Passivity and Passification for Delay Fuzzy System Based on Delay Partitioning Approach

Abstract

A delay partitioning approach is introduced to solve problems of passivity and passification for continuous T-S fuzzy system with time delay. Our aim is to design a state feedback controller such that the resulting closed system is passive. By constructing a Lyapunov-Krasovskii functional, delay-dependent passivity/passification performance conditions are formulated in terms of linear matrix inequalities (LMIs). Finally, numerical examples are used to illustrate the effectiveness of the proposed approaches which can further reduce conservatism and become more obvious with partitioning getting thinner.

1. Introduction

The passivity concept was introduced by Willems [1] and developed further by Hill and Moylan [2]. Passivity of nonlinear systems has attracted great interest in the control area mainly because of the link between stability and passivity. The passivity theory provides a nice tool for analyzing the stability of systems and has found applications in diverse areas such as stability, signal processing, chaos control, and synchronization.

Since most physical systems in real world are nonlinear, researchers have been devoting their efforts to the control of nonlinear systems. Among the methods, the fuzzy control has been proven to be effective in dealing with the analysis of nonlinear systems. especially the T-S fuzzy control [3, 4]. It is denoted by a group of IF-THEN rules that the conventional linear system theory can be applied to the analysis of the class of nonlinear systems, and numerous nonlinear analysis problems have been studied based on this T-S fuzzy model, such as [5–7] reported the problem of stability analysis, and [8–10] investigated the 𝐻_∞ control designs. References [11–13] mentioned the fault detection of the T-S fuzzy systems. 𝐻_∞ model reduction is addressed in [14]. The fuzzy controller was carried out via Parallel Distributed Compensation (PDC) technique [15]. Based on the PDC technique, the fuzzy controller can also be designed to guarantee the passivity of T-S fuzzy systems.

On the other hand, the time delay exists naturally in various control systems. Time delays often degrade the system’s performance and even cause instability. Therefore, time delays have received great attention in recent years and many researchers have studied various analytical techniques and developed many synthesis methods for time-delay systems. For instance, model reduction is addressed in [16] and filtering problems are investigated in [17, 18]. So, the passivity and passification analysis of nonlinear systems with time delays is worth to be discussed and researched. To date, researches have gained many results in passivity control of T-S fuzzy systems; the passivity of delayed neural networks is considered in [19], passivity of fuzzy time-delay systems is investigated in [20] which adopt delay moom’s inequality, and passive controller design for T-S fuzzy systems is addressed in [21]. The passivity of uncertain fuzzy systems is considered in [22]. However, the above methods still have strong conservation, and it is necessary for us to further study.

In this paper, we adopt a delay partitioning approach to study the passivity and passification of T-S fuzzy systems with time delay. Based on this idea, we can further reduce the conservatism, and it becomes even less conservative when the partitioning goes finer. The results of this paper are given in terms of LMIs. The rest of the paper is organized as follows. In Section 2, the problem to be studied is stated and some preliminaries are presented. Passivity analysis results are presented in Section 3. Based on the results obtained in Section 3, we design the controller in Section 4. In Section 5, numerical examples are given to demonstrate the effectiveness of the theoretical results. Finally, conclusions are drawn in Section 6.

Notations. Throughout the paper, 𝐴⁻¹ and 𝐴^𝑇 denote the inverse and transpose of a square matrix 𝐴. 𝑅^𝑛 denotes the 𝑛-dimensional Euclidean apace and ‖⋅‖ refers to the Euclidean vector norm. The notation 𝐴>0 is used to define a symmetric positive definite matrix and sym (𝐴) is defined as 𝐴+𝐴^𝑇. Matrices are assumed to have compatible dimensions.

2. Problem Statement and Preliminaries

Consider the T-S fuzzy system with time delay has the following form.

Plant Rule 𝑖:

IF 𝑍₁(𝑡) is 𝑀_𝑖1 and … and 𝑍_𝑝(𝑡) is 𝑀_𝑖𝑝 THEN 𝑥̇(𝑡)=𝐴_𝑖𝑥(𝑡)+𝐴_𝑑𝑖𝑥(𝑡−ℎ)+𝐵_𝑖𝑢(𝑡)+𝐵_1𝑖𝜔(𝑡),𝑦(𝑡)=𝐶_𝑖𝑥(𝑡)+𝐶_𝑑𝑖𝑥(𝑡−ℎ)+𝐷_𝑖𝜔(𝑡),𝑥(𝑡)=𝜑(𝑡),𝑡∈[−ℎ,0],𝑖=1,…,𝑟, (1) where 𝑥(𝑡)∈𝑅^𝑛 is the state vector; 𝑢(𝑡) is the control input vector; ℎ is a time delay; 𝜑(𝑡) is the initial condition.

Controller Rule 𝑖:

