(ti) e(t)=e(ti) w(t)=w(ti)thedeterministiccostcanbe expressed as:Jd=e(tN+ 1)1r Pfe(tN+l)+N E (e(ti) 1r Wxx(ti)e(ti) + u(ti) 1r Wuu(ti)u(ti)i=O+ W(ti)1rWWW(ti)W(ti) + 2e(ti)1rWXU(ti)U(ti) + 2e(ti)1rWXW(ti)W(ti) +2u(ti)1rWUW(ti)W(ti)}(10) (11) (12) (13) (14) (15) (16)

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104RecallthatJs(t)=E{Jd(t)II(t)}andthatw(.,.) is zeromeananduncorrelatedwitheithereoru. Sinceiftwo variables x&yareuncorrelatedthenE{x,y}=E{x}E{y}.Therefore,E{2e(ti)T Wxw (ti)W(ti)}=2E{e(ti)T w (ti)}E{w(ti)}=0 E{2u(ti)TWxw(ti)W(ti)}=2E{u(ti)T w (ti)}E{w(ti)}=0and(17)Jt t.1(18)Jti+1=tr[Q(t){t.1Jt s(r)L(r) T Tdr}dt)t.1(19)Consequently,forconsiderationinJ sJd=e(tN+l)TPf e(tN+l)N+ E (e(ti)TWxx(ti)e(ti) + U(ti)TWuu(ti)U(ti) +Jw(ti)1=0(20)Wxu(ti) Wuu(ti) (21)

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105Forsmall sample times, the weighting functionscanbeapproximatedby:WxxQ(t).M Wxu(I/2)Q(t)G(t)(.6.t)2 Wuu-[R(t)+( I13)G( t) T Q(t)G(t)(.6.t)2](.6.t) (22)(23)(24)Inorderto relate the valuesofthe sampleddataweighting terms on the systemerrorto the state variables (using discrete variable notation), let: P(ti)=[H 0 -C]Tpr[ H0-C](25)Q(ti)=[H 0-C]T Wxx(ti)[ H0-C](26)R(ti)=Wuu (27) S(ti)=[H 0-C]T Wxu (28)Therefore,withthe augmented state variableX=[xTr ]T:ExpandingQandST](29)[ -CTWXX(ti)Hoo o ] CTWXX(ti)C(30)

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106With non-coincident sampling and control, the penalty terms are modifiedasfollows: e(ti')=[H0-C] +rti' This modification adds additional terms:[DE][U]ati'(31 ) (32){[HOC] +[DE][U]}T.r ti' a ti'C] [DE][U]}r ti' a ti' AgainwithWxx(ti)=Wxx(ti') and Wxu(ti')=Wxu(ti), the additional terms can be grouped withRandS to generate:R(ti')=R(ti)+DTWxx(ti)D+DTWxu(ti) S(ti')=S(ti)+DTWxx(ti)[ H0-C](33) (34)

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APPENDIXDLQGCONTROLLERDECOMPOSITIONThecombinationofthesystem, targetdisturbance,andthereference, result in a19state controller.However,thedecoupledstructure,symmetry,andthe zerosinthe controlinputandcost matricescanbeexploitedto streamline the calculations. In general, theoptimalsolution to theLQGtrackercanbe expressed as: whereand G*(ti) =[R(ti) +GT(ti)P(ti+l)G(ti)rl [GT(ti)P(ti+l)cI>(ti+l,ti) + ST(ti)] P(ti)=Q(ti) + cI>T(ti+}.ti)P(ti+l)cI>(ti+l,ti) [GT(ti)P(ti+l)cI>(ti+}.ti) + ST(ti)]TG*(ti)(I)(2)(3)Toreduce thenumberofcalculations,partitionthe gainandRiccatiequationsuchthat: Evaluating terms:(4)GT (ti)P( ti+ 1 )G(til=[ =[=[GT10I0][PIIPl2P13 ][G]P21P22 P23 0P31P32 P33 0GTPIIIGTPI2IGTP13J [g] GTPIIG]107(5)

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108Therefore,Now consider, GT (ti)P(ti+ 1 I ,ti) 0=[GT1010][PIIP12P13j 00P21P22 P230 0P31P32 P3300 0=[GTPIIIGTPl21GTP13 ] 00 0 0 00 (6) (7)[ I + I oFromAppendix C, S=[H0-C]TWxu Substituting, the required terms for the gain computation become:,ti) + ST(ti))= + WxuTHI + o(8)Consequently, with(9)the optimal control can be expressedas:G1(ti)=GI + WxuTH] G2(tj)=GI + 0G3(ti)=GI WxuTC](10)(11)(12)

