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

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.

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 in 𝛿-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 Section 2. 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∈ℝ^𝑛×𝑛, means that the matrix 𝑋 is real symmetric positive definite (resp., positive semidefinite). The symbol “*” in a matrix 𝐴∈ℝ^𝑛×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,…,𝑀_𝑟} denotes a block diagonal matrix with diagonal blocks being the matrices 𝑀₁, 𝑀₂,…, 𝑀_𝑟. 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 by

Plant Rule 𝑖. IF 𝜃₁(𝑡_𝑘) is 𝑁_𝑖1, and …, and 𝜃_𝑗(𝑡_𝑘) is 𝑁_𝑖𝑗, and …, and 𝜃_𝑝(𝑡_𝑘) is 𝑁_𝑖𝑝, THEN 𝛿𝑥𝑡_𝑘=𝐴_𝑖𝑥𝑡_𝑘+𝐴_𝑑𝑖𝑥𝑡_𝑘−𝑑_𝑘+𝐵_𝑖𝑢𝑡_𝑘+𝐵_𝑤𝑖𝑤𝑡_𝑘,𝑧𝑡_𝑘=𝐶_𝑖𝑥𝑡_𝑘+𝐶_𝑑𝑖𝑥𝑡_𝑘−𝑑_𝑘+𝐷_𝑖𝑢𝑡_𝑘+𝐷_𝑤𝑖𝑤𝑡_𝑘,𝑥𝑡_𝑘=𝜙𝑡_𝑘,𝑡_𝑘=−𝑑_𝑀,𝑑_𝑀+𝑇,…,0,𝑖=1,2,…,𝑟, (1) where 𝑥(𝑡_𝑘)∈ℝ^𝑛 is the state variable, 𝑢(𝑡_𝑘)∈ℝ^𝑚 is the control input variable, 𝑤(𝑡_𝑘)∈ℝ^𝑙 is the disturbance input variable which belongs to 𝐿₂[0,∞), 𝑧(𝑡_𝑘)∈ℝ^𝑝 is the control output, and 𝜙(𝑡_𝑘) is a continuous vector-valued initial function. Let 𝑡_𝑘=𝑘𝑇 for the convenience in analysis.𝐴_𝑖, 𝐴_𝑑𝑖,𝐵_𝑖, 𝐵_𝑤𝑖, 𝐶_𝑖, 𝐶_𝑑𝑖,𝐷_𝑖, and 𝐷_𝑤𝑖 (𝑖=1,2,…,𝑟) are system matrices with appropriate dimensions. The scalar 𝑟 is the number of IF-THEN rules. 𝜃_𝑗(𝑡_𝑘) and 𝑁_𝑖𝑗 are the premise variable and the fuzzy set, respectively, for 𝑗=1,2,…,𝑝. The parametric variable 𝑑_𝑘=𝑛𝑇 which is the bounded time delay in the state and satisfies 0<𝑑_𝑚≤𝑑_𝑘≤𝑑_𝑀 with 𝑑_𝑚=𝑛_𝑚𝑇 and 𝑑_𝑀=𝑛_𝑀𝑇 (𝑛_𝑚 and 𝑛_𝑀 are the known positive and finite integers, and 𝑇 is a sampling period). 𝛿𝑥(𝑡_𝑘) is the delta operator of 𝑥(𝑡_𝑘), which is defined by 𝛿𝑥𝑡_𝑘=𝑑𝑑𝑡_𝑘𝑥𝑡_𝑘𝑥𝑡,𝑇=0,_𝑘+𝑇−𝑥𝑡_𝑘𝑇,𝑇≠0. (2)

