Mao Wang，Yan－Ling Wei，Zhao－Lan He，Le Xiao

(1.Space Control and Inertial Technology Research Center，Harbin Institute of Technology，Harbin 150001，China; 2.School of Automation，Harbin University of Science and Technology，Harbin 150080，China)

1 Introduction

Itis known thattime-delays are frequently encountered in many practical engineering systems，such as biological systems，automatic control systems，neural networks and nuclear reactors and so on［1－4］.It has been well recognized that time-delay is one of the main sources of instability， oscillation， and performance degradation.Therefore， the stability analysis of time-delay system has been widely studied in the past decades，and many sophisticated results have been obtained in Refs.［5－7］.

On the other hand，neutral systems，as a special case of time-delay systems，have received considerable attention due to theirwide applicationsin many dynamic systems.As we know，nonlinear perturbations appear owing to slowly varying parameters of systems，or noise of the environment，which can also incur instability and poor performance of systems as timedelays.Correspondingly，a number of robust stability analysis conditions，including delay-independent and delay-dependent types，have been developed for this class of systems with time-varying delay and nonlinear perturbations［8－16］.To mention a few， Goubet-Batholomeus et al.［11］investigated the stability analysis problem for neutral systems by using matrix properties and decomposition technique.It is noted that the matrix measure should be negative which may bring to much conservatism.Cao and Lam［12］considered the stability analysis problem for neutral systems with time-varying delays and nonlinear perturbations based on a model transformation technique.It is also noted that the model transformations often introduce additionaldynamics which lead to relatively conservative results.Han and Yu［13］addressed the delay-dependent robust stability analysisproblem for linearneutralsystems with nonlinear parameter perturbations.Although these works promote the studies of neutral systems，the approach developed still has room for improvement in terms of conservatism.In fact，conservatism reduction of the stability analysis criteria is a critical issue.It has been shown that a bounding technique［17］or model transformation［18］may increase conservatism of stability analysis conditions.By contrast，the free-weighting matrix method contributes to the conservatism reduction of the analysis and synthesis of neutral systems［19］.In the meantime，choosing a proper Lyapunov-Krasovskii functional and estimating the upper bound of its time derivative are also crucial in developing the stability analysis criteria.

In this paper，the delay-dependent robust stability analysis problem will be investigated for neutral systems with interval time-varying delays and nonlinear perturbations.The nonlinear perturbations are with time-varying but norm-bounded characteristics.Based on a Lyapunov-Krasovskii functional together with a free-weighting matrices technique，new delay-rangedependent stability criteria are formulated in terms of LMIs.Finally，the simulation studies are provided to illustrate the effectiveness and superiority ofthe proposed method over the existing ones.Compared with the existing results on the stability analysis criteria for neutral systems with time-varying delay and nonlinear perturbations，the main contributions of the work in this paper are twofold:1)The time-delay is assumed to vary in a range，and its lower bound is not restricted to be 0，which is expected to be more general and thus more practicable.2)By applying a free-weighting matrices technique to estimate the upper bound of the derivative of Lyapunov functional，which do not ignore any useful terms，new delay-dependent conditions with less conservatism are obtained for the stability analysis of neutral systems with time-varying delay.

The notation used throughout this paper is as follows.Rnis the n－dimensional Euclidean space;Rn×mis the space of all n×m－dimensional matrix.For a matrix A，the transpose is denoted by AT.P ＞ 0 (respectively，P≥ 0)，means that P is positive definite (respectively，positive semi-definite).I denotes the identity matrix.An asterisk(*)is used to represent the elements below the main diagonal of a symmetric matrix.Matrices，if not clearly stated，are assumed to have compatible dimensions.

