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Extended space method for parameter identifiability of DAE systems Li Chen a, Xi Chen a,n, Jixin Qian a, Zhen Yao b a b
State Key Laboratory of Industrial Control Technology, Department of Control Science & Engineering, Zhejiang University, Hangzhou 310027, China State Key Laboratory of Chemical Engineering, Department of Chemical & Biological Engineering, Zhejiang University, Hangzhou 310027, China
art ic l e i nf o
a b s t r a c t
Article history: Received 27 July 2013 Received in revised form 19 September 2013 Accepted 14 December 2013 This paper recommanded for publication by prof. A.B. Rad
Mathematical models of physical systems often have parameters that must be identified from physical data. This makes the analysis of the parameter identifiability of the given model system an essential prerequisite. Thus far, several methods have been proposed for analyzing the parameter identifiability of ordinary differential equation (ODE) systems. But, to the best of our knowledge, the parameter identifiability of differential algebraic equation (DAE) systems has scarcely been analyzed as a specific topic. Traditional differential algebraic (DA) methods developed for ODE systems are often applied directly on DAE systems. These methods, however, are not always applicable, e.g., when the prime ideal condition is not satisfied by a DAE system. In this paper, we propose a novel method to analyze the identifiability of DAE systems, based on the concept of space extension, through which the algebraic and differential variables can be decoupled. Furthermore, an inherent, low-dimensional, regular ODE system can be obtained, which is the external equivalent of the original DAE system. Subsequently, the differential algebraic (DA) method can then be used to analyze the identifiability of the lowdimension ODE system. Theoretical analysis is also presented for the proposed method. Two examples, including a simplified interaction model and an isothermal reactor system, are presented to illustrate the detailed steps and effectiveness of the proposed method. & 2013 ISA. Published by Elsevier Ltd. All rights reserved.
Keywords: Parameter identifiability Differential algebraic equation Differential algebra Extension
1. Introduction During the past decades, rapid advancements have been achieved in the understanding of dynamic systems characterized by differential-algebraic equations (DAEs) [1,2]. DAE systems arise in many areas of science and engineering, particularly in constrained mechanical systems, electronic circuits, and chemical engineering. Such systems often include a number of unknown physical parameters. As the behavior of a dynamic system strongly depends on its physical parameters, identifying them from experimental input–output (I/O) data is a challenging task. There are several open issues regarding parameter identification in dynamic systems. One of them is parameter identifiability [3–6], which characterizes the one-to-one property of the map from the parameters to the I/O variables of the systems. In other terms, parameter identifiability is concerned with the investigation of the relationship between the unknown parameters and the known I/O data under ideal conditions, with noise-free observation, error-free model structure, and continuous data. Thus far, several approaches have been proposed to understand the parameter identifiability of linear and non-linear dynamical
n
Corresponding author. Tel.: þ 86 571 87953966; fax: þ86 571 87951068. E-mail address: [email protected] (X. Chen).
systems. These approaches include transfer function method [6,7], similarity transformation method [8,9], power series expansion [10], and Volterra and generating power series approaches [11]. In particular, differential algebraic (DA) theory [12,13] has been applied successfully to study parameter identifiability [14–18]. These methods mainly focus on dynamic systems, described by ODEs. In fact, DAE systems are not identical to ODE systems because of the existence of algebraic equations [1,2]. For instance, DAE systems have been well known to possess the differentiation index concept, while the ODE systems do not. Therefore, identification of the apt method to check the parameter identifiability of DAE systems seems to be an interesting area of research. In the current paper, a novel method based on extended space is proposed to check the identifiability of DAE systems. The key idea of the proposed method is to find an inherent ODE system, the input– output behavior of which matches that of the DAE system. The externally equivalent ODE system has lower dimension than the original DAE system. In contrast to the classical DA method, the method proposed in this study does not require the ideal generated by the DAE system to be prime. In addition, it has been proven that the local identifiability of the inherent low-dimension ODE system can be considered as a sufficient condition for the local identifiability of the original DAE system. Section 2 introduces the key ideas from the study of DAE systems. In Section 3, the novel DA method is introduced to check the identifiability of DAE system. In Section 4, a
0019-0578/$ - see front matter & 2013 ISA. Published by Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.isatra.2013.12.014
Please cite this article as: Chen L, et al. Extended space method for parameter identifiability of DAE systems. ISA Transactions (2014), http://dx.doi.org/10.1016/j.isatra.2013.12.014i
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2
simplified interaction model and an isothermal reactor system are provided to illustrate the effectiveness of the new method. Section 5 summarizes the key conclusions to be derived from this study.
characteristic decomposition of the inherent regular ODE system can be computed by the DA method. 3.1. Extension
2. Problem formulation Consider a parameterized DAE system, ∑, described by DAEs as follows: 8 _ > < x ¼ f ðu; p; x; yÞ 0 ¼ gðu; p; x; yÞ ð1Þ > : z ¼ hðxÞ where u A Rnu , x A Rnx , y A Rny , z A Rnz and p A Θ D Rnp . Vectors x, y z denote differential variables, algebraic variables, and output variables of ∑, respectively. Input vector u is the differential functions of time t ðt A DÞ and vector p is the unknown parameters. f, g, and h are polynomial or rational functions defined on open sets, except for the subsets of Lebesgue measure zero. The differential index of ∑ is not greater than 1. As the purpose of the new method is mainly to look for an externally equivalent inherent ODE system from the set of equations of ∑, system ∑ is assumed to be locally solvable. Definition 1. The input-output behavior of system ∑ is the vectorvalued function ioΣ ðpÞ mapped from the accessible input u to the output z, i.e. ioΣ ðpÞ : u-z. Thus, according to the input-output behavior, the external equivalence and the identifiability of ∑ are defined as: Definition 2. Two system ∑ and ∑1 are externally equivalent if they have the same input-output behavior, i.e. ioΣ ðpÞ ¼ ioΣ 1 ðpÞ. Definition 3. System ∑ is globally identifiable if there exists for at least a generic point set pn A Θ, (at least) one accessible input u, such that equation ioΣ ðp; uÞ ¼ ioΣ ðpn ; uÞ has a unique solution p ¼ pn . n
Definition 4. System ∑ is locally identifiable at p A Θ if (at least) one accessible input u and open neighborhood U pn of pn exist, such that equation ioΣ ðp; uÞ ¼ ioΣ ðpn ; uÞ has a unique solution p ¼ pn . Definition 5. System ∑ is unidentifiable at pn A Θ if the solutions to the equation ioΣ ðp; uÞ ¼ ioΣ ðpn ; uÞ are not denumerable. These definitions of identifiability are equivalent to “weak” identifiability proposed by Tunali and Tarn [19], where the initial conditions can be manipulated by the experimenter to help identify the system parameters.
