Robust 𝐻_∞ Control of Uncertain T-S Fuzzy Time-Delay System: A Delay Decomposition Approach

Abstract

This paper is concerned with the problem of robust 𝐻_∞ control for a class of uncertain time-delay fuzzy systems with norm-bounded parameter uncertainties. By utilizing the instrumental idea of delay decomposition, the decomposed Lyapunov-Krasovskii functional is introduced to uncertain T-S fuzzy system, and some delay-dependent conditions for the existence of robust controller are formulated in the form of linear matrix inequalities (LMIs). When these LMIs are feasible, a controller is presented. A numerical example is given to demonstrate the effectiveness of the proposed method.

1. Introduction

It is well known that time delay is built-in features in various nonlinear systems such as tandem mills, remote control systems, long transmission lines in pneumatic systems, and chemical system. The time delay is recognized to be a source of instability and performance deterioration of control systems. Therefore, stability analysis and controller synthesis for time-delay system have been one of the most hot research area in the control community over the past years [1–14].

Fuzzy systems in the form of the Takagi-Sugeno (T-S) model have attracted rapidly growing interest in recent years. It has been shown that the T-S model method is a simple and effective way to represent complex nonlinear systems by a set of simple local linear dynamic systems with their linguistic description [12, 15–19]. Over the past few years, most work has been devoted to analysis and synthesis of T-S fuzzy control systems. See the survey papers [16, 17] and the reference citied therein for the most recent advances on this topic. The appeal and superiority of T-S fuzzy models is that the analysis and synthesis of the overall fuzzy systems can be carried out in the Lyapunov-function-based framework. To mention a few, by using LMI, Cao and Frank presented controller design for a class of fuzzy dynamic systems with time delay in both continuous and discrete cases in [20, 21]. Wu et al. studied the model approximation problem and 𝐿₂-𝐿_∞ control problem for nonlinear time-delay systems in [22, 23]. Moreover, great attention from researchers has been drawn to the study of stability analysis and controller design for T-S fuzzy systems with time delays [24–28]. On the other hand, type-2 fuzzy mode are considered in [29, 30].

Recently, many scholars studied the stability problem based on the piecewise Lyapunov-Krasovskii functional [31–33]. Reference [31] investigated the linear continuous/discrete systems with time-varying delay and divided the variation interval of the time delay into several subintervals. based on this method, [32]addressed the problem of the robust 𝐻_∞ filtering for singular linear parameter varying (LPV). Reference [33] researched the stability of linear time-invariant systems and divided the delay interval into 𝑁 subintervals. The simulations show these methods can lead to much less conservative results than those in the existing references.

Motivated by the above observations, in this paper, we will investigate the problem of robust 𝐻_∞ control of uncertain T-S fuzzy systems with constant delay. Attention is focused on the design of robust 𝐻_∞ controllers via the parallel distributed compensation scheme such that the closed-loop fuzzy time-delay system is asymptotically stable and the 𝐻_∞ disturbance attenuation is below a prescribed level. Based on delay decomposition approach [33], the decomposed Lyapunov-Krasovskii functional is introduced, and some delay-dependent conditions have been obtained. These conditions are formulated in the form of LMIs, and the controller design is cast into a convex optimization problem subject to LMI constraints, which can be readily solved via standard numerical software. Finally, a numerical example is provided to show the effectiveness and less conservatism of the proposed results.

The rest of this paper is organized as follows. In Section 2, the model description and problem are first formulated. The main results for delay-dependent robust 𝐻_∞ controller are presented in Section 3. Illustrative examples are given in Section 4, and the paper is concluded in Section 5.

Notations. The notations used throughout this paper are fairly standard. The superscript “𝑇” stands for matrix transpose, and the notation 𝑃>0(𝑃≥0) means that matrix 𝑃 is real symmetric and positive (or being positive semidefinite). 𝐼 and 0 are used to denote appropriate dimensions identity matrix and zero matrix, respectively. The notation ∗ in a symmetric always denotes the symmetric block in the matrix. The parameter diag{⋯} denotes a block-diagonal matrix. Matrices, if not explicitly stated, are assumed to have compatible dimensions for algebraic operations.

2. System Descriptions and Preliminaries

Consider the uncertain nonlinear system with state delay that is described by the following T-S model with uncertain parameter matrices.

Plant Rule 𝑖. IF 𝑠₁(𝑡) is 𝐹_𝑖1 and 𝑠₂(𝑡) is 𝐹_𝑖2 and … and 𝑠_𝑛(𝑡) is 𝐹_𝑖𝑛 THEN𝑥̇(𝑡)=𝐴_𝑖+Δ𝐴_𝑖𝑥(𝑡)+𝐴_𝑑𝑖+Δ𝐴_𝑑𝑖𝑥()+𝐵𝑡−𝜏_1𝑖+Δ𝐵_1𝑖𝑢(𝑡)+𝐵_2𝑖𝜔(𝑡),𝑧(𝑡)=𝐶_𝑖𝑥(𝑡),𝑥(𝑡)=𝜙(𝑡),𝑡∈[−𝜏,0],𝑖=1,2,…,𝑟, (1) where 𝑠₁(𝑡), 𝑠₂(𝑡), …, 𝑠_𝑛(𝑡) are the premise variables that are measurable and each 𝐹_𝑖𝑗 (𝑗=1,2,…,𝑛) is fuzzy set. 𝑥(𝑡)∈𝑅^𝑛 is the state vector and 𝑢(𝑡)∈𝑅^𝑚 is the control input vector. 𝑧(𝑡) is the output vector. 𝜔(𝑡)∈𝑅^𝑞 is the disturbance input vector belongs to 𝐿₂[0,∞). 𝑟 is the number of IF-THEN rules, 𝜏 is the constant delay in the state. 𝜙(𝑡) is a vector-valued initial continuous function.

