Stability Analysis and 𝐻_∞ Output Tracking Control for Linear Systems with Time-Varying Delays

Abstract

The problem of stability analysis and 𝐻_∞ output tracking control for linear systems with time-varying delays is studied. First, by construction of a newly augmented Lyapunov-Krasovskii functional, a delay-dependent stability criterion for nominal systems with time-varying delays is established in terms of linear matrix inequalities (LMIs). Second, based on the 𝐻_∞ sense, the proposed method is extended to solve the problem of designing an 𝐻_∞ output tracking controller to track the output of a given reference model. Finally, three examples are included to show the validity and effectiveness of the presented delay-dependent stability and the 𝐻_∞ output tracking controller design.

1. Introduction

The problem of output tracking control has received a great deal of attention since this issue is an important requirement in many systems [1–5]. In detail, output tracking controller design methods are utilized in robot systems [1, 2], flight systems [3, 4], and so on. Moreover, in [5], the problem of output tracking control is derived in 𝐻_∞ sense for time-delayed systems with nonlinear perturbations. The main purpose of output tracking control is to design a closed-loop feedback controller such that the output of plant tracks the output of a given reference model as close as possible. Moreover, disturbances should be considered in designing output tracking control since disturbances can lead to adverse effects on the performance of systems. To minimize the effects of the disturbances on systems, one possible approach is to design a tracking controller in 𝐻_∞ sense. Since 𝐻_∞ control which has been used to minimize the effects of the disturbances was firstly introduced by Zames [6], the study on 𝐻_∞ controller design has been attracted by many researchers [7–10]. Similarly, 𝐻_∞ output tracking controller has an objective of designing a controller such that the closed-loop system is asymptotically stable and the tracking error by the effects of disturbances does not exceed a prescribed level.

On the other hand, time-delay is one of the sources of instability and poor performance in various systems such as physical and chemical systems and industrial and engineering systems. The reason is that time-delays frequently occur in various systems due to the finite capabilities of information processing and transmission. For this reason, during the past several decades, stability analysis for time-delayed systems has been widely studied by many researchers [11–23]. One of the aims of analysis for time-delayed systems is to establish a less conservative stability criterion which can find maximum upper bounds of time-delays for guaranteeing the asymptotic stability of systems. Stability analysis for time-delayed systems can be classified into delay-independent criteria and delay-dependent ones. Since delay-dependent stability criteria use information of time-delays such as lower bounds and upper bounds of delays while delay-independent ones do not have them, delay-dependent stability criteria are generally less conservative than delay-independent ones especially when the sizes of time-delays are small.

In line with this thinking, according to Zhang and Yu [5], it has been well acknowledged that output tracking control design for time-delayed system is more general and more difficult than stabilization. However, most existing results on output tracking control have focused on systems without time-delays. From the practical point of view, it is worth designing output tracking controllers for time-delayed systems since it is well known that there exist time-delays in many practical systems. Therefore, to establish less conservative stability criteria and design output tracking controllers for time-delayed systems are still challenging.

Motivated by the matters mentioned above, this paper investigates the problem of stability analysis and 𝐻_∞ tracking controller designing for linear systems with time-varying delays and disturbances. First, in Theorem 5, by constructing a newly augmented Lyapunov-Krasovskii functional, an improved delay-dependent stability criterion is derived by utilizing reciprocally convex approach [15] and some new zero equalities with LMI framework [24] which can be formulated as convex optimization algorithms. Second, based on the results of Theorem 5, an 𝐻_∞ tracking controller design method for linear systems with time-varying delays and disturbances will be proposed in Theorem 7. Finally, through three examples, validity and effectiveness of the proposed theorem will be verified.

Notation. ℝ^𝑛 is the 𝑛-dimensional Euclidean space and ℝ^𝑚×𝑛 denotes the set of 𝑚×𝑛 real matrix. For symmetric matrices 𝑋 and 𝑌, 𝑋>𝑌 (resp., 𝑋≥𝑌) means that the matrix 𝑋−𝑌 is positive definite (resp., nonnegative). 𝑋^𝑇 denotes the transposition of 𝑋. 𝐼_𝑛 denotes the 𝑛×𝑛 identity matrix. 0_𝑛 and 0_𝑛×𝑚 denote the 𝑛×𝑛 zero matrix and the 𝑛×𝑚 zero matrix, respectively. ‖⋅‖ refers to the Euclidean vector norm and the induced matrix norm. diag{⋯} denotes the block diagonal matrix, respectively. ∗ represents the elements below the main diagonal of a symmetric matrix. ℒ₂[0,∞) is the space of square integrable vector. 𝑋_[𝑓(𝑡)]∈ℝ^𝑚×𝑛 means that the elements of the matrix 𝑋_[𝑓(𝑡)] include the value of 𝑓(𝑡); for example, 𝑋_[𝑓0]≡𝑋_{[𝑓(𝑡)=𝑓0]}.

2. Problem Statements

Consider the following system with time-varying delays: 𝑥̇(𝑡)=𝐴𝑥(𝑡)+𝐴_𝑑(𝑡)𝑥(𝑡−ℎ(𝑡))+𝐵𝑢(𝑡)+𝐵_𝑤𝑤(𝑡),𝑧(𝑡)=𝐶₁𝑥(𝑡)+𝐶₂𝑥(𝑡−ℎ(𝑡))+𝐵₁𝑢(𝑡), (1) where 𝑥(𝑡)∈ℝ^𝑛 is the state vector, 𝑢(𝑡)∈ℝ^𝑚 is the control input, 𝑤(𝑡)∈ℝ^𝑟 is the disturbance input which belongs to ℒ₂[0,∞), 𝑧(𝑡)∈ℝ^𝑠 is the vector of controlled output, and 𝐴∈ℝ^𝑛×𝑛, 𝐴_𝑑∈ℝ^𝑛×𝑛, 𝐵∈ℝ^𝑛×𝑚, 𝐵_𝑤∈ℝ^𝑛×𝑟, 𝐶₁∈ℝ^𝑠×𝑛, 𝐶₂∈ℝ^𝑠×𝑛, and 𝐵₁∈ℝ^𝑠×𝑚 are known real constant matrices.  ℎ(𝑡) means a time-delay satisfying time-varying continuous function as follows: 0≤ℎ(𝑡)≤ℎ_𝑀,ℎ̇(𝑡)≤ℎ_𝑑, (2) where ℎ_𝑀>0 and 0≤ℎ_𝑑<1 are known scalars.

Designing the output tracking control has an objective to design a controller such that the output of the system tracks a reference signal. Therefore, the reference signal 𝑧_𝑟(𝑡) is assumed to be generated by reference model as follows: 𝑥̇_𝑟(𝑡)=𝐹𝑥_𝑟(𝑡)(𝑡),𝑧+𝑟_𝑟(𝑡)=𝐻𝑥_𝑟(𝑡), (3) where 𝑧_𝑟(𝑡)∈ℝ^𝑠 is the reference signal which has the same dimension as 𝑧(𝑡), 𝑥_𝑟(𝑡)∈ℝ^𝑝 is the reference state, and 𝑟(𝑡)∈ℝ^𝑝 is the energy bounded reference input. 𝐺∈ℝ^𝑝×𝑝 and 𝐻∈ℝ^𝑠×𝑝 are known real constant matrices with the assumption that the matrix 𝐺 is Hurwitz. The state-feedback controller is considered as the following form: 𝑢(𝑡)=𝐾₁𝑥(𝑡)+𝐾₂𝑥_𝑟(𝑡), (4) where 𝐾₁∈ℝ^𝑚×𝑛 and 𝐾₂∈ℝ^𝑚×𝑝 are gain matrices of the state-feedback controller. Let us define 𝜉(𝑡)=[𝑥^𝑇(𝑡),𝑥^𝑇_𝑟(𝑡)]^𝑇∈ℝ^𝑞, 𝑒(𝑡)=𝑧(𝑡)−𝑧_𝑟(𝑡), and 𝑣(𝑡)=[𝑤^𝑇(𝑡),𝑟^𝑇(𝑡)]^𝑇. Then the following augmented systems can be obtained as 𝜉̇(𝑡)=𝐴+𝐵𝐾𝜉(𝑡)+𝐴_𝑑(𝑡)𝜉(𝑡−ℎ(𝑡))+𝐵_𝑤𝑣(𝑡),𝑒(𝑡)=𝐶₁𝜉(𝑡)+𝐶₂𝜉(𝑡−ℎ(𝑡)), (5) where 𝐴=𝐴0_𝑛×𝑝0_𝑝×𝑛𝐹,𝐵0𝐵=_𝑝×𝑚,𝐾=𝐾₁,𝐾₂,𝐴_𝑑=𝐴_𝑑0_𝑛×𝑝0_𝑝×𝑛0_𝑝×𝑝,𝐵_𝑤=𝐵_𝑤0_𝑛×𝑝0_𝑝×𝑟𝐼_𝑝,𝐶₁=𝐶₁+𝐵₁𝐾₁,−𝐻+𝐵₁𝐾₂,𝐶₂=𝐶₂,0_𝑠×𝑝. (6)

From the 𝐻_∞ sense, let us consider the two requirements of the output tracking controller as follows.(i)With 𝑣(𝑡)=0, the closed-loop system (5) with control input 𝑢(𝑡) is asymptotically stable.(ii)The effects of 𝑤(𝑡) and 𝑟(𝑡) on the tracking error 𝑒(𝑡) are attenuated below a desired level in the 𝐻_∞ sense; that is, 𝐽=^∞₀𝑒^𝑇(𝑠)𝑒(𝑠)−𝛾²𝑣^𝑇(𝑠)𝑣(𝑠)𝑑𝑠⩽0,i.e.sup_{𝑣(𝑡)(𝑡)≠0,𝑣∈ℒ2[)0,∞}‖𝑒(𝑡)‖₂‖𝑣(𝑡)‖₂≤𝛾, (7)where 𝛾>0 is a prescribed scalar. Then, if the obtained controller 𝑢(𝑡) is satisfied with two requirements, then it is said to be an 𝐻_∞ output tracking controller.

The objective of this paper is to design a state-feedback controller (4) such that system (5) achieves the 𝐻_∞ output tracking below a prescribed level and the 𝐻_∞ output tracking performance 𝛾 is minimized.

Before proceeding further, the following lemmas will be utilized in deriving main results.

Lemma 1 (see [17]). For a positive matrix 𝑍, scalars ℎ₂>ℎ₁>0 such that the integrations are well defined; then −ℎ₂−ℎ₁^𝑡−ℎ1_𝑡−ℎ2𝑥^𝑇(𝑠)𝑍𝑥(𝑠)𝑑𝑠≤−^𝑡−ℎ1_𝑡−ℎ2𝑥(𝑠)𝑑𝑠^𝑇𝑍^𝑡−ℎ1_𝑡−ℎ2𝑥(𝑠),−ℎ𝑑𝑠₂₂−ℎ²₁2^𝑡−ℎ1_𝑡−ℎ2^𝑡_𝜃𝑥^𝑇(𝑠)𝑍𝑥(𝑠)𝑑𝑠𝑑𝜃≤−^𝑡−ℎ1_𝑡−ℎ2^𝑡_𝜃𝑥(𝑠)𝑑𝑠𝑑𝜃^𝑇𝑍^𝑡−ℎ1_𝑡−ℎ2^𝑡_𝜃𝑥(𝑠),−ℎ𝑑𝑠𝑑𝜃₃₂−ℎ³₁6^𝑡−ℎ1_𝑡−ℎ2^𝑡_𝜃^𝑡_𝜆𝑥^𝑇(𝑠)𝑍𝑥(𝑠)𝑑𝑠𝑑𝜆𝑑𝜃≤−^𝑡−ℎ1_𝑡−ℎ2^𝑡_𝜃^𝑡_𝜆𝑥(𝑠)𝑑𝑠𝑑𝜆𝑑𝜃^𝑇×𝑍^𝑡−ℎ1_𝑡−ℎ2^𝑡_𝜃^𝑡_𝜆𝑥(𝑠)𝑑𝑠𝑑𝜆𝑑𝜃. (8)

Lemma 2 (see [15]). For a scalar 𝛼 in the interval (0,1), a given matrix 𝑅∈ℝ^𝑛×𝑛>0, two matrices 𝑊₁∈ℝ^𝑛×𝑚 and 𝑊₂∈ℝ^𝑛×𝑚, for all vector 𝜉∈ℝ^𝑚, let one define a function Θ(𝛼,𝑅) given by Θ()=1𝛼,𝑅𝛼𝜉^𝑇𝑊^𝑇₁𝑅𝑊₁1𝜉+𝜉1−𝛼^𝑇𝑊^𝑇₂𝑅𝑊₂𝜉. (9) Then, if there exists a matrix 𝑋 in ℝ^𝑛×𝑛 such that ^𝑅𝑋_∗𝑅>0, then the following inequality holds min_()𝛼∈0,1Θ()≥𝑊𝛼,𝑅¹𝜉𝑊₂𝜉^𝑇𝑅𝑋𝑊∗𝑅¹𝜉𝑊₂𝜉. (10)

Lemma 3 (Finsler’s lemma [25]). Let 𝜁∈ℝ^𝑛, Φ=Φ^𝑇∈ℝ^𝑛×𝑛, and 𝐵∈ℝ^𝑚×𝑛 such that rank(𝐵)<𝑛. The following statements are equivalent:(1)𝜁^𝑇Φ𝜁<0,∀𝐵𝜁=0,𝜁≠0. (2)(𝐵^⟂)^𝑇Φ𝐵^⟂<0, where 𝐵^⟂ is a right orthogonal complement of 𝐵.

