Improved Robust Stability Criteria of Uncertain Neutral Systems with Mixed Delays

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Volume 2009, Article ID 294845,18pages doi:10.1155/2009/294845

Research Article

Improved Robust Stability Criteria of Uncertain Neutral Systems with Mixed Delays

Zixin Liu,

^{1, 2}

Shu L ¨u,

¹

Shouming Zhong,

¹

and Mao Ye

³

1School of Applied Mathematics, University of Electronic Science and Technology of China, Chengdu 610054, China

2School of Mathematics and Statistics, Guizhou College of Finance and Economics, Guiyang 550004, China

3School of Computer Science and Engineering, University of Electronic Science and Technology of China, Chengdu 610054, China

Correspondence should be addressed to Zixin Liu,[email protected] Received 1 March 2009; Revised 5 August 2009; Accepted 1 September 2009 Recommended by Wolfgang Ruess

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 inequalitiesLMIs. Compared with some previous publications, our results are less conservative. Simulation examples are presented to illustrate the improvement of the main results.

Copyrightq2009 Zixin Liu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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

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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 categories21: that is, delay-independent criteria and delay-dependent criteria. As pointed out in12, 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 LMItechnique 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 xt,xt, xt˙ − τ₁, xt − τ₂, ˙xt − τ₂, _t

t−τ1xsds,˙ _t

t−τ2xsds. 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 delays4as follows:

Σ:

⎧⎪

⎪⎪

⎨

⎪⎪

⎪⎩

xt˙ −Cxt˙ −τ₂ A ΔAtxt B ΔBtxt−τ₁ D ΔDt

_t

t−hxsds, t >0, xt φt, t∈−τ,0,

2.1

wherext∈ Rⁿis the state vector;A, B, C∈ R^n×nrepresent the weighting matrices;τ₁, τ₂are discrete delays;his distributed delay;φtis initial condition which is continuous on interval

−τ,0, where τ max{τ1, τ2, h}; ΔAt,ΔBt,ΔDtdenote the time-varying structured uncertainties which are of the following form:

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ΔAt,ΔBt,ΔDt KFtEa, Eb, Ed, 2.2

where K, E_a, E_b, E_d are the known constant matrices with appropriate dimensions;Ft is unknown continuous time-varying matrix function satisfyingF^TtFt≤I,for allt≥0.

The nominalΣ0ofΣcan be defined as

Σ0:

⎧⎪

⎨

⎪⎩

xt˙ −Cxt˙ −τ2 Axt Bxt−τ1 D _t

t−hxsds, t >0, xt φt, t∈−τ,0.

2.3

For further discussion, we first introduce the following lemmas.

Lemma 2.1see22. Given constant symmetric matricesΣ1,Σ2,Σ3whereΣ^T₁ Σ1and 0<Σ2

Σ^T₂, thenΣ1 Σ^T₃Σ⁻¹₂ Σ3<0 if and only if

Σ1 Σ^T₃ Σ3 −Σ2

<0, or

−Σ2 Σ3

Σ^T₃ Σ1

<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

QHFEE^TF^TH^T <0, 2.5

for allFsatisfyingF^TF≤Rif and only if there exists a positive numberε >0, such that

Qε⁻¹HH^TεE^TRE <0. 2.6

Lemma 2.3. For any real vectorX, Yand positive definite matrixΣ>0 with appropriate dimensions, it follows that

2X^TY ≤X^TΣXY^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

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2.3withΔAt 0,ΔBt 0,ΔDt 0. In order to introduce free-weighting matrix, we can use the following fact:

M

xt−xt−τ1− _t

t−τ1

xsds˙

0, 3.1

whereMis an arbitrary matrix with appropriate dimensions. Substituting zero equation3.1 into system2.3, the original system can be transformed into the following form:

xt˙ −Cxt˙ −τ2 A−Mxt BMxt−τ1 D

_t

t−hxsdsM _t

t−τ1

xsds,˙ t >0, xt φt, t∈−τ,0.

3.2

For the asymptotic stability of system3.2, we can obtain the following results.

