*3.3. Characteristics of SIFOFLC*

In this section, the SIFOFLC and T-S FLC used in the simulation research of Section 4 are proposed and the superiority of SIFOFLC is presented through comparison.

To the SIFOFLC, the fuzzy sets of *d* and *u*<sup>0</sup> are both {NL, NM, NS, Z, PS, PM, PL} and the membership functions are shown in Figure 5. The membership functions of *d* include both S shape and triangular shape membership functions, with a singleton value membership function for *u*0. In this case, the rule table is reduced to the one-dimensional vector as shown in Table 2.

**Figure 5.** (**a**) Input membership function. (**b**) Output membership function.

**Table 2.** Rule table for the above SIFOFLC.


Based on the work mentioned above, the proposed SIFOFLC has the following advantages compared to conventional FLC:

(1) Simplified design process.

For the two-dimensional FLC, the membership functions for error input and its derivative are required simultaneously, which means a lengthy complex tuning process. With the only input, *d*, the parameter tuning process for SIFOFLC is significantly reduced. Further, in a two-dimensional FLC, the number of fuzzy rules to be inferred is the square of n, which is the size of the fuzzy set. The distinguishing feature of SIFLC is that it requires only n rules, which is an exponential decay. Typically, better performance can be obtained with more a complicated control algorithm, such as the increase of fuzzy sets and rules, which further reveals the superiority of SIFOFLC.

(2) Reduced computation burden.

The control surfaces for the SIFOFLC and conventional T-S FLC are respectively shown in Figures 6 and 7. Compared with the complex curved surface of T-S FLC, the control surface of SIFOFLC has been simplified to a linear with different slopes. Thus, the computation burden of controller operation, which includes fuzzification, fuzzy inference and defuzzification, has been significantly reduced.

**Figure 6.** T-S FLC control surface.

**Figure 7.** SIFLC control surface.
