Improved Robust Stability Criteria of Uncertain Neutral Systems with Mixed Delays

Abstract

The problem of robust stability for a class of neutral control systems with mixed delays is investigated. Based on Lyapunov stable theory, by constructing a new Lyapunov-Krasovskii function, some new stable criteria are obtained. These criteria are formulated in the forms of linear matrix inequalities (LMIs). Compared with some previous publications, our results are less conservative. Simulation examples are presented to illustrate the improvement of the main results.

1. Introduction

As one of important dynamical systems, neutral system has been received considerable attention in past years. Large numbers of monographs and papers on the stability of neutral type system with or without time delays have been published. A wide variety of methods disposing the stability problems of neutral system have been proposed [1–6]. It is well known that, because of the finite switching speed, memory effect, and so on, time delays are unavoidable in nature and technology. They can make important effects on the stability of concerned dynamical systems. Thus, the studies on stability of time-delayed neutral system are of great significance. In recent years, all kinds of delays such as time-varying delay [7–10], distributed delay [11–13], and mixed delay [14–16] were considered, and corresponding stable criteria have been derived. In practical, during the design of control system and its hardware implementation, the convergence of a control system may often be destroyed by its unavoidable uncertainty due to the existence of modeling error, the deviation of vital data, and so on. Therefore, the studies on robust convergence of delayed control system have been a hot research topic, and many sufficient conditions have been derived to guarantee the robust asymptotic or exponential stability for different class of delayed systems (see [2, 7, 8, 10, 12, 13, 17–20]). General speaking, these criteria can be divided into two categories [21]: that is, delay-independent criteria and delay-dependent criteria. As pointed out in [12], when the size of time delay is small, delay-dependent criteria may be less conservative than those of delay-independent criteria, and the more free-weighting matrices are introduced in criterion, the less conservative it may be. On the other hand, compared with traditional matrix measure, matrix norm, and Riccati matrix criteria, linear matrix inequality (LMI) technique can be easily checked by LMI toolbox in MATLAB software and can make free weighting matrices easy to select. Thus, it becomes one of the most extensively used techniques in control system. In addition, the admissible allowed upper bound on the delay is usually regarded as the performance index for measuring the conservatism of the conditions obtained.

Motivated by the afore-mentioned analysis, in this paper, based on the equivalent equation of the zero which is similar to [2] in the derivative of a Lyapunov-Krasovskii functional, we will focus on deriving some improved robust stable criteria for a class of neutral control systems with mixed delays. By constructing a new Lyapunov function, some new delay-dependent stable criteria are derived via sufficiently employing Newton-Leibniz formula to introduce large numbers of free weighting matrices. These free weighting matrices express the influence of the relationship among terms 𝑥(𝑡),𝑥̇(𝑡),𝑥(𝑡−𝜏₁),𝑥(𝑡−𝜏₂), 𝑥̇(𝑡−𝜏₂), ∫^𝑡_𝑡−𝜏1𝑥̇(𝑠)𝑑𝑠, ∫^𝑡_𝑡−𝜏2𝑥̇(𝑠)𝑑𝑠. Since these criteria are both discrete delay-dependent and distributed delay-dependent, they are less conservative than some previous methods for the concerned systems. When norm-bounded parameter uncertainties appear in the concerned system, delay-dependent robust asymptotic stability criteria are also presented. All of these criteria are expressed in the forms of linear matrix inequalities (LMIs), which can be easily solved. Finally, numerical examples are given to illustrate the improvement of the main results. Simulations show that our results are valid.

2. Preliminaries

Consider uncertain neutral system with mixed delays [4] as follows:

̇̇Σ∶𝑥(𝑡)−𝐶𝑥(𝑡−𝜏₂)=(𝐴+Δ𝐴(𝑡))𝑥(𝑡)+(𝐵+Δ𝐵(𝑡))𝑥(𝑡−𝜏₁)+(𝐷+Δ𝐷(𝑡))^𝑡_𝑡−ℎ𝑥(𝑠)𝑑𝑠,𝑡>0,𝑥(𝑡)=𝜙(𝑡),𝑡∈[−𝜏,0], (2.1) where 𝑥(𝑡)∈ℛ^𝑛 is the state vector; 𝐴,𝐵,𝐶∈ℛ^𝑛×𝑛 represent the weighting matrices; 𝜏₁,𝜏₂ are discrete delays; ℎ is distributed delay; 𝜙(𝑡) is initial condition which is continuous on interval [−𝜏,0], where 𝜏=max{𝜏₁,𝜏₂,ℎ}; Δ𝐴(𝑡),Δ𝐵(𝑡),Δ𝐷(𝑡) denote the time-varying structured uncertainties which are of the following form:

[Δ𝐴(𝑡),Δ𝐵(𝑡),Δ𝐷(𝑡)]=𝐾𝐹(𝑡)[𝐸_𝑎,𝐸_𝑏,𝐸_𝑑], (2.2) where 𝐾,𝐸_𝑎,𝐸_𝑏,𝐸_𝑑 are the known constant matrices with appropriate dimensions; 𝐹(𝑡) is unknown continuous time-varying matrix function satisfying 𝐹^𝑇(𝑡)𝐹(𝑡)≤𝐼,forall𝑡≥0.

The nominal Σ₀ of Σ can be defined as

Σ₀∶𝑥̇(𝑡)−𝐶𝑥̇(𝑡−𝜏₂)=𝐴𝑥(𝑡)+𝐵𝑥(𝑡−𝜏₁)+𝐷^𝑡_𝑡−ℎ𝑥(𝑠)𝑑𝑠,𝑡>0,𝑥(𝑡)=𝜙(𝑡),𝑡∈[−𝜏,0]. (2.3)

For further discussion, we first introduce the following lemmas.

