Recursive Output Error Identification Algorithm for Switched Linear systems with Bounded Noise

The International Federation of Automatic Control The International Federation of Congress Automatic Control Proceedings of the 20th World Toulouse, France, 2017 Proceedings of the theJuly 20th9-14, World Congress Proceedings of 20th World Toulouse, France, July 9-14, 2017 The International Federation of Congress Automatic Control Available online at www.sciencedirect.com The of The International International Federation of Automatic Automatic Control Control Toulouse, France,Federation July 9-14, 2017 Toulouse, France, France, July July 9-14, 9-14, 2017 2017 Toulouse,

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Recursive Output Error Identification IFAC PapersOnLine (2017) 14112–14117 Recursive Output 50-1 Error Identification Algorithm Algorithm Recursive Output Error Identification Algorithm for Switched Linear systems with Bounded Noise Recursive Output Error Identification Algorithm for Switched Linear systems with Bounded Noise for Switched ∗Linear systems∗ with Bounded Noise ∗ ∗ OlivierNoise for Switched Linear systems with Bounded Abdelhak Goudjil Mathieu Pouliquen Eric Pigeon ∗ ∗ ∗ Abdelhak Goudjil Mathieu Pouliquen Eric Pigeon Olivier Gehan Gehan ∗

∗ Boubekeur Targui ∗ Eric ∗ Pigeon ∗ Olivier Gehan ∗ Abdelhak Goudjil ∗∗∗ Mathieu Pouliquen Boubekeur Targui ∗ ∗ Abdelhak Goudjil Goudjil Mathieu Mathieu Pouliquen Pouliquen Eric Eric Pigeon ∗∗ Olivier Olivier Gehan Gehan ∗∗ ∗ Pigeon Abdelhak Boubekeur Targui ∗ ∗ Laboratoire d’Automatique ∗ Boubekeur Targui de Normandie Boubekeur Targui ∗ Laboratoire d’Automatique de Caen, Caen, Normandie Univ, Univ, UNICAEN, UNICAEN, ENSICAEN, LAC, 14000 Caen, France. ∗ ENSICAEN, LAC, 14000 Caen, France. Laboratoire d’Automatique de Caen, Normandie Univ, UNICAEN, ∗ ∗ Laboratoire d’Automatique de Univ, Laboratoire ENSICAEN, d’Automatique de Caen, Caen, Normandie Univ, UNICAEN, UNICAEN, LAC, 14000Normandie Caen, France. ENSICAEN, LAC, LAC, 14000 14000 Caen, Caen, France. France. ENSICAEN, Abstract: The problem considered in this paper is the identification from input/output Abstract: The problem considered in this paper is the identification from input/output data data of of Switched Switched linear systems with Switched Output Error (SOE) models. The identification is carried out the linear systems with Switched Output Error (SOE) models. The identification is carrieddata outofin inSwitched the case case Abstract: The problem considered in this paper is the identification from input/output Abstract: The problem considered in this paper is the identification from input/output data of Switched where an upper bound on the noise is assumed to be known. This problem includes the estimation of Abstract: The problem considered in this paper is the identification from input/output data of Switched linear systems Switched (SOE) TheThis identification is carried in the case where an upperwith bound on the Output noise isError assumed to models. be known. problem includes theout estimation of linear systems with Switched Output Error (SOE) models. The identification is carried out in the case the discrete state that corresponds to the active sub-model and the estimation of the parameters of all linear systems with Switched Output Error (SOE) models. The identification is carried out in the case where an upper bound on the noise is assumed to be known. This problem includes the estimation of the discrete state that corresponds to the active sub-model and the estimation of the parameters of all where an bound on noise assumed to be known. problem includes the estimation of sub-models. The proposed algorithm stages. The stability and convergence where an upper upper on the the noisetois isproceeds assumed totwo be successive known. This problem the of the discrete state that corresponds the activein andThis the estimation of the parameters of all sub-models. Thebound proposed algorithm proceeds insub-model two successive stages. Theincludes stability andestimation convergence the discrete state that corresponds to the active sub-model and the estimation of the parameters of all proprieties are established. the discrete state that corresponds to the active sub-model and the estimation of the parameters of all sub-models.are The proposed algorithm proceeds in two successive stages. The stability and convergence proprieties established. sub-models. The proposed algorithm proceeds in in twomodels, successive stages.noise, The convergence stability and and properties. convergence sub-models. The proposed algorithm proceeds two successive stages. The stability convergence Keywords: Switched systems, Switched Error bounded proprieties are established. © 2017, IFAC (International Federation Automatic bynoise, Elsevier Ltd. All rights reserved. Keywords: systems, SwitchedofOutput Output ErrorControl) models,Hosting bounded convergence properties. proprieties are proprieties Switched are established. established. Keywords: Switched systems, Switched Output Error models, bounded noise, convergence properties. Keywords: Switched systems, systems, Switched Switched Output Output Error Error more, models, bounded bounded noise, noise, convergence convergence properties. et al., 2016b] Keywords: Switched models, properties. 1. 1. INTRODUCTION INTRODUCTION more, the the methods methods ([Bako ([Bako et et al., al., 2011], 2011], [Goudjil [Goudjil et al., 2016b] and [Goudjil et al., 2016a] parameterize the linear 1. INTRODUCTION more, the methods al., 2011], [Goudjil et al., 2016b] and [Goudjil et al.,([Bako 2016a]et parameterize the switched switched linear This paper presents a recursive output error identification alINTRODUCTION more, the methods ([Bako et al., 2011], [Goudjil et al., 2016b] systems as a SARX models. Another common disadvantage is This paper presents 1. a recursive output error identification al1. INTRODUCTION more, theasmethods al., 2011], [Goudjil et al., 2016b] and [Goudjil et al.,([Bako 2016a]et Another parameterize the switched linear systems a SARX models. common disadvantage is gorithm for switched linear systems with bounded noise. Such and [Goudjil et al., 2016a] parameterize the switched linear that most of them perform well in the noise free case or with gorithm for presents switchedalinear systems witherror bounded noise. Such This paper recursive output identification aland [Goudjil et al., 2016a] parameterize the switched linear systems as a SARX models. Another common disadvantage is that most of them perform well in the noise free case or with This paper presents a recursive output error identification alsystem is defined as set of finite number of dynamical linear This paper presents recursive output error al- systems as aa SARX models. Another common disadvantage is the usual noise assumption except ([Goudjil et al., system isfor defined as aaalinear set of aasystems finite number of identification dynamical linear gorithm switched with bounded noise. Such systems asstochastic SARXperform models. Another common disadvantage is the usual stochastic noise assumption except ([Goudjil et al., that most of them well in the noise free case or with gorithm for switched linear systems with bounded noise. Such sub-systems. The switching mode between sub-systems can be gorithm for switched linear systems with bounded noise. Such that most of them perform well in the noise free case or with 2016b] and [Goudjil et al., 2016a]). In many cases the stochassystem is defined as a set of a finite number of dynamical linear sub-systems. The switching mode between sub-systems can be that most of them perform well in the noise free case or with the usual stochastic noise assumption except ([Goudjil et al., 2016b] and [Goudjil et al., 2016a]). In many cases the stochassystem defined as aa set aa finite number of dynamical linear arbitrary. the last years, aa sizable literature switched system is is In defined as set of of finitebetween number sub-systems ofon linear the usual stochastic noise assumption except ([Goudjil et al., al., tic assumption not satisfied the only information arbitrary. In the last years, sizable literature ondynamical switchedcan linear sub-systems. The switching mode be the usual stochastic except ([Goudjil et 2016b] and [Goudjil noise etis 2016a]). many the stochastic noise noise assumption is al., notassumption satisfiedInand and the cases only information sub-systems. The switching mode between sub-systems can be systems has been developed. This is due to the increasing in sub-systems. The switching mode between sub-systems can be 2016b] and [Goudjil et al., 2016a]). In many cases the stochasavailable is an upper bound on the noise. Thus, we have chosen systems has been developed. This literature is due toon theswitched increasing in 2016b] arbitrary. In the last years, a sizable linear and [Goudjil et al., 2016a]). In many cases the stochastic noise is assumption is not on satisfied andThus, the only information available an upper bound the noise. we have chosen arbitrary. In the last years, a sizable literature on switched linear importance such system many engineering fields, such as arbitrary.has In of the