Proceedings of the 40th IEEE Conference on Decision and Control Orlando, Florida USA, December 2001
ThM06-3
A new differential algebra algorithm to test identifiability of nonlinear systems with given initial conditions Maria Pia Saccomani Dip. di Elettronica ed Informatica, Univ. di Padova via Gradenigo 6/a, 35131 Padova, Italy
[email protected] Stefania Audoly Dip. di Ingegneria Strutturale, Univ. di Cagliari piazza D’Armi, 09100 Cagliari, Italy
[email protected] Giuseppina Bellu, Leontina D’Angi`o Dip. di Matematica, Univ. di Cagliari viale Merello, 09100 Cagliari, Italy
[email protected] [email protected] from ideal, noise-free, input-output experiments. Assuming that measured input-output variables are available in the absence of noise, one would like to recover a unique model (i.e. a unique parametric structure) from an experiment. For nonlinear models very few results have been obtained and no standard algorithm exists for testing a priori global identifiability. The early efforts have not been very successful; in particular, the method based on power series [11] leads to an infinite number of nonlinear algebraic equations, the similarity transformation method of [4, 16], although leading to a finite number of equations, is very difficult to implement, etc. The introduction of concepts of differential algebra in control and system theory, mainly due to Fliess [5], has led to a better understanding of the nonlinear identifiability problem. In particular, Ollivier [10] and Ljung and Glad [6, 9] have shown that the concept of characteristic set of a differential ideal introduced by Ritt [12] is a very useful tool in identifiability. Although differential algebra methods has been an important factor for addressing the identifiability problem for nonlinear models, the construction of an efficient algorithm still remains a difficult task [9, 13]. In [2] we have presented a new algorithm to test global identifiability based on differential algebra. The algorithm has been implemented in a computer program making use of the symbolic language Reduce. It allows, for the first time, a fully automatic check of identifiability of nonlinear systems of realistic order (up to five or six) in a reasonably short time (typically 50-100 secs on a Pentium III PC). This result is based on several conceptual improvements on the methods existing in the literature, in
Abstract A priori global identifiability is a fundamental prerequisite for model identification. It concerns uniqueness of the parametric structure of a dynamic model describing given input and output functions measured during an experiment. Assessing a priori global identifiability is particularly difficult for nonlinear dynamic models. Various approaches have been proposed in the literature, but no solution exists in the general case. The introduction of concepts of differential algebra and in particular the concept of characteristic set of a differential ideal introduced by Ritt in ’50 have proven very useful tools in identifiability analysis. Yet the construction of an efficient algorithm still remains a difficult task. An improvement on existing algorithms has been recently published by the authors. Unfortunately this algorithm, like all other algorithms based on differential algebra, may run into difficulties for systems which are started at certain specific initial conditions. We argue in a companion publication that this difficulty may be present only for initial states from which the system is not accessible, and propose here a new version of the algorithm which gives the correct answer even if the system is started at special states from which the accessibility property is not guaranteed. 1 Introduction A priori global identifiability is a fundamental prerequisite for model identification. It concerns unique solvability for the parametric structure of a dynamic model 0-7803-7061-9/01/$10.00 © 2001 IEEE
3108
relation of the system in implicit form, i.e. as a set of m polynomial differential equation in the variables (y, u). The coefficients of these differential equations will depend, in general polynomially, on the parameter p [10]. The availability of these r input-output differential equations greatly simplifies the analysis. In fact, in order to analyse the a priori identifiability of the model (2.1) one has just to define a proper “canonical” set of coefficients of the polynomial differential equations, say c(p). One refers to this family of functions of p as the exhaustive summary of the model [10, 17]. After the exhaustive summary is found, to study a priori global identifiability of the model, one has to check if the map c(p) is injective, i.e. to check if the system of nonlinear algebraic equations:
particular a suitable ranking of the variables has been introduced which greatly improves on computational efficiency. This algorithm, like other algorithms based on differential algebra, analyses identifiability of systems assuming generic initial conditions. It has been pointed out that it may give wrong answers in special cases when the initial condition is fixed to some special value, a situation frequently encountered in identification of biological and medical systems. Our recent work has been devoted to extend the applicability of the algorithm to systems started at specific initial conditions [14]. 2 Background and definitions 2.1 The system Consider a parameterised nonlinear system described in state space form ½ ˙ x(t) = f [x(t), p] + G[x(t), p]u(t) (2.1) y(t) = h[x(t), u(t), p]
