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ON IDENTIFIABILITY OF LINEAR
INFINITE DIMENSIONAL SYSTEMS
Yury Orlov
Electr. Dept. CICESE Research Center
Km. 107 Carretera Tijuana-Ensenada Ensenada B.C. Mexico 22860
Abstract
Identifiability analysis is developed for linear dynamic systems evolv-
ing in a Hilbert space. Finite-dimensional sensing and actuation are
assumed to be only available. Identifiability conditions for the trans-
fer function of such a system is constructively addressed in terms of
sufficiently nonsmooth controlled inputs . The introduced notion of a
sufficiently nonsmooth input does not relate to a system and it can
therefore be verified independently of any particular underlying system.
Keywords: Identifiability, Hilbert space, Markov parameters, sufficiently rich input.
Introduction
A standard approach to identifying a linear system implies that the
structure of the system is deduced by using physical laws and the prob-
lem is in finding the values of parameters in the state equation. The
ability to ensure this objective is typically referred to as parameter iden-
tifiability. For complex systems, however, it may not be possible to
model all of the system and a black box approach should be brought
into play. Here knowledge of the input-output map comes from con-
trolled experiments. The questions then arise as to if the input-output
map and the transfer function of the system have a one-to-one relation
and in which sense a sta te space model, if any, is unique.
Many aspects of the identifiability of linear finite-dimensional systems
are now well-understood. In this regard, we refer to [I],
[ 5 ]
and [7] to
name a few monographs.
Recently [8], [9], he identifiability analysis was proposed in an infinite-
dimensional setting for linear time-delay systems with delayed states,
control inputs and measured outputs, all with a finite number of lumped
delays. It was demonstrated that the transfer function of such
a
system
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172
S Y S T E M M O D E L IN G A N D O P T I M I ZA T I O N
can be re-constructed on-line whenever a sufficiently nonsmooth input
signal is applied to the system. The present work extends this result
to linear dynamic systems evolving in a Hilbert space. The transfer
function identifiability conditions for these systems are also addressed in
terms of sufficiently nonsmooth input signals which are constructively
introduced in the Hilbert space. The notion of a sufficiently nonsmooth
input does not relate to a system and it can therefore be verified inde-
pendently of any particular underlying system.
Similar to linear finite-dimensional systems, the parameter identifia-
bility in a Hilbert space requires the system with unknown parameters to
be specified in a form such that if confined to this form all the unknown
parameters are uniquely determined by the transfer function. In con-
trast to [2],[3], and
[lo]
where the parameter identifiability is established
for linear distributed parameter systems with sensing and actuation dis-
tributed over the entire state space, the present development deals with
a practical situation where finite-dimensional sensing and actuation are
only available.
1. Basic Definitions
We shall study linear infinite-dimensional systems
i A x + Bu, x(0) xO (1)
y Cx, (2)
defined in a Hilbert space H, where E H is the state, x0 is the initial
condition, u E Rm is the control input, y E RP is the measured output,
A is an infinitesimal operator with a dense domain V( A), B is the input
operator, C is the measurement operator. All relevant background ma-
terials on infinite-dimensional dynamic systems in a Hilbert space can
be found, e.g., in [4].
The following assumptions are made throughout:
1 A generates an analytical semigroup S A ( t )and has compact resol-
vent
;
2 B E C(Rm,H ) and C E C(H,Rp);
3 X(O) xo E n:==,D ( A ~ ) ;
4 R(B) E
or=,
(An).
Hereafter, the notation is fairly standard. The symbol C(U,
H)
stands
for the set of linear bounded operators from a Hilbert space U to H ;
V(A) is for the domain of the operator
A; R B )
denotes the range of
the operator B.
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On Identifiability of L inear Infinite-D imensiona l Systems
173
T h e above assum ptions are m ade for technical reasons. I t is well-
known that under these assumptions, the Hilbert space-valued dynamic
system ( I ) , dr iven by a local ly in tegrable inp ut u ( t ) , has a unique stron g
solut ion x ( t ) , globally defined for all t
>
0.
T h e s p e c t r u m a ( A )
{Xn)r=lf the operator A is discrete,
A, -
-cm as
n
-+ cm , an d th e
sol uti on of (1) can be represented in the form of the Fourier series
w ritt en in term s of th e eigenvectors r,,
n
1 , . . of the operator A a nd
the inner product
+ ., >
in the Hilber t space
H.