lastdeveloped. years, sizable literature on switched linear tic noise assumption is satisfied and the information in this paper the bounded noise assumption. importance of such systema in in many engineering fields, such as systems been This is due to the increasing in tic noise assumption is not not satisfied andThus, the only only information in this paper the bounded noise assumption. available is an upper bound on the noise. we have chosen systems has been developed. This is due to the increasing in computer vision, automotive control, air traffic control, etc. (see systems has developed. This isengineering due to the increasing in available is upper bound on noise. we importance ofbeen such system incontrol, many fields, such as computer vision, automotive air traffic control, etc. (see available is an anthe upper bound on the the noise.inThus, Thus, we have have chosen chosen in this paper bounded noise assumption. The contribution of this paper consist the adaptation importance of such system in many engineering fields, such as [Sun and Ge, 2005b], [Sun and Ge, 2005a], [Garulli et al., 2012] importance of such system in many engineering fields, such as The contribution of this paper in the adaptation of of an an in this this paper the the bounded bounded noise consist assumption. [Sun and Ge, 2005b], [Sun and Ge, 2005a], [Garulli et al., 2012] computer vision, automotive control, air traffic control, etc. (see in paper noise assumption. Outer Bounding Ellipsoid (OBE) type algorithm presented in computer vision, automotive control, air traffic control, etc. (see and references therein). Examples of existing methods in of this paper consist the adaptation of an computer vision, automotive control, traffic control, etc. (see Outercontribution Bounding Ellipsoid (OBE) type in algorithm presented in and references therein). Examples ofairthe the existing methods in The [Sun and Ge, 2005b], [Sun and Ge, 2005a], [Garulli et al., 2012] The contribution of this paper consist in the adaptation of an ([Goudjil et al., 2016b] and [Goudjil et al., 2016a]) for the The contribution of this paper consist in the adaptation of an [Sun and Ge, 2005b], [Sun and Ge, 2005a], [Garulli et al., 2012] literature for the identification of such system are: the algebraic [Sun and Ge, 2005b], [Sun and Ge, 2005a], [Garulli et al., 2012] ([Goudjil et al., 2016b] and [Goudjil et al., 2016a]) for the Outer Bounding Ellipsoid (OBE) type algorithm presented in literature for thetherein). identification of such system are: the algebraic and references Examples of the existing methods in Outer Bounding Ellipsoid (OBE) type algorithm presented in recursive identification of switched linear output error models. Outer Bounding Ellipsoid (OBE) type algorithm presented in and references therein). Examples of the existing methods in method ([Vidal et al., 2003]), the clustering-based methods ([Goudjil et al., 2016b] and [Goudjil et al., 2016a]) for the and references therein). Examples of the existing methods in recursive identification of switched linear output error models. literature for the identification of such system are: the algebraic method ([Vidal et al., 2003]), the clustering-based methods This ([Goudjil et al., 2016b] and [Goudjil et al., 2016a]) for the model has the advantage that the disturbance model ([Goudjil et al., 2016b] and [Goudjil et al., 2016a]) for the literature for the identification of such system are: the algebraic ([Ferrari-Trecate et al., 2003] and [Nakada et al., 2005]), the recursive identification of switched linear output error models. literature for the identification of such system are: the algebraic model has the advantage that the disturbance model is is ([Ferrari-Trecate et al., al., 2003]), 2003] and [Nakada et al., 2005]), the This method ([Vidal et the clustering-based methods recursive identification of linear output models. not incorporated with the process model because the noise recursive identification of switched switched linear output error error models. method et([Juloski al., 2003]), the clustering-based methods Bayesian method et 2005]), the based method ([Vidal ([Vidal al., 2003]), the clustering-based methods not incorporated withadvantage the process model because themodel noise is is This model has the that the disturbance Bayesian methodetet ([Juloski et al., al.,and 2005]), the optimization optimization based ([Ferrari-Trecate al., 2003] [Nakada et al., 2005]), the This model model has the output. advantage that the the of disturbance model is added directly to An convergence is has advantage that disturbance model is ([Ferrari-Trecate et al., and [Nakada et al., 2005]), the methods ([Bako, 2011], [Lauer et al., 2011] et al., ([Ferrari-Trecate et al., 2003] 2003] and [Nakada etand al.,[Ozay 2005]), the This added directly to the the output. An analysis analysis of convergence is preprenot incorporated with the process model because the noise is methods ([Bako, 2011], [Lauer et al., 2011] and [Ozay et al., Bayesian method ([Juloski et al., 2005]), the optimization based not incorporated with the process model because the noise is sented. The use of an OBE type algorithm is suitable for system not incorporated with the process model because the noise is Bayesian method ([Juloski et al., 2005]), the optimization based 2015]), the bounded error method ([Bemporad et al., 2005]) An algorithm analysis ofisconvergence is preBayesian method ([Juloski etmethod al., et 2005]), the optimization based sented.directly The usetoofthe an output. OBE type suitable for system methodsthe ([Bako, 2011], [Lauer al., 2011] andet[Ozay et al., added 2015]), bounded error ([Bemporad al., 2005]) added directly to output. An analysis of convergence is bounded noise. Some that the added directly toofthe the output. An analysis convergence is prepremethods ([Bako, 2011], [Lauer et 2011] and [Ozay et and the methods (( [Vidal, 2008], [Bako al., 2011], sented. The usewith an OBE type algorithm suitable system methods ([Bako, 2011], [Lauer et al., al., 2011] andetet [Ozay et al., al., identification identification with bounded noise. Someofisworks works thatforused used the and the recursive recursive methods [Vidal, 2008], [Bako et al., 2005]) 2011], 2015]), the bounded error method ([Bemporad al., sented. The use of an OBE type algorithm is suitable for system OBE type algorithm are presented in ([Fogel and Huang, 1982], sented. The use of an OBE type algorithm is suitable for system 2015]), the bounded error method ([Bemporad et 2005]) [Goudjil et 2016b] and et al., 2015]), the bounded error method ([Bemporad etet al., al., 2005]) OBE type algorithm are presented inSome ([Fogel and Huang, 1982], identification with bounded noise. works that used the [Goudjil et al., al., 2016b] and ([Goudjil [Goudjil et al., 2016a]). 2016a]). and the recursive methods [Vidal, 2008], [Bako al., 2011], identification with bounded noise. Some works works that that used the [Dasgupta and Huang, 1987], [Canudas-De-Wit and Carrillo, identification with bounded noise. Some used the and the recursive methods (([Goudjil [Vidal, 2008], [Bako et al., 2011], and the recursive methods [Vidal, 2008], [Bako et al., 2011], [Dasgupta and Huang, 1987], [Canudas-De-Wit and Carrillo, OBE type algorithm are presented in ([Fogel and Huang, 1982], [Goudjil et al., 2016b] and et al., 2016a]). To the best of our knowledge, the existing methods menOBE type algorithm are presented in ([Fogel and Huang, 1982], 1990], [Boutayeb et al., 2002], [Pouliquen et al., 2011] and OBE type algorithm are presented in ([Fogel and Huang, 1982], [Goudjil et al., 2016b] and [Goudjil et al., 2016a]). To the best of our knowledge, the existing methods men[Dasgupta and Huang, and2011] Carrillo, [Goudjil et al., 2016b] and [Goudjil et al., 2016a]). 1990], [Boutayeb et al.,1987], 2002],[Canudas-De-Wit [Pouliquen et al., and tioned previously parameterize the switched linear systems as [Dasgupta and Huang, [Canudas-De-Wit and Carrillo, et al., 2016]). The low computational complexity [Dasgupta and Huang, 1987], [Canudas-De-Wit and Carrillo, tioned previously parameterize thethe switched linear systems as [Pouliquen To the best of our knowledge, existing methods men1990], [Boutayeb et al.,1987], 2002], [Pouliquen et al., 2011] and [Pouliquen et al., 2016]). The low computational complexity To the best of our knowledge, the existing methods menaationed Switched AutoRegressive eXogenous (SARX) models. Only ToSwitched the previously bestAutoRegressive of our knowledge, existing methods men1990], [Boutayeb et al., 2002], [Pouliquen et al., al., complexity 2011] and motivates the use of such methods. The proposed recursive parameterize thethe switched linear systems as 1990], [Boutayeb et al., 2002], [Pouliquen et 2011] and eXogenous (SARX) models. Only [Pouliquen et al., 2016]). The low computational motivates the use of such methods. The proposed recursive tioned previously parameterize the linear systems as few methods have been for the identification of tioned previously the switched switched systems as [Pouliquen [Pouliquen et al., 2016]). The low computational complexity output error identification algorithm realizes at each time a Switched AutoRegressive eXogenous