Delay-Dependent Stability Criteria for Systems with Interval Time-Varying Delay

Abstract

This paper is concerned with robust stability of uncertain linear systems with interval time-varying delay. The time-varying delay is assumed to belong to an interval, which means that the derivative of the time-varying delay has an upper bound or a restriction. On other occasions, if we do not take restriction on the derivative of the time-varying delay into consideration, it allows the delay to be a fast time-varying function. The uncertainty under consideration includes a polytopic-type uncertainty and a linear fractional norm-bounded uncertainty. In order to obtain much less conservative results, a new Lyapunov-Krasovskii functional, which makes use of the information of both the lower and upper bounds of the interval time-varying delay, is proposed to derive some new stability criteria. Numerical examples are given to demonstrate the effectiveness of the proposed stability criteria.

1. Introduction

Time delays are frequently encountered in many fields of science and engineering, including communication network, manufacturing systems, biology, economy, and other areas. During the last two decades, the problem of stability of linear time-delay systems has been the subject of considerable research efforts. Many significant results have been reported in the literature. For the recent progress, the reader is referred to Gu et al. [1] and the references therein.

With the development of networked control technology, increasing attention has been paid to the study of stability analysis and controller design of networked control systems (NCSs) due to their low cost, simple installation and maintenance, and high reliability. For the NCSs, the sampling data and controller signals are transmitted through a network. As a result, it leads to a network-induced delay in a networked control closed-loop system. The existence of such kind of delay in a network-based control loop may induce instability or poor performance of NCSs. As pointed out by Yue et al. [2], NCSs are typical systems with interval time-varying delay. In fact, we consider the following system controlled through a network 𝑥̇(𝑡)=𝐴𝑥(𝑡)+𝐵𝑢(𝑡),(1) where 𝑥(𝑡)∈𝑅^𝑛 is the state vector, and 𝑢(𝑡)∈𝑅^𝑝 is the input vector. In the presence of the control network, which is shown in Figure 1, data transfer between the controller and the remote system, for example, sensors and actuators in a distributed control system will induce network delay in addition to the controller proceeding delay. First, since there exists the communication delay 𝜏^sc between the sensor and the controller and computational delay 𝜏^𝑐 in the controller, which is shown in Figure 1, the following control law is employed for the system (1) 𝑢𝑡⁺=𝐾_𝑝𝑥𝑡−𝜏^sc_𝑘−𝜏^𝑐_𝑘,𝑡∈𝑘ℎ+𝜏^sc_𝑘+𝜏^𝑐_𝑘,,∀𝑘∈𝑁,(2) where ℎ is the sampling period, 𝑁 is the set of nonnegative integers, and 𝐾 is the controller gain to be determined. Second, substituting (2) into (1) yields the closed-loop system by considering the communication delay 𝜏^ca between the controller and the actuator 𝑥̇(𝑡)=𝐴𝑥(𝑡)+𝐵𝐾_𝑝𝑥(𝑘ℎ),𝑡∈𝑘ℎ+𝜏_𝑘,(𝑘+1)ℎ+𝜏_𝑘+1,∀𝑘∈𝑁,(3) where the time-delay 𝜏_𝑘=𝜏^sc_𝑘+𝜏^𝑐_𝑘+𝜏^ca_𝑘>0 denotes the time from the instant 𝑘ℎ when sensor nodes sample sensor data from a plant to the instant when actuators transfer data to the plant. Since 𝑥(𝑘ℎ)=𝑥(𝑡−(𝑡−𝑘ℎ)), defining 𝜏(𝑡)=𝑡−𝑘ℎ, 𝑡∈[𝑘ℎ+𝜏_𝑘,(𝑘+1)ℎ+𝜏_𝑘+1),forall𝑘∈𝑁, rewrite (3) as 𝑥̇(𝑡)=𝐴𝑥(𝑡)+𝐵𝐾_𝑝𝑥(𝑡−𝜏(𝑡)),(4) where 𝜏(𝑡) is piecewise linear with derivative .𝜏(𝑡)=1 for 𝑡≠𝑘ℎ+𝜏_𝑘, and 𝜏(𝑡) is discontinuous at the points 𝑡=𝑘ℎ+𝜏_𝑘,forall𝑘∈𝑁. It is clear that 𝜏_𝑘≤𝜏(𝑡)≤ℎ+𝜏_𝑘+1 for 𝑡∈[𝑘ℎ+𝜏_𝑘,(𝑘+1)ℎ+𝜏_𝑘+1), for  all 𝑘∈𝑁.

Figure 1 

A networked control system.

Let 𝜏₁∶=min_𝑘∈𝑁𝜏_𝑘>0 and 𝜏₂∶=max_𝑘∈𝑁{ℎ+𝜏_𝑘+1}. Then we have 0<𝜏₁≤𝜏(𝑡)≤𝜏₂.(5)

The system (3) is equivalent to the linear system (4) with interval time-varying delay described by (5). It should be pointed out that 𝜏₁>0 is essential for NCSs.

To cover the routine case of 0≤𝜏(𝑡)≤𝜏₂, we consider 𝜏(𝑡) as a uniformly continuous time-varying function satisfying 0≤𝜏₁≤𝜏(𝑡)≤𝜏₂.

Throughout this paper, we will analyze the following two scenarios of the time-varying delay 𝜏(𝑡).

Case 1. 𝜏(𝑡) is a differentiable function satisfying 0≤𝜏₁≤𝜏(𝑡)≤𝜏₂,|||.𝜏(𝑡)|||≤𝜇,∀𝑡≥0.(6)

Case 2. 𝜏(𝑡) is a continuous function satisfying 0≤𝜏₁≤𝜏(𝑡)≤𝜏₂,∀𝑡≥0,(7) where 𝜏₁ and 𝜏₂ are the lower and upper delay bounds, respectively, and 𝜏₁,𝜏₂ and 𝜇 are constants.