Theorem 3.1. For any given matrixM, scalarsτ1 > 0, τ2 > 0, h > 0, the nominal systemΣ0 is asymptotically stable if C < 1,and there exist positive definite matrices P₁, Q₁, Q₂, Q₃, Q₄, Q₅, arbitrary matricesP_ii 2,3, . . . ,25,F_ii 1,2, . . . ,8with appropriate dimensions such that the following linear matrix inequality is feasible:

Ξ1

⎛

⎜⎜

⎝

Ξ11 Ξ12 Ξ13 Ξ14 Ξ15 Ξ16 Ξ17 Ξ18 Ξ19

∗ Ξ22 Ξ23 Ξ24 Ξ25 Ξ26 Ξ27 Ξ28 Ξ29

∗ ∗ Ξ33 Ξ34 Ξ35 Ξ36 Ξ37 Ξ38 Ξ39

∗ ∗ ∗ Ξ44 Ξ45 Ξ46 Ξ47 Ξ48 Ξ49

∗ ∗ ∗ ∗ Ξ55 Ξ56 Ξ57 Ξ58 Ξ59

∗ ∗ ∗ ∗ ∗ Ξ66 Ξ67 Ξ68 Ξ69

∗ ∗ ∗ ∗ ∗ ∗ Ξ77 Ξ78 Ξ79

∗ ∗ ∗ ∗ ∗ ∗ ∗ Ξ88 Ξ89

∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ Ξ99

⎞

⎟⎟

⎠

<0, 3.3

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where

Ξ11 P₂^TA−M P₁₀^T P₁₈^T A−M^TP2P10P18hQ3Q1, Ξ12 P₁^T−P₂^T A−M^TP3P11P19,

Ξ13 P₂^TBM−P₁₈^T A−M^TP4P12P20, Ξ14 −P₁₀^T A−M^TP₅P₁₃P₂₁,

Ξ15 P₂^TC A−M^TP₆P₁₄P₂₂,

Ξ16 P₂^TM−P₁₈^T A−M^TP₇P₁₅P₂₃−F₁^T, Ξ17 −P₁₀^T A−M^TP₈P₁₆P₂₄−F₁^T, Ξ18 P₂^TD A−M^TP9P17P25−F₁^T, Ξ19 F₁^T,

Ξ22 −P₃^TP₃^Tτ1Q4τ2Q5Q2,

Ξ23 P₃^TBM−P₁₉^T −P₄, Ξ24 −P₁₁^T −P₅^T, Ξ25 P₃^TC−P₆^T, Ξ26 P₃^TM−P₁₉^T −P7−F₂^T, Ξ27 −P₁₁^T −P₈−F^T₂, Ξ28 P₃^TD−P₉−F₂^T, Ξ29 F₂^T,

Ξ33 P₄^TBM−P₂₀^T BM^TP4−P20−Q1, Ξ34 −P₁₂^T BM^TP5−P21,

Ξ35 P₄^TC BM^TP₆−P₂₂,

Ξ36 P₄^TM−P₂₀^T BM^TP₇−P₂₃−F₃^T, Ξ37 −P₁₂^T BM^TP₈−P₂₄−F₃^T, Ξ38 P₄^TD BM^TP9−P25−F₃^T, Ξ39 F₃^T,

Ξ44 −P₁₃^T −P₁₃,

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Ξ45 P₅^TC−P14,

Ξ46 P₅^TM−P₂₁^T −P15−F₄^T, Ξ47 −P₁₃^T −P₁₆−F₄^T, Ξ48 P₅^TD−P₁₇−F₄^T, Ξ49 F₄^T,

Ξ55 P₆^TCC^TP₆−Q₂, Ξ56 P₆^TM−P₂₂^T C^TP7−F^T₅, Ξ57 −P₁₄^T C^TP8−F₅^T, Ξ58 P₆^TDC^TP₉−F^T₅, Ξ59 F₅^T,

Ξ66 P₇^TM−P₂₃^T M^TP7−P23−F₆^T−F6, Ξ67 −P₁₅^T M^TP₈−P₂₄−F₆^T−F₇^T, Ξ68 P₇^TDM^TP9−P25−F₆^T−F^T₈, Ξ69 F₆^T,

Ξ77 −P₁₆^T −P16−F₇^T−F7, Ξ78 P₈^TD−P₁₇−F₈^T−F^T₇, Ξ79 F₇^T,

Ξ88 P₉^TDD^TP9−F₈^T−F^T₈, Ξ89 F₈^T,

Ξ99 − 1

hQ3 1 τ1Q4 1

τ2Q5

.