2 Problem Statements

Considering the following neutral system with time-varying delays and nonlinear perturbations

where x(t)∈Rnis the state vector;{A0，A1，A2}∈Rn×nare constant matrices;φ(t)and φ(t)are the initial condition functions that are continuously differentiable on［－max(hM，τM)，0］，and the timevarying delays h(t)and τ(t)satisfy

where hm，hM，hD，τMand τDare constants;Δf0(t，x(t))，Δf1(t，x(t－h(huán)(t)))and Δf2(t，˙x(t－τ(t))) are known nonlinear perturbations， satisfying Δf0(t，0)=0，Δf1(t，0)=0，Δf2(t，0)=0 and the following properties:

(A1)Nonlinear time-varying perturbations，

where αi(i∈ {0，1，2})aregiven nonnegative constants.

(A2)Parametric norm-bounded perturbations，

where G，Ei，i∈ {0，1，2}are some given constant matrices;F(t)is an unknown and maybe time-varying realmatrix with Lebesgue measurable elements satisfying

In order to obtain the main results，the lemmas are firstly introduced as follows:

Lemma 1(Ref.［20］)Let A，G，E and F(t)be real matrices satisfying FT(t)F(t)≤I.Then for any symmetric positive definite matrix P＞0 and a scalar δ＞0 such that δI－GTPG＞0 and

Lemma 2(Ref.［21］)Let Q=QT，H，E and F(t)satisfying FT(t)F(t)≤ I areappropriate dimensional matrices，then the inequality as follows:

is true，if and only if the following inequality holds for any ε＞0

3 Main Results

With nonlinear time-varying perturbations(4)，now a new delay-dependent stability analysis condition for neutral delay systems(1)is proposed.

Theorem 1 For given scalars α0，α1，α2，system (1)with mixed time-varying delays and uncertainties satisfying Eq.(4)is asymptotically stable，if‖A2‖+ α2＜1，and there exist symmetric matrices Pi＞0，X≥0，Y≥0，Z≥0 and proper dimensioned matrices Lj，Mj，Nj，Sj，Tj，and scalars ε0，ε1，ε2，i=0，1，…，9，j=1，2，…，6，such that the following LMIs hold:

where

Proof Choosing the Lyapunov-Krasovskii functional candidate for the system(1)as follows:

For notation simplification，Δf0(t，x(t))，Δf1(t，x(t－h(huán)(t)))and Δf2(t，˙x(t－τ(t)))are replaced by Δf0，Δf1and Δf2，respectively.Taking the time derivative of V(t)along the trajectories of system(1) yields，

Then by using Newton-Leibniz formula，for any matrices L，M，N，S and T with appropriate dimensions，the following equations hold:

Moreover， for matrices X，Y and Z with appropriate dimensions，we have

On the other hand，for any scalars ε0＞0，ε1＞0，ε2＞0，it follows from Eq.(4)that

Then combining Eqs.(10)－(21)yields

where

By applying the Schur complement，it is easy to see that ξT(t)Θξ(t)+ζT(t)ΓTRΓζ(t) ＜ 0 is equivalent to Eq.(7).Thus，from Theorem 1.6 in Ref.［3］，if‖A2‖+α2＜1，and Eqs.(7)－(8)hold，then Eq.(22)implies that there exists a scalar λ＞0 such that≤λ‖x(t)‖2.Therefore，system(1) with nonlineartime-varyingperturbations(4)is asymptotically stable based on the Lyapunov stability theory.This completes the proof.

Remark 1 In Ref.［6］，the derivative of(hM－dsdθ is estimated，and theis neglected，which may lead to more orless conservatism.By contrast，a new estimation on the upper bound of the time derivative of V(t)without ignoring any useful terms is employed in the proof of Theorem 1，which may reduce the conservatism of the stability analysis criteria.

Next，we propose a delay-dependent condition for the asymptotic stability of system(1)with normbounded uncertainties Eq.(5).

Theorem 2 System(1)with mixed timevarying delays and uncertainties satisfying Eq.(5)is asymptotically stable if there exist symmetric matrices Pi＞ 0，X≥ 0，Y≥ 0，Z≥ 0，and appropriate dimensioned matrices Lj，Mj，Nj，Sj，Tj，and scalars δ＞0，ε＞0，i=0，1，…，9，j=1，2，3，such that the following LMIs hold:

where

and d，Ξ，Σ，R are defined in Eq.(7).