Remark 1. When 𝜏₁=0, the interval delay becomes routine delay. When 𝜇 is zero, that is, 𝜏₁=𝜏₂, the time-varying delay becomes constant delay, Case 1 is a special case of the Case 2. We may obtain a less conservative result using Case 1 than that using Case 2. However, if the time-varying delay is not differentiable, or information about derivative of time-varying delay is absent, only Case 2 can be used to handle the situation.

Notation 1. 𝑅^𝑛 denotes the 𝑛-dimensional Euclidean space, 𝑅^𝑛×𝑚 is the set of 𝑛×𝑚 real matrices, 𝐼 is the identity matrix of appropriate dimensions, the notation 𝑋>0 (resp.,  𝑋≥0), for 𝑋∈𝑅^𝑛×𝑛 means that the matrix 𝑋 is real positive definite (resp., positive semidefinite). For an arbitrary matrix 𝐵 and two symmetric matrices 𝐴 and 𝐶, ^𝐴𝐵_∗𝐶 denotes a symmetric matrix, where ∗ denotes a block matrix entry implied by symmetry.

2. System Descriptions and Preliminaries

Let 𝐵=𝐵𝐾_𝑝 in (4), we have the following linear system with interval time-varying delay 𝑥̇(𝑡)=𝐴𝑥(𝑡)+𝐵𝑥(𝑡−𝜏(𝑡)),𝑥(𝑡)=𝜙(𝑡),𝑡∈−𝜏₂,−𝜏₁,(8) where 𝐴 and 𝐵 are the constant matrices with appropriate dimensions, 𝜙(𝑡) is the initial condition of the system.

In this paper, we will study stability criteria of system described by (8) satisfying (6) or (7), employing the following new Lyapunov-Krasovskii functional: 𝑉𝑡,𝑥_𝑡=𝑥^𝑇(𝑡)𝑃𝑥(𝑡)+^𝑡_{𝑡−𝜏(𝑡)}𝑥^𝑇(𝜃)𝑍𝑥(𝜃)𝑑𝜃+^𝑁_𝑖=1^{𝑡−(𝑖−1)𝛿}_{𝑡−𝑖𝛿}𝑥^𝑇(𝜃)𝑄_𝑖𝑥(𝜃)𝑑𝜃+^𝐾_𝑖=1^{𝑡−𝜏1−(𝑖−1)𝜌}_{𝑡−𝜏1−𝑖𝜌}𝑥^𝑇(𝜃)𝐺_𝑖𝑥(𝜃)𝑑𝜃+^𝑁_𝑖=1^{−(𝑖−1)𝛿}_−𝑖𝛿^𝑡_𝑡+𝜃𝑥̇^𝑇(𝑣)𝛿𝑅_𝑖𝑥̇(𝑣)𝑑𝑣𝑑𝜃+^𝐾_𝑖=1^{−𝜏1−(𝑖−1)𝜌}_{−𝜏1−𝑖𝜌}^𝑡_𝑡+𝜃𝑥̇^𝑇(𝑣)𝜌𝐻_𝑖𝑥̇(𝑣)𝑑𝑣𝑑𝜃+^−𝜏1_−𝜏2^𝑡_𝑡+𝜃𝑥̇^𝑇(𝑣)(𝛾𝑊)𝑥̇(𝑣)𝑑𝑣𝑑𝜃,(9) where 𝑃>0,  𝑍≥0,  𝑄_𝑖>0,  𝑅_𝑖>0(𝑖=1,…,𝑁),  𝐺_𝑖>0,  𝐻_𝑖>0(𝑖=1,…,𝐾),  𝑊>0 of appropriate dimensions, and 𝜏₁=𝑁×𝛿,  𝜏₂−𝜏₁=𝐾×𝜌,  𝛾=𝜏₂−𝜏₁,  𝑁,  𝐾 are positive integer numbers of divisions of intervals [−𝜏₁,0] and [−𝜏₂,−𝜏₁], respectively, and 𝛿, 𝜌 are the length of each division. When 𝜏₁=0 and 𝜏₁=𝜏₂, we assume that 𝑁=0 and 𝐾=0, respectively. Moreover, we will consider a poly-topic uncertainty and a linear fractional norm-bounded uncertainty which includes a routine norm-bounded uncertainty as a special case. Some numerical examples will be given to show the improvement over some previous results.

For any delay satisfying Case 1 or Case 2, our objective of this study is to develop new stability criteria which guarantee that system (8) is asymptotically stable and the system (8) subject to some uncertainties is robustly stable. For this purpose, the following lemmas are introduced.

Lemma 1 (see [3]). For any constant matrix 𝑈∈𝑅^𝑛×𝑛,𝑈=𝑈^𝑇>0, scalar 𝜎>0, and vector function 𝑥̇∶[−𝜎,0]→𝑅^𝑛 such that the following integration is well defined, then it holds that −𝜎^𝑡_𝑡−𝜎𝑥̇^𝑇(𝑤)𝑈𝑥̇(𝑤)𝑑𝑤≤⎡⎢⎢⎣𝑥(𝑡)𝑥(𝑡−𝜎)⎤⎥⎥⎦^𝑇⎡⎢⎢⎣−𝑈𝑈∗−𝑈⎤⎥⎥⎦×⎡⎢⎢⎣𝑥(𝑡)𝑥(𝑡−𝜎)⎤⎥⎥⎦.(10)
Applying the Lemma 1 yields the following new integral inequality for cross-product term.