Lemma 4 (see [21]). For a positive matrix Λ, a symmetric matrix Ξ, and a matrix Γ, the following two statements are equivalent:(1)Ξ−Γ^𝑇ΛΓ<0; (2)There exists a matrix of appropriate dimension 𝑈 such that Ξ+𝑈^𝑇Γ+𝑈^𝑇Γ𝑈^𝑇𝑈−Λ≤0. (11)

3. Main Results

This section consists of two subsections. The first section introduces an improved stability criterion for system (1) with 𝑢(𝑡)=0 and 𝑤(𝑡)=0 as Theorem 5. The second subsection will investigate the problem of an 𝐻_∞ output tracking controller design method for the augmented system (5) based on the results of Theorem 5.

3.1. Stability Analysis

Let us consider the system (1) with 𝑤(𝑡)=0 and 𝑢(𝑡)=0 given by 𝑥̇(𝑡)=𝐴𝑥(𝑡)+𝐴_𝑑𝑥(𝑡−ℎ(𝑡)). (12) In this subsection, a delay-dependent stability criterion for system (12) is derived. For simplicity of matrix and vector representation, 𝑒_𝑖(𝑖=1,…,10)∈ℝ^10𝑛×𝑛 are defined as block entry matrices which will be used in Theorem 5. For example, 𝑒₁=[𝐼_𝑛,0_𝑛×9𝑛]^𝑇 and 𝑒₅=[0_𝑛×4𝑛,𝐼_𝑛,0_𝑛×5𝑛]^𝑇. The other notations are defined as 𝜁^𝑇(𝑡)=𝑥^𝑇(𝑡),𝑥^𝑇(𝑡−ℎ(𝑡)),𝑥^𝑇𝑡−ℎ_𝑀,𝑥̇^𝑇(𝑡),𝑥̇^𝑇𝑡−ℎ_𝑀,^𝑡_{𝑡−ℎ(𝑡)}𝑥^𝑇(𝑠)𝑑𝑠,_{𝑡−ℎ(𝑡)𝑡−ℎ𝑀}𝑥^𝑇(𝑠)𝑑𝑠,^𝑡_{(𝑡)𝑡−ℎ}^𝑡_𝑠𝑥^𝑇(𝑢)𝑑𝑢𝑑𝑠,_{𝑡−ℎ(𝑡)𝑡−ℎ𝑀𝑡𝑠}𝑥^𝑇(𝑢)𝑑𝑢𝑑𝑠,𝑥̇^𝑇(𝑡−ℎ(𝑡)),Θ=𝐴𝑒^𝑇₁+𝐴_𝑑𝑒^𝑇₂−𝑒^𝑇₄,Ψ₁₁=𝑒₁,𝑒₂,𝑒₃,𝑒₆,𝑒₇,𝑒₈,𝑒₉,Ψ₁₂=𝑒₁,𝑒₂,𝑒₃,𝑒₄,𝑒₅,𝑒₆,𝑒₇,𝑒₁₀,Ψ₂₁=𝑒₁,𝑒₄,0_10𝑛×𝑛,0_10𝑛×𝑛,Ψ₂₂=𝑒₃,𝑒₅,𝑒₁−𝑒₃,𝑒₆+𝑒₇,Ψ₂₃=𝑒₆+𝑒₇,𝑒₁−𝑒₃,ℎ_𝑀𝑒₁−𝑒₆−𝑒₇,𝑒₈+𝑒₉,Ψ₂₄=0_10𝑛×𝑛,0_10𝑛×𝑛,𝑒₄,𝑒₁,Ψ₃₁=𝑒₁,𝑒₄,0_10𝑛×𝑛,0_10𝑛×𝑛,Ψ₃₂=𝑒₂,𝑒₁₀,𝑒₁−𝑒₂,𝑒₆,Ψ_{33[ℎ(𝑡)]}=𝑒₆,𝑒₁−𝑒₃,ℎ(𝑡)𝑒₁−𝑒₆,𝑒₈,Ψ₃₄=0_10𝑛×𝑛,0_10𝑛×𝑛,𝑒₄,𝑒₁,Ψ₃₅=𝑒₂,𝑒₁₀,𝑒₁−𝑒₂,𝑒₆,Ψ₃₆=𝑒₃,𝑒₅,𝑒₁−𝑒₃,𝑒₆+𝑒₇,Ψ_{37[ℎ(𝑡)]}=𝑒₇,𝑒₂−𝑒₃,ℎ_𝑀−ℎ(𝑡)𝑒₁−𝑒₇,𝑒₉,Ψ₃₈=0_10𝑛×𝑛,0_10𝑛×𝑛,𝑒₄,𝑒₁,Ψ₄₁=𝑒₁,𝑒₄,Ψ₄₂=𝑒₆,𝑒₁−𝑒₂,𝑒₇,𝑒₂−𝑒₃,Ψ₅=𝑒₁,𝑒₄,Ψ₆=ℎ²_𝑀2𝑒₁−𝑒₈−𝑒₉,Υ₁=𝑄₁+𝑃₃𝑃₁𝑃₁0_𝑛𝑆₁𝑆^𝑇₁𝑄₁+𝑃₄𝑃₂𝑃₂0_𝑛,Υ₂=𝑄₂+4ℎ_𝑀0_𝑛𝑃₃𝑃₃0_𝑛𝑆₂𝑆^𝑇₂𝑄₂+4ℎ_𝑀0_𝑛𝑃₄𝑃₄0_𝑛,Γ_{̇1[ℎ(𝑡),ℎ(𝑡)]}=0_𝑛×3𝑛,𝐼_𝑛,0_𝑛×4𝑛0_𝑛×7𝑛,1−ℎ̇(𝑡)𝐼_𝑛0_𝑛×4𝑛,𝐼_𝑛,0_𝑛×3𝑛𝐼_𝑛,−1−ℎ̇(𝑡)𝐼_𝑛,0_𝑛×6𝑛0_𝑛,1−ℎ̇(𝑡)𝐼_𝑛,−𝐼_𝑛,0_𝑛×5𝑛ℎ(𝑡)𝐼_𝑛,0_𝑛×4𝑛,−1−ℎ̇(𝑡)𝐼_𝑛,0_𝑛×2𝑛ℎ_𝑀−ℎ(𝑡)𝐼_𝑛,0_𝑛×4𝑛,−ℎ̇(𝑡)𝐼_𝑛,0_𝑛×2𝑛,Γ_2[ℎ(𝑡)]=0_𝑛×3𝑛,𝐼_𝑛,0_𝑛ℎ(𝑡)𝐼_𝑛,−𝐼_𝑛,0_𝑛×3𝑛0_𝑛×4𝑛,𝐼_𝑛ℎ_𝑀−ℎ(𝑡)𝐼_𝑛,0_𝑛,−𝐼_𝑛,0_𝑛×2𝑛𝑒^𝑇₁𝑒^𝑇₆𝑒^𝑇₇𝑒^𝑇₈𝑒^𝑇₉,Ξ_{̇1[ℎ(𝑡),ℎ(𝑡)]}=Ψ₁₁Γ_{̇1[ℎ(𝑡),ℎ(𝑡)]}𝑅Ψ^𝑇₁₂+Ψ₁₂𝑅Γ^𝑇_{̇1[ℎ(𝑡),ℎ(𝑡)]}Ψ^𝑇₁₁,Ξ₂=Ψ₂₁𝑁Ψ^𝑇₂₁−Ψ₂₂𝑁Ψ^𝑇₂₂+Ψ₂₃𝑁Ψ^𝑇₂₄+Ψ₂₄𝑁Ψ^𝑇₂₃,Ξ_{̇3[ℎ(𝑡),ℎ(𝑡)]}=Ψ₃₁𝐺Ψ^𝑇₃₁−1−ℎ̇(𝑡)Ψ₃₂𝐺Ψ^𝑇₃₂+Ψ_{̇33[ℎ(𝑡),ℎ(𝑡)]}𝐺Ψ^𝑇₃₄+Ψ₃₄𝐺Ψ^𝑇_{̇33[ℎ(𝑡),ℎ(𝑡)]}+1−ℎ̇(𝑡)Ψ₃₅𝑀Ψ^𝑇₃₅−Ψ₃₆𝑀Ψ^𝑇₃₆+Ψ_{̇37[ℎ(𝑡),ℎ(𝑡)]}𝑀Ψ^𝑇₃₈+Ψ₃₈𝑀Ψ^𝑇_{̇37[ℎ(𝑡),ℎ(𝑡)]},Ξ_{zero[ℎ(𝑡)]}=ℎ_𝑀𝑒₁𝑃₁𝑒^𝑇₁−𝑒₂𝑃₁𝑒^𝑇₂+𝑒₂𝑃₂𝑒^𝑇₂−𝑒₃𝑃₂𝑒^𝑇₃+ℎ_𝑀ℎ(𝑡)𝑒₁𝑃₃𝑒^𝑇₁−ℎ_𝑀−ℎ(𝑡)𝑒₁𝑃₄𝑒^𝑇₁,Ξ_4[ℎ(𝑡)]=ℎ²_𝑀Ψ₄₁𝑄₁Ψ^𝑇₄₁−Ψ₄₂Υ₁Ψ^𝑇₄₂+Ξ_{zero[ℎ(𝑡)]},Ξ₅=ℎ²_𝑀2²Ψ₅𝑄₂Ψ^𝑇₅,Ξ₆=ℎ³_𝑀6²𝑒₄𝑄₃𝑒^𝑇₄−Ψ₆𝑄₃Ψ^𝑇₆,Ξ_{̇[ℎ(𝑡),ℎ(𝑡)]}=Ξ_{̇1[ℎ(𝑡),ℎ(𝑡)]}+Ξ₂+Ξ_{̇3[ℎ(𝑡),ℎ(𝑡)]}+Ξ_4[ℎ(𝑡)]+Ξ₅+Ξ₆,𝐿_{̇[ℎ(𝑡),ℎ(𝑡)]}=Θ^⟂𝑇Ξ_{̇[ℎ(𝑡),ℎ(𝑡)]}Θ^⟂+Θ^⟂𝑇Γ^𝑇_2[ℎ(𝑡)]Λ+Λ^𝑇Γ_2[ℎ(𝑡)]Θ^⟂,Φ_{̇[ℎ(𝑡),ℎ(𝑡)]}=𝐿_{̇[ℎ(𝑡),ℎ(𝑡)]}Λ^𝑇Λ−Υ₂, (13) where Θ^⟂ is the right orthogonal complement of Θ.

Now, a delay-dependent stability of the system (12) is given as Theorem 5.