3. An extended space method Ever since Glad and Ljung [15,16] and Ollivier [17] have shown the concept of the characteristic set of a differential ideal [13] to be a very useful tool for identifiability, DA methods have been widely used to check the identifiability of ODE systems. These methods are suitable to state-space-description systems, which generate the prime ideals. However, the ideals generated by DAE systems are not necessarily prime ideals, and determination of whether the ideal is a prime ideal or not is still an open issue. In this paper, a novel extended space method is presented to analyze the parameter identifiability of DAE systems. This method does not require the prime ideal condition. It decouples the algebraic variables and differential variables from the model equations. As a result, an inherent, low-dimensional, regular ODE system can be obtained, which is externally equivalent to the original DAE system. The
To investigate the input-output behavior of system ∑, it is useful to decouple the algebraic variables and differential variables from the model equations. The concept of extension is thus introduced to deal with the algebraic variables by purely algebraic manipulation. Here, for the given accessible input u, the following definition is given: Definition 6. [20]: The based space J0 of ∑ is the space (t, x, y, z) of independent variable t and dependent variables x, y, z. The extended space J1 of ∑ is the space ðt; x; y; z; x_ Þ of independent variable t and dependent variables x; y; z; x_ . Remark. x_ are not derivatives of solutions of ∑, but rather formal coordinates on extended space J1 of ∑. Definition 7. Space J1 is the extension of space J0 if there is a projection π : J 1 -J 0 such that the following diagram commutes: j1
J 1 -D Rnx Rnx Rny Rnz π↓ 0 j0
ϕ↓ nx
J -R
R
ny
Rnz
where j0, j1, ϕ are the corresponding maps. System ∑ can be transformed into system ∑1 described by the “algebraic” equations on space J1 as, 8 > < v ¼ f ðu; p; x; yÞ 0 ¼ gðu; p; x; yÞ ð2Þ > : z ¼ hðxÞ where v represents the derivative of x. Lemma 1. If DAE system ∑ is locally solvable for all p A Θ on J0, then system ∑ is equivalent to system ∑1 on J1. Proof. Suppose Sol∑ to be denoted as the total of solutions of Eq. (1), and SolΣ1 to be denoted as the total of solutions of ∑1. Given the input u, as DAE system ∑ is locally solvable for all p A Θ, for arbitrary solution η ¼ ðx; y; zÞT A SolΣ , there is a map φ1 : tx; π 0 : t-η, (t A D) such that π 0 ðtÞ satisfies Eq. (1). According to the Cartan–Kähler theorem [21], there is a map φ2 : x-x_ , such that (ðφ2 ðφ1 ðtÞÞ; π 0 ðtÞÞT ) satisfies Eq. (2). Therefore, SolΣ DSolΣ 1 .□ On the other hand, for arbitrary solution η1 ¼ ðx; y; zÞT A SolΣ 1 , as v ¼ x_ and J 1 is the extension of J 0 , according to Definition 7, there is a projection map π : ðt; η1 ; vÞ-ðt; η1 Þ. So η1 satisfies Eq. (1), and is the solution of Eq. (1). Thus, η1 A SolΣ 1 DSolΣ . Therefore, SolΣ ¼ SolΣ1 . According to Sussmann [22], system ∑ is equivalent to system ∑1. As the two systems, ∑ and ∑1, are equivalent, the identifiability of ∑ is equivalent to that of ∑1. It is possible to use the purely algebraic manipulations to eliminate algebraic variablesy, and not alter the intrinsic identifiability of ∑. In this work, the elimination method [23] is adopted to deal with algebraic variables. 3.2. The elimination method Let the corresponding algebraic set of ∑1 be S ¼ fS1 ; S2 ; S3 g, where S1 ¼ v f ðu; p; x; yÞ; S2 ¼ gðu; p; x; yÞ; S3 ¼ z hðxÞ. The solution set SolΣ1 is the total zeros of S, which is denoted as zeroðSÞ. Choosing the elimination rank u ! z ! x ! v ! y, the fine
Please cite this article as: Chen L, et al. Extended space method for parameter identifiability of DAE systems. ISA Transactions (2014), http://dx.doi.org/10.1016/j.isatra.2013.12.014i
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irreducible triangular set T of S is computed as: 2 3 T 1 ðy; v; x; z; u; pÞ 6 7 6 T 2 ðv; x; z; u; pÞ 7 7: T ¼6 6 T 3 ðx; z; u; pÞ 7 4 5 T 4 ðz; u; pÞ
expressed as ðA1 ðu; zÞ; ⋯; Am ðu; zÞÞ ð3Þ
Notably, the subset {T 2 ; T 3 ; T 4 } does not depend on the algebraic variables y. In fact, the subset is obtained after the elimination of algebraic variables y from the algebraic set T, denoted as T^ 2 3 T 2 ðv; x; z; u; pÞ 6 7 ð4Þ T^ ¼ 4 T 3 ðx; z; u; pÞ 5 T 4 ðz; u; pÞ If Σ T^ is denoted as the regular system described by equations T^ ¼ 0: Lemma 2. System Σ T^ is externally equivalent to system ∑1. Proof. For T^ is the subset of T by eliminating the algebraic vari^ D zeroðTÞ. ables y, according to the elimination theory [23], zeroðTÞ On the other hand, T is one of the irreducible sets of S, and ^ D zeroðSÞ. For the given input u, zeroðTÞ D zeroðSÞ. Therefore, zeroðTÞ system Σ T^ has the same output z as system Σ 1 , and so, Σ T^ is externally equivalent to Σ 1 . As symbol v is the derivative function of x with respect to t, the parameter identifiability of Σ T^ cannot be conducted on the extended space J 1 and should be further pulled back to the based space J 0 to analyze parameter identifiability.
3.3. Parameter identifiability Let regular system ∑2 be described by Eq. (5) on J0 as: 8 > < T 4 ðu; z; pÞ ¼ 0 T 3 ðu; z; x; pÞ ¼ 0 > : T ðu; z; x; x_ ; pÞ ¼ 0 2
3
ð5Þ
Theorem. If system ∑2 is identifiable at p A Θ, then system ∑ is identifiable. Proof. As system Σ T^ on J1 is the extension of system ∑2 on J0, according to the Cartan-Kähler theory, and Definition 7, ∑2 is equivalent to Σ T^ . From Lemmas 1, and 2, system Σ T^ is externally equivalent to system ∑. Thus, system ∑2 is the external equivalent to system ∑. Hence, system ∑2 can be used to analyze the identifiability of system ∑. In other terms, if ∑2 is identifiable, then ∑ is identifiable too.□ Remark. Only on the condition of persistence of excitation [24] is the identifiability of ∑2 equivalent to that of ∑. Otherwise, the identifiability of ∑2 is a sufficient condition of that of ∑. The classical DA method [14,16] can be applied directly to analyze the parameter identifiability of regular ODE system ∑2. It mainly uses the Ritt0 s pseudo-reduction algorithm to computer the characteristic set of the differential ideal which is generated by the polynomial system. Then the relationship between I/O and unknown parameters can be obtained. The identifiability of unknown parameters can be analyzed from this relationship. Some essential theory about differential algebra has been provided in Appendix A. According to Ljung and Glad [16], the characteristic set of ∑2 with the rank u ! uðαÞ ! z ! zðβÞ ! p ! x ! xðγÞ can be
ðB1 ðu; z; p1 Þ; ⋯; Bq ðu; z; p1 ; ⋯; pq ÞÞ
ð6Þ
ðC 1 ðu; z; p; xÞ; ⋯; C n ðu; z; p; xÞÞ; where uðαÞ ; zðβÞ ; xðγÞ denote the derivatives of u; z; x, respectively; A ¼ fAi ; 1 r ir mg represents the I/O relationships; B ¼ fBj ; 1 r jr qg represents the relationship between I/O and unknown parameters; and C ¼ fC e ; 1 r e rng represents the relationship among I/O, unknown parameters, and state variables. If all Bj ; ð1 r j r qÞ, are of order zero and degree one in pj , then Σ 2 is globally identifiable. If all Bj ; ð1 r jr qÞ, are of order zero in pj and some Bj are of degree greater than 1 in pj , then Σ 2 is locally identifiable. In practice, an exhaustive summary [17] τ : Θ-Rκ of Σ 2 is used to analyze the parameter identifiability of Σ 2 , where κ is the set of total indices running over the monomial indices of Bj . For the arbitrary parameter vector pn A Θ, if the corresponding algebraic equations τðpÞ ¼ τðpn Þ have only one solution p ¼ pn , system Σ 2 is globally identifiable. On the other hand, if equations τðpÞ ¼ τðpn Þ have finite solutions, system Σ 2 is locally identifiable. 4. Case studies To illustrate the features and effectiveness of the proposed method, we have presented two examples to analyze the parameter identifiability of DAE systems. In the algebraic cases, the fine irreducible triangular sets are computed with the trisys [23] package. In the differential cases, characteristic sets are computed in Maple with the diffalg [25] package. Buchberger0 s algorithm [3] is used to compute the corresponding algebraic equations of the exhaustive summary of Σ 2 . Example 1. Consider the non-linear DAE system, ∑, that has been derived to model the interaction of Escherichia coli and somatic cells during persistent and acute bovine mastitis [26]: 8 x_ 1 ¼ p1 x1 xp12 xþ1 px2 > > > 3 > > < x_ ¼ 1 x2 x ðp þ p x Þ 2 2 7 8 1 y ð7Þ > > y ¼ p p5 ðp4 p6 Þ > 4 > x þ p 1 5 > : z1 ¼ x1 ; z2 ¼ x2 where x1,x2 are the differential variables, y is the algebraic variable, z1, z2 are the output variables, and p1, p2, p3, p4, p5, p6, p7, p8 are the unknown parameters. p ðp p6 Þ ∂g ¼ 1; ; rank As gðx1 ; yÞ ¼ y p4 þ 5 4 x1 þ p5 ∂y the differential index of system ∑ is 1. Introducing based space J0 and extended space J1, system ∑ is described by Eq. (7) on J0, and system ∑1 is described by Eq. (8) on J1 as: 8 v1 ¼ p1 x1 xp12 xþ1 px2 > > > 3 > > < v ¼ 1 x2 x ðp þ p x Þ 2 2 7 8 1 y ; ð8Þ > p ðp p Þ >y¼p 5 4 6 > 4 > x1 þ p5 > : z1 ¼ x1 ; z2 ¼ x2 where symbols v1, v2 represent the derivatives of x1, x2 with respect to t. With the elimination rank z1 ! z2 ! x1 ! x2 ! v1 ! v2 ! y, the fine irreducible triangular subset T^ of ∑1, described by polynomial set (9), is computed by trisys package as follows: T 1 : ðp3 þ z1 Þv1 p1 z21 p1 p3 z1 þ p2 z1 z2 T 2 : ðp4 z1 þ p5 p6 Þv2 þ p8 z21 z22 þ p5 p8 z1 z22
Please cite this article as: Chen L, et al. Extended space method for parameter identifiability of DAE systems. ISA Transactions (2014), http://dx.doi.org/10.1016/j.isatra.2013.12.014i
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4
þ p7 z1 z22 þ p5 p7 z22 p4 p8 z21 z2 p5 p6 p8 z1 z2 p4 p7 z 1 z 2 p5 p6 p7 z 2 T 3 : z 1 x1 ;
T 4 : z 2 x2 :
ð9Þ 1
Let u ¼ yA;in ðtÞ, x ¼ ½C A ðtÞ; C D ðtÞ; C L ðtÞ; C N ðtÞT ,y ¼ ½yA ðtÞ; yD ðtÞ; yL ðtÞ; yN ðtÞ; yN;in ðtÞT , p ¼ ½k1 ; k2 andz ¼ C L ðtÞ. The system ∑ is solvable, as shown by Ben–Zvi et al. [27]. As rankð∂g=∂yÞ ¼ 5, the differential index of ∑ is 1.
