An extended instrument variable approach for nonparametric LPV model identification

2nd IFAC Workshop on Linear Parameter Varying Systems 2nd IFAC Workshop Linear Parameter Systems Florianopolis, Brazil,on September 3-5, 2018Varying 2nd IFAC on Linear Varying Systems 2nd IFAC Workshop Workshop Linear Parameter Parameter Varying Systems Available online at www.sciencedirect.com Florianopolis, Brazil,on September 3-5, 2018 Florianopolis, Brazil, September 3-5, 2018 2nd IFAC Workshop on Linear Parameter Varying Systems Florianopolis, Brazil, September 3-5, 2018 Florianopolis, Brazil, September 3-5, 2018

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IFAC PapersOnLine 51-26 (2018) 81–86

An extended instrument variable approach for An extended instrument variable approach for An extended instrument variable approach Annonparametric extended instrument variable approach for for LPV model identification Annonparametric extended instrument variable approach for LPV model identification nonparametric LPV model identification nonparametric LPV model identification nonparametric LPV model∗∗ Paulo identification Marcelo M. L. Lima ∗∗ Rodrigo A. Romano Lopes dos Santos ∗∗ ∗∗

Marcelo M. L. Lima ∗∗ Rodrigo A. Romano dos Santos ∗ Paulo Lopes ∗∗ Marcelo A. Felipe Pait ∗∗∗ ∗∗∗ ∗ Paulo Marcelo M. M. L. L. Lima Lima ∗ Rodrigo Rodrigo A. Romano Romano Paulo Lopes Lopes dos dos Santos Santos ∗∗ Felipe Pait ∗ ∗∗∗ Marcelo M. L. Lima Rodrigo A. Romano Paulo Lopes dos Santos ∗∗ Felipe Pait Felipe Pait ∗∗∗ ∗∗∗ ∗ Escola de Engenharia do Instituto Felipe Paita de Tecnologia, Prac¸a Mau´a n.1 ∗ Escola de Engenharia do Instituto Mau´ Mau´ a de Tecnologia, Prac¸¸ aa Mau´ a n.1 ∗∗ Escola de Engenharia do Instituto Mau´ 09580-900, S˜ ao Caetano do Tecnologia, Sul, Brazil, Prac de Engenharia do Instituto Mau´aa de de Tecnologia, Prac¸a Mau´ Mau´aa n.1 n.1 09580-900, S˜ aaoo Caetano do Sul, Brazil, ∗ Escola Escola de Engenharia do Instituto Mau´ a de Tecnologia, Prac ¸ a Mau´ a n.1 09580-900, S˜ Caetano do Sul, Brazil, ([email protected]; [email protected]) 09580-900, S˜ a o Caetano do Sul, Brazil, ([email protected]; [email protected]) ∗∗ Faculdade 09580-900, da S˜aoUniversidade Caetano do Sul, Brazil,Portugal, Rua do ([email protected]; [email protected]) de Engenharia do Porto, ∗∗ Faculdade ([email protected]; [email protected]) de Engenharia da Universidade do Porto, Portugal, Rua do ∗∗ ([email protected]; [email protected]) ∗∗Dr Faculdade de Engenharia da Universidade do Porto, Portugal, Roberto Frias s/n 4200–465 Porto, Portugal, ([email protected]) Faculdade de Engenharia da Universidade do Porto, Portugal, Rua Rua do do Roberto Frias s/n 4200–465 Porto, Portugal, ([email protected]) ∗∗Dr ∗∗∗ Faculdade de Engenharia da Universidade do Porto, Portugal, Rua do Dr Roberto Frias s/n 4200–465 Porto, Portugal, ([email protected]) Polit´ e cnica da Universidade de S˜ a o Paulo, Av Luciano Gualberto ∗∗∗ Escola Dr Roberto Frias s/n 4200–465 Porto, Portugal, ([email protected]) Escola Polit´ e cnica da Universidade de S˜ a o Paulo, Av Luciano Gualberto ∗∗∗ Escola Dr Roberto s/n Porto, Portugal, ([email protected]) ∗∗∗ Polit´ eeFrias cnica da Universidade de S˜ Av n.380 05508–010, S˜ao Paulo, ([email protected]) Polit´ cnica da 4200–465 Universidade de Brazil, S˜aaoo Paulo, Paulo, Av Luciano Luciano Gualberto Gualberto n.380 05508–010, S˜ aaoo Paulo, Brazil, ([email protected]) ∗∗∗ Escola Escola Polit´ e cnica da Universidade de S˜ a o Paulo, Av Luciano Gualberto n.380 05508–010, S˜ Paulo, Brazil, ([email protected]) n.380 05508–010, S˜ao Paulo, Brazil, ([email protected]) n.380 05508–010, S˜ao Paulo, Brazil, ([email protected]) Abstract: Linear parameter varying models (LPV) have proven to be effective to describe non–linearities Abstract: Linear parameter varying models (LPV) have proven to be effective to describe non–linearities Abstract: Linear models (LPV) have be to non–linearities and time–varying behaviors. varying In this work, a new non-parametric estimation algorithm for state-space LPV Abstract: Linear parameter parameter varying models (LPV) have proven proven to to be effective effective to describe describe non–linearities and time–varying behaviors. In this work, a new non-parametric estimation algorithm for state-space LPV Abstract: Linear parameter varying models (LPV) have proven to be effective to describe non–linearities and time–varying behaviors. In this work, a new non-parametric estimation algorithm for state-space LPV models based on support vector machines is presented. This technique allows the functional dependence and time–varying behaviors. In this work, aisnew non-parametric estimation algorithm for state-space LPV models based on support vector machines presented. This technique allows the functional dependence and time–varying behaviors. In this work, a new non-parametric estimation algorithm for state-space LPV models based on support vector machines is presented. This technique allows the functional dependence betweenbased the model coefficients and the scheduling signal to be “learned” from the input dependence and output models on support vector machines is presented. This technique allows the functional between the model coefficients and the scheduling signal to be “learned” from the input and output models based on support vector machines is presented. This technique allows the functional dependence between the model coefficients and the scheduling signal to be “learned” from the input and output data. The proposed algorithm is formulated in the context of instrumental (IV) estimators, in order to between the model algorithm coefficients and the scheduling signal of to instrumental be “learned”(IV) fromestimators, the input and output data. The proposed is formulated in the context in order to between the model coefficients and the scheduling signal to be “learned” from the input and output data. The proposed algorithm is formulated in the context of instrumental (IV) estimators, in order obtain consistent estimates for general noise conditions. The method is based on a canonical state–space data. The proposed algorithm is formulated in the context ofmethod instrumental (IV) estimators, in order to to obtain consistent estimates for general noise conditions. The is based on a canonical state–space data. The proposed algorithm is formulated in the ofmethod instrumental in order obtain consistent estimates general noise conditions. The is on canonical state–space representation and admits afor predictor form hascontext shown be suitable for(IV) system identification, asto it obtain consistent estimates for general noisethat conditions. Theto method