Type-2 fuzzy wavelet networks (T2FWN) for system identification using fuzzy differential and Lyapunov stability algorithm

We propose a novel method for the identification of non-linear system by utilizing some of the important properties of wavelets like denoising, compression, multiresolution along with the concepts of fuzzy logic. Two new type-2 fuzzy wavelet networks (T2FWNs) are proposed here. These T2FWNs can handle rule uncertainties in a better way because of using the type-2 fuzzy sets in modeling and fuzzy differential (FD) and Lyapunov stability during learning. Lot of work has been done in the identification of non-linear system by using the models based on type-1 fuzzy logic system (FLS). But in practice they are unable to handle uncertainties in the rules. The robustness of the system is assured by Lyapunov stability (LS). Also we have explored the properties of wavelets and FLS to handle the uncertainties efficiently. As the stability of the model is highly dependent on the learning of the system we use Lyapunov stability in combination with fuzzy differential. FD gives the range of variation of parameters having lower and upper bound in which the system is stable. The performance of T2FWN is compared with type-1 FLS, FWN [D.W.C. Ho, P.-A. Zhang, J. Xu, Fuzzy wavelet networks for function learning, IEEE Trans. Fuzzy Syst. 9 (February (1)) 2000] and FWNN [S. Srivastava, M. Singh, M. Hanmandlu, A.N. Jha, New fuzzy wavelet neural networks for system identification and control, Intl. J. Appl. Soft Comput. 6 (November (I)) 2005, 1–17]. It is shown that noise and disturbance in the reference signal are reduced with wavelets. A comparison of three learning algorithms: (i) gradient descent (GD) (ii) a combination of Lyapunov stability and fuzzy differential (LSFD) and, (iii) a combination of (i) and (ii) is done.