Algebraic Theory for Multivariable Linear Systems

381 Pages·1983·4.717 MB·English

by  Hans Blomberg and Raimo Ylinen (Eds.)

Algebraic Theory for
Multivariable Linear Systems
This is Volume 166 in
MATHEMATICS IN SCIENCE AND ENGINEERING
A Series of Monographs and Textbooks
Edited by RICHARD BELLMAN, Uniuersity of Southern California
The complete listing of books in this series is available from the Publisher upon
request.
Algebraic Theory for
Multivariable Linear Systems
Hans Blomberg and Raimo Ylinen
Helsinki University of Technical Research
Technology Centre of Finland
1983
Academic Press
A Subsidiary of Harcourf Brace Jovanovich, Publishers
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No p_art of this hook may he reproduced in any form
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British Library Cataloguing in Publication Dara
momberg, H.
Algebraic theory for multivariahle linear
systems.-(Mathematics in science and engineering)
1. Control theory 2. Polynomials
I. Title 11. Ylinen. R. 111. Series
629.8’32 QA402.3
ISBN 0- 12-107150-2
LCCCN 82-72337
Typeset and printed in Great Britain by Page Bros, (Norwich) Ltd.
Preface
The present book has evolved out of research work done by a great many
people in the course of the last two decades. The work of the older of the
present authors, HB, started back in the late fifties with a series of lecture
notes on control theory. A safe method was sought for the determination
of the correct characteristic polynomial for various feedback and other
configurations with the aid of the then so popular transfer function tech-
nique. The considerations led to the introduction of an “uncancelled” form
of the ordinary transfer function, but the result was not encouraging. In
fact, it became apparent that there was something seriously wrong with
the traditional form of the whole transfer function representation.
It was an easy matter to isolate and explore the weak point in the transfer
function technique-bviously the source of all the trouble was the fact
that the method was based on the assumption of zero initial conditions.
On the other hand, if nonzero initial conditions were to be cared for too,
then the nice transfer function algebra-essentially a field structure-would
no longer work. The classical transfer function technique evidently needed
to be improved, but it was certainly not so easy to see how this should and
could be done. The improved technique should, in the first place, also
make it possible to deal with nonzero initial conditions, but at the same
time it should be based on some suitable algebraic structure in order to
make the calculations easy to perform.
Fortunately, HB had at the time two bright young assistants, Sampo
Ruuth (formerly Salovaara) and Seppo Rickman. Above all they were
mathematically oriented, and they pointed out that quite a lot of highly
developed algebraic structures existed other than just fields and vector
spaces. They also suggested that it might be possible to replace, in a sense,
the field structure utilized in the transfer function technique by some other
suitable algebraic structure. As a result, an extensive study of set theory,
topology, and abstract algebra was taken up within the research group
working with HB. At the same time the qualitative properties of linear
time-invariant differential and difference equations were being explored.
V
Preface
But it was still not quite clear how these qualitative properties should be
interpreted in terms of abstract algebra in order to arrive at the desired
result. The key, a suitable “system” concept, was still missing.
Then things started to happen. With respect to the present subject the
most important events were the appearance of the early papers by Rudolf
Kalman (e.g. Kalman, 1960; notations of this kind refer to the list of
references at the end of the book) and the book by Lotfi Zadeh and Charles
Desoer: Linear System Theory, The State Space Approach, 1963.
Through Kalman’s eminent works the power of the “state-space
approach” as a means for structural studies of the properties of linear
time-variant and time-invariant differential equations was convincingly
shown. The algebraic structure was here induced by the vector space
structure of the state space-in the finite dimensional real case an Euclidean
vector space. Moreover, the state-space representation also suggested very
natural “state” and “system” concepts. The state-space approach made it
possible to solve many problems in a rigorous and elegant fashion using
known vector space methods. However, the methodology thus obtained
does not show much resemblance to the transfer function technique-for
instance, the treatment of systems of interacting differential equations by
means of state-space methods turns out to be strikingly awkward, and it
lacks the simple elegance of the transfer function technique.
