# A Reconstruction Procedure Associated with Switching Lyapunov Function for Relaxing Stability Assurance of T-S Fuzzy Mode

Authors: Yau-Tarng Juang; Chih-Peng Huang; Chung-Lin Yan
Journal: Mathematical Problems in Engineering
 (2015)

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## Abstract

This paper proposes a novel reconstruction procedure to lessen the conservatism of stability assurance of T-S Fuzzy Mode. By dividing the state variables into some bounded regions, the considered T-S fuzzy model can be first transferred to an alternative form via a reconstructing procedure. Thus, we can attain some relaxing stability criteria based on the switching quadratic Lyapunov function (SQLF) method. Notably, these proposed conditions are explicitly formulated by linear matrix inequality (LMI) form and can handily be evaluated by current software tools. Finally some illustrative examples are given to experimentally demonstrate the validity and merit of the proposed method.


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## Body

## 1. Introduction

A class of empirical control architecture, fuzzy logic control (FLC), is originally proposed in the past two decades [1]. And, it then has become one of the most active and fruitful areas in this research community. Many researchers have dedicated a lot of time and great effort to both theoretical research and implementation techniques for FLC. The basic idea behind FLC is to incorporate the “expert experience” of a human operator into a system. Notably, a complicated dynamic model is not necessary any more. However, model-free type of fuzzy control is useful and practical, but this control technique is insufficient of mathematical support. In order to systematically analyze the stability of the fuzzy system, Takagi-Sugeno (T-S) fuzzy model [2, 3] was presented to open a different viewpoint from the traditional fuzzy system. Based on the T-S fuzzy model, a complicated system can be represented as a set of fuzzy IF-THEN rules, each of which represents the local linear subsystem in a different state-space region. By the fuzzy blending operating, they thus can approximately represent some uncertain or/and nonlinear systems [4–8].The stability issues of T-S fuzzy model were first analyzed by Tanaka and Sugeno [3]. Based on Lyapunov direct method, they presented some sufficient conditions that could systematically achieve the stability analysis and stabilization of fuzzy systems [9, 10]. Moreover, the stability conditions could be cast into linear matrix inequality (LMI) forms [1, 10–13]. For earlier research, most works are devoted to common quadratic Lyapunov function (CQLF) methods. Thus, the proposed criteria require finding a common positive definite matrix P of a CQLF to satisfy all the subsystems [3, 14–19]. But, if the number of fuzzy rules is large, a common matrix P may hardly be obtained or does not exist in all the subsystems. Instead of CQLF, a switching quadratic Lyapunov function (SQLF) was involved [20–24]. This SQLF method beforehand divides the state space into several regions, and, in each region we can try to find the individual positive-definite matrix Pi for the SQLF. Furthermore, we need to guarantee the SQLF Vixt be the same in the region boundary but did not expect that all Pi are equal. In previous works, the stability criteria need to involve bilinear matrix inequality (BMI) conditions, which cannot be directly evaluated by traditional LMI solver [25].In this work, based on SQLF method, we devote the stability criteria of T-S fuzzy model to the relaxing. Firstly, we use a distinct region-division approach for the state space. When assuming that the state variables of the considered T-S fuzzy model can be correspondingly formulated by single membership function at the region boundary, we thus propose a reconstructing procedure for substituting. Thus, based on the SQLF method, more relaxed stability criteria can be derived and expressed by LMI manner. Finally, an illustrative example is given to verify the efficiency and superiority of the proposed method.


## 2. Problem Description and Preliminaries

T-S fuzzy model can approximately represent some intractable systems with parametric uncertainty or nonlinearity via the fuzzy IF-THEN rules interpretation. This mode mainly characterizes the local dynamics behaviors of a system by the fuzzy rules where their consequent parts are represented by some locally linear models. Thus, an overall system can be achieved via the fuzzy “blending” for these linear models. Ther rules of a T-S fuzzy model are expressed as follows:(1)ModelRulei—MRi:IFx1tisMi1andx2tisMi2⋯andxntisMin,THENx˙t=Aixti=1,2,…,r,where MRi denotes the model rule i; Mij denotes the fuzzy set corresponding to the state variable xjt of MRi; r is the number of IF-THEN fuzzy model rules; xt=x1tx2t⋯xntT∈Rn is the state vector; Ai∈Rn×n denotes the system matrix of the subsystem.The T-S fuzzy model in (1) is assumed to be inferred by center average defuzzification (CAD) and can be described by(2)x˙t=∑i=1rwixtAixt∑i=1rwixt=∑i=1rhixtAixt,where(3)wixt=∏j=1nMijxjt,∑i=1rwixt>0,wixt≥0,i=1,2,…,r.Mij(xj(t)) is the grade of membership of xjt in membership function Mij. And, the inferred grade function is normalized as(4)hixt=wixt∑i=1rwixt,∑i=1rhixt=1,hixt≥0,i=1,2,…,r.Based on Lyapunov stability theory, a candidate Lyapunov function with a constant matrixP is given by (5)Vxt=xTtPxtwith P>0. Thus, for stability assurance, the chosen Lyapunov function needs to satisfy the following four conditions:(1)
V is C1,(2)
xt=0⇔Vxt=0,(3)
xt≠0⇔Vxt>0,(4)
xt→∞⇒Vxt→∞,where C1 is the set of all real-valued functions that have a continuous first derivative on -∞,∞ and · denotes the Euclidean norm for the considered vectors.Based on (5), Tanaka and Sugeno [3] originally proposed a stability criterion as follows.Lemma 1 (see [3]).
The equilibrium of the autonomous T-S fuzzy system in (2) is asymptotically stable in the large, if there exists a common positive-definite symmetric matrix P such that(6)AiTP+PAi<0,i=1,2,…,r.Remark 2.
Lemma1 shows that we need to find a common matrix P to satisfy all the conditions in (6) for global stability assurance. Based on Lemma 1, some less conservative results [12, 16] with a common positive-definite symmetric matrix P were achieved by involving some extra slack matrices in criteria.


