Table Of ContentINTERNATIONAL CENTRE FOR MECHANICAL SCIENCES
C 0 U R S E SAN D LEe T U RES - No. 153
ANTONIO RUBERTI
Al.BERTO ISIDORI
PAOLO D'ALESSANDRO
UNIVERSITY OF ROME
THEORY OF BILINEAR DYNAMICAL SYSTEMS
COURSE HELD AT THE DEPARTMENT
FOR AUTOMATION AND INFORMATION
JULY 1972
lJDINE 1972
SPRINGER-VERLAG WIEN GMBH
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whether the whole or part of the material is concerned
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or similar means, and storage in data banks.
© 1972 by Springer-Verlag Wien
Originally published by Springer -Verlag Wien -New York in 1972
ISBN 978-3-211-81206-8 ISBN 978-3-7091-2979-1 (eBook)
DOI 10.1007/978-3-7091-2979-1
Note to the Reader.
The following system of numbering and cross-1'eferencing is used in these
notes. Each item (definition, theorem, remark etc.) is given a pair of numbers in the
left lumd margin, the first one indicating the section, the second one is used to num
ber consecutively within each section. When we refer in a section to an item within
the same chapter both item numbers are given. For cross references a third number
is added on the left to indicate the chapter.
Moreover preceding the text a list ofs ymbols is git/en.
List of symbols and abbreviations.
..
..
implies
...
is implied by
implies and is implied by
v for all
r. It. s right hand side
l.h.s left hand side
~ is defmed to be
is equivalent to
<l end of discussion
is an element of
c
is a subset
c
is a subset or coincides
contains
contains or coincides
:R(.) range space
R(.) null space
n dimensional euclidean space
C(O,T) space of continuous functions defmed on [O,T]
I. I norm
* (super script) coniugate transpose
1 orthogonal complement
direct sum
® tensor product
o
special product of matrices defined in section 1.4.
PREFACE
The term bilinear systems is usually applied to systems in which the right
=
hand side of the input-state differential equation x(t} /{x(t}, u(t}} is defined by sum
ming, to a linear function, a bilinear function of the state x(t} and the input u(t};
the output of such systems is obtained by means of a linear transformation of the
state.
These systems, whose introduction into literature is of fairly recent date,
aroused a great deal of interest among researchers, this on account of two principal
reasons. The first of these is to be found in the fact that they constitute a rather
satisfactory model of many natural and artificial systems, including processes of po
pulation generation (from the ones concerning the populations in biological species
to the ones concerning neutron population in nuclear fission), many regulation pro-
• cesses at the physiological level (for example the regulation of CO in the respiratory
2
system), many physical processes (such as thermal exchange, chemical reactions) and
finally many economic processes (such as control of capital by varying the rate of in
terest). The second is to be found in the relative ease with which they lend them
selves to analytical treatment, thanks to the particular type of non-linearity that cha
racterizes them.
From an analytical point of view, the bilinear systems can be introduced
by considering the right hand side /{x(t}, u(t}} of an arbitrary non-linear differential
equation and its possible approximations. The simplest approximation of this func
tion, in fact, is the one that leads to linear systems, and it is interesting to note that
the forcing action in this case is additive. Immediately afterwards one would natural
ly consider the bilinear approximation, obtaining a multiplicative forcing term in ad
dition to the additive one. This multiplicative term can be interpreted as the effect
of the linear control applied to the parameters and, therefore, to the structure of the
system. For this reason the bilinear systems can be considered as variable structure
systems. This characteristic explains the great potential of these systems as models of
natural and artificial processes.
Research on bilinear systems has been concerned with system theoretic
problems in the strict sense, as well as with problems of modelling and identification,
control and optimization. These notes will confine themselves t~ the former, and
This wor. was partially supported by Fo. .n dGziorae Ugo BORDONI.
8 Pre&ce
will therefore deal with the analysis of the mathematical descriptions (Chapter I), the
structure of the state space (Chapter II), the analysis of the properties of the minimal
outline of some specific methods (Chapter IV).
