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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.: T/ T7 p. ?. ~$ F' x* z) I: v) F& w. [
LINEAR SYSTHNIE
2 b- C2 r2 H3 f3 E5 D" JTHOMAS KAILATH
C. K- U! w" x$ \5 n" QDepartment of Electrical Engineering7 L( D5 F G; }5 c B4 y- q
Stanford University! z e8 u5 o4 i7 g9 U- _
鹏T只.严积死堅您: c$ I# ^% O; b. Y1 `7 u: x& t( e
PRENTICE-HALL, INC,, Englewood Cliffs, N.J. 07632
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Library of Congress Cataloging in Publication Data1 ]. }4 S P2 G( E9 B( Y0 K4 r' @
af systems.6 }* U, v1 y7 s8 y7 r
Includes bibliographies and index.
5 u4 [+ X. E6 c+ aQA402K29598000379-14928% w8 @9 o4 s. ~2 V8 f
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. k$ K3 b, c" b0 G$ E V& Ccher moffa and lori opre. G" P' V' X; Y2 l# W& C+ z
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C 1980 by Prentice-Hall, Inc; H# o' B' h/ Q9 R5 D: I: e( H) g
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WEITEHALL BOOKS LI2 [4 s0 i. R6 u0 o" [( E
FElli
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SarAh
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% K$ Y* o& _* E7 x! a! k4 gstyle and taste
+ E1 J2 D) g# J4 b0 Elove and support; }$ Z9 u; W' s
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1 _6 Z& Z$ b7 v0 C& p0 R
CONTHE
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PREFACE; i2 }8 L! F- c; M
XI[I
0 S1 |" x7 R/ XCHAPTER 1 BACKGROUND MATERIAL
/ o2 o, J1 v- \: B! {I I Some Subtleties in the Definition of linearity,2 E- R$ I7 B6 \& z
1.2 Unilateral Laplace Transforms and a generalized2 i( u2 o7 d% t l
1.3 Impulsive Functions, Signal Representations,* O$ ?5 C% }7 k9 R% [
and Input-Output Relations, I4% b& E' @5 x% s! ~
1.4 Some remarks on the Use of matrices, 27
7 c+ |0 c1 q1 `+ t2 n1 ICHAPTER 2 STATE-SPACE DESCRIPTIONS-SOME/ c1 b9 I: j9 H. U- k8 s( _
BASIC CONCEPTS
6 X6 s* G( j* q% h: M2 t20
! N) n, m3 S9 @# n$ Q8 K9 P) kal realizations. 35
7 e% E" g0 P( B7 m5 c5 a1 p* p. `! t/ S2.1.I Some remarks on analog computers, 356 o# D& c( a; H4 ~+ z
2.1 2 Four canonical realizations, 37
) p. o+ E: A" O/ L4 d4 k- _7 U6 G* c2.1.3 Parallel and cascade realizations, 45& M* N( V) |- n, d( S$ x5 \5 y! m! B
Sections so marked throughout Contents may be skipped8 F- ?6 P$ ^7 K( w0 {
: f! F/ G' U! L1 e
Yll
. J' V D1 e% E/ Z# H2.2 State Equations in the Time and fre
9 k' u) A9 m/ s4 h3 D6 v( B22. i Matrix notation and state space equations, 50
0 c0 t) S! ?+ X7 H; t22.2 obtuinii
9 Q$ ]( w7 ^$ @, U9 S1 ~乎 lions directy--s0me8 J/ ^+ M+ f2 n( L: A
examples, linearization, 55
! m' y( V. P% s6 A) c2. 2.3 A definition of state, 628 X8 v* ?9 Y' R2 J( D
a2.4 More names and definitions,66
. ^8 @( P6 C( r% G5 G: z2.3 Initial Conditions for Analog-Computer simulation
$ Y& f- b2 f8 n" t, pObservability and Controllability for Continuous
5 j2 S. _9 _1 R; Gand discrete- Time realizations , 79
" |/ g9 \2 A D! S- B/ _# U- z2.3I Determining the initial conditions,szare
; x0 L. j$ z3 _. y23. 2 Setting up initial conditions, state controllability, &4- b& h5 [5 I4 S% H# c& \
23.3D
# X5 H: q$ W& }# p7 Y( kbility, 90
$ o+ \/ o( Y6 i0 @ P" [, W1 h# _23.4S
H& V1 _3 a" m8 _0 Y2. 4 Further Aspects of Controllability and obs" E9 L* h% ~* m% O# r* N
ty,20
! \. y! h9 ]2 ^0 B9 u: E; [2.4I Joins observability and controlability, the uses! _( |9 C7 m+ ?0 x5 T, ]
diagonal forms, 120
; R1 R3 t, b% O: S2 Standard forms for noncontrollable andor5 u3 x8 U; F q# J
nonobservable sy stems, 728
5 M( {4 B4 J: P# D a* c# h( v2.4.3 The PapoyBelevirch-Hautus tests for
3 l" i l6 l: v6 w9 a( a0 Y& jcontrollability and observability, 13.5
; }; [! J' U, [+ v1 sR2.4.4 Some tests for relatively prime polynomials, 140
& }0 m; F c( x6 M( H2 @: l7 V2. 4,5 Some worked exan% n' l5 X* E3 G; ?