Lemma 2 (see [4]). For any constant matrix 𝑈∈𝑅^𝑛×𝑛,𝑈=𝑈^𝑇>0, scalars 𝜎₁≤𝜏(𝑡)≤𝜎₂ and vector function 𝑥̇∶[−𝜎₂,−𝜎₁]→𝑅^𝑛 such that the following integration is well defined, then it holds that −𝜎₂−𝜎₁^𝑡−𝜎1_𝑡−𝜎2𝑥̇^𝑇(𝑤)𝑈𝑥̇(𝑤)𝑑𝑤≤𝜂^𝑇(𝑡)Ω𝜂(𝑡),(11) where 𝜂(𝑡)=⎡⎢⎢⎢⎢⎣𝑥𝑡−𝜎₁𝑥(𝑡−𝜏(𝑡))𝑥𝑡−𝜎₂⎤⎥⎥⎥⎥⎦,Ω=⎡⎢⎢⎢⎢⎣−𝑈𝑈0∗−2𝑈𝑈∗∗−𝑈⎤⎥⎥⎥⎥⎦.(12)

3. New Stability Criteria

We first consider asymptotic stability for the nominal system (8). Employing Lyapunov-Krasovskii functional (9), we have the following result.

Theorem 1. For some given scalars 0≤𝜏₁≤𝜏₂ and 𝜇, the nominal system (8) satisfying (6) is asymptotically stable if there exit real symmetric matrices 𝑃>0,  𝑍≥0,  𝑄_𝑖>0,  𝑅_𝑖>0(𝑖=1,…,𝑁),  𝐺_𝑖>0,  𝐻_𝑖>0(𝑖=1,…,𝐾),  𝑊>0, and any matrix 𝑆=[𝑆^𝑇₁𝑆^𝑇₂⋯𝑆^𝑇_𝑁+𝐾+3]^𝑇 of appropriate dimensions such that the following LMI holds: Ξ=Ξ₁+𝑆𝐴_𝑑+𝑆𝐴_𝑑^𝑇<0,(13) where Ξ₁=⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣Ξ₁₁0𝑅₁0⋯0000⋯00𝑃∗Ξ₂₂00⋯0𝑊00⋯0𝑊0∗∗Φ₁𝑅₂⋯0000⋯000∗∗∗Φ₂⋯0000⋯000⋮⋮⋮⋮⋱⋮⋮⋮⋮⋮⋮⋮⋮∗∗∗∗⋯Φ_𝑁−1𝑅_𝑁00⋯000∗∗∗∗⋯∗Φ_𝑁𝐻₁0⋯000∗∗∗∗⋯∗∗Ψ₁𝐻₂⋯000∗∗∗∗⋯∗∗∗Ψ₂⋯000⋮⋮⋮⋮⋯⋮⋮⋮⋮⋱⋮⋮⋮∗∗∗∗⋯∗∗∗∗⋯Ψ_𝐾−1𝐻_𝐾0∗∗∗∗⋯∗∗∗∗⋯∗Ψ_𝐾0∗∗∗∗⋯∗∗∗∗⋯∗∗Υ⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦,(14)Ξ₁₁=𝑍+𝑄₁−𝑅₁,Ξ₂₂=−(1−𝜇)𝑍−2𝑊,𝐴_𝑑=⎡⎢⎣𝐴𝐵00⋯0_{𝑁+𝐾}−𝐼⎤⎥⎦,Φ_𝑖=𝑄_𝑖+1−𝑄_𝑖−𝑅_𝑖−𝑅_𝑖+1(𝑖=1,2,…,𝑁−1),Φ_𝑁=−𝑄_𝑁−𝑅_𝑁−𝑊+𝐺₁−𝐻₁,Ψ_𝑖=𝐺_𝑖+1−𝐺_𝑖−𝐻_𝑖−𝐻_𝑖+1(𝑖=1,2,…,𝐾−1),Ψ_𝐾=−𝐺_𝐾−𝐻_𝐾−𝑊,Υ=^𝑁_𝑖=1𝛿²𝑅_𝑖+^𝐾_𝑖=1𝜌²𝐻_𝑖+𝛾²𝑊.(15)

