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Robust Stabilization for Uncertain Takagi-Sugeno Fuzzy Continuous Model with Time-Delay Based on Razumikhin Theorem

Written By

Yassine Manai and Mohamed Benrejeb

Submitted: November 27th, 2011 Published: September 27th, 2012

DOI: 10.5772/48422

From the Edited Volume

Fuzzy Controllers

Edited by Sohail Iqbal, Nora Boumella and Juan Carlos Figueroa Garcia

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1. Introduction

Fuzzy control systems have experienced a big growth of industrial applications in the recent decades, because of their reliability and effectiveness. Many researches are investigated on the Takagi-Sugeno models [1], [2] and [3] last decades. Two classes of Lyapunov functions are used to analysis these systems: quadratic Lyapunov functions and non-quadratic Lyapunov ones which are less conservative than first class. Many researches are investigated with non-quadratic Lyapunov functions [4]-[6], [7].

Recently, Takagi–Sugeno fuzzy model approach has been used to examine nonlinear systems with time-delay, and different methodologies have been proposed for analysis and synthesis of this type of systems [1]-[11], [12]-[13]. Time delay often occurs in many dynamical systems such as biological systems, chemical system, metallurgical processing system and network system. Their existences are frequently a cause of infeasibility and poor performances.

The stability approaches are divided into two classes in term of delay. The fist one tries to develop delay independent stability criteria. The second class depends on the delay size of the time delay, and it called delay dependent stability criteria. Generally, delay dependent class gives less conservative stability criteria than independent ones.

Two classes of Lyapunov-Razumikhin function are used to analysis these systems: quadratic Lyapunov-Razumikhin function and non-quadratic Lyapunov- Razumikhin ones. The use of first class brings much conservativeness in the stability test. In order to reduce the conservatism entailed in the previous results using quadratic function.

As the information about the time derivatives of membership function is considered by the PDC fuzzy controller, it allows the introduction of slack matrices to facilitate the stability analysis. The relationship between the membership function of the fuzzy model and the fuzzy controllers is used to introduce some slack matrix variables. The boundary information of the membership functions is brought to the stability condition and thus offers some relaxed stability conditions [5].

In this chapter, a new stability conditions for time-delay Takagi-Sugeno fuzzy systems by using fuzzy Lyapunov-Razumikhin function are presented. In addition, a new stabilization conditions for Takagi Sugeno time-delay uncertain fuzzy models based on the use of fuzzy Lyapunov function are presented. This criterion is expressed in terms of Linear Matrix Inequalities (LMIs) which can be efficiently solved by using various convex optimization algorithms [8],[9]. The presented methods are less conservative than existing results.

The organization of the chapter is as follows. In section 2, we present the system description and problem formulation and we give some preliminaries which are needed to derive results. Section 3 will be concerned to stability and stabilization analysis for T-S fuzzy systems with Parallel Distributed Controller (PDC). An observer approach design is derived to estimate state variables. Section 5 will be concerned to stabilization analysis for time-delay T-S fuzzy systems based on Razumikhin theorem. Next, a new robust stabilization condition for uncertain system with time delay is given in section 6. Illustrative examples are given in section 7 for a comparison of previous results to demonstrate the advantage of proposed method. Finally section 8 makes conclusion.

Notation: Throughout this chapter, a real symmetric matrix S>0 denotes S being a positive definite matrix. The superscript ‘‘T’’ is used for the transpose of a matrix.


2. System description and preliminaries

Consider an uncertain T-S fuzzy continuous model with time-delay for a nonlinear system as follows:

IF z1(t) is Mi1 andand zp(t) is MipTHEN    {x˙(t)=(Ai+ΔAi)x(t)+(Di+ΔDi)x(tτi(t))+(Bi+ΔBi)u(t)x(t)=ϕ(t),t[τ,0]E1

whereMij(i=1,2,,r,j=1,2,,p)is the fuzzy set and r is the number of model rules; x(t)nis the state vector, u(t)mis the input vector,Ain×n ,Din×n ,Bin×m , andz1(t),,zp(t)are known premise variables,ϕ(t) is a continuous vector-valued initial function on[τ,0]; the time-delayτ(t)may be unknown but is assumed to be smooth function of time.. ΔAiΔDiand ΔBiare time-varying matrices representing parametric uncertainties in the plant model. These uncertainties are admissibly norm-bounded and structured.