ˆ c(p) = c ˆ is a family of given values, has (generically) a where c unique solution in p. 2.3 Differential algebra For a formal description of the fundamentals of differential algebra, the reader is referred to [12]. Here only some basic definitions are briefly recalled. Let z be a vector of smooth functions of the variable t; the totality of polynomials in the variables zi and their derivatives, with coefficients in a field K is a differential polynomial ring and will be denoted K[z]. Consider a set Σ of differential polynomials belonging to K[z]. The totality of polynomials that can be formed by elements in Σ by addition, multiplication by elements in K[z] and differentiation, is a differential ideal I having the elements of Σ as generators. A differential ideal is prime if Ai Aj ∈ I implies that either Ai ∈ I or Aj ∈ I. In order to handle differential ideals, a ranking on the variables and their derivatives must be introduced, according to a system that satisfies the following relations
where x is the n-dimensional state variable; u the mdimensional input vector made of smooth functions; y is the r-dimension output; p ∈ P is the ν-dimensional parameter vector. If initial conditions are specified, the relevant equation x(t0 ) = x0 is added to the system. Although this is not strictly necessary we have assumed the system affine in the control variable. This simplifies some technical checks of the algorithm. The essential assumption here is that the entries of f , G = [g1 , . . . , gm ] and h are polynomial functions of their arguments. 2.2 A Priori Identifiability Different definitions have been given in the literature [1, 2, 9, 16, 17]. Here we adopt the one used in [1, 2]. Let y = Φ(p, u) be the input-output map of the system (2.1). Although Φ(p, u) obviously depends also on the initial state of the system (2.1) we shall not write this dependence explicitely. Definition We say that (2.1) is a priori globally (or uniquely) identifiable if and only if, for at least a generic set of points p∗ ∈ P the equation Φ(p∗ , u) = Φ(p, u)
(ν)
zi
(ν+k)
< zi (µ)
;
(µ)
zi
(ν)
< zj
(µ+k)
⇒ zi
(ν+k)
< zj
(ν)
where zi , zj are arbitrary derivatives. The leader, uj , of a polynomial Aj is the highest ranking derivative of the variables appearing in that polynomial (in particular it can be a derivative of order zero). The polynomial Aj is said to be of lower rank than Ai if uj < ui or uj = ui and deg(uj ) < deg(ui ). A polynomial Ai will be said reduced with respect to a polynomial Aj if it contains neither the leader of Aj with equal or greater degree nor its derivatives. If Ai is not reduced with respect to Aj it can be reduced by using a pseudodivision algorithm according to the following: if Ai contains the (k) k th -derivative, uj (it can be k = 0) of the leader of Aj , Aj is differentiated k times so its leader becomes (k) uj . Find two differential polynomials Ijν (of smallest
(2.2)
has only one solution p∗ = p for at least one input function u; the system is locally (or nonuniquely) identifiable if and only if, for at least a generic set of points p∗ ∈ P, the system (2.2) has more than one, but at most a finite number of solutions for all input functions u; the system is non-identifiable if, for at least a generic set of points p∗ ∈ P, the system (2.2) has an infinite number of solutions for all input functions u.
(k)
degree ν) and Q such that Ijν Ai = QAj + R where R
As we shall review in detail later in this paper, the use of differential algebra permits to write the input-output
(k)
is reduced with respect to Aj . The polynomial R is
3109
3.2 Choice of the ring The characteristic set of the differential polynomials (3.3) (3.4) will be considered in the ring R(p)[u, y, x,], where R(p) is the field of rational functions of the parameter p. The variables are the states, inputs and outputs. In this way, once the characteristic set is obtained, the coefficients of the differential polynomials are rational functions (or polynomials) in p and the differential polynomials themselves depend only on the u, y and x variables and, possibly, on their derivatives.
called the pseudoremainder of the pseudodivision. The polynomial Ai is replaced by the pseudo-remainder R (k−1) (k) in place of Aj and the process is iterated using Aj and so on, until the pseudoremainder is reduced with respect to Aj . A set of differential polynomials A1 , A2 , . . . , Ar that are all reduced with respect to each other, is called an autoreduced set. Two autoreduced sets, A = { A1 , A2 , . . . , Ar } and B = { B1 , B2 , . . . , Bs } are ranked according to the following principle: if there is an integer k ≥ min(s, r) such that rank Ai = rank Bi , i = 0, . . . , k − 1, rank Ak < rank Bk then A is said to be of lower rank than B; if r > s and rank Ai = rank Bi , i = 0, . . . , s, then A is also said to be of lower rank than B.
