System Identification Based on Dynamical Training for Recurrent Interval Type-2 Fuzzy Neural Network

Introduction

In the past decades, fuzzy sets and their associated fuzzy logic have supplanted conventional technologies in many scientific applications and engineering systems, especially in control systems, pattern recognition and system identification. We have also witnessed a rapid growth in the use of fuzzy logic in a wide variety of consumer products and industrial systems. Since 1985, there has been a strong growth in their use for dealing with the control of, especially nonlinear, time varying systems. For instance, fuzzy controllers have generated a great deal of excitement in various scientific and engineering areas, because they allow for ill-defined and complex systems rather than requiring exact mathematical models (Narendra & Parthasarathy, 1990; Wang 1993; Wang, 1994; Rovithakis & Christodoulou, 1994; Castro et al., 1995; Chen et al., 1996). The most important issue for fuzzy control systems is how to deal with the guarantee of stability and control performance, and recently there have been significant research efforts on the issue of stability in fuzzy control systems (Spooner & Passino, 1996; Ma & Sun, 2000).

Quite often, the information that is used to construct the rules in a fuzzy logic system (FLS) is uncertain. At least, there are four possible ways of rule uncertainties in type-1 FLS (Mendel 2004; Liang & Mendel, 2000): (1) Being words mean different things to different people, the meanings of the words that are used in antecedents and consequents of rules can be uncertain; (2) consequents may have a histogram of values associated with them, especially when knowledge is extracted from a group of experts who do not all agree; (3) measurements that activate a type-1 FLS may be noisy and therefore uncertain; (4) finally, the data that are used to tune the parameters of a type-1 FLS may also be noisy. Therefore, antecedent or consequent uncertainties translate into uncertain antecedent or consequent membership functions (MFs). Type-1 FLSs are unable to directly handle rule uncertainties, since their membership functions are type-1 fuzzy sets. On the other hand, type-2 FLSs involved in this paper whose antecedent or consequent membership functions are type-2 fuzzy sets can handle rule uncertainties (Hagras, 2004; Hagras, 2007; Martinez et al., 2009; Sepulveda et al., 2009; Castro et al., 2009). A type-2 FLS is characterized by IF-THEN rules, but its antecedent or consequent sets are type-2. Hence, type-2 FLSs can be used when the circumstances are too uncertain to determine exact membership grades such as when training data is corrupted by noise.

Type-2 FLSs have been applied successfully to deal with decision making (Yager, 1980), time-series forecasting (Karnik & Mendel, 1999), time varying channel equalization (Liang & Mendel, 2000), fuzzy controller designs (Wang, 1994; Lin et al., 2009, 2010; Lin, 2010b), VLSI fault diagnosis (Lin, 2010a) and control of mobile robots (Wu, 1996), due to the type-2 FLSs ability to handle uncertainties. Further, genetic algorithm (GA) was adopted to fine tune the Gaussian MFs in the antecedent part of type-1 FNN (Wang et al., 2001). The dynamical optimal training algorithm for the two-layer consequent part of interval TFNN (Wang et al., 2004), was proposed to learn the parameters of the antecedent type-2 MFs as well as of the consequent weighting factors of the consequent part of the T2FNN. The back propagation (BP) equations proposed by Wang et al. (2004) are not correct and were modified by Hagras (2006).