# Robust FuzzyH∞ Output Feedback Control for a Class of Nonlinear Uncertain Systems with Mixed Time Delays

Authors: Xiaona Song; Shanzhong Liu
Journal: Mathematical Problems in Engineering
 (2013)

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## Abstract

This paper studies the problem of delay-dependent robustH∞ output feedback control for a class of uncertain fuzzy neutral systems with both discrete and distributed delays. The system is described by a state-space Takagi-Sugeno fuzzy model with distributed delays and norm-bounded parameter uncertainties. The purpose is to design a fuzzy dynamic output feedback controller which ensures the robust asymptotic
stability of the closed-loop fuzzy neutral system and satisfies an H∞ norm bound constraint for all admissible uncertainties. In terms of linear matrix inequalities, sufficient conditions for the solvability of this problem are presented. Finally, a numerical example is included to demonstrate the effectiveness of the proposed method.


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## Body

## 1. Introduction

Fuzzy control, as a promising way to approach nonlinear control problems, has had an impact on the control community [1–4]. Furthermore, the Takagi-Sugeno (T-S) fuzzy dynamic model [5–9] is nonlinear system described by fuzzy IF-THEN rules which give local linear representations of the underlying systems [10, 11]. It has been shown that such models can describe a wide class of nonlinear systems. Hence it is important to investigate their stability analysis and controller design problems, and in the past two decades many stability and control issues related to the T-S fuzzy systems have been studied; see, for example, [12–15] and the references cited therein.On the other hand, time delays exist commonly in dynamic systems due to measurement, transmission, transport lags, and so forth [16], which have been generally regarded as a main source of instability and poor performance. Thus the analysis of time delay systems and controller design for them is very important [17–20]. Recently, T-S fuzzy systems with time delays have attracted a great deal of interests. For example, in [21], the stability analysis and stabilization problems for T-S fuzzy delay systems were considered, and state feedback fuzzy controllers and fuzzy observers were designed. The robust H∞ control problem for T-S fuzzy systems with time delays was investigated in [22, 23], and state feedback fuzzy controllers were designed; the corresponding results for the discrete case can be found in [24, 25], while in [26, 27], the robust H∞ output feedback controllers were designed for the continuous and discrete fuzzy time-delay systems, respectively.Quite recently,T-S fuzzy time-delay systems of neutral type were introduced in [28], where both the stabilization and H∞ control problems were studied. It should be point out that distributed delays were not taken into account. While in [29], authors considered the problems of robust stabilization and robust H∞ control for uncertain T-S fuzzy neutral systems with both discrete and distributed time delays. However, the results in [28, 29] were all delay-independent. It is known that delay-dependent results are less conservative than delay-independent ones, especially in the case when the size of the delay is small. On the other hand, these obtained results, however, are mainly dealt with through a state feedback controller design method that requires all state variables to be available. In many cases, this condition is too restrictive. So it is meaningful to investigate the output feedback control method. To the best of our knowledge, so far, there are no results of delay-dependent robust H∞ output feedback control for uncertain fuzzy neutral systems with both discrete and distributed delays. This motives the present studies.In this paper, we consider the delay-dependent robustH∞ output feedback control problem for fuzzy neutral systems with both discrete and distributed delays. The system to be considered is described by a state-space T-S fuzzy model with mixed delays and norm-bounded parameter uncertainties. The distributed delays are assumed to appear in the state equation, and the uncertainties are allowed to be time varying but norm bounded. The aim of this paper is to design a full-order fuzzy dynamic output feedback controller such that the resulting closed-loop system is robustly asymptotically stable while satisfying an H∞ norm condition with a prescribed level irrespective of the parameter uncertainties. A sufficient condition for the solvability of this problem is proposed in terms of linear matrix inequalities (LMIs). When these LMIs are feasible, an explicit expression of a desired output feedback controller is also given.Notation. Throughout this paper, for real symmetric matrices X and Y, the notation X≥Y (resp., X>Y) means that the matrix X-Y is positive semidefinite (resp., positive definite). I is an identity matrix with appropriate dimension. ℕ is the set of natural numbers. ℒ2[0,∞) refers to the space of square summable infinite vector sequences. ∥·∥2 stands for the usual ℒ2[0,∞) norm. The notation MT represents the transpose of the matrix M. Matrices, if not explicitly stated, are assumed to have compatible dimensions. “*” is used as an ellipsis for terms induced by symmetry.


