Identifiability of time-dependent linear systems

Identifiability

of lime-Dependent

Linear Systems

STAVROS N. BUSENBERG Depurtment of Muthemutics, Hurvey Mudd College, Cluremont, Culiforniu 9171 I SUZANNE

M. LENHART*

Department of Muthemutlcs, Knoxville, Tennessee 37996

University of Tennessee,

AND

CURTIS C. TRAVIS He&h und Sufefetv Division, Ouk Ridge Nutionul Lahorutoty, Ouk Ridge* Tennessee

37831

Received

1985; revised I9 April I987

18 December

ABSTRACT

dent

This paper establishes a direct method for determining the identifiability of time-depenlinear systems. Contrary to most other results on identifiability, we do not require

system controllability. We apply this method to a specific example time-dependent system may be identifiable on a finite interval without on a longer time interval.

1.

and show that a being identifiable

INTRODUCTION

In this note we derive a criterion for the structural identifiability of linear time-dependent systems. The identifiability problem consists of determining if the unknown of a system can be uniquely identified from observations of the results of input-output experiments. For time-independent systems, this problem has received considerable attention in recent years [l, 2, 4, 5, g-211. Very little attention has been given to identifiability of time-dependent systems [3, 71. One reason for the lack of attention is that the Laplace transform method of identifiability introduced by Bellman and Astrom [l] does not generalize to time-dependent equations. Our aim is to extend a

*Address all coz$%&nce to Suzanne M. Lenhart, University of Tennessee, Knoxville, TN 37996.

MATHEMATICAL

BIOSCIENCES

87:63-71

QElsevier Science Publishing Co., Inc., 1987 52 Vanderbilt Ave., New York, NY 10017

Department

of Mathematics,

63

(1987)

0025-5564/87,‘$03.50

64 STAVROS

N. BUSENBERG,

SUZANNE

M. LENHART

AND CURTIS

recently obtained direct method for determining the identifiability invariant linear systems [19] to time-dependent linear systems. 2.

IDENTIFIABILITY Consider

OF TIME-DEPENDENT

the dynamical

LINEAR

C. TRAVIS

of time-

SYSTEMS

system

dx(t)/dr=A(t)x(r)+B(t)u(t) Y(l) =C(t)x(t)

(2.1)

with x(t) E Iw”, and (A,B,C) E H = {(A,B,C): A(.),B(.),C(.) are matrixvalued functions with dimensions n X n , n X k, m X n, which are piecewise continuous on [0, CQ)}.We denote the system (2.1) by S(A,B,C). We say that two systems S,(A, ,B,,C,) and &(A, ,B,,C,) are equivalent if and only if, given an initial state xl0 and a starting time s > 0 for system S,, there exists an initial state x2() for system S, such that, for all u E C([O, co),R”) Y2(f)

=

=Yl(t) C*(tb2( ~,S,Xzo)

=C,(~)x,(t,s,x,,),

and the same condition holds with the indices 1 and 2 interchanged. Here, x, (t, s,x& denotes the solution of the differential equation of S, with initial state x,(s,s,x,~) = x,~. In many applications, a priori knowledge of the structure and parameters of a system provide a substantial reduction in the collection of systems eligible for modelling a given biological situation. If we denote by K the subfamily of all systems of the form (2.1) which satisfy these a priori conditions, then we are interested in the parameter identifiability of the subfamily K. Let K c H and consider the class S, of all systems S(A,B,C) with (A,B,C) E K. We say that the class S, is (globally) identifiable if and only if any two systems in S, that are equivalent are also identical. Another concept of equivalence and, consequently, of identifiability requires that the initial states xl0 and xZO be the zero vector. This notion was 0 introduced by Bellman and Astrom [l], and we distinguish it from our definitions by calling it zero-state equivalence and zero-state identifiability. (See [16, 211 for a discussion of alternate definitions of identifiability used in the literature.) The identifiability problem consists of finding criteria that characterize those classes S, which are identifiable. To this end we recall the following notion. Two systems S, and S, in H are algebraically equivalent if and only if there exists a piecewise differentiable matrix-valued function P: [0, co) + L(R “, II3“) such that P(t) is invertible for all t E [0,00) and A,(t)

= P( t)A2( t)P-‘(

t) +P’( r)P-‘( t),

(2.2)

IDENTIFIABILITY

OF TIME-DEPENDENT

where P’(t) is the derivative

LINEAR

65

SYSTEMS

of P with respect to time, assumed to exist, B,(t)

=P(t)B,(t),

(2.3)

C,(I)

= C,( t)P-‘( t).

