# Stability Analysis and Stabilization of T-S Fuzzy Delta Operator Systems with Time-Varying Delay via an Input-Output Approach

Authors: Zhixiong Zhong; Guanghui Sun; Hamid Reza Karimi; Jianbin Qiu
Journal: Mathematical Problems in Engineering
 (2013)

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## Abstract

The stability analysis and stabilization of Takagi-Sugeno (T-S) fuzzy delta operator systems with time-varying delay are investigated via an input-output approach. A model transformation method is employed to approximate the time-varying delay. The original system is transformed into a feedback interconnection form which has a forward subsystem with constant delays and a feedback one with uncertainties. By applying the scaled small gain (SSG) theorem to deal with this new system, and based on a Lyapunov Krasovskii functional (LKF) in delta operator domain, less conservative stability analysis and stabilization conditions are obtained. Numerical examples are provided to illustrate the advantages of the proposed method.


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## Body

## 1. Introduction

The T-S fuzzy modeling approach, as a simple and effective tool for nonlinear control systems, has been widely accepted and extensive studied for a few decades [1–8]. In addition, it is well known that time delay is a source of instability or performance degradation [9]. Hence, analysis and synthesis of time-delay systems and other relative studies have attracted much attention during the past years [10–17]. Moreover, high-speed digital processing methods are of increasing importance in modern industrial applications. However, most traditional signal processing and control algorithms are inherently ill-conditioned when data are taken at high sampling rates [18]. The delta operator model can be applied as a useful approach to deal with discrete-time systems under high sampling rates through the analysis methods of continuous-time systems [19–22]. In view of the above considerations, both T-S fuzzy modeling approach and delta operator modeling approach have been extended to tackle the analysis and synthesis of nonlinear systems with time delay [23–25].Recently, some works on analysis and design of T-S fuzzy systems via delta operator approach were developed [26–28]. However, to the authors' best knowledge, few results on the stability analysis and stabilization for Takagi-Sugeno (T-S) fuzzy delta operator systems with time-varying delay are proposed.In this paper, an indirect approach, namely, theinput-output (IO) approach is introduced to deal with the stability analysis and control design of T-S fuzzy delta operator systems with time-varying delay. The main contribution of paper is that the stability analysis and stabilization problems for fuzzy delta operator systems with time-varying delay are investigated by the IO approach. A model approximation method is employed to transform the original system into an equivalent interconnected system, which is comprised of a forward subsystem with constant time delays and a feedback one with delayed uncertainties. The scaled small gain (SSG) method is applied and an LKF in delta domain is constructed to analyze and synthesize this system. Furthermore, a frequency sweeping method [9] is suggested to guarantee the internal stability for the forward subsystem, such that less conservative results are ensured. Finally, some comparisons are made with the existing results and control of a truck-trailer model is also presented to illustrate the effectiveness of our method.This paper is organized as follows. A model transformation method and the proof of the SSG theorem for T-S fuzzy delta operator systems with time-varying delay are presented in Section2. In Section 3, the stability analysis and stabilization results are provided. The simulation studies are given in Section 4 to illustrate the effectiveness of the proposed method. Finally, conclusions are drawn in Section 5.Notations. The notations used throughout this paper are standard. ℝn and ℝn×m represent the n-dimensional Euclidean space and n×m real matrices, respectively. G1∘G2 represents the series connection of mapping G1 and G2. The notation P>0(≥0) means that the matrix P is positive (semi) definite, In denotes an identity matrix with dimension n, and diag{⋯} denotes a block-diagonal matrix. The symbol “*” in a matrix stands for the transposed elements in the symmetric positions.


## 2. Model Description and Problem Formulation

In the following, we consider a fuzzy delta operator system with time-varying delay, which can be described by the following T-S fuzzy model.Plant Rule
i. IFθ1(t) is Mi1 and θ2(t) is Mi2 and … and θp(t) is Mip, THEN(1)∂x(t)=Aix(t)+Adix(t-nT)+Biu(t),x(t)=ϕ(t),t∈[-h2,0],i=1,2,…,r,
where x(t)∈ℝnx is the state variable; u(t)∈ℝm is control input; n is a time-varying integer; T is the sampling period; the bounded time-varying delay nT satisfies 0<h1≤nT≤h2; ϕ(t)∈ℝnx is the vector-valued initial condition; Mij is the fuzzy set; r is the number of IF-THEN rules; θ(t)=[θ1(t),θ2(t),…,θp(t)] are the premise variables which do not depend on the control input; Ai,Adi, and Bi are known constant matrices with appropriate dimensions; ∂x(t) is the delta operator of x(t), which is defined by
(2)∂x(t)={ddtx(t),T=0,x(t+T)-x(t)T,T≠0.The overall T-S fuzzy delta operator system with time-varying delay is inferred as follows:(3)∂x(t)=∑‍ri=1λi(θ(t))[Aix(t)+Adix(t-nT)+Biu(t)],
where  ∑i=1rλi(θ(t))=1, λi(θ(t))=ωi(θ(t))/∑i=1rωi(θ(t))≥0, and  ωi(θ(t))=∏j=1rMij(θj(t)) with  Mij(θj(t)) represent the grade of membership of  θj(t) in  Mij.The following control law is employed to deal with the problem of stabilization via state feedback, where the controller rule shares the same fuzzy sets with the T-S model.Controller Rule  i. IFθ1(t) is Mi1 and θ2(t) is Mi2 and … and θp(t) is Mip, THEN(4)u(t)=K1ix(t)+12K2ix(t-h1)+12K3ix(t-h2),i=1,2,…,r.The overall T-S fuzzy state feedback control law is inferred as(5)u(t)=∑‍ri=1λi(θ(t))[K1ix(t)+12K2ix(t-h1)+12K3ix(t-h2)].Remark 1.
It is noted that the controller given in (5) covers the special cases of the memoryless controller when K2i=K3i=0 and the purely delayed controller when K1i=0, respectively.
Combining system (3) with the control law (5), the resulting closed-loop system can be expressed as follows:
(6)∂x(t)=∑‍ri=1∑‍rj=1λi(θ(t))λj(θ(t))×[12(Ai+BiK1j)x(t)+Adix(t-nT)+12BiK2jx(t-h1)+12BiK3jx(t-h2)].
Before ending this section, we introduce the following lemmas as to be used to prove our main results in the following sections.Lemma 2 (see [9]).
Consider an interconnected system with two subsystems𝒮~1 and𝒮~2:
(7)𝒮~1:z(t)=Gω(t),𝒮~2:ω(t)=Δz(t),
where the forward subsystem 𝒮~1 is known, the feedback subsystem 𝒮~2 is unknown and time-varying, and assume that 𝒮~1 is internally stable. The closed-loop system formed by 𝒮~1 and 𝒮~2 is asymptotically stable for all Δ∈D≜{Δ:∥Δ∥∞≤1} if there exist matrices {Tw,Tz}∈𝕋 satisfied:
(8)𝕋≜{∥Tw∘Δ∘Tz-1∥∞{Tw,Tz}∈ℝw×w×ℝz×z:Tw,Tznonsigular;∥Tw∘Δ∘Tz-1∥∞≤1ℝz×z{Tw,Tz}},
such that the following SSG condition holds:
(9)∥Tz∘G∘Tw-1∥∞≤1.Lemma 3 (see [29]).
For any constant positive semidefinite symmetric matrixW, two positive integers r and r0 satifying r≥r0≥1, the following inequality holds:
(10)[∑‍ri=r0x(i)]TW[∑‍ri=r0x(i)]≤(r-r0+1)∑‍ri=r0xT(i)Wx(i).Lemma 4 (see [30]).
The property of delta operator: for any time functionx(t) and y(t), it holds that
(11)∂(x(t)y(t))=∂x(t)y(t)+x(t)∂y(t)+T∂x(t)∂y(t),
where T is the sampling period.