3.3 Ranking of variables The standard ranking used in the literature defines the inputs as the smallest components, followed by the outputs, and the state variables. Normally we choose the following:
Definition A lowest rank autoreduced set that can be formed with polynomials from a given set Σ of differential polynomials, is called a characteristic set of Σ.
< u˙ 1 < . . . < u2 < u˙ 2 < . . . < y1 < y˙ 1 < . . . < y˙ 2 < . . . < x1 < x˙ 1 < . . . < x2 < x˙ 2 < . . . (3.5) With respect to this ranking, the characteristic set of the polynomials (3.3) (3.4) has the following form: u1 < y2
The characteristic set of a differential ideal has been introduced by Ritt in ’50 [12] who also proposed an algorithm to construct it. The peculiarity of a characteristic set is that it can be used to generate the differential ideal by means of a finite number of polynomials.
A1 (u, y) . . . Ar (u, y) Ar+1 (u, y, x1 ) Ar+2 (u, y, x1 , x2 ) .. .
3 Identifiability and characteristic sets
(3.6)
Ar+n (u, y, x1 , . . . , xn )
3.1 The problem With the basic definitions at hand, we can now return to the dynamic system of (2.1). The model (2.1) can be looked upon as a set of n + r differential polynomials:
We will refer to the corresponding first r differential polynomial equations
˙ x(t) − f [x(t), p] + G[x(t), p ]u(t)
(3.3)
y(t) − h[x(t), u(t), p ]
(3.4)
of (3.6) as the input-output relations. These polynomial differential equations are the implicit description of the input-output map of the system (2.1) that we were referring to in section 2. In fact these polynomials are obtained after elimination of the state variables x from the set (3.3) (3.4) and hence represent exactly the pairs (u, y) which are described by the original system. Clearly only the input-output relations contain information on model identifiability.
A1 (u, y) = 0 A2 (u, y) = 0 . . . Ar (u, y) = 0
Polynomials (3.3)(3.4) are the generators of a differential ideal I in a differential ring. It is known that the state space description ensures the primality of the ideal generated by (2.1) [9]. The characteristic set of the ideal I is a finite set of n + r nonlinear differential equations which describes the same solution set of the original system. Its special structure allows to construct the exhaustive summary of the model used to test identifiability. The problems now are: 1) to construct, in an algorithmic way, the characteristic set starting from the model equations, 2) to solve the algebraic nonlinear equations of the exhaustive summary. This can be solved by a computer algebra method, e.g. the Buchberger algorithm [3] to calculate the Gr¨ obner basis. To solve the first problem, we have observed [13, 2] that the computational complexity is strongly influenced by the choice of the ring of multipliers of the differential ring K[z] and by the ranking of the variables of the differential polynomials.
(3.7)
The equations (3.7) are considered in the differential ring R(p)[u, y]. Note that the characteristic set is in general non-unique. However it can be rendered unique by normalization. In the following we shall assume that a suitable normalization has been introduced. Also, allowing the coefficients to be rational functions of the parameter permits to normalize the input-output relation by making the highest degree coefficient of the leading variable in each polynomial Ak , k = 1, . . . , r equal to one. Now, making each polynomial of the input-output set monic in the above sense, fixes uniquely the coefficients cij ∈ R(p), i = 1, . . . , r, j = 1, . . . , v(i), (here j is an index running over the monomial indices of the polynomial Ai , the
3110
teristic set is calculated:
monomials being ordered, say, in a lexicographic ordering). It follows that the functions cij (p) are uniquely attached to the input-output relation of the system. They constitute the exhaustive summary of the parameterisation. Remark 1. The triangular form of the characteristic set allows to extract easily information from the equations containing the state variables. For example, if the derivatives of the state components do not appear in the last n equations, the dynamic system (2.1) isalgebraically observable [6]. Thus algebraic observability of the system is easy to check from the characteristic set. Remark 2. If the system is of high dimension, the calculation of the characteristic set can become very complex. In this case, to decrease the computational complexity a different suitable ranking may be chosen.
