Open access peer-reviewed chapter

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

By Yassine Manai and Mohamed Benrejeb

Submitted: November 27th 2011Reviewed: May 10th 2012Published: September 27th 2012

DOI: 10.5772/48422

Downloaded: 1539

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>0denotes Sbeing 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)=i=1rhi(z(t)){(Ai+ΔAi)x(t)+(Di+ΔDi)x(tτi(t))+(Bi+ΔBi)u(t)}E3
x(t)=ϕ(t),   t[τ,0],E4

where

z(t)=[z1(t)z2(t)zp(t)]E5

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

Since

{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:

h˙i(z(t))=hiz(t)z(t)x(t)dx(t)dt=l=1sυilξil×dx(t)dtE7

We have the following property:

k=1rh˙k(z(t))=0E8

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

u(t)=i=1rhi(z(t))Fix(t)m=1rh˙m(z(t))Kmx(t)E9

The fuzzy controller design consists to determine the local feedback gains Fi,andKmin 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

where

AΔi=Ai+ΔAi;DΔi=Di+ΔDiandBΔi=Bi+ΔBiE11

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

{ΔAi=DaiFai(t)EaiΔBi=DbiFbi(t)EbiE14

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

{FaiT(t)Fai(t)I,tFbiT(t)Fbi(t)I,tE15

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

Ω1+Ω3TΩ21Ω30E16

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
Q+HF(t)E+ETFT(t)HT0E18

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

Q+λ1HHT+λETE0E19

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:×nand a continuous non-decreasing function p(s)sfor s0such 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 x0of is uniformly asymptotically stable.

Lemma. 3 [6]

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

2ΩaT(α)Nb(α)dαΩ[a(α)b(α)]T[XYNYTNTZ][a(α)b(α)]dα,E22

where

[XYYTZ]0E23
.

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:

[PAi+AiTP+τ(Xai+Xdi)+(2τ+1)P+Yi+YiTPDiYiTDiTPSi]0E24
SiPE25
AjTZaijAjPE26
DjTZdijDjPE27
[XaiYiYiTZaij]0E28
[XdiYiYiTZdij]0E29

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.

x˙(t)=i=1rhi(z(t))Aix(t)E30

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=RTsuch that the following LMIs hold.

Pk+R0,k{1,,r}E31
Pj+μR0,j{1,,r}E32
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

Proof.

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

x˙(t)=i=1rhi(z(t))hi(z(t))Giix(t)+2i=1rijhi(z(t))hj(z(t)){Gij+Gji2}x(t),E34

where

Gij=AiBiFjand
Gii=AiBiFiE35

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,,Frsuch that the following LMIs hols.

Pk+R0,k{1,,r}E36
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

where

Gij=AiBiFj,
Gii=AiBiFiE40

And

Pϕ=k=1rϕk(Pk+R)E41

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:

x˙^(t)=j=1rhi(z(t)){Aix^(t)+Biu(t)+Lj(Cix^(t)y(t))}u(t)=j=1rhj(z(t))Fjx^(t)E42

The closed-loop fuzzy system can be represented as:

x˙(t)=i=1rj=1rhi(z(t))hj(z(t)){(AiBiFj)ρ=1rh˙ρ(z(t))(Hρ+R)}x(t)+i=1rj=1rhi(z(t))hj(z(t)){BiFj+ρ=1rh˙ρ(z(t))(Hρ+R)}e(t)E43
e˙(t)=i=1rj=1rhi(z(t))hj(z(t)){AiKiCj}e(t)E44

The augmented system is represented as follows:

x˙a(t)=i=1rj=1rhi(z(t))hj(z(t))Gijxa(t)=j=1rhi(z(t))hj(z(t))Giixa(t)+2i=1rijrhi(z(t))hj(z(t)){Gij+Gji2}xa(t)E45

where

xa(t)=[x(t)e(t)]Gij=[AiBiFjρ=1rh˙ρBi(Hρ+R)BiFj+ρ=1rh˙ρBi(Hρ+R)0AiKiCj]E46

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,,Frsuch that the following LMIs hols.

