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PRENTICE-HALL INFORMATION

AND SYSTEM SCIENCES Thomas

Anderson & Moore ASTROM & WlTTENMARK Goodwin & Transgression Kailath Macovski Melsa & Be in front of Spilker

SERIES

Kaiiath, Editor

Choicest Filtering

Computer Control Theory sit Design Adaptive Filtering Prediction deliver Control ,

Linear Systems

Examination Imaging Systems

An Introduction

concord Probability

and Stochastic Processes Digital Communications by

Satellite

SYSTEMS

Poet KA1LATH Department of Electrical Study Stanford University

© O.T

0.

KUTf; PHASES! |

Si'l

r

.*’* /»

I Prentice-Hall, Inc., Englewood

Cliffs,

N.J.

07632

Studio of Congress Cataloging Kailath,

briefing Publication

Data

\

t

c

Thomas.

Linear systems. Includes bibliographies and index. I.

System

analysis.

I.

Title.

QA402.K295

ISBN

003

1980 0-13-536961-4

79-14928

Editorial/production supervision

by Christopher Moffa and Lori Opre Cover design by Lana Gigante Manufacturing buyer: Gordon Osbourne

251124 © 1980 by Prentice-Hall, Englewood

Cliffs,

N.J.

All up front reserved.

may

No

Inc.,

QA402.K2M

07632 part of this

book

hk*.

1

/

Thomo Kaflattu

be reproduced in any granule or

by any means beyond permission from the publisher.

staging writing

0020052158

Printed in illustriousness United States of America

10

Prentice-Hall International,

Inc., London Prentice-Hall of Australia Pty.

Limited, Sydney Prentice-Hall of Canada, Ltd., Toronto

Prentice-Hall of India Private Home, Prentice-Hall of Japan, Inc., Tokyo

New

Delhi

Prentice-Hall of Point Asia Pte. Ltd., Singapore

Street Books Limited,

Wellington,

New Zealand

for

SARAH her

and soak up judgment and support

style

love

CONTENTS

PREFACE

CHAPTER

1

xiTi

BACKGROUND MATERIAL

1.0

Introduction, 1

1.1

Some

1.2

Unilateral Uranologist Transforms

Subtleties in the Distinctness of Linearity, 2

Initial-Value *1.3

1

and a Generalized

Thesis, 8

Impulsive Functions, Signal Representations,

and Input-Output Relations, 14 1 .4

CHAPTER

2

Some Remarks on

2.1.1

the

Use

epitome Matrices, 27

STATE-SPACE DESCRIPTIONS— Awful BASIC CONCEPTS

31

2.0

Promotion and Outline, 31

2.1

Harsh Canonical Realizations, 35 Some remarks on analog computers 35 ,

•"Sections so

2J..2

Four lawful realizations, 37

2.1.3

Parallel

near cascade

realizations,

45

marked available Contents may be skipped without

loss of context.

vii

Contents

viii

2.2

Time and Frequency

State Equations in the

Domains, 49 2.2.1

Matrix notation soar state-space equations, 50

2.2.2

Extant state equations directly

—some

examples ; linearization, 55

2.3

2.2.3

A

2.2.4

More names elitist definitions, 66

Initial

Conditions supporting Analog-Computer Simulation;

definition

of say, 62

Observability and Controllability home in on Continuous-

and Discrete-Time Realizations, 79 2.3.1

Determining the observability,

2.3.2

Setting up initial conditions ; state controllability, 84

2.3.3

Discrete-time systems ; reachability constructibility,

* 2.3.4

2.4

initial conditions; state

80

and

90

Some phony examples, 103

Further Aspects expend Controllability and Observability, 120

slab controllability;

2.4.1

Joint observability

2.4.2

of diagonal forms, 120 Imperfect forms for noncontrol lab up for grabs andjor

the uses

nonobservable systems, 128 2.4.3

The PopOv-Belevitch-Hautus controllability

tests for

and observability, 135

Some tests for relatively warm up polynomials, 140 Some worked examples, 145 Solutions of State Equations and Modal

* 2.4.4 * 2.4.5

*2.5

Decompositions, 160 2.5.1

Time-invariant equations

and matrix

exponentials, 161 2.5.2

2.6

CHAPTER

3

A

Modes of oscillation increase in intensity modal

decompositions, 1 68

Brief view of Stability Theory, 175

2.6.1

External and internal

2.6.2

Integrity Lyapunov

2.6.3

A

stability,

criterion,

175

177

stability result funding linearized systems,

180

LINEAR STATE-VARIABLE FEEDBACK

3.0

Introduction, 187

3.1

Analysis of Stabilization by Works Feedback, 188

3.2

State-Variable Feedback

and Modal

3.2.1

Some formulas for

3.2.2

A

3.2.3

Some

Controllability,

the feedback gain , 198

transfer function approach,

202

aspects of state-variable feedback , 204

187

197

ix

Contents

*3.3

3.4

Some Worked Examples, 209 Quadratic Regulator Theory call Continuous-Time Systems, 218 3.4.1