The matrices Δ𝐴_𝑖, Δ𝐴_𝑑𝑖, and Δ𝐵_1𝑖 denote the parameters uncertainties, which are assumed of the form Δ𝐴_𝑖,Δ𝐴_𝑑𝑖,Δ𝐵_1𝑖=MF(𝑡)𝐸_𝑖,𝐸_𝑑𝑖,𝐸_1𝑖, (2) where 𝑀, 𝐸_𝑖, 𝐸_𝑑𝑖, and 𝐸_1𝑖 are known constant matrices and 𝐹(𝑡) is an unknown time-varying matrix function satisfying 𝐹^𝑇(𝑡)𝐹(𝑡)≤𝐼.

For simplicity, introduce the following notations: 𝐴_𝑖=𝐴_𝑖+Δ𝐴_𝑖𝐴_𝑑𝑖=𝐴_𝑑𝑖+Δ𝐴_𝑑𝑖𝐵_1𝑖=𝐵_1𝑖+Δ𝐵_1𝑖. (3)

By using a center-average defuzzier, product fuzzy inference, and a singleton fuzzifier, the following global T-S fuzzy model can be obtained: 𝑥̇(𝑡)=^𝑟_𝑖=1𝛼_𝑖(𝑠(𝑡))𝐴_𝑖+Δ𝐴_𝑖𝑥(𝑡)+𝐴_𝑑𝑖+Δ𝐴_𝑑𝑖𝑥()+𝐵𝑡−𝜏_1𝑖+Δ𝐵_1𝑖𝑢(𝑡)+𝐵_2𝑖𝜔(𝑡×_𝑟𝑖=1)𝛼_𝑖(𝑠(𝑡))⁻¹=^𝑟_𝑖=1𝜇_𝑖(𝑠(𝑡))𝐴_𝑖+Δ𝐴_𝑖𝑥(𝑡)+𝐴_𝑑𝑖+Δ𝐴_𝑑𝑖𝑥()+𝐵𝑡−𝜏_1𝑖+Δ𝐵_1𝑖𝑢(𝑡)+𝐵_2𝑖𝜔(𝑡,𝑧(𝑡)=_𝑟𝑖=1)𝜇_𝑖(𝑠(𝑡))𝐶_𝑖𝑥(𝑡)(𝑡)𝑥(𝑡)(𝑡)(𝑡)[],=𝐶,𝑥=𝜙𝑡∈−𝜏,0 (4) where 𝛼_𝑖(𝑠(𝑡))=∏^𝑛_𝑗=1𝐹_𝑖𝑗(𝑠_𝑗(𝑡)), 𝜇_𝑖(𝑠(𝑡))=𝛼_𝑖(𝑠(𝑡))/∑^𝑟_𝑖=1𝛼_𝑖(𝑠(𝑡)), and 𝐹_𝑖𝑗(𝑠_𝑗(𝑡)) is the grade of membership of 𝑠_𝑗(𝑡) in 𝐹_𝑖𝑗, and it is assumed that 𝛼_𝑖(𝑠(𝑡))≥0,𝑖=1,2,…,𝑟∑^𝑟_𝑖=1𝛼_𝑖(𝑠(𝑡))>0 for all 𝑡. Therefore, 𝜇_𝑖(𝑠(𝑡))≥0 and ∑^𝑟_𝑖=1𝜇_𝑖(𝑠(𝑡))=1 for all 𝑡.

In this paper, employing the idea of parallel distributed compensation (PDC), the T-S fuzzy-model-based controller via the PDC can be constructed as follows.

Controller Rule 𝑖. IF 𝑠₁(𝑡) is 𝐹_𝑖1 and 𝑠₂(𝑡) is 𝐹_𝑖2 and … and 𝑠_𝑛(𝑡) is 𝐹_𝑖𝑛 THEN𝑢(𝑡)=𝐾_𝑖𝑥(𝑡), (5) where 𝐾_𝑖(𝑖=1,2,…,𝑟) are the controller gains of (5) to be determined.

Then, the overall output of the controller rules is given by 𝑢(𝑡)=^𝑟_𝑖=1𝜇_𝑖(𝑠(𝑡))𝐾_𝑖𝑥(𝑡). (6) Substituting (6) into (4), the closed-loop system can be given as 𝑥̇(𝑡)=^𝑟𝑟_𝑖=1𝑗=1𝜇_𝑖(𝑠(𝑡))𝜇_𝑗(𝑠(𝑡×))𝐴_𝑖+𝐵_1𝑖𝐾_𝑗𝑥(𝑡)+𝐴_𝑑𝑖𝑥(𝑡−𝜏)+𝐵_2𝑖𝜔(𝑡,𝑧(𝑡)=_𝑟𝑖=1)𝜇_𝑖(𝑠(𝑡))𝐶_𝑖𝑥(𝑡), (7) with its compact form 𝑥̇(𝑡)=𝐴(𝑡)+𝐵₁(𝑡)𝐾(𝑡𝑥(𝑡)+)𝐴_𝑑(𝑡)𝑥(𝑡−𝜏)+𝐵₂(𝑡)𝜔(𝑡),𝑧(𝑡)=𝐶(𝑡)𝑥(𝑡), (8)(9) where 𝐴(𝑡)=𝐴(𝑡)+Δ𝐴(𝑡)=^𝑟_𝑖=1𝜇_𝑖(𝑠(𝑡))𝐴_𝑖+Δ𝐴_𝑖(𝑡),𝐴_𝑑(𝑡)=𝐴_𝑑(𝑡)+Δ𝐴_𝑑(𝑡)=^𝑟_𝑖=1𝜇_𝑖(𝑠(𝑡))𝐴_𝑑𝑖+Δ𝐴_𝑑𝑖(𝑡),𝐵₁(𝑡)=𝐵₁(𝑡)+Δ𝐵₁(𝑡)=^𝑟_𝑖=1𝜇_𝑖(𝑠(𝑡))𝐵_1𝑖+Δ𝐵_1𝑖(𝑡,𝐵₂)(𝑡)=^𝑟_𝑖=1𝜇_𝑖(𝑠(𝑡))𝐵_2𝑖,𝐶(𝑡)=^𝑟_𝑖=1𝜇_𝑖(𝑠(𝑡))𝐶_𝑖,𝐾(𝑡)=^𝑟_𝑖=1𝜇_𝑖(𝑠(𝑡))𝐾_𝑖. (10)

Before ending this section, we introduce the following definitions and lemmas, which will be used in the derivation of our main results.

