Lecture Notes in Control and Information Sciences Edited by M.Thoma and A.Wyner
115 H. J. Zwart
Geometric Theory for Infinite Dimensional Systems
Springer-Verlag Berlin Heidelberg New York London Paris Tokyo
Series Editors M. Thoma • A. Wyner
Advisory Board L. D. Davisson • A. G..I. MacFarlane • H. Kwakernaak J. L. Massey. Ya Z. Tsypkin • A. J. Viterbi
Author Hans J. Zwart Faculty of Applied Mathematics University of Twente P. O. B o x 217 7500 AE Enschede The Netherlands
ISBN 3-540-50512-1 $pdnger-Verlag Berlin Heidelberg New York ISBN 0-387-50512-1 Springer-Verlag New York Berlin Heidelberg This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microfilms or in other ways, and storage in data banks. Duplication of this publication or parts thereof is only permitted under the provisions of the German Copyright Law of September g, 1965, in its version of June 24, 1965, and a copyright fee must always be paid. Violations fall under the prosecution act of the German Copyright Law. © Springer-Verlag Bedin, Heidelberg 1989 Printed in Germany Offsetprinting: Mercedes-Druck, Berlin Binding: B. Helm, Berlin 216113020-543210
Geometry may sometimes appear to take the lead over analysis but in fact precedes it only as a servant goes before the master to clear the path and
light
him on his way.
Ja~ms Joseph Sylvester
PREFACE AND ACKNOWLEDGEMENT
In the spring of 1984 I started with my research on geometric theory for infinite
dimensional
systems.
The research
topic was suggested to
me by
Ruth Curtain~ who had done some preliminary investigations on this topic. Many questions were at that time still open and a more fundamental theory was still missing. We knew that the key-concept in geometric theory for finite
dimensional
systems,
that
is (A~B)-invariance,
has
lost
its
strength
for infinite dimensional systems. So I began to look for different concepts which would be more appropriate for infinite dimensional systems. It turned out that
these were the concepts of open-loop invariance and
frequency
invariance. Although the concept of frequency invariance had already been introduced for finite dimensional systems by Hautus, he did not give it any special
name. I have chosen this name,
since this expresses in a concise
way that this is an invariance concept in the frequency domain. Once the equivalence between open-loop, established, came
the
solvability
relatively
problems
are
easy.
of
In
studied:
frequency, and closed loop invariance was various
this
the
disturbance
monograph
Disturbance
three
decoupling
problems
disturbance
Decoupling
Problem
decoupling (DDP),
the
Disturbance Decoupling Problem with Measurement Feedback (DDPM) and the Disturbance
Decoupling
Problem
with
Measurement
Feedback
and
Stability
(DDPMS). The theory can easily be extended to other disturbance decoupling problems,
with
the
notable
exception
studied in the finite dimensional
of
the
'almost' version,
which are
case by Willems and Trentelman, see e.g.
[39]. The theory for the almost disturbance decoupling problems is one of the main still
open problems in geometric theory for
infinite
dimensional
systems. The monograph is addressed to researchers in the field of geometric theory of infinite dimensional the
infinite
dimensional
controllability,
initial
third
of
chapter
systems. In this book I shall use basic concepts of system
theory
as
observabUity, which are
Curtain
and
C0-semigroup~ covered
Pritchard [9]. This
in the
book
is
approximate second and self-contained
with respect to the notions of the geometric theory~ although sometimes we shall refer to the references for the finite dimensional case.
VI Although it may seem that writing a monograph and doing research is a solo occupation, in reality it is a team occupation and I owe my team members of the Groningen System Theory Group a great debt of gratitude. First of all I want to thank Ruth Curtain who found always the time and the patience to listen to my ideas. Her guidance made sure that my research would not wander off in queer directions. During the past four years it has been a great pleasure to share the office with Jan Bontsema. As a room-mate he always had a lending ear to listen to my (sometimes
obscure)
problems and his relativizing
way of
looking at
these problems really meant a lot to me. Furthermore I would like to thank the other members of the System Theory Group in Groningen; Harry Croon, Christiaan
Hey,
Hans
Nieuwenhuis,
Paula
Rocha,
Siep
Weiland
and
Willems, for the privilege of working with them. They all contributed
Jan in
their own way to this research and made our lunch breaks a very cosy hour. I also express my gratitude
to
Hans Schumacher
whose
insight
into
the
problem plus his remarks and ideas helped me to get my research started. Special thanks go to Erik Thomas, Malo Hautus and Luciano Pandolfi for the careful way they read this monograph. Their discussions and interest from different mathematical backgrounds all contributed to this research. I also want to university
of
thank
Groningen
the for
office of their
the mathematics
help
during
the
department last
years.
of the Special
thanks go to Janieta Schlukebir for typing part of this monograph. This research was sponsored by the Netherlands
Organization for Scientific
Research (N.W.O.), under grant no. 10-64-06 for which I am grateful. Hans Zwart
July, 1988
CONTENTS
Introduction Disturbance Decoupling Problem for Finite Dimensional Systems
1
4
Disturbance Decoupling Problem for Infinite Dimensional Systems
11
Outline of this Monograph
I,l
Chapter h Invarlance Concepts
15
1.1: A- and TA(t)-Invaziance
15
1.2: The Relation between TA(t)-Invariance and the Spectrum of A
18
Chapter Ih System Invarlance Concepts
20
II.l: System Invariance Concepts
21
II.2: Open Loop Invariance
23
II.3: Frequency Invariance
32
II.4: Equivalence
41
Chapter IIh Disturbance Deeoupling Problem
47
III.h DDP in Frequency Domain
48
III.2: DDP in Time-Domain
57
IiI.3: Properties of Controlled Invaxiant Subspaces
59
Chapter IV: Controlled Invarlance f o r Discrete Spectral Systems
62
IV.l: Discrete Spectral Operators
64
IV.2: Zeros and Invariance
69
IV.3: Characterization of all Invariant Subspaces for Spectra] Systems
72
IV.4: Examples
78
Chapter V: The Disturbance Decoupltng Problem with Measurement Feedback
87
V.I: Conditioned Invariance and (C,A,B)-Pairs
88
V.2: Disturbance Decoupling Problem with Measurement Feedback
97
Vlll Chapter Vh The Disturbance Decoupllng Problem with Measurement Feedback and Stability VLI: Stability, Stabilizability and Stabilizability Subspaces
107 107
VI.2: Disturbance Decoupling Problem with Measurement Feedback and Stability
116
Appendix E: Examples
127
E.I: Spectral Realisations of Delay Equations E.2: The Relation between
V*(K), Voz(K) and Vz(K)
E.3: On the Sum of two Controlled Invaxiant Subspaces
127 133 138
Conclusions
144
References
147
List of all Invarlance Concepts and their Relation
153
Notation
154
Index
156
INTRODUCTION
The aim of this monograph is to present a geometric approach to disturbance decoupling
problems
for
infinite dimensional
systems. Before
we
go
into
details we shall give an outline of the disturbance decoupling problems. By a disturbance type. Let E
decoupling
problem we mean
a problem
of the following
denote a s3~stem for which we can distinguish two
inputs, u(.) and
q(.), and
two
classes of outputs, y(.) and
classes of z(.), as
is
schematized in figure 1 by a signal flow graph. q
Z
.
E
U
,7
f~re In
this system
(disturbance)
on
outputs. Now
a
we
regard
the
system.
disturbance
the In
1
input general
decoupling
q(.) as
an
this input
problem
undesired
influence
will influence both
amounts
to constructing
a
second system El, which takes as input y(.) and gives as output a control input u(.), such that z(.) has become
independent of the disturbance
input
q(.). Pictorially we have
q
q~
Z
Z D
E 3
,Y
i
! i
I
[
E/
,
i..............................................
r.~z
figure 2 So
if we
system, Ed, disturbance
regard
the interconnection of the systems,
as schematized
in figure 2, then we
E
and
EI, as one
see the input q(.) as a
signa~ which should not influence the output z(.), and
the measurement
we
use
y(.) in order to design a control input u(.) which cancels
the effect of the q(.).
Therefore
disturbance
we shall
q(.) on z(.),
call u(.)
the
i.e. which decouples z(.)
control
input,
q(.)
the
from
disturbance
input, y(.) the measured output or measurement and z(.) the to be decoupled output.
The
next
example
shows that
this
problem
is solvable
for
some
systems ,~.
Example 1. In this example we consider the binary distilation column as studied by Takamatsu,
Hashimoto and Nakai [38]. The system is assumed to be in an
equilibrium and we want to make the composition of tile distilate composition of the
reboiler
independent of changes
in the
and the
composition of
the feed stream. As a model for this distilation column we take
f
(1)
x(t) =Ax(t)+Bu(t)+Eq(t)
E
y(t) =x(t) z(t) = Cx(t)
where x(.) = [xl(. ),x2(.),.. ,xll (.)~ t
xd.)
denotes
the
difference
between
the
liquid composition on the i-th tray and its equilibrium value,
u(.)=(ut(.),u2(.)]t; ul(.)
is the difference of the flow rate
of the liquid
stream and its equilibrium value, u2(. ) is the same, but now for the vapor stream,
z(. ) = (zl(.),z2(. )] t; zx(.) = xl(. ), z2(. ) =
x11(.).
Furthermore A=(ai,y ) is a
tri-diagonal
matrix,
of
which the
upper
diagonal,
the
diagonal and the lower diagonal are given by respectively, {aj,/+l} = (0.105, 0.469, O.529, 0.596, O.569, 0.718, 0. 799, 0. 901,1.021~ 1.142),
{aj,j} = - (0.174, 0. 943, 0. 991,1.051, i. 118,1.584,1.64,1.721,1.823,1.943, 0.171 ), {aj+l, j } = ( 0. 522, 0. 522~0. 522, 0. 522, 0. 522, O.922, O.922, O.922, 0.922, 0.115), B is given by
L0 -244
C=
-288 -304 -280 -232 -312 -382 -412 -396 -42
10000000000} OOOO00OO0O
and E - - ( 0
J
0 0 0 0 0.4 0 0 0 0 0 ) t .
1
It is shown in [35] that there exists a simple system
E!
which makes the
output z(.) independent of the disturbances. This system is given by
(2)
L'I: u(t)fFx(t),
where F is given by, (correct to five digits) 0 0 -330.06 0 0 0 0 0 0 470.17 0
F-~
0 0 -251.47 0 0 0 0 0 0 632.04 0 1
So Ul(.) = -330.06"x3(.) + 470.17"xm(.) and u=(.) = -251.47"x3(,) +632.04"xlo(.).
In figure 3 the output z(.) is drawn for the system ,U and for the system Ed. In both systems the same disturbance signal was used. 102
10-1
10-4
10-7
10-1o
lO-lS
10-18 /:" ° - ...........................................
- ....................................
~.!~[ .....................
e-
10-19
(
0
t
I
I
l
I
I
i
L
i
20
40
60
80
100
120
140
160
180
the output signal has become a
factor
1016
order
So
this
200
time - >
figure 3 Note that smaller
for
and
is
the system Ed of
the
same
as
the
machine accuracy.
in
example we can reject the effect of disturbances
A
naturaJ
question
decoupling possible
now and
axises:
how
can
under one
which
construct
C:]
conditions the
is
feedback
disturbance system
•17
Before we can solve this question we should specify which systems we want to
consider. A simple, but not
unimportant, class of
systems is the
class
4
of
linear
time-invariant,
finite
dimensional
systems.
For
this
class
we
shall give the solution of the di.qturbance decoupling problem as presented by Basile /o Marro [1] and Woaham [42].
Disturbance DecoupUng Problem f o r Finite Dimensional Systems The class of systems that we shall consider in this section is the class of systems that have the following representation
x(t) = Ax(t) +Bu(t) +Eq(t);
(3)
y(t)---Cx(t); z(t)=Dx(t); t > 0 ; x ( 0 ) = x 0
where x(t), R', ~ ,
u(t),
~,
~
q(t),
y(t) and z(t) are time trajectories
in respectively
and R~, and A, B, C, D and E are matrices of appropriate
size. Notice that the system of example 1 is of this class. In order to obtain some insight into the disturbance decoupling problem we assume that we measure the full state of the system, i.e. we assume that
y(t)=x{t), or equivalently C=I,. Furthermore we want the system ,U/ to be as simple
as
possible,
and
so
we
assume
it
to
be
time
invariant
and
memoryless. So we assume that (4)
u(t)=Fx(t)
/~I:
with F a matrix. components of
Or in other words, u(t) is a linear combination of the
the
state
at
time
feedback system (4) such that
t.' The problem
after
at
interconnecting
hand
is
(3) and
to
(4)
find
a
we have
that the disturbance input q(.) has no influence on the output z(.) for all disturbance
signals.
This
problem
is
commonly known
as
th__~e disturbance
decoupling problem. Definition: Disturbance Decoupling Problem. The
Disturbance
Decoupling Problem
is
to
find,
if
possible,
for
the
system (3) a feedback system of the form (4) such that in the closed loop system
the
disturbance
disturbance signals.
input
q(.)
has
no
influence
on
z(.)
for
all
5 It is standard to refer to this problem by its initials, DDP, and we shall continue this tradition. For the class of systems defined in (3) we can give a precise formula for the
closed
loop
behaviour
of
the
system.
This
solution
is given
by
the
well-known variation of constant formula, t
z(t) =De(A+BF)txo+I De(A+BF)(t-S)Eq(s)ds
(5)
0 Since we assumed no prior knowledge about the disturbance must have
that
the
input q(.), we
DeiA+BF)tE" " is identically zero on [O,co), in
function
order to make z(.) independent of q(.). So DDP is solvable if and only if we can find a feedback law F such that This
last
problem
is
very
hard
De(A+BF)tE-O; t~O.
to
solve
directly.
Since
one
h~m
to
calculate e(.A+BF)t for all F. However there is a simple necessary condition which we can deduce from it, namely for t = O we have that De(A+BF)UE=I)E,--and tttis must be zero. So DDP is solvable, only if DE=O. This condition shows in particular
that
DDP is not
solvable for
every system in our
class. In
DE=O is not sufficient. De(A+BF}tE=o; t~O in the following system theoretic way, namely all trajectories of the system x(.)=(A+BF)x(.) that start in ImE will remain in the kernel of D. Let V denote the reachable subspace for the system (A+BF,E), where F is the feedback law that solves example 2 we shall see that the condition
We can interpret the condition
DDP. This subspace is defined as
(6)
V = span { e(A+BF)tEq ); where the span is taken over t>_O and q~Rq
By the solvability of DDP we must have
(7.a)
Ira E cV cKer D
ImE and KerD denote the image of E and the kernel of D respectively, and the semigroup property of e(A+BF}t implies that V also
where
satisfies
(7.b)
e(A+BF)tvcv; t>O
Ou the other
hand,
suppose that there exists a subspace VcR",
(7.a) and (7.b). Then DDP is solvable with e(A+BF)tlmEce(A+BF)tv cVcKerD, and thus De(A+BF)tE-O.
some F satisfies the conditions this F, since
whictt for
So in conclusion we
can
exists a subspace F and could argue
say that DDP
is solvable if and
only if there
a feedback F which satisfy (7.a) and (7.b). One
that this result makes
the problem
more
difficult; not
only
must one construct a feedback F, but also one must construct a subspace
VcR n with the properties (7.a) and (7.b). The next theorem which can be found in Basile & Marro [1] and W o r d m m
[42, p.88] shows that one only has
to construct a subspace V c R n with special properties which
are easy
to
check.
Theorem be a linear subspace of I~n. Then
Let V
the following properties are
equivalent:
i)
There exists an F such that
ii) iii)
There exists an F such that
e(A+BF)tvcv. (A+BF)VcV.
A VeV+Im B
Some remarks must be made about this theorem; first the feedbacks in i) and ii) can
be
feedback F
chosen
to be
the same,
and
second
the construction of the
iii) to ii) or i) is done by solving linear equations,
from
which is a relatively easy problem, see example 2. Before
we
concepts
the
continue
differential
the
DDP
There
we
shall briefly refer to
invariance
]c(t)=Ax(t)+Bu(t). Since the solution of
~c(t)= (A+BF)x(t);
equation
x(t)=e(A+BF)txo, i')
with
related with the system
x(0) = x0
is
given
by
the
differential
we can replace assertion i) by exists
equation
an
F
such
:~(t)=(A+BF)x(t)
that
all
solutions
of
which start in V will remain in V.
We can now pose the question whether one would gain more if one were to allow
for
'arbitrary'
inputs
instead
of
inputs
generated
by
feedback.
In
other words are there subspaces V which do not satisfy i'), but do satisfy: iv)
For every solution of
xoeV there exists x(t)=Ax(t)+Bu(t);
a continuous input u(t) such that the x(O)=x 0 remains in V.
The answer to this question is negative. Since if x(t) is in V for all t_>O, then x(t) is in F for all t>O. So for t=O we have that Ax0=Ax(O)=x(O)-Bu(O) e I/+ImS. Thus by the equivalence of i') and ii~) we have that there exists an F such that the solutions of
x(t)=(A+BF)x(t); x(O)=xo
remain in g.
7 From the theorem and the argument above we see that property iii) is of great importance. This property has been given a special name. Definition: (A,B)-lnvariance A subspace V is (A,B)-invariant if
A VcV+Im B
(S) So
with
respect to
the DDP
we
see that this problem
is equivalent
to
finding an (A,B)-invariant subspace V with property (7.a). Let ~(A,B;KerD) denote the class of all subspaces that axe (A,B)-invariant and contained in the kernel of D. Note that we are looking for an element in ~(A,B;KerD) which contains Im E. Trivially the zero subset is an element of ~(A,B;KerD).
Furthermore
it is
an easy exercise to prove that the sum of two elements in ~(A,B;KerD) again
an
element
of
~(A,B;KerD).
So
by
the
finite dimensionality
is of
KerDcK n there will exists a supremal element in ~(A,B;KerD), which we shall denote by P*(KerD). Thus for every element P' in ~(A,B;KerD) we have that VcP*(Ker.D). From this we have the following theorem as an easy corollary.
Theorem The DDP is solvable
if
and
only
if
/mEcP*(KcrD).
Furthermore the feedback that solves the DDP can be any feedback F that satisfies
(A+BF)I~*(KerD) cP~'(Ker D).
We have mentioned that calculating the feedback F is a linear problem. Now we shall see that
the calculation of P*(KerD)
is also fairly easy. Define
the sequence P~ according to (9)
p°=KerD; ll~=KerD n A-~(ImB+P~-I~; #--1,2,..
where A-t(X) is the set consisting of all elements y such that
AyeX.
By induction it is easy to show that I f c l ~ -t, and for some k <_dim(Ker D) we
have that PJ~=PJ~+I--lZ~÷2. . . . .
Furthermore we have that Vt--P"(KerD).
A precise proof of this can be found in e.g. Wonham [42, p.gt]. So we have obtained an algorithm which calculates the supremal
(A,B)-invariant subsp~e
contained
in finitely many steps
in the kernel of D. Some
8 authors call this algorithm the Invariant Subspace Algorithm (ISA). Now we illustrate the previous theory by a simple example.
Example 2 Consider the following system on •3 x(t)=
(zo)
I 0-26 0 1 -9
x(t)+
[-il [*} u(t)+
3 0
q(t);
z(t)= (0 o Ox(t)
We want to solve the DDP for this system. Note that the necessary condition of
Im E--span{
3
}c xs=0
--Ker D is satisfied, so to check whether or not
0 DDP is solvable we have to calculate V*(KerD). This can be done by means of algorithm (9), yielding:
V°://er D {x3 0}, =
=
Lx3j Now
Pt+ImBffi{x3=O},
so P2fVt and this implies that
0
v*(gerD)f~/lfspa
So pictorially we have the following situation.
o) 1
i I
Since
/mE is
not
contained
in
P*(gerD)
we have
that
the
DDP is not
solvable for this example. We shall
complete
this
example
(A+BF)V*(KcrD)cY*(Ker D).
with
the
calculation
of
a
F
such
that
9
Let v =
be an element of I)*(KerD). Then by the (A,B)-invaxiance of
I)*(K~r D) we can decompose Av as an element of Im B and P*(Ker D), yielding (11)
Av=
1 0 -26
=
1 =
+xl
0 1 -9
(fl, f2,fa)
If we take for the feedback F any F =
fl=-l,
with
then F
=-x D
and
(A+BF)v=Av+BFv=v+xtB+BFv=v+x t
(12)
=veP*(KerD).
+(-xl)
azly feedback F= ~fl,[~,fa) with [1 = -1 (A+BF)I~'(KerD)cPC'(KerD). From equation (12) we also see that 1 an eigenvalue of A+BF. From this it seems that there exists no which keeps ])'(KerD) invaxiant and is also stable (all eigenvalues
So
in
the
open
left
half
plane}.
We
shall
see
in
the
sequel
satisfies is always feedback
A+BF
of
that
this
is
indeed true.
Q
We shall conclude this section with some results of Hautus [19], which give in
the
frequency
domain
a
necessary
and
sufficient
condition
for
the
solvability of the DDP. The
disturbance
decoupling
De(A+BF)tE=-O; t>O. De(A+BF)tE--O; t~O if
By
problem taking
is to the
and only if
find
a
Laplace
feedback
transform
D(s-A-BF}-IEEO.
F
such
that
we
see
that
We can rewrite this
equation as
O=-D(s-A-BF)-IE=D(s-A)'I(s-A)(s-A-BF)-IEffi
(13)
D(s-A)-I(s-A-BF+BF)(s-A-BF)-tE=D(s-A)-IE+D{s-A)'IBF(s-A-BF)'IE. Since
F(s-A-BF)'IE
that
the
is a strictly proper rational function we see from (13)
DDP is solvable only
if
there
exists
a
strictly
proper
rational
by
strictly
function X(s) which satisfies
D(s-A)'IE-I-D(s-A)-IBX(s)
(14) It
was
proved
by
--0
Hautus [19] that
the
solvability of
(14)
a
10 proper rational function is also sufficient.
D(s-A)'IE and D(s-A)-IB are
Notice that the the functions transfer
function
of
the
disturbance
to
the
output
respectively the
and
the
transfer
function of the input to the output. Furthermore we see for one dimensional disturbances,
inputs
and
outputs
that
a
solution
of
(14)
is
given
by
X(s)~--D(s-A)-IE/D(s-A)'18. So if s o is a zero of D(s-A)-18 and not a zero of D(s-A)'XE, then s o is a pole for every solution of (14}. Thus if D(s-A)'X8 has zeros in the right half plane, then no stable solution of (14) exists, see also example 2. We close this section
with the following remark.
In this section we have
mainly concentrated on the question whether or not disturbance decoupling was solvable with a system E! of the form (4). It has been shown by Basfle & Marro [1] that for the case that one measures the state of the system (3), i.e.
CfIm
the feedback system ,~! can always be chosen of this form.
So if disturbance decoupling is solvable for some system ,U.f which takes as input z(.), then there exists a system ,~, of the form (4) which solves the DDP
In the former section we have tried to give an introduction into the ideas behind
the
geometric
approach
to
disturbance
decoupling
problems.
The
disturbance decoupling problem and its relatively easy and nice solution in terms of subspaces which one can calculate marked the starting point of a totally
new
theory,
dimensional
systems.
now In
known terms
as
of
the
this
geometric theory
a
theory lot
of
for
finite
disturbance
decoupling problems were solved, as well as many related problems. Out of this theory arose a deeper insight into
the system theoretic structure of
the state space. As is to be excepted after almost twenty years there is a huge
amount
[34],
[35],
an
of
[36],
interested
literature [39],
reader
[40],
on
this
[42] and
who is not
subject, [43]
familiar
see
e.g.
[1],
[2]~ [19],
[33],
and
the
references
there.
For
with
the
geometric
theory
the
first article of Basfle and Marro [I] and the book by Wonham [42] will give a good introduction to this theory. For the
class
of
finite
dimensional
systems the
decoupling problems is very well understood.
solvability
of disturbance
Inspired by these nice results
authors have tried to extend these results to other classes of systems, see
11 Hautus [20] for systems over rings, Pandolfi [27] for
systems described
by
delay equations and Schmidt & Stern [32] and Curtain [6], [7] and
[8] for
infinite
needed
extra
dimensional conditions
systems.
on
their
All
these
cla.~
of
authors
systems
found
in
that
order
to
they
make
the
nice
results of the finite dimensional case go through. As is
clear
from
the
theory
for
infinite
title
that
this monograph
dimensional systems.
In
the
is about
next
the
section
geometric
we give an
introduction to this problem similar to the finite dimensional case.
Disturbance Decoupllng Problem for Infinite Dhnensional Systems. The class of systems that
shall consider in this section is the class of
we
systems that have the representation ~¢(t) =Ax(t)+Bu(t)+Eq(t);
y(t) = Cx(t);
(14)
5>0; x(O)=xo
z(t)=Dx(t); where x(.),
u(.),
q(.),
y(.)
and
z(.)
axe
time
trajectories
in
respectively
X, Rm, ~, :Y and Z, where A~, ~, y and Z are general Banach spaces, B, C, D and E are bounded linear operators, A is a generator of a C0-semigroup TA(t ). These
systems
dimensionality
axe
of
called
the
state
infinite
space
dimensional
X. The notion
after
of
the
infinite
C0-semigroup
is the
generalization of eAt from the finite dimensional case. Namely we have that x(t): =TA(t)x o Furthermore
is
the
the solution
continuous of
solution
{14) is given
by,
of
x(t) =Ax(t);
see Curtain
and
x(O) =Xo. Pritchard
[9, p.31]
X(t) ~- T A($)Xo+ ftTA(t-8)BU(8)dS-I- ftTA(t-8)Eq{8)d,.¢; o o t t
(is)
y(t) = Cx(O =
CTA(t)xo+f CTA(t-s)Bu(s)es+f CTA(t-slEq(s)ds; 0
0
zl t) = OX(t) = DTAlt)xo+ ftDT A(t- .)B~l s)ds + f'DT A(~- s)Eq( s)ds; o
As in the finite
dimensional part
o
of this introduction
we assume that
we
12 measure
the
full state
of
the
system,
i.e.
we assume that
y(t)=x(t),
or
C=I. Furthermore we want again that the system ~7! is as simple as possible, we therefore assume it to be time invariant and memory less. So we assume that
~:s: u(t)=Fx(t)
(16) for some
F. In the finite dimensional a
matrix.
In
case this feedback
infinite dimensional
can
respresented
by
exists many
different types of feedbacks that are linear and memory
For mathematical convenience we shall make
systems
always be
however
there less.
the extra assumption that F a
bounded operator from the state space to Km. The problem at hand is to find a
feedback
have
system
This problem we From
(16) such that after interconnecting (15) and
(16) we
that the disturbance input q(.) has no influence on the output z(.).
Curtain
shall again define as th..~edisturbance decoupling problem.
and
Pritchard [9, p.31] we
have
that
the
solution of
the
closed loop system is given by a formula similar to (5) t
z(t)=DTa+BF(t)Xo+ f DTA+BF(t-s)Eq(s)d,~;
(17)
0
So the
DDP is solvable if
and
only if there
exists a
bounded
feedback
operator such that DTA,BF(.)E-O. We can now apply a similar argument as in the finite dimensional case to link the DDP with invariant subspaces of the state space. For we have that DDP is solvable if and only if there exists a closed subspace V of the state space X with the properties (18.a)
lm Ec V cKer D,
(18. b)
TA+BF(t,)VcV;t >_0
Once
again
equivalent to
we
have
that
solving
finding subspaces
that
the
disturbance
satisfy
(18),
decoupling
only now
we
problem have
is the
extra condition that V must be a closed subspaxe. In the finite dimensional case every subspace is closed. Thus it would be nice to have in our class a theorem similar to that on page 6. However since A and A+BF are unbounded operators with the same domain we have that Av is not well defined for a general element v e X ,
but only for veD(A), the domain of A. So the notions
ii) and iii) in the theorem on page 6 must be modified to
13
(A+BF)(VnD(A))cV for A (VoD(A))cV+ImB
ii) iii)
some bounded F.
We can pose the question whether these two concepts are equivalent,
and
whether one of them is equivalent to
i) It
Ta+BF(t)VcV;t>O is
easy
iii).
implies been
to
prove
for some bounded F
that
for
closed
subspaces
i)
implies
ii)
and
ii)
Unfortunately the converse does not hold in general. This has
proved
by
Pandolfi [27]
for
the
implication
ii)
to
iii),
and
by
Schmidt and Stern [32] for the implication ii) to i). So
already
dimensional
in
the
beginning
systems
between i) and
there
iii)
are
of
the
negative
is very unfortunate,
geometric results.
the
dimensional case, geometric
develop
a
theory
geometric
finite
theory
dimensional
we
have
loss
of
infinite
equivalence
between i) and
has made condition iii) in
The
for
since iii) is a simple concept to
work with. This combined with the equivalence finite
theory
to
iii)
in the
into the key concept for
systems.
So
if
develop
concepts
we
want
other
to than
(A~B)-invariance. As in the finite dimensional case we can pose the question if there exists a
supremal
subspace
in
the
kernel
subspace will again be denoted by
of
D,
which
V*(KcrD).
satisfies
(18.b).
This
The use of such a subspace
will become clear from the next theorem by Curtain [6]. Theorem
If
P*(KerD)
Furthermore
exists, then DDP is solvable if and only if the
feedback
feedback F that satisfies
that
solves
the
ImEcP*(KerD).
DDP can
be
any
bounded
TA+Br(t)P*(KerD)cP*(KerD).
llere again we have the negative result that
1)*(KerD)
does not necessarily
exist, as was shown by Pandolfi [27] and Zwaxt [47], [48]. In this monograph we shall try to gain insight into these negative results. We introduce new concepts of invariance and we investigate whether they are equivalent to i). In particular the follow new concept of invariance proves very important. iv)
For every solution of
xo~V there exists x(t)=Ax(t)+Bu(t);
a continuous input
u(t)
such that the
x ( 0 ) = x 0 remains in V for all t_>0.
]4 Outline o f this Monograph This work consists of 6 chapters and one appendix. The first chapter will be of an introductory nature and contains results on the relation
between
invariant subspaces of the system operator A and of its semigroup Ta(t ). In chapter II the main result of this monograph will be proved, equivalence
between
Furthermore
we
the
shall
concepts
prove
i)
that
iv)
and
these
for
systems
closed
that
is the
subspaces
invariance
concepts
V. are
equivalent to the concept of frequency invariance, which may be regarded, by
Laplace transformation,
concept
was
introduced
as
for
the
finite
frequency dimensional
domain
version
systems by
of
iv}. This
Hautus [19] and
the generalization of it to the infinite dimensional case constituted starting chapter
point
for
the
geometric
theory
for
this
class
of
a new
systems.
