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A self-contained, highly motivated and comprehensive account of basic methods for analysis and application of linear systems that arise in signal processing problems in communications, control, system identification and digital filtering.1 @4 X: J. x5 U1 i
LINEAR SYSTHNIE8 s% k8 H4 S2 A7 T9 `2 q
THOMAS KAILATH" m+ E" B( R7 a. k1 G* J5 D
Department of Electrical Engineering9 q: s3 } _' @. L1 l h
Stanford University
; ^5 f; t1 B9 w" E+ ^; L鹏T只.严积死堅您: J/ d4 i) I3 f# w3 l& `, K2 x- Y
PRENTICE-HALL, INC,, Englewood Cliffs, N.J. 076322 ~9 i% r7 s2 m! j
% z8 o5 t3 {* x9 [( SLibrary of Congress Cataloging in Publication Data2 |# @0 I& o8 O) M$ e
af systems.
, |! g- `0 }- k L% V8 r" i) a% tIncludes bibliographies and index.; t( E6 L8 y& } O9 L6 ~4 @
QA402K29598000379-14928
, F( Y8 z1 b/ A& n, i% j& vEditorial/production supervision) p3 U# D; E; U9 R7 f- T
cher moffa and lori opre( \1 R0 k9 M+ D0 B$ M
Cover dcsign by Lana gigante
' u! U6 Q" q: T6 q) _) ZManufacturing buyer Gordon Osbourne
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C 1980 by Prentice-Hall, Inc0 C( j5 j7 D6 c) J1 M. o
Englewood Clifts, N. 07632
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8 ~' X4 q- G+ W. q; O' YFElli6 w7 }, w, e4 }2 ~' Y. z
/ `1 X3 T2 w" u- w$ ~SarAh
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7 R; Z+ X$ L& ^7 i; B, \+ o1 [. a/ istyle and taste1 e. r4 G1 A+ U7 M
love and support
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A
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CONTHE
. I, ^: _1 a) H! y, b# G# Hiut' { ]! ~9 z% E) X/ O, j4 [
PREFACE
3 y: I& h" Q/ ` ZXI[I$ @7 m7 N4 o6 n A" R' b
CHAPTER 1 BACKGROUND MATERIAL
0 h5 O- i5 X' ` U8 _+ V+ ]; B6 YI I Some Subtleties in the Definition of linearity,
& G) _# w+ }( T% o* ^( s% r7 @1.2 Unilateral Laplace Transforms and a generalized
. [2 J' ]& c# e& T9 ^- w% y4 i8 e1.3 Impulsive Functions, Signal Representations,5 C) T6 P2 g W, R/ N
and Input-Output Relations, I4
0 ~5 I( M8 ?- e1 O4 N6 J1.4 Some remarks on the Use of matrices, 27( `# ?/ ?# u5 L1 q" |' o1 K
CHAPTER 2 STATE-SPACE DESCRIPTIONS-SOME6 h4 W, c* k/ M" U
BASIC CONCEPTS
$ E n- B' Z/ \) s; ?/ \ ?6 ^6 c207 D/ p5 z6 b& F& M
al realizations. 35
$ y3 l! H8 m$ C1 p. I( s& q2.1.I Some remarks on analog computers, 35! Q, E% y% a2 v/ C0 h
2.1 2 Four canonical realizations, 37
1 c, d; U T9 p) ^2.1.3 Parallel and cascade realizations, 45* Z4 \+ h+ z9 w( N+ |4 A' W
Sections so marked throughout Contents may be skipped, _% P: A" b! a
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Yll5 N2 F' R3 G& I3 w3 g4 C
2.2 State Equations in the Time and fre! n+ W+ d- Z S6 V) E7 k
22. i Matrix notation and state space equations, 50
1 h( U6 H( L( x6 B7 f22.2 obtuinii& G; `' a! f1 n. @: {6 u, g
乎 lions directy--s0me' z+ _- x# c9 y( j2 H* W
examples, linearization, 55, e2 \* O5 a6 F! l
2. 2.3 A definition of state, 627 x% {* U' N G3 I
a2.4 More names and definitions,663 \$ A8 z; @- |6 I& w( D0 c8 W
2.3 Initial Conditions for Analog-Computer simulation0 M. i& m U' G
Observability and Controllability for Continuous
9 H7 L" x5 e1 [9 w/ |( \and discrete- Time realizations , 79. [6 d1 L# m w v0 A
2.3I Determining the initial conditions,szare
, [ F/ r5 I' [$ w+ ^23. 2 Setting up initial conditions, state controllability, &4+ M3 l4 m) E8 X
