**1. Introduction**

420 Fuzzy Inference System – Theory and Applications

Ross, T.J. (2010). *Fuzzy Logic with Engineering Applications*, John Wiley and Sons, ISBN

Being an artificial neural network inspired by the cerebellum, the cerebellar model articulation controller (CMAC) was firstly developed in (Albus, 1975a, 1975b). With the advantages such as fast learning speed, high convergence rate, good generalization capability, and easier hardware implementation (Lin & Lee, 2009; Peng & Lin, 2011), the CMAC has been successfully applied to many fields; for example, identification (Lee et al., 2004), image coding (Iiguni, 1996), ultrasonic motors (Leu et al., 2010), grey relational analysis (Chang et al., 2010), pattern recognition (Glanz et al., 1991), robot control (Harmon et al., 2005; Mese, 2003; Miller et al., 1990), signal processing (Kolcz & Allinson, 1994), and diagnosis (Hung & Wang, 2004; Wang & Jiang, 2004). However, there are three main drawbacks of Albus' CMAC, i.e., larger required computing memory (Lee et al., 2007; Leu et al., 2010; Lin et al., 2008)), relatively poor ability of function approximation (Commuri & Lewis, 1997; Guo et al., 2002; Ker et al., 1997), and difficulty of adaptively selecting structural parameters (Hwang & Lin, 1998; Lee et al., 2003).

In order to tackle these disadvantages, several methods, such as online-based clustering (Kasabov & Song, 2002; Tung & Quek, 2002) for the above-mentioned first drawback, Bspline functions (Lane et al., 1992; Wu & Pratt, 1999) and fuzzy concepts (Jou, 1992; Chen, 2001; Guo et al., 2002; Ker et al., 1997; Lai & Wong, 2001; Zhang & Qian, 2000) for the second one, and competitive learning (Chow & Menozzi, 1994), clustering (Hwang & Lin, 1998) and Shannon's entropy and golden-section search (Lee et al., 2003) for the third one, were proposed. Among these approaches, further improvements were implemented by Lin et al. (2008) with self-constructing algorithm and Gaussian basis functions.

The rest of this chapter is organized as follows. Starting from the first CMAC model in 1975 the development processes, related learning algorithms and system identification examples of the fuzzy CMACs are briefly reviewed in section 2. Sections 3 and 4 respectively discuss the self constructing FCMAC (SC-FCMAC) and the powerful parametric FCMAC (P-FCMAC). Lastly, section 5 concludes this chapter, with suggested directions of further researches.

<sup>\*</sup> Corresponding Author

System Identification Using Fuzzy Cerebellar Model Articulation Controllers 423

firstly considered, while the developments of other five training schemes (i.e., credit assignment, gray relational, error norm, active deformable and Tikhonov ones) were mentioned as well. In order to reduce relative memory usages, proposed approaches of hierarchical and self-organizing CMACs were reasoned, whereas the fuzzy variation of the self-organizing CMAC will be further presented in the following section of this chapter.

Many researchers have integrated the fuzzy concept into the CMAC network, such as in (Chen 2001; Dai et al., 2010; Guo et al., 2002; Jou, 1992; Ker et al., 1997; Lai & Wong, 2001; Lee & Lin, 2005; Lee et al., 2007a; Lee et al., 2007b; Lin & Lee, 2008; Lin & Lee, 2009; Lin et al., 2008; Peng & Lin, 2011; Wang, 1994; Zhang & Qian, 2000). In general, they use membership functions rather than basis functions, and the resulting structures are then

In addition, the work of (Mohajeri et al., 2009b) provides a review of FCMACs, including over 23 relative aspects such as membership function, memory layered structure, defuzzification and fuzzy systems, was provided. Even FCMACs have originally reduced memory requirement for the CMAC, further discussions of clustering (such as fuzzy Cmean, discrete incremental clustering and Bayesian Ying-Yang) and hierarchical approaches for reducing memory sizes of FCMACs themselves were overviewed in (Mohajeri et al., 2009b) as well. Furthermore, as divided in (Dai et al., 2010), there are two classes of FCMACs architectures, i.e., forward and feedback fuzzy neural networks, which is useful

In the following sections, being the example models in this chapter the self-constructing FCMAC (SC-FCMAC, Lee et al., 2007a) and the powerful parametric FCMAC (P-FCMAC, Lin & Lee, 2009) are reviewed, in order to provide readers the insight knowledge of how these FCMAC work. Companied by their corresponding architectures and learning schemes,

From relative architectures to learning algorithms this section provides a brief review and

As illustrated in Fig. 2, the SC-FCMAC model (Lee et al., 2007a) consists of the input space partition, association memory selection, and defuzzification. Similar to the traditional CMAC model, the SC-FCMAC model approximates a nonlinear function *y fx* ( ) by

: *SX A* (1)

 : *PA D* (2) where *X* is an s-dimensional input space, *A* is an *NA*-dimensional association space, and *D* is a 1-dimensional (1-D) output space. These two mappings are realized by using fuzzy

for beginners to have a big picture of the basic concept for the FCMACs.

illustrative examples of system identification are provided as well.

discussions of the self-constructing FCMAC (SC-FMAC, Lee et al., 2007a).

**3. The self-constructing fuzzy CMAC** 

**3.1 Architecture of the SC-FCMAC model** 

applying the following two primary mappings:

**2.2 Fuzzy CMAC models** 

called fuzzy CMACs (FCMACs).