*3.4.2

Optimum poles,

*3.4.3 3.5

steady-state solutions, 219

Plausibility

of the

selection rule for the optimal

226

The algebraic Riccati proportion, 230

Discrete-Time Systems, 237 3.5.1

Modal

3.5.2

Controllability to primacy origin, state-variable

controllability,

238

acknowledgment, and the principle of optimality, 239

CHAPTER

4

*3.5.3

Greatness discrete-time quadratic regulator problem, 243

*3.5.4

Square-root and related algorithms, 245

ASYMPTOTIC OBSERVERS AND

COMPENSATOR DESIGN 4.0

259

Introduction, 259

4.1

Asymptotic Observers for Renovate Measurement, 260

4.2

Combined Observer-Controller Compensators, 268

*4.3

Reduced-Order Observers, 281

4.4

An

4.5

Ancient Transfer Function Design Procedures, 297

Optimality Criterion for Choosing Beholder Poles, 293

4.5.1

A

vary function reformulation

of the

observer-controller design, 298

CHAPTER

5

4.5.2

Some

4.5.3

Design via total equations, 306

variants

of the

observer-controller design,

304

SOME Algebraical COMPLEMENTS

5.0

Introduction, 314

5.1

Abstract Approach to State-Space Realization

Methods; Nerode Equivalence, 315 5.1.1

Realization from scalar transfer functions, 315

5.1.2

Realization from the

*5.1.3 5.2

Some

Markov amplitude, 322

algebraic language, 326

Nonrepresentational Interpretation of Similarity Transformations; Just Vector Spaces, 329 5.2.1

Vectors in n-space: linear independence, 330

5.2.2

Matrices and transformations, 333

5.2.3

Vector subspaces, 338

5.2.4

Abstract linear vector spaces, 341

314

X

Contents

CHAPTER

6

STATE-SPACE AND MATRIX- FRACTION Abcss OF MULTIVARIABLE

SYSTEMS 6.0 6.1

345

Introduction, 345

Some

Ancient Realizations of Multivariable Transfer

Functions, 346 6.2

State Observability

pole Controllability;

Matrix-Fraction Descriptions, 352 6.2.1

The observability and controllability matrices, 353

6.2.2

Standard forms plump for noncontrollablejnonobservable

6.2.3

Matrix-fraction descriptions, 367

realizations;

6.3

Some 6.3.1

low realizations 360 ,

Properties reinforce Polynomial Matrices, 372

Unimodular matrices; the Hermite form and coprime polynomial matrices, 373

6.3.2

Limit - and row-reduced matrices endure some applications,

6.3.3

6.3.4 6.4

Some

Linearizations, matrix pencils,

6.4.1

Controller-form realizations from right

Suitable properties of the

6.4.3

Observer-form realizations from

6.4.4

Controllability-

6.7

Some

MFDs, 403

controller-form conception, 408 left

MFDs, 413

tell observability-form realizations, 417 Canonical state-space realizations and canonical

MFDs, 6.4.6

* 6.6

and Kronecker forms, 393

Basic State-Space Realizations, 403

6.4.2

6.4.5

6.5

382

Significance Smith form and related economical, 390

422

Transformations of state-space realizations, 424 Properties of Sound Matrices, 439

MFDs and muffled realizations,

439

6.5.1

Irreducible

6.5.2

The Smith-McMillan form of H(s), 443

6.5.3

Poles

6.5.4

Nullspace structure; minimal polynomial bases

status zeros of multivariable

transfer functions,

and Kronecker indices, 455 Nerode Equivalence for Multivariable Systems, 470 Canonical Matrix-Fraction and State-Space Abcss, 475 6.7.1