models. Only et al., 2016]). The low computational complexity few methods haveparameterize been developed developed for(SARX) the linear identification of motivates theidentification use of suchalgorithm methods. realizes The proposed output error at each recursive time the the aswitched AutoRegressive eXogenous (SARX) models. Only systems with switched output model. The a Switched Switched AutoRegressive eXogenous (SARX) models. Only motivates the use of such methods. The proposed recursive estimation of the discrete state and the update of the parameters motivates the use of such methods. The proposed recursive switched systems with switched outputforerror error model. The methmethfew methods have been developed the identification of output error identification algorithm realizes at each time the estimation of the discrete state and the update of the parameters few methods have been developed for the identification of ods in ([Juloski and Weiland, 2006] and [Canty et few presented methods have been developed for themodel. identification of vector output error algorithm at each time the associated to the estimated state two succesoutput errorofidentification identification algorithm realizes atvia each time the ods presented in ([Juloski and Weiland, 2006] and [Canty et al., al., switched systems with switched output error The methestimation the discrete state anddiscrete therealizes update of the parameters vector associated to the estimated discrete state via two successwitched systems with switched output error model. The meth2012]) can be applied for the identification of the PieceWise switched systems with switched output error model. The methestimation of the discrete state and the update of the parameters sive stages. The algorithm guarantees the identification error to ods presented in ([Juloski and Weiland, 2006] and [Canty et al., estimation of the discrete state and the update of the parameters 2012]) can be applied for the identification of the PieceWise vector associated to the estimated discrete state via two successive stages. The algorithm guarantees the identification error to ods in ([Juloski and Weiland, 2006] and [Canty et Affine Output Error (PWAOE) models. This is aa particular ods presented presented inapplied ([Juloski and Weiland, 2006] and [Canty et al., al., vector vector associated to the estimated discrete state via two succesbe bounded by the upper bound on the noise. associated to the estimated discrete state via two succesAffine Output Error (PWAOE) models. This is particular 2012]) can be for the identification of the PieceWise sive stages. The algorithm guarantees the identification error to be bounded by the upper bound on the noise. 2012]) can be for the of the switched models where the switching is defined 2012]) Output canlinear be applied applied for the identification identification ofmode thea PieceWise PieceWise stages. The algorithm guarantees the identification error sive stages. The algorithm guarantees the identification error to to switched linear models where the switching mode is defined sive Affine Error (PWAOE) models. This is particular The rest of the paper is organized in the following manner: be bounded by the upper bound on the noise. Affine Output Error (PWAOE) models. This is a particular by aa polyhedral partition of space. and rest of by thethe paper is bound organized in noise. the following manner: Affine Output Error (PWAOE) models. This isIn particular be bounded upper on the be section bounded by the upper bound on the noise. by polyhedral partition of the the regression regression space. Ina[Wang [Wang and The switched linear models where the switching mode is defined In 2 a description of the switched output error models switched linear models where the mode is Chen, 2011] an online of linear output rest of2 the paper is organized in theoutput following In section a description of the switched errormanner: models switched linear modelsidentification where the switching switching mode is defined defined by a polyhedral of the regression space. In [Wang and The Chen, 2011] an partition online identification of switched switched linear output The of paper is organized in the following manner: and aarest formulation of the are presented. Thesection rest of2 the the paper is identification organized in problem theoutput following manner: by aa polyhedral partition of the regression space. In [Wang and error models is presented. and formulation of the identification problem are presented. In a description of the switched error models by polyhedral partition of the regression space. In [Wang and error models is presented. Chen, 2011] an online identification of switched linear output In section description ofstages the switched switched outputare error models sections 44ofthe identification algorithm and In section 2233aaand description of the output error models Chen, 2011] an online identification of switched linear output and a formulation thetwo identification problem presented. In sections and the two stages identification algorithm and Chen, 2011] an online identification of switched linear output error models is presented. Unfortunately, most of the existing methods present some limand a formulation of the identification problem are presented. the convergence analysis are provided. Section 5 gives some and a formulation of the identification problem are presented. error models is presented. Unfortunately, most of the existing methods present some lim- In sections 3 and analysis 4 the twoare stages identification algorithm and the convergence provided. Section 5 gives some error models is presented. iting most of be applied for In sections 33 and 44Section the identification algorithm and simulation results. 66stages concludes the paper. In sections and analysis the two twoare stages identification algorithm and iting features. features. Indeed, Indeed, most of them them cannot cannot bepresent appliedsome for recurrecurthe convergence provided. Section 5 gives some Unfortunately, most of the existing methods limsimulation results. Section concludes the paper. Unfortunately, most of the existing methods present some limsive identification and they have high computational complexity the convergence analysis are provided. Section 5 gives some Unfortunately, most of the existing methods present some limthe convergence analysis are provided. Section 5 gives some sive identification and most they have highcannot computational complexity simulation results. Section 6 concludes the paper. iting features. Indeed, of them be applied for recuriting features. Indeed, most of them cannot be for recur2. FORMULATION except recursive methods. However, no analysis simulation results. results. Section 66 concludes concludes the paper. paper. itingidentification features. Indeed, most of them cannot be applied applied for recur- simulation Section the 2. PROBLEM PROBLEM FORMULATION except recursive methods. However, no convergence convergence analysis sive and they have high computational complexity sive identification and they have high computational complexity has been done for the recursive methods ([Bako et al., 2011], sive identification and they have high computational complexity except recursive methods. However, no convergence analysis 2. PROBLEM FORMULATION has been done for the recursive methods ([Bako et al., 2011], Consider a discrete time linear 2. PROBLEM PROBLEM FORMULATION except recursive methods. However, no convergence analysis [Wang and Chen, 2011] and [Goudjil et al., 2016b]). Further2. FORMULATION except recursive methods. However, no analysis Consider a discrete time switched switched linear system system parameterized parameterized [Wang and Chen, 2011] and [Goudjil et convergence al., 2016b]). Furtherhas been done for the recursive methods ([Bako et al., 2011], by a Switched Output Error (SOE) model as follows: has been done for the recursive methods ([Bako et al., 2011], Consider a discrete time switched linear system parameterized has been done for the recursive methods ([Bako et al., 2011], by a Switched Output Error (SOE) model as follows: and Chen, 2011] and [Goudjil et al., 2016b]). FurtherB Consider aa discrete time switched linear system parameterized 1[Wang λk (q) Consider discrete time switched linear system parameterized and Chen, [Goudjil et al., 2016b]). FurtherThis work has been 2011] partiallyand supported by the French program InvestisseB (q) [Wang and Chen, 2011] and [Goudjil et al., 2016b]). Further1[Wang by a Switched Output Error (SOE) model as follows: λ y u + e (1) = k k + ek as follows: k = (SOE)umodel This work has been partially supported by the French program Investisseby a Switched Output Error y (1) by a Switched Output Error (SOE) model as follows: d’Avenir. k k k A B (q) 1ment λ 1 This λ k ment d’Avenir. work has been partially supported by the French program InvestisseA (q) k Bλλλkk (q) uk + ek yk = B (1) 11 This work has been partially supported by the French program InvestisseThis work has been partially supported by the French program Investisse+ eekk (1) = Aλkk (q) uukk + ment d’Avenir. yykk = (1) k (q) ment d’Avenir. A ment d’Avenir. Aλλkk (q) Copyright © 2017 IFAC 14677