In
llI. the concept of frequency invariance is used to investigate the
solvability of the DDP. The relation
between
the zeros of a single input
system and its system invariant subspaces is used in chapter IV to give a complete
characterization
of
all
closed subspaces
contained
in the
kernel
of D and satisfying i). It turns out that the problem of the existence of 12*(KerD I is equivalent to the solvability of a pole placement problem. In chapter V we shall investigate disturbance decoupling for the case that we do not measure the whole state, i,e. C # I . cousidering
stability.
to
properties
certain
In
the
that
appendix hold
in
longer in the infinite dimensional case.
Chapter VI is analogous, but also
E we shall the
finite
present
dimensional
countcrexamples case,
bLJt not
CHAPTER I: INVARIANCE CONCEPTS
In this chapter we shall consider the uncontrolled system
~(t)=Ax(t);
(1.1) where
A
is
the
x(O)-- xo
infinitesimal generator
of
the
TA(t )
C0-semigrou p
on
a
Banach space X and x0 is an arbitrary element of 2d. For standard theory concerning [9, ch.2].
these In
generators
this
chapter
we
refer
the
we discuss some
reader
to
concepts
Curtain of
&
Pritchard
invariance
for
the
to
the
concepts
of
system (1.1).
S e c t i o n LI: A - and
TA(t)-Invariance.
In this section we shall investigate invariance concepts system
(1.1).
invariance: space
For
the
semigroup
system
(1.1)
invariance
X
is
finite
dimensional,
However,
if
X
is
infinite
then
they
and then
dimensional are
in
we
shall
generator these (this
general
define
is not
two
invariance.
two
concepts the
case
interested
in),
generators
of C0-semigroups , example 1.6. We shall first
related If are we
equivalent
for
the
state
equivalent. are
mainly
unbounded
define semigroup-
and generator invariance. D e f i n i t i o n I.l: Semlgroup Invariance or TA(t)-Invariance A closed linear subspace V of X will be called semigroup invariaat or
TA(t)-invariant
if
TA(t}VcV,
for all t_>0.
D e f i n i t i o n 1.2: G e n e r a t o r I n v a r i a n c e o r A - I n v a r l a n c e A closed linear subspace V of X will be called generator or A-invariant if
A(VnD(A)}cV
So
TA(t)
where D(A) is the domain of A.
invaxiance tells you that if we start in V, then the solution
of (1.1) remains in V for all t_>0.
16 We shall
recall
most important
some basic facts
about
these
definitions.
One of the
questions is of course if these two concepts of invariance
are equivalent. The next lemma will give a partial answer to this problem. Lemma 1.3.
TA(t }.
Let A be the generator of the semigroup a)
If a closed subspace
is semigroup invariant,
then it is also generator
operator
semigroup
invariant. b)
If
A
is
a
bounded
on
X,
then
and
generator
invariance are equivalent for closed subspaces.
Proof: See Schmidt and Stern [32]. So
the
question
remains
[3 whether
or
not
semigroup
and
generator
invariance are in general equivalent. The following lemma is very useful in order to answer this question. Define p~ as
the
largest
connected
subset
of
p(A) that contains an
interval of the form [r,+oo). Since A generates a C0-semigrou p this set is nonempty, see Curtain and Pritchard [9, p.17]. Lemma 1.4. Let V be a closed linear subspace of A' , then the following concepts are equivalent a)
V is semigroup invariant.
b) (M-A)'~VcV for a R in p~. (AI-A)-IVcV for all A in p~. d) The range of (M-A) restricted to V is V for all Aep®.
c)
Proof: a) ~. b):see Pazy [29, p.121]. b) .* c):see Kurtz [23]. c) ~ a):see Pazy [29, p.121]. a) ~=~ d):see Kurtz [23].
D
We remark here that the lemma is in general false (even for bounded generators) if p~ in b), c) or d) is replaced by the next example.
p(A), as can be seen from
17 Example 1.5. This
example
will
show
that
TA(t)-invariance
is
in
general
no__~t
equivalent to ( A - A ) - l - i n v a r i a n c e for all A~p(A).
,t"
Let
be
ec°(Z),
the
space
consisting
of
all
x:Z~-+C
such
that
suplx(i)l
vector,
ei(j)=Sij ,
spectrum
of
A
V : = s - ~ { e i ; i~N},
then is
Ae i=ei+ I.
equal
then
to
This
the
is
unit
a
hounded
circle.
If
operator we
define
and
the
tim
set
V is obviously A-invariant. Since A is a bounded
operator we have from lernma 1.3 that V is TA(t)-invariant. However A-1 is the left shift and since A - l e 0 = % , V is not A-1 invariant. For closed subspaces we have
that
O
semigroup invariance always implies
generator invariance, but the converse is in general not true for unbounded generators, .as the next example will show. Example 1.6. Let X
be the
Hilbert space
LZ([0,1]);
the space
of
all functions
f(.)
1
from
[0,1] to R such that
flf(s)12ds
Furthermore
let A be
the
"heat
0
operator",
the second derivative on X with domain equal
f(s) that are zero for s = 0
and s = l
to all functions
and such that the second derivative is
in ,Y. This operator has a discrete spectrum, so p®=p(A). We define V to be tile following subspace:
V={feL2([0,1])[ f(s)=O almost everywhere on [0,~]} Then
V is A-invariant. However,
if l[½,~](s) denotes the
indicator
function
of the interval [½,1], then
A-I ~hh,fl(s)~r sin(Its) =
- s + l - s i n ( ~ s ) ; s~[w, 1]
is not in V. Now lemma 1.4 gives the desired result. So we have that for unbounded generators A - and TA(t)-invariance is not in general equivalent. However, there is a class of subspaces where the two concepts of invariance are equivalent.
]8 Lemma I.'f. If V is a closed linear subspace contained in the domain of A, then generator invariance is equivalent to semigroup invariance. Proof: see Curtain [6, lemma 2.3.]. The
next
lemma
will
t2
give
an
important
property
of
TA(t)-invariant
subspaces. Lemma 1.8. Let
V
be
a
closed
linear
subspace
which
is
TA(t)-'mvariant,
then
VnD(A) •V. Proof: From page 37 of Davies [14] we have that every element x in X is the limit of A ( A / - A ) - l x , as A -* +co. Assume that xeV, then by lemma 1.4. we have, for ,X sufficiently large, that (AI-A)'~x is in VnD(A). Combining these results we have that x is the limit of a sequence in
VnD(A). So Vr~D(A)=V.
El
As can be seen from example 1.6 we have that A-invariance is weaker than TA(t}-invariance
even if we assume the e x t r a condition that griD(A) is
dense in V.
S e c t i o n 1.2: The R e l a t i o n b e t w e e n T ~ ( t ) - I n v a r i a n c e and t h e s p e c t r u m o f A In this section we shall investigate the relation between
the
spectrum
of A, a(A), and the semigroup invariant subspaces. We remark that if V is a closed
A-invariant
subspace,
then
the
restriction
of
A
to
V
is
a
well
defined closed o p e r a t o r on V. We shall denote this operator by AV. Again let p® denote
the largest connected subset of the
resolvent set
of A that contains an interval of the form Iv,co). Now we have the next lemma.
19 [,emma 1.9. Let A generate
the C0-semigrou p TA(t ) and
let F be
a
closed linear
subspace of X. Then the following assertions are equivalent: i)
V is T~(t)-invariant.
ii) V is A-invariant and p®cp(Av). Proof: t)~it) This follows directly from lemma 1.3.a) and 1.4.d).
tt)..i) Let V be an A-invaxiant subspace with p®cp(Av). Then for all A in
p(Av) we have that the range of AIv-A v is equal to V, but by the definition of Av we have that this implies that the range of A - A restricted to V is equal to V. Lemma 1.4 gives the desired result. This result is the strongest one that
D
we can obtain, since in example
1.5 we have that p®=p(Av)={seC: Isl >1}, see Kato [22, p.210]. From this lemma we have the following corollary. C o r o l l a r y 1.10. Suppose that the resolvent set of A is connected and that V is a closed A-invariant suhspace. Then V is TA(t)-invariant if and only if a(Av)ca(A). Proof: This
is
a
direct
consequence
of
the
above
lemma.
Since
connected, then p(A) = p®. So for example 1.6 this corollary implies that there a(Av)=C.
if p(A)
is D
CHAPTER Ih SYSTEM INVARIANCECONCEPTS
The theory of controlled invariance of a subspace has been investigated in detail
in the
[1],
[35]
[19],
dimensional [27]
and
case that
and
there
[32],
[42]. have
but
shall
investigate
case
of
finite
state the
been
many
this
the For
some
questions
invariance
rank
inputs.
is finite dimensional, see
that
the
preliminary remain
for
We
space
case
space
investigations
unanswered.
infinite
shall
state
In this
dimensional
consider
the
e.g.
is
infinite
in
[6]-[8],
chapter
systems
controlled
we
for
the
version
of
system (1.1):
(2.1)
x(t)=Ax+Bu; x(O)=x0, x~X, u~U,
where X and U are Banach spaces, A is a generator of a Co-semigroup, Ta(t), and furthermore
we shall impose the condition that B is a bounded linear
operator with ImB finite dimensional, so without loss of generality we may assume that ht=R m and B is injective. The reason for only considering finite rank of
inputs operators view
a
natural
Secondly there
is twofold. First of all it is from a practical point choice,
one
can
is the mathematical
only
implement
reason;
finitely many
we believe
that
inputs.
only
a
small
part of the theory, as will be presented here, will remain valid if ImB is not finite dimensional. For
the
invariance loop The
system
in section
invariance concept
finite infinite
of
and
dimensions.
invariance
concepts open
that
for
II.2.)
frequency
dimensions,
we
II.1, and
(section
subspaces, prove
(2.1)
In is
it
shall
discuss
we shall pay and
frequency
invariance turns
section
was
introduced concept
11.4.
the
relation
and
there
frequency invaxiance.
subspaces
closed
loop
we
of
attention
invariance
this
investigated,
kinds
particular
that
and closed loop invariance are these
various
open II.3.).
Hautus [19] a
key
between
the
show
to
(section
by
plays
system
that
for
role
is
in
various closed
equivalent. Furthermore invariance
for
equivalent
we to
21 Section ILl: System Invariance Concepts The theory of system invariance entails many definitions. Here we shall summarize some of them and give some important properties. We shall start with the strongest. By
TA+s~(t) we
A+BF.
shall denote the semigroup generated by
Since F is
a bounded operator we have from Curtain and Pritchard [9, p.38] that always generates a C0-semigroup and the domain of
A+BF
A+BF
is equal to the
domain of A.
Definition ILl: Closed Loop Invariance A subspace V of A' is called closed loop invariant
if there
exists a
bounded feedback law F such that
TA+BF(~)VcV
(2.2}
for all t in [0,oo).
Remark: So a
subspace
V is
invariant for the system
called
closed
x(t)= (A+BF)x(t)
loop
invariant
for some
if
it
is semigroup
Fe£(X,U).
Lemma II.2.
VeX is TA+BFl(t)-invariant, for a T~.BF2(t)-invariant for a bounded ImB(Fz-F=)]VnD(A)CV.
Assume that a closed linear subspace certain
bounded operator
operator F 2 if and only if
F 1. Then V is
Proof: See lemma 4 in Curtain [7].
D
C o r o l l a r y II.3. Let 8 be any subspace of If
a
closed
subspace
V
is
ImB
such that
closed
loop
B+(ImBnV)=lmB. invariant~
then
bounded feedback law F such that V is Ta.sF(t)-invariant and
there
ImBF
exists
a
VnD(A)CB"
Proof: Assume that V is Ta+B~.(t)-invariant , then from the fact that the range of
F
is
finite-dimensional
and
F
is
bounded,
BF
can
be
written
as,
22
(Kato [22, p.160]),
~bi<.,fi>+ i--I
~ bi<.,fi>,
where
i~q+l
span {bi}=/~ and i~l,
Defining F = t..,,['<"f~;>]q " " we obtain that
,q
{bi}cImBng.
span
i - - q + l , . ,m
ImB(F-P)IVnD(A ) = v
and
BF}vnD(A)c~.
i.,1
Thus lemma II.2 concludes the assertion. Now we shall define
the
[]
generator
invaxiance Corresponding to
loop invariance. In the sequel of this chapter convenient to
use a
larger class of
it will turn out that
closed it is
feedback operators in this definition.
We shall use the class of A-bounded operators. Definition II.4: A-Bounded O p e r a t o r Let A be an element of
p(A).
An o p e r a t o r F from X to 12 will be A-bounded if D(A)cD(F) and F ( A - A ) -~ is a bounded operator from 9:' to U.
It is easy to see that if F is a bounded operator, then it is also an A-bounded operator. So the class of A-bounded operators is larger than the class of bounded operators. If A is a bounded operator itself, then these classes are equal. Furthermore, if F(A0-A) "1 is a bounded operator from X to U, then by the resolvent identity F ( A - A ) -l is a bounded operator from X to
U for
all Aep(A). So definition II.4
is independent
of
the
particular
choice of Aep(A)
Definition H.5: Feedback Invariance A subspace
V of
2¢ is
called
feedback
A-bounded feedback law F such that
(A +BF)(VnD(A))c V
(2.3)
Definition II.6:
(A,B)-Invarianee
A subspace g of X is called (A,B) invariant if
(2.4)
A(Vf~D(A))cV+Im B.
invariant
if
there
exists
an
23 Remark:
Let B be any subspace of ImB such that 8+(ImBnV)=ImB.
Then by a
proof similar to that in corollary II.3, it can be shown that if a subspa~:e is
feedback
invariant,
it
is
also
feedback
invariant
for
an
A-bounded
feedback satisfying Im BF[ VnD(A) ¢ ~ ' see Zwart [48]. In the next two sections we shall define two other invariance concepts which-are invariant domain
strongly related with the solutions of equation concepts
version
description of
are
of
closely
related,
invariance
the
same
and
property.
i.e.
the
the
second
As is to
first is
(2.1). These two one
the
expected
is
the
frequency
these
time
domain
concepts
have
many properties in common.
Section II.2- Open Loop I n v a r i a n c e Here we shall make precise what we mean if we say that the solution of (2.1)
remains
in V. Furthermore
we shall
give
some
interesting
properties
of this concept, that are useful in the sequel. By the solution of (2.1} for u(.):[0,oo)~4,/ we mean the mild solution: t
(2.5)
x(t) = T A(t)x o+f T,~(t- s)Bu(s)ds 0
D e f i n i t i o n H.7: Open Loop I n v a r i a n c e A subspace V e X there
exist~
a
is said to be open loop invariant if for every xoeV
continuous
u(.):[0,o0)~4,/
such
that
the
solution
of
{2.1)
it is standard
to
allow
remains in g.
Remark:
If
the state
arbitrary
space
measurable
is finite dimensional, then
functions
in
the
definition
of
open
loop
invariance.
As a result one obtains there that the input function could be chosen to be continuous.
However
if
the
state
space
is
infinite
dimensional,
then
this
result no longer holds, see Zwart [48]. So we have to restrict the class of input functions
in this definition and here we have
restricted ourselves to
24 continuous
functions. A
hold, is that we
result that is still unproved,
could also use piecewise continuous
but
is believed
functions. Then
to our
definition of open loop invariance would be the same as the one used by J.v.d. Woude [43].
Example II.8, In this example we shall study the delay system
~(t)=y(t-1)+u(t); t>o and y(t)=yo(t); - 1 s t < 0
(2.6)
This system can be rewritten in the form (2.1} if we make the following definitions.
M2((-1,0);R) = R ~/2((_1,0);R) A :D(A)r--~b/2(( - 1, 0);R) with
O(A)={(~oo,~o)eM2((-1,O);R)l~ooeR,~o such that ~o'eL2((-1,0);R)} A(~,o,~o) = (~( - i),~')
B:R~-,.M2((-1,0);IR)with Bu=(~,o). It can be shown that A generates a semigroup on Mz((-1,0);R), see Curtain &
Pritchard [9, p.48]. We
shall show
that
the
subspace
V: ={0} • {~oELz((-1,0);R)I ~o is
continuous on
[-L0]
artg
to(0)=0}
is open loop invaxiant. Let (0,~o) be an element of V, then by defining u(t)=-q~(t-1) for 0_
~(t)=y(t-D+u(t)=0
for o < t < l and
y(0)=0
This implies that y(t)ffi0 for O<_t<__l. N o w we define u(t) to be zero for t21, and
with (2.6) this gives that y(t)-0
(y(t), y(t-r);
0
for t~0. The
state at time t is
and this is in V for all t>_.0. Furthermore
continuity of y(.) for t > - I
we have that st(.) is continuous. So V is open
loop invaxiant. In
the
sequel
by the
[] of
this
section
we shall
prove
some
technical
results
25 concerning open loop invaxiance. We advise the reader who is not interested in details to proceed to section II.3. In
the
useful to
proof have
of
the
properties
of
some results concerning
open
loop
invariance
the convolution
it
is
very
in (2.5}. This will
be given in the next technical lemma. Lemma II.9. Let ~ function
denote
the
with values
continuous
convolution
product,
in Rn*'* (the
function
kmv(.)
such
and
let
k(.)
be
n by n matrices}. Then that
if
1(.),
a
continuous
there
g(.)eC([O,oo);Rn)
exists a
satisfy
the
following convolution equation g(O = / ( t ) +k(t)./(t),
(2.7)
then they satisfy also
f(t) = g(t) - k'u(t),g(t), Proof:
See Doetsch [16, ch.40].
c:l
Let V be a subspace of 9:'. By B°(V) and BI(V) we shall denote subspaces of X such that
Im B = BO(V) • BI(V), B°(V) n V = {0} and
(2.8)
B'(V) = V n I m B We
remark
that
(2.8)
is
possible,
since
B°(V) is a subspace contained in l m B
lmB
is
finite
which has zero
dimensional.
and
is of maximal dimension under
this restriction. Throughout
we
shall
a
fix
B°(V).
So
if
X
is
Hilbert
space,
So
intersection with V then
this section
we
can
take
B ° ( V ) = I m B n ( I m B n V ) ±. If there cannot he any doubt about V, then we shall simply use /30 and B 1. With this loop invariant
notation for
the
we shall prove system
(A,B),
for the system (A,B°), where ImB°=6 °.
that then
if a closed subspace it
is also open
loop
is open invariant
28 By C([O,oo),//)
we
shall
denote
the
space
of
all
continuous
functions
from [O,co) to U. We stress that we do not impose any norm on this space.
[,emma II.lO. If V is a closed subspace of X that is open loop invariant, then for every x 0 in V there exists one and only one u(.)~C([0,oo);~) such that the solution of (2.1) remains in I/ and Bu(.)~8 °. Remark: This omitted.
result Or
choosing
our
tells
from
us
a
that
control
different
control
action
actions
perspective; is
in
13I.
with
the
So
if
values
only
in
freedom
ul(.)
and
13~ can we
u2(.)
be
have are
in
both
controls that keep xo in V, then B { u l ( . ) - u 2 ( . ) } e B 1. This result is intuitively not hard
to understand. Since if we start
in
zero at t=O and the control action Bu(.) is in V, then since x(0)=Bu(0) we shall remain in V. Otherwise if
the
control
action
is not
in V, then
we
shall leave V. Proof: Let {bl,..,b~,0} be a basis for
B° and let {b~,o+l,..,bm} be a basis for
B 1. We shall begin with showing that for every bj; mo+l<_j<_m,
there exists
an input ~j(.) such that B~j(.)~B ° and if we start in bi (x0=bj), then the solution of (2.1} with this ~j(.} remains in V. Since V is open loop invariant we have for every bi in 81 the existence of
a
uj(.)=[ltl, i,..,Um,j]Wec([O, oo);l~) SUCh that if one starts in b) at time
instant zero,
then one remains in V. Now rewriting this solution of
(2.1)
gives
xj(t)=T,4(t)b i +
Ta(t-s)biui,j(s)ds i
0
+
Ta(t-s}biui,i(s)ds, i=m°+l
0
is in V for m0+l<_j<m. Let .
denote the convolution product,
then this equation can be rewritten
in the following form
xj(t)=TA{t)bj + [~,l,j(.),..,zt,%,~(.)l, / } / + LrA( .)b~oJ
[r.I.Ibm0,]1
[~"o÷',-"(" )'"" ! ~",-J(" )1" ] i ; m+l~j_<m L TA('}b.,
27 Writing this in matrix n o t a t i o n for j = m o + l , . . , m ,
x~(t) J
we h a v e
u,,~,(.)
ra
u.0:.,<.) J Lr.d :)bmoj
By introducing the following n o t a t i o n we can simplify the a b o v e equation.
Let
U.,o: =
i 1.o.lc,.0 o l,,1 "
'
and
U.~_.,o: =
L ~,,~(.).. ~-o:-(.)
"
"
L ~0÷,,m(.).. ~:,m(.) j
So these matrices axe named a f t e r their number of colomns. Rewriting the a b o v e equation with this n o t a t i o n gives
LrA(')amoJ L TA(t)bm J
L x'.,Ct) j From
lemma
IL9
we
have
that
there
exists
a
L TA(1)b=J function
Ur,_moft),, in ~nu
C([O,co);Rm - m o , m - m 0 ) such that
"TA(t!bmo+t"= Ixmo+l(t)l"
(2.9)
TA(t)b.,
ITA(." )bl1
-
Umo(')*]
_
L x,.it) J v~'_"~o(
"
)
-
Umo(.)*[rA('
"
Rearranging this equation gives
L x,,,it) J
L x.,(t) j
L TA(i)b., J
I-],,o(.) - Urn_toO(.)*U~o(.
•
.
For j=mD+1,..,m let xi(') and ui(.) denote respectively j-m o th element of :
L ~.i~) J
-
~:_":0(~).I
:
[
L ~.it) j
an~ the transpose o~ the J-'~0 t~ ~ow o~
28
inv ~,ll Then the equation above gives Urea- Um_mo.vmo. (2.1o)
£'j(. ) =
)fTA(']lb~
TAb )bj+~(.)T.]
.
.
Lr~(:)bm~J Since V is a closed subspace and xj(.)~V we have that ~cj(.)~V, j=mo+l,..m. inv
inv
Furthermore Um_mo(.)*Umo(.) is by the continuity of Urn_m0(.) and U=0(.) and by standard convolution theory an element of C([0,~);R "'m°'m°) and so ~ ( . ) is an element of C([0,oo);Rm°). With this input function we can define an input function in C([0,co)iRre) by just adding m - m o rows with the zero function. This extended input function we shall again denote by £(.). So from equation
(2.10) we have
fij(QeC([O,co);R ra) such B~j(t)eB ° for all t>O.
that
that for every bj in 131 there exists an the
solution
of
(2.1)
remains
in
V
and
Thus we have proved our assertion for xoeB 1. Let x o be an arbitrary element of V, then there exists an input u(t) such that t I" x(t) = TA(t)x0+ [ Ta(t- s)Bu(s)ds d
o
remains in V. Using the fact that u(t)=[.ul(t),..,um(t)] T we have that mo
x(t)=TA(t)xo + ~
t
~
ft
I TA(t-s)b'u'(s)ds +
TA(t-8)b,u,(8)ds
i=l 0 i=m°+l 0 Or in matrix notation with . again denoting the convolution product
x(t) =Ta(t)x o
+
[ul(.l,..,v~0(.)],
"
+
[Tai. )b,~oJ [um0+t(.),..,um(.)]* ITA ( )bmo+t"
( TAi.)b: Equation {2.9) gives that
L TA(tl~m J
I x~itl J
L x~Ctl J frA(.)b~ 1
/
I ra(.)b.oJ
29
r+,.o.,.,,,,11 L ~,.,,<~Ijj
L L ~,,,ctl; gives
ITA(')b' I
<++.11>
x(,>: ~,,,)xo +{t,,,c.),..,,,,,,o<.>]_~.)"+}.i ? i+~,> [r+t.)b,.,d it'tu
By the continuity of u(.), Umo(.) and Urn_too(.) plus standard convolution theory we have that ~1(.)eC([O,co);R) for 1
xj(.)eV;j=mo+l,..,m
we have that ~(.)eV.
ITA(')b' 1 x(t)-~it}=TAit)x° + [~'~{'}-~(')'"'%0('}-~=0(')]*I
/
i
LT~(.)b.oJ We
x(t)-~(t)is
have that
an element of V mad ui(.)-~i(.); 1_
continuous function. If we define ~i(.)=ui(.)-~i(.) for for
mo+l<_i<_m, then +~(.)=[~l(.),..,~m(.)]eC([O,oo);R'), B~(.)eB 0 me
t
i=1 So
we
have
proved
~{t}eC([O,oo};U) such that It
1<_i<_m°and
remains
to
prove
m
0 for
every
x°
in
and
t
/=m°+l
that
~i(.)=O
0
V
there
exists
an
B~{t}eB °.
that
system it is sufficient to
this
input
prove that
is the
unique.
By the
linearity
of
the
input which holds xo=O in V is
identically zero. So
suppose
that
t
x(t)=lTA(t-s)Bu(s)as @
is
in
v
and
Bu(.)el~ °.
Let
,/
0 {bx,..,b,0 } be a basis for B 0. Then by the Hahn-Banach theorem there exist functionals
x(t)
9jeX'
such that
=¢5ij and
a/lv=o.
Operating these
gives
L..J L~,%i.)J
gj
on
3O
Since
is in t = 0 equal to the identity on
g.,o,Ta(. )b 1 > • " Rra° we
have
from
standard
convolution
theory
that
u(t)
is
identically
zero.
D
Lemma IX.11. Let V be a closed subspace that is open loop invariant and suppose that BI(V)={O}. Then for every x 0 in V, there exists a u(.)eC([O, oo);/A) which is exponentially bounded and such that the solution of (2.1) remains in V. By lemma we already have that this input is unique. Proof: From lemma IX.tO we have u(.)~C([O, oo);U) such that
that
all xoeV
for
there
exists an
unique
the solution of (2.1} remains in V. Define Q to
be the operator that assigns to x 0 in V this input restricted to [0,1]. The linearity
of
this
operator
follows
obviously
from
the
uniqueness
of
the
input. If we denote by C[0,1} the Banach space of all continuous
functions
from
we shall
prove,
[0,1] to Q
H with
is linear
as norm
bounded
the
sup-norm
operator
from
on
the
[0,1],
then,
as
Banach
space
V to
C[0,11.
Since Q is defined on the whole of V it is sufficient to prove that Q is a closed operator.
By xu(t ) we shall denote
C[0,U to u(.).
V and let un=Qx n converge in
Let x n converge to x i n
the solution of
(2.1) for
x n and
u.(.). So I"
t
x,(t) = Ta(t)x,+ I T4(t-s)Bu,(s)ds ~t
0
Then
we
exists and
have
that
satisfies
for
all
t~[0,1]
x(t)=TAt)x+ [
the
pointwise
limit,
x(t),
of
xn(t}
t
Ta(t-s)Bu(s)ds.
Furthermore
since V is
0
a closed subspace x(t) is in V. So Qx=u, and Q is closed, thus bounded. Now let xo be an a r b i t r a r y element of V and let u(.) be the input that keeps the solution x(.) of (2.1) in V. So /. t
x(t) = TA(t)Xo+ I TA(t- s)Bu(s)ds
(2.12)
,d
0 We shall
prove
that
this
input
u(.)
is exponentially
x(i), q= HQ[I and suppose that ][TA(t)U<Me"t, then
bounded.
Let
x i be
31 1
Ilx(1)ll
(2.13)
0
< Me~'{l+llBIIq}llxoll. Since suplluls)ll = IIQxll---qllxll. 0_<8_<1
Me~{l+llBIIq}, then we shall show IIx,+lll_
Let
k
be
sup
that
Ilu(s}ll<_qllxdl and
i<_s_
system
we have
that
for
t
larger than O: t
x(t+i)=TA(t)xi+ I Ta(t-s)Bu(i+s)ds So
Qxi=u(i÷. )
0
and thus
sup
Ilu(s)ll= sup Ilu(i+s)ll=llQxd]<_qllxdl
i < , <_i+ 1
and with a
O_<s .~1
Ilxi+tll
similar argument as in (2.13) we have that [i,i+1]
we
have
that
proves that u(.) is exponentiMly bounded.
C]
With the above lemma one can easily prove that if a closed subspace is open
loop
invariant, then
it
is also c l o s e d loop
invariant. Here
we shall
only give a sketch of this proof since in section II.4 we shall present a different proof. With the same notation as in the proof of II.11 we define the feedback law F as the composition of the bounded operator O and the
Au=u(O).
bounded operator z~:C[0,t]~,// given by operator
and satisfies
we have
Fx(t)=u(t),
Fx(O)=u(O). where u(.)
Then this F is a bounded
With the time invariance of
the system
is the unique continuous input that
holds
x(.) in V.
So
the
above
lernma
exponentially bounded. In this closed
subspaces.
In
section
implies
that
the
Iemma we have I]I.3
we
shall
u(.)
in
restricted give
a
lemma our
II.10
is
attention
to
sufficient
condition
such that if V is open loop invariant, then the same holds for the closure of V.
32 S e c t i o n II.3: F r e q u e n c y I n v a r i a n c e In this section we shall give a frequency domain version of open loop invariance,
this
Hautus [19]
concept
the
time. Only the
concept
will
be
of .frequency
called
frequency
invariance
invariance.
appeared
finite dimensional case was discussed there.
for
the
In first
In this section
we shall discuss this concept for infinite dimensional systems. The
next
definition
will
generalize
the
concept
of
the
space
of
all
rational functions.
Definition II.12: Y(s) and Y_~(s) Let Y be a subspace, function
[(.)
is an
element
not necessarily closed~ of a Banach space Z. A of
Y(s} if f(.) is defined and continuous on
some interval of the form [r,co) and [(So) is an element of Y for all s o in this interval. By
Y_~(s) we shall denote all functions in Y(s) that are of the order s "1
for s-co i.e.
Y_l(s) = {feY(s)J l i m sf(s) exists} S-~30
These functions will be called
strictly proTer.
We remark that
instead working with these definitions we could also follow
Zwart [48] where
Y(s) was defined as the set of all functions which a r e
meromorphic in some right half-plane and with values in Y, and Y_L(s) as those meromorphic functions with values in Y w i t h
l im
sf(s) exists. We have
$-,~c0 S~R
chosen the definitions above, since these will shorten some proofs.
Definition II.13: (~,w)-Representatlon If x 0 6 X , ~(.)~D(A)(s) and w(.)e//_l(s), then the expression
(2.Z4)
xo = ( s - A ) ~ ( s ) - B ~ ( s )
is called a (~,w)-representation of x o.
33 We
remark
that
(2.14)
Furthermore we have
~(s)=(s-A)-lXo
that
is only defined
for
s ~ [ r , oo), for
every x 0 in X has
and w(s)=0, for
a
some
r>0.
(~,w)-representation: take
sep(A)nR.
The (~,w)-representation has the following important property.
[ e m m a II.14.
If
w(s)
~(s),
is a
(~[,~)-representation of
x,
,¢.(s)ED(A)_l(s)
then
and
lira s~(s)=x. s-,~0 Proof: By lim
the
fact
s(s-A)-lx=x
that
A
generates
a
Co-semigroup
we
have
that
for all x in X. By the (~,~)-representation
S-)~O
x = (s-A)~(s)-Bw(s) s~(s) =
Thus I im S-~O
1 im
s(s-A)-Ix -
,,ca-CO
or
s~(s)=s(s-A}-lx-s(s-A)-IBw(s).
