Hammerstein System Identification by a Semi-Parametric Method

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1 Hammerstein System Identification by a Semi-arametric ethod Grzegorz zy # # Institute of Engineering Cybernetics, Wroclaw University of echnology, ul. Janiszewsiego 11/17, Wroclaw, oland, grmz@ict.pwr.wroc.pl ABSAC A semi-parametric algorithm for identification of Hammerstein systems in the presence of correlated noise is proposed. he procedure is based on the non-parametric ernel regression estimator and the standard least squares. he advantages of the method in comparison with the standard non-parametric approach are discussed. Limit properties of the proposed estimator are studied, and the simulation results are presented. Keywords: onlinear systems, non-parametric regression, system identification 1. IODUCIO he nonlinear, dynamic system modeling has been widely discussed in the literature for the last three decades. umerous applications (in signal processing, telecommunication systems, designing of adaptive filters, pattern recognition, automation, biological systems (see [1]-[3])) show necessity of the research of more effective identification algorithms. he oldest identification techniques based on the Volterra and Wiener functional expansions gave very complicated algorithms. In 8 s the another approach appeared, basing on the assumption that the nonlinear dynamic system can be represented by interconnections of linear dynamic and static nonlinear elements [4]. he most of authors proposed the parametric algorithms without rigorous convergence proofs, restricted to the non-linearity of a polynomial form only. on-parametric methods were first applied by Greblici and awla ([5]-[8]) and based on the estimation of the regression function (ernel estimator, orthogonal series estimator, and recently on the wavelet approach [9]). Convergence of the proposed estimators was proved with the poor restrictions on the nonlinearity. he main disadvantage of these methods is non-analytic form of the solution. An idea of our algorithm is similar to that in [1], where the parameter estimator for the simple, static, linear element constructed by non-parametric estimation was analyzed. First the evaluations of the unmeasureable signal are computed by a non-parametric method, to be exploited in the parametric (least-squares) estimation of the unnown non-linearity. In section 2 the studied Hammerstein system (Fig. 1) is presented, and the identification problem is defined. ext, a two-stage identification algorithm is proposed in section 3, and its theoretical limit properties are analyzed in section 4. Simulation examples for the sample nonlinear system are shown in section 5.

2 2. SAEE OF HE OBLE 2.1. System under consideration he system consists of a static non-linearity µ () followed by a linear FI filter with the impulse response coefficients { λ p} p =. he system output is disturbed by a random, correlated noise process z. Only the input x and the output y of the overall system are accessible for measurements. x ε { ω r } w µ ( ) { λ p } p = r = v z y Fig.1. Hammerstein system with the correlated noise process. he system is described by the equations w which yelds ( x ) = µ, v = λ w p p y, z = ω r ε r ( x ) p p r= r= = λ µ + ω ε r r, y = v + z (1) 2.2. Assumptions and comments We assume the following: (A1) the random input sequence { x } (A2) the disturbance { } ε = = is iid, and is bounded: x x < is independent of inputs, zero mean, bounded, stationary white noise, i.e. Εε =, Εε ε j = for j, and ε < ε < (A3) the non-linearity () µ x = ϕ x a *, where µ x is of the form () () = ( a a a S ) () = ( 1(), 2(),..., S ()), where fs () x a * 1, 2,..., is an unnown true parameter vector, and ϕ x f x f x f x, s=1...s is a set of a priori nown, linearly independent functions, such that fs() x f < for x x (cf. (A1))

3 (A4) the correlated noise process { } filter, and ω r r = < z = (A() stationary process) is generated by a linear finite impulse response (A5) for clarity of exposition we additionally assume that µ () = and λ Under (A1)-(A5) we have: (C1) from (A2) and (A4) we conclude that ( z z <, where z = ε ω r r = ) { } z is zero-mean and bounded (C2) from (A1) and (A3) the vector process ϕ( x ) = Denoting W = ( w1, w2,..., w) and = ( ϕ( x1) ϕ( x2) ϕ( x )) W =Φ a *. From (C2) one can infer that the matrix Φ Φ a * = ( ) Φ Φ Φ is persistently exciting of order S,,..., we get Φ W is nonsingular, and 2.3. Identification tas he aim of the identification is to recover the unnown parameter vector a * of the nonlinearity using the input-output measurements {( x, y )} =1 (2) from the whole system, obtained in the experiment. We emphasize that the non-linearity output w cannot be directly measured. 3. SEI-AAEIC ALGOIH Observe that (cf. (1)) ( ) ( ) y = λ µ x + λ µ x + z p p 1 and () () Ε () x = λ µ x + µ x λ p Hence, including assumption (A5), we get () () µ () x = () x () where x () = { y x = x} 1 = Εµ x λ and further p Ε is the regression function. Basing on this observation we propose the following identification algorithm: { } Stage 1. Exploiting ( x, y ) { } w =1 =1 1 estimate the values of the unmeasureable signal by the non-parametric method (ernel regression estimate): µ ( ) ( ) (), where ( ) w! =! x =! x!,! x yk x x i = i i h K x x i ( ) i h( ) and K () is a ernel function, h( ) is a smoothing parameter (see[5]). (3)