Theorem 5. For given scalars 0≤ℎ_𝑀 and 0≤ℎ_𝑑<1, the system (12) is asymptotically stable for 0≤ℎ(𝑡)≤ℎ_𝑀 and |ℎ̇(𝑡)|≤ℎ_𝑑, if there exist positive definite matrices 𝑅∈ℝ^7𝑛×7𝑛, 𝑁∈ℝ^4𝑛×4𝑛, 𝐺∈ℝ^4𝑛×4𝑛, 𝑀∈ℝ^4𝑛×4𝑛, 𝑄₁∈ℝ^2𝑛×2𝑛, 𝑄₂∈ℝ^2𝑛×2𝑛, and 𝑄₃∈ℝ^𝑛×𝑛, symmetric matrices 𝑃_𝑖∈ℝ^𝑛×𝑛(𝑖=1,…,4), and any matrices 𝑆₁∈ℝ^2𝑛×2𝑛, 𝑆₂∈ℝ^2𝑛×2𝑛, and Λ∈ℝ^4𝑛×9𝑛, satisfying the following LMIs: Φ_𝑗,𝑘,Υ<0,𝑗,𝑘=1,2_𝑖≥0,(𝑖=1,2), (14) where Φ_𝑗,𝑘 are the four vertices of Φ_{̇[ℎ(𝑡),ℎ(𝑡)]} with the bounds of ℎ(𝑡) and ℎ̇(𝑡), that is, ℎ_𝑀 and ℎ_𝐷 when 𝑗=𝑘=1, ℎ_𝑀 and −ℎ_𝐷 when 𝑗<𝑘, 0 and ℎ_𝐷 when 𝑗>𝑘, and 0 and −ℎ_𝐷 when 𝑗=𝑘=2.