0
Pulling extended space J back to based space J , regular system ∑2 can be obtained as: 8 2 _ > < ðp3 þ z1 Þx1 ¼ p1 z1 þ p1 p3 z1 p2 z1 z2 ðp4 z1 þ p5 p6 Þx_ 2 ¼ p8 z21 z22 p5 p8 z1 z22 p7 z1 z22 p5 p7 z22 þ p4 p8 z21 z2 þ p5 p6 p8 z1 z2 þ p4 p7 z1 z2 þ p5 p6 p7 z2 > : z1 ¼ x1 ; z2 ¼ x2 It is worth noting that the exhaustive summary τ of ∑2 can be easily deduced from Eq. (10), sincez_ 1 ¼ x_ 1 , z_ 2 ¼ x_ 2 . The computation of the characteristic sets of ∑2 is unnecessary and is hence skipped here. For arbitrary parameter values, pn ¼ [c1,c2,c3,c4,c5,c6, c7,c8]T p3 ¼ c3 ; p1 ¼ c1 ; p1 p3 ¼ c1 c3 ; p2 ¼ c2 ; p5 p6 ¼ c5 c6 ; p4 ¼ c4 ; p8 ¼ c8 ; p5 p8 ¼ c5 c8 ; p7 ¼ c7 ; p5 p7 ¼ c5 c7 ; p4 p8 ¼ c4 c8 ; p5 p6 p8 ¼ c5 c6 c8 ; p4 p7 ¼ c4 c7 ; p5 p6 p7 ¼ c5 c6 c7 ð11Þ Solving Eq. (11) using Buchberger0 s algorithm yields the following results:
ð10Þ
Extending J0 to J1, system ∑ described by Eq. (12) is equivalent to system ∑1, described by Eq. (13) as: 8 V ν1 ¼ uF k1 x1 V Fy1 > > > > > V ν2 ¼ k1 x1 V k2 x2 V Fy2 > > > > > V ν3 ¼ k2 x2 V Fy3 > > > > > V ν ¼ y5 F Fy4 > > > 4 > > < y1 þy2 þy3 þ y4 1 ¼ 0 V ð13Þ ¼0 y1 x1 ntotal > > > > V > y2 x2 ntotal ¼ 0 > > > > > V > > > y3 x3 ntotal ¼ 0 > > > > > y5 þu 1 ¼ 0 > > :z ¼x 3 By choosing the elimination rank u ! z ! x ! v ! y, the fine irreducible triangular set T^ of ∑1, described by polynomial set (14) is computed by trisys package as follows:
p1 ¼ c1 ; p2 ¼ c2 ; p3 ¼ c3 ; p4 ¼ c4 ; p5 ¼ c5 ; p6 ¼ c 6 ; p7 ¼ c 7 ; p8 ¼ c 8 : Therefore, according to Definition 3, DAE system ∑ is globally identifiable. The result is in accordance with the result obtained by Margaria et al. [26].
T 2 : ntotal k1 x1 ntotal v2 ðF þ ntotal k2 Þx2
Example 2. Consider the gas-phase reactions in a continuous isothermal reactor [27]
T 4 : ðntotal V k2 þ VFÞx2 ntotal V ðv3 þ v4 Þ þ VFx1 ntotal Fu
k1
A-D
T 1 : ntotal Vv1 þ ðVF þ ntotal Vk1 Þx1 ntotal Fu T 3 : ntotal k2 x2 ntotal v3 Fz T 5 : z x3
k2
D-L
ð14Þ 1
Then, by pulling J back to J , regular system ∑2 can be obtained as described in Eq. (15): 8 ntotal V x_ 1 ¼ ðV F þ ntotal Vk1 Þx1 þntotal Fu > > > > > _ > < ntotal x2 ¼ ntotal k1 x1 ðF þ ntotal k2 Þx2 ntotal x_ 3 ¼ ntotal k2 x2 Fz ð15Þ > > > > ntotal V ðx_ 3 þ x_ 4 Þ ¼ ðntotal Vk2 þ VFÞx2 þ VFx1 ntotal Fu > > : z ¼ x3
to have been modeled using the following DAE system ∑: 8 dC A ðtÞ V dt ¼ yA;in ðtÞF k1 C A ðtÞV FyA ðtÞ > > > > > dC D ðtÞ > > > V dt ¼ k1 C A ðtÞV k2 C D ðtÞV FyD ðtÞ > > > dC ðtÞ > > V dtL ¼ k2 C D ðtÞV FyL ðtÞ > > > > dC N ðtÞ > > < V dt ¼ yN;in ðtÞF FyN ðtÞ yA ðtÞ þ yD ðtÞ þyL ðtÞ þ yN ðtÞ 1 ¼ 0 > > V > ¼0 > yA ðtÞ C A ðtÞntotal > > > > > y ðtÞ C D ðtÞ V ¼ 0 > D > ntotal > > > V > ¼0 yL ðtÞ C L ðtÞntotal > > > > : y ðtÞ þ y ðtÞ 1 ¼ 0 N;in A;in
0
ð12Þ
where C i and yi (with i taking on values in the set fA; D; L; Ng) are the volumetric concentrations and mole fractions, respectively, of the jth species. The volume is V, and the unknown parameters are k1 ; k2 . The input is the inlet mole fraction A, yA;in ðtÞ. The inlet contains only inert gas N and feed A. The inlet flow rate F is known and constant. ntotal is calculated according to the ideal gas law. We assume that the pressure and the temperature are known. In addition, we assume that ntotal is a positive integer, and k1 ; k2 , and V are all positive real numbers. Finally, we assume that only the measurement of C L ðtÞ is available.
Obviously, system ∑2 is a reduced regular ODE system. Subsequently, the diffalg package is used to calculate the characteristic sets of ∑2. As far as the parameter identifiability is concerned, only the polynomials that include the relationship between the I/O variables u; z and unknown parameters k1 ; k2 are listed as follows: ::: ð16Þ z þδ2 z€ þ δ1 z_ þ δ0 z þ ι0 u here, ι0 ¼ k1 k2 VF ; δ0 ¼ k1 k2 ζ þ ðk1 þ k2 Þζ 2 þ ζ 3 ; δ1 ¼ 2ðk1 þk2 Þζ þ k1 k2 þ 3ζ 2 ; δ2 ¼ k1 þ k2 þ3ζ; ζ¼
F ; ntotal
ð17Þ
Please cite this article as: Chen L, et al. Extended space method for parameter identifiability of DAE systems. ISA Transactions (2014), http://dx.doi.org/10.1016/j.isatra.2013.12.014i
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The exhaustive summary τ of ∑2 is described as Eq. (18) at symbolic values of pn ¼[c1,c2]T: k1 k2 VF ¼ c1 c2 VF ; k1 k2 ζ þ ðk1 þ k2 Þζ 2 þ ζ 3 ¼ ðc1 c2 ζ þðc1 þ c2 Þζ 2 þζ 3 Þ; 2ðk1 þ k2 Þζ þk1 k2 þ 3ζ 2 ¼ ð2ðc1 þ c2 Þζ þ c1 c2 þ 3ζ 2 Þ; k1 þ k2 þ 3ζ ¼ ðc1 þc2 þ3ζÞ
ð18Þ
0
Solving Eq. (18) using Buchberger s algorithm yields the following: ( ( k1 ¼ c 1 k1 ¼ c1 or : k2 ¼ c2 k2 ¼ c 2 According to Definition 4, DAE system ∑ is locally identifiable, which agrees with the results obtained by Ben–Zvi et al. [27].