*2.5 Solutions of state Equations and modal
e: p1 N- ~" `7 G" hDecompositions, 160* Q& h2 I+ }, y0 D
2.5.1 Time-invariant equations and matrix
' d# O* S C9 o0 ~, Hexponentials, 162
* V3 C+ J! u- O3 A! I5 c4 R. E9 F2.5.2 Modes of oscillation and modal decompositions, 168
' Y( a. a# j( W9 |2.6 A Glimpse of Stability Thcory, 17
& u$ t8 a% E4 o! P9 R2, 6.2 The Lyapunov criterion, I77
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l stability
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2.5.3 A stability resuit for linearised systems, 180
& ~5 w/ w# |& B' ]4 Q3 I5 _% P, }CHAPTER 3 LINEAR STATE-VARIABLE FEEDBACK- J7 Y, O+ @2 j& W
3.0 Introduction, 187
- Z' R* e# a' e$ T3.1 Analysis of Stabilization by Output Feedback 188' w' G$ t: o3 V
3. 2 State-Variable Feedback and Modal Controllability, 7972 n0 R4 Y2 _- b( ~% T
3. 2. Some formulas for the
+ B8 _5 e# ]. p- x) j$ J; O2 y( B3.2.2 A transfer function approach, 2021 M: g6 j8 C( N) f, U% g+ l
3 Some aspects of state-yariable feedback, 204
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3, 3 Some worked Examples, 2097 S5 a; x- ]) V @2 j
4 Quadratic Regulator Theory for ContinuousTime
5 x, E! q+ z) X- d7 e8 r- A2 C( ZSystems, 218% {9 B0 B0 w. n+ f; h
3.4,7O; z% ^" U- A. K; [' l7 k
ite solut4 z5 ?2 [4 w6 w: S2 V
29
+ B' }7 S% ^. }8 \# S8 d) t*3.4.2 Plausibility of the selection rule for the optima- L* [; U7 _/ c; U" H- ?* V
*3.4.3 The algebraic Riccati equation, 230
, ?( W5 r( J% _1 f- ^3 ]3.5 Discrcto-Time Systems, 2
/ b, e, }. y$ J, b }3.5.1 Modal controlability, 238* c% T2 D2 ?5 v+ u- D: {; O& q
3.5.2 Controllability to the origin, state-pariable
# w. L: ]1 u7 U; ?: Fe principle of optimality, 239. Y) s' C8 s6 w4 b
*3.5.3 The diserele-lime quadratic regulator problem, 243
$ P" f' ^" |+ t$ u( a; t; ^, r6 t*3.5.4 Square-root and related algorithms, 245+ j+ H! \4 s W- t2 n; Y
CHAPTER 4 ASYMPTOTIC OBSERVERS AND
4 O" ]( M& {" s' y% aCOMPENSATOR DESIGN* t* p* K2 n- t+ h
259% o. i# R( n4 \2 o
4.0 Introduction, 259
; z7 r+ |6 j) o( @; S: R* l# g4.1 Asymptotic Observers for State Measurcment, 250+ o I$ f) U; m1 X. P
4.2 Combined Obscrvcr-Controllcr Compensators, 268
7 t- r1 a$ P9 n/ G# M4.3 Rcduced-order observers 281
( T% U, j3 E! g2 Q1 y4.4 An Optimality Critcrion for Choosing Observer Poles, 293
/ l- q A% i# c0 ?4 @4.5 Direct Transfer Function Design Procedures, 297- R- P& l% t% Q7 W% Q
4.5,I A transfer function reformulation of the
9 m M3 Z& N7 S' s& nzer design, 298: [" z4 y0 W s4 {
4.5.2 Some variants of the observer-controller design, 304, P5 h/ \. V6 `3 f' U
4.5.3 Design via polynomial equationf, 3069 f- c& a/ ]' u8 O6 M
CHAPTER 5 SOME ALGEBRAIC COMPLEMENTS& h2 V5 D6 n- h( f$ i& r9 Y
314
6 t$ Q9 i" Y7 q* }, h/ d5.0 Introduction. 314
% ^# Q9 Z7 s/ N0 K: |5.1 Abstract Approach to State -space Realization8 P1 r- z) K- T/ B* a3 I8 F
Methods; Nerode Equivalence, 315
2 P! d3 j# N! ]5.1.1 Realis$ F% x) T" q2 F O
