Computer control of stochastic distributed systems with applications to very large electrostatically figured satellite antennas

APPLICATIONS TO VERY LARGE ELECTROSTATICALLY FIGURED

SATELLITE ANTENNAS

by

JEFFREY HASTINGS LANG

B.S., Massachusetts Institute of Technology

( 1975)

M.S.,

Massachusetts Institute of Technology

( 1977)

SUBMITTED IN PARTIAL FULFILL~ENT

OF THE REQUIREMENTS FOR THE

DEGREE OF

DOCTOR OF PHILOSOPHY

a~

the

MASSACH'. ':: TTS INSTITUTE OF TECHNOLOGY

January

1980

©

Jeffrey H. Lang

1979

The author hereby grants to M.I.T. permission to reproduce and to

distribute copies of this thesis dJcument in whole or

in

part.

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SATELLITE ANTENNAS

by

JEFFREY HASTINGS LANG

Submitted to the Department of Electrical Engineering and Computer Science on November 26, 1979 in partial fulfillment of the

requirements for the Degree of Doctor of Philosophy in

Electrical Engineering ABSTRACT

A satellite antenna concept is described that permits very large diameter

reflectors to be deployed from a single Space Shuttle payload. Antenna diameters reaching one kilometer are sought for a very low mass wire mesh reflector stretched across a hoop and distended electrostatically into

parabolic shape. The figure control of such a multimode structure motivates the present theoretical and experimental study of the computer control of stochastic distributed systems.

The theoretical control study considers linear time-invariant stochastic distributed systems which are driven by zero mean white Gaussian noise

processes and which evolve within finite bounds. Computer-based controllers are defined through limits placed upon the controller's processing speed, accuracyand supply of actuators and sensors which couple to the system. A

scalar quadratic performance cost, involving the distributed state and control, represents the desired closed-loop performance criteria in proportion to

their importance. A controller design procedure based upon a limited normal

mode distributed system description is developed which nearly minimizes the cost and which includes controller processing rate, controller sampling rate and sensor accuracy requirements sufficient to guarantee adequate

performance. An estimate of the operating point along the trade off between controller sampling rate and dimension, in cases of fixed controller

processing speed, is also developed.

Two methods of near-optimal controller design are presented. The simpler method applies finite dimensional optimal control theory to a truncated, or primary, set of modes yielding a full dimension controller. The second method yields a reduced dimension controller which allows the inclusion of additional modes into the primary mode set without sacrificing sampling rate as required by the simpler method when controller processing speed is fixed. The performance costs associated with both controller designs are calculated.

Both design methods assume that the ignored, or secondary, modes do not

couple to the actuators and sensors, and consequently overlook the shift in natural frequencies and performance cost caused by the unmodeled spillover

Simple perturbation expressions are developed for the shifts. It is observed that spillover effects are best minimized with the elimination of spillover to secondary modes with discrete time natural frequencies near the closed-loop primary natural frequencies or the unit circle. The effects are also minimized with the elimination of spillover to secondary modes driven by

noise sources which are highly correlated to the primary system noise sources. The experimental study, intended to test the computer-based controllers of the theoretical study, is executed on a hyperbolic distributed system which consists of a square meter of flexible tensioned wire mesh with a

flat equilibrium. The mesh supports normal deflections which are destabilized by electrostatic forces and controlled via perturbations to those forces.

Deflection measurements are available through measurements of the capacitance between the mesh and fixed plates. The deflection measurements and controls are connected in a closed-loop through a minicomputer.

The open-loop distributed system is internally destabilized and

computer-based controllers are successfully implemented to impose stability up to an electric pressure four times that required to destabilize the mesh.

A maximum of three open-loop unstable modes is stabilized and this is done

with information from only two sensors. The limits of controller success in achieving stability are explored in terms of sampling rate, degree of

open-loop stability, the number of primary modes and the number of available actuators and sensors. The limits, and the closed-loop system performance within these limits, are found to be consistent with theory. Finally, closed-loop primary, and open-loop secondary, natural frequencies are observed to shift and destabilize as a result of spillover.

The results of the experimental and theoretical studies are applied to the electrostatically figured satellite antenna concept. Diffraction-limited beamwidths of ten arc seconds appear possible for antennas reaching one

kilometer in diameter.

Thesis Co-Supervisors: James R. Melcher

Prbfessor of Electrical Engineering David H. Staelin

I wish to express my thanks to all members of the Continuum

Electromechanics Group and the Radio Astronomy Group for their help and friendship during our association. A special thanks is extended to

my co-advisors, Professor J.R. Melcher and Professor D.H. Staelin, for their guidance during the course of this thesis; they have certainly had a significant influence on my approach to engineering problems. Professors L.A. Gould and T.L. Johnson are gratefully acknowledged

for their interest in my thesis and their many helpful suggestions.

I am grateful to my family, particularly to my wife Betsy, for

their support during my thesis work. I want to thank my father, in particular, for sharing his interest in engineering while developing mine.

The Hertz Foundation is gratefully acknowledged for accepting the financial responsibility of my doctoral education. Additional financial support, to cover the cost of experimental materials and computer processing, was made available through the U.S. Joint Services Electronics Program Contract DAAG-29-78-C-0020.

TABLE OF CONTENTS ABSTRACT ... ACKNOWLEDGEMENTS ... LIST OF FIGURES... Page 2 4 7 CHAPTER 1. CHAPTER II. INTRODUCTION... A. Introduction... B. References... COMPUTER CONTROL OF STOCHASTIC DISTRIBUTED SYSTEMS.

A. Introduction... B. A Review of Finite Dimensional LQG Control.... C. Stochastic Distributed Systems and Modal

Decomposi tion... D. _{Ideal Control of Stochastic Distributed}

Systems.. ...

E. Computer Control of Stochastic Distributed Systems... 1. Preliminaries...

2. Near Optimal Truncated Modal Controllers...

3. More Optimal Reduced Order Controllers...

4. On Issues Related to Controller Resource Requirements and Allocation... F. Skew Correlated Noise Sources...

G. Summary ... H. References... CHAPTER III. EXPERIMENTS...

A. Introduction... B. Experimental System and Model...

1. Experimental System... 2. System Dynamics... 13 13 30 32 32 34 41 45 48 48 56 84 94 105 112 118 120 120 122 122 127

CHAPTER IV. CHAPTER V. APPENDIX A. APPENDIX B. APPENDIX C.

