Journal of Control Science and Engineering

Volume 2016, Article ID 9614167, 12 pages

http://dx.doi.org/10.1155/2016/9614167

## Combined Parameter and State Estimation Algorithms for Multivariable Nonlinear Systems Using MIMO Wiener Models

University of Sfax, National Engineering School of Sfax (ENIS), Laboratory of Sciences and Technique of Automatic Control and Computer Engineering (Lab-SAT), BP 1173, 3038 Sfax, Tunisia

Received 29 November 2015; Revised 20 April 2016; Accepted 12 May 2016

Academic Editor: James Lam

Copyright © 2016 Houda Salhi and Samira Kamoun. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

This paper deals with the parameter estimation problem for multivariable nonlinear systems described by MIMO state-space Wiener models. Recursive parameters and state estimation algorithms are presented using the least squares technique, the adjustable model, and the Kalman filter theory. The basic idea is to estimate jointly the parameters, the state vector, and the internal variables of MIMO Wiener models based on a specific decomposition technique to extract the internal vector and avoid problems related to invertibility assumption. The effectiveness of the proposed algorithms is shown by an illustrative simulation example.

#### 1. Introduction

Over the last years, modeling, identification, and parameter estimation theories have received much attention by various research teams [1–4]. Blocks-oriented nonlinear models, in particular, which consist of interconnected linear dynamic subsystems and memory less nonlinear elements, have been widely used for modeling a large variety of nonlinear systems in such different fields as mechanical dynamics [5], chemical process [6], biotechnologies [7], signal filtering [8], and so on. In fact, this class of nonlinear models is able to describe the dynamic of complex systems, with a relatively simple structure. They can even simplify the identification, control, or diagnostic problems [9–12]. Added to that, several approaches developed in the linear case can be applied with an appropriate practical implementations [13–19]. In recent years, many identification methods have been studied for blocks-oriented systems and a large amount of works have been published in the literature. For example, Vörös [20] proposed a least squares based iterative algorithm for Hammerstein-Wiener systems with a backlash output; Hu et al. [21] developed an extended least squares parameter estimation algorithm for Wiener systems based on the overparameterization method; Ding et al. [22] used the hierarchical identification principal to identify Hammerstein systems; Mao and Ding [8] proposed a multi-innovation stochastic gradient algorithm for Hammerstein systems using the key term separation principal; Li [23] studied the maximum likelihood estimation algorithm for Hammerstein CARARMA systems; Guo and Bretthauer [24] proposed a recursive identification method for Wiener models, based on the prediction error method. Furthermore, Chaudhary et al. [25] explored an adaptive algorithm based on fractional signal processing for parameter estimation of Hammerstein autoregressive models; Wu et al. [26] presented a robust Hammerstein adaptive filtering algorithm based on the Maximum Correntropy Criterion, which aims at maximizing the similarity between the model output and the reel response; Falck et al. [27] proposed an identification method of NARX Wiener-Hammerstein models using kernel-based estimation technique; Kibangou and Favier [28] developed a new approach for estimating a Parallel-Cascade Wiener System using a joint diagonalization of the th-order Volterra kernel slices to identify linear subsystems and using the least square algorithm to identify nonlinear subsystems. This approach has been extended to other blocks-oriented models with polynomials nonlinearities [29].

Recently, much attention has been paid to blocks-oriented state-space systems which have been successfully used for control algorithms, identification schemes, and signal filtering [30, 31]. However, the parameter estimation has become more difficult because the blocks-oriented models not only include the unknown parameter of linear and nonlinear subsystems but also include the unmeasurable state variables [32–35]. In this framework, Wang and Ding proposed a recursive parameter and state estimation for Hammerstein state-space systems [36] and for Hammerstein-Wiener state-space systems [37], using the hierarchical principal; Wang et al. [38] discussed an iterative identification algorithm Hammetstein state-space system, by combining the iterative least square and the hierarchical identification method. However, for Wiener state-space models, there is a little contribution in the literature that addresses the parameter estimation problems or the state estimation problems. In fact, Westwick and Verhaegen [39] proposed a subspace identification method for MIMO Wiener subsystems with odd and even nonlinearities and a Gaussian input system; Bruls et al. [40] derived separable least squares algorithms for a state-space Wiener model with Chebyshev polynomials nonlinearity; Lovera et al. [41] developed a recursive subspace identification method for Wiener state-space models using the singular-value decomposition technique; Glaria Lopez and Sbarbaro [42] proposed an observer design for a Wiener model with known parameters.

The main difficulty in the identification of Wiener models is that the internal variables, acting between linear and nonlinear blocks, are almost unavailable and the input-output available data are not enough to provide all information on these unknown variables. To overcome this difficulty, most published works, addressing the identification of Wiener systems, assume one of these assumptions: the invertibility of the unknown nonlinear element [43], an* a priori* knowledge of the nonlinearity [44], an approximation of the nonlinearity as a piecewise linear function [44], and a specific input signal [39]. However, these assumptions, and especially the invertibility assumption, severely limit the applicability of Wiener models because the output nonlinearity, in several real cases, is noninvertible or is quite complicated to find the inverse nonlinearity especially for multivariable systems.

This paper introduces a recursive identification method for MIMO Wiener model. This model is characterised by a linear dynamic block as an observer state-space model and a nonlinear block as combined and arbitrary (reversible or irreversible) nonlinearities. A recursive algorithm which combines the least squares technique, the adjustable model, and the Kalman filter principle is developed to resolve the parameters and state estimation problem with less computational effort and a fast convergence rate. Indeed, in the proposed method, the parameters of the linear part and nonlinear part of the MIMO Wiener model are estimated separately in order to decrease the dimension of the unknown parameters matrices and reduce the parameter redundancy. Moreover, a modified Kalman filter and a specific decomposition technique are developed to extract and estimate the unknown internal vector without any research of the inverse nonlinear functions.

The remainder of this paper is organized as follows. Section 2 describes the problem formulation for MIMO Wiener state-space models. The least squares based and adjustable model based recursive parameter estimation algorithm and a new recursive state estimation algorithm based on Kalman filter theorem are presented in Section 3. Section 4 provides an illustrative example to show the efficiency of the proposed algorithms. Finally, some concluding remarks are given in Section 5.

#### 2. Problem Formulation

Consider the MIMO discrete-time Wiener model Figure 1 where the linear dynamic part is given by the following state-space equation:where , , and are, respectively, the state vector, the input vector, and the internal vector at the discrete-time , and are two noise vectors, , , and and are defined, respectively, by