(2.4)

This notion of algebraic equivalence is clearly reflexive. Finally, recall that a system S is observable if and only if, with II = 0, C(t)x( t, s,xO) = C(t)x(t, s,x,) for t > s implies x0 = xi. Note that this notion of system observability is more general than the standard notion of an observable system where knowledge of C(t)x(t, s,xO) for t > s uniquely determines x0. In fact, any linear system which is observable in the standard sense must also be observable in the more general sense of the above definition, but the converse need not be true. Consequently, our results, which require system observability, also hold when the standard definition of this notion is imposed. The following theorem on identifiability requires observability but not controllability of the systems S,. This differs from results in the system theory literature which require both controllability and observability for zero-state equivalence. The reason why we can dispense with the requirement of controllability and can assume a more general notion of observability is that we have departed from the requirement of zero-state equivalence and identifiability. This departure leads to simpler hypotheses for our main result but has the disadvantage that in the identification problem both the impulse response and the initial state of the system need to be determined rather than just the impulse response. This can present experimental difficulties in dealing with some compartmental systems. TIfEOREM

.?.I

Assume that the dimension of the desired system representation is part of the a priori knowledge. Suppose that every system in the set S, is observable. Then S, is identifiable if and only if any two systems of S, that are algebraical[v equivalent are also identical. Proof: First we show that, if S, is identifiable, then any two algebraically equivalent systems are identical. Let Q, (t, s) denote the fundamental matrix of S, with u(t) = 0, and note that @, satisfies the initial value problem a~,(t,s)/at=A,(t)~,(t,s),

Q,(S,S)

= I,

where I is the identity on Rnx” (See [6]). It is also known that @(t, s)-l = cP(s, t). So, if systems S, and S, are algebraically equivalent, we have

66 STAVROS N. BUSENBERG, SUZANNE M. LENHART AND CURTIS C. TRAVIS and &P(t)@,(r,s)P-t(s)

=P(t)A,(t)Q,(t,s)P-‘(s)+P’(t)Q(t,s)P-’(s) =P(t)A,(t)P-‘(t)[P(t)Q&,s)P-‘(s)] +P’(t)P~‘(l)[P(t)Q,,(t,s)P-‘(s)]

with P(t)~Dz(t,s)P-l(s)l,_, = I. So, QI(t, s) and P(r)Qz(t, the same initial value problem, and we must have $(t,s)

=cl(~>~l(~~~>xlo

satisfy

=P(t)Qz(t,s)P-l(s).

Since S, and S, are algebraically of equations: Yl(r)

s)P-‘(s)

+

equivalent,

we have the following sequence

dr J‘Cl(r>~,(r,r)Bl(r)u(r)

=C,(t)P-l(r)P(r)B,;r,s)P~‘(s)x,, +

fCZ( t)P-‘( /s

r)P( r)Q,(

r, r)P-‘(

r)P( r)B2( r)u( r) dr

(2.5)

=Cz(r)~,(r,s)[P-1(s)x,ol+~fC2(r)~2(frr)B2(r)u(r)dr =c,(r)x,(r,s,P-‘(s)x,,)

=y2;fJ.

Since algebraic equivalence is reflective, we can interchange the subscripts 1 and 2 in Equation (2.5). That is, S, and S, are equivalent systems. Since by hypothesis S, is identifiable, S, and S, must be identical, and the first half of the theorem is established. Now, suppose that S, has the property that any two of its systems which are algebraically equivalent are identical. We need to show that S, is identifiable. So, let S, and S, belong to S, and suppose that they are equivalent systems. Then, given xIO and a starting time s and taking u(r) = 0, there exists xzO such that Cl(~)~l(~,~)xlo=C2(~)~‘2(~,~)xZO~

The map (s,xlo)

r E[s,m).

(2.6)

above is single-valued because, if --) x2o defined the fact that S, is observable implies C,(0~&.%, = C,(4@2(t9.9x,o, x20 = x30. So, there exists a function P: [0, co) x R” -+ Iw” such that xzo = A similar argument using (2.6) shows P is one-to-one. The fact that, P(~h. for each s E [0, co), P(s) is a linear map from 88” to R” follows directly from Equation (2.6) and the single-valuedness of the map (s,xIo) + xzo.