Page

3. The Actuators ... 135 4. The Sensors...140

C. Model Verification and Parameter Identification. 151 D. The Experimental Control Problem and Controller. 163 E. Experimental Results...181

1. Preliminaries...181

2. One Mode Experiments. . ... 183

3. Three Mode Experiments...204

4. Spillover Experiments... 219

F. Summary ... 243

G. References...245

VERY LARGE SATELLITE ANTENNAS... 246

A. Introduction ... 246

B. Antenna Model... ... 247

C. Antenna Performance... ... 261

D. References ... 276

SUMMARY...277

PERFORMANCE STUDIES OF FIRST AND SECOND ORDER MODES. 285 A. The Performance Studies...285

B. References ... 342

INSTRUMENTATION... 343

A. Instrumentation ... 343

B. References... 359

LIST OF FIGURES

Figure

Page

1.1 A functional diagram of the antenna concept... 15

1.2 Antenna equilibrium geometry... 18

1.3 Antenna perturbations... 19

1.4 Antenna parameters... 22

2.1 A diagrammatic summary of a computer controlled stochastic distributed system... 55

2.2 Necessary conditions for significant stable mode performance improvement... 99

2.3 Sufficient conditions for near ideal modal performance.. 101

3.1 A functional diagram of the distributed control experimental apparatus... 123

3.2 Control plate conventions and dimensions... 124

3.3 Sensor plate conventions and dimensions... 126

3.4 Distributed control experiment construction details... 128

3.5 Distributed control experiment construction details... 129

3.6 Conventions for the experimental distributed system model... 130

3.7 Modal actuator gains... 139

3.8 Sensor parameters and modal sensor gains... 142

Figure Page

3.10 Actuator ganging control commands to address single modes. 153

3.11 An example of an IV step response...155

3.12 An example of an IV step response decay...157

3.13 An example of the IV measurement of w2 + B2/4M2 versus V2.159

mn

3.14 An example of the IV determination of F and F...161

3.15 A schematic of the nonlinear modal deflection integrators. 176 3.16 A flow chart of the experimental controller... 178

3.17 Closed loop experimental system summary...180

3.18 Single mode experiment system parameters...0184

3.19 Single mode experiment measurements of w2+ B2/4M

mn

versusV2...185

3.20 Single mode experimental combinations of V, T and sensor

usage...187

3.21 Single mode experiment open 3.22 Single mode experiment mean estimate versus T; E sensor 3.23 Single mode experiment mean

estimate versus T; E sensor 3.24 Single mode experiment mean

estimate versus V; E sensor 3.25 Single mode experiment mean estimate versus V; E sensor 3.26 Single mode experiment mean

loop natural frequencies. 188 square modal deflection

and V = 2.5 KV...191

square modal deflection

and V = 3.5 KV...192

square modal deflection

and T = 10 mS...193

square modal deflection

and T = 20 mS...194

square modal deflection

Page

3.27 Single mode experiment mean square modal deflection

estimate versus V; H sensor ...

3.28 Single mode experiment mean square modal deflection

estimate distribution ... Figure 196 198 200 201 205 206 209 210 211

3.29 Single mode experiment J versus T...

3.30 Single mode experiment J versus V...

3.31 Triple mode experiment system parameters...

3.32 Triple mode experiment measurements of o + B2/4M mn

versus V ...

3.33 _{Triple mode experiment open loop natural frequencies;}

m = 1 and n = 1 ...

3.34 Triple mode experiment open loop natural freouencies; m = 1 and n = 2...

3.35 Triple mode experiment open loop natural frequencies;

m = 2 and n = 1. ...

3.36 Triple mode experiment mean square modal deflection

estimate versus V; m = 1land n = ... 212

3.37 Triple mode experiment mean square modal deflection estimate versus V; m = 1 and n = 2...213

3.38 Triple mode experiment mean square modal deflection estimate versus V; m = 2 and n = ... 214

3.39 Triple mode experiment J versus V...216

3.40 Spillover experiment system parameters...220

Figure 3.42 3.43 3.44 3.45 3.46 3.47 4.1 4.2 4.3 4.4 4.5 A. 1 A.2 A.3 A.4 A. 5 A.6 A.7 A.8 A.9 A. 10

I

2

1

2

1

2

Antenna equilibrium geometry ... Antenna perturbations ... Antenna oarameters ...

1000 meter antenna open loop natural frequencies...

100 meter antenna open loop natural frequencies...

First order mode normalized variable definitions... First order mode normalized equivalent discrete time

system coefficients ... First order integrator Jn versus an...

A A

First order stable mode

Jn

versus Tn;

cc

= 1...

First order stable mode Jn versus Tn; an= 100...

4

First order stable mode Jn versus Tn; a = 10 ...

n

nn

First order unstable mode J versus

T

;

&

= ?10...

n n n

First order unstable mode

3

versus

t

;

&

=

I...

n

n n

First order unstable mode in versus Tn; an

=

102...

Second order mode normalized variable definitions...

Spillover m =

I

and Spillover m = 1 and Spillover m = 1 and Spillover m = 1 and Spillover m = 1 and Spillover m = 1 and step response; p = 0... step response; p = 0... step response; p = -3... step response; p = -3... step response; p = +3... step response; p = +3... Page 234 235 237 238 240 241 248 249 262 266 267 287 292 296 299 300 301 303 304 305 308

Figure Page

A.ll Second order mode normalized system coefficients...314

A.12 Second order integrator Jn versus a.n...317

A.13 Second order stable double pole mode 3n versus Tn; a = 1. 319 A.14 Second order stable double pole mode Jn versus Tn;

an

= 100... ... ... ...

320

A.15 _{Second order stable double pole mode Jn versus Tn;}

n=1...321

anA

A

A.16 Second order unstable double pole mode Jn versus Tn;

an -2...323

n = 1

A.17 Second order unstable double pole mode Jn versus Tn;

a =1'..'...324

A.18 Second order unstable double pole mode Jn versus Tn;

an = 100... ... 325

A.19 Second order imaginary mode s)

n

versus T n; an = 10 ... 32

232

A.20 Second order imaginary mode i versus T n; an = 10~...32

A.21 Second order imaginary mode 3 versus T ; a = 1...327

n

n n

A.22 Second order imaginary mode in versus Tn; an = 100...328 A.23 Second order stable complex mode Jn

n

versus T ; a =100

and p = /10 .. ... 3 3 A.24 Second order stable complex mode Jn versus T ;

&

= 100

n n n

and

p

= 10 l... n6 A.25 _{Second order stable complex mode Jn versus Tn; an = l}

nn

n

335

and p = 10T... ...