IF 𝑧₁(𝑡) is 𝑀_𝑖1 and… and 𝑧_𝑝(𝑡) is 𝑀_𝑖𝑝 THEN 𝑢(𝑡)=𝐾_𝑖𝑥(𝑡),𝑖=1,…,𝑟, (2) where 𝑀_𝑖𝑗 is the fuzzy set; 𝑟 is the number of IF-THEN rules; 𝑧(𝑡)=[𝑧₁(𝑡),𝑧₂(𝑡),…,𝑧_𝑝(𝑡)] is the premise variables vector; 𝐾_𝑖, 𝑖=1,…,𝑟 are constant matrices representing state-feedback control gains. Let 𝜆_𝑖(𝑡) be the normalized membership function of the fuzzy set 𝜆_𝑖(𝑡)=∏^𝑝_𝑗=1𝑀_𝑖𝑗𝑧_𝑗(𝑡)∑^𝑟_𝑖=1∏^𝑝_𝑗=1𝑀_𝑖𝑗𝑧_𝑗(𝑡), (3) where 𝑀_𝑖𝑗(𝑧_𝑗(𝑡)) is the grade of membership function of 𝑧_𝑗(𝑡) in 𝑀_𝑖𝑗(𝑡). It is assumed that ∏^𝑝_𝑗=1𝑀_𝑖𝑗(𝑧_𝑗(𝑡))≥0,  𝑖=1,…,𝑟, and ∑^𝑟_𝑖=1{∏^𝑝_𝑗=1𝑀_𝑖𝑗(𝑧_𝑗(𝑡))}>0 for all 𝑡. Therefore, 𝜆_𝑖(𝑡)≥0 and ∑^𝑟_𝑖=1𝜆_𝑖(𝑡)=1 for all 𝑡. By applying (3) into (1), the fuzzy system can be expressed as 𝑥̇(𝑡)=𝐴(𝑡)𝑥(𝑡)+𝐴_𝑑(𝑡)𝑥()(𝑡)𝑢(𝑡)𝑡−ℎ+𝐵+𝐵₁(𝑡)𝜔(𝑡),𝑦(𝑡)=𝐶(𝑡)𝑥(𝑡)+𝐶_𝑑(𝑡)𝑥(𝑡−ℎ)+𝐷(𝑡)𝜔(𝑡) (4) with 𝐴(𝑡)=^𝑟_𝑖=1𝜆_𝑖(𝑡)𝐴_𝑖,𝐴_𝑑(𝑡)=^𝑟_𝑖=1𝜆_𝑖(𝑡)𝐴_𝑑𝑖,𝐵(𝑡)=^𝑟_𝑖=1𝜆_𝑖(𝑡)𝐵_𝑖,𝐵₁(𝑡)=^𝑟_𝑖=1𝜆_𝑖(𝑡)𝐵_1𝑖,𝐶(𝑡)=^𝑟_𝑖=1𝜆_𝑖(𝑡)𝐶_𝑖,𝐶_𝑑(𝑡)=^𝑟_𝑖=1𝜆_𝑖(𝑡)𝐶_𝑑𝑖,𝐷(𝑡)=^𝑟_𝑖=1𝜆_𝑖(𝑡)𝐷_𝑖. (5) By applying (2) into (4), we can get the following closed-loop system: 𝑥̇(𝑡)=^𝑟𝑟_𝑖=1𝑗=1𝜆_𝑖(𝑧(𝑡))𝜆_𝑗(𝑧(𝑡×𝐴))_𝑖+𝐵_𝑖𝐾_𝑗𝑥(𝑡)+𝐴_𝑑𝑖𝑥(𝑡−ℎ)+𝐵_1𝑖𝜔(𝑡,𝑦(𝑡)=_𝑟𝑖=1)𝜆_𝑖(𝑧(𝑡))𝐶_𝑖𝑥(𝑡)+𝐶_𝑑𝑖𝑥(𝑡−ℎ)+𝐷_𝑖𝜔(𝑡). (6) Before formulating the main problem, we first give the following definition.

Definition 1 (Li et al. [20]). The fuzzy system (1) is called passive if there exists a scalar 𝛾≥0 such that 2^𝑇₀𝜔(𝑠)^𝑇𝑦(𝑠)𝑑𝑠≥−𝛾^𝑇₀𝜔^𝑇(𝑠)𝜔(𝑠)𝑑𝑠 (7) for all 𝑇≥0.

3. Passivity Analysis

The problems to be addressed in this paper can be expressed as follows.

Problem 1  (passivity analysis). Given the feedback controller gain matrices 𝐾_𝑖,  𝑖=1,…,𝑟 in (2), determine under what conditions the closed-loop system (6) is passive for all 𝑇≥0 in the sense of Definition 1.

Problem 2  (passification). Determine the feedback controller gain matrices, 𝐾_𝑖, 𝑖=1,…,𝑟 in (2), such that the closed-loop system (6) is passive for all 𝑇≥0 in the sense of Definition 1.

In this section, we will present a sufficient condition in terms of LMIs, under which the closed-loop system (6) is passive.

Theorem 2. Given matrices 𝐾_𝑖, an integer 𝑚≥1 and a scalar ℎ>0, if there exist symmetric positive definite matrices 𝑃, 𝑄_𝑖, 𝑍_𝑖, 𝑅_𝑖 and matrices 𝑆_1𝑖, 𝑆_2𝑖 and scalar 𝛾>0, satisfying Θ_{𝑖𝑖𝑙𝑘}+𝜎𝑊^𝑇_𝜎𝑊_𝜎𝑆_1𝑖𝑚∗−ℎ𝑍_𝑖Θ<0,𝑖,𝑙,𝑘=1,…,𝑟_{𝑖𝑗𝑙𝑘}+𝜎𝑊^𝑇_𝜎𝑊_𝜎𝑆_1𝑖𝑚∗−ℎ𝑍_𝑖+Θ_{𝑗𝑖𝑙𝑘}+𝜎𝑊^𝑇_𝜎𝑊_𝜎𝑆_1𝑗𝑚∗−ℎ𝑍_𝑗𝑍<01≤𝑟<𝑗≤𝑟,𝑙,𝑘=1,…,𝑟,_𝑖<𝑅_𝑗,𝑖,𝑗=1,…,𝑟, (8)(9)(10) where Ω_{𝑖𝑗𝑙𝑘}=𝑊𝑝𝑃^𝑇∧ℎ𝑊𝑝+𝑚𝑊𝑟^𝑇𝑅_𝑖𝑊𝑟+𝑊𝑞^𝑇₁𝑄_𝑖𝑊𝑞₁−𝑊𝑞^𝑇₂𝑄_𝑙𝑊𝑞₂+sym𝑆_𝑖𝑊𝑠_𝑗𝑘,Θ_{𝑖𝑗𝑙𝑘}=Ω_{𝑖𝑗𝑙𝑘}−𝑊₁𝐶^𝑇_𝑗𝑊₂+𝑊^𝑇₂𝐶_𝑗𝑊^𝑇₁+𝑊^𝑇₂𝐷^𝑇_𝑗𝑊₂+𝑊^𝑇₂𝐷_𝑗𝑊₂+𝑊^𝑇₂𝐶_𝑑𝑗𝑊₃+𝑊^𝑇₃𝐶^𝑇_𝑑𝑗𝑊₂+𝛾𝑊^𝑇₂𝑊₂,𝐼𝑊𝑝=^𝑛𝑂_{𝑛,𝑚𝑛}𝑂_𝑛,𝑛𝑂_𝑛,𝑛𝑂_𝑛,𝑛𝑂_{𝑛,𝑚𝑛}𝐼_𝑛𝑂_𝑛,𝑛𝑂_𝑛,𝑛𝑂_{𝑛,𝑚𝑛}𝑂_𝑛,𝑛𝑂_𝑛.𝑛∧,𝑃=𝑂_𝑛,𝑛𝑃𝑂_𝑛,𝑛𝑃𝑂_𝑛,𝑛𝑂_𝑛,𝑛𝑂_𝑛,𝑛𝑂_𝑛,𝑛𝑂_𝑛,𝑛,𝑊_𝜎=𝐼_𝑛,𝑂_{𝑛,𝑚𝑛+2𝑛},𝑊𝑟=𝑂_{𝑛,𝑚𝑛+𝑛},𝐼_𝑛,𝑂_𝑛,𝑛,𝑊𝑞₁=𝐼_𝑚𝑛,𝑂_{𝑚𝑛,3𝑛},𝑊𝑞₂=𝑂_{𝑚𝑛,𝑛},𝐼_𝑚𝑛,𝑂_{𝑚𝑛,2𝑛},𝑆_𝑖=𝑆_1𝑖𝑆_2𝑖,𝑊_𝑆𝑗𝑘=𝐼_𝑛−𝐼_𝑛𝑂_{𝑛,𝑚𝑛+𝑛}𝐴_𝑗+𝐵_𝑗𝐾_𝑘𝑂_{𝑛,𝑚𝑛−𝑛}𝐴_𝑑𝑗−𝐼_𝑛𝐵_1𝑗,𝑊₁=𝐼_𝑛𝑂_{𝑚𝑛+2𝑛,𝑛},𝑊₂=𝑂_{𝑛,𝑚𝑛+2𝑛},𝐼_𝑛𝑊₃=𝑂_{𝑛,𝑚𝑛},𝐼_𝑛,𝑂_𝑛,2𝑛. (11) Then the fuzzy system (1) is passive in the sense of Definition 1 for the time delay 0≤𝜏≤ℎ.