is based based on aaestimators, canonical state–space representation and admits a predictor form that has shown to be suitable for system identification, as it obtain estimates general noise conditions. Theto method on afiltering canonical state–space representation and admits aafor predictor that has shown be suitable for system identification, it leads toconsistent a convenient regression form.form In addition, this predictor has is anbased inherent feature. Inas the representation and admits predictor form that has shown to be suitable for system identification, as it leads to a convenient regression form. In addition, this predictor has an inherent filtering feature. In the representation and admits a predictor form that has shown to be suitable for system identification, as it leads to a convenient regression form. In addition, this predictor has an inherent filtering feature. In the context of vector support machines, such filtering mechanism leads to two–dimensional data processing, leads to a convenient regression form. In addition, this predictor has an inherent filtering feature. In the context of vector support machines, such filtering mechanism leads to two–dimensional data processing, leads to a convenient regression form. In addition, this predictor has an inherent filtering feature. In context of vector support machines, such filtering mechanism leads to two–dimensional data processing, which can be used to decrease the variance of estimates due to noisy data. The performance of the contextcan of vector support machines, such filtering mechanism leads to two–dimensional data processing, which be used to decrease the variance of estimates due to noisy data. The performance of the contextcan ofapproach vector support machines, filtering mechanism to two–dimensional data which be decrease the variance of estimates to data. The of proposed isto from simulated subjectdue toleads different noise Theprocessing, technique which can be used used toevaluated decrease the such variance ofdata estimates due to noisy noisy data. scenarios. The performance performance of the the proposed approach is evaluated from simulated data subject to different noise scenarios. The technique which can be used to decrease the variance of estimates due to noisy data. The performance of the proposed approach is evaluated from simulated data subject to different noise scenarios. The technique was able to reduce the error due from to thesimulated variance of thesubject estimator in most of the analyzed proposed approach is evaluated data to different noise scenarios.scenarios. The technique was able to reduce the error due to the variance of the estimator in most of the analyzed scenarios. proposed approach is evaluated from simulated data subject to different noise scenarios. The technique was able to reduce the error due to the variance of the estimator in most of the analyzed scenarios. was able IFAC to reduce the error due to the variance of theControl) estimator in most the analyzed © 2018, (International Federation of Automatic Hosting by of Elsevier Ltd. Allscenarios. rights reserved. was able toNon–parametric reduce the error identification, due to the variance of the estimator most of the analyzed scenarios. Keywords: time–varying systems,inlearning algorithms, system Keywords: Non–parametric identification, time–varying systems, learning algorithms, system Keywords: Non–parametric identification, time–varying systems, systems, learning learning algorithms, algorithms, system system identification and estimationidentification, algorithms. time–varying Keywords: Non–parametric identification and estimation algorithms. Keywords: Non–parametric identification, time–varying systems, learning algorithms, system identification and estimation algorithms. identification and estimation algorithms. identification and estimation algorithms. parametric reconstruction of the function which characterizes 1. INTRODUCTION 1. INTRODUCTION parametric reconstruction of the function which characterizes 1. INTRODUCTION parametric reconstruction of characterizes the model parameters dependence on thewhich scheduling signal. 1. INTRODUCTION parametric reconstruction of the the function function which characterizes the model parameters dependence on the scheduling signal. 1. INTRODUCTION parametric reconstruction of the function which characterizes the model parameters dependence on the scheduling signal. The LS–SVM was applied in the LPV system identification the model parameters dependence on the scheduling signal. Linear parameter varying (LPV) models provide an effective The LS–SVM was applied in the LPV system identification the model parameters dependence on the scheduling signal. The LS–SVM was applied in the LPV system identification Linear parameter varying (LPV) models provide an effective context using input–output model descriptions, though most The LS–SVM was applied in the LPV system identification Linear parameter varying (LPV) models provide an effective approach for modeling of a large class of nonlinear and time– context using input–output model descriptions, though most Linear parameter varying (LPV) models provide anand effective The LS–SVM was applied in the LPV system identification context using input–output model descriptions, though most approach for modeling of a large class of nonlinear time– of the literature of parameter–varying filter and control design using input–output model descriptions, thoughdesign most Linear parameter varying anand effective approach for aa(LPV) large class of nonlinear time– varying systems (T´ oth, of 2010; dosmodels Santos et al., 2012, among of the literature of parameter–varying filter and control approach for modeling modeling of large class of provide nonlinear and time– context context using input–output model descriptions, though most of the literature of parameter–varying filter and control design varying systems (T´ o th, 2010; dos Santos et al., 2012, among focuses on state–space representations. Previous works based on of the literature of parameter–varying filter and control design approach for modeling of a large class of nonlinear and time– varying systems (T´ o th, 2010; dos Santos et al., 2012, among others). systems The development of efficient algorithms to estimate focuses on state–space representations. Previous works based on varying (T´ o th, 2010; dos Santos et al., 2012, among of the literature of parameter–varying filter and control design focuses on state–space representations. Previous works based on others). The development of efficient algorithms to estimate state-space representations (e.g., Rizvi