Proof. Taking the time derivative of 𝑉(𝑡,𝑥_𝑡) with respect to 𝑡 along the trajectory of (8) yields 𝑉̇‌𝑡,𝑥_𝑡≤2𝑥^𝑇(𝑡)𝑃𝑥̇(𝑡)+𝑥^𝑇(𝑡)𝑍𝑥(𝑡)−(1−𝜇)𝑥^𝑇(𝑡−𝜏(𝑡))𝑍𝑥(𝑡−𝜏(𝑡))+^𝑁_𝑖=1𝑥^𝑇(𝑡−(i−1)𝛿)𝑄_𝑖𝑥(𝑡−(𝑖−1)𝛿)−^𝑁_𝑖=1𝑥^𝑇(𝑡−𝑖𝛿)𝑄_𝑖𝑥(𝑡−𝑖𝛿)+^𝐾_𝑖=1𝑥^𝑇𝑡−𝜏₁−(𝑖−1)𝜌𝐺_𝑖𝑥𝑡−𝜏₁−(𝑖−1)𝜌−^𝐾_𝑖=1𝑥^𝑇𝑡−𝜏₁−𝑖𝜌𝐺_𝑖𝑥𝑡−𝜏₁−𝑖𝜌+𝑥̇^𝑇(𝑡)⎛⎜⎝^𝑁_𝑖=1𝛿²𝑅_𝑖+^𝐾_𝑖=1𝜌²𝐻_𝑖+𝛾²𝑊⎞⎟⎠𝑥̇(𝑡)−^𝑁_𝑖=1^{𝑡−(𝑖−1)𝛿}_{𝑡−𝑖𝛿}𝑥̇^𝑇(𝜃)𝛿𝑅_𝑖𝑥̇(𝜃)𝑑𝜃−^𝐾_𝑖=1^{𝑡−𝜏1−(𝑖−1)𝜌}_{𝑡−𝜏1−𝑖𝜌}𝑥̇^𝑇(𝜃)𝜌𝐻_𝑖𝑥̇(𝜃)𝑑𝜃−^𝑡−𝜏1_𝑡−𝜏2𝑥̇^𝑇(𝜃)(𝛾𝑊)𝑥̇(𝜃)𝑑𝜃.(16)
Applying the Lemma 1, the following inequalities hold: −^𝑁_𝑖=1^{𝑡−(𝑖−1)𝛿}_{𝑡−𝑖𝛿}𝑥̇^𝑇(𝜃)𝛿𝑅_𝑖𝑥̇(𝜃)𝑑𝜃≤^𝑁_𝑖=1⎡⎢⎢⎣𝑥(𝑡−(𝑖−1)𝛿)𝑥(𝑡−𝑖𝛿)⎤⎥⎥⎦^𝑇⎡⎢⎢⎣−𝑅_𝑖𝑅_𝑖∗−𝑅_𝑖⎤⎥⎥⎦⎡⎢⎢⎣𝑥(𝑡−(𝑖−1)𝛿)𝑥(𝑡−𝑖𝛿)⎤⎥⎥⎦,−^𝐾_𝑖=1^{𝑡−𝜏1−(𝑖−1)𝜌}_{𝑡−𝜏1−𝑖𝜌}𝑥̇^𝑇(𝜃)𝜌𝐻_𝑖𝑥̇(𝜃)𝑑𝜃≤^𝐾_𝑖=1⎡⎢⎢⎣𝑥(𝑡−𝜏₁−(𝑖−1)𝜌)𝑥𝑡−𝜏₁−𝑖𝜌⎤⎥⎥⎦^𝑇⎡⎢⎢⎣−𝐻_𝑖𝐻_𝑖∗−𝐻_𝑖⎤⎥⎥⎦×⎡⎢⎢⎣𝑥𝑡−𝜏₁−(𝑖−1)𝜌𝑥𝑡−𝜏₁−𝑖𝜌⎤⎥⎥⎦.(17)
Applying the Lemma 2, the following inequality holds: −^𝑡−𝜏1_𝑡−𝜏2𝑥̇^𝑇(𝜃)(𝛾𝑊)𝑥̇(𝜃)𝑑𝜃≤⎡⎢⎢⎢⎢⎣𝑥𝑡−𝜏₁𝑥(𝑡−𝜏(𝑡))𝑥𝑡−𝜏₂⎤⎥⎥⎥⎥⎦^𝑇⎡⎢⎢⎢⎢⎣−𝑊𝑊0∗−2𝑊𝑊∗∗−𝑊⎤⎥⎥⎥⎥⎦×⎡⎢⎢⎢⎢⎣𝑥𝑡−𝜏₁𝑥(𝑡−𝜏(𝑡))𝑥𝑡−𝜏₂⎤⎥⎥⎥⎥⎦.(18)
For any matrix 𝑆=[𝑆^𝑇₁𝑆^𝑇₂⋯𝑆^𝑇_𝑁+𝐾+3]^𝑇 of appropriate dimensions, the following equality holds: 2𝜉^𝑇(𝑡)𝑆[𝐴𝑥(𝑡)+𝐵𝑥(𝑡−𝜏(𝑡))−𝑥̇(𝑡)]=0,(19) where is defined in the top of next page, 𝜉^𝑇(𝑡)=𝑥^𝑇(𝑡)𝑥^𝑇(𝑡−𝜏(𝑡))𝜉^𝑇₁(𝑡)𝜉^𝑇₂(𝑡)𝑥̇(𝑡),𝜉^𝑇₁(𝑡)=𝑥^𝑇(𝑡−𝛿)𝑥^𝑇(𝑡−2𝛿)⋯𝑥^𝑇(𝑡−(𝑁−1)𝛿)𝑥^𝑇𝑡−𝜏₁,𝜉^𝑇₂(𝑡)=𝑥^𝑇𝑡−𝜏₁−𝜌𝑥^𝑇𝑡−𝜏₁−2𝜌⋯𝑥^𝑇𝑡−𝜏₁−(𝐾−1)𝜌𝑥^𝑇𝑡−𝜏₂.(20)
Considering (16)–(19) together, we have 𝑉(̇‌𝑡,𝑥_𝑡)≤𝜉^𝑇(𝑡)Ξ𝜉(𝑡). If (13) is satisfied, then 𝑉(̇‌𝑡,𝑥_𝑡)≤−𝜆𝑥^𝑇(𝑡)𝑥(𝑡) for some scalar 𝜆>0, from which we conclude that the nominal system (8) is asymptotically stable. This completes the proof.

When the restriction on the derivative of the interval time-varying delay is removed, that is, choosing 𝑍≡0 in Theorem 1, we can obtain a delay variety rate-independent criterion for a delay that only satisfies (7).

Corollary 1. For some given scalars 0≤𝜏₁≤𝜏₂, the nominal system (8) satisfying (7) is asymptotically stable, if there exit real symmetric matrices 𝑃>0, 𝑄_𝑖>0,  𝑅_𝑖>0(𝑖=1,…,𝑁), 𝐺_𝑖>0,  𝐻_𝑖>0(𝑖=1,…,𝐾), 𝑊>0 and any matrix 𝑆 of appropriate dimensions such that the LMI (13) holds with 𝑍≡0.

