Structural identifiability of linear mamillary compartmental systems

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1 entrum voor Wiskunde en Informatica REPORTRAPPORT Structural identifiability of linear mamillary compartmental systems JM van den Hof Department of Operations Reasearch, Statistics, and System Theory S-R

2 Report S-R952 ISSN WI PO ox G Amsterdam The Netherlands WI is the National Research Institute for Mathematics and omputer Science WI is part of the Stichting Mathematisch entrum (SM), the Dutch foundation for promotion of mathematics and computer science and their applications SM is sponsored by the Netherlands Organization for Scientific Research (NWO) WI is a member of ERIM, the European Research onsortium for Informatics and Mathematics opyright Stichting Mathematisch entrum PO ox 9479, 9 G Amsterdam (NL) Kruislaan 43, 98 SJ Amsterdam (NL) Telephone Telefax

3 Structural Identiability of Linear Mamillary ompartmental Systems JM van den Hof WI, PO ox 9479, 9 G Amsterdam, The Netherlands Abstract In biology and mathematics compartmental systems are frequently used System identication of systems based on physical laws often involves parameter estimation efore parameter estimation can take place, we have to examine whether the parameters are structurally identiable In this paper we propose a test for the structural identiability of linear mamillary compartmental systems The method is based on the similarity transformation approach AMS Subject lassication (99): 933, 935 Keywords and Phrases: system identication, compartmental systems, structural identiability, positive linear systems, realization Note: This paper will appear in the Proceedings of the Third European ontrol onference, E95, Rome, September, 995 Introduction In this paper we shall investigate the structural identiability of a certain class of compartmental systems, the so-called mamillary systems The class of compartmental systems is a frequently used class of models in biology and mathematics Such a system consists of several compartments with more or less homogeneous concentrations of material The compartments interact by processes of transportation and diusion Linear compartmental systems consist of inputs, states, and outputs, which are positive, so these systems are in system theory called positive linear systems In many models we can use some prior knowledge on the system parameters, derived from physical laws and, in biological systems, from anatomical structure, biochemistry, and physiology We can incorporate this structure in a system and obtain a class of structured linear dynamic systems efore estimating the parameters, we have to examine whether the parameters are structurally identiable, ie, whether we can in principle determine the parameters uniquely from the data In much of the literature on identiability only local identiability is treated, see [9, 8] There are several methods to test local identiability, but only a few can test global identiability These include the Laplace transform approach, described in for example [, 8,, 5], the Taylor series expansion approach, in for example [8, 5], and the exhaustive modeling or similarity transformation approach, see for example [3, 7, 8,, 4] The last two approaches can also be applied, in an adapted version, to nonlinear systems, see [4, 9] In this paper we mean global structural identiability, when we speak about structural identiability The outline of this paper is as follows In section 2 the problem is formulated and the theorems we need are given In section 3 the results for a class of mamillary systems are given

4 2 2 Definitions and problem formulation In this section the problem is posed We shall use the notation and denitions given in [7, section 3], but we will consider the continuous-time case, which can easily be derived from the discrete-time version For linear dynamic systems of the form _x(t) = Ax(t) + u(t); x(t ) = x ; y(t) = x(t) + Du(t); (2) with A 2 R nn, 2 R nm, 2 R kn, D 2 R km, the impulse response function W is dened by W (t) = e At ; t > ; and will be extended at t = to W () = D It can completely be characterized by the Markov parameters, dened as follows: M() = D; M(j) = dj? dt j? W (t)j t= = A j? ; j = ; 2; : : : : We can view the Markov parameters M(); M(); M(2); : : : as a function M : NI! R km The set of Markov parameters M : NI! R km will be denoted by M(k; m) Let LP(n; m; k) = R nn R nm R kn R km consist of the elements (A; ; ; D), ie the system parameters, as dened above Dene the map that associates the system parameters of a linear dynamic system with the Markov parameters g : LP(n; m; k)! M(k; m) by g((a; ; ; D))(j) = D; j = ; A j? ; j > : (22) Structured linear dynamic systems are linear dynamic systems in which the system parameters A,,, and D are polynomial maps, dened on a parameter set P R r In this paper the initial condition x is assumed to be zero or known In practice x is often (partially) unknown This problem is much more complicated, and is treated in [2], together with structural identiability from the input-output observations A subclass of the linear dynamic systems is formed by the positive linear systems Positive linear systems are linear dynamic systems in which the state, input, and output space are X = R n +, U = R m +, Y = R k +, respectively A special class of structured positive linear dynamic systems is formed by the linear compartmental systems For a general text on this class of systems, see [] Linear compartmental systems are positive linear systems of the form (2), for which conservation of mass holds This corresponds with the following requirements on the matrices of continuous-time linear systems all elements of,, and D are nonnegative; for A = (a ij ) i;j=;:::;n, we have a ij ; i; j 2 f; : : : ; ng; i 6= j; a ii? nx a ji ; or nx j=;j6=i j= a ji ; i 2 f; : : : ; ng: The model studied in this paper belongs to this class of systems The problem in this paper is whether the parameters of a structured linear system are structurally identiable For the denition of this concept we refer to [7, section 4] Intuitively, structural