et al. (2015b,a)) assume focuses on state–space representations. Previous works based on varying systems (T´ o th, 2010; dos Santos et al., 2012, among others). The development of efficient algorithms to estimate such models has attracted considerable research interest within state-space representations (e.g., Rizvi et al. (2015b,a)) assume others). The development of efficient algorithms to estimate focuses on state–space representations. Previous works based on state-space representations (e.g., Rizvi et al. (2015b,a)) assume such models has attracted considerable research interest within the availability of the state measurements. More recently, this state-space representations (e.g., Rizvi et al. (2015b,a)) assume others). The development of efficient algorithms to estimate such models has attracted considerable research interest within the system identification community (T´ o th et al., 2007, 2012; the availability of the state measurements. More recently, this such modelsidentification has attracted community considerable(T´ research interest within state-space representations (e.g., Rizvi et al. (2015b,a)) assume the availability of the state measurements. More recently, this the system o th et al., 2007, 2012; limiting factor has been overcome by Rizvi et al. (2018) using the availability of the state measurements. More recently, this such models has 2017). attracted considerable research interest within the community (T´ o et al., 2007, 2012; Alkhoury etidentification al., The identification models can limiting factor has been overcome by Rizvi et al. (2018) using the system system identification community (T´ oth thof etLPV al., 2007, 2012; the availability of the state measurements. More this limiting factor has been overcome by et (2018) using Alkhoury al., 2017). The identification models can nonlinear canonical correlation analysis. In Romano et al. (2016), factor has been overcome by Rizvi Rizvi et al. al. recently, (2018) using the systemet identification community (T´ocategory, thof etLPV al., 2007, 2012; Alkhoury et al., 2017). The identification of LPV can be classified in local or global. In the first amodels parameternonlinear canonical correlation analysis. In Romano et al. (2016), Alkhoury et al., 2017). The identification of LPV models can limiting limiting factor has been overcome by Rizvi et al. (2018) using nonlinear canonical correlation analysis. In Romano et al. (2016), be classified in local or global. In the first category, a parameteran LPV model estimation procedure based on a particular canonical correlation analysis. In Romano al. (2016), Alkhoury et in al., 2017). The of LPV“local” can nonlinear be or In aamodels parametervarying model islocal obtained by identification interpolating various linear an LPV model estimation procedure based on aet particular be classified classified in local or global. global. In the the first first category, category, parameternonlinear canonical correlation analysis. In Romano al. (2016), an LPV model estimation procedure based on particular varying model is obtained by interpolating various “local” linear state–space parameterization that can be written inaaetregression an LPV model estimation procedure based on particular be classified in local or global. In the first category, a parametervarying model is obtained by interpolating various “local” linear time–invariant (LTI) models (Vizer et al., 2015). This work state–space parameterization that can be written in regression varying model is(LTI) obtained by interpolating various “local” linear an LPV model estimation procedure based on a particular state–space parameterization that can be written in regression time–invariant models (Vizer et al., 2015). This work form enables model estimation to be addressed using LS–SVM. state–space parameterization that can be written in regression varying model obtained by interpolating various “local” linear time–invariant models (Vizer et 2015). This work focuses on theis(LTI) global approach, in which an LPV model is form enables model estimation to be addressed using LS–SVM. time–invariant (LTI) models (Vizer et al., al., 2015). This work state–space that can written inoutput, regression form enables model estimation to addressed using LS–SVM. focuses on the global approach, in which an LPV model is The relies on samples ofbethe input, and formmethod enablesparameterization modelonly estimation to be be addressed using LS–SVM. time–invariant (LTI) (Vizer et al., 2015). This work focuses the global approach, in which an LPV model is estimatedon directly frommodels input–output data. method relies only on samples of the input, output, and focuses on the global approach, in which an LPV model is The form enables model estimation to be addressed using LS–SVM. The method relies only on samples of the input, output, and estimated directly from input–output data. scheduling variables data. In addition, that method provides The method relies only on samples of the input, output, andaa focuses on the global approach, in which an LPV model is estimated directly from input–output data. scheduling variables data. In addition, that method provides estimated directly from input–output data. The method relies only on samples of the input, output, andaa scheduling variables data. In addition, that method provides The coefficients of LPV models are functions of a time–varying built–in low–pass filtering mechanism that can be used to cope variables data. mechanism In addition,that thatcan method provides estimated directly from input–output data. of a time–varying scheduling The coefficients of LPV models are functions built–in low–pass filtering be to The coefficients of LPV models are functions of aaistime–varying scheduling variables data. the In addition, thatcan method provides built–in low–pass filtering mechanism that can be used used to cope cope signal, the so–called scheduling signal, which assumed to with noisy data. However, effectiveness of this feature maya The coefficients of LPV models are functions of time–varying built–in low–pass filtering mechanism that be used to cope signal, the so–called scheduling which assumed to with noisy data. However, the effectiveness of this feature may The coefficients of LPV models aresignal, functions of ais signal, the so–called scheduling signal, which is assumed to built–in low–pass filtering mechanism that can be used to cope noisy data. However, the effectiveness of feature may be known measurable). Parametric methods require the be restricted to limited noise conditions. signal, the (or so–called scheduling signal, which istime–varying assumed to with with