(α, β)Pythagorean Fuzzy Numbers Descriptor Systems

*Chuan-qiang Fan, Wei-he Xie and Feng Liu*

### **Abstract**

By using pythagorean fuzzy sets and T-S fuzzy descriptor systems, the new (*α*, *β*)-pythagorean fuzzy descriptor systems are proposed in this paper. Their definition is given firstly, and the stability of this kind of systems is studied, the relation of (*α*, *β*)-pythagorean fuzzy descriptor systems and T-S fuzzy descriptor systems is discussed. The (*α*, *β*)-pythagorean fuzzy controller and the stability of (*α*, *β*)-pythagorean fuzzy descriptor systems are deeply researched. The (*α*, *β*)-pythagorean fuzzy descriptor systems can be better used to solve the problems of actual nonlinear control. The (*α*, *β*)-pythagorean fuzzy descriptor systems will be a new research direction, and will become a universal method to solve practical problems. Finally, an example is given to illustrate effectiveness of the proposed method.

**Keywords:** Pythagorean fuzzy sets, T-S fuzzy descriptor systems, stability

#### **1. Introduction**

Pythagorean fuzzy sets [1–4] were proposed by Yager in 2013, are a new tool to deal with vagueness. Pythagorean fuzzy sets maintain the advantages of both membership and non-membership, but the value range of membership function and non-membership function is expanded from triangle to quarter circle. The expansion of the value area makes the amount of information of pythagorean fuzzy sets expand 1.57 times that of the intuitionistic fuzzy sets, and ensures that intuitionistic fuzzy sets are all pythagorean fuzzy sets. They can be used to characterize the uncertain information more sufficiently and accurately than intuitionistic fuzzy sets. Pythagorean fuzzy sets have attracted great attention of a great many scholars that have been extended to new fields and these extensions have been used in many areas such as decision making, aggregation operators, and information measures. Due to theirs wide scope of description cases are very common in diverse real-life issue, pythagorean fuzzy sets have given a boost to the management of vagueness caused by fuzzy scope. Pythagorean fuzzy sets have provided two novel algorithms in decision making problems under Pythagorean fuzzy environment.

Takagi-Sugeno (T-S) fuzzy systems [5–9] has been applied on intelligent computing research and complex nonlinear systems. T-S fuzzy systems have also been extended to new fields and these extensions have been used in many areas by a great many scholars. However, the membership functions of T-S fuzzy systems cannot make full use of the all uncertain message in the premise conditions. So we decide to study the new (*α*,*β*)-pythagorean fuzzy descriptor systems in order to solve practical control problems more easily and feasible.

The advantages of (*α*, *β*)-pythagorean fuzzy descriptor systems are the following:


The rest of this paper is organized as follows: In Section 1, the basic concepts of T-S fuzzy descriptor systems are introduced. In Section 2, (*α*,*β*)-pythagorean fuzzy descriptor systems are firstly proposed. Then the relationship of T-S fuzzy descriptor systems and (*α*,*β*)-pythagorean fuzzy descriptor systems are discussed in Section 3. (*α*,*β*)-pythagorean fuzzy controller and the stability of (*α*,*β*)-pythagorean fuzzy descriptor systemsare deeply researched in Section 4. In Section 5, a numbers examples is given to show the corollaries are corrected. We discussed in detail the effects of controls in several cases. Through this practical example, we find that the selection of pythagorean fuzzy membership functions in the premise conditions of the rules has a great influence on the control effect. Therefore, the choice of pythagorean fuzzy membership functions must be determined after more tests, and we can not completely believe the original given functions. Finally, the conclusion is given in Section 6**.**

Notations: Throughout this paper, *R<sup>n</sup>* and *R<sup>n</sup><sup>m</sup>*denote respectively the *n* dimensional Euclidean space and *n m* dimensional Euclidean space. PFS denotes pythagorean fuzzy set.

### **2. Preliminaries**

This section will briefly introduce some baisc definitions and theorems on pythagorean fuzzy sets and T-S fuzzy descriptor systems.