Proof. Choose a Lyapunov-Krasovskii functional as 𝑉(𝑡)=𝑉₁(𝑡)+𝑉₂(𝑡)+𝑉₃(𝑡) with 𝑉₁(𝑡)=𝑥^𝑇(𝑡)(𝑡),𝑉𝑃𝑥₂(𝑡)=⁰_−ℎ/𝑚^𝑡_𝑡+𝛽𝑥̇^𝑇(𝛼)𝑅(𝛼)𝑥̇(𝛼)𝑉𝑑𝛼𝑑𝛽,₃(𝑡)=^𝑡_{𝑡−ℎ/𝑚}𝛾^𝑇(𝛼)𝑄(𝛼)𝛾(𝛼)𝑑𝛼, (12) where 𝛾(𝑡)=𝑥^𝑇(𝑡)𝑥^𝑇ℎ𝑡−𝑚⋯𝑥^𝑇𝑡−𝑚−1𝑚ℎ^𝑇. (13) The time-derivative of 𝑉(𝑡) along the trajectory of the system in (6) is given by 𝑉̇₁=𝑥̇^𝑇(𝑡)𝑃𝑥(𝑡)+𝑥^𝑇(𝑡)𝑃𝑥̇(𝑡),𝑉̇₂=ℎ𝑚𝑥̇^𝑇(𝑡)𝑅(𝑡)𝑥̇(𝑡)−^𝑡_{𝑡−ℎ/𝑚}𝑥̇^𝑇(𝛼)𝑅(𝛼)𝑥̇(𝛼)𝑉̇𝑑𝛼₃=𝛾^𝑇(𝑡)𝑄(𝑡)𝛾(𝑡)−𝛾^𝑇ℎ𝑡−𝑚𝑄ℎ𝑡−𝑚𝛾ℎ𝑡−𝑚. (14) Define 𝜉(𝑡)=𝛾^𝑇(𝑡)𝑥^𝑇(𝑡−ℎ)𝑥̇^𝑇(𝑡)𝜔^𝑇(𝑡)^𝑇,𝜔(𝑡)=𝜔₁(𝑡)⋯𝜔_𝑛(𝑡)^𝑇 (15) and according to the Newton-Leibniz formula and the system in (6), we have Π₁=2𝜉^𝑇(𝑡)𝑆₁(𝑡)×𝑥(𝑡)ℎ−𝑥𝑡−𝑚−^𝑡_{𝑡−ℎ/𝑚}𝑥̇(𝛼)Π𝑑𝛼=0,₂=2𝜉^𝑇(𝑡)𝑆₂(𝑡)(𝐴(𝑡)+𝐵(𝑡)𝐾(𝑡))𝑥(𝑡)+𝐴_𝑑(𝑡)𝑥(𝑡−ℎ)+𝐵₁(𝑡)𝜔(𝑡)−𝑥̇(𝑡Π)=0,₃=ℎ𝑚𝜉^𝑇(𝑡)𝑆₁(𝑡)𝑍⁻¹(𝑡)𝑆^𝑇₁(𝑡)𝜉(𝑡)−^𝑡_{𝑡−ℎ/𝑚}𝜉^𝑇(𝑡)𝑆₁(𝑡)𝑍⁻¹(𝑡)𝑆^𝑇₁(𝑡)𝜉(𝑡)𝑑𝛼=0. (16) Therefore𝑉̇(𝑡)≤𝑉̇₁(𝑡)+𝑉̇₂(𝑡)+𝑉̇₃(𝑡)+Π₁+Π₂+Π₃=𝑥̇^𝑇(𝑡)𝑃𝑥(𝑡)+𝑥^𝑇(𝑡)𝑃𝑥̇(𝑡)+ℎ𝑚𝑥̇^𝑇(𝑡)𝑅(𝑡)𝑥̇(𝑡)−^𝑡_{𝑡−ℎ/𝑚}𝑥̇^𝑇(𝛼)𝑅(𝛼)𝑥̇(𝛼)𝑑𝛼+𝛾^𝑇(𝑡)𝑄(𝑡)𝛾(𝑡)−𝛾^𝑇ℎ𝑡−𝑚𝑄ℎ𝑡−𝑚𝛾ℎ𝑡−𝑚+2𝜉^𝑇(𝑡)𝑆₁(𝑡)×𝑥(𝑡)ℎ−𝑥𝑡−𝑚−^𝑡_{𝑡−ℎ/𝑚}𝑥̇(𝛼)𝑑𝛼+2𝜉^𝑇(𝑡)𝑆₂(𝑡)×(𝐴(𝑡)+𝐵(𝑡)𝐾(𝑡))𝑥(𝑡)+𝐴_𝑑(𝑡)𝑥(𝑡−ℎ)+𝐵₁(𝑡)𝜔(𝑡)−𝑥̇(𝑡+ℎ)𝑚𝜉^𝑇(𝑡)𝑆₁(𝑡)𝑍⁻¹(𝑡)𝑆^𝑇₁(𝑡)𝜉(𝑡)−^𝑡_{𝑡−ℎ/𝑚}𝜉^𝑇(𝑡)𝑆₁(𝑡)𝑍⁻¹(𝑡)𝑆^𝑇₁(𝑡)𝜉(𝑡)(𝑡)+ℎ𝑑𝛼=Λ𝑚𝜉^𝑇(𝑡)𝑆₁(𝑡)𝑍⁻¹(𝑡)𝑆^𝑇₁(𝑡)𝜉(𝑡)−^𝑡_{𝑡−ℎ/𝑚}𝑥̇^𝑇(𝛼)𝑅(𝛼)𝑥̇(𝛼)𝑑𝛼−2𝜉^𝑇(𝑡)𝑆₁(𝑡)^𝑡_{𝑡−ℎ/𝑚}𝑥̇(𝛼)−𝑑𝛼_{𝑡𝑡−ℎ/𝑚}𝜉^𝑇(𝑡)𝑆₁(𝑡)𝑍⁻¹(𝑡)𝑆^𝑇₁(𝑡)𝜉(𝑡)𝑑𝛼. (17) If 