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109Partitioningequation3:[PHPI2Pl3] [ 0-HTWXX('i)C](13) P21 P22 P23=0 (ti) P31 P32 P33(ti)-CTWXX(ti)H 0 4>n 0 T 4>n 0 4>x [ PI IP12Pl31 4>x 0 0 0 0 + 0 4>T 0 P21 P22P23 0 4>T 0 0 0 4>r P31 P32P33 (ti+ 1) 0 0 4>r4>n 0 -([ GTI0101[PIIPI2Pl3] 4>x 0 0P2IP22 P23 0 4>T 0P3IP32 P33ti+I)0 0 4>r +[WxuTHI01-WxuTC]}T [GIlG21G3]=[ -CTWxXCti)Ho oo ] CTWXX(ti)C (14)+ {4>xTPll}{[4>n TOJPll+ 4>xTP2t>{4>rTP3t>{4>x TPI2} {4>x TPI3} {[4>n T OJPI2+4>TTP22}{[4>n T 0]P13+4> T T P23 }{4>rTP32}{4>rTP33}[;X4>n )('i+l)0 4>T 0 GTPI2I GTPl31 [;x4>nil -{[GTPIII0 4>T 0 + [WxuTHI0I-WxuTCnT [GIIG21G3]

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= -cT Wxx(ti)H 110 g oCTWXX(ti)C (15) o + o o TH}T [GIIG21G3]o (ti+l)ForthepropagationoftheRiccatiequationonly these termsarerequiredtogeneratethecontrol gain:Pll(ti)=HTWxx(tj)H + TH}T][GIlPI2(ti)= + o-[[G2] P13(ti)=-HTWxx(ti)C + -(16)(17)(18)

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APPENDIX E CONTROLLER ANDFILTERPARAMETERS Controller Control Delay=.21seconds Target Maneuver Correlation time=.21seconds Riccati initialization (Pf) Linear controller::Pf=IE+2 for sample times<.01secondPf=lEIfor sampletimes>.01second Roll rate controller:Pf=1.0Quadratic Cost Terms (Continuous) Linear Accelerations Q=320.R=l.Roll Control (Angle) (Rate)QIlQ22RHI GAIN IE+lIE-4lE-3LOGAINlE-4lE-3Sample Time=integration (not sample) stepforthe continuous-discrete filter System Disturbance Input=(AT AM) (Tf-TO)2/ 2.III

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112Filter Target Correlation Time = 2.0 seconds Riccati initialization (Pf)Positions:PII= 2500 Velocities: P44 = 2.0E6 Covariance:Pl4= 5.0E4 Maneuver Excitation Matrix (2a2) = 5120000 Seeker Measurement Noise Azimuth&Elevation R= (SITH*SITH/(RNG HAT*RNGHA T)+SOTH*SOTH +SITHI*SITHI*RNG HAT**4)/MEAS ST Range R= (SOR*SOR+SIR*SIR*RNG HAT**4)/MEAS ST Range Rate R= (SODR*SODR+SIDR*SIDR*(RNG HAT**4) )/MEAS ST With SITH = SIPH =1.5SOTH = SOPH = .225E-4 SITHI= SIPHI= 0.0 SOR = SODR = 3.0 SIR =IE-8SIDR = .2E-IO

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LISTOFREFERENCES [1] A. Arrow, "StatusandConcerns for Bank-To-TurnControlofTactical Missiles," GACIACPR-85-01,"Proceedingsofthe Workshop onBank-To-TurnControlled Terminal Homing Missiles," Vol1,Joint ServiceGuidanceandControl Committee, January 1985. [2] J.R. McClendon and P.L. Verges, "ApplicationsofModern Control and Estimation Theory to the GuidanceandControlofTacticalAir-to-AirMissiles," Technical ReportRG-81-20,"Research onFurureArmyModular Missile," USArmyMissile Command, Redstone Arsenal, Alabama, March 1981. [3] N.K. Gupta, J.W. Fuller,andT.L. Riggs, "Modern ControlTheoryMethodsforAdvanced Missile Guidance," Technical ReportRG-81-20,"Research onFurureArmy Modular Missile," USArmyMissile Command, Redstone Arsenal, Alabama, March 1981. [4] N.B. Nedeljkovic, "New Algorithms for Unconstrained Nonlinear Optimal Control Problems," IEEE Transactions on Automatic Control, Vol.AC-26,No.4,pp868-884, August 1981. [5] W.T. Baumann and W.J.Rugh,"Feedback ControlofNonlinear Systems byExtendedLinearization," IEEE Transactions on Automatic Control, Vol.AC-31,No.1,pp. 40-46, January 1986. [6] E.D. Sontag, "Controllability and Linearized Regulation,"DepartmentofMathematics, Rutgers University, New Brunswick, NJ (unpublished),14February 1097.[7]A.E. Bryson and Y.C. Ho, Applied Optimal Control, Blaisdell Publishing Company, Waltham, Massachusetts, 1969. [8] Y.-S. Lim, "LinearizationandOptimizationofStochastic Systems with Bounded Control," IEEE Transactions on Automatic Control, Vol.AC-15,No.1,pp49-52, February 1970. [9] P.-O.GutmanandP.Hagander,"ANew DesignofConstrained ControllersforLinearSystems," IEEE Transactions on Automatic Control, Vol.AC-30,No. 1,pp22-33, January 1985. [10] R.L. Kousut, "Suboptimal ControlofLinearTime-InvariantSystems Subject to Control Structure Constraints," IEEE TransactionsonAutomatic Control, Vol.AC-15,No.5,pp 557-562, October 1970.113