Definition 1 (𝐻_∞ performance). Given a scalar 𝛾>0 and under zero initial condition, the system (1) is said to be asymptotically stable with 𝛾-disturbance attenuation if the system (4) is asymptotically stable and the output 𝑧(𝑡) satisfies ‖𝑧(𝑡)‖₂≤𝛾‖𝜔(𝑡)‖₂, (11) that is, ^∞₀𝑧^𝑇(𝑡)𝑧(𝑡)−𝛾²𝜔^𝑇(𝑡)𝜔(𝑡)𝑑𝑡≤0, (12) for all nonzero 𝜔(𝑡)∈𝐿₂[0,∞).

Lemma 2 (see [34]). For any constant matrix 𝑊∈𝑅^𝑛×𝑛,𝑊=𝑊^𝑇>0, scalar 𝑟>0, and vector-valued function 𝑥̇∶[−𝑟,0]→𝑅^𝑛 such that the following integration is well defined; then −^𝑡_𝑡−𝑟𝑥̇^𝑇(𝑠)(𝑟𝑊)𝑥̇(𝑠)ds≤𝑥^𝑇(𝑡)𝑥^𝑇()𝑥(𝑡)𝑥()𝑡−𝑟.−𝑊𝑊𝑊−𝑊𝑡−𝑟 (13)

Lemma 3 (see [35]). Given a symmetric matrix 𝑀 and matrices 𝐷, 𝐹(𝑡), and 𝐸 of compatible dimensions, then, for 𝐹^𝑇(𝑡)𝐹(𝑡)≤𝐼, the inequality 𝑀+𝐷𝐹(𝑡)𝐸+(𝐷𝐹(𝑡)𝐸)^𝑇<0 (14) holds if and only if there exists a scalar 𝜀>0 such that 𝑀+𝜀𝐷𝐷^𝑇+𝜀⁻¹𝐸^𝑇𝐸<0. (15)

3. Main Results

In this section, some delay-dependent sufficient conditions on the existence of robust 𝐻_∞ controller for T-S fuzzy system (7) will be presented. A Lyapunov-Krasovskii functional, based on the idea of delay decomposition approach, will be introduced, which can potentially reduce the conservatism of the results.

To this end, we first consider the following nominal closed-loop system: 𝑥̇(𝑡)=𝐴(𝑡)+𝐵₁(𝑡)𝐾(𝑡)𝑥(𝑡)+𝐴_𝑑(𝑡)𝑥(𝑡−𝜏)+𝐵₂(𝑡)𝜔(𝑡),𝑧(𝑡)=𝐶(𝑡)𝑥(𝑡). (16)(17) Firstly, the sufficient condition of 𝐻_∞ performance analysis for the unforced case of system (16) is established in Proposition 4.

Proposition 4. For some prescribed 𝛾>0 and 𝜏>0, the unforced case of system (16) is asymptotically stable with a guaranteed 𝐻_∞ performance 𝛾, if there exist matrices 𝑃>0, 𝑅_𝑙>0, and 𝑄_𝑙>0(𝑙=1,2,…,𝑁) such that the following LMIs hold for 𝑖=1,2,…,𝑟: ΣΣ=⁽¹⁾Σ⁽²⁾Σ⁽³⁾∗−𝛾²𝐼Σ⁽⁴⁾∗∗Σ⁽⁵⁾<0, (18) where Σ⁽¹⁾=Σ⁽¹⁾₁₁𝑅₁0⋯0𝑃𝐴_𝑑𝑖∗Σ⁽¹⁾₂₂𝑅₂⋯00∗∗Σ⁽¹⁾₃₃⋯00⋮⋮⋮⋱⋮⋮∗∗∗⋯Σ⁽¹⁾_𝑁𝑁𝑅_𝑁∗∗∗⋯∗Σ⁽¹⁾_𝑁+1𝑁+1,Σ⁽¹⁾₁₁=𝑄₁−𝑅₁+𝐴^𝑇_𝑖𝑃+𝑃𝐴_𝑖+𝐶^𝑇_𝑖𝐶_𝑖,Σ⁽¹⁾₂₂=𝑄₂−𝑄₁−𝑅₁−𝑅₂,Σ⁽¹⁾₃₃=𝑄₃−𝑄₂−𝑅₂−𝑅₃,⋮Σ⁽¹⁾_𝑁𝑁=𝑄_𝑁−𝑄_𝑁−1−𝑅_𝑁−1−𝑅_𝑁,Σ⁽¹⁾_𝑁+1𝑁+1=−𝑄_𝑁−𝑅_𝑁,Σ⁽²⁾=𝑃𝐵_2𝑖0⋮0,Σ⁽³⁾=ℎ𝐴^𝑇_𝑖𝑅₁ℎ𝐴^𝑇_𝑖𝑅₂⋯ℎ𝐴^𝑇_𝑖𝑅_𝑁00⋯000⋯0⋮⋮⋱⋮00⋯0ℎ𝐴^𝑇_di𝑅₁ℎ𝐴^𝑇_di𝑅₂⋯ℎ𝐴^𝑇_di𝑅_𝑁Σ⁽⁴⁾=ℎ𝐵^𝑇_2𝑖𝑅₁ℎ𝐵^𝑇_2𝑖𝑅₂⋯ℎ𝐵^𝑇_2𝑖𝑅_𝑁,Σ⁽⁵⁾=diag−R₁,−R₂,…,−R_N. (19)