𝑍(𝑡)<𝑅(𝛼). (18) Then 𝑍⁻¹(𝑡)>𝑅⁻¹(𝛼),−^𝑡_{𝑡−ℎ/𝑚}𝜉^𝑇(𝑡)𝑆₁(𝑡)𝑍⁻¹(𝑡)𝑆^𝑇₁(𝑡)𝜉(𝑡)𝑑𝛼<−^𝑡_{𝑡−ℎ/𝑚}𝜉^𝑇(𝑡)𝑆₁(𝑡)𝑅⁻¹(𝛼)𝑆^𝑇₁(𝑡)𝜉(𝑡)𝑑𝛼. (19)(20) So, we can obtain 𝑉̇(𝑡)≤Λ(𝑡)+ℎ𝑚𝜉^𝑇(𝑡)𝑆₁(𝑡)𝑍⁻¹(𝑡)𝑆^𝑇₁(𝑡)𝜉(𝑡)−^𝑡_{𝑡−ℎ/𝑚}𝑥̇^𝑇(𝛼)𝑅(𝛼)+𝜉^𝑇(𝑡)𝑆₁(𝑡)×𝑅⁻¹(𝛼)×𝑅(𝛼)𝑥̇(𝛼)+𝑆^𝑇₁(𝑡)𝜉(𝑡)𝑑𝛼, (21) where Λ(𝑡)=𝑥̇^𝑇(𝑡)𝑃𝑥(𝑡)+𝑥^𝑇(𝑡)𝑃𝑥̇(𝑡)+ℎ𝑚𝑥̇^𝑇(𝑡)𝑅(𝑡)𝑥̇(𝑡)+𝛾^𝑇(𝑡)𝑄(𝑡)𝛾(𝑡)−𝛾^𝑇ℎ𝑡−𝑚𝑄ℎ𝑡−𝑚ℎ×𝛾𝑡−𝑚+2𝜉^𝑇(𝑡)𝑆₁(𝑡𝑥(𝑡)ℎ)−𝑥𝑡−𝑚+2𝜉^𝑇(𝑡)𝑆₂(𝑡)×(𝐴(𝑡)+𝐵(𝑡)𝐾(𝑡))𝑥(𝑡)+𝐴_𝑑(𝑡)𝑥(𝑡−ℎ)+𝐵₁(𝑡)𝜔(𝑡)−𝑥̇(𝑡). (22) Besides 𝑥̇(𝑡)=𝑂_{𝑛,𝑚𝑛+𝑛},𝐼_𝑛,𝑂_𝑛,𝑛𝜉(𝑡),𝑥(𝑡)=𝐼_𝑛,𝑂_{𝑛,𝑚𝑛+2𝑛}𝜉(𝑡),𝛾(𝑡)=𝐼_𝑚𝑛,𝑂_{𝑚𝑛,3𝑛}𝜉(𝑡),𝛾ℎ𝑡−𝑚=𝑂_{𝑚𝑛,𝑛},𝐼_𝑚𝑛,𝑂_{𝑚𝑛,2𝑛}𝜉(𝑡). (23) Then Λ(𝑡)=𝜉^𝑇(𝑡)Ω(𝑡)𝜉(𝑡),Ω(𝑡)=𝑊𝑃^𝑇_𝑝^∧𝑊_𝑝+ℎ𝑚𝑊^𝑇_𝑟𝑅(𝑡)𝑊_𝑟+𝑊^𝑇_𝑞1𝑄(𝑡)𝑊_𝑞1−𝑊^𝑇_𝑞2𝑄ℎ𝑡−𝑚𝑊_𝑞2+sym𝑆(𝑡)𝑊_𝑆(𝑡),𝑂𝑃=^𝑛,𝑛𝑃𝑂_𝑛,𝑛𝑃𝑂_𝑛,𝑛𝑂_𝑛,𝑛𝑂_𝑛,𝑛𝑂_𝑛,𝑛𝑂_𝑛,𝑛,𝑆(𝑡)=𝑆₁(𝑡)𝑆₂(𝑡𝐼)𝑊𝑝=^𝑛𝑂_{𝑛,𝑚𝑛}𝑂_𝑛,𝑛𝑂_𝑛,𝑛𝑂_𝑛,𝑛𝑂_{𝑛,𝑚𝑛}𝐼_𝑛𝑂_𝑛,𝑛𝑂_𝑛,𝑛𝑂_{𝑛,𝑚𝑛}𝑂_𝑛,𝑛𝑂_𝑛.𝑛,𝑊𝑟=𝑂_{𝑛,𝑚𝑛+𝑛},𝐼_𝑛,𝑂_𝑛,𝑛,𝑊𝑞₁=𝐼_𝑚𝑛,𝑂_{𝑚𝑛,3𝑛},𝑊𝑞₂=𝑂_{𝑚𝑛,𝑛},𝐼_𝑚𝑛,𝑂_{𝑚𝑛,2𝑛},𝑊_𝑆(𝑡)=𝐼_𝑛−𝐼_𝑛𝑂_{𝑛,𝑚𝑛+𝑛}𝐴(𝑡)+𝐵(𝑡)𝐾(𝑡)𝑂_{𝑛,𝑚𝑛−𝑛}𝐴_𝑑(𝑡)−𝐼_𝑛𝐵₁(𝑡). (24) Besides 2𝜔^𝑇(𝑡)𝑦(𝑡)+𝛾𝜔^𝑇(𝑡)𝜔(𝑡)=𝑥^𝑇(𝑡)𝐶^𝑇_𝑖𝜔(𝑡)+𝜔^𝑇(𝑡)𝐶_𝑖𝑥(𝑡)+𝜔^𝑇(𝑡)𝐷^𝑇_𝑖+𝐷_𝑖𝜔(𝑡)+𝜔^𝑇(𝑡)𝐶_𝑑𝑖𝑥(𝑡−ℎ)+𝑥^𝑇(𝑡−ℎ)𝐶^𝑇_𝑑𝑖𝜔(𝑡)+𝛾𝜔^𝑇(𝑡)𝜔(𝑡)=𝜉^𝑇(𝑡)𝑊₁𝐶^𝑇_𝑖𝑊₂+𝑊^𝑇₂𝐶_𝑖𝑊^𝑇₁+𝑊^𝑇₂𝐷^𝑇_𝑖+𝐷_𝑖𝑊₂+𝑊^𝑇₂𝐶_𝑑𝑖𝑊₃+𝑊^𝑇₃𝐶^𝑇_𝑑𝑖𝑊₂+𝛾𝑊^𝑇₂𝑊₂𝜉(𝑡), (25) where 