A1 A2 A3 A4
p21 = c1 p12 + p21 + p13 = c2 p12 p13 + p21 p13 = c3 p21 p13 = c4
Example. Consider the system p13 x3 + p12 x2 − p21 x1 + u −p12 x2 + p21 x1 −p13 x3 x2
(4.10)
Equations (4.10) have a unique solution and hence all the parameters p12 , p21 and p13 seem to be globally identifiable. However, with zero initial conditions it is immediate to check that the transfer function of the model (6.14) depends only on the parameters p12 and p21 so that p13 cannot be identifiable! What goes wrong? Three observations are in order. 1. The characteristic set does not take into account the specific initial conditions. 2. The reachability subspace of the system (6.14) is the two dimensional subspace {x3 = 0 }. Said in another language, the system is non-accessible from initial states of the form x(0) = [x1 x2 0]T . 3. The exhaustive summary is the one provided by the transfer function method without performing the cancellations on the transfer function due to nonminimality. Since x3 (0) = 0 ⇒ x3 (t) = 0 for all times (as the third compartment is evolving autonomously in time) the state variable x3 can be set equal to zero in the polynomials of the charateristic set (4.9) where A4 becomes
As we have seen, the construction of the characteristic set ignores the initial conditions. In particular, the input-output relation (3.7) represents the input-output pairs of the system for “generic” initial conditions. Often, however, physical systems have to be started at special initial conditions, e.g. all radiotracer kinetics experiments in humans [2] are necessarily started at the initial state x(0) = 0. Thus, the problem arises if some specific initial conditions can change the inputoutput relations. To better understand the problem a very simple example is presented below. The example will be linear, since identifiability of linear system can be checked by standard transfer function methods. The system will be a three-compartment model describing the dynamics of a drug in a tissue. For this model the initial conditions have to be set to zero, since no drug quantities are present before the exogenous injection u.
= = = =
(4.9)
Note that only the differential polynomial A1 contains information on model identifiability, in fact it does not contain as variables neither x nor its derivatives. This polynomial represents the input-output relation of the model. By extracting the coefficients from A1 (which is already monic) and setting them equal to known symbolic values, the following exhaustive summary equations are obtained
4 The question of initial conditions
x˙ 1 x˙ 2 x˙ 3 y
...
≡ up ˙ 21 + y +¨ y (p12 + p21 + p13 )+ +y(p ˙ 12 p13 + p21 p13 ) + up21 p13 ≡ y˙ + yp12 − x1 p21 ≡ y − x2 ≡ y¨ + y(p ˙ 12 + p21 ) + up21 − x3 p13 p21
˙ 12 + p21 ) + up21 = 0 ∀t A¯4 ≡ y¨ + y(p
(4.11)
Note that the input-output relation A1 of (4.9) can be written as: (4.12) A1 ≡ A¯˙4 + p13 A¯4 = 0
x1 (0) = 0 x2 (0) = 0 x3 (0) = 0
Thus, the set formed by A1 , A2 , A3 of (4.9) and A¯4 is no longer a characteristic set. To take account initial conditions, the ideal generated by the original differential polynomial system plus A¯4 has to be computed. The new characteristic set is
(4.8) where x = [x1 , x2 , x3 ] is the state vector, e.g. x1 , x2 , x3 are drug masses in compartment 1, 2 and 3 respectively; u is the drug input; y is the measured drug output; p = [p12 , p21 , p13 ] is the rate parameter vector (assumed constant). The question is: are all the unknown parameters p12 , p21 , p13 globally identifiable from the input-output experiment? With the standard ranking (3.5), the following charac-
A¯4 A2 A3 A4
3111
≡ y¨ + y(p ˙ 12 + p21 ) + up21 ≡ y˙ + yp12 − x1 p21 ≡ y − x2 ≡ −x3 p13 p21
(4.13)
5 A computer algebra algorithm
Only p12 and p21 result uniquely identifiable while p13 has disappeared. This is in agreement with the transfer function method. Thus we have observed that the ideal generated by polynomials describing the dynamic system (2.1) may change when initial conditions are taken into account. Thus the above example gives a hint on why the method may fail, in particular that lack of reachability from the initial state (i.e. accessibility) may be the cause of troubles.