23.3D6 r- }) L& A* c
bility, 90
. O. t) l7 q) }+ g& c$ a# y! s23.4S
: }8 ^" q" d8 T# Y2. 4 Further Aspects of Controllability and obs9 z4 o7 r* I3 n) f3 P: D) a
ty,20
, I. {) f5 ^8 Z4 v" d4 z9 t' R2.4I Joins observability and controlability, the uses
$ C6 }/ F: _ |$ ]9 \diagonal forms, 120+ C7 U5 P! d! E0 [, z/ Q
2 Standard forms for noncontrollable andor$ ]/ ~8 F3 S' C0 C
nonobservable sy stems, 7284 S9 l5 d& E9 Q% z! V5 R) d6 E
2.4.3 The PapoyBelevirch-Hautus tests for( w9 _( H' K& q$ _; h* @& _* G
controllability and observability, 13.5/ @$ b4 g# Y5 D- f# E
R2.4.4 Some tests for relatively prime polynomials, 140
* d% k3 O7 c( N2. 4,5 Some worked exan
& @ X$ z: k, t* J& k$ ?1 q*2.5 Solutions of state Equations and modal
; G6 E7 s; l, [0 zDecompositions, 1605 z: ?1 N, P$ T
2.5.1 Time-invariant equations and matrix0 v6 L* H7 K5 n0 n8 z, X
exponentials, 162' E# z! W* Y E9 R3 J* `1 ^( i8 Y
2.5.2 Modes of oscillation and modal decompositions, 168
9 q6 D8 n/ U3 U9 P b& w `" ]2.6 A Glimpse of Stability Thcory, 17
" ^' E5 |) s$ k: L) }2, 6.2 The Lyapunov criterion, I77
% V9 i: ?5 b3 g7 M8 V8 [$ n8 J26.1E
- h% B! v8 ^9 ^: [: Rl stability4 p, \" z! h' c$ ~
75
0 j l" G8 s7 ]: W7 }$ S2.5.3 A stability resuit for linearised systems, 180/ S M: J& [ I. x$ Q. V
CHAPTER 3 LINEAR STATE-VARIABLE FEEDBACK
7 [3 L7 }4 }1 i& N7 ^" i6 K* l M; e3.0 Introduction, 187& z6 j3 E# A/ |/ U0 b2 E2 C
3.1 Analysis of Stabilization by Output Feedback 188+ Q# ~' v% o+ t( [
3. 2 State-Variable Feedback and Modal Controllability, 797; z2 J( u9 Z# ]8 N" ]$ l
3. 2. Some formulas for the
; V( a. _( ]/ R+ f4 }3.2.2 A transfer function approach, 2020 O1 T# Q" C7 j' Q6 B1 O
3 Some aspects of state-yariable feedback, 204
$ T! c$ |. k$ q, C. T0 B8 c8 B# B8 L, S1 ^( f' A0 X0 h! h# T
3, 3 Some worked Examples, 209
3 A' E f* K1 W7 \& x' K! ]4 Quadratic Regulator Theory for ContinuousTime- @) m, P: H! D. n4 k7 z% f
Systems, 218
2 m3 ?1 ^* ?* v3 W8 _+ f* `8 ? ?3.4,7O
4 R6 K7 a1 I& `0 ]2 J5 xite solut# o. r: @! g/ e- B5 e
29
1 ]1 @( v3 V6 N2 d*3.4.2 Plausibility of the selection rule for the optima3 _, w+ \& N3 V7 ^8 h
*3.4.3 The algebraic Riccati equation, 230
4 ?: L0 i5 x5 ^+ c3 m n: J3.5 Discrcto-Time Systems, 2! e+ M1 L- u* c% \3 ?9 R8 I
3.5.1 Modal controlability, 2383 P" w1 P# K! Q( N. V+ ]1 P
3.5.2 Controllability to the origin, state-pariable% C' Z3 V! C8 O. e; d
e principle of optimality, 239
, Z% ]. @) Y2 {0 g. g3 M- f*3.5.3 The diserele-lime quadratic regulator problem, 243
) s1 y8 ^& P, v: {5 s*3.5.4 Square-root and related algorithms, 245
% l/ t! _' d# ~' E0 j2 a8 LCHAPTER 4 ASYMPTOTIC OBSERVERS AND; y) B! b/ S- G! d1 I
COMPENSATOR DESIGN
* X& ~1 _" I9 F% Y; I$ y, ?259# c+ R5 c5 L. E6 q- Q# d: L
4.0 Introduction, 2599 {4 J0 l+ t5 R5 h( k
4.1 Asymptotic Observers for State Measurcment, 250# Y6 L/ V. Y$ c5 [* e- ?. ]# k
4.2 Combined Obscrvcr-Controllcr Compensators, 268
7 [& D0 I7 X5 V/ x4.3 Rcduced-order observers 281
" ]" B; a$ H! V7 l4.4 An Optimality Critcrion for Choosing Observer Poles, 293
7 Z" A4 J- P) P) |2 c+ I+ ]4.5 Direct Transfer Function Design Procedures, 2976 |7 l$ U" d6 m s# s5 @7 W
4.5,I A transfer function reformulation of the
; m3 i8 Z; { |6 H% M* b2 izer design, 298 }- f3 ?6 i$ D* l5 r1 K2 j
4.5.2 Some variants of the observer-controller design, 304