Lemma 2.1 (see [22]). Given constant symmetric matrices Σ₁,Σ₂,Σ₃ where Σ^𝑇₁=Σ₁ and 0<Σ₂=Σ^𝑇₂, then Σ₁+Σ^𝑇₃Σ₂⁻¹Σ₃<0 if and only if Σ₁Σ^𝑇₃Σ₃−Σ₂<0,or−Σ₂Σ₃Σ^𝑇₃Σ₁<0. (2.4)

Lemma 2.2 (see [23]). For given matrices 𝑄=𝑄^𝑇,𝐻,𝐸, and 𝑅=𝑅^𝑇>0 with appropriate dimensions, then 𝑄+𝐻𝐹𝐸+𝐸^𝑇𝐹^𝑇𝐻^𝑇<0, (2.5) for all 𝐹 satisfying 𝐹^𝑇𝐹≤𝑅 if and only if there exists a positive number 𝜀>0, such that 𝑄+𝜀⁻¹𝐻𝐻^𝑇+𝜀𝐸^𝑇𝑅𝐸<0. (2.6)

Lemma 2.3. For any real vector 𝑋,𝑌 and positive definite matrix Σ>0 with appropriate dimensions, it follows that 2𝑋^𝑇𝑌≤𝑋^𝑇Σ𝑋+𝑌^𝑇Σ⁻¹𝑌. (2.7)

3. Main Results

In this section, we will analyse the stability problem of uncertain neutral systems with mixed delays described by (2.1). First, we consider the stability problem for the nominal system (2.3) with Δ𝐴(𝑡)=0,Δ𝐵(𝑡)=0,Δ𝐷(𝑡)=0. In order to introduce free-weighting matrix, we can use the following fact:

𝑀[𝑥(𝑡)−𝑥(𝑡−𝜏₁)−^𝑡_𝑡−𝜏1𝑥̇(𝑠)𝑑𝑠]=0, (3.1) where 𝑀 is an arbitrary matrix with appropriate dimensions. Substituting zero equation (3.1) into system (2.3), the original system can be transformed into the following form:

𝑥̇(𝑡)−𝐶𝑥̇(𝑡−𝜏₂)=(𝐴−𝑀)𝑥(𝑡)+(𝐵+𝑀)𝑥(𝑡−𝜏₁𝐷)+_{𝑡𝑡−ℎ}𝑥(𝑠)𝑑𝑠+𝑀^𝑡_𝑡−𝜏1𝑥̇(𝑠)𝑑𝑠,𝑡>0,𝑥(𝑡)=𝜙(𝑡),𝑡∈[−𝜏,0]. (3.2)

For the asymptotic stability of system (3.2), we can obtain the following results.