The starting point of these notes is constituted by a paper [1] in which we
presented a systematic theory of the realization of bilinear systems. Results obtained
in the course of further research have enable us to generalize the theory to the case
of bilinear systems with multidimensional input and outputs, and to present it in a
unique form that covers both the continuous and the discrete-time case. Sources and
references are given in greater detail in the introductions to each of the chapters.
Before concluding this briefp reface, it is interesting to note that the theo
ry here developed for bilinear systems is, quite naturally, a generalization of the the
ory of linear systems. A simple particularization leads back to the familiar results of
the linear theory. For the sake of brevity, however, the reader is generally left to
perform these particularizations on his own account.
Chapter I
MATHEMATICAL DESCRIPTIONS
1. Introduction
In this chapter we present systematically the mathematical description of
bilinear dynamical systems. More precisely we consider both the descriptions through
input-state-output equations, which form the basis of the structure theory, and the
descriptions through input-output maps, the starting point of the realization theory.
Moreover, a unique formal treatment of continuous and discrete-time sys
tems is made possible thanks to a suitable choice of parameters for describing both
kinds of systems.
The chapter is organized as follows. In section 2 the most general bilinear
differential equation is introduced and a recursive procedure for computing the solution
is given. This solution, through a suitable formalism, can be expressed through Vol
terra-series expansions. In section 3 the results of a similar analysis for difference
equations are presented. Section 4, which is particularly important for the subsequent
developments, is devoted to discussing the choice of the most convenient parameters
for identifying a given input-state-output description of a bilinear system. Further
more, a relationship is established between the kernels characterizing the zero-state
input-output model for both continuous - and discrete-time systems.
As regards the sources of the material contained in this Chapter ,Theorem
(2.5) and the procedure for arriving at expression (2.19) were presented in [2]; the
remaining part is original.
10
2. Bilinear differential equations
In this section we present some basic results concerning the equations
(2.1.) Definition - We call "bilinear differential equation" the equation
(2.2) x(t) = Ax(t) + N[x(t)®u(t)] + Bu(t)
where u(t)€ RP, x(t)€Rn. The tensor product ® is defmed in such a way that
x(t)®u(t)€Rnp, and A,N,B are constant matrices of proper dimensions. <l
Note that the second term in the r.h.s. of equation (2.2) is the most general
p n n
bilinear map R x R ~ R .
If u(.)€C[O,T], the solution x(.) of the equation (2.2) on [O,T] with given
initial condition x(O) = i can usefully be obtained by considering the sequence
{xJ)} of solutions on [O,T] of the equations
(2.3) Xo (t) == Axo (t) + Bu(t)
(2.4)
(i = 1, 2, ... )
In fact we have the following
(2.5) Theorem - If u(.)€C[O,T], then
(2.6) lim x.(.) = x(.)
i+- 1
in the sense of the norm of C[O,T].
Proof - Consider the sequence of functions
(2.7) (i = 0, 1, 2, ... )
Bilinear differential equations 11
Since
X(t) = e At X + f t e A(t-r) {N[x(r)®u(r)] + Bu(r)}dr (2.8)
o
and
xi(t) = e At X + f t e A (t-r) {N[xi_1 (r)®u(r)] + Bu(r)} dr (i = 1,2, ... ) (2.9)
o
then we have
(i = 1, 2, ... ) (2.10)
>
Since u(.)eC[O,T], a constant M 0 exists such that
11/ (t-r)N[a®u(r)) 11< M na II Vt, re[O,T] VaeRD (2.11)
Then, for each te[O,T], we have
(2.12)
By subsequent substitution from (2.12) it is easily deduced that
II z.(t) II < M i f t f Tl f T2 ... £T1o -1 II z (r.) II dr. dr. 1 • •• dr 1 (2.13)
1 - 0 0 0 0 0 1 1 1-
>
Since u(.)eC[O,T], then also x(.), Xo (.), Zo (.)eC[O,T] and therefore a constant K 0
exists such that
Vte[O,T] (2.14)
and from this we conclude
MiK. (MT)i
Hz.(t) n~-r~ - K (i = 0, 1, 2, ...) (2.15)
1 ., .,
1. 1.