## 3. Switching T-S Fuzzy Model and Stability Issues

### 3.1. Switching T-S Fuzzy Model

Consider(7)RegionModelRuleq—RMRq:IFxt∈RegionqTHENLocalModelRulei—LMRi:IFx1tisMqi1andx2tisMqi2⋯andxntisMqin,THENx˙t=Aqixt+Βqiuti=1,2,…,rq,q=1,2,…,2n,where RMRq denotes the region model rule q. LMRi denotes the local model rule i. Mqij denotes the fuzzy set corresponding to the state variable xjt of LMRi in Region q. rq is the number of IF-THEN fuzzy local model rules in Region q. 2n is the number of region. n is the number of state variables. r=∑q=12nrq is the total number of IF-THEN fuzzy model rules. Aqi∈Rn×n denotes the system matrix of the subsystem. Bqi∈Rn×m denotes the input matrix of the subsystem.Each linear consequent equation,Aqixt+Βqiut, is called a subsystem.Given a pair ofxt,ut, the final output of the T-S fuzzy model (7) is inferred by CAD and can be represented as follows [23]: (8)x˙t=∑q=12n∑i=1rqηqxthqixtAqixt+Βqiut,where(9)ηqxt=1,xt∈Rq0,xt∉Rq.hqixt is the normalized membership function of the LMRi in Region q and satisfies(10)∑q=12n∑i=1rqηqxthqixt=1,∑q=12n∑i=1rqηqxthqixt≥0,i=1,2,…,rq.In theqth region, we define(11)Rqs1q,s2q,…,snqskq=1,xkt≥00,xkt<0k=1,2,…,n,where the number of the state variables is equal to n and its corresponding state space can be divided into 2n region. For clarification, assume that the considered state space has three dimensions and each separated region corresponds to a quadrant. Figure 1 depicts that the trajectory of a given initial state passes over the different quadrants in a three-dimensional space. The state in Figure 1 passes through Region 2, Region 1, Region 5, Region 7, Region 3, Region 4, and Region 2 in turn. To guarantee the stability of T-S fuzzy system, we suppose that the trajectory of a system cannot be motionless on the region boundaries.Figure 1
Divided state space and change of state.


### 3.2. Switching Quadratic Lyapunov Function

Consider(12)Vxt=xTtP1xt,xt∈Region1xTtP2xt,xt∈Region2⋮⋮xTtP2nxt,xt∈Region2n,where Pq∈Rn×n∀q are a set of positive-definite symmetric matrices. This function is continuous on region boundaries if Pq satisfies the following constraint [23]:(13)Pq=p11ζ1,qp12ψ1,2,q⋯p1nψ1,n,qp12ψ1,2,qp22ζ2,q⋯p2nψ2,n,q⋮⋮⋱⋮p1nψ1,n,qp2nψ2,n,q⋯pnnζn,q,ζk,q=2-skq,ψk,l,q=-skq-2slq+4.Since the involved property is crucial for discussing the SQLF, an example is illustrated as follows.Example 3 (see [23]).
As in Figure1, consider a system with three state variables x1t, x2t, and x3t. The state space can be divided into the following eight regions, eight quadrants:(14)R11,1,1,R20,1,1,R31,0,1,R40,0,1;R51,1,0,R60,1,0,R71,0,0,R80,0,0.Based on the notation of a region, that is,Rq(s1q,s2q,…,snq), each Pq is represented as follows:(15)P1=p111p121p131p121p221p231p131p231p331,P2=p112p122p132p122p221p231p132p231p331,P3=p111p123p131p123p222p232p131p232p331,P4=p112p124p132p124p222p232p132p232p331,P5=p111p121p133p121p221p233p133p233p332,P6=p112p122p134p122p221p233p134p233p332,P7=p111p123p133p123p222p234p133p234p332,P8=p112p124p134p124p222p234p134p234p332.On the region boundary between Regions 5 and Regions 7, a plane A in Figure1 for xt=x1t0x3tT, the SQLF is calculated as(16)Vxt=xTtP5xt=x1t0x3tp111p121p133p121p221p233p133p233p332x1t0x3t=p111x12t+2p133x1tx3t+p332x32t=x1t0x3tp111p123p133p123p222p234p133p234p332x1t0x3t=xTtP7xt.Furthermore, the illustrative results can be similarly extended to the other region boundaries of x(t).Remark 4.
Based on the SQLF, (13), we have shown that the state trajectory crossed a plane; that is, xt=x1t0x3tT is continuous. In addition, it is also continuous when the trajectory crosses a line, such as xt=0x2t0T [23].


