Functionally Pooled Models for the Global Identification of Stochastic Systems Under Different Pseudo Static Operating Conditions

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1 Functionally Pooled Models for the Global Identification of Stochastic Systems Under Different Pseudo Static Operating Conditions JS Sakellariou, and SD Fassois,2 Stochastic Mechanical Systems&Automation (SMSA) Laboratory Department of Mechanical Engineering & Aeronautics, University of Patras, GR Patras, Greece {sakj,fassois}@mechupatrasgr, Internet: wwwsmsaupatrasgr 2 Department of Mechanical Engineering, Khalifa University of Science, Technology & Research, (KUSTAR), PO Box 27788, Abu Dhabi, UAE July 28, 25 Abstract The problem of identifying a single global model for a stochastic dynamical system operating under various conditions is considered within a novel Functionally Pooled (FP) identification framework Within it the operating conditions have a pseudo static effect on the dynamics, with each one being characterized by a value of a proper measurable scheduling variable Identification is based on excitation response signal pairs obtained under a number of sample conditions, each corresponding to a specific value of the scheduling variable The FP framework incorporates suitable & parsimonious FP models capable of fully accounting for cross correlations among the operating conditions, functional pooling for the simultaneous treatment of all data records, and statistically optimal estimation Within the FP framework Least Squares and Maximum Likelihood based estimators are postulated Unlike seemingly related Linear Parameter Varying (LPV) model identification techniques treating the problem in two sequential steps, separately operating on each excitation response pair, and thus leading to suboptimal accuracy, the postulated FP model estimators are analytically shown and numerically confirmed to achieve optimal statistical accuracy An application case study pertaining to the global modal parameter estimation for a simulated railway vehicle under various mass loading conditions is also presented Through this, the high achievable accuracy of FP modelling and the significant improvements over local models employed in the context of LPV type identification are demonstrated Keywords: Stochastic identification, global model identification, functional models, ARX models, LPV models, asymptotic analysis, Monte Carlo experiments, railway suspension modelling Copyright c 25 by JS Sakellariou and SD Fassois All rights reserved PLEASE ADDRESS ALL CORRESPONDENCE TO: JS Sakellariou, Department of Mechanical Engineering & Aeronautics, University of Patras, GR Patras, Greece Tel/Fax (direct): (+ 3) , ; Tel/Fax (central): (+ 3) sakj@mechupatrasgr

2 Important Conventions Bold-face upper/lower case symbols designate matrix/column-vector quantities, respectively Matrix transposition is indicated by the superscript T A functional argument in brackets designates function of an integer variable; for instance x[t] is a function of normalized discrete time (t =,2,) The conversion from discrete normalized time to analog time is based on(t )T s, witht s standing for the sampling period A hat designates estimator/estimate of the indicated quantity; for instance ˆθ is an estimator/estimate of θ Tilde designates sample quantity; for instance σ 2 designates sample variance For simplicity of notation, no distinction is made between a random variable and its value(s) Important Symbols k x k [t] y k [t] w k [t] e k [t] N na,nb pa,pb E{ } γ w (k,l) σw 2 a i,j,b i,j G j θ θ M N B o plim N d scheduling variable excitation signal for thek operating condition response signal for the k operating condition model innovations for the k operating condition one-step-ahead prediction error (residuals) for thek operating condition normal distribution AutoRegressive (AR) and exogenous (X) orders dimensionality of AR and X functional subspaces (equal to p if pa = pb) statistical expectation innovations cross correlation between operating conditions k and l innovations variance as a function of the scheduling variable AR, X coefficients of projection j basis function coefficients of projection vector augmented parameter vector including innovations variance number of excitation-response signal pairs used for FP-ARX identification signal length in samples for each individual operating condition backshift operator (B j u[t] = u[t j]) Kronecker product subscript designating actual (true) system probability limit operator convergence in distribution p convergence in probability Cov{, } covariance between two random quantities o(x) function that tends to zero faster than x Acronyms AR, X : AutoRegressive, exogenous RSS : Residual Sum of Squares ARX : AutoRegressive with exogenous excitation SSS : Signal Sum of Squares FP ARX : Functionally Pooled ARX BIC : Bayesian Information Criterion SPP : Samples per Parameter AIC : Akaike Information Criterion LS : Least Squares OLS : Ordinary Least Squares ML : Maximum Likelihood WLS : Weighted Least Squares 2

