# RobustH∞ Control for a Class of Uncertain Switched Fuzzy Time-Delay Systems Based on T-S Models

Authors: Yang Cui; Kaiqing Liu; Yang Zhao; Xue Wang
Journal: Mathematical Problems in Engineering
 (2013)

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## Abstract

The problem of robustH∞ control for a class of uncertain switched fuzzy time-delay systems is discussed for system described by T-S fuzzy model with Lyapunov stable theory and linear matrix inequality approach. A sufficient condition in terms of the LMI is derived such that the stability of the closed-loop systems is guaranteed. The continuous state feedback controller is built to ensure the asymptotically stable closed-loop system for all allowable uncertainties, with the switching law designed to implement the global asymptotic stability of uncertain switched fuzzy time-delay systems. In this model, each and every subsystem of the switched systems is an uncertain fuzzy one to which the parallel distributed compensation
(PDC) controller of each sub fuzzy system system is proposed with its main condition given in a more solvable form of convex combinations. Such a switched control system is highly robust to varying parameters. A simulation shows the feasibility and effectiveness of the design method.


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## Body

## 1. Introduction

In recent years, as intelligent control method, the research of the fuzzy system has been paid extensive attention [1–3]. The T-S model which is a fuzzy system is the most effective system model. This paper is focused on a class of uncertain switched time-delay systems, in which each subsystem is T-S fuzzy model. The T-S fuzzy model is a kind of fuzzy system proposed by Sugeno et al. [4, 5], which is described by a set of fuzzy IF-THEN rules representing local linear input-output relations of a nonlinear system. The main idea of T-S fuzzy model is to express the local dynamics of each fuzzy rule by a linear system model and to express the overall system by fuzzy “blending” of the local linear system models. The stability studies based on this model fuzzy system have had yielded fruitful results [6–10]. Recently, considerable effort [11–13] has been contributed to the problem of H∞ robust fuzzy control for a class of nonlinear systems that can be represented by T-S fuzzy models.On the other hand, switched systems are an important class of hybrid systems. A switched system consists of a number of subsystems, both continuous time and discrete time dynamic systems, and a switching law, which orchestrates the switching between the subsystems. The applications in robot control systems, computer disc drives, and other engineering systems indicate that switched systems have extensive practice background. Therefore, it has the important theoretical significance and practical value, which has yielded fruitful research results [14–20]. A switched system is called a switched fuzzy system if all subsystems are fuzzy systems. This class of systems can often more precisely describe continuous dynamics and discrete dynamics as well as their interactions in actual systems. Compared with the results on stability of switched systems and those of fuzzy control systems, the results on switched fuzzy systems are very few. Reference [21] presents a novel switched T-S fuzzy control design approach based on control Lyapunov function. Reference [22] advances a fuzzy-basis-dependent and mode-dependent Lyapunov function to study the problems of stability analysis and H∞ controller design for discrete-time switched fuzzy systems.This paper will study the problem of designing state feedbackH∞ controllers for continuous-time T-S switched fuzzy systems. A new type of state feedback controllers, namely, switched parallel distributed compensation (PDC) controllers, are proposed, which are switched based on the values of membership functions. The problem begins with the representation models for switched systems. The design method inherits some hybrid features and presents the information of fuzzy system. The delay-independent sufficient condition in terms of the LMI is derived such that the quadratic stability of the closed-loop systems is guaranteed. The state feedback controller is built to ensure the asymptotically stable closed-loop system for all allowable uncertainties, with the switching law designed to implement the global asymptotic stability of uncertain fuzzy time-delay switched systems. In this model, each and every subsystem of the switched systems is an uncertain fuzzy one to which the PDC controller of each sub fuzzy system system is proposed with its main condition given in a more solvable form of convex combinations. Numerical example is given to illustrate the effectiveness of the proposed method.


