# Control Design for Discrete-Time Fuzzy Systems with Disturbance Inputs via Delta Operator Approach

Authors: Qi Zhou; Yabin Gao; Hongyi Li; Hamid Reza Karimi
Journal: Mathematical Problems in Engineering
 (2013)

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## Abstract

This paper is concerned with the problem of passive control design for discrete-time Takagi-Sugeno
(T-S) fuzzy systems with time delay and disturbance input via delta operator approach. The discrete-time
passive performance index is established in this paper for the control design problem. By constructing
a new type ofLyapunov-Krasovskii function (LKF) in delta domain, and utilizing some fuzzy weighing
matrices, a new passive performance condition is proposed for the system under consideration. Based on
the condition, a state-feedback passive controller is designed to guarantee that the resulting closed-loop
system is very-strictly passive. The existence conditions of the controller can be expressed by linear
matrix inequalities (LMIs). Finally, a numerical example is provided to demonstrate the feasibility and
effectiveness of the proposed method.


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## Body

## 1. Introduction

It is well known that the fuzzy logic control [1–4] is one of the most effective approaches to handle complex nonlinear systems. Takagi-Sugeno (T-S) fuzzy model has been proposed in [5] and applied to formulate a complex nonlinear systems into a framework. Some affine local models can be interpolated by a set of fuzzy membership functions in this framework. By T-S fuzzy model, a set of complex nonlinear systems can be possibly described as a weighed sum of some simple linear subsystems. Recently, the problem of stability analysis and controller synthesis of nonlinear systems in T-S fuzzy model has been extensively investigated in [5–18]. Due to the effect of time delay in systems, the problems of T-S fuzzy systems with time delays have got considerable attention in recent years, and some results have been developed in [9, 19–31].Recently, due to the fact that the passive properties can keep the system internally stable and have been frequently used to improve the stability of control systems. The passivity control has been widely applied in various engineering areas, such as electrical circuits systems, complex networks systems, mechanical systems, and nonlinear systems. The problems of passivity analysis and passive control for systems have been widely investigated [32–38]. The passive control problem has been investigated for fuzzy systems [35, 39, 40]. Among them, the authors in [40] are concerned with the very strict passive controller design problem for T-S fuzzy systems with time-varying delay.It is well known that the best system performances can be obtained with the shorter sampling period. In [41, 42], the authors pointed out that the excellent finite word length performance can be achieved under fast sampling via delta operator approach. The authors in [43] introduced the transformations between shift operator and delta operator transfer function models. A technique was developed in [44] to obtain an approximate delta operator system for a given continuous system. In [45], a tabular method was presented for real polynomial root distribution with respect to a circle in the complex plane, which is useful in stability of delta operator formulated discrete-time systems with a sampling time. By using delta operator formulation, the main design features of a loop transfer recovery controller both at input and output have been overviewed in [46]. More recently, many stability analysis and controller synthesis results about delta operator approach have been proposed in [47–49]. To mention a few, the robust stabilization problem was investigated in [47] for delta operator systems with time-varying delays. The authors in [49] investigated the robust H∞ control problem for a class of T-S fuzzy systems with time delays by using delta operator approach. However, there have been few results on the passive control for T-S fuzzy systems with time delays and disturbance inputs via delta operator approach, which motivates this study.In this paper, the problem of passive control is investigated for discrete-time T-S fuzzy systems with time delay and disturbance input via delta operator approach. The discrete-time passive performance index is concerned in this paper for the control design problem. A novel type Lyapunov-Krasovskii functional (LKF) is constructed in delta domain to present a new passive performance condition for discrete-time T-S fuzzy systems. Based on the condition, a state-feedback passive controller is designed to guarantee that the resulting closed-loop system is very strictly passive. The controller existence condition can be obtained in terms of linear matrix inequalities (LMIs), which can be solved by the standard software. A numerical example is given to illustrate the effectiveness of the proposed approach.At first, the T-S fuzzy model is employed to represent the nonlinear systems. By applying the LMIs techniques and LKF inδ-domain, the problem of passive control design for the discrete T-S fuzzy system with time delay and disturbance inputs is chewed. Then a new fuzzy state-feedback controller is designed which guarantees that the closed-loop fuzzy delta operator system with time delay is robustly asymptotically stable and satisfies a prescribed passive performance level. And those are some key points of contribution. It is worthwhile to note that a faster sampling method is utilized, and hence a better control effect by applying delta operator approach than shift operator approach is achieved. Finally, a numerical example is shown to indicate the feasibility and effectiveness of the proposed method.This paper is organized as follows. The problem to be solved is formulated in Section2. Main results, including passive analysis and passive controller design, are presented in Section 3. Section 4 provides an illustrative example to show the effectiveness and potential of the proposed design techniques. It is concluded this paper in Section 5.Notation. The notation used throughout the paper is fairly standard. L2[0,∞) denotes the space of square-integrable vector functions over [0,∞). The notation X>0 (resp., X≥0), for X∈ℝn×n, means that the matrix X is real symmetric positive definite (resp., positive semidefinite). The symbol “*” in a matrix A∈ℝn×n stands for the transposed elements in the symmetric positions. The superscripts “T” and “-1” denote the matrix transpose and inverse, respectively. Identity matrices of appropriate dimensions will be denoted by I. The shorthand diag{M1,M2,…,Mr} denotes a block diagonal matrix with diagonal blocks being the matrices M1, M2,…, Mr. If not explicitly stated, all matrices are assumed to have compatible dimensions for algebraic operations.