Figure Paje

A.26 _{Second order unstable complex mode}

₃

versus n =an

andp = /l0 ... 337

A.27 Second order unstable complex mode 3 versus Tn;

n

n

^-4^ _-2 33 p n o10 and = 10...338

B.1 Command interface between computer and experiment... 344

B.2 Data interface between computer and experiment... 346

B.3 An actuator D/A converter...347

B.4 Actuator addressing circuitry...348

B.5 An actuator high voltage amplifier...350

B.6 A sensor oscillator ... 351

B.7 The crystal time reference oscillator...353

B.8 An oscillator frequency measurement counter... 354

B.9 The bias voltage measurement A/D converter... 356

B.10 Measurement addressing circuitry...357

C.l Idealized cardboard standoff layout and dimensions... 361

CHAPTER I

INTRODUCTION

A. Introduction

The promise of the NASA Space Shuttle Program has initiated the development of very large reflecting satellite antennas. These

antennas, reaching hundreds of meters in diameter, will greatly improve the technologies of deep space vehicle tracking and telemetry, radio astronomy, satellite communications, surveillance, and remote sensing of the earth. Additionally, they could encourage new technologies such as a national video communications network or solar energy collection. In this spirit of antenna development, a novel antenna design [1] is now considered which could lead to antennas reaching a kilometer in diameter.

One practical and economic limit to the size of an orbiting antenna is determined by the mass required per unit reflecting area to yield the desired reflector figure tolerance. An elastic wire mesh of several grams mass per square meter could, for example, form a one thousand meter diameter reflecting surface with a mass under five thousand kilograms, well below the thirty thousand kilogram launch capability of the Space Shuttle. Such a flexible reflector, however, would require extensive external surface control. This thesis

investigates one possible approach to this control problem and in the process presents new results concerning the control of stochastic

distributed systems.

The surface control method explored here is the use of

electrostatic forces arising from charge distributions continuously and rapidly manipulated by a scanning electron beam. The control charges reside on a second surface, or command surface, located a short distance behind the reflector. The command surface is pulled into an approximate paraboloid by a network of guy wires. The tolerance of the command surface would be a fraction of the separation between the two surfaces. Optical measurements of the reflector shape provide the data necessary for effective surface control. The system is

portrayed in figure 1.1.

The command surface is composed of an insulating film which supports many conducting segments individually addressable by the electron beam. To charge a segment negatively, the beam is run so as to deposit electrons within the conductor. If the beam is more

energetic the electrons will travel through the command surface and generate secondary electrons upon exiting. Both secondaries and primaries are swept to the reflector by the bias field, thus charging the command surface positively. Although positive charging is

alternatively achieved by secondaries produced upon beam entrance, this would require very low beam energies, making directional control of the beam, in the face of a magnetic environment, more difficult. Furthermore, entrance side secondaries would likely return to the command surface with an undesirable charging effect.

Optical Deflection and Velocity Measurement Sys tem ... ... .... ... .... ... Posittiion

I

I

I

Reflector

I

Bias Computer Controller Command Surface Electron Beam Electron Gun

Position, Energy and Current Density

Figure 1.1: A functional diagram of the antenna. Dashed lines indicate data flow.

via the electron beam, in accordance with the data provided by the measurement system, by redistributing the command surface charge and hence the electrostatic forces acting on the reflector. Furthermore, if the command surface is, for instance, constructed from aluminum segments, two microns thick, deposited on a three micron layer of plastic, the total mass of a one thousand meter reflector and accompanying command surface would not exceed twelve thousand kilograms. _{The result is a potentially very large high precision} reflecting satellite antenna.

It should be noted at this point that several studies [2-5] have reported the development efforts of large satellite antennas. Further studies [6 and 7] have reported the development of electrostatically distended mirrors. None of these, however, have considered kilometer dimensions, the use of an electron beam as a control element or the electromechanical dynamics of the reflector.

In trading structural rigidity for low mass, the need for active control of the reflecting surface arises. Indeed, the electrically biased reflector may become unstable. The surface control system must, then, stabilize the reflector, correct for imperfections in the command surface shape and reflector elasticity, and extinguish reflector

figure disturbances caused by external noise sources. A preliminary examination of this control problem follows. A more extensive analysis

is given in Chapter IV.

The reflector, tensioned by a rigid rim at the perimeter, is distended toward the command surface into parabolic equilibrium by a bias field. The reflector is treated as a thin tensioned membrane

are obtained by balancing the distending electric pressure, given approximately by EV2/2H212, against the restoring pressure of surface tension, 2F/L, exterted via the equilibrium reflector curvature, figure 1.2. Thus, in equilibrium

EV2 2F 2H

where V denotes the bias potential, L, the reflector radius of curvature, H, the surface separation and s, the permittivity of free space.

Linear perturbations about this equilibrium are considered in a flattened polar geometry, concentric with the antenna axis, with radial coordinate r and tangential coordinate

e,

figure 1.3. The flexible equipotential reflector supports normal perturbation

deflections h(r,G,t). The command surface supports Ohmic conservation of charge, characterized by an average surface conductivity a, and therefore supports the potential distribution -v(re,t) about the bias potential -V. _{Electron beam control of the command surface charge,} and thus the potential v, is represented by the perturbation current density i(r,e,t). These perturbations are assumed to vanish at the reflector perimeter, r = D/2, where D is the antenna diameter.

The three state variables, h, ah/3t and v, and the control current i are expressed as the sum of spatial Fourier-Bessel modes with

F L

Bias

Focus V H _D Reflectorl Cmand Surface

D/2

I

F

e--- d~

H

mob'

M

Reflector

I

I

Th(r,6,t)/3t h(r,e,t)

+-@-AX

e

a

Command Surface -V-v(r,6,t)

i(r,6,t)

Figure 1.3: Perturbational Analysis of the Antenna. Bias

V

-t

h(r,O,t) = 3h ( , = v(r,O,t) = i(r,O,t) = 00

2

m=-o 00

m=-co

2 20 m=y M .0 00 m=I 00 00

n=1

co

n=l

n=l

n=l

J (k r)ejme h (t) m mnD/2) dhn (t) Jm(k r)eime dt Jm+i(kmnD/2) J (k r)ejme mn ( m+(kmnD/2₎₁ )Jm(kmn r)ejme in ( m+t)mnD/2)

where Jm is the m th Bessel function of the first kind and kmn > 0

satisfies the boundary condition Jm(k D/2) = 0. The electromechanics

of the reflector and the command surface are coupled via the

electroquasistatic Maxwell's equations. The resulting linearized electromechanical dynamics for each mode are given by

0 SV2 Knk FH2 2

H

M

0

+ 1

O0

0 -EV

km

HM sinh(kmnH) -kmnH -ak2 V e mn H (k +Ym

0

0

-l

run,+ ru

imn(t)

(1.2a) (l.2b)

(1

.2c) (1 .2d) d

atf

hn (t) dh _(t) dt v (t) hmn (t) dh _mn(t) dt

Vmn

(t)

(1.3)

where ymn = k coth(k H). Again, the details behind (1.3) are presented in Chapter IV.