cov parame
# G, Q: ^/ o/ i% H! E8 f. w; n5.2 Geometric Interpretation of Similarity Transformations( J- @, P7 t# K O& D
Linear vector Spaces, 329
. p) B$ x5 T v: D% C5.2. Vectors in n-space: linear independence, 330
0 m/ [/ _9 U5 N! L2 H1 g5.2.2 Matrices and transformations, 333
8 |- R) g- w4 S5.2.3 Vector subspaces, 338( f, j- U$ R+ [6 B0 ^
5.2 4 Abstract linear vector spaces, 3411 W# J# \3 a' L) ~2 F
! c2 g+ Z }5 |1 j& i3 W; S, ?CHAPTER 6 STATE-SPACE AND MATRIX-FRACTION
B( N9 }$ p, G( ?% c1 eDESCRIPTIONS OF MULTIVARIABLE
# [1 ?8 V1 w, {" x4 q, z1 n3 | @5 rSYSTEM
7 B3 ^) o+ F2 @; X; S2 ]+ E& W345
: ^3 t4 O Z4 S6.0 Introd
l) M6 ^9 d }4 {* e61 Some Direct realizations of multivarable transfer# Y. y4 _, x5 y4 w
6.2 State Observability and Controllability. i, w5 i P }3 c) p
Matrix-Frection Descriptions, 352
$ N$ @: D8 x3 ^+ V0 C62. The observability and controllability matrices, 358 N/ s; M% W* D. y3 d
6.2.2 Standard forms for noncontrollable nonobservable
' _. [! d1 l) v' } MIns minimal realizations. 360
2 [6 {! S6 `* n, I9 f, v2 ~6. 2. 3 Mairixfrcction description, 367
$ o: d$ X" ?+ y" r1 k7 I; ?6.3 Some pr
- c- |% f& U. q6.3.2 Unimodular matrices, the Hermite form and
+ l; @0 \3 r8 dcoprime polynomial matrices, 373. H' z: X0 E2 [2 m6 Y
632- K. M- d. N8 }1 Q' }* b
d some2 a p0 [! J( s4 r1 U* r$ } k
application, 382
7 x& r# k: l2 @63.3 The Smith form and related results, 390
1 B) P0 ]% j. {4 s( }' Q- vrix pencils, and Kronecker form, 393
$ \8 l8 o# K7 [, T/ B6.4 Some Basic Sta: e-Space Realizations, 4036 p: l8 [7 w& }' S4 o& c) A
64.i' `( c7 T+ y* d) m" `; u1 i( ~
form realizations from right MFDs, 403
5 d3 ~% V& K' z$ p, i# uiler fo
# s8 N8 N6 }8 {% @5 O" s; [/ C3 Z0 xlization 408" M4 P$ a7 Y: k! f0 p$ d- p$ g
6.4.5 Observer form realizations from left MFDs, 413) @/ n" [( z n( _% o
6.4.4 Controllability- and observability form realizations, 4.17
+ N* n( {/ r% |( q6.4.5 Canonical state-space realizations and canonical
6 l" Q1 ^ B; `" f. t- }6.4.6 Transformations of state-space realizations, 4243 M0 `4 K6 `- j& \ f5 V; V( l
6.5 Some Properties of Rational Matrices, 439
. N6 H* I+ T; Y `6.5.1 Irreducible MFDs and minimai realizations, 4399 ~" }( @5 O& b0 W
6.5.2 The Smith-McMillan form ofH(s), 443. g6 f; @* I' i5 d) h* M
6.5.3 Poles and zeros of multivariable transfer functions, 446! V0 O0 f) M) ]6 f0 I; R
6.5.4 Nullspace structure, minimal polynomial bases
9 L1 m; z- O1 G i: Q' l2 E9 cand Kronecker indices, 4550 F [( j& v N0 n$ j
*6.6 Ncode eg
/ {/ o+ V- l9 {ble Systems, 4704 {( Z1 {& c2 Q Y8 H; @$ H
6.7 Canonical Matrix-Fraction and State-space" @* I$ G6 {" u
6.7.1 Hermite-form MFDs and scheme I siaie-space
$ |7 V7 ]; _/ }1 L6 }6 b" M67,2Pc
( H7 \0 b& h0 C! S" I2 L k" Cnialechelon mFd
0 Q5 `& Y0 o* y- T$ lScheme i realizations, 48.7
5 U2 v' D _" y1 K4 f& T ^# U*6.7.3 The formal definition of canonical form, 492
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