• TABLE OF CONTENTS
HIDE
 Copyright
 Title Page
 Dedication
 Acknowledgement
 Table of Contents
 List of Figures
 Key to symbols
 Abstract
 Introduction
 Background
 Constrained control
 Constrained control with unmodeled...
 Reachable set control example
 Reachable set control for preferred...
 Results and discussion
 Appendix A. Simulation results
 Appendix B. Sampled-data conve...
 Appendix C. Sampled data cost...
 Appendix D. LQG controller...
 Appendix E. Controller paramet...
 List of references
 Biographical sketch






Title: Reachable set control for preferred axis homing missiles
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Title: Reachable set control for preferred axis homing missiles
Series Title: Reachable set control for preferred axis homing missiles
Physical Description: Book
Creator: Caughlin, Donald J.,
Publisher: Donald J. Caughlin
Place of Publication: Gainesville, Fla.
Publication Date: 1988
Copyright Date: 1988
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Bibliographic ID: UF00082279
Volume ID: VID00001
Source Institution: University of Florida
Holding Location: University of Florida
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Resource Identifier: oclc - 19761693
alephbibnum - 001102747

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Table of Contents
    Copyright
        Copyright
    Title Page
        Page i
        Page ii
    Dedication
        Page iii
    Acknowledgement
        Page iv
    Table of Contents
        Page v
        Page vi
    List of Figures
        Page vii
        Page viii
    Key to symbols
        Page ix
        Page x
        Page xi
        Page xii
        Page xiii
    Abstract
        Page xiv
        Page xv
    Introduction
        Page 1
        Page 2
        Page 3
    Background
        Page 4
        Page 5
        Page 6
        Page 7
        Page 8
        Page 9
        Page 10
        Page 11
        Page 12
    Constrained control
        Page 13
        Page 14
        Page 15
        Page 16
        Page 17
        Page 18
        Page 19
        Page 20
        Page 21
        Page 22
        Page 23
        Page 24
    Constrained control with unmodeled setpoint and plant variations
        Page 25
        Page 26
        Page 27
        Page 28
        Page 29
        Page 30
        Page 31
        Page 32
        Page 33
        Page 34
        Page 35
        Page 36
        Page 37
        Page 38
        Page 39
        Page 40
    Reachable set control example
        Page 41
        Page 42
        Page 43
        Page 44
        Page 45
        Page 46
        Page 47
        Page 48
        Page 49
        Page 50
        Page 51
        Page 52
        Page 53
        Page 54
    Reachable set control for preferred axis homing missiles
        Page 55
        Page 56
        Page 57
        Page 58
        Page 59
        Page 60
        Page 61
        Page 62
        Page 63
        Page 64
        Page 65
        Page 66
        Page 67
        Page 68
        Page 69
        Page 70
        Page 71
        Page 72
        Page 73
        Page 74
        Page 75
    Results and discussion
        Page 76
        Page 77
        Page 78
        Page 79
        Page 80
        Page 81
        Page 82
        Page 83
        Page 84
        Page 85
        Page 86
        Page 87
    Appendix A. Simulation results
        Page 88
        Page 89
        Page 90
        Page 91
        Page 92
        Page 93
    Appendix B. Sampled-data conversion
        Page 94
        Page 95
        Page 96
        Page 97
        Page 98
        Page 99
        Page 100
    Appendix C. Sampled data cost functions
        Page 101
        Page 102
        Page 103
        Page 104
        Page 105
        Page 106
    Appendix D. LQG controller decomposition
        Page 107
        Page 108
        Page 109
        Page 110
    Appendix E. Controller parameters
        Page 111
        Page 112
    List of references
        Page 113
        Page 114
        Page 115
        Page 116
    Biographical sketch
        Page 117
Full Text





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AUTHOR: Caughlin, Donald
TITLE: Reachable Set Control for Preferred Axis Homing Missles (record
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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




































Copyright 1988

By

DONALD J. CAUGHLIN JR.


