5. Conclusions In this paper, we have proposed a new method for analyzing the parameter identifiability of DAE systems. It decouples the algebraic and differential variables from the model equations by space extension. An inherent, regular, low-dimensional ODE system ∑2 can be obtained, which is the external equivalent to the original DAE system ∑. Subsequently, the DA method is used to analyze the identifiability of the regular ODE system. The proposed method does not require the ideal generated by the DAE system to be prime, which makes the method a general one for DAE systems. Eventually, examples are presented to illustrate its effectiveness. The proposed method maybe limited by the development of computer algebra, and may encounter difficulties in dealing with large-scale problems because of limitation in symbolic computational techniques. Acknowledgments We gratefully acknowledge the financial support of 973 Program (No.2012CB720500) and National Natural Science Foundation of China (Nos. 61074148 & 21206149). Appendix A The appendix contains some essential differential algebra theory [12,13,23] as follows: A differential ring ℜ½X is a set of all polynomials in indeterminate X and its derivative X_ with coefficients in the field ℜ. For a set S A ℜ½X, the differential ideal IðSÞ is a subset of ℜ½X containingS, which satisfies S_ i A IðSÞ and Σ ni¼ 1 f i Si A IðSÞ, (f i A ℜðXÞ; Si A IðSÞ).IðSÞ is prime if Si Sj A IðSÞ implies Si A IðSÞ or Sj A IðSÞ, and is radical if Sm i A IðSÞ implies Si A IðSÞ. To compute the characteristic set of IðSÞ, a rank is introduced, as denoted by symbol ! . The leader of a polynomial is the indeterminate with the highest rank polynomial. Polynomials are ranked in terms of their leaders. A polynomial S1 with leader lvðS1 Þ is said to be of lower rank than polynomial S2 with leader lvðS2 Þ, if lvðS1 Þ ! lvðS2 Þ or if lvðS1 Þ ¼lvðS2 Þ and degðS1 ; lvðS1 ÞÞ o degðS2 ; lvðS2 ÞÞ, (where lvðSi Þ and degðSi ; lvðSi ÞÞ are the leader and degree of polynomial Si (i ¼ 1; 2), respectively). A set S ¼ ½S1 ; S2 ; ⋯; Sr is a triangular set iflvðS1 Þ ! lvðS2 Þ ! ⋯ ! lvðSr Þ. Algebraic triangular set S is irreducible if each Si is irreducible overℜðlvðS1 Þ; ⋯; lvðSi 1 ÞÞ, where lvðS1 Þ; ⋯; lvðSi 1 Þ are adjoined as algebraic elements by the equations Si ¼ 0; ⋯; Si 1 ¼ 0 in the case i 4 1. An algebraic triangular set S is fine if premðiniðSi Þ; SÞ a 0
5
(prem means pseudo-remainder). A polynomial Si is reduced with respect to another polynomial Sj if the former contains neither the leader of Sj with equal or greater degree nor its derivatives. A set of polynomials S1 ; ⋯; Sr , reduced with respect to one another, is called an auto-reduced set. The lowest-rank auto-reduced set is called a characteristic set. The peculiarity of the characteristic set is that it summarizes all information contained in the same ideal into a finite number of polynomials. References [1] Brenan KE, Campbell SL, Campbell SLV, Petzold LR. Numerical solution of initial-value problems in differential-algebraic equations. Soc Ind Appl Math Philadelphia 1996. [2] Riaza R. Differential-algebraic systems: analytical aspects and circuit applications. Singapore: World Scientific Publishing Co. Pte. Ltd; 2008. [3] Audoly S, D0 Angio L, Saccomani MP, Cobelli C. Global identifiability of linear compartmental models-a computer algebra algorithm. IEEE Trans Biomed Eng 1998;45(1):36–47. [4] Nazarian P, Haeri M, Tavazoei MS. Identifiability of fractional order systems using input output frequency contents. ISA Trans 2010;49:207–14. [5] Mei H, Li S. Decentralized identification for multivariable integrating processes with time delays from closed-loop step tests. ISA Trans 2007;46:189–98. [6] Godfrey KR, DiStefano JJ. Identifiability of model parameters. In: Walter E, editor. Identifiability of Parametric Models. Oxford: Pergamon Press; 1987. [7] Jacquez JA, Greif P. Numerical parameter identifiability and estimability: Integrating identifiability, estimability, and optimal sampling design. Math Biosci 1985;77(1–2):201–27. [8] Evans ND, Chapman MJ, Chappell MJ, Godfrey KR. Identifiability of uncontrolled nonlinear rational systems. Automatica 2002;38(10):1799–805. [9] Vajda S, Godfrey KR, Rabitz H. Similarity transformation approach to identifiability analysis of nonlinear compartmental models. Math Biosci 1989;93 (2):217–48. [10] Pohjanpalo H. System identifiability based on the power series expansion of the solution. Math Biosci 1978;41(1–2):21–33. [11] Lecourtier Y, Lamnabhi-Lagarrigue F, Walter E. Volterra and generating power series approaches to identifiability testing. In: Walter E, editor. Identifiability of parametric models. Oxford: Pergamon Press; 1987. p. 50–66. [12] Kolchin ER. Differential algebra and algebraic groups. New York: Academic Press; 1973. [13] Ritt JF. Differential algebra. New York: American Mathematical Society; 1950. [14] Audoly S, Bellu G, D0 Angio L, Saccomani MP, Cobelli C. Global identifiability of nonlinear models of biological systems. IEEE Trans Biomed Eng 2001;48 (1):55–65. [15] Glad ST. Differential algebraic modelling of nonlinear systems. In: Kaashoek JHvS MA, Ran ACM, editors. Proceedings of the international symposium MTNS-89: realization and modelling in system theory. Basel: Birkhäuser; 1990. p. 97–105. [16] Ljung L, Glad T. On global identifiability for arbitrary model parametrizations. Automatica 1994;30(2):265–76. [17] Ollivier F. Le problème de l0 identifiabilité structurelle globale: approche théorique. Méthodes effectives et bornes de complexité. Paris, France: École polytechnique; 1990 ([PhD thesis]). [18] Saccomani MP, Audoly S, D0 Angiò L. Parameter identifiability of nonlinear systems: the role of initial conditions. Automatica 2003;39(4):619–32. [19] Tunali ET, Tarn TJ. New results for identifiability of nonlinear-systems. IEEE Trans Autom Control 1987;32(2):146–54. [20] Chen L, Chen X, Qian J,Yao Z. A new method for parameter identifiability of DAE systems. In: Proceedings of the 10th IEEE international conference on control and automation. IEEE, Hangzhou, China; July 12–14 2013. [21] Kuranishi M, On E. Cartan0 s prolongation theorem of exterior differential systems. Am J Math 1957;79(1):1–47. [22] Sussmann HJ. Existence and uniqueness of minimal realizations of nonlinear systems. Math. Syst. Theory 1977;10:263–84. [23] Wang D. Elimination methods. Austria: Springer; 2001. [24] Glad S, Ljung L. Model structure identifiability and persistence of excitation. Decision and control 1990. In: Proceedings of the 29th IEEE conference. IEEE, Honolulu, Hawali; 1990. p. 3236–40. [25] Boulier F, Lazard D, Ollivier F., Petitot M. Representation for the radical of a finitely generated differential ideal. In: Proceedings of the 1995 International symposium on symbolic and algebraic computation. ACM; 1995. p. 158–66. [26] Margaria G, Riccomagno E, Chappell MJ, Wynn HP. Differential algebra methods for the study of the structural identifiability of rational function state-space models in the biosciences. Math Biosci 2001;174:1–26. [27] Ben-Zvi A, McLellan PJ, McAuley KB. Identifiability of linear time-invariant differential-algebraic systems. 2. The differential-algebraic approach. Ind & Eng Chem Res 2004;43:1251–9.
