A priori parameter identifiability in models with non-rational functions

Automatica 109 (2019) 108513

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A priori parameter identifiability in models with non-rational functions✩ ∗

Rishabh Jain a , Sridharakumar Narasimhan a,c , , Nirav P. Bhatt b,c a

Department of Chemical Engineering, Indian Institute of Technology Madras, Chennai, 600036, India Department of Biotechnology, Indian Institute of Technology Madras, Chennai, 600036, India Initiative for Biological Systems Engineering and Robert Bosch Centre for Data Science and Artificial Intelligence, Indian Institute of Technology Madras, Chennai, 600036, India b c

article

info

Article history: Received 27 January 2018 Received in revised form 21 March 2019 Accepted 11 July 2019 Available online 13 August 2019 Keywords: Identifiability Differential algebra Padé approximation Power series Reaction networks Systems with non-rational functions

a b s t r a c t Differential algebra based approaches are used to study a priori parameter identifiability of nonlinear systems with polynomial or rational functional forms. However, these methods cannot be applied to state-space models which have non-rational functions (e.g., exponential, sinusoidal etc.) of state variables. In this paper, we propose a method to test identifiability for systems with non-rational functions using Padé and power series approximations and differential algebra. In particular, for a certain class of systems, we show that if the approximation of a certain order is used and the resulting system is identifiable, then higher order approximations will also result in identifiable systems. The proposed approach is illustrated using a non-isothermal reaction system. © 2019 Elsevier Ltd. All rights reserved.

1. Introduction Identification of a unique and reliable dynamic model of the underlying system is a central theme in the field of system identification. The problem of identifying a unique model of the underlying system from data depends on three aspects: (A1) model identifiability or a priori parameter identifiability (A2) input characteristics, and (A3) estimation methods. A priori identifiability or model identifiability is a property of the model alone and is concerned with answering the following question: ‘‘Does there exist a unique one-to-one map between the model and parameters being identified?’’ (Tangirala, 2015). A priori parameter identifiability analysis is an important tool to determine to answer the above question. If and only if a priori parameter identifiability provides a positive response for the model structure (Aspect A1), does the question of generating informative experimental data arise through appropriate choice of inputs arise (A2). Finally, the question of the estimation method which ✩ The financial support to Nirav P. Bhatt from Department of Science & Technology, India through INSPIRE Faculty Fellowship No IFA 12-ENG-34 is acknowledged. The material in this paper was not presented at any conference. This paper was recommended for publication in revised form by Associate Editor Chunjiang Qian under the direction of Editor André L. Tits. ∗ Corresponding author at: Department of Chemical Engineering, Indian Institute of Technology Madras, Chennai, 600036, India. E-mail addresses: [email protected] (R. Jain), [email protected] (S. Narasimhan), [email protected] (N.P. Bhatt). https://doi.org/10.1016/j.automatica.2019.108513 0005-1098/© 2019 Elsevier Ltd. All rights reserved.

allows one to estimate the true parameters in the given model from the informative data needs to be answered (A3). In this work, the problem of a priori identifiability of parameters in a model structure (A1) is addressed. Further, this analysis is done at a purely theoretical level where the input and experiment conditions and estimation methods are assumed to be ideal. Differential algebra has been used to address global as well as local identifiability properties of systems represented by linear or nonlinear differential equations (Audoly, Bellu, D’Angio, Saccomani, & Cobelli, 2001; Ljung & Glad, 1994). The methods based on differential algebra use Ritt’s pseudo-division algorithm to find a characteristic set based on the input–output equations. These methods in the literature require state-space equations to be in polynomial or rational form. Hence, these methods are of limited use when the model equations contain non-rational functions such as exponential terms. In modeling of (bio-)chemical reaction networks, the unknown parameters are estimated using input– output data. Here, it is often imperative to know a priori if the parameters in the model can be uniquely identified, given the input–output data (Audoly et al., 2001; Evans et al., 2013; Fliess, Join, & Sira-Ramirez, 2008). The Arrhenius equation used in modeling of non-isothermal reaction systems and biological reactions following Tessier kinetics are examples of reaction models with non-rational functions (Wang & Witarsa, 2016). The resulting reaction model structures cannot be accommodated within the current framework. Recently, data from reactors operated in a non-isothermal manner has been used for kinetic model identification (Kopyscinski, Schildhauer, Vogel, Biollaz, & Wokaun, 2010).