## 3.1. Switching T-S Fuzzy Model

Consider(7)RegionModelRuleq—RMRq:IFxt∈RegionqTHENLocalModelRulei—LMRi:IFx1tisMqi1andx2tisMqi2⋯andxntisMqin,THENx˙t=Aqixt+Βqiuti=1,2,…,rq,q=1,2,…,2n,where RMRq denotes the region model rule q. LMRi denotes the local model rule i. Mqij denotes the fuzzy set corresponding to the state variable xjt of LMRi in Region q. rq is the number of IF-THEN fuzzy local model rules in Region q. 2n is the number of region. n is the number of state variables. r=∑q=12nrq is the total number of IF-THEN fuzzy model rules. Aqi∈Rn×n denotes the system matrix of the subsystem. Bqi∈Rn×m denotes the input matrix of the subsystem.Each linear consequent equation,Aqixt+Βqiut, is called a subsystem.Given a pair ofxt,ut, the final output of the T-S fuzzy model (7) is inferred by CAD and can be represented as follows [23]: (8)x˙t=∑q=12n∑i=1rqηqxthqixtAqixt+Βqiut,where(9)ηqxt=1,xt∈Rq0,xt∉Rq.hqixt is the normalized membership function of the LMRi in Region q and satisfies(10)∑q=12n∑i=1rqηqxthqixt=1,∑q=12n∑i=1rqηqxthqixt≥0,i=1,2,…,rq.In theqth region, we define(11)Rqs1q,s2q,…,snqskq=1,xkt≥00,xkt<0k=1,2,…,n,where the number of the state variables is equal to n and its corresponding state space can be divided into 2n region. For clarification, assume that the considered state space has three dimensions and each separated region corresponds to a quadrant. Figure 1 depicts that the trajectory of a given initial state passes over the different quadrants in a three-dimensional space. The state in Figure 1 passes through Region 2, Region 1, Region 5, Region 7, Region 3, Region 4, and Region 2 in turn. To guarantee the stability of T-S fuzzy system, we suppose that the trajectory of a system cannot be motionless on the region boundaries.Figure 1
Divided state space and change of state.


## 3.2. Switching Quadratic Lyapunov Function

Consider(12)Vxt=xTtP1xt,xt∈Region1xTtP2xt,xt∈Region2⋮⋮xTtP2nxt,xt∈Region2n,where Pq∈Rn×n∀q are a set of positive-definite symmetric matrices. This function is continuous on region boundaries if Pq satisfies the following constraint [23]:(13)Pq=p11ζ1,qp12ψ1,2,q⋯p1nψ1,n,qp12ψ1,2,qp22ζ2,q⋯p2nψ2,n,q⋮⋮⋱⋮p1nψ1,n,qp2nψ2,n,q⋯pnnζn,q,ζk,q=2-skq,ψk,l,q=-skq-2slq+4.Since the involved property is crucial for discussing the SQLF, an example is illustrated as follows.Example 3 (see [23]).
As in Figure1, consider a system with three state variables x1t, x2t, and x3t. The state space can be divided into the following eight regions, eight quadrants:(14)R11,1,1,R20,1,1,R31,0,1,R40,0,1;R51,1,0,R60,1,0,R71,0,0,R80,0,0.Based on the notation of a region, that is,Rq(s1q,s2q,…,snq), each Pq is represented as follows:(15)P1=p111p121p131p121p221p231p131p231p331,P2=p112p122p132p122p221p231p132p231p331,P3=p111p123p131p123p222p232p131p232p331,P4=p112p124p132p124p222p232p132p232p331,P5=p111p121p133p121p221p233p133p233p332,P6=p112p122p134p122p221p233p134p233p332,P7=p111p123p133p123p222p234p133p234p332,P8=p112p124p134p124p222p234p134p234p332.On the region boundary between Regions 5 and Regions 7, a plane A in Figure1 for xt=x1t0x3tT, the SQLF is calculated as(16)Vxt=xTtP5xt=x1t0x3tp111p121p133p121p221p233p133p233p332x1t0x3t=p111x12t+2p133x1tx3t+p332x32t=x1t0x3tp111p123p133p123p222p234p133p234p332x1t0x3t=xTtP7xt.Furthermore, the illustrative results can be similarly extended to the other region boundaries of x(t).Remark 4.
Based on the SQLF, (13), we have shown that the state trajectory crossed a plane; that is, xt=x1t0x3tT is continuous. In addition, it is also continuous when the trajectory crosses a line, such as xt=0x2t0T [23].