I im
s(s-A)-tBw(s) = x+O,
since B is bounded and
S-tO0
l i m w(s)=O,
r3
3--*.¢0
With
this
(~,w)-representation we can
introduce
the
concept
of
frequency
invariance. D e f i n i t i o n II.15: F r e q u e n c y I n v a r i a n c e . Let V be frequency
a,
not
invariant
necessarily closed, if
every
x0
in
subspace of X.
V
has
a
V is said
to
(~,w)-representation
be with
~(.)~V(s).
Remark: With the above result we have that every x o in V has a (~,w)-representation with
if V is frequency invariant, then
~(.)eV_l(s ).
Example ILl6. Let arbitrary
(A,B)
satisfy the assumption made on equation (2.1) and let A be an
element
VA.u: =span{(A-A)-lBu},
of
resolvent
set
of
A
and
u • U.
Define
then g.~,, is frequency invariant.
By a simple calculation we obtain that
(s-A)(A-A)'aBu=(s-A)(A-A)-1Bu+Bu,
so
34
(A-A)'IBu= (s-A)((sl_-~A) A-A)-tBu}- B((sl-~A)u}
(2.15)
Thus Vx.~ is frequency invariant. These subspaces together with the eigenvectors of A are the one-dimensional frequency invariant subspaces
{::1
Example II.17. Let
(A,B)
satisfy the assumption made on equation (2.1) and let C be a
bounded operator from X to Rp. Let g be the following subset of X.
V={xeX]there exists art U(s)eH_t(s) such that
(2.16)
C(s-A)-lBU(s)=C(s-A)-lx for all s>_sx;s~p(A)} We shall prove that this subspace is frequency invariant. Define for x e V
the function ~x{s) to be
(s-A)-tx-(s-A)-'BU(s).
Then (x(-)
is continuous and for fixed s we have
C(A-A)-I(,,(s) =C(A-A)-I((s-A)-Ix-(s-A)-'BU(s)}
(2.17)
=
C(~-A)-'B{(-~_s) (~)+(~I_-%~s) (s)} So ~,(.) is in V and by the definition of V we have that
x=(s-A)~x(s)+BU(s); s>_sx
(2.18)
So V is frequency invariant.
[]
In the sequel of this section we shall prove some technical
properties of
frequency invariant subspaces, which we shall need to prove the equivalence in section II.4. We advice
the
reader
who is not
interested
in details to
proceed to section II.4. Lemma
II.18
unique if
and
lemma
Ita B•V={O}.
B°(V) is a subspace of
II.19
will
imply
that
the
(~,w)-representation
is
First we recall some notation of the previous section.
ImB
that
has zero intersection with V and
is of
35 maximal dimension under this restriction, for short we shall write /~o. Lemma II.18. If a subspace V of g
is frequency invariant, then every element of V
BoJ(.)eB°_l(s).
has a (~,¢o)-representation with ~(.)eV_t(s ) ~nd Proof:
By our assumption we know that every x in V has a (~,~) representation such that l im sw(s) exists. s-~o Let bl,..,bm ° be a basis for
bmo+l...,b~ be a basis for 6'(V): =ImBnV. Then, since V is a subspace , bl,..,bm is a basis for lraB. 4 Since V is frequency invariant and BI(V)cV, we have that there exist pairs (~i(s),wi(s) ) such that ~i(.)~V_t(s), wi(.)eU_1(s ) and and let
bi=(s-A)~i(s)-Bwi(s),
(2.19)
If we write o~i(s) =
(2.20)
/3°(V),
i =m0+ 1,..,ra
, then (2.19) becomes:
bi=(s-A)~i(s )- ~ biwo(s);
i=mo+l,..,m
j=l
We can write (2.18) in matrix notation, and obtain
(2.21)
"
=Q(s)
bm
"
+ R(s); tuhere Q~i(s)=-=~j(s)
bm
a n d R~(s)= ( s - A ) ~ i ( s ) -
m0 ~ b j ~ i i ( s ). j=l
Since ~i(.) is in U_l(s); Q(s) is a continuous function and
limQ(s)=0. S-~.O0
So
( I - Q ( s ) ) -t
(2.21) gives
exists
for
s
sufficiently
large,
and
rearranging
equation
36
bm°.+1 = (I-Q.(s))-IR(8)=
I
(2.22)
mo
.fE I bjWmo+l,.~(8 ) "~,.o + l ( s)
(I-Q(s))-I(s-A)
-
(l.O(s)) -I mo
~,.(s)
i Elbj~,,,j(s)
- .~m0bi(I-Q(s))-' I
(s-A)(I-Q(s))-'
1
!
,
by simple linear a2gebra. This
last
formula
representation
with
implies ~i(-)
to be the i-th row of
mo of
bl,
rao+l ~ i _<m,
(I-Q.(s))-l[
~m°÷l(s)"
by
a
taking
" have
w~,i(s)
(~,w) ~(s)
and Btoi(. ) to be the i-th row
Furthermore
since Q(e~)=O,
j
lira
sCai(s)
=
s-co
, and by assumption this limit exists. proved
the
ImBnV=B~(V). Let x (~(s),w(s)), ~(.)~V_l(s )
assertion
of
lemma
H.18
for
all
elements
in
be an arbitrary element of V, then there exists a pair and
w(.)~l~_l(s ) such
that mo
(2.23)
has
[ Wmo+~,]j(s)[ (,.(s)
bi(l_Q(s)) -I j=lw,,l(s) 1
we
each
in V_l(s ) arid B~i(. ) i_rl B°I(V),
~
lira s So
that
m
x={S-A)~(S)-Dw(s)=(s-A)~(s)- ~ bi.~i(s)- ~ b~i(s) i=l m
i=mo+l mo
i =mo.t
ill
m
i =mo+l
m
i=
nlO
O+
~,(S), ~,(s)~V_,(S) and B~,(. )eB_1(s); o eo =span{bl,..,bmo}.
i=l
37
Thus x has a (~,w) representation with
~(s)=~(s)- ~ ~i(s)w,(s)eV.,(s), i =mo+l
B~ls) =
BC~,(s) wi(s)- ~,, b+w~ls) e B°.I(s) and i=mo+l
lemma
i=!
lira s ~(s) exists. S-t-GO
I"1
11.19.
If V is closed subspace
that
is frequency invariant,
such that
+or a . + L
Proof:
bt,...,b,+°
Let
I
l r er
there
exists
,+.
be a basis for B °. Then by the Hahn-Banach theorem and
{g+mX',i=l,..,mo}
the fact that B°nY={O}, there exist functionals
=6ij
then
such that
gilv=O. Define for sep(A)nR the following matrix valued function S(s): =(). By the continuity of the resolvent operator we have that this function is continuous on p(A)nR. Furthermore we have that S+j(oo): =limSo(s)= =$+j. So S(oo) is nonsingular. Then by and
the continuity of that
S(s)
S(s) in
plus infinity there exists an interval
[?,co) such
is invertible on that interval.
On this interval we have that
(s-A)-tB°nV={O},
since otherwise there
ra 0
would exists a vector ul+O and a so>~ such that ~ {s0-A }-1 b+uj 1 is contained j=l
in V, and so mo
ra 0 I
-I,L
L
%uj>
j=l S(so)u I
= m o
m 0
-1
!
I
bjuj >
~
(So-A)-Zbjuj
>
=l
rn 0
1
=80
0
= m0
~ (so -A)-~'bju} >
0
j=l
Thus det(S(s0) ) =0, providing the contradiction. The following leman is the frequency domain version of leman II.10.
12
38 [,emma II.20. If a closed subspace V of X is frequency invaxiant, then every x in V
has a unique and
there
((s)
and
(~,w)-representation
exists an
w(s)
interval
with Bw(.}
[~,oo),
axe continuous
on
in B°_l(s) and ~(.) in V.l(s )
which is independent
this
interval,
of x,
this .~ is the
such same
that as
in
lemma II.19. Proof: Let ~ be the constant of lemma II.19 and suppose that g i e X '
= 6 i i , where {bj}m°1"= is a basis for B °. Furthermore let
gi[ v = 0 and be
an
satisfies
arbitrary
representation
element
with Bw(.}
of
V,
contained
t~en
by
lemma
in g°_l(s)
II.18
it
has
a
X
((,w)
. Thus oil an interval [rx,~ )
x can be written as
(2.24)
x=(s-A)~(s)-
mO
~, b#oi(s), ~(.)~V_I(s ) i=l
(2.24) implies: mo
(2.25)
(s-A)-tx=((s)-
~, (s-A)-tbioai(s) /=,
Calculating < g i , ( s - A ) - t x > ; i = 1,..,m o gives mO
(2.26)
= -~
wj(s)
mo
= - F, wj(s), since ((s) is in V. If we premultiply (2.26) by s we obtain; ml]
(2.27)
= F.
wj(s).
i=1
Or in matrix notation:
(2.2s)
=
-S(s)
< g,.0,s( s - A ) " x >
,
w.,o(s) j
where Sift s) = < g,, s(s - A )-'bj >. Note
that
S(s)
and
< g o s ( s - A ) - 1 x>
are
continuous
on
the
interval
39 [~,oo)¢p(A), and from lemma II.19 we have interval.
So
on
[~,oo), {wi(.)},
i=l,..,m
that
S(s)
is invertible on this
is
the
tmique
o
solution
of
equation (2.28). By (2.25) we have that there is only one choice for ~(s), that is: mo
(s-A)-lx+(s-A)-t( ~ biwi(s) ) []
i--I
Let us remark that (2.28) implies that if ,1" is finite dimensional, and so A is a matrix,
wds)
then
is a
rational function, and with the
last line of
the proof of lemma I/.20 ~(s) is a rational function too. So if X is finite dimensional, then
definition H.15
is
the
same
as
if
we
were
to
restrict
ourselves to strictly proper rational functions, as in Hautus [19].
Before we can prove the equivalence between frequency and and closed loop invariance we need
some
properties
of
the
set
of
all
possible values
of
~(s) for x in V, which we shall denote by ~s. D e f i n i t i o n II.21: ~"
s
If
V
is
a
~t~VnD(A) such
frequency that
there
invariant exists a
subspace, x
then
in V with a
x=(s-A)~(s)-Bw(s), ~(. )~V_t(s), w(. )~ll_t(s),
Zatconsists (~,w)
of
all
representation;
such that ~(st)=~t.
Lemma II.22. If V is a closed subspace that then
we
have
that
there
exists
is frequency invariant and
a
real
.~ such
that
for
any
lmBnV={O}, st>~ , the
equalities (2.29)
I x=(sl-A)~t-Bwt
and
[ x=(sl-A)~2-Bw2,
when x, ~x and ~2 in V,
imply that ~t = ~2. Proof: Since V is a closed subspace of ,¥ and simple corollary lemma II.19
of
lemma II.19, and
ImB•V=(O},
this lemma is a
.~ in this lemma is the
same
as
in []
40 Lemma II.23. Let g be a closed frequency invaxia~t subspace with Im BoV= {0}, then:
S1 - - ~ s 2 )
for all s:,s2~[~,oo), where ~ is as in lemma II.19. Proof: Let ~l be an element of --h' then there exists a x in V with
x = ( s l - A ) ~ l - Bw(sl); ~t = ~(sl) Rewriting this equation gives x = (sL-s2+s2-A)~l-Bw(sx) , or
(2.30)
(s~-sl)~l+ x = (s2-A)~ 1-Bw(sl)
(s~-sl)~l+x is an element, of g thus it has a (~,w) representation. So there exists a pair (~,w) such that
(s2-sl)~,+x=(s-A)~(s)-O~(s)
(2.31)
From lemma II.20 we have that equation (2.31) holds on [.~,oo). Now relations (2.30}
and
(2.31)
with
lemma H.22
imply that
~(s2)=~r So _41c_sz.--~ By
symmetry we conclude that _ 41 _ - s2. - -
[]
_
Lemma II.24. Let V be a closed frequency invaxiant subspace with ImBnV={O}, : =~'-
4
is
closed
in
the
graph
norm
of
A,
for
definition
see
then Davies
[14, lemma 1.6.], where ~ is defined as in lemma II.19. Proof: Let ~n be a sequence in ~
such thai; ~n-~ y and A~n-~ z. Since A is a
closed operator) we have that yeD(A) and Ay=z. for ImB,
then since ImBnV={O}
Let {bl,..,bm0} be a basis
and V is a closed subspace there exist gieX' such that 9i[ =0 and <9i, bi> =6~i. Since ~,, is an element of ~, there V exist xn in V and wn in // such that m O
(2.32)
x. =
(~-A)~.-Bo2. = ( . ~ - A ) ~ , - ~ b ~ . # j=l
Since xn~V , we have that
41
O= < g , , x . > = < g i , ( ~ - A ) ~ n - B w . >
and thus
= - < g i , A ~ . > - wni
=-u,r= ~
So ~ni converges as n ÷ ¢0, i = l , . . , m 0 . Thus t¢n converges to say w e / / and since x n = ( ~ - A ) ~ n - B w n we have that x n converges to x. Since V is closed we have that x ~ V and so there exist ~{s) and w(s) such that x = (s-A)~(s)-B~(s).
By definition x is also equal to (.~-A)y-Bw. From lemma II.22 we have that y=~(.~), and thus y ~ S .
[]
S e c t i o n II.4: Equivalence In
the
closed
theory
loop
of
and
system
open
equivalence tells us that, stays in a subspace, trajectory this
stays
dimensional.
If
equivalence
was
bounded.
but
We
Furthermore frequency between
is
the
of
equivalence
great
between
importance.
This
the trajectory
then we can also find a feedback law such that subspace.
was
proved
the
state
proved
by
In Basfle & Maxro [1] and
in
the
case
space
is
Schmidt
that
the
infinite
and
state
space
dimensional,
Stern [32]
the
Wonham [42] is
finite
then
provided
that
the A
is
However, the interesting case in infinite dimensions is when A is
unbounded, section.
subspaces
invariance
if we can find an input such that
in this
equivalence
invariant
loop
generates
formulate
and
we
prove
shall
invariance open
a
and
C0-semigroup; prove that
for
closed
closed
loop
this
this
is
equivalence
closed
linear
and loop
subspaces
invariance
is lost
the
for
the
invariance and
focus
that
if the
of
system is
the
this (2.1).
equal
to
equivalence
subspace
is not
closed. We shall begin by showing that there is equivalence between (A,B)- and feedback (A,B)-
invariance.
and
However
closed loop
there
is
invariance even
in
general
no
equivalence
if we impose the
that VnD(A) is dense in V, as shown in Schmidt and Stern [32].
extra
between condition
42 Lemma II.25. If V~cD(A) is a linear subspace, closed with respect to the graph norm of A and V2c9:' is a closed linear subspace with (2.33)
AV, c
Vz+ImB ,
then there exists an A-bounded feedback law F such that (2.34)
(A + BF)V l c V~
Proof.. If X
is finite dimensional, then the
proof
can
be found in Basile &
Marro [1] and Wonham [42, p.88]. The general proof
that
will be presented
here is an adaptation of the proof given by Pandolfi [27]. Define X A to be the graph of A, with the graph
norm [[(x, Ax)[[A
=
[[x[[+[IAx[[ , where [[.[[ is the norm of X. A is a bounded operator from X a to X, and V1 is by definition closed in X A. If v l e V l , are
then there exists v 2 e V 2 and u e U such that A v l = v 2 + B u , and u and v2 J_ uniquely determined if we assume that u e [ K e r B ] (the annihilator of
Ker B) and B u e B ° ( V 2 ) .
Let F be defined by F v l = - u ;
V v l e V ~. The operator F is linear, since u is
uniquely determined. We shall show that F is a closed operator n
us assume that (vl,Fv~) converges to ( v L , - u ) , must prove that Fv I = - u . n
n
in ,~'A. Let n
thus v l - ~ v 1 and A v l - ~ Av 1. We
This is obvious since
rA
w : -- Av I - Bun = Av~ + BFv n converges to A v 1 - B u = : w,
and v l e V ,
since V1 is closed and B u e B ° ( V s ) , since B°(V2) is closed.
Thus F is a closed operator from the whole of V~, with induced norm of [[.[[4, to /L By the Closed Graph Theorem F is a bounded operator on VI, with norm [[-[[a, by the Hahn-Banach Theorem and the fact that, since /l is finiLe dimensional, F has finite dimensional range, F has a bounded extension on X A. From Kato [22, p.191 and 245] we have that F is A-bounded on X.
O
T h e o r e m II.26. If V is a closed subspace of X, then (A,B) and feedback invariance are equivalent.
43 Proof: This is a easy corollary of lemma ]:I.25 since if V is a closed subspace and A is a closed operator, then Vr~D(A) is closed with respect to Lhe graph norm of A.
[]
Now we have proved all the ingredients for our main result.
Theorem II.27. Let V be a closed linear subspace of X, then the following concepts of invariance are equivalent: a)
V is closed loop invaria~t
b)
V is open loop invariant
c)
V is frequency invariant
We remark that lemma 1.4 can be seen as a special case of this theorem i.e. no control action thus B =0. If X is finite dimensional, then the equivalence between these concepts are known, see for a) ,~ b) e.g. Basile & Maxro [1] and for a) ,~ c) Hautus [19]. Since we have equivalence between these invariance concepts we introduce a new concept that we shall use if a subspace satisfies II.27 a),b), or c).
D e f i n i t i o n II.28 Controlled Invariance A closed
linear subspace
V of
X
is called
controlled
invariant
if
it
satisfies II.27 a),b) or, equivalently c).
As we shall see in the next chapter the equivalence between open loop and closed loop invariance as well as the equivalence between frequency and closed loop invariance is lost in general if the subspace V is not closed. From the
previous chapter
between controlled and
we see
that
there
is no
hope
for
equivalence
(A,B) invariance in general. However there is one
case were this equivalence holds.
44 Lemma I1.29. If
V is a
closed
subspace
contained
in the
domain
of
A,
then
the
following assertions axe equivalent: a)
V is closed loop invariant
b)
V is open loop invariant
c)
V is frequency
d)
V is
invariant
e)
V is feedback invariant
(A,B) invariant
Proof: The equivalence between a) and e) is a consequence of lemma 1.7, the other equivalences follow from theorem 1/.26 and II.27. We shall prove theorem II.27 by showing a) =~ b) =~ c) =~ a). P r o o f o f T h e o r e m II.27:
a) => b)
TA+SF(t)V c V. x(t) as TA+BF(t)X0 and u(t) as FTA+BF(t)Xo, then Curtain and
Let F be the feedback law such that Defining
Pritchard [9, th.2.31] gives the desired result.
b)
=>
c)
Let x o be an element of V. Then from lemma II.11 we have that there exists an u(.)~C([0,oo);U) such that t
x(t}=T~(t)xo+ I TA(t-s)Bu(s)ds
(2.35)
0
is in V and I I ~ ( t ) l l _< 2~" ~eoa , f o r some H and ~ in R. The exponential boundedness of u(.) implies the same for x(.), and so we can take the Laplace transform of
equation
w(.)
the
(2.35).
Laplace
Define
transform
~(.)
of
to
be
the
u(t). Then
Laplace transform
equation
(2.35)
of
gives
x(t)
on
and
an
interval [r%,0o) the following relation (2.36)
~(s)=(s-A)-tXo + (s-A)-'Bw(s)
Since V is a closed subspace we have that Zemanian [46] we have that l i r a s-,.oo
(~,w) representation of x o.
f(s)eV, and with theorem 8.6-1 of
sw(s) = lira u(t) -- u(O). So ~(.), w(.) is a t¢ 0
45
c) => a) Since V is a closed subspace,
we see,
by lemma II.20, that
restrict our input operator B to B, such that B is injective, and ~ l ( V ) = I m B n V = { O } the
system
(A,B).
we may
B°(V)~.~°(V)
(see (2.8)). Then V is also frequency invariant for
So
we
may
assume
~tithout
loss
of
generality
that
Im Bn V = {0}. By assumption we have that all x in V admit the following decomposition.
(2.37) If s
x = (s-A)~(s)-Bw(s).
is larger
than
~ (see
lemma II.19 and
definition
II.21),
then
since
~{s)eEj,(2.37) implies that (2.38)
A ~ , c V + lm B.
From lemma II.24 we have that - = , = 3 . So (2.38) implies that (2.39)
A---cV+Im B.
5" is closed in the graph norm of A, so from lemma II.25 we have
the
existence of an A-bounded F such that (A+BF)EcV. Rearranging equation (2.37) gives x = (s-A)~(s)-Bw(s) = (s-A-BF)~(s)-B(w(s)-F~(s)) Thus B ( w ( s ) - F ~ ( s ) ) e V ,
so
by the assumption made in the beginning of this
proof w(s)=F~(s). So (2.40)
x = (s - A - BF)~(s)
It remains to prove that this feedback law is bounded. We shall begin by showing that the mapping x~-*sw(s) is a bounded operator from V to U for se[~,oo)np(A). Let for se[~,oo)np(A), F~ be defined as Fsx: =sw(s), where w(s) is the (unique)
input
from equation (2.37). It is easy to show that
these
46 operators
are
closed,
and
thus,
since
they
axe
defined
on
the
closed
subspace V, bounded. By the definition of //_l(s) we have that lira sw(s} exists, so for every s-~o x in V we can define the operator Fx by F x = l i m F,x= lira sw(s). Then by the $-t00
uniqueness
of w(.) ,~ is a linear
Boundedness theorem,
operator
3=)~0
defined
on V. By the Uniform
and since V is closed we have that F is a bounded
operator on V. The Hahn Banach Theorem gives that F can be extended ms a bounded operator to the whole of X. We shall show that on ~. F--F. Let ~0 be an element :7, then there exists a x0, ~(.) and w(.) such that
xo = (s-A)~(s)-Bw(s) and ~(~)=~o
(2.41)
Rearranging this equation gives that
[. o~(s)-~(,~} (2.42)
-e
~o= ( s - A )
.q-$
~(S ) -,% Define ~t(.)
and
wl(.)
to
be respectively
Then F~0=lim swl(s)=w(~)=F~(~)=F~o. operator
2-)¢o
F.
So
equation
(2.40)
and I .~-s ]" ~-s So on &" F is equal to the bounded
implies
that
for
(s-A-BF)-tx=~(s)eV, and lemma 1.4 concludes the proof.
sufficiently
large
s C:]
CHAPTER II1: DISTURBANCE DECOUPLING PROBLEM
tn
this
chapter
we
shall
consider
the
disturbance
decoupling problem
(DDP): given the system
(3.U
x(t)=Ax(t)+Bu(t)+Eq(t),
z(t)=Dx(t),
where A and B are the same as in (2.I) and E and D are bounded linear operators from respectively Q to ,1' and ,l" to Z, find an bounded feedback law such that, in the closed loop system, z(.) does not depend on q(.). So pictorially we have the following situation;
x = Ax U
+ B u + Eq
z = Dx
Thus the Disturbance Decoupling Problem is to design a feedback law F such that the transfer from q to z is zero i.e. D ( s - A - B F ) - I E - - O . Tile next theorem gives the link between controlled invariance and DI)P.
Lemma III.1. The
Disturbance Decoupling Problem
is solvable
if
and
only
if
there
exists a controlled invariant subspace V such that I m E c V c K e r D . Proof: See Curtain [6].
In
the
finite
dimensional
theory
it
(:outrolled invariant subspace contained in the calculated.
Here
we shall also
turns
out
kernel of
introduce this subspace,
that
the
D can but
largest
ahvays be
tile calculation
is a difficult problem in general. $
D e f i n i t i o n III.2: V (K). Let K be a closed subspace in X. Then we shall denote by I)*(K) the largest controlled invariant subspace contained in K.
48 If this subspace exists, then we have the following nice result. T h e o r e m III.3. If V*(KerD) exists, then DDP is solvable if and only if 12*(Ker D)~ Im E.
{3.2) Proof: See Curtain [6] In
Curtain [6]
Y•(K)
the
is posed.
ID question
of
We remark
sufficient
here
that
conditions
for
the
existence
~'*(K) need
not
always
of
exist, see
Pandolfi [27] for a counter example for delay equations, or see appendix E for
a
Y*(K)
counter exists,
example then
it
for
partial
must
be
differential
closed,
since
problem
will
equations.
Note
that
TA+sF(t)VcV implies
if that
rA+sr(t)VcV. The
Disturbance
Decoupling
be
frequency-domain as well as in the time-domain respectively.
In
section
HI.3
we
investigate
investigated
in
the
in section III.1 and III.2
some
properties
of
closed
loop invariant subspaces using the results from llI.1 and Ill.2.
Section III.l: DDP In Frequency Domain Keeping the equivalence
between closed loop-
and frequency invariance
in mind we define the natural candidate for 12*(K).
Definition I~.4: V~(K) We assume that K is a closed linear subspace. Let
PE(K)
be
the
subset
of
X
which
contains
all
x~X
with
a
(~,w)
representation with ~(.) in K(s), (see definitions II.12 and H.13).
The next lemma will show that P u(K) is the supremal frequency invariant subspace contained in the closed subspace K.
49
I.emma III.5 a)
Every
frequency
invariant
subspace
contained
in
K
is
contained
in
VE(K). b) Every closed loop invarianL subspace contained in K is contained in VE(K)c)
V£(K) c K.
d)
1)•(K) is the supremal frequency invariant subspace, contained in K.
Proof: a)
Obvious,
with the
definition of V u(K ) and the definition of
frequency
invaxiance. b)
Obvious,
with
theorem
II.27, since if V is closed loop
invaxiant,
then
]7 is it too. c)
Let (~(s),w(s)) and
x=lim
be
a (~,w) representation of x
with lemma TI.14 and the fact
in V F.(K),
then
E,(s)~K
that K is closed we conclude that
s~(s) is an element of K.
S-~O0
d)
Let
x
be
an
element
functions ~(s) and w(s)
of
I)~(K),
so
that
there
exist
strictly
proper
which are continuous on an interval [rx,oo ) and
such that 5(s) is in K and
(3.3)
for s>r~
x=(s-A)((s)-Bw(s);
(see
definition
II.15). Let
s o be
an
arbitrary,
but
fixed
point
in R
with s0>rx, then by using (3.3) we get
(3.4)
0 = ( s - A ) ~ ( s ) + ( s o - A ) ( - ~ ( s o ) ) - B w ( s ) +Bw(so)
,s A, i ,s, ,so,l +,s
=
,so, 1
So if s ~ So, then (3.4) implies that
(3.5)
~(so)
= (s-A)
I(~(s)-~(s°))lJ -s I (w(s)-w(s°)) 1 -
--
.[ -
B
S O
SO-S
Define for s > s o {l(s): =
and wl(s): = S o --S
It
is
not
hard
to
show
that
. S o -S
(~l,wl)
is
a
(~,w)-representation
of
5O ~=~(so)
limswl(s)=w(so).
with
Since ~(s) is in K, (l(s)
is in K. Thus by
8-~.¢O
the
definition
of
V u(K),
~(so)
is
in Vx:(K ). Since
so was
an
arbitrary
element of [r~,oo) we may conclude that x has a (~,w)-representation
with
~{.)eY]:(K). Now since x was an arbitrary element of 1)~(K) we have that V~(K) is frequency invariant. By c) we obtain
I;$(K)cK
and a} implies
that
Vx(K)
is the
largest
frequency invariant subspace with this property.
[]
With this lemma we can easily prove the next theorem. T h e o r e m III.6.
Let K be a closed subspace of k'. If
VF(K)
is closed, then
V"(K) exists
and is equal to
VF(K).
Proof: If conclude
~)U(K) is that
closed, VE(K ) is
then closed
by
lemma
loop
III.5.a) and c) it must be the largest and is equal to
III.5.d)
invariant.
and
theorem
Furthermore
with this property. So
VF(K).
II.27
from
we
lemma
V*(K) exists 0
Remark:
We remark that the converse of the theorem does not hold, i.e. if exists, then
VF_(K)
So the question which arises is under what conditions is
12E(K)
turns out that these conditions can he formulated in terms of
IZE(K),
V*(K)
need not be closed, see example E.10. closed. It
ImB
and
which we do in the next lemma. In this lenLma we shall use the
following notation. If V is a subspace of X, then
VJ={fe2d'[
such that
f(V)=O}.
V'cX'
this sequence is weak * convergent to f if
will denote the annihilator of V i.e.
Furthermore if {)¢,,} is a sequence in ,V', then
~
for all
xeX,
where
denotes the operation of the functional f on x. See for more details
Kato [22, p.136] and Yosida [44].
51 Lemma III.7. Let
13°(l;E(ll)) denote the largest subspace of IrnB that with llE(K), let {bD..,bm0} be a basis of 13o(Vz(K)) Irn BnV. Then the following assertions are equivalent. VE(K ) is closed.
has
intersection
BI(V) i)
be
ii)
(VE(K)) atnD(A')
iii)
There
exist
=6ij,
is weak * dense in ~xJE(K)).~ and m 0
(f~}i=t
functionals
zero let
BIO;~(K))= ~ I ( V - - ~ ) .1.
in
and
[V~:(K)) nD(A')
such
that
where D(A') is the domain of the dual operator of A, see
Yosida [44]. Remark: Before proving this theorem, we remark that the result of this lemma is independent
of
8°(V~(K)). condition
the
basis
Furthermore in
the
third
it
of
B°(]/E(K))
and
is
clear
simple
assertion
of
by
this
of
the
linear
lemma
is
actual
choice
of
algebra
that
the
to
the
equivalent
&
existence of functionals f i e (V~:(K)) nD(A') such that the matrix
S®=
is nonsingular. P r o o f o f lemma III.7:
0=>~0 .I.
So we must show that the annihilator of V~(K), lYZ(K) intersected with D(A') is sufficiently rich. If I)E(K ) is a closed subspace, then by theorem ILI.6 it is closed loop invariant. So by lemma 1.4.c) there exists a bounded feedback law such that
(A-A-BF)-IVE(K)cV~(K)
for all AeR sufficiently large. at
Let y be an arbitrary
(A_ A" _ F'B')-ty
element of
is an element of
[VE(K)) , then we shall show that
[VE(K)]~'nD(A ") and A(A_A'_F°B')-ly
is weak
J.
* convergent to y, A-,co. This will prove that is 0;~(K)] nD(A'} weak * dense .L
in ( v E ( g ) ) . Let x be an arbitrary element of VE(K), then since Vz:(K) is invariant we have that
<(A-A'-F'B')-Iy, x> =
=0
(A-A-BF) -1
52
(A-A'-F'B')'ty
Thus
is an element of (I,)$(K))±nD(A').
Now let x be an a r b i t r a r y element of X, then
lira A..t,~
= lim = . ),.÷¢0
This proves the assertion.
ii)=>iil) 13*(PZ(K))=BI(VZ~O)
Since
we have from the well known Hahn-Banach mQ
theorem
the
matrix S ' =
existence
of
functionals
{gi}i= l
<9~,bj>
S=
such
that
the
is non singular. Since (VE(K)) nD(A') is weak * dense in m0
there
/.