Proof Firstly，based on Lemma 1 we can get

If Eq.(23)issatisfied，wehave(A2+ GF(t)E2)T(A2+GF(t)E2)≤ I，which implies‖A2+GF(t)E2‖ ＜1.

Next，by removing the 4－6 rows and 4－6 columns of Θ in Eq.(7)and replacing A0，A1and A2by A0+ GF(t)E0，A1+GF(t)E1and A2+GF(t)E2respectively，then the inequality in Eq.(7)for system (1)with norm-bounded uncertainties Eq.(5)is equivalent to the following condition

where

Using Lemma 2， a sufficient condition guaranteeing Eq.(26)is that there exists a scalar ε＞0 satisfying

By applying Schur complement，Eq.(27)is equivalent to Eq.(24)，and the condition form(25)is similar to the condition form(8).Therefore，we can conclude that the system(1)with mixed time-varying delays and norm-bounded uncertainties Eq.(5)is asymptotically stable based on the conditions(23)－(25).The proof is completed.

Remark 2 When P8=P9=0，˙τ(t)≤τD＜1 and unknown τMin Theorem 2，the stability analysis criterion of neutral system(1)with norm-bounded uncertainties Eq.(5)and time-varying delays condition (2)can be obtained from the conditions(23)-(25) as Corollary 1 in the following.

Corollary 1 System(1)with mixed timevarying delays satisfying condition(2)，˙τ(t)≤τD＜1 and uncertainties Eq.(5)is asymptotically stable if there exist symmetric matrices Pi＞0，X≥0，Y≥0， and appropriate dimensioned matrices Lj，Mj，Nj，and scalars δ＞0，ε＞0，i=0，1，…，7，j=1，2，3，such that the condition(23)and the following LMIs hold:

where

4 Numerical Examples

Example 1 Consideringsystem (1)with nonlinear time-varying perturbations Eq.(4)and the following parameters(the study in Ref.［13］):

where 0≤|c|＜1 and ε0≥0，ε1≥0，ε2≥0.With c=0.1，ε1= 0.1，τM= 1，τD= 0，hD= 0.5，to compare with Refs.［5，13－14］，it is assumed that hm=0，for different value of ε2，and apply Theorem 1 to calculate the maximal allowable value hMwhich guarantees the asymptotic stability of the system.Table 1 lists the numerical results for different ε2，ε0= 0 and ε0=0.1，respectively.From Table 1，it can be seen that the allowable delay hMdecreases as ε2increases.Moreover，it is easy to see the stability criteria proposed in this paper are less conservative than those in Refs.［5，13－14］.

Table 1 Maximum upper bound of hMwith τD=0，hD= 0.5，and different values of ε2

Example 2 Considering the system(1)with perturbations Eq.(5)as follows(the study in Ref.［13］):

where 0≤|c|＜1 and δ1，δ2，δ3and δ4are unknown parameters satisfying:

Case 1 With hD=0.1，τD=0，unknown τM，the maximum values of hMare listed in Table 2 for different values of c by applying the criteria in Refs.［5－6，13，22］and Corollary 1.It can be observed from Table 2 that the maximum allowable delay hMdecreases as c increases.

Table 2 Maximum upper bound of hMwith different c

Case 2 With c=0.1，τD=0.1，unknown τM，the maximum values of hMis listed in Table 3 for different values of hDby using the criteria in Refs.［5－6］and Corollary 1.It can be observed from Table 3 that the results obtained in this paper are less conservative than those in Refs.［5－6］.

Table 3 Maximum upper bound of hMwith τD=0.1 and different hD

5 Conclusions

The problem of delay-dependent robust stability analysis for a class of neutral systems with interval time-varying delays and nonlinear perturbations has been studied in this paper. Such nonlinear perturbations are with time-varying but norm-bounded characteristics.Based on a novel Lyapunov-Krasovskii functionaltogetherwith a free-weighting matrices technique，the improved delay-dependentstability criteria have been developed in terms of linear matrix inequalities(LMIs).Numerical examples have been given to demonstrate the effectiveness and less conservatism of the proposed approach over the recent literatures.

