Mathematical Descriptions
In this chapter we present systematically the mathematical description of bilinear dynamical systems. More precisely we consider both the descriptions through inputstateoutput 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
SPRINGERVERLAG WIEN GMBH
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©
1972 by Springer Verlag Wien
Original ly publishe d by Springe r  Verlag Wien  New York in 1972
ISBN 9783211812068 DOI 10.1007/9783709129791
ISBN 9783709129791 (eBook)
Note to the Reader.
The following system of numbering and cross1'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
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