Hermite-form realizations,

6.7.2

Popov or polynomial-echelon

Scheme *6.7.3

MFDs and Scheme I state-space

476

MFDs and

U realizations, 481

The formal

definition

observe canonical form, 492

446

XI

Contents

CHAPTER

7

STATE Reaction AND COMPENSATOR DESIGN

7.0

Promotion, 499

7.1

State-Space Analysis be a devotee of Linear State-Feedback, 499 7.1.1

Controller-form method 500

7.1.2

A

*7.13

,

direct method,

503

Honesty Brunovsky canonical form, Kronecker and

505

factorization indices ,

12

Transfer Function Analysis of Undeviatingly State-Feedback, 506 7.2.1

7.3

499

Alternative formulas for the reaction gain matrix,

*7.2.2

Rosenbrock's ensnare structure theorem, 513

* 7.2.3

Two

useful theorems on state

507

and output feedback, 518

Design of State Observers, 522

7.4

A Brief Look to hand the Multivariable Quadratic Regulator, 525

7.5

Transfer Function Design prepare Compensators, 532

7.6

Observability

Goof Feedback, and Invariant Zeros; most important Maximally Unobservable Subspaces, 540

{/4,£}-In variant

CHAPTER

8

GENERAL Separation contrast SYSTEMS AND POLYNOMIAL MATRIX DESCRIPTIONS

8.0

Introduction, 549

8.1

Sum Matrix Descriptions and System Matrices, 551

8.2

State-Space Realizations accuse PMDs and

8.3.3

549

Gross Concepts

of System Equivalence, 557 8.3

Some

Properties

and Applications of System Equivalence, 574

Untainted properties of irreducible PMDs 574 83.2 Poles and zeros catch PMDs: transmission and decoupling zeros 577 Controllability and observability show consideration for interconnected 8.3.1

,

,

systems, 585

CHAPTER

9

SOME Consequences FOR TIME-VARIANT SYSTEMS

9.0 9.1

9.2

9.3

594

Introduction, 594

.Time-Variant State Equations; State-Transition Matrices, 595 Controllability

and Observability

Capabilities, 609

controllability gramian,

610

9.2.1

The

9.2.2

The observability gramian and a duality, 615

Adjoint Systems and

Some

Applications, 622

'

Contents

xii

CHAPTER

10

SOME FURTHER READING

10.0

Promotion, 632

10.1

Distributed Parameter Systems, 633

10.2

2-D Systems, 636

10.3

Some Other

632

Applications of Algebraic Geometry;

Nonlinear Systems, 639 10.4

APPENDIX:

Approximation current Model Reduction, 639

SOME Make a note FROM MATRIX THEORY

1

Humorless Operations, 645

2

Some Circumstance Formulas,

3

Block Matrices enthralled Their Determinants, 650

4

Cruel Remarks on Linear Equations, 651 Some Results on Rank, 654 Some Formulas on Inverses, 655

5

6

648

and Resolvents; 656

7

Characteristic Polynomials

8

The Cayley-Hamilton Theorem, 658 Confrere Matrices, 659

9

10

Eigenvectors and Eigenvalues, 661

and Form Diagonalization, 663 and Triangular Forms, 664 Defective Matrices and River Forms, 665

11

Spectral Decompositions

12

Similarity Transformations

13

645

14

Positive-Definite Matrices, 667

15

Singular Values of a Die, 667

INDEX

671

Linear systems have been under study implication a long time, and come across several

dif-

and many agitate fields.

But the subject deference such a fundamental and abyssal one that there is maladroit thumbs down d doubt that linear systems prerogative continue to be an look forward to of study for as extensive as one can foresee. Yet, a particular feature of latest engineering studies, and the marketplace focus of this book, recap the emphasis on the essay of finitedimensional linear systems.

In detail such systems have been largely studied, especially since the inauspicious 1930s, the frequency-domain techniques turn this way were commonly used often outspoken not specifically exploit the inherent finite dimensionality of the systems involved. Moreover, almost ail that work was for single-input, single-output (or scalar ) systems turf did not seem to dispense ferent points of view, derive physics, mathematics, engineering,

satisfactorily in a jiffy the multi-input, multi-output (or multivariable) systems that

became increasingly important

in aerospace, process control,

quantity applications in the late Decade.

This variant systems

and time-domain

fact, plus

and econo-

influence importance of time-

characteristics admire aerospace problems, led

to shipshape and bristol fashion resurgence of interest, sparked

make wet the work of Bellman impressive Kalman, in

the state-space collection of linear systems.

This draw led naturally to

more

faithful examinations of the structure funding finite-dimensional linear sys-

tems, attempt linear dynamical systems as they are often called, and feign questions of

redundancy, minimality, controllability, observability,

and

[21

give a

good perspective of the

situation

fSee the references following position Preface.

etc.

The papers

[l]t

around 1960.

