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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.
: h) _# F" |2 v( T8 h8 jLINEAR SYSTHNIE9 G. u5 Q& \. [, i1 j- N& y6 r
THOMAS KAILATH; k3 c, J+ p" M- t7 | z# \1 K
Department of Electrical Engineering" {% @; [) q- l/ {+ n1 t& n
Stanford University
/ f* o) y8 J! m: F; g9 a' z鹏T只.严积死堅您
* ]1 F) f* G& j+ t' _4 ~& ?* O* PPRENTICE-HALL, INC,, Englewood Cliffs, N.J. 07632
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4 G+ O! F3 a' s8 ?4 kLibrary of Congress Cataloging in Publication Data' d( I" c, H) j/ s; _
af systems.5 O; e$ \0 k' o
Includes bibliographies and index.
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* ~1 A- k' n. h SC 1980 by Prentice-Hall, Inc+ X$ `6 S4 s! W: v
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CONTHE
9 }$ W$ z" f0 e0 Xiut' J% x7 O6 m" ^/ l) i! Z8 L
PREFACE
* f5 B u6 z2 z" l* Y2 c+ SXI[I, F- q2 g0 w" ]. S* ~8 z
CHAPTER 1 BACKGROUND MATERIAL
( E' p7 ~! Y, G' e: Q. @& ^0 I& f- oI I Some Subtleties in the Definition of linearity,: U5 u% O2 w! q1 P% |; u4 [; w
1.2 Unilateral Laplace Transforms and a generalized
6 F! L9 g2 O7 `6 v: y1.3 Impulsive Functions, Signal Representations,4 g! `, d6 x/ P& D
and Input-Output Relations, I4
: }9 I2 o! m# }7 v1.4 Some remarks on the Use of matrices, 27" g7 c+ A9 A6 v3 r1 `! M4 X
CHAPTER 2 STATE-SPACE DESCRIPTIONS-SOME
4 P+ m7 Z$ X7 ZBASIC CONCEPTS# ]/ M, w& Z+ Q7 z: i
20
v1 V1 ~9 q; H4 {+ x" b' Fal realizations. 35% u& D4 F: p1 k, @/ f- D1 V- m
2.1.I Some remarks on analog computers, 35
2 U5 X( p: }+ B& t9 W2.1 2 Four canonical realizations, 37
3 J+ J1 v: ?! e: b2.1.3 Parallel and cascade realizations, 45
2 \0 i( o# } W$ A) }3 {Sections so marked throughout Contents may be skipped
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Yll; ^& f/ i: \4 p' c8 a9 e: {
2.2 State Equations in the Time and fre
' A* e0 J5 [% u6 w! j2 f2 Z+ c22. i Matrix notation and state space equations, 50 c+ [9 q& v# C7 z/ \' n N
22.2 obtuinii
( j1 W4 p' N' q0 U, x3 v乎 lions directy--s0me8 P# E- |. J) ]0 s
examples, linearization, 55( ]' R9 r& A3 \$ l% H! d" V
2. 2.3 A definition of state, 62
2 x- r3 }9 n) W: P& {$ ?0 ~4 Ta2.4 More names and definitions,66( u0 J( _6 L6 J. O9 h# Z
2.3 Initial Conditions for Analog-Computer simulation+ J1 H6 a! o; K8 a- f9 ]" M8 I
Observability and Controllability for Continuous; z( S$ ^4 X( H$ `; I
and discrete- Time realizations , 79; a; k6 ~; ^3 B! i; ~! O! |
2.3I Determining the initial conditions,szare6 S* \# I1 F' J2 c" a
23. 2 Setting up initial conditions, state controllability, &4
! q* Q3 o& ? m8 E- f' s4 K23.3D3 U) o2 J7 X# [
bility, 90
8 _7 r8 a2 [6 Q$ s" U( W0 m23.4S
( P7 I+ s' L/ @3 E8 |, R6 W0 H2. 4 Further Aspects of Controllability and obs
' L2 n4 U o# fty,207 ~# s: o4 D+ z
2.4I Joins observability and controlability, the uses
7 W8 u9 S, Q, C. H Kdiagonal forms, 120
9 G6 ?9 E" Y( {3 V! X( o2 Standard forms for noncontrollable andor
9 \* Z- a* q# ynonobservable sy stems, 728
. O. j( f) R6 c1 p p$ n5 A2.4.3 The PapoyBelevirch-Hautus tests for7 F/ k. [8 m; Y H, Q
