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3.4. INFERENCE WITH A RULE SET

Considering a Single-Input Single-Outout (SISO) system, given the relationship "(X --> Y) isr R’" and the information that X belongs to a fuzzy set A’(x) is observed, then the problem of inference is finding the fuzzy value for Y. According to Zadeh's Compositional Rule of Inference (CRI) (1973), the membership function of the output is defined as:

F(y) = [inverted wedge] x[T“(A’(x), R(x,y))]

where [inverted wedge]x means the maximum for all values of x [element of] X and T“ is a t-norm.

It is known that generally two different methods of computing, i.e., FITA or FATI with a rule base, R, lead to two different reasoning results (Türks¸enand Lucas, 1991). Whether it is executed with Mamdani's Reasoning, or with Formal Logical Myopic Reasoning, i.e.:

FC(y) = [inverted wedge] x[T“[A’(x), Rc(x,y)]], FD(y) = [inverted wedge] x[T“[A(x), Rd(x,y)]], respectively.

The reason for this is that triangular norm and co-norm operators are not distributive. Hence, it is necessary first to combine all the rules to get the overall relation R of the set of rules and then compose it with the fuzzy input A’(x). This method, called First-Aggregate-Then-Infer (FATI), is quite inefficient in terms of computational time and memory (Türks¸enand Tian, 1993).

However, in most applications of fuzzy modeling and control, the input x* is crisp and therefore, the input fuzzy set A’(x*) is a fuzzy singleton. In this case, it can be shown that the distributivity property holds for any type of t-norm and t-conrom family. Hence, it is possible to fire each single rule first and calculate individual fuzzy reasoning output Fi and finally aggregate fuzzy outputs of all the rules to obtain the inferred fuzzy output F(y). This method, called First-Infer-Then-Aggregate (FITA), is computationally more efficient than FATI but equivalent to it.

For multi-input single-output systems, the antecedent of each rule i is a conjunction of r fuzzy sets Ai1, Ai2, ..., Air. For crisp input , given a rule i:

Ri - IF X1 isr Ai1 AND ... AND Xr isr Air THEN Y isr Bi

we need to compute:

where [tau]i is called the Degree Of Firing (DOF) of the rule i. By using the arguments presented above, it is clear that the fuzzy output membership function may be computed either by FITA or by FATI, producing the same result for both of the reasoning mechanisms as:

Note that there is no need to aggregate over all x, since x = x* is a singleton, i.e., A’(x*) = 1. Therefore, it is computationally more efficient to do the execution by the FITA method.

With these preliminaries, the unified reasoning mechanism is introduced as a linear combination of the two extremes, i.e., the myopic approaches of Mamdani's reasoning and formal logical reasoning:

E(y) = [beta]FD(y) + (1 - [beta])FC(y), 0 [less than or equal to] [beta] [less than or equal to] 1

3.4.1. Defuzzification

Yager and Filev (1994) suggest a general defuzzification method, based on the probabilistic nature of the selection process among the values of a fuzzy set, called Basic Defuzzification Distribution (BADD) method:

As we can see, BADD method is essentially a family of defuzzification methods parameterized by parameter [alpha]. By varying [alpha] continuously in the real interval, it is possible to have more appropriate mappings from the fuzzy set to the crisp value, depending on the system behavior.

It should be clear that the unified reasoning mechanism that is combined with BADD method of defuzzification, allows us to train the parameters p, q, [alpha], and [beta] of this model with a given set of data in order to minimize the error between the model output results and the actual system outputs; and hence this approach provides us with a better fiting of the system model.

4. CASE STUDY

In this section, two textbook case study results are presented first; then a real-life industrial process model is discussed.

4.1. A NONLINEAR SYSTEM

In this example, we re-analyze the fuzzy modeling process of the nonlinear system discussed in Sugeno and Yasukawa (1993) and Nakanishi, Türks¸en,and Sugeno (1993). After deriving the output fuzzy clusters and performing the classification for the entire output space, significant input variables are identified as the first two input variables,i.e., x1 and x2; the dummy input variables were discarded as expected. The fuzzy system identification is based on 50 input-output data. Two dummy input variables x3 and x4 have been added to check the input selection strategy. Instead of Sugeno-Yasukawas's approach, which divides the data into two sets and uses a time-demanding combinatorial strategy, we apply the straightforward strategy described in the sequel where input membership functions are assigned through fuzzy line clustering. After approximating the input and output fuzzy clusters by suitable trapezoidal functions, the first-step forms the fuzzy model of the system. Note that no inference mechanism is required up to this point. The next step is to select and tune the set of parameters of the fuzzy model, which consists of the inference parameters (p, q, [alpha], [beta]), and the input and output membership function parameters. At this step of parameter identification, the optimum inference paramters are identified as:

p = 19.9959; q = 0.4536; [alpha] = 5.3715; [beta] = 0.0255.

By optimizing the inference paramters, the fuzzy model performance index, PI, is determined to be PI = 0.171. The second step of parameter identification is to adjust the input and output membership parameters with supervised training. After five iterations, the error is reduced to PI = 0.0106 and after five more iterations, the performance index becomes PI = 0.0040. The performance index of Sugeno-Yasukawa's fuzzy model of the same type (position type) starts from PI = 0.318, and after 20 iterations, it reaches to PI = 0.079.


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