Refined Instrumental Variable Methods for Identification of Hammerstein Continuous-time Box Jenkins Models

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1 Proceedings o the 47th IEEE Conerence on Decision and Control Cancun, Mexico, Dec. 9-, 8 TuC4.5 Reined Instrumental Variable Methods or Identiication o Hammerstein Continuous-time Box Jenkins Models V. Laurain, M. Gilson, H. Garnier, P.C. Young Abstract This article presents instrumental variable methods or direct continuous-time estimation o a Hammerstein model. The non-linear unction is a sum o known basis unctions and the linear part is a Box Jenkins model. Although the presented algorithm is not statistically optimal, this paper urther shows the perormance o the presented algorithms and the advantages o continuous-time estimation on relevant simulations. I. INTRODUCTION The need or non-linear identiication grows as the studied system complexity increases and non-linear models apply in many ields [3]. Many dierent approaches were developed to deal with black-box model identiication, whether they are non parametric, using Volterra series approach [], semi-parametric using neural network methods and support vector machine classiication [5], [8], or parametric such as state dependent parameters [3] or extended Kalman ilter [4]. Other reerences can be ound in e.g. [7]. Semiparametric approaches, even i perorming eiciently, lack the possibility o giving an a posteriori physical representation o the studied system. On the other hand, transer unction models provide a generic approach to data-based modelling o linear systems, encompass both discrete-time and continuous-time applications and are in an ideal orm to interpret serial, parallel connections o sub-systems which oten have a physical signiicance. Hammerstein block diagram model is widely represented or modelling non-linear systems [5], [3], [8]. The nonlinear block can be represented as a piecewise linear unction [] or as a sum o basis unctions [4], [6]. The available methods are oten designed or discrete-time (DT) model estimation and usually, extended least squares are used to minimize a prediction error [5]. Even i acquired data are sampled, the underlying dynamic o a real system is continuous. Thereore, direct continuoustime model identiication methods regained interest in the recent years [9]. A survey by Rao and Unbehauen [7] shows that CT model identiication methods applied to Hammerstein models are poorly represented in literature, and to the best o our knowledge, no method uses instrumental variable (IV) techniques to handle Hammerstein CT model identiication. Instrumental variables have the advantage o Centre de Recherche en Automatique de Nancy (CRAN), Nancyuniversité, CNRS, BP 39, 5456 Vandoeuvre-les-Nancy Cedex, France vincent.laurain@cran.uhp-nancy.r Centre or Research on Environmental Systems and Statistics,Lancaster University, UK; and Integrated Catchment Assessment and Management Centre, School o Resources, Environment and Society, Australian National University, Canberra. p.young@lancaster.ac.uk e ξ H o (q) u(t) ū(t) x(t) x y (.) G o (p) Fig.. Hammerstein block representation producing an unbiased estimation independently on the noise model assumed with an acceptable variance in parameter estimation in many practical cases. Moreover, an optimal choice o these instruments leads to a minimal variance estimator [9], []. Thereore, the main contribution o this paper is to present IV estimation methods or Hammerstein CT models where the non-linear unction is a sum o known basis unctions γ, γ,..., γ l given as: ū(t) = α i γ i (u(t)). () i= The proposed algorithms are based on the multi-input single-output reined instrumental variable or CT systems (MISO RIVC) irst introduced by Young and Jakeman [6] and recently extended to handle the case o Box Jenkins models [4], [5], []. II. PROBLEM DESCRIPTION Consider the Hammerstein system represented