Mathematical Problems in Engineering

Volume 2016 (2016), Article ID 1794921, 10 pages

http://dx.doi.org/10.1155/2016/1794921

## Subspace Identification of Hammerstein Model with Unified Discontinuous Nonlinearity

^{1}National Higher Engineering School of Tunis (ENSIT), University of Tunis, 5 Av. Taha Husein, BP 56, 1008 Tunis, Tunisia^{2}Laboratoire d’Ingenierie des Systèmes Industriels et des Energies Renouvelables (LISIER), University of Tunis, ENSIT, Tunis, Tunisia

Received 10 June 2016; Revised 8 October 2016; Accepted 31 October 2016

Academic Editor: Qingsong Xu

Copyright © 2016 Borhen Aissaoui et al. 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

The main aim of this study is to handle the case where the structures of nonlinear systems are unknown. In the many works, the parametric identification of nonlinear systems represented by Hammerstein model, with discontinuous and asymmetric nonlinearity, considers the structures of the nonlinear and linear blocks are known, especially the nonlinear bloc. To solve this problem, a unified form of nonlinearity representing eight cases of nonlinearities can be used. The parameters of both blocks, linear and nonlinear, are estimated using an iterative subspace approach. More importantly, in an attempt to show the extent to which this method is efficient, we apply it to experimental data obtained from the electropneumatic system. As a result, the numerical and experimental examples confirm a good conditioning and computational efficiency.

#### 1. Introduction

Many physical and biological systems have been modelled as a Hammerstein structure, such as the Stretch Reflex at the human ankle [1], the mechanical behaviour of lung tissue [2] and the electropneumatic system. In the Hammerstein model, there are two cascaded linear and nonlinear blocks which are an interconnection between the static structure nonlinearity and the Linear Time-Invariant (LTI) dynamic subsystem. Indeed, a variety of methods have been developed to identify Hammerstein systems. Narendra and Gallman [3] developed an iterative identification algorithm. They separate the parameters into two sets: one corresponding to the nonlinear component and the second from the linear element. In addition, the cross correlation-based methods are applied in order to estimate the linear dynamic subsystem [4] where the algorithm updates the nonlinearity’s output estimation at each step using the inverse dynamic of the linear component. Finally, they estimate and predict the inverse dynamics and identify the output nonlinearity. In other studies, the structures of the nonlinear and linear blocks are known in the case of continuous nonlinearity [5], discontinuous nonlinearities (DN) such as saturation with hysteresis (SH) [6], hysteresis-backlash [7], and hysteresis-relay nonlinearities [8].

In the real world, the identification of static nonlinearity subsystem remains the major problem in the Hammerstein model. Indeed, the structures of the nonlinear block are unknown and their estimate is important. Rejeb et al. [9] used the least squares method to estimate the parameters of unknown nonlinear block. Later, the same authors [10] proposed a general model combining discontinuous and asymmetric nonlinearities which generates all the possible combinations of the nonlinearities discontinuous elements. In some context, Vörös [11] developed a system with the general backlash which is based on appropriate switching and internal functions.

The identification using parametric approaches requires a minimal information priority about the system structure, that is, system order, noise model, and so forth [12]. Then, the LTI state model cannot be estimated through input-output measurements. Hence, the subspace method is introduced in order to overcome this problem. The subspace methods estimate the state space models for linear system with no a priori knowledge about the system [13]. In other words, those approaches are characterized by the simplicity of implementation including its effective results. In many cases, these methods provide a good alternative to the classical nonlinear optimization-based prediction-error methods [14].

Subspace identification is quite a well-accepted method for the identification of multivariable LTI systems. This method estimates an LTI state space model directly from input-output measurements. Moreover, Verhaegen and Varga [15] as well as Verhaegen and Westwick [16] proposed the Multivariable Output Error State Space identification model (MOESP) algorithm to identify Hammerstein models. In [1, 17, 18], the authors have treated the case of Single Input Single Output (SISO) Hammerstein model. Other authors proposed some subspace methods adapted to the Hammerstein-Wienner [19, 20]. Moreover, many researchers have developed a new identification subspace-based algorithm for the linear model [21–23].

Nevertheless, the structure of nonlinear block is used as known in those methods. Indeed, we developed an identification algorithm for a SISO Hammerstein model with discontinuous nonlinearities such as hysteresis, saturation, preload, and dead-zone. The proposed method uses an iterative subspace algorithm. The determination of the structure of nonlinearity is obtained based on Unified Discontinuous Nonlinearity (UDN) method. The parameters of both blocks are estimated simultaneous using the proposed iterative subspace approach.

This paper is organized as follows. Section 2 presents the problem formulation. Section 3 describes the proposed method. A numerical simulation validates the proposed algorithm and then applying to estimate the linear and nonlinear blocks model of electropneumatic system are shown in Section 4. Finally, this paper will be concluded in Section 5.

#### 2. Problem Formulation

Figure 1 presents a discrete-time SISO Hammerstein system, which is described by the following forms: where , , , , and are the LTI system output, state, output of the nonlinearity, process noise, and output measurement noise vectors at discrete time . , , , and are the unknown system matrices of the state space.