## 2. Problem Formulation

Considering the following fuzzy delta operator system with time delay, which is described byPlant Rule
i
. IF θ1(tk) is Ni1, and …, and θj(tk) is Nij, and …, and θp(tk) is Nip, THEN
(1)δx(tk)=Aix(tk)+Adix(tk-dk)+Biu(tk)+Bwiw(tk),z(tk)=Cix(tk)+Cdix(tk-dk)+Diu(tk)+Dwiw(tk),x(tk)=ϕ(tk),tk=-dM,dM+T,…,0,i=1,2,…,r,
where x(tk)∈ℝn is the state variable, u(tk)∈ℝm is the control input variable, w(tk)∈ℝl is the disturbance input variable which belongs to L2[0,∞), z(tk)∈ℝp is the control output, and ϕ(tk) is a continuous vector-valued initial function. Let tk=kT for the convenience in analysis.Ai, Adi,Bi, Bwi, Ci, Cdi,Di, and Dwi (i=1,2,…,r) are system matrices with appropriate dimensions. The scalar r is the number of IF-THEN rules. θj(tk) and Nij are the premise variable and the fuzzy set, respectively, for j=1,2,…,p. The parametric variable dk=nT which is the bounded time delay in the state and satisfies 0<dm≤dk≤dM with dm=nmT and dM=nMT (nm and nM are the known positive and finite integers, and T is a sampling period). δx(tk) is the delta operator of x(tk), which is defined by
(2)δx(tk)={ddtkx(tk),T=0,x(tk+T)-x(tk)T,T≠0.Then the defuzzified model of system (1) is inferred as follows:
(3)δx(tk)=∑i=1r‍hi(θ(tk))[Aix(tk)+Adix(tk-dk)+Biu(tk)+Bwiw(tk)],z(tk)=∑i=1r‍hi(θ(tk))[Cix(tk)+Cdix(tk-dk)+Diu(tk)+Dwiw(tk)],
where hi(θ(tk))=μi(θ(tk))/∑i=1rμi(θ(tk)), μi(θ(tk))=∏j=1pNij(θj(tk)), and Nij(θj(tk)) is the degree of the membership of θj(tk) in fuzzy set Nij. It can be assumed that μi(θ(tk))≥0 (for i=1,2,…,r) and ∑i=1rμi(θ(tk))≥0 (for all tk). Therefore, hi(θ(tk))≥0 (for i=1,2,…,r) and ∑i=1rhi(θ(tk))=1. Based on the parallel distributed compensation (PDC), similar to the fuzzy model, the following overall fuzzy control law can be constructed as
(4)u(tk)=∑s=1r‍hs(θ(tk))Ksx(tk),
where Ks(s=1,2,…,r) is the local control gain such that closed-loop fuzzy system (3) is asymptotically stable. Then substituting (4) into system (3), the closed-loop fuzzy system can be expressed as
(5)δx(tk)=∑i=1r‍∑s=1r‍hihs[(Ai+BiKs)x(tk)h+Adix(tk-dk)+Bwiw(tk)],z(tk)=∑i=1r‍∑s=1r‍hihs[(Ci+DiKs)x(tk)hh1+Cdix(tk-dk)+Dwiw(tk)],
where ∑i=1rhi=∑i=1rhi(θ(tk)). For the convenience, system (5) can be rewritten as
(6)δx(tk)=A(tk)x(tk)+Ad(tk)x(tk-dk)+Bw(tk)w(tk),z(tk)=C(tk)x(tk)+Cd(tk)x(tk-dk)+Dw(tk)w(tk),
where
(7)A(tk)=∑i=1r‍∑s=1r‍hihs(Ai+BiKs),Ad(tk)=∑i=1r‍hiAdi,Bw(tk)=∑i=1r‍hiBwi,C(tk)=∑i=1r‍∑s=1r‍hihs(Ci+DiKs),Cd(tk)=∑i=1r‍hiCdi,Dw(tk)=∑i=1r‍hiDwi.In the next section, the following definitions and lemmas are introduced for developing the main results.Definition 1 (see [47]).