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Proceedings of the 20th IFAC World Congress Abdelhak Goudjil et al. / IFAC PapersOnLine 50-1 (2017) 14112–14117 Toulouse, France, July 9-14, 2017

uk ∈ IR and yk ∈ IR are respectively the input and the output of the system. ek ∈ IR is an unknown but bounded noise. An upper bound δe on its magnitude is assumed to be known: | ek |≤ δe , ∀ k (2) λk ∈ {1, · · · , s} is the discrete state that indicates the active submodel at time k, s is the number of sub-models. Aλk (q), Bλk (q) are the unknown polynomial of the sub-model λk { (1) (n ) Aλk (q) = 1 + aλ q−1 + · · · + aλ a q−na k k (3) (1) (n ) Bλk (q) = bλ q−1 + · · · + bλ b q−nb k

k

where na and nb are the orders of the system, q−1 denotes the delay operator. The orders of the system are assumed to be known and equal for all sub-models. We suppose that there is no problem of stability on the system (no instability caused by switching). The one-step-ahead output predictor is defined bearing in mind the optimal predictor structure ([Ljung, 1999]), the system equation in (1) implies ([Wang and Chen, 2011] ,[Zhu, 2001]): Bλ (q) yk = yˆk + ek = k uk + e k (4) Aλk (q) yˆk is the non-measurable noise-free part of the output. The parameter vectors to be identified of the switched output error ] [ model are: (na ) (1) (n ) θλk = a(1) (5) bλ · · · bλ b λ · · · aλ k

