Realization Theory of Continuous-Time Dynamical Systems

318 Pages·1981·6.588 MB·English

by  Dr. Tsuyoshi Matsuo (eds.)

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Berlin Heidelberg New kroY 1891
Series Editors
A.V. Balakrishnan - M. Thoma
Advisory Board
L D. Davisson • A. G. .J MacFarlane • H. Kwakernaak
.J L Massey • Ya. Z. Tsypkin • A. .J Viterbi
Author
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ISBN 3-540-10682-0 Springer-Verlag Berlin Heidelberg NewYork
ISBN 0-38?-10682-0 Springer-Verlag NewYork Heidelberg Berlin
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III
PREFACE.
In this monograph, I shall present a basis of realization theory of
dynamical system. Realization problem is to determine an intrinsic
mathematical model (canonical dynamical system) from the input-output
relations of a given causal black box, i.e., to understand fully the
internal behavior of the black box from the experimental data of it.
Morerover, if possible, we want to make the mathematical model real.
Realization theory was mainly developed in the field of discrete-time
linear systems as the algebraic system theory. On the other hand, it has
been recognized that the highly developed automata theory has close
relations with realization theory. However~ there exist serious gaps
between discrete-time linear systems and automata, for example, the way of
action of inputs on states are quite different. In contlnuous-tlme linear
system, the situation is worse. There is no established way to think what
is the set of input functions which act on states.
Here, I propose the axioms of a set ~ of inputs (experiments), namely,
a concatenation monoid. Then we can describe the causal input-output
relations of a black box by an input response map (Theorem (B,3.2)). I use
rather automata style idea to define dynamical systems. A dynamical system
is defined to be a collection which consists of a transformation monoid
over ~ (state space)~ an initial state~ and a readout map. Since we have
introduced concatenation monoids, we define teachability and
distinguishability, hence canonicality of dynamical systems naturally.
In chapter B~ it is shown that, for any input response map a, there is
a uniquely determined (up to isomorphisms) canonical dynamical system such
that its behavior is a. If the set U of input values is finltej completely
steady finite dynamical systems are considered as asynchronous digital
circuits, and they can be constructed concretely. Exisistence and
uniqueness theorems of natural partial realizations are also given.
In chapter C, we consider linear representation systems which are
special dynamical systems such that the state spaces have linear structure.
A linear representation system is canonical if it is quasi-reachable and
distinguishable. It is shown that the class of linear representation
systems is complete in the sense that any input response map can be
Vi
realized by a uniquely determined (up to Isomorphisms) canonical linear
representation system. If the set U of input values is finite, finite-
dimensional differential linear representation systems can be constructed
concretely by analog computer circuits. Real-tlme partial realization
problem is also discussed.
In chapter D, linear concatenation monoids and (algebraic) linear
time-constant systems are defined. A canonical linear system is a canonical
dynamical system. It is shown that any linear stationary input response
map can be realized uniquely (up to isomorphisms) by a canonical linear
system. For three kind of linear concatenation monoids~ detailed
discussions are given. Finite-dimensional differential linear systems
can be constructed by analog computer circuits. Note that linear (time-
constant) systems are special affine dynamical systems which are introduced
Section D,2. The detailed theory of afflne dynamical systems will be
appear soon.
This monograph may be considered as the contlnuous-time version of my
lecture "Realization theory of discrete-time systems" at the graduate course
in the School of Engineering, Nagoya University, 1976~ 1980.
I am particularly indebted to Prof. R. E. Kalman who taught me the
realization theory of discrete-time linear systems and gave the opportunity
to publish the material in this form. Dr. Yasumichi Hasegawa read the
draft carefully and corrected many minor errors. The manuscript typing was
undertaken by Miss Motoko Ohashl.
Nagoya, Japan
December, 1980 Tsuyoshi Matsuo.