Remark 2. For system (8) with the routine delay case described by 𝜏₁=0 and 𝜏₂>0, that is, 𝑁=0,𝐾>0, the corresponding Lyapunov-Krasovskii functional reduces to 𝑉𝑡,x_𝑡=𝑥^𝑇(𝑡)𝑃𝑥(𝑡)+^𝑡_{𝑡−𝜏(𝑡)}𝑥^𝑇(𝜃)𝑍𝑥(𝜃)𝑑𝜃+^𝐾_𝑖=1^{𝑡−(𝑖−1)𝜌}_{𝑡−𝑖𝜌}𝑥^𝑇(𝜃)𝐺_𝑖𝑥(𝜃)𝑑𝜃+^𝐾_𝑖=1^{−(𝑖−1)𝜌}_−𝑖𝜌^𝑡_𝑡+𝜃𝑥̇^𝑇(𝑣)𝜌𝐻_𝑖𝑥̇(𝑣)𝑑𝑣𝑑𝜃+⁰_−𝜏2^𝑡_𝑡+𝜃𝑥̇^𝑇(𝑣)(𝛾𝑊)𝑥̇(𝑣)𝑑𝑣𝑑𝜃.(21)
And for system (8) with the constant delay case described by 𝜏₁=𝜏(𝑡)=𝜏₂>0, that is, 𝑁>0,𝐾=0, the corresponding Lyapunov-Krasovskii functional reduces to 𝑉𝑡,𝑥_𝑡=𝑥^𝑇(𝑡)𝑃𝑥(𝑡)+^𝑁_𝑖=1^{𝑡−(𝑖−1)𝛿}_{𝑡−𝑖𝛿}𝑥^𝑇(𝜃)𝑄_𝑖𝑥(𝜃)𝑑𝜃+^𝑁_𝑖=1^{−(𝑖−1)𝛿}_−𝑖𝛿^𝑡_𝑡+𝜃𝑥̇^𝑇(𝑣)𝛿𝑅_𝑖𝑥̇(𝑣)𝑑𝑣𝑑𝜃.(22)

Similar to the proof of the Theorem 1, one can easily derive a less conservative results than some exiting ones, respectively, which will be shown through numerical examples in the next section. For the sake of simplicity, the results are omitted.

In what follows, we consider robust stability of the system (8) satisfying (6) or (7) subject to a poly-topic uncertainty and a linear fractional norm-bounded uncertainty. For the poly-topic uncertainty, that is, the matrices 𝐴 and 𝐵 in (8) can be expressed as [𝐴𝐵]=^𝑟_𝑗=1𝜁_𝑗𝐴_𝑗𝐵_𝑗,(23) where ∑^𝑟_𝑗=1𝜁_𝑗=1,0≤𝜁_𝑗≤1. Based on Theorem 1, we can easily obtain the following result.

Theorem 2. For some given scalars 0≤𝜏₁≤𝜏₂ and 𝜇, the system described by (8) satisfying (6) subject to polytopic uncertainty (23) is robustly stable, if there exit real symmetric matrices 𝑃_𝑗>0,𝑍_𝑗≥0,𝑄_𝑖𝑗>0,𝑅_𝑖𝑗>0(𝑖=1,…,𝑁), 𝐺_𝑖𝑗>0,𝐻_𝑖𝑗>0(𝑖=1,…,𝐾), 𝑊_𝑗>0, and any matrix 𝑆 of appropriate dimensions such that the following LMIs hold for 𝑗=1,2,…,𝑟: Ξ^(𝑗)=Ξ^(𝑗)₁+𝑆𝐴^(𝑗)_𝑑+𝑆𝐴^(𝑗)_𝑑^𝑇<0,(24) where Ξ^(𝑗)₁=⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣Ξ^(𝑗)₁₁0𝑅_1𝑗0⋯0000⋯00𝑃_𝑗∗Ξ^(𝑗)₂₂00⋯0𝑊_𝑗00⋯0𝑊_𝑗0∗∗Φ^(𝑗)₁𝑅_2𝑗⋯0000⋯000∗∗∗Φ^(𝑗)₂⋯0000⋯000⋮⋮⋮⋮⋱⋮⋮⋮⋮⋮⋮⋮⋮∗∗∗∗⋯Φ^(𝑗)_𝑁−1𝑅_𝑁𝑗00⋯000∗∗∗∗⋯∗Φ^(𝑗)_𝑁𝐻_1𝑗0⋯000∗∗∗∗⋯∗∗Ψ^(𝑗)₁𝐻_2𝑗⋯000∗∗∗∗⋯∗∗∗Ψ^(𝑗)₂⋯000⋮⋮⋮⋮⋯⋮⋮⋮⋮⋱⋮⋮⋮∗∗∗∗⋯∗∗∗∗⋯Ψ^(𝑗)_𝐾−1𝐻_𝐾𝑗0∗∗∗∗⋯∗∗∗∗⋯∗Ψ^(𝑗)_𝐾0∗∗∗∗⋯∗∗∗∗⋯∗∗Υ^(𝑗)⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦,(25)𝑆 is defined in Theorem 1, and Ξ^(𝑗)₁₁=𝑍_𝑗+𝑄_1𝑗−𝑅_1𝑗,Ξ^(𝑗)₂₂=−(1−𝜇)𝑍_𝑗−2𝑊_𝑗,𝐴^(𝑗)_𝑑=⎡⎢⎣𝐴_𝑗𝐵_𝑗00⋯0_{𝑁+𝐾}−𝐼⎤⎥⎦,Φ^(𝑗)_𝑖=𝑄_𝑖+1,𝑗−𝑄_𝑖𝑗−𝑅_𝑖𝑗−𝑅_𝑖+1,𝑗(𝑖=1,2,…,𝑁−1),Φ^(𝑗)_𝑁=−𝑄_𝑁𝑗−𝑅_𝑁𝑗−𝑊_𝑗+𝐺_1𝑗−𝐻_1𝑗,Ψ^(𝑗)_𝑖=𝐺_𝑖+1,𝑗−𝐺_𝑖𝑗−𝐻_𝑖𝑗−𝐻_𝑖+1,𝑗(𝑖=1,2,…,𝐾−1),Ψ^(𝑗)_𝐾=−𝐺_𝐾𝑗−𝐻_𝐾𝑗−𝑊_𝑗,Υ^(𝑗)=^𝑁_𝑖=1𝛿²𝑅_𝑖𝑗+^𝐾_𝑖=1𝜌²𝐻_𝑖𝑗+𝛾²𝑊_𝑗.(26)