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Extended space method for parameter identifiability of DAE systems Li Chen a, Xi Chen a,n, Jixin Qian a, Zhen Yao b a b
State Key Laboratory of Industrial Control Technology, Department of Control Science & Engineering, Zhejiang University, Hangzhou 310027, China State Key Laboratory of Chemical Engineering, Department of Chemical & Biological Engineering, Zhejiang University, Hangzhou 310027, China
art ic l e i nf o
a b s t r a c t
Article history: Received 27 July 2013 Received in revised form 19 September 2013 Accepted 14 December 2013 This paper recommanded for publication by prof. A.B. Rad
Mathematical models of physical systems often have parameters that must be identified from physical data. This makes the analysis of the parameter identifiability of the given model system an essential prerequisite. Thus far, several methods have been proposed for analyzing the parameter identifiability of ordinary differential equation (ODE) systems. But, to the best of our knowledge, the parameter identifiability of differential algebraic equation (DAE) systems has scarcely been analyzed as a specific topic. Traditional differential algebraic (DA) methods developed for ODE systems are often applied directly on DAE systems. These methods, however, are not always applicable, e.g., when the prime ideal condition is not satisfied by a DAE system. In this paper, we propose a novel method to analyze the identifiability of DAE systems, based on the concept of space extension, through which the algebraic and differential variables can be decoupled. Furthermore, an inherent, low-dimensional, regular ODE system can be obtained, which is the external equivalent of the original DAE system. Subsequently, the differential algebraic (DA) method can then be used to analyze the identifiability of the lowdimension ODE system. Theoretical analysis is also presented for the proposed method. Two examples, including a simplified interaction model and an isothermal reactor system, are presented to illustrate the detailed steps and effectiveness of the proposed method. & 2013 ISA. Published by Elsevier Ltd. All rights reserved.
Keywords: Parameter identifiability Differential algebraic equation Differential algebra Extension
1. Introduction During the past decades, rapid advancements have been achieved in the understanding of dynamic systems characterized by differential-algebraic equations (DAEs) [1,2]. DAE systems arise in many areas of science and engineering, particularly in constrained mechanical systems, electronic circuits, and chemical engineering. Such systems often include a number of unknown physical parameters. As the behavior of a dynamic system strongly depends on its physical parameters, identifying them from experimental input–output (I/O) data is a challenging task. There are several open issues regarding parameter identification in dynamic systems. One of them is parameter identifiability [3–6], which characterizes the one-to-one property of the map from the parameters to the I/O variables of the systems. In other terms, parameter identifiability is concerned with the investigation of the relationship between the unknown parameters and the known I/O data under ideal conditions, with noise-free observation, error-free model structure, and continuous data. Thus far, several approaches have been proposed to understand the parameter identifiability of linear and non-linear dynamical
n
Corresponding author. Tel.: þ 86 571 87953966; fax: þ86 571 87951068. E-mail address: [email protected] (X. Chen).
systems. These approaches include transfer function method [6,7], similarity transformation method [8,9], power series expansion [10], and Volterra and generating power series approaches [11]. In particular, differential algebraic (DA) theory [12,13] has been applied successfully to study parameter identifiability [14–18]. These methods mainly focus on dynamic systems, described by ODEs. In fact, DAE systems are not identical to ODE systems because of the existence of algebraic equations [1,2]. For instance, DAE systems have been well known to possess the differentiation index concept, while the ODE systems do not. Therefore, identification of the apt method to check the parameter identifiability of DAE systems seems to be an interesting area of research. In the current paper, a novel method based on extended space is proposed to check the identifiability of DAE systems. The key idea of the proposed method is to find an inherent ODE system, the input– output behavior of which matches that of the DAE system. The externally equivalent ODE system has lower dimension than the original DAE system. In contrast to the classical DA method, the method proposed in this study does not require the ideal generated by the DAE system to be prime. In addition, it has been proven that the local identifiability of the inherent low-dimension ODE system can be considered as a sufficient condition for the local identifiability of the original DAE system. Section 2 introduces the key ideas from the study of DAE systems. In Section 3, the novel DA method is introduced to check the identifiability of DAE system. In Section 4, a
0019-0578/$ - see front matter & 2013 ISA. Published by Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.isatra.2013.12.014
Please cite this article as: Chen L, et al. Extended space method for parameter identifiability of DAE systems. ISA Transactions (2014), http://dx.doi.org/10.1016/j.isatra.2013.12.014i
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2
simplified interaction model and an isothermal reactor system are provided to illustrate the effectiveness of the new method. Section 5 summarizes the key conclusions to be derived from this study.
characteristic decomposition of the inherent regular ODE system can be computed by the DA method. 3.1. Extension
2. Problem formulation Consider a parameterized DAE system, ∑, described by DAEs as follows: 8 _ > < x ¼ f ðu; p; x; yÞ 0 ¼ gðu; p; x; yÞ ð1Þ > : z ¼ hðxÞ where u A Rnu , x A Rnx , y A Rny , z A Rnz and p A Θ D Rnp . Vectors x, y z denote differential variables, algebraic variables, and output variables of ∑, respectively. Input vector u is the differential functions of time t ðt A DÞ and vector p is the unknown parameters. f, g, and h are polynomial or rational functions defined on open sets, except for the subsets of Lebesgue measure zero. The differential index of ∑ is not greater than 1. As the purpose of the new method is mainly to look for an externally equivalent inherent ODE system from the set of equations of ∑, system ∑ is assumed to be locally solvable. Definition 1. The input-output behavior of system ∑ is the vectorvalued function ioΣ ðpÞ mapped from the accessible input u to the output z, i.e. ioΣ ðpÞ : u-z. Thus, according to the input-output behavior, the external equivalence and the identifiability of ∑ are defined as: Definition 2. Two system ∑ and ∑1 are externally equivalent if they have the same input-output behavior, i.e. ioΣ ðpÞ ¼ ioΣ 1 ðpÞ. Definition 3. System ∑ is globally identifiable if there exists for at least a generic point set pn A Θ, (at least) one accessible input u, such that equation ioΣ ðp; uÞ ¼ ioΣ ðpn ; uÞ has a unique solution p ¼ pn . n
Definition 4. System ∑ is locally identifiable at p A Θ if (at least) one accessible input u and open neighborhood U pn of pn exist, such that equation ioΣ ðp; uÞ ¼ ioΣ ðpn ; uÞ has a unique solution p ¼ pn . Definition 5. System ∑ is unidentifiable at pn A Θ if the solutions to the equation ioΣ ðp; uÞ ¼ ioΣ ðpn ; uÞ are not denumerable. These definitions of identifiability are equivalent to “weak” identifiability proposed by Tunali and Tarn [19], where the initial conditions can be manipulated by the experimenter to help identify the system parameters.