IDENTIFIABILITY

OF TIME-DEPENDENT

From Equation

67

SYSTEMS

(2.6) we now get for any xi0 E R” = C,(~)%(~,~)P(~)X,,

= C,(~)%(S~S)XlO

Cits)%

LINEAR

since Qi(s, s) = $(s, s) = I. Consequently, hold for any xi0 belonging to R ‘,

= C,(~)P(~)X,,,

since the above equation

must

Cl(S)=C,(s)fTs). Substituting

(2.7) that P-‘(s)

Equation (2.7) in (2.6) and noting we obtain

exists because

P(s) is one-to-one,

Since the above relationship system S, implies

must be true for any t, the observability

holds for all xi0 E R”. So, $(s,

t)P-‘(t)Q2(t,~)P(s)

of the

= I, and we have

(2.8) Now, from Equation

(2.8) we immediately P(r)

obtain

= Q2( r,s)P(s)Ql(s,

r)

and fixing s and letting r vary, we see that P is continuous differentiable function of r. Differentiating Equation (2.8) with respect to r, we get

and substituting

Now,

we

for $(r,

multiply

on

s) from Equation

the

right

of

and a piecewise

(2.8), we have

both

sides

of

this

equality

by

68 STAVROS N. BUSENBERG,

P(s)@~ (s, t)P-t(t)

SUZANNE M. LENHART

AND CURTIS C. TRAVIS

to get A*(t)

=P(t)A,(t)P-‘(~)+P’(r)P’(t).

Finally, since S, and S, are equivalent C([O, co),R”) we have

(2.9)

systems, for any arbitrary

u E

+ ‘C,(t)Q2(t,r)B2(r)u(r)dr. =C?.(t)(P*(t,S)P(s)xlo JJ

Since this must be true for any u, it must be so when u is identically which implies

Subtracting

zero,

this equality from the previous one, we get

/s

‘C,(l>~~(t,r>Bl(r>u(r) dr =

‘C1( t)P-‘(

IT

t)P( t)@t( t, r)P-‘(

r)B2( r)u( r) dr

must hold for all u E C([O, co),W’). Consequently,

and by the observability

of the system S,

B,(r) =P-‘(r)&(r),

r E[O,c9).

From this relation and from Equations (2.7) and (2.9) we conclude that S, and S, are algebraically equivalent; hence, they must be identical. This concludes the proof of the theorem. n 3.

APPLICATION

Consider the pharmacokinetic model shown in Figure 1. Input is into the first compartment, and the product of the content of the second compartment with an unknown nonvanishing function c(t) is observed. We assume that c( t)X,, (t) # 0 and that X,,(t) is a known fixed function. Note h,,, Xi2, and c need not be nonnegative functions although they usually are in

IDENTIFIABILITY

OF TIME-DEPENDENT

input

LINEAR

x2l

U

x1

<

,'

to compartmental

X’( t) =

observed l

model

I [I

h,(t)

1

1 0

X(t)+

-~uo)--cdt)

h, ( t)

u(t)

(3.1)

Y(t) = [o,c(t>lx(t>. regarding

-u(t)

K,= i[

a(t)

>

systems. The model can be given mathemati-

-b(t)

A priori knowledge

xo2

x2

\

FIG. 1. Pharmacokinetic

application cally as

69

SYSTEMS

u(t), II

the system structure implies that

b(r) -b(t)-e(t)

b(t)

with e(t)

arbitrary

known

1’

K,= {[1,01’} ad Kc= {[o,f(t)llf(t) arbitrary). Since c(t)a(t) # 0, it is easily checked that every system in K = {(A,B,C): A E K,,B E K,,C E Kc} is observable. Suppose now that there exists a nonsingular, differentiable matrix P(t) such that P(t)B E K, and CP-‘(t) E Kc. Then B = P( t)B, and together with CP-‘( t) E Kc, it follows that P(t) must have the form P(t)

P-‘(t) Thus PAP-’

+ P’P-’

=

= 1 p,*(t) [ 0 p,*(t)

l [0

I

- 4,(W,_,‘(t) P,;;‘(t)

1 ’

70 STAVROS

N. BUSENBERG,

Restrictions

on class K, imply

SUZANNE

M. LENHART

AND CURTIS

C. TRAVIS

b, - PnL = P22h. Since X,,(t)

> 0, 1 = P,, + Pz2

(3.2)

- P& = Pi2.

(3.3)

and hence that

We also have

P,,X,, + A,, - P;2 P$ = P,, &‘A,, -

+[A12- Pl,h -

KG1 + plipii’.

