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Structural identifiability and identification of systems under output couplings
17th IFAC Symposium on System Identification 17th Symposium on Identification 17th IFAC IFAC Symposium on System SystemCenter Identification Beijing International Convention 17th IFAC Symposium on System Identification Beijing International Convention Center Available online at www.sciencedirect.com Beijing International Convention Center October 19-21, 2015. Beijing, China Beijing International Convention Center October 19-21, 2015. Beijing, China October October 19-21, 19-21, 2015. 2015. Beijing, Beijing, China China
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Structural identifiability and identification Structural identifiability and identification Structural identifiability and identification of systems under output couplings of systems under output couplings of systems under output couplings Safa JEDIDI ∗∗∗ Romain BOURDAIS ∗∗∗ Jean BUISSON ∗∗∗ Safa JEDIDI ∗ Romain Romain BOURDAIS BUISSON ∗ ∗ Jean Safa Jean BUISSON LEFEBVRE Safa JEDIDI JEDIDI Marie-Anne Romain BOURDAIS BOURDAIS Jean BUISSON ∗ ∗ ∗ Marie-Anne LEFEBVRE ∗ Marie-Anne LEFEBVRE Marie-Anne LEFEBVRE ∗ Equipe ASH Automatique des Systémes Hybrides ∗ ∗ Equipe ASH Automatique des Systémes Hybrides ∗ Equipe ASH Automatique des Systémes SUPELEC - IETR UMR 6164, de Hybrides la Boulaie Equipe ASH Automatique desAvenue Systémes Hybrides SUPELEC -- IETR UMR 6164, Avenue de la Boulaie SUPELEC IETR UMR 6164, Avenue de Boulaie CS 47601, F-35576 Cesson-Sévigné Cedex, SUPELEC IETR UMR 6164, Avenue de la la France Boulaie CS 47601, 47601,- F-35576 F-35576 Cesson-Sévigné Cedex, France CS Cesson-Sévigné Cedex, France [email protected]). CS 47601,(e-mail: F-35576 Cesson-Sévigné Cedex, France (e-mail: [email protected]). (e-mail: (e-mail: [email protected]). [email protected]). Abstract: This paper deals with the identification of large-scale systems that can be decomAbstract: This paper deals with the identification of large-scale systems that can be decomAbstract: paper deals with identification of systems can be posed into aThis collection of subsystems that are coupled by their outputs. Itthat is first shown that Abstract: This paper of deals with the the that identification of large-scale large-scale systemsIt that canshown be decomdecomposed into a collection subsystems are coupled by their outputs. is first that posed into a collection subsystems that are coupled by their outputs. It first shown that if the global system is of structurally identifiable, then all the subsystems areis also structurally posed into a collection of subsystems that are coupled by their outputs. It is first shown that if the global system is structurally identifiable, then all the subsystems are also structurally if the system the are structurally identifiable considering the couplingidentifiable, outputs asthen new all inputs. This property is then used to if the global globalconsidering system is is structurally structurally the subsystems subsystems are also also structurally identifiable the coupling couplingidentifiable, outputs as asthen new all inputs. This property property is then then used to to identifiable considering the outputs new inputs. This is used propose a decentralized identification procedure. The inputs. efficiency of the proposed approach is identifiable considering the coupling outputs as new This property is then used to propose a decentralized identification procedure. The efficiency of the proposed approach is propose a decentralized identification procedure. The efficiency of the proposed approach emphasized on an academical example. procedure. The efficiency of the proposed approach is propose a decentralized identification is emphasized on academical example. emphasized on an an example. emphasized an academical academical example. © 2015, IFACon (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. Keywords: Identifiability, Identification, Large scale systems, Coupling models, Subsystems. Keywords: Identifiability, Identification, Large scale scale systems, Coupling Coupling models, Keywords: Identifiability, Identifiability, Identification, Identification, Large models, Subsystems. Subsystems. Keywords: Large scale systems, systems, Coupling models, Subsystems. 1. INTRODUCTION 1. 1. INTRODUCTION INTRODUCTION 1. INTRODUCTION There is an increasing interest in large scale systems, such There is interest scale such There is an an increasing increasing interest in in large large systems, scale systems, systems, such as network systems, transportation buildings, There is an increasing interest in large scale systems, such as network systems, transportation systems, buildings, as network systems, transportation systems, buildings, . . . .network Moreover, for years, the performance requirements as systems, transportation systems, buildings, .. .. .. .. Moreover, for the requirements Moreover, for years, years, the performance performance requirements for large scale systems are increasingly enhanced. Systems .for . . .large Moreover, for years, the performance requirements scale systems are increasingly enhanced. Systems for large scale systems are increasingly enhanced. Systems must be scale considered inaretheir totality,enhanced. considering their for large systems increasingly Systems must be considered in their totality, considering their must be considered in their totality, considering their interactions with the environment and others systems. must be considered in their totality, considering their interactions with the and systems. interactions withsystems the environment environment and others others systems. The size of such leads to various open challenges interactions with the environment and others systems. The of such systems to open The size size of such systems leads leads to various various open challenges challenges such as of thesuch elaboration of models in order to derive The size systems leads to various open challenges such as the elaboration of models in order to derive such as the elaboration of models in order topropose derive efficient control lows. In this context many works such as the elaboration of models in order to derive efficient control lows. In this context many works propose efficient control control lows. In Inthe thiscomplexity context many many works propose approaches to reduce by works rewriting the Fig. 1. Example of a global system composed with M efficient lows. this context propose approaches to reduce the complexity by Fig. 1. 1. Example Example of of aa global global system system composed composed with with M approaches to into reduce the smaller complexity by rewriting rewriting the the Fig. global problem several problems. subsystems coupled by their outputs approaches to reduce the complexity by rewriting the Fig. subsystems 1. Examplecoupled of a global system composed with M M global problem into several smaller problems. by their their outputs global problem into several smaller problems. subsystems coupled by outputs global problem into several smaller problems. subsystems coupled by their outputs Identification (Bellman and Astrom (1970)) is a crucial studied the closed loop identification problems of discrete Identification (Bellman and Astrom (1970)) is crucial Identification (Bellman and Astrom (1970)) is a crucial studied the the closed closed loop loop identification identification problems problems of of discrete discrete point if someone wants to develop a model based on con- studied Identification (Bellman and Astrom (1970)) is aa on crucial time systems that operate in a network structure and point if someone wants to develop aa model based conthe closed identification problems of discrete point if someone wants to develop model based on con- studied time systems systems thatloop operate in aa network network structure and trol strategy. From a practical point of view, parameters point if someone wants to develop a model based on contime that operate in structure and propose generalized tools to deal with this type of complex trol strategy. From a practical point of view, parameters time systems that operate in a network structure and trol strategy. strategy. is From practical point of view, view, parameters propose generalized to deal deal with this this type of of complex identification not an exceptionpoint and also suffers from the propose trol From aa practical of parameters generalized tools tools to networks. identification is and generalized tools to deal with with this type type of complex complex identification is not not an an exception exception and also also suffers suffers from from the the propose networks. size of the identification problem.and identification is not an exception also suffers from the networks. size of the identification problem. networks. size of the identification problem. In this paper, the global system is supposed structurally size of is the identification problem. In this this paper, paper, the the global global system system is is supposed supposed structurally structurally There a common feature to all the works that deal with In identifiable. Inthe a few words, structural identifiability is There is a common feature to all the works that deal with In this paper,In global system is supposed structurally Therescale is aa common common feature to alltothe the worksthe that deal with with identifiable. a few words, structural identifiability is large systems: feature they allto tryall exploit structure of identifiable. There is works that deal In aa few structural identifiability is a crucial property for words, parameters identification, because large scale systems: they all try to exploit the structure of identifiable. In few words, structural identifiability is large scale systems: they all try to exploit the structure of a crucial property for parameters identification, because the global system tothey propose simpler waysthe to structure identify the large scale systems: all try to exploit of aait crucial property for identification, guarantees the uniqueness of the parameters because (Walter the global system to propose simpler ways to identify the crucial property for parameters parameters identification, because the global system to propose simpler ways to identify the it guarantees the uniqueness of the parameters (Walter system. In system (Guinzy and Sagesimpler (1973)),ways the to global system the global to and propose identify the it guarantees the uniqueness of (Walter (1987)). In (Van den Hof (1998)), theparameters author focuses on system. In (1973)), the global system guarantees the den uniqueness of the thethe parameters (Walter system. In (Guinzy (Guinzy and Sage Sage and (1973)), the globalvariables system it (1987)). In (Van (Van Hof (1998)), (1998)), author focuses focuses on is supposed to be hierarchical then the coupling system. In (Guinzy and Sage (1973)), global system (1987)). In den Hof the author on compartmental systems to(1998)), study the structural identifiais supposed to be hierarchical and then coupling variables (1987)). In (Van den Hof the author focuses on is supposed to be hierarchical and then coupling variables compartmental systems systems to to study study the the structural structural identifiaidentifiaaresupposed introduced and an iterative method is proposed for compartmental is to be hierarchical andmethod then coupling variables bility of the global systems, and inthe (Gerdin et al.identifia(2007)) are and iterative is for systems to study structural are introduced introduced andInan an(Massioni iterative and method is proposed proposed for compartmental bility of of the the global global systems, and in in (Gerdin (Gerdin et al. al. (2007)) (2007)) the identification. Verhaegen (2008)), are introduced and an iterative method is proposed for bility systems, and et the global identifiability can be checked under sufficient the identification. In (Massioni and Verhaegen (2008)), bility of the global systems, and in (Gerdin et al. (2007)) the identification. identification. In (Massioni (Massioni and Verhaegen (2008)), the global global identifiability identifiability can can be be checked checked under under sufficient sufficient authors deal with circulant and systems (D’Andreaand the In Verhaegen (2008)), the conditions on the interaction signals between thesufficient subsysthe authors deal with circulant systems (D’Andreaand the global on identifiability can signals be checked under the authors deal with circulant systems (D’Andreaand conditions the interaction between the subsysand Dullerud (2003)), and they use their property to the authors deal with and circulant systems (D’Andreaand conditions interaction signals subsystems. Here,on a the particular structure ofbetween a modelthe composed and Dullerud (2003)), they use their property to conditions on the interaction signals between the subsysand Dullerud (2003)), and they use their property to tems. Here, Here, aa particular particular structure structure of of aa model model composed composed defineDullerud an original identification procedure. (Haber and and (2003)), and they use theirIn to tems. from a collection of linearstructure systems coupled by their outdefine identification procedure. In property (Haber and and tems. a particular a model define an an original original identification procedure. (Haber from aaHere, collection of linear linear systems systems of coupled by composed their outoutVerhaegen (2014)), the authorsprocedure. propose aIn decentralized define an original identification In (Haber and from collection of coupled by their puts, ascollection shown onofthelinear example in Figure 1, is considered. Verhaegen the authors propose aa decentralized from systems coupled byconsidered. their outVerhaegen (2014)), (2014)), the based authors propose decentralized puts, aas as shown shown on on the the example example in Figure Figure 1, is is identification algorithm on the decomposition of the puts, Verhaegen (2014)), the authors propose a decentralized in 1, considered. Many processes belong to this in class of systems, such as identification algorithm based decomposition of the shown onbelong the example Figure 1, is considered. identification algorithm based on onofthe thesubsystems decomposition of the the puts, Many asprocesses processes to this this class class of systems, systems, such as as global systemalgorithm into a collection where identification based on the decomposition of Many to of such thermalprocesses systems belong in buildings that can be structured in global system into aa collection of subsystems where the Many belong to this class of systems, such as global system into collection of subsystems where the thermal systems in buildings that can be structured in local state interaction between subsystems can be replaced global system into a collection of subsystems where the thermal systems in buildings that can be structured in such a way (Morosan et al. (2010)). local state interaction between subsystems can be replaced thermal systems in buildings that can be structured in localthe state interaction between subsystems can be be replaced replaced such aa way way (Morosan et et al. al. (2010)). (2010)). by sequence of local inputs and outputs in their such local state interaction between subsystems can by the sequence of local inputs and outputs in a way (Morosan (Morosan et paper al. (2010)). by the sequence of local inputs and outputs in their their such The contribution of this is twofold: it is first proved neighborhood. Another technique exploits a closed-loop by the sequence of local inputs and outputs in their The contribution contribution of of this this paper paper is is twofold: twofold: it it is is first proved proved neighborhood. Another technique exploits closed-loop neighborhood. Another technique exploitsof a adecentralized closed-loop The that contribution if the global of system is structurally identifiable, then decoupling property to define a collection The this paper is twofold:identifiable, it is first first proved neighborhood. Another technique exploits a closed-loop that if if the the global global system system is is structurally structurally then decoupling property to define aa collection of decentralized decoupling property to define collection of decentralized that identifiable, then all theif subsystems are also isstructurally identifiable, as soon observers for interconnected nonlinear systems (Tsai et al. that the global system structurally identifiable, then decoupling property to define a collection of decentralized all the the subsystems subsystems are are also also structurally structurally identifiable, identifiable, as as soon soon observers for nonlinear systems (Tsai et observersand for interconnected interconnected nonlinear systemsthe (Tsai et al. al. all as the we consider theare coupling outputs asidentifiable, new local as inputs. (2010)), in (Van den Hof et al. (2013)) authors all subsystems also structurally soon observers for interconnected nonlinear systems (Tsai et al. as we consider the coupling outputs as new local inputs. (2010)), and in (Van den Hof et al. (2013)) the authors (2010)), and and in in (Van (Van den den Hof Hof et et al. al. (2013)) (2013)) the the authors authors as as we we consider consider the the coupling coupling outputs outputs as as new new local local inputs. inputs. (2010)), 2405-8963 © 2015, IFAC (International Federation of Automatic Control) Copyright IFAC 2015 1415Hosting by Elsevier Ltd. All rights reserved. Copyright IFAC 2015 1415 Copyright © IFAC 2015 1415 Peer review© of International Federation of Automatic Copyright ©under IFAC responsibility 2015 1415Control. 10.1016/j.ifacol.2015.12.331
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From this result, a decentralized identification procedure is developed, that allows to reduce the complexity of the global identification. This paper is organized as follows. The section 2 recalls some results on structural identifiability and present the global system and its structure. The first contribution is presented in section 3 in where a sufficient condition is given for the structural identifiability of the subsystems. In section 4, the decentralized identification scheme is defined and its efficiency regarding the number M of subsystems into consideration is illustrated on an academical example. Conclusions and future works are given in section 5. 2. PROBLEM FORMALIZATION 2.1 Structural identifiability Testing the structural identifiability of a system is an important step in the validation of a theoretical model. Intuitively the real values of the parameters of a system can be identified if it is theoretically possible to uniquely compute these parameters from the observations of inputs and outputs. Literature on structural identifiability and methods to verify structural identifiability is extensive (Chappell et al. (1990)),(Ljung and Glad (1994)),. . . . The reader may be referred to the books (Walter (1982)) and (Walter (1987)) which treat the subject thoroughly. Its formal definition is recalled in the following definition:
matrices will be omitted: for instance, A(p) will be denoted with A. For another parametrization p˜, A(˜ p) will be ˜ The n-by-n identity matrix will be denoted denoted by A. by Idn . To use the similarity approach, the system (1) has to be structurally reachable and structurally observable: Definition 2. The system (1) is said to be structurally reachable if for almost all p ∈ P, we have: � � rank B AB . . . Anx −1 B = nx Definition 3. The system (1) is said to be structurally observable if for almost all p ∈ P, we have: C CA = nx rank .. . CAnx −1
The results of this paper are based on the following theorem: Theorem 1. (Van den Hof (1998))
The system (1), structurally reachable and observable, is structurally globally identifiable if and only if, given two parameterizations p and p˜ and a non-singular matrix T ∈ Rnx ×nx , such that:
Definition 1. : Global Structural Identifiability (Walter (1987)) Let us consider a linear parametrized system (1), with p ∈ P ⊆ Rnp , under its state space representation: � x˙ = A(p)x + B(p)u , x(0) = x0 (p) ∈ Rnx (1) y = C(p)x in which x ∈ Rnx is the state vector, u ∈ Rnu is the input vector and y ∈ Rny is the output vector. A parameter p0 ∈ P0 ⊆ R is said to be structurally globally identifiable (s.g.i.) if and only if, for almost any p�0 ∈ P0 and an input class U: y(� p, t) ≡ y(p, t), ∀t ∈ R+ , ∀u ∈ U =⇒ p�0 = p0 (2)
then T = Idnx
Several methods are available to test structural identifiability such as the Laplace transformation approach, the Markov parameters method explained in (Profos and Delgado (1995)) which is based on the Bond Graph representation of the system, the Taylor series expansion method, in (Norton (1982)) for example. In this paper, we will use a powerful tool to check the structural identifiability for a linear model under a state space description: the similarity transformation approach, which is described in (Van den Hof (1998)).