k

k

k

Using this definition of the parameter vectors and from the definition of regressor vector ϕk ∈ IRn as: ϕkT = [−yˆk−1 · · · −yˆk−na uk−1 · · · uk−nb ] (6) with n = na + nb are the number of parameters of each submodel, the non-measurable noise-free part of the output yˆk can be written as follow: yˆk = ϕkT θλk = yk − ek (7) T Thus: yk = ϕ θλ + ek (8) k

k

Note that, the optimal predictors yˆk−i introduced in the regressor vector ϕk are unknown and non-measurable. In the identification algorithm, these predictors will be substituted by their estimates. Furthermore, at each time k, at least one parameter vector θi is such that | yk − ϕkT θi |≤ δe . This θi corresponds to the active sub-model at time k (i = λk ). If at the time k the output yk is generated by the ith sub-model, i.e. λk = i, then: yk = ϕkT θi + ek yk can also be expressed from the other parameter vectors as:  T T ϕ y =  k k θ1 + ek + ϕk (θi − θ1 )    .    .. yk = ϕkT θi + ek   .  ..     yk = ϕkT θs + ek + ϕkT (θi − θs ) Thus (9) can be rewritten as:     ϕTθ   ek + ϕkT (θi − θ1 ) yk k 1  ..   ..  ..     .   .   .        T    ek  yk  =  ϕk θi  +    .   .    ..  .   .    . . . T T yk ek + ϕk (θi − θs ) ϕk θs

(9)

(10)

We define the global parameter vector Θ∗ ∈ IRns×1 , the extended noise vector Vk ∈ IRs×1 and the extended output vector Yk ∈ IRs×1 as:

  θ1  ..  ∗ Θ =  . ,

θs

14113



 ek + ϕkT (θi − θ1 )   ..   .     ek Vk =     ..   . T ek + ϕk (θi − θs )

  yk  ..  Yk =  .  , yk

Thus, (10) can be rewritten as: (11) Yk = φkT Θ∗ +Vk with φk = Is ⊗ ϕk , where ⊗ denotes the Kronecker product.

Note that, there are s! possible variants of Θ∗ from s! different combinations between parameter vectors {θi }si=1 . Θ∗ can be any combination among the s! possible combinations.

Objective: Given a set of measurements {uk ,Yk }Nk=1 , the number of sub-models s and their orders na and nb . The objective is to estimate the discrete state {λk }Nk=1 and the global pa� must satisfy: rameter vector Θ∗ . These estimations λˆ k and Θ ˆ ˆ | εk (λk ) |≤ δe , ∀ k, where εk (λk ) denotes the λˆ kth component � of εk = Yk − φkT Θ. 3. IDENTIFICATION ALGORITHM

Before bringing the algorithm let us introduce some definitions: � k is an λˆ k is the estimate of the discrete state λk at time k. Θ estimation of the global parameter vector Θ∗ at time k. The � k is as follows: structure of Θ �  θ1/k .  �k =  Θ (12)  ..  θ�s/k

where, for any i ∈ {1, · · · , s}, θ�i/k is an estimate of θi . We also define the a priori and the a posteriori predictors as:  T�  T�   ϕˆk θ1/k−1 ϕˆk θ1/k    . .  T T � � .. Yˆk/k−1 = φ�k Θk−1 =   , Yˆk/k = φ�k Θk =  ..  ϕˆkT θ�s/k−1 ϕˆkT θ�s/k with φ�k is an estimate of the matrix φk obtained by substuting the unknown noise-free part of the output yˆk−i by their a posteriori predictors Yˆk−i/k−i (λˆ k−i ): φ�k = Is ⊗ ϕˆk with [ ϕˆkT = −Yˆk−1/k−1 (λˆ k−1 )· · ·−Yˆk−n

ˆ

(λk−na ) a /k−na

uk−1· · ·uk−nb

]

Considering the definition of ϕˆk , we define the a priori prediction error εk/k−1 and the a posteriori prediction error εk/k as:

εk/k−1 = Yk − Yˆk/k−1 , εk/k = Yk − Yˆk/k (13) The Recursive Switched Output Error (R-SOE-OBE) identification algorithm for switched linear systems with bounded noise is described in Table 1. The algorithm proceeds in two successive stages at each time k. Stage 1: Detection of the discrete state λk . The estimated current sub-model λˆ k is the index i of the smallest component | εk/k−1 (i) | with i ∈ {1, · · · , s}. This value is denoted ζk . Stage 2: Update the estimation of Θ∗ This stage is an adapted Outer Bounding Ellipsoid (OBE)

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• If ζk ≤ δ , then σk = 0, consequently we get | εk/k (λˆ k ) |=| εk/k−1 (λˆ k ) |. This gives | εk/k (λˆ k ) | δ • If ζk > δ , then using the value of σk in (17) for σk ̸= 0 we get: εk/k = Ck−1 εk/k−1 . It follows δ εk/k (λˆ k ) = ε (λˆ k ) ζk k/k−1 Thus from (14) we get | ε (λˆ k ) |= δ .

R-SOE-OBE ALGORITHM Initialization: � 0 is randomly initialized. Θ P0 = p0 Ins with p0 ≫ 1. max{n ,n } max{n ,n } {Yˆk/k−1 }k=1 a b = {Yˆk/k }k=1 a b = 0s (0s is a vector of length s with all entries = 0) Stage 1 : Detection of the discrete state λk  T�   εk/k−1 = Yk − φ�k Θk−1 ,

λˆ k = argmin | εk/k−1 (i) | i=1,··· ,s

ζk = min | εk/k−1 (i) |=| εk/k−1 (λˆ k ) |

 

k/k

(14)

This result is consistent with the study objective. The estimation � k provided by the R-SOE-OBE algorithm is such that, of Θ at each time k: | εk/k (λˆ k ) |≤ δ . Thus, δ is a bound on the a posteriori error of the estimated active sub-model. Its value is specified taking into account the upper bound on the noise δe .