3.
.L
0?~(K}~ ,
(VZ(K))
in
also
exisl;
..k
{fi}i=lc(Y,F,(K))nD(A')
functionals
is non singular. Taking linear combinations of
these
such
that
fi's
gives
the desired result. iii)=>i)
V u(K ) is frequency invariant and by lemma II.18, we obtain that every x ra 0
in Y ~ { K ) h a s
a (~,w) representation with
argument
in
as
representation. So
(3.61
the
proof
wi(s)
is given by (2.27)
"
of
= -S(s)
lemma
- 1
Bw(s)= ~ b,~J,(s). II.20
we
can
calculate
this
(~,w)
and
.
~w,,:(s)J
Using the same
ra 0
,(s)=($-A)'Ix-F (s-A)-I I ~ biw/($)]) i=1 where
Sifts)=
Let x be a element of V~(K), then there exists a sequence {xn} in Y~(R) which
converges
to
x. xn is
an
element
of
YECK) so
it
has
r=
representation with w " ( s ) =
wmo(s) rl
w.,(s)
, where m is the dimension of
a
ImB.
(~,w)
With
53
the above we may choose wm0+t(s)=,.. , =w~(s)~-0 and
"
as in (3.6) with
l W~o(S) n x
replaced
by
x n.
With
this choice
[ ~l(s)]
converges to x, then ~ ( s )
•
is the
right
it can
easily be
converges, and wi(s):=
hand
side
of
(3.6).
o-~o
seen
that
if x n
n
lira wi(s)--O if i > m o and
From
(3.6)
it
is
easily
seen
~omo(s)J that
w(s) is continuous
on an
interval [r,oo) and since for
= + 1 im s-~o
sw(s) =
since 1 im
1 im S ( s ) ' l l im s-*oo s-~o
=0+
"
all x c V £ ( K ) , we have that
=
, < A'/,~o, x >
SO(s) = 64i
S-~O0
Define ~(s)=(s-A)-Xx-(s-A)-XBw(s); x=(s-A)~(s)-B~(s) and ~(s) is the limit
then of
~(s)
is
continuous,
~n(s)=(s-A)-lx,~-(s-A)-lB~o'~(s),
as n ÷ oo, thus ~(s) is in K. So x has a (~,w) representation, with f(.) in K(s), thus x is in Y,u(K). The
importance
results,
not
in
of its
case of A bounded
this
theorem
direct
lies in its usefulness
application
to
specific
in deriving
systems.
Even
El further for
the
it is not clear if the conditions are fulfilled. However
in the next lemma we show that if A is a bounded operator, then
YE(K)
is
closed. Lemma 111.8. If A is a bounded linear operator, then 1)"(K) exists and it is equal to
vz( l. Proof: Suppose first that the following holds: (3.7) Then
A[ Y~--~] with
Schmidt
and
lemma III.5 ))•(K)=V*(K),
c
])E(K)+ImB.
Stern [32], but
V£(K) is controlled
V*(K)cV£(K),
invariant.
So
with
so VE(K)=VE(K). Thus it remains
to prove (3.7). Let
x
be
an
element
of
I/,u(K), then
there
exists
~(.)~K(s)
and
54 ~(.)e//_l(s) such that
(3.8)
x= (s-A)~(s)-Bw(s)
With lemma II.14 we have that x = l i m
s((s). Rearranging equation (3.8) gives
As~(s) =s2((s)-sx-Bsw(s) From lemma ffL5.d) it follows that ~(s)~VI:(K ).
So As~(s)eVz(K)+ImBcV,u(K)+ImB.
Since A is a
bounded
operator
and
s~(s)-*x as s ÷ 0% we have that Ax~VI~(K)+Im B=VE(K)+ImB , because lmB is finite dimensional. Thus AV~(K) ¢])z(K) +Ira B. Using once again the fact that A is a bounded operator and V~(K)+ImB is a closed subspace we have proved that ,4 lenm~a.
VI~IK) [--1
c VE(K)+ImB and thus this D
The next lemma will give sufficient conditions for theorem I]].6, which are easy
to
verify.
These
conditions
first
occurred
in
Curtain [6],
where
the
aim was to give sufficient conditions for the existence of V*(K). Lemma III.9. Let KerD be the kernel of a bounded linear operator D. Then Vz(KerD) is closed if either of the following conditions holds. a)
There exists a q in Nu{0} such that DA~ has a bounded extension defined on the whole of ?d (denoted by DA ~) for 0<_i<_q+l, DA~B=O for O
DAqBu # 0 for all u ~ 0 in ~'. b) D(s-A)-IB=O for all s in [s, oo) for some .~ in •. Proof:
a) Suppose that x~l)z(K~rD ) and condition a) holds. From lemma II.14 we have that ~ = l i m S-aq~o
s~(s) if
55
(3.9)
x
Using the
fact
=
( , - A ) ~ ( ~ ) - e~(*).
that w(s)
is strictly proper,
lemma II.14 and equation (3.9)
we obtain
(3.10)
lim A~(s}=O. S->O0
If we premultiply (3.9) by D we get since ~(s)¢KerD emd DB=O
(3.11)
DX = D( s - A )~(s ) -DBw( s ) -- sD~(s) - DA~(s) -DBo:(s ) = -DA~( s )
Using the fact that A~(s) is strictly proper and D is a bounded operator we get Dx=O and DA~(s)=O. By induction it is easy to prove that DAix=O and
DAi+t~(s)=O for O<_i
(3.12)
DAqx = DAq(s-A)~(s)-DAq'Bo:(s) = sDAq~(s)-DAq(A~(s))
-DAqBw(s) = O-DAq(A~(s))-DAqBw(s}.
Using the strict properness of A~(s) and w(s) we get that DAqx=O for all x
in YE(KerD). With the fact that DA q is a bounded operator we obtain that DAqx=O for all xcVs,(Ker D). Let z be an element of the dual space of Z,
with this functional we can
define a bounded functional on X, i.e. . This functional is zero on
))s(KerD) and since DAq+' has a bounded extension DAq. is an element of D(A'). The uniformity of the zero structure gives that one can find z i such m is a basis for ImB. Thus by lemma III.7 that =5ii, where {bi}j=1
VE(Ker D) is closed. b) In the cane that D(s-A)'IB=O, VE(KerD) is equal to the subset in X of all x
in KerD
with D(s-A)'lx=O.
This follows since for
x
in Vs(KerD),
56
(s-A)-lx = ~(s)-(s-A)'lOw(s), holds and thus D(s-A)-lx = O((s}-D(s-A)-tBta(s) = 0-0 -- O. D(s-A)-tx=O, then x=(s-A)(s-A)'tx-BO is a (~,w) representation with ~(s)=(s-A}-lx is in KerD. So ImBcll~s(KerD), which implies that Yf(KerD)
Suppose
is closed by lemma III.7. Let us
remark
that
[]
the
condition
condition that the transfer function in plus
infinity. This
holds
even
a)
in lemma HI.9
is weaker
than
the
D(s-A)-IB has only finitely many zeros if the
input and
output
space
are
one
dimensional, as can been seen from the next example: Let X be a Hilbert space and A a generator of a C0-semigrou p on X. Assume
b,d~X with tieD(A*), (the domain of the adjoint of A), and < d, b > x ~ 0. Then < d,A. > x does not have a bounded extension from Pg to C, but
furthermore that the
multiplicity
of
the
zeros
in
plus
infinity
of
the
transfer
function
is one. Using the characterization of III.6 we derive
the
following
l~E(tierD) given in lemma III.5 and theorem
equivalent
statements
for
the
solvability of
DDP. Theorem III.10.
Assume
that
YE(KerO) is closed,
then
the
following
statements
are
equivalent. a)
DDP is solvable.
b)
lm Ec l),u(Ker D).
c)
For every
q in Q
there
exists a w(.) in U_a(s ) (see definition II.12),
such that (3.13) d)
D(s- A)'tEq = - D(s- A)-IBw{s)
There exists a U ( s ) i n
(3.14)
I£,(Q,bl)]_~s)such that
D(s-A)-IBU(s) = D(s-A)-tE
57 e)
There exists a U I s ) i n I/:(Q,//)] (s} and X ( s ) i n ~ I I / : ( Q ' X ) ] - I ( s ) s u c h that
13.15/
-D
0
.
-U(s)J
0
Proof: This
follows
trivially
from
theorems
III.3
and
]]].6,
and
definitions
III.4 and I1.13.
E1
Remark:
Theorem ITI.10 is the same as theorem 3.4 and 3.6 in Hautus [19] for the finite dimensional case. We see from this theorem that under an extra condition on the system the
solvability of DDP is equivalent
matrix
equation.
One
can
easily
to
show
the
solvability of
that
the
solvability
a
meromorphic of
(3.14)
is
always a necessary condition for the solvability of DDP. However it can be seen from example E.10 in appendix E that
it is not sufficient; of course
in this case P~(Ker D) will not be closed.
Section III.2: DDP in T i m e - D o m a i n In
this
section
we
shall
state
similar
results
as
in
the
previous
section, however now we shall work in the time-domain. Analogous to r E ( K ) we can define the largest open loop invaxiant subspace.
D e f i n i t i o n III.11: Let that
Pol(K)
there
Vol(K )
be the subset of X which contains all
exists a continuous
input u(.)
such that
xoeX
with the property
the mild solution x(.}
t
of (2.1) i.e.
x(t)=TA(t)xo+fTA(t-s)Bu(s)ds
is in K for all t > 0 .
0
Remark:
If K is the kernel of a bounded operator D~ then this subspace contains
z(.)=O , ~(t)=Ax(t)+Bu(t); x(O)=zo; z(t)=Dx(t).
all initial values x 0 such that for some continuous u(.) the output where z is the output of the system
58
For this subspace we shall give similar results as for lZE(K ) in section ]:[I.1. Since
the
proofs
of
these
results
are
very
similar
we
shall
omit
them. Theorem III.12. Let Poz(K) be the subspace defined by ITI.11, then it has the following properties. a)
V~(K) is the largest open loop invariant subspace in K.
b)
If Vo~{K) is closed, then Y*(K) exists and is equal to Pot(K).
c}
Let
B°(12oz(K)) denote
intersection V~(K)
is
the
with Vo~(K) and closed
if
and
largest
let
only
subspace
of
[roB
{bl,..,b~,0} be a basis if
there
exist
that
has
zero
of /~°(Vol(K)). Then
functionais
{fi}
m0
in
D(A')
i =l
such that < f~, bi > = ~ii and fi[ d)
Let
BI(V)
denote
v~(K)
-- 0.
ImBnV.
Then
(Vo,(g)) ± nD(A') is weak * dense in e)
If A is
a
bounded
operator,
)J~(K)
is
dosed
if
and
only
if
(Vo,(K))x and B I (Yo,(K))=]3'(P--~ then
V*(K) exists and it is equal
to
Yo~(K). Remark: As in the frequency domain it is possible that ~2*{K) exists, but VodK) is not closed, see example E.12. Remark:
As an easy corollary of theorem III.12.d) one has the following result. If V is open loop invariant,
then 17 is controlled
invariant
provided
that
VXnD(A') is weak * dense in Vl and BI{V)=131(17). Lemma III.13. Let KerD be the kernel of a bounded linear operator D. Then V~(KerD) is closed if either of the following conditions holds. a)
There exists a q in Nu{0} such that DAi has a bounded extension from ,k' to Z (denoted by DA ~) for 0 < i < q + l , DAiB=O for O
b}
D(s-A)'IB=O for all s in [~,¢o) for some ~ in R.
59 So we see that with the definition
Vr(K )
and P~(K) have the same properties, this together
of these subspaces suggests that P~:(K) and Yoj(K) are
equal. However this is presently still unknown. It is even an open problem whether V~:(K) is closed if and only if
Vol(K) is
closed.
Section III.3: Properties of Controlled Invarlant Subspaces From
the
previous
sections
we
obtain
very
nice
results
concerning
controlled invariant subspaces. We
mv.ke
the
following
important
remark.
subspace that is controlled invariant, then
If
V is
a
Y~(V)=Yo~(V)=V.
closed
lineax
This is obvious
from lemma I]1.5, theorem HI.6 and theorem III.12.a) and b). In finite dimensions use is made of the fact that controlled invariance is closed infinite
under
subspace
dimensions
addition.
in general.
Unfortunately
The
next
lemma
this
no
gives
longer
sufficient
holds
in
conditions
for this to hold. Lemma III,14. Let V1 and
V~ be closed,
controlled
invariant
subspaces
of
X. Then
Vt+V ~ is controlled invariant provided that one of the following conditions holds.
a) Vt+V2 is b)
There
closed, so in particular g 1 or g 2 is finite dimensional.
exist
funetionals
{fi}m0
in
D(A')
such
=8ii, where bi is a balls:for B°(Vl+V2). c) (V,+V2)±nD(A') is weak * dense in ~VI+V~) ~ and
that
f/Jvl.%=O
Bt(VI+V2)
and
=Bt(~.
Proof: a)
If Vt and V2 are controlled invariant,
then they are by definition open
loop invaxiant. It is not hard to prove that the sum of these two subspaces is
Mso
open
loop
invariaz~t.
If
one
of
these
subspaces
is
finite
dimensional, then the sum is a closed subspace, and by theorem II.27 we may conclude that this sum is controlled invariant b)
Consider
YE(Vl+V 2),
V2cPF(Vt+V2). Yy(Vl+V2).
So by the
By lemma III.5.c)
then
by
linearity
lemma
III.5.b)
of P!:(Vx+V2},
VE(VI+V2)cVx+V2,
thus
VI c y E ( V l +V2)
VI+V2
is contained
PE(Vt+V2)=VI+V2.
and in
60 Let {bl,..,b=l } be a basis for B°(1)E(VI+V2) ) then since FI+V z is contained in i)~:(V~+V2) we can extend this basis to a basis of B°(VI+V2). Denote this basis
functionals f]
By assumption
{bt,..bmp..,br% }.
by
{f~}cD(A')
~
such
we
have
=6ii
that
that
there
exist
and f i l v t + v 2 = f i l ~
=
From lemma lll.7.iii) we have that I)~(Vt+Vz)
=0.
is a closed
V27,(Vl÷V 2 )
subspace, and thus V~(V l +V 2) = V1+V 2 is controlled invariant. e)
The
proof
of
c)
is similar
to
that
of
b)
and
therefore
it
will be
omitted.
[::1
Remark:
It will be shown in exaznple E.15 that there exist controlled invaxiant subspaces
whose
sum
is
no
longer
controlled
invariant,
contrary
to
the
finite-dimensional case. We conclude with a generalization of lemma III.14 and give conditions under which the union of an infinite nest of controllcd invariant subspaces will be controlled invariant. Lemma III.15. Let Vn; n>_l be a nest of closed, linear and controlled invariant
a)
subspaces.
Then
V: = 13 V n
is
controlled
invariant
if
there
exist
n_>l
functionals {fi}7°1 ha D(A') such that / j [ v = 0
and
=6#,
where bi is
a basis for B°( O V,). n>l
Consider the same nest Vn. Let B be one dimensional and suppose
b)
there exists a closed subspace K such that VncK and the controllability subspace,
, is not contained in K. Then V:= U v,, is controlled
invariant
if
n_>l
and
only
if there
exists
a
functional f
in D(A') such
that
f l y = 0 and = l . Proof: a)
Let V be U Vn, then V, is closed loop invariant and contained in V for n~'t
all n, nest,
thus by iemma 1TI.5.b) Vn is contained we
have
U vncyE(v). n>_l
By
definition
in YE(V). of
Since {Vn} is a
FE(V) we
have
that
61
V£(V)cV = U v. , U V,,CVF(V)cV. Thus V~:(V) = V. n>l
n>l
Let {bl,..bml } be
a basis for
B°(PE(V))
then
since
U V,, is contained n>l
in I)E(V ) we can
extend
this basis to
a
basis of
B°(UV,).
Denote this
n_>l
basis
by
{bw.bml,..,bmo }.
functionals {/~}¢D(A')
By
assumption
we
have
=6~j and lily=
such that
that
f,[
there v ~c( v)
=0.
exist From
lemma IlI.7.iii) we have that VE(V ) is a closed subspace, and thus ~)~(V)=V is controlled invariant.
[]
b) (if): see a) (only if): Suppose V is controlled invariant TA(t)-invaxiant.
Furthermore is
the
since
smallest
K
is
closed
Ta(t)-invaxiant
b~V, then V is also
and
we
subspace
have
that
containing
VcK. Im B
=
span{b}. So c g ¢ K. So 13°(V)=span{b}, and III.7 gives the desired result
O
In this chapter we have analyzed the problem of the existence of the largest controlled invariant
subspace. As we have
not
when
necessarily
largest
open
difficult to
exist
loop
or
and the
verify when
it
largest
does
exist
frequency
it exists. In the
seen, it
is
invariant
this subspace does not
necessarily
subspace
following chapter
and
the it
is
we consider
a
special class of systems for which we can give readily verifiable necessary and
sufficient
conditions
invariant subspace.
for
the
existence
of
the
largest
controlled
CHAPTER IV: CONTROLLED INVARIANCE FOR DISCRETE SPECTRAL SYSTEMS
In
this
chapter
we shall
consider
~gain
the
following
in a
general
linear
controlled
system described by the set of equations
(4.1a)
x(t) = Ax(t) +Bu(t)
(4.1b)
z(t)=Dx(t)
but
instead
assume the
of
that
input
considering
the state
space
chapter
A is a
operator
from
this system
space ~/ is a separable
U is assumed generator
7/ to
to
be
one
Banach
space
Hilbert space
dimensional.
and
As
the
l]anach
space Z
and
since
the
furthermore
in
of a C0-semigrou p on H, D is a
we shall
the
previous
bounded
linear
space
is one
input
dimensional we h a v e Bu=bu with b ~ . In this
chapter
chapter
2
ill
tile
of
of
kernel
this
the we
introduced
shall
characterization
zeros of
we
we
of
derive
all D.
the a
controlled This
concept class
of
of
invariant
characterization
controlled
spectral
is
given
necessary
and
sufficient
invariance.
systems
subspaces
function D(s-A)-1B of system
transfer derive
for
of in
a
(4.1)
the
In
complete contained
terms
of
the
(4.1). As a consequence
conditions
for
the
existence
of
/ ) ' ( K e r D), the largest controlled invariant subspace in the kernel of D That
there
illustrated What
is
exists
in KerD and
subspaces
by the
the form
a
close
following of
relationship
zeros is well known
all
problem.
one
We
dimensional
between
controlled
in finite dimensions
can
ask
a
controlled
very
invariant and
simple
invariant
can
be
question.
subspaces
in
the kernel o f D? l,et span{v}
be
such
a subspace,
A+BF invariant,
also lenmaa
1.3.a).
Thus
for v
is
some an
then
since
feedbzLck
eigenvector
it
law of
is controlled F,
see
A+BF.
invariant
del'iuitioa We
shall
it
is
11.2~ and distinguish
between two cases olle;
FV = 0
In this case v is an e i g e n v e c t o r of A and in the kernel of D. two: Fv ~ 0 Without loss of generality we may assume that Fv=l. So that (A+BF)v=c~'u implies (c~l-A)v=b. If we assume that (~ is in p(A), then v=(c~l-A)-Ib. Since v is in the kernel of D we obtain D(cd-A)'tb=O. Thus c~ is a zero of the
63 transfer
function.
So
we
see
that
a
one-dimensional
controlled
invariant
subspace is either an eigenvector of A in the kernel of D or it is span{v}
Dv=O and (cd-A)v=b for some ¢~ in C. If this
where v satisfies the equations c~ is an
element
of
the
resolvent
set
of
A,
then
it
is a
zero
of
the
transfer function. For
finite
understood,
dimensional
see
e.g.
systems
Davison
&
the
concept
Wang [15].
In
of
zeros
infinite
is
very
dimensions
well
however
only a few articles have been published, see Pohjolalnen [30].
P*(KerD) is based on the result on pole
The proof of the existence of placement
from
Sun [37].
To
get
an
between the existence of Y*(KerD)
idea that
there
exists
a
relationship
and the .problem of pole placement we I
refer and then
to
the
113])
finite
that
dimensional
if the
a(A+BF[ , In
)
this
It
is well-known
single input system is
V (KerD)
V-(KerD).
case.
fixed
paper
we
for
all
shall
{4.1) F
prove
(Wonham [42, p.112
is controllable
D#0,
(A+BF)Y*(KerD)c
satisfying a
and
similar
result
for
spectral
systems, see theorem IV.5. Otherwise we have from the results of Sun [37] a
a(A+BF), for some F. Hence especially on a(A+BFIv.(K~.o)) , and this will give a condition for the existence of l)*(KerD).
restriction on
all possible sets
the fixed part of
The organization of this chapter wiU be as follows. In
section
spectral
IV.1
we
operators
and
characterization of all Properties of
shah
recall
some
facts
this
class
of
for
and
properties
operators
we
of
shall
discrete derive
a
given
in
presented.
In
TA(t ) invariant subspaces.
the zeros of
the class of spectral systems wiU be
section IV.2. In section this
IV.3
section
subspaces
the
we
main
shall
in the
theorems
give
kernel
a
of
full
of
these theorems
chapter
description
/9. In particular
sufficient conditions for the existence of Application of
this
will be
of
will be all
controlled
we shall give
invariant
necessary
and
I}*(Ker D). given in section IV.4. The
examples
in this section were partly calculated by L. Nooitgedagt [26]. We restricted and
remark our
here
that
attention
initially observable,
in to
order systems
(see Curtain
to
improve
which
are
the
readability
approximately
& Pritchaxd [9, p.60
more general theory we refer the reader to Zwart [47].
and
we
have
controllable 69])
for
a
64 S e c t i o n IV.l: D i s c r e t e S p e c t r a l O p e r a t o r s
In this section we shall give the definition and some properties of discrete spectral operators.
For more detail about these operators we refer
the reader to Dunford & Schwartz [18]. D e f i n i t i o n IV.l: D i s c r e t e O p e r a t o r A linear o p e r a t o r A from H to H is discrete if there exists a number A in its resolvent set for which the resolvent R(2;AI:
=(AI-A) -1
is compact.
Lemma IV.2. If A is discrete, then a)
its spectrum,
a(A),
is a denumerable set of points with no finite limit
point; b)
The resolvent R(A,A) is compact for every A not in
c)
Every A0 in
e(A)
a(A).
is a pole of finite order 0(20) of the resolvent and
if, for some positive integer k, x satisfies the equation
(A-AoI)kx = 0 then x satisfies the equation
(A-AoI}O(Ao)x=O The set of all vectors x satisfying the equation
(A-AoI)O(A°)x=O is &
finite
space
dimensional
linear
space,
called
the
of
generalized
eigenvectors of A corresponding to the eigenvalue A0; d)
If
¢4.21
P¢201=
rI21- Al-ld2
where F is a small closed curve surrounding only the eigenvalue 20 and P is traversed
once in the positive sense, then P(20)
projects 7f onto
the space of generalized eigenvectors corresponding to 20. Proof: See Dunford & Schwartz [18, lemma XIX 2.2.].
t::l
65 Remark: The spectrum of A shall be denoted by {An} n > 1. Definition IV.3. Discrete Spectral Operator. A
discrete
operator
is
spectral
if
the
spectral
projections
P(),j)
defined by (4.2) satisfy a)
The
family
of
sums
of
finite
collections
of
b)
No non zero x in ~ satisfies all of the equations P(Ai)x=0, Aj in
P(Aj)
projections
is
uniformly bounded and
a(A).
Remark: The spectral projections P(Aj) are not necessarily selfadjoint. Lemma IV.4. If
A
is
{P(Aj), Aj in
a
discrete
a(A)
spectral
operator,
then
the
spectral
projections
} generate an uniformly bounded Boolean algebra with the
completeness property:
~ P(Aj)=I j=l
(4.3)
where the convergence is in the strong topology. Proof: See Dunford 8, Schwartz [18, XVIII.I.].
[]
If a subspace V of I~n is invariant with respect to a diagonal matrix, then V must be of the form span{ci,
ieJc{1..n)
vector
discrete
of
Rn.
For
the
class
of
} where ei is the i'th basis
spectral
operators
a
similar
theorem holds. First we shall recall a lemma of Dunford and Schwartz [18] that gives some invariance properties of the spectral projections, P(A~). Lemma IV.5.
a(A). AP(Aj)x=P(Ai)Ax
Let A be a discrete spectral operator and Aj an element of Then all x
D(A)~P(Aj)H, the subspace P(Aj)H in D(A) and a(AIP(Aj)H)={Aj}.
is A-invariant,
for
Proof: See Dunford & Schwartz [18, p.2294].
El
66 With these lemmas we can now give a complete characterization
of all
TA(t)-invariant subspaces for the class of discrete spectral generators. T h e o r e m IV.6. Let
tile
discrete
spectral
operator
A generate
the
C0-semigrou p
TA(t),
then a closed linear subspace V of 2-/ is TA(t)-invariant if and only if OO
V= ~ |Vi 4=1
(4.4) where
Wi
is
a
subspace
of
H
which
is
contained
in
P(Ai)H
and
is
A-invariant. 'File summation (4.4)
is in the strong topology,
i.e. for all x e l '
th(u'c exist
{wi;ieN, with wieWi} such that ~ wi convergcs to x for n -~ co. Oil th(' other i-----l
hand, if {wi;ieN, with wieWi} is such that ~ wi converges to x for n ÷ co, i=l
then the limit is in V. Furthermore
the spectrum of A restricted
to V is equal
to
the
set of
all A i e a ( A } such that the corresponding Wi is not the zero subspace. Proof; (if): Since the be
dimension of P(Ai)?/ is finite, the
finite. So IVi is a
contained
in
D(A),
closcd
and
with
linear subspace lemma
1.7
dimension of Wi must also
of
we
7/. Furthermore may
conclude
P(Ai)H is
that
Wi
is
TA(t)-invariant. Every x in V is the limit of a sequence xn, with xne ~ Wv St) with the i=l
above we have Ta(t)x n is in ~ Wi. TA(t ) is a bounded linear o p e r a t o r thus i=1 TA(t)x n converges to an element in R, but also to a n element of V since
TA(t)x n is in V. So V is Ta(t)-invariant. ( o n l y if): Let V be a TA(t)-invariant subspace. Since a(A)={Ai} , i e N we have that the
resolvent
set,
p(A),
is connected.
With lemma
1.4 this
iml)lies that
V
is also ( A I - A ) -t invariant for all A in the resolvent set of A. So
P(Ai)V=~----~[ (AI-A)-IV dAcV; see (4.2) 2 So
F
P(Ai)Vc(P(AI)~I)nV. Using the fact that
P(Ai)
is
a
projection
we
get
67
(P(Ai)H) n V = (P(Ai)H)n (P(Ai)V) c P(Ai)V. Thus P(Ai)V= (P(Ai)7/)nV. If we set Wi equal to P(Ai)V and xi:
xEV,
=P(Ai)x;
then A(Wi) = AP(Ai)V
= AP(Ai)P(Ai)V = AP(Ai)(P(Ai)~lnV} c AP(Ai){VnD(A)}=P(Ai)A(VnD(A)} AWicW i.
c P ( A i ) V = W fi see lemma IV.5. So By (4.3) ~ xi. ~=~
and definition IV.3 every
x in H can
be uniquely written
as
[ Wi" iF'~
Hence V=
We shall now prove the last assertion. Let d denote than
zero.
Ai~a(A )
the index set of all
WicP(Ai)l~
Since
is
finite
such that dim(W/) is larger
dimensional and
A-invariant
it
i~J } c a ( A ] v ). (M-A)-I]V is a bounded
must
contain an eigenvector corresponding to ~i, thus {Ai; From lemma 1.4 we have that for all ,~ep®,
linear
operator from V to V and since A is discrete it, is also a compact operator. Furthermore
it
is the
(M-AIr).
inverse of
A]v
So
is a
discrete
operator
Air is a pure point spectrum, l.'urthermore we have from corollary 1.10 that a(A]v)ca(A ). Let Ao he an element from a(A]v), then there exists a v in V such that A]v(v)=)~oV. From the above we have that Wo=P(Ao)V. We shall show that Wo=P{Ao)V is non zero, but this is obvious since P(Ao)v=v. So it is shown that { Ai; i~d }=a(AIv). rn and hence the spectrum of
With aheady
this
theorenl
known
but
we
our
can
prove
proof
some
is much
interesting
si,npler,
see
corollaries
Curtain
that
are
and
Pritchard
with
{(p,} an
[9, p. 6l]. OO I
Let
A be
of
the
following
A= ) i Az<"Oi>tl g)"
form
i=l orthonormal basis of
7-l, AiER , Ai~,~1
-2
and
~ Ai
<e0. If we set the domain of
t=| a
equal
to
all
x
in It stlch
that
~ IAi<x,q~i>HI 2
exists,
then
A is
a
i=1
discrete
spectral
operator
with
spectral
projections
P(-'~i)= < ., g~i> ~i.
l'urthermore if s u p { A i l i ~ N }
Tz(t),
see
Curtain [5] and
TA(t)=e At= ~ eAit<.,CPi>Hc/)i Let D be a bounded linear operator operator
can
be
seen
as
an
from 7-/ to a llilbert sl)ace Z. This
observation.
An important
concept
in system
68 theory
is
the
nonobservable
subspace
DTA(t)xo=O;
conditions x 0 such that
V0
i.e.
the
set
of
all
initial
Vt>O.
C o r o l l a r y IV.7. If
A
and
D
satisfy
the
properties
as
stated
above,
then
V0
is
span(¢iID(~i)--0 }. So V0={0 } if and only if D ( ~ i ) ~ 0 for all i in t~. Proof: From Curtain [6] we obtain that V0 is the largest subspace of 7/ that is semigroup invariant and
in the
kernel of
special structure of A we have that
D. From theorem
IV.6 and
Vo=s~(d)i[ieJcN;D(¢i)=O }.
the
Combining
these results for V0 we conclude this corollary.
[]
If the nonobservable subspace equals the zero set, then the pair is
initially
observable (A,B)
observable, is
the
see
concept
is approximately
[9, p.69]. of
Dual
to
approximately
controllable if
the
the
concept
controllable,
set
of
all states
reached from zero with an arbitrary input is dense in H, or
of
i.e.
(D,A)
initially
the
system
that
can
by
{xEH[3t>_O,3u(.)
t
s.t x=lTA(t-s)Bu(s)ds
} is a
dense subset
of
~/. The
system
(A,B)
is
0
approximately
controllable
if
and
only
if
the
system
(B*,A*)
is
illitially
observable. For our special system we now have the following corollary.
C o r o l l a r y IV.8. If A satisfies element of only if
the
7~, then
#0
same
the
properties
system
(A,b)
as
in corollary
IV.7 and
b
is an
is approximately controllable if and
for all i in t~.
Proof: This is the dual of corollaxy IV.7.