## 2. Problem Formulation

In this section, we consider the continuous uncertain switched fuzzy time-delay model; namely, every subsystem of switched systems is uncertain fuzzy time-delay system. Consider the following:R
σ
l: if ξ1(t) is Mσ1l⋯ and ξp(t) is Mσpl, then
(1)x˙(t)=[Aσl+ΔAσl(t)]x(t)+[Adσl+ΔAdσl(t)]x(t-d)+Bσlu(t)+B1σlw(t),z(t)=Cσlx(t),x(t)=φ(t),∀t∈[-d,0],l=1,2,…,Nσ,
where σ=σ(t):[0,+∞)→{1,2,…,N}=N- is the switching signal to be designed (Figure 1).Figure 1
The switching signal.M
σ
1
l
,
…
,
M
σ
p
l denote fuzzy sets in the σth switched subsystem. Rσl denotes the lth fuzzy inference rule in the σth switched subsystem. Nσ is the number of inference rules in the σth switched subsystem, and fuzzy rules are selected in every switched subsystem. x(t)∈Rn is the state variable vector, u(t) is the input variable, z(t) is output variable vector, w(t) is external disturbance of the switched systems, and satisfies w(t)∈L2[0,T]. d is the delay constant, and satisfies d≥0. Aσl, Adσl, Bσl, B1σl, and Cσl are known constant matrices of appropriate dimensions of the σth switched subsystem. ΔAσl and ΔAdσl are the uncertain matrices corresponding to the σth switched subsystem with appropriate dimensions. φ(t) denotes a differentiable vector-valued initial function. ξ1(t),ξ2(t),…,ξn(t) are the premise variables.In this paper, the switching signalσ=σ(t) is state dependent; namely, σ∈N-, the switching signal σ=σ(t)=i, and the switching signal is totally described by vi={1,x(t)∈Ω~i0,x(t)∉Ω~i, that is, if and only σ=σ(t)=i, vi(x(t))=1. Ω~1,Ω~2,…,Ω~n will be designed later.Through the functionvi(x(t)), the global model of the ith switched subsystem is described by
(2)x˙(t)=∑i=1nvi(t)∑l=1Niηil(ξ(t))[Ail+ΔAil(t)]x(t)+[Adil+ΔAdil(t)]x(t-d)+Bilu(t)+B1ilw(t),z(t)=∑i=1nvi(t)∑l=1Niηil(ξ(t))Cilx(t),x(t)=φ(t),∀t∈[-d,0],i=1,2,…,n,l=1,2,…,Ni,
where 0≤ηil(ξ(t))≤1, and ∑l=1Niηil(ξ(t))=1.Also,ωil(ξ(t))=∏ρ=1pMiρl(ξρ(t)), and ηil=ωil(ξ(t))/∑l=1Niωil(ξ(t)), where Miρl(ξρ(t)) denotes the membership function, and ξρ(t) belongs to the fuzzy set Miρl.Definition 1.
TheH∞ control problem for the switched fuzzy system (1) is stated as follows.
Let a constantγ>0 be given. Find a continuous state feedback controller u=u(x) for each subsystem and a switching law i=σ(t) such that(1)
the closed-loop system is asymptotically stable whenw(t)=0,(2)
the outputz satisfies ∥z∥L2[0,T]<γ∥w∥L2[0,T] under the zero initial condition.