2

R. Jain, S. Narasimhan and N.P. Bhatt / Automatica 109 (2019) 108513

The kinetic rate constants vary non-linearly in an exponential manner with temperature and hence the existing methods cannot be directly applied to determine if the systems are identifiable. To overcome this problem, we approximate the non-rational functions using rational and polynomial functions and test identifiability. A Padé approximation is usually a better rational approximation than Taylor series approximation of the same order, particularly in functions having poles (Moler & Van Loan, 2003). The general idea is to convert the model with non-rational functions to a model with completely rational functions. Then, differential algebra methods can be applied to test identifiability. Further, we show that we can deduce the identifiability of the original model with non-rational functions based on the identifiability analysis of model with approximated rational functions. While approximation using rational or polynomial functions is straightforward, it is not immediately clear if identifiability of the approximated system can resolve the identifiability of the original system. We provide a partial answer in this regard. In particular, for a certain class of systems, we show that if approximation of a certain order is used and the resulting system is identifiable, then higher order approximations will also result in identifiable systems. The paper is organized as follows. In Section 2, the differential algebraic approach for parameter identifiability and Padé approximation are introduced. In Section 3, the proposed approach for parameter identifiability of systems with non-rational functional forms is described. These approaches are demonstrated through examples of a non-isothermal reaction system. Conclusions are presented in Section 4.

2.2. Padé Approximations Padé approximation of a function is a specific type of rational fraction approximation of a function. Consider a function f (x) and integer M , N ≥ 1 and M ≤ N. Then, the [M/N] Padé approximation of f (x) is given by f (x) =

AM (x) BM (x)

=

a0 + a1 x + · · · + aq x M b0 + b1 x + · · · + bN xN

,

(2)

where ai and bi are the co-efficients. Usually, b0 = 1. The values of ai and bi are obtained from Padé tables (Baker, Jr., 1975). Furthermore, if the Padé approximation exists, the [M/N] Padé approximation of f (x) is unique (Baker, Jr., 1975). A [q/0] approximation is simply equal to the qth order Taylor series expansion. For example, the [2/2] Padé approximation of an exponential function is given as follows: e

−x

=

1−

x 2

+

x2 12

1+

x 2

+

x2 12

.

(3)

3. Identifiability in models with non-rational functions The test described in the previous section based on differential polynomials cannot be applied to systems with f (·) containing non-rational functions. In this section, we first introduce an example of reaction kinetics under non-isothermal condition having exponential dependency on the temperature, and, subsequently discuss how to handle non-rational functional forms for testing a priori identifiability.

2. Preliminaries

3.1. Motivation: Non-isothermal reaction system

2.1. Differential algebra approach to identifiability

Example 4. Let us consider an elementary reaction involving two components A and B in a non-isothermal batch reactor as follows:

A nonlinear model in state-space form can be written as: x˙ (t) = f (x(t), u(t), p) y(t) = h(x(t), u(t), p)

A → B, (1)

where x is the n-dimensional state variable; u is the m-dimensional input vector made of piece-wise smooth functions; y is the r-dimensional output; p is the p-dimensional vector of unknown parameters. It is assumed that f and h are smooth functions. When the nonlinear functions in Eq. (1) are rational or polynomial, identifiability of the parameters can be determined as follows. A set of r differential polynomials in terms of input variables u, output variables y, and derivatives of inputs, outputs and parameters p are determined. These differential polynomials are denoted as input–output relations of the system (1) which are determined from Eq. (1) using the Ritt’s pseudo-division algorithm (Saccomani, Audoly, & D’Angio, 2003). Let c(p) denote the vector of coefficients in the input–output relationships. Then, we can check the injectivity of the coefficients map c(p) of the r differential polynomials for determining a priori parameter identifiability by evaluating c(p) for an arbitrary vector with symbolic values p∗ . Depending on the nature of solutions of c(p) = c(p∗ ), we can classify the system as follows. Definition 1 (Global Identifiable). The model (1) is globally identifiable from the input–output data if and only if for any arbitrary p∗ , c(p) = c(p∗ ) has a unique solution p = p∗ . Definition 2 (Local Identifiable). If there exist multiple but finite number of distinct solutions for c(p) = c(p∗ ), then the model (1) is locally identifiable. Definition 3 (Unidentifiability). If there are infinite number of solutions for c(p) = c(p∗ ), then the model (1) is unidentifiable.