## 3. Model Transformation

In this paper, the T-S fuzzy delta operator system with time-varying delay is investigated by an IO approach. By this method, the termx(t-nT) is approximated and the error is written into the feedback path. The recent work in [31] proposed a two-term approximation method (1/2)[x(t-h1)+x(t-h2)] for x(t-nT), which results in a smaller approximation error bound. Inspired by this method, the approximation error of time-varying delay can be expressed as
(12)ωt(t)=x(t-nT)-12[x(t-h1)+x(t-h2)]=T2∑‍-n-1i=-h2/T∂x(t+iT)-T2∑‍-(h1/T)-1i=-n∂x(t+iT)=T2∑‍-(h1/T)-1i=-h2/Tk(i)∂x(t+iT),
where ∂x(t) is defined in (2), and
(13)k(i)={1,i<-n,-1,i≥-n.


### 3.1. Open-Loop Case

Considering the fuzzy delta operator system (3) and setting u(t)=0, we have
(14)∂x(t)=∑‍ri=1λi(θ(t))[Aix(t)+Adix(t-nT)].Employing the two-term approximation method to pull out the uncertainties of time-varying delay, the open-loop system can be written as an interconnected system with a forward subsystem and a feedback one, which is described by(15)𝒮1:[∂x(t)z(t)]=∑‍ri=1λi(θ(t))[Θ1h122AdiX-1XΘ1Xh122AdiX-1]×[ζ(t)ω(t)],𝒮2:ω(t)=XΔX-1z(t),
where Θ1=[Ai(1/2)Adi(1/2)Adi], ζ(t)=col{x(t)x(t-h1)x(t-h2)}, h12=h2-h1, ω(t)=(2/h12)Xωt(t), the scaling matrix {X,X}∈𝕋 has the appropriate dimensions, and the operator Δ is the maping z(t)→ω(t).For convenience, we denoteω(t)=Xω~(t) and z(t)=Xz~(t). The system (15) can be rewritten as
(16)𝒮3:[∂x(t)z~(t)]=∑‍ri=1λi(θ(t))[Θ1h122AdiΘ1h122Adi][ζ(t)ω~(t)],𝒮4:ω~(t)=Δz~(t).Now, the uncertainties of the time-varying delay have been pulled out from the system (14). Furthermore, the system has been transformed into the interconnection by the forward subsystem and the feedback subsystem. The following result shows that this reformulated system satisfies the following SSG condition.Lemma 5.
The operatorΔ:z(t)→ω(t) in system (15) satisfies the SSG theorem if there exists the nosingular matrix {X,X}∈𝕋, such that
(17)∥XΔX-1∥≤1.Proof.
Following the notations in (12), under the zero initial condition, we have the following inequalities by using the discrete Jensen inequality in Lemma 3:
(18)∑‍∞t=0ωT(t)ω(t)=(Th12)2∑‍∞t=0[∑‍-(h1/T)-1i=-h2/Tk(i)∂x(t+iT)]T×XTX[∑‍-(h1/T)-1i=-h2/Tk(i)∂x(t+iT)]≤Th12∑‍-(h1/T)-1i=-h2/T∑‍∞t=0[∂xT(t+iT)XTX∂x(t+iT)]≤Th12∑‍-(h1/T)-1i=-h2/T∑‍∞t=0[∂xT(t)XTX∂x(t)]=∑‍∞t=0zT(t)z(t),
which implies that ∥XΔX-1∥≤1. The proof is completed.


