Controllability and Optimization: Lectures Held at the Department for Automation and Information September – October 1969

81 Pages·1969·4.007 MB·English

by  Angelo Marzollo (auth.)

INTERNATIONAL CENTRE FOR MECHANICAL SCIENCES
C 0 U R S E S AND LECTURES - No. 17
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ANGELO MARZOLLO
UNIVERSITY OF TRIESTE
CONTROLLABILITY AND OPTIMIZATION
LECTURES HELD' AT THE DEPARTMENT
FOR AUTOMATION AND INFORMATION
SEPTEMBER - OCTOBER 1969
UDINE 1969
SPRINGER-VERLAG WIEN GMBH
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© 1972 by Springer-Verlag Wien
Originally published by Springer-Verlag Wien New York in 1972
ISBN 978-3-211-81123-8 ISBN 978-3-7091-2959-3 (eBook)
DOI 10.1007/978-3-7091-2959-3
P r> e f a c e
In Chapter> I of these Zectur>e notes
functionaZ analysis methods wiZl be used to der>ive in
detail the conditions of contr>ollability of continu
ous Zinear> time-var>ying systems~ for> the case in which
the contr>oZs ar>e bounded in nor>m by a given constant~
the time inter>vaZ is not only finite but aZso fixed~
and the contr>oZZabiZity is intended as the possibiZi
ty of tr>ansfer>r>ing the state vector> fr>om one given
point ~0 to an other> given point ! 1 of state space.
These conditions ar>e both mor>e gener>aZ and someway of
mor>e pr>acticaZ inter>est than the usuaZ contr>oZZabiZi
ty conditions in which initiaZ and finaZ points~ as
welZ as the time inter>vaZ~ ar>e fixed~ and the contr>oZs
ar>e not nor>m-Zimited.
Fr>om the contr>ollability
conditions~
as exposed her>e~ it is easy to der>ive conditions of
optimality (both in time and in nor>m)~ as it will be
done in Chapter> II~ wher>e for> the special case of con
tr>ols belonging to an Hilber>t space the completely
expZicit for>m of optimaZ contr>oZs wiZZ be given. An
exer>cize wiZZ cZar>ify some deZicate points of the th£
or>y~ par>ticuZar>Zy for> the case in which the system is
not contr>oZZabZe in the usuaZ sense of the ter>m~ in
which case time optimaZ contr>oZs may be not nor>m mi
nimaL
The main r>esult of Chapter> III is the
formal solution of the problem of "controllability in
the presence of noise"~ which is a problem analogous
to the one of Chapter I~ but with noise present (and
the point ~ 1 substituted with a given region A of
the state space).
Chapter I and II~ as weZZ as the first
part of Chapter III are essentially an elaboration of
Professor H.A. Antosiewiez's materiaZ~ already ap
peared in [1] .
Only the formal solution of the prob
lem of controllability in the presence of noise~
which constitutes the second part of Chapter III~ is
original; it should be pointed out~ however~ that the
method and the main tool used (separation theorem for
convex compact sets) were inspired by Professor
Antosiewiez's idea for the solution of the controZla
biZity probZem in the absence of noise.
I wish to acknoZedge Dr. S. De JuZio
for the very useful exchange of ideas on the materiaZ
exposed in Chapter I and II; I wish also to thank
CISM for the invitation to held this course and its
students for their attention and heZpfuZ suggestions.
AngeZo Marzollo
Udine~ October 1969
Linear control systems 5
chapter 1
CONTROLLABILITY
Let us consider a linear control system, described
by a linear differential equation of the type
~= A(t)! +B(t)u. I (1)
In I(l), and throughout these notes, x(t) is ann-component
vector, A(t)is an n x n matrix and 'B (t) is an nx Ill matrix
such that on the bounded interval K = [t0 , t1] the elements
a~k(t) of A(t)are Lebesgue measurable and the elements b~/t)
of B ( t) are Lebesgue integrable with their 9th power ( 1 :E q < oo ) ,
and ~ (t) is an m-component vector whose components are L-i~
tegrable with t h e1. r pth power ( ·~• ~p <oo ) on the same 1. nt erva l ,",
Moreover, the class U of "admissible control functions" will
consist for the moment of functions ~such that ~(t)~ Q a.e.
in K , where Q is an appropriate convex compact set in the m
dimensional space to which ~(t) belongs. Further requirements
on U will be made later. As it is well known, given K • [t0, t1 J
and any X E. R" , for each ~E. U the equation I(l) has ink
-0
an unique (and absolutely continuous) solution ~(t), with ~ ( t )=
0
=
~ 0 , which is given by
ft
!!" ( t ) = v( t ) !: 0 + v( t ) v- -t ( ~) !:!- ( ~) d.t~ I ( 2)
to
where V (t) is the solution of the matrix equation