The natural frequencies, smn' of each modal system are determined from the characteristic equation of the state matrix of (1.3) with (1.1).

ak2

7Fk

2

7ak

4 F mn 3 2 mn

Kmn4

4kmn 4m

mn

S

K=s

kmn+FTJT

+5S

H

In e m kmnmn + n Mt Lknj(k n+Yn) (1.4)

Several important observations concerning the control of the reflector are now made for the two antenna configurations described by the parameters listed in figure 1.4. In the figure, the command

surface conductivity, a, has been chosen as low as reasonably allowed

by contamination considerations in order to prevent command surface

charge relaxation. The tension, F, has been chosen just high enough to smooth undesirable wrinkles and 100 kilovolts has been set as a practical upper limit for V, which, through (1.1), sets a maximum for H. Maximum H is desirable because it reduces the number of unstable modes and, by the nature of Laplace's equation, renders the equilibrium shape of the reflector less sensitive to the fine structure of the command surface shape.

From (1.4) it is seen that modal instability is approximately governed by the inequality

4 coth(k H)

c kmn (1.5a)

Parameter Diameter

Surface

Separation Radius of

Curvature

Aspect Ratio Tension Reflector Surface Mass Density Symbol

13

H

L

F

MN

1000 Meter

Antenna

1000

5

2000

1

1 0.005 100 Meter

Antenna

100 2 200

Units

Meters

Meters

Meters

1

I 0.005 Newton/Meter Ki 1 ograms/Meter2 Bias Command Surface Conductivity Number of Unstable Modes Growth Rate

of the Most

Unstable Mode

V

75 1 -a 15 0.18 95 1 -Kilovolts Mho Squares

3

1.1 Seconds 1

Typical Antenna Parameters. Figure 1.4:

which simplifies to

1< 42 (1.5b)

LHkmn

with the reasonable assumption that H << L. The periodicity of Bessel zeroes can be used to approximate kmn > 0, defined by Jm(kD/2) = 0,

by k ~i(2n +

ImI)/D.

With this approximation and (1.5), an

approximation for the number of unstable modes is determined as D/5Hr2 where 6 is the antenna aspect ratio L/2D. The 1000 meter antenna of figure 1.4 would have approximately 20 unstable modes. The 100 meter antenna would have approximately 5 unstable modes. The actual numbers are 15 and 3.

Using the typical parameters, the growth rate of the most unstable mode, m = 0 and n = 1, is calculated as 0.18/second for the

1000 meter antenna and as 1.1/second for the 100 meter antenna. If

each unstable mode is controlled with a common sampling rate equal to 10 times the most unstable growth rate, then antenna stability requires a total control capability of 27 and 33 modes/second. A similar requirement is imposed on the measurement system. It seems clear that a contemporary minicomputer is quite capable of regulating the unstable modes as well as many stable modes, and, that the

measurement system is not overtaxed.

for the proportional control law i (t) = -Nh (t). Substitution of

the control law into (1.3) yields the approximate stability requirement

2

2

N > aVkmn(1 - LHk2/4)/H. The maximum required gain occurs for

Mnk

mn

k = 2/HL where N = aV/H2L. This corresponds to control current mn

densities of 1.5 * 10-8 and 1.2 * 10-6 amps/meter2/meter deflection for

the 1000 meter and 100 meter antennas, respectively. During normal operation, millimeters of deflection are expected; thus, a 20 kilovolt electron beam [1] and an 85 kilovolt bias combined would require a maximum of near one watt per mode for both antennas. The power demands are quite reasonable.

In conclusion, on the basis of these and other considerations [L], it appears that very large electrostatically figured reflecting satellite antennas could be built, and that the requirements for the electrostatic charging system, the deflection measurement system and the controller would be reasonable. Furthermore, it appears that the available

capability of these subsystems exceeds the basic requirements imposed by reflector stability. Having thus demonstrated feasibility, the natural subsequent study should determine how to apply the excess subsystem capability in order to achieve the most precise reflector figure, and,

to determine the figure tolerance which can reasonably be achieved. This thesis will attempt to answer these questions by examining the

reflector control problem. Specifically, this thesis will investigate:

1. the electrical power requirements,

2. the controller computation requirements, including speed and precision,

3. the deflection measurement requirements, including rate,

quality and spatial location, and

4. the electrostatic charging requirements, including quantitative and positional accuracy,

all with respect to the design goal of improved surface tolerance. In order to control the reflector figure, surface deflection, and possibly velocity, measurements are made by an optical system. The instantaneous displacement of one point on the surface is determined

by triangulation. Velocity could be determined by a Doppler shift.

The command surface charge distribution, or equivalently the perturbation potential v, however, is not easily measured, and, since the net

secondary electron yield of the command surface can not be predicted accurately over the lifetime of the antenna, due to contamination and local charging effects, the control system must continuously generate a map of command surface charge, or equivalently potential, from which the instantaneous yield.can be calculated for use in the next control cycle. Additionally, both measurements and control are subject to corruption by noise. Consequently, the need for a complex computer based controller is expected. In view of this conclusion, further

examination of the antenna design will first require a basic examination of the time-sampled control of stochastic distributed systems.

The area of distributed control is not a new field of research. Chemical processes, thermal continua, electromechanical continua, fluids and plasmas have all been the subject of research in the field of

these papers are well reviewed by the books and several extensive

surveys [15-21].

Just as with the theory of finite dimensional control, the theory of distributed control has progressed from a classical treatment to an optimal strategy, typically expressed as a set of necessary and

sufficient conditions. General stability criteria have been presented and the types of instabilities have been classified. The concept of a stochastic control problem has been introduced, and from it, a separation principle for linear systems has emerged. Several

simplifying methods of analysis, intended to reduce a distributed system into one or more finite dimensional systems, have also appeared. Modal analysis is probably the most popular and experimentally successful. Others include finite element and finite difference analysis.