in Figure and assume that both input and output signals, u(t) and y(t) are uniormly sampled at a constant sampling time T s over N samples. Notice irst that this Hammerstein system produces the same input-output data or any pair (β(u), Go(p) β ). Thereore, to get a unique parametrization, one o the gains o (u) or G o (p) has to be ixed [5], []. Hence, the irst coeicient o the unction (.) equals, i.e. α = in (). Moreover, ū(t) in () is an internal variable and is actually not directly accessible. The Hammerstein system S, is described by the ollowing input-output relationship: where x(t) = l i= G o,i(p)γ i (u(t)) S ξ = H o (q)e, y = x + ξ, () G o,i (p) = B o,i(p) A o (p) = α ib o (p) A o (p). (3) /8/$5. 8 IEEE 386

2 47th IEEE CDC, Cancun, Mexico, Dec. 9-, 8 TuC4.5 B o (p) and A o (p) are polynomials in dierential operator p (p i x(t) = di x(t) dt ) o respective degree n i b and n a (n a n b ). The method presented is based on the identiication o a Box Jenkins model, where the linear and the noise models are not constrained to have common polynomials. Given the discrete-time, sampled nature o the data, an obvious assumption is that the model o the basic dynamic system is in CT, dierential equation orm while the coloured noise associated with the sampled output measurement y has rational spectral density and can be represented by a discretetime autoregressive moving average ARMA model ([7], [8]): ξ = H o (q)e = C o(q ) D o (q ) e(t k) (4) where C o (q ) and D o (q ) are polynomials in shit operator q (q r x = x(t k r )) with respective degree n c and n d. e is a zero-mean, normally distributed, discrete-time white noise sequence: e N(, σe ). Consequently, the Hammerstein model estimation problem can be treated under the previous assumptions using a MISO RIVC algorithm where all inputs have a common denominator A o (p) and B o,i (p) = α i B o (p). This method will be called Hammerstein RIVC (HRIVC). A. Reined IV or Hammerstein CT Models The model set to be estimated, denoted as M with system (G) and noise (H) models parameterized independently, then takes the orm, M : {G i (p, ρ), H(q, η)}, i =...l (5) where ρ and η are parameter vectors that characterise the system and noise model, respectively. In particular, the system model is ormulated in CT terms: G : G i (p, ρ) = B i(p, ρ) A(p, ρ), = α i(b p n b + b p n b + + b nb ) p na + a p na + a na, (6) with i =...l. The associated model parameters are stacked columnwise in the parameter vector, a a b α b ρ =.. a, a = Rnρ.. b.., b = Rna Rn b+, α l b a na b nb (7) with n ρ = n a +l(n b +) while the noise model is in discretetime orm H : H(q, η) = C(q, η) D(q, η) = + c q + + c nc q nc + d q + + d nd q n d (8) where the associated model parameters are stacked columnwise in the parameter vector, η = [ c c nc d d nd ] T R n c+n d (9) Consequently, the noise transer unction takes the usual ARMA model orm: ξ = C(q, η) D(q, η) e(t k). () More ormally, there exists the ollowing decomposition o the parameter vector θ or the whole hybrid model, ( ρ θ = () η) such that the model can be written in the orm y = B i (p, ρ)γ i (u) + C(q, η) A(p, ρ) D(q, η) e(t k), i= with B i (p, ρ) = α i B(p, ρ). The HRIVC method derives rom the RIV algorithm or DT systems. This was evolved by converting the maximum likelihood estimation equations to a pseudo-linear orm involving optimal preilters [], [6]. A similar analysis can be utilised in the present situation since the problem is very similar, in both algebraic and statistical terms. Following the usual prediction error minimisation approach in the present hybrid situation, a suitable error unction ε, at kth sampling instant, is given as: ε = D(q, η) C(q, η) { y i= B i (p, ρ) A(p, ρ) γ i(u) () which can be written as { [ ε = D(q, η) C(q A(p, ρ)y, η) A(p, ρ) ]} B i (p, ρ)γ i (u), (3) i= where the DT preilter D(q, η)/c(q, η) will be recognised as the inverse o the ARMA(n c,n d ) noise