## 3. Main Results

Assumption 2.
The uncertainties can be represented and emulated as(3)[ΔAil,ΔAdil]=EilΣil(t)[F1il,F2il],i=1,2,…,n,l=1,2,…,Ni,
where, Eil, F1il, and F2il are known constant matrices, and Σil(t) is an unknown time-varying matrix satisfying ΣilT(t)Σil(t)<I,  for all  t≥0.Here, the PDC fuzzy controller design method is used for each sub fuzzy system namely, fuzzy controller and system (2) have the same fuzzy inference premise variables. Consider the following:R
j
l: if ξ1(t) is Mj1l⋯ and ξp(t) is Mjpl, then
(4)u(t)=Kjlx(t),j=1,2,…,n,l=1,2,…,Nj.Thus, the global controller is(5)u(t)=∑i=1nvi(t)∑l=1Niηil(ξ(t))Kilx(t).The global closed-loop system can be expressed as follows:(6)x˙(t)=∑i=1nvi(t)∑l=1Niηil(ξ(t))×∑h=1Niηih(ξ(t))[(Ail+BilKih+ΔAil(t))x(t)+(Adil+ΔAdil(t))x(t-d)+B1ilw(t)],z(t)=∑i=1nvi(t)∑l=1Niηil(ξ(t))Cilx(t),x(t)=φ(t),∀t∈[-d,0],i=1,2,…,n,h,l=1,2,…,Ni.Lemma 3 (see [23]).
LetR1 and R2 be real matrices of appropriate dimensions, with ΣT(t)Σ(t)≤I; then, one has that for any scalar α>0,
(7)R1Σ(t)R2+R2TΣT(t)R1T≤αR1R1T+α-1R2TR2.Lemma 4 (Schur complement).
The matrix[QRRTS]<0 is symmetrical matrix, and then the following two functions are equivalent:
(8)S<0,Q-RS-1RT<0,orQ<0,S-RQ-1RT<0.Theorem 5.
Let a constantγ>0 be given. If there exist the symmetric positive definite matrices X,Qi∈Rn×n, scalar ε,ε1,ε2>0, αi(αi∈[0,1],∑i=1nαi=1), and the matrices Wih satisfying (9)∑i=1nαi[(AilX+BilWih)T+(AilX+BilWih)+Qi*****0EilT-εI00000F1ilX0-ε1-1I0000CilX00-I000B1ilT000-γ2I0000000-ε2-1I*XAdilT0000XF2ilT-Qi]<0,then system (6) is globally asymptotically stable, and H∞ robust control problem is solvable under the switching law σ(t), where ε-1=ε1-1+ε2-1.
So, the gain matrices of switched fuzzy controllers and the switching law are given as follows:(10)Kih=WihX-1,(11)σ(t)=argmini∈N-maxl,h{xT[Rilh+γ-2PB1ilB1ilTP+CilCilTPAdilAdilTP-Si+ε2F2ilTF2il]x,l,h=1,2,…,Ni{[Rilh+γ-2PB1ilB1ilTP+CilCilTPAdilAdilTP-Si+ε2F2ilTF2il]x,}},where
(12)P=X-1,Rilh=(Ail+BilKih)TP+P(Ail+BilKih)+Si+ε-1PEilTEilP+ε1F1ilTF1il.Proof.
We defineKih=WihX-1 and Si=X-1QiX-1.
So, choose the Lyapunov function as follows:(13)V(t)=xT(t)Px(t)+∫t-dtxT(τ)Six(τ)dτ.
We now compute the time derivative of the function (13):
(14)V˙(t)=∑i=1nvi(t)∑l=1Niηil(ξ(t))×∑i=1nηih(t){xT[(Ail+BilKil)TP+P(Ail+BilKil)+ΔAilT(t)P+PΔAil(t)+Si]x+xTPB1ilw+wTB1ilTPx+x(t-d)TAdilTPx+xTPAdilx(t-d)+x(t-d)TΔAdilTPx+xTΔPAdilx(t-d)-x(t-d)TSix(t-d)}.
It follows from Lemma3 that for any of scalars ε1>0 and ε2>0, we have that
(15)ΔAilT(t)P+PΔAil(t)=F1ilTΣiT(t)EilTP+PEilΣi(t)F1il≤ε1F1ilTF1il+ε1-1PEilEilTP,xT(t-d)ΔAilT(t)Px+xTPΔAil(t)x(t-d)=xT(t-d)F2ilTΣijT(t)EilTPx+xTPEilΣil(t)F2ilx(t-d)≤ε2xT(t-d)F2ilTF2ilx(t-d)+ε2-1xTPEilEilTPx.
When w(t)=0, the function (14) can be given as follows: (16)V˙(t)≤∑i=1nvi(t)∑l=1Niηil(ξ(t))×∑i=1nηih(ξ(t)){xT[(Ail+BilKil)TP+P(Ail+BilKil)+ε1F1ilTF1il+ε-1PEilEilTP+Si(Ail+BilKil)T]x+x(t-d)TAdilTPx+xTPAdilx(t-d)-xT(t-d)(Si-ε2F2ilTF2il)x(t-d)(Ail+BilKil)T}=∑i=1nvi(t)∑i=1nηil(ξ(t))×∑i=1nηih(ξ(t))[xTxT(t-d)][RilhPAdilAdilTP-Si+ε2F2ilTF2il]×[xx(t-d)].That is,