Proof. For positive definite matrices 𝑅∈ℝ^7𝑛×7𝑛, 𝑁∈ℝ^4𝑛×4𝑛, 𝐺∈ℝ^4𝑛×4𝑛, 𝑄₁∈ℝ^2𝑛×2𝑛, 𝑄₂∈ℝ^2𝑛×2𝑛, and 𝑄₃∈ℝ^𝑛×𝑛, let us consider the following Lyapunov-Krasovskii functional candidate as 𝑉(𝑡)=⁶_𝑖=1𝑉_𝑖(𝑡), (15) where 𝑉₁(𝑡)=Ω^𝑇₁(𝑡)𝑅Ω₁(𝑡),𝑉₂(𝑡)=^𝑡_{𝑡−ℎ𝑀}Ω^𝑇₂(𝑡,𝑠)𝑁Ω₂()𝑉𝑡,𝑠𝑑𝑠,₃(𝑡)=^𝑡_{𝑡−ℎ(𝑡)}Ω^𝑇₂(𝑡,𝑠)𝐺Ω₂()+𝑡,𝑠𝑑𝑠_{𝑡−ℎ(𝑡)𝑡−ℎ𝑀}Ω^𝑇₂(𝑡,𝑠)𝑀Ω₂()𝑉𝑡,𝑠𝑑𝑠,₄(𝑡)=ℎ_𝑀^𝑡_{𝑡−ℎ𝑀}^𝑡_𝑠𝑥(𝑢)𝑥̇(𝑢)^𝑇𝑄₁𝑥(𝑢)𝑥̇(𝑢)𝑉𝑑𝑢𝑑𝑠,₅(𝑡)=ℎ²_𝑀2^𝑡_{𝑡−ℎ𝑀}^𝑡_𝑠^𝑡_𝑢𝑥(𝑣)𝑥̇(𝑣)^𝑇𝑄₂𝑥(𝑣)𝑥̇(𝑣)𝑉𝑑𝑣𝑑𝑢𝑑𝑠,₆(𝑡)=ℎ³_𝑀6^𝑡_{𝑡−ℎ𝑀}^𝑡_𝑠^𝑡_𝑢^𝑡_𝑣𝑥̇^𝑇(𝜆)𝑄₃𝑥̇(𝜆)𝑑𝜆𝑑𝑣𝑑𝑢𝑑𝑠, (16) with Ω₁(𝑡)=[𝑥^𝑇(𝑡),𝑥^𝑇(𝑡−ℎ(𝑡)),𝑥^𝑇(𝑡−ℎ_𝑀),∫^𝑡_{𝑡−ℎ(𝑡)}𝑥^𝑇(𝑠)𝑑𝑠,∫^𝑡_{𝑡−ℎ𝑀}𝑥^𝑇(𝑠)𝑑𝑠,∫^𝑡_{𝑡−ℎ(𝑡)}∫^𝑡_𝑠𝑥^𝑇(𝑢)𝑑𝑢𝑑𝑠, ∫^𝑡_{𝑡−ℎ𝑀}∫^𝑡_𝑠𝑥^𝑇(𝑢)𝑑𝑢𝑑𝑠]^𝑇 and Ω₂(𝑡,𝑠)=[𝑥^𝑇(𝑠),𝑥̇^𝑇(𝑠),∫^𝑡_𝑠𝑥̇^𝑇(𝑢)𝑑𝑢,∫^𝑡_𝑠𝑥^𝑇(𝑢)𝑑𝑢]^𝑇.
Now, calculating the time-derivative of 𝑉₁(𝑡) yields 𝑉̇₁(𝑡)=2Ω₁(𝑡)𝑅×𝑥̇(𝑡)ℎ̇(𝑡)𝑥̇((𝑡𝑥̇1−𝑡−ℎ))𝑡−ℎ_𝑀𝑥(𝑡)−1−ℎ̇(𝑡)𝑥𝑡−ℎ_𝑀1−ℎ̇(𝑡)𝑥(𝑡−ℎ(𝑡))−𝑥𝑡−ℎ_𝑀ℎ(𝑡)𝑥(𝑡)−1−ℎ̇(𝑡)^𝑡_{(𝑡)𝑡−ℎ}𝑥(𝑠)ℎ𝑑𝑠_𝑀−ℎ(𝑡)𝑥(𝑡)+1−ℎ̇(𝑡)^𝑡_{(𝑡)𝑡−ℎ}𝑥(𝑠)−𝑑𝑠_{𝑡(𝑡)𝑡−ℎ}𝑥(𝑠)𝑑𝑠−^𝑡_{𝑡−ℎ𝑀}𝑥(𝑠)𝑑𝑠=𝜁^𝑇(𝑡)Ξ_{̇1[ℎ(𝑡),ℎ(𝑡)]}𝜁(𝑡). (17) Also, 𝑉̇₂(𝑡) can be represented as 𝑉̇₂(𝑡)=𝑥(𝑡)𝑥̇(𝑡)0_𝑛×10_𝑛×1^𝑇𝑁𝑥(𝑡)𝑥̇(𝑡)0_𝑛×10_𝑛×1−𝑥𝑡−ℎ_𝑀𝑥̇𝑡−ℎ_𝑀𝑥(𝑡)−𝑥𝑡−ℎ_{𝑀𝑡𝑡−ℎ𝑀}𝑥(𝑢)𝑑𝑢^𝑇𝑥×𝑁𝑡−ℎ_𝑀𝑥̇𝑡−ℎ_𝑀𝑥(𝑡)−𝑥𝑡−ℎ_{𝑀𝑡𝑡−ℎ𝑀}𝑥(𝑢)𝑑𝑢+2^𝑡_{𝑡−ℎ𝑀}̇𝑥(𝑠)𝑥(𝑠)𝑥(𝑡)−𝑥(𝑠)_𝑡𝑠𝑥(𝑢)𝑑𝑢^𝑇0𝑑𝑠𝑁^𝑛×10_𝑛×1𝑥̇(𝑡)𝑥(𝑡)=𝜁^𝑇(𝑡)Ξ₂𝜁(𝑡). (18) The time-derivative of 𝑉₃(𝑡) can be given as follows: 𝑉̇₃(𝑡)=̇0𝑥(𝑡)𝑥(𝑡)_𝑛×10_𝑛×1^𝑇𝐺𝑥(𝑡)𝑥̇(𝑡)0_𝑛×10_𝑛×1−ℎ̇(𝑡)̇1−𝑥(𝑡−ℎ(𝑡))𝑥(𝑡−ℎ(𝑡))𝑥(𝑡)−𝑥(𝑡−ℎ(𝑡))_{𝑡𝑡−ℎ(𝑡)}𝑥(𝑠)𝑑𝑠^𝑇𝑥((𝑡𝑥̇((𝑡𝑥(𝑡)((𝑡×𝐺𝑡−ℎ))𝑡−ℎ))−𝑥𝑡−ℎ))_{𝑡𝑡−ℎ(𝑡)}𝑥(𝑠)𝑑𝑠+2^𝑡_{𝑡−ℎ(𝑡)}̇𝑥(𝑠)𝑥(𝑡)_𝑡𝑠𝑥̇(𝑢)𝑑𝑢_𝑡𝑠𝑥(𝑢)𝑑𝑢^𝑇0𝑑𝑠𝐺^𝑛×10_𝑛×1𝑥̇(𝑡)𝑥(𝑡)+ℎ̇(𝑡)̇1−𝑥(𝑡−ℎ(𝑡))𝑥(𝑡−ℎ(𝑡))𝑥(𝑡)−𝑥(𝑡−ℎ(𝑡))_{𝑡𝑡−ℎ(𝑡)}𝑥(𝑠)𝑑𝑠^𝑇𝑥((𝑡𝑥̇((𝑡𝑥(𝑡)((𝑡×𝑀𝑡−ℎ))𝑡−ℎ))−𝑥𝑡−ℎ))_{𝑡𝑡−ℎ(𝑡)}𝑥(𝑠)−𝑑𝑠𝑥(𝑡−ℎ_𝑀)𝑥̇(𝑡−ℎ_𝑀)𝑥(𝑡)−𝑥(𝑡−ℎ_𝑀)_{𝑡𝑡−ℎ𝑀}𝑥(𝑠)𝑑𝑠^𝑇𝑀𝑥𝑡−ℎ_𝑀𝑥̇𝑡−ℎ_𝑀𝑥(𝑡)−𝑥𝑡−ℎ_{𝑀𝑡𝑡−ℎ𝑀}𝑥(𝑠)𝑑𝑠+2^{𝑡−ℎ(𝑡)𝑡−ℎ𝑀}𝑥(𝑠)𝑥(𝑡−ℎ(𝑡))−𝑥𝑡−ℎ_𝑀ℎ_𝑀−ℎ(𝑡)𝑥(𝑡)−^{𝑡−ℎ(𝑡)}_{𝑡−ℎ𝑀}𝑥(𝑠)𝑑𝑠_{𝑡−ℎ(𝑡)𝑡−ℎ𝑀𝑡𝑠}𝑥(𝑢)𝑑𝑢𝑑𝑠^𝑇0×𝑀^𝑛×10_𝑛×1𝑥̇(𝑡)𝑥(𝑡)=𝜁^𝑇(𝑡)Ξ_{̇3[ℎ(𝑡),ℎ(𝑡)]}𝜁(𝑡). (19) Calculating 𝑉̇₄(𝑡) gives 𝑉̇₄(𝑡)=ℎ²_𝑀̇𝑥(𝑡)𝑥(𝑡)^𝑇𝑄₁𝑥(𝑡)𝑥̇(𝑡)−ℎ_𝑀^𝑡_{𝑡−ℎ𝑀}̇𝑥(𝑠)𝑥(𝑠)^𝑇𝑄₁𝑥(𝑠)𝑥̇(𝑠)𝑑𝑠. (20) Inspired by the work in [22], the following four zero equalities with symmetric matrices 𝑃₁, 𝑃₂, 𝑃₃, and 𝑃₄ are considered: 0=𝑥^𝑇(𝑡)ℎ_𝑀𝑃₁𝑥(𝑡)−𝑥^𝑇(𝑡−ℎ(𝑡))ℎ_𝑀𝑃₁𝑥(𝑡−ℎ(𝑡))−2ℎ_𝑀^𝑡_{𝑡−ℎ(𝑡)}𝑥^𝑇(𝑠)𝑃₁𝑥̇(𝑠)𝑑𝑠,0=𝑥^𝑇(𝑡−ℎ(𝑡))ℎ_𝑀𝑃₂𝑥(𝑡−ℎ(𝑡))−𝑥^𝑇𝑡−ℎ_𝑀ℎ_𝑀𝑃₂𝑥𝑡−ℎ_𝑀−2ℎ_𝑀^{𝑡−ℎ(𝑡)}_{𝑡−ℎ𝑀}𝑥^𝑇(𝑠)𝑃₂𝑥̇(𝑠)𝑑𝑠,0=𝑥^𝑇(𝑡)ℎ_𝑀ℎ(𝑡)𝑃₃𝑥(𝑡)−ℎ_𝑀^𝑡_{𝑡−ℎ(𝑡)}𝑥^𝑇(𝑠)𝑃₃𝑥(𝑠)𝑑𝑠−2ℎ_𝑀^𝑡_{(𝑡)𝑡−ℎ}^𝑡_𝑠𝑥^𝑇(𝑢)𝑃₃𝑥̇(𝑢)𝑑𝑢𝑑𝑠,0=𝑥^𝑇(𝑡)ℎ_𝑀ℎ_𝑀−ℎ(𝑡)𝑃₄𝑥(𝑡)−ℎ_𝑀^𝑡_{𝑡−ℎ(𝑡)}𝑥^𝑇(𝑠)𝑃₄𝑥(𝑠)𝑑𝑠−2ℎ_𝑀^{𝑡−ℎ(𝑡)}_{𝑡−ℎ𝑀}^𝑡_𝑠𝑥^𝑇(𝑢)𝑃₄𝑢̇(𝑠)𝑑𝑢𝑑𝑠. (21) By adding (21) into (20), 𝑉̇₄(𝑡) can be represented as follows: 𝑉̇₄(𝑡)=ℎ²_𝑀̇𝑥(𝑡)𝑥(𝑡)^𝑇𝑄₁𝑥(𝑡)𝑥̇(𝑡)−ℎ_𝑀^𝑡_{𝑡−ℎ𝑀}̇𝑥(𝑠)𝑥(𝑠)^𝑇𝑄₁𝑥(𝑠)𝑥̇(𝑠)𝑑𝑠+𝑥^𝑇(𝑡)ℎ_𝑀𝑃₁𝑥(𝑡)−𝑥^𝑇((𝑡×ℎ𝑡−ℎ))_𝑀𝑃₁𝑥(𝑡−ℎ(𝑡))−2ℎ_𝑀^𝑡_{𝑡−ℎ(𝑡)}𝑥^𝑇(𝑠)𝑃₁𝑥̇(𝑠)𝑑𝑠+𝑥^𝑇(𝑡−ℎ(𝑡))ℎ_𝑀𝑃₂𝑥(𝑡−ℎ(𝑡))−𝑥^𝑇𝑡−ℎ_𝑀ℎ_𝑀𝑃₂𝑥𝑡−ℎ_𝑀−2ℎ_𝑀^{𝑡−ℎ(𝑡)}_{𝑡−ℎ𝑀}𝑥^𝑇(𝑠)𝑃₂𝑥̇(𝑠)𝑑𝑠+𝑥^𝑇(𝑡)ℎ_𝑀ℎ(𝑡)𝑃₃𝑥(𝑡)−ℎ_𝑀^𝑡_{𝑡−ℎ(𝑡)}𝑥^𝑇(𝑠)𝑃₃𝑥(𝑠)𝑑𝑠−2ℎ_𝑀^𝑡_{(𝑡)𝑡−ℎ}^𝑡_𝑠𝑥^𝑇(𝑢)𝑃₃𝑥̇(𝑢)𝑑𝑢𝑑𝑠+𝑥^𝑇(𝑡)ℎ_𝑀ℎ_𝑀−ℎ(𝑡)𝑃₄𝑥(𝑡)−ℎ_𝑀^𝑡_{𝑡−ℎ(𝑡)}𝑥^𝑇(𝑠)𝑃₄𝑥(𝑠)𝑑𝑠−2ℎ_𝑀^{𝑡−ℎ(𝑡)}_{𝑡−ℎ𝑀}^𝑡_𝑠𝑥^𝑇(𝑢)𝑃₄𝑢̇(𝑠)𝑑𝑢𝑑𝑠=ℎ²_𝑀̇𝑥(𝑡)𝑥(𝑡)^𝑇𝑄₁𝑥(𝑡)𝑥̇(𝑡)−ℎ_𝑀^𝑡_{𝑡−ℎ𝑀}̇𝑥(𝑠)𝑥(𝑠)^𝑇𝑄₁+𝑃₃𝑃₁𝑃₁0_𝑛𝑥(𝑠)𝑥̇(𝑠)𝑑𝑠−ℎ_𝑀^{𝑡−ℎ(𝑡)}_{𝑡−ℎ𝑀}̇𝑥(𝑠)𝑥(𝑠)^𝑇𝑄₁+𝑃₄𝑃₂𝑃₂0_𝑛𝑥(𝑠)𝑥̇(𝑠)𝑑𝑠+𝜁^𝑇(𝑡)Ξ_{zero[ℎ(𝑡)]}𝜁(𝑡)−2ℎ_𝑀^𝑡_{(𝑡)𝑡−ℎ}^𝑡_𝑠𝑥^𝑇(𝑢)𝑃₃𝑥̇(𝑢)𝑑𝑢𝑑𝑠−2ℎ_𝑀^{𝑡−ℎ(𝑡)}_{𝑡−ℎ𝑀}^𝑡_𝑠𝑥^𝑇(𝑢)𝑃₄𝑥̇(𝑠)𝑑𝑢𝑑𝑠. (22) From Lemma 1, an upper-bound of 𝑉̇₄(𝑡) can be calculated as 𝑉̇₄(𝑡)≤ℎ²_𝑀̇𝑥(𝑡)𝑥(𝑡)^𝑇𝑄₁𝑥(𝑡)𝑥̇(𝑡)−ℎ_𝑀ℎ(𝑡)^𝑡_{𝑡−ℎ𝑀}̇𝑥(𝑠)𝑥(𝑠)^𝑇𝑄₁+𝑃₃𝑃₁𝑃₁0_𝑛𝑥(𝑠)𝑥̇(𝑠)−ℎ𝑑𝑠_𝑀ℎ_𝑀(𝑡)×−ℎ_{𝑡−ℎ(𝑡)𝑡−ℎ𝑀}̇𝑥(𝑠)𝑥(𝑠)^𝑇𝑄₁+𝑃₄𝑃₂𝑃₂0_𝑛𝑥(𝑠)𝑥̇(𝑠)𝑑𝑠+𝜁^𝑇(𝑡)Ξ_{zero[ℎ(𝑡)]}𝜁(𝑡)−2ℎ_𝑀^𝑡_{(𝑡)𝑡−ℎ}^𝑡_𝑠𝑥^𝑇(𝑢)𝑃₃𝑥̇(𝑢)𝑑𝑢𝑑𝑠−2ℎ_𝑀^{𝑡−ℎ(𝑡)}_{𝑡−ℎ𝑀}^𝑡_𝑠𝑥^𝑇(𝑢)𝑃₄𝑥̇(𝑠)𝑑𝑢𝑑𝑠=ℎ²_𝑀̇𝑥(𝑡)𝑥(𝑡)^𝑇𝑄₁𝑥(𝑡)𝑥̇(𝑡)+^{𝑡𝑡−ℎ(𝑡)}𝑥(𝑠)𝑑𝑠𝑥(𝑡)−𝑥(𝑡−ℎ(𝑡))_{𝑡−ℎ(𝑡)𝑡−ℎ𝑀}𝑥(𝑠)𝑑𝑠𝑥(𝑡−ℎ(𝑡))−𝑥(𝑡−ℎ_𝑀)^𝑇×^{1𝜙1(𝑡)𝑄1+𝑃3𝑃1𝑃10𝑛02𝑛}_{02𝑛11−𝜙1(𝑡)𝑄1+𝑃4𝑃2𝑃20𝑛}×^{𝑡𝑡−ℎ(𝑡)}𝑥(𝑠)𝑥(𝑡)((𝑡𝑑𝑠−𝑥𝑡−ℎ))_{𝑡−ℎ(𝑡)𝑡−ℎ𝑀}𝑥(𝑠)𝑥((𝑡𝑑𝑠𝑡−ℎ))−𝑥𝑡−ℎ_𝑀+𝜁^𝑇(𝑡)Ξ_{zero[ℎ(𝑡)]}𝜁(𝑡)−2ℎ_𝑀^𝑡_{(𝑡)𝑡−ℎ}^𝑡_𝑠𝑥^𝑇(𝑢)𝑃₃𝑥̇(𝑢)𝑑𝑢𝑑𝑠−2ℎ_𝑀^{𝑡−ℎ(𝑡)}_{𝑡−ℎ𝑀}^𝑡_𝑠𝑥^𝑇(𝑢)𝑃₄𝑥̇(𝑠)𝑑𝑢𝑑𝑠, (23)(24) where 𝜙₁(𝑡)=ℎ(𝑡)/ℎ_𝑀 which satisfies 0≤𝜙₁(𝑡)≤1. It should be noted that if ℎ(𝑡) is zero, then the term of ∫^𝑡_{𝑡−ℎ(𝑡)}𝑥(𝑠)𝑑𝑠 and 𝑥(𝑡)−𝑥(𝑡−ℎ(𝑡)) are zero. Also, if ℎ(𝑡) is ℎ_𝑀, then the term of ∫^{𝑡−ℎ(𝑡)}_{𝑡−ℎ𝑀}𝑥(𝑠)𝑑𝑠 and 𝑥(𝑡−ℎ(𝑡))−𝑥(𝑡−ℎ_𝑀) are zero. Thus, inequality (24) still holds.
