orlov, yuri - "on identifiability of linear infinite-demensional systems"

8/17/2019 Orlov, Yuri - "On Identifiability of Linear Infinite-Demensional Systems"

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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

[email protected]

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



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.



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.



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.



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,



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) .

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orlov, yuri - "on identifiability of linear infinite-demensional systems"

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