3 Introduction Many dynamical systems operate under different conditions that significantly affect their dynamics Oftentimes, the operating conditions are characterized by one or more measurable variables and remain constant or vary slowly over time, thus having a pseudo static effect on the dynamics Typical examples include structural systems vibrating under different loading conditions, such as bridges, sea vessels and trains [, 2], structures vibrating under different environmental (for instance temperature) or boundary conditions [3, 4], rotating machinery dynamics under different speeds [5], aircraft dynamics at various altitudes or flight conditions [6, 7], and many more In such cases the problem of identifying a single global model of the system, that is a model capable of representing the dynamics under any operating condition based on available excitation response signal pairs, each one corresponding to a sample operating condition, is of particular interest and the subject of the present study This problem is typically tackled via Linear Parameter Varying (LPV) models [5, 8, 9] These are dynamical models with parameters expressed as functions of the measurable variable(s) referred to as the scheduling variable(s) designating the operating condition In this context model identification is based on the so called local approach [,, 2], the rationale of which is simple and is based on a two step approach that effectively splits the problem into two distinct subproblems: First a number of local (or else frozen) models each corresponding to a single operating condition for which excitation response signal pairs are available are identified using conventional identification techniques [3, ch 7] (step ) Second, the identified models are interpolated, typically using orthogonal interpolation functions, in order to provide a single global model [2, pp 25 25], [4, 5] (step 2) This approach seems reasonable, as a straightforward extension of classical identification Yet, when viewed within a stochastic framework in which the excitation response signals are stochastic, it leads to suboptimal accuracy The intuitive explanation for this is simple, and may be readily understood from the fact that the signal pairs are not treated as a single entity, but rather in complete isolation of each other in the process of obtaining each local (conventional) model (step ) This not only neglects potential cross correlations among the signal pairs, thus resulting into information loss, but additionally leads to an unnecessarily high number of estimated parameters, thus violating the principle of statistical parsimony [3, p 492] and further leading to increased estimation variance and thus reduced accuracy (lack of efficiency in statistical terminology) [3, pp ] To these one should also add the errors involved in the subsequent (step 2) interpolation of the obtained local models when constructing the LPV (global) model The end result is a final, global, LPV model characterized by reduced that is suboptimal accuracy Recognizing the aforementioned problems that arise within a stochastic context, the present authors and their co workers have postulated a novel class of stochastic global models, referred to as Functionally Pooled (FP) models, for the proper global representation of systems and the remedy of the aforementioned weaknesses [6, 7] The class of FP models resembles the form of LPV models, with some of the important differences being that the signal pairs are treated as a single entity, the number of estimated parameters is minimal, potential cross correlations among the signal pairs are accounted for, and the estimation is accomplished in a single step (instead of two subsequent steps) as necessary for achieving optimal accuracy The optimal achievable accuracy is analytically established as well The FP identification framework is based on three entities (also see Figure ): (a) The concept of data pooling, in its functional pooling form This allows for the simultaneous treatment of all excitation response signal pairs and is important for accounting for cross correlations and for achieving optimal estimation accuracy (b) The class of Functionally Pooled (FP) stochastic models These have a structure resembling that of their LPV counterparts, but additionally include proper cross correlation terms [see Eq (b)] This leads to more complete model forms, but also paves the way for accurate estimation (c) Statistically optimal estimation Equal to the number of local models times the number of model parameters 3

4 System Condition S Ṃ S j S Operating Conditions xk j yk j [ t] [ t] Data Records Input Output t Input Output Input j S System S j System Output j Input M Output M S M System Operating Condition Operating Condition j Operating Condition M kkj km Measurable Variable t Data Pooling Functionally Pooled Model Data Pooling Estimation (a) (b) Figure : (a) Schematic representation of data collection from a system operating under different conditions characterized by a measurable (scheduling) variable; (b) estimation of a global model within the Functional Pooling framework based on data from different operating conditions FP models have been thus far used in their AutoRegressive with exogenous excitation (ARX) form and mainly from an application point of view using simple estimators For instance a global model for composite beam dynamics under different temperatures has been obtained in [3] and FP models have been used for representing aircraft dynamics under different flight conditions and configurations in [6, 7] The main use of FP ARX models has, thus far, been in the context of damage localization and magnitude estimation within the broader Structural Health Monitoring (SHM) framework [6, 7, 8, 9, 2, 2] In all these studies the simplest possible (Least Squares, LS) type estimators have been employed, without any claims on estimator optimality or any analysis on related issues a preliminary discussion on estimator properties was briefly presented in an early conference paper [22] The aims of the present study thus are: (i) the formal introduction of FP models in their complete form, (ii) the postulation of estimation methods based on the Least Squares (LS) and Maximum Likelihood (ML) principles, (iii) the theoretical investigation of estimator properties, such as consistency and efficiency which refer to estimated model optimality In addition, (iv) the comparison, via numerical Monte Carlo experiments, of the postulated estimation methods, and (v) the presentation of an application case study where FP ARX modelling is used for estimating the dynamics and global modal characteristics (natural frequencies and damping ratios) of a simulated railway vehicle under various mass loading conditions (different numbers of passengers) Comparisons demonstrating the estimation accuracy improvement over that of local models used in the context of LPV type identification are also made The rest of the article is organized as follows: The FP ARX model structure and the identification problem are presented in section 2 The LS and ML based estimation methods are presented in section 3, while estimator asymptotic properties (consistency and efficiency) are analysed in section 4 The methods performance is assessed in section 5 via numerical Monte Carlo experiments The application case study pertaining to the estimation of the dynamics and the global modal characteristics (natural frequencies and damping ratios) of a simulated railway vehicle under various mass loading conditions is presented in section 6, along with a demonstration of the improvement achieved over local models used in LPV model identification The conclusions of the study are finally summarized in section 7 4

