Mathematical Descriptions

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

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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

This work is subject to copyright.

All rights are reserved, whether the whole or part of the material is concerned specifically those of translation. reprinting, re-use of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks.

©

1972 by Springer -Verlag Wien

Original ly publishe d by Springe r - Verlag Wien - New York in 1972

ISBN 978-3-211-81206-8 DOI 10.1007/978-3-7091-2979-1

ISBN 978-3-7091-2979-1 (eBook)

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 number 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 of symbols is git/en.

List of symbols and abbreviations.

.. ...

implies and is implied by

v

for all

r. It. s

right hand side

l.h.s

left hand side

~

is defmed to be

implies is implied by

is equivalent to

0

exists such that Vte[O,T]

(2.14)

and from this we conclude

MiK. (MT)i Hz.(t) n~-r~ K 1 ., ., 1.

1.

(i = 0, 1, 2, ... )

(2.15)

12

Bilinear differential equations

Inequality (2.15) shows that the sequence {Izj(t) I}, for each tE[O,T], has, as an upper bound, the sequence {(MT)iK/i! } which converges to zero for i ~

00.

Thus

{zJ)} converges uniformly to zero on [O,T], and {x j(.) }converges uniformly to x(.) on the same interval.

1)

(4.8)

This completes the proof. l). instead of (4.4) and (4.6). it will be necessary to consider p equations for fmding F(i)(t) and. respectively. p equations for finding H(i)(t). i

= 1•...• p.ln these equations there should

appear suitable partitions of the matrix corresponding to Sm,m + I' Before concluding this section. we observe that &om this procedure emerges directly the possibility of proving the following (4.9) Theorem - A factorizable sequence of kernels {WI (t l specified by the sequence Wi(t l

•...•

ti)~mo +

I.

•••.•

tin; is uniquely

where mo is the dimension of its

minimal factorization. (4.10) Remark - The importance of this result from the point of view of the modelling of bilinear systems should be clear. We note that this is valid under quite general hypotheses: the only assumption is that the sequence {Wi(t l

•... ,

t i )}; is factoriza-

ble. One may therefore conjecture the existence of a stronger result valid for bilinear systems, since in this case F(t). G(t), H(t) have a proper rational Laplace transform. In effect it is possible to prove that a sequence of kernels realizable by means of a fmite dimensiona