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4.2. GAS FURNACE MODEL

The proposed algorithm was applied to a famous example of the system identification given by Box and Jenkins (Box, 1970). The process is a gas furnace with single input u(t) (gas flow rate) and single output y(t) (CO2 concentration). For this dynamic system, 10 input candidates y(t - 1), ..., y(t - 4), u(t - 1), ..., u(t - 6) are considered. We use the same 296 data as in the Sugeno and Yasukawa (1993) analysis. With the ETG (i.e., Emami, Türks¸en,Goldenberg) method, it is found that m = 2.5 and c = 6 are suitable for the system. The significant input variables are determined as y(t - 1), u(t - 2), and u(t - 3). The optimum inference parameters for the gas furnace system are derived as:

p = 102.0147; q = 8.5918; [alpha] = 27.9338; [beta] = 0.3782

After 10 iterations, the performance index reduces to a value of PI = 0.1584, which is less than both the identified linear model (PI = 0.193) and the Sugeno-Yasukawa position-gradient fuzzy model (PI = 0.190).

4.3. INDUSTRIAL PROCESS MODEL

A recent case study is briefly discussed in this section to demonstrate the effectiveness of the integrated knowledge representation and inference model presented in the previous sections of this chapter.

Real-life process data was provided by an industrial company. It consists of 93 input-output data vectors. Each data vector has 46 input variables and 1 output variable.

First, the improved clustering method discussed in Section 2.4 and identified as ETG method is applied with the data. In Figure 1, the trace of Fuzzy Total Scatter Matrix is shown as a function of "m" for various "c" values. After proper analyses of the cluster validity index as a function of c within the reliable domain, it is found that m = 2.5 with c = 7 are suitable values for this industrial process.

Next, the most important input variables that affect the output significantly are determined among all of the 46 input variables. It is found that only 13 variables, i.e., x4, x9, x10, x11, x12, x17, x18, x20, x21, x28, x34, x44, and x45, have a significant effect on the output variable, which is the throughput efficiency measure of the process. The fuzzy rule base for this industrial process is shown in Figure 2. It is observed that there are seven effective rules, rule 0 to rule 6, that control the hidden behavior of this industrial process.

In Figure 2, it should be noted that the four numbers shown on the right-hand side of each trapezoid are the four values that define the trapezoid, e.g., in rule 3, variable x4 has a trapezoid membership function defined by ·218.95, 229.00, 310.72, 997.95Ò, etc.

The integrated fuzzy inference model of this industrial system is developed next. The combined Mamdani and Logical myopic model with parameters p = 1.6190, q = 0.8438, [alpha] = 8.2935, and [beta] = 0.001 produces the best fit for this industrial process with PI = 0.0886 for the test data. The comparison of the combined model output and the actual system output are shown in Figure 3.


FIGURE 1 Trace of fuzzy total scatter matrix as a function of "m" for an industrial process.

5. RESEARCH ISSUES

In this chapter, we have proposed and discussed a fuzzy-neural system development schema. For this purpose, we have identified three knowledge representation and approximate reasoning approaches. For the Type I fuzzy theory, we have described the extraction of fuzzy sets and fuzzy rules with the application of an improved fuzzy clustering technique called the ETG method, which is essentially an unsupervised learning of the fuzzy sets and rules from a given input-output data set. Next we have described how this set of rules and its fuzzy sets may be adapted and/or modified for known target sets with supervised learning within a fuzzified neural network architecture. Finally, we have introduced a unified (fuzzy) approximate reasoning formulation for fuzzy modeling and control. For this purpose, approximate reasoning parameters are introduced as p, q, [alpha], and [beta]. In particular, we have unified two common but myopic knowledge representation and approximate reasoning methods in our proposed schema.

Unification of myopic reasoning methods still leave a lot to be desired from a theoretical point of view. It is left for future research to develop a unified knowledge representation and reasoning method with the developments of Type II fuzzy set theory. It is conjectured that with developments in Type II fuzzy theory, fuzzy-neural system models will provide much more powerful results for real-life applications. In such a future development, FDNF(·) and FCNF(·) limits of the Type II fuzzy sets will have to be integrated for a unified optimal system modeling and reasoning with it.


FIGURE 2a Fuzzy logic model for an industrial process.


FIGURE 2b Fuzzy logic model for an industrial process.


FIGURE 2c Fuzzy logic model for an industrial process.


FIGURE 2d Fuzzy logic model for an industrial process.


FIGURE 3 Comparison of the combined Mamdani and logical fuzzy model (- - -) and real data (---) of an industrial process.


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