Neither theoretical nor experimental research is lacking. Yet, despite all the attention, research into the control of distributed systems has not nearly been exhausted. Non-linear systems, for example, have received relatively little attention. Linear systems, too, present many unsolved problems. In experimental studies, for instance, first

order, ordiffusive,systems have received most of the attention. Successful stabilization of higher order systems exhibiting multimode instabilities is lacking. _{In the theoretical domain, there is little work on the} nature of controllers restricted to a finite dimension and a finite number of actuators and sensors. Finally, there is virtually no work

on the nature of computer control, or time-sampled control, of

distributed systems. Beyond these there are certainly other unanswered questions.

This thesis will be primarily concerned with the time-sampled control of distributed systems and with the effects of controller limitations. Specifically this thesis will:

1. investigate the problems specific to computer control of distributed systems, including the temporal delay, numerical discretization and discontinuity of action inherent in

computer control,

2. examine the nature of controllers and the resulting system performance in situations where the controller dimension and the number of actuators and sensors are limited, and

3. experimentally demonstrate the stabilization of a hyperbolic distributed system exhibiting multimode instabilities.

The results of this control theory study will be applied to the antenna problem.

For the purposes of the control theory study, we shall focus on those linear systems which allow a normal mode description of spatial dependence. _{We are ultimately concerned with the computer control of} distributed systems. Computer controllers, however, are inherently limited from the viewpoint of distributed systems because they can not handle infinite dimensionality. As a result, we are faced with the problem of assigning the finite capability of a computer to an infinite dimensional task and assessing the resulting performance relative to

that of unrestricted control. Since the control of a single mode is a finite dimensional task, we utilize a modal description of the linear distributed system to manage the assignment of computer capability. The assumption of a modal spatial dependence is not considered to be a

severe restriction on the class of applications since many linear or linearized physical systems with finite bounds satisfy the requirements of the assumption.

The stochastic nature, the complexity, and the nature of the design goal of the antenna application suggest the use of linear quadratic Gaussian (LQG) control theory as the guideline for the control theoretic development because it offers a flexible, yet structured, approach to the design of complex control systems. In designing a control for a complex system, one is often faced with a conflicting set of performance criteria. By incorporating all the

relevant criteria into one mathematically expressed performance index, optimal control theory enables the designer to systematically balance the various trade-offs and arrive at an optimal solution.

The performance indices described here will be referred to as costs, indicating a desired minimization. It is possibla, and equally common in practice, to consider the negatives, that is, profits, which one seeks to maximize. The choice is mathematically arbitrary and usually coincides with the nature of the application.

Due primarily to the application which motivates this research, we will be outwardly concerned only with the steady state regulation of linear time invariant stochastic distributed systems. The nominal

linear, even deterministic or time varying distributed control problems. Finally, it is assumed throughout that the reader is familiar

with the results of optimal stochastic linear finite dimensional control theory. These results are used frequently and are quickly reviewed in Chapter II.

B. References

1. Lang, J.H., Gersh, J.R. and Staelin, D.H., "Electrostatically Controlled Wire Mesh Antenna," Electronics Letters, Vol. 14, No. 20, September 28, 1978, page 665.

2. Powell, R.V., Programmatic Considerations for Large Space Antennas, Jet Propulsion Laboratory Report 760-178, June 15, 1977,

Pasadena, California.

3. Freeland, R.E., Industry Capability for Large Space Antenna

Structures, Jet Propulsion Laboratory Report 710-12, May 25,

1978, Pasadena, California.

4. Creedon, J.F. and Lindgren, A.G., "Control of the Optical Surface of a Thin Deformable Primary Mirror with Application to an Orbiting Astronomical Observatory," Third I.F.A.C. Symposium on Automatic Control in Space, Toulouse, France, March, 1970, page 339.

5. Perkins, C.W. and Rohringer, G., Controlled Flexible Membrane Reflector, United States Patent 4093351, June 6, 1978.

6. Grosso, R.P. and Yellin, M., "The Membrane Mirror as an Adaptive Optical Element," Journal of the Optical Society of America, Vol. 67, 1977, page 399.

7. Hardy, J.W., "Active Optics: A New Technology for the Control of Light," I.E.E.E. Proceedings, Vol. 66, 1978, page 651.

8. Butkovskiy, A.G. and Lerner, A.Y., "The Optimum Control of Systems with Distributed Parameters," Automation and Remote Control, Vol. 21, No. 6, June 1960.

9. Wang, P.K.C., "Control of Distributed Parameter Systems," Chapter Three of Advances in Controls Systems, Volume 1, C.T. Leonedes, editor, Academic Press, 1964.

10. Brogan, W.L., "Optimal Control Theory Applied to Systems Described

by Partial Differential Equation," Chapter Four of Advances

in Control Systems, Volume 6, C.T. Leonedes, editor, Academic Press, 1968.

11. Butkovskiy, A.G., Distributed Control Systems, American Elsevier Publishing Company, New York, 1969.

12. Chu, T.K. and Hendel, H.W., Feedback and Dynamic Control of

Plasmas, AIP Conferen ceedings, No. 1XAmerican nstitute of Physics, New York, 1970.

13. Lions, J.L., Optimal Control of Systems Governed by Partial Differential Equations, Springer-Verlag, New York, 1971. 14. Ray, W.H. and Lainiotis, D.G., Distributed Parameter Systems,

Identification, Estimation, and Control, Control and Systems Theory Volume 6, Marcel Dekker, Inc., New York, 1978.

15. Butkovskiy, A.H. and Egerov, A.I. and Lurie, K.A., "Optimal Control of Distributed Systems -- A Survey of Soviet Publication," SIAM Journal on Control, Vol. 6, No. 3, 1968.

16. Wang, P.K.C., "Theory of Stability and Control for Distributed Paramter Systems -- A Bibliography," International Journal of Control, Vol. 7, No. 2, 1968, page&TI.

17. Robinson, A.C., A Survey of Optimal Control of Distributed

Parameter Systems, Aerospace Research Labs. Report ARL-69-0177,

1969.

18. Athans, M., Mitter, S.K., Wang, P.K.C., and Willems, J.C., Control of Distributed Parameter Systems, Joint Automatic

Control Conference, Boulder, CO, 1969.

