Open access peer-reviewed chapter

# (α, β)−Pythagorean Fuzzy Numbers Descriptor Systems

Written By

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

Reviewed: 12 November 2020 Published: 12 January 2022

DOI: 10.5772/intechopen.95007

From the Edited Volume

## Fuzzy Systems - Theory and Applications

Edited by Constantin Volosencu

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## 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, 2, 3, 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, 6, 7, 8, 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:

1. 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. They can be used to characterize the uncertain information more suffificiently and accurately than intuitionistic fuzzy sets.

2. The membership function and non-membership function of pythagorean fuzzy sets can be easy to be defined. The value ranges of membership function and non-membership function are also more consistent with objective reality and many hesitant problems and people’s thinking.

3. Pythagorean fuzzy sets can ensure that intuitionistic fuzzy sets are all pythagorean fuzzy sets, i.e. intuitionistic fuzzy sets are the special examples of pythagorean fuzzy sets. So intuitionistic fuzzy control systems can be changed into (0,1)-pythagorean fuzzy control systems.

4. (α,β)-pythagorean fuzzy descriptor systems are a broader generalization of T-S fuzzy descriptor systems i.e. T-S fuzzy descriptor systems are the special examples of (α, β)-pythagorean fuzzy descriptor systems.

5. We can judge the degree of weight in the control process according to the value of membership function and non-membership function of the rules. By setting the values of α and β, we decide whether the rules will participate in the final calculation, thereby reducing the calculation process and improving the control efficiency and effectiveness.

6. In fact, (α, β)-pythagorean fuzzy descriptor systems are consistent with the control methods of human being. This method is to imitate the control process of people and also solves the most difficult problem for humans.

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, Rn and Rn×mdenote 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, 2, 3, 4] Let X be a universe of discourse. A PFS P in X is given by.

P=<xμPxνPx>xX,

where μP: X → [0,1] denotes the degree of membership and νP: X → [0,1] denotes the degree of non-membership of the element xX to the set P, respectively, with the condition that 0 ≤ (μP (x))2 + (νP (x))2 ≤ 1. The degree of indeterminacy πP (x) = 1 − (μP (x))2 − (νP (x))2.

For convenience, a pythagorean fuzzy number (μP (x), νP (x)) denoted by p = (μP, νP).

Definition 1.2 [10, 11] T-S fuzzy descriptor systems are as follows:

Rule i: if x1t is F1i and…and xnt is Fni, then.

Eẋt=Axit+Bμit
yt=Cxit+Diμt

Where xt=xt1x2txntTRnand μtRmare the state and control input, respectively; Ai, Bi, Ci and Di are known real constant matrices with appropriate dimension;

Eis a singular matrix; F1i, F2i,, Fni(i=1,2,,r) are the fuzzy sets.

By fuzzy blending, the overall fuzzy model is inferred as follows.

Eẋt=Atxt+Btμt
yt=Ctxt+Dtμt

where

At=i=1rhixtAi,Bt=i=1rhixtBi,Ct=i=1rhixtCi,Dt=i=1rhixtDi,

and hixt is the normalized grade of membership, given as.

hixt=ωixti=1rωxti,ωixt=Πi=1nμijxjt,

which is satisfying

0hixt1,i=1rhixt=1,

μijxjt is the grade of membership function of xjt in Fji.

## 3. αβ−pythagorean fuzzy descriptor systems

As T-S fuzzy descriptor systems are very familiar to us, and pythagorean fuzzy sets are a new tool to deal with vagueness. So we decide to study the new (α, β)-pythagorean fuzzy descriptor systems in order to solve practical control problems more easily and feasible. Next, the related definitions of αβpythagorean fuzzy descriptor systems are gradually given.

Definition 2.1αβpythagorean fuzzy descriptor systems are as follows:

Rule i: if x1t is P1i and…and xnt is Pni, then.