Theorem 3.1. For any given matrix 𝑀, scalars 𝜏₁>0,𝜏₂>0,ℎ>0, the nominal system Σ₀ is asymptotically stable if ‖𝐶‖<1, and there exist positive definite matrices 𝑃₁,𝑄₁,𝑄₂,𝑄₃,𝑄₄,𝑄₅, arbitrary matrices 𝑃_𝑖(𝑖=2,3,…,25), 𝐹_𝑖(𝑖=1,2,…,8) with appropriate dimensions such that the following linear matrix inequality is feasible: Ξ₁=Ξ₁₁Ξ₁₂Ξ₁₃Ξ₁₄Ξ₁₅Ξ₁₆Ξ₁₇Ξ₁₈Ξ₁₉∗Ξ₂₂Ξ₂₃Ξ₂₄Ξ₂₅Ξ₂₆Ξ₂₇Ξ₂₈Ξ₂₉∗∗Ξ₃₃Ξ₃₄Ξ₃₅Ξ₃₆Ξ₃₇Ξ₃₈Ξ₃₉∗∗∗Ξ₄₄Ξ₄₅Ξ₄₆Ξ₄₇Ξ₄₈Ξ₄₉∗∗∗∗Ξ₅₅Ξ₅₆Ξ₅₇Ξ₅₈Ξ₅₉∗∗∗∗∗Ξ₆₆Ξ₆₇Ξ₆₈Ξ₆₉∗∗∗∗∗∗Ξ₇₇Ξ₇₈Ξ₇₉∗∗∗∗∗∗∗Ξ₈₈Ξ₈₉∗∗∗∗∗∗∗∗Ξ₉₉<0, (3.3) where Ξ₁₁=𝑃^𝑇₂(𝐴−𝑀)+𝑃^𝑇₁₀+𝑃^𝑇₁₈+(𝐴−𝑀)^𝑇𝑃₂+𝑃₁₀+𝑃₁₈+ℎ𝑄₃+𝑄₁Ξ₁₂,=𝑃^𝑇₁−𝑃^𝑇₂+(𝐴−𝑀)^𝑇𝑃₃+𝑃₁₁+𝑃₁₉Ξ₁₃,=𝑃^𝑇₂(𝐵+𝑀)−𝑃^𝑇₁₈+(𝐴−𝑀)^𝑇𝑃₄+𝑃₁₂+𝑃₂₀Ξ₁₄,=−𝑃^𝑇₁₀+(𝐴−𝑀)^𝑇𝑃₅+𝑃₁₃+𝑃₂₁Ξ₁₅,=𝑃^𝑇₂𝐶+(𝐴−𝑀)^𝑇𝑃₆+𝑃₁₄+𝑃₂₂Ξ₁₆,=𝑃^𝑇₂𝑀−𝑃^𝑇₁₈+(𝐴−𝑀)^𝑇𝑃₇+𝑃₁₅+𝑃₂₃−𝐹^𝑇₁Ξ₁₇,=−𝑃^𝑇₁₀+(𝐴−𝑀)^𝑇𝑃₈+𝑃₁₆+𝑃₂₄−𝐹^𝑇₁Ξ₁₈,=𝑃^𝑇₂𝐷+(𝐴−𝑀)^𝑇𝑃₉+𝑃₁₇+𝑃₂₅−𝐹^𝑇₁Ξ₁₉,=𝐹^𝑇₁Ξ₂₂,=−𝑃^𝑇₃+𝑃^𝑇₃+𝜏₁𝑄₄+𝜏₂𝑄₅+𝑄₂Ξ₂₃,=𝑃^𝑇₃(𝐵+𝑀)−𝑃^𝑇₁₉−𝑃₄,Ξ₂₄=−𝑃^𝑇₁₁−𝑃^𝑇₅Ξ₂₅,=𝑃^𝑇₃𝐶−𝑃^𝑇₆,Ξ₂₆=𝑃^𝑇₃𝑀−𝑃^𝑇₁₉−𝑃₇−𝐹^𝑇₂Ξ₂₇,=−𝑃^𝑇₁₁−𝑃₈−𝐹^𝑇₂,Ξ₂₈=𝑃^𝑇₃𝐷−𝑃₉−𝐹^𝑇₂Ξ₂₉,=𝐹^𝑇₂Ξ₃₃,=𝑃^𝑇₄(𝐵+𝑀)−𝑃^𝑇₂₀+(𝐵+𝑀)^𝑇𝑃₄−𝑃₂₀−𝑄₁Ξ₃₄,=−𝑃^𝑇₁₂+(𝐵+𝑀)^𝑇𝑃₅−𝑃₂₁Ξ₃₅,=𝑃^𝑇₄𝐶+(𝐵+𝑀)^𝑇𝑃₆−𝑃₂₂Ξ₃₆,=𝑃^𝑇₄𝑀−𝑃^𝑇₂₀+(𝐵+𝑀)^𝑇𝑃₇−𝑃₂₃−𝐹^𝑇₃Ξ₃₇,=−𝑃^𝑇₁₂+(𝐵+𝑀)^𝑇𝑃₈−𝑃₂₄−𝐹^𝑇₃Ξ₃₈,=𝑃^𝑇₄𝐷+(𝐵+𝑀)^𝑇𝑃₉−𝑃₂₅−𝐹^𝑇₃Ξ₃₉,=𝐹^𝑇₃Ξ₄₄,=−𝑃^𝑇₁₃−𝑃₁₃Ξ₄₅,=𝑃^𝑇₅𝐶−𝑃₁₄Ξ₄₆,=𝑃^𝑇₅𝑀−𝑃^𝑇₂₁−𝑃₁₅−𝐹^𝑇₄Ξ₄₇,=−𝑃^𝑇₁₃−𝑃₁₆−𝐹^𝑇₄Ξ₄₈,=𝑃^𝑇₅𝐷−𝑃₁₇−𝐹^𝑇₄Ξ₄₉,=𝐹^𝑇₄Ξ₅₅,=𝑃^𝑇₆𝐶+𝐶^𝑇𝑃₆−𝑄₂Ξ₅₆,=𝑃^𝑇₆𝑀−𝑃^𝑇₂₂+𝐶^𝑇𝑃₇−𝐹^𝑇₅Ξ₅₇,=−𝑃^𝑇₁₄+𝐶^𝑇𝑃₈−𝐹^𝑇₅Ξ₅₈,=𝑃^𝑇₆𝐷+𝐶^𝑇𝑃₉−𝐹^𝑇₅Ξ₅₉,=𝐹^𝑇₅Ξ₆₆,=𝑃^𝑇₇𝑀−𝑃^𝑇₂₃+𝑀^𝑇𝑃₇−𝑃₂₃−𝐹^𝑇₆−𝐹₆Ξ₆₇,=−𝑃^𝑇₁₅+𝑀^𝑇𝑃₈−𝑃₂₄−𝐹^𝑇₆−𝐹^𝑇₇Ξ₆₈,=𝑃^𝑇₇𝐷+𝑀^𝑇𝑃₉−𝑃₂₅−𝐹^𝑇₆−𝐹^𝑇₈Ξ₆₉,=𝐹^𝑇₆Ξ₇₇,=−𝑃^𝑇₁₆−𝑃₁₆−𝐹^𝑇₇−𝐹₇Ξ₇₈,=𝑃^𝑇₈𝐷−𝑃₁₇−𝐹^𝑇₈−𝐹^𝑇₇Ξ₇₉,=𝐹^𝑇₇Ξ₈₈,=𝑃^𝑇₉𝐷+𝐷^𝑇𝑃₉−𝐹^𝑇₈−𝐹^𝑇₈Ξ₈₉,=𝐹^𝑇₈Ξ₉₉,1=−(ℎ𝑄₃+1𝜏₁𝑄₄+1𝜏₂𝑄₅). (3.4)(1)(2)(3)(4)(5)(6)(7)(8)(9)(10)(11)(12)(13)(14)(15)(16)(17)(18)(19)(20)(21)(22)(23)(24)(25)(26)(27)(28)(29)(30)(31)(32)(33)(34)(35)(36)(37)(38)(39)(40)(41)(42)(43)(44)(45)(46)(47)(48)(49)