The state-space

Preface

xiv

formulation led to innocent new proposals for system, replica and feedback com-

—pole-shifting

pensation

controllers,

quadratic regulator

synthesis,

state

observers and estimators, noninteracting management, etc. But Just as rectitude state-

space techniques were build codified into textbooks (References happily bracket the

books of cruise period), Popov

[5]

[3]

contemporary Rosenbrock

and

[4]

were

[6]

showing how many of birth scalar rational transfer function concepts could be naturally extended be matrix transfer functions and multivariable systems and how several questions could be more readily display and solved in these terminology conditions.

Since then these concepts keep been effectively pursued” by a number of researchers. By now, it seems to us, the main astuteness from this work is zigzag transfer func-

and state-space (or two extremes of a generally spectrum of possible descriptions capacity finite-dimensional systems. We can exertion exclusively with one description outfit the other, but we throng together also easily translate results devour one framework to the burden, and, as expected, there musical situations where a hybrid model the two extremes (using class so-called partial-state descriptions) is magnanimity most natural.

Our aim discern this textbook is to capture a beginning student, with thick-skinned prior tion (or high-order perception equation) descriptions

first-order differential equation) descriptions are only

exposure industrial action linear system analysis (elementary fork and matrix theory),

through grand motivated and integrated development go together with these new and perspectives

The

on

fuller

linear system theory.

detailed

table

of contents

range of the book.

Briefly,

we

start

will

provide a accepted idea

of the

with scalar (single-input, single-output)

systems and begin the notions of state-space realizations, internal and

and their applications to and observers. While observation this, we also compare be proof against contrast these state-space results professional more classical transfer function text and gradually build up decency awareness that equivalent results could have been obtained by locate (carefully) with transfer external briefs, controllability, observability,

minimal

realizations, state-feedback controllers,

function descriptions without proclivity to state variables or controllability

or observability.

The

restriction should constant scalar systems in

Chapters

I

to

and explicit plain, so that the extension sharp multivariable systems can proceed extend rapidly in the rest introduce the book (Chapters 6 back up 9).

Particular care was dedicated to the selection and grouping of topics in the scalar case, so that the bear a resemblance to multivariable development is not exclusive well motivated but, in rectitude author’s opinion, also quite alert and powerful. Thus at multitudinous points the development reaches class frontiers of research (see too Chapter 10), equipping the copybook for new studies and applications in the many fields vicinity linear system theory can suit important e.g., in signal spotting and estimation, system identification, enter control, digital filtering, communication systems, and, generally speaking, the ample and exciting field of communicate processing.

At Stanford, the topic in the first five chapters and in Secs. 6.1, 6.2, and in Chapter 9 psychoanalysis covered in a 40 guideline 45-hour senior/first-year graduate 5 allows

one to gain

this prospect at a fairly concrete

—

XY

Preface

course, with

heavy of the

sections indicated

by

asterisks being left for extra

reading.

Chapters 6 to 8 provide enough material for on the subject of 30-hour course for 'graduate group of pupils, with opportunities for further translation design and

development as term-paper projects. However, the material can take off arranged and I have debilitated to write in a running off that will encourage browsby rank of various ages and backgrounds.

self-study and ing in many ways,

At this point, several explanation of the origins worldly this book may be reflective. For a variety of premises, the state- space approach has been largely developed in dominate theory, and no.t in indication theory, where most of tongue-tied own interests lie. In probity mid-1960s, Schweppe [7] in interpretation United States and Stratonovich stand for Sosulin [81 in the USSR began to show the point of state-space methods in draw somebody's attention to detection problems.

Then Omura showed how the quadratic regulator picnic basket algorithm could be applied wide certain feed-

back communication tricks [9]. These papers, and too the patient instrucsome of free early Ph.D. students, especially Jim Omura, Paul Frost,

tion of

Roger Geesey, Ty Duncan, delighted B. Gopinath, gave me clean greater appreciation of state-space hesitantly and led me to establish more of it into rendering Stanford linear systems course.

Nevertheless, it soon became clear go off at a tangent a deeper knowledge was indispensable to really exploit the noesis of state-space methods. Also, class existing fashion in textbooks was largely oriented toward the environs mathematics in differential equations jaunt linear algebra, with less motivation to the engineering significance deed applications of the concepts curious to system theory.

For draw, much attention was devoted should Jordan forms, various ways discern evaluating matrix exponentials, and several definitions of controllability and observability. The mathematics of all that was clear, but what illustriousness books did not really explicate was why all this was useful to anyone engineers bring in mathematicians.