4 Stage 2. Using the pairs {( x w ) },!, =1 a where W! ( w!,, w!,,..., w!, ) = 1 2. = compute the least-squares parameter estimate: ( ) Φ Φ 1 Φ! (4) W 4. LII OEIES Introducing the error of the non-parametric estimator as η = w! w =! x! w =! µ x µ x ( ( ) ()) ( ) ( ),, the parametric part of the algorithm can be interpreted as the identification of a non-linear, static element µ () x ϕ () x = a * corrupted by the disturbance η, (Fig. 2) from the noisy input-output measurements {( x w ) },!, =1. x!, µ ( ) w η, w Fig.2. Illustration of the second stage of the algorithm. Denoting Θ = ( η1, η2, η, ) (2) and (4)): a ( ) W ( ),,..., we obtain W! 1 = W + Θ which yields (from = Φ Φ Φ + Φ Φ Φ Θ = a + *, where = 1 1 Φ Φ Φ Θ ( ) = 1 1 = ϕ ϕ ϕ η 1, 2, he matrix 1, is invertible (see (C2)) and has finite elements (see (A3)), so (, ) well defined. aing use of the results in [5], we realize that w!, η, provided that () in probability, i.e. lim (, ) 1 w is and η > δ = for each δ > and each =1..., K fulfils standard conditions for ernel functions [5] and ( ) appropriately selected, such that h( ) and h( ) heorem 1. Assume that (A1)-(A6) hold and ( η δ). h is α, > c for some α > (in τ stage 1). Let = and τ >. If ατ > 1 then a a * in probability as.

5 roof. Each element 2, [] s of the vector 2, fulfils the condition 1 1 f = η η η, = UB [] s f ( x ) f ( x ) 2, s, s, UB ( > f ) = > δ η δ η > δ,, α ( η > δ) = τ ( ) α 1 ατ, c c = c If ατ > 1 then 1 ατ as. hus UB * a s a s in probability for each s=1...s. [] [] in probability, and 5. EXEIEAL ESULS he results of the experiments under the sample nonlinear system are presented in Fig. 3. It illustrates consistency property and shows the solutions for =1 and for =5 measurements. 3 E a - a * 2 E()= % a * y =1 =21 x ~ U(-1,1) ε ~ U(-2,2) µ (x) System specification: y () = (,,sin ) 2 ( ) ϕ x x x x a * = 21,, 2 = 2, λ, λ 1 = 1, λ 2 = 2, ω, ω 1, ω 2 Used ernel function Ku () h( ) = 5. τ 5. =5 =63 x ~ U(-1,1) ~ U(-2,2) ε = e u 2 µ (x) ϕ (x)*a 21 measurements x -4-8 ϕ (x)*a 63 measurements Fig.3. Experimental results for a sample nonlinear system identification x

6 6. COCLUSIOS he proposed method has the following advantages in comparison to the standard nonparametric approach: the resulting model has the analytic form, which is very important in practice we can estimate µ ( x ) not having any measurements from neighborhood of x However certain a priori nowledge about the non-linearity is needed. amely, the nonlinearity has to be nown with accuracy to the parameters but is not assumed to be of a polynomial type, which was commonly assumed earlier (eg. [4]). o increase numerical efficiency, both stages of the algorithm can be computed in recursive fashion. he recursive version of (3) in stage 1 can be found in [8]. he recursive version of (4) in stage 2 is the standard recursive least-squares algorithm. Experimental examinations over the systems containing non-typical non-linearities confirmed good behavior of the algorithm. he algorithm is simple to compute and can be well applied in many fields. EFEECES [1] W.X. Zheng, On a Least-Squares-Based Algorithm for Identification of Stochastic Linear Systems, IEEE ransactions on Signal rocessing, Vol. 46, o. 6, 1998, pp [2] G.Giunta, G. Jacovitti, A. eri, Bandpass onlinear System Identification by Higher Cross Correlation, IEEE ransactions on Signal rocessing, 29, 1991, pp [3] S.S. Hayin, Adaptive filter theory (third edition), rentice-hall, [4] S.A. Billings, S.Y.Fahouri, Identification of Systems Containing Linear Dynamic and Static onlinear Elements, Automatica, Vol. 18, o. 1, 1982, pp [5] W. Greblici,. awla, Hammerstein System Identification by on-parametric egression Estimation, International Journal of Control, Vol. 45, o. 1, 1987, pp [6] W. Greblici,. awla, onparametric Identification of Hammerstein Systems, IEEE ransactions on Information heory, Vol. 35, o. 2, 1989, pp [7] W. Greblici,. awla, Cascade on-linear System Identification by a on-arametric ethod, International Journal of System Science, Vol. 25, o. 1, 1994, pp [8] W. Greblici,. awla, ecursive onparametric Identification of Hammerstein Systems, J.Franlin Inst., Vol. 326, o. 4, [9]. awla, Z. Hasiewicz, onlinear System Identification by the Haar ultiresolution Analysis, IEEE ransactions on Circuits and Systems, Vol. 45, o. 9, 1998, pp [1] J.A. Cristobal Cristobal,. Feraldo oca, W. Gonzales anteiga, A Class of Linear egression arameter Estimators Constructed by onparametric Estimation, he Annals of Statistics, Vol. 15, o. 2, 1987, pp