The identification of neuro-fuzzy based MIMO Hammerstein model with separable input signals

Abstract

A novel identification method of neuro-fuzzy based MIMO Hammerstein model by using the correlation analysis method is presented in this paper. A special test signal that contains independent separable signals and uniformly random multi-step signal is adopted to identify the MIMO Hammerstein process, resulting in the identification problem of the linear model separated from that of nonlinear part. As a result, the identification of the dynamic linear element can be separated from the static nonlinear element without any redundant adjustable parameters. Moreover, it can circumvent the problem of initialization and convergence of the model parameters discussed in the existing iterative algorithms used for identification of MIMO Hammerstein model. Examples are used to illustrate the effectiveness of the proposed method.

Introduction

The Hammerstein model is a block-oriented nonlinear model consisting of the cascade structure of a static nonlinear element followed by a dynamic linear element. It has been shown that such a model can efficiently represent and approximate many industrial processes. For example, pH neutralization processes [1], heat exchangers [2], [3], distillation columns [4], [5], dryer process [6], polypropylene grade transition [7], and continuous stirred tank reactor (CSTR) [8] have been modeled with Hammerstein model. Various system identification methods have been extended to identify the Hammerstein model. Some approaches assumed that the nonlinearity is a two segment piecewise linear or multi-segment piecewise-linear function, a discontinuous function or hard nonlinearities [9]. Moreover, some methods assumed that the nonlinearity is monotonous and odd, or invertible [9]. From the perspective of immeasurable intermediate variable of Hammerstein model and combined with the basic processes and characteristics of Hammerstein model identification, this paper surveys the relevant theories and methods of Hammerstein model by synchronous and separate step identification methods of static nonlinear element and dynamic linear element. Considering immeasurable intermediate variables in the Hammerstein model, the synchronous method and the separate step method are two different ways to identify the static nonlinear element and the dynamic linear element of the Hammerstein model [10]. The synchronous method identifies the parameters of the Hammerstein model by directly constructing a hybrid model of the static nonlinear element and the dynamic linear element, such as over-parameterization method [11], [12], subspace method [13], modulation function method [14] and direct identification method [15]. The separate step method decouples the identification problems of the dynamic linear element and the static nonlinear element by estimating the immeasurable intermediate variables, such as iterative method [16], separable least squares method [17], stochastic method [18], blind identification method [19], frequency domain method [20] and multi-signal based method [21].

In addition, neural networks and fuzzy systems have been explored to model the static nonlinearity of Hammerstein processes owing to their ability to model a nonlinear function to any arbitrary accuracy. Xiang et al. presented a model-predictive control strategy based on hybrid neural networks for the multi-input and multi-output Hammerstein model [22]. Wang et al. presented a Hammerstein recurrent neurofuzzy network associated with an online minimal realization learning algorithm for dealing with nonlinear dynamic applications [23]. Jia et al. used a neuro-fuzzy-based model to describe the nonlinearity of the Hammerstein process without any prior process knowledge [24]. Furthermore, it is worth pointing out that the Hammerstein model is very frequently used into controller design [25], [26], [27].

In general, two possible structures can be used to describe a MIMO Hammerstein model depending on whether the nonlinearities are separate or combined [28], [29], as depicted in Fig. 1, Fig. 2. The combined nonlinearity case is more general, but it can cause a very challenging parameter estimation problem because of the large number of parameters to be estimated [30]. Chan et al. used the multivariable cardinal spline functions to model nonlinearities of the MIMO Hammerstein process, which is relatively easy to develop control strategies [31]. Goethals et al. used the least-squares support vector machines to identify MIMO Hammerstein systems [32]. In [33], a recursive stochastic identification method for MIMO Hammerstein process was presented with a series of assumptions on the signals, which limits its application in industrial processes. Lee et al. extended the binary signal and random multi-step signal based method to decouple the cascade elements of the MIMO Hammerstein model [34]. However, the parameters to be identified increases with the number of input variables, which causes a burden on parameter estimation. Based on this work, Jyh-Cheng et al. further designed special multiple input signals to decouple the parameters of the cascade elements without any redundant adjustable parameters [35]. However, no efficient method is currently available for the special signals based identification of MIMO Hammerstein model.

Bussgang׳s classic theorem [36] about Gaussian signals presents a useful condition that $R_{vu}\left(\tau \right)=b_0{R}_u\left(\tau \right)$ holds for an arbitrary static nonlinearity $f(\cdot )$ if the input signal is separable. Here $R_{vu}\left(\tau \right)=E\left(v\left(k\right)u(k-\tau )\right)$ is the cross-correlation function between output $v\left(t\right)$ and input $u\left(t\right)$, $R_u\left(\tau \right)=E\left(u\left(k\right)u(k-\tau )\right)$ is the auto-correlation function of the input and $b_0=E\left(f^{`}\left(u\left(k\right)\right)\right)$. Nuttall generalized the theorem to the class of separable processes, such as sine signals, random binary signals and several kinds of modulated signals [37]. Based on above theorem, Enqvist et al. identified generalized Single-input Single-output (SISO) Wiener–Hammerstein systems by utilizing separable signals [38], [39]. Motivated by the previous works, a novel identification method for neuro-fuzzy based MIMO Hammerstein model by using the correlation analysis method is presented in this paper. We employ the extended version of Bussgang׳s theorem for MIMO Hammerstein system. The property of the separable signal of MIMO system is analyzed further. Moreover, the conditions how Bussgang׳s theorem can be used for the identification of the generalized MIMO Hammerstein model are given. In this work, a special test signal is adopted to identify the Hammerstein process, resulting in the identification problem of the linear model separated from that of nonlinear part. The signal satisfies the following conditions: 1) it contains independent separable signals, such as binary signal, sine signal and Gaussian signal; and 2) it contains a uniformly random multi-step signal. As a result, the identifications of the liner model and the static nonlinear function are carried out independently by using the separable input signals. And the identification of the dynamic linear element can be separated from the static nonlinear element without any redundant adjustable parameters.

The rest of this paper is organized as follows. The MIMO Hammerstein process identification problem and neuro-fuzzy based MIMO Hammerstein model are given in Section 2. A correlation analysis and neuro-fuzzy based identification method is presented in Section 3. Simulation examples are given in Section 4, followed by the concluding remarks given in Section 5.