6 Reachable region
d. V(t)
--= A(t)V(t)
d.t
with initial value V ( t0) = I
To stress the dependance of ~ on~· we shall sometimes write¥!!-(t)
instead of ~(t). The initial time will always be understood
to be t even when not explicitely mentioned.
0
Let us define as "region reachable X_ o " the set
n.
Rx ( K) of points !E'R such that ~!:!- given by I(2) in
_o
correspondence to an admissible control ~ is ! at t1
is therefore defined by
U}
R~JI<) = { !!! (t1) 1 y. E
If we consider a non empty set ACR TI , the distance aA between
the sets 'Rx ( k.) and A is defined as
_o
4 lRx(K) AJ = ~nf d(x.~)
1
L 0 xt:R (K)uE:A -
- !o J JL
where d(x.~)is the usual Euclidean distance between the points
:X and ~ of R . It is clear by the definition of Rx (K)that
0
the distance A is equal to the infimum
I(3)
which means that !l. is the infimum of the distance 0 from A
of all admissible trajectories at t1 , starting from ! 0 • The
Distance of solutions from the reachable region 7
first problem we are going to face is the following: does there
exist in the stated hypothesis an admissible control function
~0 such that the corresponding infimum of I (3) is also a
minimum? In other words, the question is whether there exists an
admissible control ~0 such that the corresponding value of "!.:Y.o(t)
at t has a minimal distance from A among all ~~(t1)1 ~ £ U
1
In Theorem 3 of this chapter it will be shown that the an-
swer to this question is in the affirmative sense, and such an
admissible control function exists. In the preliminary Theorems
I 1 and I 2 the convexity and compactness of the reachable set
1\! (k) will be proved, under the already stated hypothesis on U .
0
These important properties of R~_o (k) are basic for the solu-
tion of an other problem, this one of great practical importance:
for sets A and classes U of special kind, is it possible to
find conditions in order that the distance I(3) is zero or, in
other words, in order that A is reachable at t 1 by an admiss-
ible trajectory?
With respect to this last problem, we shall
consider J::!. to belong to specific linear normed spaces and con-
sist of controls ~ bounded in norm by a given constant Q ,
and we shall consider A to be either the closed ball Af.. (~1) of
radius f. around a given point ~1 , or the convex compact set
A~ ( ~.,) which we shall define in chapter I II for treating the
case of presence of noise as input, or to be simply a given
8 Convexity of the reachable region
point ~1. Although this last case may be obviously considered
as a particular case of the other ones, it will be treated first,
and Theorem I,4 which will appear at the end of this chapter
will give necessary and sufficent conditions on the system I(1),
and on ~ 0 , ~ 1 , L< = [ t 0' t ~ , e for x1 to be reachable by an
admissible trajectory.
A generalization of Theorem I,4 will be given in
Theorem I I I , 1 , where the case A = AE . (! 1) will be cons ide red, and
the result will be reached with a slightly different proof, which
could indeed be used also to prove Theorem I,4.
The following Theorem I,1 and I,2 may be consider
ed as preliminary and will be used also in Chapter 3.
T he o r em I, 1
In the already stated hypothesis for system I(l)
and admissible class U of controls, the set 'Rx ( k) is convex.
-0
P r o o f
From I(2), it is clear that the set 'R x ( k) is
_o
given by
ft1
R!o(k) ={ V(t1);0 + V(t1) v-1(~) B(~) ~ (o) ol~ ~f. u}
I
to
I (4)
If we define the map 1\ ~from U to ~nas follows:
Proof of convexity 9
ft 1
At<(~ao)• V(t1)V-1(~) B(~) Y:(~) d.-:> Vu. t U I ( 5)
to
we need to prove only that the set 1\ K (U) is convex, since
and V (t1) '!o is a fixed
point.
To show that A.l< (U) is convex, suppose !;!. 1, and ~2.
to be two arbitrary points of U , so that 1\ K (!:!-1) and A K (~2.)
are two arbitrary points 1\I<(U). Then, for 0 E p. E 1 we
have
since the map 1\ k is obviously linear.
Furthermore, since the functions Y:1, and Y:2. belong
to U 1 ~1 (t) and !;!:2(t)belong to Q for t E k a.e., and by the
convexity of Q also p. U. ( t ) + ( 1 - p. ) 1.\.2. ( t) belongs to Q for
t € k a.e., and therefore )L ~1 + (1-p.) ~2 belongs
to U.
J
[P.
Hence AK ~1 + ( 1 - P.) ~ 2. belongs to 1\K(U)
and 1\ K (U.) is convex, since 1\t<(\!oi) and 1\ K c~l) were
arbitrary, and the proof is completed.
Let us now take the usual Euclidean norm for the
control vector U