5 2 The FP-ARX model structure and the identification problem The Functionally Pooled AutoRegressive with exogenous (FP-ARX) excitation model structure is defined as: na nb y k [t]+ a i y k [t i] = b i x k [t i]+w k [t], i= i= E{w k [t]w l [t τ]} = γ w (k,l) δ[τ], γ w (k,k) = σ 2 w k R a i = pa a i,j G j, b i = b i,j G j pb j= j= (a) (b) (c) with t designating normalized discrete time, k the measurable scheduling variable characterizing each operating condition,x k [t],y k [t] the corresponding excitation and noise corrupted response signals for thek operating condition, na, nb the AutoRegressive (AR) and exogenous (X) orders w k [t] is the model innovations Gaussian zero-mean white process with variance σ 2 w, that is N(,σ 2 w), which is uncorrelated with x k [t] k and potentially cross-correlated with its counterparts corresponding to different operating conditions E{ } designates statistical expectation and δ[τ] Kronecker delta (equal to unity for τ = and equal to zero for τ ) As Eq (c) indicates, the AR and X parameters a i, b i, are modelled as explicit functions of the scheduling variable k belonging to corresponding functional subspaces spanned by the mutually independent functions G j (functional basis): F a i = {G,G 2,,G pa }, F b i = {G,G 2,,G pb } with F designating functional subspace and pa, pb the dimensionality of the AR and X functional subspaces, respectively The constants a i,j,b i,j designate the corresponding AR and X, coefficients of projection The representation of () is referred to as an FP-ARX model of orders (na,nb) and functional subspaces of dimensionalitiespa,pb, or in short an FP-ARX(na,nb) [pa,pb] (or FP-ARX(na,nb) p forp = pa = pb) and is further designated as M( θ) parameterized in terms of the parameter vector θ = [a i,j b i,j c i,j σ 2 w] T i,j 2 Three cases are distinguished: Case (i): γ w (k,l) = σ 2 w δ(k l) (groupwise homoscedastic innovations) Case (ii): γ w (k,l) = σ 2 w δ(k l) (groupwise heteroscedastic innovations) Case (iii): γ w (k,l) of general form (contemporaneously correlated innovations) withδ(k l) equal to unity if k = l and equal to zero ifk l Thus the FP-ARX estimation problem may be stated as follows: Given M pairs of excitation response signals, each signal being of lengthn samples, for the sample values of the scheduling variablek,k 2,,k M : Z = {xk [t],y k [t] k = k,k 2,,k M, t =,,N}, (2) determine an estimate of the parameter vector θ Before proceeding with model parameter estimation it is worth noting that: 2 Bold-face upper/lower case symbols designate matrix/column-vector quantities, respectively 5

6 (a) The FP ARX model structure allows for the representation of contemporaneous cross correlations among different excitation response pairs via theγ w (k,l) term Moreover, such information is fully accounted form in the parameter estimation phase and is vital for obtaining statistically optimal models (see section 3) (b) The projection of the parameters a i, b i on the functional subspaces F a i, F b i allows for models capable of representing the dynamics everywhere within [k min,k max ] R and not only at the distinct {k,k 2,,k M } values (c) The form of functional dependence is important Physical insight may be used, while experience [3, 6, 7, 9, 2, 2] indicates that orthogonal polynomials, such as shifted Type II Chebyshev polynomials or trigonometric functions, are often sufficient Using the backshift operator B (B j x k [t] = x k [t j]) the main expression of the FP-ARX model may be compactly written as: A[B,k] y k [t] = B[B,k] x k [t]+w k [t] (3a) with A[B,k] = na + a i B i, B[B,k] = nb b i B i i= i= (3b) In analogy to conventional models this representation is assumed to satisfy the following conditions: A Stability condition: The poles of the AR polynomial lie inside the unit circle k [k min,k max ] A2 Irreducibility condition: The polynomials A[B,k],B[B,k] are coprime (have no common factors) A3 Each excitation signalx k [t] is stationary, ergodic, and persistently exciting of sufficiently high order [3, pp 42-44] withe{x k [t] w l [t]} = k,l 3 FP-ARX model estimation A model, corresponding to an actual underlying system of the form (), is now to be estimated based on the available data of Eq (2): na nb y k [t]+ a i y k [t i] = b i x k [t i]+e k [t] i= i= E{e k [t]e l [t τ]} = γ e (k,l) δ[τ], γ e (k,k) = σ 2 e k R a i = pa a i,j G j, b i = b i,j G j pb j= j= (4a) (4b) (4c) with e k [t] designating the model s one-step-ahead prediction error (residual), that like the w k [t] s is N(, σ 2 e) Without any loss of generality it is assumed that p = pa = pb, thus the main model expression above is written in a linear regression form as follows ( designates Kronecker product [23, pp 27-28]): y k [t] = [ ϕ T k [t] gt ] θ +e k [t] = φ T k [t] θ + e k[t] (5) with: ϕ k [t] = [ ] T y k [t ] y k [t na] x k [t]x k [t nb] [(na+nb+) ] 6 (6)

7 [ ] T g = [G G p ] T [p ], θ = a, a na,p b, b nb,p [(na+nb+)p ] (7) The substitution of the values (fort =,2,,N) for a single signal pair characterized byk to the above leads to: y k [] φ T k [] e k [] = θ + = y k = Φ k θ +e k (8) y k [N] φ T k [N] e k [N] Pooling together (see [24, ch 3] for the concept of pooling in a linear regression framework in econometrics) the signal pairs corresponding to the sample values k,k 2,,k M leads to: y = Φ θ +e (9a) with: y = y k y k2 y km Φ = Φ k Φ k2 [ ] Φ km e = e k e k2 [ p(na+nb+)] e km (9b) [ ] Based on the above functional pooling (the first term emphasizing the functional dependence of each equation on the scheduling variable k) the estimation of the parameter vector θ may be achieved through Least Squares (LS) or Maximum Likelihood (ML) principles 3 Least Squares (LS) type estimation The estimation of the parameter vector θ may be achieved by minimizing the following Least Squares criterion: J(θ,Z ) = et R w e () where e is given by Eq (9a) and R w designates the covariance of w (of the actual system) defined similarly toe In case (i) of groupwise homoscedasticity that corresponds to innovations with equal variances,e{wk 2[t]} = σw, 2 for = k,,k M,R w = σwi 2 (I : identity matrix of dimension ), the criterion of Eq () leads to the Ordinary Least Squares (OLS) estimator: ˆθ = ( Φ T Φ ) Φ T y = while the residual variance may be estimated as: k=k t= φ k [t]φ T k [t] k=k t= φ k [t]y k [t] () ˆσ 2 w = N e 2 k [t, ˆθ] for k = k,k 2,,k M (2) t= In case (ii) of groupwise heteroscedasticity, with different variances among innovations of the considered operating conditions, E{wk 2[t]} = σ2 w, fork = k,,k M, and covariance matrix given by: σ 2 w(k )I N σw(k 2 2 )I N R w = (3) σw(k 2 M )I N 7