By using Lemma 2 with Υ₁≥0 which is defined in (13), a new upper-bound of 𝑉̇₄(𝑡) can be obtained as 𝑉̇₄(𝑡)≤ℎ²_𝑀̇𝑥(𝑡)𝑥(𝑡)^𝑇𝑄₁𝑥(𝑡)𝑥̇(𝑡)−^{𝑡𝑡−ℎ(𝑡)}𝑥(𝑠)𝑑𝑠𝑥(𝑡)−𝑥(𝑡−ℎ(𝑡))_{𝑡−ℎ(𝑡)𝑡−ℎ𝑀}𝑥(𝑠)𝑑𝑠𝑥(𝑡−ℎ(𝑡))−𝑥(𝑡−ℎ_𝑀)^𝑇×Υ₁^{𝑡𝑡−ℎ(𝑡)}𝑥(𝑠)𝑥(𝑡)((𝑡𝑑𝑠−𝑥𝑡−ℎ))_{𝑡−ℎ(𝑡)𝑡−ℎ𝑀}𝑥(𝑠)𝑥((𝑡𝑑𝑠𝑡−ℎ))−𝑥𝑡−ℎ_𝑀×^{𝑡𝑡−ℎ(𝑡)}𝑥(𝑠)𝑥(𝑡)((𝑡𝑑𝑠−𝑥𝑡−ℎ))_{𝑡−ℎ(𝑡)𝑡−ℎ𝑀}𝑥(𝑠)𝑥((𝑡𝑑𝑠𝑡−ℎ))−𝑥𝑡−ℎ_𝑀+𝜁^𝑇(𝑡)Ξ_{zero[ℎ(𝑡)]}𝜁(𝑡)−2ℎ_𝑀^𝑡_{(𝑡)𝑡−ℎ}^𝑡_𝑠𝑥^𝑇(𝑢)𝑃₃𝑥̇(𝑢)𝑑𝑢𝑑𝑠−2ℎ_𝑀^{𝑡−ℎ(𝑡)}_{𝑡−ℎ𝑀}^𝑡_𝑠𝑥^𝑇(𝑢)𝑃₄𝑥̇(𝑠)𝑑𝑢𝑑𝑠=𝜁^𝑇(𝑡)^ℎ2_𝑀^Ψ₄₁^𝑄₁^Ψ𝑇₄₁^−Ψ₄₂^Υ₁^Ψ𝑇₄₂^+Ξ_{zero[ℎ(𝑡)]Ξ4[ℎ(𝑡)]}𝜁(𝑡)−2ℎ_𝑀^𝑡_{(𝑡)𝑡−ℎ}^𝑡_𝑠𝑥^𝑇(𝑢)𝑃₃𝑥̇(𝑢)𝑑𝑢𝑑𝑠−2ℎ_𝑀^{𝑡−ℎ(𝑡)}_{𝑡−ℎ𝑀}^𝑡_𝑠𝑥^𝑇(𝑢)𝑃₄𝑥̇(𝑠)𝑑𝑢𝑑𝑠. (25)
The 𝑉̇₅(𝑡) is calculated as𝑉̇₅(𝑡)=ℎ²_𝑀2²̇𝑥(𝑡)𝑥(𝑡)^𝑇𝑄₂𝑥(𝑡)𝑥̇(𝑡)−ℎ²_𝑀2^𝑡_{𝑡−ℎ𝑀}^𝑡_𝑠𝑥(𝑢)𝑥̇(𝑢)𝑄₂𝑥(𝑢)𝑥̇(𝑢)𝑑𝑢𝑑𝑠−2ℎ_𝑀^𝑡_{(𝑡)𝑡−ℎ}^𝑡_𝑠𝑥^𝑇(𝑢)𝑃₃𝑥̇(𝑢)𝑑𝑢𝑑𝑠+2ℎ_𝑀^𝑡_{(𝑡)𝑡−ℎ}^𝑡_𝑠𝑥^𝑇(𝑢)𝑃₃𝑥̇(𝑢)𝑑𝑢𝑑𝑠₀−2ℎ_𝑀^{𝑡−ℎ(𝑡)}_{𝑡−ℎ𝑀}^𝑡_𝑠𝑥^𝑇(𝑢)𝑃₄𝑥̇(𝑠)𝑑𝑢𝑑𝑠+2ℎ_𝑀^{𝑡−ℎ(𝑡)}_{𝑡−ℎ𝑀}^𝑡_𝑠𝑥^𝑇(𝑢)𝑃₄𝑥̇(𝑠)𝑑𝑢𝑑𝑠₀=ℎ²_𝑀2²̇𝑥(𝑡)𝑥(𝑡)^𝑇𝑄₂𝑥(𝑡)𝑥̇(𝑡)−ℎ²_𝑀2^𝑡_{(𝑡)𝑡−ℎ}^𝑡_𝑠𝑥(𝑢)𝑥̇(𝑢)𝑄₂+4ℎ_𝑀0_𝑛𝑃₃𝑃₃0_𝑛𝑥(𝑢)𝑥̇(𝑢)−ℎ𝑑𝑢𝑑𝑠_2𝑀2^{𝑡−ℎ(𝑡)}_{𝑡−ℎ𝑀}^𝑡_𝑠𝑥(𝑢)𝑥̇(𝑢)𝑄₂+4ℎ_𝑀0_𝑛𝑃₄𝑃₄0_𝑛𝑥(𝑢)𝑥̇(𝑢)𝑑𝑢𝑑𝑠+2ℎ_𝑀^𝑡_{(𝑡)𝑡−ℎ}^𝑡_𝑠𝑥^𝑇(𝑢)𝑃₃𝑥̇(𝑢)𝑑𝑢𝑑𝑠+2ℎ_𝑀^{𝑡−ℎ(𝑡)}_{𝑡−ℎ𝑀}^𝑡_𝑠𝑥^𝑇(𝑢)𝑃₄𝑥̇(𝑠)𝑑𝑢𝑑𝑠. (26)Applying Lemma 1 to (26), the upper-bound of 𝑉̇₅(𝑡) can be derived as 𝑉̇₅(𝑡)≤ℎ²_𝑀2²̇𝑥(𝑡)𝑥(𝑡)^𝑇𝑄₂𝑥(𝑡)𝑥̇(𝑡)−ℎ_𝑀ℎ(𝑡)^2𝑡_{𝑡−ℎ(𝑡)}^𝑡_𝑠𝑥(𝑢)𝑥̇(𝑢)𝑑𝑢𝑑𝑠^𝑇×𝑄₂+4ℎ_𝑀0_𝑛𝑃₃𝑃₃0_𝑛^{𝑡𝑡−ℎ(𝑡)𝑡𝑠}𝑥(𝑢)𝑥̇(𝑢)−ℎ𝑑𝑢𝑑𝑠_2𝑀ℎ²_𝑀−ℎ(𝑡)^{2𝑡−ℎ(𝑡)𝑡−ℎ}_𝑀^𝑡𝑠𝑥(𝑢)𝑥̇(𝑢)𝑑𝑢𝑑𝑠^𝑇×𝑄₂+4ℎ_𝑀0_𝑛𝑃₄𝑃₄0_𝑛^{𝑡−ℎ(𝑡)𝑡−ℎ𝑀𝑡𝑠}𝑥(𝑢)𝑥̇(𝑢)𝑑𝑢𝑑𝑠+2ℎ_𝑀^𝑡_{(𝑡)𝑡−ℎ}^𝑡_𝑠𝑥^𝑇(𝑢)𝑃₃𝑥̇(𝑢)𝑑𝑢𝑑𝑠+2ℎ_𝑀^{𝑡−ℎ(𝑡)}_{𝑡−ℎ𝑀}^𝑡_𝑠𝑥^𝑇(𝑢)𝑃₄𝑥̇(𝑠)=ℎ𝑑𝑢𝑑𝑠_2𝑀2²̇𝑥(𝑡)𝑥(𝑡)^𝑇𝑄₂𝑥(𝑡)𝑥̇(𝑡)−^{𝑡(𝑡)𝑡−ℎ𝑡𝑠}𝑥(𝑢)𝑑𝑢𝑑𝑠_{𝑡𝑡−ℎ(𝑡)𝑡𝑠}𝑥̇𝑑𝑢𝑑𝑠_{𝑡−ℎ(𝑡)𝑡−ℎ𝑀𝑡𝑠}𝑥(𝑢)𝑑𝑢𝑑𝑠_{𝑡−ℎ(𝑡)𝑡−ℎ𝑀𝑡𝑠}𝑥̇(𝑢)𝑑𝑢𝑑𝑠^𝑇×^{1𝜙2(𝑡)𝑄2+4ℎ𝑀0𝑛𝑃3𝑃30𝑛02𝑛}_{02𝑛11−𝜙2(𝑡)𝑄2+4ℎ𝑀0𝑛𝑃4𝑃40𝑛}×^{𝑡(𝑡)𝑡−ℎ𝑡𝑠}𝑥(𝑢)𝑑𝑢𝑑𝑠_{𝑡𝑡−ℎ(𝑡)𝑡𝑠}𝑥̇𝑑𝑢𝑑𝑠_{𝑡−ℎ(𝑡)𝑡−ℎ𝑀𝑡𝑠}𝑥(𝑢)𝑑𝑢𝑑𝑠_{𝑡−ℎ(𝑡)𝑡−ℎ𝑀𝑡𝑠}𝑥̇(𝑢)𝑑𝑢𝑑𝑠+2ℎ_𝑀^𝑡_{(𝑡)𝑡−ℎ}^𝑡_𝑠𝑥^𝑇(𝑢)𝑃₃𝑥̇(𝑢)𝑑𝑢𝑑𝑠+2ℎ_𝑀^{𝑡−ℎ(𝑡)}_{𝑡−ℎ𝑀}^𝑡_𝑠𝑥^𝑇(𝑢)𝑃₄𝑥̇(𝑠)𝑑𝑢𝑑𝑠, (27) where 𝜙₂(𝑡)=(ℎ(𝑡)/ℎ_𝑀)² satisfies 0≤𝜙₂(𝑡)≤1. Note that when ℎ(𝑡) is zero, the terms of ∫^𝑡_{𝑡−ℎ(𝑡)}∫^𝑡_𝑠𝑥(𝑢)𝑑𝑢𝑑𝑠 and ∫^𝑡_{𝑡−ℎ(𝑡)}∫^𝑡_𝑠𝑥̇𝑑𝑢𝑑𝑠 are zero, and when ℎ(𝑡) is ℎ_𝑀, the terms of ∫^{𝑡−ℎ(𝑡)}_{𝑡−ℎ𝑀}∫^𝑡_𝑠𝑥(𝑢)𝑑𝑢𝑑𝑠 and ∫^{𝑡−ℎ(𝑡)}_{𝑡−ℎ𝑀}∫^𝑡_𝑠𝑥̇(𝑢)𝑑𝑢𝑑𝑠 are zero. Thus, inequality (27) still holds.
By the use of Lemma 2 to (27) with Υ₂≥0 defined in (13), the following inequality can be obtained: 𝑉̇₅(𝑡)≤ℎ²_𝑀2²̇𝑥(𝑡)𝑥(𝑡)^𝑇𝑄₂𝑥(𝑡)𝑥̇(𝑡)−^{𝑡(𝑡)𝑡−ℎ𝑡𝑠}𝑥(𝑢)𝑑𝑢𝑑𝑠ℎ(𝑡)𝑥(𝑡)−^𝑡_{𝑡−ℎ(𝑡)}𝑥(𝑠)𝑑𝑠_{𝑡−ℎ(𝑡)𝑡−ℎ𝑀𝑡𝑠}𝑥(𝑢)𝑑𝑢𝑑𝑠ℎ(𝑡)−ℎ_𝑀𝑥(𝑡)−^{𝑡−ℎ(𝑡)}_{𝑡−ℎ𝑀}𝑥(𝑠)𝑑𝑠^𝑇Υ₂×^{𝑡(𝑡)𝑡−ℎ𝑡𝑠}𝑥(𝑢)ℎ(𝑡)𝑥(𝑡)−𝑑𝑢𝑑𝑠_{𝑡𝑡−ℎ(𝑡)}𝑥(𝑠)𝑑𝑠_{𝑡−ℎ(𝑡)𝑡−ℎ𝑀𝑡𝑠}𝑥(𝑢)ℎ(𝑡)𝑑𝑢𝑑𝑠−ℎ_𝑀𝑥(𝑡)−^{𝑡−ℎ(𝑡)}_{𝑡−ℎ𝑀}𝑥(𝑠)𝑑𝑠+2ℎ_𝑀^𝑡_{(𝑡)𝑡−ℎ}^𝑡_𝑠𝑥^𝑇(𝑢)𝑃₃𝑥̇(𝑢)𝑑𝑢𝑑𝑠+2ℎ_𝑀^{𝑡−ℎ(𝑡)}_{𝑡−ℎ𝑀}^𝑡_𝑠𝑥^𝑇(𝑢)𝑃₄𝑥̇(𝑠)𝑑𝑢𝑑𝑠=𝜁^𝑇(𝑡)Ξ₅−Γ^𝑇_2[ℎ(𝑡)]Υ₂Γ_[2ℎ(𝑡)]𝜁(𝑡)+2ℎ_𝑀^𝑡_{(𝑡)𝑡−ℎ}^𝑡_𝑠𝑥^𝑇(𝑢)𝑃₃𝑥̇(𝑢)𝑑𝑢𝑑𝑠+2ℎ_𝑀^{𝑡−ℎ(𝑡)}_{𝑡−ℎ𝑀}^𝑡_𝑠𝑥^𝑇(𝑢)𝑃₄𝑥̇(𝑠)𝑑𝑢𝑑𝑠. (28) The time-derivative of 𝑉₆(𝑡) is obtained as follows: 𝑉̇₆(𝑡)=ℎ³_𝑀6²𝑥̇^𝑇(𝑡)𝑄₃𝑥̇(𝑡)−ℎ³_𝑀6×^𝑡_{𝑡−ℎ𝑀}^𝑡_𝑠^𝑡_𝑢𝑥̇^𝑇(𝑣)𝑄₃𝑥̇(𝑣)𝑑𝑣𝑑𝑢𝑑𝑠. (29) Using Lemma 1 to (29) yields a new upper-bound of 𝑉̇₆(𝑡) as 𝑉̇₆(𝑡)≤ℎ³_𝑀6²𝑥̇^𝑇(𝑡)𝑄₃𝑥̇(𝑡)−ℎ²_𝑀2𝑥(𝑡)−^𝑡_{𝑡−ℎ𝑀}^𝑡_𝑠𝑥(𝑢)𝑑𝑢𝑑𝑠^𝑇×𝑄₃ℎ²_𝑀2𝑥(𝑡)−^𝑡_{𝑡−ℎ𝑀}^𝑡_𝑠𝑥(𝑢)𝑑𝑢𝑑𝑠=𝜁^𝑇(𝑡)Ξ₆𝜁(𝑡). (30) By summation of (17)–(30), an upper-bound of 𝑉̇(𝑡) is obtained as follows: 𝑉̇(𝑡)≤𝜁^𝑇(𝑡)Ξ_{̇[ℎ(𝑡),ℎ(𝑡)]}−Γ^𝑇_2[ℎ(𝑡)]Υ₂Γ_2[ℎ(𝑡)]𝜁(𝑡). (31) From the dynamic equation of the system (12), zero equality 0=Θ𝜁(𝑡) can be obtained. Therefore, from Lemma 3, a stability criterion for the system (12) can be obtained as follows: Θ^⟂𝑇Ξ_{̇[ℎ(𝑡),ℎ(𝑡)]}−Γ^𝑇_2[ℎ(𝑡)]Υ₂Γ_2[ℎ(𝑡)]Θ^⟂<0. (32) By using Lemma 4 with any matrix 𝑈∈ℝ^4𝑛×9𝑛, inequality (32) is equivalent to Φ_{̇[ℎ(𝑡),ℎ(𝑡)]}=Θ^⟂𝑇Ξ_{̇[ℎ(𝑡),ℎ(𝑡)]}Θ^⟂+Θ^⟂𝑇Γ^𝑇_2[ℎ(𝑡)]Λ+Λ^𝑇Γ_2[ℎ(𝑡)]Θ^⟂Λ^𝑇Λ−Υ₂=𝐿_{̇[ℎ(𝑡),ℎ(𝑡)]}Λ^𝑇Λ−Υ₂<0, (33) where 𝐿_{̇[ℎ(𝑡),ℎ(𝑡)]} is defined in (13). Note that inequality (33) is affinely dependent on ℎ(𝑡) and ℎ̇(𝑡) where 0≤ℎ(𝑡)≤ℎ_𝑀 and |ℎ̇(𝑡)|≤ℎ_𝑑<1. So, Φ_{̇[ℎ(𝑡),ℎ(𝑡)]}<0 is satisfied if Φ_{̇[ℎ(𝑡)=0,ℎ(𝑡)=−ℎ𝑑]}Φ<0,_{̇[ℎ(𝑡)=0,ℎ(𝑡)=ℎ𝑑]}Φ<0,_{[ℎ(𝑡)=ℎ𝑀,̇ℎ(𝑡)=−ℎ𝑑]}Φ<0,_{[ℎ(𝑡)=ℎ𝑀,̇ℎ(𝑡)=ℎ𝑑]}<0. (34) This completes our proof.