To Barbara

Amy

Jon















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.
















TABLE OF CONTENTS

ACK N OW LEDG M ENTS.................................................................................................... iv

LIST OF FIG U RES............................................................................................................... vii

K EY TO SYM BOLS............................................................................................................. ix

ABSTR A CT......................................................................................................................... xiv

CHAPTER

I INTRODU CTION ....................................................................................................... 1

II BA CK G RO U N D ................................................................................................... ........ 4
M missile D ynam ics ............................................................................................... 5
Linear A ccelerations...................................................................................... 6
M om ent Equations....................................................................................... 6
Linear Q uadratic G aussian Control Law ............................................................. 7

III CONSTR A INED CONTROL.................................................................................... 13

IV CONSTRAINED CONTROL WITH UNMODELED SETPOINT
AN D PLA NT V A RIATIONS.................................................................................... 25
Linear Optimal Control with Uncertainty and Constraints ...........................31
Control Technique........................................................................................... 32
D discussion ........................................................................................................... 36
Procedure ........................................................................................................ 37

V REA CH A BLE SET CONTROL EX A M PLE.............................................................41
Performance Comparison Reachable Set and LQG Control...........................41
Sum m ary...........................................................................................................54

VI REACHABLE SET CONTROL FOR PREFERRED AXIS
H OM IN G 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
A pplication................................................................................................. 72



















VII RESU LTS AN D DISCU SSION ................................................................................. 76
Sim ulation....................................................................................................... 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
Sam pled Data Equations................................................................................... 96
System ........................................................................................................... 96
Target Disturbance.................................................................................... 98
M inim um Control Reference.................................................................... 99
Sum m ary.......................................................................................................... 100

C SAM PLED DATA COST FU NCTIONS................................................................... 101

D LQG CONTROLLER DECOM POSITION ...............................................................107

E CONTROLLER PARAM ETERS.............................................................................. 11
Control Law ...................................................................................................... 111
Filter................................................................................................................. 112

LIST OF REFERENCES ............................................................................................... 113

BIOG RAPHICAL SK ETCH ............................................................................................. 117















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 ..............................................................................45

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 Disturbance processes................................................... 60

6.2. Roll Angle Error Definition from Seeker Angles..................................................63

6.3. Roll Control Zones................................................................................................65

6.4 Target Missile System............................................................................................74

6.5 Command Generator/Tracker..............................................................................75

7.1 RMS Missile Acceleration .....................................................................................76

7.2 Engagement Geometry ..........................................................................................77

7.3 Deterministic Results............................................................................................80

7.4 Stochastic R esults..................................................................................................... 81

7.5 Measured vs Actual Z Axis 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 Missile Acceleration Reachable Set Control.......................................................91

A.6 Missile 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 Missile Roll Angle Error ........................................................................................93

viii
















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.

eg 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(.)




L(.)


LO

LOwt

Lu


Loc


Loc


Lq

LO

LSe


LO

LB

LB

Lp

Lr

L8a


L6r


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.

MO Stability parameter Equilibrium pitch rate.

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

Mac 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.

N3 Stability parameter Change in yaw rate due to sideslip angle.

N3 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.

NSr 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


IVtotI

X(.)

X,Y,Z

YO

YOwt

YB

YB


Yp

Yr

Ye

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









Ax Target model correlation time.

Oa Target elevation aspect angle.

Og Seeker elevation gimbal angle.

Oba Target azimuth aspect angle.

VOg Seeker azimuth gimbal angle.















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.









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.














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.









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 [11].

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 [13]. 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









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.















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 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.





















z

Figure 2.1 Missile Reference System.