Proof. Construct the following parameter-dependent Lyapunov-Krasovskii functional: 𝑉_𝑢𝑡,𝑥_𝑡=^𝑟_𝑗=1𝑥^𝑇(𝑡)𝜁_𝑗𝑃_𝑗𝑥(𝑡)+^𝑟_𝑗=1^𝑡_{𝑡−𝜏(𝑡)}𝑥^𝑇(𝜃)𝜁_𝑗𝑍_𝑗𝑥(𝜃)𝑑𝜃+^𝑟_𝑗=1^𝑁_𝑖=1^{𝑡−(𝑖−1)𝛿}_{𝑡−𝑖𝛿}𝑥^𝑇(𝜃)𝜁_𝑗𝑄_𝑖𝑗𝑥(𝜃)𝑑𝜃+^𝑟_𝑗=1^𝐾_𝑖=1^{𝑡−𝜏1−(𝑖−1)𝜌}_{𝑡−𝜏1−𝑖𝜌}𝑥^𝑇(𝜃)𝜁_𝑗𝐺_𝑖𝑗𝑥(𝜃)𝑑𝜃+^𝑟_𝑗=1^𝑁_𝑖=1^{−(i−1)𝛿}_−𝑖𝛿^𝑡_𝑡+𝜃𝑥̇^𝑇(𝑣)𝜁_𝑗𝛿𝑅_𝑖𝑗𝑥̇(𝑣)𝑑𝑣𝑑𝜃+^𝑟_𝑗=1^𝐾_𝑖=1^{−𝜏1−(𝑖−1)𝜌}_{−𝜏1−𝑖𝜌}^𝑡_𝑡+𝜃𝑥̇^𝑇(𝑣)𝜁_𝑗𝜌𝐻_𝑖𝑗𝑥̇(𝑣)𝑑𝑣𝑑𝜃+^𝑟_𝑗=1^−𝜏1_−𝜏2^𝑡_𝑡+𝜃𝑥̇^𝑇(𝑣)𝜁_𝑗𝛾𝑊_𝑗𝑥̇(𝑣)𝑑𝑣𝑑𝜃,(27) where 𝑃_𝑗>0,  𝑍_𝑗≥0,  𝑄_𝑖𝑗>0,  𝑅_𝑖𝑗>0(𝑖=1,…,𝑁),  𝐺_𝑖𝑗>0,  𝐻_𝑖𝑗>0(𝑖=1,…,𝐾),  𝑊_𝑗>0 are unknown matrices of appropriate dimensions. Then the proof follows a linear similar to the proof of Theorem 1, we can obtain the following inequality: 𝑉_𝑢̇‌𝑡,𝑥_𝑡≤^𝑟_𝑗=1𝜉^𝑇(𝑡)𝜁_𝑗Ξ^(𝑗)𝜉(𝑡),(28) where 𝜉(𝑡) is defined in the formula (19), then 𝑉_𝑢̇‌(𝑡,𝑥_𝑡)≤−𝜆𝑥^𝑇(𝑡)𝑥(𝑡) for some scalar 𝜆>0, from which we conclude that the system described by (8) satisfying (6) subject to polytopic uncertainty (23) is robustly stable. This completes the proof.

Similar to Corollary 1, we can easily obtain a delay variety rate-independent criterion for a delay that only satisfies (7).

Corollary 2. For some given scalars 0≤𝜏₁≤𝜏₂, the system described by (8) satisfying (7) subject to polytopic uncertainty (23) is robustly stable if there exit real symmetric matrices 𝑃_𝑗>0,  𝑄_𝑖𝑗>0,  𝑅_𝑖𝑗>0(𝑖=1,…,𝑁),  𝐺_𝑖𝑗>0,  𝐻_𝑖𝑗>0(𝑖=1,…,𝐾),  𝑊_𝑗>0, and any matrix 𝑆 of appropriate dimensions such that the LMIs (24) hold with 𝑍_𝑗≡0 for 𝑗=1,2,…,𝑟.

Remark 3. We succeed to separate the system matrices and Lyapunov matrices in Theorem 1, so we can easily use parameter-dependent Lyapunov-Krasovskii functional method. Different Lyapunov matrices are used in Theorem 2 for 𝑟 different LMIs, which are distinguished with Proposition 7 in Jiang and Han [6], in which used fixed Lyapunov matrices for 𝑟 different LMIs. In fact parameter-dependent Lyapunov-Krasovskii functional method can reduce stability criteria conservatism significantly. Numerical example will be given to show the improvement with Jiang and Han [6] in the next section.

Next we address the linear fractional norm-bounded uncertainty. Suppose that matrices 𝐴 and 𝐵 have parameter perturbations as Δ𝐴(𝑡) and Δ𝐵(𝑡), which are in the form of [Δ𝐴(𝑡)Δ𝐵(𝑡)]=𝐷(𝐼−𝐹(𝑡)𝑀)⁻¹𝐹(𝑡)𝐸₁𝐸₂,(29) where 𝐷,𝑀,𝐸₁, and 𝐸₂ are known real constant matrices of appropriate dimensions, and 𝑀^𝑇𝑀<𝐼; 𝐹(𝑡) is an unknown matrix function with Lesbesgue measurable elements satisfying 𝐹^𝑇(𝑡)𝐹(𝑡)≤𝐼.

For system (8) with uncertainty (29), we can establish the following result by considering Theorem 1 and applying 𝑆 procedure [9].