## 2. Problem Formulation

A continuousT-S fuzzy neutral model with distributed delays and parameter uncertainties can be described by the following.Plant Rulei: if s1(t) is μi1 and ⋯ and sp(t) is μip, then
(1)x˙(t)=[Ai+ΔAi(t)]x(t)+[A1i+ΔA1i(t)]x(t-τ1)+[A2i+ΔA2i(t)]x˙(t-τ2)+[A3i+ΔA3i(t)]∫t-τ3t‍x(s)ds+[Bi+ΔBi(t)]u(t)+D1iw(t),y(t)=[Ci+ΔCi(t)]x(t)+C1ix(t-τ1)+D2iw(t),z(t)=Eix(t)+E1ix(t-τ1)+Giu(t),x(t)=ϕ(t)∀t∈[-τ-,0],i=1,2,…,r,
where μij is the fuzzy set, r is the number of IF-THEN rules, and s1(t),…,sp(t) are the premise variables. Throughout this paper, it is assumed that the premise variables do not depend on control variables; x(t)∈ℝn is the state; u(t)∈ℝm is the control input; y(t)∈ℝs is the measured output; z(t)∈ℝq is the controlled output; w(t)∈ℝp is the noise signal; τi>0  (i=1,2,3) are integers representing the time delay of the fuzzy systems; τ-=max{τ1,τ2,τ3}; Ai, A1i, A2i, A3i, Bi, Ci, C1i, D1i, D2i, Ei, E2i, and Gi are known real constant matrices; ΔAi(t), ΔA1i(t), ΔA2i(t), ΔA3i(t), ΔBi(t), and ΔCi(t) are real-valued unknown matrices representing time-varying parameter uncertainties and are assumed to be of the form
(2)[ΔAi(t)ΔA1i(t)ΔA2i(t)ΔA3i(t)ΔBi(t)ΔCi(t)]=MiFi(t)[N0iN1iN2iN3iN4iN5i]i=1,2,…,r,
where Mi, N0i, N1i, N2i, N3i, N4i, and N5i are known real constant matrices and Fi(·):ℕ→ℝl1×l2 are unknown time-varying matrix function satisfying
(3)Fi(t)TFi(t)≤I,∀t.
The parameter uncertainties ΔAi(t), ΔA1i(t), ΔA2i(t), ΔA3i(t), ΔBi(t), and ΔCi(t) are said to be admissible if both (2) and (3) hold.Then the final output of the fuzzy neutral system is inferred as follows:(4)x˙(t)=∑i=1r‍hi(s(t)){∫t-τ3t‍[Ai+ΔAi(t)]x(t)+[A1i+ΔA1i(t)]x(t-τ1)+[A2i+ΔA2i(t)]x˙(t-τ2)+[A3i+ΔA3i(t)∫t-τ3t‍x(s)ds]∫t-τ3t‍+[Bi+ΔBi(t)]u(t)+D1iw(t)},y(t)=∑i=1r‍hi(s(t)){[Ci+ΔCi(t)]x(t)+C1ix(t-τ1)+D2iw(t)},z(t)=∑i=1r‍hi(s(t))[Eix(t)+E1ix(t-τ1)+Giu(t)],
where
(5)hi(s(t))=ϖi(s(t))∑i=1rϖi(s(t)),ϖi(s(t))=∏j=1p‍μij(sj(t)),s(t)=[s1(t)s2(t)⋯sp(t)],
in which μij(sj(t)) is the grade of membership of sj(t) in μij. Then, it can be seen that
(6)ϖi(s(t))≥0,i=1,…,r,∑i=1r‍ϖi(s(t))>0,
for all t. Therefore, for all t,
(7)hi(s(t))≥0,i=1,…,r,∑i=1r‍hi(s(t))=1,∀t.
Now, by the parallel distributed compensation (PDC), the following full-order fuzzy dynamic output feedback controller for the fuzzy neutral system in (4) is considered.Control Rulei: if s1(t) is μi1 and ⋯ and sp(t) is μip, then
(8)x^.(t)=Akix^(t)+Bkiy(t),(9)u(t)=Ckix^(t),i=1,2,…,r,(10)x^(t)=0,t≤0,
where x^(t)∈ℝn is the controller state and Aki, Bki, and Cki are matrices to be determined later. Then, the overall fuzzy output feedback controller is given by
(11)x^.(t)=∑i=1r‍hi(s(t))[Akix^(t)+Bkiy(t)],(12)u(t)=∑i=1r‍hi(s(t))Ckix^(t),i=1,2,…,r.
From (4) and (11), one can obtain the closed-loop system as
(13)e˙(t)=∑i=1r‍∑j=1r‍hi(s(t))hj(s(t))×{[Aij+ΔAij(t)]e(t)∫t-τ3t‍+[A1ij+ΔA~1i(t)]e(t-τ1)+[A~2i+ΔA~2i(t)]e˙(t-τ2)+[A~3i+ΔA~3i(t)]×∫t-τ3t‍e(s)ds+Dijw(t)},(14)z(t)=∑i=1r‍∑j=1r‍hi(s(t))hj(s(t))[Eije(t)+E~1ie(t-τ1)],
where
(15)e˙(t)=[x˙(t)x^.(t)],Aij=[AiBiCkjBkjCiAkj],ΔAij(t)=[ΔAi(t)ΔBi(t)CkjBkjΔCi(t)0],A1ij=[A1i0BkjC1i0],ΔA~1i(t)=[ΔA1i(t)000],A~2i=[A2i000],ΔA~2i(t)=[ΔA2i(t)000],A~3i=[A3i000],ΔA~3i(t)=[ΔA3i(t)000],Dij=[D1iBkjD2i],Eij=[EiGiCkj],E~1i=[E1i0].
Then, the problem of robust fuzzy H∞ control problem to be addressed in the previous is formulated as follows: given an uncertain distributed delay fuzzy neutral system in (4) and a scalar γ>0, determine a dynamic output feedback fuzzy controller in the form of (8) and (9) such that the closed-loop system in (13) and (14) is robustly asymptotically stable when w(t)=0 and
(16)∥z∥2<γ∥w∥2,
under zero-initial conditions for any nonzero w(t)∈ℒ2[0,∞) and all admissible uncertainties.