### 3.2. Closed-Loop Case

Employing the two-term approximation method to pull out the uncertainties of time-varying delay, the closed-loop system (6) can also be written as an interconnected system with a forward subsystem and a feedback one, which is described by
(19)𝒮5:[∂x(t)z(t)]=∑‍ri=1∑‍rj=1λi(θ(t))λj(θ(t))[Θ2h122AdiX-1XΘ2Xh122AdiX-1]×[ζ(t)ω~(t)],𝒮6:ω(t)=XΔX-1z(t),
where Θ2=[(Ai+BiK1j)(1/2)(Adi+BiK2j)(1/2)(Adi+BiK3j)], and ζ(t), ω(t), and z(t) are defined as the same as the open-loop case.For convenience, we denoteω(t)=Xω~(t) and z(t)=Xz~(t). The system (19) can be rewritten as
(20)𝒮7:[∂x(t)z~(t)]=∑‍ri=1∑‍rj=1λi(θ(t))λj(θ(t))[Θ2h122AdiΘ2h122Adi][ζ(t)ω~(t)],𝒮8:ω~(t)=Δz~(t).Remark 6.
The definitions ofω(t) andz(t) for the closed-loop system are the same as the open-loop system, so it is easy to see that the closed-loop system (19) also satisfies the SSG condition.Now the reformulated systems have been shown to satisfy the SSG condition in both the open-loop and closed-loop cases. Then the systems in (15) and (19) are asymptotically stable if both the forward subsystems are internally stable. Indeed, a frequency sweeping method is often used to check this condition [9].Lemma 7 (see [9]).
Consider the following system:(21)𝒮~1:z(t)=Gω(t),𝒮~2:ω(t)=Δz(t).
The aforementioned system is internally asymptotically stable if there exist a scalarε>0 and a Lyapunov Krasovskii functional V(t) satisfying
(22)V(t)>ε∥x(t)∥2,
such that the functional
(23)w(t)=V˙(t)+zT(t)z(t)-ωT(t)ω(t)
statisfies
(24)w(t)≤-ε∥x(t)∥2-ε∥ω(t)∥2.


## 3.1. Open-Loop Case

Considering the fuzzy delta operator system (3) and setting u(t)=0, we have
(14)∂x(t)=∑‍ri=1λi(θ(t))[Aix(t)+Adix(t-nT)].Employing the two-term approximation method to pull out the uncertainties of time-varying delay, the open-loop system can be written as an interconnected system with a forward subsystem and a feedback one, which is described by(15)𝒮1:[∂x(t)z(t)]=∑‍ri=1λi(θ(t))[Θ1h122AdiX-1XΘ1Xh122AdiX-1]×[ζ(t)ω(t)],𝒮2:ω(t)=XΔX-1z(t),
where Θ1=[Ai(1/2)Adi(1/2)Adi], ζ(t)=col{x(t)x(t-h1)x(t-h2)}, h12=h2-h1, ω(t)=(2/h12)Xωt(t), the scaling matrix {X,X}∈𝕋 has the appropriate dimensions, and the operator Δ is the maping z(t)→ω(t).For convenience, we denoteω(t)=Xω~(t) and z(t)=Xz~(t). The system (15) can be rewritten as
(16)𝒮3:[∂x(t)z~(t)]=∑‍ri=1λi(θ(t))[Θ1h122AdiΘ1h122Adi][ζ(t)ω~(t)],𝒮4:ω~(t)=Δz~(t).Now, the uncertainties of the time-varying delay have been pulled out from the system (14). Furthermore, the system has been transformed into the interconnection by the forward subsystem and the feedback subsystem. The following result shows that this reformulated system satisfies the following SSG condition.Lemma 5.
The operatorΔ:z(t)→ω(t) in system (15) satisfies the SSG theorem if there exists the nosingular matrix {X,X}∈𝕋, such that
(17)∥XΔX-1∥≤1.Proof.
Following the notations in (12), under the zero initial condition, we have the following inequalities by using the discrete Jensen inequality in Lemma 3:
(18)∑‍∞t=0ωT(t)ω(t)=(Th12)2∑‍∞t=0[∑‍-(h1/T)-1i=-h2/Tk(i)∂x(t+iT)]T×XTX[∑‍-(h1/T)-1i=-h2/Tk(i)∂x(t+iT)]≤Th12∑‍-(h1/T)-1i=-h2/T∑‍∞t=0[∂xT(t+iT)XTX∂x(t+iT)]≤Th12∑‍-(h1/T)-1i=-h2/T∑‍∞t=0[∂xT(t)XTX∂x(t)]=∑‍∞t=0zT(t)z(t),
which implies that ∥XΔX-1∥≤1. The proof is completed.


## 3.2. Closed-Loop Case

Employing the two-term approximation method to pull out the uncertainties of time-varying delay, the closed-loop system (6) can also be written as an interconnected system with a forward subsystem and a feedback one, which is described by
(19)𝒮5:[∂x(t)z(t)]=∑‍ri=1∑‍rj=1λi(θ(t))λj(θ(t))[Θ2h122AdiX-1XΘ2Xh122AdiX-1]×[ζ(t)ω~(t)],𝒮6:ω(t)=XΔX-1z(t),
where Θ2=[(Ai+BiK1j)(1/2)(Adi+BiK2j)(1/2)(Adi+BiK3j)], and ζ(t), ω(t), and z(t) are defined as the same as the open-loop case.For convenience, we denoteω(t)=Xω~(t) and z(t)=Xz~(t). The system (19) can be rewritten as
(20)𝒮7:[∂x(t)z~(t)]=∑‍ri=1∑‍rj=1λi(θ(t))λj(θ(t))[Θ2h122AdiΘ2h122Adi][ζ(t)ω~(t)],𝒮8:ω~(t)=Δz~(t).Remark 6.
The definitions ofω(t) andz(t) for the closed-loop system are the same as the open-loop system, so it is easy to see that the closed-loop system (19) also satisfies the SSG condition.Now the reformulated systems have been shown to satisfy the SSG condition in both the open-loop and closed-loop cases. Then the systems in (15) and (19) are asymptotically stable if both the forward subsystems are internally stable. Indeed, a frequency sweeping method is often used to check this condition [9].Lemma 7 (see [9]).
Consider the following system:(21)𝒮~1:z(t)=Gω(t),𝒮~2:ω(t)=Δz(t).
The aforementioned system is internally asymptotically stable if there exist a scalarε>0 and a Lyapunov Krasovskii functional V(t) satisfying
(22)V(t)>ε∥x(t)∥2,
such that the functional
(23)w(t)=V˙(t)+zT(t)z(t)-ωT(t)ω(t)
statisfies
(24)w(t)≤-ε∥x(t)∥2-ε∥ω(t)∥2.