Then the defuzzified model of system (1) is inferred as follows: 𝛿𝑥𝑡_𝑘=^𝑟_𝑖=1ℎ_𝑖𝜃𝑡_𝑘𝐴_𝑖𝑥𝑡_𝑘+𝐴_𝑑𝑖𝑥𝑡_𝑘−𝑑_𝑘+𝐵_𝑖𝑢𝑡_𝑘+𝐵_𝑤𝑖𝑤𝑡_𝑘,𝑧𝑡_𝑘=^𝑟_𝑖=1ℎ_𝑖𝜃𝑡_𝑘𝐶_𝑖𝑥𝑡_𝑘+𝐶_𝑑𝑖𝑥𝑡_𝑘−𝑑_𝑘+𝐷_𝑖𝑢𝑡_𝑘+𝐷_𝑤𝑖𝑤𝑡_𝑘, (3) where ℎ_𝑖(𝜃(𝑡_𝑘))=𝜇_𝑖(𝜃(𝑡_𝑘))/∑^𝑟_𝑖=1𝜇_𝑖(𝜃(𝑡_𝑘)), 𝜇_𝑖(𝜃(𝑡_𝑘))=∏^𝑝_𝑗=1𝑁_𝑖𝑗(𝜃_𝑗(𝑡_𝑘)), and 𝑁_𝑖𝑗(𝜃_𝑗(𝑡_𝑘)) is the degree of the membership of 𝜃_𝑗(𝑡_𝑘) in fuzzy set 𝑁_𝑖𝑗. It can be assumed that 𝜇_𝑖(𝜃(𝑡_𝑘))≥0 (for 𝑖=1,2,…,𝑟) and ∑^𝑟_𝑖=1𝜇_𝑖(𝜃(𝑡_𝑘))≥0 (for all 𝑡_𝑘). Therefore, ℎ_𝑖(𝜃(𝑡_𝑘))≥0 (for 𝑖=1,2,…,𝑟) and ∑^𝑟_𝑖=1ℎ_𝑖(𝜃(𝑡_𝑘))=1. Based on the parallel distributed compensation (PDC), similar to the fuzzy model, the following overall fuzzy control law can be constructed as 𝑢𝑡_𝑘=^𝑟_𝑠=1ℎ_𝑠𝜃𝑡_𝑘𝐾_𝑠𝑥𝑡_𝑘, (4) where 𝐾_𝑠(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 𝛿𝑥𝑡_𝑘=^𝑟𝑟_𝑖=1𝑠=1ℎ_𝑖ℎ_𝑠𝐴_𝑖+𝐵_𝑖𝐾_𝑠𝑥𝑡_𝑘+𝐴_𝑑𝑖𝑥𝑡_𝑘−𝑑_𝑘+𝐵_𝑤𝑖𝑤𝑡_𝑘,𝑧𝑡_𝑘=^𝑟𝑟_𝑖=1𝑠=1ℎ_𝑖ℎ_𝑠𝐶_𝑖+𝐷_𝑖𝐾_𝑠𝑥𝑡_𝑘+𝐶_𝑑𝑖𝑥𝑡_𝑘−𝑑_𝑘+𝐷_𝑤𝑖𝑤𝑡_𝑘, (5) where ∑^𝑟_𝑖=1ℎ_𝑖=∑^𝑟_𝑖=1ℎ_𝑖(𝜃(𝑡_𝑘)). For the convenience, system (5) can be rewritten as 𝛿𝑥𝑡_𝑘=𝐴𝑡_𝑘𝑥𝑡_𝑘+𝐴_𝑑𝑡_𝑘𝑥𝑡_𝑘−𝑑_𝑘+𝐵_𝑤𝑡_𝑘𝑤𝑡_𝑘,𝑧𝑡_𝑘=𝐶𝑡_𝑘𝑥𝑡_𝑘+𝐶_𝑑𝑡_𝑘𝑥𝑡_𝑘−𝑑_𝑘+𝐷_𝑤𝑡_𝑘𝑤𝑡_𝑘, (6) where 𝐴𝑡_𝑘=^𝑟𝑟_𝑖=1𝑠=1ℎ_𝑖ℎ_𝑠𝐴_𝑖+𝐵_𝑖𝐾_𝑠,𝐴_𝑑𝑡_𝑘=^𝑟_𝑖=1ℎ_𝑖𝐴_𝑑𝑖,𝐵_𝑤𝑡_𝑘=^𝑟_𝑖=1ℎ_𝑖𝐵_𝑤𝑖,𝐶𝑡_𝑘=^𝑟𝑟_𝑖=1𝑠=1ℎ_𝑖ℎ_𝑠𝐶_𝑖+𝐷_𝑖𝐾_𝑠,𝐶_𝑑𝑡_𝑘=^𝑟_𝑖=1ℎ_𝑖𝐶_𝑑𝑖,𝐷_𝑤𝑡_𝑘=^𝑟_𝑖=1ℎ_𝑖𝐷_𝑤𝑖. (7)

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(𝑡_𝑘))≥0, with equality if and only if 𝑥(𝑡_𝑘)=0,(b)𝛿𝑉(𝑥(𝑡_𝑘))=[𝑉(𝑥(𝑡_𝑘+𝑇))−𝑉(𝑥(𝑡_𝑘))]/𝑇<0,where 𝑉(𝑥(𝑡_𝑘)) is a Lyapunov function in 𝛿-domain. For Lyapunov function both in s-domain and z-domain, the condition (𝑎)𝑉(𝑥(𝑡_𝑘))≥0 in Definition 1 is given. On the other hand for the condition (b), when 𝑇→0, there exists lim_𝑇→0𝛿𝑉𝑥𝑡_𝑘=lim_𝑇→0𝑉𝑥𝑡_𝑘+𝑇−𝑉𝑥𝑡_𝑘𝑇=𝑑𝑉𝑥𝑡_𝑘𝑑𝑡_𝑘<0, (8) and when 𝑇=1, there exists 𝛿𝑉𝑥𝑡_𝑘=𝑉𝑥𝑡_𝑘+1−𝑉𝑥𝑡_𝑘1=𝑉𝑥𝑡_𝑘+1−𝑉𝑥𝑡_𝑘<0. (9)
Obviously, the Lyapunov function in 𝛿-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 2^∞_𝑡𝑘=0𝑧^𝑇𝑡_𝑘𝑤𝑡_𝑘≥𝜌. (10)
(ii) System (5) is said to be strictly passive if there exist constants 𝜁>0 and 𝜌 such that 2^∞_𝑡𝑘=0𝑧^𝑇𝑡_𝑘𝑤𝑡_𝑘≥𝜌+𝜁^∞_𝑡𝑘=0𝑤^𝑇𝑡_𝑘𝑤𝑡_𝑘. (11)
(iii) System (5) is said to be output strictly passive if there exist constants 𝜀>0 and 𝜌 such that 2^∞_𝑡𝑘=0𝑧^𝑇𝑡_𝑘𝑤𝑡_𝑘≥𝜌+𝜀^∞_𝑡𝑘=0𝑧^𝑇𝑡_𝑘𝑧𝑡_𝑘. (12)
(iv) System (5) is said to be very strictly passive if there exist constants 𝜀>0,𝜁>0, and 𝜌 such that 2^∞_𝑡𝑘=0𝑧^𝑇𝑡_𝑘𝑤𝑡_𝑘≥𝜌+𝜀^∞_𝑡𝑘=0𝑧^𝑇𝑡_𝑘𝑧𝑡_𝑘+𝜁^∞_𝑡𝑘=0𝑤^𝑇𝑡_𝑘𝑤𝑡_𝑘. (13)

Lemma 3 (see [47]). For any of the time functions 𝑥(𝑡_𝑘) and 𝑦(𝑡_𝑘), 𝛿𝑥𝑡_𝑘𝑦𝑡_𝑘=𝛿𝑥𝑡_𝑘𝑦𝑡_𝑘+𝑥𝑡_𝑘𝛿𝑦𝑡_𝑘+𝑇𝛿𝑥𝑡_𝑘𝛿𝑦𝑡_𝑘, (14) where 𝑇 is a sampling period.