## 3. Main Results

In this section, an LMI approach will be developed to solve the problem of robust output feedbackH∞ control of uncertain distributed delay fuzzy neutral systems formulated in the previous section. We first give the following results which will be used in the proof of our main results.Lemma 1 (see [30]).
Let𝒜, 𝒟, 𝒮, 𝒲, and F be real matrices of appropriate dimensions with 𝒲>0 and F satisfying FTF≤I. Then one has the following.(1)
For any scalarϵ>0 and vectors x, y∈ℝn,
(17)2xT𝒟ℱ𝒮y≤ϵ-1xT𝒟𝒟Tx+ϵyT𝒮T𝒮y.(2)
For any scalarϵ>0 such that 𝒲-ϵ𝒟𝒟T>0,
(18)(𝒜+𝒟ℱ𝒮)T𝒲-1(𝒜+𝒟ℱ𝒮)≤𝒜T(𝒲-ϵ𝒟𝒟𝒯)-1𝒜+ϵ-1𝒮T𝒮.Lemma 2 (see [31]).
Given any matrices𝒳, 𝒴, and 𝒵 with appropriate dimensions such that 𝒴>0, then, one has
(19)𝒳T𝒵+𝒵T𝒳≤𝒳T𝒴𝒳+𝒵T𝒴-1𝒵.Theorem 3.
The uncertain fuzzy neutral delay system in (13) and (14) is asymptotically stable, and (16) is satisfied if there exist matrices P>0, P1>0, P2>0, Q1>0, Q2>0, R1, and R2 and scalars ϵij>0, 1≤i≤j≤r, such that the following LMIs hold:(20)[Π1iiΠ2iiPA~2iPA~3iPDiiτ1R1EiiTΓiiPMiiNiiT*J1000τ1R2E1iTΓ1ii00**-P20000Γ2i00***-Q2000Γ3i00****-γ2I00Γ4ii00*****-τ1Q10000******-I000*******-J2TJ2TMii0********-ϵiiI0*********-ϵii-1I]<0,1≤i≤r,[Λ1ijΛ2ijΛ3ij*Λ4ijΛ5ij**Λ6ij]<0,1≤i<j≤r,where
(21)Λ1ij=[Π1ij+Π1jiΠ2ij+Π2jiPA~2i+PA~2jPA~3i+PA~3j*2J100**-2P20***-2Q2],Λ2ij=[PDij+PDji2τ1R1EijT+EjiTΓij+Γji02τ1R2E~1iT+E~1jTΓ1ij+Γ1ji000Γ2i+Γ2j000Γ3i+Γ3j],Λ3ij=[PMijPMjiNijTNjiT00N~1iTN~1jT00N~2iTN~2jT00N~3iTN~3jT],Λ4ij=[-2γ2I00Γ4ij+Γ4ji*-2τ1Q100**-2I0***-2J2T],Λ5ij=[000000000000J2TMijJ2TMji00],Λ6ij=[-ϵijI000*-ϵjiI00**-ϵij-1I0***-ϵji-1I],Υ1=HT(P1+τ32Q2-R1-R1T)H,Π1ij=Υ1+PAij+AijTP,Υ2=HT(R1-R2T),Π2ij=Υ2+PA1ij,Γij=AijT(P2+τ1Q1),Γ1ij=A1ijT(P2+τ1Q1),Γ2i=A~2iT(P2+τ1Q1),Γ3i=A~3iT(P2+τ1Q1),Γ4ij=DijT(P2+τ1Q1),J1=-P1+R2+R2T,J2=P2+τ1Q1,Mij=[Mi00BkjMi],F~i(t)=[Fi(t)00Fi(t)],Nij=[N0iN3iCkjN4i0],N~1i=[N1i000],N~2i=[N2i000],N~3i=[N3i000].Proof.
To establish the robust stability of the system in (13), we consider (13) with w(t)≡0; that is,
(22)e˙(t)=∑i=1r‍∑j=1r‍hi(s(t))hj(s(t))×{[Aij+ΔAij(t)]e(t)∫t-τ3t‍+[A1ij+ΔA~1i(t)]e(t-τ1)+[A~2i+ΔA~2i(t)]e˙(t-τ2)+[A~3i+ΔA~3i(t)]∫t-τ3t‍e(s)ds}.
For this system, we define the following Lyapunov function candidate:
(23)V(t)=V1(t)+V2(t)+V3(t)+V4(t)+V5(t)+V6(t),
where
(24)V1(t)=e(t)TPe(t),V2(t)=∫t-τ1t‍e(s)TP1e(s)ds,V3(t)=∫t-τ2t‍e˙(s)TP2e˙(s)ds,V4(t)=∫-τ10‍∫t+βt‍e˙(α)TQ1e˙(α)dαdβ,V5(t)=∫t-τ3t‍[∫st‍e(θ)Tdθ]Q2[∫st‍e(θ)dθ]ds,V6(t)=∫0τ3‍ds∫t-βt‍(θ-t+β)e(α)TQ2e(α)dαdβ.
The time derivative of V(t) is given by
(25)V˙(t)=V˙1(t)+V˙2(t)+V˙3(t)+V˙4(t)+V˙5(t)+V˙6(t),
where
(26)V˙1(t)=2e(t)TPe˙(t),V˙2(t)=e(t)TP1e(t)-e(t-τ1)TP1e(t-τ1),V˙3(t)=e˙(t)TP2e˙(t)-e˙(t-τ2)TP2e˙(t-τ2),V˙4(t)=τ1e˙(t)TQ1e˙(t)-∫t-τ1t‍e˙(α)TQ1e˙(α)dα,V˙5(t)=2∫t-τ3t‍(θ-t+τ3)e(t)TQ2e(θ)dθ-[∫t-τ3t‍e(θ)Tdθ]Q2[∫t-τ3t‍e(θ)dθ],V˙6(t)=12τ32e(t)TQ2e(t)-∫t-τ3t‍(θ-t+τ3)e(θ)TQ2e(θ)dθ.
Now, by Lemma 1, it can be shown that
(27)2e(t)TQ2e(θ)≤e(t)TQ2e(t)+e(θ)TQ2e(θ).
Therefore
(28)V˙5(t)≤12τ32e(t)TQ2e(t)+∫t-τ3t‍(θ-t+τ3)e(θ)TQ2e(θ)dθ-[∫t-τ3t‍e(θ)Tdθ]Q2[∫t-τ3t‍e(θ)dθ].
This together with (26) implies
(29)V˙5(t)+V˙6(t)≤τ32e(t)TQ2e(t)-α(t)TQ2α(t),
where
(30)α(t)=∫t-τ3t‍e(θ)dθ.
Then, there holds
(31)V˙(t)≤2e(t)TPe˙(t)+e(t)TP1e(t)-e(t-τ1)TP1e(t-τ1)+e˙(t)TP2e˙(t)-e˙(t-τ2)TP2e˙(t-τ2)+τ1e˙(t)TQ1e˙(t)-∫t-τ1t‍e˙(α)TQ1e˙(α)dα+τ32e(t)TQ2e(t)-α(t)TQ2α(t)+2e(t)TR1∫t-τ1t‍e˙(α)dα-2e(t)TR1[e(t)-e(t-τ1)]+2e(t-τ1)TR2∫t-τ1t‍e˙(α)dα-2e(t-τ1)TR2[e(t)-e(t-τ1)].
It follows from (22) and Lemma 2 that
(32)2e(t)TPe˙(t)=2e(t)TP∑i=1r‍∑j=1r‍hi(s(t))hj(s(t))×{[Aij+ΔAij(t)]e(t)∫t-τ3t‍+[A1ij+ΔA~1i(t)]e(t-τ1)+[A~2i+ΔA~2i(t)]e˙(t-τ2)+[A~3i+ΔA~3i(t)]∫t-τ3t‍e(s)ds}≤2∑i=1r‍∑j=1r‍hi(s(t))hj(s(t))e(t)TPA-ijβ(t)+∑i=1r‍∑j=1r‍hi(s(t))hj(s(t))×[ε1ij-1e(t)TPMijMijTPe(t)+ε1ijβ(t)TN~ijTN~ijβ(t)],
where
(33)A-ij=[AijA1ijA~2iA~3i],β(t)=[e(t)Te(t-τ1)Te˙(t-τ2)Tα(t)Te˙(α)T],N~ij=[NijN~1iN~2iN~3i].
It also can be verified that
(34)e˙(t)TP2e˙(t)=12∑i=1r‍∑j=1r‍∑u=1r‍∑v=1r‍hi(s(t))hj(s(t))hu(s(t))hv(s(t))β(t)T×{[A-ij+MijF~i(t)N~ij]T×P2[A-uv+MuvF~u(t)N~uv]+[A-uv+MuvF~u(t)N~uv]T[A-ij+MijF~i(t)N~ij]T×P2[A-ij+MijF~i(t)N~ij]}β(t)≤∑i=1r‍∑j=1r‍hi(s(t))hj(s(t))β(t)T×[A-ij+MijF~i(t)N~ij]TP2[A-ij+MijF~i(t)N~ij]≤∑i=1r‍∑j=1r‍hi(s(t))hj(s(t))β(t)T×[A-ijT(P2-1-ε2ij-1MijMijT)-1A-ij+ε2ijN~ijTN~ij]β(t),τ1e˙(t)TQ1e˙(t)≤∑i=1r‍∑j=1r‍hi(s(t))hj(s(t))β(t)T×{A-ijT[(τ1Q1)-1-ε3ij-1MijMijT]-1A-ij+ε3ijN~ijTN~ijA-ijT[(τ1Q1)-1-ε3ij-1MijMijT]-1A-ij}β(t).