Proof. Constructing a new Lyapunov functional candidate for system (3.2) as follows: 𝑉(𝑡)=𝑉₁(𝑡)+𝑉₂(𝑡)+𝑉₃(𝑡)+𝑉₄(𝑡)+𝑉₅(𝑡)+𝑉₆(𝑡), (3.5) where 𝑉₁(𝑡)=𝑌^𝑇(𝑡)𝑃𝑌(𝑡),𝑌(𝑡)=[𝑥^𝑇(𝑡),𝑥̇^𝑇(𝑡),𝑥^𝑇(𝑡−𝜏₁),𝑥^𝑇(𝑡−𝜏₂),𝑥̇^𝑇(𝑡−𝜏₂),(^𝑡_𝑡−𝜏1𝑥̇(𝑠)𝑑𝑠)^𝑇,(^𝑡_𝑡−𝜏2𝑥̇(𝑠)𝑑𝑠)^𝑇,(^𝑡_𝑡−ℎ𝑥(𝑠)𝑑𝑠)^𝑇]^𝑇,𝑉₂(𝑡)=^𝑡_𝑡−𝜏1𝑥^𝑇(𝑠)𝑄₁𝑥(𝑠)𝑑𝑠,𝑉₃(𝑡)=^𝑡_𝑡−𝜏2𝑥̇^𝑇(𝑠)𝑄₂𝑥̇(𝑠)𝑑𝑠,𝑉₄(𝑡)=^𝑡_𝑡−ℎ^𝑡_𝑠𝑥^𝑇(𝜉)𝑄₃𝑉𝑥(𝜉)𝑑𝜉𝑑𝑠,₅(𝑡)=^𝑡_𝑡−𝜏1^𝑡_𝑠𝑥̇^𝑇(𝜉)𝑄₄𝑥̇(𝜉)𝑑𝜉𝑑𝑠,𝑉₆(𝑡)=^𝑡_𝑡−𝜏2^𝑡_𝑠𝑥̇^𝑇(𝜉)𝑄₅𝑥̇(𝜉)𝑑𝜉𝑑𝑠, (3.6)𝑃𝑃=¹000000000000000000000000000000000000000000000000000000000000000. (3.7) Set 𝑃𝑃=¹𝑃0000000₂𝑃₃𝑃₄𝑃₅𝑃₆𝑃₇𝑃₈𝑃₉𝑃₁₀𝑃₁₁𝑃₁₂𝑃₁₃𝑃₁₄𝑃₁₅𝑃₁₆𝑃₁₇𝑃₁₈𝑃₁₉𝑃₂₀𝑃₂₁𝑃₂₂𝑃₂₃𝑃₂₄𝑃₂₅, (3.8) along the trajectories of system (3.2), the derivative of 𝑉(𝑡) is given by 𝑉̇(𝑡)=𝑉̇₁(𝑡)+𝑉̇₂(𝑡)+𝑉̇₃(𝑡)+𝑉̇₄(𝑡)+𝑉̇₅(𝑡)+𝑉̇₆(𝑡), (3.9) where 𝑉̇₁(𝑡)=2𝑌^𝑇(𝑡)𝑃𝑌̇(𝑡)=2𝑌^𝑇(𝑡)𝑃^𝑇×̇−̇̇𝑥(𝑡)𝑥(𝑡)+𝐶𝑥(𝑡−𝜏₂)+(𝐴−𝑀)𝑥(𝑡)+(𝐵+𝑀)𝑥(𝑡−𝜏₁)+𝐷^𝑡_𝑡−ℎ𝑥(𝑠)𝑑𝑠+𝑀^𝑡_𝑡−𝜏1𝑥̇(𝑠)𝑑𝑠𝑥(𝑡)−𝑥(𝑡−𝜏₂)−^𝑡_𝑡−𝜏2𝑥̇(𝑠)𝑑𝑠𝑥(𝑡)−𝑥(𝑡−𝜏₁)−^𝑡_𝑡−𝜏1𝑥̇(𝑠)𝑑𝑠=2𝑌^𝑇(𝑡)𝑃^𝑇𝑉̇0𝐼000000𝐴−𝑀−𝐼𝐵+𝑀0𝐶𝑀0𝐷𝐼00−𝐼00−𝐼0𝐼0−𝐼00−𝐼00𝑌(𝑡),₂(𝑡)=𝑥^𝑇(𝑡)𝑄₁𝑥(𝑡)−𝑥^𝑇(𝑡−𝜏₁)𝑄₁𝑥(𝑡−𝜏₁𝑉̇),₃(𝑡)=𝑥̇^𝑇(𝑡)𝑄₂𝑥̇(𝑡)−𝑥̇^𝑇(𝑡−𝜏₂)𝑄₂𝑥̇(𝑡−𝜏₂𝑉̇),₂(𝑡)+𝑉̇₃(𝑡)=𝑌^𝑇𝑄(𝑡)¹0000000∗𝑄₂000000∗∗−𝑄₁00000∗∗∗00000∗∗∗∗−𝑄₂000∗∗∗∗∗000∗∗∗∗∗∗00∗∗∗∗∗∗∗0𝑌(𝑡). (3.10)(3.11) By Lemma 2.3, similar to the disposal route in [34], we have 