𝑊₁=𝐼_𝑛𝑂_{𝑚𝑛+2𝑛,𝑛},𝑊₂=𝑂_{𝑛,𝑚𝑛+2𝑛},𝐼_𝑛,𝑊₃=𝑂_{𝑛,𝑚𝑛},𝐼_𝑛,𝑂_𝑛,2𝑛. (26) So, we have 𝑉̇(𝑡)−2𝜔^𝑇(𝑡)𝑦(𝑡)−𝜆𝜔^𝑇(𝑡)𝜔(𝑡)≤𝜉^𝑇(𝑡)Θ(𝑡)𝜉(𝑡)+ℎ𝑚𝜉^𝑇(𝑡)𝑆₁(𝑡)𝑍⁻¹(𝑡)𝑆^𝑇₁(𝑡)𝜉(𝑡)−^𝑡_{𝑡−ℎ/𝑚}𝑥̇^𝑇(𝛼)𝑅(𝛼)+𝜉^𝑇(𝑡)𝑆₁(𝑡)×𝑅⁻¹(𝛼)×𝑅(𝛼)𝑥̇(𝛼)+𝑆^𝑇₁(𝑡)𝜉(𝑡)𝑑𝛼 (27) with Θ(𝑡)=Ω(𝑡)−𝑊₁𝐶^𝑇_𝑖𝑊₂+𝑊^𝑇₂𝐶_𝑖𝑊^𝑇₁+𝑊^𝑇₂𝐷_𝑖+𝐷^𝑇_𝑖𝑊₂+𝑊^𝑇₂𝐶_𝑑𝑖𝑊₃+𝑊^𝑇₃𝐶^𝑇_𝑑𝑖𝑊₂+𝛾𝑊^𝑇₂𝑊₂. (28) If Θ(𝑡)+𝜎𝑊^𝑇_𝜎𝑊_𝜎+ℎ𝑚𝑆₁(𝑡)𝑍⁻¹(𝑡)𝑆^𝑇₁(𝑡)<0 (29) then Θ(𝑡)+ℎ𝑚𝑆₁(𝑡)𝑍⁻¹(𝑡)𝑆^𝑇₁(𝑡)<−𝜎𝑊^𝑇_𝜎𝑊_𝜎,𝑉̇(𝑡)−2𝜔^𝑇(𝑡)𝑦(𝑡)−𝛾𝜔^𝑇(𝑡)𝜔(𝑡)<𝜉^𝑇(𝑡)Θ(𝑡)𝜉(𝑡)+ℎ𝑚𝜉^𝑇(𝑡)𝑆₁(𝑡)𝑍⁻¹(𝑡)𝑆^𝑇₁(𝑡)𝜉(𝑡)<−𝜉^𝑇(𝑡)𝜎𝑊^𝑇_𝜎𝑊_𝜎𝜉(𝑡)<−𝜎‖𝑥(𝑡)‖²<0. (30) We can obtain 𝑉̇(𝑡)<2𝑦^𝑇(𝑡)𝜔(𝑡)+𝛾𝜔^𝑇(𝑡)𝜔(𝑡). (31) It follows by integrating (31) with respect to 𝑡 over the time period 0∼𝑇 that (7) holds, and hence the delayed fuzzy system (1) is passive in the sense of Definition 1. Our next objective is to convert the inequalities in (18) and (29) to some finite LMIs, then (29) can be rewritten as 𝜂_{𝑖𝑗𝑙𝑘}=𝜂_{𝑖𝑖𝑙𝑘}+𝜂_{𝑖𝑗𝑙𝑘(𝑖<𝑗)}+𝜂_{𝑗𝑖𝑙𝑘(𝑖<𝑗)}𝜂<0,_{𝑖𝑗𝑙𝑘}=^𝑟_𝑖=1𝜆_𝑖(𝜃(𝑡))^𝑟_𝑗=1𝜆_𝑗(𝜃(𝑡×))_𝑟𝑙=1𝜆_𝑙𝜃ℎ𝑡−𝑚^𝑟_𝑘=1𝜆_𝑘(𝜃(𝑡×Ω))_{𝑖𝑗𝑙𝑘}−𝑊₁𝐶^𝑇_𝑗𝑊₂+𝑊^𝑇₂𝐶_𝑗𝑊^𝑇₁+𝑊^𝑇₂𝐷_𝑗+𝐷^𝑇_𝑗𝑊₂+𝑊^𝑇₂𝐶_𝑑𝑗𝑊₃+𝑊^𝑇₃𝐶^𝑇_𝑑𝑗𝑊₂+𝛾𝑊^𝑇₂𝑊₂+𝜎𝑊^𝑇_𝜎𝑊_𝜎+ℎ𝑚𝑆_1𝑖𝑍_𝑖⁻¹𝑆^𝑇_1𝑖. (32) Equation (18) can be rewritten as 𝑍_𝑖<𝑅_𝑗. (33) From the Schur complement, we can get the inequalities (8), (9), and (10), and the proof is completed.