Proof. Choose a Lyapunov-Krasovskii functional candidate as 𝑉(𝑡)=𝑉₁(𝑡)+𝑉₂(𝑡)+𝑉₃(𝑡),𝑉₁(𝑡)=𝑥^𝑇(𝑡)(𝑡),𝑉𝑃𝑥₂(𝑡)=^𝑁_𝑙=1^{𝑡−(𝑙−1)ℎ}_{𝑡−𝑙ℎ}𝑥^𝑇(𝑠)𝑄_𝑙𝑥(𝑠)𝑉𝑑𝑠,₃(𝑡)=^𝑁_𝑙=1^{−(𝑙−1)ℎ}_−𝑙ℎ^𝑡_𝑡+𝜃𝑥̇^𝑇(𝑠)ℎ𝑅_𝑙𝑥̇(𝑠)𝑑𝑠𝑑𝜃, (20) where ℎ=𝜏/𝑁 and 𝑁 is the partitioning number of time delay 𝜏.
Taking the derivative of 𝑉_𝑖(𝑡), for 𝑖=1,2,3, with respect to 𝑡 along the trajectory of unforced case of system (16) yields 𝑉̇₁(𝑡)=𝑥̇^𝑇(𝑡)𝑃𝑥(𝑡)+𝑥^𝑇(𝑡)𝑃𝑥̇(𝑡)=𝐴(𝑡)𝑥(𝑡)+𝐴_𝑑(𝑡)𝑥(𝑡−𝜏)+𝐵₂(𝑡)𝜔(𝑡)^𝑇×𝑃𝑥(𝑡)+𝑥^𝑇(𝑡)𝑃(𝐴(𝑡)𝑥(𝑡)+𝐴_𝑑(𝑡)𝑥(𝑡−𝜏)+𝐵₂(𝑡)𝜔(𝑡,𝑉̇₂)(𝑡)=^𝑁_𝑙=1𝑥^𝑇(𝑡−(𝑙−1)ℎ)×𝑄_𝑙𝑥(𝑡−(𝑙−1)ℎ)−𝑥^𝑇(𝑡−𝑙ℎ)𝑄_𝑙𝑥()=𝑡−𝑙ℎ_𝑁𝑙=1𝑥^𝑇(𝑡−(𝑙−1)ℎ)𝑄_𝑙𝑥(()ℎ)−𝑡−𝑙−1_𝑁𝑙=1𝑥^𝑇(𝑡−𝑙ℎ)𝑄_𝑙𝑥(),𝑉̇𝑡−𝑙ℎ₃(𝑡)=^𝑁_𝑙=1ℎ𝑥̇^𝑇(𝑡)ℎ𝑅_𝑙𝑥̇(𝑡)−^𝑁_𝑙=1^{𝑡−(𝑙−1)ℎ}_{𝑡−𝑙ℎ}𝑥̇^𝑇(𝑠)ℎ𝑅_𝑙𝑥̇(𝑠)𝑑𝑠. (21)
By using Lemma 2, we obtain that𝑉̇₃(𝑡)≤^𝑁_𝑙=1ℎ𝑥̇^𝑇(𝑡)ℎ𝑅_𝑙𝑥̇(𝑡)+^𝑁_𝑙=1𝑥^𝑇(𝑡−(𝑙−1)ℎ)𝑥^𝑇()×𝑡−𝑙ℎ−𝑅_𝑙𝑅_𝑙𝑅_𝑙−𝑅_𝑙𝑥(()ℎ)𝑥()=𝑡−𝑙−1𝑡−𝑙ℎ_𝑁𝑙=1𝐴(𝑡)𝑥(𝑡)+𝐴_𝑑(𝑡)𝑥(𝑡−𝜏)+𝐵₂(𝑡)𝜔(𝑡)^𝑇×ℎ²𝑅_𝑙𝐴(𝑡)𝑥(𝑡)+𝐴_𝑑(𝑡)𝑥(𝑡−𝜏)+𝐵₂(𝑡)𝜔(𝑡)+^𝑁_𝑙=1𝑥^𝑇(𝑡−(𝑙−1)ℎ)𝑥^𝑇()×𝑡−𝑙ℎ−𝑅_𝑙𝑅_𝑙𝑅_𝑙−𝑅_𝑙𝑥(()ℎ)𝑥()=𝐴(𝑡)𝑥(𝑡)𝑡−𝑙−1𝑡−𝑙ℎ+𝐴_𝑑(𝑡)𝑥(𝑡−𝜏)+𝐵₂(𝑡)𝜔(𝑡)^𝑇×^𝑁_𝑙=1ℎ²𝑅_𝑙𝐴(𝑡)𝑥(𝑡)+𝐴_𝑑(𝑡)𝑥(𝑡−𝜏)+𝐵₂(𝑡)𝜔(𝑡)+^𝑁_𝑙=1𝑥^𝑇(𝑡−(𝑙−1)ℎ)𝑥^𝑇(𝑡−𝑙ℎ)−𝑅_𝑙𝑅_𝑙𝑅_𝑙−𝑅_𝑙𝑥(()ℎ)𝑥()=𝑥(𝑡)𝑥()𝜔(𝑡)𝑡−𝑙−1𝑡−𝑙ℎ𝑡−𝜏^𝑇×𝐴^𝑇(𝑡)∑^𝑁_𝑙=1ℎ²𝑅_𝑙𝐴(𝑡)𝐴^𝑇(𝑡)∑^𝑁_𝑙=1ℎ²𝑅_𝑙𝐴_𝑑(𝑡)𝐴^𝑇(𝑡)∑^𝑁_𝑙=1ℎ²𝑅_𝑙𝐵₂(𝑡)𝐴^𝑇_𝑑(𝑡)∑^𝑁_𝑙=1ℎ²𝑅_𝑙𝐴(𝑡)𝐴^𝑇_𝑑(𝑡)∑^𝑁_𝑙=1ℎ²𝑅_𝑙𝐴_𝑑(𝑡)𝐴^𝑇_𝑑(𝑡)∑^𝑁_𝑙=1ℎ²𝑅_𝑙𝐵₂(𝑡)𝐵^𝑇₂(𝑡)∑^𝑁_𝑙=1ℎ²𝑅_𝑗𝐴(𝑡)𝐵^𝑇₂(𝑡)∑^𝑁_𝑗=1ℎ²𝑅_𝑙𝐴_𝑑(𝑡)𝐵^𝑇₂(𝑡)∑^𝑁_𝑙=1ℎ²𝑅_𝑙𝐵₂(𝑡)×𝑥(𝑡)𝑥()𝜔(𝑡)+𝑡−𝜏_𝑁𝑙=1𝑥^𝑇(𝑡−(𝑙−1)ℎ)𝑥^𝑇()×𝑡−𝑙ℎ−𝑅_𝑙𝑅_𝑙𝑅_𝑙−𝑅_𝑙𝑥(()ℎ)𝑥().