19. Butkovsky, A.G., "Control in Distributed Systems -- A Review,"

Proceedings of the IFAC Symposium on the Control of

Distributed Parameter Systems, Vol. 2, Banff, Canada 1971. 20. Thomassen, K.I., "Feedback Stabilization in Plasmas," Nuclear

Fusion, Vol. 11, No. 2, 1971, page 175.

21. Berge, G., "Equilibrium and Stability of MHD Fluids by Dynamic

CHAPTER II

COMPUTER CONTROL OF STOCHASTIC DISTRIBUTED SYSTEMS

A. Introduction

Chapter II develops the theoretical contributions of this thesis to the linear quadratic Gaussian (LQG) computer control of stochastic distributed systems. The study is organized into five parts which consider the essential aspects of the control problem and follow a brief formula oriented review of the LQG control of finite dimensional systems. The first part presents a general description of the

distributed system to be studied and reviews the concept of modal decomposition. _{In the second part, under the availability assumption} of perfect distributed state measurements, an ideal control strategy

is determined which minimizes a standard quadratic cost criterion. The resulting performance represents a lower bound on achievable cost below which no realizable controller can operate. The third section studies the effects of computer-based control. These effects include limitations on the dimension, processing rate and processing precision of the

controller. Again, optimal and near optimal control strategies are formulated for the standard cost and the resulting performances are evaluated. A fourth section treats the special case of temporally

skew correlated actuator and sensor noise, a common occurence with time averaging sensors. A final section serves to summarize the results

stochastic distributed systems.

In this chapter, and in the sequel, E(-) will &i;ote the

statistical expected value of the argument and a superscript prime will

B. _{A Review of Finite Dimensional LQG Control}

This section presents a basic review of the LQG regulation of linear time invariant finite dimensional systems. Since the results are generally well known, [1-3], the review is brief and formula oriented. _{Both continuous and discrete time systems are examined.}

Consider first the continuous time system with state dynamics given by

dx(t) _{= A x(t) + B u(t) + G C(t)}

(2.la)

dt

where x denotes the state vector, u denotes the control vector and ( denotes a zero mean white Gaussian vector noise process. Observations

of the state are available through

y(t) = C x(t) +

e(t)

(2.lb)

where y denotes the observation vector and 6 denotes a second zero mean

white Gaussian vector noise process. A, B, C and G are constant matrices of appropriate dimension with (A,B) and (A,G) controllable and (A,C) observable. The noise statistics are assumed to take the form

E(E(t)) = 0 (2.2a)

E(((t

1

) t'(t

2

))

= 6(t1 -

t

₂)

(2.2b)

E(e(t)

e'(t9))

=

e

6(t1 - t2) (2.2d) E( (t1

) e'(t

2)) = Y 6(t1 - t2) (2.2e)

with E and e constant positive definite matrices of appropriate dimension. In many physical cases, the matrix W vanishes; it is

included here for completeness.

An optimal control over (2.1) is sought which minimizes the steady state quadratic cost

J = Lim E[x'(t)Qx(t) + 2x'(t)Su(t) + u'(t)Ru(t)] (2.3)

t-I-with R positive definite, Q positive semidefinite and (A,Q5)

observable. Here, the matrix Q serves to penalize the expected state excursions from the origin, the matrix R serves to penalize the expected

control excursions from the origin and the matrix S serves to penalize the expected cross product. Physically, the first term in (2.3)

penalizes the expected energy within the system (2.la) and the last term in (2.3) penalizes the expected control energy expended in the course of regulating the state. Like T, (2.2e), in many physical cases, S vanishes.

According to theory, [1 and 2], the optimal control is given by

u(t) = -R~I(S' + B'K) z(t) (2.4a)

dz(t) = Az(t) + Bu(t) + (EC' + Gw)

®~

(y(t) - Cz(t)) . (2.4b)

The feedback matrix K in (2.4a), and the filter error covariance E

in (2.4b), are the unique positive definite solutions to the matrix Riccati equations

0 = Q + KA + A'K - (KB + S) R~1(S' + B'K) (2.5a)

0 = GEG' +

AE

+ EA' - (EC' + GT)

&0

1

(t'G'

+ CE) . (2.5b)

The minimized cost is given by the three equivalent expressions

J = Trace[QE + GEG'K + KAE + EA'K] (2.6a)

J = Trace[GBG'K + (KB + S) R~1(S' + B'K)E] (2.6b)

J = Trace[QE + (EC' + GW)

&~

1

i('G'

+ CE)K] (2.6c)

and stability of the closed loop system is guaranteed.

To put the improvement of the controlled system performance in perspective, one can compare (2.6) to the cost associated with no

control. Assuming A to be stable, the cost of the uncontrolled system, u = 0, is given by

where X, the uncontrolled state covariance matrix, is given by the unique positiv3 definite solution to

0

= AX + XA' + GEG'. (2.8)

If A is unstable, no such solution to (2.8) exists and the system cost is infinite. Note that (2.6) is strictly less than (2.7).

Consider now the discrete time system with state dynamics given

by

x(k + 1) = A x(k) + B u(k) + G (k) (2.9a)

where x denotes the state vector, u denotes the contrl vector and denotes _a zero mean white Gaussian vector noise process. Observations of the state are available through

y(k) = C x(k) + e(k) (2.9b)

where y denotes the observation vector and 6 denotes a second zero mean white Gaussian vector noise process. A, B, C and G are constant

matrices of appropriate dimension with (A,B) and (A,G) controllable and (A,C) observable. The noise statistics are assumed to take the form

E( (k)) = 0 (2.1Oa)

E(t(k₁

)

'(k2)) =

E

6(k1 - k2) (2.10b)

E(6(k)) = 0 (2.10c)

E(e(k1

) e'(k

2)) =

e

6(k, - k2) (2.10d)

E(t(kI) 6'(k2)) = Y 6(k1 - k2) . (2.10e)

with E and 0 constant positive definite matrices of appropriate dimension. Again, in many cases, the matrix T vanishes. It is

included here both for completeness and for use in Section F in treating the skew correlation exhibited between actuator noise sources and the noise sources of time averaging sensors.

An optimal control over (2.9) is sought which minimizes the steady state quadratic cost

J = Lim E[x'(k)Qx(k) + 2x'(k)Su(k) + u'(k)Ru(k)] (2.11)

k-+c

with R positive definite, Q positive semidefinite and (A,Q2₎

observable. Here, the matrices,

Q,

S and R assume the same physical

meaning as in (2.3).