4









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,









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 /IVtotI

B = R Poc + Ayb / IVtotI



Linear Accelerations


1
U= RV- QW -- (D+ DOwt)
M

DuU- Dococ- Dococ- DqQ- DO0 DSe6e + Ax/M


1
V= PW- RU + -- {YO+ YOwt)
M

+ YBB+ Y3B8+ YpP+ YrR+ YoO + YSa6a + Y6r6r


1
W= QU- PV + (LO + LOwt}
M

LuU- Lococ- Lococ- LqQ- LOO L6e6e



Moment Equations




Q = Mo/Iyy + MuU + Mcocc + M0oc + MqQ + M6e6e



+ Izz-xx) PR ( P2 R2) -x
Iyy Iyy












R NO/Izz + NBB+ NB8+ NpP+ NrR+ NSa6a + N6r6r (7)


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



P = LO/Ixx + LfBB+ Lf3B + LpP+ LrR+ L^a6a + LSr6r (8)


(Ixx-I^yy) xz
+ 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









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 [7], 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 [18]. Given the finite dimensional linear system:



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


where

x
y Ax
x = z u = Ay
Vx Az
Vy
Vz

and


F = G=
0 0 -I










with the cost functional:

tf
J=xfPfxf + t uTRudr (10)
JtO


P= R=I
[o 0]


Application of the Maximum principle results in a linear optimal control law:


3(Tgo) 3(Tgo)2 (11)
ui(t) = +(Tgo)3 xi(t) + +(Tgo)3 Vi(t)
3-" +(Tgo)3 3"1 +(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 (0). 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









10

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

vehicle must roll, define the roll angle error:



eo = tan-l{ 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 I1 << 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)
eg = (15)
(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:


oc= Q PB + Azb / IVtot (17)



B = R Poc + Ayb / IVtotI (18)


(Izz Ixx)
Q = Mocc + MqQ + MseSe + PR (19)
Iyy









11

(Iy Ixx)
R = NfBB + NrR + Nr6rr + PQ (20)
Izz


P = LpP + L6a6a (21)


where


Az = Z oc + Z6q6q (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.









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.















CHAPTER III
CONSTRAINED CONTROL

In Chapters I and II, 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 clas-

sical 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.









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 00J


is minimized. The weighting matrix Q is assumed to be positive semidefinite and

A > 0. Dynamic programming leads to Bellman's equation:


min ({ tr[LTVxx(x)L] + (Fx + Gu)TVx(x) + xTQx + Au2 = cx* (3)
Iul|
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 } A 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
z
4D(z) = (2/7r), exp { -1y2) dy
J0
for (5b): k = (2/7r) - (gTwg)-1 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 + AkkT = 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.








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:

t1
J = {((/2)llx(t)2i2Q(t)+llx(t)IIs(t)dt (12)
to
+ (1/2)llx(t1)ll2p(t)


With the constraint

Ilu(t)I|R(t) < 1

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)
A(t) = Q(t)x(t) + FT(t)A(t) A(tl) = -P1x(tl) (14)
IIx(t)||s(t)

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

(12) can be expressed as

R- (t)GT(t)A(t)
u(t) = (15)
II(t) = JR- (t)GT(t)A(t)lIR(t) (15)


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 l|xils + 0, P will be the solution of


PGR-lGTP S
P + PF + -- Q= +- FTP (18)
IIR- lGTPxIR I|xlls


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 [9,28]. 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 < ui 5 hi i = 1,...,m


where gi and hi are the control constraints. Consider an n x m matrix


L [ 11 12 I I m ] (20)
such that

Fc = Fc(L) (F + GLT) (21)


is a stability matrix.









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 = F(L) n { (eFct)-1 E } (23)
te[0,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 O(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 0 by choosing c in (24) such that D C 0 C E:

sup xTpx < c < min xTpx (25)
xeD xESE









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 ] (26)

where K is defined

ki 0 '

K= [ ki > 0, i = 1,2,...,m (27)

0 km.

and tune the parameters ki by simulations.