5 3 identiability is whether we can uniquely determine the parameter values from the observations of inputs and outputs Structural identiability has been introduced by ellman and Astrom in [] Since then structural identiability of linear and nonlinear structured systems has been studied by many authors The similarity approach for the test of structural identiability is based on realization theory For time-invariant nite-dimensional linear systems the realization problem has been solved Hence equivalent conditions for structural identiability may be formulated For other classes of dynamic systems, such as positive linear systems, the realization problem is still open, so conditions for structural identiability may not be known For the realization of positive linear systems, see [2, 22] Theorem 2 onsider the structured linear dynamic system _x(t) = A(p)x(t) + (p)u(t); x(t ) = x ; y(t) = (p)x(t) + D(p)u(t); with parametrization (P; f), f : P! SLP (n; m; k), p 7! (A(p); (p); (p); D(p)) a This system is structurally minimal if and only if it is structurally reachable and structurally observable b Assume that the system is structurally minimal Then this parametrization is structurally identi- able from the Markov parameters if and only if for all p; q 2 P outside an algebraic set and T 2 R nn, nonsingular, the equations A(p) = T A(q)T? ; (p) = T (q); (p) = (q)t? ; D(p) = D(q); (23) imply that p = q Under the assumption stated and for all q 2 P outside an algebraic set, the system of equations (23) for the pair (p; T ) 2 P R nn ; with T nonsingular, has the unique solution (p; T ) = (q; I) Proof This result follows directly from the main result of realization theory for time-invariant nite-dimensional linear systems For references see [2, 2, 3, 6] Remark 22 A structured linear dynamic system is structurally reachable (structurally observable) if it is reachable (observable) for all p 2 P outside an algebraic set To obtain equations that are not too complex, we will rewrite condition (23) as follows: A(p)T = T A(q); (p) = T (q); (p)t = (q); D(p) = D(q): (24) Then we obtain polynomial equations that are linear in the elements of T, which are easier to handle For positive linear systems, ie, with input, state, and output positive, the conditions given in Theorem 2 are only sucient, not necessary Proposition 23 onsider the structured positive linear system _x(t) = A(p)x(t) + (p)u(t); x(t ) = x ; y(t) = (p)x(t) + D(p)u(t); with parametrization (P; f), f : P! SLP (n; m; k) The parametrization (P; f) is structurally identiable from the Markov parameters if the following conditions are satised: this system is structurally reachable and structurally observable in the sense of ordinary linear dynamic systems; 2 ondition (23) implies p = q for all p; q 2 P, outside an algebraic set, and T 2 R nn, nonsingular Proof If a linear dynamic system is structurally minimal as an ordinary linear system, it is denitely minimal as a positive linear system 2