Eẋt=Aixt+BμitE1
yt=Cxit+DiμtE2

where xt=xt1,x2t,,xnt]TRnand μtRmare the state vector and the control input vector, respectively;ytis the measurable output vector; Ai, Bi, Ci and Diare known real constant matrices with appropriate dimension;Eis a singular matrix; P1i,P2i,…,Pni(i=1,2,,r) are all pythagorean fuzzy sets.

By fuzzy blending, the overall fuzzy model is inferred as follows.

Eẋt=Atxt+Btμt
yt=Ctxt+Dtμt

where

At=i=1rhixtAi,Bt=i=1rhixtBi,
Ct=i=1rhixtCi,Dt=i=1rhixtDi,

and hixtis the normalized grade of membership, given as.

hixt=hiαβxti=1rhiαβxt,i=1,2,3,,r;

where

hiαβxt=hi1xtwhen hi1xtαorhi2xtβ0else,α+β1,i=1,2,3,,r;
hi1xt=μPixti=1rμxtPi,hi2xt=νpixti=1rνxtpi,

where hi1xt and hi2xt are respectively positive and negative membership functions.

i=1rhi1xt=1,i=1rhi2xt=1;
μPixjt=j=1rμPjixjt,νPixjt=j=1rνPjixjt;

μPjixjtand νPjixjtis the membership function value of xjtthat belongs and does not belong to the intuitionistic fuzzy numbers set Pji.

Remark 2.1:

1. We can judge the degree of weight in the control process according to the value of the positive and negative membership functions of the rules. By setting the values of α and β, we decide whether the rules will participate in the final calculation, thereby reducing the calculation process and improving the control efficiency and effectiveness.

2. In fact, (α,β)-pythagorean fuzzy descriptor systems are consistent with the control methods of human being. People generally proceed appropriate control at one point by the past experience, i.e. people’s decisions are decided and implemented at roughly one point. This method is to imitate the control process of people

3. The relations between (α,β)-pythagorean fuzzy descriptor systems and T-S fuzzy descriptor systems

Firstly, the relation of T-S fuzzy descriptor systems and (α,β)-pythagorean fuzzy descriptor systems is studied through an example.

When α=0,β=1, then

hixt=hiαβxt=hi1xt=μiMxti=1rμiMxt,hi2xt=0,μiMxt=j=1nμijMxjt.

Then the special (0,1)-pythagorean fuzzy descriptor systems are T-S fuzzy descriptor systems. In other words, T-S fuzzy descriptor systems are all the special (0,1)-pythagorean fuzzy descriptor systems. Therefore, it is easy to get the following Theorem 3.1.

Theorem 3.1 T-S fuzzy descriptor systems are all the (α,β)-pythagorean fuzzy descriptor systems.

Proof:It is so easy, so omit.

## 4. αβ−pythagorean fuzzy numbers controller

Now we continue to study the feedback control and stability of pythagorean fuzzy descriptor systems according to the traditional research path of the control systems.

Suppose.

Rule i: if x1t is P1ix1t and … and xnt is Pnixnt, then.

uxt=i=1rhixt)GixtE3

where Gi(i = 1,2,…, r) are the state feedback-gains matrices.

hixt=hiαβxti=1rhiαβxt,i=1,2,3,,r;

where

hiαβxt=hi1xtwhenhi1xtαorhi2xtβ0else,α+β1,i=1,2,3,,r;
hi1xt=μPixti=1rμxtPi,hi2xt=νpixti=1rνxtpi,

where hi1xt and hi2xtare respectively positive and negative membership functions.

i=1rhi1xt=1,i=1rhi2xt=1;
μPixjt=j=1rμPjixjt,νPixjt=j=1rνPjixjt;

μPjixjt and νPjixjt is the membership function value of xjt that belongs and does not belong to the intuitionistic fuzzy numbers set Pji.

If we take (3) into (1, 2), we can get.

Eẋt=i=1rj=1rhixthjxtAi+BiGjxtE4
yt=i=1rj=1rhixthjxtCi+DiGjxtE5

The system stability is guaranteed by determining the feedback gains Gj.