## 4. Main Result

Considering in general fuzzy inference rules, a single state variable triggers no more than two fuzzy sets. Thus, fuzzy sets for a premise variablexjt can be assumed to have two different patterns, defined in Figures 2 and 3, respectively, where their differences are at xjt=0 including single membership function or two membership functions.Figure 2
State variablexj(t) triggers one fuzzy set at xj(t)=0.Figure 3
State variablexj(t) triggers two fuzzy set at xj(t)=0.For generalization and unification, our proposed approach merely focuses on addressing the pattern withxjt=0 with single fuzzy set, that is, as in Figure 2. Thus, if the pattern is like Figure 3, we need to beforehand perform a reconstruction procedure described in the following.By illustrating from the pattern in Figure3, the reconstruction procedure has two stages. Stage 1 is to delete the two Mj,p, Mj,p+1 regarding xjt=0 and retain the other fuzzy sets. Stage 2 is to establish three new fuzzy sets Mj,p′, Mj,0, and Mj,p+1′ for xjt=0 include single membership function. The achieved pattern can thus be depicted by Figure 4. The membership function Mj,p′(xj(t)), Mj,0(xj(t)), and Mj,p+1′(xj(t)) can be, respectively, constructed by(17)Mj,p′xjt=Mj,pxjt,forϕj,p-1≤xjt≤ϕj,pMj,pxjt-Mj,p0Mj,p+1xjtMj,p+10,forϕj,p≤xjt≤0(18)Mj,0xjt=Mj,p+1xjtMj,p+10,forϕj,p≤xjt≤0Mj,pxjtMj,p0,for0≤xjt≤ϕj,p+1(19)Mj,p+1′xjt=Mj,p+1xjt-Mj,p+10Mj,pxjtMj,p0,for0≤xjt≤ϕj,p+1Mj,p+1xjt,forϕj,p+1≤xjt≤ϕj,p+2.Figure 4
Reconstruction of fuzzy sets with state variablexj(t).


### 4.1. Reconstruction of Switching Takagi-Sugeno Fuzzy Model via Vertex Expression

Consider(20)RegionModelRuleq1,q2,…,qn—RMRq1,q2,…,qn:IFx1tisN1,q1andx2tisN2,q2⋯andxntisNn,qn,THENLocalModelRulei1,i2,…,in—LMRi1,i2,…,in:IFx1tisMq1,1,i1andx2tisMq2,2,i2⋯andxntisMqn,n,in,THENx˙t=Aq1,q2,…,qni1,i2,…,inxt+Βq1,q2,…,qni1,i2,…,inutqj=0,1,ij=1,2,…,rjqj,j=1,2,…,n,where RMRq1,q2,…,qn denotes the regionmodelruleq1,q2,…,qn, LMRi1,i2,…,in denotes the localmodelrulei1,i2,…,in, and Nj,qj denotes the crisp set corresponding to the state variable xjt in divisional region qj.The membership functionNj,qj(xj(t)) of crisp set Nj,qj is defined as below:(21)Nj,0xjt=1forxjt<00forxjt≥0Nj,1xjt=1forxjt≥00forxjt<0.Mqj,j,ij denotes the fuzzy set corresponding to the state variable xjt of LMRi1,i2,…,in in divisional region qj. rjqj is the number of xjt’s fuzzy sets in divisional region qj. rq1,q2,…,qn=∏j=1nrjqj is the number of IF-THEN region model rule (q1,q2,…,qn) r=∑q1=01∑q2=01⋯∑qn=01rq1,q2,…,qn is the total number of IF-THEN fuzzy model rules. Aq1,q2,…,qni1,i2,…,in∈Rn×n denotes the system matrix of the subsystem. And, Bq1,q2,…,qni1,i2,…,in∈Rn×m denotes the input matrix of the subsystem.Each linear consequent equationAq1,q2,…,qni1,i2,…,inxt+Βq1,q2,…,qni1,i2,…,inut is called a subsystem. Given a pair of xt,ut, the final output of the T-S fuzzy model (20) inferred by CAD can be represented as follows:(22)x˙t=∑q1=01∑q2=01⋯∑qn=01∑i1=1r1q1∑i2=1r2q2⋯∑in=1rnqnηq1,q2,…,qnxt×hq1,q2,…,qni1,i2,…,inxt×Aq1,q2,…,qni1,i2,…,inxt+Βq1,q2,…,qni1,i2,…,inut,where the normalized membership function of the RMRq1,q2,…,qn is(23)ηq1,q2,…,qnxt=∏j=1nNj,qjxjt,and the fire strength of the LMRi1,i2,…,in in Rq1,q2,…,qn is(24)wq1,q2,…,qni1,i2,…,inxt=∏j=1nMqj,j,ijxjtfor all t. And the term Mqj,(j,ij)(xj(t)) is the grade of membership of xjt in Mqj,(j,ij).The normalized membership function of theLMRi1,i2,…,in in Rq1,q2,…,qn can be described by(25)hq1,q2,…,qni1,i2,…,inxt=wq1,q2,…,qni1,i2,…,inxt∑i1=1r1q1∑i2=1r2q2⋯∑in=1rnqnwq1,q2,…,qni1,i2,…,inxtwith(26)∑i1=1r1q1∑i2=1r2q2…∑in=1rnqnhq1,q2,…,qni1,i2,…,inxt=1,hq1,q2,…,qni1,i2,…,inxt≥0,ij=1,2,…,rjqj,j=1,2,…,n.Example 5.
Consider a system with two state variablesx1t and x2t. The T-S fuzzy model is assumed to have 16 subsystems and four fuzzy sets for each state variable. A state space with two dimension can be divided into the following four regions: R0,0, R0,1, R1,0, and R1,1 as shown in Figure 5. The T-S fuzzy model that uses the process of reconstruction can obtain the following new T-S fuzzy model as shown in Figure 6. The new T-S fuzzy model has 25 subsystems and five fuzzy sets for each state variable(27)A0,1=M1,2x1tA2,1+M1,3x1tA3,1,A0,2=M1,2x1tA2,2+M1,3x1tA3,2,A0,3=M1,2x1tA2,3+M1,3x1tA3,3,A0,4=M1,2x1tA2,4+M1,3x1tA3,4,A1,0=M2,2x2tA1,2+M2,3x2tA1,3,A2,0=M2,2x2tA2,2+M2,3x2tA2,3,A3,0=M2,2x2tA3,2+M2,3x2tA3,3,A4,0=M2,2x2tA4,2+M2,3x2tA4,3,A0,0=M1,2x1tM2,2x2tA2,2+M1,3x1tM2,2x2tA3,2+M1,2x1tM2,3x2tA2,3+M1,3x1tM2,3x2tA3,3.
For the new T-S fuzzy model in Figure6, it can be separated at region boundaries and depicted as Figure 7.Figure 5
Rules illustration of Example5.Figure 6
Rules illustration of Example5 by the first process of reconstruction.Figure 7
Rules illustration of Example5 by the second process of reconstruction.