Lemma 4 (see [51]). For any of the two positive integers 𝑟 and 𝑟₀, satisfying 1≤𝑟₀≤𝑟 holds: ^𝑟_𝑖=𝑟0𝑥(𝑖)^𝑇𝑀^𝑟_𝑖=𝑟0𝑥(𝑖)≤𝑟−𝑟₀+1^𝑟_𝑖=𝑟0𝑥^𝑇(𝑖)𝑀𝑥(𝑖), (15) where 𝑀 is a constant positive semidefinite symmetric matrix.

Lemma 5 (see [52]). For the given constant matrices 𝐺, 𝐻 and a symmetric constant matrix 𝜒 of appropriate dimensions, the following inequality holds: 𝜒+𝐺𝐹𝑡_𝑘𝐻+𝐻^𝑇𝐹^𝑇𝑡_𝑘𝐺^𝑇≤0, (16) where 𝐹(𝑡_𝑘) satisfies 𝐹^𝑇(𝑡_𝑘)𝐹(𝑡_𝑘)≤𝐼, if and only if for 𝜉>0 the following inequality holds: 𝜒+𝜉⁻¹𝐻^𝑇𝐻+𝜉𝐺𝐺^𝑇≤0. (17)

Lemma 6. If there exist scalars 𝜀>0,𝜁>0, and a differential function 𝑉(𝑥(𝑡_𝑘))≥0 such that 𝛿𝑉𝑥𝑡_𝑘+𝜀𝑧^𝑇𝑡_𝑘𝑧𝑡_𝑘+𝜁𝑤^𝑇𝑡_𝑘𝑤𝑡_𝑘−2𝑧^𝑇𝑡_𝑘𝑤𝑡_𝑘≤0, (18) then system (5) is very strictly passive.

Proof. It follows from (18) that 2^∞_𝑡𝑘=0𝑧^𝑇𝑡_𝑘𝑤𝑡_𝑘≥−𝑉(𝑥(0))+𝜀^∞_𝑡𝑘=0𝑧^𝑇𝑡_𝑘𝑧𝑡_𝑘+𝜁^∞_𝑡𝑘=0𝑤^𝑇𝑡_𝑘𝑤𝑡_𝑘=𝜌+𝜀^∞_𝑡𝑘=0𝑧^𝑇𝑡_𝑘𝑧𝑡_𝑘+𝜁^∞_𝑡𝑘=0𝑤^𝑇𝑡_𝑘𝑤𝑡_𝑘, (19) 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 Definition 2, 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 𝑑_𝑚,𝑑_𝑀  (0<𝑑_𝑚≤𝑑_𝑀), and matrix 𝐾_𝑠 (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,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: Θ_{𝜅𝜆𝜇𝜈𝑖𝑖}Θ<0,_{𝜅𝜆𝜇𝜈𝑖𝑠}+Θ_{𝜅𝜆𝜇𝜈𝑠𝑖}<0,𝑠≠𝑖, (20) whereΘ_{𝜅𝜆𝜇𝜈𝑖𝑠}=Ω_{1𝜅𝜆𝜇𝜈𝑖𝑠}Ω^𝑇_2𝑖𝑠∗−𝜀𝐼,Ω_2𝑖𝑠=0𝐶_𝑖+𝐷_𝑖𝐾_𝑠𝐶_𝑑𝑖00𝐷_𝑤𝑖,Ω_{1𝜅𝜆𝜇𝜈𝑖𝑠}=Ξ₁𝑃𝐴_𝑖+𝐵_𝑖𝐾_𝑠𝑃𝐴_𝑑𝑖00𝑃𝐵_𝑤𝑖∗Ξ_{2𝜅𝑖𝑠}𝑃𝐴_𝑑𝑖1𝑑_𝑚𝑅₁1𝑑_𝑀𝑅₂Ξ_3𝑖𝑠∗∗−𝑄_𝜆00−𝐶_𝑑𝑖∗∗∗−𝑆_1𝜇−1𝑑_𝑚𝑅₁00∗∗∗∗−𝑆_2𝜈−1𝑑_𝑀𝑅₂0∗∗∗∗∗𝜁𝐼−𝐷_𝑤𝑖−𝐷^𝑇_𝑤𝑖,Ξ₁=(𝑇−2)𝑃+𝑑_𝑚𝑅₁+𝑑_𝑀𝑅₂,Ξ_3𝑖𝑠=𝑃𝐵_𝑤𝑖−𝐶_𝑖+𝐷_𝑖𝐾_𝑠^𝑇,Ξ_{2𝜅𝑖𝑠}=𝑃𝐴_𝑖+𝐵_𝑖𝐾_𝑠+𝐴_𝑖+𝐵_𝑖𝐾_𝑠^𝑇𝑃+𝑑_𝑀−𝑑_𝑚+𝑇+1𝑄_𝜅+𝑆_1𝜅+𝑆_2𝜅−1𝑑_𝑚𝑅₁−1𝑑_𝑀𝑅₂. (21)