0τ(t)τ,    τ˙(t)d1,E2

whereτ0anddare two scalars.

The final outputs of the fuzzy systems are:

x(t)=ϕ(t),   t[τ,0],E4



hi(z(t))=wi(z(t))/i=1rwi(z(t)),wi(z(t))=j=1pMij(zj(t)) for all t.

The term Mi1(zj(t)) is the grade of membership of zj(t) in Mi1


{i=1rwi(z(t))0wi(z(t))0,           i=1,2,,rE6

we have {i=1rhi(z(t))=1hi(z(t))0,         i=1,2,,rfor all t.

The time derivative of premise membership functions is given by:


We have the following property:


Consider a PDC fuzzy controller based on the derivative membership function and given by the equation


The fuzzy controller design consists to determine the local feedback gains Fi,andKm in the consequent parts. The state variables are determined by an observer which detailed in next section.

By substituting into , the closed-loop fuzzy system without time-delay can be represented as:

x˙(t)=i=1rj=1rhi(z(t))hj(z(t)){[AΔiBΔiFjm=1rh˙m(z(t))BΔiKm]x(t)+DΔix(tτi(t))}x(t)=ϕ(t),   t[τ,0],E10



The system without uncertainties is given by equation

x˙(t)=i=1rj=1rhi(z(t))hj(z(t)){[AiBiFjm=1rh˙m(z(t))BiKm]x(t)+Dix(tτi(t))}x(t)=ϕ(t),   t[τ,0],E12

The open-loop system is given by the equation ,

x˙(t)=i=1rhi(z(t))(AΔix(t)+DΔix(tτi(t)))x(t)=ϕ(t),  t[τ,0],E13

Assumption. 1

The time derivative of the premises membership function is upper bounded such that|h˙k|ϕk, fork=1,,r, where, ϕk,k=1,,rare given positive constants.


Assumption. 2

The matrices denote the uncertainties in the system and take the form of


where Dai,Dbi,Eai and Ebiare known constant matrices and Fai(t)andFbi(t)are unknown matrix functions satisfying :


where I is an appropriately dimensioned identity matrix.

Lemma 1 (Boyd et al. Schur complement [16])

Given constant matrices Ω1,Ω2 and Ω3with appropriate dimensions, where Ω1=Ω1TandΩ2=Ω2T, then


if and only if

[Ω1Ω3T*Ω2]0 or [Ω2Ω3*Ω1]0E17

Lemma 2 (Peterson and Hollot [2])

Let Q=QT,H,E and F(t)satisfying FT(t)F(t)Iare appropriately dimensional matrices then the follow-ing inequality

is true, if and only if the following inequality holds for any λ0


Theorem 1 (Razumikhin Theorem)[5]

Suppose u,v,w:++are continuous, non-decreasing functions satisfying u(s)0, v(s)0 and w(s)0fors0,u(0)=v(0)=0, andvstrictly increasing. If there exist a continuous function V:×n and a continuous non-decreasing function p(s)s for s0 such that

u(|x|)V(t,x)v(|x|),       t,  xn,E20
V˙(t,x)w(|x|)    if V(t+σ,x(t+σ))p(V(t,x)),    σ[τ,0],E21

then the solution x0 of is uniformly asymptotically stable.


Lemma. 3 [6]

Assume thatana,bnb ,Nna×nb are defined on the intervalΩ. Then, for any matricesXna×na,Yna×nb andZnb×nb, the following holds:




Lemma. 4 [9]

The unforced fuzzy time delay system described by with u = 0 is uniformly asymptotically stable if there exist matricesP0,Si0,Xai,Xdi,Zaij,Zdij,andYi, such that the following LMIs hold:


3. Basic stability and stabilization conditions

In order to design an observer for state variables, this section introduce two theorem developed for continuous TS fuzzy model for open-loop and closed-loop. First, consider the open-loop system without time-delay given by equation.