### 4.2. Relaxed Stability Conditions Based on Switching Quadratic Lyapunov Function

Consider(28)RegionFunctionRuleq1,q2,…,qn—RFRq1,q2,…,qn:IFx1tisN1,q1andx2tisN2,q2⋯andxntisNn,qn,THENVxt=xTtPq1,q2,…,qnxt,qj=0,1,j=1,2,…,n,where RFRq1,q2,…,qn denotes the region function rule q1,q2,…,qn. Pq1,q2,…,qn∈Rn×n is a set of positive-definite symmetric matrices.Based on the CAD for theRFRq1,q2,…,qn. The overall SFQLF is given as(29)Vxt=∑q1=01∑q2=01⋯∑qn=01ηq1,q2,…,qnxtxTtPq1,q2,…,qnxt.This function is continuous on region boundaries ifPq1,q2,…,qn satisfies the following constraint:(30)Pq1,q2,…,qn=p11q1p122q1+q2⋯p1n2q1+qnp122q1+q2p22q2⋯p2n2q2+qn⋮⋮⋱⋮p1n2q1+qnp2n2q2+qn⋯pnnqn.The autonomous switching T-S fuzzy system via vertex expression with (22) can be represented as(31)x˙t=∑q1=01∑q2=01⋯∑qn=01∑i1=1r1q1∑i2=1r2q2⋯∑in=1rnqnηq1,q2,…,qnxt×hq1,q2,…,qni1,i2,…,inxtAq1,q2,…,qni1,i2,…,inxt.By reconstruction of the switching T-S fuzzy model, we can obtain relaxed stability conditions for autonomous switching T-S fuzzy systems via vertex expression (31) based on SFQLF.Theorem 6.
The equilibrium of the reconstructive autonomous switching T-S fuzzy system (31) is globally asymptotically stable if there exist a set of matrices Pq1,q2,…,qn such that (30) and (32):(32)Pq1,q2,…,qn>0,qj=0,1,j=1,2,…,n,Aq1,q2,…,qni1,i2,…,inTPq1,q2,…,qn+Pq1,q2,…,qnAq1,q2,…,qni1,i2,…,in<0,qj=0,1,ij=1,2,…,rjqj,j=1,2,…,n.Proof.
Consider a candidate of Lyapunov function (29):(33)V˙xt=∑q1=01∑q2=01⋯∑qn=01ηq1,q2,…,qnxt×x˙TtPq1,q2,…,qnxt+xTtPq1,q2,…,qnx˙t.By (22), we manipulate the above equation and it then yields (34) (see below). Thus, from (30) and (32), the considered T-S fuzzy system (31) is asserted to be globally asymptotically stable.(34)V˙xt=∑q1=01∑q2=01…∑qn=01ηq1,q2,…,qnxt×∑i1=1r1q1∑i2=1r2q2⋯∑in=1rnqnhq1,q2,…,qni1,i2,…,inxtxTtAq1,q2,…,qni1,i2,…,inT×Pq1,q2,…,qnxt+xTtPq1,q2,…,qn×∑i1=1r1q1∑i2=1r2q2⋯∑in=1rnqnhq1,q2,…,qni1,i2,…,inxtAq1,q2,…,qni1,i2,…,inxt=∑q1=01∑q2=01⋯∑qn=01∑i1=1r1q1∑i2=1r2q2⋯∑in=1rnqnηq1,q2,…,qnxthq1,q2,…,qni1,i2,…,inxt×xTtAq1,q2,…,qni1,i2,…,inTPq1,q2,…,qnxt+Pq1,q2,…,qnAq1,q2,…,qni1,i2,…,inxt.