(3)
Now that the main definitions are given, the structure of the global system will be studied in the next section. 2.2 From local description to global description In this paper, a collection of M dynamical systems is considered. Each system is described by the following state equation:
(Σi ) :
The system model (1) is s.g.i. if and only if all its parameters are s.g.i.. In this paper, without loss of generality, the initial condition is supposed known and not parametrized. The extension is straightforward.
� T A = AT � TB = B � CT = C and p = p˜.
�
x˙ i = Ai xi + Bi ui +
�
Kij yj
(4)
j∈Ni
yi = Ci x i
In which the matrices Ai , Bi , Ci and Kij are parametrized by the vector pi ∈ Rnpi (i = 1 . . . M ). In this equation, xi ∈ Rnxi is the local state vector, ui ∈ Rnui is the local input vector and yi ∈ Rnyi is the local output vector. Ni denotes the neighborhood of the system (Σi ). A system (Σj ) is in the neighborhood of (Σi ) if its output yj influences x˙ i . By definition, Kij = 0 if j∈ / Ni . From these local descriptions, the global system can be defined, which is a composition of all these systems:
Before recalling the main theorem of this approach, some notations are introduced. The p dependence of all the 1416
x1 x2 x= ... , u = xM
u1 y1 u2 y2 ,y = .. ... . uM yM
p1 p2 ,p = . .. pM
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A1,no A1,12 ⋆ A1,o ⋆ , A= 0 0 K21 C1,o A2
The global state equation is then: (Σ) : in which
�
x˙ = Ax + Bu y = Cx
K12 C2 A1 A2 K21 C1 A(p) = .. .. . . K M 1 C 1 KM 2 C 2
(5)
Assumption 1. We assume that the global system is structurally reachable and observable. 3. A SUFFICIENT CONDITION FOR THE STRUCTURAL IDENTIFIABILITY OF THE LOCAL SYSTEMS
C CA CA 2 = . .
=Γ
The original idea is to consider each local system i independently from the others and the outputs yj of their neighbors will be seen as virtual inputs u ¯ij . This leads to the virtual local system: �
Kij u ¯ij
j∈Ni
yi = Ci x i
(6)
A1 =
A1,no A1,12 0 A1,o
�
,
C1 0 C2 0 C 1 A1 0 C 2 A2 0 2 0 C 1 A1 C 2 A2 0 2 . . .
0
0 C2 K 21 Γ= C1 K12 C2 K21 C2 A2 K21 . . .
. . .
(7)
0 ... ... Id Id 0 ... C1 K12 0 0 Id 0 C1 A1 K12 0 C1 K12 Id C2 K21 C1 K12 C2 K21 0 0 . . .
. . .
. . .
. . .
... ... ... 0 Id . . .
... ... ... ... ...
0 0 0 0 0
. . . Id
(8)
Then, we have:
0 C1 0 C2 C C1 A1 0 CA 0 C A 2 2 2 rank CA = rank C1 A2 0 1 .. 0 C2 A2 . 2 .. .. . .
Lemma 1. The global system (Σ) is structurally� observ� ¯ i are able if and only if all the virtual local systems Σ structurally observable. Proof. Let us suppose that the global system is structurally observable. To prove that all the virtual local systems are structurally observable, we will argue by contradiction. For the sake of clarity, the proof will be given for the case of two subsystems, the generalization is based on the same principles. Let us suppose that the first subsystem is not structurally observable, and let us rewrite this subsystem into its observability staircase form:
C1 0 0 C2 C 1 A1 C1 K12 C2 C2 K21 C1 C 2 A2 2 C1 A1 K12 C2 + C1 K12 C2 A2 C1 A1 + C1 K12 C2 K21 C1 C2 K21 C1 K12 C2 + C2 A2 C2 K21 C1 A1 + C2 A2 K21 C1 2 . . . . . .
Where Γ is a block lower triangular matrix with identity on its diagonal: Id 0 0 ... ... ... ... 0
In order to use the similarity approach on these virtual local systems, we first have to check whether they are structurally observable and structurally reachable. These points are presented in the following two lemmas.
�
�
As previously, we only consider the 2 subsystems case, the generalization is based on the same principles. Let us consider that the couples (A1 , C1 ) and (A2 , C2 ) define observable systems. By construction, the first lines of the observability matrix of the global system are:
.
x˙ i = Ai xi + Bi ui +
0 C1,o 0 0 0 C2
Let us now prove that this condition is also sufficient.
In this paper, without loss of generality, the following assumption which is common when dealing with structural identifiability is made:
�
C=
�
Then, the observability matrix of the global system cannot be a full rank matrix. This leads to a contradiction. Then all the subsystems have to be structurally observable.
. . . K1M CM . . . K2M CM .. .. . . . . . AM
B(p) = diag (Bi )i=1,M , C(p) = diag (Ci )i=1,M
� � ¯i : Σ
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As the observability matrix of the each system is full rank by assumption, then the global observability matrix is also full rank. This concludes the proof. The following lemma focuses on the reachability, but contrary to the observability, the condition is not an equivalence.
C1 = ( 0 C1,o )
Let us consider the second subsystem, with its matrices A2 and C2 . As the two systems can only be coupled by their outputs, the interaction of 1 on 2 : K21 C1 has the following structure: K21 C1 = ( 0 K21 C1,o ) Then the global system has the following matrices A and C:
Lemma 2. If the global system (Σ) is reach� structurally � ¯ i are structurally able then all the virtual local systems Σ reachable. Proof. As for the observability, we will argue by contradiction, and we only consider the 2 subsystems case. Let us suppose that the global system is structurally reachable,
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� � ¯ 1 is not structurally reachand the first virtual system Σ able, and let us rewrite this subsystem into its reachability staircase form: � � � 0 0 A1 = , B1 = , K12 = B1,c K12,c � � � � ¯ 1 is given by: ¯ 2 on Σ The interaction of Σ �
A1,nc 0 A1,12 A1,c
�
�
K12 C2 =
�
0 K12,c C2
�
Then the global system has the following matrices A and B: 0 0 A1,nc 0 0 A = A1,12 A1,c K12,c C2 , B = B1,c 0 ⋆ ⋆ A2 0 B2
These � �two matrices exhibit the fact the unreachable state ¯ 1 remains unreachable in the global system. Then, of Σ the reachability matrix of the global system cannot be a full rank matrix. This leads to a contradiction. Then all the virtual subsystems have to be structurally reachable. The main result of the paper can be stated as follows: Theorem 2. If the global system (Σ) under the assumption 1 is structurally globally identifiable, then all the ¯ i ) are also structurally globally virtual local systems (Σ identifiable. Proof. For all i = 1, . . . , M , let us suppose that we have parametrizations pi and p˜i and a matrix Ti such that: Ti Ai = A˜i Ti ˜i Ti Bi = B
(9) (10)
˜ ij ∀j �= i, Ti Kij = K C˜i Ti = Ci
(11) (12)
supposed to be globally identifiable, using theorem 1, then T = Idnx and p = p˜. Consequently, for all i = 1, . . . , M , Ti = Idnxi and pi = p˜i . Then using lemma 2 and lemma ¯ i ) is structurally reachable 1, each virtual local system (Σ and structurally observable. Consequently, using theorem 1, all these systems (Σi ) are structurally identifiable. This concludes the proof. Theorem 2 will be used to develop a decentralized identification procedure. This point is explained in the following section. 4. A DECENTRALIZED IDENTIFICATION PROCEDURE AND APPLICATION This section is composed from two parts: first the decentralized identification scheme is defined then its efficiency is illustrated on an academic example. 4.1 A decentralized identification procedure Theorem 2 implies that if the large system composed of subsystems coupled by their outputs is globally identifiable then the subsystems are also globally identifiable. Based on this theorem, we propose to replace the global parameters identification by a decentralized parameters identification of each subsystem. The input signals for the identification of a subsystem are its input signals and output signals of its neighborhood. 4.2 Academic example To illustrate the effectiveness of the theorem 2 on the identification, the study will be made on a global system structurally identifiable, composed of M subsystems under output couplings. We analyze the influence of three indicators to compare the results between the global and the decentralized identification: the parametric error, the time of identification and the fit of outputs.