The main approach for T-S fuzzy model stability is given in theorem follows. This approach is based on introduction of εparameter which influences the stability region.


Theorem. 2 [17]

Under assumption 1 and for0ε1, the Takagi Sugeno fuzzy system is stable if there exist positive definite symmetric matricesPk,k=1,2,,r, matrix R=RT such that the following LMIs hold.

Pϕ+12{AiT(Pj+μR)+(Pj+μR)Ai   +AjT(Pi+μR)+(Pi+μR)Aj}0,  ijE33

where i,j=1,2,,rand Pϕ=k=1rϕk(Pk+R)and μ=1ε



The proof of this theorem is given in detailed in article published in [17].

The closed-loop system without time delay is given by equation




In this section we define a fuzzy Lyapunov function and then consider stability conditions. A sufficient stability condition, for ensuring stability is given follows.


Theorem. 2[18]

Under assumption 1, and assumption 2 and for given0ε1, the Takagi-Sugeno system is stable if there exist positive definite symmetric matricesPk,k=1,2,,r, andR, matrices F1,,Fr such that the following LMIs hols.

Pj+μR0,   j=1,2,,rE37
Pϕ+{GiiT(Pk+μR)+(Pk+μR)Gii}0,  i,k{1,,r}E38
{Gij+Gji2}T(Pk+μR)+(Pk+μR){Gij+Gji2}0, for i,j,k=1,2,,r such that ijE39





4. Observer design for T-S fuzzy continuous model

In order to determine state variables of system, this section gives a solution by the mean of fuzzy observer design.

A stabilizing observer-based controller can be formulated as follow:


The closed-loop fuzzy system can be represented as:


The augmented system is represented as follows:




By applying Theorem 2[18] in the augmented system we derive the following Theorem.

Theorem. 3

Under assumption 1 and for given0μ1, the Takagi-Sugeno system is stable if there exist positive definite symmetric matricesPk,k=1,2,,r, andR, matrices F1,,Fr such that the following LMIs hols.

Pj+μR0,   j=1,2,,rE48
Pϕ+{GiiT(Pk+μR)+(Pk+μR)Gii}0,  i,k{1,,r}E49
{Gij+Gji2}T(Pk+μR)+(Pk+μR){Gij+Gji2}0, for i,j,k=1,2,,r such that ijE50




The result follows immediately from the Theorem 2[18].

5. Stabilization of continuous T-S Fuzzy model with time-delay

The aim of this section is to prove the asymptotic stability of the time-delay system based on the combination between Lyapunov theory and the Razumikhin theorem [5].

Theorem. 4

Under assumption 1 and for given0ε1, the unforced fuzzy time delay system described by withu=0is uniformly asymptotically stable if there exist matrices Pk0,k=1,2,,r, Si0,Xaij, Xdi,Zaij,Zdij,Yi, andX, such that the following LMIs hold:

[         Pβ+(Pk+εX)Gij+GijT(Pk+εX)+τ(Xaij+Xdi)+(2τ+1)(Pk+εX)+Yi+YiT(Pk+εX)DiYiTDiT(Pk+εX)Si]0E53




Let consider the fuzzy Lyapunov function as


Given the matrix property, clearly,


where λmin(max)denotes the smallest (largest) eigenvalue of the matrix.

Finding the maximum value of k=0rhkxT(t)(Pk+εX)x(t)is equivalent to determining the maximum value ofk=0rhkλmax(Pk+εX).

Finding the minimum value of k=0rhkxT(t)(Pk+εX)x(t)is equivalent to determining the minimum value ofk=0rhkλmin(Pk+εX).


κ1=minkk=0rhkλmax(Pk+εX)  for 0kr,κ2=maxkk=0rhkλmin(Pk+εX)  for 0kr.E62



In the following, we will prove the asymptotic stability of the time-delay system based on the Razumikhin theorem [5].