Pk+R0,k{1,,r}E47
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

where

Gij=[AiBiFjρ=1rh˙ρBi(Hρ+R)BiFj+ρ=1rh˙ρBi(Hρ+R)0AiKiCj]E51
And
Pϕ=k=1rϕk(Pk+R)E52

Proof

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

where

Pβ=k=1rβk(Pk+εX)Gij=AiBiFjE54
Si(Pk+εX)E55
GijTZaijGij(Pk+εX)E56
DjTZdijDj(Pk+εX)E57
[XaijYiYiTZaij]0E58
[XdiYiYiTZdij]0E59

Proof.

Let consider the fuzzy Lyapunov function as

V(x)=xT(t)Vk(x)x(t)Vk(x)=k=1rhk(Pk+εX)E60

Given the matrix property, clearly,

λmin(Pk+εX)x(t)2xT(t)(Pk+εX)x(t)λmax(Pk+εX)x(t)2,E61

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).

Define

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

Then,

κ1x(t)2k=1rxT(t)(Pk+εX)x(t)κ2x(t)2E63

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

Since

x(t)x(tτi(t))=tτi(t)tx˙(s)ds,E64

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

x˙(t)=i=1rj=1rhihj[(Gij+Di)x(t)Ditτi(t)tx˙(s)ds]E65

where

Gij=AiBiFjE66

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

V˙=xT(t)k=1rh˙k(Pk+εX)x(t)+2xT(t)i=1rhi(Pi+εX)x˙(t)=ϒ1(x,t)+ϒ2(x,t)ϒ1(x,t)=xT(t)k=1rh˙k(Pk+εX)x(t)ϒ2(x,t)=2xT(t)k=1rhk(Pk+εX)x˙(t)=2xT(t)k=1rhk(Pk+εX)×i=1rj=1rhihj[(Gij+Di)x(t)Ditτi(t)tx˙(s)ds].E67

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

ϒ1(x,z)k=1rβkx(t)T(Pk+εX)x(t)E68

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

tτi(t)t2xT(t)(Pk+εX)Di×ν=1rς=1rhν(s)hς(s)Gνςx(s)dsτi(t)xT(t)Xaix(t)+2xT(t)(Yi(Pk+εX)Di)×tτi(t)tν=1rς=1rhν(s)hς(s)Gνςx(s)ds+tτi(t)tν=1rς=1rhν(s)hς(s)xT(s)GνςTZaiνGνςx(s)dsE70

For any matrices Xaν,Yν and Zaiνsatisfying

[XaνYνYνTZaiν]0E71

Similarly, it holds that

tτi(t)t2xT(t)(Pk+εX)Dij=1rhj(s)Djx(sτj(s))dsτi(t)xT(t)Xdix(t)+2xT(t)(Yi(Pk+εX)Di)tτi(t)tj=1rhj(s)Djx(sτj(s))ds+tτi(t)tj=1rhj(s)xT(sτj(s))DjTZdijDjx(sτj(s))dsE72

For any matrices Xdi,Yi and Zdijsatisfying

[XdiYiYiTZdij]0E73

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

Li(δ)=Pβ+(Pk+εX)Gij+GijT(Pk+εX)+Yi+YiT+τ(Xaij+Xdi)+(Yi(Pk+εX)Di)Si1(Yi(Pk+εX)Di)Tx(t)+(2τ+1+τδ)(1+δ)(Pk+εX)E76

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,,Frsuch that the following LMIs hols.

Pk+R0,k{1,,r}E78
Pj+μR0,   j=1,2,,rE79
[Φ1(Pk+μR)Dai(Pk+μR)Dbi(Pk+μR)(DdiΔdiEdi)λI00λI00]0i,k{1,,r}E80

with

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
with
Φ2=(G¯ij+G¯ji2)T(Pk+μR)+(Pk+μR)(G¯ij+G¯ji2)+λ(Pk+μR)[(Eai+Eaj)T(Eai+Eaj)+(EbiFj+EbjFi)T(EbiFj+EbjFi)T]E82

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

Proof

Let consider the Lyapunov function in the following form:

V(x(t))=k=1rhk(z(t))Vk(x(t))E83

with

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

where

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 talong the trajectory of the system is given by:

V˙(x(t))=k=1rh˙k(z(t))Vk(x(t))+k=1rhk(z(t))V˙k(x(t))E86

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,

V˙(x(t))=ϒ1(x,z)+ϒ2(x,z)+ϒ3(x,z)E88

where

ϒ1(x,z)=xT(t)(k=1rh˙k(z(t))(Pk+μR))x(t)E89
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:

ϒ1(x,z)k=1rϕkx(t)T(Pk+μR)x(t)E92

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

Adding R¯to , then

ϒ1(x,z)k=1rϕkx(t)T(Pk+R)x(t)E93

Then,

V˙(x(t))k=1rϕkxT(t)(Pk+R)x(t)+ϒ2(x,z)+ϒ3(x,z)E94

If

[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 H11obtained as:

k=1rϕk(Pk+R)+G¯iiT(Pk+μR)+(Pk+μR)G¯ii+λ1(Pk+μR)[DaiDbi][DaiTDbiT]+λ[EaiT(EbiFi)T][EaiEbiFi](Pk+μR)0E96

by Schur complement, we obtain,

[Φ1(Pk+μR)Dai(Pk+μR)Dbi**λI0*λI]0E97

with

Φ1=Pϕ+G¯iiT(Pk+μR)+(Pk+μR)G¯ii+λ(Pk+μR)[EaiTEai+(EbiFi)TEbiFi]{(G¯ij+G¯ji2)T(Pk+μR)+(Pk+μR)(G¯ij+G¯ji2)}+{([Dai+DajDbi+Dbj][Δai+Δaj00Δbi+Δbj]×[Eai+EajEbiFjEbjFi])T(Pk+μR)+(Pk+μR)×([Dai+DajDbi+Dbj][Δai+Δaj00Δbi+Δbj]×[Eai+EajEbiFjEbjFi])}0E98

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

(G¯ij+G¯ji2)T(Pk+μR)+(Pk+μR)(G¯ij+G¯ji2)+λ1(Pk+μR)[Dai+DajDbi+Dbj][DaiT+DTajDbiT+DTbj]+λ[(Eai+Eaj)T(EbiFjEbjFi)T]×[Eai+EajEbiFjEbjFi](Pk+μR)0E99

by Schur complement, we obtain,

[Φ2(Pk+μR)(Dai+Daj)(Pk+μR)(Dbi+Dbj)**λI0*λI]0E100

with

Φ2=(G¯ij+G¯ji2)T(Pk+μR)+(Pk+μR)(G¯ij+G¯ji2)+λ(Pk+μR)[(Eai+Eaj)T(Eai+Eaj)+(EbiFj+EbjFi)T(EbiFj+EbjFi)T]E101

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

7. Numerical examples

Consider the following T-S fuzzy system:

x˙(t)=i=1rhi(z(t))Aix(t)E102

with:

r=2E103

the premise functions are given by:

h1(x1(t))=1+sinx1(t)2;h2(x1(t))=1sinx1(t)2;A1=[5412];
A2=[24202]E104

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

P1=[37.786426.805826.805836.2722];P2=[98.555928.757728.757722.9286];
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:

x˙(t)=i=12hi(z(t)){Aix(t)+Dix(tτi(t))},E106

with:

A1=[2.10.10.20.9],A2=[1.900.21.1],D1=[1.10.10.80.9],
D2=[0.901.11.2],E107

with the following membership functions :

h1=sin2(x1+0.5);h2=cos2(x1+0.5).E108

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].

Methodsτmax
Lemma 2.10.6308
Theorem 3.1+

Table 1.

Comparison results of maximumτfor 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

P1=[1.51210.18010.18011.1057],P2=[1.4510.1780.1780.883],S=[1.0210.0640.0640.664],Y=[0.6110.1690.2430.421],Xa=[2.5230.7070.7072.155],Xd=[1.4480.0940.0942.353],Za=[0.2010.0870.0870.369],Zd=[0.8490.2270.2270.246],E109

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.

© 2012 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution 3.0 License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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Yassine Manai and Mohamed Benrejeb (September 27th 2012). Robust Stabilization for Uncertain Takagi-Sugeno Fuzzy Continuous Model with Time-Delay Based on Razumikhin Theorem, Fuzzy Controllers - Recent Advances in Theory and Applications, Sohail Iqbal, Nora Boumella and Juan Carlos Figueroa Garcia, IntechOpen, DOI: 10.5772/48422. Available from:

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