Let us introduce T = diag (Ti )i=1,M . Then
T1 A1 T2 K21 C1 TA = .. .
T1 K12 C2 T2 A2 .. . TM KM 1 C1 TM KM 2 C2
A˜1 T1 K ˜ 21 C1 = .. . ˜ M 1 C1 K
˜ 12 C2 K A˜2 T2 .. . ˜ M 2 C2 K
A˜1 T1 K ˜ 21 C˜1 T1 = .. . ˜ KM 1 C˜1 T1
... ... .. . ...
• The maximum parametric error (MPE) of the used approach (global or decentralized) is the maximum relative error of parametric estimation:
. . . T1 K1M CM . . . T2 K2M CM .. .. . . . . . TM AM ˜ 1M CM K ˜ 2M CM K .. . ˜ AM TM
˜ 12 C˜2 T2 K A˜2 T2 .. . ˜ KM 2 C˜2 T2
˜ 1M C˜M TM K ˜ 2M C˜M TM K .. . ˜ . . . AM TM
... ... .. .
� � � estimated value (pi ) − real value (pi ) � � � (P E)i = 100 � � real value (pi ) which pi is a parameter of the studied system. • The minimum fit is the minimum fit calculated on the various outputs. These fits show the resemblance between estimated outputs and measured outputs
• The time of identification calculated as the sum of identification time of all the subsystems that compose the global system (not parallelized calculation).
˜ = AT (13) Straightforwardly, as B, C and T are block diagonal ma˜ and CT ˜ = C. And as the global system is trices, BT = B
Description of the System The chosen example to illustrate the approach comes from building thermal dynamics. Let us consider a large system representing a building composed of M parts: the coupling model is a coupling through outputs. The equivalent electrical model of each subsystem is described in Fig 2.
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the order of 20% of the real values. For the identification of parameters, we have used the PEM function (Prediction Error Estimate) of Matlab with the Trust-RegionReflective Algorithm which is an optimization algorithm of a cost function differentiable. The simulations are made on 18000 points. Identification To analyze the results of the identification, we have treated two cases: the first case is the identification of parameters in an ideal framework without noise on the outputs. For the second case we have added noise on the outputs.
Fig. 2. Equivalent electrical schema of subsystem i The inputs are the heating flow Pi and the outside temperature Text . The output of each subsystem is the temperature Ti of the room. The wall temperature Tmi is not measured. For this example, we will treat the case where the coupling outputs is only between the nearest subsystems as it is shown in Fig 3. For subsystems 1 and M there is only one coupling. We can shown that this structure is s.g.i. regardless the size of the system ( M ).
Identification without noise on outputs: The results of identification for the global system composed of M sub-systems:the minimum fit (mF), the maximum parametric error (MPE) and the time of identification (T) for global identification (GI) and decentralized identification(DI) are given in table 2. Table 2. Influence of M on the fit, the MPE and the time of identification M M=2 M=3 M=4 M=5 M=6 M=7
Fig. 3. Coupling between neighboring subsystems
M=10
This coupling between the subsystems is made by the coupling resistors Ri,j (j = i − 1 or j = i + 1). The parameters to estimate are the resistors and the capacitors of each subsystem: Cmi , Ci , Ri , R0i and Ri,j . The value of these parameters is different from one subsystem to another, but still close to the values given in table 1. Table 1. Average values of parameters pi real value
Cmi 5
Ci 4
Ri 3
R0i 0.4
Ri,j 6
Each subsystem i is described by the equation (4), in which:
Ai =
Bi =
0 1 Ci
1 1 1 −1 Cmi ( R0i + Ri ) Cmi R i 1 −1 1 1 Ci R i Ci ( R i + Ri,j ) j∈Ni
1 Cmi R0i
0
xi =
Tmi Ti
Kij =
0 1 Ci Ri,j
Ui =
Pi Text
mF(%)
mF(%)
MPE(%)
MPE(%)
T(min)
T(min)
(GI)
(DI)
(GI)
(DI)
(GI)
(DI)
100 100 100 99.76 99.54 98.24 N/A
100 100 100 100 100 100 100
0 0 0 0.64 0.98 1.2 N/A
0 0 0 0 0 0 0
2.65 5.77 7.10 9.15 14.20 19.30 N/A
1.7 2.57 4.20 5.14 6.28 7.16 11.25
By using the decentralized approach (identification of each subsystem), we have obtained good results(fit=100% and parametric error = 0%). The results show also a further interest of theorem 2 which is the reduction of the identification time by comparing it to that of the global approach. However, from these results, we can see too that as the number of subsystems increases, the identification becomes more complicated: for example, for M = 10 (20 states and 49 parameters), the identification procedure fails to provide a result. We have also noted that in the global case, the percentage of parametric errors increases as the size of the system increases and the minimum value of fits of outputs decreases, showing the deterioration of the quality of the identification. This simple academic example, highlights the interest of theorem 2 and the identification procedure that follows. Identification with noise on outputs:
Ci = ( 0 1 )
Experimental conditions To proceed with the identification, we have used as inputs various signals: PRBS signal (Pseudo Random Binary Sequence), RGS signal (Random Gaussian Signal) and RBS signal (Random Binary Signal). The initial values of the parameters are in
For illustrating the interest of theorem 2, white measurement noise has been added to all the 6 outputs of a global system. We have gradually increased the signal-tonoise ratio (SNR) and we have calculated the maximum parametric error (MPE) and the minimal fit of outputs for the global and decentralized approaches as shown in Fig 4 and 5. Following this experience, we can notice that the parametric errors for the decentralized approach are lower than those for the global approach and the fits on the outputs are better in the case of the decentralized approach.
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impact of this potential loss in wealth of inputs will be studied soon. Moreover, the virtual input signals depend on the state of the system, then we can see them as a feedback loop. it will be then interesting to study the techniques of closed loop identification in order to improve the quality of identification. REFERENCES
Fig. 4. Comparison of maximum parametric errors (MPE) .
Fig. 5. Comparison of minimal fits. This experience attests that the proposed approach is less sensitive to measurement noise . This will be used in future work, where a sensitivity analysis will be conducted. 5. CONCLUSION The identification of large scale systems is always considered as a complicated problem. In this article, we have showed how a model structure (coupling of subsystems by their outputs) can simplify and reduce the complexity of the identification of large scale systems. This technique has been illustrated by an example inspired from thermal systems of buildings. In the problem addressed in this article, it is assumed that the system could be represented as a collection of several subsystems coupled by their outputs. However, this model is not always appropriate. In future work, it would be interesting to study the extension of this approach to other types of coupling (coupling by inputs, by the states . . . ). On the other hand, from a practical viewpoint, identifying the parameters of a system requires sufficiently rich input signals. But in our approach, the signals associated to virtual inputs of subsystems ( u ij ) are actually the outputs of our process which can filter certain frequencies. The
Bellman, R. and Astrom, K. (1970). On structural identifiability. Mathematical Biosciences, 7(34), 329 – 339. Chappell, M., Godfrey, K., and Vajda, S. (1990). Global identifiability of the parameters of nonlinear systems with specified inputs: A comparison of methods. Mathematical Biosciences, 102(1), 41 – 73. D’Andreaand, R. and Dullerud, G. (2003). Distributed control design for spatially interconnected systems. IEEE Transactions On Automatic Control, 48, 1478– 1495. Gerdin, M., Glad, T., and Ljung, L. (2007). Global identifiability of complex models, constructed from simple submodels. Technical report, Automatic Control at Linkopings universitet(Sweden). Guinzy, N. and Sage, A. (1973). System identification in large scale systems with hierarchical structures. Computers and Electrical Engineering, 1(1), 23 – 42. Haber, A. and Verhaegen, M. (2014). Subspace identification of large-scale interconnected systems. IEEE transaction on automatic control, 59. Ljung, L. and Glad, T. (1994). On global identifiability for arbitrary model parametrizations. Automatica, 30(2), 265 – 276. Massioni, P. and Verhaegen, M. (2008). Subspace identification of circulant systems. Automatica, 44(11), 2825– 2833. Morosan, P., Bourdais, R., Dumur, D., and Buisson, J. (2010). Building temperature regulation using a distributed model predictive control. Energy and Buildings, 42(9), 1445–1452. Norton, J. (1982). An investigation of the sources of nonuniqueness in deterministic identifiability. Math Biosciences, 60, 89–108. Profos, G. and Delgado, M. (1995). Identifiability of linear systems represented by Bond Graphs. In Proceedings of Bond Graph modeling and simulation. Tsai, J., Hu, N., Yang, P., Guo, S., and Shieh, L. (2010). Modeling of decentralized linear observer and tracker for a class of unknown interconnected large-scale sampleddata nonlinear systems with closed-loop decoupling property. Computers and Mathematics with Applications, 60(3), 541–562. Van den Hof, J. (1998). Structural identifiability of linear compartmental systems. IEEE Transactions on Automatic Control, 43(6), 800–818. Van den Hof, M., Dankers, A., Heuberger, P.S., and Bombois, X. (2013). Identification of dynamic models in complex networks with prediction error methods-basic methods for consistent module estimates. Automatica, 49(10), 2994 – 3006. Walter, E. (1982). Identification of State Space Models. Spring Verlag. Walter, E. (1987). Identifiability of parametric models. Pergamon Press,Oxford.