6 4 3 A general mamillary system In this section we shall derive results on the structural identiability of a class of compartmental systems, in which there is a central compartment and only exchange between this central compartment and the other compartments In the literature this kind of system is called a mamillary system, and the central compartment the mother compartment In [5] also results on identiability of mamillary systems are given, but these only concern local identiability The reason that the system in this section is globally identiable, is that there is some prior knowledge about some parameter values We will study the following mamillary system, in which there is an input u in the mother compartment and we can take measurements in the mother compartment, as shown in Figure There is excretion to the environment from only one non-mother compartment, say, without loss of generality, from compartment x xn?2 hn?2 n?2 xj j hn? xn? hj n? input u y = xn h x exchange 2 h 2 with environment x 2 Figure : General mamillary model The model is given by the following dierential equations _x =?( + )x + h x n ; _x 2 =? 2 x 2 + h 2 x n ; _x 3 =? 3 x 3 + h 3 x n ; _x n? =? n?x n? + h n?x n ; _x n = n? X i= i x i? h n x n + u; in which x i denotes the concentration or amount of a substance in compartment i, i denotes the rate of exchange of the substance from compartment i into the mother compartment, is the excretion rate, and h i, for i = ; : : : ; n?, is the rate of exchange from the mother compartment

7 P n? into compartment i Finally, h n = h i= i The values of h ; h 2 ; : : : ; h n are assumed to be known, which is realistic: we have several examples for which this assumption holds We assume these values h i to be nonzero and mutually unequal The unknown parameters are ; 2 ; : : : ; n?, and If we write the system in the compact form of a structured linear system, we will obtain _x = A(p)x + (p)u; x(t ) = x ; y = (p)x;? T with x = x x 2 x n, p = ( ; 2 ; : : : ; n?; ),?? h? 2 h 2 A(p) =? n? h n? 2 n??h n A ; (p) = A ; (p) =? Formally, this system is parametrized by the parametrization (P; f), with P = f( ; 2 ; : : : ; n?; ) 2 R n + Rn g f(p) = (A(p); (p); (p)) for p 2 P: Theorem 3 The parametrization (P; f) of the structured linear dynamic system described above is structurally identiable from the Markov parameters as a positive linear system For the proof of Theorem 3 we need the following lemma Lemma 32 onsider the linear dynamic system 5 (3) _x = Ax + u; x(t ) = x ; y = x; (32) with x 2 R n, u 2 R, y 2 R, A = 2 2 n? n? A ; = 2 n? =? Then A ; i 6= ; i = ; : : : ; n? ; (A; ) is reachable if and only if i 6= j ; i; j = ; : : : ; n? ; i 6= j: i 6= ; i = ; : : : ; n? ; 2 (A; ) is observable if and only if i 6= j ; i; j = ; : : : ; n? ; i 6= j: Proof of Lemma 32 We will use the reachability condition of Hautus, []: (A; ) is reachable if and only if rank? I? A = n; for all 2 :

8 6 For (32) we have? rank I? A = rank = rank(d ) + ;??? 2? 2? n?? n??? 2? n?? A with D :=??? 2? 2 A 2 R(n?)n :? n?? n? (() For 6= i, i = ; : : : ; n?, we have rank(d ) = n? For = i, we have i 6= ; i = ; : : : ; n? ; j 6= i ; i; j = ; : : : ; n? ; j 6= i imply rank(d ) = n? : ()) Assume i = Then, for = i, the ith row of D is equal to zero, so rank(d ) < n? Assume i = j for some i; j = ; : : : ; n? ; i 6= j Then, for = i, the ith and the jth column of D are equal to zero, so rank(d ) < n? i 6= ; i = ; : : : ; n? ; It follows that (A; ) is reachable if and only if i 6= j ; i; j = ; : : : ; n? ; i 6= j: 2 This follows from the fact that (A; ) is observable if and only if (A T ; T ) is reachable 2 Remark 33 The system (32) is an example of a general input-output hierarchical system, dened by Davison [6] In that paper Davison proves that such a system is controllable and observable for almost all interconnection gains (in Lemma 32 the i and i ) and almost all output gain matrices (in Lemma 32 the i and ) ut this does not say anything about what happens if for example j = j for some j This can just be the exception which lies in the algebraic set, or the hyper-surface in [6] Proof of Theorem 3 For the problem of structural identiability we will look at this positive linear system as an ordinary linear system For Theorem 2, we rst have to check structural reachability and structural observability From Lemma 32 it follows that (A(p); (p)) is reachable if and only if 8 < : h i 6= ; i = ; 2; : : : ; n? ; i 6= j ; i; j = 2; 3; : : : ; n? ; i 6= j; + 6= i ; i = 2; 3; : : : ; n? ; and (A(p); (p)) is observable if and only if 8 < : i 6= ; i = ; 2; : : : ; n? ; i 6= j ; i; j = 2; 3; : : : ; n? ; i 6= j; + 6= i ; i = 2; 3; : : : ; n? ; It follows that the system will be structurally reachable and structurally observable, if we take for the algebraic set