3. An extended space method Ever since Glad and Ljung [15,16] and Ollivier [17] have shown the concept of the characteristic set of a differential ideal [13] to be a very useful tool for identifiability, DA methods have been widely used to check the identifiability of ODE systems. These methods are suitable to state-space-description systems, which generate the prime ideals. However, the ideals generated by DAE systems are not necessarily prime ideals, and determination of whether the ideal is a prime ideal or not is still an open issue. In this paper, a novel extended space method is presented to analyze the parameter identifiability of DAE systems. This method does not require the prime ideal condition. It decouples the algebraic variables and differential variables from the model equations. As a result, an inherent, low-dimensional, regular ODE system can be obtained, which is externally equivalent to the original DAE system. The
To investigate the input-output behavior of system ∑, it is useful to decouple the algebraic variables and differential variables from the model equations. The concept of extension is thus introduced to deal with the algebraic variables by purely algebraic manipulation. Here, for the given accessible input u, the following definition is given: Definition 6. [20]: The based space J0 of ∑ is the space (t, x, y, z) of independent variable t and dependent variables x, y, z. The extended space J1 of ∑ is the space ðt; x; y; z; x_ Þ of independent variable t and dependent variables x; y; z; x_ . Remark. x_ are not derivatives of solutions of ∑, but rather formal coordinates on extended space J1 of ∑. Definition 7. Space J1 is the extension of space J0 if there is a projection π : J 1 -J 0 such that the following diagram commutes: j1
J 1 -D Rnx Rnx Rny Rnz π↓ 0 j0
ϕ↓ nx
J -R
R
ny
Rnz
where j0, j1, ϕ are the corresponding maps. System ∑ can be transformed into system ∑1 described by the “algebraic” equations on space J1 as, 8 > < v ¼ f ðu; p; x; yÞ 0 ¼ gðu; p; x; yÞ ð2Þ > : z ¼ hðxÞ where v represents the derivative of x. Lemma 1. If DAE system ∑ is locally solvable for all p A Θ on J0, then system ∑ is equivalent to system ∑1 on J1. Proof. Suppose Sol∑ to be denoted as the total of solutions of Eq. (1), and SolΣ1 to be denoted as the total of solutions of ∑1. Given the input u, as DAE system ∑ is locally solvable for all p A Θ, for arbitrary solution η ¼ ðx; y; zÞT A SolΣ , there is a map φ1 : tx; π 0 : t-η, (t A D) such that π 0 ðtÞ satisfies Eq. (1). According to the Cartan–Kähler theorem [21], there is a map φ2 : x-x_ , such that (ðφ2 ðφ1 ðtÞÞ; π 0 ðtÞÞT ) satisfies Eq. (2). Therefore, SolΣ DSolΣ 1 .□ On the other hand, for arbitrary solution η1 ¼ ðx; y; zÞT A SolΣ 1 , as v ¼ x_ and J 1 is the extension of J 0 , according to Definition 7, there is a projection map π : ðt; η1 ; vÞ-ðt; η1 Þ. So η1 satisfies Eq. (1), and is the solution of Eq. (1). Thus, η1 A SolΣ 1 DSolΣ . Therefore, SolΣ ¼ SolΣ1 . According to Sussmann [22], system ∑ is equivalent to system ∑1. As the two systems, ∑ and ∑1, are equivalent, the identifiability of ∑ is equivalent to that of ∑1. It is possible to use the purely algebraic manipulations to eliminate algebraic variablesy, and not alter the intrinsic identifiability of ∑. In this work, the elimination method [23] is adopted to deal with algebraic variables. 3.2. The elimination method Let the corresponding algebraic set of ∑1 be S ¼ fS1 ; S2 ; S3 g, where S1 ¼ v f ðu; p; x; yÞ; S2 ¼ gðu; p; x; yÞ; S3 ¼ z hðxÞ. The solution set SolΣ1 is the total zeros of S, which is denoted as zeroðSÞ. Choosing the elimination rank u ! z ! x ! v ! y, the fine
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irreducible triangular set T of S is computed as: 2 3 T 1 ðy; v; x; z; u; pÞ 6 7 6 T 2 ðv; x; z; u; pÞ 7 7: T ¼6 6 T 3 ðx; z; u; pÞ 7 4 5 T 4 ðz; u; pÞ
expressed as ðA1 ðu; zÞ; ⋯; Am ðu; zÞÞ ð3Þ
Notably, the subset {T 2 ; T 3 ; T 4 } does not depend on the algebraic variables y. In fact, the subset is obtained after the elimination of algebraic variables y from the algebraic set T, denoted as T^ 2 3 T 2 ðv; x; z; u; pÞ 6 7 ð4Þ T^ ¼ 4 T 3 ðx; z; u; pÞ 5 T 4 ðz; u; pÞ If Σ T^ is denoted as the regular system described by equations T^ ¼ 0: Lemma 2. System Σ T^ is externally equivalent to system ∑1. Proof. For T^ is the subset of T by eliminating the algebraic vari^ D zeroðTÞ. ables y, according to the elimination theory [23], zeroðTÞ On the other hand, T is one of the irreducible sets of S, and ^ D zeroðSÞ. For the given input u, zeroðTÞ D zeroðSÞ. Therefore, zeroðTÞ system Σ T^ has the same output z as system Σ 1 , and so, Σ T^ is externally equivalent to Σ 1 . As symbol v is the derivative function of x with respect to t, the parameter identifiability of Σ T^ cannot be conducted on the extended space J 1 and should be further pulled back to the based space J 0 to analyze parameter identifiability.
3.3. Parameter identifiability Let regular system ∑2 be described by Eq. (5) on J0 as: 8 > < T 4 ðu; z; pÞ ¼ 0 T 3 ðu; z; x; pÞ ¼ 0 > : T ðu; z; x; x_ ; pÞ ¼ 0 2
3
ð5Þ
Theorem. If system ∑2 is identifiable at p A Θ, then system ∑ is identifiable. Proof. As system Σ T^ on J1 is the extension of system ∑2 on J0, according to the Cartan-Kähler theory, and Definition 7, ∑2 is equivalent to Σ T^ . From Lemmas 1, and 2, system Σ T^ is externally equivalent to system ∑. Thus, system ∑2 is the external equivalent to system ∑. Hence, system ∑2 can be used to analyze the identifiability of system ∑. In other terms, if ∑2 is identifiable, then ∑ is identifiable too.□ Remark. Only on the condition of persistence of excitation [24] is the identifiability of ∑2 equivalent to that of ∑. Otherwise, the identifiability of ∑2 is a sufficient condition of that of ∑. The classical DA method [14,16] can be applied directly to analyze the parameter identifiability of regular ODE system ∑2. It mainly uses the Ritt0 s pseudo-reduction algorithm to computer the characteristic set of the differential ideal which is generated by the polynomial system. Then the relationship between I/O and unknown parameters can be obtained. The identifiability of unknown parameters can be analyzed from this relationship. Some essential theory about differential algebra has been provided in Appendix A. According to Ljung and Glad [16], the characteristic set of ∑2 with the rank u ! uðαÞ ! z ! zðβÞ ! p ! x ! xðγÞ can be
ðB1 ðu; z; p1 Þ; ⋯; Bq ðu; z; p1 ; ⋯; pq ÞÞ
ð6Þ
ðC 1 ðu; z; p; xÞ; ⋯; C n ðu; z; p; xÞÞ; where uðαÞ ; zðβÞ ; xðγÞ denote the derivatives of u; z; x, respectively; A ¼ fAi ; 1 r ir mg represents the I/O relationships; B ¼ fBj ; 1 r jr qg represents the relationship between I/O and unknown parameters; and C ¼ fC e ; 1 r e rng represents the relationship among I/O, unknown parameters, and state variables. If all Bj ; ð1 r j r qÞ, are of order zero and degree one in pj , then Σ 2 is globally identifiable. If all Bj ; ð1 r jr qÞ, are of order zero in pj and some Bj are of degree greater than 1 in pj , then Σ 2 is locally identifiable. In practice, an exhaustive summary [17] τ : Θ-Rκ of Σ 2 is used to analyze the parameter identifiability of Σ 2 , where κ is the set of total indices running over the monomial indices of Bj . For the arbitrary parameter vector pn A Θ, if the corresponding algebraic equations τðpÞ ¼ τðpn Þ have only one solution p ¼ pn , system Σ 2 is globally identifiable. On the other hand, if equations τðpÞ ¼ τðpn Þ have finite solutions, system Σ 2 is locally identifiable. 4. Case studies To illustrate the features and effectiveness of the proposed method, we have presented two examples to analyze the parameter identifiability of DAE systems. In the algebraic cases, the fine irreducible triangular sets are computed with the trisys [23] package. In the differential cases, characteristic sets are computed in Maple with the diffalg [25] package. Buchberger0 s algorithm [3] is used to compute the corresponding algebraic equations of the exhaustive summary of Σ 2 . Example 1. Consider the non-linear DAE system, ∑, that has been derived to model the interaction of Escherichia coli and somatic cells during persistent and acute bovine mastitis [26]: 8 x_ 1 ¼ p1 x1 xp12 xþ1 px2 > > > 3 > > < x_ ¼ 1 x2 x ðp þ p x Þ 2 2 7 8 1 y ð7Þ > > y ¼ p p5 ðp4 p6 Þ > 4 > x þ p 1 5 > : z1 ¼ x1 ; z2 ¼ x2 where x1,x2 are the differential variables, y is the algebraic variable, z1, z2 are the output variables, and p1, p2, p3, p4, p5, p6, p7, p8 are the unknown parameters. p ðp p6 Þ ∂g ¼ 1; ; rank As gðx1 ; yÞ ¼ y p4 þ 5 4 x1 þ p5 ∂y the differential index of system ∑ is 1. Introducing based space J0 and extended space J1, system ∑ is described by Eq. (7) on J0, and system ∑1 is described by Eq. (8) on J1 as: 8 v1 ¼ p1 x1 xp12 xþ1 px2 > > > 3 > > < v ¼ 1 x2 x ðp þ p x Þ 2 2 7 8 1 y ; ð8Þ > p ðp p Þ >y¼p 5 4 6 > 4 > x1 þ p5 > : z1 ¼ x1 ; z2 ¼ x2 where symbols v1, v2 represent the derivatives of x1, x2 with respect to t. With the elimination rank z1 ! z2 ! x1 ! x2 ! v1 ! v2 ! y, the fine irreducible triangular subset T^ of ∑1, described by polynomial set (9), is computed by trisys package as follows: T 1 : ðp3 þ z1 Þv1 p1 z21 p1 p3 z1 þ p2 z1 z2 T 2 : ðp4 z1 þ p5 p6 Þv2 þ p8 z21 z22 þ p5 p8 z1 z22
Please cite this article as: Chen L, et al. Extended space method for parameter identifiability of DAE systems. ISA Transactions (2014), http://dx.doi.org/10.1016/j.isatra.2013.12.014i
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4
þ p7 z1 z22 þ p5 p7 z22 p4 p8 z21 z2 p5 p6 p8 z1 z2 p4 p7 z 1 z 2 p5 p6 p7 z 2 T 3 : z 1 x1 ;
T 4 : z 2 x2 :
ð9Þ 1
Let u ¼ yA;in ðtÞ, x ¼ ½C A ðtÞ; C D ðtÞ; C L ðtÞ; C N ðtÞT ,y ¼ ½yA ðtÞ; yD ðtÞ; yL ðtÞ; yN ðtÞ; yN;in ðtÞT , p ¼ ½k1 ; k2 andz ¼ C L ðtÞ. The system ∑ is solvable, as shown by Ben–Zvi et al. [27]. As rankð∂g=∂yÞ ¼ 5, the differential index of ∑ is 1.