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114[11] D.J. Caughlin, "Bank-To-TurnControl," Master's Thesis, UniversityofFlorida, 1983. [12] J.F. Frankena and R. Sivan,"Anon-linear optimal control law for linear systems," INT. J. CONTROL, Vol 30, NoI,pp159-178, 1979. [13] P.S. Maybeck, Stochastic Models. Estimation. and Control, Volume 3, Academic Press, New York, 1982. [14] S.A. Murtaugh and H.E. Criel, "FundamentalsofProportional Navigation," IEEESpectrum, pp.75-85, December 1966. [15] L.A. Stockum, and I.C. Weimer, "Optimal and Suboptimal Guidance for a Short Range Homing Missile," IEEE Trans. on Aerospace and Electronic Systems, Vol. AES-12,No.3,pp 355-361, May 1976. [16]B.Stridhar, and N.K. Gupta, "Missile Guidance Laws Based on Singular Perturbation Methodology," AIAA JournalofGuidance and Control, Vol 3,No.2,1980. [17] R.K. Aggarwal and C.R. Moore, "Near-Optimal Guidance LawforaBank-To-TurnMissile," Proceedings 1984 American Control Conference, Volume 3, pp. 1408-1415, June 1984. [18] P.H. Fiske, "Advanced Digital Guidance and Control Concepts forAir-To-AirTactical Missiles,"AFATL-TR-77-130,Air Force Armament Laboratory, United States Air Force, Eglin Air Force Base, Florida, January 1980. [19] USAF Test Pilot School, "Stability and Control Flight Test Theory,"AFFTC-77-I,revised February 1977. [20] L.C. Kramer andM.Athans, "On the ApplicationofDeterministic Optimization MethodstoStochastic Control Problems," IEEE Transactions on Automatic Control, Vol. AC-19,No.I,pp22-30, February 1974. [21]Y.Bar-Shalom and E. Tse, "DualEffect,Certainty Equivalence, and Separation in Stochastic Control," IEEE Transactions on Automatic Control, Vol. AC-19,No.5,pp 494-500, October 1974. [22] H. Van DE Water and J.C. Willems, "The Certainty Equivalence Property in Stochastic Control Theory," IEEE Transactions on Automatic Control, Vol. AC-26,No.5,pp1080-1086, October 1981. [23] "Bank-To-TurnConfiguration Aerodynamic Analysis Report" Rockwell International Report No. C77-142Ij034C, date unknown.