8 the criterion of Eq () leads to the Weighted Least Squares (WLS) estimator: ˆθ = [ Φ T R w Φ ] [ Φ T R w y ] = k M σ 2 φ k [t]φ T k k=k w [t] t= k=k σ 2 w φ k [t]y k [t] As the innovations covariance matrix R w is practically unavailable, the unknown σ 2 w may be estimated by Eq (2) with ˆθ obtained through the previous OLS estimator The estimation procedure may be iterated until convergence in the coefficients of projection vector θ is achieved In case (iii) with E{w 2 k [t]} = σ2 w and contemporaneously correlated innovations, E{w k [t]w l [t]} = γ w (k,l), the criterion to be minimized becomes: t= (4) J(θ,Z ) = N et R w e (5) withr w = R w[t] I N and: R w[t] = E{w[t]w T [t]} = σw(k 2 ) γ w (k,k 2 ) γ w (k,k M ) γ w (k 2,k ) σw(k 2 2 ) γ w (k M,k ) σw(k 2 M ) M M (6) wherew[t] = [w k [t]w km [t]] T M Then minimization of the LS criterion leads to the estimator: ˆθ = [ Φ T R w Φ ] [ Φ T R w y ] (7) As R w is not available, an initial estimate R w is initially obtained through the OLS or WLS estimator and the final residual covariance matrix is estimated as: R w[t] = N e[t, ˆθ]e T [t, ˆθ] (8) t= 32 Maximum Likelihood (ML) estimation The estimation of the augmented parameter vector θ (see section 2) for the more general cases of groupwise heteroscedastic and contemporaneously correlated residuals based on the ML principle [25, pp 98-99] is considered Thus in case (ii) the log-likelihood function corresponding to an FP-ARX model is given by: k M N L( θ/e) = lnp(e/ θ) = ln p(e k [t]/ θ) = ln 2π N k M lnσ 2 2 w 2 k M 2 σ 2 e 2 k k=k t= k=k k=k w [t,θ] t= (9) with p( ) designating probability density By setting λ k = σw, 2 σ e(k,θ) 2 = N N t= e2 k [t,θ], Eq (9) may be rewritten as follows (tilde designates sample quantity): L( θ/e) = 2 ln 2π N 2 The first two derivatives oflwith respect to λ k are: k=k ln λ k N 2 k M σ2 k=k e(k,θ) (2) λ k ϑl(θ,λ k /e) ϑλ k = N + N 2λ k 2λ 2 σ e(k,θ), 2 k ϑ 2 L(θ,λ k /e) ϑλ 2 k = N 2λ 2 N k λ 3 σ e(k,θ) 2 (2) k 8

9 Maximization of L with respect to λ k leads to λ k = σ 2 e(k,θ), while the right expression is negative This provides the estimate ofλ k for givenθ In order to obtain the estimate ofθ,λ k is inserted into Eq (2) giving: L(θ/e) = 2 (ln 2π +) N k M 2 ln e(k,θ) k=k σ 2 = = 2 (ln2π +) N σ 2 e(k,θ) 2 lndet σ e(k 2 M,θ) }{{} R(θ):M M (22) The ML estimator ofθ then is: with: ˆθ ML = argmin θ det R(θ) (23) ˆσ 2 w = σ 2 e(k,ˆθ ML ) = N e 2 k [t, ˆθ ML ] (24) t= Remarks: (a) Evidently the above specializes to case (i) for det R(θ) = [ σ 2 e(k,θ)] M for any k = k,,k M (b) Based on Eq (9) the estimation of θ based on the ML principle coincides with the WLS estimator of Eq (4) provided that consistent estimates of the innovations variancesσ 2 w for k = k,,k M are available In case (iii) the log-likelihood function of (jointly) normally distributed residuals is for all k s given by: N L(θ,R/e[t ],,e[t N ]) = ln p(e[t]/θ,r) = e T [t]r e[t] ln2π N lndetr (25) t= withrreplacingr w[t] for simplicity By setting: t= Eq (25) becomes: Λ(θ) = N e[t,θ]e T [t,θ] (26) t= L(θ,R/e) = N 2 Tr { Λ(θ)R } N 2 lndetr 2 ln 2π (27) The ML estimator then is obtained as: ˆθ ML = argmin θ det Λ(θ) (28) R w[t] = Λ(ˆθ ML ) = N e[t, ˆθ ML ]e T [t, ˆθ ML ] (29) t= 33 Model structure selection and validation Model structure selection (model identification) includes order determination for the AR and X polynomials [26] as well as the determination of the functional subspace dimensionality for a given basis function family such as Chebyshev, Laguerre, Jacobi and so on [27, pp 77-82] Thus models of various structures, forms and subspace dimensionalities are estimated, and subsequently each one is validated A model is rejected if validation fails Among those successfully validated, the best one is selected according to minimization of the Akaike Information Criterion (AIC) or the Bayesian Information Criterion (BIC) [3, pp 55-57] 9