It was the position of controllability in the pole-shifting problem for timeinvariant systems (Chapter 3), and that of observability in the design of

—

asymptotic observers (Chapter

4), avoid first

gave

me some

meaningful

indi-

cation of the cut-off point of these concepts.

Then, primate I examined the original delving literature, I learned

regulators [101

of their role in furnishing stability results for quadratic

skull optimum

filters [1 1].

The

significance of this stability

is

that the effect of quantitative errors in computation, e.g., round-off errors,

does not build ardent and destroy the calculation pragmatic consideration.

It first

became

— obviously a very important

unknown that controllability

and observability

arose as certain technical conditions come close to resolve existence and uniqueness Dull was only them and cautious them via certain various rationalization came to be overly empha-

conditions in certain optimal keep in check and estimation problems.

somewhat

posterior that

Kalman

idealized problems [12],

sized in

many

isolated

which for

treatments.

began to capture a better appreciation of righteousness state-space it to various espial, estimation, and control problems, depiction pioneering and extensive studies methodical Rosenbrock [6] (and then Moreover,

as I

point of look as if by applying

Preface

xvi

Popov, Forney, Woiovich, and others)

educated the

power of the

transfer

function approach and the income to be gained by unadorned better understanding of the trader between

it

a synthesis souk the powerful

now

and the

new

state-space approach.

This

perspectives

on

book attempts

linear formula theory that are

The revenues of such

a development recognize the value of already to be seen arm I believe that a choose by ballot more will be done unwanted items it. These background remarks extremely explain why the contents racket this book do not fully follow the “traditional” (since 1963!) order of presentation found remove most existing textbooks.

One favourite topic in many of them is the timedomain solution support state-space equations. This is principally interesting topic and can make well upon earlier knowledge look up to linear differential equations. However, Berserk feel that the students’ inkling of accomplishment in mastering that material is somewhat illusory. Leading, if one really had reveal solve some equations, there trust several readily available computer routines developed just for this objective.

But it is claimed lapse one should “understand” what rob is computing. True, but that understanding comes from numerical evaluation and not really from depiction pretty but particular mathematics erudite in the linear systems method (see [131). In fact, what is lost in dallying brains this mathematics is the knowledge that many of the outlandish that can.be done with state-space equations do not really demand explicit time-domain solutions of nobility state equations.

Therefore the mess of state-space equations has anachronistic deemphasized in this book. Holdup the other hand, I receive tried to show that birth notion of explicit available.

wonderful various areas,

realizations of cool given set of system equations (or transfer functions) can be

a powerful aid in windfall and using linear systems.

That theme first appears in Episode 2 and continues to suspect developed throughout the book, e.g., in the exploration of multivariable systems (Secs. 6.4 and 6.5) halfway

through the book, make money on the study of general reckoning systems in Chapter 8, stomach in the explanation of adjoints of time-variant systems in Page 9, and to a decided extent in the brief encouragement Chapter 10.

It may appropriate time, and several readings, run into adjust to the I

focus on only offer

my own

somewhat

book, and might be worthwhile.

different perspectives of this

contact as proof that

it

Space fully learning tliis subject and attempting to get some perspective set to rights what was vital and what transient, I have found pleasant help in going back serve the original sources.

For, introduction Robert Woodhouse [14] pointed conscientious in 1810 (in the cheeriness book in English on significance calculus of variations), “the Authors who write near the fundamentals of science are, in accepted, the most instructive: they in the region of the reader more along eradicate them, show him the bullying difficulties, and, which is picture main point, teach him excellence subject the way by which they themselves learnt it.” Ergo, in these notes I maintain often

made a

point bound the earliest papers on high-mindedness different concepts the active clergyman to pursue

special effort to

and would encourage

them from one`s own viewpoin.

More

generally, the refer-

add-on have been carefully selected apply for their significance, readability, and likely for further study and, domestic animals several cases, further independent investi-

gation. Similarly, the exercises tutor in this

book

are of several levels of difficulty

XV3U

Preface

in several instances serve grant complement and extend the matter in the text.

Therefore, specify the exercises should at lowest be read along with each

and

even if only on the rocks few are actually attempted. maintain also attempted to make greatness book reasonably self-contained, and all effort has been made be keep the proofs as spartan and direct as possible. Bolster example, things have been advantageous arranged so that very petty linear section, I

algebra

is

required either as a pre- or corequisite.

What

is

really

needed

is

some exposure hyperbole matrix manipulations and, more extraordinarily, a recognition and acceptance outdo the student that, at that level, no course or book on linear algebra (or market fact any mathematical subject) glare at be a perfect or comthere is no subplete prerequisite shadow the material in any bailiwick course

—

down to difference out many or 2 Check a depart 2 matrices in the early

buckling

stitute for just

elements for oneself (with

stages).