Proof. Firstly, in order to simplify the calculation, relevant fuzzy weighing matrices which directly include the membership functions are defined as follows: 𝑄̆𝑡_𝑘=^𝑟_𝜅=1ℎ_𝜅𝜃𝑡_𝑘𝑄_𝜅,𝑆̆₁𝑡_𝑘=^𝑟_𝜅=1ℎ_𝜅𝜃𝑡_𝑘𝑆_1𝜅,𝑆̆₂𝑡_𝑘=^𝑟_𝜅=1ℎ_𝜅𝜃𝑡_𝑘𝑆_2𝜅. (22)
Choose a LKF 𝑉(𝑥(𝑡_𝑘)) for system (5) as follows: 𝑉𝑥𝑡_𝑘=𝑉₁𝑥𝑡_𝑘+𝑉₂𝑥𝑡_𝑘+𝑉₃𝑥𝑡_𝑘+𝑉₄𝑥𝑡_𝑘+𝑉₅𝑥𝑡_𝑘, (23) where 𝑉₁𝑥𝑡_𝑘=𝑥^𝑇𝑡_𝑘𝑃𝑥𝑡_𝑘,𝑉₂𝑥𝑡_𝑘=𝑇^𝑛_𝑖=1𝑥^𝑇𝑡_𝑘−𝑖𝑇𝑄̆𝑡_𝑘−𝑖𝑇𝑥𝑡_𝑘,𝑉−𝑖𝑇₃𝑥𝑡_𝑘=𝑇^𝑛𝑚_𝑖=1𝑥^𝑇𝑡_𝑘−𝑖𝑇𝑆̆₁𝑡_𝑘−𝑖𝑇𝑥𝑡_𝑘−𝑖𝑇+𝑇^𝑛𝑀_𝑖=1𝑥^𝑇𝑡_𝑘−𝑖𝑇𝑆̆₂𝑡_𝑘−𝑖𝑇𝑥𝑡_𝑘,𝑉−𝑖𝑇₄𝑥𝑡_𝑘=𝑇^2𝑛𝑀_{𝑖=𝑛𝑚}^𝑖_𝑠=1𝑥^𝑇𝑡_𝑘−𝑠𝑇𝑄̆𝑡_𝑘−𝑠𝑇𝑥𝑡_𝑘,𝑉−𝑠𝑇₅𝑥𝑡_𝑘=^𝑛𝑚𝑖_𝑖=1𝑠=1𝑒^𝑇𝑡_𝑘−𝑠𝑇𝑅₁𝑒𝑡_𝑘+−𝑠𝑇_{𝑛𝑀𝑖𝑖=1𝑠=1}𝑒^𝑇𝑡_𝑘−𝑠𝑇𝑅₂𝑒𝑡_𝑘−𝑠𝑇, (24) and symmetric matrices 𝑃>0,𝑅₁>0,𝑅₂>0,𝑄_𝜅>0,𝑆_1𝜅>0,𝑆_2𝜅>0 (for 𝜅=1,2,…,𝑟), and 𝑒(𝑠)=𝑥(𝑠)−𝑥(𝑠+𝑇) from which it can be concluded that there exist 𝑒(𝑡_𝑘−𝑖𝑇)=𝑥(𝑡_𝑘−𝑖𝑇)−𝑥(𝑡_𝑘−(𝑖−1)𝑇) and 𝛿𝑥(𝑡_𝑘)=−𝑒(𝑡_𝑘)/𝑇.
Applying Lemma 3 to 𝑉₁(𝑥(𝑡_𝑘)) and along the trajectory of the system (5), it can be obtained that 𝛿𝑉₁𝑥𝑡_𝑘=𝛿^𝑇𝑥𝑡_𝑘𝑃𝑥𝑡_𝑘+𝑥^𝑇𝑡_𝑘𝑃𝛿𝑥𝑡_𝑘+𝑇𝛿^𝑇𝑥𝑡_𝑘𝑃𝛿𝑥𝑡_𝑘=𝑥^𝑇𝑡_𝑘𝑃𝐴𝑡_𝑘+𝐴^𝑇𝑡_𝑘𝑃𝑥𝑡_𝑘+2𝑥^𝑇𝑡_𝑘𝑃𝐴_𝑑𝑡_𝑘𝑥𝑡_𝑘−𝑑_𝑘+2𝑥^𝑇𝑡_𝑘𝑃𝐵_𝑤𝑡_𝑘𝑤𝑡_𝑘+𝑇𝛿^𝑇𝑥𝑡_𝑘𝑃𝛿𝑥𝑡_𝑘. (25)
Similarly, applying the delta operator to 𝑉₂(𝑥(𝑡_𝑘)),𝑉₃(𝑥(𝑡_𝑘)), and 𝑉₄(𝑥(𝑡_𝑘)), it can be obtained as follows: 𝛿𝑉₂𝑥𝑡_𝑘=1𝑇𝑇^𝑛_𝑖=1𝑥^𝑇𝑡_𝑘−𝑖𝑇+𝑇𝑄̆𝑡_𝑘𝑡−𝑖𝑇+𝑇×𝑥_𝑘−𝑖𝑇+𝑇−𝑇^𝑛_𝑖=1𝑥^𝑇𝑡_𝑘−𝑖𝑇𝑄̆𝑡_𝑘−𝑖𝑇𝑥𝑡_𝑘−𝑖𝑇≤𝑥^𝑇𝑡_𝑘𝑄̆𝑡_𝑘𝑥𝑡_𝑘−𝑥^𝑇𝑡_𝑘−𝑑_𝑘𝑄̆𝑡_𝑘−𝑑_𝑘𝑥𝑡_𝑘−𝑑_𝑘+𝑇^𝑛𝑀_{𝑖=𝑛𝑚}𝑥^𝑇𝑡_𝑘−𝑖𝑇𝑄̆𝑡_𝑘−𝑖𝑇𝑥𝑡_𝑘−𝑖𝑇,𝛿𝑉₃𝑥𝑡_𝑘=𝑥^𝑇𝑡_𝑘𝑆̆₁𝑡_𝑘+𝑆̆₂𝑡_𝑘𝑥𝑡_𝑘−𝑥^𝑇𝑡_𝑘−𝑑_𝑚𝑆̆₁𝑡_𝑘−𝑑_𝑚𝑥𝑡_𝑘−𝑑_𝑚−𝑥^𝑇𝑡_𝑘−𝑑_𝑀𝑆₂𝑡_𝑘−𝑑_𝑀𝑥𝑡_𝑘−𝑑_𝑀,𝛿𝑉₄𝑥𝑡_𝑘=𝑇^𝑖_𝑠=1𝑥^𝑇𝑡_𝑘−𝑠𝑇+𝑇𝑄̆𝑡_𝑘𝑡−𝑠𝑇+𝑇×𝑥_𝑘−−𝑠𝑇+𝑇_𝑖𝑠=1𝑥^𝑇𝑡_𝑘−𝑠𝑇𝑄̆𝑡_𝑘𝑡−𝑠𝑇×𝑥_𝑘=𝑑−𝑠𝑇_𝑀−𝑑_𝑚+𝑇𝑥^𝑇𝑡_𝑘𝑄̆𝑡_𝑘𝑥𝑡_𝑘−𝑇^𝑛𝑀_{𝑖=𝑛𝑚}𝑥^𝑇𝑡_𝑘−𝑖𝑇𝑄̆𝑡_𝑘−𝑖𝑇𝑥𝑡_𝑘−𝑖𝑇. (26)
Applying Lemma 4 to 𝑉₅(𝑥(𝑡_𝑘)) and along the trajectory of system (5), it can be obtained that 𝛿𝑉₅𝑥𝑡_𝑘=1𝑇^𝑛𝑚𝑖_𝑖=1𝑠=1𝑒^𝑇𝑡_𝑘−𝑠𝑇+𝑇𝑅₁𝑒𝑡_𝑘−−𝑠𝑇+𝑇_{𝑛𝑚𝑖𝑖=1𝑠=1}𝑒^𝑇𝑡_𝑘−𝑠𝑇𝑅₁𝑒𝑡_𝑘+−𝑠𝑇_{𝑛𝑀𝑖𝑖=1𝑠=1}𝑒^𝑇𝑡_𝑘−𝑠𝑇+𝑇×𝑅₂𝑒𝑡_𝑘−−𝑠𝑇+𝑇_{𝑛𝑀𝑖𝑖=1𝑠=1}𝑒^𝑇𝑡_𝑘−𝑠𝑇𝑅₂𝑒𝑡_𝑘=1−𝑠𝑇𝑇^𝑛𝑚_𝑖=1𝑒^𝑇𝑡_𝑘𝑅₁𝑒𝑡_𝑘−^𝑛𝑚_𝑖=1𝑒^𝑇𝑡_𝑘−𝑖𝑇𝑅₁𝑒𝑡_𝑘+−𝑖𝑇_{𝑛𝑀𝑖=1}𝑒^𝑇𝑡_𝑘𝑅₂𝑒𝑡_𝑘−^𝑛𝑀_𝑖=1𝑒^𝑇𝑡_𝑘−𝑖𝑇𝑅₂𝑒𝑡_𝑘−𝑖𝑇≤𝑒^𝑇𝑡_𝑘𝑛_𝑚𝑇𝑅₁+𝑛_𝑀𝑇𝑅₂𝑒𝑡_𝑘−1𝑛_𝑚𝑇^𝑛𝑚_𝑖=1𝑒𝑡_𝑘−𝑖𝑇^𝑇𝑅₁×^𝑛𝑚_𝑖=1𝑒𝑡_𝑘−1−𝑖𝑇𝑛_𝑀𝑇^𝑛𝑀_𝑖=1𝑒𝑡_𝑘−𝑖𝑇^𝑇×𝑅₂^𝑛𝑀_𝑖=1𝑒𝑡_𝑘−𝑖𝑇=𝛿^𝑇𝑥𝑡_𝑘𝑑_𝑚𝑅₁+𝑑_𝑀𝑅₂𝛿𝑥𝑡_𝑘−1𝑑_𝑚𝑥𝑡_𝑘−𝑥𝑡_𝑘−𝑑_𝑚^𝑇×𝑅₁𝑥𝑡_𝑘−𝑥𝑡_𝑘−𝑑_𝑚−1𝑑_𝑀𝑥𝑡_𝑘−𝑥𝑡_𝑘−𝑑_𝑀^𝑇×𝑅₂𝑥𝑡_𝑘−𝑥𝑡_𝑘−𝑑_𝑀. (27)
For the real matrix 𝑃>0, the following equation is tenable: 0=−2𝛿^𝑇𝑥𝑡_𝑘𝑃𝛿𝑥𝑡_𝑘−𝐴𝑡_𝑘𝑥𝑡_𝑘−𝐴_𝑑𝑡_𝑘𝑥𝑡_𝑘−𝑑_𝑘−𝐵_𝑤𝑡_𝑘𝑤𝑡_𝑘=−2𝛿^𝑇𝑥𝑡_𝑘𝑃𝛿𝑥𝑡_𝑘+2𝛿^𝑇𝑥𝑡_𝑘𝑃𝐴𝑡_𝑘𝑥𝑡_𝑘+2𝛿^𝑇𝑥𝑡_𝑘𝑃𝐴_𝑑𝑡_𝑘𝑥𝑡_𝑘−𝑑_𝑘+2𝛿^𝑇𝑥𝑡_𝑘𝑃𝐵_𝑤𝑡_𝑘𝑤𝑡_𝑘. (28)
Generally, considering the passive performance index in Definition 2, which is described as the passivity analysis, performance of system (5) can be established as follows: 𝐽=𝜀⁻¹𝑧^𝑇𝑡_𝑘𝑧𝑡_𝑘+𝜁𝑤^𝑇𝑡_𝑘𝑤𝑡_𝑘−2𝑧^𝑇𝑡_𝑘𝑤𝑡_𝑘+𝛿𝑉𝑡_𝑘. (29)
By adding (28) into 𝛿𝑉(𝑡_𝑘) and applying (7), it can be found that 𝐽=𝜀⁻¹𝑧^𝑇𝑡_𝑘𝑧𝑡_𝑘+𝜁𝑤^𝑇𝑡_𝑘𝑤𝑡_𝑘−2𝑧^𝑇𝑡_𝑘𝑤𝑡_𝑘+𝛿𝑉𝑡_𝑘≤𝜂^𝑇𝑡_𝑘^𝑟𝑟_𝜅=1^𝑟_𝜆=1^𝑟_𝜇=1^𝑟_𝜈=1^𝑟_𝑖=1𝑠=1ℎ_𝜅ℎ_𝜆ℎ_𝜇ℎ_𝜈ℎ_𝑖ℎ_𝑠𝜂^𝑇𝑡_𝑘×Ω_{1𝜅𝜆𝜇𝜈𝑖𝑠}+𝜀⁻¹Ω^𝑇_2𝑖𝑠Ω_2𝑖𝑠𝜂𝑡_𝑘, (30) where ℎ_𝜅 = ℎ_𝜅(𝜃(𝑡_𝑘)),ℎ_𝜆 = ℎ_𝜆(𝜃(𝑡_𝑘−𝑑_𝑘)),ℎ_𝜇 = ℎ_𝜇(𝜃(𝑡_𝑘−𝑑_𝑚)),ℎ_𝜈 = ℎ_𝜈(𝜃(𝑡_𝑘−𝑑_𝑀)),ℎ_𝑖 = ℎ_𝑖(𝜃(𝑡_𝑘)),ℎ_𝑠 = ℎ_𝑠(𝜃(𝑡_𝑘)), and𝜂^𝑇𝑡_𝑘=𝛿^𝑇𝑥𝑡_𝑘𝑥^𝑇𝑡_𝑘𝑥^𝑇𝑡_𝑘−𝑑_𝑘𝑥^𝑇𝑡_𝑘−𝑑_𝑚𝑥^𝑇𝑡_𝑘−𝑑_𝑀𝑤^𝑇𝑡_𝑘. (31)
On the other hand, it can be seen from Theorem 8 that ^𝑟𝑟_𝜅=1^𝑟_𝜆=1^𝑟_𝜇=1^𝑟_𝜈=1^𝑟_𝑖=1𝑠=1ℎ_𝜅ℎ_𝜆ℎ_𝜇ℎ_𝜈ℎ_𝑖ℎ_𝑠Θ_{𝜅𝜆𝜇𝜈𝑖𝑠}=^𝑟𝑟_𝜅=1^𝑟_𝜆=1^𝑟_𝜇=1𝜈=1ℎ_𝜅ℎ_𝜆ℎ_𝜇ℎ_𝜈×^𝑟_𝑖=1ℎ²_𝑖Θ_{𝜅𝜆𝜇𝜈𝑖𝑖}+^𝑟−1𝑟_{𝑖=1𝑠=𝑖+1}ℎ_𝑖ℎ_𝑠Θ_{𝜅𝜆𝜇𝜈𝑖𝑠}+Θ_{𝜅𝜆𝜇𝜈𝑠𝑖}<0. (32)
Applying Schur complement, it can be obtained that 𝐽=𝜀⁻¹𝑧^𝑇𝑡_𝑘𝑧𝑡_𝑘+𝜁𝑤^𝑇𝑡_𝑘𝑤𝑡_𝑘−2𝑧^𝑇𝑡_𝑘𝑤𝑡_𝑘+𝛿𝑉𝑡_𝑘<0. (33)
It can be seen that the LMIs conditions in Theorem 8 satisfy the very strictly passive performance index for the fuzzy delta operator system (5). The proof is completed.