The state equation of with u=0 can be rewritten as




The derivative of V along the solutions of the unforced system with u=0is thus given by


Then, based on assumption 1, an upper bound of ϒ1(x,z) obtained as:


and for ϒ2(x,t)we can written as,

ϒ2(x,t)=2i=1rj=1rhihjxTk=1rhk(Pk+εX)(Gij+Di)x(t) i=1rj=1rhihjtτi(t)t{2xT(t)k=1rhk(Pk+εX)Di×ν=1rς=1rhν(s)hς(s)[Gνςx(s)+Dςx(sτj(s))]ds}ϒ2(x,t)=2i=1rj=1rk=1rhihjhkxT{(Pk+εX)(Gij+Di)}x(t) i=1rj=1rk=1rhihjhktτi(t)t{2xT(t)(Pk+εX)Di×ν=1rς=1rhν(s)hς(s)[Gνςx(s)+Dςx(sτj(s))]ds}E69

Using the bounding method in, by setting a=x(t)andb=Gijx(s), we have


For any matrices Xaν,Yν and Zaiνsatisfying


Similarly, it holds that


For any matrices Xdi,Yi and Zdijsatisfying


Hence, substituting and into , we have

V˙Pβ+i=1rj=1rk=1rhihjhkxT(t)[2(Pk+εX)(Gij+Di)+τ(Xai+Xdi)]x(t)       +i=1rk=1rhihk2xT(t)(Yi(Pk+εX)Di)×tτi(t)tν=1rς=1rhν(s)hς(s)[Gνςx(s)+Dνx(sτν(s))]ds       +i=1rk=1rhihktτi(t)tν=1rς=1rhν(s)hς(s)xT(s)GνςTZaiνGνςx(s)ds      +i=1rk=1rhihktτi(t)tj=1rhj(s)xT(sτj(s))DjTZdijDjx(sτj(s))dsE74
Pβ+i=1rj=1rhihjxT(t)[(Pk+εX)Gij+GijT(Pk+εX)+Yi+YiT+τ(Xaij+Xdi)]x(t)       +i=1rhi[xT(t)(Yi(Pk+εX)Di)Si1(Yi(Pk+εX)Di)Tx(t)+xT(tτi(t))Six(tτi(t))]       +i=1rhi×tτi(t)tν=1rς=1rhν(s)hς(s)xT(s)GνςTZaiνGνςx(s)ds       +i=1rhitτi(t)tj=1rhj(s)xT(sτj(s))DjTZdijDjx(sτj(s))dsE75

Note that, by Shur complement, the LMI in impliesLi(δ)0for a sufficiently small scalarδ0, where


In order to use the Razumikhin Theorem, suppose V(x(t+σ))(1+δ)V(x(t)) forσ[τ,0]. Then, if the LMIs in – also hold, we have from that

V˙i=1rj=1rhihjxT(t)[(Pk+εX)Gij+GijT(Pk+εX)+Yi+YiT+τ(Xaij+Xdi)]x(t)       +i=1rhi[xT(t)(Yi(Pk+εX)Di)Si1(Yi(Pk+εX)Di)Tx(t)+xT(t)(1+δ)(Pk+εX)x(t)]      +i=1rhiτi(t)xT(t)(1+δ)(Pk+εX)x(t)+i=1rhitτi(t)txT(s)(1+δ)(Pk+εX)x(s)ds     i=1rhixT(t)[(Pk+εX)Ai+AiT(Pk+εX)+Yi+YiT+τ(Xai+Xdi)]x(t)       +i=1rhi[xT(t)(Yi(Pk+εX)Di)Si1(Yi(Pk+εX)Di)Tx(t)+xT(t)(1+δ)(Pk+εX)x(t)]      +τxT(t)(1+δ)(Pk+εX)x(t)+τxT(t)(1+δ)2(Pk+εX)x(t)     =i=1rhixT(t)Li(δ)x(t)     0E77

which shows the motion of the unforced system with u = 0 is uniformly asymptotically stable. This completes the proof.

6. Robust stability condition with PDC controller

Consider the closed-loop system . A sufficient robust stability condition for Time-delay system is given follow.

Theorem. 5

Under assumption 1, and assumption 2 and for given0ε1, the Takagi-Sugeno system is stable if there exist positive definite symmetric matricesPk,k=1,2,,r, andR, matrices F1,,Fr such that the following LMIs hols.