The starting point of the algorithm is the differential polynomials (3.3) (3.4) defining the dynamical system. The principal steps of the algorithm are: 1. The accessibility Lie Algebra of the system is constructed and the set of zero measure, if exists, where the accessibility rank condition does not hold is calculated. Let φ(x) = 0 be its equation; 2. if φ(x) exists and φ(x(0)) = 0, φ(x) is added to polynomials (3.3) (3.4); 3. a ranking is introduced; 4. the leaders of each polynomial are found; 5. the polynomials are ordered. Each polynomial is compared with the previous ones and, if it is of equal or higher rank, is reduced with respect to them. This step is repeated until the autoreduced set of minimum rank is reached. This is the characteristic set; 6. the input-output relations are made monic; 7. the coefficients belonging to the field R(p) are extracted from the input-output relations; 8. a random numerical point pˆ from the parameter space is chosen and the exhaustive summary of the system is calculated; 9. the Buchberger algorithm is applied to solve the equations and the number of solutions for each parameter is provided. The algorithm has been implemented in Reduce 3.5 and runs on a Pentium PC.
4.1 The role of accessibility In the following we shall refer to the concept of accessibility as defined in [15]. A full understanding of the identifiability problem with specific initial conditions requires to study the role of accessibility in the structure of the characteristic set [14]. For reasons of space in this paper we shall only present the main ideas leaving the details to a future publication. In [14] we have shown that, when the system is accessible from x0 , adding the specific initial condition x(0) = x0 as a constraint, cannot change the characteristic set. This is so since the variety where the motion of the system takes place has the same dimension of the initial variety [7], and the order of the system can not drop. Conversely, suppose that the system is algebraically observable, generically accessible (i.e. accessible from all points except from a “thin” subvariety) and assume that x0 belongs to the “thin” set from which the system is non accessible. There is an invariant subvariety where the motion of the system takes place when started at the initial condition x0 [8]. This subvariety can be calculated by the construction of the accessibility Lie Algebra, see [15, p.154]. Let φ(x) = 0 be the equation of the invariant subvariety of non-accessible states. This algebraic equation must be added to the characteristic set in order to get a reduced representation of the system dynamics. In order to compute the new characteristic set we note that the polynomials representing the state dynamics alone (3.3), with a suitable ranking, are a characteristic set and that φ(x) is reduced w.r.t. it. For a known property [12, p.5] the characteristic set of the ideal generated by polynomials (3.3) plus φ(x) is of lower order than the original one. Once the output polynomial (3.4) is added, a new characteristic set is calculated which still remains of the (lower) order obtained in the previous elimination step. A new algorithm, based on these ideas has been implemented in Reduce 3.5 and runs on a Pentium PC. A proof that the conceptual procedure sketched above guarantees a correct answer to the identifiability analysis will be reported in a future publication. Note that in the elimination steps of the procedure described above, the primality of the ideal needs to checked. If the ideal is not prime a factorisation needs to be performed.
6 Example Consider the nonlinear model discussed in [4]. It is a two compartment model which describes the kinetics of a drug in the human body. The drug is injected into the blood where it exchanges linearly with the tissues; the drug is irreversibly removed with a nonlinear saturative characteristic from the blood and with a linear one from the tissue. The system is x˙ 1 = −(k21 + VM /(Km + x1 ))x1 + k12 x2 + b1 u x1 (0) = 0 x ˙ = k x − (k + k )x x 2 21 1 02 12 2 2 (0) = 0 y = c1 x1 (6.14) where x1 , x2 are drug masses in blood and tissues respectively, u is the drug input, y the measured drug output in the blood, k12 , k21 and k02 are the constant rate parameters, VM and Km are the classical Michaelis-Menten parameters, b1 and c1 are the input and output parameters respectively. The question is whether the unknown vector p = [k21 , k12 , VM , Km , k02 , c1 , b1 ] is globally identifiable from the input-output experiment. Let the ranking of the variables be u < y < x1 < x2 . The accessibility Lie
3112
Algebra ALA is computed: since rank ALA = 2, the system is accessible from every point. The reduction procedure is started and the characteristic set is calculated. Here only the input-output relation is reported
els of biological and physiological systems. IEEE Trans. Biomed. Eng., 48:, 2001.
y¨y 2 + k21 k02 y 3 − (k21 c1 b1 + k02 c1 b1 )y 2 u+ ˙ 2 − c1 b1 y 2 u˙ − Km c1 3 b1 u+ ˙ +(k21 + k12 + k02 )yy +(2k21 Km k02 + k12 VM k02 VM )c1 y 2 + −2(k12 + k02 )c1 2 b1 Km yu+ +2(k21 Km c1 + k12 Km c1 + k02 Km c1 )y y˙ + 2Km c1 y y¨+ −2Km c1 2 b1 y u˙ + Km c1 2 (k21 k02 + k12 VM + k02 VM )y+ −(k12 Km 2 b1 c1 3 + k02 Km 2 b1 c1 3 )u + Km 2 c1 2 y¨+ +(k21 Km 2 c1 2 + k12 Km 2 c1 2 + VM Km c1 2 + k02 Km 2 c1 2 )y˙ (6.15) Coefficients are extracted (these are the exhaustive summary of the model) and evaluated at a numerical point pˆ randomly chosen in the parameter space P. Each coefficient, in its polynomial form, is then set equal to its corresponding numerical value. The Gr¨ obner basis is calculated by Buchberger’s algorithm and applied to solve the obtained equations. The obtained system has an infinite number of solutions, thus the model is a priori non identifiable. Note that if the input parameter is assumed to be known, i.e. b1 = 1, the model becomes a priori globally identifiable.