Hence, with the support of the above conditions, we have
(35)V˙(t)≤2e(t)TPe˙(t)+e˙(t)TP2e˙(t)+τ1e˙(t)TQ1e˙(t)+e(t)T(P1+τ32Q2)e(t)-e(t-τ1)TP1e(t-τ1)-e˙(t-τ2)TP2e˙(t-τ2)-∫t-τ1t‍e˙(α)TQ1e˙(α)dα-α(t)TQ2α(t)+2e(t)TR1∫t-τ1t‍e˙(α)dα-2e(t)TR1[e(t)-e(t-τ1)]+2e(t-τ1)TR2∫t-τ1t‍e˙(α)dα-2e(t-τ1)T×R2[e(t)-e(t-τ1)]=∑i=1r‍∑j=1r‍hi(s(t))hj(s(t))β(t)T×[Ξij+A-ij(P2-1-ε2ij-1MijMijT)-1A-ij+A-ij(τ1-1Q1-1-ε3ij-1MijMijT)-1A-ij+δijN~ijTN~ij]β(t),
where
(36)Ξij=[ΣijPA1ij+R1-R2TPA~2iPA~3iτ1R1*-P1+R2+R2T00τ1R2**-P200***-Q20****-τ1Q1],Σij=PAij+AijTP+P1+τ32Q2-R1-R1T+ε1ij-1PMijMijTP,
for 1≤i≤j≤r. On the other hand, by applying the Schur complement formula to (20), we have that for 1≤i<j≤r,
(37)Γij<0,
and from this and (7), we have that V˙(t)<0 for all β(t)≠0. Therefore, the system in (13) is robustly asymptotically stable.
This completes the proof.Next, we will establish the robustH∞ performance of the system in (13) and (14) under the zero initial condition. To this end, we introduce
(38)J(t)=∫0t‍[z(s)Tz(s)-γ2w(s)Tw(s)]ds,
where t>0. Noting the zero initial condition, it can be shown that for any nonzero w(t)∈ℒ2[0,∞) and t>0,
(39)J(t)=∫0t‍[z(s)Tz(s)-γ2w(s)Tw(s)+V˙(s)]ds-V(t)≤∫0t‍[z(s)Tz(s)-γ2w(s)Tw(s)+V˙(s)]ds,
where V(s) is defined in (23), and then we have
(40)V˙(t)=∑i=1r‍∑j=1r‍hi(s(t))hj(s(t))×{2e(t)TP[Aij(t)e(t)+A1ij(t)e(t-τ1)+A2i(t)e˙(t-τ2)+A3i(t)α(t)+Dijw(t)]}+e(t)TP1e(t)-e(t-τ1)TP1e(t-τ1)+e˙(t)TP2e˙(t)-e˙(t-τ2)TP2e˙(t-τ2)+τ1e˙(t)TQ1e˙(t)-∫t-τ1t‍e˙(t)TQ1e˙(α)dα+τ32e(t)TQ2e(t)-α(t)TQ2α(t)+2e(t)TR1∫t-τ1t‍e˙(α)dα-2e(t)T×R1[e(t)-e(t-τ1)]+2e(t-τ1)TR2∫t-τ1t‍e˙(α)dα-2e(t-τ1)T×R2[e(t)-e(t-τ1)].
Then, by noting (14) and using Lemma 2, we have
(41)z(s)Tz(s)=∑i=1r‍∑j=1r‍∑u=1r‍∑v=1r‍hi(s(s))hj(s(s))hu(s(s))×hv(s(s))ζ(s)TE~ijTE~uvζ(s)≤∑i=1r‍∑j=1r‍hi(s(s))hj(s(s))ζ(s)TE~ijTE~ijζ(s).
It can be deduced that
(42)z(s)Tz(s)-γ2w(s)Tw(s)+V˙(s)≤∑i=1r‍∑j=1r‍hi(s(s))hj(s(s))ζ(s)TΠ1ijζ(s)=∑i=1r‍hi(s(s))2ζ(s)TΞ1iiζ(s)+2∑i,j=1,i<jr‍hi(s(s))hj(s(s))ζ(s)TΞ1ij+Ξ1ji2ζ(s),