## 4. Stability Analysis

The previous section presents a model transformation for the original system (3). The open-loop system has been converted into an interconnected system in (15), and the closed-loop system has been converted into (19). In this section, we investigate the asymptotic stability of the system in (15). First, we present the following result for T-S fuzzy delta system with time-varying delay.Theorem 8.
Consider T-S fuzzy delta operator system in (14). Then given scalars h2>h1>0 and the sampling period T>0, the fuzzy delta operator system (14) with time-varying delay is asymptotically stable if there exist positive definite symmetric matrices U, P, R1, R2, Q1, Q2, and Z, such that the following LMIs hold for i=1,2,…,r: (25)Φi=[Φi(1,1)PAi12PAdi12PAdih122PAdi*Φi(2,2)12PAdi+R1h112PAdi+R2h2h122PAdi**-Q1-14Z-R1h1-14Z-h124Z***-Q2-14Z-R2h2-h124Z****-h1224Z-U]<0,
where
(26)Φi(1,1)=TP+h1R1+h2R2-2P+U,Φi(2,2)=PAi+AiTP+Z+h12TZ+Q1+Q2-R1h1-R2h2.Proof.
Firstly, choosing a Lyapunov-Krasovskii functional candidate in delta domain,(27)V(t)=V1(t)+V2(t)+V3(t)+V4(t),
where
(28)V1(t)=xT(t)Px(t),V2(t)=T∑‍h2/Ti=h1/T∑‍ij=1xT(t-jT)Zx(t-jT),V3(t)=T∑‍h1/Ti=1xT(t-iT)Q1x(t-iT)+T∑‍h2/Ti=1xT(t-iT)Q2x(t-iT),V4(t)=∑‍h1/Ti=1∑‍ij=1eT(t-jT)R1e(t-jT)+∑‍h2/Ti=1∑‍ij=1eT(t-jT)R2e(t-jT),
and T is the sampling period, e(j)=x(j)-x(j+T), so that ∂x(j)=-e(j)/T and e(t-jT)=x(t-jT)-x(t-(j-1)T).
Taking the delta operator manipulations ofV1(t) along the trajectory of systems 𝒮1 and 𝒮2, and using Lemma 4, it can be obtained that
(29)∂V1(t)=∂T(x(t))Px(t)+xT(t)P∂(x(t))+T·∂T(x(t))P∂(x(t))=[∂x(t)x(t)x(t-h1)x(t-h2)ω~(t)]T×[TP0000*PAi+AiTP12PAdi12PAdih122PAdi**000***00****0]×[∂x(t)x(t)x(t-h1)x(t-h2)ω~(t)].
Taking the delta operator manipulation ofV2(t), we have
(30)∂V2(t)=T·1T∑‍h2/Ti=h1/T∑‍ij=1xT(t-(j-1)T)×Zx(t-(j-1)T)-T·1T∑‍h2/Ti=h1/T∑‍ij=1xT(t-iT)Zx(t-iT)=∑‍h2/Ti=h1/TxT(t)Zx(t)-∑‍h2/Ti=h1/TxT(t-iT)Zx(t-iT)≤(h12T+1)xT(t)Zx(t)-xT(t-nT)Zx(t-nT).
Substituting (12) into (30), we have
(31)∂V2(t)=[x(t)x(t-h1)x(t-h2)ω~(t)]T×[Z+h12TZ000*-14Z-14Z-h124Z**-14Z-h124Z***-h1224Z]×[x(t)x(t-h1)x(t-h2)ω~(t)].
Taking the delta operator manipulation ofV3(t), we have
(32)∂V3(t)=1T·T[∑‍h1/Ti=1xT(t-(i-1)T)×Q1x(t-(i-1)T)+∑‍h2/Ti=1xT(t-(i-1)T)×Q2x(t-(i-1)T)-∑‍h1/Ti=1xT(t-iT)Q1x(t-iT)-∑‍h2/Ti=1xT(t-iT)Q2x(t-iT)]=xT(t)(Q1+Q2)x(t)-xT(t-h1)Q1x(t-h1)-xT(t-h2)Q2x(t-h2).
Taking the delta operator manipulation ofV4(t) and using Lemma 3, we have
(33)∂V4(t)=1T[∑‍h1/Ti=1∑‍ij=1eT(t-(j-1)T)×R1e(t-(j-1)T)+∑‍h2/Ti=1∑‍ij=1eT(t-(j-1)T)×R2e(t-(j-1)T)-∑‍h1/Ti=1∑‍ij=1eT(t-jT)R1e(t-jT)-∑‍h2/Ti=1∑‍ij=1eT(t-jT)R2e(t-jT)]≤h1T2eT(t)R1e(t)-1h1(∑‍h1/Ti=1e(t-iT))TR1(∑‍h1/Ti=1e(t-iT))+h2T2eT(t)R2e(t)-1h2(∑‍h2/Ti=1e(t-iT))TR2(∑‍h2/Ti=1e(t-iT))=h1∂xT(t)R1∂x(t)-1h1(x(t-h1)-x(t))T×R1(x(t-h1)-x(t))+h2∂xT(t)R2∂x(t)-1h2(x(t-h2)-x(t))TR2(x(t-h2)-x(t)).
For the positive definite symmetric matrixP, we have the following equation from (16):
(34)0=-∑‍ri=1λi(θ(t))2∂T(x(t))P×[h122∂x(t)-Aix(t)-12Adi(x(t-h1)+x(t-h2))-h122Adiω~(t)].
Substituting (34) into ∂V(t), we have
(35)∂V(t)=∑‍ri=1λi(θ(t))ξTΣ1iξ,
where