𝑡−𝑙−1𝑡−𝑙ℎ (22)Define the variable 𝜂^𝑇(𝑡)=[𝑥^𝑇(𝑡),𝑥^𝑇(𝑡−ℎ),…,𝑥^𝑇(𝑡−(𝑁−1)ℎ),𝑥^𝑇(𝑡−𝜏)] and by simple manipulation, we have 𝑉̇(𝑡)=𝑉̇₁(𝑡)+𝑉̇₂(𝑡)+𝑉̇₃(𝑡)≤𝜉^𝑇(𝑡)Π₁+Π₂+Π₃𝜉(𝑡), (23) whereΠ₁=𝐴^𝑇(𝑡)𝑃+𝑃𝐴(𝑡)⋯𝑃𝐴_𝑑(𝑡)𝑃𝐵₂(𝑡)𝐴⋮⋱⋮⋮_𝑇𝑑(𝑡)𝐵𝑃⋯00_𝑇2(𝑡),Π𝑃⋯00₂=𝑄₁0⋯00∗𝑄₂−𝑄₁⋯00⋮⋮⋱⋮⋮∗∗⋯𝑄_𝑁−𝑄_𝑁−10∗∗⋯∗−𝑄_𝑁,Π₃=−𝑅₁𝑅₁0…000∗−𝑅₁−𝑅₂𝑅₂…000∗∗−𝑅₂−𝑅₃…000⋮⋮⋮⋱⋮⋮⋮∗∗∗…−𝑅_𝑁−1−𝑅_𝑁𝑅_𝑁0∗∗∗…∗−𝑅_𝑁0−Σ∗∗∗…∗∗0₍₃₎Σ⁽⁴⁾Σ⁽⁵⁾⁻¹Σ⁽³⁾Σ^(4)𝑇,𝜉(𝑡)=𝜂^𝑇(𝑡)𝜔^𝑇(𝑡)^𝑇. (24)
First, we prove the asymptotic stability of the system in (16). To this end, assume 𝜔(t)≡0, and thus Π₁+Π₂+Π₃ in (23) reads Π₁+Π₂+Π₃, (25) where Π₁=𝐴^𝑇(𝑡)𝑃+𝑃𝐴(𝑡)⋯𝑃𝐴_𝑑(𝑡),Π⋮⋱⋮∗⋯0₂=𝑄₁0⋯0∗𝑄₂−𝑄₁⋯0⋮⋮⋱0∗∗⋯𝑄_𝑁−𝑄_𝑁−1,Π₃=−𝑅₁𝑅₁0⋯00∗−𝑅₁−𝑅₂𝑅₂⋯00∗∗−𝑅₂−𝑅₃⋯00⋮⋮⋮⋱⋮⋮∗∗∗⋯−𝑅_𝑁−1−𝑅_𝑁0∗∗∗⋯∗−𝑅_𝑁−Σ⁽³⁾Σ⁽⁵⁾⁻¹Σ^(3)𝑇. (26)
From (18), we know that Π₁+Π₂+Π₃<0, (27) which guarantees 𝑉̇(𝑡)<0 for all non-zero 𝜂(𝑡). Thus one can always find a sufficiently small 𝜐>0 such that V̇‌(𝑡)<−𝜐‖𝑥(𝑡)‖². The asymptotic stability of the considered system is proved.
Next, assuming that 𝜔(𝑡)≠0 and 𝜙(𝑡)=0,𝑡∈[−𝜏,0], we consider 𝐻_∞ performance of the system in Definition 1.
Considering 𝜇_𝑖(𝑠(𝑡))≥0,∑^𝑟_𝑖=1𝜇_𝑖(𝑠(𝑡))=1, we obtain that 𝑧^𝑇(𝑡)𝑧(𝑡)−𝛾²𝜔^𝑇(𝑡)𝜔(𝑡)+𝑉̇(𝑡)≤^𝑟_𝑖=1𝜇_𝑖(𝑠(𝑡))𝜉^𝑇(𝑡)Ψ𝜉(𝑡), (28) where ΣΨ=⁽¹⁾Σ⁽²⁾∗−𝛾²𝐼−Σ⁽³⁾Σ⁽⁴⁾Σ⁽⁵⁾⁻¹Σ⁽³⁾Σ^(4)𝑇. (29) Applying Schur complement, guarantees Ψ<0. From (28), we can get that 𝑧^𝑇(𝑡)𝑧(𝑡)−𝛾²𝜔^𝑇(𝑡)𝜔(𝑡)+𝑉̇(𝑡)≤0. (30) Integrating the preceding inequality from 0 to ∞, it is easy to get that ^∞₀𝑧^𝑇(𝑡)𝑧(𝑡)𝑑𝑡≤^∞₀𝛾²𝜔^𝑇(𝑡)𝜔(𝑡)𝑑𝑡+𝑉𝑥₀−𝑉𝑥_∞. (31)
Since 𝑉(𝑥₀)=0 and 𝑉(𝑥_∞)≥0, we have ^∞₀𝑧^𝑇(𝑡)𝑧(𝑡)𝑑𝑡≤^∞₀𝛾²𝜔^𝑇(𝑡)𝜔(𝑡)𝑑𝑡. (32) Then, according to Definition 1, the 𝐻_∞ performance of the system in (16) is established. This completes the proof.