where(43)Ξ1ij=[ΣijPA1ij(t)+R1-R2TPA2i(t)PA3i(t)PDijτ1R1EijTAijT(t)(P2+τ1Q1)*J1000τ1R2E~1iTA1ijT(t)(P2+τ1Q1)**-P20000A2iT(t)(P2+τ1Q1)***-Q2000A3iT(t)(P2+τ1Q1)****-γ2I00DijT(P2+τ1Q1)*****-τ1Q200******-I0*******-(P2+τ1Q1)T],ζ(s)=[e(t)Te(t-τ1)Te˙(t-τ2)Tα(t)Tw(t)Te˙(α)T]T,E~ij=[EijE~1i0000],for 1≤i<j≤r. Similar to the previous section, applying the Schur complement formula to the LMI in (20) results in Ξ1ij<0, 1≤i<j≤r, which together with (39) gives J(t)<0 for any nonzero w(t)∈ℒ2[0,∞) and t>0. Therefore, we have ∥z∥2<γ∥w∥2.Now, we are in a position to present a solution to the robust output feedbackH∞ control problem.Theorem 4.
Consider the uncertain fuzzy neutral delay system in (1), and let γ>0 be a prescribed constant scalar. The robust H∞ problem is solvable if there exist matrices X>0, Y>0, Q1, Q2, Φi, Ψi, and Ωi and scalars ϵij, 1≤i≤j≤r, such that the following LMIs hold:(44)[ΣiiΣ1iiΣ2iiΣ3iiR5*-R11R2ii00**R3ii0R6ii***-R40****-R7]<0,1≤i≤r,(45)[Σij+ΣjiΣ1ij+Σ1jiΣ~2ijΣ4ijΣ5ijΣ6ij0*-2R11R~2ij0000**R~3ij000R~6ij***-I000****-R800*****-I0******-I]<0,1≤i<j≤r,(46)[-Y-I-I-X]<0,where
(47)Θij=[AiX+XAiT+BiΨj+ΨjTBiTAi+ΩiTAiT+ΩiYAi+AiTY+ΦjCi+CiTΦjT]-R1-R1T,Θ1ij=[A1iXA1i0YA1i+ΦjC1i]+R1-R2T,Θ2i=[A2i0YA2i0],Θ3i=[A3i0YA3i0],Θ4ij=[D1iYD1i+ΦjD2i],Θ5ij=[XEiT+ΨjTGiTEiT],Θ6ij=[XAiT+ΨjTBiTΩjTAiTAiTY+CiTΦjT],Θ7ij=[Mi0YMiΦjMi],Θ8ij=[XN0iT+ΨjTN4iTXN5iTN0iTN5iT],J3i=[XE1iTE1iT],J4ij=[XA1i0A1iA1iY+C1iTΦjT],J5i=[XN1iT0N1iT0],J6i=[A2iTA2iTY00],J7i=[A3iTA3iTY00],J8ij=[D1iTD1iTY+D2iTΦjT],J9=Π2T+Π2-P2-τ1Q1,J10ij=[Mi0YMiΦjMi],Σij=[ΘijΘ1ijΘ2i*-U+R2+R2T0**-P2],Σ1ij=[Θ3iΘ4ijτ1R100τ1R2000],Σ2ij=[Θ5ijΘ6ijΘ7ijΘ8ijJ3iJ4ij0J5i0J6i0N~2iT],Σ~2ij=[Θ5ij+Θ5jiΘ6ij+Θ6jiΘ7ijΘ7jiΘ8ijΘ8jiJ3i+J3jJ4ij+J4ji00J5iJ5j0J6i+J6j00N~2iTN~2jT],Σ3ij=[Π1Tτ3Π1TTij000000],Tij=[0YA1i+ΦjC1i],R5=[000T1T10000],T1=[x0],Σ4ij=[T2ijT3ijT4ijT5ij00000000],T2ij=[(Cix-Cjx)T0],T3ij=[0Φj-Φi],T4ij=[(Ψj-Ψi)T0],T5ij=[0YBi-YBj],Σ5ij=[Π1Tτ3Π1TTij0000T10000],Σ6ij=[Tji0T4T20T1000000],R11=diag(Q2γ2IV),R2ij=[0J7i0N~3iT0J8ij000000],R~2ij=[0J7i+J7j00N~3iTN~3jT0J8ij+J8ji0000000000],R3ij=[-I000*-J9J10ij0**-ϵijI0***-ϵij-1I],R~3ij=[-2I00000*-2J9J10ijJ10ji00**-ϵijI000***-ϵjiI00****-ϵij-1I0*****-ϵji-1I],R4=diag(P1-1Q2-1I),R6ij=[00000T6ij000000],R~6ij=[00T5ijT3ij00000000],T6ij=[0YA1iT+ΦjC1i],R7=diag(III),R8=diag(12P1-112Q2-1II),Π1=[XIX0],Π2=[XIX0].
Furthermore a desired robust dynamic output feedback controller is given in the form of (11) with parameters as follows:
(48)AKi=(X-1-Y)-1(Ωi-YAiX-YBiΨi-ΦiCiX)X-1,BKi=(X-1-Y)-1Φi,CKi=ΨiX-1,1≤i≤r.Proof.
Applying the Schur complements formula to (44) and (45) and using Lemma 2 result in
(49)[Σ~iiΣ1iiΣ^2ii*-R11R2ii**R3ii]<0,1≤i≤r,[Σ^ijΣ1ij+Σ1jiΣ-2ij*-2R11R~2ij**R~3ij]<0,1≤i<j≤r,
where(50)Σ~ij=[Θij+Π1TP1Π1+(τ3Π1)TQ2(τ3Π1)Θ1ij+[00(YA1i+ΦjC1i)X0]Θ2i*-U+R2+R2T0**-P2],Σ^2ij=[Θ5ijΘ6ijΘ7ijΘ8ijJ3iJ4ij+[0X(A1iY+C1iTΦjT)00]0J5i0J6i0N~2iT],Σ^ij=Σ~ij+Σ~ji+[0(CiX-CjX)T(Φj-Φi)T+(Ψj-Ψi)T(YBi-YBj)T000*0000**000***00****0],Σ-2ij=Σ~2ij+[0(ΨjT-ΨiT)(BiTY-BjTY)+(XCiT-XCjT)(ΦjT-ΦiT)0000000000000000].Note that
(51)Π2T+Π2-P2-τ1Q1≤Π2T(P2+τ1Q1)-1Π2.
Then, by this inequality, it follows from (49) that
(52)[Σ~iiΣ1iiΣ^2ii*-R11R2ii**R^3ii]<0,1≤i≤r,[Σ^ijΣ1ij+Σ1jiΣ-2ij*-2R11R~2ij**R-3ij]<0,1≤i<j≤r,
where(53)R^3ij=[-I000*-Π2T(P2+τ1Q1)-1Π2J10ij0**-ϵijI0***-ϵij-1I],R-3ij=[-2I00000*-2Π2T(P2+τ1Q1)-1Π2J10ijJ10ji00**-ϵijI000***-ϵjiI00****-ϵij-1I0*****-ϵji-1I].Now, set
(54)P~=Π2Π1-1.
Then, by (46), it can be verified that P~>0. Noting the parameters in (48) and pre- and postmultiplying (52) by
(55)diag(Π1-T,Π1-T,I,I,I,Π1-T,I,Π2-T,I,…,I)
and its transpose, respectively, we have(56)[Π~1iiΠ~2iiPA~2iPA~3iPDiiτ1R~1EiiTΓiiPMiiNiiT*J1000τ1R~2E1iTΓ1ii00**-P20000Γ2i00***-Q2000Γ3i00****-γ2I00Γ4ii00*****-τ1Q10000******-I000*******-J2TJ2TMii0********-ϵiiI0*********-ϵii-1I]<0,1≤i≤r,[Λ1ijΛ2ijΛ3ij*Λ4ijΛ5ij**Λ6ij]<0,1≤i<j≤r,where
(57)Λ~1=HT(P1+τ32Q2-R~1-R~1T)H,Π~1ij=Λ~1+PAij+AijTP,Λ~2=HT(R~1-R~2T),Π~2ij=Λ~2+PA1ij,R~1=Π1TR1Π1.
Finally, by Theorem 3, the desired result follows immediately.Remark 5.
Theorem4 provides a sufficient condition for the solvability of the robust H∞ output feedback control problem for uncertain fuzzy neutral systems with both discrete and distributed delays. It is worth pointing out that the result in Theorem 4 can be readily extended to the case with multiple delays.Remark 6.
In Theorem4, if we set A21=0, Mi=0, and N2i=0  (i=1,2,…r),then the results in [32] are included in our paper. Also, the controller design method in our paper can be the reference for designing the observer-based output feedback controllers and so forth.