**Definition 1.1** [1–4] Let *X* be a universe of discourse. A PFS *P* in *X* is given by.

$$P = \{ <\infty, \mu\_P \: (\varkappa), \nu \rho \ (\varkappa) > | \: \varkappa \in X \},$$

where *μP*: *X* ! [0,1] denotes the degree of membership and *νP*: *X* ! [0,1] denotes the degree of non-membership of the element *x* ∈ *X* to the set *P*, respectively, with the condition that 0 ≤ (*μ<sup>P</sup>* (*x*))<sup>2</sup> + (*ν<sup>P</sup>* (*x*))<sup>2</sup> ≤ 1. The degree of indeterminacy *<sup>π</sup><sup>P</sup>* (*x*)=1 � (*μ<sup>P</sup>* (*x*))<sup>2</sup> � (*ν<sup>P</sup>* (*x*))<sup>2</sup> .

For convenience, a pythagorean fuzzy number (*μ<sup>P</sup>* (*x*), *ν<sup>P</sup>* (*x*)) denoted by *p* = (*μP*, *νP*).

**Definition 1.2** [10, 11] T-S fuzzy descriptor systems are as follows: Rule *<sup>i</sup>*: if *<sup>x</sup>*1ð Þ*<sup>t</sup>* is *<sup>F</sup><sup>i</sup>* <sup>1</sup> and...and *xn*ð Þ*<sup>t</sup>* is *<sup>F</sup><sup>i</sup> <sup>n</sup>*, then.

$$E\dot{x}(t) = A\_i x(t) + B\_i \mu(t)$$

$$\mathcal{y}(t) = C\_i x(t) + D\_i \mu(t)$$

Where *x t*ðÞ¼ *<sup>x</sup>*1ð Þ*<sup>t</sup>* , *<sup>x</sup>*2ð Þ*<sup>t</sup>* , <sup>⋯</sup>, *xn*ð Þ*<sup>t</sup>* � �*<sup>T</sup>* <sup>∈</sup>*R<sup>n</sup>*and *<sup>μ</sup>*ð Þ*<sup>t</sup>* <sup>∈</sup> *Rm*are the state and control input, respectively; *Ai*, *Bi*, *Ci* and *Di* are known real constant matrices with appropriate dimension;

*E*is a singular matrix; *F<sup>i</sup>* 1, *F<sup>i</sup>* 2,⋯, *F<sup>i</sup> <sup>n</sup>*(*i* ¼ 1, 2, … ,*r*) are the fuzzy sets. By fuzzy blending, the overall fuzzy model is inferred as follows.

$$E\dot{\mathbf{x}}(t) = A(t)\mathbf{x}(t) + B(t)\mu(t)$$

$$\mathbf{y}(t) = \mathbf{C}(t)\mathbf{x}(t) + D(t)\mu(t)$$

where

$$\begin{aligned} A(t) &= \sum\_{i=1}^r h\_i(\mathbf{x}(t)) A\_i, B(t) = \sum\_{i=1}^r h\_i(\mathbf{x}(t)) \\ B\_i, \mathbf{C}(t) &= \sum\_{i=1}^r h\_i(\mathbf{x}(t)) \mathbf{C}\_i, D(t) = \sum\_{i=1}^r h\_i(\mathbf{x}(t)) D\_i, \end{aligned}$$

and *hi*ð Þ *x t*ð Þ is the normalized grade of membership, given as.

$$h\_i(\mathbf{x}(t)) = \frac{o\_i(\mathbf{x}(t))}{\sum\_{i=1}^r o\_i(\mathbf{x}(t))},\\ o\_i(\mathbf{x}(t)) = \Pi\_{i=1}^n \mu\_{ij}(\mathbf{x}\_j(t)),$$

which is satisfying

$$\mathbf{0} \le h\_i(\mathbf{x}(t)) \le \mathbf{1}, \sum\_{i=1}^r h\_i(\mathbf{x}(t)) = \mathbf{1}, \mathbf{0}$$

*<sup>μ</sup>ij <sup>x</sup> <sup>j</sup>*ð Þ*<sup>t</sup>* � � is the grade of membership function of *<sup>x</sup> <sup>j</sup>*ð Þ*<sup>t</sup>* in *<sup>F</sup><sup>i</sup> j* .
