Operator Approach to Linear Control Systems

411 Pages·1996·10.172 MB·English

by  A. Cheremensky, V. Fomin (auth.)

Operator Approach to Linear Control Systems
Mathematics and Its Applications
Managing Editor:
M. HAZEWINKEL
Centre for Mathematics and Computer Science, Amsterdam, The Netherlands
Volume 345
Operator Approach to
Linear Control Systems
by
A. Cheremensky
Institute of Mechanics,
Bulgarian Academy of Sciences,
Sofia, Bulgaria
and
V.Fomin
Department of Mathematics and Mechanics,
St Petersburg University,
St Petersburg-Petrodvoretz, Russia
SPRINGER-SCIENCE+BUSINESS MEDIA, B.V.
A C.I.P. Catalogue record for this book is available from the Library of Congress.
ISBN 978-94-010-6544-3 ISBN 978-94-009-0127-8 (eBook)
DOI 10.1007/978-94-009-0127-8
Printed on acid-free paper
All Rights Reserved
© Springer Science+ Business Media Dordrecht
Originally published by Kluwer Acedemic Publishers 1996
Softcover reprint of the hardcover 1st edition 1996
No part of the material protected by this copyright notice may be reproduced or
utilized in any form or by any means, electronic or mechanical,
including photocopying, recording or by any information storage and
retrieval system, without written permission from the copyright owner.
Contents
Preface xi
1 Introduction 1
1.1 Basic notions of systems theory . . . . . 1
1.1.1 Plants as input-output mapping 2
1.1.2 "Free" and "external" variables . 3
1.1.3 Controllers . . . . . . . . 3
1.1.4 Transfer system operators 5
1.1.5 Optimal control . . . 5
1.1.6 Stochastic control . . . . 7
1.1.7 Separation principle . . . 8
1.1.8 Uncertainty in control problems. 8
2 Introduction to systems theory 11
2.1 Linear system and its transfer operators . . . . . . . . . . . . . 11
2.2 Example: one-dimensional time-invariant linear system . . . . . 13
2.3 System operator enlargements and parameterizations of the set
of system solutions . . . . . . . . . . . . . . . . . . . . 15
2.4 Remark . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.5 Example: one-dimensional control plant of first order . 20
2.6 Some optimization problems . 23
2.6.1 Projection lemmas 23
2.6.2 Robustness 24
3 Resolution spaces 27
3.1 Hilbert space . . . . . . . . . . . . . . . . . . 27
3.1.1 Extended and equipped spaces . . . . 27
3.1.2 Classes of transforms in Hilbert space 30
3.1.3 Stochastic elements with their values in an extended space 33
v
vi Contents
3.1.4 Stochastic processes as generalized elements of Hilbert
space . . . . . . . . . 34
3.2 Hilbert resolution space . . . . . . . . . . . 36
3.2.1 Resolution of identity . . . . . . . . 36
3.2.2 Structure of Hilbert resolution space 37
3.2.3 Examples of discrete and absolute continuous resolution
Hilbert space . . . . . . . . . . . . . . . . . . . . . . . . 39
3.2.4 Functional representation of an abstract Hilbert resolu-
tion space . . . . . . . . . . . . . . . . 41
3.3 Space extension endowed with time structure 43
3.3.1 Equipped Hilbert resolution space . . 43
3.3.2 Equipped separable resolution space 44
3.3.3 Resolution structure of extended space . 45
3.3.4 "Integral" representation of elements in extended Hilbert
resolution space . . . . . . . . . . . . . . . . . . . 48
3.3.5 Localized elements of extended resolution space . . . . . 49
3.3.6 Example of localized elements in L2 . . . . . . . . . . . 50
3.3.7 Example of localized elements in discrete resolution space 51
3.3.8 "Frequency" representation of elements of an absolutely
continuous Hilbert resolution space . . . . . . . . . . . . 52
3.3.9 "Frequency" representation of discrete Hilbert resolu-
tion space . . . . . . . . . . . . . . . . 54
3.4 Operators in resolution spaces . . . . . . . . . 56
3.4.1 Operators in Hilbert resolution space . 56
3.4.2 Linear integral operators in L (R) . . 57
2
3.4.3 Additive operators . . . . . . . . . . . 58
3.4.4 Causal operators in extended resolution space . 59
3.4.5 Block representation of linear operators in extended causal
space ....................... . 61
3.4.6 Block representation of integral operator in L 63
