Numerical Methods for Linear Control Systems - Design and Analysis

710 Pages·2004·25.044 MB·English

by  Datta, Biswa Nath

NUMERICAL METHODS
FOR LINEAR CONTROL
SYSTEMS
Design and Analysis
BISWA NATH DATTA
Department of Mathematical Sciences
Northern Illinois University
DeKalb, IL 60115
Amsterdam * Boston (cid:12)9H eidelberg * London * New York (cid:12)9O xford
Paris (cid:12)9S an Diego (cid:12)9S an Francisco (cid:12)9S ingapore (cid:12)9S ydney (cid:12)9T okyo
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Library of Congress Cataloging-in-Publication Data
Datta, Biswa Nath
Numerical methods for linear control systems design and analysis/B.N. Data.
p. cm.
Included bibliographical references and index.
ISBN 0-12-203590-9
1. Control theory. 2. System analysis. 3. Linear control systems. I. Title.
QA402.3D368 2003
629.8'32--dc22
2003058331
British Library Cataloguing in Publication Data
A catalogue record for this book is available from the British Library
ISBN: 0-12-203590-9
For all information on all Academic Press publications
visit our website at www.academicpress.com
Printed in the United States of America
03 04 05 06 07 08 9 8 7 6 5 4 3 2 1
PREFACE
Remarkable progress has been made in both theory and applications of all impor-
tant areas of control theory. Theory is rich and sophisticated. Some beautiful
applications of control theory are presently being made in aerospace, biomedi-
cal engineering, industrial engineering, robotics, economics, power systems, etc.
Unfortunately, the same assessment of progress does not hold in general for
computations in control theory.
Many of the methods described in earlier control and systems theory text books
were developed before the computer era and were based on approaches that are
not numerically sound. Most of these methods, for example, require reduction of
the system matrices to some condensed forms, such as a companion form or the
Jordan canonical form, and it is well-known that these forms cannot, in general,
be achieved in a numerically stable way.
The situation is, however, changing quite fast. In the last 20 years or so, numeri-
cally viable algorithms have been developed for many of the common linear control
problems. Softwares based on these methods have been developed and are still
being built.
Unfortunately, these methods and softwares do not seem to be widely known
and easily accessible to broad groups of applied mathematicians, control theo-
rists, and practicing control engineers. They are still largely confined in reprints
and preprints (in this context it is noted that a reprint book on "Numerical
Linear Algebra Techniques for Systems and Control" edited by R.V. Patel, A.
Laub, and E Vandooren containing a large number of important published papers
in this area has recently been published by IEEE/CRC Press). The primary
reason for the inaccessibility of these algorithms and the softwares, in my opin-
ion, is that an understanding, efficient implementations, and making possible
modifications of these methods needed for some applications of special inter-
ests, require an interdisciplinary knowledge of linear algebra, numerical linear
algebra, control theory, and computer science; and such a combined expertise is
hard to find.
What is, therefore, needed is a book that makes these algorithms accessible to
a wide variety of users, researchers, and students.
~176176
XXIII
xxiv PREFACE
For practicing users, it is important that the algorithms are described in a manner
that is suitable for easy implementation on a wide range of computers, that impor-
tant aspects of implementations are discussed, and a clear comparative study of
one algorithm over the other for a given problem with respect to efficiency, storage,
numerical stability, etc., is presented. The latter will help the users to choose the
one most suitable for his or her applications. Furthermore, for the students and
researchers, it is important that the mechanism of the development of the algo-
rithms is clearly explained and aspects of perturbation analysis of the problems and
round-off error analyses and convergence properties of the algorithms, whenever
available, are included in some details.
Of course, all these need to be accomplished requiring a minimal amount of
background in the areas mentioned above. This is certainly a difficult and an
ambitious task. But the present book aspires to do that and aims at reaching out to
a broad spectrum of audience in a number of disciplines including mathematics,
control and systems engineering, and other applications areas such as vibrations,
aerospace, space-sciences, and structural and manufacturing engineering.
