APPLICATION OF INSTRUMENTAL VARIABLE METHOD TO THE IDENTIFICATION OF HAMMERSTEIN-WIENER SYSTEMS

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1 APPLICAIO OF ISRUMEAL VARIABLE MEHOD O HE IDEIFICAIO OF HAMMERSEI-WIEER SYSEMS GREGOR MYK Wrocław University of echnology, Institute of Engineering Cybernetics, ul. Janisewsiego /7, 57 Wrocław, Poland, grm@diuna.ict.pwr.wroc.pl Abstract. Application of least squares and instrumental variables to recovering parameters of nonlinear complex dynamic bloc-oriented systems is examined. For a system with the Hammerstein-Wiener structure the instrumental variable algorithm is designed and compared with the least squares algorithm for estimating system parameters. he advantages of the proposed instrumental variable estimator are discussed and in particular its wea consistency, even in the presence of correlated noise, is shown. he problem of generating optimal values of instrumental variables is analysed, rate of convergence of the proposed estimator is evaluated and simulation examples are included. he paper provides an extension of the results introduced in []. Key Words. System identification, complex systems, parameter estimation, nonlinear models.. IRODUCIO Instrumental variable method is one of the most popular and universal methods used for determining parameters of single-element linear dynamic objects. he main aim of this paper is applying of the instrumental variable method to identify parameters of nonlinear dynamic systems with composite structure, and comparing it with the traditional least squares method. Applicability analysis is made on the example of the system with Hammerstein-Wiener structure described in detail in section. he Hammerstein-Wiener system can be viewed as a serial connection of two typical cascade structures: the Hammerstein system (Fig. and the Wiener system (Fig.. In spite of occuring in many fields ([6],[7],[9],[],[3], the Hammerstein-Wiener structure has received little attention in the literature. In section 3 the least squares algorithm proposed recently in [] for the white noise case is presented and its asymptotic bias is shown in the presence of correlated noise. o reduce the bias, the new estimator based on the instrumental variable method is proposed in section 4. he procedure is based on the ideas presented in [] and [] for singleelement linear systems. he proposed method is next compared with the least squares algorithm given in []. Asymptotic behaviour is studied, and the advantages of the proposed instrumental variable estimator are discussed. In particular its wea consistency, even in the presence of correlated noise, is shown. Also the problem of generating optimal values of instrumental variables is analysed, and rate of convergence of the proposed estimator is evaluated. Comparative simulation study results are presented. u ( B( A( Fig.. Hammerstein system. u w w B( A( ( Fig.. Wiener system. y y

2 u y ( b +b +...+b n n a... a p p ( Fig. 3. Hammerstein-Wiener system containing one linear dynamic element and two nonlinear static subsystems.. SAEME OF HE PROBLEM.. System under consideration We consider a single input - single output asymptotically stable discrete-time nonlinear complex system described by the equation (see [] and Fig. 3: p n ( j ( j y a y + b u + i i i m j where ( u c f ( u and t t ( y d g ( y l l t q l (. Signals y, u and are the system output, input and disturbance at time, respectively. We assume that the input u is bounded, white noise, disturbance is random, bounded ( < max, ero-mean, correlated and independent of u. Let us also assume that g (...g q ( and f (...f m ( are a priori nown smooth nonlinear functions and orders p, q, n, m are nown as well. Let a ( a,..., a p b ( b,..., b n c ( c,..., c m d ( d,..., d q ( denote the unnown system parameter vectors. System ( is more general than classical Hammerstein system often met in applications (Fig. and differs from widely discussed in the literature Wiener-Hammerstein system, where two linear dynamical objects surround one static nonlinear element ([],[3],[4],[5]... Identification tas he aim of the identification is to recover the true parameter vectors a, b, c and d of the system (given by (, using only the overall system input-output measurements (u, y (... obtained in the experiment. ote that parametriation of system ( is not unique. For some nonero constants α and β systems with parameter vectors a, b, c, d and βa, αb, c/α, d/β are indistinguishable from the input-output point of view (see (. o obtain a unique parametriation let us tae the following assumptions ([]: (a Θ ad ad and bc are not both ero; (b a and b, where denotes the Euclidean norm; (c signs of the first nonero elements of a and b are positive. Define: θ ( bc,..., bc m,..., bc n,..., bc n m, ad,..., ad,..., a d,..., a d q p p q ( θ,..., θ(, θ(,..., θ n m n m ( n m pq the aggregated parameters of the system, and let φ be the generalied input (regressor vector: ( ( ( ( ( f u,..., f u,..., f u,..., f u, g y,..., g y,..., g y,..., g y φ m n m n ( q( ( p q( p (3 Equation ( can be rewritten in the form y φ θ+, and the general measurement equation can be written in the form: Y θ + (4 where ( y,..., y, ( ( φ φ Y,...,.,...,, 3. LEAS SQUARES IDEIFICAIO For further comparison with the instrumental variable method we start from presenting of the two stage identification algorithm, based on the least sqares method and singular value decomposition (see []. 3.. he algorithm Step : Calculate the least squares estimate:! ( θ LS ( Y (5