(36)Σ1i=[Σ1i(1,1)PAi12PAdi12PAdih122PAdi*Σ1i(2,2)12PAdi+R1h112PAdi+R2h2h122PAdi**-Q1-14Z-R1h1-14Z-h124Z***-Q2-14Z-R2h2-h124Z****-h1224Z],Σ1i(1,1)=TP+h1R1+h2R2-2P,Σ1i(2,2)=PAi+AiTP+Z+h12TZ+Q1+Q2-R1h1-R2h2,ξT=[∂T(x(t))xT(t)xT(t-h1)xT(t-h2)ω~T(t)].
Therefore if∂V(t)<0, there always exists a sufficiently small scalar ε, for x(t)≠0, such that ∂V(t)≤-ε∥x(t)∥2, which indicates that the systems 𝒮1 and 𝒮2 under ω(t)=0 are asymptotically stable.
Next, to consider the conditionω(t)≠0, we denote U=XTX>0 and it can be expanded in Lemma 7 as
(37)𝒲≜∂V(t)+zT(t)z(t)-ωT(t)ω(t)=∑‍ri=1λi(θ(t))ξTΣ1iξ+z~T(t)XTXz~(t)-ω~T(t)XTXω~(t)=∑‍ri=1λi(θ(t))ξTΣ2iξ,
where Σ2i=Φi. The proof is completed.To compare the results obtained by IO approach, we give the following corollary, which is obtained by a direct LKF-based method.Corollary 9.
Consider T-S fuzzy delta operator system in (14). Then given scalars h2>h1>0 and the sampling period T>0, the fuzzy delta operator system (14) with time-varying delay is asymptotically stable if there exist positive definite symmetric matrices P, R1, R2, Q1, Q2, and Z, matrices N=[N1N2], M=[M1M2], S=[S1S2],  X~=[X~11X~12X~12TX~22], and Y=[Y11Y12Y12TY22], such that the following LMIs (38)-(39) hold:
(38)Ψ1=[-X~11-X~12N1*-X~22N2**-R2T]<0,Ψ2=[-X~11-X~12M1*-X~22M2**-R2T]<0,Ψ3=[-Y11-Y12S1*-Y22S2**-R1T]<0,(39)Ψ4i=[Ψ4i(1,1)PAiPAdi00*Ψ4i(2,2)Ψ4i(2,3)-S1-M1**Ψ4i(3,3)-S2-M2***-Q10****-Q2]<0,i=1,2,…,r,
where
(40)Ψ4i(1,1)=TP+h1R1+h2R2-2P,Ψ4i(2,2)=PAi+AiTP+Z+h12TZ+Q1+Q2+N1+N1T+S1+S1T+h2TX~11+h1TY11,Ψ4i(2,3)=PAdi-N1+N2T+M1+S2T+h2TX~12+h1TY12,Ψ4i(3,3)=-Z-N2-N2T+M2+M2T+h2TX~22+h1TY22.Proof.
To make a fair comparison, we choose the same LKF candidate as in the proof of Theorem8.
Taking the delta operator manipulations ofV1(t), V2(t), V3(t), and V4(t) along the trajectory of system (14), we have
(41)∂V1(t)=∂T(x(t))Px(t)+xT(t)P∂(x(t))+T·∂T(x(t))P∂(x(t))=[∂x(t)x(t)x(t-nT)]T[TP00*PAi+AiTPPAdi**0]×[∂x(t)x(t)x(t-nT)],∂V2(t)≤(h12T+1)xT(t)Zx(t)-xT(t-nT)Zx(t-nT),∂V3(t)=xT(t)(Q1+Q2)x(t)-xT(t-h1)Q1x(t-h1)-xT(t-h2)Q2x(t-h2),∂V4(t)=h1∂T(x(t))R1∂(x(t))-1T∑‍h1/Ti=1eT(t-iT)R1e(t-iT)+h2∂T(x(t))R2∂(x(t))-1T∑‍h2/Ti=1eT(t-iT)R2e(t-iT),
where e(t-iT) is defined in (27).
For a positive definite symmetric matrixP, we have the following equation from (14):
(42)0=-∑‍ri=1λi(θ(t))2∂T(x(t))P×[∂x(t)-Aix(t)-Adix(t-nT)].
From the definition ofe(t-iT), the following equations hold for any matrices N, M, and S with appropriate dimensions:
(43)0=2ΥT(t)N[x(t)-x(t-nT)+∑‍ni=1eT(t-iT)],0=2ΥT(t)M[∑‍h2/Ti=n+1eT(t-iT)x(t-nT)-x(t-h2)+∑‍h2/Ti=n+1eT(t-iT)],0=2ΥT(t)S[x(t)-x(t-h1)+∑‍h1/Ti=1eT(t-iT)],
where ΥT(t)=[xT(t)xT(t-nT)].
For any appropriate dimensions matricesX~=X~T and Y=YT, we have
(44)0=∑‍h2/Ti=1ΥT(t)X~Υ(t)-∑‍h2/Ti=1ΥT(t)X~Υ(t)=h2TΥT(t)X~Υ(t)-∑‍ni=1ΥT(t)X~Υ(t)-∑‍h2/Ti=n+1ΥT(t)X~Υ(t),0=∑‍h1/Ti=1ΥT(t)YΥ(t)-∑‍h1/Ti=1ΥT(t)YΥ(t)=h1TΥT(t)YΥ(t)-∑‍h1/Ti=1ΥT(t)YΥ(t).
Substituting (42)–(44) into ∂V(t), we have
(45)∂V(t)=∑‍ri=1λi(t)ξ1TΣ3iξ1+∑‍ni=1Υ1T(t)Σ4Υ1(t)+∑‍h1/Ti=1Υ1T(t)Σ5Υ1(t)+∑‍h2/Ti=n+1Υ1T(t)Σ6Υ1(t),
where ξ1T=[∂xT(t)xT(t)xT(t-nT)xT(t-h1)xT(t-h2)], Υ1T(t)=[xT(t)xT(t-nT)eT(t-iT)], Σ3i=Ψ4i, Σ4=Ψ1, Σ5=Ψ3, and Σ6=Ψ2. Since Σ3i<0, Σ4<0, Σ5<0, and Σ6<0 hold, then∂V(t)<0. The proof is completed.