10 These may be shown to be adapted to Functionally Pooled models as follows: AIC = N k=k lnˆσ 2 w+2 dimθ, BIC = lnˆσ w+dimθ 2 lnn N k=k (homoscedastic/heteroscedastic innovations) (3a) AIC = N lndet R w[t] +2 dimθ, BIC = lndet R w[t] + dimθ lnn N (contemporaneously correlated innovations) (3b) The validation procedure is based on the obtained residual sequences ê k [t] = e k [t,ˆθ] k, which should, for an accurate model, be serially (over time) uncorrelated, uncorrelated with the excitation k, and cross uncorrelated over the different operating conditions Testing these hypotheses may be based on typical statistical tests which employ the sample auto and cross correlation functions [3, pp 53-54], [25, pp ], [28, p 49] 4 Identification Accuracy: Asymptotic properties of the estimators The achievable identification accuracy is now examined by considering the asymptotic (as the signal length N ) properties of the presented estimators The considered properties are consistency (examining the limit value of an estimate as N ) and asymptotic distribution of the estimates Estimator optimality then refers to a consistent estimate with minimal variance in the asymptotic distribution (statistical efficiency) For the sake of brevity the special case (ii) of heteroscedastic residuals is considered Theorem : LS Estimator Consistency Letθ o be the true (actual system) projection coefficient vector,w k [t] a zero mean white process withe{wk 2[t]} = σ2 w, ande{φ k [t]φ T k [t]} a nonsingular matrix Then: ˆθ p θ o (N ) (3) with p designating convergence in probability [29, p 94] Proof: See Appendix A Remark I: Note that, using the Kolmogorov theorem [3, p 32], it is easily verified that: with wp designating convergence with probability one [29, p 93] ˆσ 2 w wp σ 2 w (N ) (32) Theorem 2: ML Estimator Consistency Let θ o = [θ o σw] 2 T be the true (actual system) parameter vector, w k [t] a zero mean white process with E{wk 2[t]} = σ2 w, E{wk 3[t]} =, E{w4 k [t]} = 3σ4 w, and E{φ k [t]φ T k [t]} a nonsingular matrix Then: Proof: See Appendix A2 θ ML p θ o (N ) (33) Theorem 3: LS Estimator Asymptotic Distribution Let θ o be the true projection coefficient vector, w k [t] a zero mean white process withe{wk 2[t]} = σ2 w, ande{φ k [t]φ T k [t]} a nonsingular matrix Then: (ˆθ θ o ) d N(,P) (N ) (34)

11 with: with P = M k M k=k σw 2 E{ϕ k[t,θ o ]ϕ T k [t,θ o]} G k d designating convergence in distribution [29, p 94] and G k = gg T Proof: See Appendix A3 (35) Remark II: Based on Eq (35), the WLS estimator of Eq (4) is efficient, that is its covariance achieves the Cramer Rao lower bound of Eq (37) This is true provided that consistent estimates of the innovations variances ˆσ w fork = k,,k M are available (also see remarks in subsection 32) In addition, the OLS estimator of Eq () is efficient only in the case of an homoscedastic (case (i)) actual system with covariance matrix given by Eq (35) by replacing σ 2 w by the common (for all operating conditions) variance σ 2 w a consistent estimate of which is obtained via Eq (2) for any k Theorem 4: ML Estimator Asymptotic Distribution&Efficiency Let θ o = [θ o σ 2 w] T be the true parameter vector,w k [t] a zero mean white process withe{w 2 k [t]} = σ2 w,e{w 3 k [t]} =,E{w4 k [t]} = 3σ4 w k, ande{φ k [t]φ T k [t]} a nonsingular matrix Then the estimate θ ML is efficient, as it follows asymptotically Gaussian distribution with mean θ o and covariance matrix equal to the Cramer Rao lower bound [25, pp ]: θ ML N( θ o,f ) (N ) (36) withf designating the Fisher information matrix which is equal to: [ ] T ϑl( θ/e) F = E ϑl( θ/e) ϑ θ θ= θ o ϑ θ θ= θ o Proof: See Appendix A4 In summary, the first two theorems establish the consistency of all estimators, while the last two provide the asymptotic distribution and efficiency of the estimates Based on these, consistent estimates are obtained by all estimators On the other hand, the ML estimator is efficient, with covariance equal to the Cramer Rao lower bound The WLS estimator is also efficient provided that consistent estimates of the innovations variances for the different operating conditions are available, while the OLS estimator is efficient only in the case of an homoscedastic (case (i)) actual system otherwise it is inefficient (37) 5 Performance assessment via Monte Carlo numerical experiments The objectives of this section are: (i) assessment of the effectiveness of the OLS, WLS, and ML estimation methods and confirmation of the asymptotic accuracy results of section 4, (ii) demonstration of functional subspace selection and validation procedures, and (iii) demonstration of the estimation accuracy improvement over that of local models used in the context of LPV type identification These objectives are achieved via Monte Carlo numerical experiments with a simulated system 5 The simulated system and the experiments The actual simulated system is of the FP-ARX(2,) 3 form with zero delay (b in the exogenous polynomial) and AR, X functional subspaces consisting of the first p = 3 shifted Chebyshev continuous polynomials of Type II [27, pp ]: a i,j G j B i y k [t] = b i,j G j B i x k [t]+w k [t] (38) i= j= i= j=