Since the following results will be used to design computer based controllers, it is convenient to impose the restriction that, due to processing delays, y(k) can not be used to determine u(k). The control is otherwise unrestricted. Then, according to theory, [1 and

u(k) = -(R + B'KB) 1(S' + B'KA)z(k) (2.12a)

where the state estimate, z(k), evolves according to the filter dynamics

z(k+l) = Az(k) + Bu(k) + (AEC' + GT)(0 + CEC')j1(y(k) - Cz(k)). (2.12b)

The feedback matrix K in (2.12a), and the filter error covariance E in (2.12b), are the unique positive definite solutions to the matrix Riccati equations

0 = Q + A'KA - K - (A'KB + S)(R + B'KB)~1(S' + B'KA) (2.13a)

0 = GEG' + AZA' -E - (AEC' + GT)(o + CEC')~1('G' + CEA') .

(2.13b)

The minimized cost is given by the three equivalent expressions

J = Trace[QE + GEG'K + AEA'K - EK] (2.14a)

J = Trace[GEG'K + (A'KB + S)(R + B'KB)-1 (S' + B'KA)E] (2.14b)

J = Trace[QE + (AEC' + GY)(o + CEC')Y~('G' + CEA')K] (2.14c)

and stability of the closed loop system is guaranteed.

To put the improvement of the controlled system performance in perspective, one can compare (2.14) to the cost associated with no control. Assuming A to be stable, the cost of the uncontrolled system, u = 0, is given by

J = Trace(QX) _(2.15)

where X, the uncontrolled state covariance matrix, is given by the

unique positive definite solution to

0

= AXA' - X + GEG' . (2.16)

If A is unstable, no such solution to (2.16) exists and the system cost

is infinite.

_{Note that (2.14) is strictly less than (2.15).}

This completes the review. These results will be used throughout

the remaining chapters. Again, references [1-3] should be consulted

for the details behind these results.

The stochastic distributed system considered in this study evolves in the spatial domain r, with boundary 9r and spatial coordinate vector y, over time t. The distributed state vector is denoted by x(y,t),

the distributed control vector is denoted by u(y,t) and the distributed vector noise drive is denoted by t(y,t). The system dynamics are

assumed to be described by

Sx(y,t)

= A(x(y,t)) + B(u(y,t)) + GE(y,t)

(2.17a)

for y in F and

x(yt) = 0 (2.17b)

for y on 3 where A and B are smooth linear spatial matrix operators and G is a constant matrix. The elements within the A and B considered here are assumed to be constant, partial differential or integral operators.

The vector noise drive (y,t) is a white Gaussian process in time and.a smooth function of y which satisfies the properties

E(E(yt)) = 0 (2.18a)

E(s((y,tl) '(y₂

,t2)

~ 3(l'Y2) 6(t1 - t2) (2.18b)

(2.18c) Y ,2) >0

for y, and Y2 in F. Here, the assumption of a white noise drive is not restrictive. In many cases, the bandwidth of the noise is sufficiently larger than the significant bandwidth of the system to justify a white noise approximation. When this is not the case, and colored noise is required, the state dynamics and the noise dynamics may be combined into a single augmented system which is then properly driven by white noise.

The distributed system description (2.17 and 2.18) is now specialized to those systems which possess a normal mode spatial dependence. Distributed systems possessing a normal mode spatial dependence are characterized by the existence of a complete set of scalar eigenfunctions, Xn(y) where n is an index, which are common to the spatial operators A and B, which satisfy the boundary conditions and which are orthonormal in r with respect to a scalar weighting

function q(y), r(y) > 0 for all y in r. Mathematically, this specifies

the existence of matrices An and Bn such that

A(a Xn(y)) = An a

X

(y) (2.19a)

B(b

Xn(y))

= Bn b X (y) (2.19b)

for every pair of vectors a and b of appropriate dimension with spatially independent elements. Furthermore,

Xn(Y)

= 0 (2.20)

(2.21)

f

n

n2(y) dy = 6(n1 - n2) r

Using the assumed completeness and orthonormality of the eigenfunctions, the distributed state, control and noise drive are

expressed in modal form.

x(y,t) = I Xn(t)

An(y)

n

u(y,t)

=

Y

un(t)

xn(y)

n

(y,t) = Yn(t) An(y)

n

(2.22a) (2.22b) (2.22c) Xn(t) = x(yt)

un(t)

=

u(yt)

n(t) = f(y,t)

r

An (y) n(y) dy n(y) n(y) dy A (y) q(y) dy

Furthermore, with the use of (2.18),

E(Qn(t)) =

0

E( (t1)

qE2(t

2)) n n 6(t1 -t2) n1I n2 1 22

0n

where (2.23a) (2.23b) (2.23c) (2.24a) (2.24b) (2.24c)

where

nIn2

JJ

y1 y2) n

(fl)

Xn(Y2) (y1 2) dydy2 (2.25)

and

2 = n n in2 n X 2) (2.26)

n 1n 2 1 2 1 2

Note that n(t) are generally not independent stochastic processes. Even the assumptions of spatially white noise or a spatial impulse dependency do not necessarily yield statistically decoupled processes

n(t).

Substitution of the modal descriptions (2.22) into the distributed system dynamics (2.17) yields an infinite set of finite dimensional modal dynamics. The state equation for each individual mode is

dx (t)

dt = An = n(t) + B u (t) + GF (t) .-(2.27)

It will be generally assumed here that the pairs (An,Bn) and (AnG) are controllable which, in turn, implies that the distributed system

D. Ideal Control of Stochastic Distributed Systems

This section investigates the ideal control of stochastic

distributed systems. No restrictions are imposed on the controller and it is assumed that perfect distributed state measurements are available. An optimal control is sought for the system (2.17 and 2.18) which

minimizes the steady state distributed quadratic cost criterion

J = Lim E (x'(y,t)Qx(y,t)+2x'(y,t)Su(y,t)+u'(y,t)Ru(y,t))n(y)dy (2.28) t-*

with R > 0 and Q > 0. This cost penalizes excursions of both the state and control from the origin according to the matrices Q, S and R. Physically, the first term in (2.28) penalizes the expected energy within the distributed system and the last term in (2.28) penalizes the expected control energy expended during the regulation of the state. In many physical cases, S vanishes. The nature of the optimal control, then, will be to return the state to the origin as quickly as possible without using excessive control, where quickly and excessive are defined by

Q

and R,respectively.