Remark 6. Compared with the existing works, the main differences of Lyapunov-Krasovskii functionals are 𝑉₂(𝑡) and 𝑉₃(𝑡). In detail, 𝑉₂(𝑡) and 𝑉₃(𝑡) have augmented vectors ∫^𝑡_{𝑡−ℎ(𝑡)}𝑥̇(𝑠)𝑑𝑠, respectively. With this consideration, some new cross terms are calculated and utilized in stability condition. Furthermore, the time-varying delayed state 𝑥(𝑡−ℎ(𝑡)) and double integral terms such as ∫^𝑡_{𝑡−ℎ(𝑡)}∫^𝑡_𝑠𝑥(𝑢)𝑑𝑢𝑑𝑠 and ∫^{𝑡−ℎ(𝑡)}_{𝑡−ℎ𝑀}∫^𝑡_𝑠𝑥(𝑢)𝑑𝑢𝑑𝑠 are utilized as an element of the augmented vectors of 𝑉₁(𝑡). In next section, the results for Theorem 5 will show larger delay bounds than the works in other literature to confirm the less conservatism of proposed criterion.

3.2. 𝐻_∞ Output Tracking Controller Design

In this subsection, an 𝐻_∞ output tracking controller design method for the system (5) will be introduced based on the result of Theorem 5. For simplicity of matrix and vector representation, the scalar 𝑞 which will be used in representing dimensions of matrices and vectors is defined as 𝑞=𝑛+𝑟. For example, block entry matrices 𝑒_𝑖(𝑖=1,…,10)∈ℝ^10𝑞×𝑞 will be used in Theorem 7. The other notations are defined as 𝜁^𝑇(𝑡)=𝜉^𝑇(𝑡),𝜉^𝑇(𝑡−ℎ(𝑡)),𝜉^𝑇𝑡−ℎ_𝑀,𝜉̇^𝑇(𝑡),𝜉̇^𝑇𝑡−ℎ_𝑀,^𝑡_{𝑡−ℎ(𝑡)}𝜉^𝑇(𝑠)𝑑𝑠,_{𝑡−ℎ(𝑡)𝑡−ℎ𝑀}𝜉^𝑇(𝑠)𝑑𝑠,^𝑡_{(𝑡)𝑡−ℎ}^𝑡_𝑠𝜉^𝑇(𝑢)𝑑𝑢𝑑𝑠,_{𝑡−ℎ(𝑡)𝑡−ℎ𝑀𝑡𝑠}𝜉^𝑇(𝑢)𝑑𝑢𝑑𝑠,𝜉̇^𝑇(𝑡−ℎ(𝑡)),Θ=𝐴+𝐵𝐾𝑒^𝑇₁+𝐴_𝑑𝑒^𝑇₂−𝑒^𝑇₄,Υ₁=𝑄₁+𝑃₃𝑃₁𝑃₁0_𝑞𝑆₁𝑆^𝑇₁𝑄₁+𝑃₄𝑃₂𝑃₂0_𝑞,Υ₂=𝑄₂+4ℎ_𝑀0_𝑞𝑃₃𝑃₃0_𝑞𝑆₂𝑆^𝑇₂𝑄₂+4ℎ_𝑀0_𝑞𝑃₄𝑃₄0_𝑞,Ξ_{̇1[ℎ(𝑡),ℎ(𝑡)]}=Ψ₁₁Γ_{̇1[ℎ(𝑡),ℎ(𝑡)]}𝑅Ψ^𝑇₁₂+Ψ₁₂𝑅Γ^𝑇_{̇1[ℎ(𝑡),ℎ(𝑡)]}Ψ^𝑇₁₁,Ξ₂=Ψ₂₁𝑁Ψ^𝑇₂₁−Ψ₂₂𝑁Ψ^𝑇₂₂+Ψ₂₃𝑁Ψ^𝑇₂₄+Ψ₂₄𝑁Ψ^𝑇₂₃,Ξ_{̇3[ℎ(𝑡),ℎ(𝑡)]}=Ψ₃₁𝐺Ψ^𝑇₃₁−1−ℎ̇(𝑡)Ψ₃₂𝐺Ψ^𝑇₃₂+Ψ_{̇33[ℎ(𝑡),ℎ(𝑡)]}𝐺Ψ^𝑇₃₄+Ψ₃₄𝐺Ψ^𝑇_{̇33[ℎ(𝑡),ℎ(𝑡)]}+1−ℎ̇(𝑡)Ψ₃₅𝑀Ψ^𝑇₃₅−Ψ₃₆𝑀Ψ^𝑇₃₆+Ψ_{̇37[ℎ(𝑡),ℎ(𝑡)]}𝐺Ψ^𝑇₃₈+Ψ₃₈𝐺Ψ^𝑇_{̇37[ℎ(𝑡),ℎ(𝑡)]},Ξ_4[ℎ(𝑡)]=ℎ²_𝑀Ψ₄₁𝑄₁Ψ^𝑇₄₁Ψ₄₂Υ₁Ψ^𝑇₄₂+ℎ_𝑀𝑒₁𝑃₁𝑒^𝑇₁−𝑒₂𝑃₁𝑒^𝑇₂+𝑒₂𝑃2𝑒^𝑇₂−𝑒₃𝑃₂𝑒^𝑇₃+ℎ_𝑀ℎ(𝑡)𝑒₁𝑃₃𝑒^𝑇₁−ℎ_𝑀−ℎ(𝑡)𝑒₁𝑃₄𝑒^𝑇₁,Ξ₅=ℎ²_𝑀2²Ψ₅𝑄₂Ψ^𝑇₅,Ξ₆=ℎ³_𝑀6²𝑒₄𝑄₃𝑒^𝑇₄−Ψ₆𝑄₃Ψ^𝑇₆,Ξ₇=𝛼𝑒₁+𝑒₄𝐴₁𝑋+𝐵𝑌𝑒^𝑇₁+𝐴_𝑑𝑒^𝑇₂−𝑒^𝑇₄+𝑒₁+𝑒₄𝐴₁𝑋+𝐵𝑌𝑒^𝑇₁+𝐴_𝑑𝑒^𝑇₂−𝑒^𝑇₄^𝑇,Ξ=Ξ_{̇1[ℎ(𝑡),ℎ(𝑡)]}+Ξ₂+Ξ_{̇3[ℎ(𝑡),ℎ(𝑡)]}+Ξ_4[ℎ(𝑡)]+Ξ₅+Ξ₆+Ξ₇,𝐿_{̇1[ℎ(𝑡),ℎ(𝑡)]}=Ξ_{̇[ℎ(𝑡),ℎ(𝑡)]}+Γ^𝑇_2[ℎ(𝑡)]Λ+Λ^𝑇Γ_2[ℎ(𝑡)],𝐿₂=𝐶₁,−𝐻𝑋𝑒^𝑇₁+𝐵₁𝑌𝑒^𝑇₁+𝐶₂,0_𝑠×𝑝𝑋𝑒^𝑇₂,Φ_{̇[ℎ(𝑡),ℎ(𝑡)]}=𝐿_{̇1[ℎ(𝑡),ℎ(𝑡)]}𝐵_𝑤Λ^𝑇𝐿^𝑇₂⋆−𝛾²𝐼_𝑝+𝑟00⋆⋆−Υ₂0⋆⋆⋆−𝐼_𝑠, (35) and other notations are the same ones as (13). Now, the following theorem is given by the second result.

Theorem 7. For given scalars 0≤ℎ_𝑀, 0≤ℎ_𝑑<1, and 𝛼>0, the augmented system (5) achieves the 𝐻_∞ output tracking performance 𝛾 for 0≤ℎ(𝑡)≤ℎ_𝑀 and |ℎ̇(𝑡)|≤ℎ_𝑑, if there exist positive definite matrices 𝑅=[𝑅_𝑖𝑗]_7×7∈ℝ^7𝑞×7𝑞, 𝑋=𝑅₁₁∈ℝ^𝑞×𝑞𝑁∈ℝ^4𝑞×4𝑞, 𝐺∈ℝ^4𝑞×4𝑞, 𝑄₁∈ℝ^2𝑞×2𝑞, 𝑄₂∈ℝ^2𝑞×2𝑞, and 𝑄₃∈ℝ^𝑞×𝑞, symmetric matrices 𝑃_𝑖∈ℝ^𝑞×𝑞(𝑖=1,…,4), and any matrices 𝑆₁∈ℝ^2𝑞×2𝑞, 𝑆₂∈ℝ^2𝑞×2𝑞, Λ∈ℝ^4𝑞×9𝑞, and 𝑌∈ℝ^𝑚×𝑞 satisfying the following LMIs: Φ_𝑗,𝑘<0,𝑗,𝑘=1,2,Υ_𝑖≥0,(𝑖=1,2), (36) where Φ_𝑗,𝑘 are the four vertices of Φ_{̇[ℎ(𝑡),ℎ(𝑡)]} with the bounds of ℎ(𝑡) and ℎ̇(𝑡), that is ℎ_𝑀 and ℎ_𝐷 when 𝑗=𝑘=1, ℎ_𝑀 and −ℎ_𝐷 when 𝑗<𝑘, 0 and ℎ_𝐷 when 𝑗>𝑘, and 0 and −ℎ_𝐷 when 𝑗=𝑘=2. Moreover, if the above conditions are feasible, a desired 𝐻_∞ tracking controller gain matrix is obtained by 𝐾=𝐾₁,𝐾₂=𝑌𝑋⁻¹.