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Structural identifiability and identification Structural identifiability and identification Structural identifiability and identification of systems under output couplings of systems under output couplings of systems under output couplings Safa JEDIDI ∗∗∗ Romain BOURDAIS ∗∗∗ Jean BUISSON ∗∗∗ Safa JEDIDI ∗ Romain Romain BOURDAIS BUISSON ∗ ∗ Jean Safa Jean BUISSON LEFEBVRE Safa JEDIDI JEDIDI Marie-Anne Romain BOURDAIS BOURDAIS Jean BUISSON ∗ ∗ ∗ Marie-Anne LEFEBVRE ∗ Marie-Anne LEFEBVRE Marie-Anne LEFEBVRE ∗ Equipe ASH Automatique des Systémes Hybrides ∗ ∗ Equipe ASH Automatique des Systémes Hybrides ∗ Equipe ASH Automatique des Systémes SUPELEC - IETR UMR 6164, de Hybrides la Boulaie Equipe ASH Automatique desAvenue Systémes Hybrides SUPELEC -- IETR UMR 6164, Avenue de la Boulaie SUPELEC IETR UMR 6164, Avenue de Boulaie CS 47601, F-35576 Cesson-Sévigné Cedex, SUPELEC IETR UMR 6164, Avenue de la la France Boulaie CS 47601, 47601,- F-35576 F-35576 Cesson-Sévigné Cedex, France CS Cesson-Sévigné Cedex, France [email protected]). CS 47601,(e-mail: F-35576 Cesson-Sévigné Cedex, France (e-mail: [email protected]). (e-mail: (e-mail: [email protected]). [email protected]). Abstract: This paper deals with the identification of large-scale systems that can be decomAbstract: This paper deals with the identification of large-scale systems that can be decomAbstract: paper deals with identification of systems can be posed into aThis collection of subsystems that are coupled by their outputs. Itthat is first shown that Abstract: This paper of deals with the the that identification of large-scale large-scale systemsIt that canshown be decomdecomposed into a collection subsystems are coupled by their outputs. is first that posed into a collection subsystems that are coupled by their outputs. It first shown that if the global system is of structurally identifiable, then all the subsystems areis also structurally posed into a collection of subsystems that are coupled by their outputs. It is first shown that if the global system is structurally identifiable, then all the subsystems are also structurally if the system the are structurally identifiable considering the couplingidentifiable, outputs asthen new all inputs. This property is then used to if the global globalconsidering system is is structurally structurally the subsystems subsystems are also also structurally identifiable the coupling couplingidentifiable, outputs as asthen new all inputs. This property property is then then used to to identifiable considering the outputs new inputs. This is used propose a decentralized identification procedure. The inputs. efficiency of the proposed approach is identifiable considering the coupling outputs as new This property is then used to propose a decentralized identification procedure. The efficiency of the proposed approach is propose a decentralized identification procedure. The efficiency of the proposed approach emphasized on an academical example. procedure. The efficiency of the proposed approach is propose a decentralized identification is emphasized on academical example. emphasized on an an example. emphasized an academical academical example. © 2015, IFACon (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. Keywords: Identifiability, Identification, Large scale systems, Coupling models, Subsystems. Keywords: Identifiability, Identification, Large scale scale systems, Coupling Coupling models, Keywords: Identifiability, Identifiability, Identification, Identification, Large models, Subsystems. Subsystems. Keywords: Large scale systems, systems, Coupling models, Subsystems. 1. INTRODUCTION 1. 1. INTRODUCTION INTRODUCTION 1. INTRODUCTION There is an increasing interest in large scale systems, such There is interest scale such There is an an increasing increasing interest in in large large systems, scale systems, systems, such as network systems, transportation buildings, There is an increasing interest in large scale systems, such as network systems, transportation systems, buildings, as network systems, transportation systems, buildings, . . . .network Moreover, for years, the performance requirements as systems, transportation systems, buildings, .. .. .. .. Moreover, for the requirements Moreover, for years, years, the performance performance requirements for large scale systems are increasingly enhanced. Systems .for . . .large Moreover, for years, the performance requirements scale systems are increasingly enhanced. Systems for large scale systems are increasingly enhanced. Systems must be scale considered inaretheir totality,enhanced. considering their for large systems increasingly Systems must be considered in their totality, considering their must be considered in their totality, considering their interactions with the environment and others systems. must be considered in their totality, considering their interactions with the and systems. interactions withsystems the environment environment and others others systems. The size of such leads to various open challenges interactions with the environment and others systems. The of such systems to open The size size of such systems leads leads to various various open challenges challenges such as of thesuch elaboration of models in order to derive The size systems leads to various open challenges such as the elaboration of models in order to derive such as the elaboration of models in order topropose derive efficient control lows. In this context many works such as the elaboration of models in order to derive efficient control lows. In this context many works propose efficient control control lows. In Inthe thiscomplexity context many many works propose approaches to reduce by works rewriting the Fig. 1. Example of a global system composed with M efficient lows. this context propose approaches to reduce the complexity by Fig. 1. 1. Example Example of of aa global global system system composed composed with with M approaches to into reduce the smaller complexity by rewriting rewriting the the Fig. global problem several problems. subsystems coupled by their outputs approaches to reduce the complexity by rewriting the Fig. subsystems 1. Examplecoupled of a global system composed with M M global problem into several smaller problems. by their their outputs global problem into several smaller problems. subsystems coupled by outputs global problem into several smaller problems. subsystems coupled by their outputs Identification (Bellman and Astrom (1970)) is a crucial studied the closed loop identification problems of discrete Identification (Bellman and Astrom (1970)) is crucial Identification (Bellman and Astrom (1970)) is a crucial studied the the closed closed loop loop identification identification problems problems of of discrete discrete point if someone wants to develop a model based on con- studied Identification (Bellman and Astrom (1970)) is aa on crucial time systems that operate in a network structure and point if someone wants to develop aa model based conthe closed identification problems of discrete point if someone wants to develop model based on con- studied time systems systems thatloop operate in aa network network structure and trol strategy. From a practical point of view, parameters point if someone wants to develop a model based on contime that operate in structure and propose generalized tools to deal with this type of complex trol strategy. From a practical point of view, parameters time systems that operate in a network structure and trol strategy. strategy. is From practical point of view, view, parameters propose generalized to deal deal with this this type of of complex identification not an exceptionpoint and also suffers from the propose trol From aa practical of parameters generalized tools tools to networks. identification is and generalized tools to deal with with this type type of complex complex identification is not not an an exception exception and also also suffers suffers from from the the propose networks. size of the identification problem.and identification is not an exception also suffers from the networks. size of the identification problem. networks. size of the identification problem. In this paper, the global system is supposed structurally size of is the identification problem. In this this paper, paper, the the global global system system is is supposed supposed structurally structurally There a common feature to all the works that deal with In identifiable. Inthe a few words, structural identifiability is There is a common feature to all the works that deal with In this paper,In global system is supposed structurally Therescale is aa common common feature to alltothe the worksthe that deal with with identifiable. a few words, structural identifiability is large systems: feature they allto tryall exploit structure of identifiable. There is works that deal In aa few structural identifiability is a crucial property for words, parameters identification, because large scale systems: they all try to exploit the structure of identifiable. In few words, structural identifiability is large scale systems: they all try to exploit the structure of a crucial property for parameters identification, because the global system tothey propose simpler waysthe to structure identify the large scale systems: all try to exploit of aait crucial property for identification, guarantees the uniqueness of the parameters because (Walter the global system to propose simpler ways to identify the crucial property for parameters parameters identification, because the global system to propose simpler ways to identify the it guarantees the uniqueness of the parameters (Walter system. In system (Guinzy and Sagesimpler (1973)),ways the to global system the global to and propose identify the it guarantees the uniqueness of (Walter (1987)). In (Van den Hof (1998)), theparameters author focuses on system. In (1973)), the global system guarantees the den uniqueness of the thethe parameters (Walter system. In (Guinzy (Guinzy and Sage Sage and (1973)), the globalvariables system it (1987)). In (Van (Van Hof (1998)), (1998)), author focuses focuses on is supposed to be