Basic LMI-based stability conditions guaranteeing the stability of the above control system in the form of (4, 5) are given in the following theorem.

Theorem 4.1 The system (3) is asymptotically stable, if there exist matrices NjRm×n (j = 1,2,3,…, r) and K=KTRn×n such that the following LMIs are satisfied:

K>0E6
ETK=KTE0E7
Qij=AiK1+K1AiT+BiNj+NjTBiT<0i,jE8

where the feedback gains are defined as Gj=NjK for all j.

Proof: Considering the quadratic Lyapunov function.

Vxt=xTtETKxt,

where 0<K=KTRn×n.

then

V̇xt=ẋTtETKxt+xTtETKẋt=EẋtTKxt+xTtKTEẋt=i=1rj=1rhihjxTtKK1ATiK+KTNjTBiTK+KTAi+KTBiNjK}K1Kxt,

let Z = Kx(t), then

V̇xt=i=1rj=1rhihjxTtKK1ATiK+KNjTBiTK+KAi+KBiNjK}K1Kxt=i=1rj=1rhihjZK1ATi+NjTBiT+AiK1+BiNjZ.

As Qij=AiK1+K1AiT+BiNj+NjTBiT<0, so the system (3) is asymptotically stable.

## 5. Simulation example

Example 5.1: Considering an inverted pendulum, subject to parameter uncertainties [12, 13, 14, 15] as the nonlinear plant to be controlled. The dynamic equation for the inverted pendulum is given by.

θ¨t=gsinθtampLθ̇t2sin2θt/2acosθtμt4L/3ampLcos2θt

Where θtis the angular displacement of the pendulum, g = 9.8 m/s2 is the acceleration due to gravity, mp[mpmin,mpmax] = [2,3]kg is the mass of the pendulum, McMminMmax= [8, 12].

Kg is the mass of the cart, a=1/mp+Mc, 2 L = 1 m is the length of the pendulum, and utis the force (in newtons) applied to the cart. The inverted pendulum is considered working in the operating domain characterized by x1=θt5π/125π/12 and x2=θ̇t[−5,5].

Rule 1: If x1t is M11, x2t is M21, then

ẋ1tẋ2t=0110.00780x1tx2t+00.1765μt;

Rule 2: If x1t is M12, x2t is M22, then

ẋ1tẋ2t=0110.00780x1tx2t+00.0261μt;

Rule 3: If x1t is M13, x2t is M23, then

ẋ1tẋ2t=0118.48000x1tx2t+00.1765μt;

Rule 4: If x1t is M14, x2t is M24, then

ẋ1tẋ2t=0118.48000x1tx2t+00.0261μt;

Next, according to the ideas based on the principles of interpolation and interval coverage, we firstly change the interval-valued T-S fuzzy model of inverted pendulum into the special (α,β)- pythagorean fuzzy descriptor systems of inverted pendulum as follows.

Rule 1: If x1t is P11x1t, x2t is P21x2t, then

ẋ1tẋ2t=0110.00780x1tx2t+00.1765μt;

Rule 2: If x1k is P12x1t, x2t is P22x2t, then

ẋ1tẋ2t=0110.00780x1tx2t+00.0261μt;

Rule 3: If x1t is P13x1k, x2t is P23x2t, then

ẋ1tẋ2t=0118.48000x1tx2t+00.1765μt;

Rule 4: If x1t is P14x1k, x2t is P24x2t, then

ẋ1tẋ2t=0118.48000x1tx2t+00.0261μt;

According to the theorem 4.1, we can get.

K=1/71/71/78/7,K1=8111,N1=N3=100100,N2=N4=10001000,

So the above (α,β)-pythagorean fuzzy descriptor systems of inverted pendulum is asymptotically stable, and the state feedback-gains matrices G1=G3=0100, G2=G4=01000.

The first case, suppose x10=11π29,x20=0.88,α=0.3,β=0.25,then take the variable x1t as the main factor of the control, and according to Table 2 we can control in three steps, i.e. x10=11π29x1t1x1t20 and 0<t1t2.