* I6 V [8 K0 D7 |9 E0 U4.5.3 Design via polynomial equationf, 306/ g7 ~2 _& f5 M2 C3 ]$ K' C
CHAPTER 5 SOME ALGEBRAIC COMPLEMENTS* L/ i8 Y# W' J0 p! w
314
8 U$ Z* F6 y) p# V% Z- o5.0 Introduction. 314
, @* G0 r# X2 A5.1 Abstract Approach to State -space Realization
; K* ?3 V+ z: ?/ iMethods; Nerode Equivalence, 315
% ^. q1 W. L' Z5.1.1 Realis+ h7 c; @ d) Z X1 q
cov parame
9 e9 }4 t" z: h5.2 Geometric Interpretation of Similarity Transformations7 ^/ c+ M9 e( F/ a# G
Linear vector Spaces, 329% r+ x5 k9 I: Q. Q- L) T3 N
5.2. Vectors in n-space: linear independence, 330
. e7 D% y3 G |$ d& h4 h4 h5.2.2 Matrices and transformations, 333
8 [) P- O/ F" q5.2.3 Vector subspaces, 338" E, l& B4 H$ T2 @3 s% X9 E
5.2 4 Abstract linear vector spaces, 341* L: u% t8 m0 c+ F. E5 \1 v
6 ]' B9 Z/ [# K5 o5 Q4 E% ^# ^) W
CHAPTER 6 STATE-SPACE AND MATRIX-FRACTION( S: |# p: @* H9 L
DESCRIPTIONS OF MULTIVARIABLE
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345% { R2 M- [+ s: Z1 P u+ ~' [
6.0 Introd
: T' |4 s" i" z9 e1 {3 ` X$ a61 Some Direct realizations of multivarable transfer
k7 K* i1 @4 ]5 L8 F# h" a6.2 State Observability and Controllability, |9 f: U t" b# ~8 u9 g
Matrix-Frection Descriptions, 352
# r: k- |6 @! z62. The observability and controllability matrices, 358 [' g! P1 `# [9 x
6.2.2 Standard forms for noncontrollable nonobservable
3 L* v1 y- Z# qIns minimal realizations. 3604 Z/ A$ B8 `. q. @( X1 C
6. 2. 3 Mairixfrcction description, 367
# n0 e( s. w$ |8 F6 R7 p/ B8 J. j) @6.3 Some pr
t$ ^. @) i; l$ X% A, e, A: l5 b6.3.2 Unimodular matrices, the Hermite form and
% B ~/ d9 Q" N, ~- xcoprime polynomial matrices, 373
0 Y. d0 N$ Z8 K3 S; {- i' y+ p632# _7 Z9 f. s+ u W' Z
d some6 M3 _" D. s& t* y7 D `7 p
application, 382
5 t" ]: T0 R; L63.3 The Smith form and related results, 390
/ B' a+ S- d+ r+ yrix pencils, and Kronecker form, 393 L. B% a2 r9 g% ?) I+ V4 C$ p
6.4 Some Basic Sta: e-Space Realizations, 403. w- H5 i6 W$ R# M+ c
64.i2 M0 G5 d& h$ Z k/ E# J# c$ \) K
form realizations from right MFDs, 403
2 o& s7 E O& E$ }8 X! @; |$ ailer fo1 i. V( T7 L2 S) z; a# P
lization 408
/ g: H( m' g% O. Y6.4.5 Observer form realizations from left MFDs, 413( h9 S: h9 F! u6 J, K# v4 S. T
6.4.4 Controllability- and observability form realizations, 4.17& [8 @3 ^ ?6 Y9 K& V
6.4.5 Canonical state-space realizations and canonical& m9 P, L* v! f/ Z1 S
6.4.6 Transformations of state-space realizations, 424! A1 z. y5 o/ F+ K5 \
6.5 Some Properties of Rational Matrices, 439
: E' ]' m' z ?/ |6.5.1 Irreducible MFDs and minimai realizations, 4397 c, h* H8 C4 n5 N- w& l
6.5.2 The Smith-McMillan form ofH(s), 4437 A% s7 _0 {. s7 ?0 `( w
6.5.3 Poles and zeros of multivariable transfer functions, 446
9 J2 z( ^7 M$ O' Q: [6.5.4 Nullspace structure, minimal polynomial bases: r! b; y1 e% y4 E6 R* @
and Kronecker indices, 455+ P3 b- ~* g* E1 {; ^' U
*6.6 Ncode eg
6 `, [/ O# K$ R4 l; F jble Systems, 470, z3 V) p R" R$ }' {. G8 D
6.7 Canonical Matrix-Fraction and State-space g$ s1 h+ t( Q+ O0 j
6.7.1 Hermite-form MFDs and scheme I siaie-space, z d1 j! r2 ^/ \/ y
67,2Pc6 n( @# W9 |, x$ b7 g4 i, v! C
nialechelon mFd
9 i/ ^+ Z- y6 U' {, s# r8 iScheme i realizations, 48.7" Y9 _; u7 Z5 T, g6 a3 \
*6.7.3 The formal definition of canonical form, 492$ `) u3 f1 Q/ v4 i1 K7 I
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