Theorem 3. For some given scalars 0≤𝜏₁≤𝜏₂ and 𝜇, the system described by (8) satisfying (6)subject to the linear fractional norm-bounded uncertainty (29) is robustly stable if there exit real symmetric matrices 𝑃>0,  𝑍≥0, 𝑄_𝑖>0,  𝑅_𝑖>0(𝑖=1,…,𝑁), 𝐺_𝑖>0,  𝐻_𝑖>0(𝑖=1,…,𝐾), 𝑊>0, any matrix 𝑆 of appropriate dimensions, and a scalar 𝜀>0 such that the following LMI holds: ⎡⎢⎢⎢⎢⎣Ξ𝑆𝐷𝜀Γ∗−𝜀𝐼𝜀𝑀^𝑇∗∗−𝜀𝐼⎤⎥⎥⎥⎥⎦<0,(30) where Γ^𝑇=^{𝐸1𝐸200⋯0}_𝑁+𝐾+1,  Ξ is defined in the formula (13), and 𝑆 is defined in Theorem 1.

Proof. Replacing 𝐴 and 𝐵 in (13) with 𝐴+𝐷(𝐼−𝐹(𝑡)𝑀)⁻¹𝐹(𝑡)𝐸₁ and 𝐵+𝐷(𝐼−𝐹(𝑡)𝑀)⁻¹𝐹(𝑡)𝐸₂, respectively, and multiply both sides of the resulting matrix by vectors 𝑥_𝑖 for 𝑖=1,2,…,𝑁+𝐾+3. Next, define 𝑝=(𝐼−𝐹(𝑡)𝑀)⁻¹𝐹(𝑡)𝐸₁𝑥₁,𝑞=(𝐼−𝐹(𝑡)𝑀)⁻¹𝐹(𝑡)𝐸₂𝑥₂. Then, we have the following condition for the admissible uncertainty 𝐹(𝑡): 𝛽^𝑇⎡⎢⎢⎣Ξ𝑆𝐷∗0⎤⎥⎥⎦𝛽<0,(31) where 𝛽^𝑇=[𝑥^𝑇₁,𝑥^𝑇₂,…,𝑥^𝑇_𝑁+𝐾+3,(𝑝+𝑞)^𝑇]. It is easy to know that 𝑝 and 𝑞 can be rewritten as 𝑝=𝐹(𝑡)𝑀𝑝+𝐹(𝑡)𝐸₁𝑥₁ and 𝑞=𝐹(𝑡)𝑀𝑞+𝐹(𝑡)𝐸₂𝑥₂, since 𝐹^𝑇(𝑡)𝐹(𝑡)≤𝐼, it is obvious that the following inequality holds: (𝑝+𝑞)^𝑇(𝑝+𝑞)≤⎡⎢⎢⎢⎢⎣𝑥₁𝑥₂𝑝+𝑞⎤⎥⎥⎥⎥⎦^𝑇⎡⎢⎢⎢⎢⎣𝐸^𝑇₁𝐸₁𝐸^𝑇₁𝐸₂𝐸^𝑇₁𝑀∗𝐸^𝑇₂𝐸₂𝐸^𝑇₂𝑀∗∗𝑀^𝑇𝑀⎤⎥⎥⎥⎥⎦⎡⎢⎢⎢⎢⎣𝑥₁𝑥₂𝑝+𝑞⎤⎥⎥⎥⎥⎦.(32) Applying 𝑆 procedure, both inequalities (31) and (32) are true if and only if there is a 𝜀>0, promising that the following condition holds: 𝛽^𝑇⎡⎢⎢⎣Ξ+𝜀ΓΓ^𝑇𝑆𝐷+𝜀Γ𝑀∗𝜀𝑀^𝑇𝑀−𝜀𝐼⎤⎥⎥⎦𝛽<0.(33) By Schur complement, for any 𝛽≠0, (33) is equivalent to (30). This completes the proof.

Similar to Corollary 1, we can easily obtain a delay variety rate-independent criterion for a delay that only satisfies (7).

Corollary 3. For some given scalars 0≤𝜏₁≤𝜏₂, the system described by (8) satisfying (7) subject to the linear fractional norm-bounded uncertainty (29) is robustly stable if there exit real symmetric matrices 𝑃>0, 𝑄_𝑖>0,  𝑅_𝑖>0(𝑖=1,…,𝑁), 𝐺_𝑖>0,  𝐻_𝑖>0(𝑖=1,…,𝐾), 𝑊>0, any matrix 𝑆 of appropriate dimensions, and a scalar 𝜀>0 such that the LMI (30) holds with 𝑍≡0.

Remark 4. It is clear to see that if we set 𝑀=0, the linear fractional norm-bounded uncertainty reduces to the routine norm-bounded uncertainty, and we can derive a corresponding results for the routine norm-bounded uncertainty from Theorem 3 and Corollary 3. For the sake of simplicity, the results are omitted.

4. Numerical Examples

In this section, two examples are given to show the effectiveness of the results derived in this paper.