0
Pulling extended space J back to based space J , regular system ∑2 can be obtained as: 8 2 _ > < ðp3 þ z1 Þx1 ¼ p1 z1 þ p1 p3 z1 p2 z1 z2 ðp4 z1 þ p5 p6 Þx_ 2 ¼ p8 z21 z22 p5 p8 z1 z22 p7 z1 z22 p5 p7 z22 þ p4 p8 z21 z2 þ p5 p6 p8 z1 z2 þ p4 p7 z1 z2 þ p5 p6 p7 z2 > : z1 ¼ x1 ; z2 ¼ x2 It is worth noting that the exhaustive summary τ of ∑2 can be easily deduced from Eq. (10), sincez_ 1 ¼ x_ 1 , z_ 2 ¼ x_ 2 . The computation of the characteristic sets of ∑2 is unnecessary and is hence skipped here. For arbitrary parameter values, pn ¼ [c1,c2,c3,c4,c5,c6, c7,c8]T p3 ¼ c3 ; p1 ¼ c1 ; p1 p3 ¼ c1 c3 ; p2 ¼ c2 ; p5 p6 ¼ c5 c6 ; p4 ¼ c4 ; p8 ¼ c8 ; p5 p8 ¼ c5 c8 ; p7 ¼ c7 ; p5 p7 ¼ c5 c7 ; p4 p8 ¼ c4 c8 ; p5 p6 p8 ¼ c5 c6 c8 ; p4 p7 ¼ c4 c7 ; p5 p6 p7 ¼ c5 c6 c7 ð11Þ Solving Eq. (11) using Buchberger0 s algorithm yields the following results:
ð10Þ
Extending J0 to J1, system ∑ described by Eq. (12) is equivalent to system ∑1, described by Eq. (13) as: 8 V ν1 ¼ uF k1 x1 V Fy1 > > > > > V ν2 ¼ k1 x1 V k2 x2 V Fy2 > > > > > V ν3 ¼ k2 x2 V Fy3 > > > > > V ν ¼ y5 F Fy4 > > > 4 > > < y1 þy2 þy3 þ y4 1 ¼ 0 V ð13Þ ¼0 y1 x1 ntotal > > > > V > y2 x2 ntotal ¼ 0 > > > > > V > > > y3 x3 ntotal ¼ 0 > > > > > y5 þu 1 ¼ 0 > > :z ¼x 3 By choosing the elimination rank u ! z ! x ! v ! y, the fine irreducible triangular set T^ of ∑1, described by polynomial set (14) is computed by trisys package as follows:
p1 ¼ c1 ; p2 ¼ c2 ; p3 ¼ c3 ; p4 ¼ c4 ; p5 ¼ c5 ; p6 ¼ c 6 ; p7 ¼ c 7 ; p8 ¼ c 8 : Therefore, according to Definition 3, DAE system ∑ is globally identifiable. The result is in accordance with the result obtained by Margaria et al. [26].
T 2 : ntotal k1 x1 ntotal v2 ðF þ ntotal k2 Þx2
Example 2. Consider the gas-phase reactions in a continuous isothermal reactor [27]
T 4 : ðntotal V k2 þ VFÞx2 ntotal V ðv3 þ v4 Þ þ VFx1 ntotal Fu
k1
A-D
T 1 : ntotal Vv1 þ ðVF þ ntotal Vk1 Þx1 ntotal Fu T 3 : ntotal k2 x2 ntotal v3 Fz T 5 : z x3
k2
D-L
ð14Þ 1
Then, by pulling J back to J , regular system ∑2 can be obtained as described in Eq. (15): 8 ntotal V x_ 1 ¼ ðV F þ ntotal Vk1 Þx1 þntotal Fu > > > > > _ > < ntotal x2 ¼ ntotal k1 x1 ðF þ ntotal k2 Þx2 ntotal x_ 3 ¼ ntotal k2 x2 Fz ð15Þ > > > > ntotal V ðx_ 3 þ x_ 4 Þ ¼ ðntotal Vk2 þ VFÞx2 þ VFx1 ntotal Fu > > : z ¼ x3
to have been modeled using the following DAE system ∑: 8 dC A ðtÞ V dt ¼ yA;in ðtÞF k1 C A ðtÞV FyA ðtÞ > > > > > dC D ðtÞ > > > V dt ¼ k1 C A ðtÞV k2 C D ðtÞV FyD ðtÞ > > > dC ðtÞ > > V dtL ¼ k2 C D ðtÞV FyL ðtÞ > > > > dC N ðtÞ > > < V dt ¼ yN;in ðtÞF FyN ðtÞ yA ðtÞ þ yD ðtÞ þyL ðtÞ þ yN ðtÞ 1 ¼ 0 > > V > ¼0 > yA ðtÞ C A ðtÞntotal > > > > > y ðtÞ C D ðtÞ V ¼ 0 > D > ntotal > > > V > ¼0 yL ðtÞ C L ðtÞntotal > > > > : y ðtÞ þ y ðtÞ 1 ¼ 0 N;in A;in
0
ð12Þ
where C i and yi (with i taking on values in the set fA; D; L; Ng) are the volumetric concentrations and mole fractions, respectively, of the jth species. The volume is V, and the unknown parameters are k1 ; k2 . The input is the inlet mole fraction A, yA;in ðtÞ. The inlet contains only inert gas N and feed A. The inlet flow rate F is known and constant. ntotal is calculated according to the ideal gas law. We assume that the pressure and the temperature are known. In addition, we assume that ntotal is a positive integer, and k1 ; k2 , and V are all positive real numbers. Finally, we assume that only the measurement of C L ðtÞ is available.
Obviously, system ∑2 is a reduced regular ODE system. Subsequently, the diffalg package is used to calculate the characteristic sets of ∑2. As far as the parameter identifiability is concerned, only the polynomials that include the relationship between the I/O variables u; z and unknown parameters k1 ; k2 are listed as follows: ::: ð16Þ z þδ2 z€ þ δ1 z_ þ δ0 z þ ι0 u here, ι0 ¼ k1 k2 VF ; δ0 ¼ k1 k2 ζ þ ðk1 þ k2 Þζ 2 þ ζ 3 ; δ1 ¼ 2ðk1 þk2 Þζ þ k1 k2 þ 3ζ 2 ; δ2 ¼ k1 þ k2 þ3ζ; ζ¼
F ; ntotal
ð17Þ
Please cite this article as: Chen L, et al. Extended space method for parameter identifiability of DAE systems. ISA Transactions (2014), http://dx.doi.org/10.1016/j.isatra.2013.12.014i
L. Chen et al. / ISA Transactions ∎ (∎∎∎∎) ∎∎∎–∎∎∎
The exhaustive summary τ of ∑2 is described as Eq. (18) at symbolic values of pn ¼[c1,c2]T: k1 k2 VF ¼ c1 c2 VF ; k1 k2 ζ þ ðk1 þ k2 Þζ 2 þ ζ 3 ¼ ðc1 c2 ζ þðc1 þ c2 Þζ 2 þζ 3 Þ; 2ðk1 þ k2 Þζ þk1 k2 þ 3ζ 2 ¼ ð2ðc1 þ c2 Þζ þ c1 c2 þ 3ζ 2 Þ; k1 þ k2 þ 3ζ ¼ ðc1 þc2 þ3ζÞ
ð18Þ
0
Solving Eq. (18) using Buchberger s algorithm yields the following: ( ( k1 ¼ c 1 k1 ¼ c1 or : k2 ¼ c2 k2 ¼ c 2 According to Definition 4, DAE system ∑ is locally identifiable, which agrees with the results obtained by Ben–Zvi et al. [27].