P,,~o,I

by Pz2 gives

Multiplying

P,,Pzh

+ Pnh

= P,,h

Using Equations (I-

P,‘2 P,-,‘&,

P,,)P,,%,

- Pi2

- P:,h,

+x,2

- P,,h

- Pnhl,

+ Ph.

(3.2) and (3.3), we obtain +(I-

P,,)&

= PA1

- Pl$&, + A,, - P,,L

- PI,%,

and then O=Pr,(A,,). Notice that if 0+%*(t) (hence,

P,,(t)=Oand

for 0
P,,(t)=lonO
for t > t*,

then the system (3.1) is identifiable on [0, t*] but not identifiable on [t*, co). The system is not identifiable on any interval [a,/?], with p > t*. Thus it is possible, in fact, for a linear time-dependent system to be identifiable on a finite interval without being identifiable on the whole positive real line. REFERENCES 1 2

R. Bellman

and K. J. Astrom,

On Structural

Identifiability,

M&h.

Biosci. 7:329-339

(1970). C. Cobelli, A. Lepscky, and R. Romanin Jacur, Comments on “On the relationship between structural identifiability and the controllability, observability properties,” IEEE Truns. Automat. Confrol AC-23:965 (1978).

IDENTIFIABILITY C. Cobelli,

OF TIME-DEPENDENT A. Lepschy,

and

G. Romanin

LINEAR

SYSTEMS

Jacur,

Identifiability

71 of Time

Invariant

4

Compartmental Models of Biological Processes, preprint. C. Cobelli and G. Romanin Jacur, On the structural

5

compartmental systems in a general input-output configuration, Math. Biosci. 30:139-151 (1976). C. Cobelli and J. J. DiStefano III, Parameter and structural identifiability concepts

6

and ambiguities: A critical review and analysis, Amer. J. Physiol. 239:R7-R24 (1980). E. A. Coddington and N. Levinson, Theory of Ordinary Differential Equations, McGraw-Hill,

8 9

10 11

identifiability

of biological

New York, 1955.

J. Delforge, The identifiability of a non-stationary linear system, Int. J. s?/s.SC?., 8:163-167 (1977). J. Delforge, The problem of structural identifiability of a linear compartmental system: Solved or not? Math. Biosci. 36:119-125 (1977). J. J. DiStefano III, On the relationship between structural identifiability and the controllability, observability properties, IEEE Trans. Automat. Control AC-22~652 (1977). J. J. DiStefano, III, Author’s reply, IEEE Trans. Automat. Control AC-23:966 (1978). J. J. DiStefano III and C. Cobelli, On parameter and structural identifiability: Nonunique observability/reconstructibility for identifiable systems, other ambiguities and new definitions, IEEE Trans. Automat. Control AC-25:830 (1980).

12

K. Glover and J. C. Willems, Parameterizations of linear dynamical systems: cal forms and identifiability, IEEE Trans. Automat. Control AC-19:640-646

13

K. R. Godfrey and J. J. DiStefano III, Identifiability of model Identification and System Parameter Estimation, Proc. 7th IFAC/IFORS Pergamon, (United Kingdom), 1985.

14

17

M. S. Grewal and K. Glover, Identifiability of linear and nonlinear dynamical systems, IEEE Trans. Automat. Control AC-21:833-837 (1976). J. A. Jacquez, Further comments on “On the relationships between structural identifiability and the controllability, observability properties,” IEEE Trans. Automat. Control AC-23:966 (1978). Y. Lecourtier and E. Walter, Comments on “On parameter and structural identifiability: Nonunique observability, reconstructibility for identifiable systems, other ambiguities, and new definitions,” IEEE Trans. Automat. Control AC-26:800-801 (1981). J. G. Reed. Structural identifiability in linear time-invariant systems, IEEE Trans.

18

Automat. Control AC-22~242-246 (1977). A. Thowsen, Identifiability of dynamical

15

16

19 20 21

systems,

Internat.

J. Sys.

Canoni(1974).

parameters, in Symp., York,

Sci. 9:163-169

(1978). C. C. Travis and G. Haddock, On Structural Identification, Math. Biosci. 56:157-173 (1981). S. Vadja, Structural equivalence of linear systems and compartmental models, Math. Biosci. 55:39-64 (1981). E. Walter, Identifiability of State Space Models, Springer-Verlag, New York, 1982.