E

r1 = k1 (T )cA = k10 e− T cA ,

(4)

where r1 is the rate of reaction, k10 and E are the unknown parameters, cA is the concentration of component A, T is temperature of the reaction mixture. Under certain assumptions, the material and energy balances for this system can be written as follows (Levenspiel, 1972): E

E

E

c˙A = −k10 e− T cA ; c˙B = k10 e− T cA ; T˙ = −h1 k10 e− T cA ,

(5)

where cB is the concentration of component B and h1 is an unknown parameter. It is assumed that the measurements of T and cA are available. By defining the state, input, output and parameter vectors, x = [cA , cB , T ]T , y = [cA , T ]T , and p = [k10 , E , h1 ]T , we can write Eq. (5) in the state-space form (1). The identifiability tests based on differential algebra proposed by Saccomani et al. (2003) cannot be applied to this system due to the presence of the exponential terms. One solution is to define E a new variable x4 = e− T and convert the system into the statespace form with the rational functions. This approach is called state augmentation approach. Then, the system in Eq. (5) can be written as follows (Audoly et al., 2001)1 : c˙A = −k10 x4 cA , c˙B = k10 x4 cA , T˙ = −h1 k10 x4 cA , x˙ 4 = −h1 k10 E(x24 cA /T 2 )

(6)

The characteristic set for Eq. (6) is given as follows: A1 ≡ −h1 y˙ 1 + y˙ 2 , A2 ≡ y¨ 1 y1 y22 − E h1 y˙ 21 y1 − y˙ 21 y22 A3 ≡ y1 − cA ,

A4 ≡ y2 − T

A5 ≡ y˙ 1 + k10 x4 y1 ,

(7)

A6 ≡ x˙ 2 + y˙ 1

1 The authors in Audoly et al. (2001) have applied a similar idea to time varying terms.

R. Jain, S. Narasimhan and N.P. Bhatt / Automatica 109 (2019) 108513

In Eq. (7), A1 and A2 are the input–output relations. Inspection of A1 and A2 shows that the only parameters E and h1 are globally identifiable with y1 and y2 measurements by the state augmentation approach. In addition if x4 is measured, k10 is identifiable, as seen from (A5) in Eq. (7). However, since x4 is a function of the state T and the unknown parameter E by definition, it is not possible to measure x4 . Furthermore, the computation of x4 is not possible without knowledge of E. Hence, the k10 is not identifiable without knowledge of E for the system (6) using this approach. However, the original system is identifiable which will be demonstrated next. The second solution is to find a nonlinear transformation of Eq. (5). By defining z1 = ln(−˙y1 ) − ln(y1 ) and z2 = ln(y˙ 2 ) − ln(y1 ) and taking natural logarithm on both sides of the equations related to cA and T in Eq. (5), the transformed equations are z1 = ln(k10 ) − E /T ;

z2 = ln(−h1 k10 ) − E /T .

(8)

Since z1 , z2 , and T are known, the transformed parameters ln(k10 ), E, and ln(−h1 k10 ) can be estimated by linear regression. Hence, k10 , E, and h1 are identifiable. Consequently, the system (5) is globally identifiable with the y1 and y2 measurements. Equivalently, the polynomials in Eq. (8) can be considered as pseudo differential polynomials in z1 , z2 , 1/y2 , y1 . △ There are several disadvantages with the approaches mentioned in Example 4. In the state augmentation approach, defining a new variable or variables increases the dimension of the state space, and hence, correspondingly the computational effort in obtaining the characteristic set. Further, it may not be possible to compute the initial condition of the new variables. For example, in Example 4, x4 (0) cannot be computed. Most importantly, identifiability analysis of the augmented system may result in incorrect conclusions. E.g., a negative result does not allow us to conclude that the original system is unidentifiable. In fact, the system (5) is identifiable which was illustrated in Example 4. In the nonlinear transformation approach, it may not always be possible to determine an appropriate transformation. In the light of these disadvantages, it is necessary to develop an alternate approach for a priori identifiability for models with non-rational (transcendental) functional terms.

