Adaptive Fuzzy Modelling and Control for Non-Linear Systems Using Interval Reasoning and Differential Evolution

Fuzzy systems have been developed to a major scientific domain since fuzzy set theory was introduced by Zadeh about four decades ago (Zadeh, 1965). There are certain particular properties of fuzzy systems that offer them better performance for specific applications. In general, fuzzy systems are suitable for uncertain or approximate reasoning, allow decision making with estimated values under incomplete information and represent descriptive or qualitative expressionswhich are easily incorporatedwith symbolic statements (Klir & Folger, 1987). However, under the general framework of typical fuzzy systems, some kinds of uncertainty cannot be handled, particularly in practical applications (Mendel & John, 2002; Ross, 2004; Hagras, 2004). Therefore, further flexibility can be obtained by considering the uncertainty in fuzzy systems which occur from qualitative knowledge and stochastic information. As mentioned in (Mendel & John, 2002; Hagras, 2004; Liu & Li, 2005a), most of uncertainties in fuzzy systems can be embodied by the information of fuzzy membership functions. In order to expand fuzzy systems to solve more complex uncertainty, some novel methods have been proposed during recent decade. Type-2 fuzzy logic system (T2FLS) was proposed to model and control further uncertainties in typical fuzzy systems by using the secondary fuzzy membership functions (Karnik & Liang, 1999; Liang & Mendel, 2000a). The T2FLS was originally inspired by the fact that the typical FLS limits introducing uncertain factors from linguistic rules through predefined membership functions. The type-2 fuzzy methods can be roughly described that their fuzzy sets are further defined by the typical fuzzy membership functions, i.e., the membership degree of belonging for each element of these sets are fuzzy sets, not a crisp number (Liang & Mendel, 2000b; Karnik & Mendel, 2001; Wu &Mendel, 2009). In comparison with the typical FLS, a type-2 FLS has the two-fold advantages as follows. Firstly, it has the capability of directly handling the uncertain factors of fuzzy rules caused by expert experience or linguistic description. Secondly, it is efficient to employ a