Example 1. Consider the following time-delay system: 𝑥̇(𝑡)=⎡⎢⎢⎣−200−0.9⎤⎥⎥⎦𝑥(𝑡)+⎡⎢⎢⎣−10−1−1⎤⎥⎥⎦𝑥(𝑡−𝜏(𝑡)).(34) Table 1 lists the maximum allowable upper bound (MAUB) of the time-varying delay for different 𝜏₁, and those in He et al. [5], Jiang and Han [6], and Shao [7].(1)If 𝜏₁=0, by using Corollary 1, we have 𝜏₂=1.3454 for 𝑁=0 and any integer 𝐾≥2, which is larger than 𝜏₂=1.34 in derived in He et al. [5] and close to 𝜏₂=1.3453 in derived in Jiang and Han [6].(2)If 0<𝜏₁≤𝜏(𝑡)≤𝜏₂ and 𝜏₁≠𝜏₂. By using Corollary 1 for 𝑁=2,𝐾=2, our results listed in the second last column is same as the results derived in Jiang and Han [6]. Moreover, by using Corollary 1 for 𝑁=3,𝐾=3, we have the results listed in the last column, which are larger than the results derived in He et al. [5] and Jiang and Han(2008).(3)By using Proposition 8 in Jiang and Han [6], we can have 𝜏₂=4.6007 for 𝜏₁=4.4697, not have 4.6009^a.(4)For the constant delay 𝜏₁=𝜏₂, that is, 𝐾=0. By using Corollary 1, we have 𝜏_max=4.4721 for 𝑁=1; 5.7175 for 𝑁=2; 5.9677 for 𝑁=3; 6.0983 for 𝑁=5; 6.1537 for 𝑁=10, which is close to analytical delay limit for stability 6.17258.

Table 1 

MAUB of the time-varying delay for different 𝜇.

Example 2. Consider the following time-delay system: 𝑥̇(𝑡)=⎡⎢⎢⎣−2+𝛿₁00−1+𝛿₂⎤⎥⎥⎦𝑥(𝑡)+⎡⎢⎢⎣−1+𝜌₁0−1−1+𝜌₂⎤⎥⎥⎦𝑥(𝑡−𝜏(𝑡)),(35) where |𝛿₁|≤1.6, |𝛿₂|≤0.05, |𝜌₁|≤0.1, |𝜌₂|≤0.3, and suppose that 𝜏(𝑡) is continuous function. We handle the uncertainty as polytopic uncertainty and norm-bounded uncertainty, respectively. First, the uncertainty is handled as polytopic uncertainty. Then(1)if 𝜏(𝑡) is a constant, applying Corollary 2 in this paper yields the result 𝜏₂=1.8130 for 𝜏₁=𝜏₂, which is less conservative than the result 𝜏₂=1 given in Fridman and Shaked [10], the result 𝜏₂=1.02 given in Han et al. [11] and the result 𝜏₂=1.3500 given in Jiang and Han [6]; (2)if 𝜏₁=0, we have 𝜏₂=1.0238, which is larger than 𝜏₂=0.7692 in derived in Fridman and Shaked [10],𝜏₂=0.8654 in derived in Jiang and Han [8] and 𝜏₂=0.8748 in derived in Jiang and Han [6], respectively;(3)if 0<𝜏₁≤𝜏(𝑡)≤𝜏₂, Table 2 lists the maximum allowable upper bounds of the time-varying delay for different lower bounds the delay by Corollary 2. From Table 2, we can see that our results are larger than the previous ones.

Table 2 

MAUB of the time varying delay for different 𝜏₁.

Next, the uncertainty is handled as norm-bounded uncertainty, and we choose 𝐷,𝐸₁ and 𝐸₂ as𝐷=⎡⎢⎢⎣0.01000.04⎤⎥⎥⎦,𝐸₁=⎡⎢⎢⎣160001.25⎤⎥⎥⎦,𝐸₂=⎡⎢⎢⎣10007.5⎤⎥⎥⎦.(36)

Table 3 lists the maximum allowable upper bound (MAUB) of the time-varying delay for different 𝜏₁, and those in Jiang and Han [6, 8].(1)If 𝜏₁=0, we have 𝜏₂=0.9442 for 𝑁=0 and any integer 𝐾≥2, which is larger than 𝜏₂=0.8654 in derived in He et.al [5] and same as 𝜏₂=0.9442 in derived in Jiang and Han [6].(2)If 0<𝜏₁≤𝜏(𝑡)≤𝜏₂ and 𝜏₁≠𝜏₂, By using Corollary 3, our results listed in the second last column for 𝑁=2,𝐾=2 and in the last column for 𝑁=3,𝐾=3, which is larger than those in derived in Jiang and Han [6].(3)If 𝜏(𝑡) is a constant, this is 𝜏₁=𝜏₂, we have 𝜏₂=1.4368 for 𝑁=2,𝐾=0; 1.4810 for 𝑁=3,𝐾=0.

Table 3 

MAUB of the time-varying delay for different 𝜇.

It is clear that proposed stability criteria in this paper can significantly improve some exiting results in the literature.

Remark 5. From Tables 2 and 3, we can find that the conservatism using the criterion derived for the poly-topic uncertainty is less than using the criterion derived for norm-bounded uncertainty, which is distinguished to the result derived in Jiang and Han [6], and there are some reverse results, for there are some fixed Lyapunov matrices in Jiang and Han [6] for different LMIs. Moreover, the conservatism can be reduced for 𝜏₁≠0 by increasing divisions of the intervals [−𝜏₁,0] or [−𝜏₂,−𝜏₁] in this paper.

5. Conclusion

This note presents some new stability criteria for the interval time-varying delay system with a poly-topic uncertainty and a linear fractional norm-bounded uncertainty. We have proposed a new Lyapunov-Krasovskii functional, which is based on dividing intervals [−𝜏₁,0] and [−𝜏₂,−𝜏₁] into 𝑁 and 𝐾 divisions, respectively, to derive some new stability criteria. Numerical examples show that our criteria would be less conservative along with 𝑁,𝐾 increased and demonstrate our criteria are less conservative than previous ones. Furthermore, based on time-delay system stability results, some future research can be focused on designing feedback controller, which can promise system robust stabilization and satisfy some system performance.

Acknowledgments

This work is supported by NSFC Project 11071087, NSFC-Guangdong Joint Foundation Key Project U0735003, and Guangdong Province Natural Science Foundation of China Project 06105413 and DGUT-NSYF Project 2010ZQ23.