## 5. Stabilization

The previous section presents the criterion for asymptotic stability of fuzzy delta operator open-loop system. In this section, we are interested in designing a controller in (5). By employing the same LKF and applying IO method, the following criteria can be obtained.Theorem 10.
Consider T-S fuzzy delta operator system (3) with the controller in (5). Then given scalars h2>h1>0 and the sampling period T>0, the fuzzy delta operator system with time-varying delay is asymptotically stable if there exist positive definite symmetric matrices G, U-, R-1, R-2, Q-1, Q-2, and Z- and matrices K~1i, K~2i, and K~3i, such that the following LMIs hold:
(46)Φii<0,(1≤i≤r),Φij+Φji<0,(1≤i<j≤r),
where
(47)Φij=[Φij(1,1)Φij(1,2)Φij(1,3)Φij(1,4)h122AdiG*Φij(2,2)Φij(2,3)Φij(2,4)h122AdiG**Φij(3,3)-14Z--h124Z-***Φij(4,4)-h124Z-****-h1224Z--U-],Φij(1,1)=TG+h1R-1+h2R-2-2G+U-,Φij(1,2)=AiG+BiK~1j,Φij(1,3)=12(AdiG+BiK~2j),Φij(1,4)=12(AdiG+BiK~3j),Φij(2,2)=AiG+BiK~1j+GAiT+K~1jTBiT+Z-+h12TZ-+Q-1+Q-2-R-1h1-R-2h2,Φij(2,3)=12(AdiG+BiK~2j)+R-1h1,Φij(2,4)=12(AdiG+BiK~3j)+R-2h2,Φij(3,3)=-Q-1-14Z--R-1h1,Φij(4,4)=-Q-2-14Z--R-2h2.
Moreover, a suitable stabilizing fuzzy state feedback controller can be chosen by(48)u(t)=∑‍ri=1λi(θ(t))[K1ix(t)+12K2ix(t-h1)+12K3ix(t-h2)],i=1,2,…,r,
where K1i=K~1iG-1, K2i=K~2iG-1, K3i=K~3iG-1.Proof.
Choosing the same LKF candidate as in the proof of Theorem8, we have
(49)∂V(t)=∑‍ri=1∑‍rj=1λi(θ(t))λj(θ(t))ξTΣ1ijξ,
where
(50)Σ1ij=[Σ1ij(1,1)Σ1ij(1,2)Σ1ij(1,3)Σ1ij(1,4)h122PAdi*Σ1ij(2,2)Σ1ij(2,3)Σ1ij(2,4)h122PAdi**Σ1ij(3,3)-14Z-h124Z***Σ1ij(4,4)-h124Z****-h1224Z],Σ1ij(1,1)=TP+h1R1+h2R2-2P,Σ1ij(1,2)=P(Ai+BiK1j),Σ1ij(1,3)=12P(Adi+BiK2j),Σ1ij(1,4)=12P(Adi+BiK3j),Σ1ij(2,2)=P(Ai+BiK1j)+(Ai+BiK1j)TP+Z+h12TZ+Q1+Q2-R1h1-R2h2,Σ1ij(2,3)=12P(Adi+BiK2j)+R1h1,Σ1ij(2,4)=12P(Adi+BiK3j)+R2h2,Σ1ij(3,3)=-Q1-14Z-R1h1,Σ1ij(4,4)=-Q2-14Z-R2h2,
and ξ is defined in (35).
Next, by applying Lemma7, we have
(51)𝒲≜∂V(t)+zT(t)z(t)-ωT(t)ω(t)=∑‍ri=1∑‍rj=1λi(θ(t))λj(θ(t))ξTΣ1ijξ+z~T(t)XTXz~(t)-ω~T(t)XTXω~(t)=∑‍ri=1∑‍rj=1λi(θ(t))λj(θ(t))ξTΣ2ijξ,
where
(52)Σ2ij=[Σ1ij(1,1)+UΣ1ij(1,2)Σ1ij(1,3)Σ1ij(1,4)h122PAdi*Σ1ij(2,2)Σ1ij(2,3)Σ1ij(2,4)h122PAdi**Σ1ij(3,3)-14Z-h124Z***Σ1ij(4,4)-h124Z****-h1224Z-U].
It can be clearly shown that(53)𝒲≜∑‍ri=1λi2(θ(t))ξTΣ2iiξ+∑‍ri=1∑‍ri<jλi(θ(t))λj(θ(t))ξT×(Σ2ij+Σ2ji)ξ.
Premultiplying and postmultiplyingΣ2ii by diag{P-1P-1P-1P-1P-1}, and letting G=P-1, R-1=P-1R1P-1, R-2=P-1R2P-1, Q-1=P-1Q1P-1, Q-2=P-1Q2P-1, U-=P-1UP-1, K~1i=K1iP-1, K~2i=K2iP-1, and K~3i=K3iP-1 yield Φii. Following a similar line in the previous process to Σ2ij and Σ2ji yields Φij and Φji.