(17)V˙(t)≤∑i=1nvi(t)∑i=1nηil(ξ(t))×∑i=1nηih(ξ(t))ζT(t)[RilhPAdilAdilTP-Si+ε2F2ilTF2il]ζ(t),
where ζ(t)=[xx(t-d)]. So, we need to prove that [RilhPAdilAdilTP-Si+ε2F2ilTF2il]<0, the closed-loop system (6) is asymptotically stable when w(t)=0.
Inequalities (9) are pre- and postmultiplied by the transformation diag{P,I,I,I,I,I,P}, respectively. In turn, we have tha following:(18)∑i=1nαi[(Ail+BilKih)TP+P(Ail+BilKih)+Si*****0EilTP-εI00000F1il0-ε1-1I0000Cil00-I000B1ilTP000-γ2I0000000-ε2-1I*AdilTP0000F2ilT-Si]<0.
Due to Lemma4, inequality (18) can be changed as follows:
(19)∑i=1nαi[Rilh+γ-2PB1ilB1ilTP+CilCilTPAdilAdilTP-Si+ε2F2ilTF2il]<0.
So, for any x∈Rn∖{0},(20)∑i=1nαi{xT[Rilh+γ-2PB1ilB1ilTP+CilCilTPAdilAdilTP-Si+ε2F2ilTF2il]x}<0.
Due to the switching law (11), for all  t≥0, the inequality is tenable as follows:
(21)xT[Rilh+γ-2PB1ilB1ilTP+CilCilTPAdilAdilTP-Si+ε2F2ilTF2il]x<0.
Let(22)Ωi={x∈Rn∣xT×[Rilh+γ-2PB1ilB1ilTP+CilCilTPAdilAdilTP-Si+ε2F2ilTF2il]x<0xT}.
Constructing the sets Ω~1=Ω1,…,Ω~n=Ωn-⋃i=1n-1Ω~i, obviously, we have ⋃i=1nΩ~i=Rn∖{0}, and Ω~i∩Ω~j=Φ,i≠j.
Now, we focus on designing a switching law as follows:(23)σ=i,x(t)∈Ω~i.
When w=0, ∑i=1nαi[RilhPAdilAdilTP-Si+ε2F2ilTF2il]<0 is correct. So, we have that V˙(t)≤∑i=1nvi(t)∑i=1nηil(ξ(t))∑i=1nηih(ξ(t))ζT(t)[RilhPAdilAdilTP-Si+ε2F2ilTF2il]ζ(t)<0; so, the system is asymptotically stable.
From the design of switching law, we can obtain thatV˙(t)<0,  for all  x(t)≠0; that is, the switching fuzzy controllers can make the system (6) asymptotically stable, when w=0.
To demonstrate that theH∞ control problem of system is solvable, we define that
(24)J=∫0∞[zT(t)z(t)-γ2wT(t)w(t)]dt.
Thus, when the initial state isx(t)=0, for any nonzero vector w(t)∈L2[0,∞), the function J can be written as follows:
(25)J=∫0∞[zT(t)z(t)-γ2wT(t)w(t)+V˙(t)]dt-[V(∞)-V(0)]=∫0∞[zT(t)z(t)-γ2wT(t)w(t)+V˙(t)]dt-V(∞)≤∫0∞[zT(t)z(t)-γ2wT(t)w(t)+V˙(t)]dt.
Due to the Lemma3, the function (14) can be changed as follows:
(26)V˙(t)≤∑i=1nvi(t)∑l=1Niηil(ξ(t))×∑i=1nηih(ξ(t)){xT[(Ail+BilKil)TP+P(Ail+BilKil)+ε1F1ilTF1il+ε-1PEilEilTP+Si+γ-2PB1ilB1ilTP]x+γ2wTw+x(t-d)TAdilTPx+xTPAdilx(t-d)-xT(t-d)×(Si-ε2F2ilTF2il)x(t-d)}.
So, the functionJ can be further expressed as follows: (27)J≤∑i=1nvi(t)∑l=1Niηil(ξ(t))∑i=1nηih(ξ(t))×∫0∞{xT[(Ail+BilKil)TP+P(Ail+BilKil)+ε1F1ilTF1il+ε-1PEilEilTP+Si+γ-2PB1ilB1ilTP+CilTCil]x+x(t-d)TAdilTPx+xTPAdilx(t-d)-xT(t-d)(Si-ε2F2ilTF2il)x(t-d)}dt=∑i=1nvi(t)∑l=1Niηil(ξ(t))∑i=1nηih(ξ(t))×∫0∞[xTxT(t-d)][Rilh+γ-2PB1ilB1ilTP+CilCilTPAdilAdilTP-Si+ε2F2ilTF2il]×[xx(t-d)]dt=∑i=1nvi(t)∑l=1Niηil(ξ(t))∑i=1nηih(ξ(t))×∫0∞ζT(t)[Rilh+γ-2PB1ilB1ilTP+CilCilTPAdilAdilTP-Si+ε2F2ilTF2il]×ζ(t)dt.From the design of switching law, we get thatJ<0; that is, for any w(t)∈L2[0,∞), ∥z∥L2[0,T]<γ∥w∥L2[0,T] holds.