As in theorem IV.6 we can pose ttle question what tile Ta+Br(t)-invariant subspaces look like. This question is in general not solvable even if A is a
discrete
operator.
spectral
operator
Furthermore
we
since are
A+BF not
need
interested
not
be
in
all
a
discrete
spectral
TA+BF(t)-invariant
subspaces but only in those which are in the kernel of D. Before we can give
a
complete
description
of
all
Tn+nf(t)-invariant
investigate the notion of the zeros of a transfer function.
subspace
we
must
69 Section IV.2: Ze r o s and I n v a r t a n c e
In this section we shall discuss the relation system (4.1) and its controlled invaxiant Although and
we
shall
approximately
only
consider
controllable
we
between the
subspaces in systems
shall
of the
the kernel of D.
that
define
zeros
axe
zeros
initially in
a
observable
more
general
setting.
Definition IV.9: Zero An element #
of C is called a zero for the system (4.1)
if there exist
nonzero xeT/ and a u e U such that
--0.0 Remark: This definition can also be found in Davison and Wang [15].
Remark: If
/.L~p(A)~ then
# is a zero if and only if
D(pI-A)-lB=O.
[,emma IV.10. Let V be a closed subspace the kernel of D. Assume further
A+BFIv ,
then X is a zero for the
TA+BF(t)-invaziant and that Aeap(A+BF]v), the point system (D,A,B). that
is
contained
in
spectrum of
Proof: By assumption t h e r e exists an x in V such that
u=-Fx,
then
=
D system
0
(A+BF)x=Ax.
=
Now define
. So ~ is a zero for the
Dx
(D,A,B).
[]
Remark: As can be seen from the proof we have not used the special structure of system
(4.1).
So
the
result
remains
true
if
our
state
space
is
a
general
Banach space.
In the sequel of additional
assumptions
this section we shall consider system and
discuss
the
concept
of
zeros
(4.1) for
with some this
special
7O kind of system. The additional ~sumptions we make on the system (4.1) are: (&l)
The
generator
A
is
a
discrete
spectral
operator
with
spectrM
decomposition A = ~ ;~P(A~) iml
where Ai#A j (for all i # j ) generality
we
may
and dimP(Ai)=l (i>__1). Without loss of
assume
that
the
P(Ai)
(i_>Á) are
selfadjoint
operators in ~/, see Wermer [40]. The normalized eigenvector of A corresponding to P(Ai) will be denoted by ¢i ( i > l ) ,
(zx2)
b ~ : = < b , O i > H ~ 0 , for all i>_l.
(A3)
For all i eN, D4~ii~ O. Let us remark
that
(zX2) is the controllability assumption and
(A3) is
the observabflity assumption, see corollary 13/.7 and IV.8. Lemma IV.11. Assume that
the system (4.1) satisfies conditions (A1),
If p is a zero of the system (4.1), then
(ZX2) and
(A3).
pep(A) and D(p-A)-IB=O.
Proof: Let # be a zero, then there exists a
xeb(A), x#O and uell such that
'1[:] 00 = If u were to be zero, then x would be an eigenvector of ,4 in the kernel of D. However this would imply by assumption contradiction. So uia0. Assume that
(ZX3) that
x=O, providing the
pea(A), then O=P(#)O=P(I.~)((p-A)x+bu}
P(#)(~-A)x+P(p)bu = (#-A)Pl#)x+P(#)bu = O+P(p)bu. Since u # 0 this implies that P ( p ) b = 0 , but this is not possible by assumption (A2). So pep(A) and the remark below definition IV.9 gives the desired result.
C]
With this lemma we can define controlled invariant subspaces associated with a zero of the system (4.1).
71 k Definition IV.12: Z~ Let the system (4.1) satisfy assumptions (A1), (A2) and (A3). For
a
zero
#
of
this
system we shall
define
the
following nest
of
subspaces, /1:
-n
Z~=span{(p-A) b; l
order of the zero of D(s-A)-lb
is the
as a meromorphic
function. This order is always finite, see Rudin [31]. Remark: k
By the previous lemma we have that pep(A), and thus Z~ is well defined. Lemma IV.13. Z~ is a .controlled invariant subspace in the kernel of D. Proof: By
the
definition
of
Z~
this
subspace
is
finite
dimensional
and
contained in D(A). So from lemma HI.29 we only have to prove that Z~ is (A,B)-invariant. Let x be an element of Z.. Then x=
Ti(~-A) b, where n--min(k, orger(#)),
i n!
n-I
n-I
and thus (#-A)x=Tlb+ ~ Ti+l(/z-A)-~b. So with z = ~ Ti+l(#-A)'ib we have that i=l k
k
iffil
Ax = p x - z - T t b , and A(Z~}cZ~+Im B.
1:3
Lemma IV.14.
Assume that the system (4.1) satisfies (A:)~ (A2) and (A3). o~ ( oj Then z . , ~ ~p,,,, ~ z ~ ~.--~o~, where S denotes the index set of all zoros J
and 8j = order(/~j). Proof: The
complete
proof
is very long and
rather
technical;
here
give the proof in the speciM ease that the order of all invariant one.
So we have to prove that (pl-A)-lb is not in s - - ~ ( ( # J - A ) - ' b ) . Suppose that this were true then there exists a sequence T~ such that U (#I-A} -lb- ~ yi(iJiI-A n )-1b []
we shall zeros is
72
]] D(pI-A)-X{ (PL/-A)-lb- ~ *'{?(]'ziI-A) -lb }l[ < ]lO(pI-A)-~jI]n
(4.5)
i=1
By the resolvent identity we have that
(pl-A)-t(pj-A)-'b=(#-pi)"{(#j-A)"b-(#I-A)-'b}
(4.6)
Using this identity we have
(#[-A)-I((#I-A)-lb- ~ T~.(p.iI-A)'lb) =
(4.7)
i-1
=(#I-A)'~b - ~ y~(i.t-#i)-lf(piI-A)-tb-(pI-A)-lb ) iffil
Notice that since # and #i are invariant zeros we have that and
D(pil-A)-lb
D(pI-A)-'b
are both zero. With this property and (4.7) we can simplify
the expression inside the norm signs of (4.5).
D(#I-A)-~( (#I-A)-~b - ~ y'~(#j-A)'~b
(4.8)
I (4-7)
i=l
= D(i.tI-A)-2b. Combining (4.8) with (4.5) implies that of order
two. This
is not in
span(
Pi~#
Section IV.3:
D(pI-A)'2b
is zero, so #
is in contradiction with the assumptions. So
(#j-A)-lb }.
Characterization
of
all
Invariant
Subspaces
is a zero
(#I-A)'Ib []
for
Spectral
Systems In
this
section
controlled
invariant
arbitrary
generator,
we
shall
subspaces but
there
give
a
KerD.
in is
a
complete This
large
characterization
is yet class,
not
possible
introduced
by
of
all
for
an
Sun [37],
where we can give a complete answer. We shall start by defining this class. We say that
the system (4.1) satisfies condition A
if it satisfies A I ,
A2 and A3, see section IV.2, and we have the following extra conditions on
73 the eigenva2ues of A
(a4)
in/. [z~-~,~[ = 6> o, s.,~,1
su
(AS)
fill 1
j---1 icj So these accumulation in their
conditions
imply that
point(A4),
and
the spectrum of A may not
furthermore
successive distances.
For
there
instance
have
must be sufficiently
if ,~i=-V~,
then
this
an
growth sequence
satisfies A4, but not A5. Before we shall state
the
main results of this chapter
we shall
give
the result of Sun [37]. Theorem IV.15. Suppose that the system (A,b) satisfies A1, A2, A4 and A5. Then for a given
sequence
of
complex
numbers
iq={ul,..,um..},
in
order
that
there
exist F~£(R,R) such that the spectrum of the operator A+bF satisfies
a(A+bF) = ap(A +bF) = it, a necessary and sufficient condition
is that
there
exists a subsequence nj
(jeN) ha N such that Z
A,,j - vj
2 < O0
jeN where
Anj
is
the
bnj ~j'th
eigenvalue
of
A and
bnJ = < ¢n1' b > ~/; ¢ni is the
eigenvector corresponding to inj. Proof: See Sun [37].
r'l
Our main results are: Theorem IV.16. Suppose
that
(4.1)
satisfies
subspace V in KerD is controlled subset of N, denoted by J, such that:
conditions invariant
A.
Then
a
closed
if and only if there
linear exists a
74
a) (4.9)
v = ~ z ~i j~'j .i
where #j are distinct zeros, b)
dim(Z~:)>0 for aU j in J and dim(Z2: ) =1 for all but finitely many j in J and
c)
there exists a subsequence
(4.10)
n I (jeJ) in hi such that
~, An-/--~uY2l
where ~ j
b.y is the n / t h
eigenvalue of A and
b,~j= <¢,~j,b>H;
r~nj
is the
eigenvector corresponding to ~ j . We shall postpone the proof of this and the next theorem to the end of this section. Remark:
From theorem IV.5 and the proof of this theorem we have that a(A+BFiv)--
ap(A+BF[v) = { # i ; j e J } . So if the subspace V in theorem IV.16 is feedback invaxiant with respect to a stable semigroup, then
{i~i;jeJ } c {seC;Re(s)<0 }.
With this theorem we can solve the existence of l~*(Ker O). Theorem IV.17. Suppose that the system (4.1) satisfies condition A. Then Y'(KerD) exists if and only if the following conditions hold: i)
For all but finitely many j's the order of the zero, pj, is one and
ii) There exists a subsequence nj (jeJ) in N such that (4.11)
where J
~ jeJ
~Z- ~j ~
is the index set of
as in theorem IV.16.
all invariant zeros and Anj, bnj are the same
75 Remark:
For the proof of the sufficient part in both theorems, condition •5
can
be omitted, see Clarke and Holland [4].
Combining this theorem with the results proved in chapter 3 gives for a class
of
systems
properties
concerning the
asymptotic
distribution
of
the
zeros, see also the next section.
C o r o l l a r y IV.18. Assume that the system (4.1) satisfies A and assume further that there exists a constant q e N such that DAi has a bounded extension from H to Z for 0<_i_
For all but finitely many j's the order of the zero, #j, is one and
ii) There exists a subsequence n i (j~J) in N such that
j~J where J
bnj
is the index set of all invariant zeros and ~Ant' bnj are
the same
as in theorem IV.16. Proof: This is a direct consequence of lemma 111.9 and IV.17.
O
Before we can prove theorem IV.16 and IV.17 we must first prove some lemmas.
Lemma IV.19. If
A satisfies
spectral operator
conditions A1,
A4
and
AS,
then
A+BF is a discrete
for all bounded feedback laws F. The spectral projections
of A+BF will be denoted by P~.(vi). Furthermore if i is sufficiently large, then dim(PF(v~)H ) -- 1. Proof: See
Sun [37]
theorem
2.1
and
page
734.
fact that ( A I - A - B F ) -1 is a compact operator.
See
Kato [22, p. 196] for
the O
76 Lernma IV.20. Suppose that the system (4.1) satisfies assumption A, If W is an subspace of PF(I~)H contained in the kernel of D which is
(A+BF)-invariant with dim(W)=k, k > 0 , then # is a zero with k
(A+BF)-invariant subspace in PF(tl}H there exists a non zero vector eI in W such that (A+BF-pl)e~=O. Thus Since IV is a
finite
dimensional,
(#-A)el-bFel=O and DelcD(W)=O.
(4.12) So
p
is
a
zero
and
thus
by
lemma
IV.11 an
element
of
p(A). So
1 el=(H-A)-IbFel. Since e l ~ 0 we have that Fel¢O, and so span{et} = Z~,. Now all eigenvectors of A+BF}w will satisfy el=(i~-A)-ZbFel and so A+BFIw has
only one eigenvector. Since W is finite dimensional, only
one
eigenvector
we
(A+BF)WcW, a(A+BFIw) = {H} and A+BF[w has
have
that
W=span{e,,..,ek)
where
ei
satisfies:
(A+BF-~)el=O and (A+BF-I~)e~=ei.,, 2<~
Equality
follows
easily
from
the
fact
equal.
that
the
dimensions
are D
P r o o f o f t h e o r e m IV.16. ( o n l y if) Let V be
T•+BF(t)-invariant, then by lemma IV.19 and theorem IV.6 V is
of the following form OO
(4.13)
V=
where Wi is a
(A+BF)-invariant subspace of P~.(#)H.
Define J
~lWi i=
to be the s e t
of indices with W~ not the zero subspace. By
i in J, dim~Wi)=dim{PF(]~)ti)=l. Since V is in KerD we have that W~ is in KerD, so by lemma IV.19 we have lemma IV.19, we have for atl but finitely
that
W~=Z~, iaJ.
Since {/1;;in J} is contained in the spectrum of that {4.10) must hold.
A+BF we have from Sun [37]
77
(~) Let • I¢j dun(Zui)
Jx
be
the
is one,
subset
then
by
of
J
that
contains
all
7 ~ IV.12 7kj _~j=_~j.
definition
indices
j
such
By condition
A
that and
(4.10} we may apply theorem IV.15. So there exists a bounded feedback law Fx such that
a(A+BF1)~{#iljeJl}.
We shall prove that Vz: =a-'~{Z~i } is closed loop invariant with respect
JaJx TA+BFI(t). Let ei be an eigenvector o f A+BF 1 with eigenvalue /zj. We shall prove that eieKerD. Since ej is an eigenvector we have that {A+BF~)e~=pjej or equivalently to
the
operator
(#iI - A) ej = bFxej
(4.14)
Since /zj is a zero we have that
#jep(A) and so we
can premultiply (4.14)
(piI-A) -1. So
with
(4.15)
(t~jI A )'l bFlej
ej =
-
Dej=D(pff-A)-~bFlej. This last expression is zero since /zj is a zero. So ejeIierD, and we can conclude from lemma IY.I9 This
equation
implies that
that span{ej} -- Z #~j . So for every j in Jx we have that Z +,j 1 is TA+BFl(t)-invaxiant. Since the l is feedback law FI is independent of #/, j e J l , we have that Vx=span{Z~fl
jaJ1 also
T A+BFl(t )-invaxiant. Let
.
J2
be
the
subset
of
J
that
contains
all
indices
j
such
that
kj
dun(Z~j) is larger than one. Then by the condition in theorem IV.16 we have that
J2
is finite. Now with lemma kj V2=span(Zj, i} is controlled invariant.
IV.13
and
IlI.14.a)
we
conclude
that
jeJ 2
Since
V=VI+V 2 and V2 is finite dimensional we can apply lemma III.14.a) to
conclude the proof.
0
P r o o f o f t h e o r e m IV,17.
(~) ej j e J , where pj are Z~j, Oj is their order. From theorem 13/.16 we have that
Define V to be the closure of the span over all the invariant zeros and this subspace
is bounded
closed loop invariant and by definition
it is the
78 largest closed subspace with this property. (only if) By heorem IV.16 we have that V*(KerD) is of the form
kj ~ Z~,f If there jal
exists a #Jo such that Jo is not in J or kj0 is smaller than 8jo; the order of
#1o' then
Zkj, ~k/o, by leman IV.14 ( ~ ~jtg_,~jO}
larger than
is
jeJ Z ~i°
PJo
is
invariant.
finite This
dimensional is
this
~j ~ Z~,j and
since
jfJ subspace
in contradiction
to
the
is
by fact
leman that
III.14.a)
controlled
kj V*(KerD)= ~ Zzj
is
jeJ the largest closed subspace with this property. So the summation is over all invariant zeros and 9j
o1 V'(KerD)= ~ Z~,j, =
where
jeJ
order(~ti). Conditions i)
and ii) are direct consequences of theorem IV.16.
Section IV.4: Examples In this section we shall discuss some examples. The example that
will be discussed in this section is the heated
rod with
various controls and observations. This can be schematized as below sensor 4
~7=0
| control
r/=l
and for the mathematical model we take
(4.16)
Ox Ot
02x 0~:
x(O,t)
= 0 = x(1,t)
zlt)
+ b(~)u(t)
1
= I g(r/)x(t,r/)dr/ 0
This can be formulated as a system of the form (4.1), where we take H=L2(0,1) and the system operator A is given by
79
A=
O2
Orl z
; D(A) = { h ~ i : h " e ~ f
and h(O)=O=h(1) }.
A is self adjoint and has eigenvalues { -n2~r2; n=l,..,oo {¢j(t]) = V2.sin(jTrrl); j=l,..,oo In of
the
this
section
interval
we
}.
take
[0,2]1r],
) and eigenvectors
b(rl)=l[0,2/~l(r]),
and
we
shall
the
characteristic
investigate
the
function
existence
of
V'(Ker D) for different measurement functions d(r/). It is easy to see that A satisfies A1, A4 and A5. Furthermore since 1
(4.17)
h i = <¢~j,b> = f l[o,=/~](r/) "t/2sin(firrl)dr I = jlr
{ cos(2j)-I
}.~O,
0
(A,B) satisfies A2. With this A and B we shall investigate the existence of V*{KerD)
for
three different measurement operators. Example
IV.21.
We shall
consider
system
(4.16)
with
control
operator
b(.)=l[o,2/@. ).
In this example we take as measurement operator, D~= <.,d~> where (4.18)
dl = l[3/,r,I] !
Since Dx¢~i = < ~i' dl >
J 113/~,11(rl) V2 sin(jlrrl)drl jlr 0 we have that the system (A,B,D) satisfies condition A. =
{ (-l)J-cos(3j)
=
}#0,
Using the same technique as in Nooitgedagt [26] we can calculate
the
transfer function of this system. The transfer function is given by
(1 - cos((2pr ):~)) (1 - cos((1 - 3/,r ),~)) , where A=-:Vs and
al(s)=
A~sin( A ) (4.19) a l ( O ) = - (1 - 31~)2/rr2
For
this
system
we want
to
investigate
the
existence
of
V'(KerDt).
From theorem IV.17 we have that the existence is determined by the zeros of
Gl(s ). The zeros of this transfer set of zeros is given by:
functions
are
easily
calculated
and
the
80
{ ~ } = { - n 2 ~ 4 ; n a n and -n2~+*( 2/pr-3))2; naN}
(4.20)
Let the zeros be numbered in decreasing order, thus From theorem IV.17 we have that the existence of the convergence of the sum
--
j~SN the
numerical
have
[:][ ,
>_
b
we
]
.. b~
achieved for respectively A.
index such that
b-
where .~j is the subsequence of
single
terms
for
in
_>
sum.
j=30,45,60,75,90
n=89,133,177,222 and .
this
By
j • {1,.., 100}/{30, 45, 60, 75, 90}
Vn~gl and for
- #j
is determined by
bnj
A~ as in theorem IV.16. We shaft first investigate calculations
#i>11i+1. ))*(NarD)
A m i . i - #j
266. Let
Vn~N.
simple that
this minimum is
mini
We then
denote the have
that
btainj _~0 A n j=l b,j
is larger
than
for
j=l
every
subsequence
bml"i
{hi}. In the next table we have listed the partial sums, Sl(m): = ~
[ A ~ , j - #j
2
In figure 4.1 the numbers of table 4.1 are plotted and we see that in this example we have (numerical) evidence that
Y*(KcrDl)
does not exist.
81
Table 4.1
S~(m)
m
.91(m)
m
1
3. 8211e÷003
20
2.8479e÷009
2
7. 5736e+004
30
1.8650e+010
3
4. 5039e+005
40
8.0862e+010
4
1. 6463e+006
50
2.3042e+011
5
4. 5795e+006
60
5 .8 6 8 1 e+ 0 1 1
6
1. 0677e+007
70
1 .2 7 2 5 e+ 0 1 2
7
2.1990e+007
80
2.3695e+012
8
4. 1308e÷007
90
4.1805e+012
9
7. 2273e+007
i00
7.2350e+012
10
1. 1949e+008
xlOZ2
Partial sums for GI(S)
A I
v
I
I
I
10
20
30
J 40
50
60
70
80
90
,00
m ->
Figure 4.1 [] For our second example we shall take two measurement functions d 2 and d3 which are relatively close to each other with ~ 0 and ~ 0 , but the
82 the first is an element of D(A') and the second is not. Example IV.22. In this second example we shall investigate the existence of ~)'(KerD) for two measurement
functions that
we shall see
sequel,
in
the
are close in the L2(0,1)-norm,
P*(KerD)
only exists for
one
of
but,
them.
as The
measurements functions we shall consider are D2 and D3, where D,= < . , d , > with (4.21)
d2(t/) = l[0,t/~l(r/)
and
=f
-1007/2 + 2Or/ 1
(4.22)
da(r])
;
0<_9_<0.1
;
0.i__.~_
2000(7/ - l/,r) 3 + 30O(Fl - t/,,)2; i/,r-O.l_
t/~
;
We remark that d 3 is an element of D(A)=D(A'). Furthermore we have that Da(b)
is unequal
))*(KerD3) chapter
zero.
So
from
lemma
III.9.a)
we have
directly
that
exists. However in this example we shall follow the line of this
and
P*(Ker Da).
to
we
shall
give
Furthermore
a
we
(numerical) shall
indication for
investigate
also
the the
existence existence
of of
P" (Ker D2). Let G2 and (73 denote
the transfer
function of respectively the system
(D2,A,b) and (Da,A,b). Then
(4.23)
( - 1/lr)A*s~n(A)- cos(A) + { 1 - cos(Apr)}*cos((1 - 2/7r)*•)
G2(s ) =
A3si~ (A) where O~A 2 = - s , (4.24) and
and
G2(O)=
11G_•14 I
--
+
g2
- t . -13
83
G3(s)= -1000 IO.O01,(l/ _o..~/e)A4sin(A)+ A ~sin(A) L
(4.25)
O.2~'At0. l*Asin(A) +cos(A) - cos(O. 9*A) +3cos(( 1 - I/~)*A) +3cos( 1.1 - i/n )*A) +
+cos( (1-2/,)'A){cos(O. l*A)-l-3cos(l/.r*A)-3cos( (0.1-'/lr)*A)}} + t
+121sin((i - i/=)*A) -sin((I. 1
m
I/Tr)*A) +
%
~ohere 0 ~ A2 = - s, and
G3(O)=l/Tr4_O.388,1/cf~_
(2.26)
3~ 5 9 , .lpr . 2-6--0--0-~1/~r 71 ~.. - 0. 00005.
For both measurement functions we have that D,@i~O , for D2 this is an easy check, for D~ however we had to make a numerical check. This we can only do on a finite number of eigenfunctions, and for this measurement we have chosen the first 1000. As
in
the
previous
example
we have
calculated
the
zeros
of
these
systems, these zeros are given in table 4.2. The last column of this table contains the eigenvalue of A which is closest to the zero.
84
T a b l e 4.2 ZCTO
--
number
G2
G3
pole
1
-2.5213e+001
-2.4597e+001
-3.9478e+001
2
-8.8340e+001
-8.8222e+001
-8.8826e+001
3
-1.4581e+002
-1.4010e+002
-1.5791e+002
4
-2.3932e+002
-2.3182e+002
-2.4674e+002
5
-3.5527e+002
-3.5488e+002
-3.5531e+002
6
-4.8216e+002
-4.8464e+002
-4.8361e+002
7
-6.1713e+002
-6.3462e+002
-6.3165e+002
8
-1.9544e+002
-7.9865e+002
-7.9944e+002
9
-9.8002e+002
-9.8268e+002
-9.8696e+002
10
-1.1821e+003
-1.1774e+003
-1.1942e+003
20
-4.3389e+003
-4.3537e+003
-4.3525e+003
30
-9.4845e+003
-9.4845e+003
-9.4847e+003
40
-1.6590e+004
-1.6591e+004
-1.6591e+004
50
-2.5669e+004
-2.5671e+004
-2.5671e+004
60
-3.6710e+004
-3.6725e+004
-3.6725e+004
70
-4.9738e+004
-4.9753e+004
-4.9753e+004
80
-6.4753e+004
-6.4754e+004
-6.4754e+004
90
-8.1730e+004
-8.1730e+004
-8.1730e+004
100
-I.0068e+005
-i.0068e+005
-i.0068e+005
Let #2,i and #a,i denote the i'th zero of respectively G2(. ) and Ga( .). For the calculated zeros, #2,~; 2
j~N
bj
hi+ l
j
m [ Ai+I - P2,i So if Sz(m ) denotes the partial sum S2(m):= V-" ] .
i=1~ [
i=1
then
bi÷t
[ Aji - P~, i
for m<100 we have that S2(m ) is smaller than
2
j z for every
bJi
subsequence {j~; 1 < i < 100 } in N. Calculating
this
partial
sum
and
the
partiaJ
sum
for
G3(.),
85 m Ai+I
--
S3(m): =
i=l
#3,i
2
; where A i + l = - ( i + l ) = ~"2 is the eigenvalue closest
bi+l
to P3,i, yields:
Tab(e m
S2(m)
4.3.
S3Cm)
1
5.7929e+002
1.5987e+003
2
7.1984e+003
1.1842e+004
3
1.6011e+004
3.0937e+004
4
1.8017e+004
3.9057e+004
5
1.8027e÷004
4.0401e÷004
6
1.8710e+004
4.0744e+004
7
3.6107e+004
4.1469e+004
8
9.1562e+004
4.3636e+004
9
1.5939e+005
6.9395e+004
i0
1.8121e÷005
1.1164e+005
20
1.2466e+006
1.9698e+005
30
2.9993e+006
2.1765e+005
40
8.1812e+006
2.3134e+005
50
1.3083e+007
2.3634e+005
60
2.4109e+007
2.4241e+005
70
3.7819e+007
2.4474e+005
80
5.3018e+007
2.4819e+005
90
8.0695e+007
2.4969e+005
100
9.8908e+007
2.5212e+005
The plots of these partial sums as function of m are given in figure 4.2 and 4.3. Thus
concerning
theorem
IV.17 we have
(numerical)
evidence
that
for
d~(rl) I)*(KerD) exists, but not for d2(r/). In this example we see that, are
close in L2-norm,
in spite of the fact that d2(~/) and da(r/)
Y*(KcrD) exists for d 3 but not for d2, and this
depends on their smoothness.
86 xt0 ? to The partial
9
sums
C2(s)
for
#
8
7 6
A
I 5 oa 5q
4 3 2 1 0 lO
0
20
I
I
I
i
i
l
T
30
40
50
60
70
80
90
I00
m->
Figure
4.2
xl0 a r
The partial
sums
f o r G3(s )
2.5
f
f
f
J
S
A I 1.5 v O~
0.5
0
,I
0
10
I
I
I
I
I
I
I
I
20
30
40
50
60
70
80
90
m ->
Figure
4.3
100
CHAPTER V: THE DISTURBANCE DECOUPLING PROBLEM WITH MEASUREMENT FEEDBACK
In
this
problem,
as
chapter in
we
chapter
III,
shall but
investigate instead
of
the using
disturbance full state
decoupling feedback
we
shall now use a compensator. In
Curtain [7]
feedback
the
Disturbance
(DDPM) was
investigated
Decoupling for
Problem
infinite
with
dimensional
Measurement
systems.
Under
~trong assumptions she was able to prove sufficient conditions for DDPM to be solvable. By examining the proofs given in Curtain [7] more deeply and using the results of the previous chapters, we can obtain weaker couditions for
the
give
solvability of
necessary
and
DDPM in
sufficient
infinite conditions
dimensions. such
Furthermore
that
the
we
shall
compensator
is
finite dimensional. Instead of working in a general Banach space, as we did for the Disturbance Decoupling Problem, we shall here use a llilbert space. This
condition
will
simplify
the
proofs,
and
it
will
improve
the
readability. We consider the following system ~ on a ttilbert space ~/:
~,:
(5.1)
x=Ax+Bu y=Cx
where A is the
infinitesimal generator
Ta(t ) on 74, B~E(Rm, H) and
of
a
strongly
continuous
semigroup
C~£(R,Rk). ]'he following lemma will prove useful
in tile sequel. Lemma V.1. Suppose that
V is invariant under the semigroup
TA+o(t), generated by
A+Q for Q~£(H), and let Q0~£(H) be such that (Q-Qo)(VnD(A))=O Then
TA+o(t)x=Ta+oo(t)x for all x in V.
Proof: The generator of
A+Q = A+Qo.
Ta+Q(t)[v is (A+Q)Iv.DCA) and on VnD(A) we have that
88 From lemma II.2. we have that A+Q o generates a semigroup on V. Since the generators of both semigroups are the same on V, the semigroups must be the same. So TA+Q(t)x=TA,Qo(t)X VxEV.
13
S e c t i o n V.I: C o n d i t i o n e d I n v a r i a n c e and (C,A,B)-pairs The
concept
of
controlled
invaxiance
was
a
property
of
the
system
operator A and the input operator B. The dual concept (see lemma V.3) is a property of the system operator and the output operator C. This concept is called conditioned invaxiance. D e f i n i t i o n V.2: C o n d i t i o n e d I n v a r i a n c e
A closed subspace 5" of 7/ is conditioned invariant
if
there
exists
a
bounded operator G such that Ta+cc(t)ScS.
As stated
above,
this
is
the
dual
concept
of
controlled
invariance.
The
next lemma will give the precise meaning of this dualism. Lemma V.3. A closed subspace V of 7/ is controlled invariant for the system (A,B) if and only if Vx is conditioned invariant for the system (B*,A*), where Q* denotes the adjoint operator of Q. Proof: This
follows trivially from
the
fact
that
TA+BF(t)VcV if and
T~t,BF(t)VZcVtand TA+BF(t)= TA*+F*B*(t). As is to be expected,
the concept of condioned invariance has
dual open loop version and a frequency domain version.
only
if
[] also a
89 Theorem V.4. Let S c H
be a closed linear subspace. Then the following assertions are
equivalent. i)
S is a conditioned invariant subspace.
ii)
S x, To(t), and a
There exists a generator (2 of a C0-semigrou p on bounded operator R : R k ~ state trajectory x(t):
such that
=TA(t)x Q and
for every x 0 e H
the output y(t):
the uncontrolled
=Cx(t)
satisfy
Ps±x(t) =To(t)PsxXo+ ftTo(t-s)Ry(s)ds, t >0
(5.2)
0 where
iii)
Ps*
is the projection on Sa'.
There exists a generator Q of a C0-semigroup on S "t, T~2(~), and a bounded operator R : R ~ S x such that frequency
domain
trajectories
for every x0e}/
~(s)=(s-A)'*xo
the uncontrolled
and O(s)=C~(s);
sep(A)
satisfy on an interval It,co)
(5.3)
PsxXo=(s-Q)Psx~(s)-R~(s)
Proof: We shall prove successively
i)~ii), ii)~iii)
and
iii)~i).
i)~u): Since S
is conditioned invariant we have
that
there
operator G such that S is TA,av(t)-invariant. Now define define
To{t ) as Ps,TA,GV(t)Psa.
It is obvious that R and
exists a
bounded
R as -Ps.~G and To(t ) are bounded
operators on b-~'. Furthermore since S is TA÷ae(t)-invariant we have that for Z E S J', PsxTA,GC(S)rA,GC(£)z
PsxrA+ac(s)Ps±rA+aC{t)z.
