Content area
Full Text
1 Introduction
Because of universal approximation property ([7] Hornik et al. , 1989), they have the ability to approximate nonlinear functions with arbitrary precision by learning methods. Owing to this property, neural networks have been vastly used for nonlinear systems identification and control ([1] Brown and Harris, 1994; [18] Polycarpou, 1996; [22] Zhihong et al. , 1998; [23] Wang et al. , 2002; [5] Ge and Wang, 2004; [8] Hayakawa et al. , 2005; [20] Xu and Tan, 2007) in many fields of science and engineering. Some different structures of neural networks have been developed in the literature, such as multilayer networks, radial basis function networks, wavelet neural networks, etc. The structure of the wavelet neural networks can be regarded as a linear combination of a set of wavelet basis functions. Since the wavelet functions are localized in space and orthonormal, the wavelet neural networks are more suitable for learning than the other neural networks ([4] El-Sousy, 2010; [12] Khan and Rahman, 2010; [20] Xu and Tan, 2007; [21] Yoo et al. , 2008; [13] Lin, L.K., 2006; [14] Lin, C.K., 2002; [9] Hsu et al. , 2006).
By utilizing the universal approximation feature of neural networks, many adaptive control methods ([20] Xu and Tan, 2007; [21] Yoo et al. , 2008; [13] Lin, L.K., 2006; [14] Lin, C.K., 2002; [9] Hsu et al. , 2006; [5] Ge and Wang, 2004; [15] Li et al. , 2004) have been proposed to achieve the control goal for nonlinear systems, and some adaptive control schemes ([2] Choi and Farrell, 2001; [3] Chang and Yen, 2005; [16], [17] Leu et al. , 2009, 1999) also have been proposed to acquire better performance for uncertain nonlinear systems under the constraint that only system output is available for measurement. In general, the neural networks combined with the design of adaptive control are used to approximate the system dynamics by appropriately tuning their internal parameters. Then, according to Lyapunov stability theorem, the stability of the closed-loop systems can be proved, and adaptive laws to tune their parameters can be derived. For neural networks with nonlinear parameterization, gradient approaches which need to find the derivative of neural networks with respect to their parameters are often utilized for the purpose of designing...