noisy data. However, the effectiveness of this this feature may be known (or measurable). Parametric methods require the be restricted to limited noise conditions. signal, the so–called scheduling signal, which is assumed to be known (or measurable). Parametric methods require the with noisy data. However, the effectiveness of this feature may be restricted to limited noise conditions. specification of these functions. If there is no prior knowledge be known (or measurable). Parametric methods require the be restricted to limited noise conditions. specification these If there no knowledge this worktowe consider same state–space structure of be known (orof require the In specification ofmeasurable). these functions. functions. If scheduling there is ismethods no prior prior knowledge be restricted limited noisethe conditions. about the dependence shape onParametric the signal, it is often In this work we consider the same state–space structure of specification of these functions. If there is no prior knowledge In this consider the state–space structure of about the dependence shape on the signal, it is often et al. we (2016), but a new algorithm is proposed using an In this work work we consider the same same state–space structure of specification functions. If scheduling there is no prior knowledge about the dependence shape scheduling signal, it is often necessary to of usethese a large seton ofthe basis functions. This leads to Romano Romano et al. (2016), but a new algorithm is proposed using an about the dependence shape on the scheduling signal, it is often In this work we consider the same state–space structure of Romano et al. (2016), but a new algorithm is proposed using an necessary to use a large set of basis functions. This leads to instrument variable (IV) estimator, in order to get accurate reRomano et al. (2016), but a new algorithm is proposed using an about the dependence shape on the scheduling signal, it is often necessary to use a large set of basis functions. This leads to overparameterized model structures and sparse true parameter variable (IV) estimator, in order to get accurate renecessary to use a model large set of basisand functions. Thisparameter leads to instrument Romano et al. (2016), but a new algorithm is proposed using an instrument variable (IV) estimator, in order to get accurate reoverparameterized structures sparse true sults under general noise scenarios. In fact, the IV approach was instrument variablenoise (IV) scenarios. estimator,Ininfact, order toIV getapproach accuratewas renecessary a model large set of variance basisand functions. Thisparameter leads to sults overparameterized structures sparse vectors. Astoa use consequence, the of the true estimates may under general the overparameterized model structures and sparse true parameter variablenoise (IV) estimator,In order getapproach results under general scenarios. the was vectors. As aa consequence, the variance of the estimates may previously to nonparametric estimation ofaccurate LPV modsults under employed general noise scenarios. Ininfact, fact, thetoIV IV approach was overparameterized and sparse true vectors. As consequence, theorder variance of (T´ theoth estimates may instrument be quite As large, evenmodel for a structures low model et parameter al., 2009; previously employed to nonparametric estimation of LPV modvectors. a consequence, the variance of the estimates may sults under general noise scenarios. In fact, the IV approach was previously employed to nonparametric estimation of LPV modbe quite large, even for a low order model (T´ o th et al., 2009; els to deal with colored noise, first for input–output structures employed to nonparametric estimation of LPV modvectors. As a consequence, theorder variance of (T´ theo estimates may previously be large, even model et Laurain al., 2012). els to deal with colored noise, first for input–output structures be quite quite et large, even for for aa low low order model (T´ oth th et al., al., 2009; 2009; previously employed to nonparametric estimation of LPV models to deal colored noise, for input–output structures Laurain et al., 2012). (Laurain et with al., 2011), and after first to state–space representations in els to deal with colored noise, first for input–output structures be quite large, even for a low order model (T´ o th et al., 2009; Laurain et al., 2012). (Laurain et al., 2011), and after to state–space representations in Laurain et al., 2012). els to deal with colored noise, first for input–output structures (Laurain et al., 2011), and after to state–space representations in Nonparametric approaches present an alternative to prevent (Rizvi et al., 2015a; Piga et al., 2015). These methods provide (Laurain et al., 2011),Piga and et after to2015). state–space representations in Laurain et al., 2012). Nonparametric approaches present an alternative to prevent (Rizvi et al., 2015a; al., These methods provide Nonparametric approaches present an alternative to prevent (Laurain et al., 2011), and after to state–space representations in (Rizvi et al., 2015a; Piga et al., 2015). These methods provide parameter dependence function selection issue. The least–square consistent estimates underetgeneral noise conditions, but at the Nonparametric approaches present an alternative to prevent (Rizvi et al., 2015a; Piga al., 2015). These methods provide parameter dependence function selection issue. The least–square consistent estimates under general noise conditions, but at the Nonparametric approaches present an alternative to prevent parameter dependence function selection issue. The least–square (Rizvi et al., 2015a; Piga et al., 2015). These methods provide consistent estimates under general noise conditions, but at the support vector machine (LS–SVM) framework enables the nonparameter dependence function selection issue. The least–square estimates under general noise conditions, but at the support vector machine (LS–SVM) framework enables the non- consistent parameter dependence selection issue. The least–square support machine (LS–SVM) framework enables the support vector vector machinefunction (LS–SVM) framework enables the nonnon- consistent estimates under general noise conditions, but at the support vector machine (LS–SVM) framework the Control) non-258 Hosting by Elsevier Ltd. All rights reserved. 2405-8963 © 2018 2018, IFAC (International Federation ofenables Automatic Copyright © IFAC Copyright © under 2018 IFAC 258 Control. Peer review responsibility of International Federation of Automatic Copyright © 258 Copyright © 2018 2018 IFAC IFAC 258 10.1016/j.ifacol.2018.11.164 Copyright © 2018 IFAC 258