12 Table : Monte Carlo estimation results for the FP-ARX(2,) 3 (5 experiments per method) Coeff True OLS Estimates WLS Estimates ML Estimates a, ± ± ± 96 a,2 7 ± 98 ± 66 ± 66 a, ±5-7±67-7±67 a 2, ± ± ± 92 a 2, ±86-95±66-95±66 a 2, ± ± ± 59 b, ± -478±45-478±45 b, ±2-4766± ±53 b, ± ± ± 83 b, ± ± -28± b, ± ±8-273±8 b, ± ± ± 82 E AR E X )% 9927± ± ±69 Sample Estimate±Sample Standard Deviation, σw % = %, σỹk : noise free response standard deviation σỹk ( RSS SSS with: E{w k [t]w l [t τ]} = σ 2 w δ[k l] δ[τ] (case (ii)) (39) F a i = F b i = { G =, G 2 = 2+4k, G 3 = 6k 2 6k +3 }, k [,5] (4) The scheduling variable k is normalized with respect to its maximum value, k max = 5, in order to lie within the range [, ] (see [27, pp ]) The system s true coefficients of projection are indicated in Table, while the variance of the heteroscedastic innovations for various values of k is shown in Table 2 5 Monte Carlo experiments per estimation method (OLS, WLS, ML), with each covering the range of k [,5] (before normalization) with an increment of δk =, are included Each experiment is based on M = 6 pairs ofn =, sample long excitation response signals In each experiment the system response is generated by using a number of mutually independent Gaussian zero mean random sequences acting as excitation and innovations Further details are provided in Table 3 Model structure selection includes functional subspace dimensionality p selection based on the Residual Sum of Squares normalized by the Series Sum of Squares (RSS/SSS): RSS/SSS = M k=k N t= e2 k [t] N t= y2 k [t] % (4) as well as the BIC and AIC criteria of section 33 Model validation is based on the whiteness examination of the residuals for each k, as well as on the normalized cross correlation function between the excitation signals and the residuals for all k, and among the residuals of the different operating conditions (also see section 33) Interval parameter estimates are recorded for all estimators, along with normalized (percentage) aggregate AR and X errors corresponding to the sample mean estimatese AR = â a a %, E X = ˆb b %, with the b subscript designating the true value of the indicated quantity, a, b the projection coefficient vectors for the AR and X polynomials, and a = i a i A convergence monitoring index is also used for the WLS and ML estimatorsδ r = ˆθ r ˆθ r, with r designating iteration number and selected threshold of 8 (also see ˆθ r Table 3) 2

13 Table 2: Monte Carlo estimation results for the residual variances of the FP-ARX(2,) 3 (5 experiments per method) Variance True OLS Estimates WLS Estimates ML Estimates σw(k 2 ) ± ± ±5759 σw(k 2 2 ) ±68 288±68 288±67 σw(k 2 3 ) ± ± ±236 σw(k 2 4 ) 4 43±28 4±28 4±28 σw(k 2 5 ) 2 22± 2± 2± σw(k 2 6 ) ±3 456±3 456±3 σw(k 2 7 ) 97 97±7 96±7 96±7 σw(k 2 8 ) ±3 656±3 656±3 σw(k 2 9 ) ±24 965±24 965±24 σw(k 2 ) ±25 979±25 979±25 σw(k 2 ) ±4 682±4 682±4 σw(k 2 2 ) 37 38±72 37±72 37±72 σw(k 2 3 ) 54 55±33 54±33 54±33 σw(k 2 4 ) 42 42±2 42±2 42±2 σw(k 2 5 ) ±7 247±7 247±7 σw(k 2 6 ) ±56 277±56 277±56 Mean Estimate±Standard Deviation Table 3: Details on the Monte Carlo experiments for FP-ARX(2,) 3 model identification No of Monte Carlo Operat conds: M = 6(k [, 5], δk = ) Func subspace: first 3 Chebyshev II polyn exps: 5 Innovs depend: groupwise heteroscedastic Data length: N =, samples FP-ARX estimation OLS method: QR decompostion ML method: Quasi-Newton nonlin optim methods WLS method: iterative; QR decomp init vars via WLS mean estims init vars via OLS mean estim max number of iterations = 4 converg index: δ r = 8 estim pars termination tol = 8 No of iterations = 2 objective function term tol = 6 Model structure selection Model orders: true values Funct subspace: criteria RSS/SSS, AIC, BIC 52 Model structure selection and parameter estimation results Model structure selection results are presented in Figures 2 and 3 The first depicts the RSS/SSS for OLS estimated models as a function of functional subspace dimensionality p for 5 experiments per p The results indicate (with very small variance) an initial drop in the RSS/SSS and a subsequent levelling off for p 3, leading to a potential selection of p = 3 which is the correct value This result is confirmed by the BIC and AIC criteria (Figure 3) The results are very similar for the WLS and ML based estimates The obtained, by all three (OLS, WLS, ML) Monte Carlo estimation results are presented in Tables, 2 and in Figure 4) They indicate excellent agreement between the estimates and their true counterparts for all estimators, with the attained standard deviations being particularly low This is also supported by the insignificant AR and X aggregate errors and the very low values of the sample mean RSS/SSS (Table ) Yet, as expected according to the asymptotic analysis of section 4, the WLS and ML estimators achieve accuracy that is considerably better (lower standard deviations) than that achieved by the OLS This is due to the inefficiency of the OLS estimator in the present (heteroscedastic) case The true FP ARX parameter trajectories versus k and their estimates, as obtained by the OLS, WLS and ML estimators (5 experiments), are shown in Figure 5 and further confirm the high accuracy achieved 3

14 RSS/SSS (%) functional basis dimensionality p Figure 2: Functional subspace dimensionality p selection via the RSS/SSS (Monte Carlo experiments each cross corresponds to a single experiment; 5 experiments per p) 5 2 BIC 5 AIC functional basis dimensionality p (a) functional basis dimensionality p (b) Figure 3: Functional subspace dimensionality p selection via the BIC (a) and AIC (b) criteria (Monte Carlo experiments each cross corresponds to a single experiment; 5 experiments per p) α 2 α α α α α b b b b OLS WLS ML OLS WLS ML OLS WLS ML b 2 b 3 Figure 4: True (red dashed lines) projection coefficients and Monte Carlo estimates (boxes indicating estimate sample mean ± sample standard deviation) by the OLS, WLS and ML estimators (Monte Carlo experiments 5 experiments per method) 4