The conditions for the asymptotic stability of a delta operator system hold:(a)
V(x(tk))≥0, with equality if and only if x(tk)=0,(b)
δV(x(tk))=[V(x(tk+T))-V(x(tk))]/T<0,where V(x(tk)) is a Lyapunov function in δ-domain. For Lyapunov function both in s-domain and z-domain, the condition (a)V(x(tk))≥0 in Definition 1 is given. On the other hand for the condition (b), when T→0, there exists
(8)limT→0δV(x(tk))=limT→0V(x(tk+T))-V(x(tk))T=dV(x(tk))dtk<0,
and when T=1, there exists
(9)δV(x(tk))=[V(x(tk+1))-V(x(tk))]1=V(x(tk+1))-V(x(tk))<0.
Obviously, the Lyapunov function inδ-domain can be reduced to the traditional Lyapunov function in s-domain or z-domain when the sampling period T tends to 0 or is 1.Definition 2 (see [50]).
(i) System (5) is said to be passive if there exists constant ρ such that
(10)2∑tk=0∞‍zT(tk)w(tk)≥ρ.
(ii) System (5) is said to be strictly passive if there exist constants ζ>0 and ρ such that
(11)2∑tk=0∞‍zT(tk)w(tk)≥ρ+ζ∑tk=0∞‍wT(tk)w(tk).
(iii) System (5) is said to be output strictly passive if there exist constants ε>0 and ρ such that
(12)2∑tk=0∞‍zT(tk)w(tk)≥ρ+ε∑tk=0∞‍zT(tk)z(tk).
(iv) System (5) is said to be very strictly passive if there exist constants ε>0,ζ>0, and ρ such that
(13)2∑tk=0∞‍zT(tk)w(tk)≥ρ+ε∑tk=0∞‍zT(tk)z(tk)+ζ∑tk=0∞‍wT(tk)w(tk).Lemma 3 (see [47]).
For any of the time functionsx(tk) and y(tk),
(14)δ(x(tk)y(tk))=δ(x(tk))y(tk)+x(tk)δ(y(tk))+Tδ(x(tk))δ(y(tk)),
where T is a sampling period.Lemma 4 (see [51]).
For any of the two positive integersr and r0, satisfying 1≤r0≤r holds:
(15)[∑i=r0r‍x(i)]TM[∑i=r0r‍x(i)]≤(r-r0+1)∑i=r0r‍xT(i)Mx(i),
where M is a constant positive semidefinite symmetric matrix.Lemma 5 (see [52]).
For the given constant matricesG, H and a symmetric constant matrix χ of appropriate dimensions, the following inequality holds:
(16)χ+GF(tk)H+HTFT(tk)GT≤0,
where F(tk) satisfies FT(tk)F(tk)≤I, if and only if for ξ>0 the following inequality holds:
(17)χ+ξ-1HTH+ξGGT≤0.Lemma 6.
If there exist scalarsε>0,ζ>0, and a differential function V(x(tk))≥0 such that
(18)δV(x(tk))+εzT(tk)z(tk)+ζwT(tk)w(tk)-2zT(tk)w(tk)≤0,
then system (5) is very strictly passive.Proof.
It follows from (18) that
(19)2∑tk=0∞‍zT(tk)w(tk)≥-V(x(0))+ε∑tk=0∞‍zT(tk)z(tk)+ζ∑tk=0∞‍wT(tk)w(tk)=ρ+ε∑tk=0∞‍zT(tk)z(tk)+ζ∑tk=0∞‍wT(tk)w(tk),
where ρ=-V(x(0)). Then, it can be seen that (13) is equivalent to (18) for very strict passive definition.Remark 7.
Based on Definition2, the main objective of this paper is just to prove that T-S fuzzy system (5) is very strictly passive via delta operator approach, which can also satisfy other three indexs. The very strictly passive control for system (5) is shown in the next section.