Proof. For positive definite matrices 𝑅∈ℝ^7𝑞×7𝑞, 𝑁∈ℝ^4𝑞×4𝑞, 𝐺∈ℝ^4𝑞×4𝑞, 𝑄₁∈ℝ^2𝑞×2𝑞, 𝑄₂∈ℝ^2𝑞×2𝑞, and 𝑄₃∈ℝ^𝑞×𝑞, let us consider the following Lyapunov-Krasovskii functional candidate as 𝑉(𝑡)=⁶_𝑖=1𝑉_𝑖(𝑡), (37) where 𝑉₁(𝑡)=Ω^𝑇₁(𝑡)𝑅Ω₁(𝑡),𝑉₂(𝑡)=^𝑡_{𝑡−ℎ𝑀}Ω^𝑇₂(𝑡,𝑠)𝑁Ω₂()𝑉𝑡,𝑠𝑑𝑠,₃(𝑡)=^𝑡_{𝑡−ℎ(𝑡)}Ω^𝑇₂(𝑡,𝑠)𝐺Ω₂()+𝑡,𝑠𝑑𝑠_{𝑡−ℎ(𝑡)𝑡−ℎ𝑀}Ω^𝑇₂(𝑡,𝑠)𝑀Ω₂()𝑉𝑡,𝑠𝑑𝑠,₄(𝑡)=ℎ_𝑀^𝑡_{𝑡−ℎ𝑀}^𝑡_𝑠𝜉(𝑢)𝜉̇(𝑢)^𝑇𝑄₁𝜉(𝑢)𝜉̇(𝑢)𝑉𝑑𝑢𝑑𝑠,₅(𝑡)=ℎ²_𝑀2^𝑡_{𝑡−ℎ𝑀}^𝑡_𝑠^𝑡_𝑢𝜉(𝑣)𝜉̇(𝑣)^𝑇𝑄₂𝜉(𝑣)𝜉̇(𝑣)𝑉𝑑𝑣𝑑𝑢𝑑𝑠,₆(𝑡)=ℎ³_𝑀6^𝑡_{𝑡−ℎ𝑀}^𝑡_𝑠^𝑡_𝑢^𝑡_𝑣𝜉̇^𝑇(𝜆)𝑄₃𝜉̇(𝜆)𝑑𝜆𝑑𝑣𝑑𝑢𝑑𝑠, (38) with Ω₁(𝑡)=[𝜉^𝑇(𝑡),𝜉^𝑇(𝑡−ℎ(𝑡)),𝜉^𝑇(𝑡−ℎ_𝑀),∫^𝑡_{𝑡−ℎ(𝑡)}𝜉^𝑇(𝑠)𝑑𝑠,∫^𝑡_{𝑡−ℎ𝑀}𝜉^𝑇(𝑠)𝑑𝑠,∫^𝑡_{𝑡−ℎ(𝑡)}∫^𝑡_𝑠𝜉^𝑇(𝑢)𝑑𝑢𝑑𝑠, ∫^𝑡_{𝑡−ℎ𝑀}∫^𝑡_𝑠𝜉^𝑇(𝑢)𝑑𝑢𝑑𝑠]^𝑇, and Ω₂(𝑡,𝑠)=[𝜉^𝑇(𝑠),𝜉̇^𝑇(𝑠),∫^𝑡_𝑠𝜉̇^𝑇(𝑢)𝑑𝑢,∫^𝑡_𝑠𝜉^𝑇(𝑢)𝑑𝑢]^𝑇. By using similar method in the proof of Theorem 5, an upper-bound of 𝑉̇(𝑡) can be obtained as follows: 𝑉̇(𝑡)≤𝜁^𝑇(𝑡)Ξ_{̇[ℎ(𝑡),ℎ(𝑡)]}−Γ^𝑇_2[ℎ(𝑡)]Υ₂Γ_2[ℎ(𝑡)]𝜁(𝑡), (39) with Υ₁Υ≥0,₂≥0. (40) Since zero equality Θ𝜁(𝑡)+𝐵_𝑤𝑣(𝑡)=0 can be obtained from the dynamic equation of the augmented system (5), the following zero equality for any matrix 𝑍∈ℝ^𝑞×𝑞 is considered: 0=2𝜁^𝑇(𝑡)𝑒^𝑇₁+𝑒^𝑇₄^𝑇𝑍Θ𝜁(𝑡)+𝐵_𝑤𝑣(𝑡). (41) For 𝑅=[𝑅_𝑖𝑗]_7×7∈ℝ^7𝑞×7𝑞 and a given scalar 𝛼>0, let us define 𝑍=𝛼𝑅₁₁. By adding (41) into (39), a new upper-bound of 𝑉̇(𝑡) can be 𝑉̇(𝑡)≤𝜁^𝑇(𝑡)Ξ_{̇[ℎ(𝑡),ℎ(𝑡)]}−Γ^𝑇_2[ℎ(𝑡)]Υ₂Γ_2[ℎ(𝑡)]𝜁(𝑡)+2𝜁^𝑇(𝑡)𝑒^𝑇₁+𝑒^𝑇₄^𝑇𝛼𝑅₁₁Θ𝜁(𝑡)+𝐵_𝑤𝑣(𝑡). (42) Since 𝑉(𝑡)|_𝑡=0=0 and 𝑉(𝑡)|_𝑡=∞≥0 are satisfied under the zero initial condition, 𝐻_∞ performance index in (7) leads to the following inequality: 𝐽=^∞₀𝑒^𝑇(𝑠)𝑒(𝑠)−𝛾²𝑣^𝑇(𝑠)𝑣(𝑠=)𝑑𝑠_∞0𝑒^𝑇(𝑠)𝑒(𝑠)−𝛾²𝑣^𝑇(𝑠)𝑣(𝑠)+𝑉̇(𝑠)𝑑𝑠−^∞₀𝑉̇(𝑠)=𝑑𝑠_∞0𝑒^𝑇(𝑠)𝑒(𝑠)−𝛾²𝑣^𝑇(𝑠)𝑣(𝑠)+𝑉̇(𝑠−𝑉(𝑡)|)𝑑𝑠_𝑡=∞+𝑉(𝑡)|_𝑡=0≤^∞₀𝑒^𝑇(𝑠)𝑒(𝑠)−𝛾²𝑣^𝑇(𝑠)𝑣(𝑠)+𝑉̇(𝑠)𝑑𝑠<0. (43) If inequality (43) is satisfied, the augmented system (5) achieves the 𝐻_∞ output tracking performance 𝛾 under the obtained 𝐻_∞ tracking controller (4). With 𝑒^𝑇(𝑡)𝑒(𝑡)=^𝜉(𝑡)_{𝜉(𝑡−ℎ(𝑡))}^{𝑇𝐶𝑇1}_𝐶𝑇2^𝐶𝑇1_𝐶𝑇2^{𝑇𝜉(𝑡)}_{𝜉(𝑡−ℎ(𝑡))}, (43) is equivalent to the following inequality: 𝜁^𝑇(𝑡)Ξ_{̇[ℎ(𝑡),ℎ(𝑡)]}−Γ^𝑇_2[ℎ(𝑡)]Υ₂Γ_2[ℎ(𝑡)]𝜁(𝑡)+2𝜁^𝑇(𝑡)𝑒^𝑇₁+𝑒^𝑇₄^𝑇𝛼𝑅₁₁Θ𝜁(𝑡)+𝐵_𝑤𝑣(𝑡+)𝜉(𝑡)𝜉(𝑡−ℎ(𝑡))^𝑇𝐶^𝑇₁𝐶^𝑇₂𝐶^𝑇₁𝐶^𝑇₂^𝑇𝜉(𝑡)𝜉(𝑡−ℎ(𝑡))−𝛾²𝑣^𝑇(𝑡)𝑣(𝑡)=𝜁(𝑡)𝑣(𝑡)^𝑇Π_{̇1[ℎ(𝑡),ℎ(𝑡)]}Π₂Π^𝑇₂Π₃𝜁(𝑡)𝑣(𝑡)<0, (44) where Π_{̇1[ℎ(𝑡),ℎ(𝑡)]}=Ξ_{̇[ℎ(𝑡),ℎ(𝑡)]}−Γ^𝑇_2[ℎ(𝑡)]Υ₂Γ_2[ℎ(𝑡)]+𝛼𝑒^𝑇₁+𝑒^𝑇₄^𝑇𝑅₁₁Θ+𝛼Θ^𝑇𝑅₁₁𝑒^𝑇₁+𝑒^𝑇₄+𝐶₁𝑒^𝑇₁+𝐶₂𝑒^𝑇₂^𝑇𝐶₁𝑒^𝑇₁+𝐶₂𝑒^𝑇₂,Π₂=𝛼𝑒^𝑇₁+𝑒^𝑇₄^𝑇𝑅₁₁𝐵_𝑤,Π₃=−𝛾²𝐼_𝑝+𝑟. (45) Therefore, inequality (44) is equivalent to the following condition: Π_{̇1[ℎ(𝑡),ℎ(𝑡)]}Π₂Π^𝑇₂Π₃<0. (46) Let us define 𝑋=𝑅⁻¹₁₁>0 and Σ_𝑖=diag{𝑋,…,𝑋_𝑖}∈ℝ^{𝑖𝑞×𝑖𝑞}. For example, Σ₁=𝑋 and Σ₂=diag{𝑋,𝑋}. And other notations are defined as 𝑅=Σ^𝑇₇𝑅Σ₇, 𝑁=Σ^𝑇₄𝑁Σ₄, 𝐺=Σ^𝑇₄𝐺Σ₄, 𝑀=Σ^𝑇₄𝑀Σ₄, 𝑄₁=Σ^𝑇₂𝑄₁Σ₂, 𝑄₂=Σ^𝑇₂𝑄₂Σ₂, 𝑄₃=Σ^𝑇₁𝑄₃Σ₁,𝑃₁=Σ^𝑇₁𝑃₁Σ₁, 𝑃₂=Σ^𝑇₁𝑃₁Σ₁, 𝑃₃=Σ^𝑇₁𝑃₃Σ₁, 𝑃₄=Σ^𝑇₁𝑃₄Σ₁, 𝑆₁=Σ^𝑇₂𝑆₁Σ₂, 𝑆₂=Σ^𝑇₂𝑆₁Σ₂, and 𝑌=𝐾𝑋. By pre- and post-multiplying (40) and (46), respectively, by Σ₄ and diag{Σ₁₀,𝐼_𝑝+𝑟}, the following inequalities can be obtained: 0≤Σ₄Υ₁Σ₄=Υ₁,0≤Σ₄Υ₂Σ₄=Υ₂,Σ₁₀0_{10𝑞×(𝑝+𝑟)}0_{(𝑝+𝑟)×10𝑞}𝐼_𝑝+𝑟Π_{̇1[ℎ(𝑡),ℎ(𝑡)]}Π₂Π^𝑇₂Π₃×Σ₁₀0_{10𝑞×(𝑝+𝑟)}0_{(𝑝+𝑟)×10𝑞}𝐼_𝑝+𝑟=Π_{̇1[ℎ(𝑡),ℎ(𝑡)]}Π₂Π^𝑇₂Π₃<0, (47)(48) where Π_{̇1[ℎ(𝑡),ℎ(𝑡)]}=Ξ_{̇[ℎ(𝑡),ℎ(𝑡)]}−Γ^𝑇_[ℎ(𝑡)]Υ₂Γ_[ℎ(𝑡)]+𝐶₁,−𝐻𝑋𝑒^𝑇₁+𝐷₁𝑌𝑒^𝑇₁+𝐶₂,0_𝑠×𝑝𝑋𝑒^𝑇₂^𝑇×𝐶₁,−𝐻𝑋𝑒^𝑇₁+𝐷₁𝑌𝑒^𝑇₁+𝐶₂,0_𝑠×𝑝𝑋𝑒^𝑇₂Π₂=𝛼𝑒^𝑇₁+𝑒^𝑇₄^𝑇𝐵_𝑤,Π₃=−𝛾²𝐼_𝑝+𝑟. (49) Also, by using Schur complements and Lemma 4 with any matrix Λ∈ℝ^{4𝑞×(10𝑞+𝑝+𝑟)}, inequality (47) is equivalent to Φ_{̇[ℎ(𝑡),ℎ(𝑡)]}=𝐿_{̇1[ℎ(𝑡),ℎ(𝑡)]}𝐵_𝑤Λ^𝑇𝐿^𝑇₂⋆−𝛾²𝐼_𝑝+𝑟00⋆⋆−Υ₂0⋆⋆⋆−𝐼_𝑠<0, (50) where 𝐿_{̇1[ℎ(𝑡),ℎ(𝑡)]} and 𝐿₂ are defined in (35). Note that inequality (50) is affinely dependent on ℎ(𝑡) and ℎ̇(𝑡). Therefore, Φ_{̇[ℎ(𝑡),ℎ(𝑡)]}<0 holds if Φ_{̇[ℎ(𝑡)=0,ℎ(𝑡)=−ℎ𝑑]}<0,Φ_{̇[ℎ(𝑡)=0,ℎ(𝑡)=ℎ𝑑]}<0,Φ_{[ℎ(𝑡)=ℎ𝑀,̇ℎ(𝑡)=−ℎ𝑑]}<0,Φ_{[ℎ(𝑡)=ℎ𝑀,̇ℎ(𝑡)=ℎ𝑑]}<0. (51) This completes our proof.