In the following, based on Proposition 4, we design robust state feedback 𝐻_∞ controller for the system (7).

Theorem 5. For some prescribed 𝛾>0, 𝜏>0, 𝛿>0, and N is a positive integer, if there exist scalar 𝜀>0, matrices 𝑋>0, 𝑉_𝑙>0(𝑙=2,3,…,𝑁), and 𝑄_𝑙>0(𝑙=1,2…𝑁), and appropriate dimension matrices 𝑊_𝑗(𝑗=1,2,…,𝑟) such that the following LMIs simultaneously hold for 𝑖,𝑗=1,2,…,𝑟: Ω_𝑖𝑖(),Ω<0,𝑖=1,2,…,𝑟_𝑖𝑗+Ω_𝑗𝑖2,𝑉<0,1≤𝑖<𝑗≤𝑟_𝑖𝑅_𝑖=𝐼,(𝑖=1,2,…,𝑁), (33) whereΩ_𝑖𝑗=Υ_𝑖𝑗𝜀𝑈𝑇^𝑇_𝑖𝑗∗−𝜀𝐼0∗∗−𝜀𝐼,Υ_𝑖𝑗=Π⁽¹⁾_𝑖𝑗Π_𝑖⁽²⁾Π⁽³⁾_𝑖𝑗Π_𝑖⁽⁶⁾∗−𝛾²𝐼Π_𝑖⁽⁴⁾0∗∗Π⁽⁵⁾0,∗∗∗−𝐼𝑈=𝐻000∗000∗∗𝑁0∗∗∗0,𝑇_𝑖𝑗=𝑆_𝑖𝑗𝐿0000000_𝑖𝑗,,𝑆0000000𝐻=𝑀00⋯00∗00⋯00∗∗0⋯00⋮⋮⋮⋱⋮⋮∗∗∗⋯00∗∗∗⋯∗0_𝑖𝑗=𝐸_𝑖𝑋+𝐸_1𝑖𝑊_𝑗00⋯0E_𝑑𝑖,𝐿∗00⋯00∗∗0⋯00⋮⋮⋮⋱⋮⋮∗∗∗⋯00∗∗∗⋯∗0_𝑇𝑖𝑗=ℎ𝑋𝐸^𝑇_𝑖+ℎ𝑊^𝑇_𝑗𝐸^𝑇_1𝑖ℎ𝑋𝐸^𝑇_𝑖+ℎ𝑊^𝑇_𝑗𝐸^𝑇_1𝑖⋯ℎ𝑋𝐸^𝑇_𝑖+ℎ𝑊^𝑇_𝑗𝐸^𝑇_1𝑖00⋯000⋯0⋮⋮⋱⋮00⋯0ℎ𝐸^𝑇_𝑑𝑖ℎ𝐸^𝑇_𝑑𝑖⋯ℎ𝐸^𝑇_𝑑𝑖,Π⁽¹⁾_𝑖𝑗=Π⁽¹⁾_11𝑖𝑗𝛿𝐼0…0𝐴_𝑑𝑖∗Π⁽¹⁾₂₂𝑅₂⋯00∗∗Π⁽¹⁾₃₃⋯00⋮⋮⋮⋱⋮⋮∗∗∗⋯Π⁽¹⁾_𝑁𝑁𝑅_𝑁∗∗∗⋯∗Π⁽¹⁾_𝑁+1𝑁+1,𝑁^𝑇=𝑀^𝑇00⋯00∗𝑀^𝑇0⋯00∗∗𝑀^𝑇⋯00⋮⋮⋮⋱⋮⋮∗∗∗⋯𝑀^𝑇0∗∗∗⋯∗𝑀^𝑇,Π⁽¹⁾_11𝑖𝑗=𝑋𝑄₁𝑋−𝛿𝑋+𝑋𝐴^𝑇_𝑖+𝑊^𝑇_𝑗𝐵^𝑇_1𝑖+𝐴_𝑖𝑋+𝐵_1𝑖𝑊_𝑗,Π⁽¹⁾₂₂=𝑄₂−𝑄₁−𝛿𝑃−𝑅₂,Π⁽¹⁾₃₃=𝑄₃−𝑄₂−𝑅₂−𝑅₃,Π⁽¹⁾_𝑁𝑁=𝑄_𝑁−𝑄_𝑁−1−𝑅_𝑁−1−𝑅_𝑁,Π⁽¹⁾_𝑁+1𝑁+1=−𝑄_𝑁−𝑅_𝑁,Π_𝑖⁽⁴⁾=ℎ𝐵^𝑇_2𝑖ℎ𝐵^𝑇_2𝑖⋯ℎ𝐵^𝑇_2𝑖,Π⁽⁵⁾=diag−𝛿⁻¹𝑋,−𝑉₂,…,−𝑉_𝑁,Π_𝑖⁽²⁾=𝐵_2𝑖0⋮0,Π⁽³⁾_𝑖𝑗=ℎ𝑋𝐴^𝑇_𝑖+ℎ𝑊^𝑇_𝑗𝐵^𝑇_1𝑖ℎ𝑋𝐴^𝑇_𝑖+ℎ𝑊^𝑇_𝑗𝐵^𝑇_1𝑖⋯ℎ𝑋𝐴^𝑇_𝑖+ℎ𝑊^𝑇_𝑗𝐵^𝑇_1𝑖00⋯000⋯0⋮⋮⋱⋮00⋯0ℎ𝐴^𝑇_𝑑𝑖ℎ𝐴^𝑇_𝑑𝑖⋯ℎ𝐴^𝑇_𝑑𝑖Π_𝑖⁽⁶⁾=𝑋𝐶^𝑇_𝑖0⋮0. (34)Then the closed-loop system (16) is asymptotically stable with the 𝐻_∞ performance index 𝛾. Moreover, if the above condition is feasible, the gain matrices of a desired controller in the form of (6) can be designed by 𝐾_𝑗=𝑊_𝑗𝑋⁻¹. (35)