3.4

Proof. Constructing a new Lyapunov functional candidate for system3.2as follows:

Vt V₁t V₂t V₃t V₄t V₅t V₆t, 3.5

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where

V₁t Y^TtP Yt,

Yt

⎡

⎣x^Tt,x˙^Tt, x^Tt−τ1, x^Tt−τ2,x˙^Tt−τ2, _t

t−τ1

xsds˙

T

,

t t−τ2

xsds˙

T

, t

t−hxsds

T⎤

⎦

T

,

V2t _t

t−τ1

x^TsQ1xsds, V3t _t

t−τ2

˙

x^TsQ2xsds, V˙ 4t _t

t−h

_t

s

x^TξQ3xξdξ ds,

V₅t _t

t−τ1

_t

s

˙

x^TξQ4xξdξ ds, V˙ 6t _t

t−τ2

_t

s

˙

x^TξQ5xξdξ ds,˙

3.6

P

⎛

⎜⎜

⎝

P₁ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

⎞

⎟⎟

⎠

. 3.7

Set

P

⎛

⎜⎜

⎜⎝

P₁ 0 0 0 0 0 0 0

P₂ P₃ P₄ P₅ P₆ P₇ P₈ P₉ P₁₀ P₁₁ P₁₂ P₁₃ P₁₄ P₁₅ P₁₆ P₁₇ P18 P19 P20 P21 P22 P23 P24 P25

⎞

⎟⎟

⎟⎠

, 3.8

along the trajectories of system3.2, the derivative ofVtis given by

V˙t V˙1t V˙2t V˙3t V˙4t V˙5t V˙6t, 3.9

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where

V˙1t 2Y^TtPYt˙ 2Y^TtP^T

×

⎛

⎜⎜

⎜⎝

xt˙

−xt ˙ Cxt˙ −τ₂ A−Mxt BMxt−τ1 D

_t

t−hxsdsM _t

t−τ1

xsds˙ xt−xt−τ₂−

_t

t−τ2

xsds˙ xt−xt−τ1−

_t

t−τ1

xsds˙

⎞

⎟⎟

⎟⎠

2Y^TtP^T

⎛

⎜⎜

⎜⎝

0 I 0 0 0 0 0 0

A−M −I BM 0 C M 0 D

I 0 0 −I 0 0 −I 0

I 0 −I 0 0 −I 0 0

⎞

⎟⎟

⎟⎠Yt,

3.10

V˙₂t x^TtQ1xt−x^Tt−τ₁Q1xt−τ₁, V˙3t x˙^TtQ2xt˙ −x˙^Tt−τ2Q2xt˙ −τ2,

V˙₂t V˙₃t Y^Tt

⎛

⎜⎜

⎜⎝

Q1 0 0 0 0 0 0 0

∗ Q2 0 0 0 0 0 0

∗ ∗ −Q1 0 0 0 0 0

∗ ∗ ∗ 0 0 0 0 0

∗ ∗ ∗ ∗ −Q2 0 0 0

∗ ∗ ∗ ∗ ∗ 0 0 0

∗ ∗ ∗ ∗ ∗ ∗ 0 0

∗ ∗ ∗ ∗ ∗ ∗ ∗ 0

⎞

⎟⎟

⎟⎠ Yt.

3.11

ByLemma 2.3, similar to the disposal route in34, we have

V˙₄t hx^TtQ3xt− _t

t−hx^TsQ3xsds

≤hx^TtQ3xt _t

t−h

−2x^TsFYt Y^TtF^TQ₃⁻¹FYt ds

hx^TtQ3xt−2 t

t−hxsds

T

FYt hY^TtF^TQ⁻¹₃ FYt,

3.12

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whereF F1, F2, F3, F4, F5, F6, F7, F8. Similarly, we have

V˙5t τ1x˙^TtQ4xt˙ − _t

t−τ1

˙

x^TsQ4xsds˙

≤τ₁x˙^TtQ4xt ˙ _t

t−τ1

−2 ˙x^TsFYt Y^TtF^TQ⁻¹₄ FYt ds

τ₁x˙^TtQ4xt˙ −2 _t

t−τ1

xsds˙

T

FYt τ₁Y^TtF^TQ⁻¹₄ FYt,

3.13

V˙6t τ2x˙^TtQ5xt˙ − _t

t−τ2

˙

x^TsQ5xsds˙

≤τ₂x˙^TtQ5xt ˙ _t

t−τ2

−2 ˙x^TsFYt Y^TtF^TQ⁻¹₅ FYt ds

τ2x˙^TtQ5xt˙ −2 _t

t−τ2

xsds˙

T

FYt τ2Y^TtF^TQ⁻¹₅ FYt.

