In most of Fuzzy Logic System (FLS) designs, human reasoning is encoded into programs to make decisions and/or control systems. Designing an optimal FLS is equivalent to an optimization problem, in which efforts are made to locate a point in fitness search-space where the performance is better than that of other locations. The number of parameters to be tuned in designing an FLS is quite large. Also, fitness search space is highly non-linear, deceptive, non-differentiable, and multi-modal in nature. Noisy data, from which to construct the FLS, may make the design problem even more difficult. This chapter presents a framework to design Type-1 (T1) and Interval Type-2 (IT2) FLSs (Liang and Mendel, 2000c, Mendel, 2001, 2007, Mendel et al., 2006) using Particle Swarm Optimization (PSO) (Eberhart and Kennedy, 1995, Kennedy and Eberhart, 1995). This framework includes the use of PSO based Nature Inspired (NI) Toolbox discussed in the chapter titled, “Nature-Inspired Toolbox to Design and Optimize Systems.”
TopIntroduction
FLSs are being used successfully in an increasing number of application areas where system response can be described far easily using linguistic variables and rules than using their mathematical models. One of the most important considerations in designing any FLS is the generation of the fuzzy rules and Membership Function (MF) for each involved Fuzzy Set (FS). In most existing applications, the fuzzy rules are extracted from the experts' knowledge (Mendel, 2001, Khosla et al., 2005). With an increasing number of variables, the possible number of rules for the system increases exponentially, which makes it difficult for experts to define a complete rule set for modeling a system with good performance. An automated way to design fuzzy systems might be preferable (Shi et al., 1999, Saini et al., 2004, Khosla et al., 2005).
The design of an FLS can be formulated as a search problem in high-dimensional search-space where each point is an FLS, i.e., collection of FSs, and rulebase, etc. Given some performance measurement criteria, the performance of the system forms a hyper-surface in the space. Developing the optimal FLS design is equivalent to finding the optimal location of this hyper-surface. The hyper-surface can be (Shi et al., 1999):
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infinitely large since the number of possible FSs for each variable can be unbounded.
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non-differentiable since changes in the FSs can be discrete and can have a discontinuous effect on the fuzzy systems performance.
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complex and noisy since the mapping from a fuzzy rule set to its performance is indirect and dependent on the evaluation method used.
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multi-modal since different fuzzy rule sets and/or membership functions may have similar performance.
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deceptive since almost similar fuzzy rule sets and membership functions may have quite different performances.
All these characteristics make evolutionary algorithms a better option to search optimal designs from the hyper-surface search-space than the conventional hill-climbing search methods.
TopFuzzy Sets And Systems
Each input is expressed linguistically and depicted in graphical shapes (S, Z, Triangular, Trapezoidal, and Gaussian, etc.) called FSs or MFs. If membership value for a particular FS is crisp (certain) for some crisp input over the Universe of Discourse (UOD), the FS is called T1 FS and denoted as 
. It may be expressed in two different mathematical formats as in (1) and (2)
(1) (2)If the membership value for a particular FS is not one crisp value for a crisp input from the UOD 
, i.e., 
, but is another single FS or are multiple FSs (called secondary MFs) of any shape, then such an FS is called T2 FS. Figure 1 shows a T2 FS, with all triangular secondary MFs, in 3D and 2D representations. Such a T2 FS is denoted, by 
, and is characterized by a T2 MF as 
, where 
and 
, where 
is the primary membership of 
where 
for all 
. 
can be expressed mathematically in either of the following forms, i.e., (3) and (4).