## 4.1. Reconstruction of Switching Takagi-Sugeno Fuzzy Model via Vertex Expression

Consider(20)RegionModelRuleq1,q2,…,qn—RMRq1,q2,…,qn:IFx1tisN1,q1andx2tisN2,q2⋯andxntisNn,qn,THENLocalModelRulei1,i2,…,in—LMRi1,i2,…,in:IFx1tisMq1,1,i1andx2tisMq2,2,i2⋯andxntisMqn,n,in,THENx˙t=Aq1,q2,…,qni1,i2,…,inxt+Βq1,q2,…,qni1,i2,…,inutqj=0,1,ij=1,2,…,rjqj,j=1,2,…,n,where RMRq1,q2,…,qn denotes the regionmodelruleq1,q2,…,qn, LMRi1,i2,…,in denotes the localmodelrulei1,i2,…,in, and Nj,qj denotes the crisp set corresponding to the state variable xjt in divisional region qj.The membership functionNj,qj(xj(t)) of crisp set Nj,qj is defined as below:(21)Nj,0xjt=1forxjt<00forxjt≥0Nj,1xjt=1forxjt≥00forxjt<0.Mqj,j,ij denotes the fuzzy set corresponding to the state variable xjt of LMRi1,i2,…,in in divisional region qj. rjqj is the number of xjt’s fuzzy sets in divisional region qj. rq1,q2,…,qn=∏j=1nrjqj is the number of IF-THEN region model rule (q1,q2,…,qn) r=∑q1=01∑q2=01⋯∑qn=01rq1,q2,…,qn is the total number of IF-THEN fuzzy model rules. Aq1,q2,…,qni1,i2,…,in∈Rn×n denotes the system matrix of the subsystem. And, Bq1,q2,…,qni1,i2,…,in∈Rn×m denotes the input matrix of the subsystem.Each linear consequent equationAq1,q2,…,qni1,i2,…,inxt+Βq1,q2,…,qni1,i2,…,inut is called a subsystem. Given a pair of xt,ut, the final output of the T-S fuzzy model (20) inferred by CAD can be represented as follows:(22)x˙t=∑q1=01∑q2=01⋯∑qn=01∑i1=1r1q1∑i2=1r2q2⋯∑in=1rnqnηq1,q2,…,qnxt×hq1,q2,…,qni1,i2,…,inxt×Aq1,q2,…,qni1,i2,…,inxt+Βq1,q2,…,qni1,i2,…,inut,where the normalized membership function of the RMRq1,q2,…,qn is(23)ηq1,q2,…,qnxt=∏j=1nNj,qjxjt,and the fire strength of the LMRi1,i2,…,in in Rq1,q2,…,qn is(24)wq1,q2,…,qni1,i2,…,inxt=∏j=1nMqj,j,ijxjtfor all t. And the term Mqj,(j,ij)(xj(t)) is the grade of membership of xjt in Mqj,(j,ij).The normalized membership function of theLMRi1,i2,…,in in Rq1,q2,…,qn can be described by(25)hq1,q2,…,qni1,i2,…,inxt=wq1,q2,…,qni1,i2,…,inxt∑i1=1r1q1∑i2=1r2q2⋯∑in=1rnqnwq1,q2,…,qni1,i2,…,inxtwith(26)∑i1=1r1q1∑i2=1r2q2…∑in=1rnqnhq1,q2,…,qni1,i2,…,inxt=1,hq1,q2,…,qni1,i2,…,inxt≥0,ij=1,2,…,rjqj,j=1,2,…,n.Example 5.
Consider a system with two state variablesx1t and x2t. The T-S fuzzy model is assumed to have 16 subsystems and four fuzzy sets for each state variable. A state space with two dimension can be divided into the following four regions: R0,0, R0,1, R1,0, and R1,1 as shown in Figure 5. The T-S fuzzy model that uses the process of reconstruction can obtain the following new T-S fuzzy model as shown in Figure 6. The new T-S fuzzy model has 25 subsystems and five fuzzy sets for each state variable(27)A0,1=M1,2x1tA2,1+M1,3x1tA3,1,A0,2=M1,2x1tA2,2+M1,3x1tA3,2,A0,3=M1,2x1tA2,3+M1,3x1tA3,3,A0,4=M1,2x1tA2,4+M1,3x1tA3,4,A1,0=M2,2x2tA1,2+M2,3x2tA1,3,A2,0=M2,2x2tA2,2+M2,3x2tA2,3,A3,0=M2,2x2tA3,2+M2,3x2tA3,3,A4,0=M2,2x2tA4,2+M2,3x2tA4,3,A0,0=M1,2x1tM2,2x2tA2,2+M1,3x1tM2,2x2tA3,2+M1,2x1tM2,3x2tA2,3+M1,3x1tM2,3x2tA3,3.
For the new T-S fuzzy model in Figure6, it can be separated at region boundaries and depicted as Figure 7.Figure 5
Rules illustration of Example5.Figure 6
Rules illustration of Example5 by the first process of reconstruction.Figure 7
Rules illustration of Example5 by the second process of reconstruction.