IFAC LPVS 2018 82 Florianopolis, Brazil, September 3-5, 2018 Marcelo M.L. Lima et al. / IFAC PapersOnLine 51-26 (2018) 81–86

expense of increased variance compared to the least-squares counterpart (Laurain et al., 2012, 2015). In our proposal, we take a hint from the LTI identification literature and combine the instrumental variable method with the filtering feature provided by the considered model representation with the aim of decreasing the variance of standard IV estimator. The paper is organized as follows. The problem formulation is presented in Section 2. An extended instrumental variable approach for LPV model identification using support vector machines is developed in Section 3. The effectiveness of the presented approach is assessed in Section 4 using a simulation example. Concluding remarks are drawn in Section 5. 2. PROBLEM FORMULATION This paper considers linear parameter varying (LPV) models of the form xk+1 = (A + L(pk )C) xk + B(pk )uk yk = Cxk

(1)

respectively, where the operator vec (·) stacks the columns of the argument on top of each other. The LPV model (1)–(2) admits a predictor of the from    yk ⊗ F(pk ) ϕk+1 = A ϕk + B (7) uk yˆk = θ  ϕk

A  {A , . . . , A } ∈ ℜ2nx n f ×2nx n f

B  {C , . . . ,C } ∈ ℜ2nx n f ×2n f and ⊗ is the Kronecker product operator.

Notice that the previous result provides an output prediction equation in the regression form (8), whose parameters are the basis functions weights and the regressor vector ϕk is built from the input–output data and from the basis functions. Though the pair (C, A) can be chosen almost arbitrarily (the eigenvalues of A must be stable and (C, A) observable), a convenient choice is

(2)

 0 · · · 0 −αnx −αnx −1  1  A=  ... ...  1 −α1 

ℜnx

where xk ∈ is the state vector, uk ∈ ℜ and yk ∈ ℜ are the model input and output respectively. The scheduling signal pk : Z → P is assumed to be known at each sampling instant k, where P denotes the scheduling space. The matrices A and C can be arbitrarily chosen, provided the pair (A,C) is observable and the eigenvalues of A are located inside the unity circle. The time–varying parameter vectors L(pk ) ∈ ℜnx and B(pk ) ∈ ℜnx are defined through arbitrary basis functions fr (pk ), for r = {1, . . . , n f } such that nf

L(pk ) = B(pk ) =

∑ Lr fr (pk )

(3)

∑ Br fr (pk ),

(4)

r=1 nf

(8)

where

(9)

C = [0 · · · 0 1] because it allows us to write the regressor vector as     −n   yk qn x  −1  x I2n f ⊗ q ⊗ F(pk ) ϕk = ··· q uk α(q) qnx  f1 (pk−nx )yk−nx · · · f1 (pk−1 )yk−1 · · · = α(q) fn f (pk−nx )yk−nx · · · fn f (pk−1 )yk−1 f1 (pk−nx )uk−nx · · · f1 (pk−1 )uk−1 · · ·  fn f (pk−nx )uk−nx · · · fn f (pk−1 )uk−1 .

r=1

where Lr and Br are the weights of the basis functions. The previous state-space LPV model structure is based on the matchable-observable LTI parameterization proposed initially in the context of adaptive control theory (Morse and Pait, 1994) and applied more recently to multivariable system identification problems Romano and Pait (2017). For the sake of simplicity, this paper considers single–output model descriptions. However, the proposed contribution can be generalized quite straightforwardly to the multi-output case, provided the model structural indices (i.e., the set of observability indices) are assumed to be time-invariant. Such an assumption implies in invariance of the (lexicography–fixed) observability structure (Shafai and Carroll, 1986) and is often considered in the linear time-varying systems literature (Guidorzi and Diversi, 2003).

where

It is important to notice that, due to the observability of the pair (C, A), the frozen (local) models achieved by fixing pk to a constant value p, ¯ the model (1)–(2) matches any linear time–invariant transfer function of order nx . In addition, such representation enables us to state the following result, which is shown in Romano et al. (2016). Proposition 1. Define a basis function vector and a parameter vector as

where lm and bm for m = {1, . . . , nx } are the elements of the parameter vectors L(pk ) and B(pk ), respectively. This is equivalent to the one–step-ahead predictor of linear parameter-varying autoregressive with exogenous inputs (LPV–ARX) models with a special dynamic dependence on the scheduling variable p, which is referred to as shifted–form 1 in (T´oth et al., 2012; Schulz et al., 2017). Hence, LPV–ARX models in the shifted– form can be seen as a particular case of the state–space model structure adopted.

  F(pk ) = f1 (pk ) · · · fn f (pk ) ∈ ℜn f   θ = vec L1 · · · Ln f B1 · · · Bn f ∈ ℜ2nx n f ,

(5) (6) 259

α(q) = qnx + α1 qnx −1 + . . . + αnx −1 q + αnx is the characteristic polynomial of the user defined matrix A. The coefficients of α(q) determine the poles of the realization (7)–(8). A data–driven procedure to tune these design variables (hyper–parameters) is described in Subsection 3.3. Remark: Consider the particular choice α(q) = qnx . Direct manipulation of (8) leads to yˆk =

1

nx

∑ (lm (pk−m ) yk−m + bm (pk−m ) uk−m ) ,

m=1

The time shift of coefficients bm and lm coincide with ones of the associated input and output signals, respectively.