3.14

From3.12–3.14, we get

V˙4t V˙5t V˙6t

≤Y^Tt

⎛

⎜⎜

⎜⎝

hQ3 0 0 0 0 −F₁^T −F₁^T −F₁^T

∗ τ₁Q₄τ₂Q₅ 0 0 0 −F₂^T −F₂^T F₂^T

∗ ∗ 0 0 0 −F₃^T −F₃^T −F₃^T

∗ ∗ ∗ 0 0 −F₄^T −F₄^T −F₄^T

∗ ∗ ∗ ∗ 0 −F₅^T −F₅^T −F₅^T

∗ ∗ ∗ ∗ ∗ −F₆^T−F6 −F₇^T−F₆^T −F₈^T−F₆^T

∗ ∗ ∗ ∗ ∗ ∗ −F₇^T−F₇^T −F₈^T−F₇^T

∗ ∗ ∗ ∗ ∗ ∗ ∗ −F₈^T−F₈^T

⎞

⎟⎟

⎟⎠ Yt

Y^TtF^T

hQ⁻¹₃ τ1Q⁻¹₄ τ2Q₅⁻¹ FYt.

3.15

Hence,

V˙t≤Y^Tt

ΞF^T

hQ⁻¹₃ τ₁Q⁻¹₄ τ₂Q₅⁻¹ F

Yt, 3.16

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where

Ξ

⎛

⎜⎜

⎜⎝

Ξ11 Ξ12 Ξ13 Ξ14 Ξ15 Ξ16 Ξ17 Ξ18

∗ Ξ22 Ξ23 Ξ24 Ξ25 Ξ26 Ξ27 Ξ28

∗ ∗ Ξ33 Ξ34 Ξ35 Ξ36 Ξ37 Ξ38

∗ ∗ ∗ Ξ44 Ξ45 Ξ46 Ξ47 Ξ48

∗ ∗ ∗ ∗ Ξ55 Ξ56 Ξ57 Ξ58

∗ ∗ ∗ ∗ ∗ Ξ66 Ξ67 Ξ68

∗ ∗ ∗ ∗ ∗ ∗ Ξ77 Ξ78

∗ ∗ ∗ ∗ ∗ ∗ ∗ Ξ88

⎞

⎟⎟

⎟⎠

. 3.17

In views ofLemma 2.1and 3.3, we have ˙Vt < 0, namely, there exists a positive scalar λ >0 such that ˙Vt≤ −λYt². According to35, system2.3is asymptotically stable, this completes the proof.

Remark 3.2. Motivated by the results obtained in 34, free-weighting matrices F_ii 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 ofV₁t, which may make the conservatism reduce further.

Remark 3.3. The transformation from system2.3to3.2enables us to utilize the information of the relationship among termsxt, xt−τ1,_t

t−τ1xsds. Combined with the arbitrariness˙ of matrixM, the conservatism of stability criterion is reduced further.

Remark 3.4. WhenM 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 matricesP_ii 2,3, . . . ,25,F_ii 1,2, . . . ,8with appropriate dimensions such that the following linear matrix inequality is feasible:

Ξ2

⎛

⎜⎜

⎝

Ξ₁₁ Ξ₁₂ Ξ₁₃ Ξ₁₄ Ξ₁₅ Ξ₁₆ Ξ₁₇ Ξ₁₈ Ξ19

∗ Ξ22 Ξ₂₃ Ξ24 Ξ25 Ξ₂₆ Ξ27 Ξ28 Ξ29

∗ ∗ Ξ₃₃ Ξ₃₄ Ξ₃₅ Ξ₃₆ Ξ₃₇ Ξ₃₈ Ξ39

∗ ∗ ∗ Ξ44 Ξ45 Ξ₄₆ Ξ47 Ξ48 Ξ49

∗ ∗ ∗ ∗ Ξ55 Ξ₅₆ Ξ57 Ξ58 Ξ59

∗ ∗ ∗ ∗ ∗ Ξ₆₆ Ξ₆₇ Ξ₆₈ Ξ69

∗ ∗ ∗ ∗ ∗ ∗ Ξ77 Ξ78 Ξ79

∗ ∗ ∗ ∗ ∗ ∗ ∗ Ξ88 Ξ89

∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ Ξ99

⎞

⎟⎟

⎠

<0, 3.18

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where

Ξ₁₁ P₂^TA−I P₁₀^T P₁₈^T A−I^TP₂P₁₀P₁₈hQ₃Q₁, Ξ₁₂ P₁^T−P₂^T A−I^TP₃P₁₁P₁₉,

Ξ₁₃ P₂^TBI−P₁₈^T A−I^TP4P12P20, Ξ₁₄ −P₁₀^T A−I^TP₅P₁₃P₂₁,

Ξ₁₅ P₂^TC A−I^TP6P14P22,

Ξ₁₆ P₂^T−P₁₈^T A−I^TP₇P₁₅P₂₃−F^T₁, Ξ₁₈ P₂^TD A−I^TP₉P₁₇P₂₅−F^T₁, Ξ₂₃ P₃^TBI−P₁₉^T −P₄,

Ξ₂₆ P₃^T−P₁₉^T −P₇−F₂^T,

Ξ₃₃ P₄^TBI−P₂₀^T BI^TP4−P20−Q1, Ξ₃₄ −P₁₂^T BI^TP₅−P₂₁,

Ξ₃₅ P₄^TC BI^TP₆−P₂₂,

Ξ₃₆ P₄^T−P₂₀^T BI^TP7−P23−F₃^T, Ξ₃₇ −P₁₂^T BI^TP₈−P₂₄−F₃^T, Ξ₃₈ P₄^TD BI^TP9−P25−F₃^T, Ξ₄₆ P₅^T−P₂₁^T −P15−F₄^T,

Ξ₅₆ P₆^T−P₂₂^T C^TP7−F₅^T, Ξ₆₆ P₇^T−P₂₃^T P7−P23−F^T₆ −F6, Ξ₆₇ −P₁₅^T P8−P24−F₆^T−F₇^T, Ξ₆₈ P₇^TDP9−P25−F^T₆ −F₈^T.

3.19

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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 ΔAt,ΔBt,ΔDtas follows.

Theorem 3.7. For any given matrixM, scalarsτ1>0, τ2>0, h >0, the original systemΣis robustly and asymptotically stable ifC<1,and there exist positive definite matricesP₁, Q₁, Q₂, Q₃, Q₄, Q₅, positive scalar δ, and arbitrary matricesP_ii 2,3, . . . ,25, F_ii 1,2, . . . ,8 with appropriate dimensions such that the following linear matrix inequality is feasible:

Ξ3

⎛

⎜⎜

⎜⎝

Ξ11δE^T_aEa Ξ12 Ξ13 Ξ14 Ξ15 Ξ16 Ξ17 Ξ18 Ξ19 P₂^TK

∗ Ξ22 Ξ23 Ξ24 Ξ25 Ξ26 Ξ27 Ξ28 Ξ29 P₃^TK

∗ ∗ Ξ33δE_b^TE_b Ξ34 Ξ35 Ξ36 Ξ37 Ξ38 Ξ39 P₄^TK

∗ ∗ ∗ Ξ44 Ξ45 Ξ46 Ξ47 Ξ48 Ξ49 P₅^TK

∗ ∗ ∗ ∗ Ξ55 Ξ56 Ξ57 Ξ58 Ξ59 P₆^TK

∗ ∗ ∗ ∗ ∗ Ξ66δE_c^TEc Ξ67 Ξ68 Ξ69 P₇^TK

∗ ∗ ∗ ∗ ∗ ∗ Ξ77 Ξ78 Ξ79 P₈^TK

∗ ∗ ∗ ∗ ∗ ∗ ∗ Ξ88 Ξ89 P₉^TK

∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ Ξ99 0

∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ −δI

⎞

⎟⎟

⎟⎠

<0.