The book by Zadeh and Desoer proved to be a real gold mine with
respect to future work on an improved transfer function technique. The
book contained new and fresh ideas concerning many concepts appearing
in the present monograph. Sampo Ruuth (Salovaara, 1967) formalized and
developed in set theoretical terms many of the suggestions presented by
Zadeh and Desoer. Our system and related concepts rely very much on
Ruuth’s work. HB and his co-workers Sampo Ruuth, Jyrki Sinervo, Aarne
Halme, Raimo Ylinen and Juhani Hirvonen combined Zadeh’s and
Desoer’s ideas concerning representations of linear differential systems
and interconnections of such systems with results obtained through their
own research work. As a result a well-founded basic mathematical machin-
ery suitable for the present purpose emerged. The main principles of this
machinery were published in a number of papers and reports during the
period 1968-1972 (e.g. Blomberg and Salovaara, 1968, Blomberg et al.,
1969; Sinervo and Blomberg, 1971; Blomberg, 1972a). These early works
dealt almost entirely with structural problems-during the years that fol-
lowed applications were also considered. It also became apparent that
further studies were required of the application of the machinery to inter-
connections of systems. The final results of these studies are published for
the first time in the present monograph.
The new transfer function technique-here called the “polynomial sys-
vi
Preface
tems theory”-thus created relies essentially on a module structure (a
module is “almost” a vector space, the difference being that the scalar set
of a module is only a ring and not a field as in the vector space case). The
set of scalars of this module is a ring formed from polynomials in the
differentiation operator p P d/dt (in the discrete time case p is replaced by
a shift operator) interpreted as a linear mapping between suitably chosen
signal spaces. It turns out that the polynomial systems theory has many
features in common with the classical transfer function technique-it is
therefore justified to call it a “generalized transfer function technique”.
It works rather nicely and it is well suited to the treatment of various
problems concerning interconnections of systems. Further, the state-space
representation has, interestingly enough, its own natural place within this
framework. Note in passing that the module structure introduced and
utilized by Kalman (Kalman, Falb, and Arbib, 1969) differs from our
module and serves different purposes.
While HB and his co-workers were busy developing a rigorously founded
mathematical machinery for the purpose at hand, Howard H. Rosenbrock
(cf. Rosenbrock, 1970, and further references therein) was creating in an
ingenious and rather direct way a methodology that closely resembled the
polynomial systems theory developed by HB and his co-workers. Rosen-
brock’s works contain far-reaching results which are also directly applicable
within the framework of the present theory. W. A. Wolovich (cf. Wolovich,
1974, and further references therein) also derived a number of new and
important results based essentially on the same polynomial interpretations
as Rosenbrock’s work. Many details in the present text are founded on
and inspired by results obtained by Rosenbrock and Wolovich.
In later years new people interested in the subject have joined the above
“pioneers” and produced new and valuable results. References to some
of them can be found later on in the main part of the text.
The present monograph presents the “state of the art” of the polynomial
systems theory and related matters based on results obtained as described
above. The organization of the material is as follows.
We start with an introductory chapter 1, which provides a quick survey
of some of the main points of the present polynomial systems theory along
with examples. Of particular interest from an application point of view is
section 1.5, where a new algorithm for the design of a feedback compensator
is outlined.
There are a few basic ideas and concepts which are essential to a true
understanding of “systems thinking”. These are introduced and discussed
in a rather concentrated manner in part I (chapters 2 and 3) of the book.
The subsequent treatment of the subject relies firmly on the material
presented here.
vii
Preface
After a very brief and superficial motivation we start. in part I1 (chapters
4 to 7), a detailed study of the polynomial systems theory as applied to
systems governed by ordinary linear time-invariant differential equations.
The mathematical machinery needed is discussed in detail in chapters 4
and 5. It is based on a module structure. A survey of the relevant part of
the abstract algebra used in this context can be found in appendices A1
and A2. In chapter 6 a number of basic system concepts are interpreted
in terms of the module structure chosen. Chapter 7 is probably the most
important chapter in the whole book from an application point of view.