hierarchical then the coupling system. In (Guinzy and Sage (1973)), global system (1987)). In den Hof the author on compartmental systems to(1998)), study the structural identifiais supposed to be hierarchical and then coupling variables (1987)). In (Van den Hof the author focuses on is supposed to be hierarchical and then coupling variables compartmental systems systems to to study study the the structural structural identifiaidentifiaaresupposed introduced and an iterative method is proposed for compartmental is to be hierarchical andmethod then coupling variables bility of the global systems, and inthe (Gerdin et al.identifia(2007)) are and iterative is for systems to study structural are introduced introduced andInan an(Massioni iterative and method is proposed proposed for compartmental bility of of the the global global systems, and in in (Gerdin (Gerdin et al. al. (2007)) (2007)) the identification. Verhaegen (2008)), are introduced and an iterative method is proposed for bility systems, and et the global identifiability can be checked under sufficient the identification. In (Massioni and Verhaegen (2008)), bility of the global systems, and in (Gerdin et al. (2007)) the identification. identification. In (Massioni (Massioni and Verhaegen (2008)), the global global identifiability identifiability can can be be checked checked under under sufficient sufficient authors deal with circulant and systems (D’Andreaand the In Verhaegen (2008)), the conditions on the interaction signals between thesufficient subsysthe authors deal with circulant systems (D’Andreaand the global on identifiability can signals be checked under the authors deal with circulant systems (D’Andreaand conditions the interaction between the subsysand Dullerud (2003)), and they use their property to the authors deal with and circulant systems (D’Andreaand conditions interaction signals subsystems. Here,on a the particular structure ofbetween a modelthe composed and Dullerud (2003)), they use their property to conditions on the interaction signals between the subsysand Dullerud (2003)), and they use their property to tems. Here, Here, aa particular particular structure structure of of aa model model composed composed defineDullerud an original identification procedure. (Haber and and (2003)), and they use theirIn to tems. from a collection of linearstructure systems coupled by their outdefine identification procedure. In property (Haber and and tems. a particular a model define an an original original identification procedure. (Haber from aaHere, collection of linear linear systems systems of coupled by composed their outoutVerhaegen (2014)), the authorsprocedure. propose aIn decentralized define an original identification In (Haber and from collection of coupled by their puts, ascollection shown onofthelinear example in Figure 1, is considered. Verhaegen the authors propose aa decentralized from systems coupled byconsidered. their outVerhaegen (2014)), (2014)), the based authors propose decentralized puts, aas as shown shown on on the the example example in Figure Figure 1, is is identification algorithm on the decomposition of the puts, Verhaegen (2014)), the authors propose a decentralized in 1, considered. Many processes belong to this in class of systems, such as identification algorithm based decomposition of the shown onbelong the example Figure 1, is considered. identification algorithm based on onofthe thesubsystems decomposition of the the puts, Many asprocesses processes to this this class class of systems, systems, such as as global systemalgorithm into a collection where identification based on the decomposition of Many to of such thermalprocesses systems belong in buildings that can be structured in global system into aa collection of subsystems where the Many belong to this class of systems, such as global system into collection of subsystems where the thermal systems in buildings that can be structured in local state interaction between subsystems can be replaced global system into a collection of subsystems where the thermal systems in buildings that can be structured in such a way (Morosan et al. (2010)). local state interaction between subsystems can be replaced thermal systems in buildings that can be structured in localthe state interaction between subsystems can be be replaced replaced such aa way way (Morosan et et al. al. (2010)). (2010)). by sequence of local inputs and outputs in their such local state interaction between subsystems can by the sequence of local inputs and outputs in a way (Morosan (Morosan et paper al. (2010)). by the sequence of local inputs and outputs in their their such The contribution of this is twofold: it is first proved neighborhood. Another technique exploits a closed-loop by the sequence of local inputs and outputs in their The contribution contribution of of this this paper paper is is twofold: twofold: it it is is first proved proved neighborhood. Another technique exploits closed-loop neighborhood. Another technique exploitsof a adecentralized closed-loop The that contribution if the global of system is structurally identifiable, then decoupling property to define a collection The this paper is twofold:identifiable, it is first first proved neighborhood. Another technique exploits a closed-loop that if if the the global global system system is is structurally structurally then decoupling property to define aa collection of decentralized decoupling property to define collection of decentralized that identifiable, then all theif subsystems are also isstructurally identifiable, as soon observers for interconnected nonlinear systems (Tsai et al. that the global system structurally identifiable, then decoupling property to define a collection of decentralized all the the subsystems subsystems are are also also structurally structurally identifiable, identifiable, as as soon soon observers for nonlinear systems (Tsai et observersand for interconnected interconnected nonlinear systemsthe (Tsai et al. al. all as the we consider theare coupling outputs asidentifiable, new local as inputs. (2010)), in (Van den Hof et al. (2013)) authors all subsystems also structurally soon observers for interconnected nonlinear systems (Tsai et al. as we consider the coupling outputs as new local inputs. (2010)), and in (Van den Hof et al. (2013)) the authors (2010)), and and in in (Van (Van den den Hof Hof et et al. al. (2013)) (2013)) the the authors authors as as we we consider consider the the coupling coupling outputs outputs as as new new local local inputs. inputs. (2010)), 2405-8963 © 2015, IFAC (International Federation of Automatic Control) Copyright IFAC 2015 1415Hosting by Elsevier Ltd. All rights reserved. Copyright IFAC 2015 1415 Copyright © IFAC 2015 1415 Peer review© of International Federation of Automatic Copyright ©under IFAC responsibility 2015 1415Control. 10.1016/j.ifacol.2015.12.331
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From this result, a decentralized identification procedure is developed, that allows to reduce the complexity of the global identification. This paper is organized as follows. The section 2 recalls some results on structural identifiability and present the global system and its structure. The first contribution is presented in section 3 in where a sufficient condition is given for the structural identifiability of the subsystems. In section 4, the decentralized identification scheme is defined and its efficiency regarding the number M of subsystems into consideration is illustrated on an academical example. Conclusions and future works are given in section 5. 2. PROBLEM FORMALIZATION 2.1 Structural identifiability Testing the structural identifiability of a system is an important step in the validation of a theoretical model. Intuitively the real values of the parameters of a system can be identified if it is theoretically possible to uniquely compute these parameters from the observations of inputs and outputs. Literature on structural identifiability and methods to verify structural identifiability is extensive (Chappell et al. (1990)),(Ljung and Glad (1994)),. . . . The reader may be referred to the books (Walter (1982)) and (Walter (1987)) which treat the subject thoroughly. Its formal definition is recalled in the following definition:
matrices will be omitted: for instance, A(p) will be denoted with A. For another parametrization p˜, A(˜ p) will be ˜ The n-by-n identity matrix will be denoted denoted by A. by Idn . To use the similarity approach, the system (1) has to be structurally reachable and structurally observable: Definition 2. The system (1) is said to be structurally reachable if for almost all p ∈ P, we have: � � rank B AB . . . Anx −1 B = nx Definition 3. The system (1) is said to be structurally observable if for almost all p ∈ P, we have: C CA = nx rank .. . CAnx −1
The results of this paper are based on the following theorem: Theorem 1. (Van den Hof (1998))
The system (1), structurally reachable and observable, is structurally globally identifiable if and only if, given two parameterizations p and p˜ and a non-singular matrix T ∈ Rnx ×nx , such that:
Definition 1. : Global Structural Identifiability (Walter (1987)) Let us consider a linear parametrized system (1), with p ∈ P ⊆ Rnp , under its state space representation: � x˙ = A(p)x + B(p)u , x(0) = x0 (p) ∈ Rnx (1) y = C(p)x in which x ∈ Rnx is the state vector, u ∈ Rnu is the input vector and y ∈ Rny is the output vector. A parameter p0 ∈ P0 ⊆ R is said to be structurally globally identifiable (s.g.i.) if and only if, for almost any p�0 ∈ P0 and an input class U: y(� p, t) ≡ y(p, t), ∀t ∈ R+ , ∀u ∈ U =⇒ p�0 = p0 (2)
then T = Idnx
Several methods are available to test structural identifiability such as the Laplace transformation approach, the Markov parameters method explained in (Profos and Delgado (1995)) which is based on the Bond Graph representation of the system, the Taylor series expansion method, in (Norton (1982)) for example. In this paper, we will use a powerful tool to check the structural identifiability for a linear model under a state space description: the similarity transformation approach, which is described in (Van den Hof (1998)).