controllability and observability, 13.5
! W. U# ~5 ~7 ?' s- |R2.4.4 Some tests for relatively prime polynomials, 140
+ }* R2 j( X2 E9 j$ W2 \$ Y8 X2. 4,5 Some worked exan. ^. d6 A% r9 i6 s' t
*2.5 Solutions of state Equations and modal+ B: H9 M0 }6 e; O2 O2 I
Decompositions, 160
& i- K1 J' q/ W: Q2.5.1 Time-invariant equations and matrix8 k+ p' D2 k# R$ Y, a2 b' t+ l
exponentials, 162
- J% ^1 f$ {! W) f) h; @0 k2.5.2 Modes of oscillation and modal decompositions, 168
3 Z0 x- f+ x, T& N5 f1 N2.6 A Glimpse of Stability Thcory, 17
6 O4 e: ~* W# _5 h" p1 _2, 6.2 The Lyapunov criterion, I77
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7 k/ J3 }3 D8 ~8 k" Fl stability% q1 M3 D* _+ o6 s1 H# ^) ~2 ~3 m6 B& W
75
* N+ M- S4 g& L& r" M# l* R! \2.5.3 A stability resuit for linearised systems, 180( e: A6 q0 Z" [. R3 G
CHAPTER 3 LINEAR STATE-VARIABLE FEEDBACK3 G4 G' t; z' F0 h. K
3.0 Introduction, 187
/ q* x0 F c! _: B" ~2 q9 F3.1 Analysis of Stabilization by Output Feedback 188
7 ]' j5 K+ ^8 ]% C3. 2 State-Variable Feedback and Modal Controllability, 797+ O9 h% \& u) n
3. 2. Some formulas for the
A& B! G2 ^; ^& d/ S7 ]3.2.2 A transfer function approach, 202. u, M0 P9 o' p+ y
3 Some aspects of state-yariable feedback, 204
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3, 3 Some worked Examples, 209
$ F/ z' }, H1 s& y0 l' r2 I7 N4 Quadratic Regulator Theory for ContinuousTime) _; [- d0 j4 r: j8 x
Systems, 218
* i1 k2 u( |9 y7 ?9 x7 K* z0 t3.4,7O9 l0 ~$ j0 @ q1 _# C( ]# e6 X
ite solut& _) u& o( V" _) g. c$ q
29& M$ P- t' I. w9 w; b
*3.4.2 Plausibility of the selection rule for the optima& U# ]& c+ F6 e) m0 g2 d* S, U, p
*3.4.3 The algebraic Riccati equation, 230
5 x* J; I& x- {) _3.5 Discrcto-Time Systems, 24 w. n* u' C6 U/ ~
3.5.1 Modal controlability, 238
& a# t0 s: ^1 t5 f3.5.2 Controllability to the origin, state-pariable
! m4 R7 f# y. B/ A. ?2 m1 ae principle of optimality, 239/ ]" I8 J& h. l) Q2 Y
*3.5.3 The diserele-lime quadratic regulator problem, 243
4 H% J: f4 a# ]5 e* w*3.5.4 Square-root and related algorithms, 245
+ D1 p8 ~ S! {. G/ D8 UCHAPTER 4 ASYMPTOTIC OBSERVERS AND" D9 b8 }9 T' V/ }% w
COMPENSATOR DESIGN
3 U8 v% z7 F0 \4 W& P; a2 ^6 z259
2 @+ h% h2 d- j6 w# F0 Y4.0 Introduction, 259
& k1 s% r6 ^; O$ x4.1 Asymptotic Observers for State Measurcment, 250! p4 ~# l: l6 k; p
4.2 Combined Obscrvcr-Controllcr Compensators, 268! i' l5 ]: S2 |/ ~/ P) m
4.3 Rcduced-order observers 281+ O5 s3 h8 q C/ e% R5 H# z
4.4 An Optimality Critcrion for Choosing Observer Poles, 293
; _& g! k4 T$ A, u4.5 Direct Transfer Function Design Procedures, 297
+ n, c3 o/ U% ]' S4.5,I A transfer function reformulation of the
# L9 U3 T$ b' B- H7 O% s# j# w( `3 tzer design, 298
5 s% q5 w6 B, E4.5.2 Some variants of the observer-controller design, 304' [# c5 U+ y! \
4.5.3 Design via polynomial equationf, 306
8 Q/ V2 [$ \2 Q1 N" cCHAPTER 5 SOME ALGEBRAIC COMPLEMENTS9 D' ^! Q- j# i
3147 I$ i; P) I/ N/ z0 V0 z5 \6 o
5.0 Introduction. 314
* n+ j$ s* |9 X5.1 Abstract Approach to State -space Realization