［1］Gu K，Kharitonov V L，Chen J.Stability of Time-delay Systems.Boston:Birkhauser，2003.

［2］Hale J K，Verduyn Lunel S M.Introduction to Functional Differential Equations.New York:Springer-Verlag，1993.

［3］ Kolmanovskii V B，Myshkis A.Applied Theory of Functional Differential Equations.Netherlands:Kluwer Academic Publishers，1992.

［4］Kwon O M，Park J H，Lee S M.On stability criteria for uncertain delay-differential systems of neutral type with time-varying delays. Applied Mathematics and Computation，2008，197(2):864－873.

［5］Qiu F，Cui B，Ji Y.Further results on robust stability of neutralsystem with mixed time-varying delays and nonlinear perturbations.Nonlinear Analysis:Real World Applications，2010，11(2):895－906.

［6］Yu K，Lien C.Stability criteria for uncertain neutral systems with interval time-varying delays.Chaos，Solitons and Fractals，2008，38(3):650－657.

［7］Yue D，Peng C，Tian G.Guaranteed cost control of linear systems over networks with state and input quantisations.IEE Proceedings-Control Theory and Applications，2006，153(6):658－664.

［8］Kwon O M，Park J H，Lee S M.On delay-dependent robust stability of uncertain neutral systems with interval timevarying delays.Applied Mathematics and Computation，2008，203(2):843－853.

［9］Park J H.Novel robust stability criterion for a class of neutral systems with mixed delays and nonlinear perturbations.Applied Mathematicsand Computation，2005，161(2):413－421.

［10］Zou Z，Wang Y.New stability criterion for a class of linear systems with time-varying delay and nonlinear perturbations.IEE Proceedings－ControlTheory and Applications，2006，153(5):623－626.

［11］Goubet-Batholomeus A，Dambrine M，Richard J P.Stability of perturbed systems with time-varying delays.Systems＆Control Letters，1997，31(3):155－163.

［12］Cao Y Y，Lam J.Computation of robust stability bounds for time-delay systems with nonlinear time-varying perturbations.International Journal of Systems Science，2000，31(3):359－365.

［13］Han Q，Yu L.Robust stability of linear neutral systems with nonlinear parameter perturbations.IEE Proceedings-Control Theory and Applications，2004，151(5):539－546.

［14］Zhang W，Yu L.Delay-dependent robust stability of neutral systems with mixed delays and nonlinear perturbations.Acta Automatica.Sinica，2007，33(8): 863－866.

［15］Zhang W，Cai X，Han Z.Robust stability criteria for systems with interval time-varying delay and nonlinear perturbations.JournalofComputationaland Applied Mathematics，2010，234(1):174－180.

［16］Lakshmanan S，Senthilkumar T，Balasubramaniam P.Improved results on robust stability of neutral systems with mixed time-varying delays and nonlinear perturbations.Applied Mathematical Modelling，2011，35(11):5355－5368.

［17］Moon Y S，Park P，Kwon W H，et al.Delay-dependent robust stabilization of uncertain state-delayed systems.International Journal of Control，2001，74(14):1447－1455.

［18］Fridman E，Shaked U.Delay-dependent stability and H∞control:constant and time-varying delays.International Journal of Control，2003，76(1):48－60.

［19］Wu M，He Y，She J.Delay-dependent criteria for robust stability of time-varying delay systems.Automatica，2004，40(8):1435－1439.

［20］Li X，de Souza C E.Criteria for robust stability and stabilization of uncertain linear systems with state delay.Automatica，1997，33(9):1657－1662.

［21］Xie L.Output feedback H∞control of systems with parameter uncertainty.International Journal of Control，1996，63(4):741－750.

［22］Zhao Z，Wang W，Yang B.Delay and its derivative dependent robuststability ofneutralcontrolsystem.Applied Mathematics and Computation，2007，187(2): 1326－1332.