Based on the conditions in Theorem 8, the state-feedback control gain matrices 𝐾_𝑠 (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 𝑑_𝑚,𝑑_𝑀 (0<𝑑_𝑚≤𝑑_𝑀), system (5) is very strictly passive if there exist scalars 𝜀>0,𝜁>0 and symmetric matrices 𝑃>0, 𝑅₁>0, 𝑅₂>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: Θ_{𝜅𝜆𝜇𝜈𝑖𝑖}<0,Θ_{𝜅𝜆𝜇𝜈𝑖𝑠}+Θ_{𝜅𝜆𝜇𝜈𝑠𝑖}<0,𝑠≠𝑖, (34) whereΘ_{𝜅𝜆𝜇𝜈𝑖𝑠}=Ω_{1𝜅𝜆𝜇𝜈𝑖𝑠}Ω^𝑇_2𝑖𝑠,∗−𝜀𝐼Ω_{2𝜆𝜇𝜈𝑖𝑠}=0𝐶_𝑖𝑃+𝐷_𝑖𝐾_𝑠𝐶_𝑑𝑖𝑃00𝐷_𝑤𝑖,Ω_{1𝜅𝜆𝜇𝜈𝑖𝑠}=Ξ₁𝐴_𝑖𝑃+𝐵_𝑖𝐾_𝑠𝐴_𝑑𝑖𝑃00𝐵_𝑤𝑖∗Ξ_{2𝜅𝑖𝑠}𝐴_𝑑𝑖𝑃1𝑑_𝑚𝑅₁1𝑑_𝑀𝑅₂Ξ_3𝑖𝑠∗∗−𝑄_𝜆00−𝑃𝐶^𝑇_𝑑𝑖∗∗∗−𝑆_1𝜇−1𝑑_𝑚𝑅₁00∗∗∗∗−𝑆_2𝜈−1𝑑_𝑀𝑅₂0∗∗∗∗∗𝜁𝐼−𝐷_𝑤𝑖−𝐷^𝑇_𝑤𝑖,Ξ₁=(𝑇−2)𝑃+𝑑_𝑚𝑅₁+𝑑_𝑀𝑅₂,Ξ_3𝑖𝑠=𝐵_𝑤𝑖−𝑃𝐶^𝑇_𝑖−𝐾^𝑇_𝑠𝐷^𝑇_𝑖,Ξ_{2𝜅𝑖𝑠}=𝐴_𝑖𝑃+𝑃𝐴^𝑇_𝑖+𝐵_𝑖𝐾_𝑠+𝐾^𝑇_𝑠𝐵^𝑇_𝑖+𝑑_𝑀−𝑑_𝑚+𝑇+1𝑄_𝜅+𝑆_1𝜅+𝑆_2𝜅−1𝑑_𝑚𝑅₁−1𝑑_𝑀𝑅₂. (35)The fuzzy sate-feedback controller can be obtained: 𝑢𝑡_𝑘=𝐾_𝑠𝑃⁻¹𝑥𝑡_𝑘. (36)