model. However, since the polynomial operators commute in this linear case, (3) can be considered in the alternative orm, by using or sake o clarity u i (t) = γ i (u(t)): ε = A(p, ρ)y } B i (p, ρ)u i (4) i= where y and u i represent the sampled outputs o the complete CT and DT preiltering operation, involving the CT iltering operations using the ilter (see [5], [4]): c (p, ρ) = A(p, ρ), (5) as well as DT iltering operations, using the inverse noise model ilter: d (q, η) = D(q, η) C(q, η). (6) Thereore, rom (4), the associated linear-in-the-parameters model then takes the orm [5]: y (na) = ϕ T ρ + ς (7) 387

3 47th IEEE CDC, Cancun, Mexico, Dec. 9-, 8 TuC4.5 where y y (na ) u ϕ =., y = y (na )., u l y (t) u (n b) i u (n b ) i u i =.., u i v (n) is the nth time derivative o v(t) sampled at the kth sampling instant and ς = A(p, ρ)ξ. O course none o A(p, ρ), B i (p, ρ), C(q, η) or D(q, η) is known and only their estimates are available. Thereore, IV estimation normally involves an iterative (or relaxation) algorithm in which, at each iteration, the auxiliary model used to generate the instrumental variables, as well as the associated preilters, are updated, based on the parameter estimates obtained at the previous iteration [4], [5]. B. Iterative HRIVC Algorithm Let us consider the jth iteration where we have access to the estimate: ˆθ j j ) (ˆρ = (8) ˆη j obtained at iteration j. The most important aspect o optimal IV estimation is the deinition o the instrumental variable. It has been shown that this instrument requires the knowledge o the noise ree regressor [9], []. Thereore, in this context, the associated optimal IV vector ˆϕ, is then an estimate o the noise-ree version o the vector ϕ in (7) and is deined as ollows: ˆx u ˆϕ =., ˆx = u l ˆx (na ) ˆx (na ). ˆx (9) where the iltered noise-ree output ˆx is obtained rom: ˆx(t, ˆρ j ) = G i (p, ˆρ j )u i (t). () i= The IV optimisation problem can now be stated in the orm [ ] ˆρ j N (N) = argmin ˆϕ ρ N ϕ T k= [ N ˆϕ N y (na) ] () k= where x = x T Qx and Q = I. This results in the solution o the IV estimation equations: [ N ] N ˆρ j (N) = ˆϕ ϕ T ˆϕ y (na) () k= k= Q where ˆρ j (N) is the IV estimate o the system model parameter vector at the jth iteration based on the appropriately preiltered input/output data Z N = {u; y} N k=. i G,i G i, HRIVC provides a consistent estimate under the N conditions: lim N N t= Eˆϕ ϕ T is ull column rank and lim N N t= Eˆϕ ς =. N An estimate o the sampled noise signal ξ, at the jth iteration, is obtained by subtracting the sampled output o the auxiliary model equation () rom the measured output y, i.e.: ˆξ = y ˆx(t k, ˆρ j ). (3) This estimate provides the basis or the estimation o the noise model parameter vector η j, using in this case the MATLAB identiication toolbox ARMA estimation algorithm. The process is iterated until a stopping criterion or a certain number o iterations is reached. At the end o the iterative process, coeicients ˆα i are not directly accessible. They are however deduced rom polynomial ˆB i (p) as B i (p, ρ) = α i B(p, ρ). The hypothesis α = guarantees that ˆB (p, ρ) = ˆB(p, ρ) and ˆα i can be computed rom: ˆα i = n b + n b k= ˆbi,k ˆb,k, (4) where ˆb i,k is the kth coeicient o polynomial term ˆB i (p, ρ) or i =...l. C. Comments A simpliied version o HRIVC algorithm named HSRIVC ollows the exact same theory or estimation o Hammerstein CT output-error models. It is mathematically described by, C(q, η j ) = C o (q ) = and D(q, η j ) = D o (q ) =. All previous given equations remain true, and it suices to estimate ρ j as θ j = ρ j. The implementation o HSRIVC is much simpler than HRIVC as there is no model noise estimation in the algorithm. The present paper considers CT model identiication. However, the DT versions o both IV-based methods can be easily developed and will be also evaluated in the next section, Even i the proposed algorithm perorms well, it is not statistically optimal as discussed in section III-C. III. NUMERICAL EXAMPLES This section presents numerical evaluation o both suggested HRIVC and HSRIVC methods. For all presented examples, the non-linear block has a polynomial orm, i.e. γ i (u(t)) = u i (t), i and ū(t) = u(t)+.5u (t)+.5u 3 (t), where u(t) ollows a uniorm distribution with values between and. To highlight the perormance o CT model IV-based methods, two simulated systems are considered. All systems are simulated with a zero order hold on the input. 388