9 7 n? n? S = n( ; 2 ; : : : ; n?; ) 2 R n j f( +? i ) i Y i=2 Y j=i+ o ( j? i )g = : Now we may check structural identiability by the similarity approach For p 2 R n, nd all ~p 2 R n and T 2 R nn, such that (p)t = (~p); (33) (p) = T (~p); (34) A(p)T = T A(~p): (35) From (33) and (34) it immediately follows that T has the following form: T = t t ;n? t n?; t n?;n? A (36) The rst row of (35) gives the following equations?( + )t =?t (~ + ); ~?( + )t k =?t k ~ k ; k = 2; 3; : : : ; n? ; (37) and h = P n? j= t jh j : (38) For i = 2; 3; : : : ; n?, the ith row of (35) gives the following equations? i t i =?t i (~ + ); ~? i t ik =?t ik ~ k ; k = 2; 3; : : : ; n? ; (39) and h i = P n? j= t ijh j : (3) Outside S, +, 2, 3 ; : : : ; n? are mutually unequal, so in every column of T there is at most one nonzero element Indeed, assume t ik and t jk, i 6= j, are both nonzero To make notation easier, we take i; j; k 2 f2; 3; : : : ; n? g, but for i; j; k 2 f; 2; : : : ; n? g the same reasoning holds Then (39) gives i t ik = t ik ~ k and j t jk = t jk ~ k, from which it follows that i = j ontradiction Moreover, from the non-singularity of T it follows that exactly one element in every row and every column of T is nonzero Assume t ij 6= for some i; j 2 f2; 3; : : : ; n? g Then it follows from (39),? i t ij =?t ij ~ j, that ~ j = i, and from the jth entry of the nth row of (35) that i t ij = ~ j So t ij = ut then (3) gives h i = h j, which, by assumption on the h i being dierent, is only possible if i = j onsequently we have t 22 = t 33 = = t n?;n? =, t ij = for i; j 2 f; 2; : : : ; n? g, i 6= j, and therefore t 6= Now (38) gives h = t h, which implies t = So T = I and ~p = p With Theorem 2 and Proposition 23 it follows that the parametrization is structurally identiable as a positive linear system 2 Assume there is not only excretion from compartment x to the environment, but also from a nonmother compartment, say, compartment x j, for some j 2 f2; 3; : : : ; n? g Then in the matrix A(p) the j; j-entry is? j? with unknown With Lemma 32 the system is structurally reachable and structurally observable, but the parametrization is not structurally identiable Indeed, T, with t = t jj =, t j = h =h j, t j = h j =h, and the other elements as above, satises (33), (34),