15 5 OLS 5 WLS 5 ML α 5 α 5 α 5 (a) (e) (i) α 2 5 α 2 5 α 2 5 (b) (f) (j) b 2 2 (c) b 2 2 (g) b b b b (d) 5 5 k series (h) 5 5 k series (l) 5 5 k series Figure 5: True (- - -) AR and X parameter trajectories versus k and Monte Carlo estimates ( ): (a)-(d) OLS based estimates; (e)-(h) WLS based estimates; (i)-(l) ML based estimates (Monte Carlo experiments 5 experiments per method) 53 Estimation accuracy improvement over local modelling The estimation accuracy improvement achieved by FP-ARX modelling over that of local models used in the context of LPV type identification is now examined In the local model approach 6 separate conventional ARX(2,) local models (again with b in the exogenous polynomial) are estimated, one for each sample value of the scheduling variable k This is in sharp contrast to FP ARX model estimation which is estimated in a single step, simultaneously using all 6 data records Purely based on this fact, one may immediately observe a very significant difference in the number of model estimated parameters between the two approaches, and, as the data records are identical, a sharp difference in the signal Samples Per estimated Parameter (SPP) the latter being a quantity that critically affects the achievable estimation accuracy Indeed, in the FP ARX approach the number of estimated parameters (the coefficients of projection) is 2, resulting into an SPP of 267 Yet, in the local approach the corresponding numbers are4 6 = 64 and SPP of 5 The detrimental effects of the lower SPP in the local (LPV type) approach are evident in the Monte Carlo comparative estimation results of Figure 6 In that Figure, parts of the true, FP ARX based and local model based trajectory estimates (sample means and standard deviations) are presented (for local models only integer values of k are used) It is evident, that the FP ARX based approach achieves high estimation accuracy that cannot be matched by that of the local model approach The conclusion is that the FP ARX framework achieves the identification of a global model that is both compact (parsimonious) and of considerable higher accuracy than that achievable via the local model approach in the context of LPV type identification 5

16 α 35 α b 38 b k k Figure 6: Comparison of FP ARX & local ARX modelling for selected values of k True model parameters versusk (- - -) contrasted to FP ARX estimates (WLS estimator; sample mean estimate: - - -; sample standard deviation: ) and local ARX estimates (sample mean estimate (x) ± sample standard deviation) 5 Monte Carlo experiments per method 6 Application case study: modelling of railway suspension dynamics under different operating conditions In this application case study the identification of simulated railway suspension dynamics and the estimation of the global modal parameters (natural frequencies and damping ratios) [26] under different operating conditions (mass loadings) is considered based on available data records The operating conditions correspond to vehicle body mass changes (which may be due to factors such as varying number of passengers, varying freight, fuel consumption, and so on), and are in the range of[ ]kg Like in the previous section,5 Monte Carlo experiments are employed, and comparisons with the local ARX modelling approach are made 6 The suspension dynamics model The six degree-of-freedom model of one (left or right) half of a railway vehicle is depicted in Fig 7 Its parameters correspond to those of a typical passenger vehicle of the Hellenic Railways southern network (MAN OSE/KAT4) and are provided in Appendix B The vehicle is assumed to run on an horizontal track with constant speedu, symmetrical loading of the two rails (small roll angle) and no wheel lift The car is modelled as a rigid body with two degrees of freedom (vertical displacementy b and pitch angleθ b ), and is connected to the bogies located at its front and rear ends (leading and trailing bogies, respectively) via the secondary suspension The secondary suspension elements (physically realized via air chambers and hydraulic dampers) are indicated as K sl, C sl (leading part), andk st, C st (trailing part) Each bogie is modelled as a rigid body with two degrees of freedom (vertical displacement y tl or y dd and pitch angleθ tl orθ tt ) and is connected to two wheelsets (modelled as massless point followers) via the primary suspension The primary suspension consists of coil springs and shock absorbers modelled as linear springdashpot elements, which are indicated as K pi, C pi, with i =,2,3,4 designating the corresponding wheelset 6

17 u L c Car body M B B A b (t) C st K st Output Acceleration y b (t) K sl C sl Secondary Suspension Trailing bogie M tt tt tt (t) M tl tl tl (t) Leading bogie y tt (t) K p4 C p4 K p3 C p3 4 th wheelset 3 rd wheelset y w4 (t) L bw y w3 (t) L y w2 (t) y tl (t) K p2 C p2 K p C p 2 nd wheelset st wheelset Velocity Input y w (t) Primary Suspension x Figure 7: Schematic representation of the model of one-half railway vehicle suspension (Fig 7) The track is assumed to be fixed and rigid, with the track vertical velocity excitation being approximated by Gaussian white noise with spectrum equal to: S vv (ω) = (2π) 2 A r u (42) withurepresenting the (horizontal) vehicle velocity and A r a roughness factor indicative of track quality [7] The vertical dynamics of the vehicle model are, for small displacements, described by the linear differential equation: M ÿ(t)+c ẏ(t)+k y(t) = B x w (t) (43) with y(t) = [y b (t) θ b (t) y tl (t) θ tl (t) y tt (t) θ tt (t)] T designating the displacement vector (see Fig 7), x w (t) the input vector, B an input shaping matrix, and M, C, K the mass, damping, and stiffness matrices, respectively (see full details in [7]) The transfer function between the vehicle body vertical acceleration at point A (right above the trailing airspring, Fig 7) and the track vertical velocity profile which due to the negligible wheel mass coincides with the vertical velocity of any wheelset is presently considered Thus, the track vertical velocity and the acceleration at point A, based on a sampling period of T s = 4s (sampling frequency f s = 25 Hz), are the excitation response signal pairs are employed for each operating condition The variability of the true Frequency Response Function (FRF) magnitude versus body mass in a range that corresponds to mass reduction in the range of % is presented in Figure 8 5 Monte Carlo experiments are performed with each experiment covering the range of k [, ]% reduction on the car body mass with an increment of δk = % that corresponds to 3284 kg Thus each experiment includes M = pairs of N = 4, 96 sample long excitation response signals, with the system response being generated by using a number of mutually independent Gaussian zero mean random sequences acting as the velocity excitation Each response signal is corrupted by random noise with noise to signal standard deviation ratio equal to % 7