4 47th IEEE CDC, Cancun, Mexico, Dec. 9-, 8 TuC4.5 A. Second-order System The linear dynamic block is irst a second-order system described by: p + 3 G o (p) = p + p + 5. (5) The sampling time equals T s =.48s. Based on this process, two dierent systems S and S are deined. S is a Hammerstein output error model and thereore H o (q) =. while S is a Hammerstein Box Jenkins model with: H o (q) = q +.q. The models considered or estimation are: b p + b G(p, ρ) = p, + a p + a M HRIVC H(q, η) = + d q + d q, (u(t)) = u(t) + α u (t) + α u 3 (t) or the HRIVC method and b p + b G(p, ρ) = p, + a p + a M HSRIVC H(q, η) =, (u(t)) = u(t) + α u (t) + α u 3 (t) (6) (7) or HSRIVC. 5 Monte Carlo simulation runs with a new noise realization or each run were realized under a signal to noise ratio (SNR) o 3dB and db with: ( ) Pe SNR = log, (8) P g being the average power o signal g. The number o samples is N =. Table I exhibits the mean value o the estimated parameters, their standard deviation and their normalised root mean square error (RMSE) deined as: RMSE(ˆθ j ) = N exp N exp i= P x ( θ o j ˆθ ) j (i), with ˆθ j the jth estimated parameter o θ. Table I shows that the HRIVC and HSRIVC methods provide similar, unbiased estimates o the model parameters with reasonable standard deviations. Results obtained using the HRIVC algorithm, have standard deviations which are always smaller than the ones produced by HSRIVC. Even though, the HSRIVC algorithm based on an output-error model is a reasonable alternative to the ull HRIVC algorithm based on a Box Jenkins model. θ o j B. Fourth-order System The aim o this paper is not to compare direct continuoustime and indirect discrete-time model estimation methods. However, authors show through a chosen example the interest o using the direct CT methods with respect to the traditional DT methods. The linear part o the second system is based on a benchmark proposed by Rao and Garnier in [6] (see also []). It is a ourth-order, non-minimum phase system with complex poles. Its transer unction is given by: G o (p) = 64p + 6 p 4 + 5p p + 46p + 6. (9) The sampling requency is chosen to be about ten times the bandwidth o the system under study which leads to T s =.34s. White noise is added to the output samples. 5 Monte Carlo simulation runs were realized with a SNR o db using the proposed HSRIVC method and its discretetime version HSRIV. The models take the orms: b o p + b G(p, ρ) = p 4 + a p 3 + a p, + a 3 p + a 4 M HSRIV C H(q, η) =, (u(t)) = u(t) + α u (t) + α u 3 (t) (3) or HSRIVC and G(q, ρ) = M HSRIV H(q, η) =, (u(t)) = u(t) + α u (t) + α u 3 (t) b q + b q + b q 3 + b 3q 4 +ã q +ã q +ã 3q 3 +ã 4q 4, (3) or HSRIV. Figures (a) and 3(a) display the magnitude Bode plots o the DT and CT estimated linear models. It can be irstly noticed that both models present similar results or low requencies whereas or high requencies, the CT method exhibits a superiority in model estimation. Both methods correctly estimate both resonance peaks. On the other side, the DT method appears to be less reliable, as or some realizations, the algorithm did not converge to acceptable values even though the initialization step is the same or both methods. By only looking at Bode diagram and considering only realizations which converged, both methods give satisactory results. However, when looking at non-linear unction estimations (Figures (b) and 3(b)), the DT method hands