Proof. The proof of this theorem is divided into two parts. First, we design the state-feedback 𝐻_∞ controller of the nominal case of closed-loop system (7).
According to Proposition 4, it is easy to know that the 𝐻_∞ performance requirement of the nominal case of closed-loop system (7) implies Σ<0, (36) where Σ is a matrix derived from (18) by changing the term 𝐴_𝑖 to 𝐴_𝑖+𝐵_1𝑖𝐾_𝑗.
Introduce the following matrix variables: 𝑋=𝑃⁻¹,𝑅₁𝑉=𝛿𝑃,₂=𝑅₂⁻¹,…,𝑉_𝑁=𝑅_𝑁⁻¹,𝐾_𝑗𝑋=𝑊_𝑗. (37) Combined with Schur complement, and pre- and post-multiply (36) by diag{𝑋,𝐼,𝛿⁻¹𝑋,𝑉₂,…,𝑉_𝑁,𝐼} and its transpose, respectively, we get Υ_𝑖𝑗<0. (38)
Next, we investigate robust state feedback 𝐻_∞ controller of the closed-loop system (7).
Replace 𝐴_𝑖, A_𝑑𝑖, and 𝐵_𝑖 with 𝐴_𝑖+Δ𝐴_𝑖, A_𝑑𝑖+ΔA_𝑑𝑖, and 𝐵_1𝑖+Δ𝐵_1𝑖, respectively; then we obtain from (2) and (38) that Υ_𝑖𝑗+𝑈𝐹(𝑡)𝑇_𝑖𝑗+𝑇^𝑇_𝑖𝑗𝐹(𝑡)𝑈^𝑇<0. (39) By Lemma 3, we can know (39) holds, if and only if the following inequality holds: Υ_𝑖𝑗+𝜀𝑈𝑈^𝑇+𝜀⁻¹𝑇^𝑇_𝑖𝑗𝑇_𝑖𝑗<0, (40) where 𝜀 is a positive scalar.
Applying Schur complement to (40), we have that 𝐸_𝑖𝑗=Υ_𝑖𝑗𝜀𝑈𝑇^𝑇_𝑖𝑗∗−𝜀𝐼0∗∗−𝜀𝐼<0. (41) Noting 𝜇_𝑖(𝑠(𝑡))≥0,∑^𝑟_𝑖=1𝜇_𝑖(𝑠(𝑡))=1, ^𝑟𝑟_𝑖=1𝑗=1𝜇_𝑖(𝑠(𝑡))𝜇_𝑗(𝑠(𝑡))𝐸_𝑖𝑗=^𝑟_𝑖=1𝜇²_𝑖(𝑠(𝑡))𝐸_𝑖𝑖+^𝑟_𝑖<𝑗𝜇_𝑖(𝑠(𝑡))𝜇_𝑗(𝑠(𝑡))𝐸_𝑖𝑗+𝐸_𝑗𝑖. (42) Therefore, condition (33) can guarantee that condition (41) holds. This completes the proof.

It should be noted that the obtained conditions in Theorem 5 are not strict LMI conditions due to the existence of nonlinear term 𝑋𝑄₁𝑋 in (33). It cannot be directly solved by standard LMI Toolbox. In the following, we present an approach to solving the condition in Theorem 5.

Introduce additional matrix variable 𝐺>0 such that 𝑋𝑄₁𝑋≤𝐺. (43) By Schur complement, it follows from (43) that −𝐺𝑋𝑋−𝑄₁⁻¹≤0. (44) Then, we readily obtain the following theorem.

Theorem 6. For some prescribed 𝛾>0, 𝜏>0, 𝛿>0, and 𝑁 is a positive integer, if there exist scalar 𝜀>0, matrices 𝐺>0,𝑃>0,𝑋>0, 𝑉_𝑙>0(𝑙=2,3,…,𝑁), 𝑄_l>0,(𝑙=1,2…,𝑁), and 𝑄₁>0, and appropriate dimension matrices 𝑊_𝑗(𝑗=1,2…𝑟) such that the following LMIs simultaneously hold for 𝑖,𝑗=1,2,…,𝑟: Ω^⌢_𝑖𝑖<0,(𝑖=1,2,…,𝑟_⌢),Ω_𝑖𝑗+Ω^⌢_𝑗𝑖2<0,1≤𝑖<𝑗≤𝑟,−𝐺𝑋𝑋−𝑄₁≤0,𝑄₁𝑄₁=𝐼,𝑃𝑋=𝐼,𝑉_𝑖𝑅_𝑖=𝐼,(𝑖=1,2,…,𝑁), (45)(46)(47) where Ω^⌢_𝑖𝑗 is a matrix derived from Ω_𝑖𝑗 by replacing the term Π⁽¹⁾_11𝑖𝑗=𝑋𝑄₁𝑋−𝑋+𝑋𝐴^𝑇_𝑖+𝑊^𝑇_𝑗𝐵^𝑇_1𝑖+𝐴_𝑖𝑋+𝐵_1𝑖𝑊_𝑗 (48) with Π⁽¹⁾_11𝑖𝑗=𝐺−𝛿𝑋+𝑋𝐴^𝑇_𝑖+𝑊^𝑇_𝑗𝐵^𝑇_1𝑖+𝐴_𝑖𝑋+𝐵_1𝑖𝑊_𝑗, (49) then the closed-loop system (25) is asymptotically stable with the 𝐻_∞ performance index 𝛾. Moreover, if the above condition is feasible, the gain matrices of a desired controller in the form of (6) can be designed by 𝐾_𝑗=𝑊_𝑗𝑋⁻¹. (50)

Remark 7. Note that the obtained conditions in Theorem 6 are not all in LMI form due to equality constraints, which cannot be solved directly using standard LMI procedures. However, via the result in [35] which has been widely used by many scholars [36, 37], we can solve these nonconvex feasibility problems by formulating them into some sequential optimization problems subject to LMIs constraints.

Now using the approach [35], we suggest the following minimization problem involving LMI conditions instead of the original nonconvex feasibility problem formulated in Theorem 6.