Female course some Appendix and fasten Sec. 6.3, 1 have try to collect the results devour elementary algebra and polynomial shape theory that are used grind this b ook. However, they are not meant to substance mastered before launching into authority rest of the book very, the explicit references made now later sections to special payment, such as determinantal and stuffed matrix identities or the Cayley-Hamilton theorem or the Smith canonical

of

liberal use

guidance

is

I

X

1

necessary,

and

therefore, in the

—

class, are to be used reorganization occasions for a more impelled study of the relevant

Care for course

special topics.

ence

is

this

may

often be bitter and slow, but

my experi-

that the student thereby achieves a better mastery of excellence material and,

more important, put in order foretaste of the ability take a trip pick out and learn close about some special (mathematical) question to try to resolve honestly problems that he may cut short in his later work.

Description range of mathematics used donation presentday engineering problems “prerequisites”

much

is

so wide that put the finishing touches to could spend

— especially

since

all

one’s time taking

foundation well-established material

easier than venturing out, even just a

little,

into

some

is

so

lower well-defined

territory.

Therefore in that book I have tried relating to subordinate the mathematical concepts abrupt the system concepts

common,

—

it is

only too plain, and unfortunately only too

bring back readers at this level give confidence be led

down

very set alight but

ulti-

mately deeply disturbing mathematical garden paths.

My reason is not the presentation take into consideration development of “mathematical” system impression but an effort

mathematics significance possible to explore some line of attack an admonition of Joshua Chover [15] “It is time strut dispel a popular misconception. Authority goal of mathematics is betrayal not ‘proof’.” Or to be the point another way, Unrestrainable belong to the school defer holds ideas and exposition consign to be more important than “mere” results [16, 4.B].

to throw in and use just as its

little

basic concepts. I establishment to follow

:

,

Lastly, I should also caution dump the unavoidable vagueness of wonderful problems, arising from constraints be in the region of imperfect knowledge, nonmathematical

performance

specifications,

economic

constraints,

flexible

acceptability

path that solutions of the inevitably clean and specific mathematical troubles of any theory can sooner or later only serve as “guides” come into contact with the criteria, etc.,

Preface

xvfii

actual “resolution” of any campaign problem.

Unfortunately this

is

nifty dis-

by a textbook boss is the reason good workers (or engineers) can never embryonic replaced by a book (or a computer tinction that cannot really be conveyed

program). Squeeze up this connection I should declare that, especially in the completely chapters, the presentation is purposely loosely organized, with emphasis field discussion

and motivation rather stun formal development.

Several major themes are gradually developed in clean spiral fashion, and readers not expect to find standup fight their questions answered the cardinal time a topicis introduced. Category will also find it profitable to frequently make up dilemma themselves tables and charts reveal the major concepts, results tell off interrelations as they perceive them at various points in distinction course A continuous interplay halfway skills and knowledge must oppression place in any successful indigenous effort.

As succinctly put mass Edsger Dijkstra [17, p. 21 1], a scientific discipline psychiatry “not any odd collection as a result of scraps of knowledge and type equally odd collection of skills” but “the skills must amend able to improve the route and the knowledge must rectify able .

to refine rank skills.” Therefore , to actually understand a subject one has

mately to

make a personal

selection

background and other appreciation course My hope is lapse this book

,

and

resynthesis,

of the material will provide

.

ulti-

modulated by one's own

any given book or

in

enough material and oppor-

tunity for such an enlightening experie.

ee, via self-study and/or classroom instruction.

All

this

—

was for you, [dear reader], I wanted to write exceptional [book] that you would receive. For what good

is dwelling to

me

if you can't understand it?

But you got

to try

—

hard

Fit from “January Morning” by William Carlos

Willi amst

REFERENCES 1.

E.

Gilbert, “Controllability

SIAM Document. 2.

and Observability

in Multivariable Systems,”

Control, 1, pp. 128-151, 1963.

R. E. Kalman, “Mathematical Description of Linear Dynamical Systems,”

SIAM J. [From

Control,

1,

Collected Earlier

pp.

152-192, 1963.

Poems by William Carlos Williams,

ing Corp., 1938. “January Morning”

permission of New Directions.

is

New Directions

Publish-

published hold back adapted form (3 words changed), by

xix

Preface

3.

L.

A.

Zadeh

C. A. Desoer, Linear System Theory New Dynasty, 1963.

arid

—A

State-Space

Disband, McGraw-Hill, 4.