IFAC LPVS 2018 Florianopolis, Brazil, September 3-5, 2018 Marcelo M.L. Lima et al. / IFAC PapersOnLine 51-26 (2018) 81–86

Using a permutation matrix P, it is introduced a new (rearranged) regressor vector φk subject to qnx Pϕk = φk , α(q) which is given by     φk  F  (pk−nx ) ⊗ z (10) k−nx · · · F (pk−1 ) ⊗ zk−1 ,

with zk  [yk uk ] .

Thus, considering a data window of length N the output prediction vector can be represented as qnx Yˆ = ΦPθ (11) α(q) where  Yˆ = [yˆnx +1 · · · yˆN ]  Φ = [φnx +1 · · · φN ] . (12) As shown in Section 3, the regression vector in the form (10) is particularly suitable for the proposed estimation algorithm.

3. EXTENDED INSTRUMENTAL VARIABLE APPROACH 3.1 Nonparametric estimation algorithm Inspired by other instrument variable approaches, namely Laurain et al. (2011) and Laurain et al. (2015), the model estimation problem is formulated as 1 γ qnx  θˆ = arg min θ  θ + Z E, (13) 2 2 α(q) θ where the regularization parameter γ ∈ ℜ∗+ adjusts the bias– variance trade–off, Z is matrix composed of instruments ζk and of the basis functions, i.e.   Z = φnivx +1 · · · φNiv with     · · · F  (pk−1 ) ⊗ ζk−1 , (14) φkiv = F  (pk−nx ) ⊗ ζk−n x and

qnx ΦPθ α(q) is the estimation error vector, according to (11). E =Y −

conditions (Laurain et al., 2011): (C1) γ → ∞ and (C2) the instrument vector must be uncorrelated to the noise, which is fulfilled by the choice (15). In order to solve the optimization problem (13) the Lagrangian function is introduced qnx qnx γ 1 ZZ  E L (E, θ , λ ) = θ  θ + E  2 2 α(q) α(q)   qnx ΦPθ , − λ  E −Y + α(q) where λ is the Lagrange multipliers vector. So the Karush–Kuhn– Tucker (KKT) conditions for the Lagrangian are λ qnx qnx ∂L = 0 ⇒E = ZZ  (16) ∂E γ α(q) α(q) qnx ∂L = 0 ⇒θ = P Φ λ (17) ∂θ α(q) ∂L q nx = 0 ⇒E = Y − ΦPθ . (18) ∂λ α(q) By substituting (16)–(17) into (18) yields   I(N−nx ) + ΞK λ = ΞY γ where qnx qnx K = ΦΦ , α(q) α(q) qnx qnx ZZ  . Ξ= α(q) α(q)

It is well known that accurate IV estimates rely on an adequate choice of instruments. Generally speaking, the instruments must be correlated to the regressor variables but uncorrelated with the noise; for a detailed argumentation the reader is referred to Section 7.6 of Ljung (1999). In this work, we employed as instruments (15) ζk = [yˆk uk ] , where yˆ is the predicted output generated from a model estimated using the LS–SVM algorithm reported in Romano et al. (2016), similarly as performed in Laurain et al. (2012). The instrument variable approach (13) provides unbiased estimates under two 260

(19)

(20) (21)

To solve λ in (19) it is first necessary to compute matrices ΦΦ and ZZ  . The structure of (10) allows us to write the (i, j)th entry of ΦΦ as    ΦΦ = φi+n φ x j+nx i, j   (F(p j ) ⊗ z j ) + . . . = F  (pi ) ⊗ z i    + F  (pi+nx −1 ) ⊗ z F(p j+nx −1 ) ⊗ z j+nx −1 i+nx −1 =

A fundamental difference in our formulation is the introduction of a filter term qnx /α(q) in the optimization problem. The motivation to consider this term is to extend the standard IV method in order to decrease the variance of the estimates due to noisy data, in a similar way as accomplished by LTI instrumental variable estimation methods. The reader is referred to (Laurain et al., 2010) for a detailed discussion about the role and the effectiveness of prefiltering in the instrumental variable context.

83

=

nx −1 

∑

m=0 nx −1

 F  (pi+m ) ⊗ z i+m (F(p j+m ) ⊗ z j+m )

∑ F  (pi+m)F(p j+m) zi+m z j+m .

m=0

ψ ( pi+m ,p j+m )

where the term ψ (pi , p j ) is a positive definite kernel function. As mentioned previously, the model estimation is addressed using a nonparametric approach, that is, instead of explicitly parameterizing the basis functions fr (pk ), a kernel function ψ(pi , p j ) is used to represent the inner product F(pi ), F(p j ). This strategy is called the “kernel trick” (Vapnik, 1998) and permits the reconstruction of vectors L(pk ) and B(pk ) from vector λ and the kernel functions. A natural choice for the kernel functions is the radial basis function (RBF) with width σ ∈ ℜ∗+   pi − p j 22 . ψ (pi , p j ) = exp − σ2 Analogously to (10), the structure of (14) leads to

IFAC LPVS 2018 84 Florianopolis, Brazil, September 3-5, 2018 Marcelo M.L. Lima et al. / IFAC PapersOnLine 51-26 (2018) 81–86



ZZ





ij

=

nx −1

∑

m=0



 ψz (pi+m , p j+m ) ζi+m ζ j+m .