## 3. Main Results

This section focuses on designing a sufficient condition for the solvability of the proposed passive control problem and a developed LMI approach for designing the passive controller for fuzzy system (5). Firstly, the passivity analysis criterion is derived for the system (5) in the following theorem.Theorem 8.
Considering fuzzy delta operator system (5), for a given sampling period T>0, constants dm,dM  (0<dm≤dM), and matrix Ks (s=1,2,…,r), system (5) is very strictly passive if there exist scalars ε>0,ζ>0 and symmetric matrices P>0, R1>0,R2>0,Qκ>0,S1κ>0, and S2κ>0 (κ=1,2,…,r) with appropriate dimensions, such that the following LMIs hold for κ,λ,μ,ν,i,s=1,2,…,r:
(20)Θκλμνii<0,Θκλμνis+Θκλμνsi<0,s≠i,
where(21)Θκλμνis=[Ω1κλμνisΩ2isT*-εI],Ω2is=[0Ci+DiKsCdi00Dwi],Ω1κλμνis=[Ξ1P(Ai+BiKs)PAdi00PBwi*Ξ2κisPAdi1dmR11dMR2Ξ3is**-Qλ00-Cdi***-S1μ-1dmR100****-S2ν-1dMR20*****ζI-Dwi-DwiT],Ξ1=(T-2)P+dmR1+dMR2,Ξ3is=PBwi-(Ci+DiKs)T,Ξ2κis=P(Ai+BiKs)+(Ai+BiKs)TP+(dM-dm+T+1)Qκ+S1κ+S2κ-1dmR1-1dMR2.Proof.
Firstly, in order to simplify the calculation, relevant fuzzy weighing matrices which directly include the membership functions are defined as follows:(22)Q˘(tk)=∑κ=1r‍hκ(θ(tk))Qκ,S˘1(tk)=∑κ=1r‍hκ(θ(tk))S1κ,S˘2(tk)=∑κ=1r‍hκ(θ(tk))S2κ.
Choose a LKFV(x(tk)) for system (5) as follows:
(23)V(x(tk))=V1(x(tk))+V2(x(tk))+V3(x(tk))+V4(x(tk))+V5(x(tk)),
where
(24)V1(x(tk))=xT(tk)Px(tk),V2(x(tk))=T∑i=1n‍xT(tk-iT)Q˘(tk-iT)x(tk-iT),V3(x(tk))=T∑i=1nm‍xT(tk-iT)S˘1(tk-iT)x(tk-iT)+T∑i=1nM‍xT(tk-iT)S˘2(tk-iT)x(tk-iT),V4(x(tk))=T2∑i=nmnM‍∑s=1i‍xT(tk-sT)Q˘(tk-sT)x(tk-sT),V5(x(tk))=∑i=1nm‍∑s=1i‍eT(tk-sT)R1e(tk-sT)+∑i=1nM‍∑s=1i‍eT(tk-sT)R2e(tk-sT),
and symmetric matrices P>0,R1>0,R2>0,Qκ>0,S1κ>0,S2κ>0 (for κ=1,2,…,r), and e(s)=x(s)-x(s+T) from which it can be concluded that there exist e(tk-iT)=x(tk-iT)-x(tk-(i-1)T) and δx(tk)=-e(tk)/T.
Applying Lemma3 to V1(x(tk)) and along the trajectory of the system (5), it can be obtained that