Remark 8. This subsection concludes the criterion of 𝐻_∞ output tracking controller design for linear systems with time-varying delays and disturbances. To obtain an 𝐻_∞ output tracking controller gain which provides enhanced feasible region, the authors introduce the improved delay-dependent stability criterion as shown in Theorem 5. Then, based on the results of Theorem 5, the designing method of 𝐻_∞ output tracking controller gain was introduced in Theorem 7.

Remark 9. Theorems 5 and 7 have some restriction such as ℎ_𝑑<1. In order to remove the restrictive condition ℎ_𝑑<1 in Theorem 5, 𝑉₁(𝑡) and 𝑉₃(𝑡) of Lyapunov-Krasovskii functional (15) should be changed as follows: 𝑉₁(𝑡)=Ω^𝑇₁(𝑡)𝑅Ω₁(𝑡),𝑉₃(𝑡)=^𝑡_{𝑡−ℎ(𝑡)}𝑥(𝑠)_𝑡𝑠𝑥̇(𝑢)𝑑𝑢_𝑡𝑠𝑥(𝑢)𝑑𝑢^𝑇𝐺𝑥(𝑠)_𝑡𝑠𝑥̇(𝑢)𝑑𝑢_𝑡𝑠𝑥(𝑢)+𝑑𝑢𝑑𝑠_{𝑡−ℎ(𝑡)𝑡−ℎ𝑀}𝑥(𝑠)_𝑡𝑠𝑥̇(𝑢)𝑑𝑢_𝑡𝑠𝑥(𝑢)𝑑𝑢^𝑇𝑀𝑥(𝑠)_𝑡𝑠𝑥̇(𝑢)𝑑𝑢_𝑡𝑠𝑥(𝑢)𝑑𝑢𝑑𝑠, (52) where 𝑅∈ℝ^6𝑛×6𝑛>0, 𝐺∈ℝ^3𝑛×3𝑛>0, 𝑀∈ℝ^3𝑛×3𝑛>0, and Ω₁(𝑡)=[𝑥^𝑇(𝑡),𝑥^𝑇(𝑡−ℎ_𝑀),∫^𝑡_{𝑡−ℎ(𝑡)}𝑥^𝑇(𝑠)𝑑𝑠,∫^𝑡_{𝑡−ℎ𝑀}𝑥^𝑇(𝑠)𝑑𝑠,∫^𝑡_{𝑡−ℎ(𝑡)}∫^𝑡_𝑠𝑥^𝑇(𝑢)𝑑𝑢𝑑𝑠, ∫^𝑡_{𝑡−ℎ𝑀}∫^𝑡_𝑠𝑥^𝑇(𝑢)𝑑𝑢𝑑𝑠]^𝑇. Then, by utilizing the following vector, 𝜁(𝑡)=𝑥^𝑇(𝑡),𝑥^𝑇(𝑡−ℎ(𝑡)),𝑥^𝑇𝑡−ℎ_𝑀,𝑥̇^𝑇(𝑡),𝑥̇^𝑇𝑡−ℎ_𝑀,^𝑡_{(𝑡)𝑡−ℎ}𝑥^𝑇(𝑠)𝑑𝑠,_{𝑡−ℎ(𝑡)𝑡−ℎ𝑀}𝑥^𝑇(𝑠)𝑑𝑠,^𝑡_{(𝑡)𝑡−ℎ}^𝑡_𝑠𝑥^𝑇(𝑢)𝑑𝑢𝑑𝑠,_{𝑡−ℎ(𝑡)𝑡−ℎ𝑀𝑡𝑠}𝑥^𝑇(𝑢)𝑑𝑢𝑑𝑠^𝑇, (53) as an augmented vector in deriving a stability condition, condition ℎ_𝑑<1 can be removed. Similarly, in Theorem 7, the restrictive condition ℎ_𝑑<1 can be also removed by using the following 𝑉₁(𝑡) and 𝑉₃(𝑡) of Lyapunov-Krasovskii functional (37) with positive matrices 𝑅∈ℝ^6𝑞×6𝑞, 𝐺∈ℝ^3𝑞×3𝑞, and 𝑀∈ℝ^3𝑞×3𝑞 as 𝑉₁(𝑡)=Ω^𝑇₁(𝑡)𝑅Ω₁(𝑡),𝑉₃(𝑡)=^𝑡_{𝑡−ℎ(𝑡)}𝜁(𝑠)_𝑡𝑠𝜁̇(𝑢)𝑑𝑢_𝑡𝑠𝜁(𝑢)𝑑𝑢^𝑇𝐺𝜁(𝑠)_𝑡𝑠𝜁̇(𝑢)𝑑𝑢_𝑡𝑠𝜁(𝑢)+𝑑𝑢𝑑𝑠_{𝑡−ℎ(𝑡)𝑡−ℎ𝑀}𝜁(𝑠)_𝑡𝑠𝜁̇(𝑢)𝑑𝑢_𝑡𝑠𝜁(𝑢)𝑑𝑢^𝑇𝑀𝜁(𝑠)_𝑡𝑠𝜁̇(𝑢)𝑑𝑢_𝑡𝑠𝜁(𝑢)𝑑𝑢𝑑𝑠, (54) where Ω₁(𝑡)=[𝜉^𝑇(𝑡),𝜉^𝑇(𝑡−ℎ_𝑀),∫^𝑡_{𝑡−ℎ(𝑡)}𝜉^𝑇(𝑠)𝑑𝑠,∫^𝑡_{𝑡−ℎ𝑀}𝜉^𝑇(𝑠)𝑑𝑠,∫^𝑡_{𝑡−ℎ(𝑡)}∫^𝑡_𝑠𝜉^𝑇(𝑢)𝑑𝑢𝑑𝑠, ∫^𝑡_{𝑡−ℎ𝑀}∫^𝑡_𝑠𝜉^𝑇(𝑢)𝑑𝑢𝑑𝑠].

4. Numerical Examples

In this section, three numerical examples demonstrate the effectiveness of the proposed criteria.

Example 1. Consider the system (12) with following parameters: 𝐴=−200−0.9,𝐴_𝑑=.−10−1−1 (55) The results for maximum upper-bound of ℎ(𝑡) with different ℎ_𝑑 are listed in Table 1 which conducts the comparison of the obtained results by Theorem 5 with the previous results. Table 1 shows that when ℎ_𝑑 is bigger, the maximum upper-bound of ℎ(𝑡) is smaller. It means that ℎ_𝑑 affects the stability region of the system. Moreover, from the results in Table 1, it can be confirmed that Theorem 5 gives less conservative results than the existing results in [11–16].

Table 1 

The maximum ℎ_𝑀 in Example 1.

Example 2. Consider the system (5) with the following parameters: 𝐴=−0.5000.1,𝐴_𝑑=,,𝐶−1−0.30.2−1𝐵=1000₁=𝐶₂=1001,𝐵_𝑤,𝐵=0.51001₁=.1001 (56) Also, the reference model is given by 𝑥̇_𝑟(𝑡)=−𝑥_𝑟(𝑡)(𝑡),𝑧+𝑟_𝑟(𝑡)=0.5𝑥_𝑟(𝑡). (57) Disturbances are assumed to be 𝑤₁(𝑡)=sin(8𝑡),5≤𝑡≤8,0,otherwise,𝑤₂(𝑡)=cos(12𝑡),6≤𝑡≤9,0,otherwise. (58) Reference signal 𝑟^𝑇(𝑡)=[𝑟^𝑇₁(𝑡),𝑟^𝑇₂(𝑡)] is defined as 𝑟₁(𝑡)=sin(0.1𝑡),𝑟₂(𝑡)=cos(0.1𝑡). (59) In this example, Theorem 7 is used to obtain feedback controller gain 𝐾 for the system (5) with (56). By applying Theorem 7, the feedback controller gain, 𝐾=,−1.8931−0.00570.498300.0455−1.9317−0.00030.5000 (60) can be obtained with minimum 𝛾=0.1068 when ℎ_𝑀=0.1, ℎ_𝑑=0.5, and the tuning parameter 𝛼=1. Figure 1 illustrates the trajectories of outputs of the system (5) with disturbances (58) and time-delay ℎ(𝑡)=0.5ℎ_𝑀(cos(2ℎ_𝑑𝑡/ℎ_𝑀)+1) in the 3D plot. The simulation result in Figure 1 shows that the 𝑧(𝑡) follows the 𝑧_𝑟(𝑡) under the obtained 𝐻_∞ output tracking controller. Also, to confirm the results in detail, Figure 2 is added to show the trajectories of 𝑧(𝑡) and 𝑧_𝑟(𝑡) with disturbances 𝑤(𝑡) in 2D plot.

Figure 1 

Trajectories of outputs 𝑧(𝑡) and 𝑧𝑟(𝑡) for Example 2.

Figure 2 

Trajectories of 𝑧(𝑡), 𝑧𝑟(𝑡), and 𝑤(𝑡) in 2D plot for Example 2.

Example 3. Let us consider the system (5) as follows: 𝐴=−0.600−0.5,𝐴_𝑑=,1,𝐶−0.1−0.1−0.3−1𝐵=−1₁=𝐶₂=11,𝐵_𝑤=10,𝐵₁=1. (61) Moreover, the reference model is given by 𝑥̇_𝑟(𝑡)=−𝑥_𝑟(𝑡)(𝑡),𝑧+𝑟_𝑟(𝑡)=0.5𝑥_𝑟(𝑡). (62) Disturbances are assumed to be 𝑤(𝑡)=1,15≤𝑡≤20,0,otherwise. (63) The reference signal 𝑟(𝑡) is as follows: 𝑟(𝑡)=1,10≤𝑡≤40,0,otherwise. (64) And assume that 𝑥(0)=[0,0]^𝑇 and 𝑥_𝑟(0)=0. By applying Theorem 7, the feedback controller gain, 𝐾=−1.9454−2.01230.4929, (65) can be obtained with minimum 𝛾=0.1020 when 𝛼=1, ℎ_𝑀=0.1, and ℎ_𝑑=0.2. Figure 3 shows the outputs of the systems when the time-delay is ℎ(𝑡)=0.05(sin(4𝑡)+1) which satisfied with 0≤ℎ(𝑡)≤0.1 and |ℎ̇(𝑡)|<0.2. By calculation, ‖𝑒(𝑡)‖₂=0.0794 and ‖𝑣(𝑡)‖₂=5.9160, which yields ‖𝑒(𝑡)‖₂/‖𝑣(𝑡)‖₂=0.0134<𝛾. This results show the effectiveness of the proposed 𝐻_∞ output tacking controller design.

Figure 3 

Outputs 𝑧(𝑡) and 𝑧𝑟(𝑡) with disturbances 𝑤(𝑡) for Example 3.

5. Conclusions

This paper has investigated the problem of stability analysis and 𝐻_∞ output tracking control for linear systems with time-varying delays and disturbances. Main results were made up of two subsections which were stability analysis criterion and 𝐻_∞ output tracking controller design criterion. First, in Theorem 5, the stability criterion for the nominal systems with time-varying delays was derived by utilizing the newly augmented Lyapunov-Krasovskii functional and by using reciprocally convex approach. Second, Theorem 5 was extended to the problem of 𝐻_∞ output tracking controller design for the augmented system by using the 𝐻_∞ sense in Theorem 7. Finally, three examples were given to illustrate the effectiveness of the presented criteria. Based on these results, future research will focus on 𝐻_∞ output tracking control for various systems such as Markovian jump systems [26, 27], fault diagnosis problem [28], and discrete-time descriptor systems [29].

Conflict of Interests

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

Acknowledgments

This research was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (2008-0062611) and by a Grant of the Korea Healthcare Technology R & D Project, Ministry of Health and Welfare, Republic of Korea (A100054).