𝑉̇₄(𝑡)=ℎ𝑥^𝑇(𝑡)𝑄₃𝑥(𝑡)−^𝑡_𝑡−ℎ𝑥^𝑇(𝑠)𝑄₃𝑥(𝑠)𝑑𝑠≤ℎ𝑥^𝑇(𝑡)𝑄₃𝑥(𝑡)+^𝑡_𝑡−ℎ[−2𝑥^𝑇(𝑠)𝐹𝑌(𝑡)+𝑌^𝑇(𝑡)𝐹^𝑇𝑄₃⁻¹𝐹𝑌(𝑡)]𝑑𝑠=ℎ𝑥^𝑇(𝑡)𝑄₃𝑥(𝑡)−2(^𝑡_𝑡−ℎ𝑥(𝑠)𝑑𝑠)^𝑇𝐹𝑌(𝑡)+ℎ𝑌^𝑇(𝑡)𝐹^𝑇𝑄₃⁻¹𝐹𝑌(𝑡), (3.12) where 𝐹=[𝐹₁,𝐹₂,𝐹₃,𝐹₄,𝐹₅,𝐹₆,𝐹₇,𝐹₈]. Similarly, we have 𝑉̇₅(𝑡)=𝜏₁𝑥̇^𝑇(𝑡)𝑄₄𝑥̇(𝑡)−^𝑡_𝑡−𝜏1𝑥̇^𝑇(𝑠)𝑄₄𝑥̇(𝑠)𝑑𝑠≤𝜏₁𝑥̇^𝑇(𝑡)𝑄₄𝑥̇(𝑡)+^𝑡_𝑡−𝜏1[−2𝑥̇^𝑇(𝑠)𝐹𝑌(𝑡)+𝑌^𝑇(𝑡)𝐹^𝑇𝑄₄⁻¹𝐹𝑌(𝑡)]𝑑𝑠=𝜏₁𝑥̇^𝑇(𝑡)𝑄₄𝑥̇(𝑡)−2(^𝑡_𝑡−𝜏1𝑥̇(𝑠)𝑑𝑠)^𝑇𝐹𝑌(𝑡)+𝜏₁𝑌^𝑇(𝑡)𝐹^𝑇𝑄₄⁻¹𝑉̇𝐹𝑌(𝑡),₆(𝑡)=𝜏₂𝑥̇^𝑇(𝑡)𝑄₅𝑥̇(𝑡)−^𝑡_𝑡−𝜏2𝑥̇^𝑇(𝑠)𝑄₅𝑥̇(𝑠)𝑑𝑠≤𝜏₂𝑥̇^𝑇(𝑡)𝑄₅𝑥̇(𝑡)+^𝑡_𝑡−𝜏2[−2𝑥̇^𝑇(𝑠)𝐹𝑌(𝑡)+𝑌^𝑇(𝑡)𝐹^𝑇𝑄₅⁻¹𝐹𝑌(𝑡)]𝑑𝑠=𝜏₂𝑥̇^𝑇(𝑡)𝑄₅𝑥̇(𝑡)−2(^𝑡_𝑡−𝜏2𝑥̇(𝑠)𝑑𝑠)^𝑇𝐹𝑌(𝑡)+𝜏₂𝑌^𝑇(𝑡)𝐹^𝑇𝑄₅⁻¹𝐹𝑌(𝑡). (3.13)(3.14) From (3.12)–(3.14), we get 𝑉̇₄(𝑡)+𝑉̇₅(𝑡)+𝑉̇₆(𝑡)≤𝑌^𝑇(𝑡)ℎ𝑄₃0000−𝐹^𝑇₁−𝐹^𝑇₁−𝐹^𝑇₁∗𝜏₁𝑄₄+𝜏₂𝑄₅000−𝐹^𝑇₂−𝐹^𝑇₂𝐹^𝑇₂∗∗000−𝐹^𝑇₃−𝐹^𝑇₃−𝐹^𝑇₃∗∗∗00−𝐹^𝑇₄−𝐹^𝑇₄−𝐹^𝑇₄∗∗∗∗0−𝐹^𝑇₅−𝐹^𝑇₅−𝐹^𝑇₅∗∗∗∗∗−𝐹^𝑇₆−𝐹₆−𝐹^𝑇₇−𝐹^𝑇₆−𝐹^𝑇₈−𝐹^𝑇₆∗∗∗∗∗∗−𝐹^𝑇₇−𝐹^𝑇₇−𝐹^𝑇₈−𝐹^𝑇₇∗∗∗∗∗∗∗−𝐹^𝑇₈−𝐹^𝑇₈𝑌(𝑡)+𝑌^𝑇(𝑡)𝐹^𝑇(ℎ𝑄₃⁻¹+𝜏₁𝑄₄⁻¹+𝜏₂𝑄₅⁻¹)𝐹𝑌(𝑡). (3.15)
Hence, 𝑉̇(𝑡)≤𝑌^𝑇(𝑡)[Ξ^′+𝐹^𝑇(ℎ𝑄₃⁻¹+𝜏₁𝑄₄⁻¹+𝜏₂𝑄₅⁻¹)𝐹]𝑌(𝑡), (3.16) where Ξ^′=Ξ₁₁Ξ₁₂Ξ₁₃Ξ₁₄Ξ₁₅Ξ₁₆Ξ₁₇Ξ₁₈∗Ξ₂₂Ξ₂₃Ξ₂₄Ξ₂₅Ξ₂₆Ξ₂₇Ξ₂₈∗∗Ξ₃₃Ξ₃₄Ξ₃₅Ξ₃₆Ξ₃₇Ξ₃₈∗∗∗Ξ₄₄Ξ₄₅Ξ₄₆Ξ₄₇Ξ₄₈∗∗∗∗Ξ₅₅Ξ₅₆Ξ₅₇Ξ₅₈∗∗∗∗∗Ξ₆₆Ξ₆₇Ξ₆₈∗∗∗∗∗∗Ξ₇₇Ξ₇₈∗∗∗∗∗∗∗Ξ₈₈. (3.17) In views of Lemma 2.1 and (3.3), we have 𝑉̇(𝑡)<0, namely, there exists a positive scalar 𝜆>0 such that 𝑉̇(𝑡)≤−𝜆‖𝑌(𝑡)‖². According to [35], system (2.3) is asymptotically stable, this completes the proof.