[3] B. Buchberger. An algorithmical criterion for the solvability of algebraic system of equation. Aequationes Mathematicae, 4:45–50, 1988. [4] M.J. Chappel. Structural identifiability of models characterizing saturable binding: comparison of pseudo-steady-state and non pseudo-steady-state models formulation. Math. Biosci., 133:1–20, 1996. [5] M. Fliess and S.T. Glad. An Algebraic Approach to Linear and Nonlinear Control. In Essays on Control: Perspectives in the Theory and its Applications, 7:223– 267, 1993. [6] S.T. Glad. Differential algebraic modelling of nonlinear systems, in Realization and Modelling in System Theory, MTNS’89. Birkhuser, 1:97–105, 1990. [7] R. Hermann and A.J. Krener. Nonlinear controllability and observability. IEEE Trans. Automatic Control, AC-22:728-740, 1977. [8] A. Isidori. Nonlinear Control Systems. 3nd ed., London: Springer, 1995. [9] L. Ljung and S.T. Glad. On global identifiability for arbitrary model parameterisations. Automatica, 30:265–276, 1994. [10] F. Ollivier. Le probl`eme de l’identifiabilit´e structurelle globale: ´etude th´eorique, m´ethodes effectives et bornes de complexit´e. Th`ese de Doctorat en Science, ´ Ecole Polyt´echnique, Paris, France, 1990.
7 Conclusions A priori identifiability is a necessary prerequisite for parameter identification. Checking a priori global identifiability, i.e. the uniqueness of the solution, is particularly difficult for nonlinear dynamical systems. In this paper, we briefly describe a new algorithm, recently developed by the authors, for testing a priori identifiability of nonlinear systems. The algorithm is based on differential algebra and, in particular, on the concept of characteristic set of the ideal generated by the polynomials defining the model. We propose a new version of the algorithm which allows to successfully deal with systems starting from initial conditions fixed to some specific value. In order to do this, the accessibility property of the system is checked by using the accessibility Lie Algebra and, whenever needed, a new ideal associated to the dynamical system which takes into account initial conditions is calculated. The algorithm has been used successfully to analyse a priori identifiability of several biological system models.
[11] H. Pohjanpalo. System identifiability based on the power series expansion of the solution. Math. Biosci., 41:21–33, 1978. [12] J.F. Ritt. Differential Algebra. Providence, RI: American Mathematical Society, 1950. [13] M.P. Saccomani, S. Audoly, G. Bellu, L. D’Angi` o and C. Cobelli. Global identifiability of nonlinear model parameters. In Proc. SYSID ’97 11th IFAC Symp. System Identification, 3:219–224, 1997. [14] M.P. Saccomani, S. Audoly and L. D’Angi` o. Testing global identifiability of nonlinear dynamic models by a differential algebra algorithm. Abstract in Proc. Academy Colloquium Contructive Algebra and Systems Theory, Amsterdam, 2000. [15] E.D. Sontag. Mathematical Control Theory, 2nd ed., Berlin: Springer, 1998. [16] S. Vajda, K. Godfrey and H. Rabitz. Similarity transformation approach to identifiability analysis of nonlinear compartmental models. Math. Biosci., 93:217–248, 1989.
References [1] S. Audoly, L. D’Angi` o, M.P. Saccomani and C. Cobelli. Global identifiability of linear compartmental models. IEEE Trans. Biomed. Eng., 45:36–47, 1998.
[17] E. Walter and Y. Lecourtier. Global approaches to identifiability testing for linear and nonlinear state space models. Math. and Comput. in Simul., 24:472– 482, 1982.
[2] S. Audoly, G. Bellu, L. D’Angi` o, M.P. Saccomani and C. Cobelli. Global identifiability of nonlinear mod-
3113