When x10=11π29,x20=0.88, and α=0.30,β=0.25, according to Table 2 we can get μP11x10 = μP12x10 = 0.69, νP11x10 = νP12x10 = 0.72,μP13x10 = μP14x10 = 0, νP13x10 = νP14x10 = 1, μP21x20 = μP22x20 = 0.02,νP21x20 = νP22x20 = 1, μP23x20 = μP24x20 = 0.40, νP23x20 = νP24x20 = 0.92, noteworthy, μP11x10+νP11x10 = μP12x10+νP12x10 = 1.41 > 1, μP23x20 + νP23x20 = μP24x20+νP24x20 = 1.32 > 1. Then according to Definition 2.1, taking it one step further, we can get h11 = 0.49, h12 = 0.22, h21 = 0.49, h22 = 0.22, h31 = 0.01, h32 = 0.28, h41 = 0.01, h42 = 0.28, so h1 = 0.5, h2 = 0.5, h3 = 0, h4 = 0, according to 4.2, so the overall fuzzy model of the (0.30,0.25)- pythagorean fuzzy descriptor systems is.

ẋ1tẋ2t=00.252.501913.929x1tx2t,

The solution of the systems is xt=1129π+0.25t0.21720.045t1.0972e13.93t

When x140.19103,x240.0372, and α=0.30,β=0.25, according to Table 2, we can get μP11x14 = μP12x14 = 0.03, νP11x14 = νP12x14 = 1,μP13x14 = μP14x14 = 0.20, νP13x14 = νP14x14 = 0.98, μP21x24 = μP22x24 = 0.50, νP21x24 = νP22x24 = 0.87, μP23x24 = μP24x24 = 0, νP23x24 = νP24x24 = 1, noteworthy, μP11x14+νP11x14 = μP12x14 + νP12x14 = 1.03 > 1, μP21x24+νP21x24 = μP22x24+νP22x24 = 1.37 > 1.Then according to Definition 2.1, taking it one step further, we can get h11 = 0.49, h12 = 0.23, h21 = 0.49, h22 = 0.23, h31 = 0.01, h32 = 0.27, h41 = 0.01, h42 = 0.27, so h1 = 0.50, h2 = 0.50, h3 = 0, h4 = 0, then according to 4.2, so the overall fuzzy model of the (0.3,0.3)- pythagorean fuzzy descriptor systems is.

ẋ1t4ẋ2t4=00.252.501913.929x1t4x2t4,

The solution of the systems is xt=0.19103+0.25t0.037540.045t0.00034e13.93t

When x14.7640.0000344, x24.7640.00323, so the overall fuzzy model of the (0.30, 0.25)-pythagorean fuzzy descriptor systems is E-asymptotic stability. But it takes a shorter time (Figure 1).

The second case(interval-valued T-S fuzzy model of inverted pendulum), suppose x10=11π29,x20=0.88, then take the variable x1t as the main factor of the control, and according to Table 1 we can control in three steps, i.e. x10=11π29x11x1t0.

Left membership functionsRight membership functions
MM111x1=1ex121.2MM113x1=10.23ex120.25
MM121x1=1ex121.2MM123x1=10.23ex120.25
MM131x1=0.23ex120.25MM133x1=ex121.2
MM141x1=0.23ex120.25MM143x1=ex121.2
MM211x2=0.5ex220.25MM213x2=ex221.5
MM221x2=1ex221.5MM223x2=10.5ex220.25
MM231x2=0.5ex220.25MM233x2=ex221.5
MM241x2=1ex221.5MM243x2=10.5ex220.25

### Table 1.

The membership functions of the IT-2 T-S fuzzy model of inverted pendulum.