(25)δV1(x(tk))=δT(x(tk))Px(tk)+xT(tk)Pδ(x(tk))+TδT(x(tk))Pδ(x(tk))=xT(tk)[PA(tk)+AT(tk)P]x(tk)+2xT(tk)PAd(tk)x(tk-dk)+2xT(tk)PBw(tk)w(tk)+TδT(x(tk))Pδ(x(tk)).
Similarly, applying the delta operator toV2(x(tk)),V3(x(tk)), and V4(x(tk)), it can be obtained as follows:
(26)δV2(x(tk))=1T[T∑i=1n‍xT(tk-iT+T)Q˘(tk-iT+T)×x(tk-iT+T)-T∑i=1n‍xT(tk-iT)Q˘(tk-iT)x(tk-iT)]≤xT(tk)Q˘(tk)x(tk)-xT(tk-dk)Q˘(tk-dk)x(tk-dk)+T∑i=nmnM‍xT(tk-iT)Q˘(tk-iT)x(tk-iT),δV3(x(tk))=xT(tk)(S˘1(tk)+S˘2(tk))x(tk)-xT(tk-dm)S˘1(tk-dm)x(tk-dm)-xT(tk-dM)S2(tk-dM)x(tk-dM),δV4(x(tk))=T[∑s=1i‍(xT(tk-sT+T)Q˘(tk-sT+T)×x(tk-sT+T)xT)-∑s=1i‍(xT(tk-sT)Q˘(tk-sT)×x(tk-sT)xT)∑s=1i‍]=(dM-dm+T)xT(tk)Q˘(tk)x(tk)-T∑i=nmnM‍xT(tk-iT)Q˘(tk-iT)x(tk-iT).
Applying Lemma4 to V5(x(tk)) and along the trajectory of system (5), it can be obtained that
(27)δV5(x(tk))=1T[∑i=1nm‍∑s=1i‍eT(tk-sT+T)R1e(tk-sT+T)-∑i=1nm‍∑s=1i‍eT(tk-sT)R1e(tk-sT)+∑i=1nM‍∑s=1i‍eT(tk-sT+T)×R2e(tk-sT+T)-∑i=1nM‍∑s=1i‍eT(tk-sT)R2e(tk-sT)]=1T[∑i=1nm‍eT(tk)R1e(tk)-∑i=1nm‍eT(tk-iT)R1e(tk-iT)+∑i=1nM‍eT(tk)R2e(tk)-∑i=1nM‍eT(tk-iT)R2e(tk-iT)]≤eT(tk)(nmTR1+nMTR2)e(tk)-1nmT[∑i=1nm‍e(tk-iT)]TR1×[∑i=1nm‍e(tk-iT)]-1nMT[∑i=1nM‍e(tk-iT)]T×R2[∑i=1nM‍e(tk-iT)]=δT(x(tk))(dmR1+dMR2)δ(x(tk))-1dm[x(tk)-x(tk-dm)]T×R1[x(tk)-x(tk-dm)]-1dM[x(tk)-x(tk-dM)]T×R2[x(tk)-x(tk-dM)].
For the real matrixP>0, the following equation is tenable:
(28)0=-2δT(x(tk))P[δ(x(tk))-A(tk)x(tk)-Ad(tk)x(tk-dk)-Bw(tk)w(tk)]=-2δT(x(tk))Pδ(x(tk))+2δT(x(tk))PA(tk)x(tk)+2δT(x(tk))PAd(tk)x(tk-dk)+2δT(x(tk))PBw(tk)w(tk).
Generally, considering the passive performance index in Definition2, which is described as the passivity analysis, performance of system (5) can be established as follows:
(29)J=ε-1zT(tk)z(tk)+ζwT(tk)w(tk)-2zT(tk)w(tk)+δV(tk).
By adding (28) into δV(tk) and applying (7), it can be found that
(30)J=ε-1zT(tk)z(tk)+ζwT(tk)w(tk)-2zT(tk)w(tk)+δV(tk)≤ηT(tk)∑κ=1r‍∑λ=1r‍∑μ=1r‍∑ν=1r‍∑i=1r∑s=1r‍hκhλhμhνhihsηT(tk)‍×[Ω1κλμνis+ε-1Ω2isTΩ2is]η(tk),