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115[24] D.E Williams andB.Friedland, "DesignofAn AutopilotforBank-To-TurnHoming Missile Using Modern Control and Estimation Theory," ProcFifthMeetingofthe Coordinating Group on Modern Control Theory, 15-27 October 1983, (Picatinny) Dover, New Jersey, October 1983, pp. 397-419. [25]B.Friedland,etaI., "On the "Adiabatic ApproximationforDesignofControl Laws for Linear, Time-Varying Systems,"IEEETransactions on Automatic Control, Vol.AC-32,No.1,pp. 62-63, January 1987. [26] D.W. Tufts and D.A. Shnidman, "Optimum Waveforms Subject to Both Energy and Peak-Value Constraints," Proceedingsofthe IEEE, September 1964. [27] M. Pontier and J. Szpirglas, "Linear Stochastic Control with Constraints, IEEE Transactions on Automatic Control, Vol.AC-29,No. 12, pp 1100-1103, December 1984. [28] P.-O.GutmanandS.Gutman,"ANote on the ControlofUncertain Linear Dynamical Systems with Constrained Control Input," IEEE Transactions on Automatic Control, Vol. AC-30,No.5,pp 484-486, May 1985. [29]M.W.Spong, J.S. Thorp, and J.M. Kleinwaks, "The ControlofRobot Manipulators withBounded Input," IEEE TransactionsonAutomatic Control, Vol. AC-31,No.6,pp 483-489, June 1986. [30]D.Feng and B.H. Krogh, "Acceleration-Constrained Time Optimal Control in n Dimensions," IEEE Transactions on Automatic Control, Vol. AC-31, No. 10, pp 955-958, October 1986. [31] B.R. BarmishandW.E. Schmitendorf, "New Results on ControllabilityofSystemsofthe Form x(t)=A(t)x(t)+F(t,u(t,"IEEETransactions on Automatic Control, Vol. AC-25,No.3,pp 540-547, June 1980. [32] W.-G. HwangandW.E. Schmitendorf, "Controllability Results for Systems with a Nonconvex Target," IEEE TransactionsonAutomatic Control, Vol. AC-29,No.9,pp 794-802, September 1984. [33] T. Kaliath, Linear Systems, Prentice-Hall, Inc., Englewood Cliffs, New Jersey, 1980. [34] M. Vidyasager, Nonlinear Systems Analysis, Prentice-Hall, Englewood Cliffs, New Jersey, 1978. [35] K. Zhou, and P. Khargonekar, "Stability Robustness Bounds for Linear State-Space Models with Structured Uncertainity," Transactions on Automatic Control, Vol.AC-32,No.7,pp 621-623, July 1987. [36] K.G. Shin and N.D. McKay, "Minimum Time ControlofRobotic Manipulators with Geometric Path Constraints," IEEE TransactionsonAutomatic Control, Vol. AC-30,No.6,pp 531-541, June 1985.

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116[37] A. Sabari and H. Khalil. "Stabilization and RegulationofNonlinear Singularly Perturbed Systems Composite Control," IEEE Transactions on Automatic Control, Vol.AC-30,No.8,pp. 739-747, August 1985. [38] M. Sampei and K.Furuta,"On Time Scaling for Nonlinear Systems: Application to Linearization." IEEE Transactions on Automatic Control, Vol.AC-31.No.5,pp 459-462, May 1986. [39] I.J. Ha and E.G. Gilbert."AComplete CharacterizationofDecoupling Control Laws for a General ClassofNonlinear Systems," IEEE Transactions on Automatic Control, Vol.AC-31,No.9,pp. 823-830. September 1986. [40] H. K wakernaak and R. Sivan, Linear Optimal Control Systems, Wiley-Interscience. New York, 1972. [41] R.A. Singer. "Estimating Optimal Tracking Filter Performance for Manned Maneuvering Targets," IEEE Transactions on Aerospace and Electronic Systems, Vol. AES6,No.4,pp 473-483, July 1970. [42] M.E Warren and T.E. Bullock, "DevelopmentandComparisonofOptimal Filter Techniques with Application toAir-to-AirMissiles," Electrical Engineering Department. UniversityofFlorida, Prepared for the Air Force Armament Laboratory, Eglin Air Force Base, Florida, March 1980.

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BIOGRAPHICAL SKETCH Donald J. Caughlin, Jr., was born in San Pedro, California on17Dec. 1946. He graduated from the United States Air Force Academy, earning aB.S.in physics, and chose pilot training insteadofan Atomic Energy Commission Fellowship. Since then he has flown over 3100 hours in over 60differentaircraft and completed one tour in Southeast Asia flying theA-ISkyraider. A Distinguished Graduateofthe United StatesAirForce Test Pilot School, Don has spent muchofhis career in research, development,andtest at both major test facilities--Eglin AFB in Florida, and Edwards AFBinCalifornia. Don Caughlinisa Lieutenant Colonel in theUnitedStatesAirForcecurrently assigned as the Assistant for SeniorOfficerManagement at HeadquartersAirForce Systems Command. In addition to theB.S.in physics,Lt.Colonel Caughlin has an M.B.A. from the UniversityofUtah, and a masters degreeinelectrical engineering from the UniversityofFlorida. Heisa memberofthe SocietyofExperimental Test Pilots and IEEE. Lt. Colonel Caughlinismarried to theformerBarbara SchultzofMontgomery, Alabama. They have two children, a daughter, Amy Marie, age eight, and Jon Andrew, age four.117