Membership functionsNon-membership functions
μP11x1=1ex121.2νP11x1=11ex121.22
μP12x1=1ex121.2νP12x1=11ex121.22
μP13x1=0.23ex120.25νP13x1=10.23ex120.252
μP14x1=0.23ex120.25νP14x1=10.23ex120.252
μP21x2=0.5ex220.25νP21x2=10.5ex220.252
μP22x2=1ex221.5νP22x2=11ex221.52
μP23x2=0.5ex220.25νP23x2=10.5ex220.252
μP24x2=1ex221.5νP24x2=11ex221.52

### Table 2.

The membership functions and non-membership functions of (α,β)-pythagorean fuzzy descriptor systems of inverted pendulum.

When x10=11π2911π290, x20=0.880.880, and λ1=λ2=12, according to Table 1, we can get h1 = 0.26, h2 = 0.58, h3 = 0.05, h4 = 0.11, according to theorem 4.1, so the overall interval-valued fuzzy model of the interval-valued fuzzy descriptor systems is

ẋ1tẋ2t=0148.56159.925x1tx2t,

The solution of the systems is xt=1.2229e0.82t+0.0319e59.1t1.0048e0.82t1.8848e59.1t;

When x11=0.53860.53860, x21=0.442500.4425, and λ1=λ2=12, according to Table 1, we can get h1 = 0.32, h2 = 0.25, h3 = 0.24, h4 = 0.19, according to 4.2, the overall interval-valued fuzzy model of the interval-valued fuzzy descriptor systems is

ẋ1tẋ2t=0144.4958.141x1tx2t,

The solution of the systems is xt=0.4654e0.78t0.0732e57.37t0.4174e0.78t+0.0251e57.37t;

When x19.60.0006, x29.60.0005, so the IT2 T-S fuzzy descriptor system of the inverted pendulum will to be stable too.

Thus the stable control time of the (0.30,0.25)-pythagorean fuzzy descriptor systems of inverted pendulum is 4.836 second shorter than the interval-valued T-S fuzzy descriptor systems of the inverted pendulum (Figure 2).

Remark 5.1: In this way, the (0.30,0.25)-pythagorean fuzzy descriptor systems can get the better effect than the control effect of interval-valued T-S fuzzy model of inverted pendulum. It is easy to see that the (0.30,0.25)-pythagorean fuzzy descriptor systems has the best control, and can reduce the number of rules and thus reduce the amount of calculations.

In this way, it can get the better effect than the control effect of interval-valued T-S fuzzy model of inverted pendulum. Because the feedback more or less needs a little time, when the system carries out feedback instructions, but the time has gone, so the feedback that have been given are also lagging and out of date. αβ pythagorean fuzzy descriptor systems can be closer to the actual, and easy to control the error range. The new control method is more convenient and feasible!

## 6. Conclusions

In this paper, the new αβpythagorean fuzzy descriptor systemsare firstly introduced, and more consistent with the human way of thinking and more likely to be set up and more convenient for popularization. The newαβpythagorean fuzzy descriptor systems is very simply and quickly. We can do not know the control principle, but we can directly achieve good control effect. The new theory can be studied in parallel to the basic framework of the original theories and easy to promote the old theories and achieve good results. In addition, we can judge the degree of weight in the control process according to the value of the positive and negative membership functions of the rules. By setting the values ofαandβ, we decide whether the rules will participate in the final calculation, thereby reducing the number of the rules and the calculation process, and improving the control efficiency and effectiveness. Otherwise, T-S fuzzy descriptor systems are the special examples of αβpythagorean fuzzy descriptor systems. αβpythagorean fuzzy controller and the stability ofαβpythagorean fuzzy descriptor systems are deeply researched. At last, a numbers example is given to show the corollaries are corrected.

But the theoretical part of the new systems need to be in-depth studied, and specific applications are also to be further developed. For example, αβ pythagorean fuzzy descriptor systems can also be used as the model of autonomous learning in order to establish intelligent control, and can be used well in unmanned driving in the future. Soαβpythagorean fuzzy descriptor systems is just to meet the reality requirements.

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Written By

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

Reviewed: 12 November 2020 Published: 12 January 2022