Problem HSFCD (𝐻_∞ state feedback controller design). Consider the following: Minimizetrace𝑃𝑋+𝐺𝐺+^𝑁_𝑗=2𝑅_𝑗𝑉_𝑗subjectto(45),(46)and𝑄𝑃𝐼𝐼𝑋≥0,₁𝐼𝐼𝑄₁𝑅≥0,^𝑗𝐼𝐼𝑉_𝑗^𝑁_𝑗=2≥0. (51)

When minimize Trace (𝑃𝑋+𝑄₁𝑄₁+∑^𝑁_𝑗=2𝑅_𝑗𝑉_𝑗)=(𝑁+1)𝑛, then the conditions in Theorem 6 are solvable. Algorithm  1 in [35] can be easily adapted to solve Problem HSFCD.

4. Numerical Example

In this section, we use an example to show the applicability of the results proposed in this paper.

Example 8. Consider the truck trailer system borrowed from [38], which can be represented by the following uncertain time-delay T-S fuzzy model.

Rule 1. If 𝜃(𝑡)=𝑥₂(𝑡)+𝑎(𝜐𝑡/2𝐿)𝑥₁(𝑡)+(1−𝑎)(𝜐𝑡/2𝐿)𝑥₁(𝑡−𝜏) is about 0, then 𝑥̇(𝑡)=𝐴₁+Δ𝐴₁𝑥(𝑡)+𝐴_𝑑1+Δ𝐴_𝑑1𝑥()+𝐵𝑡−𝜏_𝑢1+Δ𝐵_𝑢1𝑢(𝑡)+𝐵_𝑤1𝜔(𝑡),𝑧(𝑡)=𝐶₁𝑥(𝑡). (52)

Rule 2. If 𝜃(𝑡)=𝑥₂(𝑡)+𝑎(𝜐𝑡/2𝐿)𝑥₁(𝑡)+(1−𝑎)(𝜐𝑡/2𝐿)𝑥₁(𝑡−𝜏) is about 𝜋 or −𝜋, then 𝑥̇(𝑡)=𝐴₂+Δ𝐴₂𝑥(𝑡)+𝐴_𝑑2+Δ𝐴_𝑑2𝑥()+𝐵𝑡−𝜏_𝑢2+Δ𝐵_𝑢2𝑢(𝑡)+𝐵_𝑤2𝜔(𝑡)𝑧(𝑡)=𝐶₂𝑥(𝑡), (53) where 𝑎=0.7,𝜐=−1,𝐴𝑡=2.0,𝐿=5.5,𝜏=0.5,₁=0.509100−0.509100−0.5091−40,𝐴_𝑑1=,𝐵0.218200−0.2182000.218200_𝑢1=00−1.4286,𝐵_𝑤1=100,𝐴₂=,𝐴0.509100−0.509100−0.8102−6.36620_𝑑2=0.218200−0.2182000.347200,𝐵_𝑢2=00,𝐵−1.4286_𝑤1=100,Δ𝐴_𝑖,Δ𝐴_𝑑𝑖,Δ𝐵_𝑢𝑖=𝑀𝐹(𝑡)𝐸_𝑖,𝐸_𝑑𝑖,𝐸_𝑢𝑖,𝐶₁=𝐶₂=,{},𝐸010𝑀=diag0.05,0.05,0.05₁=0.509100−0.5091000.509100,𝐸₂=,𝐸0.509100−0.5091000.810700_𝑑1=0.218200−0.2182000.218200,𝐸_𝑑2=,𝐸0.218200−0.2182000.347200_𝑢1=00−0.3571,𝐸_𝑢2=00.−0.3571 (54)
For this example, the prescribed 𝐻_∞ performance level is chosen as 𝛾=0.5. In order to design a robust 𝐻_∞ state feedback controller for the given T-S fuzzy model, choose 𝛿=1, 𝑁=4. According to Theorem 5, the gain matrix of controller is given as 𝐾₁=,𝐾25.6229−29.90058.0667₂=23.8882−31.92438.1041. (55)
According to [38], take the membership function as ℎ₁=11−((𝜃(𝑡)×11+exp−3−0.5𝜋))((𝜃(𝑡),ℎ1+exp−3+0.5𝜋))₂=1−ℎ₁. (56)
Let disturbance input 𝜔(𝑡)=sin(2𝑡)𝑒^−0.05𝑡 and initial condition 𝜙(𝑡)=[111]^𝑇,𝑡∈[−0.5,0], and simulation time is 100 s.
Figure 1 shows the controlled output 𝑧(𝑡) and the disturbance input 𝜔(𝑡). According to Figure 1, the resulting output energy of the robust 𝐻_∞ controller is ∫₀¹⁰⁰𝑧²(𝑡)𝑑𝑡=0.832, while the input energy is ∫₀¹⁰⁰𝜔²(𝑡)𝑑𝑡=5. Simulation result for the ratio of the output energy to the disturbance energy is 0.1664, and the 𝑙₂-norm is √0.1664=0.41<𝛾=0.5 (due to the fact that the state has been stable for a long time, we can regard the value 0.41 as the 𝑙₂-norm).
State response of the closed-loop system and controller input are shown in Figure 2.
The simulation results show that the algorithm proposed in this paper is effective for robust 𝐻_∞ control problem of the truck trailer system with time delay.

Figure 1 

Controlled output 𝑧(𝑡) and the disturbance input 𝑤(𝑡).

(a)

(b)

(c)

(d)

Figure 2 

Response of the closed-loop system and controller input.

5. Conclusion

The problem of robust 𝐻_∞ controller design has been addressed for a class of T-S fuzzy-model-based systems with constant delay and norm-bounded parameter uncertainty. Based on the Lyapunov-Krasovskii functional approach, a sufficient condition for the existence of the robust 𝐻_∞ controller, which robustly stabilizes the T-S fuzzy-model-based uncertain systems and guarantees a prescribed level on disturbance attenuation, has been obtained in an LMI form. The given numerical example has shown the effectiveness of the proposed method. In addition, the filtering problems of T-S fuzzy delayed systems by using the delay decomposition approach are also challenging, which could be our further work.

Acknowledgments

This work was supported in part by National Natural Science Foundation of China (61203005) and in part by Harbin Engineering University Central University foundation Research Special Fund (HEUCFR1024).