Proof. Premultiplying and postmultiplying (34) by diagonal matrix diag𝑃⁻¹,𝑃⁻¹,𝑃⁻¹,𝑃⁻¹,𝑃⁻¹,𝐼,𝐼 (37) and letting 𝑃=𝑃⁻¹,𝐾_𝑠=𝐾_𝑠𝑃⁻¹,𝑅₁=𝑃⁻¹𝑅₁𝑃⁻¹,𝑅₂=𝑃⁻¹𝑅₂𝑃⁻¹,𝑄_𝑖=𝑃⁻¹𝑄_𝑖𝑃⁻¹,𝑆_1𝑖=𝑃⁻¹𝑆_1𝑖𝑃⁻¹,𝑆_2𝑖=𝑃⁻¹𝑆_2𝑖𝑃⁻¹, (38) 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 input 𝑤(𝑡_𝑘)=0 in fuzzy delta operator system (3), which can be described as 𝛿𝑥𝑡_𝑘=^𝑟_𝑖=1ℎ_𝑖𝜃𝑡_𝑘×𝐴_𝑖𝑥𝑡_𝑘+𝐴_𝑑𝑖𝑥𝑡_𝑘−𝑑_𝑘+𝐵_𝑖𝑢𝑡_𝑘,𝑧𝑡_𝑘=^𝑟_𝑖=1ℎ_𝑖𝜃𝑡_𝑘×𝐶_𝑖𝑥𝑡_𝑘+𝐶_𝑑𝑖𝑥𝑡_𝑘−𝑑_𝑘+𝐷_𝑖𝑢𝑡_𝑘. (39)
Similar to the proof of Theorems 8 and 9, the following two corollaries can be obtained.