=
PsxTA+ac(s){PsTa+ac(t)Z+Ps,ra,ac(t)z } =
O+
Thus TO(t ) is a semigroup on S J'. Let Q denote the
generator of this semigroup. From Curtain h Pritchard [9, p.38] we have that f
(5.4)
Ta(t)x o =Ta+ac(t)x o - ] T A+oc( t s )GCTa (S )XodS o
Since
S is rA+ac(t)-invariant
we have
Ps,Ta+ac(t)xoffiPs.Ta+ac(t)Ps,xo using this equality gives
that
for all x 0 in H. Operation Ps* on (5.4) and t
(5.5)
Ps±TA(t)xo = Ps±TA+ac(t)PsxXo - f Ps,Ta+ac(t- S)Ps±GCTA(s)xods 0
90 With
the
definitions
of
x(.),
y(.),
R
and
Q
this
proves
the
assertion
~)~:i).
ii)-~iii): Since
the
state
PsxX(t) and the "input" trajectory y(t) axe
trajectory
exponentially bounded we may take the Laplace transform of equation (5.2}. Let £ denote this transform, then
£ (Ps±X(.)) (s) = (s - Q)-lPsJX o + (s - Q)-lR £, (y(.)) (s) for s • [r, oo)
Since
)
L(es TA(.)xo)(
)
and c (y(.))(,)=
C(s-A)-Ixo we have proved (5.3). iii)~i): We shall prove that S x is frequency invaxiant for the system (A*,C*). Then theorem I/.27 and lemma V.3 conclude this assertion.
Let is±:S±~.-,H denote the inclusion mapping i.e. isj.(z)=z. Let z s S ~', x~Fl and se[r,oo)np(A)np(Q). Then -<(s-A*)-lz, x> = =
and {5.3) implies that this is equal to + *
-1
*
-1
*
*
*
=
-1
+ < ( s - A ) C R (s-O. ) z , x > .
Since this holds for all x eH we have that *
-1
*
-1
( s - A ) Z=isx(S-Q ) z+(s-A*)-lC*R*{s-Q*)-lz;
se[r, oo)np(A)np(Q).
Thus
z=(s-A*)is~(S-Q* )- 'z + C*R*(s-Q * )" l z,
(5.6}
and so z has a strictly proper is
contained
in
S x.
By
•
*
(~,w) representation with ~ ( s ) = : s z ( S - Q )
definition
this
implies
that
Sx
is
-1
z
frequency
invariant for the system (A*,C*).
O
Following Curtain [7] and Schumacher [34] we can combine the concepts of
controlled
invariance
and
conditioned
invaxiance
concept of (C,A,B)-pairs for the system /: in (5.1).
to
introduce
the
91
(C,A,B)-palr
Definition V.5:
A pair
of
closed
ScV, S is (C,A,B} pair.
properties called a
The concept of
subspaces
(S,V)
of
the
Hilbert
space
~
with
the
conditioned invaxiant and V is controlled invaxiant is
(C,A,B)-palr
is related
to A¢-invariance,
where
A* is
the extended system operator of the system ~' on the extended state space h~=TfsW,
where the Hilbert space W is the state space for the feedback
processor. The feedback processor is given by (5.7)
~b=N~+My
u=L~+Ky
where M,
L,
K axe
generator
of
a
xe=(x,w),
we obtain the closed loop operator
(5.8)
bounded
strongly
A' =LrA+BKCMc
bounded perturbation of diag the
operators
and
semigroup,
N is the
TN(t),
on
infinitesimal I¢.
Defining
BNL1
A~ is also the infinitesimal
We prove
linear
continuous
generator
of a semigroup TAr(t} since it
is a
(A,N) with D(A')=D(A),~D(N).
following modification of the
finite
dimensional
result
from
Schumacher [34] Lemma V.6.
Suppose
that V¢ is a
closed
TAe(t)-invariant subspace
following subspaces (5.9)
S~,h:=
(5.1o)
s-=
and
rt.~j1o
somo o . ,
and
define
the
92
V={xeH: [:] eVe for
some ~0eW}
(5.11) := P~I f
Then S ~ h
is frequency
invariant
for
the
system (A*,C*),
V is frequency
invariant for the system (A,B) asld S---{xe?t] [ol eV~} and is contained in V. Proof: W e shall begin by showing that V is frequency invaxiant for the system
(A,B). Since Ve is Tae(t}-invariant we have that (s-Ae)=lVecVe for all s larger than some soeR.
element of I/e.
[MC (s-A~)-lI~]" So
x=¢s-A)Pn(s-
(5.12)
) [~j -
By the invariance of P~ we have that
.
(s-A)
w e V , thus
,sAe,.ix w]is
in V. kWj
operators we have that
(5.13)
@-~CO
So by system
Since
ka,j
definition we may
conclude that
V is frequency
invariant
for
the
(A,B).
V•J-
So~t~ is
*
is TA~(Q-invariant, we can apply a similar argument to show that frequency
left to the reader.
invariant
with
respect
to
the
system
(A*,C*). This is
93 show that
= l It . Let
e
be an arbitrary element of Vt , then
z~Sorth, and thus
<(o] [:1> then there exists a ~ W
[:] •
such that
[;] [:]
eV ~ . So 0 = <
,
> =
<x,z>, and thus
X E ~oorth"
[]
Remark: With a similar proof one can show that g is open loop invariant for the system
(A,B) and Sorth is open loop invariant for the system (A',C'}.
Remark: The
projection
of
a
dosed
linear
subspace
is
not
necessarily
closed
subspace. This is easy to see from the next example. Let A:H~-,7/ be a
bo~mded linear operator
range is unequal to 7/. Then the graph of A; ~e~,
with dense range,
and
this
Ve:={(x, Ax)[xe~}, is closed in
but the projection of this graph on the second coordinate gives the
range of A, which was by assumption not closed.
Remark: If Soreh and V axe lemma V.3.
The finite
both closed subspaces,
then
by
theorem
II.27 and
(S,V) is a (C~A,H}-pair.
next
lemma will show
dimensional
i.e.
14/ is
that a
if the
finite
feedback
processor
dimensional space~
then
in (5.7} S
defined by (5.10) and (5.11) are closed subspaces and so in this case is a
and
is V
(S,V)
(C~A,B)-pair.
Lemma V.7. Suppose
that
Vt
is a
closed Ta~(t)-invariant subspace and
Iq is finite
(S,V) of subspaces defined by (5.10) and (5.11} (C,A,B)-pair and dim(VnS')
dimensional, then the pair is a
Proof: With lemma Y.6 and the remark made above, we only have to prove that
Sorth and V are closed subspaces. We shall begin by showing that V is
94 closed. By definition:
V = { x e l ' l I [ : ] e V e, for some ~eh~} and
This equality gives that S~{O}=V~n[~/~{O}]. of ~ 1 4
Since V~ is a closed subspace
we have that S~{0} is a closed subspace, .and this easily implies
that S is a closed subspace too. So every x in ~ can be uniquely decomposed as the sum of two elements with the first in S and the second in S x. Since S is contained in
V, we have that every element v, in V can be uniquely
decomposed as
v=s+F
(5.14)
s e S and ~=Psl(v)eVnS ". We shall show that the finite dimensionality of
with
hJ implies the finite dimension~lity of V n S ~. Let l~ be the subspace of W containing all w in W such that there exists an x in h¢, with IX]eVe. We shall prove that there exists a linear map from onto V n S ~. Let ~ be an element of W, then there exists a x e •
such that
[ X ] e V e"
From the definition of V we get that this x is an element of V and with equation (5.14)we have that [ ~ ] = [ 0 ] + of S
we have
different
that
F is uniquely
ellement with the
[ ~ ] ; s E S ; ~ e V n 5 ~t. From the definition
determined by 9.
property that
is an
Since
if x ' ~ /
element of
is a
V~, then
So there exists a linear map from I~ to VnS"~ and by the definition of V this map is onto. Thus (5.z5)
dim(Vn S~) _
or VA5 ~ is finite
dimensional Since
V = S ~ ( V n S ~) we have that V is a
closed subspace. A similar
argument
So~th = { x ~ ~l ]Bw ~ W, (~j
as
that
used
above
can
be
used
to
show
that
[]
95 This
result
shows
that
a
semigroup
invariant subspace
for
the
extended
system introduces a (C~A,B)-pair. We shall now show that conversely every
(C,A,B)-pair
brings
on
an
extended
system
and
its
associated
invariant
subspace. Before
we
can
prove
this
result
we
need
the
following
result
of
Curtain [7].
Lemma V.8. Given a
(C,A,B)-pair,
(S,V), there exist F~£(H,i~"~),
G~£(Rk, H), and
K e £(Rk, Rra) such that TA+BF(t)VcV, TA+GC($)Sc S, F=KC+Fo, G=BK+Go KcrFo~S and ImGocV
~) iii)
and
Proof: See Curtain [7].
El
L e m a n V.9. If
(S,V)
W~V/S,
is
bounded
(C,A,B)-pair,
a
linear
then
operators
M,
there L,
K
exists and
a
an
extension
generator
C0-semigroup Tlv(t } on I~ and a closed subspace V~ of ~lc :=H$)¢
N
space of
a
which is
invariant under the semigroup generated by A e. Furthermore the subspace Ve satisfies V =p~I~ and S=Ven{H ~ {O}}. Proof: Since
V
and
S
are
closed
and
VmS,
we
have
the
representation
V ~=S ~ V/S. Let ~ ~ V/S and define He=~/@ W. Let R be a bounded linear map from V to W such that KerR=S and ImR=W. Then there exists a map R+e~()q,V) such that R R +=Iw. Furthermore this mapping satisfies R+Rx=O if and only if ~ES. Define the subspace Ve of 7/e by
By definition S=V~c~H and V=PltI~. Let F, G, Fo and GO be the same as in l e m a n V.8. We shall investigate the semigroup TA+stz+aoc(t) more closely.
a)
Ta+BF+~oc(t)gc g This follows easily from Leman II.2. with leman V.8. i), iii).
96
TA+sf+%c(t)S=TA+ac(t)seS;
b)
for all s in S.
This is clear from lemma V.1. and lemma V.8. Now we shall consider the operator
TN(t)
:=
R T,~+Bv+aoc(t)R+.
By a) this is a well defined operator on W. Furthermore since element of S, for
xeV,
R+Rx-x
is an
we have by b) that
(5.17)
RITA+Bv+%c(t)R'Rx-L T,~+gp+aoc(t)x])cR(S)={O},
So
Tn(t)Rx :=TA+BF+GoC(t)R Rx=TA+Bp+%c(t)x,
÷
for all x in V.
for all x in V, and
T~/(t)Tlv(s) = TN(t)R TA+BF+GoG(S)R+= R TA+BF+GoG(t)TA+BF+GoC($)R+= R TA+BF+%c(t+ s)R ÷= Ttd t + s), and thus T~v(.) is Co-semigrou p on N. Following Curtain [7]
and
Schumacher [34]
we
construct
N,
M
and
L as
follows
IM=-R Go, L=FoR+
(5.18)
I N is the generator of
Clearly We
TN(t }
Me£(Rk, W), Le£(W, Rm).
shall
show
that
Ve
is
TA¢(t)-invaxiant
(TAJt)
is
the
Co-semigroup
generated by At). Let [R:] be an element of I f , then ( s . x2) m
0
=
Ta(t )
"TA+BF+%c( t)x ] RTA+BF+aoC( t ) x
[RTA+BF+aoc(t)R+RxJ
~V e by
a).
97 Thus V¢ is invaxiant under the semigroup generated by
O] =
"A+BF+GoC
"A+ BKC + BF 0 + GoC O]
NJ
0
NJ
0
+ From
lemma
II.2.
-MC
R
we
=A*+ have
that
lie
is
Te(t )
invaxiant
if
and
only
if
~V e for all x in V. This assertion will be proved using
the other properties of GO and F o.
-~¢c
L
- Mc~
j
[B(Fox - FoR* Rx) +GoCx]
[
) . Since S c R e r Fo we have that this equals + R GoCx
)
G°Cx ], and by the inclusion ImGocV, this is an element of Ve. RG o CxJ
S e c t i o n V.2:
D i s t u r b a n c e Decoupling Problem with M e a s u r e m e n t F e e d b a c k
We now feedback,
[3
consider
DDPM which
the
disturbance
was
the
decoupling problem with measurement
subject
of
Schumacher [34],
WiUems
&
Commault [41] and Curtain [7]. Consider the system on the Hilbert space H.
x = Ax + Bu + Eq (5.19)
y=Cx, z=Dx where A, B, C axe as in section V.1. and Ee£(Rq, H), DE£(H, Rd), q is the disturbance input and z the output to be decoupled. DDPM is to design a feedback processor of the form (5.7) such that the output z is decoupled from the disturbance q, i.e.
DJTAe(t- s)E~q(s)ds = O, t >_0 t
(5.2o)
0
98
where TAb(t) is the semigroup generated by the closed-loop system operator Ac,
(5.8)D'~-(D,O)and
Ee=[0E]. L
figure 5.1.
Pictorially our configuration
is given in
.$
d:(t) ~-. Ax( t) + Bu( t) + Eq(t)
q(t)
• z(t)
y ( t ) = Cx(t)
--u( t)
y(:)
z(t) = Dx( t)
~(t)
= Nw(t) + My(t)
u(t)
= L ~ ( t ) + Ky(t)
figure 5.1 Before we shall prove the DDPM in terms of subspaces we shall prove it in terms of transfer functions. Theorem V.lO. The DDPM is solvable with the compensator (N,M,L,K)
if and
only if
there exists a X(s)~ (£(Y,//))(s) such that l imX(s) exists, $-~¢O
D(s-A)-tE + D(s-A)-tBX(s)C(s-A)-IE =- 0 on [r,c~)
(5.21)
and X(s)[ I + C(s-A)-tBX(s) ]'t=K+L(s-N)'tM Remark: Since C(s-A)-tB an
interval
[rDoo)
is strictly proper and X(.) is proper we have that on the
inverse
in
the
last
line
of
this
theorem
always
exists. Proof: We shall begin with deriving fundamental equalities. In order
to
on p(A)np(A e)
derive
equation
(5.21)
we
shall need
the
following equalities
i
T
A
-t.-
II
!
T
T
3' I
-b
II
I
0
•
I
I
I
~
U
T
~
o
~
T
I
I
II
I
+
v
v
+
I
4"
T
•
I
II
A
II
I
+
Y
T
~
~
~
~
+
i
~-'
~
e
T
4-
cr
~
"~
T
I
II
I
o
A
A
II
!
I
I
~b
4-
I
I
4-
v
?
v
4-
II
100
=K +
L(s-N)"H + KC(s-A)-tBK + KC(s-A)"BCKC, L](s_A') 'I~h] + + L(s-N)-tMCC, O)(s-AC)-I[ BK] =
F(s)
where
F(s)=K+L(s-N)-tM.
is the transfer function of the compensator so
Using a similar axgument as above one can show that
(C,O)(s-Ae)-IIBt~ = C(s-A)-~B X(s)
KC(s-A)-tB X(s)+L(s-N)-IMC(s-A)-'B X(s) X(s)=K
Since 1 i m
and ! i m
8-*~0
C(s-A)-tB=9
= F(s)[
I + C(s-A)-'B X(s) ].
we h~ve that on an interval [r0,cc )
S-~O0
the inverse of [I+C(s-A)-tB X(s)] exists, and so
]'
E
X(s) I+C(s-A)-XBX(s)
(5.24)
= F(s) = K+L(s-N)-~M.
The equalities (5.23) and (5.24) prove the theorem.
If
we
do
realisation
not
know
(IV,hI, L,K)
that by
the (5.24),
solution X(.) then
we
Q of
must
(5.21)
is
assume
related
to
the
conditions on
the
system (5.1). Furthermore as for the DDP in chapter 3 to obtain necessary and sufficient conditions for DDPM in terms of invariant subspaces, we need to
assume
(A,B) for
that
the
largest
in KerD, P(A,•)(KerD) ])(A*c*)(KerE ). Then
frequency
conditioned
with
respect
to
is closed and furthermore that the same holds we
have
controlled invaxiant subspace contained smallest
invaxiant subspace
from
chapter
3
that
in KerD, V*(KerD),
as
the
and
l)(a,B)(KerD) = Y*(tler D), {P(a* c*)(Ker E
invariant subspace *
&
containing
)} = S*(Im
E).
the exists
largest as
well
lm E, S*(Im E),
101
Theorem Y.ll. If
P(A,B)(KerD)
and
1)(A*c*)(KerE ) are
closed
subspaces,
then
the
following azsertions are equivalent i)
DDPM is solvable
ii)
S'(Im E) c Y'(Ker D) where S*(Im E) = (V(A*,c*)(Ker E )) and V*(Ker O) = I;(A,B)(Ker D).
iii)
There exists a X(s)~ (£(y,ll))(s) such that lira X(s) exists and S-!,¢~
D(s-A)-tE + D(s-A)-tBX(s)C(s-A)-tE =- 0 on [r, oo) Proof: We shall prove i)**ii), i)-,iii) and iii):~ii). Furthermore we shall use S*
and V* instead
of S*(ImE)
and V* (Ker D),
respectively.
i)~ti): If
DDPM is
TAt (t)-invariant,
solvable,
then
there
exists
a
subspace
V'
that
is
lm Ee c V~c Ker De, namely the space containing all x e such
t
that x e = f TC(t-s)E~q(s)ds for some t and q(.). 0
From lemma V.6. we have that
Sorth:={XeHlL~JeVex:"~ for some weW} is frequency invariant
for the system (A*,C*) and by the definition of V~,
So~,hcKer E*. So Sor~hcV(,t*,C*)(Ker E*), and by definition of S and S* we have (5.25)
o
S = [V(A*.C*)(Ker E )]xcS~orth =S.
Furthermore we have that V:= {xeHI L~J eve for some ~0ehJ} is frequency invariant, and by definition we have that VcKerD. So we have that
102
$*cScYcl~*
(5.26)
ii~i} Since
V(A,s)IKerD)
and
V ( A ' e ' ) ( K e r E °)
axe
closed,
we
have
with
ii)
that (8*,P*) is a (C,A,B)-pair. By lemma V.9. we can construct an extended system operator Ae on He=HO hi,
hi~- V*/S*
and
a
subspace
extended system I: e on
V~
which
is
Tae(t)-invariant.
Consider
the
1"/e
~e = Aex e + Deue + Eeq (5.27)
y = Cexe~ z = Dex ~
Now ~q*~ l m E and V* c Ker D implies that
Ira EecVecKer De
(5.28)
Now Ve is Tae(t)-invariant and so from (5.28) we obtain Tae (t) Im Ee c TAr ( t )Ve c Ve c Ker De and (5.20) holds.
i)~iii): This is a direct consequence of theorem V.10.
iii~ii): Since we do not know that X(s)[ I + C ( s - A ) - I B X ( s )
]-'
has a realisa~ion
(N,M~L~K) we cannot apply theorem V.10. Let X(s)
be the solution of (5.21), then we shall define two subspaces~ .,t
V and So,-~h, of R such that I m E c S o r t h c V c K e r D , for
the
system
(A,B)
and
Sorth
is
frequency
V is frequency invaxiant invaxiant
(A* C*). We note that equation (5.21) always implies that I m E c R e r D . Now define V as
for
the
system
103 (5.29)
V={xeT/I
there exists
s e [ r , oo)
and qeQ such that
x = (s-A)-lEq + (s-A)-IBX(s) C(s-A)-lEq} Now we trivially have that V is contained in
l{erD.
Thus it remains to show
the frequency invariance. Let s be an arbitrary element of [r,co) and let x be an element of V, then
(s-A)"x=(s-A)-'((so-A)-'E q + (so-A)-'BX(su)C(su-A)-'Eq}; for
(5.30)
some s0~[r, eo)
and qEQ.
Using the resolvent equation, equation (5.30) gives:
(s-A)"x=(sol_~s) (so-A)-'E q + (so-Al-lBX(so) C(so-A)-'Eq} +
(5.31)
(~oi_----~,~(s-A)-'eq ) + (s-a)-'. X(.o>C(~0-A)-'t'q} =
=~x+~,-A,-'Eq+(~-A)-', (~ol_~=~X(~'oC'(so-A,-'~:q ) } = - ( ~1x
+ (sol_~s) (S-A)-lEq + (s-A)-tBX(s)C(s-A)-ll','q} +
+ (s-A)-'. (~_~)x(~.)C<~o-A)-% -X(s)C(s-,~)-%} Defining 5(s) as (5.32)
sX(s)=~x
+
~(s-A)-tEq+
(s-A)-'BX(s)C(s-A)-'Eq}
w(s) as
and
(5.33)
-X(s)Cls-Al-'Eq}
w(s) = ( s 0 _ ~ s ) X ( s ° ) C l s o - A ) - l E q
Then
~(s)
is contained
gives
that
this
~(.)
and
in
V, w(s) w(.)
frequency invariant for the system
is a
is strictly
proper
((,~)-representation
and of
equation
(5.31)
x. Thus
1' is
(A,B).
S'o,.th we take tile largest frequency invariant subspace with respect to (A*,C*), contained in KerE*. Then we only have to check that V2cSor~tv Define for xeV" and se[r,c~) For
104 w(s)=X (s)B ( s - A ) ' x
(5.34) • * ~(s)=(s-A $ }- 1 X+(s-A * )"C to(s)
Then
this
is a strictly
proper
(~,w)-representation
of x,
with
respect
to
the system (A*,C*}. In order to prove that x is an element of S~rtn it is sufficient to prove that ~(.)~Ker E*, see definition III.4. Let q be an arbitrary element of Q, then
<E*~(s),q> = < (s-A')-tx +(s-A*)-tC'w(s),Eq> = < (s - A* )'Ix + (s - A* )-SC*X*(s)B*(s - A*
)-tx, Eq >
=
<x,(s-A)-XEq + (s-A)-'BX(s)C(s-A)-aEq> =0, since x e V ~. Since q was an arbitrary element of Q, we have that E*~(s)=O. So summarizing we have the existence of a pair of subspace, S~-~h and V, such that V is frequency invariaat for the system (A,B) and VcKerD, So~h is frequency invariant for the system (A*,C*) and S~thcKerE* , S'~orthc V Combining these results give that
S* (lm £) = S~orthc V c l)* (Ker D)
Now we shall investigate under which conditions we obtain the existence of a finite dimensional compensator. T h e o r e m V.12. The following assertions are equivalent i)
DDPM is solvable with a finite dimensional compensator
ii)
There exists a (C,A,B)-pair (S,V) with dim(VnS~)
105
iii)
There exists a X(s)e (£(y,ld))(s) such that l i r a X(s) exists, S-*~O
D(s-A)'IE + D ( s - A ) - I B X ( s ) C ( s - A ) - t E
(s.3s)
[
a n d F(s):=X(s)
= 0 on it, co)
]_1
I + C(s-A)-tBX(s)
is a proper rational function.
Proof:
i)~ii): Let Iff be
the
closure of
the space
consisting of
all
x c such
that
t
x• =
f TAe(t-s)Eeq(s)ds for some t a n d q(s). 0
This
subspace
is
TAe(t)-invaxiant and
contained
in
the
kernel
of
De. By
lemma V.7. the pair (S,V) is a (C,A,B}-pair, where
V = { x ~ R [ [ ~ o J e V •, for some toel¢) and furthermore dim(VnS ~t) < co. From the definition of S and V we have
Im E c S c V c K e r D.
ii)#i) From
lemma
compensator
V.9
we
have
the
and a subspace I f f c R $ W
existence
of
a
finite dimensional
which is invariant under the extended
system TA•{t). ' Furthermore we have that
Se(O}cVecV~W.
So
ImEcS
and
VcKcrO
implies
that Im E~cP~ cKer D •
Thus
TAc(t)ImE~cTAe(t)VeclffcKerDe,and
so i) holds.
O,~m) This is a direct consequence of theorem V.10.
0
106
Remark: As
in
finite
dimensions we
have
that
the
minimal compensator
state
dimension is:
min{dim(V/S)[(S,V) is a (C,A,B)-pair
with
ImEcScVcKer D}.
So direct output feedback is possible if and only if there exists a V which is controlled and conditioned invaxiant with
ImEcVcKerD.
CHAPTER VI: THE DISTURBANCE DECOUPLING PROBLEM WITH MEASUREMENT FEEDBACK AND STABILITY
As
in
the
previous
conditions such
that
chapter
we
shall
derive
necessary
and
sufficient
the Disturbance Decoupling Problem with Mea~surement
feedback and Stability is solvable and once again the main aim is to obtaiu a
finite
dimensional
compensator.
Whereas
in
the
previous
chapter
closely followed the work of Curtain [7] and thus of Schumacher [34],
we now
we shall follow the line in Willems and Commault [41]. Again we shall study the system (5.1) on a Hilbert space H, which was given by (6.1)
~.:
x = Ax + Bu y = Cx
where
A is the
infinitesimal generator
of
a
strongly continuous semigroup
TA(t ) on ~/, BE£(Rm,7/) and C e £ ( T l , R k ) .
We
shall
start
with
refel'ing
briefly
the
stability
theory
for
infinite
dimensional systems.
Section VI.1. Stability, Stabilizability and Stabilizability Subspaces. In this section we shall introduce the concept
we shall associate a
notion of stability and
new subspace. With the
to
this
basic results derived
in this section we can give a geometric characterization of
the solvability
of DI)PMS. We remark
that
in this monograph we shall consider only exl~onemial
stability.
D e f i n i t i o n VI.I: S t a b i l i t y . We say that Ta(t ) or A is stable if and only if there exists a 6 > 0 and M> 1 such that
(6.2)
JJTa(t)JI<Me TM
108
Definition VI.2: S t a b i l l z a b i l i t y The system (A,B) is stabilizable if there exists a bounded feedback law F such that A+BF is stable.
From Jacobson & Nett [21], Nefedov & Sholokhovich [25] and Datko [13] we have the following characterization of stabilizable systems. T h e o r e m VI,3. For the system (6.1) the following assertions are equivalent i)
The system (A,B) is stabilizable
ii)
There exists a 6 > 0 such that the state space 7/ can be decomposed
as H=Hu$~s which are both T,t(t)-invariant, Hu is finite dimensional, a(A[llu)c{seC[Re s> -6}, [[TA(t)[~ [[ <Me -st and the system restricted to ~/u is controllable.
iii)
For all x o in ~/, there exists a u(,)eL2([O, oo);U) such that
x(t) := T A(t)Xo+ [tT A(t-s)Bu(s)ds O
satisfies x(.) ~ L2([0,oo);7/).
iv)
For all x o in 71 there exists a w(.)eH~(U) and a ((.)eHZ(D(A)) such that
xo= (s-A)~(s) - B~(s) holds for all seC÷: = { s e C [ ~
s>0}.
Proof:
i)**ii), see Jacobson & Nett [21] or Nefedov & Sholok.hovich [25]. i)¢,iit), see Datko [13]. ili),~iv), This is a direct consequence of the fact that H2(U) isomorphic to L2([0,co);U). From this result we have the following corollary.
is Q
109 C o r o l l a r y VI.4. Suppose
that
the
system
(A,B)
is
stabilizable.
Then
we
have
the
following results.
i)
TA(t )
is
a(A)c{s~C[Re
stable
if
and
only
if
there
exists
a
6>0
such
that
s< -6}.
ii)
Let B'E£(RP;H). Then the system
the
system
(A[~iu,P~luB' )
-is
(A,B')
controllable,
is stabilizable
where
~u
is
if and only if
the
span
of
all
imply tha•
A satisfies
the
stabilizable,
there
exist
(generalized) unstable eigenvectors. Remark: ~) of
Corollary VI.4. does
not
necessarily
spectrum determined growth assumption. Proof:
i)
if):
Since
Ta(t)-invariant
the subspaces
C_~,+:={seC]Res>-6} TA(t)-invariant
system
(A,B)
Ha and
~/u such
for
subspace
C_6,_:={seC[Res<-g }.
some we
6>0.
have
So ~ = { 0 }
is Since
from and
that
dim(?'/u)
~u
is
lemma
1.9
thus
7/=Hj,
a
finite
a(A]H..)c dimensional
a(A]Hu)ca(A)c VI.3.ii) gives the
that and
desired result.
only if):
(A,B)
Since all
See Curtain and Pritchard [9, p. 17]. is stabilizable we have that the dimension of the space of
generalized
exists a ~
unstable
which is
eigenvectors
TA(t)-invariant
for some M>_I and 6>0. ~ in ~Q. The result of The
next
ii)
result
is
finite
dunensional.
such that H=~u$~a and
So
there
[[TA(t)[~_ls][<Me-6t
is H~ plus all stable (generalized) eigenvectors
follows now easily from theorem VI.6. will show that
a
similar
decomposition
as
in
6)
a
closed
of
theorem VI.3 holds for all semigroup invariant subspaces. Lemma YI.5. Suppose
that
(A,B)
is
stabilizable.
Then
if
V
is
Ta(t)-invariant subspace, then V can be decomposed in
V=Vu~V, where V~ and V~ are both TA(~)-invariant, dim(Vu)
TA(t)[Vs
is stable.
110
Proof: Since (A,B) and
~fs
such
that
dimensional and jordan
curve
Pn:=l-Pu
is stabilizable we can decompose the state htu
and
7/s
~re
both
space ~/ in ~/u
Ta(t)-invariant,
Hu
is
finite
TA(t)]Hs is stable. Let Pu=o-~Tf(A-A)-~dA, where I' is
which
encircles only the
F eigenvalues of
a
A[Hu. Then Pu and
are respectively the projection on ~/u along ~s and the projection
on ~/s along 7/u. Since ~/u and Ha are both TA(t)-invariant we have that
e,,Ta(t)=Ta(t)P,,.
(6.3)
Define Vu: =PuV and Vs: =PsV, then dim(Vu)
V~cV, and this implies that V=Vu@V s. It remains to prove the stability of Ta(t)[vs , but this follows trivially from the stability of Ta(t)lHs. O We define detectability as the dual concept of stabilizability.
D e f i n i t i o n VI.6: Detectable. This system (C,A)
is detectable if there exists an operator
(;e/.:(Rk, H)
such that A+GC is stable.
D e f i n i t i o n VI.7: Stabilizable-Detectable. The
system
(6.1)
is
stabilizable-detectable
if
(A,B)
is
stabilizable
and (C,A) is detectable.