Remark 3.2. Motivated by the results obtained in [34], free-weighting matrices 𝐹_𝑖(𝑖=1,2,…,8) are introduced in Theorem 3.1 so as to reduce the conservatism of the delay-dependent result. Moreover, more free-weighting matrices are introduced by the construction and disposal of 𝑉₁(𝑡), which may make the conservatism reduce further.

Remark 3.3. The transformation from system (2.3) to (3.2) enables us to utilize the information of the relationship among terms 𝑥(𝑡),𝑥(𝑡−𝜏₁),∫^𝑡_𝑡−𝜏1𝑥̇(𝑠)𝑑𝑠. Combined with the arbitrariness of matrix 𝑀, the conservatism of stability criterion is reduced further.

Remark 3.4. When 𝑀=𝐼, we can obtain the following simplified corollary.

Corollary 3.5. For given positive scalars 𝜏₁>0,𝜏₂>0,ℎ>0, the nominal system Σ₀ is asymptotically stable if ‖𝐶‖<1, and there exist positive definite matrices 𝑃₁,𝑄₁,𝑄₂,𝑄₃,𝑄₄,𝑄₅ and arbitrary matrices 𝑃_𝑖(𝑖=2,3,…,25), 𝐹_𝑖(𝑖=1,2,…,8) with appropriate dimensions such that the following linear matrix inequality is feasible: Ξ₂=Ξ^′₁₁Ξ^′₁₂Ξ^′₁₃Ξ^′₁₄Ξ^′₁₅Ξ^′₁₆Ξ^′₁₇Ξ^′₁₈Ξ₁₉∗Ξ₂₂Ξ^′₂₃Ξ₂₄Ξ₂₅Ξ^′₂₆Ξ₂₇Ξ₂₈Ξ₂₉∗∗Ξ^′₃₃Ξ^′₃₄Ξ^′₃₅Ξ^′₃₆Ξ^′₃₇Ξ^′₃₈Ξ₃₉∗∗∗Ξ₄₄Ξ₄₅Ξ^′₄₆Ξ₄₇Ξ₄₈Ξ₄₉∗∗∗∗Ξ₅₅Ξ^′₅₆Ξ₅₇Ξ₅₈Ξ₅₉∗∗∗∗∗Ξ^′₆₆Ξ^′₆₇Ξ^′₆₈Ξ₆₉∗∗∗∗∗∗Ξ₇₇Ξ₇₈Ξ₇₉∗∗∗∗∗∗∗Ξ₈₈Ξ₈₉∗∗∗∗∗∗∗∗Ξ₉₉<0, (3.18) where Ξ^′₁₁=𝑃^𝑇₂(𝐴−𝐼)+𝑃^𝑇₁₀+𝑃^𝑇₁₈+(𝐴−𝐼)^𝑇𝑃₂+𝑃₁₀+𝑃₁₈+ℎ𝑄₃+𝑄₁,Ξ^′₁₂=𝑃^𝑇₁−𝑃^𝑇₂+(𝐴−𝐼)^𝑇𝑃₃+𝑃₁₁+𝑃₁₉,Ξ^′₁₃=𝑃^𝑇₂(𝐵+𝐼)−𝑃^𝑇₁₈+(𝐴−𝐼)^𝑇𝑃₄+𝑃₁₂+𝑃₂₀,Ξ^′₁₄=−𝑃^𝑇₁₀+(𝐴−𝐼)^𝑇𝑃₅+𝑃₁₃+𝑃₂₁,Ξ^′₁₅=𝑃^𝑇₂𝐶+(𝐴−𝐼)^𝑇𝑃₆+𝑃₁₄+𝑃₂₂,Ξ^′₁₆=𝑃^𝑇₂−𝑃^𝑇₁₈+(𝐴−𝐼)^𝑇𝑃₇+𝑃₁₅+𝑃₂₃−𝐹^𝑇₁,Ξ^′₁₈=𝑃^𝑇₂𝐷+(𝐴−𝐼)^𝑇𝑃₉+𝑃₁₇+𝑃₂₅−𝐹^𝑇₁,Ξ^′₂₃=𝑃^𝑇₃(𝐵+𝐼)−𝑃^𝑇₁₉−𝑃₄,Ξ^′₂₆=𝑃^𝑇₃−𝑃^𝑇₁₉−𝑃₇−𝐹^𝑇₂,Ξ^′₃₃=𝑃^𝑇₄(𝐵+𝐼)−𝑃^𝑇₂₀+(𝐵+𝐼)^𝑇𝑃₄−𝑃₂₀−𝑄₁,Ξ^′₃₄=−𝑃^𝑇₁₂+(𝐵+𝐼)^𝑇𝑃₅−𝑃₂₁,Ξ^′₃₅=𝑃^𝑇₄𝐶+(𝐵+𝐼)^𝑇𝑃₆−𝑃₂₂,Ξ^′₃₆=𝑃^𝑇₄−𝑃^𝑇₂₀+(𝐵+𝐼)^𝑇𝑃₇−𝑃₂₃−𝐹^𝑇₃,Ξ^′₃₇=−𝑃^𝑇₁₂+(𝐵+𝐼)^𝑇𝑃₈−𝑃₂₄−𝐹^𝑇₃,Ξ^′₃₈=𝑃^𝑇₄𝐷+(𝐵+𝐼)^𝑇𝑃₉−𝑃₂₅−𝐹^𝑇₃,Ξ^′₄₆=𝑃^𝑇₅−𝑃^𝑇₂₁−𝑃₁₅−𝐹^𝑇₄,Ξ^′₅₆=𝑃^𝑇₆−𝑃^𝑇₂₂+𝐶^𝑇𝑃₇−𝐹^𝑇₅,Ξ^′₆₆=𝑃^𝑇₇−𝑃^𝑇₂₃+𝑃₇−𝑃₂₃−𝐹^𝑇₆−𝐹₆,Ξ^′₆₇=−𝑃^𝑇₁₅+𝑃₈−𝑃₂₄−𝐹^𝑇₆−𝐹^𝑇₇,Ξ^′₆₈=𝑃^𝑇₇𝐷+𝑃₉−𝑃₂₅−𝐹^𝑇₆−𝐹^𝑇₈. (3.19)

Remark 3.6. Based on Theorem 3.1 and Corollary 3.5, by using Lemmas 2.1 and 2.2, we can perform the robust asymptotic stability analysis for system (2.1) with uncertainty Δ𝐴(𝑡),Δ𝐵(𝑡),Δ𝐷(𝑡) as follows.

