software for solving identification and identifiability problems, e.g. in compartmental systems

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  • 490 Mathematics and Computers in Simulation XXIV (1982) 490-493 North-Holland Publishing Company

    SOFTWARE FOR SOLVING IDENTIFICATION AND IDENTIFIABILITY PROBLEMS, E.G. IN COMPARTMENTAL SYSTEMS

    H. POHJANPALO and B. WAHLSTROM Technical Research Centre of Fin~n~ VTT/SAH, SF-02150 Espoo 1~ Fin~nd

    The identif ication problem for a system with a known structure is considered. The mathematical background of the ident i f i ab i l i ty problem is shortly discussed. Two minicomputer programs which have been used in several studies to solve the identi f ia- b i t i ty and the identif ication problem for biomedical systems are described.

    1. INTRODUCTION 2. IDENTIFICATION AND IDENTIFIABILITY

    Compartmental systems (cf. /7/) have been used for the modelling of several types of bio- medical systems. The structure of the models are obtained using balance equations where a physical in terpretat ion of the state variables and the parameters often could be found. The problem of constructing a model of the system could then be reduced to the ident i f i ca t ion of set of unknown parameters. The ident i f i ca t ion could be solved using the model reference approach where the parameters of the model are adjusted to obtain the best f i t with respect to a sui table object ive function.

    The identification problem could be divi- ded into three different problems: the identifia- bility problem, the problem of obtaining the best estimates for the parameters and the problem of judging the quality of the obtained estimates.

    The identifiability problem is to deter- mine if the structure of the model allows the parameters to be determined with a proposed ex- periment. It is clear that the identifiability should be investigated before costly experiments are carried through. The identifiability has been attacked by programming the algorithms uti- lizing the Taylor series expansion of the solu- tion of the differential system /1/.

    The best estimates of the parameters could be obtained using standard methods for numerical optimization where the objective function is calculated by comparing a simulated response to the response obtained in the experi- ment. The problem has been solved using a simu- lation language i~,a repetitive mode within the optimization program.

    The quality of the best estimate could be judged by evaluating the parameter sensiti- vities i.e. the second order derivative matrix of the objective function with respect to the parameters. The quality judgement have not been included in the present programs but the method and examples have been described in the refe- rences /2/ and /3/.

    The starting point is the differential system /4/ represented as

    x(t) = f(x,u,t,O), x(O)= x (@), u(-)cU o

    y(t) = g(x,O) (1)

    h(O) = 0

    x(t)gX, tc[O,T]

    where x(-) is the state vector with values in the set of feasible states XcR n, u(-) is the control vector in the set U of feasible functions with values in R n, y(-) is the measurement vector re- corded over the time interval [O,T] with values in R p, and 0 is the vector of unknown parameters in the open set of possible values ~--Rq. The functions f, g, and h are explicitly given for each analysis. The optional function h represents auxiliary binding equations between the components of 0.

    The collection of objects (f,g,h,U,X,~) is called the structure S imposed by the system (1).

    Suppose first that the underlying system to be simulated obeys Eq. (1) with the parameter values 0 = 0 . The identifiability problem can be defined a~ finding out whether or not the parameter vector@ocould be uniquely solved from Eq. (1). The identification problem is to find an estimate of 0, say ~o' so that some objective function J(y,~) measuring the difference of the responses y(.) and ~(-) is minimized, where ~ is the trajectory resulting from 0 = 0 o. It should be noticed that no stochasticity is accepted in the state equation nor in the measurements.

    By associating the nominal parameter values 0 with the structure we have the model (S,O). Identifiability concepts for models (S,@) are meaningless without the introduction of the experi- ments Cx~(O), ui(-)), denoted as E i for short. Let the vector of the experiments E i, i = 1 .... ,s, be called the experimentation E. Each triple (S,@,E) implies a unique vector Y or measurement trajectories Yi, i = 1, .... s (each of which is a Pi-dimensionat vector of functions [O,T]~R. By introducing the mapping F we can write this as

    F(S,O,E) = Y(G) (2)

    0378-4754/82/0000-0000/$02.75 1982 IMACS/North-Holland

  • H. Pohjanpalo, B. Wahlstram / Solving identification and identifiability problems 491

    Thus, given any structure S and experi- mentation E the resulting vector of trajectories is a function of @, only.

    Auxiliary definition 1. Model (S,@_) is identi- fTaS~e-wTth-the-experTmentation E i~ F(S,O ,E) = Y(@o ) implies 0 = 0 o.