Furthermore, th is
solut ion is regular enough in the sense that x( t)
E
D (A n) fo r a l l
t 0.
T h e identif iability concept is based on th e com parison of system l ) ,
(2) and its reference model
defined on a Hilbert space H with the initial condition
2
and oper-
a tors
A, B, C,
subst i tu ted in
l ) ,
2) for xO and A,
B,
C , respectively.
Certainly, Assu mp tions 1-4 rem ain in force for the reference m odel
4),
(5) .
T he transfer function T(X)
C(XI
A)-'B of sys tem
(I) ,
(2) is com-
pletely determined by means of the Markov parameters CAn-'B,
n
1 , 2 , . . . through expanding in to th e Laurent ser ies
and the identifiability of the transfer function is introduced as follows.
D EF IN IT IO N The transfer func tion of (I) , (2), or equivalently, the
M arkov par am ete rs of system ( I) , (2) are sa id to be identifiable on
n:==,
D ( A n )
28
there exists a locally integrable input function u(t) to be
suficiently rich fo r the system in the sense that the identity y ( t) (t )
implies that
c A n - '
B
(?An- B,
n
1 , 2 ,
(6)
(and consequently T(X)
T (A ) ) ,
regardless of a choice of the initial
conditions x0 E
nrZ1
( A n ) , i?O
E
n;?, D (A n ). In that case the identi-
fiabil ity is sai d to be enforced b y the input u ( t ) .
T he identif iability of th e system itself ( rat he r th an its transfer func-
tion) is addressed in a similar manne r.
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174 S Y S T EM M O D EL IN G AN D O PT IMIZ AT IO N
DEFINITION Sys t em (I), ( 2 ) is said to be identifiable o n
nrZl
( A n )
iff there e xists a locally integrable in pu t functio n u ( t ) such that the id en-
t i ty y ( t ) G ( t ) implies that
regardless of a choice of the initial conditions x
E
nnY1o D ( A n ) , i E
nrYl
n ) .
For later use, we also define sufficiently nonsmooth inputs, which form
an appropriate subset of sufficiently rich inputs. Indeed, while being
applied to a finite-dimensional system, a nonsmooth input has enough
frequencies in the corresponding Fourier series representation and hence
it turns out to be sufficiently rich in the conventional sense
[6].
Let u ( t ) C , = l ui (t )e ibe a piece-wise smooth input, expanded in the
basis vectors ei
E R m ,
i 1 , .
. .
m, and let D i be the set of discontinuity
points
t 2
0 of u i ( t ) .
DEFINITION The input u ( t ) I&ui ( t )e i i s sa id to be su f i c ien t l y
disco ntinuo us iff for any i 1 , . .
,m
there exists
ti
E Di such that
t i
$
j or all
j
i
DEFINITION T h e i np u t u ( t ) C z l u i ( t ) e i is said t o be s u f ic i en t l y
nonsmooth of class C f f there exists l- th order derivative of the in pu t
u ( t ) and u( )t ) i s su f i c ien t l y d i scontinuous .
It is worth noticing that the notion of a sufficiently nonsmooth (dis-
continuous) input does not relate to a system and it can therefore be
verified independently of any particular underlying system.
2.
Iden t ifiability Analysis
Once the infinite-dimensional system
I ) ,
( 2 ) is enforced by a suffi-
ciently nonsmooth input, its transfer function is unambiguously deter-
mined by the input-output map. The transfer function identifiability in
that case is guaranteed by the following result.
THEOREMConsider a Hilber t space-valued sys tem ( I ) , ( 2 ) wi th the
assumpt ions above. Th en the Markov parameters C A n - ' B , n 1 , 2 , .
. .
of (I), ( 2 ) are identifiable and the ir identifiability can be enforced by a n
arbitrary suficiently nonsmooth (particularly, suf iciently discontinuous)
input ~ ( t ) C z l i ( t ) e i .
Proof: For certainty, we assume that the input u ( t ) is sufficiently
discontinuous. The general proof in the case where u ( t ) is sufficiently
nonsmooth is nearly the same and it is therefore omitted.