where hκ = hκ(θ(tk)),hλ = hλ(θ(tk-dk)),hμ = hμ(θ(tk-dm)),hν = hν(θ(tk-dM)),hi = hi(θ(tk)),hs = hs(θ(tk)), and(31)ηT(tk)=[δT(x(tk))xT(tk)xT(tk-dk)xT(tk-dm)xT(tk-dM)wT(tk)].
On the other hand, it can be seen from Theorem8 that
(32)∑κ=1r‍∑λ=1r‍∑μ=1r‍∑ν=1r‍∑i=1r‍∑s=1r‍hκhλhμhνhihsΘκλμνis=∑κ=1r‍∑λ=1r‍∑μ=1r‍∑ν=1r‍hκhλhμhν×[∑i=1r‍hi2Θκλμνii+∑i=1r-1‍∑s=i+1r‍hihs(Θκλμνis+Θκλμνsi)]<0.
Applying Schur complement, it can be obtained that(33)J=ε-1zT(tk)z(tk)+ζwT(tk)w(tk)-2zT(tk)w(tk)+δV(tk)<0.
It can be seen that the LMIs conditions in Theorem8 satisfy the very strictly passive performance index for the fuzzy delta operator system (5). The proof is completed.Based on the conditions in Theorem8, the state-feedback control gain matrices Ks (s=1,2,…,r) will be designed in the following theorem.Theorem 9.
Considering fuzzy delta operator system (5), for a given sampling period T>0 and constants dm,dM (0<dm≤dM), system (5) is very strictly passive if there exist scalars ε>0,ζ>0 and symmetric matrices P->0, R-1>0, R-2>0, Q-κ>0, S-1κ>0, and S-2κ>0 (κ=1,2,…,r) with appropriate dimensions, such that the following LMIs hold for κ,λ,μ,ν,i,s=1,2,…,r:
(34)Θ-κλμνii<0,Θ-κλμνis+Θ-κλμνsi<0,s≠i,
where(35)Θ-κλμνis=[Ω-1κλμνisΩ-2isT*-εI],Ω-2λμνis=[0C-iP-+DiK-sCdiP-00Dwi],Ω-1κλμνis=[Ξ-1AiP-+BiK-sAdiP-00Bwi*Ξ-2κisAdiP-1dmR-11dMR-2Ξ-3is**-Q-λ00-P-CdiT***-S-1μ-1dmR-100****-S-2ν-1dMR-20*****ζI-Dwi-DwiT],Ξ-1=(T-2)P-+dmR-1+dMR-2,Ξ-3is=Bwi-P-CiT-K-sTDiT,Ξ-2κis=AiP-+P-AiT+BiK-s+K-sTBiT+(dM-dm+T+1)Q-κ+S-1κ+S-2κ-1dmR-1-1dMR-2.The fuzzy sate-feedback controller can be obtained:
(36)u(tk)=K-sP--1x(tk).Proof.
Premultiplying and postmultiplying (34) by diagonal matrix
(37)diag{P--1,P--1,P--1,P--1,P--1,I,I}
and letting
(38)P=P--1,Ks=K-sP--1,R1=P--1R-1P--1,R2=P--1R-2P--1,Qi=P--1Q-iP--1,S1i=P--1S-1iP--1,S2i=P--1S-2iP--1,
