System identification is a vital work in industry applications such as control design, plant diagnosis, and system monitoring. Recently, identification of nonlinear MIMO systems have been used widely in various fields (Chan et al., 2006, Felici et al., 2007, Goethals et al., 2005a, Goethals et al., 2005b, Huang et al., 2006, Majhi and Panda, 2010, Rouss and Charon, 2008 and Vieira et al., 2005). However, it should be pointed out that structural identification and parameter estimation of nonlinear MIMO systems are rather difficult issues in system identification. Therefore, experts have put much effort in this research field. Cardinal spline functions to model MIMO Hammerstein systems have been adopted (Goethals et al., 2005a and Goethals et al., 2005b). Wang, Ding, and Liu (2007) have introduced a hierarchical least squares algorithm for identifying MIMO ARX-like systems based on the hierarchical identification principle. An artificial neural network model for system identification by expanding the input pattern by Chebyshev polynomials has been proposed (Purwar, Kar, & Jha, 2007). A systematic way that SVR integrating least squares regression has been proposed to identify MIMO systems (Fu, Wu, Jeng, & Ko, 2009). A neural inverse dynamic NARX model has been adopted to perform MIMO system identification (Anh & Phuc, 2010).
In the neural network, RBFNs have received considerable applications, such as function approximation, prediction, recognition, etc. (Chuang et al., 2004, Sing et al., 2007, Xu et al., 2003 and Yu et al., 2000). Since RBFNs have only one hidden layer and have fast convergence speed, they are widely used for nonlinear system identification recently (Apostolikas and Tzafestas, 2003, Chen et al., 2009, Falcao et al., 2006, Fu et al., 2009 and Li and Zhao, 2006). Besides, the RBFNs are often referred to as model-free estimators since they can be used to approximate the desired outputs without requiring a mathematical description of how the outputs functionally depend on the inputs (Kosko, 1992).
When utilizing RBFNs, the number of hidden layer nodes, the initial parameters of the kernel, and the initial weights of the networks must be determined first. However, a systematic way to determine the initial structure of RBFNs has not been established yet. In many cases, these parameters are determined according to the experience of the designer or just chosen randomly. For example, in Falcao et al., 2006 and Manrique et al., 2006, and Sarimveis, Alexandridis, Mazarakis, and Bafas (2004), the number of hidden layer nodes is fixed according to the choice of the designer first. Then different kinds of algorithms such as the least squares method, the gradient descent method, and the genetic algorithm are used to optimize the parameters. However, such kind of improper initialization usually results in slow convergence speed and poor performance of the RBFNs. Meanwhile, a learning rate serves as an important role in the procedure of training RBFNs. Generally, the learning rate is selected as a time-invariant constant by trial and error (Chuang et al., 2004, Chuang et al., 2002, Fu et al., 2009 and Hsieh et al., 2008). However, there still exist several problems of unstable or slow convergence. Some researchers have engaged in exploring the learning rate to improve the stability and the speed of convergence (Song et al., 2008, Yoo et al., 2007 and Yu, 2004).
Recently, support vector machine (SVM) has been successfully used in various fields due to the potential capability of handling classification tasks in the case of high dimensionality and sparsity of sampling data (Camps-Valls et al., 2009, Goethals et al., 2005a, Goethals et al., 2005b, Manel et al., 2006, Suykens, 2001 and Vapnik, 1998). In some research (Espinoza et al., 2005, Fu et al., 2009, Gao et al., 2007 and Lima et al., 2007), SVR algorithm has been adopted for nonlinear system identification. In this paper, in order to overcome the above problems of training RBFNs, first, an SVR method with Gaussian kernel function (Gao et al., 2007, Hua and Zhang, 2006 and Vapnik, 1995) is adopted to determine the initial structure of the RBFNs for identifying nonlinear systems. This means that the proposed method is to use the SVR method to determine the number of hidden layer nodes and the initial parameters of the kernel. After initialization, an annealing robust concept (Chuang et al., 2002 and Fu et al., 2009) with dynamical learning algorithm (ADLA) is then applied to train the RBFN (ADLA-RBFN), in which PSO method is adopted to find optimal learning rates during learning procedure. Two simulation examples will be given to illustrate the feasibility and efficiency of the proposed SVR-based ADLA-RBFNs (SVR-ADLA-RBFNs) for identification of MIMO systems.
This paper is organized as follows. Section 2 describes the RBFNs for identification of nonlinear MIMO systems. In Section 3, an ADLA based on SVR is introduced to train RBFNs, in which a nonlinear time-varying evolution concept is induced. In Section 4, a population-based stochastic searching method and a fitness function evaluating populations of PSO are presented. Section 5 provides the proposed algorithm and flowchart for SVR-ADLA-RBFNs using the PSO approach. Simulation results of system identification for two MIMO examples are illustrated to evaluate the SVR-ADLA-RBFNs in Section 6. Section 7 brings conclusions for the main contributions of this paper.