Pj+μR0,   j=1,2,,rE79


Φ1=Pϕ+G¯iiT(Pk+μR)+(Pk+μR)G¯ii+λ(Pk+μR)[EaiTEai+(EbiFi)TEbiFi][Φ2**λI0*λI]0 [Φ2(Pk+μR)(Dai+Daj)(Pk+μR)(Dbi+Dbj)(Pk+μR)(DdiΔdiEdi)λI00λI00]0i,k{1,,r}for i,j,k=1,2,,r such that ijE81

whereG¯ij=[AiBiFjm=1rh˙m(z(t))BiKm],Gii=[AiBiFim=1rh˙m(z(t))BiKm] , μ=1ε,and Pϕ=k=1rϕk(Pk+R)


Let consider the Lyapunov function in the following form:



Vk(x(t))=xT(t)(Pk+μR)x(t),  k=1,2,,rE84


Pk=PkT,R=RT, 0ε1,μ=1ε, and (Pk+μR)0,   k=1,2,,rE85

The time derivative of V(x(t))with respect to t along the trajectory of the system is given by:


The equation can be rewritten as,

V˙(x(t))=xT(t)(k=1rh˙k(z(t))(Pk+μR))x(t)+x˙T(t)(k=1rhk(z(t))(Pk+μR))x(t)            +xT(t)(k=1rhk(z(t))(Pk+μR))x˙(t)E87

By substituting into , we obtain,



ϒ2(x,z)=xT(t)k=1ri=1rhk(z(t))hi2(z(t))×{G¯iiT(Pk+μR)+(Pk+μR)G¯ii}x(t)+xT(t)k=1ri=1rhk(z(t))hi2(z(t))×{([DaiDbi][Δai00Δbi][EaiEbiFi])T(Pk+μR)+(Pk+μR)([DaiDbi][Δai00Δbi][EaiEbiFi])}x(t)+xT(tτi(t))k=1ri=1rhi(z(t))hk(z(t)){(DdiΔdiEdi)T(Pk+μR)}x(t)+xT(t)k=1ri=1rhi(z(t))hk(z(t))(Pk+μR)(DdiΔdiEdi)x(tτi(t))ϒ2(x,z)=k=1ri=1rhk(z(t))hi2(z(t))×ηTΣiiηwhereηT=[xT(t)xT(tτi(t))]Σii=[Π1(Pk+μR)(DdiΔdiEdi){(DdiΔdiEdi)T(Pk+μR)}0]withΠ1={G¯iiT(Pk+μR)+(Pk+μR)G¯ii}+{([DaiDbi][Δai00Δbi][EaiEbiFi])T(Pk+μR)+(Pk+μR)([DaiDbi][Δai00Δbi][EaiEbiFi])}x(t)where  G¯ii=[AiBiFim=1rh˙m(z(t))BiKm]E90
ϒ3(x,z)=x(t)Tk=1ri=1rijhk(z(t))hi(z(t))hj(z(t))×{(G¯ij+G¯ji2)T(Pk+μR)+(Pk+μR)(G¯ij+G¯ji2)}]x(t)+ x(t)Tk=1ri=1rijhk(z(t))hi(z(t))hj(z(t))×{([DaiDbi][Δai00Δbi][EaiEbiFj])T(Pk+μR)+(Pk+μR)([DaiDbi][Δai00Δbi][EaiEbiFj])}x(t)x(t)Tk=1ri=1rijhk(z(t))hi(z(t))hj(z(t))×{([DajDbj][Δaj00Δbj][EajEbjFi])T(Pk+μR)+(Pk+μR)([DajDbj][Δaj00Δbj][EajEbjFi])}x(t)+xT(tτi(t))k=1ri=1rhi(z(t))hk(z(t)){(DdiΔdiEdi)T(Pk+μR)}x(t)+xT(t)k=1ri=1rhi(z(t))hk(z(t))(Pk+μR)(DdiΔdiEdi)x(tτi(t))where  G¯ij=[AiBiFjm=1rh˙m(z(t))BiKm]ϒ3(x,z)=k=1ri=1rijhk(z(t))hi(z(t))hj(z(t))×ηTΣijηwhereηT=[xT(t)xT(tτi(t))]Σij=[Π2(Pk+μR)(DdiΔdiEdi){(DdiΔdiEdi)T(Pk+μR)}0]withΠ2={(G¯ij+G¯ji2)T(Pk+μR)+(Pk+μR)(G¯ij+G¯ji2)}]+{([DaiDbi][Δai00Δbi][EaiEbiFj])T(Pk+μR)+(Pk+μR)([DaiDbi][Δai00Δbi][EaiEbiFj])}+{([DajDbj][Δaj00Δbj][EajEbjFi])T(Pk+μR)+(Pk+μR)([DajDbj][Δaj00Δbj][EajEbjFi])}E91