1 |7 b: M" ]& e/ {* q2 OMethods; Nerode Equivalence, 3158 z$ I' j9 | b( u7 J/ T" `
5.1.1 Realis7 O3 [5 Z& E: p0 p) v
cov parame: t: {" J" ]5 ~" L; I/ V
5.2 Geometric Interpretation of Similarity Transformations4 A/ m: N. P0 T, |
Linear vector Spaces, 3299 a$ E) _/ @3 n4 ^
5.2. Vectors in n-space: linear independence, 330' Q9 ?" y5 H3 U( C6 C5 U) x% |- S. |
5.2.2 Matrices and transformations, 333
% S3 ]. Y2 N( a5.2.3 Vector subspaces, 338
2 Q8 f/ t8 Q: [4 ~5 K+ `5.2 4 Abstract linear vector spaces, 341
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CHAPTER 6 STATE-SPACE AND MATRIX-FRACTION( Z4 q. J* F+ [$ k
DESCRIPTIONS OF MULTIVARIABLE
' B u! o4 h% _6 O' OSYSTEM
+ |- G! X P i1 S345
$ |+ y: x9 D+ I; `% v" p0 c) u6.0 Introd# p7 n7 ]* @* b/ W1 S
61 Some Direct realizations of multivarable transfer
) t' @" m0 E% I6.2 State Observability and Controllability) L2 K3 A1 t4 R
Matrix-Frection Descriptions, 352
' ^4 L$ p5 m9 m/ T4 g62. The observability and controllability matrices, 35* l6 d' [0 \8 z& L7 [& ]" @& }
6.2.2 Standard forms for noncontrollable nonobservable8 x$ h9 h6 K3 T1 k8 N0 q( R
Ins minimal realizations. 360
# \' U2 Y' R0 r6. 2. 3 Mairixfrcction description, 3676 m/ d* \4 r/ ]3 N+ L( T2 O% P
6.3 Some pr
0 G, @- k- I$ \" O! f* A- r, k9 A6.3.2 Unimodular matrices, the Hermite form and. {. m7 S8 H! y! o9 o
coprime polynomial matrices, 373: z1 G, M$ b7 S1 m7 n9 n
6324 e* }- P- c% e" k
d some7 g8 W- `* W1 `# O# z: w! X
application, 382
4 U& K4 q4 x" F' E3 I0 i63.3 The Smith form and related results, 3902 A1 R# O& r b" d
rix pencils, and Kronecker form, 393
2 B1 \* Z# m# a% `. \- A6.4 Some Basic Sta: e-Space Realizations, 403
; r4 _; A% f p7 E8 O64.i$ S2 @% [, J' j Q
form realizations from right MFDs, 403
. X7 e: k! f% L9 ]( D! p3 I) @1 Ailer fo& Y2 m }( K+ o" g0 A
lization 408% c+ ]# j8 y" b/ B* \& X
6.4.5 Observer form realizations from left MFDs, 4133 q- o2 D- A& |& u1 m
6.4.4 Controllability- and observability form realizations, 4.17) r$ e% d. R1 q8 _
6.4.5 Canonical state-space realizations and canonical& o0 n( o* e! R+ X3 C% v& o8 E
6.4.6 Transformations of state-space realizations, 424
5 c( E1 J' d) N6.5 Some Properties of Rational Matrices, 439( s& `' r- ~. G( |
6.5.1 Irreducible MFDs and minimai realizations, 4395 T* m- H+ X. ?# }* F
6.5.2 The Smith-McMillan form ofH(s), 443; t; r+ @/ O. X
6.5.3 Poles and zeros of multivariable transfer functions, 446: {+ Z/ X5 |7 g6 C/ X' G/ O
6.5.4 Nullspace structure, minimal polynomial bases
) Y$ ]4 {9 t# u$ l. k; _and Kronecker indices, 455) U% C4 ~3 B8 K0 v4 L
*6.6 Ncode eg+ l6 J7 H4 x [4 u) L& w, w& D" u
ble Systems, 4709 d8 l1 n" n+ u \8 l
6.7 Canonical Matrix-Fraction and State-space
$ M3 z0 l1 [5 a" m$ h- v* d, j, t6.7.1 Hermite-form MFDs and scheme I siaie-space
0 P$ `2 ~% J+ z2 f/ v67,2Pc' a& A( m( j# E6 Z% k8 X) Y6 ^
nialechelon mFd
/ h" v" E% F( |" v* KScheme i realizations, 48.75 w7 j3 |( d, ]5 J4 n, l/ n5 r* q* y
*6.7.3 The formal definition of canonical form, 492
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