(15)

4. ANALYSIS OF R-SOE-OBE ALGORITHM

i=1,··· ,s

Stage 2 : Update the estimation of Θ∗ with

� k−1 + Γk εk/k−1 �k = Θ Θ

  Γk = Pk−1 φ�k σk (γ Is + φ�kT Pk−1 φ�k σk )−1 1 (16)  Pk = (Ins − Γk φ�kT )Pk−1 γ where   γ (φ�kT Pk−1 φ�k )−1 (Ck − Is ) σk = if φ�kT Pk−1 φ�k > 0 and ζk > δ (17)  0 otherwise and λˆ kth   1 ··· ··· ··· 0  .. . . . . . . ..  . . . . .  . (18) . .  . Ck =  .. . . ζk . . ..  λˆ kth    . . .δ . .   .. . . . . . . ..  0 ··· ··· ··· 1 • δ is a user defined parameter. • 0 < γ < 1 is the forgetting factor. It’s used to limit the weight of past data.

algorithm presented in ([Canudas-De-Wit and Carrillo, 1990], [Tan et al., 1997], [Pouliquen et al., 2011] and [Goudjil et al., 2016a]). σk is a switching flag. The value of ζk is compared to δ as shown in (17). This leads to the two following cases: � k , thanks to the • if ζk ≤ δ , then no updating is done for Θ switching flag σk = 0 which gives Γk = 0. � k , only the parameter vector θ�ˆ • if ζk > δ , then in Θ λk associated to λˆ k is updated. This is established thanks to the particular structure of the switching flag σk ∈ IRs x s . There is only one value different from zero in σk . Its position corresponds to the λˆ kth line and the λˆ kth column. This value acts on the adaptation gain Γk so as to update only the parameter vector associated to λˆ k . From Tab. 1, εk/k can be written as:

If

φ�kT Pk−1 φ�k

εk/k = γ (γ Is + φ�kT Pk−1 φ�k σk )−1 εk/k−1

> 0, then the two following cases occur:

The estimation of the discrete state has a significant role in the convergence of the R-SOE-OBE algorithm. Two cases occur: • The discrete state can be correctly estimated, then ”λˆ k = λk ” and the output yk is written as: yk = ϕkT θλk + ek with | ek |≤ δe

• The discrete state can be badly estimated, then ”λˆ k ̸= λk ” and the output yk is written as: yk = ϕkT θλˆ + ϕkT (θλk − θλˆ ) + ek k

k

We define the impact { of the estimation of the discrete state as: 0, if λˆ k = λk (20) ℓk = ϕkT (θλk − θλˆ ), if λˆ k ̸= λk k

ϕkT (θλk

Since, − θλˆ ) is finite and bounded. Thus, there is an k upper bound δℓ such that at each time k: (21) | ϕkT (θλk − θλˆ ) |≤ δℓ k

Table 1. The proposed R-SOE-OBE algorithm

εk/k = εk/k−1 − φ�kT Γk εk/k−1 = (Is − φ�kT Γk )εk/k−1 Using the expression of Γk :

It has been shown that the R-SOE-OBE algorithm provides an estimation consistant with the objective stated in section 2. � k? What about Θ

(19)

In the following, 0 ≤ Lk ≤ 1 denotes the proportion of ℓk on the time interval [1; k] as: 1 k | ℓi | (22) Lk = ∑ k i=1 δℓ

The fact that φk is replaced by its estimate φ�k has a strong impact on the stability and convergence conditions of the algorithm. This replacement leads to the following assumptions used in Theorem 4 below. Assumption 1. To ensure the stability of the R-SOE-OBE algorithm, some sufficient conditions are required: A.1 {Ai (q)}si=1 are polynomials with all their zeros located in the unit circle , they are such that: ∥ 1 − Ai (q) ∥1 < 1 where ∥ . ∥1 is the l1 induced norm. A.2 δ is such that: ∥ Ai (q) ∥1 δ≥ δe 1− ∥ 1 − Ai (q) ∥1 Remark 2. These assumptions are only sufficient conditions and the algorithm can work well in some cases where these assumptions are not satisfied. Similar assumptions are used in

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OBE type algorithm for the identification of output error models ([Pouliquen et al., 2011], [Pouliquen et al., 2014] [Pouliquen et al., 2016]). Before presenting the main result, recall the persistent excitation condition [Ljung, 1999]: {φk }Nk=1 is a persistently exciting sequence of order oe ≥ ns if there exist α > 0 and β > 0 such that for all k:

α Ins ≤

oe −1

T ≤ β Ins ∑ φk−i σk−i φk−i

(23)

i=0

Similar persistent excitation condition is used in ([Boutayeb et al., 2002] and [Pouliquen et al., 2016]). Remark 3. The persistent excitation condition requires that all sub-models are regularly visited. If only few sub-models are visited, then the positions of the non-zero value in the matrix σk remains the same over time and the other diagonal positions corresponding to unvisited sub-models remain to zero. It oe −1 T is block diagonal with some follows that ∑i=0 φk−i σk−i φk−i blocks equal to zero, condition (23) can not be satisfied in this context. Theorem 4. The class of systems defined in section (2) is considered. Assume that the sufficient conditions for stability (A.1) and (A.2) hold. If furthermore {φk }Nk=1 is a persistently exciting sequence of order oe ≥ ns, then the R-SOE-OBE algorithm described in Tab.1 is such that, while σk ̸= 0:  k = Θ∗ − Θ k • For all k ≥ oe +1, the parameter error vector Θ satisfies: kLk  k ∥2 ≤ η1 γ k ∥ Θ  0 ∥2 + + η2 1 − γ (24) ∥Θ 1−γ with η1 > 0 and η2 ≥ 0 • Finally if the proportion of bad decision on the estimation of the discrete state Lk tends toward zero over time k, then  k is such that: Θ  0 ∥2  k ∥2 ≤ η1 γ k ∥ Θ (25) ∥Θ 

Proof.

IV.1) Proof that there exist Ssup and Sin f such that: 0 < Sin f Ins ≤ Pk−1 ≤ Ssup Ins From (16) and the matrix inversion lemma we have: Pk−1

−1 + φk σk φkT = γ Pk−1

= γ k P0−1 +

k−1

∑

i=0

T γ i φk−i σk−i φk−i

(26)

From (26), after oe iterations on Pk−1 we have: −1 + Pk−1 = γ oe Pk−o e

T ∑ γ i φk−i σk−i φk−i

i=0 oe −1 i T . Taking Pk−1 ≥ ∑i=0 γ φk−i σk−i φk−i

then 1, thus:

Pk−1 ≥ γ oe −1

oe −1 i=0

(28)

into account that γ <

T ∑ φk−i σk−i φk−i

In order to show (24) let us consider the Lyapounov function  T P−1 Θ  k = Θ∗ − Θ  k . From (15), (16) and (19),  k with Θ Wk = Θ k k  k−1 = Θ  k + 1 Pk−1 φk σk εk/k . Developing  k−1 is written as: Θ Θ γ the expression of Wk−1 we get: 1 T 2 T  T P−1 Θ  k σk εk/k + 2 εk/k Wk−1 = Θ σk φkT Pk−1 φk σk εk/k k k−1 k + Θk φ γ γ 1 −1 The matrix inversion lemma states: Pk−1 = (Pk−1 − φk σk φkT ). γ Using this expression in the preceding form of Wk−1 , we get Wk = γ Wk−1 + Qk (30) with  T φk σk φT Θ  T k σk εk/k − 1 ε T σk φT Pk−1 φk σk εk/k  Qk = Θ k k k − 2Θk φ k γ k/k • If σk = 0 then Qk = 0 and Wk = γ Wk−1  k , from (19) Qk can be • If σk ̸= 0, let us define Sk = φkT Θ rewritten as: T Qk = (εk/k − Sk ) σk (εk/k − Sk )− T γεk/k−1 σk (γ Is + φkT Pk−1 φk σk )−1 εk/k−1