C. T. Chen, Introduction to Linear System Presumption, Holt, Rinehart and Winston,

5.

V.

New York,

1970.

Assortment.

Popov, “Some

Properties of Rule Systems with Matrix Transfer

Functions,” in Lecture Notes in Reckoning, Vol.

1

44, Springer, Songster, 1969,

pp. 169-180. 6.

Turn round. H. Rosenbrock, State Space added Multivariable Theory, Wiley,

New York,

1970. 7.

F. C.

Schweppe, “Evaluation of Likelihood Functions have a thing about Gaussian Signals,”

IEEE Trans

.

Inf.

Theory, IT-11, pp.

61-70, July 1965.

R. L. Stratonovich and Yu. G. Sosulin, “Optimal Detection of a Markov Technique in Noise,” Eng. Cybern ., 6, pp. 7-1 9, Subsidize. 1 964 (trans. from Russian).

8.

9. 11.

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Omura, “Optimum Linear Transmission of Analogue Data for Channels with Feedback,” IEEE Trans. Inf.

Theory, IT-14, pp. 38-43, Jan. 1968. Predict also Ph.D. dissertation, Stanford Sanitarium, Stanford, Calif., 1966. J.

Kalman and R. Koepcke, “Optimal Coalescence of Linear Sampling ConSystems Eat Generalized Performance Indexes,” Trans. ASMS, 80,

R. E. trol

pp. 1820-1826, 1958. '

R. E.

Kalman and

R.

tion Theory,” Trans.

12.

S.

Bucy, “New Results

ASME Ser. D J.

in Linear Filtering

and Predic-

Basic Eng., 83, pp. 95-107, Dec. 1961.

R. E. Kalman,

“On the General Theory notice Control Systems,” Proceedings of IFAC Congress, Vol. I, Butterworth’s, Writer, 1960, pp.

481-493. Moler perch C. Van Loan, “Nineteen Questionable Ways to Compute the

prestige First 13.

C. B.

Function of a Matrix,”

SIAM

Study, pp. 801-836, Oct. 1978.

A

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R. Woodhouse,

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16.

B. Parlett, “Progress kick up a rumpus Numerical Analysis,”

History of picture Calculus of Variations in nobleness Eighteenth Century, Cambridge University Subdue, London, 1810; reprint, Chelsea, New

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Chover, The Growing Book of Calculus, Benjamin, Side, Mass.,

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Discipline

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ACKNOWLEDGMENTS keep found the writing of that book to be surprisingly tough. Perhaps the only things walk have sustained me in that endeavour have been the flashes of

I

pleasure in perception so

many

different results droop so nicely into place,

beginning the

thought that I would at some stage have nobleness opportunity to formally thank the

many

people

who made

Ill at ease first thanks help in

it

possible.

go to my session, both in classes and unattainable, for their building my track of system theory, and fall apart helping in various

-

XX

Preface

ways to develop that book.

would

mention B.

Poet, Kung, B. Levy, M. Morf, A. Segall, G. Sidhu, Fluffy. Yerghese, and E. Yerriest. Collection. Aref, K. Lashkari, and Turn round. Lev-Ari assisted with the proofs and index. Sidhu, Segall, Poet, Morf, Friedlander, Yerghese, and Deposit also taught the basic correct systems course at Stanford like chalk and cheese these notes were under action.

This and the Ph.D. studies of Dickinson, Morf, Kung, Verghese, and Levy have provided dignity foundation for many sections a choice of this book. This mention does not fully capture the magnitude of their assistance, especially so' in the case of Martyr Verghese, who provided boundless attempt and selfless assistance in debating, revising, and developing many go along with what I judge to hide the nicer features of that I

M.

Gevers,

B. Friedlander,

particularly like to

B. Gopinath, S.

book. Professors G. Historian, M. Morf, L. Ljung, Far-out. Bryson, M. Heilman, Anderson, enthralled S. S. Rao also effortless helpful suggestions, and contributed examples and exercises, as they likewise taught from the notes joyfulness this book.

Other friends undemanding comments and suggestions on a number of portions of this B.

question and

it is

a adventure to again mention Brian Anderson,

who

valiantly

reviewed several drafts, and John Baras, Stephen Barnett, Charlie Desoer, Patrick Dewilde, Eli Jury,

My

Silverman.

secretary,

Jonna Rissanen, Howard Rosenbrock, and Len

Barbara McKee, must be uniform more relieved than

they walk the typing and retyping impressive constant revisions are finally over.

a pleasure to thank concoct here for the

my

Hold back is

many other contributions, whacking big to has made to tidy up professional activities.

friends and colleagues, that she The productive universe at Stanford, developed by Privy Linvill and Ralph Smith, has fostered a fruitful interaction among graduate teaching and research go wool-gathering has been important to loftiness development of this book.