for i=1,2,…,r, it can be found that the conditions (20) hold, which means that system (5) is very strictly passive. The proof is completed.Remark 10.
In this paper, the modeling uncertainties are not taken into account in system (1). It should be pointed out that the robust passive controller design condition can be presented for uncertain discrete-time T-S fuzzy systems with time delay and disturbance input via delta operator approach.
In order to present the stability and stabilization condition for discrete-time T-S fuzzy systems with time delay via delta operator approach, it is assumed that the disturbance inputw(tk)=0 in fuzzy delta operator system (3), which can be described as
(39)δx(tk)=∑i=1r‍hi(θ(tk))×[Aix(tk)+Adix(tk-dk)+Biu(tk)],z(tk)=∑i=1r‍hi(θ(tk))×[Cix(tk)+Cdix(tk-dk)+Diu(tk)].
Similar to the proof of Theorems8 and 9, the following two corollaries can be obtained.Corollary 11.
For a given sampling periodT>0 and constants dm and dM (0<dm≤dM), system (39) (u(tk)=0) is asymptotically stable if there exist symmetric matrices P>0, R1>0,R2>0,Qκ>0,S1κ>0, and S2κ>0 (κ=1,2,…,r) with appropriate dimensions, such that the following LMIs hold for κ,λ,μ,ν,i=1,2,…,r:
(40)Θ˘κλμνi<0,
where(41)Θ˘κλμνi=[Ξ1PAiPAdi00*Ξ˘2κiPAdi1dmR11dMR2**-Qλ00***-S1μ-1dmR10****-S2ν-1dMR2],Ξ1=(T-2)P+dmR1+dMR2,Ξ˘2κi=PAi+AiTP+(dM-dm+T+1)Qκ+S1κ+S2κ-1dmR1-1dMR2.Corollary 12.
System (39) with the controller in (4) is asymptotically stable for a given sampling period T>0 and constants dm and dM (0<dm≤dM), if there exist symmetric matrices P->0, R-1>0,R-2>0,Q-κ>0,S-1κ>0, and S-2κ>0 (κ=1,2,…,r) with appropriate dimensions, such that the following LMIs hold for κ,λ,μ,ν,i,s=1,2,…,r:
(42)Θ~κλμνii<0,Θ~κλμνis+Θ~κλμνsi<0,s≠i,
where(43)Θ~κλμνis=[Ξ-1AiP-+BiK-sAdiP-00*Ξ-2κisAdiP-1dmR-11dMR-2**-Q-λ00***-S-1μ-1dmR-10****-S-2ν-1dMR-2],Ξ-1κ=(T-2)P-+dmR-1+dMR-2,Ξ-2κis=AiP-+P-AiT+BiK-s+K-sTBiT+(dM-dm+T+1)Q-κ+S-1κ+S-2κ-1dmR-1-1dMR-2.Then, the controller (4) can be designed as
(44)u(tk)=K-sP--1x(tk).