## 4.2. Relaxed Stability Conditions Based on Switching Quadratic Lyapunov Function

Consider(28)RegionFunctionRuleq1,q2,…,qn—RFRq1,q2,…,qn:IFx1tisN1,q1andx2tisN2,q2⋯andxntisNn,qn,THENVxt=xTtPq1,q2,…,qnxt,qj=0,1,j=1,2,…,n,where RFRq1,q2,…,qn denotes the region function rule q1,q2,…,qn. Pq1,q2,…,qn∈Rn×n is a set of positive-definite symmetric matrices.Based on the CAD for theRFRq1,q2,…,qn. The overall SFQLF is given as(29)Vxt=∑q1=01∑q2=01⋯∑qn=01ηq1,q2,…,qnxtxTtPq1,q2,…,qnxt.This function is continuous on region boundaries ifPq1,q2,…,qn satisfies the following constraint:(30)Pq1,q2,…,qn=p11q1p122q1+q2⋯p1n2q1+qnp122q1+q2p22q2⋯p2n2q2+qn⋮⋮⋱⋮p1n2q1+qnp2n2q2+qn⋯pnnqn.The autonomous switching T-S fuzzy system via vertex expression with (22) can be represented as(31)x˙t=∑q1=01∑q2=01⋯∑qn=01∑i1=1r1q1∑i2=1r2q2⋯∑in=1rnqnηq1,q2,…,qnxt×hq1,q2,…,qni1,i2,…,inxtAq1,q2,…,qni1,i2,…,inxt.By reconstruction of the switching T-S fuzzy model, we can obtain relaxed stability conditions for autonomous switching T-S fuzzy systems via vertex expression (31) based on SFQLF.Theorem 6.
The equilibrium of the reconstructive autonomous switching T-S fuzzy system (31) is globally asymptotically stable if there exist a set of matrices Pq1,q2,…,qn such that (30) and (32):(32)Pq1,q2,…,qn>0,qj=0,1,j=1,2,…,n,Aq1,q2,…,qni1,i2,…,inTPq1,q2,…,qn+Pq1,q2,…,qnAq1,q2,…,qni1,i2,…,in<0,qj=0,1,ij=1,2,…,rjqj,j=1,2,…,n.Proof.
Consider a candidate of Lyapunov function (29):(33)V˙xt=∑q1=01∑q2=01⋯∑qn=01ηq1,q2,…,qnxt×x˙TtPq1,q2,…,qnxt+xTtPq1,q2,…,qnx˙t.By (22), we manipulate the above equation and it then yields (34) (see below). Thus, from (30) and (32), the considered T-S fuzzy system (31) is asserted to be globally asymptotically stable.(34)V˙xt=∑q1=01∑q2=01…∑qn=01ηq1,q2,…,qnxt×∑i1=1r1q1∑i2=1r2q2⋯∑in=1rnqnhq1,q2,…,qni1,i2,…,inxtxTtAq1,q2,…,qni1,i2,…,inT×Pq1,q2,…,qnxt+xTtPq1,q2,…,qn×∑i1=1r1q1∑i2=1r2q2⋯∑in=1rnqnhq1,q2,…,qni1,i2,…,inxtAq1,q2,…,qni1,i2,…,inxt=∑q1=01∑q2=01⋯∑qn=01∑i1=1r1q1∑i2=1r2q2⋯∑in=1rnqnηq1,q2,…,qnxthq1,q2,…,qni1,i2,…,inxt×xTtAq1,q2,…,qni1,i2,…,inTPq1,q2,…,qnxt+Pq1,q2,…,qnAq1,q2,…,qni1,i2,…,inxt.