Since ΦΦ and ZZ  have been computed by exploiting a Kronecker product property, according to (20) and (21), matrices K and Ξ are computed using bi–dimensional (2D) filtering operations, i.e. Ki, j = − α1 Ki−1, j − α1 Ki, j−1 nx

−

nx

∑ ∑

ml =1 mr =1

αml αmr Ki−ml , j−mr + ΦΦ

Ξi, j = − α1 Ξi−1, j − α1 Ξi, j−1 nx

−

nx

∑ ∑

ml =1 mr =1





αml αmr Ξi−ml , j−mr + ZZ





so that





i, j

i, j

ψ1y (p∗ )  Ψy (p∗ ) =  ... ψ y (p∗ )  nux ∗ ψ1 (p )  .. u ∗ Ψ (p ) =  . ψnux (p∗ ) 

··· .. . ··· ··· .. . ···

 y ψN−n (p∗ ) x  ..  . y ψN−1 (p∗ )  u ψN−n (p∗ ) x  .. , . u ψN−1 (p∗ )

  y ∗  L(p∗ ) Ψ (p ) ˆ λ. = B(p∗ ) Ψu (p∗ )

3.3 Derivative–free optimization filter ,

where Ki, j and Ξi, j , for i, j = {nx + 1, . . . , N}, are the entries of matrices K and Ξ, respectively. Thus, the basis function inner products are preserved. 3.2 Model parameters reconstruction

As discussed before the determination of the eigenvalues of A leads to the filter poles of the 2D filtering systems (20)–(21). As a way of introducing a physical interpretation to the filter, α(q) is parameterized as a discretized version of a Butterworth polynomial, i.e. nx α(q) = ∏ q − e−sm Ts , m=1

The last step in the model identification algorithm is to recover the model parameters L(pk ) and B(pk ) from λ and the kernel functions. Let   L = L1 · · · Ln f ∈ ℜnx ×n f   B = B1 · · · Bn f ∈ ℜnx ×n f . From (3), we have

  L(pk ) = LF(pk ) = F  (p∗ ) ⊗ Inx vec(L).

Also using (4) similarly, the model parameters for a particular scheduling signal value pk = p∗ can be expressed as      vec(L) L(p∗ ) = I2 ⊗ F  (p∗ ) ⊗ Inx . ∗ vec(B) B(p )    =θ

Next, substitute the estimate of λ , denoted as λˆ , into (17), and then by replacing the obtained θ into the previous equation leads to     qnx L(p∗ ) = I2 ⊗ F  (p∗ ) ⊗ Inx P Φ r λˆ . (22) ∗ B(p ) α(qr )

Thus, (10) and (12) enable us to write (22) as   ψ( p, ¯ p1 )y1 · · · ψ( p, ¯ pN−nx )yN−nx .. ..     . .     L(p∗ ) ¯ pnx )ynx · · · ψ( p, ¯ pN−1 )yN−1  qnr x ˆ ψ( p, = λ. ∗ B(p ) ¯ p1 )u1 · · · ψ( p, ¯ pN−nx )uN−nx   α(qr )  ψ( p,   .. ..   . .

where

sm = ωc e j

(2m+nx −1) 2nx

with a sampling period Ts . A derivative–free optimization method was used to tune the filter by an evaluation based on a representation metric J. Establishing J(ωc , D) as a cost function whose parameters are the data– set D = {y1 , u1 , p1 , . . . , yN , uN , pN } and ωc is chosen from the candidates group Ω = {ω1 , . . . , ωnω } whose elements are called curiosity points. The filter cutoff frequency will be the barycenter of the weighted elements of Ω calculated by ωc∗ =

n

ω ωυ e−µJ(ωυ ,D) ∑υ=1 . nω ∑υ=1 e−µJ(ωυ ,D)

(23)

where µ ∈ ℜ∗+ controls the aggressiveness of the weighting terms e−µJ(ωυ ,D) . Hence, better curiosities are more weighted in comparison with the values that lead to worse performance. The performance functional can be freely chosen. In this work J is chosen as 

Y −Ys (ωυ )2 J(ωυ , D) = min , 1 Y − Y¯ 2 where Ys (ωυ ) is the estimation of the output within the cutoff frequency set as ωυ . For a presentation of the barycenter method for direct optimization, see Pait (2018). 4. CASE STUDY The performance of the proposed algorithm is assessed using data generated from a state–space LPV model inspired in the ˚ om system (Ljung, 1999). The data-generating system is Astr¨ described as     a (p ) 1 b (p ) xk + 1 k u k xk+1 = 11 k a21 (pk ) 0 b2 (pk ) yk = [1 0] xk + vk ,

ψ( p, ¯ pnx )unx · · · ψ( p, ¯ pN−1 )uN−1

Finally, it is defined

q nx ψ(p∗ , pk )yk α(q) qnx ψ(p∗ , pk )uk . ψku (p∗ ) = α(q) ψky (p∗ ) =

where

and the Hankel matrices 261

IFAC LPVS 2018 Florianopolis, Brazil, September 3-5, 2018 Marcelo M.L. Lima et al. / IFAC PapersOnLine 51-26 (2018) 81–86

  a11 (pk ) = 0.35 sinc π 2 pk + 1.4

85

Table 1. Results for the system output corrupted by white–noise.

a21 (pk ) = 5p2k − 0.8  1.5 , for pk > 0.125 b1 (pk ) = 1 + 4pk , for |pk | ≤ 0.125 0.5 , for pk < −0.125  0 , for pk > 0.125 b2 (pk ) = 0.5 − 4pk , for |pk | ≤ 0.125 1 , for pk < −0.125.