(3)
Now that the main definitions are given, the structure of the global system will be studied in the next section. 2.2 From local description to global description In this paper, a collection of M dynamical systems is considered. Each system is described by the following state equation:
(Σi ) :
The system model (1) is s.g.i. if and only if all its parameters are s.g.i.. In this paper, without loss of generality, the initial condition is supposed known and not parametrized. The extension is straightforward.
� T A = AT � TB = B � CT = C and p = p˜.
�
x˙ i = Ai xi + Bi ui +
�
Kij yj
(4)
j∈Ni
yi = Ci x i
In which the matrices Ai , Bi , Ci and Kij are parametrized by the vector pi ∈ Rnpi (i = 1 . . . M ). In this equation, xi ∈ Rnxi is the local state vector, ui ∈ Rnui is the local input vector and yi ∈ Rnyi is the local output vector. Ni denotes the neighborhood of the system (Σi ). A system (Σj ) is in the neighborhood of (Σi ) if its output yj influences x˙ i . By definition, Kij = 0 if j∈ / Ni . From these local descriptions, the global system can be defined, which is a composition of all these systems:
Before recalling the main theorem of this approach, some notations are introduced. The p dependence of all the 1416
x1 x2 x= ... , u = xM
u1 y1 u2 y2 ,y = .. ... . uM yM
p1 p2 ,p = . .. pM
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A1,no A1,12 ⋆ A1,o ⋆ , A= 0 0 K21 C1,o A2
The global state equation is then: (Σ) : in which
�
x˙ = Ax + Bu y = Cx
K12 C2 A1 A2 K21 C1 A(p) = .. .. . . K M 1 C 1 KM 2 C 2
(5)
Assumption 1. We assume that the global system is structurally reachable and observable. 3. A SUFFICIENT CONDITION FOR THE STRUCTURAL IDENTIFIABILITY OF THE LOCAL SYSTEMS
C CA CA 2 = . .
=Γ
The original idea is to consider each local system i independently from the others and the outputs yj of their neighbors will be seen as virtual inputs u ¯ij . This leads to the virtual local system: �
Kij u ¯ij
j∈Ni
yi = Ci x i
(6)
A1 =
A1,no A1,12 0 A1,o
�
,
C1 0 C2 0 C 1 A1 0 C 2 A2 0 2 0 C 1 A1 C 2 A2 0 2 . . .
0
0 C2 K 21 Γ= C1 K12 C2 K21 C2 A2 K21 . . .
. . .
(7)
0 ... ... Id Id 0 ... C1 K12 0 0 Id 0 C1 A1 K12 0 C1 K12 Id C2 K21 C1 K12 C2 K21 0 0 . . .
. . .
. . .
. . .
... ... ... 0 Id . . .
... ... ... ... ...
0 0 0 0 0
. . . Id
(8)
Then, we have:
0 C1 0 C2 C C1 A1 0 CA 0 C A 2 2 2 rank CA = rank C1 A2 0 1 .. 0 C2 A2 . 2 .. .. . .
Lemma 1. The global system (Σ) is structurally� observ� ¯ i are able if and only if all the virtual local systems Σ structurally observable. Proof. Let us suppose that the global system is structurally observable. To prove that all the virtual local systems are structurally observable, we will argue by contradiction. For the sake of clarity, the proof will be given for the case of two subsystems, the generalization is based on the same principles. Let us suppose that the first subsystem is not structurally observable, and let us rewrite this subsystem into its observability staircase form:
C1 0 0 C2 C 1 A1 C1 K12 C2 C2 K21 C1 C 2 A2 2 C1 A1 K12 C2 + C1 K12 C2 A2 C1 A1 + C1 K12 C2 K21 C1 C2 K21 C1 K12 C2 + C2 A2 C2 K21 C1 A1 + C2 A2 K21 C1 2 . . . . . .
Where Γ is a block lower triangular matrix with identity on its diagonal: Id 0 0 ... ... ... ... 0
In order to use the similarity approach on these virtual local systems, we first have to check whether they are structurally observable and structurally reachable. These points are presented in the following two lemmas.
�
�
As previously, we only consider the 2 subsystems case, the generalization is based on the same principles. Let us consider that the couples (A1 , C1 ) and (A2 , C2 ) define observable systems. By construction, the first lines of the observability matrix of the global system are:
.
x˙ i = Ai xi + Bi ui +
0 C1,o 0 0 0 C2
Let us now prove that this condition is also sufficient.
In this paper, without loss of generality, the following assumption which is common when dealing with structural identifiability is made:
�
C=
�
Then, the observability matrix of the global system cannot be a full rank matrix. This leads to a contradiction. Then all the subsystems have to be structurally observable.
. . . K1M CM . . . K2M CM .. .. . . . . . AM
B(p) = diag (Bi )i=1,M , C(p) = diag (Ci )i=1,M
� � ¯i : Σ
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As the observability matrix of the each system is full rank by assumption, then the global observability matrix is also full rank. This concludes the proof. The following lemma focuses on the reachability, but contrary to the observability, the condition is not an equivalence.
C1 = ( 0 C1,o )
Let us consider the second subsystem, with its matrices A2 and C2 . As the two systems can only be coupled by their outputs, the interaction of 1 on 2 : K21 C1 has the following structure: K21 C1 = ( 0 K21 C1,o ) Then the global system has the following matrices A and C:
Lemma 2. If the global system (Σ) is reach� structurally � ¯ i are structurally able then all the virtual local systems Σ reachable. Proof. As for the observability, we will argue by contradiction, and we only consider the 2 subsystems case. Let us suppose that the global system is structurally reachable,
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� � ¯ 1 is not structurally reachand the first virtual system Σ able, and let us rewrite this subsystem into its reachability staircase form: � � � 0 0 A1 = , B1 = , K12 = B1,c K12,c � � � � ¯ 1 is given by: ¯ 2 on Σ The interaction of Σ �
A1,nc 0 A1,12 A1,c
�
�
K12 C2 =
�
0 K12,c C2
�
Then the global system has the following matrices A and B: 0 0 A1,nc 0 0 A = A1,12 A1,c K12,c C2 , B = B1,c 0 ⋆ ⋆ A2 0 B2
These � �two matrices exhibit the fact the unreachable state ¯ 1 remains unreachable in the global system. Then, of Σ the reachability matrix of the global system cannot be a full rank matrix. This leads to a contradiction. Then all the virtual subsystems have to be structurally reachable. The main result of the paper can be stated as follows: Theorem 2. If the global system (Σ) under the assumption 1 is structurally globally identifiable, then all the ¯ i ) are also structurally globally virtual local systems (Σ identifiable. Proof. For all i = 1, . . . , M , let us suppose that we have parametrizations pi and p˜i and a matrix Ti such that: Ti Ai = A˜i Ti ˜i Ti Bi = B
(9) (10)
˜ ij ∀j �= i, Ti Kij = K C˜i Ti = Ci
(11) (12)
supposed to be globally identifiable, using theorem 1, then T = Idnx and p = p˜. Consequently, for all i = 1, . . . , M , Ti = Idnxi and pi = p˜i . Then using lemma 2 and lemma ¯ i ) is structurally reachable 1, each virtual local system (Σ and structurally observable. Consequently, using theorem 1, all these systems (Σi ) are structurally identifiable. This concludes the proof. Theorem 2 will be used to develop a decentralized identification procedure. This point is explained in the following section. 4. A DECENTRALIZED IDENTIFICATION PROCEDURE AND APPLICATION This section is composed from two parts: first the decentralized identification scheme is defined then its efficiency is illustrated on an academic example. 4.1 A decentralized identification procedure Theorem 2 implies that if the large system composed of subsystems coupled by their outputs is globally identifiable then the subsystems are also globally identifiable. Based on this theorem, we propose to replace the global parameters identification by a decentralized parameters identification of each subsystem. The input signals for the identification of a subsystem are its input signals and output signals of its neighborhood. 4.2 Academic example To illustrate the effectiveness of the theorem 2 on the identification, the study will be made on a global system structurally identifiable, composed of M subsystems under output couplings. We analyze the influence of three indicators to compare the results between the global and the decentralized identification: the parametric error, the time of identification and the fit of outputs.