V
TABLE OF CONTENTS
CHAPTER A. INTRODUCTION. " .............................................. 1
CA~TER B. REALIZATION THEORY OF (GENERAL) DYNAMICAL SYSTEMS. • ......... 7
.I.B Orientation. • ................................................... 7
B.2. Concatenation Monoids : Sets of Experiments. • ................... 9
B.3. Causal Input/Output Maps and Input Response Maps. • .............. 14
B.4. a-modules : Structure of State sets. • ........................... 16
B.5. Initial States, (Posltive-Time) Input Maps, and Reachability. -.. 32
B.6. Readout Maps~ Observation Mapsj and Distinguishability. • ........ 47
B.7. (General) Dynamical Systems. • ................................... 63
B.8. Reallzation Theorems of Dynamical Systems. • ..................... 68
B.9. u°-equillbrium States. • ......................................... 81
B.10. Control Semi-Problems and Controllability. • ..................... 89
B.ll. Historical Notes and Concluding Remarks. • ....................... 99
C~PTER C. REALIZATION THEORY OF LINEAR REPRESENTATION SYSTEMS. • ....... 105
C.I. Orientation. • ................................................... 105
C.2. Linear Representation Systems in Naive Form. • ................... 108
C.3. A(~)-modules. • .................................................. Iii
C.4. Pointed A(fl)~modules and Linear Input Maps. • .................... 121
C.5. A(fl)-modules with Readout Map and Observation maps. • ............ 136
C.6. Linear Representation Systems in Sophisticated Form. • ........... 147
C.7. Reallzation Theorems of Linear Representation Systems. - ......... 154
C.8. Historical Notes and Concluding Remarks. • ....................... 169
CHAPTER D. REALIZATION THEORY OF (ALGEBRAIC) LINEAR (TIME-CONSTANT)
SYSTEMS. • .................................................... 172
D.I. Orientation. • ................................................... 172
D.2. Affine Dynamical Systems. • ...................................... 175
D.3. Linear Concatenation Monoids and Stationary Linear Input
Response Maps. • ................................................. 181
D.4. (Algebraic) Linear Time-Constant Systems. • ...................... 187
VI
D.5. Basic Properties and Examples of Dynamical Spaces. " ............... 192
D.6. Dynamical Spaces with Input Map. • ................................. 207
D.7. Step Motions and Impulses. • ....................................... 222
7.a. Concrete Concatenation Monoids. • ............... • ................ 222
7.b. Concatenation Monoids m((R_~-~,{Qt}),(U,Qu)), • ................ 223
7.c. Concatenation Monolds of Additive Set Functions. " ............... 235
D.8. Output Maps and Observabillty. - ................................... 258
D.9. Linear Systems in Sophisticated Form. • ............................ 270
D.10. Realization Theorems of Linear Systems. • .......................... 276
D.II. Canonical Realizations for Various Linear Concatenation Monoids. "" 283
++
ll.a. Canonical Realizations for m((~ ,{~t}),(K,@K)) ............ 284
ll.b. Canonical Realizations for S(R~-~,K). • .......................... 290
ll.c. Canonical Realizations for ((-~,Sr),(F,S~),f). • ................ 296
D.12. Historical Notes and Concluding Remarks. • ......................... 301
REFERENCES. • ................................................................. 306
INDEX OF NOTATIONS. • ......................................................... 313
INDEX OF SUBJECTS. • .......................................................... 315
A. INTRODUCTION.
The realization problem that we treat here is as follows.
Consider a black box ~ with fixed input terminals whose inputs can be
handled arbitrarily as we wish and with fixed measurable output terminals.
We think it as a controlled process which we want to control, or a physical
object or a machlne which we want to describe the internal behavior.
"Black box" means that we do not know anything about the internal structure,
however, we assume that the black box ~ is causal and that we can get any
input/output data by performing arbitrary allowed experiments. Then we
want to describe the internal structure of B by so-called "intrinsic
state-model" which we call a canonical system~ and want to insist that
this model is the essential internal structure of B, i.e., the model has
no redundancy and it is unique in a definite sense. In this volume, we
only consider the continuous-time case, i.e., input functions and output
functions are defined on subsets of the time set T = = , + R where R--_ + is the set
of all non-negatlve real numbers, and we do not endow any topological
structure with the "state-model". For the discrete-time ease, i.e.,
T = N - the set of all non-negative integers, and topological considerations,
see our coming works. Hence~ from the mathematical view point~
we only use set theory and algebra. This volume clears up various system
concepts and builds up a form of the realization theory. The realization
problem seems very natural from human desires, hence we expect to have a
long history. However, as we will mention later, it is a rather recent
problem from theoretical view points, and it comes from the field of
control theory.
Now we consider the control problem in the field of lienar feedback
theory. See, for instance, TRUXAL [1955]. The transfer function G(s)
[or system function] of a linear controlled process is given, usually
in a rational function form of complex variables. The problem is to change
the transfer function G(s) to a desired transfer function Gd(s) which
satisfies the specifications of total performance for the control system,
by inserting additional circuits of instruments which are called compensation
circuits or controllers. The transfer function G(s) of a controlled process
is usually supposed to be given by considering
what kinds of physical phanom~n~ are in the process and writting out the
physical laws from them. But, by this procedure, each person can get
different transfer functions for a controlled process, hence it is almost
impossible to have the exact transfer function in rational function form by
this procedure, and it is hard to determine which transfer function is the
best to describe the controlled process.
There are other procedures to have a transfer functions of a
controlled process. In the field of circuits theory, step responses,
impulse responses, and frequency responses are used, and these and
transfer functions are considered almost equivalent. See GARDNER and
BARNES [1942]o (However, we cannot find out anywhere exact statements about
the relationships between step ~esponses, impulse responses, transfer functions
and frequency responses. We consider step responses and impulse responses
independently, and we will state the relation between step responses
and impulse responses in another volume). Step responses are easily
considered to be obtainable by experiments, and frequency responses are
considered to be obtainable by experiments in engineering sense.