3.20

Proof. ReplacingA, B, Din3.3withAKFtEa,BKFtEb,andDKFtEd, respectively, 3.3for system2.1is equivalent to the following form:

Ξ Π^T₁F^TtΠ2 Π^T₂FtΠ1<0, 3.21

where Π1 K^TP₂, K^TP₃, K^TP₄, K^TP₅, K^TP₆, K^TP₇, K^TP₈, K^TP₉, Π2 Ea,0, E_b,0,0, E_c, 0,0,0. FromLemma 2.2, a suﬃcient condition for3.20is that there exists a positive scalar δ >0 such that

Ξ δ⁻¹Π^T₁Π1δΠ^T₂Π2 <0. 3.22

In views ofLemma 2.1, we can easily obtain this conclusion, this completes the proof.

Corollary 3.8. For given positive scalarsτ1 > 0, τ2 > 0, h > 0, the original systemΣis robustly and asymptotically stable ifC<1, and there exist positive definite matricesP₁, Q₁, Q₂, Q₃, Q₄, Q₅,

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Table 1: Stability bounds of time delaysExample 4.1.

24 25 26 27 28 12 Ours τ1 0< τ1<0.4991 0< τ1<0.7602 0< τ1<0.4991 0< τ1<1.6965 τ1>0 τ1>0 τ1>0

Table 2: Some comparison for allowable upper bounds onτ1Example 4.2.

29 30 31 24 28 32 13 12 33 Ours τ1, τ2 0.3 0.71 0.74 0.8844 1.3718 1.6525 1.6525 1.6525 2.2254 2.2254

positive scalar δ, and arbitrary matricesPii 2,3, . . . ,25, Fii 1,2, . . . ,8 with appropriate dimensions such that the following linear matrix inequality is feasible:

Ξ4

⎛

⎜⎜

⎜⎝

Ξ₁₁δE^T_aEa Ξ₁₂ Ξ₁₃ Ξ₁₄ Ξ₁₅ Ξ₁₆ Ξ₁₇ Ξ₁₈ Ξ19 P₂^TK

∗ Ξ22 Ξ₂₃ Ξ24 Ξ25 Ξ₂₆ Ξ27 Ξ28 Ξ29 P₃^TK

∗ ∗ Ξ₃₃δE_b^TE_b Ξ₃₄ Ξ₃₅ Ξ₃₆ Ξ₃₇ Ξ₃₈ Ξ39 P₄^TK

∗ ∗ ∗ Ξ44 Ξ45 Ξ₄₆ Ξ47 Ξ48 Ξ49 P₅^TK

∗ ∗ ∗ ∗ Ξ55 Ξ₅₆ Ξ57 Ξ58 Ξ59 P₆^TK

∗ ∗ ∗ ∗ ∗ Ξ₆₆δE_c^TEc Ξ₆₇ Ξ₆₈ Ξ69 P₇^TK

∗ ∗ ∗ ∗ ∗ ∗ Ξ77 Ξ78 Ξ79 P₈^TK

∗ ∗ ∗ ∗ ∗ ∗ ∗ Ξ88 Ξ89 P₉^TK

∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ Ξ99 0

∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ −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 Kwon28:

xt˙ −Cxt˙ −τ₂ Axt Bxt−τ₁ D _t

t−hxsds, 4.1

with

A

−3 −2

1 0 , B

−0.5 0.1

0.3 0 , C 0, D 0. 4.2

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−10

−5 0 5 10 15

xt

0 500 1000 1500 2000 2500 3000 3500 4000 t

x1 x2

Figure 1: State trajectories of system12withcc 0.4,h τ1 τ2 1.60,Ea Eb Ed 0.2I, K I.

−5

−4

−3

−2

−1 0 1 2 3 4 5

xt

0 500 1000 1500 2000 2500

t x1

x2

Figure 2: State trajectories of system12withcc 0.4,τ1 τ2 0.3,h 1.94,Ea Eb Ed 0.2I, K I.

−5

−4

−3

−2

−1 0 1 2 3 4 5

xt

0 500 1000 1500 2000 2500 3000 3500 4000 t

x1 x2

Figure 3: State trajectories of system12withcc 0.4,τ1 τ2 0.3,h 2.61,Ea Eb Ed 0.