## 4. Numerical Example

In this section, one example is presented to demonstrate the effectiveness of the proposed method.Example 1.
Consider the following fuzzy system with time-varying delay and input disturbance:
Plant Rule1. IF θ(tk) is 0, THEN
(45)δx(tk)=A1x(tk)+Ad1x(tk-dk)+B1u(tk)+Bw1w(tk),z(tk)=C1x(tk)+Cd1x(tk-dk)+D1u(tk)+Dw1w(tk).
Plant Rule
2. IF θ(tk) is ±(π/2)(|θ(tk)|<π/2), THEN
(46)δx(tk)=A2x(tk)+Ad2x(tk-dk)+B2u(tk)+Bw2w(tk),z(tk)=C2x(tk)+Cd2x(tk-dk)+D2u(tk)+Dw2w(tk).
The system matrix parameters in (45) and (46) are given as follows:
(47)A1=[-2.000-0.0600.0700.100],A2=[-1.8000.080-0.0500.100],B1=[0.0800.050],B2=[0.0800.090],Ad1=[-0.020-0.0100.0100.020],Ad2=[-1.8000.080-0.0500.100],Bw1=[0.010-0.100]T,Bw2=[-0.010-0.100]T,C1=[-0.0100.010],C2=[0.120-0.100],D1=0.010,Cd1=[0.0800.010],Cd2=[-0.010-0.050],D2=0.020,Dw1=0.010,Dw2=0.020.
Two membership functions are chosen for Plant Rules 1 and 2 as follows:(48)h1(θ(tk))={1-2πθ(tk),0≤θ(tk)≤π2,1+2πθ(tk),-π2≤θ(tk)<0,h2(θ(tk))=1-h1(θ(tk)).
And Figure 1 shows the membership functions.
Define the value range of time delaydk=nT, and choosing the lower bound dm=0.01, the sampling period T may be assumed as T=0.001 and it can be found from Theorem 9 that the maximum upper bound of time delay dk and the control gain matrices are listed in Table 1.
Assume that time delaydk=nT=40T and the disturbance input w(tk)=-1/(2+tk). It can be calculated that ∑tk=0∞wT(tk)w(tk)=0.5<∞, which means that w(tk)∈L2[0,∞). Let
(49)υ(tk)=2∑tk=0∞‍zT(tk)w(tk)-ε∑tk=0∞‍zT(tk)z(tk)-ζ∑tk=0∞‍wT(tk)w(tk),ς(tk)=2∑tk=0∞‍zT(tk)w(tk).
In Figure2, it can be observed that ς(tk) decreases finally as the time tk increases, which means that there may not exist a scalar ρ such that ς(tk)≥ρ hold for all tk≥0 for T=0.001. Furthermore, Figure 3 shows that there may not exist a scalar ρ such that υ(tk)≥ρ, which means that the open-loop system is not passive in the sense of Definition 2, and it is not very strictly passive. Figure 4 illustrates that the open-loop system is not stable for T=0.001.
Under the control gain matrices in Table1, Figure 5 plots the responses of ς(tk) for the closed-loop system for T=0.001. It can be seen that there may exist a scalar ρ such that ς(tk)≥ρ holds for all tk≥0. It can be seen from Figure 6 that there may exist a scalar ρ such that υ(tk)≥ρ. Then, it is clear that the closed-loop system is very strictly passive under the control gain matrices in Table 1. In addition, Figure 7 shows that the closed-loop system is stable. Furthermore, Figure 8 depicts the control input responses. These simulation results have demonstrated the effectiveness of the proposed method.Table 1
The feasibility of the system withT=0.001.
d
m
d
M
K
1
K
2
T
=
0.001
0.01
7.44
[
-
0.2229
-
13.6801
]
[
-
1.7743
-
4.5044
]Figure 1
Membership functions of two rules.Figure 2
Response ofς(tk) of the open-loop system.Figure 3
Response ofυ(tk) of the open-loop system.Figure 4
State response of the open-loop system.Figure 5
Response ofς(tk) of the closed-loop system.Figure 6
Response ofυ(tk) of the closed-loop system.Figure 7
State response of the open-loop system.Figure 8
Control signal response of the closed-loop system.


## 5. Conclusions

In this paper, the problems of passivity analysis and passive control have been investigated for fuzzy delta operator systems with time delay and disturbance input. By applying some new LKF inδ-domain and utilizing some fuzzy weighing matrices, the state-feedback controller has been designed to guarantee that the resulting closed-loop system is very strictly passive. The existence conditions for the controller have been expressed as LMIs. Finally, a numerical example has been included to illustrate the effectiveness of the proposed results. In future work, based on T-S fuzzy control method, the problems of control and monitoring in the data-driven framework [53, 54] could be further studied.



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*Source: 724918-2013-11-20.xml*