The recent book on "Computational Methods for Linear Control Systems" by
E H. Petkov, N.D. Christov, and M. M. Konstantinov also aims to fulfill that need
to some extent. The scope of this book is, however, much more limited than that
of the present book.
The current book is an outgrowth of lecture notes compiled by the author over
several years for a graduate course in numerical methods in control theory taught
at Northern Illinois University (almost all students of this course have been math-
ematics students with no prior background in control theory). The book has also
been used in several short courses given by the author including the SIAM short
course on Numerical Methods in Control, Signal, and Image Processing, Seattle,
August 15, 1993 and, the short course on Numerical Methods for Linear Control
and Systems at the International Conference on Mathematical Theory of Networks
and Systems, St. Louis, 1996. The audience of these short courses had varying
backgrounds.
The book covers most important and relevant problems arising in control sys-
tem design and analysis with a special emphasis on computational aspects. These
include:
(cid:12)9 Numerical solutions of state equations and frequency response computations
(cid:12)9 Controllability, observability, and distance to controllability
(cid:12)9 Stability, inertia, robust stability, and distance to instability
(cid:12)9 Numerical solutions and conditioning of Lyapunov, Sylvester, and algebraic
Riccati equations
(cid:12)9 Numerical algorithms for feedback stabilization, eigenvalue and robust
eigenvalue assignment and conditioning of the eigenvalue assignment
problem
PREFACE xxv
(cid:12)9 Numerical algorithms for full-order and reduced-order observer design and
Kalman filtering
(cid:12)9 Realization and subspace algorithms for model identification
(cid:12)9 Algorithms for balanced realization and model reduction
(cid:12)9 Large-scale solutions of control problems
(cid:12)9 H2 and H~ control
The numerical algorithms described in the book have the following desirable
features:
(cid:12)9 Efficiency. Algorithms are of order O (n3).
(cid:12)9 Numerical Stability. Algorithms are either numerically stable or composed
of numerically stable computations.
(cid:12)9 State-of-the-art Algorithms. The state-of-the-art algorithms for all prob-
lems have been included.
(cid:12)9 Comparative Study and Recommendations. Whenever possible, a com-
parison of various algorithms for the same problem with respect to effi-
ciency, numerical stability, and accuracy has been given and based on this
comparative study, recommendation for practicing engineers has been made.
(cid:12)9 Step by Step Explanation. All algorithms have been explained step by step
with illustrative examples illustrating each step of the algorithm.
(cid:12)9 Software and Implementations. Important selected software for each topic
has been included.
(cid:12)9 MATLAB Toolkit. There exists a MATLAB toolkit called MATCONTROL,
implementing major algorithms in the book.
(cid:12)9 Algorithms for both Continuous-time and Discrete-time systems.
Algorithms are described both for continuous-time and discrete-time
systems.
The discussions on theoretical aspects of control theory have been kept to a mini-
mum, only the relevant facts have been mentioned. However, the importance and
applications of the problems have been discussed to an extent to motivate the
readers in mathematics and other areas of science and engineering who are not
familiar with control problems. Numerical Linear Algebra techniques needed to
understand and implement the algorithms have been developed in the book itself
in a concise manner without going into too much details and attempts have been
made to make the techniques understandable to the readers who do not have a
prior background in numerical linear algebra and numerical analysis. Of course,
people having a background in numerical analysis or numerical algebra and/or
control theory will have a definite advantage.
A special emphasis has been given to the clear understanding of the distinction
between a "bad" algorithm and a "numerically effective" algorithm.
xxvi PREFACE
Some discussions on large-scale computing in control have been included
too. The research in this area is still in its infancy, but some aspects of current
research have been included to give the readers a flavor. There is an urgent need
for an expanded research in this area as outlined in the 1988 NSF panel report:
"Future Directions in Control Theory: A Mathematical Perspective" It is
hoped our short coverage in this area will provide enough incentive and motivation
to beginning researchers, both from control theory and applied and computational
mathematics, to work in the area.