Modal decomposition is used to obtain the optimal control. Substitute (2.22) into (2.28) and utilize (2.21).

j = Jn(2.29a) n

jn = Lim EK(t)Qxn (t)+2x (t)SuA(t)+u(t)Run( t) . (2.29b)

The cost of distributed regulation is now clearly the sum of the costs of regulating each mode. Since both the modal dynamics (2.27) and the modal cost contributions (2.29) decouple, each mode may be treated as an individual finite dimensional system. The cost of distributed regulation is then minimized when the cost of regulating each mode is minimized. _{Thus, modal decomposition has reduced the infinite dimensional} control problem to an infinite number of finite dimensional control

problems each solvable by the theory of section B.

It is generally assumed that (A,Q2_{) is observable. Applying}

(2.4a), (2.5a) and (2.6a) to the modal control problem (2.27) and (2.29b) in the perfect observation limit, 0 + 0 hence E-+ 0 and z(t) + x(t),

results in the optimal modal control strategy. Each control is given

by

un (t) = -R~1(S' + BbKn)

Xn(t)

(2.30)

where Kn satisfies the Riccati equation

0 = Q + KnAn + AIKn - (KnBn + S) R~I(S' + B'Kn) . (2.31)

The resulting cost is given by

in= Trace(GHnnG'Kn) _(2.32)

control systems because they represent a firm upper bound on the performance achievable with a realizable controller. In particular, the cost of distributed regulation given by (2.29a) and (2.32) is always exceeded in practice.

E. Computer Control of Stochastic Distributed Systems

1. Preliminaries

The stochastic distributed control problem of section D is now

subjected to the restrictions of computer control. A computer controller

is defined as a controller which

1. receives a finite number of single valued state measurements at discrete times,

2. issues a finite number of single valued state controls at discrete times, and

3. suffers a limited data processing rate and precision. This study will be limited to computer controllers with the following additional properties:

4. the controller is linear and time invariant;

5. measurements and controls are received and issued

simultaneously and at regular intervals (The control interval, or sampling period, is denoted by T.);

6. measurements are noisy state samples and controls are piecewise constant.

Note that a controller with limited processing precision is actually

highly nonlinear. In the linear framework of this thesis, the

precision limitation is assumed to be weak enough to be successfully treated as a linear source of noise.

Within the restrictions outlined above, an optimal control is sought for (2.17 and 2.18) which minimizes the standard quadratic cost criterion. Due to the time sampling restriction, however, a

computer controlled distributed system can never achieve the steady state implied by (2.28). Thus, it is necessary to transform (2.28) into an equivalent cost averaged over the control interval. Denote the

discrete time sampling index by k. The distributed cost is reformulated as

kT+T

J = Lim E

'(x'(y,t)Qx(yt)+2x'(yt)Su(y,t)+u'(y,t)Ru(yt))r(y)dydt

L

kT F

(2.33)

It should be emphasized that (2.33) penalizes both state and control excursions from the origin during the entire control interval and that

(2.33) converges to (2.28) as T approaches zero. Now, an optimal

computer control is sought for (2.17 and 2.18) which minimizes (2.33). Modal decomposition is again used to initiate analysis.

Substitute (2.22) into (2.33) and utilize (2.21):

j= Jn (2.34a)

n

KT+T

SLim E (x'n(t)Qxn(t)+2x'(t)sun (t)+u'(t)Run (t)) dt. (2.34b)

ke kT

Again the modal cost contributions decouple. Next, (2.34) and the decoupled modal dynamics, given by (2.27), are transformed into the discrete time domain. Integrating (2.27) from the initial time kT yields

xt)= An (tk) (kT)

t

An(t-s)

x ()=eAn(Bxn(kT) +(e n (Bnun(s) + GC (s))ds . (2.35)

T

Equation (2.35) is known as the variation of constants formula in which exp(Ant) denotes the matrix exponential of Ant, see references [1,4]. Using a superscript bar to denote discrete time variables, define

-n (k) = xn(kT) (2.36a)

un(k) = un(t), kT < t < kT+T (2.36b)

kT+T

tn(k)

= exp(A (kT + T - t)) G-n(t) dt. (2.36c)

n J kTnn

Note that (2.36b) incorporates the piecewise constant output

restriction of computer control. Then, substitution of (2.36) into

(2.35) into (2.34) yields the distributed cost

= 3 n i(2.37a)

n

J= Lim Exn\) x (k)+2' (k)YSnUn (k)+Wi' (k) Rnun ( k + Trace (ECn) (2. 37b)

k-+n Tf _{A't At} Q = e n Qen dt > 0 (2.37c) T 0 =

j

ntS +

J

0

Qe n Bnds dt (2.37d) +0 0 = R + Tft B' e Ans 2S + (t Qe An rBndr dsdt > 0 (2.37e) ~1

T rt A s A'1s

LEn=t

J

en

GEnnG'

e n dsdt . (2.37f)

Since _{n (2.37c) is the observability Gramian of the observable pair} (A,Q ) it is positive definite, as indicated.

in

(2.37e) is positive definite because R is positive definite.

Using (2.35) and (2.36), the modal dynamics (2.27) are written as discrete time systems.

xn(k+l) = Ann (k) + Uin (k) + Cn (k) (2.38a)

AnT

An =en (2. 38b)

r

T Ant

n =

J

e B dt . (2.38c)

0

Note that

An

is defined as a matrix exponential and therefore its inverse necessarily exists. This fact will be useful later. Using (2.36c) and (2.24), the statistics for

tn(k)

are derived as

E(- (k)) = 0 (2.39a) E(U (k₁) nK(k2 n 6(k1 - k2) (2.39b)

2~1

2

T Anlt An2 t = fT e GEn G' e t dt (2.39c)

n

n

2 0

mnn

2 > 0 . ~ ~jj nn

_(2/.39d)

Recall that is positive definite. Thus <Z exists and is invertable.

This, combined with the controllabilityof the pair (AnG), guarantees the controllability of the pair (A2,GE' 2). Since is the

n n

nn

controllability Gramian of (An,G2), it is therefore positive definite, as indicated in (2.39d).

Finally, in accordance with the requirements of computer control, the limited state measurements received by the controller, denoted by

the vector 5(k), and the limited state controls issued from the controller, denoted by the vector U(k), assume the form

y(k) =

UI

-n(k) + H(k) (2.40a)

n

iln(k)

= D n(u(k) + p(k)) (2.40b)

where 6(k) is a white Gaussian observation vector noise process with the statistics

E(6(k)) = 0 (2.41a)

E(Y(kI)

&'(k

2)) =

U

6(k

1 - k2) (2.41b)

6 > 0 (2.41c)