10 8 and (35) for a suitable ~p So it is necessary to restrict to only one non-mother compartment with excretion to the environment At the other hand, if there is besides from compartment x also excretion from the mother compartment, x n, to the environment, we obtain the following structured linear system, _x = A(p)x + (p)u; x(t ) = x ; y = (p)x; with x as above, p = ( ; 2 ; : : : ; n?; ; ), A(p) =?? h? 2 h 2? n? h n? A ; 2 n??h n? and (p) and (p) as above Formally, this system is parametrized by the parametrization (P ; f ), with P = f( ; 2 ; : : : ; n?; ; ) 2 R n+ + Rn+ g f (p) = (A(p); (p); (p)) for p 2 P : Theorem 34 The parametrization (P ; f ) of the structured linear dynamic system described above is structurally identiable from the Markov parameters as a positive linear system Proof ecause in Lemma 32 there exist no restrictions on, which is equivalent to?h n? in the case of the parametrization (P ; f ) above, the pair (A(p); (p)) will be reachable and (A(p); (p)) will be observable outside S dened in the proof of Theorem 3 Now we may check structural identiability by the similarity approach For p 2 R n+, nd all ~p 2 R n+ and T 2 R nn, such that (p)t = (~p); (3) (p) = T (~p); (32) A(p)T = T A(~p): (33) From (3) and (32) it immediately follows that T is of the form (36) Now the nth entry of the nth row of (33) gives?h n? =?h n? ~; so ~ = The remaining part of the proof is equivalent to the proof of Theorem onclusions For mamillary systems with input and observations in the mother compartment and excretion only from one non-mother compartment, it is shown that the unknown parameters are (globally) structurally identiable from the Markov parameters Also some results on structural reachability and structural observability are given References R ellman and K J Astrom On structural identiability Math iosciences, 7:329{339, 97 2 F M allier and A Desoer Linear System Theory Springer Texts in Electrical Engineering Springer-Verlag, New York, 99 3 M J hapman and K R Godfrey On structural equivalence and identiability constraint ordering In E Walter, editor, Identiability of parametric models, pages 32{4, Oxford, 987 Pergamon Press

11 9 4 M J happell, K R Godfrey, and S Vajda Global identiability of the parameters of nonlinear systems with specied inputs: A comparison of methods Math iosciences, 2:4{73, 99 5 obelli, A Lepschy, and G R Jacur Identiability results on some constrained compartmental systems Math iosciences, 47:73{95, E J Davison onnectability and structural controllability of composite systems Automatica, 3:9{23, K Glover and J Willems Parametrizations of linear dynamical systems: anonical forms and identiability IEEE Trans Automatic ontrol, 9:64{645, K R Godfrey and J J DiStefano III Identiability of model parameters In E Walter, editor, Identiability of parametric models, pages {2, Oxford, 987 Pergamon Press 9 M S Grewal and K Glover Identiability of linear and nonlinear dynamical systems IEEE Trans Automatic ontrol, 2:833{837, 976 M L J Hautus ontrollability and observability conditions of linear autonomous systems Indag Math, 3, 969 J A Jacquez ompartmental Analysis in iology and Medicine The University of Michigan Press, Ann Arbor, R E Kalman Mathematical description of linear dynamical systems Journal SIAM on ontrol, :52{92, R E Kalman, P L Falb, and M A Arbib Topics in mathematical systems theory McGraw-Hill ook o, New York, Y Lecourtier and A Raksanyi The testing of structural properties through symbolic computation In E Walter, editor, Identiability of parameter models, pages 75{84, Oxford, 987 Pergamon Press 5 J P Norton An investigation of the sources of nonuniqueness in deterministic identiability Math iosciences, 6:89{8, E D Sontag Mathematical ontrol Theory Number 6 in Texts in Applied Mathematics Springer-Verlag, erlin, 99 7 I G Sterlina and J H van Schuppen Structural identiability of dynamic systems operating under feedback with application to economic systems Report S-R943, WI, Amsterdam, S Vajda Deterministic identiability and algebraic invariants for polynomial systems IEEE Trans Automatic ontrol, 32:82{84, S Vajda, K R Godfrey, and H Rabitz Similarity transformation approach to identiability analysis of nonlinear compartmental models Math iosciences, 93:27{248, J M van den Hof Realization of positive linear systems Preprint, Submitted to a journal, J M van den Hof Structural identiability of linear compartmental systems Preprint, submitted to a journal, J M van den Hof and J H van Schuppen Realization of positive linear systems using polyhedral cones In Proceedings of the 33rd onference on Decision and ontrol, pages 3889{3893 IEEE, 994

System theory and system identification of compartmental systems Hof, Jacoba Marchiena van den

University of Groningen System theory and system identification of compartmental systems Hof, Jacoba Marchiena van den IMPORTANT NOTE: You are advised to consult the publisher's version (publisher's PDF)