The MATLAB toolkit MATCONTROL will help the students and the users under-
stand the merits and drawbacks of one algorithm over the others and possibly help
a user to make a right decision in choosing an ideal algorithm for a particular
application.
Organization of the Book:
The book has fifteen chapters. These fifteen chapters have been organized into
four parts; each part consisting of several chapters, grouped together (roughly)
with a common theme.
Part I. REVIEW OF LINEAR AND NUMERICAL LINEAR ALGEBRA
Chapter 2. A Review of Some Basic Concepts and Results from
Theoretical Linear Algebra
Chapter 3. Some Fundamental Tools and Concepts from Numerical
Linear Algebra
Chapter 4. Canonical Forms Obtained via Orthogonal Transformations
Part II. CONTROL SYSTEM ANALYSIS
Chapter 5. Linear State Space Models and Solutions of the State Equations
Chapter 6. Controllability, Observability and Distance to Uncontrollability
Chapter 7. Stability, Inertia, and Robust Stability
Chapter 8. Numerical Solutions and Conditioning of Lyapunov and
Sylvester Equations
Part III. CONTROL SYSTEMS DESIGN
Chapter 9. Realization and Subspace Identification
Chapter 10. Feedback Stabilization, Eigenvalue Assignment, and Optimal
Control
Chapter 11. Numerical Methods and Conditioning of the Eigenvalue
Assignment Problems
Chapter 12. State Estimation: Observer and the Kalman Filter
Chapter 13. Numerical Solutions and Conditioning of Algebraic Riccati
Equations
Chapter 14. Internal Balancing and Model Reduction
PREFACE xxvii
Part IV. SPECIAL TOPICS
Chapter 15. Large-scale Matrix Computations in Control: Krylov Subspace
Methods
Heading: Intended Audienee
The book can be used as a textbook for an advanced graduate course in con-
trol engineering such as Computational Methods for Control Systems Design and
Analysis and Computer-aided Control System Design or for an advanced gradu-
ate topic course on Numerical Linear Algebra Techniques in Control and Systems
in applied mathematics and scientific computing. Far more material than can be
covered in one semester has been included, so professors can tailor material to par-
ticular courses and develop their own course syllabi out of the book. Above all, the
book is intended to serve as a reference book for practicing engineers and applied
scientists, researchers, and graduate students. The book is also very suitable for
self-study.
LIST OF ALGORITHMS
3.3.1 Back Substitution Method for Upper Triangular System 38
3.4.1 The Cholesky Algorithm 51
3.4.2 LU Factorization of an Upper Hessenberg Matrix 52
3.6.1 Givens QR Factorization 62
3.8.1 Least Squares Solution Using QR Factorization 65
3.9.1 Least Squares Solutions Using the SVD 72
4.5.1 Complete QZ Algorithm for Reduction to Generalized
Schur Form 99
5.3.1 Pad6 Approximation to e A using Scaling and Squaring 132
5.3.2 Schur Algorithm for Matrix Exponential 135
5.3.3 Computing Integrals involving Matrix Exponential 137
5.5.1 Hessenberg Algorithm for the Frequency Response Matrix 144
6.7.1 Staircase Algorithm 174
6.9.1 Newton's Algorithm for Distance to Uncontrollability 186
6.9.2 An Algorithm for Computing #(A, B) 189
7.2.1 Computing the Hz-Norm 212
7.5.1 Computing Inertia and Stability 219
7.6.1 Bisection Algorithm for Estimating Distance to
Continuous-time Instability 225
7.6.2 Bisection Algorithm for Estimating Distance to