As and
for
finite
dimensional
systems
we
have
detectability is necessary and sufficient for
dimensional compensator such that is tile extended system operator
that
joint
stabilizability
tire existence of
a
finite
tile closed loop system .4e is stable. A e of
tile system Ze oil tile extel~ded state
space ~/e=/f~ }9, where the finite dimensional space ~9 is the state space for the feedback processor. The feedback processor is given by
(6.4)
~b=Nw+My
u=Lw+Ky
111
where N, 24, L~ K are matrices. Defining x e = ( x , ~ ) , we obtain the closed loop operator
(6.5)
A'= L Mc
Theorem VI.8. The following statements are equivalent: i)
The system (6.1) is stabilizable-detectable.
ii)
The
exists
a
compensator
(N,M,L,K) with finite-dimensional state
K=O such that the closed loop system (6.4) is stable.
space and Proof:
See Jacobson and Nett [21]. Now we shall
13
combine the
notion
of
invariance
with stability
as
in
Curtain [8] and Hautus [19]. Definition VI.9: Stabtlizabillty Subspace. A subspace V of ~/ is stabilizability subspace if there exists a bounded
F, such that
feedback law
a)
Ta÷BF(t)VcV and
b)
36>0 such that
IIT,t÷nr(t)lvll
Vt>_0
Remark: In stable.
Bazile,
Combining
Marro
the
and
Piazzi [2]
equivalence
this
between
notion
open
loop,
was
called
frequency
internally
domain
and
following assertions
are
closed loop invariance with theorem VI.3. gives Theorem VI.IO. Let V be a closed subspace equivalent. i)
V
is
a
ii)
For every
stabilizability
of 7/. Then
the
subspace.
x 0 in V there
exists
a
u(t)~C([O, co);Rm]nL~([O,oo);Rm}
112
such that t
x(t) =TA(t)xo+f
(6.6)
Ta(t-s)Bu(s)ds
o
is contained in V and x(.)eL2([O,~);V). For every x o e V there exists a ((.)~tI2(V) and w(.)e//z(N m) such that
iiil
lira swls) exists, ((s)~VnD(A) and
(6.7)
seE+: = {se(:l/~e(s ) >0}.
x o = (s-A)~(s)-Bw(s);
Proof:
i)~ii)From definition VI.9 we have the existence of a bounded feedback law F such
that. the
closed
loop
system
satisfies
Ta+Br(t ) V c V
and
A+/3FIv
is
stable. From Curtain & Pritchard theorem 2.31 we have that t
(6.8)
TA +BF(t )Xo = Tx(t )Xo + ] TA(t - s)BFT A +13~'(s)xods 0
Defining x(t)=Ta+tjt.(t)x o and u ( t ) = F l ' a + ~ . ( t ) x o gives the desired result.
Taking the Laplace transform of (6.6) gives (6.9)
~ ( S ) ----(8 - A ) - l x o
+ (.5 - A)-IB~(s)
where ~(s) and w(s) are respectively the Laplace transfornl of x(.) and of u(.).
Since
x(.)
and
u(.)
are
square
integrable,
~(.)
and
w(.)
are
//2
functions. By the definition of the Laplace transform we have that ~ ( s ) e V and equation (6.9) gives that ~(s)eD(A). Rewriting equation (6.9) gives
xo= (s-A)~(s)-Bo4s)
(6.10)
and since ~(.) and w(.) are H2 functions we have that (6.10) holds on C+. The
strictly
properness
of
continuous, Doetsch [16, p.226].
w(.)
follows
from
the
fact
that
u(.)
is
113
iii)~i): From theorem 1I.27 we have the existence of a bounded feedback law Fl such that
TA÷BFI(t)VcV.
So we have the invariance, but no further stability
properties yet. From equation (6.7) we have that
Xo=(s-A-BF,)~(s)-B{w(s)-Fl~(s)}.
(6.11)
TA÷BFI(t) we
By the invariance of V under
B(ta(s)-FI~(s)}~V.
Consider now the system there
VI.3.iv)
exists
with
a
(s-A-BF1)(VtaD(A))cV.
Let 121 denote the subset of Rm such that
{(A+BFI))B }
(6.11)
bounded
So
B(Ui)=Pv ImB.
{w(s)-Fl~(s)}eU 1.
Then by definition of 121 we have that
Theorem
have that
with state space V and input space 121.
implies that
feedback
law
this
system
F~:V~/21
such
is stabflizable. Thus that
(A+BFI+BF2)
generates a stable semigroup on V. Now we have to construct a feedback law on the
whole state
space such that
V is a stabilizability subspace.
Define
Fas
F[v=FI+F 2 and FIv, =Ft (BF-BF1)x=BFzPvxeV, since F2(V)cUt. So V is TA+BF(t) invariant, furthermore BF(V)= (BFI+BF2~(V). So with lemma V.1 we have that TA+Bf(t)Iv=TA+BFx+BF2(t)Iv and so V is stabilizabflity subspace. []
Then
The
next
stabilizability under
rather
technical
subspaces
in
lemma
relation
will with
show
the
maintaining
finite dimensional extensions. We shall need this for
usefulness
of
stabilizability the
disturbance
decoupling problem. [,emma VI.11. Let
(A,B)
be
stabllizable, and let V be a stabilizability subspace. For
a space )¢ with dim(lq)
114
If
Vc is
closed
loop
invariant
for
the
system
(A,B),
then
the
(A+BFIv ~,B) with state space V~ and input space /'/1: =B=l stabilizable, where F is any feedback law that satisfies: T.~+~(t)VecVe.
system
~
is
Proof: Since (A,B) is stabilizable, we have from the nice form of ,4 and B that
(A,B)
is
stabilizable
too.
Furthermore
since
effected by feedback we have that the system
stabilizabflity
(A+BF,B)
we have from lemma VI.5 that If can be decomposed
is
not
is stabilizable. Thus
as V =V,$V~
where
dim(VCu)<0o, V~ and V~ are both semigroup (A+BF) invariant and A+BFIv~ is stable.
We
shall
(A +BFIVe,PveB)
begin
by
proving
in
V, (s-A-BF)-~xeV
{seC[Res>-6}. x
the
finite
dimensional
system
is controllable.
Let F e £ ( ~ ; R " ) be such that x
that
Define
and
T4,~(t)VcV this
is
and
an
~(s)=(s-A-BF)-lxeV;
0
LR~(s)J
IJTA+~F(t)Iv u <MeTM, then
analytic then if F =
function
on
for
C_&+=
, we have ~ ~ F21 F2z
-sR~(s)+Rx
(6a2)
= "s-A-BFn-B~'Izl[ -P2~
~(s)I+[Bo
s -F22J LR~(s)J
OIlI(~'n-F)~(s)+~'12R~(s) 1 L~v2~( s)+Pz2R~(s)-sR~(s)+RxJ
Since Ve is ( s - A - B F ) invariant we have that
[ (~'n-F)~(s)+~12R~(s)
leV e
l.P2 ~,~ ( s } +g~2R ,~(s ) - sR,~(s) +RxJ So every
(6.13)
Rx IX]my"
can be written as
Rx
~(s)~ and we(s)~/~l are analytic on C_&+. Let IX Rx] be in V~u, where ~e(s)= [.R~(s)J the finite dimensionoJ, unstable part of A+BFIIf, then
115
IRxX) = PV: [X )Rx= Pv:(S -~t-'F)IV~,'(s)+Pv~B w'(s) or (6.14)
So to every
I Rx x ] e V : there exists a pair (~(.),wc(.)) which
(A+BF[v~)
C_6,+. Since the spectrum of
analytic
are
Oil
is contained in C_6,+ we have from
the well-known Hautus test that equation (6.14)
shows
(~I-BF[ve,Pv~B)
that
is controllable. Now we can consider the system ( , 4 - B F J I ~ , P I f B ) with state space V~ and input space /4z. Since we can decompose this state space into a stable part and a finite dimensional
controllable
part
we have
from
theorem
VI.3.
that
system is stabilizable. With
the
[3
concept
notion of a stable
this
of
stabflizability
subspace
we
can
introduce
the
(C,A,B)-pair.
(C,A,B)-pair A (C,A,B)-pair (S,V} is a (F,G)~£(FI;~') ~ £(R~;H) such that
D e f i n i t i o n VI.12: S t a b l e
a)
TA+BF(t)VcV~ TA+Gc(QScS
b)
36>0, M>_I such that
(6.15)
llT,+~At)ivlI<_~4e"6'
(6.16)
IITa,ac(t)
Since
there
is
a
stable
(C,A,B)-pair
if
there
exists
and
raod Sll -< M= -6'
close
relationship
between
modulo
spaces
orthogoaal complement, we have the following result. Lemma YI.13.
Let (2e£(7/) and let V be a closed invariant subspace of (2, then
IIQmo~ v11= Itt'v*QPv'il= UQ*I v'il
and
the
116 Proof: See Kato [22] and Yosida [44]. With this lemma we have
Lemma VI.14.
(S,V)
A pair
of closed subspaces is a stable
(C,A,B)-pair
if and only
if
0
ScV
~)
V is a stabilizability subspace for the system
iii)
S~ is a stabilizability subspace for the
(A,B) system (A*,C*).
Proof: This follows easily from definition VI.9 and lemma VI.13.
S e c t i o n VI.2: D i s t u r b a n c e Decoupllng P r o b l e m w i t h M e a s u r e m e n t F e e d b a c k
and Stability In
this
section
we
shall
give
necessary
and
sufficient
conditions
the solvability of DDPMS. As with DDP and DDPM these conditions
for
will be
stated in terms of the existence of certain subspaces. Consider the system on the HUbert space H.
ic=Ax+Bu+Eq (6.17) y = Cx,
z = Dx
where A, B 7 C, D, E are the same as in section V.2, q is the disturbance and z is the output output
feedback
of
to be decoupled. The DDPMS is to design a dynamic the
form
(6.4)
such
that,
in closed loop,
z(t) is decoupled from the disturbance q(0. i.e.
fffTA~lt-sIE~q(s)ds=O,Vt_>0 t
(6.18)
0
and Tae(t ) is stable, i.e. there exists M, 6>0, such that
the
output
117 (6.19)
IITA~Ct)II-<Me -6'
where Dt=(D,O), Et= [0 E] and TAC(t) is the semigroup generated by the closed loop system operator A * (6.5). Pictorially our
configuration is given in figure 6.1.
~ ( t ) = Ax( t) + Bu( t) + Eqlt)
q(t}
, z(t)
y(t) = Cx(t)
-u(t). ,
z(t)
y(t)
= Dx( t )
6(t ) = N ~ ( t ) + My(t} u(t)
= L ~ ( t ) + Ky(t)
figure 6.1 Now we can state finite
dimensional
our main result which is completely similar to
case.
For
finite
dimensional
systems
the
the
condition
dim(V/S)<eo is trivially satisfied, and thus is omited there. T h e o r e m VI.15. DDPMS is solvable with a finite dimensional compensator if and only if i)
(A,B) is stabilizable
ii) iii)
There exists a stable (C,A,B)-pair, (S,V) such that dim(V/S)
(C,A) is detectable
Im EcScVcKer D Proof: only if) This proof is essentially the same as the only if part of theorem V.IO. So suppose that DDPMS
is solvable. Thus there exists a finite dLmensionM
compensator such that DtTAt(t)E ~= 0 and TAr(t ) is stable. Define ~ that
to be the closed span of the space consisting of all x t such
118 t
x~=|TA~(t-s):q(s)ds
for some t and
q(s)
0
This
subspace
satisfies S c V and
TAe(t)-invariant dim(V/S)
and
by
lemma
V.6.
the
pair
(S,V)
V = { x ~ R I I x ] eVe for some ~¢W}. LW) We shall prove that (S,V) is a stable
(C,A,B)-pair.
V, then there exists a w e W such that I x l
Let x be an element of
is an element of Ve. From theorem
2.31 of Curtain and Pritchard [9] we have that
T e,t,i :
(t)
L o
r . ( ~ - s)j L Mc
0 So
o
Since 7 ~ ( t ) ( : l E V ~ , we have that exists a
P~iT~(t}x~V and
so for every x in V, there
u(t)={KCP~I+LPw}TA~(t)Ix~IeC([O,oo);R"~)nL2([O,oo);Rm) such
that
x(t) = Ta(t)x +f'TA(t- s)Bu(s)ds 0
is in V and
x(t)~L2([O,oo);V).
So V is a stabilizability subspace.
With the
same reasoning one can prove that .~t is a stabilizability subspace for the system
(A*,C*).
So (S,V) is a stable
(C,A,B)-pair with d~n(V/S)
the definition of S and V we have that
and ii) are a direct consequence of theorem VI.8.
119
(~) In
this sufficiency part
we shall
follow closely the line of Willeras
and Commault [41]. Step 1. Disturbance decoupling From theorem
V.12. we have
the
existence
of
a
finite
dimensional
compensator with state space W1 ~ V/S and a closed subspace ~ = { Rx IraR=~t; KerR=S which is Tae,~(t)-invariant, where Ta~,~(¢) is i-
the
"1
semigroup of the closed loop system Ae'l= ]A+BEIc BLip. Furthermore
[ MXC N~ J Im Ee'l¢ V~IcKer De't
(6.20)
where EC'~=I0~ So the
and D~4=(D,O).
system
is disturbance
decoupled but
enjoys no
further
stability
properties as yet. Later on in this proof we shall solve the DDPMS. We shall consider the extended system (Ae'I,B,C,DC'I,Ee'l) 7/e'l, where ?/~,l =~/(~ )91;
on the
state
space
Ae, l: ~i~,t H~,l, A~,,=[A+BK1C BL'] D(Ae, t) =D(A) ~9W1;
[ ~'c
N'J;
Thus the system (Ae'X,B,(:) is the closed loop system from step 1 with extra inputs and outputs. In formula this system is given by
{6.21)
y~,l = ~x,a
whoro ., Ix1o,
z = Deax ",I
1
In signal flow graph the system is given by
120
2(t)=Ax(t)+Bu(t)+Eq(t) y(t)=Cx(t) z(t)=Dx(t)
q(t) u2 , ÷ ~ - u ( t )
--11,
•
, z(t)
~oI ( t ) = N s w I ( t ) + M l y ( t )+w( ua( t )=Llwl ( t )+Kay( t )
I ,~ ~1 ~
• ye,l
y(t)
t) (
I//
figure 6.2. Then
since
satisfies
the
Wa is finite same
dimensional,
conditions
as
we have the
that
tile
(C,A,B)
system
system and
(C,Ae'~,B) furthermore
(Ae'l,B) is stabilizable and (C,Ae'a) is detectable. Later on in the proof we shall construct a stabilizing compensator for the system {Ae'I,~,C I under which the disturbance This
compensator
is
based
on
the
following
decoupling is maiutained.
decomposition
of
the
state
space, input space and output space of the system (C,Ae'a,B)
)¢'~= v~ • v;" [ ~1
~-''
VI~B* V~ , where B Vl is the set of all " ' e w ' = B~-a~ elements u in Rm~i)h/t such that BueV]
(6.22)
We remark that the decomposition of the input space and output space holds since
B~-~--- ~*
~"
and
C-'V~ =~*-a V~ .
Furthermore
we
define
U I = B Va, //2= B* V~ , Ya=CVI and In follows.
this
basis
we
can
decompose
the
system
(Ae'I,B,C, De'I,E e'l)
as
121
Ail~,l :V,. nD(A)v--,.VI; A,i I =A,,IIV ~, . . .
22 :Vl nV(A22 )v-+V1 ;
A22
is the
dual
of
generators
of
A2~l
in the
c -L
space Vi (6.23)
PV~Blu2
B22:U2~V~; B~2 = Cn:Vtv--*Yl; Cu=CIv~
e a.
C22:V~ ~ Y 2 ; C22 = l'y2CIV~t ~
We
remark
that
All ,
~'
are
the
TAe,I(t)Iv ~ and
T a,~,t(t )1V~" respectively. So heuristically we have decomposed the system (6.21) into
x2
:
ell
0
A22J x2
+
o ~22
+lo J"
u2
(6.24)
=
2 wnere
At2 =
[ 21I 1 _ 0 C22
rV,a iv, el
; z=
(0,~ e'')
x2
I
x2
, and the zero
in
the
A e't is caused
by
the
I
inwtriance of Vl, the zero in B and C is a consequence of the d(,composition of
the
input
and
output
space
and
the
zero
in
De,I
and
Ee.l
is
a
consequence of (6.20). The signal flow graph of (6.24) may be visualized as shown in figure 6.3.
122
li 1
xt--~lxx x ~ +A 12 x2+Bx~ul+Bl 2u2+E e'lq
q
"Yl
y,=Clix l +C l 2x2
~cz =A22 x2+B2~u~
¢,1 Y 2 =C2~x2 z~O e , IX2
~
,z~
1
~Z
figure 8.3 In step 2 we shah and detectable, Let
(N~,M2,L2,0)
respectively
the system (A~il,BmCI~) is stabilizable
and in step 3 we shall prove the same for (A22,B22,C22). and
(N3,M3,L3,0)
(A~II,BmCll)
6.4) the state
prove that
Ixl] x2
and
denote
(A~l,B22,C22).
the
stabilizing
Then
in
compensator
closed
loop
for
(figure
is exponentially decaying. In step 4 we shall present a
precise proof. Now we shall only give an intuitively one. The decay of x 2 is
obvious
e,l ~
~
(A22 ,B22,C22)
since
the
compensator
(Ns,Ms,Ls,O)
and
this
system
not
is
stabilised
influenced
by
the
system
other
inputs.
Furthermore we have that u2 is exponentially decaying, blow in closed loop the upper part of figure 6.4 is a stable system with exponentially decaying inputs, and thus x I is exponentially decaying.
123
• ~x (
w2=N2w2+M~Yt
uI=Lz~ 2
! 1 "-~2
xl=A u xl+A12 xz+Bnu1+Bl~u2+E" q
I
YlfCIIxI'I'CIzxz
• ¢)1 x2.=A22 x2+B22u2 y2=C22x2 z=De,l:r.z
)X 2
'
)Y2
~23=N31//3-I'M3y 2
722.~
2L2~---~3'U) 3
figure 6.4 Step 2: D i s t u r b a n c e loop s t a b i l i z a t i o n We
shall
prove
that
the
system
(All,BmCu),
with
state
space
I/~1,
input space U1 and output space Yl is stabilizable-detectable. The stabilizability of
the
system (An ,Bn)
is a
direct
consequence of
lemma VI.11. The
detectability
of
esl (Ca, All ) follows
easily
from
the
detectability
of (C)A) and is left to the reader. Now from theorem YI.8 we have that there exists a finite dimensional compensator with state space 1¢2 and no feed-through term such that is stable on the state space ~ ~ Wz.
M2CIt
N2
124 Step 3: Controlled output stabilization With a similar argument
as used
in step 2, one can prove that
system ID r ~ 2 2 *t~e,l~* ,~,~22 I C2~ ) with state space ~ , space //2 is stabilizable/detectable. So there
input space Y2 and output exmts
state
space
V~l~Ca.
So the
|L:~2:
L" is also
dual
a
compensator
C22.M3
(A22 ) state space W3, d/m(Wz)<eo such that
the
N3
" stable~
/
is stable
with
on the
J this
dual
is given
by
[ A ~ 1 B22L3'
Step 4: D D P M S It remains
to
show
that
the
connection
of
the
tb_tee constructed
compensators solves the DDPMS.
Applying both
compensator.q to
the system (6.21)
gives for
the
closed
loop system the following system operator
(6.25)
A" I B ivtL 2 B iu2L3 MzPylC Nz 0
A~ =
M~Py2C
0
N3
with D(A~)--D(A}~w 15W 2 ~ W 3. and i denotes
TAe(t)
denote
V~:--I~tsW2${0}.
the
semigroup
We shall
easily from the fact that
i001
generated
prove
that
the
by
inclusion
Ac
and
operator. define
Ve is TAe(t)-invaxiant.
V"
Let as
This follow
Ve is invariant for the semigroup generated
by
N20 /V3 and furthermore from the definition of /il and Y2 we have that
0 ~ ivlL z ~ iv2L3 M2Py~C 0 0
(6.26)
MsPy2C
o
e o
0
0
u
ooe
LUJ
D~=~D,O,O,O), we have ImE~cV~cKcrD e and thus we have still disturbance
125 decoupling. It remains to show that A¢ is stable. From the definition of Ae and lemma V.1. we have that
to V¢ is the same as the semigroup generated by
shall denote
by
Tae,2(t).
From the
TAe(t)
~
[Mzc,, N, j
disturbance loop
restricted
, which we
stabilization we have
that this is stable. In the same way one can prove that
TA~(t)*
.
T a,,3(t),
the
semigroup
generated
by
restricted to V¢" is the same as
(A~2)
C22,M3
|L'B2,*
N3
L"
stable.
(
which
was
also
J
"
With these stable semigroups we can show that
(A,B)
/'
TAe(t )
is stable. Since
is stabilizable and Wl, W2 and W3 are finite dimensional the system 00 0 0
IO 0 I
'
0 0 affect
) is stabilizable too,
and since feedback does not
0 0
stabiliT.ability we have that
(Ae,
I 0 0 I
) is stabilizable. Thus in
0 0 order to prove the stability of Ae it is sufficient to prove that Ae has no unstable eigenvalues. Suppose
that
IITAe,Z(t)ll<Mze -62t
and
Tae,2(t ) and [[Tae,3(t)ll<Mae "~at
Tae,3(t) for
some
satisfies 62 and
respectively d/3 larger
than
zero, note that these constants exist since both semigroups axe stable. Let
leeA>raax(-6z,-63) x ~ H ~ = H ~ W] ~ Wz ~ W~ such
AeC with
TA~(t)X~=eXtX~
be an unstable eigenvalue, then there exists a that
~Pv~tTa~(t)x,~feXtPv~XX" ,
since I f is Tae(t)-invariant, we have that
PvelT ae( t )Pv¢~Xc = eXtpz¢Xxe Pve~X~#0, then eat is an element of the point spectrum of PvcXTae(t)Pve ~, and thus eat is in the point spectrum of Pvea(TAe(t))*Pv ea (Tae(t))*lVe~=(Ta,,z(t)) *. By the stability of Ta~,3(t)* this is not true, so PveXxe=O, or equivalently xe~Ve. However for all t>_0 we have that
Thus
if
=
Ile~tx~ll =
IITA.(t)x'll = IITae(t)lv.x'll = IITA.a(t)x'll <<-M2e-62' Ilx'l[
126 So x=m0, which implies that
a(Ae)¢C_
and coroUaxy VI.4. gives that Ae is
stable. Thus the DDPMS is solved with the compensator (/V*,h/*,L=,/~) with finite
)¢*=Wi¢Wz~W 3 and Nt Pwt iulL 2 Pw~ N~ = M2Pv t iv~,l N2 flaPv 2iye,l 0 N:3
dimensional state space
iu2L3 ] J
Me= bl2PYlil~,l ; .MaPr2ty*,O
(6.27)
L*=
"LI,PRmiu1L2, P miv,La) and
K*=Kr where ye,1 : = ~ ¢ h j l .
[]
Remark: In I2] Basile, Marro and Piazzi solve the DDPMS in a very simple way, however
they
constructed This
become
dimensional. The
dimensional if
the
for
compensator
original system. infinite
is
a
condition
longer satisfied. However for
us
with the
undesirable since compensator
case
the
would
dim(V/S)
same
of
dimension as compensator
also
theorem
we could always prove
similar to theorem V.11, but then we have to assume that
~;(A*,c*)(KerE*)
become
are closed subspaces, see theorem V.11.
VI.15 a
the
would infinite is
no
theorem
])(A,B)(KerD)
and
APPENDIX E: EXAMPLES
In this appendix we shall give exaznples of some facts that
are stated
in the previous chapters. Many of the examples that will be discussed here are based on spectral realisations of delay equations. The reason for using this
realisation
is
that
on
tile
one
hand
we
have
that
these
transfer
functions are of a simple form and on the other hand we have from chapter IV a complete characterization of all controlled invariant subspaces. So we have the best of both worlds. In
the
realisations
first for
section
delay
we
systems
shall as
recapitulate
presented
the
in Curtain
theory &
of
spectral
Zwart [10] and
[12]. In the subsequent sections we shall give many (counter) examples of which we shall present a list.
Fact
Example
1]*(K) need not necessarily exist
E.9
1,:'(K) may exists, and be unequal to NE(K)
E.IO
V*(K) may exists, and be unequal to I)ot(K )
I'.11
The closed sum of two controlled invariant subspaces need not renlain controUed iuvariant
E.15
List o f examples.
Section E.I: Spectral Realisations of Delay Equations.
The delay transfer functions that will be considered are of the form
fls)= p(s'e-~) q(s, e -~ )
(e. 1)
where p(x,y)
and q(x,y)
stlall
that
assume
the
are lowest
polynomials in two variables. Furthermore we power
of
y
in
q(x,y}
polynomial in two variables we define the distribution diagram.
is
zero.
With
a
128
Definition E.l= Distribution Diagram
Let q(s,e-8)
by
a
polynomial
in two
variables~ then we
can
rewrite it
in the following form
q(s,e -s) = ~ qi(s)e -~s
(e.2)
where
q~(s) is a polynomial of degree ~ and O=~0<~t<.. <8,,.
With the
points
(fli,m~) we can
define the distribution
diagram
of q(.,.);
this is the polygonal line L which is the upper boundary part of the convex hull of the points (0,0), (~i,mi), (#,,0).
Since this definition is rather
technical we sham give a simple example to
illustrate it.
Example E.2. Consider
the
function
q(s,e-s): =s(s+e-s).
Then
(fl0,mo) - (0,2)
and
(fit, mr)=(1,1). So the distribution diagram has the following form.
I
0
The
distribution
diagram
I
1 2 figure e.1. is
closely
[]
related
with
the
asymptotic
q(s,e-*). To every slope of the distribution diagram there are infinitely many zeros of q(s,c-a).
distribution
of the zeros of the
function
Lemma E.3. Assume that there is no horizontal slope in the distribution diagram of the function
q(s,e "m) and assume further that on every segment of L there
lie
two
exactly
points.
Let
Li be
a
line
segment
of
the
diagram L with left end point (ill,mr) and right end point ( # r , ~ ) .
distribution So L has
129
a
kink
at
these
points.
Then
the
asymptotic
distribution
zn,i=Re(z,,i)+i:Dn(zn,j),
related with this segment is given by
of
the
zeros
pi,m i
is the
where
(e.3)
m~-mr"
and
wj
is j'th complex mt-mr'th root of
--~TDm r
~ ; Pl,m I
leading coefficient in pi(s). Proof: See Bellman and Cooke [3, th.12.10.].
Tbe next lemma will play a key role in obtaining spectral factorizations of delay
transfer
functions.
Before
we
can
state
this
Icnmla we
tim
need
concept of vertical distance.
Definition E.4: Vertical Distance. Let L denote the distribution diagram of the vertical distance of the
pair
(c~,d) to
denote the number r e R such that
function
q(s,e-~),
lly the
tile distribution diagram L we shall
(c~,d+r)~L.
When such a ntnnber does not
exist, then the vertical distance is by definition minus infinity.
Remark: By the definition of L we have that ~- is well defined.
For example if L is the distribution diagram of in example
E.2,
then
the
vertical
distance
of
q(s,e ), (0,2)
to
where L is
q(.)
is as
0 and
the
vertical distance of (2,0) is -co.
With
the
concept
of
Curtain and Zwart [10].
vertical
distance
we
can
state
the
main
result
of
130 Lemma E.5.
Let
q(s,e -s)
be the sa~ne as in (e.2)
diagram of q. Suppose that
and let L denote the distribution
on each segment of L lie exactly
two points
(/3.m~). np
p(s,e-s)= ~ p j s ) c -°i~
Let
pi(s)
with
a
polynomial of
degree
d,,
and
i=1
assume ttmt (c~i,d~) to
p and
q are
L is larger
coprime. If
than
zero
the
vertical distance of
l
for
the
f(s)= p(s'e-~) q(s,e "~)
then
points has
a
stun
is
partial fraction expansion, given by
~ p(zj) / q'(zj) (e.4)
/(s) =
"J
J
j
where
zj
are
C i ,j
(s-z~) =m+
the
j=0 i=0
]
zeros
of
q(s)
with
(s-zJ)
multiplicity n j,
attd
this
uniformly convergent in s in any compact subset of C/{zj}. Furthermore we have that
Ip(zj)/q'(zi)l<_CIz~)-~,
where C is independent
of zj and v is the minimal vertical distance of the points ( ~ / , d i ) to L. Proof:
S e e C u r t a i n and Zwart [10, p.73 and 74].
[]
This theorem enables us to make spectral realisations of delay transfer functions. In order properties of
to
improve tile
readability
we
shall
list
tile
required
f(s) = P(s'e-S). q(s,e )
D e f i n i t i o n E.6: P r o p e r t y P. n
Let
q(s,e-s)= ~ qi(s)e -l~is,
where
qi(s)
is
a
polynomial
of
degree
m,
i=O
and 0=f0
p(s,e-S)= ~ pi(s)e -~'i8
Let
with
pi(s) a
polynomial of degree
di,
and assume
i=l
that p and q are coprime i.e. no cotmnon zeros. Then
[(s)= p(s'e-s} q(s,e -~ )
satisfies
property P if it satisfies P1, P2 and P3, where Pl.
On each segment of L lie exactly two points (fi,m,).
P2.
if (fl, m~) and (fir, mr) are two succeeding points on L with
~t
then m r - m r = I. P3. Then
Let
vi
denote
the
vertical
distance
of
the minimum of these v,'s is larger than one.
the
point
(c~,dz)
to
L.
131
Note
that
P2
furthermore
implies
that
there
it implies that
axe
no
horizontal
to every segment of
segments
L there
in
L,
and
is one string of
zeros, see (e.3). With the property P we can make a spectral realisation. In order to shorten the notation we shall assume that q has no multiple zeros. We stress that this assumption
is made only to
improve the readability. The same results
as stated in the next theorem holds if this condition would be omitted.
Theorem E.T. If f(s)= P(s'e-a} has property P and assume that all the poles of f have q(s,e'*) multiplicity one, (see the remark above), then there exists a spectral realisation (D,A,B) with state sp~ce 62 such that f(s)=D(s-A)-XB,
where
rlml
D ( A ) = { x e ¢ 2 [ ~ Izn[2i <x, en> Iz
B~=bu
and n x =
b=a={(p(z.)/q'(z.)] ~}
<x,a>;
Furthermore this realisation has the following properties from chapter IV: (A1)
The generator A is a discrete spectral operator.
(A2)
hi: = # 0 , for all i > l .
(A3)
For all iEN,
(A4)
inflzi-zil = 6 > 0 , i~j
Dei#O.
O0
1
2
j=l We remark that a ( A ) = {zj}. Proof: From property P3 and lemma E.5 we have that
Ip(zn)/q'(zn)l
So b mad d are elements of d ~. That (D,A,B) is a spectral realisation
r>l.
of [(s)
follows now easily from lemma E.5. Now we shall prove
that
this
realisation satisfies A1 up to AS. The resolvent of A is given by
(e.5)
(A-A)"= ~
~l_--~n<.,en>en
niN
and from (e.3) we have that
Iz, l
132 so we have proved A1. ~ 2 and A3 follow easily from the definition of b and d and the condition that p and q are coprime. Furthermore we have from (e.3) that condition A4 holds and we have that
]zi-zil>c*j; i#j
fixed i e N
with c > 0
independent of
i. So A5 is also
satisfied
Remark: The properties A1, A 4 and /x5 are properties of the system matrix. So if b and d is another pair such that f(s)ffi<(s-A)-Ib,d> and b satisfies /x2 and d satisfies A3, then this realisation satisfies also A1 up to AS.