18 FRF Magnitude (db) No mass reduction % mass reduction 2 % mass reduction 3 % mass reduction 4 % mass reduction 5 % mass reduction 6 % mass reduction 7 % mass reduction 8 % mass reduction 9 % mass reduction % mass reduction Frequency (Hz) Figure 8: Application case study: FRF magnitude under different operating conditions Table 4: No of estimated parameters and SPP for the FP ARX and local models na nb p No of estim param SPP FP-ARX model local ARX models = FP ARX modelling & comparisons Like in the previous section, a levelling off is noted in the RSS/SSS for p 3, while both the AIC and BIC attain minimum values for the same value ofp = 3 [LS estimator of Eq ()] Thus an FP ARX(2,4) 3 model (b = in the exogenous polynomial) with groupwise homoscedastic residuals (case (i)) and AR, X subspaces consisting of the following functions: F a i = F b i = {G =,G 2 = k,g 3 = k 2 }, k [295236,3284]kg is selected Indicative global modal parameter (natural frequency and damping ratio) interval estimation results are presented in Figures 9 and (solid lines indicate sample mean estimates and dashed lines sample standard deviations) as obtained based on the FP ARX(2,4) 3 model (5 Monte Carlo experiments) The estimates are evidently very accurate, characterized by low standard deviations In the same figures comparisons with the local model approach (used in the context of LPV type identification) are also made ( local ARX(2,4) models are obtained based on standard identification procedures [3, pp 55-57]) Like in the previous section, the number of estimated parameters is considerably lower in the FP ARX case, leading to a high SPP (see Table 4) As a consequence, the interval estimates corresponding to the FP ARX model are much narrower (more accurate) than those of the local modelling approach 7 Concluding remarks The problem of identifying a global model for a stochastic dynamical system operating under different pseudo static conditions from excitation response signal pairs corresponding to those conditions was considered and a 8

19 ω (Hz) ω 2 (Hz) 3 2 ω 3 (Hz) Mass reduction k (%) Figure 9: Application case study: Estimates of the first three natural frequencies (x: true values) within the considered range of vehicle body mass based on: (i) local ARX models for certain masses (mean estimate: o; sample standard deviation ; 5 experiments), (ii) the FP ARX model (mean estimate: ; sample standard deviation ; 5 experiments) Zooms of certain results are also shown Functionally Pooled (FP) framework was postulated The FP framework incorporates parsimonious FP models which are also capable of fully accounting for cross correlations among the operating conditions, functional pooling for the simultaneous treatment of all data records, and statistically optimal estimation Within it Least Squares and Maximum Likelihood based estimators were postulated, and were analytically shown to be consistent and (under proper conditions) optimal in the sense of achieving minimal asymptotic variance (statistical efficiency) Their optimal accuracy was also demonstrated via numerical Monte Carlo experiments, while their superiority over local modelling used in an LPV type context was also confirmed An application case study pertaining to global identification of the suspension dynamics, under different mass loadings, for a simulated railway vehicle was also considered and further served in confirming the high achievable accuracy and superiority of the FP ARX approach Acknowledgement The authors wish to thank Mr K Petsounis of Mentor Hellas for fruitful discussions and support in the simulation of the railway vehicle suspension References [] S Bruni, R Goodall, TX Mei and Tsunashima H, Control and monitoring for railway vehicle dynamics, Vehicle System Dynamics: Int Journal of Vehicle Mechanics and Mobility, 45 (7-8) (27)

20 ζ 3 2 ζ ζ Mass reduction k (%) Figure : Application case study: Estimates of the first three damping ratios (x: true values) within the considered range of vehicle body mass based on: (i) local ARX models for certain masses (mean estimate: o; sample standard deviation ; 5 experiments), (ii) the FP-ARX model (mean estimate: ; sample standard deviation ; 5 experiments) Zooms of certain results are also shown [2] RE Kim, F Moreu, BF Spencer Jr, System identification of an in service railroad bridge using wireless smart sensors, Smart Structures and Systems, 5 (3) (25) [3] JD Hios and SD Fassois, Stochastic identification of temperature effects on the dynamics of a smart composite beam: assessment of multi-model and global approaches, Smart Materials and Structures, 8 (35) (29) [4] K Worden, H Sohn and CR Farrar, Novelty detection in a changing environment: regression and interpolation approaches, Journal of Sound and Vibration, 258 (4) (22) [5] V Verdult, M Lovera and M Verhaegen, Identification of linear parameter-varying state-space models with application to helicopter rotor dynamics, Int Journal of Control, 77 (3) (24) [6] PA Samara, JS Sakellariou, GN Fouskitakis, JD Hios and SD Fassois, Aircraft virtual sensor design via a time-dependent functional pooling NARX methodology, Aerospace Science and Technology, 29 () (23) [7] DG Dimogianopoulos, JD Hios, and SD Fassois, FDI for aircraft systems using stochastic pooled NAR- MAX representations: Design and assessment, IEEE Transactions on Control Systems Technology, 7 (6) (29) [8] J De Caigny, JF Camino and J Swevers, Interpolating model identification for SISO linear parametervarying systems, Mechanical Systems and Signal Processing, 23 (8) (29) [9] A Almeida Gonalves Siqueira, R Nicoletti, N Norrick, K Lucchesi Cavalca, H Fiori De Castro, J Bauer, F Dohnal, Linear parameter varying control design for rotating systems supported by journal bearings, Journal of Sound and Vibration, 33 () (22)

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