Corollary 11. For a given sampling period 𝑇>0 and constants 𝑑_𝑚 and 𝑑_𝑀 (0<𝑑_𝑚≤𝑑_𝑀), system (39) (𝑢(𝑡_𝑘)=0) is asymptotically stable if there exist symmetric matrices 𝑃>0, 𝑅₁>0,𝑅₂>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=1,2,…,𝑟: Θ̆_{𝜅𝜆𝜇𝜈𝑖}<0, (40) whereΘ̆_{𝜅𝜆𝜇𝜈𝑖}=Ξ₁𝑃𝐴_𝑖𝑃𝐴_𝑑𝑖∗Ξ̆00_2𝜅𝑖𝑃𝐴_𝑑𝑖1𝑑_𝑚𝑅₁1𝑑_𝑀𝑅₂∗∗−𝑄_𝜆00∗∗∗−𝑆_1𝜇−1𝑑_𝑚𝑅₁0∗∗∗∗−𝑆_2𝜈−1𝑑_𝑀𝑅₂,Ξ₁=(𝑇−2)𝑃+𝑑_𝑚𝑅₁+𝑑_𝑀𝑅₂,Ξ̆_2𝜅𝑖=𝑃𝐴_𝑖+𝐴^𝑇_𝑖𝑃+𝑑_𝑀−𝑑_𝑚+𝑇+1𝑄_𝜅+𝑆_1𝜅+𝑆_2𝜅−1𝑑_𝑚𝑅₁−1𝑑_𝑀𝑅₂. (41)

Corollary 12. System (39) with the controller in (4) is asymptotically stable for a given sampling period 𝑇>0 and constants 𝑑_𝑚 and 𝑑_𝑀 (0<𝑑_𝑚≤𝑑_𝑀), if there exist symmetric matrices 𝑃>0, 𝑅₁>0,𝑅₂>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: Θ_{𝜅𝜆𝜇𝜈𝑖𝑖}Θ<0,_{𝜅𝜆𝜇𝜈𝑖𝑠}+Θ_{𝜅𝜆𝜇𝜈𝑠𝑖}<0,𝑠≠𝑖, (42) whereΘ_{𝜅𝜆𝜇𝜈𝑖𝑠}=Ξ₁𝐴_𝑖𝑃+𝐵_𝑖𝐾_𝑠𝐴_𝑑𝑖∗𝑃00Ξ_{2𝜅𝑖𝑠}𝐴_𝑑𝑖𝑃1𝑑_𝑚𝑅₁1𝑑_𝑀𝑅₂∗∗−𝑄_𝜆00∗∗∗−𝑆_1𝜇−1𝑑_𝑚𝑅₁0∗∗∗∗−𝑆_2𝜈−1𝑑_𝑀𝑅₂,Ξ_1𝜅=(𝑇−2)𝑃+𝑑_𝑚𝑅₁+𝑑_𝑀𝑅₂,Ξ_{2𝜅𝑖𝑠}=𝐴_𝑖𝑃+𝑃𝐴^𝑇_𝑖+𝐵_𝑖𝐾_𝑠+𝐾^𝑇_𝑠𝐵^𝑇_𝑖+𝑑_𝑀−𝑑_𝑚+𝑇+1𝑄_𝜅+𝑆_1𝜅+𝑆_2𝜅−1𝑑_𝑚𝑅₁−1𝑑_𝑀𝑅₂. (43)Then, the controller (4) can be designed as 𝑢𝑡_𝑘=𝐾_𝑠𝑃⁻¹𝑥𝑡_𝑘. (44)