Theorem 5. If the non-rational function ω(T , pT ) in the model (9) is approximated by Eq. (10) and the approximated system is found to be identifiable, then, any higher order q′ th approximation (of the following form) will also result in an identifiable system: ′

ω(T , pT ) = a0 + a1 T + · · · + aq + · · · + aq′ T q , where q > q and a0 , . . . , aq′ are functions of parameter pT .

Proof. When the non-rational function ω(T , pT ) is approximated by the Padé approximation in Eq. (10), we obtain a rational approximation of the state space equations. Let xq denote the state variables in this approximation, where the subscript q is used to denote the fact that a qth order approximation is used. x˙ q (t) = (a0 + a1 T + · · · aq T q )f 1 (xq , u, p) + f 2 (xq , u, p) y q (t) = xq (t)

In this section, an approach for testing identifiability of models with non-rational functions using polynomial and rational approximations is proposed. We decompose the state vector as x = [xTa , T ]T and assume that all states are measured. The state-space equations can be written as: y(t) = x(t)

(9)

where x and u are the n– and m–dimensional vectors of states and inputs made of piece-wise smooth functions, respectively. f 1 (x, u, p) and f 2 (x, u, p) are the n–dimensional vectors of the rational functions of the states while ω(T , pT ) is a non-rational function of T . We assume this specific form of nonlinearity only in the interest of clarity and exposition and this is not a limitation. y is the n–dimensional output. p is the p–dimensional vector of unknown parameters. pT is the pT –dimensional vector of the unknown parameters in the non-rational functions which is also a subset of p. A [q/0] order Padé approximation (or simply qth order Taylor series approximation) of the non-rational function ω can be given by

ω(T , pT ) = a0 + a1 T + · · · + aq T q , where a0 , . . . , aq are functions of parameters pT .

(10)

(12)

E.g., consider x˙ qi , the ith component of xq : x˙ qi (t) = (a0 + · · · + aq T q )

n1 (xq , u, p) d1 (xq , u, p)

+

n2 (xq , u, p) d2 (xq , u, p)

(13)

Eq. (13) can be written as follows: d1 (·) d2 (·)x˙ qi (t) = (a0 + · · · + aq T q )n1 (·)d2 (·) + n2 (·)d1 (·)

(14)

n1 (xq , u, p) and d1 (xq , u, p), and n2 (xq , u, p) and d2 (xq , u, p) are the numerators and denominators of the ith component of the functions f 1 (·) and f 2 (·), respectively. Since all states are measured, the coefficients of the characteristic set, c q (p), are readily obtained by inspection of Eq. (14) and they are: γ (p), aj (pT )Φ (p) and Ψ (p), j = 0, . . . , q where γ (p), Φ (p) and Ψ (p) are vectorvalued functions of the parameters p arising from d1 (.)d2 (.), n1 (.)d2 (.) and n2 (.)d1 (.) terms, respectively. Let S q denote the set of solutions of c q (p) = c q (p∗ ). If the system is globally identifiable, then S q = {p∗ }. Similarly, after replacing ω(T , pT ) with the approximation in Eq. (9), x˙ q′ i can be rewritten as: d1 (·) d2 (·)x˙ q′ i (t) = (a0 + · · · aq T q +

. . . + aq′ T q )n1 (·)d2 (·) + n2 (·)d1 (·)

3.2. The proposed approach for identifiability

(11)

′

′

x˙ (t) = ω(T , pT )f 1 (x, u, p) + f 2 (x, u, p)