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Table 3: Calculated allowable size of distributed delayhΔAt ΔBt ΔDt 0 Example 4.3.

Liu12 cc 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4

τ1 τ2 h 0.67 0.62 0.56 0.51 0.46 0.41 0.36 0.32 0.28

τ1 τ2 0.3 0.79 0.78 0.73 0.66 0.58 0.50 0.41 0.37 0.21

Ours

τ1 τ2 h 2.15 2.13 2.11 2.09 2.07 2.04 2.02 1.99 1.97

τ1 τ2 0.3 2.61 2.61 2.61 2.61 2.61 2.61 2.61 2.61 2.61

Table 4: Calculated allowable size of distributed delayhK I, Ea Eb Ed 0.2I Example 4.3.

Liu12 cc 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4

τ1 τ2 h 0.67 0.62 0.56 0.51 0.46 0.41 0.36 0.32 0.28

τ1 τ2 0.3 0.79 0.78 0.73 0.66 0.58 0.50 0.41 0.37 0.21

Ours 0.3

τ1 τ2 h 1.82 1.79 1.77 1.75 1.72 1.70 1.67 1.63 1.60

τ1 τ2 0.3 1.94 1.94 1.94 1.94 1.94 1.94 1.94 1.94 1.94

SetM _{−1 0}

0 −2

. For comparisons, we calculate the allowable upper bound ofτ1for 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 in12,24–28.

Example 4.2. Consider the following neutral system studied in He et al.32:

xt˙ −Cxt˙ −τ2 Axt Bxt−τ1 D _t

t−hxsds, 4.3

with

A

−0.9 0.2 0.1 −0.9 , B

−1.1 −0.2

−0.1 −1.1 , C

−0.2 0

0.2 −0.1 , D 0. 4.4

SetM _{−1 0}

0 −2

.

For this example, we calculate the allowable upper bound of τ1 for which the asymptotic stability is guaranteed. Table 2 shows that Theorem 3.1obtained in this paper is less conservative than the related results obtained in12,13,24,28–33.

Example 4.3. Consider the neutral system studied in Liu et al.12:

xt˙ −Cxt˙ −τ2 A ΔAtxt B ΔBtxt−τ1 D ΔDt _t

t−hxsds, 4.5 with

A

−2 0 0 −15 , B

1 3

−3 1 , C

cc 0 0 cc , D

1 0.5

−0.5 1 , 0≤cc <1. 4.6

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−5

−4

−3

−2

−1 0 1 2 3 4 5

xt

0 500 1000 1500 2000 2500 3000 3500 4000 t

x1 x2

Figure 4: State trajectories of system12withcc 0.4,τ1 τ2 h 1.97,Ea Eb Ed 0.

Table 5: Comparison ofτmaxusing diﬀerent methodsExample 4.4.

cc 0 0.05 0.10 0.15 0.20 0.25 0.30 0.35

Han33 3.13 2.98 2.83 2.66 2.49 2.31 2.12 1.93

Han36 1.77 1.63 1.48 1.33 1.16 0.98 0.79 0.59

He et al.32 2.39 2.05 1.75 1.49 1.27 1.08 0.91 0.76

Theorem 3.7 3.45 3.21 3.02 2.88 2.62 2.54 2.51 2.23

For the convenience of comparison, letΔAt ΔBt ΔDt 0. SetM _{−1 0}

0 −2

. The comparative results betweenTheorem 3.1with the result obtained in12are given inTable 3.

When

ΔAt,ΔBt,ΔDt KFtEa, Eb, Ed, 4.7

whereK I, Ea Eb Ed 0.2I,Table 4shows that the robust stability criterion obtained in this paper is also less conservative than the related result obtained in 12. From the simulation figuressee Figures1–4, one can see that the results derived in this paper are valid.

Example 4.4. Consider the following uncertain neutral system32,33,36:

xt˙ −Cxt˙ −τ AKFtEaxt BKFtEbxt−τ, 4.8

whereA ₋₂ ₀

0 −0.9

,B _{−1 0}

−1−1

,C _c₀

0c

,0≤c <1,L _{0.2 0}

0 0.2

,E_a E_b _{1 0}

0 1

.Table 5 gives out the related comparative results, from which one can see thatTheorem 3.7obtained in this paper is less conservative than those established in32,33,36.

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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.

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