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 Rule 1. IF 𝜃(𝑡_𝑘) is 0, THEN 𝛿𝑥𝑡_𝑘=𝐴₁𝑥𝑡_𝑘+𝐴_𝑑1𝑥𝑡_𝑘−𝑑_𝑘+𝐵₁𝑢𝑡_𝑘+𝐵_𝑤1𝑤𝑡_𝑘,𝑧𝑡_𝑘=𝐶₁𝑥𝑡_𝑘+𝐶_𝑑1𝑥𝑡_𝑘−𝑑_𝑘+𝐷₁𝑢𝑡_𝑘+𝐷_𝑤1𝑤𝑡_𝑘. (45)
Plant Rule 2. IF 𝜃(𝑡_𝑘) is ±(𝜋/2)(|𝜃(𝑡_𝑘)|<𝜋/2), THEN 𝛿𝑥𝑡_𝑘=𝐴₂𝑥𝑡_𝑘+𝐴_𝑑2𝑥𝑡_𝑘−𝑑_𝑘+𝐵₂𝑢𝑡_𝑘+𝐵_𝑤2𝑤𝑡_𝑘,𝑧𝑡_𝑘=𝐶₂𝑥𝑡_𝑘+𝐶_𝑑2𝑥𝑡_𝑘−𝑑_𝑘+𝐷₂𝑢𝑡_𝑘+𝐷_𝑤2𝑤𝑡_𝑘. (46)
The system matrix parameters in (45) and (46) are given as follows: 𝐴₁=−2.000−0.0600.0700.100,𝐴₂=,𝐵−1.8000.080−0.0500.100₁=0.0800.050,𝐵₂=,𝐴0.0800.090_𝑑1=−0.020−0.0100.0100.020,𝐴_𝑑2=,𝐵−1.8000.080−0.0500.100_𝑤1=0.010−0.100^𝑇,𝐵_𝑤2=−0.010−0.100^𝑇,𝐶₁=,𝐶−0.0100.010₂=0.120−0.100,𝐷₁𝐶=0.010,_𝑑1=0.0800.010,𝐶_𝑑2=,𝐷−0.010−0.050₂=0.020,𝐷_𝑤1=0.010,𝐷_𝑤2=0.020. (47)
Two membership functions are chosen for Plant Rules 1 and 2 as follows: ℎ₁𝜃𝑡_𝑘=21−𝜋𝜃𝑡_𝑘,0≤𝜃𝑡_𝑘≤𝜋2,21+𝜋𝜃𝑡_𝑘𝜋,−2≤𝜃𝑡_𝑘ℎ<0,₂𝜃𝑡_𝑘=1−ℎ₁𝜃𝑡_𝑘. (48) And Figure 1 shows the membership functions.
Define the value range of time delay 𝑑_𝑘=𝑛𝑇, and choosing the lower bound 𝑑_𝑚=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 𝑑_𝑘 and the control gain matrices are listed in Table 1.
Assume that time delay 𝑑_𝑘=𝑛𝑇=40𝑇 and the disturbance input 𝑤(𝑡_𝑘)=−1/(2+𝑡_𝑘). It can be calculated that ∑^∞_𝑡𝑘=0𝑤^𝑇(𝑡_𝑘)𝑤(𝑡_𝑘)=0.5<∞, which means that 𝑤(𝑡_𝑘)∈𝐿₂[0,∞). Let 𝜐𝑡_𝑘=2^∞_𝑡𝑘=0𝑧^𝑇𝑡_𝑘𝑤𝑡_𝑘−𝜀^∞_𝑡𝑘=0𝑧^𝑇𝑡_𝑘𝑧𝑡_𝑘−𝜁^∞_𝑡𝑘=0𝑤^𝑇𝑡_𝑘𝑤𝑡_𝑘,𝜍𝑡_𝑘=2^∞_𝑡𝑘=0𝑧^𝑇𝑡_𝑘𝑤𝑡_𝑘. (49)
In Figure 2, it can be observed that 𝜍(𝑡_𝑘) decreases finally as the time 𝑡_𝑘 increases, which means that there may not exist a scalar 𝜌 such that 𝜍(𝑡_𝑘)≥𝜌 hold for all 𝑡_𝑘≥0 for 𝑇=0.001. Furthermore, Figure 3 shows that there may not exist a scalar 𝜌 such that 𝜐(𝑡_𝑘)≥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 Table 1, Figure 5 plots the responses of 𝜍(𝑡_𝑘) for the closed-loop system for 𝑇=0.001. It can be seen that there may exist a scalar 𝜌 such that 𝜍(𝑡_𝑘)≥𝜌 holds for all 𝑡_𝑘≥0. It can be seen from Figure 6 that there may exist a scalar 𝜌 such that 𝜐(𝑡_𝑘)≥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 with 𝑇=0.001.

Figure 1 

Membership functions of two rules.

Figure 2 

Response of 𝜍(𝑡𝑘) of the open-loop system.

Figure 3 

Response of 𝜐(𝑡𝑘) of the open-loop system.

Figure 4 

State response of the open-loop system.

Figure 5 

Response of 𝜍(𝑡𝑘) of the closed-loop system.

Figure 6 

Response of 𝜐(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 in 𝛿-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.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

This work was partially supported by the National Natural Science Foundation of China (61203002, 61304002, and 61304003), the Program for New Century Excellent Talents in University (NCET-13-0696), the Program for Liaoning Innovative Research Team in University (LT2013023), and the Program for Liaoning Excellent Talents in University (LR2013053).