*3.3. Fuzzy Inference*

In control, it is common practice to follow the convention shown in Table 1 of [10] or Table 2 of [40] for fuzzy rule table construction, where an equal number of input and output linguistic values exist in each set *O* ¯ *i* and *Z* ¯ . In this paper, however, this convention is not followed since the input linguistic value set contains five elements, while the output linguistic value set consists of three elements. This is done to reduce the number of decision variables in the optimization problem and speed up the overall computation. Table 1 represents the fuzzy rule table that is used in this paper. Abbreviations are used to represent the linguistic values: *N* for *Negative*, *P* for *Positive*, *Z* for *Zero*, *S* for *Small*, and *B* for *Big*. Rules are interpreted as follows: "if Speed Error is Negative-Big and Speed Error Derivative is Negative-Big, then Fuzzy Error is Negative", and so on.


**Table 1.** Fuzzy Rules.

**Table 2.** Optimization search area.


Using Table 1, it is possible to calculate the rule firing strength matrix *W*, shown in Equation (16), whose elements are calculated using Equation (17). This matrix is used to obtain the output control value in the defuzzification process.

$$\mathbf{W} = \begin{bmatrix} w\_{11} & \dots & w\_{1m} \\ \vdots & \ddots & \vdots \\ w\_{n1} & \dots & w\_{nm} \end{bmatrix} \tag{16}$$

$$w\_{nm} = \min(\mu\_{\bullet\_1^n}(o\_1), \mu\_{\bullet\_2^m}(o\_2)), \quad n, m \in [1, 5] \tag{17}$$