Then, based on assumption 1, an upper bound of ϒ1(x,z) obtained as:


Based on , it follows that k=1rh˙k(z(t))εR=R¯=0 where R is any symmetric matrix of proper dimension.

Adding R¯to , then





[H11(Pk+μR)DdiΔdiEdiEdiTΔdiTDdiT(Pk+μR)0]0where     H11=k=1rϕk(Pk+R)+G¯iiT(Pk+μR)+(Pk+μR)G¯ii+{([EaiEbiFi])T([DaiDbi])T(Pk+μR)+(Pk+μR)[DaiDbi][Δai00Δbi][EaiEbiFi]}E95

Then, based on Lemma 2, an upper bound of H11 obtained as:


by Schur complement, we obtain,




Then, based on Lemma 2, an upper bound of ϒ1(x,z) obtained as:


by Schur complement, we obtain,




If and holds, the time derivative of the fuzzy Lyapunov function is negative. Consequently, we have V˙(x(t))0 and the closed loop fuzzy system is stable. This is complete the proof.

7. Numerical examples

Consider the following T-S fuzzy system:




the premise functions are given by:

h1(x1(t))=1+sinx1(t)2;h2(x1(t))=1sinx1(t)2; A1=[5412];

It is assumed that|x1(t)|π2. Forξ11=0,ξ12=0.5, ξ21=0.5,andξ22=0,we obtain

R=[-1.2760 -2.2632-2.2632-0.6389]E105

Figure 1.

State variables

Figure 3 shows the evolution of the state variables. As can be seen, the conservatism reduction leads to very interesting results regarding fast convergence of this Takagi-Sugeno fuzzy system.

In order to show the improvements of proposed approaches over some existing results, in this section, we present a numerical example, which concern the feasibility of a time delay T-S fuzzy system. Indeed, we compare our fuzzy Lyapunov-Razumikhin approach (Theorem 3.1) with the Lemma 2.2 in [9].

Example 2. Consider the following T-S fuzzy system with u=0:




with the following membership functions :


Assume that τi(t)=0.5|sin(x1(t)+x2(t)+1)| wherex(t)=[x1(t),x2(t)]T. Then,τi(t)τ=0.5. Table 1. shows that our approach is less conservative than Lemma 2.2. given in [9].

Lemma 2.10.6308
Theorem 3.1+

Table 1.

Comparison results of maximumτ for Example 1

The LMIs in - are feasible by choosing Xai=Xa,Xdi=Xd,Yi=Y,Zaij=Za,Zdij=Zd, and Si=S,i,j=1,2, and for τ=0.5a feasible solution is given by


8. Conclusion

This chapter provided new conditions for the stabilization with a PDC controller of Takagi-Sugeno fuzzy systems with time delay in terms of a combination of the Razumikhin theorem and the use of non-quadratic Lyapunov function as Fuzzy Lyapunov function. In addition, the time derivative of membership function is considered by the PDC fuzzy controller in order to facilitate the stability analysis. An approach to design an observer is derived in order to estimate variable states. In addition, a new condition of the stabilization of uncertain system is given in this chapter.

The stabilization condition proposed in this note is less conservative than some of those in the literature, which has been illustrated via examples.


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Written By

Yassine Manai and Mohamed Benrejeb

Submitted: November 27th, 2011 Published: September 27th, 2012