2
3.4.7 Closing of operators in extended resolution space 64
3.4.8 Adjoint operators in extended resolution space 65
3.4.9 Linear time-invariant differential system operators 66
3.4.10 Linear operator factorization in an extended resolution
space . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
3.4.11 Linear operator separation in an extended resolution space 75
3.5 Linear time-invariant differential system operators . . . . . . . 75
3.5.1 Frequency description of time-invariant closed-loop sys-
tems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
3.5.2 Transfer matrix function of linear time-invariant system 77
3.5.3 Robustness of time-invariant differential linear systems . 79
3.5.4 Generalized Fourier transform . . . . . . . . 80
3.6 Stationary operators . . . . . . . . . . . . . . . . . 84
3.6.1 Stationary operators in "continuous" time . 84
Contents vii
3.6.2 Symbol of stationary operator . . . . . . . . 85
3.6.3 Stationary operator factorization . . . . . . 86
3.6.4 Generalized Fourier transform in L2(n, R+) 87
3.6.5 Stationary operators in discrete resolution space 88
3.6.6 Time-invariant (stationary) operators acting from one
resolution space to another . . . . . . . . . . . . . . . . 91
4 Linear control plants in a resolution space 93
4.1 Some control problems . . . . . . . . 93
4.1.1 A linear control plant . . . . 93
4.1.2 Admissible control strategies 97
4.1.3 Control aims . . . . . . . . . 100
4.2 Feedback problem . . . . . . . . . . 102
4.3 Feedback in linear structured systems 106
4.3.1 Gauss method application . . . 107
4.3.2 Separation principle . . . . . . 110
4.4 Design of time-invariant systems with fixed space variables 114
4.4.1 Examples of time-invariant systems with fixed space
variables . . . . . . . . . . . . . . . . . . . . . . . . . . 114
4.5 Constructing transfer operators for some time-invariant plants . 118
4.5.1 Corona problem . . . 118
4.5.2 Polynomial technique . . . . . . 123
4.5.3 Larin compensator . . . . . . . . 125
4.5.4 Lagrange-Sylvester interpolation 126
4.5.5 Neutral systems . . . . . . . . . 128
4.6 Robustness of stationary differential closed loop systems 130
4.6.1 Degenerate systems . . . . . . . . . . . . . . 131
4.6.2 Time-invariant difference differential system . 133
4.6.3 Estimation of robustness domain . . . . . . . 134
5 Linear quadratic optimization in preplanned control class 137
5.1 Preplanned optimal controls . . . . . 137
5.1.1 Setting of problem . . . . . . . . . . . . . . . . . . . . 138
5.1.2 Lagrange multiplier method . . . . . . . . . . . . . . . 145
5.2 Linear-quadratic game problem of optimal preplanned control . 150
5.3 Feedback form of preplanned stochastic optimal control . 153
5.3.1 Refining the optimal control problem setting ...... 153
5.3.2 Solving the problem of preplanned optimal control ... 156
5.3.3 Problem ofpreplanned optimal control with time-structure157
5.3.4 Operator Bellman equation . . . . . . . . . . 160
5.3.5 Riccati equation . . . . . . . . . . . . . . . . 161
5.3.6 Example: Riccati equation in Markovian case 164
5.3. 7 Optimal control and Riccati equation . 167
5.3.8 Stationary feedback and Lur'e equation . . . 167
viii Contents
5.4 Special representation of control criteria . . . . . . . . . . 169
5.4.1 General assertion . . . . . . . . . . . . . . . . . . . 169
5.4.2 Outlines of the general assertion proof and some re-
marks . . . . . . . . . . . . . . . . . . . . . . . . . 170
5.4.3 Optimal feedback for known external disturbances 172
5.4.4 Factorization of the weight operator . . . . . . . 17 4
5.5 Design of the preplanned optimal control . . . . . . . . . 175
5.5.1 Statement of the problem of preplanned control 175
5.5.2 Necessary condition of the optimal control problem solv
ability . 0 . . . . . . . . . . . 0 . . . . . . . 0 . 0 . . 176
5.503 Solving of the problem of preplanned optimal control 177
5o5.4 Limit-optimal preplanned control 0 .... 0 0 0 0 179
6 Linear quadratic optimization in feedback control class 181