4. Controller Design

In this section, fuzzy state feedback controllers will be designed based on the result developed in the previous section.

Theorem 3. Given an integer 𝑚>1 and scalars ℎ>0, 𝜏₁,𝜏₂,…,𝜏_(𝑚+3), there exists a fuzzy state feedback controller such that the closed-loop system (4) is passive if there exist symmetric positive definite matrices 𝑃, 𝑄_𝑖, 𝑍_𝑖, 𝑅_𝑖, and matrices 𝑋, 𝑆_1𝑖, 𝑀_𝑖 and scalars 𝛾>0, 𝜎>0, satisfying 𝜑_𝑖𝑖𝑙𝑆_1𝑖𝑊^𝑇_𝜎𝑋^𝑇𝑊^𝑇₂𝑋^𝑇𝑊^𝑇₂𝑋^𝑇𝑚∗−ℎ𝑍_𝑖000∗∗−𝜎⁻¹𝐼_𝑛00∗∗∗𝐷_𝑖^−𝑇+𝐷_𝑖⁻¹0∗∗∗∗𝛾⁻¹𝐼_𝑛𝜙<0𝑖,𝑙=1,…,𝑟,_𝑖𝑗𝑙𝑆_1𝑖𝑊^𝑇_𝜎𝑋^𝑇𝑊^𝑇₂𝑋^𝑇𝑊^𝑇₂𝑋^𝑇𝑚∗−ℎ𝑍_𝑖000∗∗−𝜎⁻¹𝐼_𝑛00∗∗∗𝐷_𝑖^−𝑇+𝐷_𝑖⁻¹0∗∗∗∗𝛾⁻¹𝐼_𝑛+𝜙_𝑗𝑖𝑙𝑆_1𝑗𝑊^𝑇_𝜎𝑋^𝑇𝑊^𝑇₂𝑋^𝑇𝑊^𝑇₂𝑋^𝑇𝑚∗−ℎ𝑍_𝑗000∗∗−𝜎⁻¹𝐼_𝑛00∗∗∗𝐷_𝑗^−𝑇+𝐷_𝑗⁻¹0∗∗∗∗𝛾⁻¹𝐼_𝑛<01≤𝑖<𝑗≤𝑟,𝑙=1,…,𝑟,𝑍_𝑖<𝑅_𝑗,𝑖,𝑗=1,…,𝑟, (34)(35)(36) where 𝜑_𝑖𝑗𝑙=𝑊𝑃^𝑇_𝑝^∧𝑊_𝑝+ℎ𝑚𝑊^𝑇_𝑟𝑅_𝑖𝑊_𝑟+𝑊^𝑇_𝑞1𝑄_𝑖𝑊_𝑞1−𝑊^𝑇_𝑞2𝑄_𝑙𝑊_𝑞2+sym𝑉_𝑖𝑊_𝑆𝑖𝑗−𝑊₁𝑋^𝑇𝐶^𝑇_𝑖𝑊₂−𝑊^𝑇₂𝐶_𝑑𝑖𝑋𝑊₃,𝑂𝑃=^𝑛,𝑛𝑃𝑂_𝑛,𝑛𝑃𝑂_𝑛,𝑛𝑂_𝑛,𝑛𝑂_𝑛,𝑛𝑂_𝑛,𝑛𝑂_𝑛,𝑛,𝑉_𝑖=𝑆_1𝑖𝑈,𝑈=𝜏₁𝐼_𝑛𝜏₂𝐼_𝑛⋯𝜏_(𝑚+3)𝐼_𝑛^𝑇,𝑊_𝑆𝑖𝑗=𝐼_𝑛−𝐼_𝑛𝑂_{𝑛,𝑚𝑛+𝑛}𝐴_𝑖𝑋+𝐵_𝑖𝑀_𝑗𝑂_{𝑛,𝑚𝑛−𝑛}𝐴_𝑑𝑖𝑋−𝑋𝐵_1𝑖. (37) If the above conditions are feasible, the gains of the controller are given by 𝐾_𝑖=𝑀_𝑖𝑋⁻¹,𝑖=1,…,𝑟. (38)