a Discrete-Unstable System 229
8.3.1 sep Estimation 261
8.5.1 The Hessenberg-Schur Algorithm for XA + B X = C 270
8.6.1 Cholesky Factor for Continuous Lyapunov Equation 286
8.6.2 Cholesky Factor for Discrete Lyapunov Equation 290
9.3.1 An SVD Algorithm for Minimal Realization 317
9.3.2 A Modified SVD Algorithm for Minimal Realization 322
9.4.1 A Deterministic Subspace Identification Algorithm 327
oo~
XXXUl
xxxiv LIST OF ALGORITHMS
9.4.2 A Subspace Stochastic Identification Algorithm 330
9.4.3 Continuous-Time Frequency-Domain Subspace Identification 333
10.5.1 The Algorithm Continuous-Time LQR Design 366
10.6.1 Bisection Algorithm for Hoe-Norm 376
10.6.2 Two-Step Algorithm for H~-Norm 379
10.7.1 Bisection Method for Complex Stability Radius 390
11.2.1 The Recursive Algorithm for Singe-Input Hessenberg
EVA Problem 411
11.2.2 An RQ Implementation of the Recursive Algorithm 417
11.2.3 A Storage-Efficient Version of the RQ Implementation 418
11.3.1 The Recursive Algorithm for Multi-Input EVA Problem 423
11.3.2 An Algorithm to Assign p(p = 1 or 2) Eigen values 430
11.3.3 The Schur Algorithm for Multi-Input EVA Problem 431
11.3.4 A Parametric Sylvester Equation Algorithm for PEVA 437
11.6.1 Robust Eigenvalue Assignment Algorithm (The KNV Algorithm) 447
12.3.1 Full-Order Observer Design via Sylvester-Observer Equation 472
12.4.1 Reduced-Order Observer Design via EVA 476
12.4.2 Reduced-Order Observer Design via Sylvester-Observer
Equation 479
12.7.1 A Recursive Algorithm for the Multi-Output
Sylvester-Observer Equation 488
12.7.2 A Recursive Block Triangular Algorithm for the
Multi-Output Sylvester-Observer Equation 492
12.8.1 An Algorithm for Constrained Sylvester-Observer Equation 497
12.9.1 State Estimation using Kalman Filter 501
12.10.1 LQG Design for Continuous time System 507
13.5.1 Schur Algorithm for the CARE 542
13.5.2 Generalized Schur Algorithm for the DARE 553
13.5.3 Inverse-Free Generalized Schur Method for the CARE 556
13.5.4 Inverse-Free Generalized Schur Algorithm for the DARE 558
13.5.5 Computing Sign (A) 560
13.5.6 Matrix Sign-Function Algorithm for the CARE 563
13.5.7 Matrix Sign-Function Algorithm for the DARE 566
13.5.8 Newton's Method for the CARE 568
13.5.9 Newton's Method with Line Search for the CARE 572
13.5.10 Newton's Method for the DARE 574
13.5.11 Newton's Method with Line Search for the DARE 577
14.2.1 Internal Balancing of a Continuous-Time Minimal Realization 604
14.2.2 Square-Root Algorithm for Internal Balancing of a
Continuous-Time Nonminimal Realization 607
LIST OF ALGORITHMS xxxv
14.4.1 Model Reduction via Balanced Truncation 614
14.4.2 Schur Algorithm for Continuous-Time Model Reduction 616
14.5.1 Hankel-Norm Approximation of Continuous-Time System 626
15.2.1 Block Arnoldi Algorithm 651
15.2.2 Nonsymmetric Lanczos Algorithm 652
15.4.1 Amoldi Algorithm for Single-input Lyapunov Equation 653
15.4.2 Block Arnoldi Algorithm for Stable Discrete-Time
Lyapunov Equation 654
15.4.3 Restarted Amoldi Algorithm for Sylvester Equation 655
15.4.4 Block Arnoldi Algorithm for Sylvester Equation 656
15.4.5 Amoldi Algorithm for Single-Output Sylvester-Observer
Equation 657
15.4.6 Arnoldi Algorithm for CARE 658
15.5.1 Projection Algorithm for Partial Pole-Placement 659
15.6.1 Lanczos Algorithm for SISO Model Reduction 660
15.6.2 Amoldi Algorithm for SISO Model Reduction 661