3

(15)

where xq′ i denotes the ith state variable when a q′ th order approximation is used. Again, the coefficients of the characteristic set, c q′ (p) are readily obtained by inspection of Eq. (15) and are: γ (p), aj (pT )Φ, (p) Ψ (p), j = 0, . . . , q′ . As before, γ (p) and Φ (p) and Ψ (p) are vector-valued functions of parameters p arising from d1 (.)d2 (.), n1 (.)d2 (.) and n2 (.)d1 (.) respectively. Let S q′ denote the set of solutions of c q′ (p) = c q′ (p∗ ). Since q′ > q, the coefficients of the input–output relations obtained from Eq. (14) are a subset of the coefficients of the input–output relation obtained from Eq. (15). Hence, a set of solutions obtained by the coefficient functions with the q′ th order approximation of the non-rational functions is a subset of the one obtained by the coefficient functions with the qth order approximation, i.e., S q′ ⊆ S q . Further, by construction p∗ ∈ S q′ , i.e., S q′ is not empty. Hence, if the model (9) is approximated by Eq. (10) and it is found to be identifiable, i.e., S q = {p∗ }, then the model (9) will be identifiable if approximated by Eq. (11), i.e., S q′ = {p∗ }. □ When a more general [q/q] order Padé approximation is used, then, a similar result can be obtained for nonlinear systems of the following class: x˙ (t) = ω(T , pT )f (xa , u, p) y(t) = x(t)

(16)

4

R. Jain, S. Narasimhan and N.P. Bhatt / Automatica 109 (2019) 108513

where f is a rational function. The Padé approximation of the non-rational function ω of order [q/q] can be given by

ω(T , pT ) =

a0 + a1 T + · · · + aq T q b0 + b1 T + · · · + bq T

, q

A2 ≡ y˙2 y2 +

Claim 6. If the non-rational function ω(T , pT ) in the model (9) is approximated by Eq. (17) and the approximated system is found to be identifiable, then, any higher order [q′ /q′ ]th approximation (q′ > q) will also result in an identifiable system. Proof. The proof is similar to Theorem 5 and only the important steps are shown below. For a [q/q]th order approximation, we have (b0 + · · · + bq T q )d1 (·)x˙ qi (t) = (a0 + · · · + aq T q )n1 (·)

(18)

By inspection of Eq. (18), the coefficients of the characteristic set, c q (p) are obtained as follows: bj (pT )γ (p) and aj (pT )Φ (p), j = 0, . . . , q and γ and Φ are vector-valued functions of the parameters p arising from n1 (·) and d1 (·). If the system is globally identifiable, then the solution set of c q (p) = c q (p ∗ ) is Sq = {p∗ }. Similarly, when the higher order Padé approximation [q′ /q′ ] is used, we have: (b′0 + · · · + b′q′ T q )d1 (·)x˙ q′ i (t) = (a′0 + · · · + a′q′ T q )n1 (·)

(19)

The coefficients of the solution set for c q′ (p) = c q′ (p∗ ), are: b′j (pT )γ (p) and a′j (pT )Φ (p), j = 0, . . . , q′ . The coefficients aj and a′j for j ≤ q are multiples of each other and likewise for bj and b′j . Hence, Sq′ ⊆ Sq = {p∗ }. Hence, if the model (16) is approximated by the qth order Padé and it is identifiable, then the model (16) approximated using any higher order q′ (> q) Padé approximation will also result in an identifiable system. □ Some remarks are as follows. Remark 7. Note that if the approximated or augmented model is not identifiable, then it does not mean that the original system is not identifiable. A negative result does not allow us to conclude that the original system is not identifiable. Theorem 5 and Claim 6 show that if the approximated system is identifiable, then higher order approximations also result in identifiable systems. Further, as q′ approaches towards infinity, our approximated model resembles the model (9) in the limit. Remark 8. In Theorem 5 and Claim 6, it is assumed that there is a single non-rational expression that is a function of only one state T . However, this assumption can be relaxed in a straightforward manner to several different non-rational expressions. Next, we revisit Example 4. Note that the system can be cast in the form of Eq. (9) with f 2 = 0. Hence, Theorem 5 (Taylor series approximation) and Claim 6 (Padé approximation) can be used to test identifiability. Example 9. The exponential term in Eq. (5) is replaced by the corresponding Padé approximation of order [1/1] as follows: 1− 2a