Proof. Assume that 𝑋 is invertible, define 𝑆=𝑋^−𝑇, and 𝐺1=diag𝑆,…,𝑆,𝐼_𝑛,𝑆,𝑆,𝑆,𝑆∈𝑅^{(𝑚𝑛+7𝑛)×(𝑚𝑛+7𝑛)},𝐺2=diag𝑆,…,𝑆,𝐼_𝑛∈𝑅^{(𝑚𝑛+3𝑛)×(𝑚𝑛+3𝑛)},𝐺3=diag{𝑆,…,𝑆}∈𝑅^{𝑚𝑛×𝑚𝑛}. (39) Premultiplying and postmultiplying (34) with 𝐺₁ and 𝐺^𝑇₁, then we obtainΔ𝐺₂𝑆_1𝑖𝑆^𝑇𝐺₂𝑊^𝑇_𝜎𝑋^𝑇𝑆^𝑇𝐺₂𝑊^𝑇₂𝑋^𝑇𝑆^𝑇𝐺₂𝑊^𝑇₂𝑋^𝑇𝑆^𝑇𝑚∗−ℎ𝑆𝑍_𝑖𝑆^𝑇000∗∗−𝜎⁻¹𝑆𝑆^𝑇𝐷00∗∗∗𝑆_𝑖^−𝑇+𝐷_𝑖⁻¹𝑆^𝑇0∗∗∗∗𝛾⁻¹𝑆𝑆^𝑇<0, (40)where Δ=𝐺₂𝜙_𝑖𝑖𝑙𝐺^𝑇₂,𝐺₂𝜙_𝑖𝑖𝑙𝐺^𝑇₂=𝑊^𝑇_𝑝𝑂_𝑛,𝑛𝑆𝑃𝑆^𝑇𝑂_𝑛,𝑛𝑆𝑃𝑆^𝑇𝑂_𝑛,𝑛𝑂_𝑛,𝑛𝑂_𝑛,𝑛𝑂_𝑛,𝑛𝑂_𝑛,𝑛𝑊_𝑝+ℎ𝑚𝑊^𝑇_𝑟𝑆𝑅_𝑖𝑆^𝑇𝑊_𝑟+𝑊^𝑇_𝑞1𝐺₃𝑄_𝑖𝐺^𝑇₃𝑊_𝑞1−𝑊^𝑇_𝑞2𝐺₃𝑄_𝑙𝐺^𝑇₃𝑊_𝑞2+sym𝐺₂𝑉_𝑖𝑊_𝑆𝑖𝑖𝐺^𝑇₂−𝑊₁𝐶^𝑇_𝑖𝑊₂−𝑊^𝑇₂𝐶_𝑖𝑊^𝑇₁−𝑊^𝑇₃𝐶^𝑇_𝑑𝑖𝑊₂−𝑊^𝑇₂𝐶_𝑑𝑖𝑊₃,𝐺₂𝑊^𝑇_𝜎𝑋^𝑇𝑆^𝑇=𝑊^𝑇_𝜎𝑆^𝑇,𝐺₂𝑊^𝑇₂𝑋^𝑇𝑆^𝑇=𝑊^𝑇₂𝑆^𝑇. (41) We defining 𝑃=𝑆𝑃𝑆^𝑇,𝑅_𝑖=𝑆𝑅_𝑖𝑆^𝑇,𝑄_𝑖=𝐺₃𝑄_𝑖𝐺^𝑇₃,𝑆_1𝑖=𝐺₂𝑆_1𝑖𝑆^𝑇,𝑍_𝑖=𝑆𝑍_𝑖𝑆^𝑇,𝑆_2𝑖=𝜏₁𝑆^𝑇𝜏₂𝑆^𝑇⋯𝜏_(𝑚+2)𝑆^𝑇𝜏_(𝑚+3)𝐼_𝑛^𝑇. (42) Then 𝐺₂𝑉_𝑖𝑊_𝑆𝑖𝑖𝐺^𝑇₂=𝐺₂𝑆_1𝑖𝐼_𝑛−𝐼_𝑛𝑂_{𝑛,𝑚𝑛+𝑛}𝐺^𝑇₂+𝐺₂𝑈𝐴_𝑖𝑋+𝐵_𝑖𝑀_𝑖𝑂_{𝑛,𝑚𝑛−𝑛}𝐴_𝑑𝑖𝑋−𝑋𝐵_1𝑖𝐺^𝑇₂=𝐺₂𝑆_1𝑖𝑆^𝑇𝐼_𝑛−𝐼_𝑛𝑂_{𝑛,𝑚𝑛+𝑛}+𝑆_2𝑖𝐴_𝑖+𝐵_𝑖𝐾_𝑘𝑂_{𝑛,𝑚𝑛−𝑛}𝐴_𝑑𝑖−𝐼_𝑛𝐵_1𝑖=𝑆_1𝑖𝑆_2𝑖𝐼_𝑛−𝐼_𝑛𝑂_{𝑛,𝑚𝑛+𝑛}𝐴_𝑖+𝐵_𝑖𝐾_𝑘𝑂_{𝑛,𝑚𝑛−𝑛}𝐴_𝑑𝑖−𝐼_𝑛𝐵_1𝑖 (43) then 𝐺₂𝜑_𝑖𝑖𝑙𝐺^𝑇₂=𝑊^𝑇_𝑝𝑂_𝑛,𝑛𝑃𝑂_𝑛,𝑛𝑃𝑂_𝑛,𝑛𝑂_𝑛,𝑛𝑂_𝑛,𝑛𝑂_𝑛,𝑛𝑂_𝑛,𝑛𝑊_𝑝+ℎ𝑚𝑊^𝑇_𝑟𝑅_𝑖𝑊_𝑟+𝑊^𝑇_𝑞1𝑄_𝑖𝑊_𝑞1−𝑊^𝑇_𝑞2𝑄_𝑙𝑊_𝑞2+sym𝐺₂𝑉_𝑖𝑊_𝑆𝑖𝑖𝐺^𝑇₂−𝑊₁𝐶^𝑇_𝑖𝑊₂−𝑊^𝑇₂𝐶_𝑖𝑊^𝑇₁−𝑊^𝑇₃𝐶^𝑇_𝑑𝑖𝑊₂−𝑊^𝑇₂𝐶_𝑑𝑖𝑊₃. (44) Thus, by the Schur complement, we can obtain (40) is equivalent to (8). Pre- and postmultiplying (35) with 𝐺₁ and 𝐺^𝑇₁, we obtain (9), pre- and postmultiplying (36) with 𝑆 and 𝑆^𝑇, we obtain (10). The proof is completed.

5. Numerical Example

Without delay and uncertainty, Example 1 designs different passive controllers by applying the theorem of our paper and the literature [22], respectively. We can compare the region of feasible solution.

Example 1. Consider a fuzzy system of the following form.
Plant Rules:
Rule 1: IF 𝑥₁(𝑡) is 𝑀₁, THEN 𝑥̇(𝑡)=𝐴₁𝑥(𝑡)+𝐵₁𝑢(𝑡)+𝐵₁₁𝜔(𝑡),𝑦(𝑡)=𝐶₁𝑥(𝑡)+𝐷₁𝜔(𝑡). (45)
Rule 2: IF 𝑥₁(𝑡) is 𝑀₂, THEN 𝑥̇(𝑡)=𝐴₂𝑥(𝑡)+𝐵₂𝑢(𝑡)+𝐵₁₂𝜔(𝑡),𝑦(𝑡)=𝐶₂𝑥(𝑡)+𝐷₂𝜔(𝑡) (46) with 𝐴₁=𝑎−0.021105123,𝐴₂=,𝐵−0.225−0.020103111₁=10𝑏,𝐵₂=102,𝐵₁₁=,𝐵100120010₁₂=100123012,𝐶₁=𝐶₂=,𝐷2.50002.50121₁=𝐷₂=[][].200020111,𝑎∈−2,3,𝑏∈6,10 (47) We can obtain Figure 1 by applying our method and obtain Figure 2 by applying method in [22]. Symbol “*” shows that feasible solution exists at that point; symbol “o” shows that feasible solution does not exist at that point.