It is difficult to consider that impulse responses can be obtained by
experiments, but there are some arguments that impulse responses can be
obtained by considering correlation functions. However, transfer functions
are always obtained indirectly, and there are only vague procedure
to have transfer functions in rational form, for instance, by the
graphical manipulations on the Bode diagrams. The problem of determining
transfer functions from input/output data was called roughly the
identification problem. See MISDLKIN and BRAUN [1961],
In the theory of optimal control processes PONTRYAGIN et al [1963]
considered that controlled prosesses are described by a differential
equation in vector form
d
A-l) -/gx(t) ~ f(x(t),~(t)),
where x(t)ER n, ~(t)~U for t~R, U is a region in R m, and f and@ i fj
i,j = l,---,n are defined and continuous on the product space Rn~u. The
optimal control problem is to find out a control which maximizes a given
performance index within a class of admissible controls, fence the problem
is a very rigorous control problem. Unfortuanetely, so far there
is no rigorous method to obtain the exact equation A-l) which describe a
control process. The usual way to get the equation A-I) is that to list
the physical phenomena in the process and writting out the physical
laws from them. We may forget or neglect some non-negllgible
phenomena in the process, and still believe that the equation is correct.
Even when experimental data do not fit with the equation, we may still
think the data were disturbed by noise. If the equation A-l) of the
controlled process is not rigorous, the optimal control process will
become non-sense.
There are differences and similarities between the realization
problem and physics. Physics attempt to have laws in a physical
phenomenon, i.e., physicists want to have rules in the physical
phenomenon which are independent from the special experiments (inputs)
and the special measurements (outputs). Hence we might say that physics
is a realization problem without (free from) inputs and outputs, and the
realization theory provides a way to have indivisual physical laws. However,
without inputs and outputs, i~ is hard to consider behaviors (input~output)
relations), hence it is difficult to decide whether the rules (systems) fit
with the experimental data or not. In fact, many physical laws were
hypotheses at the beginning of their proposals. By many experiments in
many years, the hypotheses could become laws if none of experimental data could
insist that they were false. Even if some experimental data do not fit
with a hypothesis, you can not insist the hypothesis is false when there is
a reasonable explanation why the data do not fit with the hypothesis.
On the other hand, experlmental physics may be considered as almost the
same probelm as =he realization problem.
The realization problem was first stated and discussed by
KALMAN [1963] in the case of differential linear time-varylng finite
dimensional systems. YOULA [1966] discussed the case of differential linear
tlme-constant finite dimensional systems. On the other hand,
HO and KALMAN [1965] proposed an algorithm to get a canonical linear
time-constant system from an impulse response for discrete-time case.
KALMAN [1969] established-the realization theory for the case of
discrete-time linea~ time-constant systems by using K[z]-module theory.
Recently, there are many attempts to have realization theories for
special non-linear systems and special infinite dimensional systems.
For instance, see BROCKETT [1976], D'ALESSANDRO et al [1974] and BARA$
and BROCKETT [1974]. These theories are not so complete compared to our
realization theory since they could not show that their definitions of
teachability are natural and the domains of their Input/output maps are
vague.
The realization problems can be formulated as follows. First, fix
the way of describing input/output relations of a black box ~, i.e., fix
the set I/0 of all input/output relations which can be the behavior of a
black box ~. Next, fix the class M of models which might describe the
internal behavior of the black box__B, inpatlcular define the subclass
CM of models which are called canonical. The definition of "canonical"
should be natural conceptually and mathematically. Then we should show
that for any input~output relation in I/O, there exists a subclass of
CM such that its input/output behavior is the given one and the subclass
can be considered as an isomorphic subclass mathematically.
In the realization problems, the choices of the canonical subclass
CM are not trivial problems. If we consider a class CMwhich is to large,
we can easily get the existence theorem, but we can not get the uniqueness
theorem. If we consider a class CM whlch is too small, we can easily
get the uniqueness theorem, but we can not ~et the existence condition
in good shape.
In Section B,2, we propose the axioms of experiments which can be
allowed to a black box_~. We call a set of experiments which satisfy the
axioms as a concatenation monold, and we think that it is the set of inputs
which can be applied to the black box Bor systems. Our realization
theory heavily depends upon the definition of concatenation monolds.
In Section B,3, we present our Theorem (B,3.2) which says that any
causal Input~Output maps which might be input/output relations of a black
box~ean be represented by input response maps. Hence we always think
that input/output relations of a black box is given by input response maps.
Our models of a black box are dynamical systems. There is a
conflict on the name "dynamical system". BIRKHOFF [1927] first used
the name "dynamical system" and recently Smale et al are working on this
field. See SMALE [1967]. "Dynamical system" is a study of the topological
properties