1+ 2a

. Simplifying Eq. (5) using the Padé approximation, we

get a following set of non-linear equations

⎡ ⎤ T − E /2 [ ] − k1o cA ⎢ ⎥ c˙A T + E /2 ⎥, x˙1 = ˙ = ⎢ ⎣ T T − E /2 ⎦ −h1 k1o cA T + E /2

A1 ≡ y˙1 y2 +

(17)

where a0 , . . . , aq and b0 , . . . , bq are functions of parameters pT .

e−a =

The characteristic set obtained for Eq. (20) is as follows2 1 2 1 2

A3 ≡ y1 − cA ,

E y˙1 + k1o y1 y2 −

1 2

E y˙2 + h1 k1o y1 y2 −

k1o E y1 , 1 2

(21)

h1 k1o Ey1 ,

A4 ≡ y2 − T

The coefficients of the input–output relations in Eq. (21) are (E /2, k1o , − k1o E /2, h1 k1o , −h1 k1o E /2). These coefficients are evaluated at parametric values of (k1o = 2, E = 1, h1 = 3) to get the following set of algebraic nonlinear equations: E = 1, k1o = k1o E = 2, h1 k1o = 6, h1 k1o E = 6

(22)

Solving the nonlinear equations (22) gives the following unique solution: E = 1, k1o = 2, h1 = 3. Hence, the model is globally identifiable. Now, if we consider the [2/2]th order the Padé 2

approximation of exponential function, e−a =

1− 2a + a12 2

1+ 2a + a12

. The char-

acteristic set obtained after simplifying the Padé approximation of Eq. (5) with two terms is as follows: A1 ≡y˙1 y22 + (1/2)E y˙1 y2 + (1/12)E 2 y˙1 + k1o y1 y22

− (1/2)k1o Ey1 y2 + (1/12)k1o E 2 y1 , A2 ≡y˙2 y22 + (1/2)E y˙2 y2 + (1/12)E 2 y˙2 + h1 k1o y1 y22

(23)

− (1/2)h1 k1o Ey1 y2 + (1/12)h1 k1o E y1 , A3 ≡y1 − ca , A4 ≡ y2 − T . 2

The coefficients of the input–output relations in Eq. (23) are (E /2, k1o , −k1o E /2, E 2 /12, k1o E 2 /12, h1 k10 , −h1 k1o E /2, h1 k1o E 2 /12). Again, these coefficients are evaluated at symbolic parametric value of (k1o = 2, E = 1, h1 = 3) to get the following set of algebraic nonlinear equations: E = 1, k1o = 2, k1o E = 2, E 2 = 1, k1o E 2 = 2, h1 k10 = 6, h1 k1o E = 6, h1 k1o E 2 = 6.

(24)

The solution of the nonlinear equations in (24) has unique solution: k1o = 2, E = 1, h1 = 3 and hence identifiable. This verifies Claim 6, i.e., the approximated system of higher order is also identifiable. Similarly, Taylor series can also be used to approximate the exponential nonlinearity. The coefficients of the input–output relations for the 1st and 2nd order Taylor series approximations are (k1o , Ek1o , h1 k1o , Eh1 k1o ) and (k1o , Ek1o , E 2 k1o /2, h1 k1o , Eh1 k1o , E 2 h1 k1o /2), respectively. These coefficients sets are evaluated at symbolic parametric value of (k1o = 2, E = 1, h1 = 3) and it is found that the solutions of both systems of nonlinear equations have a unique solution. Hence, the conditions of Theorem 5 are satisfied and all higher order approximations result in identifiable systems. Although the coefficient sets obtained by the Padé and Taylor series approximation are different, the implications on identifiability are the same. △ 4. Conclusion In this paper, we have shown the limitations of current differential algebra based methods for testing identifiability of systems with non-rational functions. It has been shown that the state augmentation approach for testing identifiability of systems with non-rational functions cannot be applied to a certain class of systems. Using Padé or polynomial approximation, the original system is approximated by a nonlinear system containing only rational functions. Hence, the identifiability tests based on differential algebra can be applied to the approximated system.

[ ] y1 =

cA T

(20)

2 In the interest of brevity, the subscripts of the order of approximation are suppressed

R. Jain, S. Narasimhan and N.P. Bhatt / Automatica 109 (2019) 108513

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