out results with a very large variance while the CT approach delivers a set o estimated unctions centered nearly exactly on the true non-linear unction. This can be explained by two acts: the DT version o the Hammerstein model (assuming the appropriate zero order hold) rises the number o parameters to be estimated or the numerator polynomial and thereore results in worse estimation. Furthermore, in the DT case, the numerator coeicients are so close to null that a small absolute error produces a large relative error. Estimated ˆα i coeicients, which are directly deduced rom ˆB (see (4)), dramatically suer rom this particular situation. 389

5 47th IEEE CDC, Cancun, Mexico, Dec. 9-, 8 TuC4.5 TABLE I ESTIMATION RESULTS FOR DIFFERENT NOISE MODELS b b a a α α d d system SNR method true value mean(ˆθ) HSRIVC std(ˆθ) RMSE S mean(ˆθ) HSRIVC std(ˆθ) RMSE true value mean(ˆθ) HSRIVC std(ˆθ) RMSE mean(ˆθ) HRIVC std(ˆθ) RMSE S mean(ˆθ) HSRIVC std(ˆθ) RMSE mean(ˆθ) HRIVC std(ˆθ) RMSE C. Discussions It can be noticed that results present a higher parameter variance than or a linear model estimation problem. This comes mainly rom the redundancy o the B(p) parameters contained in θ and by the higher number o estimated parameters: when the Hammerstein model relies on only n a + l + n b parameters, the proposed algorithm needs to estimate n a + l(n b + ) parameters. Hence, even i not optimal, this algorithm can produce a very good starting value or statistically optimal prediction error methods. However, the low variance in estimated parameters makes it an interesting method or practical data. An alternative RIV approach that can handle other types o nonlinearity, including nonlinear terms in variables other than the input, is statedependent parameter (SDP) estimation (e.g. []). Here, the parameters in the nonlinear unction are estimated by a nonlinear, iterative optimization procedure in which the RIV estimation algorithm is incorporated to estimate the linear TF parameters, based on the nonlinearly transormed input. Although computationally less eicient, this is statistically more eicient than the method proposed in the present paper. Some urther research about introducing constraint to avoid the parameters redundancy might be thereore relevant. IV. CONCLUSION The theory o multi-input single-output reined instrumental variable or CT systems has been applied to a non linear Hammerstein model composed o a linear dynamic CT Box- Jenkins transer unction and a non-linear unction deined as the sum o known basis unctions. The perormance and consistency or both HSRIVC and HRIVC methods have been highlighted. Finally, some advantages o using the suggested CT method with respect to its DT version have been illustrated. REFERENCES [] E-W. Bai. A blind approach to the Hammerstein-Wiener model identiication. Automatica, 38, Issue 6: ,. [] E-W. Bai. Identiication o linear systems with hard input nonlinearities o known structure. Automatica, 38, Issue 5:853 86, May. [3] E-W. Bai and K-S. Chan. Identiication o an additive nonlinear system and its applications in generalized Hammerstein models. Automatica, 44, Issue :43 436, February 8. [4] C. Bohn and H. Unbehauen. The application o matrix dierential calculus or the derivation o simpliied expressions in approximate non-linear iltering algorithms. Automatica, 36, Issue :553 56, October. [5] F. Ding and T. Chen. Identiication o Hammerstein nonlinear ARMAX systems. Automatica, 4, Issue 9: , September 5. [6] F. Ding, Y. Shi, and T. Chen. Auxiliary model-based least-squares identiication methods or Hammerstein output-error systems. Systems & Control Letters, 56, Issue 5:373 38, 7. [7] G. B. Giannakis and E. Serpedin. A bibliography on nonlinear system identiication. Signal Processing, 8, Issue 3:533 58, March. [8] I. Goethals, K. Pelckmans, J. A. K. Suykens, and B. De Moor. Identiication o MIMO Hammerstein models using least squares support vector machines. Automatica, 4, Issue 7:63 7, July 5. [9] H. Garnier and L. Wang (Eds). Identiication o continuous-time models rom sampled data. Springer-Verlag, London, March 8. [] Z. Q. Lang, S. A. Billings, R. Yue, and J. Li. Output requency response unction o nonlinear volterra systems. Automatica, 43, Issue 5:85 86, May 7. 39