We shall illustrate this theorem by the following easy example.
Example
E.8.
Consider the transfer function f ( s ) =
1
. Then by theorem E.7
4 s ( - 4 s + e -4~' )
we have that the spectral realisation of f is given by
AX=~ Zn<X,en>en; n=l
O(A)={xe~21~ {z.[2t <x,e.> I~<~}; z., ~>i are the zeros of ( - 4 s + e - 4 s ) ; Z l = 0 and
b = a = {1/2,1/(t6~=(
Furthermore this realisation satisfies A1 up to theorem
IV.16
a
~,.. ).
-: - 4z=)) ~,.., 1/(:6z#( - :- 4zj))
complete
characterization
A5, of
and so we have from all
controlled
invariant
subspaces in the kernel of D. Since the transfer function has no zeros we have
from
theorem IV.16 that
the
unique controlled invariant subspace
in
the kernel of D is the zero subspace. So we have for this realisatioa that
P*(KerD)
exists and it is equal to the zero subspace. Note that we do not
state anything about the more usual realisation in the space h r . In
the
next
section
equations in order
the
state
space
to
we
shall
use
spectral
obtain counter examples for
is finite dimensional
infinite dimensional case.
these
but
do
not
f~
realisations properties
alway
of
that
hold
delay hold if
for
the
133
Section E.2: The Relation between g (K), Vol(K ) and V2~(K ) The next example will show that V'(K) need not exist. Example E.9. Let ~/ be L2(0,1) and A is the "heat operator",
A = d2 with domain, dz 2
D(A) = {x~L2(O,1)lx"~L2(O,1); x(O)=x(1)=O},
8:=b(z)= [
o
; ze[0,½]
sin(27r z); ze[½,1] K = {xeL2(O,1}l x(z)=O; a.e. or~ [0,½]}. Notice that K ~ I m B . So B°(K)={0}, and from lemma II.19 we may conclude that, if K is controlled (or equivalently frequency} invariant, then every x in K has
an
unique
(~,w)
representation
with
Bw(s)--0,
which
means
that
(s-A)-IK c K. Thus K would be TA(t)-invaxiant , but this is in contradiction with example 1.6. So K cannot be controlled invariant. Define for n > 1 0 en( z ) = .
1
{sin(21rn z) + n( - 1)nsin(2zr z)}; z ~ [~, 1 ]
4ff2n(-l)n(n2- 1)
and /~. in C by /zn =-47r2n 2. Then
1)
span{e~} is controlled invariant for all n a n i=2..n
2) span {ci} is K.
i.~/{l}
P r o o f o f D" By a simple calculation one can 8how that ( A - p ~ ) e , = b . So
(c.6) Thus
(s-A)e.f-b+(~-~.)~. span{e,}
is
controlled
o,.
=.=(~-A)se~.
invariant.
By
-
lemma
span{ei} is controlled invariant for all h e N . i~2. ,n
P r o o f o f 2): 0
on [0, I
If x a K , then x =
, with £~L~(,1). So on [~, 1]
b
i
pn-S
HI.14
we
have
that
134
~ <~(z),2sin(21rnz)> 2 t 2sin(27rnz) and .=x L (~,t)
~(z)=
II~ll 2
~
.
2
[<x(z),2sln(21rnz)> L2(~,1) I [ < oo
=
Let x . l . e , V n > l ~
<x,e,>
2t
L (;,1)
=0
< ~(z),sin(21rn z) +n( - 1)'~sin(27r z) > L2t! 1~ = 0. So ~2,*/
(e.7)
<£,sin(27rnZ)>L2({,1)=-n(-l)n<~,sin(27rZ)>L2(~,l)
(e.8)
~ 2> oo> Ilxll
,1)]2= ~ n2l<~,sin(2~rz)>L2(~ ,1)[2 I<~,sin(21rnz)>L2(~
~ n=2
(e.8)
implies
that
V n>l
n=2
<~,sin(2rrz)> 2 1
L (~,1)
=0,
and
(e.7)
with
we
have
that
<~,sin(21rnz)>L2(~,l)=O , for all u in N/{0}, so ~ = 0 . Thus x -
0 is the only vector
in K perpendicular oil all en, so
span {e~}
is K. If I)*(K) were
to exist,
then it would necessarily
in K and by 1) it must contain
be closed,
span {e~}, for all h e N . i=2..
contained
This together with
n
2) would imply that 1)*(K) equals K. This contradicts the fact that K is not controlled invariant. Thus Y*(K) c a n n o t exist in this example.
"l'he next example will show that unequal
to
IIu(K ). Note
that
it is possible that
theorem
H
I;*(K) exists
III.12 implies that
but
it is
VE(li ) cam lot
he
closed then.
Example E.10. In
this
example
we
shall
study
the
delay
transfer
function
as
introduced in example E.8. So
(e.9)
f(s) =
1 4s( - 4s+e -4" )
From example
E.8 we have
that
this system
has a
spectral
realisation
(D,A,B) which satisfies the conditions A1 up to A5 and we have that, D is
135
d={i/2,1/[16z2(-1-4z2))g~,.,I/[16zj(-l-4zj))~,.};
given by D = < . , d >
where
zi, j > 2
th zero of
is the j - I
- 4 s + e -as. Furthermore we have proved that
]2*(KerD) is the zero subspace and thus by theorem ITI.3 the DDP is only solvable if E = 0 , i.e. no disturbances. Now we shall show that there exists a bounded operator E and a strictly proper
D(s-A)'IBU(s) = D(s-A)'IE.
(e. 10) The
existence of
Firstly;
since
equations solvability
such
the
disturbance
D.D.P.
(3.13), of
a
(3.14), these
is
non
and
input
solvable,
(3.15)
equatiorm
(3.14)
or
(3.15)
is
m.lo.
but
axe
is
in
solvability of DDP. So the condition that omitted i n - t h e o r e m
operator
a
necessary
the
solvable,
general
meromorphic we
not
two
have
facts. matrix
that
the
to
the
equivalent
l)£(KerD) is closed can not be
It easy to prove
always
will prove
that
condition
the solvability of for
the
{3.13),
solvability
of
unequal
to
DDP.
Secondly,
we
have
that
I)"(KerD)
can
exists,
and
it
is
P~(KerD). Namely from equation (e.10) we have that Eq is contained in VE(Kcr D), since
(e.11)
D(s-A)-IEq=D(s-A)-IBU(s)q, so (s -A)-lEq = ~(s) + (s-A)-lBU(s)q, with ~(s) e Kcr D.
Thus
(¢.12)
Eq = (s-A)~(s)+BU(s)q
and by definition this shows that
Eqe))r(KerD), and thus {O}~)~(KerD).
Now we shall define this operator E. Let E:C~-~ 2 be defined by
Eq=eq~ where
(e.13) 3
From (e.3) we have that the j th component of e is of the order j-~, so e ~ $ 2, and thus E is a bounded operator. Furthermore
it
is
from
lemma
E.5
easy
to
see
that
(D,A,E)
is
a
136 realisation of the transfer function -$
D(s_A)-tE=
(e.14)
e 4s( - 4s + e -4~ )
If
we
define
U(s)
as
e"s,
then
the
operator
E
and
the
function
U(s)
satisfies equation (e.lO) and so we have constructed the counter example.
Similar
P*(KerD)
to
this
exists,
example
but
it
subspace in the kernel of
we
is
shall
unequal
to
construct the
an
largest
example open
13
such
loop
that
invariant
D, Yoz(KerD).
Example E.1L Again we shall consider the function
f(s)=
1 . 4s( - 4 s + e "4s) realisation of f is given by
By
spectral realisation of example
E.8
we
have
the that
delay transfer the
spectral
m x = ~ Zn <X~en>en] D(A)---{x~d21~
Iz.121 <x,e.> 12<~}; z.,
n > l are the zeros of ( - 4 s + e - 4 ° ) ;
n
z 1 = 0 and b = d = {1/2,1/(16z~(
- i - 4z~
I) ~,..,
1/C16zj( - 1 - 4z~ I~ v~,..
Now we shall construct an initial value continuous input function
u(t)
such that
xoeKerD
}.
such that
the solution of
there
exists a
~c(t)=Ax(t)+bu(t);
x(0) =x0 remains in the kernel of D. As xQ we take
(e.14)
Xo={(1+2a)/2, (-(4z2)~-2a(z2)V~-a))/((z~-1)(16z2(-1-4z~))V'~,. ., ( - { 4 z S ' -2a{zj) ~ - a~) / (lz~-1) (l~zjc -1 - 4zj)) ~ ,... }
1#here
(e. 15)
a=
sinh( - 1) sinh( 2 )
and with u(t) defined as
137
f (e.16)
u(t)=
asiuh(t) sinh(t-1)+a sinh(t)
;0
0
;t__.2
we claim that this input will keep x 0 in the kernel of D. So we have to prove that t
(e. 17)
O=Dx(t)=trra(t)xo+DfTa(t-s)Bu(s)ds; Vt>_0. 0
First we remark that,
by the definition of a, u(.)
is a continuous function
on [0,oo). Furthermore the Laplace transform of u(.) is given by (e.18)
£(u)(s):=w(s)= e-s+ae-2s+a 2
S -1 Taking the Laplace transform of (e.17) yields
(e.19)
O=D(s-A)-lxo+D(s-A) "Bw(s)
So from our realisation it thus remains to show that (e.20)
e-S+ae-2s+a
1
D(s-A)-txo = -
4s( - 4 s + e -4s)
(D,A,xo)
Or, equivalently, we must prove that -S
{e.21)
ga(s) = -
s 2-1 is a realisation of
-2.'I
e +ae +a 4s(s2-1 ) ( -4s+ e-4s)
This result follows directly from the next
lemma. Thus
XoePc~(KerD),
so we have constructed our example.
and [::]
Lemma E.12. Consider the function g=(s)= on C the
- e -, - a e -2s - a , with 4 s ( s 2 - 1 ) ( - 4 s + e -4")
following equality holds"
ga(s)=D(s-A)-txo,
sinh(-l) a=s~nh { 2)
where
" Then
D, A are
a~
defined in example E.8 and xo is defined by (e.14). Proof: Since t h e numerator and the denominator have by the special value of a, common zeros we can not apply theorem E.7 directly. However if we define the sequence { a , } c R such that ~ ÷ a and
-e'S-ane'~-an
has no common zeros
138
ga(.).
with the denominator of
Then we can construct a spectral realisation
of
yah(s):=
(e. 22)
-e
-,1
-25
-ane
4s(s 2- 1 }(-4s+e As
state
space
is obviously
for
these
isometric
realisation
isomorph
-a, -4s) we
choose
12({-1,0,1..}).
This
space
but
we shall need
this
space
with ~2(N),
for notational convenience. From theorem E.7 we have that a realisation of
g%(s)
is given by
Aex= ~ z n<x,e n>e n;
D(A)={xe~Z({-1,O,..})I~ Iz.l~l <x,e.> t2
Zn, n>2
z _ l = - 1 , z 0 = l , Zl=O and
are the zeros of ( - 4 s + e - 4 ~ ) ;
de = {X, 1,1/2,1/[16zz( - 1- 4zz)) ~ , . . , 1 / 0 6 z j { xn=
e+ane:+an) (e-l+ane-2+an) 8(4+e4)
'
8(4-e-')
- 1 - 4z j)) ~ , . . }.
(l+2an) ' 2
-(4z2)¼-2an(Z2 i -a,,
'4(z~-l)
(z2(-1-4z2))
v~''"
- ( 4 z i)W-2a,~( z i ) ~ - a ,
"" ( z~_l) Ct 6zi(_l_4z i))V,'"} If n converges to infinity, then gan(s ) converges for fixed s e C
to
-#a(S).
Furthermore it is easy to see that x" converges to x~: ={O,O,x0} in the norm of
~2({-1,0,1,..}).
So
De(s-Ae)-1x~=D(s-A)-lxo.
for
fixed
seC
D~(s-A~)-tx" converges g~(s)=D(s-A)-lxo.
So for fixed s e e we have that
to
O
S e c t i o n E.3: On t h e Sum o f two C o n t r o l l e d I n v a r l a n t Subspaces In this section we shall show by means of a counter example that the closure of the sum of two controlled invariant subspaces is not necessarily controlled
invariant.
In this
example we shall use the
spectral
realisation
of a delay equation an derived in section E.1. The delay equation that will be considered is given by
F(s)=(se"
(e.23)
~+s2+l )
(se-S+s2
-1)
(s~-I)(e-'+s)(e -2"+ s) We shall
prove
that
the
spectral
realisation of this equation
satisfies
the
139 conditions
of
associated
with
the
with
zeros
of
the
Ilowcver
the
chapter
sum
4.
zeros
Purtherniore
se-S+sZ+l
of
se-S+s2-1 of
these
is
as
controlled
subspaces
will
pole of F(s) associated
since to e v e r y
we
shall
l)rove
well as
that
the
subspace
invariant, riot
(see
remain
with (e-a+s)
the
associated
theorem
coutrolled
there
subspace
are
two
IV.16).
invariant, zeros of
the system> see theorem IV.16. Before we can p r o v e our exanlple we need some results on the distribution of the poles and zeros of F(s).
Lemma E.13. Let h(.) and r/(.) satisfies on £ the following relation
(e.24)
h( s ) = h( so) + ( s - so)h'( so) + ( s - So)Tl(s ).
where h'(so) # O.
Ih(so)l Assume
further
inequality holds
that
on
tim circle C : = { z :
17/(z)l
Iz-sol=2
} the
following
Ih'(su)l Then h(.) has exactly one zero inside C.
Proof: Define / ( s ) : = h ( s ) - ( s - s o ) h ' ( s u )
and g ( s ) : = ( s - s o ) h ' ( s o ) .
Then
for z e C
we
have that }g(z)l = Iz-s01 Ih'(s0)l = v'~lz-sol Ih'(so)l +l/21z-sol Ih'(so)] =
= Ih(s0)l
+v21z-sol Ih'(so)l
> Ih(s0)l +
Ih(so)+(z-so)~(z)l---Iflz)l,
Iz-so1171(z)l >
by (e.24) and tlle definition of [(. ).
Now with Rouch6 theorem, Rudin [31] we have that g(.) and f + g = h
h a v e the
same mind)or of zeros inside C. So h(. } has exactly one zero inside C.
[]
From
this
lemma we obtain
the
implication
concerning
the
distance
between
e-~+s with norm sufficiently large. Then
inside a
the zeros and poles of F(s).
C o r o l l a r y E.14. Let s o be a zero circle
with
se-S+ s2+ l.
centre
of
so and
radius
2 is0(s0+l)l
there
is exactly
one
zero
of
140 Proof:
h(s):=se-S+s2+l
Defining
and r/(s):
=
se-S+s2-(s-s°)s~(s°+l)
gives that
(S-So)
this pair satisfies (e.6). Furthermore we have from (e.6) and the fact that h and 7/ are entire function that
k=2
h("l(so) gives; h ( s o ) = l ; h'(sol=so(so+ l); h"(So)= -s~+ 2so+2 and h(")(So)=n(-1)"so+(-1)n+zso2 ;n>2. So
Calculating
Oo
E (--J'/¢' .
(~.~6)
~($).----
,k+l
O0
2
SO(S--so)k-I+
k=2
k=2
O0
s] ~
E ~S-s°)k-l+(s--$°)----
O0
.k+so ~, ~ s - sk +ol ) k + ( S - S o ) .
~(-s -z s)ko )
k=l
k=l
With equation (e.26) we have that ¢0
(e. 27)
Co
~ Is-s°l~ [sol ~-" ---ET--'-. IS-Solk k=l k=l
[t/(s)l
+ I s - sol =
O0
oo =IS-Sol[Sol 2 ~,, (-fi4WF IS-Solk +ls-sollsol ~ (-W4-i Is-s°[~ -------------~-. +Is-sol < k=o k=l
< [s-sel( ½[so[2els-sol+ [so[els-s°l+l }. So on the circle
[z-sol
Ih(so)l
=-2
2
= [so(so+l)[ we have that
Ih'(so)l 2
(e.28)
] [77(z)[<jSo(So+l)[ {
and for
[so[
sufficiently
2
[80[2cl'°(a°+l) I large
+2[So,e''o('o+I) I +2
this is smaller than
}.
~[so(so+l)l=~lh'(so)l.
So lemma E.13 gives the desired result.
0
Remark:
With
se-S+s2-1
a
similar
proof
one
can
show that
the
zeros
of
the
function
have the same property.
Now we have proved the necessary lenmlas to give a counter example for
141
the fact that the closure of the sum of two controlled
invariant
subspace
is not necessarily controlled invariaat. Example E.15. Consider the delay transfer function
F(s) = (se- s +s2 + l ) (se'S-i-s 2 - I ) ( s 4 - I ) ( e - ' + s ) ( e - 2 ~ + s) We
shall
begin
by
constructing
a
spectral
realisation
of
this
function. Let {zj;j>0} denote the zeros of e'a+s, and { z i ; j < - 3 }
denote the
z_l=-l, z_2=i, z_3=-i. So {zj; j e Z } are the
zeros of e-Z'+s, and % = 1 , poles of
F(s).
The
spectral
realisation
this space is isometrically
of
F(s) will be constructed on ~2(Z). Since
isomorph with ~2(N) we have that
a realisation
on ~2(Z) gives by renumbering of the indices also a realisation So the ~2(N)
transfer
realisation i.e.
if
on
the
~2(Z)
transfer
has
similar
function
properties
satisfies
the
E.7., then the realisation on ~2(Z) will satisfy A1,..,AS. On ~2(Z) we have the following realisation of F(s);
mx= ~Zn<X~en>en; n~Z D(A) = { x ~ 2 ( Z ) I
~
Iz.121 <x,e.> Iz<eo);
n~Z
z., j < - 3
are the zeros of
e-Z'+s;
z_ 3 = - i ,
z_~=i, z_ 1 = - 1 ,
z., n ~ l
axe the zeros of (s+e-~);
b = {bj}i,z;
1
j N,
bj=l; j=-3,..,0 1
j<-3
zo=l~ and
as
the
conditions
on ~2(N).
realisation in
on
theorem
142 -1 D = < . , d > ; d = {dj}; d.i = ( z ~ - l ) ( z 2i + z j )
j_>l;
do= (e-l+2)(e-1) , d - l = ( - e l + 2 ) ( - e l ) , 4(e-l+l)(e-2+l) - 4 ( e l - 1)( e 2 - 1 )
d_z= ( ie-i ) ( ie-i - 2 )
, d-a=
( - iei ) ( - iei - 2 ) - 4 i (e i - i ) (e 2 i _ i ) '
4i(e -i+i)(e-2i+i)
dj =
z~ ~(-zj )'/~+zi) 2-1 j< (z~-i) ((-z ~ )~+z j)
From theorem E.7 we have that (D,A,b)
is a spectral realisation of F(.) and
furthermore it has the properties AI~..,A5 from chapter IV. So we have in this
realisation
a
complete
characterization
of
all
controlled
invariant
which
are
controlled
invariant,
subspaces. Now we
shall
construct
two
subspaces
but whose closed sum is no longer controlled invariant. From chapter IV we have of
that
the
controlled
transfer
inv~riant subspaces are
function. Corollary
closely related
E.14 gives
that
outside
to a
the
zeros
sufficiently
(e-S+s) exactly one zero of se'~+s2+l and also one zero of se-S+s2-1. Let {tzj}je N and {t,j}.mt~ denote these zeros of respectively se-S+sZ+l and se-S+s2-1. With these zeros we
large
circle F
there
is with every
zero
of
define the following closed subspaces
(e. 20)
V 1 = s-~d~{ (/~i - A)-Ib} jell
(e.30)
V2=s--~{(vj-
A)-lb}
jetJ Now we shall show that
V1 and V2 are controlled invariant. This we shall
only do for V1, the proof for V2 is similar. By
definition
satisfies
condition
IV.12 we a)
of
have
that
thcorcm
V,
is
of
the
IV.16, furthermore
form it
(4.9). also
So
V1
satisfies
condition b). Thus it remains to prove the existence of a subsequence nj such that
inside the circle /' and 2
Izj÷k-~jl < [zj,k(zj+k+li[.
from
143
Znj bnj
S°jeaN
2
j <~,
bJ+k
I
~
j
--
1
zj+k-k 1
by (e.3).
J
Thus V1 m controlled invaxiant. Now we shall prove that Vl+V = is not controlled invariant. As a consequence of theorem IV.16, A 4 and the fact that b , j converges to
zero
circle to
if j
goes
to
infinity,
we have
that
outside
a
sufficiently large
every pole of F(s) there is one or none zero of the controlled
invaxiant s u b s p ~ e .
So if V:=VI+V 2 were to be controlled invariant,
there would exists to the poles zi; j e N
then
, for j sufficiently large, one or
no zero. However we have that to every such pole there exists a zero of
se-a+s2+l and one of se-m+sZ-1. Thus V cannot be controlled invariant.
O
CONCLUSIONS
In
this
monograph
geometric
we
have
presented
a
fundamental
treatment
of
the
theory for infinite dimensional systems in Banach spaces. We have
proved
that
closed
loop
for
closed subspaces
invariance
condition that
the
examples
of
invariant,
and the
and
the
concepts of
frequency
invariance
open
loop invariance,
are
equivalent.
subspace must be closed is really essential;
open
loop
invariant
subspaces
which
are
not
The
there closed
are
loop
same holds for frequency invariance. There is another
fundamental difference between the concepts of frequency invariance, open loop
invariance
and
the
concept
of
closed loop
invariance.
Namely
the
largest frequency invariant subspace contained in a given closed subspace K, Y~;(K), and the largest open loop invariant subspace contained
in K,
l~(K) both exist, but in general there may not exist a largest closed loop invariant subspace contained in K, ]2*(K). There can concur even stranger things. It has been shown by means of an example that it is possible that both the largest closed loop invariant and the largest frequency invariant subspace exist, but are unequal. This is very unsatisfactory, since an easy test for the solvability of the DDP in terms of the transfer functions from the input to the measurement and from the disturbance to the measurement, which
was
valid
in
infinite dimensional of for
the
systems.
(~,0J)-representation. finite
dimensional
finite
dimensional
dimensional
This test
This
concept,
systems,
is
case,
loses
it
validity
for
is essentially based on the concept although of
great
introduced use
for
by
Hautus [19] the
infinite
case. For example with this concept we proved the equivalence
between open loop and closed loop invariance, but it can also be used to obtain strong results in the theory of stabilizability, Zwart [49]. As we have seen, if the largest frequency invariant subspace, Y~(K), equals the largest closed loop invariant subspace, Y*(K), then the theorems form a complete analogy to
the
finite dimensional
V~;(K) and Y*(K) is essential.
case. So the equaLity between
For applications it is however very hard to
check whether YE(K) equals Y°(K). In chapter four we have focussed on the existence of Y~(K) for the class of discrete spectral systems on a Hilbert space. There we proved that the existence of this subspace is equivalent to a pole placement problem, where the new poles must be placed on the zeros of the system. However since the existence of Y°(K) does not imply that
145
this s u b s p ~ e
is equal to PE(K), it is again hard to show that the DDP is
solvable if this pole placement problem is solvable. From all these
results
we see that there is a gap between the nice mathematical theory and the applications.
One
could
think
of
several
ways
to
investigate
the
solvability of the DDP in practical applications. One way of attacking this problem would be to approximate the given system by
a
finite
decoupling
problem,
solution for decoupling
dimensional the
and
one, then
for to
which use
this
general system. By the
problem
it
is not
clear
one
solves
solution
the
disturbance
a s - an
approximate
non-robustness
that
this
of the
approach
disturbance
should
work.
The
non-robustness of the DDP can easily be seen from the following remark. For every positive number ¢ there exists a D c such that [ID-Dc[[<¢ and Drg~0. So by
making
only a
small change
in norm
one
can
violate the necessary
condition for the solvability of the DDP. A
better approach
to
the
general
solvability of
the DDP
would
investigate its almost version as in Trentelman [39] in the hope solution will be
more
robust. In
the
infinite dimensional
covered the special case that the generator A time systems
the condition that A
interesting examples.
However
to
that this
case
we
also
is bounded. For continuous
is bounded
if one
be
is too strong
to
include
considers discrete time systems, then
most generators will be bounded. It can easily be deduced from the results in this monograph
that for this case the geometric
analogous
finite
to
the
dimensional
case.
One
theory
can
state
is completely this
as
"the
geometric theory is a theory for bounded operators". The "almost" version of the DDP for continuous time systems is still completely open, however. So we still have
many open problems if we want
to solve the DDP for
applications. However since we axe working in an infinite dimensional state space the assumption in the DDP that very
reasonable.
In
most
the whole state is measured
applications the
output
will be
a
is not
part
of
the
state, and this brings us to the problem of DDPM or DDPMS if we also want to
guarrantee
we
have
internal
derived
stability. For
necessary
and
a
system
sufficient
defined
conditions
in
in a
Hilbert space
terms
of
system-
invariance subspaces for the solvability of the DDPH and the DDPMS with a finite dimensional compensator In
these
condition that 1;~;(K) equals 1/"(K), but the
to-be-constructed
compensator
conditions we did not this only holds for
is finite
dimensional.
results are (still) not applicable in a general Banach space.
needed
the case
Furthermore
the that these
146 Concluding we can see that at this moment it is not clear what one can do about general
disturbance
decoupling problems for
systems for
which the
equality between ~E(K) and P*(K) is unknown. The
equality between PE(K)
examples this
and
P*(K)
holds ff P~(K)
we have seen that it is possible that ~ ( K )
property
is
realisation
dependent.
This
is closed.
From
is not closed, but
motivates
the
following
(still)
open problem: under what conditions does the transfer function f(s)
have
a
realisation
(D,A,B) such that f(s)=D(s-A)-IB and ~r~(KerD) is
closed. In order to solve this one would probably need the concepts of open loop invariance, closed loop invaxiance
and frequency
general systems, for instance with an unbounded there
is reason
to
believe
that
the
following
invaxiance
for more
input operator B. However conjecture
will hold; let
(A,B) be a general systems with possibly unbounded input operator B, and let the feedback laws F be elements of some class of linear operators such that A+BF generates a Co-semigroup, subspace,
with
respect
to
our
class
then the largest closed loop invariant of
feedback
operators,
does
not
necessarily exist. So geometric theory for infinite dimensional continue to p r o d u c e s o m e
systems has brought and will
nice mathematical and system theoretic
but it will be very hard to apply these results to applications.
results~
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an application to the TW-272, Mathematisch
LIST OF ALL INVARIANCE CONCEPTS AND THEIR RELATION
Definitions: (A, B)-invariant
A(VnD(A))cV. A(VnD(A)) c V + l m B.
closed loop invaxiant
there exists a bounded F such that TA+BF(t)VcV Vt>0.
A- invariant
conditioned invariant
there exists a bounded G such that TA+Gc(t)VcV Vt>0.
controlled invariant
closed loop invariant = open loop invariant = frequency invariant for closed subspa£es V.
feedback invaxiant
there exists an A-bounded F such that
frequency
(A+BF)(VnD(A))cV. For every xoeV there exist ~(.)eD(A)(s) w(.)eU_l(s ) such that xo=(s-A)~(s)-Bw(s ).
invariant
and
generator invariant
A-invariant.
open loop invariant
For every xoeV there exists a continuous u(.) such that
the
solution
of
x(.)=Ax(.}+Bu(.);
x(O)--x o
remains in V. semigroup invariant
TA(Q-invariant.
TA(t)-invariant
Ta(t)VcV;
Vt __.O.
Relations: For a closed subspace V we have that the following concepts are equivalent:
i)
closed loop invariance.
ii) iii) iv)
frequency invariance.
open loop invariance. controlled invariance.
Furthermore
the
following two
concepts
are
also
equivalent
for
a
closed
subspace V. v)
(A, B)-invaxiance.
vi)
feedback invariance.
Note: For a closed subspace V i) always implies v), but the converse does
not hold in general.
NOTATION
A
The
system
operator,
which
is
assumed
to
generate
the
C0-semigroup T A(t ). B
The input operator,
which is assumed to be bounded and to
have finite dimensional range.sional range.
B°W)
A subspace of I m B
Bl(V) C D D(A) g
lm B n V .
which has zero intersection with V and is
of maximal dimension under this restriction, restriction. An output operator, which is assumed to be bounded. An output operator, which is assumed to be bounded. The domain of the operator A. The
disturbance
input
operator,
which
is
assumed
to
be
bounded. F
A feedback operator from the state space to the input space.
G
A feedback operator from an output space to the state space.
H2(v)
The space of Hardy functions with values in V. V is assumed
H
The state space, if this state space is a Hilbert space.
iv
The inclusion map, i.e. if V c X , then iv:V~-,X and i v ( x ) = x .
lm B
The image of the operator B.
$ .~( z )
The imaginary part of the complex number z.
K
This
to be a closed subspace of a Hilbert space.
is
arbitrary
the
feed
through
term
in
the
compensator,
or
an
closed subset of the state space. The meaning will
be clear from the context. Ker D
The kernel of the operator D.
L
The output operator from the compensator, or the distribution diagram (only in appendix E).
t2([a,b);Q) 1=_
The Laplace transform.
£(x)
The space consisting of all bounded linear operators from 2'
z(2",z)
The space consisting of all bounded linear operators from 2"
The space of all square integrable functions from [a~b) to Q.
to 2". to Z.
M
The input operator in the compensator.
155 N
The system operator for the compensator, which is assumed to generate a C0-semigroup.
(N,M,L,K) P Pv Q Q
The disturbance input space.
~e(z)
The real part of the complex number z.
S S*(lm E)
A subspace of the state space.
The compensator. A projection operator. The projection operator on V. An a r b i t r a r y operator.
The
smallest
conditioned
invariant
subspace
that
containes
ImE
YA!t)
The semigroup generated by A.
U
The controlled input space.
V
A subspace of the state space.
~;+(K)
The
largest
controlled
invariant
subspace
contained
in
the
subspace
contained
in
the
subspace
contained
in
the
closed subspace K.
vo~(K )
The
vE( )
The
largest
open
loop
invariant
closed subspace K. largest
frequency
invariant
closed subspace K.
~(A,B)(K)
Pz.{K) for the system (A,B).
X
The
The state space of the compensator. state
space
of
the
system
if
this
state
space
is
an
a r b i t r a r y Banach space.
y
Ttle output space.
Y(s)
The space consisting of all function f(.) from R to Y which are continuous on some interval of the from [r,o0).
Y-l(s)
{feY(s)[ lira sf(s) exists}
7.
The output space of the to be controlled output.
~3
All states
8-~00
that
can
be
reached
at
frequency
s,
under
condition that the state trajectory has to stay in V. Zs for s sufficiently large. 0
The * will denote the Hilbert space dual of an operator. The ' will denote the Banach space dual.
lia, b]
The indicator function of the interval In,b].
the