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On Identifiability
of
Linear Infinite-Dimensional Systems 175
According to Definition 1, we need to prove that the output identity
implies the equivalence (6) of the Markov parameters.
By differentiating (8) along the solutions of (1) and 4),we obtain
that
CAx(t)+ C B C z l ~ i ( t ) e i CAk(t)+ C B C ~ ~ U ~ ( ~ ) ~ ~ .
9)
It follows that
CB
CB
because otherwise identity (9) could not remain true at the discontinuity
instants ti, 1 , . .
,
m. Indeed, in spite of the discontinuous behavior
of the input u(t ) , the solution (3) of the differential equation (1) is
continuous for all t > 0. Thus, by taking into account tha t u(t ) is
sufficiently discontinuous (see Definition 3) , [CB cB]ui(t)ei is the
only term in (9), discontinuous at t ti. Hence, [CB CB]ei 0 for
each 1 , .
.
,m, thereby yielding (10).
Now (9) subject to (10) is simplified to
and differentiating (11) along the solutions of (1) and (4) , we arrive at
Following the same line of reasoning as before, we conclude from (12)
that
CAB CAB. (13)
Finally, the required equivalence (6) of the Markov parameters for
n
> 3 is obtained by iterating on the differentiation. Theorem 5 is thus
proven.
In a fundamental term, Theorem
5
says that the input-output map
(I ) , 2) and Markov parameters of the Hilbert space-valued system have
a one-to-one relation. In general, the identifiability of the Markov pa-
rameters does not imply the identifiability of the system. Indeed, let
Q
E
L(H) be such that D(A) is invariant under Q and Q-
E
L(H) .
Then the system
has the same Markov parameters CAnB, n 1,2,
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176
S Y S T EM MO D EL IN G AN D O PT IMIZ AT IO N
Thus, in analogy to the finite-dimensional case, the identifiability of
the infinite-dimensional system is guaranteed if it is in a canonical form
such that if confined to this form the operators A, B,C are uniquely
determined by the Markov parameters. The major challenge will be an
explicit definition of the canonical form of a linear Hilbert space-valued
system
( I ) ,
(2) .
References
[I] B. As trom , K .J . an d W it ten ma rk. Adaptive Control. Addison-Wesley, Readin g,
MA, 1989.
[2]
J.
Bentsm an a nd Y. Orlov. Reference adapt ive control of spat ia l ly varying
distr ibuted p aram eter systems of parabol ic an d hyperbol ic types. Inte rna t .
J.
Adaptive Co ntrol and Sig nal Processing, 15:679-696, 2001.
[3] M. Bolm, M.A. Demet r iou , S . Re ich , and
I.G.
Rosen.
Model reference adap-
t ive contro l of d i s t r ibuted pa ram ete r sys tems. SIA M Journa l on Cont ro l and
Op timiz ation , 35:678-713, 1997.
[4] R. F. Cur t a in a nd H.
J.
Zwart . An Introduct ion to Infini te-Dimensional Linear
System s Theory. Springer-V erlag, New Yo rk, 1995.
[5] Y. D. Landau. Adaptive Control
-
The Model Reference Approach. Marcel
Dekker, New York, 1979.
[6] R.
K .
Miller an d A. N. Michel . An invariance theorem with appl icat ions to
adapt ive control , IEEE Tra ns. Au tom at. C on tr., 35:744-748, 1990.
[7] K.
S.
Naren dra an d A. Annaswamy. Stable Adaptive Systems. Prent ice Hal l,
Englewood Cliffs, NJ, 1989.
[8] Y. Orlov , L. Belkoura, J .P . Richard , and M . Dam brine . O n identi f iabi l i ty of
linear t ime-delay sys tem s. I E E E Trans. Autom at. Co ntr. , 47:1319-1323, 2002.
[9] Y. Orlov, L. Belkoura , J .P . Richard , an d M . Dambrine . Adapt ive ident if ica t ion
of l inear t ime-delay systems. I nter nat .
J.
Robust and Nonlinear Control, 13:857-
872 , 20023.
[ lo] Y. Orlov an d J . Bentsman. Adapt ive d i s t r ibuted p a ram ete r sys tems identi fica -
tion with enforceable identifiabili ty conditions and reduced spatial differentia-
tion.
IEEE
Tran s. Au tom at. Co ntr ., 45:203-216, 2000.