6 TuC Magnitude(dB) Magnitude(dB) 47th IEEE CDC, Cancun, Mexico, Dec. 9-, true model true model Frequency(rad/sec) (a) Magnitude Bode plots o the identiied DT true unction (u) (u) s HSRIVC models together with the true system. s true unction -4 Frequency(rad/sec) (a) Magnitude Bode plots o the identiied CT HSRIV models together with the true system u u.5.5 (b) Non-linear unction estimated with DT HSRIV (b) Non-linear unction estimated with CT HSRIVC models together with the true non-linear unction. models together with the true non-linear unction. Fig.. DT model identiication Fig. 3. [] L. Ljung. Initialisation aspects or subspace and output-error identiication methods. In European Control Conerence (ECC 3), Cambridge (U.K.), September 3. [] K. Mahata and H. Garnier. Identiication o continuous-time BoxJenkins models with arbitrary time-delay. In 46th Conerence on Decision and Control (CDC 7), New Orleans, LA, USA, -4 December 7. [3] O. Nelles. Nonlinear system identiication. Springer-Verlag, Berlin,. [4] H. J. Palanthandalam-Madapusi, B. Edamana, D. S. Bernstein, W. Manchester, and A. J. Ridley. Narmax identiication or space weather prediction using polynomial radial basis unctions. 46th IEEE Conerence on Decision and Control, New Orleans, LA, USA, 7. [5] A. S. Poznyak and L. Ljung. On-line identiication and adaptive trajectory tracking or nonlinear stochastic continuous time systems using dierential neural networks. Automatica, 37, Issue 8:57 68, August. [6] G. P. Rao and H. Garnier. Numerical illustrations o the relevance o direct continuous-time model identiication. In 5th Triennial IFAC World Congress on Automatic Control, Barcelona (Spain),. [7] G. P. Rao and H. Unbehauen. Identiication o continuous-time systems. IEE Proceedings on Control Theory Appl., 53(), March 6. [8] J. Schoukens, W. D. Widanage, K. R. Godrey, and R. Pintelon. Initial estimates or the dynamics o a Hammerstein system. Automatica, 43, Issue 7:96 3, July 7. [9] T. So derstro m and P. Stoica. Instrumental Variable Methods or System Identiication. Springer-Verlag, New York, 983. CT model identiication [] P. C. Young. The identiication and estimation o nonlinear stochastic systems, in A. I. Mees (Ed), Nonlinear Dynamics and Statistics, pages Birkhauser: Boston,. [] P. C. Young. Some observations on instrumental variable methods o time-series analysis. International Journal o Control, 3:593 6, 976. [] P. C. Young. Recursive Estimation and Time-series Analysis. SpringerVerlag, Berlin, 984. [3] P. C. Young and H. Garnier. Identiication and estimation o continuous-time, data-based mechanistic (DBM) models or environmental systems. Environmental Modelling & Sotware,, Issue 8:55 7, August 6. [4] P. C. Young, H. Garnier, and M. Gilson. An optimal instrumental variable approach or identiying hybrid continuous time Box Jenkins models. 4th IFAC Symposium on System Identiication, Newcastle, Australia:5 3, March 6. [5] P. C. Young, H. Garnier, and M. Gilson. Reined instrumental variable identiication o continuous-time hybrid Box-Jenkins models, in H. Garnier and L. Wang (Eds), Identiication o continuous-time models rom sampled data, pages 9 3. Springer-Verlag, London, March 8. [6] P. C. Young and A. Jakeman. Reined instrumental variable methods o recursive time-series analysis - part III. extensions. International Journal o Control, 3, Issue 4:74 764, 98. [7] R. Johansson. Identiication o Continuous Time Models. IEEE Transactions on Signal Processing, 4, Issue 4: , 994. [8] R. Pintelon, J. Schoukens and Y. Rolain Box Jenkins continuous time modeling. Automatica, 36, Issue 7:983 99,. 39