5. Conclusions In this paper, we have proposed a new method for analyzing the parameter identifiability of DAE systems. It decouples the algebraic and differential variables from the model equations by space extension. An inherent, regular, low-dimensional ODE system ∑2 can be obtained, which is the external equivalent to the original DAE system ∑. Subsequently, the DA method is used to analyze the identifiability of the regular ODE system. The proposed method does not require the ideal generated by the DAE system to be prime, which makes the method a general one for DAE systems. Eventually, examples are presented to illustrate its effectiveness. The proposed method maybe limited by the development of computer algebra, and may encounter difficulties in dealing with large-scale problems because of limitation in symbolic computational techniques. Acknowledgments We gratefully acknowledge the financial support of 973 Program (No.2012CB720500) and National Natural Science Foundation of China (Nos. 61074148 & 21206149). Appendix A The appendix contains some essential differential algebra theory [12,13,23] as follows: A differential ring ℜ½X is a set of all polynomials in indeterminate X and its derivative X_ with coefficients in the field ℜ. For a set S A ℜ½X, the differential ideal IðSÞ is a subset of ℜ½X containingS, which satisfies S_ i A IðSÞ and Σ ni¼ 1 f i Si A IðSÞ, (f i A ℜðXÞ; Si A IðSÞ).IðSÞ is prime if Si Sj A IðSÞ implies Si A IðSÞ or Sj A IðSÞ, and is radical if Sm i A IðSÞ implies Si A IðSÞ. To compute the characteristic set of IðSÞ, a rank is introduced, as denoted by symbol ! . The leader of a polynomial is the indeterminate with the highest rank polynomial. Polynomials are ranked in terms of their leaders. A polynomial S1 with leader lvðS1 Þ is said to be of lower rank than polynomial S2 with leader lvðS2 Þ, if lvðS1 Þ ! lvðS2 Þ or if lvðS1 Þ ¼lvðS2 Þ and degðS1 ; lvðS1 ÞÞ o degðS2 ; lvðS2 ÞÞ, (where lvðSi Þ and degðSi ; lvðSi ÞÞ are the leader and degree of polynomial Si (i ¼ 1; 2), respectively). A set S ¼ ½S1 ; S2 ; ⋯; Sr is a triangular set iflvðS1 Þ ! lvðS2 Þ ! ⋯ ! lvðSr Þ. Algebraic triangular set S is irreducible if each Si is irreducible overℜðlvðS1 Þ; ⋯; lvðSi 1 ÞÞ, where lvðS1 Þ; ⋯; lvðSi 1 Þ are adjoined as algebraic elements by the equations Si ¼ 0; ⋯; Si 1 ¼ 0 in the case i 4 1. An algebraic triangular set S is fine if premðiniðSi Þ; SÞ a 0
5
(prem means pseudo-remainder). A polynomial Si is reduced with respect to another polynomial Sj if the former contains neither the leader of Sj with equal or greater degree nor its derivatives. A set of polynomials S1 ; ⋯; Sr , reduced with respect to one another, is called an auto-reduced set. The lowest-rank auto-reduced set is called a characteristic set. The peculiarity of the characteristic set is that it summarizes all information contained in the same ideal into a finite number of polynomials. References [1] Brenan KE, Campbell SL, Campbell SLV, Petzold LR. Numerical solution of initial-value problems in differential-algebraic equations. Soc Ind Appl Math Philadelphia 1996. [2] Riaza R. Differential-algebraic systems: analytical aspects and circuit applications. Singapore: World Scientific Publishing Co. Pte. Ltd; 2008. [3] Audoly S, D0 Angio L, Saccomani MP, Cobelli C. Global identifiability of linear compartmental models-a computer algebra algorithm. IEEE Trans Biomed Eng 1998;45(1):36–47. [4] Nazarian P, Haeri M, Tavazoei MS. Identifiability of fractional order systems using input output frequency contents. ISA Trans 2010;49:207–14. [5] Mei H, Li S. Decentralized identification for multivariable integrating processes with time delays from closed-loop step tests. ISA Trans 2007;46:189–98. [6] Godfrey KR, DiStefano JJ. Identifiability of model parameters. In: Walter E, editor. Identifiability of Parametric Models. Oxford: Pergamon Press; 1987. [7] Jacquez JA, Greif P. Numerical parameter identifiability and estimability: Integrating identifiability, estimability, and optimal sampling design. Math Biosci 1985;77(1–2):201–27. [8] Evans ND, Chapman MJ, Chappell MJ, Godfrey KR. Identifiability of uncontrolled nonlinear rational systems. Automatica 2002;38(10):1799–805. [9] Vajda S, Godfrey KR, Rabitz H. Similarity transformation approach to identifiability analysis of nonlinear compartmental models. Math Biosci 1989;93 (2):217–48. [10] Pohjanpalo H. System identifiability based on the power series expansion of the solution. Math Biosci 1978;41(1–2):21–33. [11] Lecourtier Y, Lamnabhi-Lagarrigue F, Walter E. Volterra and generating power series approaches to identifiability testing. In: Walter E, editor. Identifiability of parametric models. Oxford: Pergamon Press; 1987. p. 50–66. [12] Kolchin ER. Differential algebra and algebraic groups. New York: Academic Press; 1973. [13] Ritt JF. Differential algebra. New York: American Mathematical Society; 1950. [14] Audoly S, Bellu G, D0 Angio L, Saccomani MP, Cobelli C. Global identifiability of nonlinear models of biological systems. IEEE Trans Biomed Eng 2001;48 (1):55–65. [15] Glad ST. Differential algebraic modelling of nonlinear systems. In: Kaashoek JHvS MA, Ran ACM, editors. Proceedings of the international symposium MTNS-89: realization and modelling in system theory. Basel: Birkhäuser; 1990. p. 97–105. [16] Ljung L, Glad T. On global identifiability for arbitrary model parametrizations. Automatica 1994;30(2):265–76. [17] Ollivier F. Le problème de l0 identifiabilité structurelle globale: approche théorique. Méthodes effectives et bornes de complexité. Paris, France: École polytechnique; 1990 ([PhD thesis]). [18] Saccomani MP, Audoly S, D0 Angiò L. Parameter identifiability of nonlinear systems: the role of initial conditions. Automatica 2003;39(4):619–32. [19] Tunali ET, Tarn TJ. New results for identifiability of nonlinear-systems. IEEE Trans Autom Control 1987;32(2):146–54. [20] Chen L, Chen X, Qian J,Yao Z. A new method for parameter identifiability of DAE systems. In: Proceedings of the 10th IEEE international conference on control and automation. IEEE, Hangzhou, China; July 12–14 2013. [21] Kuranishi M, On E. Cartan0 s prolongation theorem of exterior differential systems. Am J Math 1957;79(1):1–47. [22] Sussmann HJ. Existence and uniqueness of minimal realizations of nonlinear systems. Math. Syst. Theory 1977;10:263–84. [23] Wang D. Elimination methods. Austria: Springer; 2001. [24] Glad S, Ljung L. Model structure identifiability and persistence of excitation. Decision and control 1990. In: Proceedings of the 29th IEEE conference. IEEE, Honolulu, Hawali; 1990. p. 3236–40. [25] Boulier F, Lazard D, Ollivier F., Petitot M. Representation for the radical of a finitely generated differential ideal. In: Proceedings of the 1995 International symposium on symbolic and algebraic computation. ACM; 1995. p. 158–66. [26] Margaria G, Riccomagno E, Chappell MJ, Wynn HP. Differential algebra methods for the study of the structural identifiability of rational function state-space models in the biosciences. Math Biosci 2001;174:1–26. [27] Ben-Zvi A, McLellan PJ, McAuley KB. Identifiability of linear time-invariant differential-algebraic systems. 2. The differential-algebraic approach. Ind & Eng Chem Res 2004;43:1251–9.
Please cite this article as: Chen L, et al. Extended space method for parameter identifiability of DAE systems. ISA Transactions (2014), http://dx.doi.org/10.1016/j.isatra.2013.12.014i