## 5. Illustrative Example

Revisit Example1 in the previous work [26]. By our proposed approach, we can beforehand use the reconstruction procedure for this T-S fuzzy model and thus can suitably perform the stability verification by Theorem 6 with SFQLF described in the sequel.Example 7.
Consider the following reconstructed switching T-S fuzzy model:(35)RMR0,0:IFx1tisN1,0andx2tisN2,0THENLMR1,1:IFx1tisM0,1,1andx2tisM0,2,1THENx˙t=A0,01,1xtLMR2,1:IFx1tisM0,1,2andx2tisM0,2,1THENx˙t=A0,02,1xtLMR1,2:IFx1tisM0,1,1andx2tisM0,2,2THENx˙t=A0,01,2xtLMR2,2:IFx1tisM0,1,2andx2tisM0,2,2THENx˙t=A0,02,2xtRMR1,0:IFx1tisN1,1andx2tisN2,0THENLMR1,1:IFx1tisM1,1,1andx2tisM0,2,1THENx˙t=A1,01,1xtLMR2,1:IFx1tisM1,1,2andx2tisM0,2,1THENx˙t=A1,02,1xtLMR1,2:IFx1tisM1,1,1andx2tisM0,2,2THENx˙t=A1,01,2xtLMR2,2:IFx1tisM1,1,2andx2tisM0,2,2THENx˙t=A1,02,2xtRMR0,1:IFx1tisN1,0andx2tisN2,1THENLMR1,1:IFx1tisM0,1,1andx2tisM1,2,1THENx˙t=A0,11,1xtLMR2,1:IFx1tisM0,1,2andx2tisM1,2,1THENx˙t=A0,12,1xtLMR1,2:IFx1tisM0,1,1andx2tisM1,2,2THENx˙t=A0,11,2xtLMR2,2:IFx1tisM0,1,2andx2tisM1,2,2THENx˙t=A0,12,2xtRMR1,1:IFx1tisN1,1andx2tisN2,1THENLMR1,1:IFx1tisM1,1,1andx2tisM1,2,1THENx˙t=A1,11,1xtLMR2,1:IFx1tisM1,1,2andx2tisM1,2,1THENx˙t=A1,12,1xtLMR1,2:IFx1tisM1,1,1andx2tisM1,2,2THENx˙t=A1,11,2xtLMR2,2:IFx1tisM1,1,2andx2tisM1,2,2THENx˙t=A1,12,2xt,where the original system matrices of the subsystem are(36)A1=-5-4-1a,A2=-4-4153b-2153a-4,A3=-3-4152b-3152a-6,A4=-2-4b-2.
By the reconstructing procedure, the considered system associated with rules illustration via vertex expression is depicted in Figure8. Thus, the systems’ matrices can be, respectively, calculated as(37)A0,02,2=A2,A0,01,2=A2+A42,A1,01,2=A2+A42,A1,02,2=A4,A0,02,1=A1+A22,A0,01,1=A1+A2+A3+A44,A1,01,1A1+A2+A3+A44,A1,02,1=A3+A42,A0,12,1=A1+A22,A0,11,1A1+A2+A3+A44,A1,11,1=A1+A2+A3+A44,A1,12,1=A3+A42,A0,02,2=A1,A0,11,2=A1+A32,A1,11,2=A1+A32,A1,12,2=A3,with (38)N1,0x1t=1,forx1t<00,forx1t≥0,N1,1x1t=1,forx1t≥00,forx1t<0,N2,0x2t=1,forx2t<00,forx2t≥0,N2,1x2t=1,forx2t≥00,forx2t<0.M0,1,2x1t=1,forx1t≤-π2-sinx1t,for-π2≤x1t≤0,M0,1,1x1t=0,forx1t≤-π21+sinx1t,for-π2≤x1t≤0M1,1,1x1t=1-sinx1t,forx1t≤π20,forπ2≤x1tM1,1,2x1t=sinx1tfor0≤x1t≤π21forπ2≤x1tM0,2,2x2t=1,forx2t≤-π2-sinx2t,for-π2≤x2t≤0M0,2,1x2t=0,forx2t≤-π21+sinx2t,for-π2≤x2t≤0M1,2,1x1t=1-sinx2t,forx2t≤π20,forπ2≤x2tM1,2,2x2t=sinx2tfor0≤x2t≤π21forπ2≤x2t.
In this example, by applying a current tool, Matlab software, we experimentally attain a widely feasible region for the parameters’ pair (a,b) via SQLF method compared to that of the CQLF method. The feasible parameter pairs (a,b) of the CQLF and the SQLF methods are shown in Figures 9 and 10, respectively. By observation, the proposed approach can dramatically lessen the conservatism of the stability assurance.Figure 8
Rules illustration of switching T-S fuzzy model via vertex expression for Example7.Figure 9
Feasible area for Example7 that uses the CQLF method.Figure 10
Feasible area for Example7 that uses the SQLF method.



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*Source: 147817-2015-06-23.xml*