From (17), we have: γ Is + φkT Pk−1 φk σk = γ Ck . As result, Qk is written as: T σkCk−1 εk/k−1 (31) Qk = (εk/k − Sk )T σk (εk/k − Sk ) − εk/k−1

Using the particular structure of σk , the only one value different from zero in σk is denoted ak ≥ 0. Using the form of the matrix Ck and ζk = | εk/k−1 (λˆ k ) |, Qk becomes: (32) (λˆ k ) | Qk = ak [ε (λˆ k ) − Sk (λˆ k )]2 − ak δ | ε k/k

k/k−1

Now, let us find the relationship between εk/k (λˆ k ) and Sk (λˆ k ), we have: ε (λˆ k ) = yk − ϕˆ T θˆ ˆ = ϕ T θ + ek − ϕˆ T θˆ ˆ k/k

k

(29)

From: {φk }Nk=1 is persistently exciting sequence of order oe , it results that Pk−1 ≥ Sin f Ins > 0 with Sin f = γ oe −1 α . kLk  k ∥2 ≤ η1 γ k ∥ Θ  0 ∥2 + + η2 1 − γ IV.2) Proof that ∥ Θ 1−γ

λk

k

λk /k

k

λk /k

+ ϕˆkT θλˆ − ϕˆkT θλˆ + ϕkT θλˆ − ϕkT θλˆ k k k k = ϕˆkT θ˜ ˆ + ℓk + ek + (ϕkT − ϕˆkT )θ ˆ λk /k

λk

Bearing in mind structures of and ϕˆkT , this gives: Sk (λˆ k ) = ˆ Aλˆ (q)εk/k (λk ) − Aλˆ (q)ek − ℓk . Replacing the expression of k k Sk (λˆ k ) in (32), Qk is rewritten as:

ϕkT

Qk = ak [(1−Aλˆ (q))εk/k (λˆ k )+Aλˆ (q)ek +ℓk ]2 −ak δ | εk/k−1 (λˆ k ) | k

From the persistent excitation assumption (23) we get: Pk−1 ≤ i γ k P0−1 + β ∑k−1 i=0 γ . We know that γ < 1, this guarantees the existence of Ssup such that: (27) Pk−1 ≤ Ssup Ins oe −1

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k

This can be rewritten as follow: Qk = ak [((1 − Aλˆ (q))εk/k (λˆ k ) + Aλˆ (q)ek )2 − δ | εk/k−1 (λˆ k ) |] k

k

+ ak [ℓ2k + 2((1 − Aλˆ (q))εk/k (λˆ k ) + Aλˆ (q)ek )ℓk ] k

k

To facilitate writing we can split Qk into two parts Qk = qk + Lk with qk = ak [((1−A ˆ (q))ε (λˆ k )+A ˆ (q)ek )2 − δ | ε (λˆ k ) |] k/k

λk

k/k−1

λk

Lk = ak ℓk [ℓk + 2((1 − Aλˆ (q))εk/k (λˆ k ) + Aλˆ (q)ek )] k

k

Now, let us prove that that qk ≤ 0, that’s means: ((1 − A ˆ (q))ε (λˆ k ) + A ˆ (q)ek )2 ≤ δ | ε k/k

λk

ˆ

k/k−1 (λk )

λk

|

(33)

From the fact that | εk/k−1 (λˆ k ) |> δ then, (33) is satisfied if: (34) ((1 − A ˆ (q))ε (λˆ k ) + A ˆ (q)ek )2 ≤ δ 2 k/k

λk

λk

From triangular inequality, (34) is satisfied if the following condition is satisfied: 1 1 (35) ((1 − A ˆ (q))2 ε 2 (λˆ k ))( 2 ) + (A2 (q)e2 )( 2 ) ≤ δ

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Let introduce the l1 induced norm ∥ · ∥1 and assume (A.1) to be true. Then (35) holds if: (36) ∥ 1 − A ˆ (q) ∥1 | ε (λˆ k ) | + ∥ A ˆ (q) ∥1 | ek |≤ δ k/k

λk

2

λk

We know that: | εk/k (λˆ k ) |≤ δ , | ek |≤ δe . Then, (36) holds if: ∥ 1 − Aλˆ (q) ∥1 δ + ∥ Aλˆ (q) ∥1 δe ≤ δ

(37)

k

This leads to the condition (A.2): δ ≥

∥ Aλˆ (q) ∥1

1 k Θ

k

1.5

δe

k

1− ∥ 1 − Aλˆ (q) ∥1 k ensuring qk ≤ 0. This gives: Qk ≤ Lk , consequently: (38) Wk ≤ γ Wk−1 + Lk � k ∥2 . Now it remains only to express the upper bound on ∥ Θ Using the l1 induced norm, we have:

0.5

0

−0.5

| (1 − Aλˆ (q))εk/k (λˆ k ) + Aλˆ (q)ek |≤ k

∥ 1 − Aλˆ (q) ∥1 δ + ∥ Aλˆ (q) ∥1 δe k

Then, using (37):

| Lk |≤ ak | ℓk | (| ℓk | +2δ ) Using (39) and (38) we get � T P−1 Θ � k ≤ γ kΘ � T P−1 Θ �0 + Θ 0 0 k k

k

(39)

k−1

∑ γ i ak (ℓ2k−i + 2 | ℓk−i | δ )

i=0

We have ak ≥ 0 and finite, so there exists ρ > 0 such that ak ≤ ρ . ℓk is such that | ℓk |≤ δℓ . By using the definition of Lk given in (22) we get kLk � T P−1 Θ � k ≤ γ kΘ � T P−1 Θ � 0 + ρ (δ 2 + 2δℓ δ ) 1 − γ Θ (40) 0 0 k k ℓ 1−γ It has been proved that 0 < Sin f Ins ≤ Pk−1 then we obtain kLk � k ≤ γ kΘ � T P−1 Θ � 0 + ρ (δ 2 + 2δℓ δ ) 1 − γ � T Sin f Θ Θ (41) 0 0 k ℓ 1−γ With P0 = po Ins , this inequality provides (24), with η1 = 1 1 > 0 and η2 = ρ (δℓ2 + 2δℓ δ ) ≥ 0 po Sin f Sin f

IV.3) Conclusion of proof if Lk → 0

Replacing Lk = 0 in (24) we get (25).