Let us introduce T = diag (Ti )i=1,M . Then
T1 A1 T2 K21 C1 TA = .. .
T1 K12 C2 T2 A2 .. . TM KM 1 C1 TM KM 2 C2
A˜1 T1 K ˜ 21 C1 = .. . ˜ M 1 C1 K
˜ 12 C2 K A˜2 T2 .. . ˜ M 2 C2 K
A˜1 T1 K ˜ 21 C˜1 T1 = .. . ˜ KM 1 C˜1 T1
... ... .. . ...
• The maximum parametric error (MPE) of the used approach (global or decentralized) is the maximum relative error of parametric estimation:
. . . T1 K1M CM . . . T2 K2M CM .. .. . . . . . TM AM ˜ 1M CM K ˜ 2M CM K .. . ˜ AM TM
˜ 12 C˜2 T2 K A˜2 T2 .. . ˜ KM 2 C˜2 T2
˜ 1M C˜M TM K ˜ 2M C˜M TM K .. . ˜ . . . AM TM
... ... .. .
� � � estimated value (pi ) − real value (pi ) � � � (P E)i = 100 � � real value (pi ) which pi is a parameter of the studied system. • The minimum fit is the minimum fit calculated on the various outputs. These fits show the resemblance between estimated outputs and measured outputs
• The time of identification calculated as the sum of identification time of all the subsystems that compose the global system (not parallelized calculation).
˜ = AT (13) Straightforwardly, as B, C and T are block diagonal ma˜ and CT ˜ = C. And as the global system is trices, BT = B
Description of the System The chosen example to illustrate the approach comes from building thermal dynamics. Let us consider a large system representing a building composed of M parts: the coupling model is a coupling through outputs. The equivalent electrical model of each subsystem is described in Fig 2.
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the order of 20% of the real values. For the identification of parameters, we have used the PEM function (Prediction Error Estimate) of Matlab with the Trust-RegionReflective Algorithm which is an optimization algorithm of a cost function differentiable. The simulations are made on 18000 points. Identification To analyze the results of the identification, we have treated two cases: the first case is the identification of parameters in an ideal framework without noise on the outputs. For the second case we have added noise on the outputs.
Fig. 2. Equivalent electrical schema of subsystem i The inputs are the heating flow Pi and the outside temperature Text . The output of each subsystem is the temperature Ti of the room. The wall temperature Tmi is not measured. For this example, we will treat the case where the coupling outputs is only between the nearest subsystems as it is shown in Fig 3. For subsystems 1 and M there is only one coupling. We can shown that this structure is s.g.i. regardless the size of the system ( M ).
Identification without noise on outputs: The results of identification for the global system composed of M sub-systems:the minimum fit (mF), the maximum parametric error (MPE) and the time of identification (T) for global identification (GI) and decentralized identification(DI) are given in table 2. Table 2. Influence of M on the fit, the MPE and the time of identification M M=2 M=3 M=4 M=5 M=6 M=7
Fig. 3. Coupling between neighboring subsystems
M=10
This coupling between the subsystems is made by the coupling resistors Ri,j (j = i − 1 or j = i + 1). The parameters to estimate are the resistors and the capacitors of each subsystem: Cmi , Ci , Ri , R0i and Ri,j . The value of these parameters is different from one subsystem to another, but still close to the values given in table 1. Table 1. Average values of parameters pi real value
Cmi 5
Ci 4
Ri 3
R0i 0.4
Ri,j 6
Each subsystem i is described by the equation (4), in which:
Ai =
Bi =
0 1 Ci
1 1 1 −1 Cmi ( R0i + Ri ) Cmi R i 1 −1 1 1 Ci R i Ci ( R i + Ri,j ) j∈Ni
1 Cmi R0i
0
xi =
Tmi Ti
Kij =
0 1 Ci Ri,j
Ui =
Pi Text
mF(%)
mF(%)
MPE(%)
MPE(%)
T(min)
T(min)
(GI)
(DI)
(GI)
(DI)
(GI)
(DI)
100 100 100 99.76 99.54 98.24 N/A
100 100 100 100 100 100 100
0 0 0 0.64 0.98 1.2 N/A
0 0 0 0 0 0 0
2.65 5.77 7.10 9.15 14.20 19.30 N/A
1.7 2.57 4.20 5.14 6.28 7.16 11.25
By using the decentralized approach (identification of each subsystem), we have obtained good results(fit=100% and parametric error = 0%). The results show also a further interest of theorem 2 which is the reduction of the identification time by comparing it to that of the global approach. However, from these results, we can see too that as the number of subsystems increases, the identification becomes more complicated: for example, for M = 10 (20 states and 49 parameters), the identification procedure fails to provide a result. We have also noted that in the global case, the percentage of parametric errors increases as the size of the system increases and the minimum value of fits of outputs decreases, showing the deterioration of the quality of the identification. This simple academic example, highlights the interest of theorem 2 and the identification procedure that follows. Identification with noise on outputs:
Ci = ( 0 1 )
Experimental conditions To proceed with the identification, we have used as inputs various signals: PRBS signal (Pseudo Random Binary Sequence), RGS signal (Random Gaussian Signal) and RBS signal (Random Binary Signal). The initial values of the parameters are in
For illustrating the interest of theorem 2, white measurement noise has been added to all the 6 outputs of a global system. We have gradually increased the signal-tonoise ratio (SNR) and we have calculated the maximum parametric error (MPE) and the minimal fit of outputs for the global and decentralized approaches as shown in Fig 4 and 5. Following this experience, we can notice that the parametric errors for the decentralized approach are lower than those for the global approach and the fits on the outputs are better in the case of the decentralized approach.
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impact of this potential loss in wealth of inputs will be studied soon. Moreover, the virtual input signals depend on the state of the system, then we can see them as a feedback loop. it will be then interesting to study the techniques of closed loop identification in order to improve the quality of identification. REFERENCES
Fig. 4. Comparison of maximum parametric errors (MPE) .
Fig. 5. Comparison of minimal fits. This experience attests that the proposed approach is less sensitive to measurement noise . This will be used in future work, where a sensitivity analysis will be conducted. 5. CONCLUSION The identification of large scale systems is always considered as a complicated problem. In this article, we have showed how a model structure (coupling of subsystems by their outputs) can simplify and reduce the complexity of the identification of large scale systems. This technique has been illustrated by an example inspired from thermal systems of buildings. In the problem addressed in this article, it is assumed that the system could be represented as a collection of several subsystems coupled by their outputs. However, this model is not always appropriate. In future work, it would be interesting to study the extension of this approach to other types of coupling (coupling by inputs, by the states . . . ). On the other hand, from a practical viewpoint, identifying the parameters of a system requires sufficiently rich input signals. But in our approach, the signals associated to virtual inputs of subsystems ( u ij ) are actually the outputs of our process which can filter certain frequencies. The
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