## 4. Simulation Example

The switched systems (1) consists of two fuzzy subsystems, and each subsystem has two fuzzy rules; that is, n=2,  and  Ni=2,  where  i=1,2. Hence, we approximate the system by the following modes:R
1
1: if ξ1(t) is M111, then
(28)x˙(t)=[A11+ΔA11(t)]x(t)+[Ad11+ΔAd11(t)]x(t-d)+B11u(t)+B111w(t),z(t)=C11x(t),R12: if ξ1(t) is M112, then
(29)x˙(t)=[A12+ΔA12(t)]x(t)+[Ad12+ΔAd12(t)]x(t-d)+B12u(t)+B112w(t),z(t)=C12x(t),R21: if ξ1(t) is M211, then
(30)x˙(t)=[A21+ΔA21(t)]x(t)+[Ad21+ΔAd21(t)]x(t-d)+B21u(t)+B121w(t),z(t)=C21x(t),R22: if ξ1(t) is M212, then
(31)x˙(t)=[A22+ΔA22(t)]x(t)+[Ad22+ΔAd22(t)]x(t-d)+B22u(t)+B122w(t),z(t)=C22x(t),
where
(32)A11=[-514-2],A12=[-6-25-3],A21=[-617-4],A22=[-4-36-2],Ad11=[0.10.60.60.2],Ad12=[0.30.40.50.3],Ad21=[0.40.30.20.1],Ad22=[0.50.10.30.6],B11=[-1-2],B12=[1-2],B21=[-1-2],B22=[1-2],B111=[0.46541.4937],B112=[0.34781.5417],B121=[0.83241.6574],B122=[0.64971.8876],C11=[-1-1],C12=[1-1],C21=[-1-1],C22=[1-1],E11=E12=E21=E22=[0.40.2],F111=F112=F121=F122=[0.30.2],F111=F212=F221=F222=[0.20.1],α1=α2=0.5,Σ11(t)=Σ12(t)=Σ21(t)=Σ22(t)=0.2sint,w(t)=e-0.5tsin(2πt).The membership functions are, respectively,(33)μM111(x1)=μM211(x1)=1-11+e-4x1,μM112(x1)=μM212(x1)=11+e-4x1.Letγ=1/2, choosing u(t)=-∑i=1nvi(t)∑l=1Niηil(ξ(t))Kilx(t),   where i=1,2andNi=2.For the linear matrix inequality (9) of Theorem 5, the following can be obtained with LMI toolbox:
(34)X=[6.3950-1.4630-1.46304.1467],Q1=[29.2473-21.0996-21.099671.6683],Q2=[26.7269-19.7250-19.725069.5959],W11=[22.563034.4393],W12=[-30.483148.9880],W21=[6.088448.0289],W22=[-14.954549.1351],ω1=0.0087,ω2=0.0086.
Because Kih=WihX-1, we have that
(35)K11=[5.904810.3884],K12=[-2.245311.0215],K21=[3.918012.9647],K22=[0.405011.9920].
Designing the switching law by (11), the system state responses with the initial condition x(0)=[1-0.8] are depicted by Figure 2 which is obtained by the Matlab simulation.Figure 2
The system state responses.Obviously, the closed-loop switched fuzzy system is stable under the corresponding switching laws and the robust fuzzy controllers in Figure3, in spite of the influences by parameter uncertainties and time-delay, and possesses the strong robustness. Comparing their results with others, the simulation results indicate that the design controllers can satisfy the robust performance requirements under the corresponding switching laws.Figure 3
The system control inputs.


## 5. Conclusions

In this paper, the problem of state feedback robustH∞ control for a class of uncertain switched fuzzy time-delay systems based on T-S models is discussed. The sufficient condition in terms of the LMI is derived such that the quadratic stability of the closed-loop systems is guaranteed. A new type of continuous state feedback PDC controllers are built to ensure the asymptotically stable closed-loop system with H∞ disturbance attenuation level γ, with the switching law designed to implement the global asymptotic stability of uncertain fuzzy time-delay switched systems. The results are also extended to the interval fuzzy time-delay systems. Simulation results demonstrate that a quality control performance has been achieved.



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