It contains a great deal of material of significance to the application of the
polynomial systems theory to various analysis and synthesis problems.
Problems concerning the design of feedback controllers and compensators,
as well as of estimators and observers, are discussed in some detail.
Considerable space has also been devoted to representation theory. in
particular some effort has been made to interpret Rosenbrock’s system
matrices (Rosenbrock, 1970) and related concepts in terms of the present
theory. All this has unfortunately led to a rather oversized chapter.
Part I11 (chapters 8 and 9) serves a number of purposes, the main one
being to give a convincing justification for the polynomial systems theory
as developed and applied earlier in the text. For this purpose a rather
delicate vector space structure is constructed, also involving generalized
functions. Appendices A3 and A4 contain some basic material of relevance
in this context. Chapter 8 devises a method for constructing a suitable
system (i.e. a family of input-output mappings) from a given input-output
relation. The method is called the “projection method” because it is based
on a certain projection mapping. In passing, a justification for the engi-
neering use of the Laplace transform method is obtained. In chapter 9 this
projection method is then utilized to build up the convincing justification
wanted for the polynomial systems theory. The considerations also lead
to a number of significant by-results, for instance concerning the so-called
realization problem.
Part IV (chapters 10 to 14) is devoted to a study of the polynomial
systems theory as applied to systems governed by ordinary linear time-
invariant difference equations. Because the theory is very similar to the
theory of differential systems we can pass many considerations only by
referring to the corresponding considerations in parts I1 and 111. There
are, however, some dissimilarities which make the algebraic theory a little
more complicated. These are caused by the fact that in the most useful
case, where our time set consists of all integers, the unit shift operators are
invertible. On the other hand, the difference systems theory is simpler than
the differential systems theory in the sense that in using the projection
method for constructing finite dimensional input-output systems we do not
...
Vlll
Prefuce
need any generalized functions.
The presentation of the material within the different paragraphs is
organized in the following way.
Each paragraph contains a main part, which includes definitions, theo-
rems, proofs, etc. of fundamental importance to the subject. This part is
written in a rather precise way. Numerous subtitles are inserted in order
to make it easier for the reader to get a general view of the material
presented. Examples are also included-often with suitable details left as
exercises for the reader. In order to make the text as readable as possible,
the number of special concepts and terms introduced has been kept at a
near-minimum. For the same reason we have as a rule avoided presenting
things in their most general setting.
The material covered by the main part of the text most often leads to
a great deal of useful consequences and implications. Results of this kind
are presented in a more informal way as “notes”. Most of the items
contained in the notes are statements given without, or with only incomplete
proofs-the proofs and other considerations relating to the statements are
left as exercises for the reader. In view of the exercises implicitly contained
in the examples and notes, it has been possible to dispense entirely with
separately formulated exercises.
There are also a number of “remarks”. These are just remarks of some
general interest but without much bearing on the actual presentation of
the subject at hand.
For referencing and numbering we have essentially adopted the very
flexible and convenient system used in Zadeh and Desoer, 1963:
Theorems, definitions, notes, remarks, as well as other items of signifi-
cance are numbered consecutively within each section. These numbers
appear in the left-hand margin. References and cross references to various
items like literature, figures, equations, etc. are given in a rather self-
explanatory way. A few examples will further illustrate the simple system.
For instance, a cross reference of the form “note (8.2.1), (ii)” refers to
item (ii) of note number 1 within section 8.2 of chapter 8. A reference of
the form “theorem (A2.69)” correspondingly refers to theorem number
69 within appendix A2. If reference is made in a section to an item within
the same section, then only the item number is indicated, for instance
“consider (2)” means that we shall consider item (equality, expression,
etc.) number 2 of the same section.
The symbol 0 indicates the end of theorems, proofs, definitions, notes.
remarks, examples etc.
Hans Blomberg and Raimo Ylinen
Helsinki. Summer 1982
ix