    Auxiliary definiton 2. Model (S,@ o) is locally ident~f~ab~e-at-O-w~th the experimentation E if @ has an c-neighbourhood N(O,c)~Q so that @'oN(@,c) and (S,O',E) = Y(@o ) together imply 0" = 0.

    The underlying unknown 0o has to be in- volved in the definitions since the identifiabi- lity properties may vary largely with its value. This is because the numerical values of the Taylor series terms vary with @o and therefore the number of possible solutions of the corresponding identi- liability equations varies /1/, /4/. Model (S,@ o) can thus be locally identifiable, if anywhere, at the points which yield the measurement trajectories Y(@o ). From the identifier's point of view all such points are equivalent. If several such points exist the identification procedure may end up with the correct solution @o or any other point @ where Y(O) = Y(@o ).

    The auxiliary definitions are useless in the sense that they are restricted to a single unknown value of the parameter vector describing the system to be identified. The modeler probably finds identifiability results based on definitions concentratinq on the structure more useful.

    Definition I. The model structure S is (almost everywhere)-Tdentifiable with the experimentation E if for (almost) all @oC~ the equation F(S,@,E) = Y(@o ) implies @ = @o-

    Definition 2. The model structure S is (almost everyw~ere)-~ocaLly identifiable with the experi- mentation E if for (almost) all 8oC~ there is a @c~, such that the model (S,@ o) is locally identi- fiable at ~.

    (S, " ,E)

    Let G(@) be a matrix function of 0. The point @'c~ is called a regular point of G if rank G(@) is at its maximum at @ = 0". The open set of regular points in Q is denoted by ~r"

    Restricting to the set of regular points in~, the following mathematically equivalent formu- lation of Definition 2 can be presented. The model structure S is locally identifiable with the experimentation E if for all OoC~ the model (S,O o) iQ locally identifiable at 0 = @o-

    Model structure imposed by Eq. (I) with OE~, is locally identifiable /4/ if and only if the infinite partial derivative matrix 3g(k)/80, k = 0 .... augmented by the lines 3h/~O is of rank q for some OcQ. If E is a vector the component matrices can be put one below the other.

    3. EVALUATING IDENTIFIABILITY

    Global identifiability can be asserted if the terms of the Taylor series of the measurement trajectory are bijective functions of the unknown parameters /1/. Computing the symbolic terms of the Taylor series by hand becomes extremely tedious and practically impossible as the order of the derivative increases, even for relatively simple models. Therefore the approach of computing the derivatives by computer has been taken. The criteria for local identifiability is straight- forward to implement on the computer once that the symbolic Taylor terms are available. Evalu- ating global identifiability with a computer still remains a dream: solving a set of highly nonlinear symbolic equations is not a task easily amenable to computerization. Attempts in this direction should be based on 'stronger' tools than general high level languages, such as LISP /8/.

    The program DER has been developed for analyzing the identifiability of linear and cer- tain types of nonlinear differential models in state space. It can be used to computer the time zero derivatives of the measurements as functions of the parameters, to be used on for evaluating either local identifiability using the same prog- ram or global identifiability by hand.

    Local ident i f iab i [~tv : DER #

    By handh (0) , / /

    , / " (k)

    {'~ I , k = 1 . . .} rank[ - -Y~T -] = fu l l7

    \, 3 t t = 0 \~ 3hl;~(~

    By hand Global identi fiabl lily: each C~ i solvable ?

    F~g. I. The uses of DER for evaluatlng local and global i dent i l i ab i l i ty .

    The computer izat ion of eva luat ing ident i - f i ab i l i ty for models of the form Eq. (1) requires res t r i c t ion to some par t i cu la r class of systems. In the present implementation the models can be as follows.

    The function f(x,u,t,@) can be polynomial - in the state vector components x., i = 1 .... i - in time t, both of these in nonnegative

    powers, and - in the parameters @i' i = 1 ..... q + q~ in arbitrary combinations. Here q is the number of unknown parameters and q' is the number of parameters with a priori known values~ as an example:

    -1 + I Xl = -3@~xI + @2tx~x3 - @ 3@4

    x2 = "'" (3)

    R3 = "'"

    The model can also be specified by the matrix A with exactly the same restrictions as those posed on f, as follows

  • 492 H. Pohjanpalo, B. WahlstrOm / Solving identification and identifiability problems

    ~(t) = A(x,t,O)x(t) + u(t) (4)

    In both of the representations only the zero input function is allowed. The initial state, instead, i

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