Model LS-SVM 2D-filter IV-SVM IV-filter

0 dB 31.91 (2.51) 71.44 (3.33) 30.10 (20.7) 79.25 (3.91)

5 dB 39.94 (2.04) 83.91 (2.27) 57.82 (19.9) 86.58 (1.78)

10 dB 48.05 (2.05) 91.79 (1.31) 79.22 (5.39) 91.16 (1.97)

20 dB 79.73 (1.51) 97.18 (0.37) 93.00 (1.15) 95.34 (1.09)

Table 2. BFR mean and standard deviation for the wide–band noise scenario.

The noise signal vk is used to investigate three different scenarios, namely, a white noise vk = e k , a wide-band noise 1 vk = ek , 1 − 0.6z−1 and a narrow-band noise 1 vk = ek , 1 − 0.9z−1 where ek is a zero–mean white noise realization whose variance is adjusted to obtain a signal–to–noise ratio (SNR) of 0, 5, 10 and 20dB. The input uk is a zero-mean white noise binary signal with length N = 1000 and the scheduling signal pk is a white–noise signal with an uniform distribution in the interval [−0.25, 0.25]. The analysis will be carried out using Monte Carlo simulations with 100 runs for each combination of SNR and noise scenario. For the sake of comparison, four algorithms are considered. The proposed algorithm is referred to as IV-filter. The standard instrumental variable algorithm is indicated as IV–SVM. It follows almost the same steps of the method described in Section 3, but the roots of α(q) are set to zero, which implies in not incorporating the filtering mechanism. The other two, denoted as LS-SVM and 2D–filter, were developed in Romano et al. (2016) and are based on least-squares support vector machines. While in LS–SVM does not provide the filtering feature, 2D–filter uses the data–driven approach of Section 3.3. The IV-SVM uses as instrument the output predicted by a model obtained using the LS–SVM algorithm and the predicted output used in IV–filter instrument vector is generated from a model achieved with 2D-filter. The accuracy of the simulated output is adopt as a performance measurement. Therefore, the best fit-rate (BFR)   Y −Ys 2 BFR(%) = 100 · max 1 − ,0 Y − Y¯ 2 is considered to assess the obtained models quality. The BFR was also considered in the determination of the so–called hyper– parameters (γ, σ ), which were obtained through a grid–search procedure. Tables 1 and 2 report the BFR mean value and in parentheses the standard deviation among the simulations for the white and wide–band colored noise scenarios, respectively. In these cases, the IV–filter algorithm presents the best results for low SNR, namely 0 and 5dB. On the other hand, for higher SNR, the 2D– filter performed better than the other methods. This indicates that under high SNR conditions in the evaluated scenarios, the IV algorithms are not the best options. If the comparison is restricted among the IV algorithms the filtering feature shown to be effective in reducing the variance of the estimates. For the narrow–band noise scenario the filtering feature of IV– filter algorithm is slightly effective in reduce the variance only 262

Model LS-SVM 2D–filter IV–SVM IV–filter

0 dB 43.20 (2.75) 66.99 (4.56) 60.83 (15.59) 77.40 (3.82)

5 dB 51.05 (2.58) 80.62 (2.96) 78.84 (4.35) 84.37 (2.31)

10 dB 63.93 (2.55) 90.06 (1.43) 87.07 (1.76) 88.55 (1.85)

20 dB 91.58 (1.04) 97.12 (0.41) 95.60 (0.63) 95.19 (0.96)

for values of SNR lower than 5dB (Table 3). In addition, both IV approaches outperformed the least-squares algorithms for all the SNR conditions. It should be argued that in this scenario, the noise spectrum is concentrated in a frequency range inside of the local (frozen) models bandwidth. As the inherent filter in our formulation has a low–pass characteristic, it is not so compelling to this particular scenario. Table 3. Narrow–band noise scenario BFR results. Model LS–SVM 2D–filter IV–SVM IV–filter

0 dB 43.86 (2.97) 50.95 (4.84) 74.57 (6.32) 76.77 (5.35)

5 dB 65.11 (2.59) 74.56 (3.18) 85.67 (2.49) 85.93 (2.43)

10 dB 82.53 (1.92) 87.87 (1.91) 92.25 (1.24) 91.96 (1.35)

20 dB 96.53 (0.48) 96.73 (0.42) 97.29 (0.35) 97.04 (0.54)

5. CONCLUSION This work presents an algorithm for estimating LPV models represented in the state-space based on support vector machines so that the functional dependence of the model parameters with respect to the scheduling signal is captured from the experimental data. The adopted model parameterization allows the formulation of the algorithm using the method of instrumental variables, which is recognized as appropriate to avoid bias errors, even for general noise conditions. At the same time, this parameterization provides a filtering structure that proved to be quite effective in reducing the noise of the estimates compared to the standard instrumental variable method, according to results with simulation data in different noise scenarios. The proposed framework enables several extensions, for example, to handle the error-in-variables problem, in which the input signal and/or the switching signal are contaminated by noise (Piga et al., 2015). Another promising possibility is to consider a predictor with a time-varying filtering structure, in the sense of the recent contribution of dos Santos et al. (2017), in the context of estimation of parametric LPV models. REFERENCES Alkhoury, Z., Petreczky, M., and Merc`ere, G. (2017). Comparing global input-output behavior of frozen-equivalent LPV statespace models. IFAC–PapersOnLine, 50, 9766–9771. dos Santos, P.L., Perdico´ulis, T.P.A., Novara, C., Ramos, J.A., and Rivera, D.E. (2012). Linear parameter-varying system identification: New developments and trends, volume 14 of Advanced Series in Electrical and Computer Engineering. World Scientific.

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