6.1 Existence of optimal feedback 181
6.1.1 Objective setting . . . . . . . . . 181
6.1.2 Solvability of LQP . . . . . . . . 182
6.1.3 Modification of the optimal LQP 184
6.1.4 Minimax optimal problem . . . . 185
6.2 Abstract variant of the Wiener problem 185
6.2.1 Statement of the operator Wiener problem 186
6.2.2 "Scalar" variant of Wiener problem . . . . 188
6o2.3 Generalized Wiener problem solving on the set of local-
ized elements . . . . . . . . . . . . . . . . . . 0 . 0 . . . 190
6.2.4 Generalized Wiener problem solving on the set of finite-
localized elements ..................... 191
6.2.5 Solving the operator Wiener problem in discrete reso-
lution space . . . . . . . . . . . . . 195
6.206 Wiener problem in stationary case . . . . . . 197
6.3 Wiener method in LQP . . . . . . . . . . . . . . . . 199
6o3.1 Objective setting of abstract control problem 199
6.3.2 Remarks on setting abstract control problem 201
6.303 Transformation of the abstract control problem into the
Wiener problem . . 0 . . . . . . . . . . . . . . . . . . 205
6.3.4 Linear quadratic problem and Pareto optimal control 207
6o3.5 Wiener problem solvability . . . . . . . . 208
6.4 Optimal design o . . . . . . o . . . . . . . . . . . . . o . o . . 211
6.401 Linear-quadratic problem in Hilbert space . . . . . . 211
6.4.2 Solving the control problem in the set of admissible
transform operators .................... 214
6.4.3 Example: finite-dimensional time-invariant control system218
6.5 Systems with incomplete and noisy measurements . . . . . . . 223
6.5.1 Example: Linear time-invariant singular input-singular
output plant . . . . . . . . . . . . . . . . . 0 0 . 0 . . . 225
Contents ix
6.6 Special representation in case of incomplete and noisy measure-
ments 228
6.6.1 Optimal control in discrete resolution space . . . . 228
6.6.2 Linear optimal filtering of stochastic time-series . 230
6.6.3 Separation principle 233
6.6.4 Luenberger observer . . . . . . . . . . . . . . . . 235
6.6.5 Kalman - Bucy filter . . . . . . . . . . . . . . . 239
6. 7 Design of the optimal stabilizing feedback for finite-dimensional
time-invariant plant . . . . . . . . . . . . . . . . . . . . . . . . 242
6.7.1 Setting of an optimal control problem .......... 242
6.7.2 Operator reformulation of the optimal control problem . 247
6.7.3 Frequency reformulation of the optimal control problem 250
6.7.4 Design of the optimal control system transfer function 251
6.7.5 Remarks on the optimal feedback design ......... 253
6.7.6 Optimal feedback design in discrete case ........ 257
6.7.7 Example: optimal stabilizing feedback for stable and
miniphase plant . . . . . . . . . . . . . 265
6.7.8 Optimal control in case of "no" noises . . . . . . . . . 266
7 Finite-dimensional LQP 271
7.1 Stochastic linear quadratic problem on a finite time interval 272
7.1.1 Setting of the stochastic linear-quadratic problem 272
7.1.2 Example: optimal control problem in case of known
states and noises . . . . . . . . . . . . . . . . . . . . . . 273
7.1.3 Synthesis of feedback in case of incomplete observation
data ............................. 274
7.1.4 Example: dependence of optimal control from the choice
of control strategies set . . . . . . . . 276
7.2 General method of optimal control synthesis . . . 277
· 7.2.1 Separation theorem ........... . 278
7.2.2 Synthesis of the optimal control strategy 283
7.2.3 Example: synthesis of the optimal control for random
initial state . . . . . . . . . . . . . . . . 284
7.2.4 Kalman - Bucy filter in optimal control 286
7.2.5 Optimal tracking problem ....... . 288
7.2.6 Nonlinear optimal feedbacks ..... . 290
7.2.7 Example: scalar plant with finite-valued disturbance 292
7.3 Time-invariant SLQP on the infinite time interval . 294
7.3.1 Stochastic optimal control problem setting .. . 294
7.3.2 Reformulation of control problem ........ . 297
7.3.3 Wiener problem connected with optimal control 298
7.3.4 Solvability of SLQP ..... . . 300
7.3.5 Wiener method of solving LQP . 301
7.3.6 Design of the optimal feedback . 302