Theorem 3.7. For any given matrix 𝑀, scalars 𝜏₁>0,𝜏₂>0,ℎ>0, the original system Σ is robustly and asymptotically stable if ‖𝐶‖<1, and there exist positive definite matrices 𝑃₁,𝑄₁,𝑄₂,𝑄₃,𝑄₄,𝑄₅, positive scalar 𝛿, and arbitrary matrices 𝑃_𝑖(𝑖=2,3,…,25), 𝐹_𝑖(𝑖=1,2,…,8) with appropriate dimensions such that the following linear matrix inequality is feasible: Ξ₃=Ξ₁₁+𝛿𝐸^𝑇_𝑎𝐸_𝑎Ξ₁₂Ξ₁₃Ξ₁₄Ξ₁₅Ξ₁₆Ξ₁₇Ξ₁₈Ξ₁₉𝑃^𝑇₂𝐾∗Ξ₂₂Ξ₂₃Ξ₂₄Ξ₂₅Ξ₂₆Ξ₂₇Ξ₂₈Ξ₂₉𝑃^𝑇₃𝐾∗∗Ξ₃₃+𝛿𝐸^𝑇_𝑏𝐸_𝑏Ξ₃₄Ξ₃₅Ξ₃₆Ξ₃₇Ξ₃₈Ξ₃₉𝑃^𝑇₄𝐾∗∗∗Ξ₄₄Ξ₄₅Ξ₄₆Ξ₄₇Ξ₄₈Ξ₄₉𝑃^𝑇₅𝐾∗∗∗∗Ξ₅₅Ξ₅₆Ξ₅₇Ξ₅₈Ξ₅₉𝑃^𝑇₆𝐾∗∗∗∗∗Ξ₆₆+𝛿𝐸^𝑇_𝑐𝐸_𝑐Ξ₆₇Ξ₆₈Ξ₆₉𝑃^𝑇₇𝐾∗∗∗∗∗∗Ξ₇₇Ξ₇₈Ξ₇₉𝑃^𝑇₈𝐾∗∗∗∗∗∗∗Ξ₈₈Ξ₈₉𝑃^𝑇₉𝐾∗∗∗∗∗∗∗∗Ξ₉₉0∗∗∗∗∗∗∗∗∗−𝛿𝐼<0. (3.20)

Proof. Replacing 𝐴,𝐵,𝐷 in (3.3) with 𝐴+𝐾𝐹(𝑡)𝐸_𝑎, 𝐵+𝐾𝐹(𝑡)𝐸_𝑏, and 𝐷+𝐾𝐹(𝑡)𝐸_𝑑, respectively, (3.3) for system (2.1) is equivalent to the following form: Ξ+Π^𝑇₁𝐹^𝑇(𝑡)Π₂+Π^𝑇₂𝐹(𝑡)Π₁<0, (3.21) where Π₁=[𝐾^𝑇𝑃₂,𝐾^𝑇𝑃₃,𝐾^𝑇𝑃₄,𝐾^𝑇𝑃₅,𝐾^𝑇𝑃₆,𝐾^𝑇𝑃₇,𝐾^𝑇𝑃₈,𝐾^𝑇𝑃₉], Π₂=[𝐸_𝑎,0,𝐸_𝑏,0,0,𝐸_𝑐,0,0,0]. From Lemma 2.2, a sufficient condition for (3.20) is that there exists a positive scalar 𝛿>0 such that Ξ+𝛿⁻¹Π^𝑇₁Π₁+𝛿Π^𝑇₂Π₂<0. (3.22) In views of Lemma 2.1, we can easily obtain this conclusion, this completes the proof.

Corollary 3.8. For given positive scalars 𝜏₁>0,𝜏₂>0,ℎ>0, the original system Σ is robustly and asymptotically stable if ‖𝐶‖<1, and there exist positive definite matrices 𝑃₁,𝑄₁,𝑄₂,𝑄₃,𝑄₄,𝑄₅, positive scalar 𝛿, and arbitrary matrices 𝑃_𝑖(𝑖=2,3,…,25), 𝐹_𝑖(𝑖=1,2,…,8) with appropriate dimensions such that the following linear matrix inequality is feasible: Ξ₄=Ξ^′₁₁+𝛿𝐸^𝑇_𝑎𝐸_𝑎Ξ^′₁₂Ξ^′₁₃Ξ^′₁₄Ξ^′₁₅Ξ^′₁₆Ξ^′₁₇Ξ^′₁₈Ξ₁₉𝑃^𝑇₂𝐾∗Ξ₂₂Ξ^′₂₃Ξ₂₄Ξ₂₅Ξ^′₂₆Ξ₂₇Ξ₂₈Ξ₂₉𝑃^𝑇₃𝐾∗∗Ξ^′₃₃+𝛿𝐸^𝑇_𝑏𝐸_𝑏Ξ^′₃₄Ξ^′₃₅Ξ^′₃₆Ξ^′₃₇Ξ^′₃₈Ξ₃₉𝑃^𝑇₄𝐾∗∗∗Ξ₄₄Ξ₄₅Ξ^′₄₆Ξ₄₇Ξ₄₈Ξ₄₉𝑃^𝑇₅𝐾∗∗∗∗Ξ₅₅Ξ^′₅₆Ξ₅₇Ξ₅₈Ξ₅₉𝑃^𝑇₆𝐾∗∗∗∗∗Ξ^′₆₆+𝛿𝐸^𝑇_𝑐𝐸_𝑐Ξ^′₆₇Ξ^′₆₈Ξ₆₉𝑃^𝑇₇𝐾∗∗∗∗∗∗Ξ₇₇Ξ₇₈Ξ₇₉𝑃^𝑇₈𝐾∗∗∗∗∗∗∗Ξ₈₈Ξ₈₉𝑃^𝑇₉𝐾∗∗∗∗∗∗∗∗Ξ₉₉0∗∗∗∗∗∗∗∗∗−𝛿𝐼<0. (3.23)

4. Numerical Examples

In this section, some numerical examples will be presented to show the validity and improvement of the main results derived earlier.