SinceΦii<0 holds for 1≤i≤r, and (Φij+Φji)<0 holds for 1≤i<j≤r,then we have 𝒲<0. Then by using Lemma 7, the system (19) is internally asymptotically stable. Furthermore, from Lemma 2, the fuzzy delta operator system (3) under the controller (5) is asymptotically stable. Finally, the explicit expression of the state feedback controller is given by K1i=K~1iG-1, K2i=K~2iG-1, and K3i=K~3iG-1. The proof is completed.To compare the results obtained by the IO approach, we give the following corollary, which is obtained by a direct LKF-based method.Corollary 11.
Consider T-S fuzzy delta operator system (3) with the controller in (5). Then given scalars h2>h1>0 and the sampling period T>0, the fuzzy delta operator system with time-varying delay is asymptotically stable if there exist positive definite symmetric matrices G, R-1, R-2, Q-1, Q-2, and Z- and matrices N-=[N-1N-2], M-=[M-1M-2], S-=[S-1S-2], X-=[X-11X-12X-12TX-22],  Y-=[Y-11Y-12Y-12TY-22] and K~1i, such that (38) and the following LMIs hold:
(54)Ψ4ii<0(1≤i≤r),Ψ4ij+Ψ4ji<0(1≤i<j≤r),
where
(55)Ψ4ij=[Ψ4ij(1,1)AiG+BiK~1jAdiG00*Ψ4ij(2,2)Ψ4ij(2,3)-S-1-M-1**Ψ4ij(3,3)-S-2-M-2***-Q-10****-Q-2],Ψ4ij(1,1)=TG+h1R-1+h2R-2-2G,Ψ4ij(2,2)=AiG+BiK~1j+GAiT+K~1jTBiT+Z-+h12TZ-+Q-1+Q-2+N-1+N-1T+S-1+S-1T+h2TX-11+h1TY-11,Ψ4ij(2,3)=AdiG-N-1+N-2T+M-1+S-2T+h2TX-12+h1TY-12,Ψ4ij(3,3)=-Z--N-2-N-2T+M-2+M-2T+h2TX-22+h1TY-22.
Moreover, a suitable stabilizing fuzzy state feedback controller is given by(56)u(t)=∑‍ri=1λi(θ(t))K1ix(t),i=1,2,…,r,
where K1i=K~1iG-1.Proof.
Choosing the same LKF candidate as in the proof of Theorem8, we have
(57)∂V(t)=∑‍ri=1∑‍rj=1λi(θ(t))λj(θ(t))ξ1TΣ3ijξ1+∑‍ni=1Υ1T(t)Σ4Υ1(t)+∑‍h1/Ti=1Υ1T(t)Σ5Υ1(t)+∑‍h2/Ti=n+1Υ1T(t)Σ6Υ1(t),
where
(58)Σ3ij=[Σ3ij(1,1)P(Ai+BiK1j)PAdi00*Σ3ij(2,2)Σ3ij(2,3)-S1-M1**Σ3ij(3,3)-S2-M2***-Q10****-Q2],Σ3ij(1,1)=TP+h1R1+h2R2-2P,Σ3ij(2,2)=P(Ai+BiK1j)+(Ai+BiK1j)TP+Z+h12TZ+Q1+Q2+N1+N1T+S1+S1T+h2TX~11+h1TY11,Σ3ij(2,3)=PAdi-N1+N2T+M1+S2T+h2TX~12+h1TY12,Σ3ij(3,3)=-Z-N2-N2T+M2+M2T+h2TX~22+h1TY22,
and ξ1T, Υ1T, Σ4, Σ5, and Σ6 are defined in (45).
It can be clearly shown that(59)∂V(t)≜∑‍ri=1λi2(θ(t))ξ1TΣ3iiξ1+∑‍ri=1∑‍ri<jλi(θ(t))λj(θ(t))ξ1T(Σ3ij+Σ3ji)ξ1+∑‍ni=1Υ1T(t)Σ4Υ1(t)+∑‍h1/Ti=1Υ1T(t)Σ5Υ1(t)+∑‍h2/Ti=n+1Υ1T(t)Σ6Υ1(t).
Premultiplying and postmultiplyingΣ3ii by diag{P-1P-1P-1P-1P-1}, premultiplying and postmultiplying Σ4,Σ5,Σ6 by diag{P-1P-1P-1}, and letting G=P-1, R-1=P-1R1P-1, R-2=P-1R2P-1, Q-1=P-1Q1P-1, Q-2=P-1Q2P-1, Z-=P-1ZP-1, N-1=P-1N1P-1, N-2=P-1N2P-1, M-1=P-1M1P-1, M-2=P-1M2P-1, S-1=P-1S1P-1, S-2=P-1S2P-1, X-11=P-1X~11P-1, X-12=P-1X~12P-1, X-22=P-1X~22P-1, Y-11=P-1Y11P-1, Y-12=P-1Y12P-1, Y-22=P-1Y22P-1, and K~1i=K1iP-1 yield Ψ4ii, Ψ1, Ψ2, and Ψ3. Following a similar line of the previous process to Σ3ij and Σ3ji yields Ψ4ij and Ψ4ji.
SinceΨ4ii<0 holds for 1≤i≤r, and  (Ψ4ij+Ψ4ji)<0 holds for 1≤i<j≤r, Σ4<0, Σ5<0, and Σ6<0,then we have ∂V(t)<0. Therefore the fuzzy delta operator system (3) under the controller (59) is asymptotically stable. Finally, the explicit expression of the state feedback controller is given by K1i=K~1iG-1. The proof is completed.