A sufficient condition for the algorithm to work is

DC 0 c E. (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)) = ( cc(t,w) e RN I a


(30)


= f(s(t)) + G(s(t))w, w E f }
with
f = { 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( xc v(s(t),yd(t),t)) }
oceCt


(31)











This is equivalent to the minimization


min { (uTGTQGu (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 un-

constrained control problem [30]. By using a trigonometric transformation, the

control is defined by an angular variable, u(t) = f{cos(13),sin(1)}, 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(.) : [0,oo) 1 e Rm where 0 is the control

restraint set. The collection of all admissible controls will be denoted by M(0).

The target set X is a specified subset in Rn. A system is defined to be

f-controllable from an initial state x(t0) = x0 to the target set X at T if there

exists U(.) e M(f) such that x(T,u(.),x0) e X. A system would be globally

O-controllable to X if it is O-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 [0,oo) (34)

with the state transition matrix 4(t,r) and solution

z(t) = 4(t0,t)Tz0 (35)

The interior B and surface S of the unit ball in Rn are defined as

B = { zo e Rn : Izoll < 1 ) (36)

S = { z0 e R : IIz011o = 1 } (37)

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

t (38)
J(x0,t,x,z0) = x0Tz0 + max [ GT(r,w)z(r) ]dr x(t)Tz(t)
0 eWfl

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

(33) to be f-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)
xeX z0eS








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)p(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) = T(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 e [0,T], z0 e 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 [32] 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)
xeX z0ES

But their control technique has two shortcomings: First; it requires the

selection of z0. The initial condition z0 is not specified but limited to i|z01l =

1. A particular z0 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(i0) 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 system;" 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.









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) [34].

At any time, the effect of a plant perturbation AP can also be

characterized as a perturbation in the target set.


If P = PO + 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) (yo(ti) + Ay(ti)) (6)

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

= Ar(ti) yo(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 dis-

placement 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 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 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 [11].

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 Technique



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 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 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 two-
dimensional missile intercept problem as an example. At time t = t1, not any
specific time during the intercept, the target is at some location T1 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

SSet



T



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 Ox for t e [0,oo). The reachable set

X(to,tf,xo,fx) = { x: x(tf) = solution to (10) (11)
with x0 for some u(.) e M(Ox) }

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(to) = 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,r0,fr),

R(to,tf,ro,fr) { r: r(tf) = solution to (13) (14)
with ro for some a(.) e M(fr) }

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

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

system:

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

nm (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+l), 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 linearizedd or reduced order), and that the reference has the

capability to change and possibly counter the control input. (This maneuverabili-

ty 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) 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 integral of its' input variable. Regardless of the initial maneu-

verability 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 = [Amin,Amax] where 0 5 Amin< Amax


(17)













Target Reachable


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.









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 un-

certainty) 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(to) = x0 (1)
dt2

with the quadratic cost

1 tf
J = xTPfXf + u(r)Tu(r)dr (2)
2


where
tf E [0,oo)

and



Application of maximum principle yields the following linear optimal control

law:

1
u = + x(tf)(t-tf) (3)



x0 + x0 tf
where x(tf) = (t-)3 (4)
(tf)3
1+ -
3-y1














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 7-. 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 from an initial position of 1000 feet and with velocity of

1000 feet/sec with 7- = 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









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)
-I
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








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+xotf)/a )} (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
Time


Figure 5.7 Unconstrained and Constrained Velocity vs Time


Position

2500 -

2000

1500 -
Constrained Position
1000 -

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
1000

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

1200 -

1000

800

600
S Uncertain Target Set
400

200 -

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.








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
1 tf
J = (x-r)fTPf(x-r)f + -y u)u()d (9)



The reachable set controller minimizes
rtf
J = [(x-r)TQ(x-r)+u(r)Tu(r)]dr (10)


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
Time

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









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.















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 (Ny,Nz,P) 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









56

additional modifications, this type of solution results in a trajectory dependent

control. As we have seen, Reachable Set Control is an LQG control implementa-

tion 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 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
z Nx
x = Vx u = Ny
Vy Nz
Vz

and

"0 1 0
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









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 B3(.,.) a Wiener process) that has the solution:


x(t) = 4(t,t0)x(t0) + 1 (t,r)G(r)u(r)dr (5)
to 1t
+ '(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.