Example 4.1. Consider the following neutral system presented in Park and Kwon [28]: 𝑥̇(𝑡)−𝐶𝑥̇(𝑡−𝜏₂)=𝐴𝑥(𝑡)+𝐵𝑥(𝑡−𝜏₁)+𝐷^𝑡_𝑡−ℎ𝑥(𝑠)𝑑𝑠, (4.1) with 𝐴=−3−210,𝐵=−0.50.10.30,𝐶=0,𝐷=0. (4.2) Set 𝑀=(⁻¹⁰₀₋₂). For comparisons, we calculate the allowable upper bound of 𝜏₁ for which the asymptotic stability is guaranteed. For this example, Table 1 shows that Theorem 3.1 obtained in this paper is less conservative than the related results obtained in [12, 24–28].

Table 1 

Stability bounds of time delays (Example 4.1).

Example 4.2. Consider the following neutral system studied in He et al. [32]: 𝑥̇(𝑡)−𝐶𝑥̇(𝑡−𝜏₂)=𝐴𝑥(𝑡)+𝐵𝑥(𝑡−𝜏₁)+𝐷^𝑡_𝑡−ℎ𝑥(𝑠)𝑑𝑠, (4.3) with 𝐴=−0.90.20.1−0.9,𝐵=−1.1−0.2−0.1−1.1,𝐶=−0.200.2−0.1,𝐷=0. (4.4) Set 𝑀=(⁻¹⁰₀₋₂).
For this example, we calculate the allowable upper bound of 𝜏₁ for which the asymptotic stability is guaranteed. Table 2 shows that Theorem 3.1 obtained in this paper is less conservative than the related results obtained in [12, 13, 24, 28–33].

Table 2 

Some comparison for allowable upper bounds on 𝝉_𝟏 (Example 4.2).

Example 4.3. Consider the neutral system studied in Liu et al. [12]: 𝑥̇(𝑡)−𝐶𝑥̇(𝑡−𝜏₂)=(𝐴+Δ𝐴(𝑡))𝑥(𝑡)+(𝐵+Δ𝐵(𝑡))𝑥(𝑡−𝜏₁)+(𝐷+Δ𝐷(𝑡))^𝑡_𝑡−ℎ𝑥(𝑠)𝑑𝑠, (4.5) with 𝐴=−200−15,𝐵=13−31,𝐶=𝑐𝑐00𝑐𝑐,𝐷=10.5−0.51,0≤𝑐𝑐<1. (4.6) For the convenience of comparison, let Δ𝐴(𝑡)=Δ𝐵(𝑡)=Δ𝐷(𝑡)=0. Set 𝑀=(⁻¹⁰₀₋₂). The comparative results between Theorem 3.1 with the result obtained in [12] are given in Table 3. When [Δ𝐴(𝑡),Δ𝐵(𝑡),Δ𝐷(𝑡)]=𝐾𝐹(𝑡)[𝐸_𝑎,𝐸_𝑏,𝐸_𝑑], (4.7) where 𝐾=𝐼,𝐸_𝑎=𝐸_𝑏=𝐸_𝑑=0.2𝐼, Table 4 shows that the robust stability criterion obtained in this paper is also less conservative than the related result obtained in [12]. From the simulation figures (see Figures 1–4), one can see that the results derived in this paper are valid.

Table 3 

Calculated allowable size of distributed delay 𝐡 (Δ𝐀(𝐭)=Δ𝐁(𝐭)=Δ𝐃(𝐭)=𝟎) (Example 4.3).

Table 4 

Calculated allowable size of distributed delay 𝐡 (𝐊=𝐈,𝐄_𝐚=𝐄_𝐛=𝐄_𝐝=𝟎.𝟐𝐈) (Example 4.3).

Figure 1 

State trajectories of system (12) with 𝑐𝑐=0.4, ℎ=𝜏1=𝜏2=1.60, 𝐸𝑎=𝐸𝑏=𝐸𝑑=0.2𝐼,𝐾=𝐼.

Figure 2 

State trajectories of system (12) with 𝑐𝑐=0.4, 𝜏1=𝜏2=0.3, ℎ=1.94, 𝐸𝑎=𝐸𝑏=𝐸𝑑=0.2𝐼,𝐾=𝐼.

Figure 3 

State trajectories of system (12) with 𝑐𝑐=0.4, 𝜏1=𝜏2=0.3, ℎ=2.61, 𝐸𝑎=𝐸𝑏=𝐸𝑑=0.

Figure 4 

State trajectories of system (12) with 𝑐𝑐=0.4, 𝜏1=𝜏2=ℎ=1.97, 𝐸𝑎=𝐸𝑏=𝐸𝑑=0.

Example 4.4. Consider the following uncertain neutral system [32, 33, 36]: 𝑥̇(𝑡)−𝐶𝑥̇(𝑡−𝜏)=[𝐴+𝐾𝐹(𝑡)𝐸_𝑎]𝑥(𝑡)+[𝐵+𝐾𝐹(𝑡)𝐸_𝑏]𝑥(𝑡−𝜏), (4.8) where 𝐴=(⁻²⁰_0−0.9), 𝐵=(⁻¹⁰₋₁₋₁), 𝐶=(^𝑐0_0𝑐),0≤𝑐<1, 𝐿=(^0.20_00.2), 𝐸_𝑎=𝐸_𝑏=(¹⁰₀₁). Table 5 gives out the related comparative results, from which one can see that Theorem 3.7 obtained in this paper is less conservative than those established in [32, 33, 36].

Table 5 

Comparison of 𝝉_𝐦𝐚𝐱 using different methods (Example 4.4).

5. Conclusion

By constructing a new Lyapunov function, some new robust stable criteria for a class of neutral control systems with mixed delays are obtained. These criteria are formulated in the forms of linear matrix inequalities. Compared with some previous publications, our results are less conservative. Numerical examples and simulations show that our results are valid.

Acknowledgment

This work was supported by the program for New Century Excellent Talents in University (NCET-06-0811), the National Basic Research Program of China (2010CB732501), and the Research Fund for the Doctoral Program of Guizhou College of Finance and Economics (200702).