*3.2. Structure Design of SIFOFLC*

A fuzzy logic controller has four components: knowledge base, inference engine, fuzzification interface, and defuzzification interface [26]. The basic structure of a twodimensional fuzzy logic controller is described as in Figure 2, where the continuous input signal, *<sup>e</sup>* and . *e*, convert to the membership degree vector of the fuzzy variables through a fuzzification interface, the inference engine carries out rule inference and actual output signal is obtained through defuzzification interface. The data base denotes membership functions of the total input and output variables and the rule base is performed using a collection of fuzzy if–then rules by expert experience, both of which make up the knowledge base.

**Figure 2.** Block diagram of two-dimensional FLC.

Compared with the linear controller, fuzzy logic control method is more robust and suitable for complex control requirements, while its complex decision-making process brings a challenge for real-time operation. Thus, we aim to adjust the controller structure to reduce computation burden and achieve better performance.

Typically, a fuzzy logic controller has two control inputs, namely error (*e*) and its derivative ( . *e*). It is common for its rule table to have the same output membership in a diagonal direction, something known as the Toeplitz structure, as shown in Table 1 [4]. In addition, each position on a diagonal line has the same distance from the main diagonal line of rule table. Thus, instead of using two-variable input sets (*e*, . *e*), the corresponding control output can be obtained using the distance between input signal and the main diagonal line. This finding was first proposed by Choi et al. and is known as the signed distance method [21]. To derive the distance, *<sup>d</sup>*, a two-dimensional space of *<sup>e</sup>* and . *e* is established as shown in Figure 3.

**Table 1.** Rule table with the Toeplitz structure.


**Figure 3.** Distance variable.

The main diagonal line of the rule table is presented as a straight line crossing over the origin, whose function is *Lz* : . *e* + *λe* = 0. In this case, the distance from point *P*(*e*1, . *e*1) to *Lz* can be obtained as *<sup>d</sup>* <sup>=</sup> . *<sup>e</sup>*1+*λ<sup>e</sup>* <sup>√</sup> <sup>1</sup> <sup>1</sup>+*λ*<sup>2</sup> .

Furthermore, in order to achieve better control performance, we extend the derivative of error signal to fractional order, thus the distance variable is described as

$$d = \frac{D\_t^\kappa \varepsilon + \lambda \varepsilon}{\sqrt{1 + \lambda^2}}\tag{9}$$

where *α* is the fractional order of error signal and it further increases degrees of freedom and complexity of the controller.

Based on the above analysis, the overall structure of SIFLC can be depicted as in Figure 4. The distance variable is obtained through linear combination of error signal and its fractional derivative, the inner fuzzy logic controller is single input single output (SISO) and the final output *u* is obtained by multiplying *u*<sup>0</sup> with the scale factor, which is denoted as *r*. We set *λ* to 1. Thus, in addition to the membership functions, there are two adjustable parameters in the SIFOFLC, namely *α* and *r*.

**Figure 4.** The SIFLC control structure.
