Extended stochastic gradient identification algorithms for Hammerstein-Wiener ARMAX systems

An extended stochastic gradient algorithm is developed to estimate the parameters of Hammerstein–Wiener ARMAX models. The basic idea is to replace the unmeasurable noise terms in the information vector of the pseudo-linear regression identification model with the corresponding noise estimates which are computed by the obtained parameter estimates. The obtained parameter estimates of the identification model include the product terms of the parameters of the original systems. Two methods of separating the parameter estimates of the original parameters from the product terms are discussed: the average method and the singular value decomposition method. To improve the identification accuracy, an extended stochastic gradient algorithm with a forgetting factor is presented. The simulation results indicate that the parameter estimation errors become small by introducing the forgetting factor. © 2008 Elsevier Ltd. All rights reserved. 1. Problem formulation All physical systems are nonlinear to some extent and it is natural to use a nonlinear model to describe a system. Commonly used nonlinear models are Hammerstein models, Wiener models and their combinations. A Wiener model is a nonlinear model with a linear dynamic block followed by a static nonlinear function, and a Hammerstein model has a nonlinear block followed by a linear dynamic block. The extension of the Hammerstein andWiener models have two forms: the L–N–L Wiener–Hammerstein (W–H) model, where a nonlinear block is embedded between two linear blocks; and the N–L–N Hammerstein–Wiener (H–W) model, namely a linear block embedded between two static nonlinear gains. There exists a large amount of work on the identification of nonlinear systems which are composed of Hammerstein models and Wiener models [1–11]. For example, Chaoui used least squares and prediction-error algorithms as well as singular value decomposition to identify the nonlinear static gain of the Hammerstein model by using a set of data [6], Vörös provided a special output equation that is linear in the parameters of all the model blocks by applying an operator decomposition technique, and proposed an iterative parameter estimation algorithm for W–H models [7], and Zhu has proposed a relaxation iteration scheme bymaking use of a model structure in which the error is bilinear in parameters [12]. Bolkvadze designed a two-stage recursive identification algorithm [13]. Vörös proposed iterative estimation of the model parameters based on the estimates of internal variables, by applying a decomposition technique [14]. This paper focuses on identification problems of Hammerstein–Wiener ARMAX models. For H–W systems, by parameterization, the parameters from the identificationmodel include the products of the original system parameters [15], so separating the original parameters from the obtained parameter estimates of the product terms I This work was supported by the National Natural Science Foundation of China (No. 60574051). ∗ Corresponding author. E-mail addresses: [email protected] (D. Wang), [email protected] (F. Ding). 0898-1221/$ – see front matter© 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.camwa.2008.07.015 3158 D. Wang, F. Ding / Computers and Mathematics with Applications 56 (2008) 3157–3164 is required. Ding, Shi and Chen proposed a simple average method of separating parameters for Hammerstein models [2,4]. Another separating parameter method is the singular value decomposition one presented by Bai [15]. This paper presents an extended stochastic gradient algorithm to estimate the parameters of the H–W ARMAX models and uses these two decomposition methods to separate the system parameter estimates. The paper is organized as follows. Section 2 describes the system formulation related to the H–W models with colored noises, and develops an extended stochastic gradient algorithm. Section 3 introduces two separating parameter methods. Simulation studies are performed in Section 4. Section 5 gives the conclusions. 2. System descriptions and basic algorithms Bai studied the identification problems of the H–W systems with white noises [15] and Ding and Chen presented several identification algorithms for Hammerstein nonlinear ARMAXmodel [2–5]. This paper considers the following H–W ARMAX systems with colored noises [1],