## 6. Simulation Examples

In this section, three examples are provided to demonstrate the effectiveness of the proposed results.Example 12 (Stability Analysis).
Consider a T-S fuzzy delta operator system with time-varying delay in the form of (1) with parameters given by
(60)A1=[-200-0.9],A2=[-10.50-1],Ad1=[-10-1-1],Ad2=[-100.1-1].
In this example, for a given delay lower boundh1=0.8, we seek for the admissible upper bound h2, which guarantees the asymptotic stability of the open-loop system. Choosing the sampling period T=0.01, the obtained results are listed in Table 1.Table 1
Comparisons of maximum allowed delay upper boundh2 for Example 12 with h1=0.8.
Method
h
2 (T=0.01)
Result of Corollary9
Infeasible
Result of Theorem8
0.933Table1 shows that the proposed result in Theorem 8 is less conservative than that in Corollary 9, which demonstrates the advantages of our method. Table 2 shows the delay upper bound h2 under different delay lower bound h1 and different sampling period T. It is obvious that the delay upper bound h2 increases gradually as the sampling rate rises, which indicates the advantage of the delta operator fuzzy system at high sampling rate.Table 2
Comparisons of maximum allowed delay upper boundh2 by different h1 and T for Example 12.
Method
T
=
0.01
T
=
0.05
T
=
0.1
h
1
0.1
0.4
0.8
0.1
0.4
0.8
0.1
0.4
0.8
Result of Theorem8
0.732
0.790
0.933
0.685
0.749
0.901
0.626
0.700
0.863Example 13 (Controller Design).
To further illustrate the effectiveness of our method for controller design, we consider the following T-S fuzzy delta operator system with time-varying delay:(61)∂V(t)=∑‍ri=1λi(θ(t))[Aix(t)+Adix(t-nT)+Biu(t)],
where Ai, Bi, and Adi (i=1,2) are given by
(62)A1=[00.601],Ad1=[0.50.902],B1=[11],A2=[1010],Ad2=[0.9011.6],B2=[11].
For different delay lower boundsh1, the allowed delay upper bounds h2 are listed in Table 3. It can be seen that the proposed results in Theorem 10 are less conservative than those in Corollary 11.
The fuzzy controller gains forT=0.01, h1=0.8, and h2=0.951 by Theorem 10 are given as
(63)K11=[1.2781-4.4103],K12=[0.0812-2.9757],K21=[-0.1010-2.0993],K22=[-1.0592-2.5416],K31=[-0.1064-1.9295],K32=[-1.0592-2.5414].Table 3
Comparisons of maximum allowed delay upper boundh2 by different h1 and T for Example 13.
Method
T
=
0.01
T
=
0.05
T
=
0.1
h
1
0.1
0.4
0.8
0.1
0.4
0.8
0.1
0.4
0.8
Result of Corollary11
0.185
—
—
0.179
—
—
0.172
—
—
Result of Theorem10
0.428
0.668
0.951
0.418
0.658
0.941
0.406
0.646
0.929Example 14.
To illustrate the application of our method, we consider the following truck-trailer system given in [32]:
(64)x˙1(t)=-cvt1Lt0x1(t)-(1-c)vt1Lt0x1(t-d(t))+vt1lt0u(t),x˙2(t)=cvt1Lt0x1(t)+(1-c)vt1Lt0x1(t-d(t)),x˙3(t)=vt1t0sinx2(t)+cvt12Lx1(t)+(1-c)vt12Lx1(t-d(t)),
where x1(t) is the angular difference between the truck and trailer, x2(t) is the angle of the trailer, and x3(t) is the vertical position of rear end of the trailer.
The model parameters are given asl=2.8, L=5.5, v=-1.0, t1=2.0, and t0=0.5, and c∈[0,1] is a retarded coefficient with limits 0 and 1 corresponding to delay-free term and to a full-delay term. The premise variable is chosen as θ(t)=x2(t)+c(vt1/Lt0)x1(t)+(1-c)(vt1/Lt0)x1(t-d(t)), and the sampling period T=0.01. The following fuzzy rules via delta operator are employed byPlant Rule1: IFθ(t)= is about 0 rad, THEN(65)∂x(t)=A1x(t)+Ad1x(t-nT)+B1u(t),Plant Rule2: IFθ(t)= is about π rad or -π rad, THEN(66)∂x(t)=A2x(t)+Ad2x(t-nT)+B2u(t).
The membership functions for Rule 1 and Rule 2 are given by(67)λ={λ1=(1-11+exp(-3(θ(t)-0.5π)))×(1-11+exp(-3(θ(t)+0.5π))),λ2=1-λ1,
and with
(68)A1=[0.50900-0.509000.509-40],Ad1=[0.21800-0.218000.21800],B1=[-1.428600],A2=[0.50900-0.509000.810-6.3660],Ad2=[0.21800-0.218000.34700],B2=[-1.428600].
Assume the time-varying delay1≤nT≤2, and the initial condition x(t0)=[-0.5π0.75π-5]T. The fuzzy delta operator controller gains by Theorem 10 are given as
(69)K11=[2.2108-2.72520.1445],K12=[2.2620-3.07760.1453],K21=[0.54230.0608-0.0046],K22=[0.56560.0503-0.0038],K31=[0.54670.0095-0.0015],K32=[0.56890.0079-0.0012].
As shown in Figure1, the states of the closed-loop system converge to zero under the obtained fuzzy delta operator state-feedback controller, which demonstrates the effectiveness of our method.Figure 1
State responses for the closed-loop system in Example14.


## 7. Conclusion

This paper proposes an input-output method to analysis and synthesis of T-S fuzzy delta operator systems with time-varying delay. The two-term approximation method has been employed to transform the fuzzy delta operator system with time-varying delay into a feedback interconnection form. Based on a Lyapunov-Krasovskii functional in delta operator domain, the SSG method is suggested for the interconnected system. Numerical examples are given to demonstrate the advantages and less-conservatism of the proposed results.



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