## 4. Simulation Example

In this section, we provide one example to illustrate the output feedbackH∞ controller design approach developed in this paper.The uncertain fuzzy system with distributed delays considered in this example is with two rules.Plant Rule 1: ifx1(t) is μ1, then
(58)x˙(t)=[A1+ΔA1(t)]x(t)+[A11+ΔA11(t)]x(t-τ1)+[A21+ΔA21(t)]x˙(t-τ2)+[A31+ΔA31(t)]∫t-τ3t‍x(s)ds+[B1+ΔB1(t)]u(t)+D1iw(t),y(t)=[C1+ΔC1(t)]x(t)+C11x(t-τ1)+D21w(t),z(t)=E1x(t)+E11x(t-τ1)+G1u(t).Plant Rule 2: ifx1(t) is μ2, then
(59)x˙(t)=[A2+ΔA2(t)]x(t)+[A12+ΔA12(t)]x(t-τ1)+[A22+ΔA22(t)]x˙(t-τ2)+[A32+ΔA32(t)]∫t-τ3t‍x(s)ds+[B2+ΔB2(t)]u(t)+D12w(t),y(t)=[C2+ΔC2(t)]x(t)+C12x(t-τ1)+D22w(t),z(t)=E2x(t)+E12x(t-τ1)+G2u(t),
where
(60)A1=[0.050-0.2-0.06],A11=[0.050.02-0.050.01],A21=[-0.0020.0080.0060],A31=[-0.050.080.060],B1=[0.0100.030.06],C1=[-0.03-0.10.20.02],C11=[-0.03-0.010.020.02],D11=[0.030.06],D21=[-0.50],E1=[-0.20],E11=[-0.020],G1=[0-0.03],A2=[-0.050-0.2-0.06],A12=[-0.050.2-0.50.01],A22=[0.008-0.002-0.0030.02],A32=[0.05-0.2-0.30.2],B2=[0.01-0.080.020.05],C2=[0.03-0.10.20.02],C12=[-0.03-0.010.020.2],D12=[-0.010.02],D22=[-0.30],E2=[-0.010],E12=[-0.010],G2=[00.02],
and ΔAi(t), ΔA1i(t), ΔA2i(t), ΔA3i(t), ΔBi(t), ΔCi(t)  (i=1,2) can be represented in the form of (2) and (3) with
(61)M1=[-0.020],N01=[-0.20.1],N11=[-0.20.2],N21=[-0.020.01],N31=[-0.020.01],N41=[-0.020.01],N51=[-0.020.01],M2=[0.030],N02=N01,N12=N11,N22=N21,N32=N31,N42=N41,N52=N51.
Then, the final outputs of the fuzzy systems are inferred as follows:
(62)x˙(t)=∑i=12‍hi(s(t))×{[Ai+ΔAi(t)]x(t)+[A1i+ΔA1i(t)]x(t-τ1)+[A2i+ΔA2i(t)]x˙(t-τ2)+[A3i+ΔA3i(t)]∫t-τ3t‍x(s)ds+[Bi+ΔBi(t)]u(t)+D1iw(t)},y(t)=∑i=12‍hi(s(t))×{[Ci+ΔCi(t)]x(t)+C1ix(t-τ1)+D2iw(t)},z(t)=∑i=12‍hi(s(t)){Eix(t)+E1ix(t-τ1)+Giu(t)},
where
(63)h1(x1(t))={13,forx1<-1,23+13x1,for|x1|≤1,1,forx1>1,h2(x1(t))={23,forx1<-1,13-13x1,for|x1|≤1,0,forx1>1.
In this example, we choose the H∞ performance level γ=3.In order to design a fuzzyH∞ output feedback controller for the T-S model, we first choose the initial condition that is x(0)=[0.3-0.1]T, and the disturbance input w(t) is assumed to be
(64)w(t)=1t+0.1,t≥0.
Then, solving the LMIs in (44), (45), and (46), we obtain the solution as follows:
(65)X=[0.98680.60710.60711.3948],Y=[12.93480.18140.18146.3384],Ω1=[-1.6069-1.6112-0.0417-3.2117],Ω2=[8.2076-3.7031-5.84615.3292],Φ1=[3.18480.0758-0.1899-2.5282],Φ2=[2.05470.1643-0.2930-0.7113],ψ1=[-140.1869-27.80015.7867-77.8562],ψ2=[-79.1853-88.2601-88.1012-95.7242].
Now, by Theorem 4, a desired fuzzy output feedback controller can be constructed as in (11) and (12) with
(66)Ak1=[87.3277-68.0397139.9266-107.2394],Ak2=[-24.985615.888011.1983-12.2840],Bk1=[10.9115-0.372516.58301.9936],Bk2=[1.4642-0.02123.19400.5922],Ck1=[-16.042312.757720.2647-14.8918],Ck2=[14.2596-10.49293.2304-1.1934].
With the output feedback fuzzy controller, the simulation results of the state response of the nonlinear system are given in Figure 1. Figure 2 shows the control input, while Figures 3 and 4 present the corresponding measured output and the controlled output, respectively.Figure 1
State responsex1(t) (continuous line) and x2(t) (dashed line).Figure 2
Control inputu1(t) (continuous line) and u2(t) (dashed line).Figure 3
Measured outputy1(t) (continuous line) and y2(t) (dashed line).Figure 4
Controlled outputz1(t) (continuous line) and z2(t) (dashed line).From these simulation results, it can be seen that the designed fuzzy output feedback controller ensures the robust asymptotic stability of the uncertain fuzzy neutral system and guarantees a prescribedH∞ performance level.


## 5. Conclusion and Future Work

The problem of delay-dependent robust output feedbackH∞ control for uncertain T-S fuzzy neutral systems with parameter uncertainties and distributed delays has been studied. In terms of LMIs, a sufficient condition for the existence of a full-order fuzzy dynamic output feedback controller, which robustly stabilizes the uncertain fuzzy neutral delay systems and guarantees a prescribed level on disturbance attenuation, has been obtained.Future work will mainly cover the problem of robust output feedbackH∞ control for uncertain fractional-order (FO) T-S fuzzy neutral systems with state and distributed delays. Firstly, investigate the robust output feedback H∞ control for FO T-S fuzzy systems with state delays; then, study the robust output feedback H∞ control for FO T-S fuzzy systems with both state and distributed delays; finally, obtain the results about the robust output feedback H∞ control for uncertain FO T-S fuzzy neutral systems with state and distributed delays.



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*Source: 197845-2013-10-26.xml*