Figure 1 

The feasible region based on Theorem 2.

Figure 2 

The feasible region based on [22].

It is obvious that Theorem 3 can relax the conditions in [22], and our method futher reduced the conservatism.

Example 2 (Li et al. [20]). Consider a delayed fuzzy system of the following form.
Plant Rules:
Rule 1: IF 𝑥₁(𝑡) is 𝑀₁, THEN 𝑥̇(𝑡)=𝐴₁𝑥(𝑡)+𝐴_𝑑1𝑥(𝑡−ℎ)+𝐵₁𝑢(𝑡)+𝐵₁₁𝜔(𝑡),𝑦(𝑡)=𝐶₁𝑥(𝑡)+𝐶_𝑑1𝑥(𝑡−ℎ)+𝐷₁𝜔(𝑡). (48) Rule 2: IF 𝑥₁(𝑡) is 𝑀₂, THEN 𝑥̇(𝑡)=𝐴₂𝑥(𝑡)+𝐴_𝑑2𝑥(𝑡−ℎ)+𝐵₂𝑢(𝑡)+𝐵₁₂𝜔(𝑡),𝑦(𝑡)=𝐶₂𝑥(𝑡)+𝐶_𝑑2𝑥(𝑡−ℎ)+𝐷₂𝜔(𝑡) (49) with 𝐴₁=−10.20−0.1,𝐴₂=,𝐴−100.1−0.5_𝑑1=0.10.60.2−0.1,𝐴_𝑑2=,𝐵−0.20.10.1−0.2₁=0.2−0.5,𝐵₂=0.30.3,𝐵₁₁=,𝐵100.20₁₂=−0.2000,𝐶₁=,𝐶10.200₂=0.1100,𝐶_𝑑1=,𝐶−0.10.200_𝑑2=0.1−0.100,𝐷₁=,𝐷0.5000₂=0.1000 (50) we let ℎ=7.5,  𝑚=2, 𝜏₁=10, 𝜏₂=0, 𝜏₃=5, 𝜏₄=10, 𝜏₅=10 and we can obtain 𝑃=,29.08041.87371.87373.7272,𝑋=1.80200.03280.13850.4057𝛾=237.5809. (51) By using the delay-dependent criterion Theorem 3 of [20], we can obtain 𝛾=604.9523 when 𝜏=7.5. And applying our result, 𝛾=237.5809 is obtained. We can see that our results is not conservative since 𝛾 is much smaller than the result of [20].

Example 3. Consider a delayed fuzzy system of the following form.
Plant Rules:
Rule 1: IF 𝑥₂(𝑡) is 𝑀₁, THEN 𝑥̇(𝑡)=𝐴₁𝑥(𝑡)+𝐴_𝑑1𝑥(𝑡−ℎ)+𝐵₁𝑢(𝑡)+𝐵₁₁𝜔(𝑡),𝑦(𝑡)=𝐶₁𝑥(𝑡)+𝐷₁𝜔(𝑡). (52) Rule 2: IF 𝑥₂(𝑡) is 𝑀₂, THEN 𝑥̇(𝑡)=𝐴₂𝑥(𝑡)+𝐴_𝑑2𝑥(𝑡−ℎ)+𝐵₂𝑢(𝑡)+𝐵₁₂𝜔(𝑡),𝑦(𝑡)=𝐶₂𝑥(𝑡)+𝐷₂𝜔(𝑡) (53) with 𝐴₁=−0.1125−0.0210,𝐴₂=,𝐴−0.1125−1.52710_𝑑1=−0.0125−0.00500,𝐴_𝑑2=,𝐵−0.0125−0.2300₁=𝐵₂=11,𝐵₁₁=𝐵₁₂=𝐼₂,𝐶₁=𝐶₂=𝐷₁=𝐷₂=𝐼₂. (54) The membership function is 𝑀₁₁𝑥₂(𝑡)𝑥=1−²²(𝑡)2.25,𝑀₂₁𝑥₂(𝑡)=𝑥²₂(𝑡).2.25 (55) The initial state is 𝑥(0)=[−1−1.2]^𝑇(𝑡)=(),𝜔sin3𝑡(𝑡)()expcos3𝑡exp(𝑡)^𝑇. (56) First, we should find the allowable maximum time delay ℎ which let the fuzzy system passive. Table 1 shows the maximum time delay obtained by [21] and our method.
It clearly shows that the method in our paper can get larger upper bound than before. It also shows that conservatism is further reduced when 𝑚 increases.
Then we let ℎ=1.5, 𝑚=4, and we can obtain 𝑃=9.49809.15479.15478.8849,𝑋=0.63510.49970.58070.4826. (57) The fuzzy controller gains by our method are given by [],[].𝐾1=150.7358−218.3595𝐾2=150.9502−219.0519 (58) Figure 3 shows the state response 𝑥₁(𝑡) of the closed-loop system with the controller gains in (58), and Figure 4 shows the state response 𝑥₂(𝑡) of the closed-loop system.

Table 1 

Allowable maximum time delay ℎ.

Figure 3 

State response ×1 of the system.

Figure 4 

State response ×2 of the system.

6. Conclusion

This paper has adopted the delay partitioning approach to analyse the passivity and passification of delay fuzzy system based on T-S model. The theorems given in this paper are all in terms of LMIs. Examples have illustrated the effectiveness of our results. The method in our paper has further reduced the conservatism and the effect has been more apparent when 𝑚 increases. In addition, the results obtained in this paper also can be extended to the fuzzy system with time-varying delay and uncertainties.

Acknowledgments

This work was supported in part by the National Natural Science Foundation of China under Grant 61273144 and 61203041, the Natural Science Foundation of Beijing under Grant 4122071, the Chinese National Post-doctor Science Foundation under Grants 2011M500217 and 2012T50036, and the Doctoral Fund of Ministry of Education of China under Grant 20120036120013.