3 of the aggregated vector θ and then construct therefrom evaluations! ( and! ( of Θ ad and (see assumption a. Θ ad LS LS Step : Perform the singular value decomposition of min( nm, evaluations! ( LS Θ ad and! ( LS :! ( LS σ!! iµν i i min( pq,! ( LS Θ!! ad δξζ i i i i estimates: i and next calculate the b! s µ! µ c! s µ σ! ν a! s! ξ ξ! d sξ δζ (6 where s µ and s ξ denote the signs of the first nonero elements in vectors µ and ξ respectively. Singular value decomposition allows to split aggregated matrix of parameters on product of two parameter vectors. It was shown in [] that ( µ σν ( LS!,! arg min Θ bc c R m, b R n 3.. Limit properties he following theorems were proved in []: : For any >, if is of full column ran and the disturbance, then:!b b!c c!a a d! d (7 : Let the disturbance be white with ero mean, finite variance and independent of input u. Let moreover φ be persistently exciting. hen if : bc F, by equations (6 will also be biased. his conclusion is confirmed by the simulation example reported below Simulation examples he system explored in the experiment had the following parameters: b c a d (.5;.5;.5;.5 f t (x cos tx g l (x sin lx t,l...4 he inputs u were generated according to the uniform distribution on the interval [-;]. Disturbance was first generated as white noise with the uniform distribution on [-.;.] (about % of the maximum noiseless signal or [-.5;.5] (about 5% of the maximum noiseless signal. Experiment was repeated for transfered by colouring filter with transfer function F( + -, where - denotes one step bacward shift operator in time domain. Aggregated estimation error was evaluated from the formula: ERR_ LS! (,!(,! b c d (,! a ( b, c, d, a Experimental results are shown in Fig ERR_LS( % disturbance correlated noise [ ] b! p. b c! p. c a! p. a d! p. d (8 4 white noise ERR_LS( 5% disturbance 3 Multiplying equation (4 by we get: Y θ + Hence the estimation error of vector θ can be written as:! ( LS θ θ ( Σ Σ φ φ φ If is white noise with ero mean and finite variance then elements of matrix are independent of elements in and as. Otherwise if is correlated, i.e.ε -i for some i, then the proposed estimator (5 of vector θ will be biased asymptotically, because of dependence between the noise and values g l (y i (l..q, i..p appearing in the matrix. In consequence, estimators given correlated noise white noise Fig. 4. Aggregated least squares estimation error versus number of measurements. For correlated noise the least squares estimator is not consistent. 4. ISRUMEAL VARIABLE MEHOD 4.. Assumptions In the following Plim X X for any random vectors X, X R S means that X X in probability as. Assume we have chosen instrumental variable matrix Ψ, satisfying for the correlated noise the following four conditions:

4 (C: dimψ dim (C: ψ and are bounded and mutually independent (C3: matrix Ψ is nonsingular with probability for all (C4: Plim( / Ψ exists and is nonsingular. Lemma. A necessary condition for the existence of an instrumental variable matrix Ψ satisfying (C4 is that the matrix / be nonsigular, i.e. φ must be persistently exciting (see Appendix A. 4.. Instrumental variable algorithm After multiplying the left hand side of equation (4 by matrix Ψ we get: Ψ Ψ Y θ + Ψ aing account of the conditions (C and (C3 above, we postulate to replace the least squares estimator (5 evaluated in step by the following instrumental variable estimator:! ( θ IV ( Ψ ΨY where ( ψ ψ (9 Ψ,...,. At the second stage we proceed analogically, by maing singular value decomposition of evaluations! ( IV Θ ad and! ( IV of Θ ad and based on! ( θ IV Limit properties Estimation error for the algorithm (9 has the form: ( IV! ( IV θ θ ( Ψ Ψ ( heorem. Under conditions (C (C4, estimator (9 is wealy consistent, independently of the noise correlation structure, i.e. IV ( P lim ( (see Appendix B. heorem. Estimation error (IV has the rate of convergence of order in probability, for all choices of instruments (see Appendix C Optimal instruments Denote ( ( ( (! ( f u,..., f u,..., f u,..., f u, φ m n m n ( q( ( p q( p Ru,..., Ru,..., Ru,..., Ru ( { } ( ( R u Ε g y u u j...q j j are the regression functions. Denote and let max ( Ψ max ( Ψ Q ( be the instruments quality index, where ( Ψ is defined in (, and is euclidean vector norm. heorem 3. Under conditions (C (C4, the value Q Ψ (see ( is minimal for of ( Ψ ( φ φ!!,...,! (see Appendix D. 5. SIMULAIO EXAMPLES For comparison, algorithms (5 and (9 were examined practically on the example system described in subsection Instrumental variables were generated by the standard method: ( ( ( ( ψ f u f m u f u n f m u n (,..,,...,,...,, g y!,..., g y!,..., g y!,..., g y! ( q( ( p q( p where p q y! a! d! g y! b! c! i l l i f u i l j t n m ( + j t t( j and a!, b!, c!, d! denote the elements of vectors i j t l abcdcomputed!,!,!,! by using least squares method described in section 3. Aggregated estimation errors for the least squares method (ERR_LS and instrumental variable method (ERR_IV, evaluated as in subsection 3.3, were computed for 5 measurement data and results are shown in Fig. 5. As we see, for large - asymptotic behaviour of instrumental variable estimator is better than of least squares estimator. Convergence is rather slow, so the fundamental problem is to select the instruments in an optimal manner. where

5 ERR_LS correlated noise ERR_IV 5 ERR_LS white noise ERR_IV Fig. 5. Comparison of asymptotic properties of least squares estimator and instrumental variable estimator in the presence of correlated and white noise. 6. COCLUSIOS Computational complexity of the proposed instrumental variable algorithm is comparable with the least squares method. he only difficulty is generation of instruments. he second step is the computation of the singular value decomposition of two matrices with fixed dimentions. Instrumental variable method can be well applied to identification of nonlinear dynamic systems with composite (Hammerstein-Wiener structure in the presence of correlated noise. his is the main advantage of the method. otice that, correlation of the noise process can be connected with the presence of structural feedbac in the complex system. We refer to [8] where successful application of instrumental variable method to identification of linear static complex systems with feedbac is considered. 7. APPEDICES 7.. Appendix A. Setch of the proof of Lemma Let AB, R α β be the matrices with finite elements, and let det( AA. hen there exist a β nonero vector ξ R, such that Aξ, thus B A det B A. We get Lemma, substituting ξ and ( A: B: Ψ. 7.. Appendix B. Setch of the proof of heorem Maing use of Slutsy theorem [], one can write ( IV ( ( ( ( P lim P lim Ψ P lim Ψ Denote G i Ψ, G ψ, i. It is easy to show that Ε i G i max var G ( ψ ω where ω( Ε i i + is the noise correlation function. So if ω( we get i lim var G. From chebychev inequality: i P lim G P lim Ψ. IV and P lim ( ( 7.3. Appendix C. Setch of the proof of heorem Denote A B C Ψ ψ φ Ψ ψ IV ( D A B. hen C j A var B i j ΕD D. It can be shown that i <, ΕB Εψ i, Ε max ( ψ ω < j < var( D < hus for all number sequences χ such that lim χ it holds j j C χ D in probability.

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