E(ws(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)[I (9)
and
Wt(t) = (2at2/Tc)[II (10)


where Tc is the correlation time, and ar is the RMS value of the disturbance

process. The Power Spectral Density of the disturbance is:

27t2/Tc
tt(w() = w2 + (1/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:

fx(t) F(t) I x(t) 1 fG(t) 1 L 01 fws(t)
0 + u(t) + (12)
T(t) O N(t) T(t) 0O O 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:









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(0) = ro = 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.









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:


0e = Tan- (sin(bg)/tan(Og)) (15)

Target (
e





SLOS






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-1(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 instan-

taneously 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 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.

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

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

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

Oer = Oec Oea (19)
SZone I

Zone II

Zone III

/






Zb


Figure 6.3. Roll Control Zones.









66

Here Ny and Nz are the constrained acceleration values. In Zone I, Oer = 0.

The 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)
O-w IO

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 ma-

neuvering 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):



x(t) F= + [ u(t) + (22)
aT(t) J N aT(t) O -N wt(t) J

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 (v), seeker elevation (0), range (r),

and range rate (dr/dt):









68

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

= +Tan-{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 (0>), 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) ] L 0 ws(t)
S = O + u(t) + (24)
T(t) O N(t) T(t) O OM wt(t)

the reference:

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

with the tracking error:

x(t)
e(t) = [ yx(t) yr(t) ] = [ H(t) 1 0 1 -C(t) ] T(t) (26)
r(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 dis-

turbances will directly impact the performance of the system.









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)


STx(t)

T(t) = Ty(t)

Tz(t)


















e(t) =


cos(0)cos(4')
A
a(t) = cos(0)sin(O) A

-sin(0)


(27)


r(t) =


x(t)

y(t)

z(t)

O(t)

0

0

0

0(t)


rx(t)

ry(t)

rz(t)

ro(t)

rx(t)

ry(t)

rz(t)

r0(t)

rx(t)

ry(t)

rz(t)

re(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 =
OO w Iw


(28)


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


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











S -w i=j=8 1 i=j=8
(Owij 0 otherwise )ij 0 otherwise


( +w i=8,j=4
(w)i otherwise


The performance objective for the LQG synthesis is to minimize an
appropriate continuous-time quadratic cost:

Js(t) = E{Jd(t)|I(t)} (29)

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

tf
Jd = efTPfef + {e(r)TQ(r)e(r)+u(r)TR(r)u(r)}dr (30)
tO

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

Jd = e(tN+ )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+ l)TP(tN+1)x(tN+ ) (32)
N x(ti)T Q(ti) S(ti) x(ti)
+ E
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)
I 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) ]
and


P(ti) = Q(ti) + QT(ti+l,ti)P(ti+i)4(ti+lti) (35)

[ GT(ti)P(ti+l)4(ti+lti) + 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 im-

provements 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.












Measurement Noise D


Reachable Set Controller


Continuous-Discrete Kalman Filter


Measurement Noise


Dynamic Disturbances
Figure 6.4 Target Missile System


Missile












Reference Variable Dynamics


Yr(t) e(t)



'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











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 fi-

delity 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 II 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.
Target






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 (bg) and 180 degree aspect (0ka), 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 Cer-

tainty 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 -
120
100
80
60 -
4 0 ...........
20

S500 Z


000


4500
Initial Range


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


i_______................--


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 (feet)
2 5 0 -............. .............
250 -
Baseline Guidance Law
200 -

150 -

10 0 -..........

50 Reachable Set Control


S500 4000 4500 5000 5500
Initial Range
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 pen-
alty 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
4 0 0 I lk ... . .,
SActual\ ." -....
2 0 0 . . .... ......
0-
-200
-400 Estimate
-600 -

0 0.5 1 1.5 2
Time


Figure 7.5 Measured vs Actual Z Axis Velocity




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