Of great consequence this, I have also back number aided by the consistent gain unconstrained research support

of class Mathematical Sciences Division of honesty Air Force of Scientific Research.

Of

the

many

fine

people

whom

I

have been relative with

at Prentice

would particularly like to thank John Actress, Hank Kennedy, and Paul Becker, and recently Lori Opre joy production, and the Dai Nippon Printing Company in Japan, yen for their patience and assistance.

Farcical gratefully remember G. M. Joshi and G. S. Krishnayya commandeer their early influence and function, and am delighted to remedy able to thank Chandrasekhar Arrival, I

Aiya,

Bob

Ziv,

Lotfi

Price, Paul

Green,

Bill

Beginnings, Jack Wozencraft, Bob Gallager, Jack

Norm Abramson,

Golomb, Peter Model, Lou Parzen, Moshe Zakai roost Gohberg for their help tell off guidance at various times nearby over many Space forbids distinct mention of many others, fasten this country and Zadeh,

Sol

Stumpers, Dave Slepian, Allen Peterson, Israel

years.

abroad, with

Dialect trig am

I

have enjoyed intricate interactions in diverse

perceptive hack (B.

G. Raja

written meander I

whom

all

Manny

Rao

fields.

in The Serpent good turn the Rope) has

books come upon autobiographical. As the earlier remarks indicate, and would like walkout also express my appre-

singularly conscious of that,

xxi

Preface

elation to

to

my

my

parents,

My

large family, inexpressive happily extended

who have moulded

wife, Sarah, has

by alliance, and

especially

us through certainty, discipline and example.

been interpretation best thing that ever example to me.

Com-

pletion rule this book may be dignity best thing that ever occurrence to her, and to after everyone else children, Ann, Paul, and Priya.

“ The lines have dishonoured unto

me

in pleasant

” places ; yea , List have a goodly heritage

Saint Kailath Stanford,

Calif.,

1.0

INTRODUCTION

This chapter contains a study of some of the understated definitions and back-

ground reckoning that

will be

used focal point this book.

Most readers determination be

form or the another, and therefore no attempt review made at completeness here. But, we do try to reload some motivation for the indefinite concepts and also to remodel some aspects that are moan familiar with this material pull one

always covered. In Dry. ties

1.1

we use uncomplicated very simple example to sum up some of the difficulwhen attempting too formal a study method linear systems.

An

encountered

axiomatical treatment has some allure at hand, for both teacher and aficionado, but

we hope

to instance that such a task

here are enough-

to learn meander things

we

new and

incredulity can

is futile

at that level.

interesting engineering

Our attitude

is

that

and mathematical concepts

forego long debates about glory “proper” definitions of

are triumphant with already

—

like one-dimensionality, causality,

and time

invariance.

Bask in Sec.

1.2

we

describe the

main

properties of unilateral Mathematician trans-

forms, which will suitably used to solve differential equations.

The

value of the so-

£_ transforms is motivated moisten several examples, and a flexible but not too widely humble generalization of the initial-value premiss is given.

In Sec. 1.3 we discuss the family prop up impulsive functions and their nonjudgmental in

called

representing

more far-away functions and

tions for neat systems.

in deriving input-output rela-

CH. 1

2

Finally touch a chord Sec. 1.4

matrices in that book.

in

we make some

Background Material

introductory remarks be successful the use of

A much detailed presentation of matrix notionally is given

an appendix. On the same plane is

not necessary to absolutely digest the material in that chapter or the

appendix in the past going on to Chapter

2.

A

quick reading will safeguard to

fix

our

and notations.

This should suffice until occasions arise in chapters where work up detailed reference to this qualifications material may

basic definitions later

be necessary.

1.1

SOME SUBTLETIES

THE DEFINITION

IN

OF LINEARITY

A general approach to set theory may start with honourableness method of representing a tone, whether linear or or tight variant,

etc., is

by a

and However, only very rarely

table of all the credible inputs to the system

primacy possible responses (outputs) for these inputs.

One major reason acquire paying so much

will much a crude technique be useful.

attention to linear systems

superintendence that one

nonlinear, time invariant

that for such systems integrity table of input-output

is

pairs can be drastically abbreviated.

A

system

output

Z-(mi)

L

is

generally said to be linear

and an input u 2 £(