 This theorem states that the error on the global estimated parameter vector is bounded even in the presence of a bad decision on the estimation of the discrete state or in the case of 1 − γ kLk an unsuitable initialization. The term η2 is the worst 1−γ case error. It’s obvious that a bad initialization can lead to a bad decision on the estimation of the discrete state and an error on the estimation of the parameters vector. However, if the initialization is not too bad, its effect can be smoothed thanks to the forgetting factor. This ensures that the proportion � k ∥2 of bad decision decreases over time, then Lk → 0, thus ∥ Θ decreases exponentially as long as σk ̸= 0 and the persistent � k is the error excitation condition is satisfied. Recall that Θ ∗ � between Θk and one of the s! variants of Θ . It follows that, � k has converged in a neighborhood of one possible Θ∗ , once Θ then the adaptation is frozen. It can be shown that the size of the neighborhood is related to the bound on the noise but this is not the purpose of this article. Remark 5. The choice of δ plays an important role to obtain an estimate close to the true parameters vector. For too large value of δ , the R-SOE-OBE algorithm will be stable but it freezes the update early. For too small value, the stability conditions

0

200

400

600

800

1000

k

k

� k (full line) towards Θ∗ (dotted line). Fig. 1. Convergence of Θ

will be not satisfied. As the choice of δ depends on the system throughout Aλk (q) which is unknown and on the upper bound on the noise δe , the following strategy is recommended: Choose a large δ in the beginning of the identification procedure and then decrease its value over time. 5. NUMERICAL EXAMPLE In order to illustrate the performance of the proposed algorithm, the following system with two sub-models is considered: Bλ (42) yk = k + ek Aλk { with: A1 (q) = 1 + 0.65q−1 + 0.2q−2 sub-model 1 −1 −2 { B1 (q) = 1.4q +−10.95q A2 (q) = 1 − 0.3q + 0.55q−2 sub-model 2 B2 (q) = 2q−1 + 1.7q−2 The orders of the system are na = 2, nb = 2 (4 poles, 2 zeros), the number of sub-models is s = 2. λk switches randomly in λk ∈ {1, 2}. The input sequence uk is a zero mean random sequence with uniform distribution on [−2 ; 2]. The additive noise sequence ek is a zero mean random sequence uniformly distributed in [−δe ; δe ], δe is adjusted so as to test different values of the Signal to Noise Ratio (SNR). The R-SOE-OBE algorithm has been applied on sequence of length N = 1000 and for SNR = 25dB. The bound δ is chosen such that δ = 1.5δe . � k toward Θ∗ is shown on Fig. 1. It is clear The convergence of Θ � that Θk converges early to the true parameters.

In order to evaluate the performance of the proposed algorithm, the following statistical indices are used: ) ( y∥ ˆ • The FIT: FIT = 100 1 − ∥y− ∥y−y∥ � − Θ∗ ∥ ∥Θ • The Normalized Parameter Error: NPE = ∥ Θ∗ ∥ where yˆ is the estimated model output, Θ∗ is the true parameter � is the estimated parameter vector. vector and Θ

The R-SOE-OBE Algorithm has been applied on sequences of length N = 1000 and with SNR= 25dB. Tab.2 reports the results (the average estimation, the standard deviation and the measure of the statistical indices) of simulation of Monte Carlo with 100 runs. The results show the high performance of the proposed R-SOE-OBE Algorithm: the estimates are close to the real parameter, the standard deviation computed on each parameter is low.

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θˆ1

θˆ2

0.6512 ± 0.0083 0.1999 ± 0.0081 1.3954 ± 0.0272 0.9525 ± 0.0335

-0.2990 ± 0.0100 0.5500 ± 0.0079 2.0026 ± 0.0220 1.7026 ± 0.0294

Statistical indices FIT = 93.6% NPE = 0.0143

Table 2. Estimation results and statistical measures 100

0.15

FIT NPE

0.1

80

0.05

FIT

NPE

90

70 10

15

20

25

30

35

40

0 45

SNR)

Fig. 2. FIT and NPE as function of SNR. Fig. 2 illustrates the evolution of the FIT and the NPE as functions of SNR. The results confirm the good performance and prove the interest of the proposed algorithm. 6. CONCLUSION A recursive output error identification algorithm for switched linear systems with Bounded Noise has been proposed. Convergence analysis and stability are established. The simulation results show the good performance of the proposed algorithm. In term of future research, the use of an adaptation filter as proposed in [Pouliquen et al., 2011], [Pouliquen et al., 2014] and [Pouliquen et al., 2016] in order to relax stability conditions A.1 and A2 and the study of conditions that allow limk→∞ Lk = 0 are under investigation. REFERENCES Bako, L. (2011). Identification of switched linear systems via sparse optimization. Automatica, 47(4), 668–677. Bako, L., Boukharouba, K., Duviella, E., and Lecoeuche, S. (2011). A recursive identification algorithm for switched linear/affine models. Nonlinear Analysis: Hybrid Systems, 5(2), 242–253. Bemporad, A., Garulli, A., Paoletti, S., and Vicino, A. (2005). A bounded-error approach to piecewise affine system identification. IEEE Transactions on Automatic Control, 50(10), 1567–1580. Boutayeb, M., Becis, Y., and Darouach, M. (2002). Recursive identification of linear multivariable systems with bounded disturbances. IFAC World Congress, Barcelona. Canty, N., O’Mahony, T., and Cychowski, M. (2012). An output error algorithm for piecewise affine system identification. Control Engineering Practice, 20(4), 444–452. Canudas-De-Wit, C. and Carrillo, J. (1990). A modified EWRLS algorithm for systems with bounded noise. Automatica, 26(3), 599–606. Dasgupta, S. and Huang, Y.F. (1987). Asymptotically convergent modified recursive least square with data-dependent updating and forgetting factor for systems with bounded noise. IEEE Transactions on Information Theory, 33(3), 383–392. Ferrari-Trecate, G., Muselli, M., Liberati, D., and Morari, M. (2003). A clustering technique for the identification of piecewise affine systems. Automatica, 39(2), 250–217.

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