Exponential stability of nonlinear neutral systems with time-varying delay

In this paper, the problem of exponential stability for a class of nonlinear neutral systems with interval time-varying delay is studied. Based on improved Lyapunov-Krasovskii functionals combine with Leibniz-Newton’s formula, new delay-dependent sufficient conditions for the exponential stability of the systems are established in terms of linear matrix inequalities (LMIs), which allows to compute the maximal bound of the exponential stability rate of the solution.

Exponential stability of nonlinear neutral systems with time-varying delay

Le Van Hien†and Hoang Van Thi††

†Hanoi National University of Education

††Hong Duc University, Thanh Hoa E-mail: Hienlv@hnue.edu.vn

Abstract

In this paper, the problem of exponential stability for a class of nonlin-ear neutral systems with interval time-varying delay is studied Based on im-proved Lyapunov-Krasovskii functionals combine with Leibniz-Newton’s for-mula, new delay-dependent sufficient conditions for the exponential stability

of the systems are established in terms of linear matrix inequalities (LMIs), which allows to compute the maximal bound of the exponential stability rate

of the solution Numerical examples are also given to show the effectiveness

of the obtained results

Keywords: Neutral systems; interval time-varying delay; nonlinear uncer-tainty; exponential stability; linear matrix inequality

1 Introduction

Time-delay occurs in most of practical models, such as, aircraft stabilization, chemi-cal engineering systems, inferred grinding model, manual control, neural network, nuclear reactor, population dynamic model, ship stabilization, and systems with lossless transmis-sion lines The existence of this time-delay may be the source for instability and bad per-formance of the system Hence, the problem of stability analysis for time-delay systems has received much attention of many researchers in recent years, see [4, 5, 7, 11, 12, 14, 17] and references therein

In many practical systems, the system models can be described by functional differ-ential equations of neutral type, which depend on both state and state derivatives Neutral system examples include distributed networks, heat exchanges, and processes involving steam Recently, the stability analysis of neutral systems has been widely investigated

by many researchers, see [3, 7] for varying delay, and [8, 10-12] for interval time-varying delay The main approach is Lyapunov-Krasovskii functional method and linear matrix inequality technique However, in most of this results, the time-varying delay is assumed to be differentiable, which makes stability conditions more conservatism

In this paper, we consider exponential stability problem for a class of nonlinear neutral systems with interval time-varying delay By using improved Lyapunov-Krasovskii functionals combined with LMIs technique, we propose new criteria for the exponential stability of the system The delay-dependent conditions are formulated in terms of LMIs, being thus solvable by utilizing Matlab’s LMI Control Toolbox available in the literature

to date Compared to the existing results, our result has its own advantages First, it deals with the neutral system considered in this paper is subjected to nonlinear uncertainties Second, the time delay is assumed to be a time-varying continuous function belonging to

a given interval, which means that the lower and upper bounds for the time-varying delay are available, but the delay function is bounded but not necessary to be differentiable This allows the time-delay to be a fast time-varying function and the lower bound is not restricted to being zero Third, our approach allows us to obtain novel exponential stability conditions established in terms of LMIs, which allows to compute the maximal bound of the exponential stability rate of the solution Therefore, our results are more general than the related previous results

The paper is organized as follows: Section 2 presents definitions and some well-known technical propositions needed for the proof of the main results Delay-dependent exponential stability conditions of the system is presented in Section 3 Numerical exam-ples are given in Section 4 The paper ends with conclusions and cited references

Notations.The following notations will be used throughout this paper R+denotes the set

of all nonnegative real numbers; Rndenotes the n−dimensional Euclidean space with the norm k.k and scalar product xTy of two vectors x, y; λmax(A) (λmin(A), resp.) denotes the maximal (the minimal, resp.) number of the real part of eigenvalues of A; ATdenotes the transpose of the matrix A and I denote the identity matrix A matrix Q≥ 0 (Q > 0, resp.) means that Q is semi-positive definite (positive definite, resp.) i.e.hQx, xi ≥ 0 for all x ∈ Rn

(resp.hQx, xi > 0 for all x 6= 0); A ≥ B means A − B ≥ 0; C1([a, b], Rn

) denotes the set of all continuously differentiable functions on[a, b] The segment of the trajectory x(t) is denoted by xt= {x(t + s) : s ∈ [−¯h, 0]}

2 Preliminaries

Consider a nonlinear neutral system with interval time-varying delay of the form

(

˙x(t) − D ˙x(t − τ ) = A0x(t) + A1x(t − h(t)) + f (t, x(t), x(t − h(t)), ˙x(t − τ )) , t ≥ 0, x(t) = φ(t), t∈ [−¯h, 0],

(2.1) where x(t) ∈ Rn

is the system state; A0, A1, D are given real matrices; time varying delay h(t) satisfies 0 ≤ hm ≤ h(t) ≤ hM, constant τ ≥ 0 and ¯h = max{τ, hM}; nonlinear uncertainty function f : R+× Rn

× Rn

× Rn

→ Rn

satisfies

kf(t, x, y, z)k2 ≤ a2

0kxk2 + a2

1kyk2+ a2

2kzk2, ∀(x, y, z), t≥ 0, (2.2)

where, a0, a1, a2are given nonnegative constants The initial function φ∈ C1([−¯h, 0], Rn

) with its normkφks= sup−¯h≤t≤0

q kφ(t)k2+ k ˙φ(t)k2 Definition 2.1 System (2.1) is said to be globally exponentially stable if there exist con-stants α > 0, γ ≥ 1 such that all solution x(t, φ) of the system satisfies the following condition

kx(t, φ)k ≤ γkφkse−αt, ∀t ≥ 0

We introduce the following technical well-known propositions, which will be used

in the proof of our results

Proposition 2.1 (Schur Complement, see Boyd et al [1]) For given matrices X, Y, Z with appropriate dimensions satisfyingX = XT, YT = Y > 0 Then X + ZTY−1Z <0

if and only if

X ZT



<0 or−Y Z

ZT X



<0

Proposition 2.2 (Completing square) LetS be a symmetric positive definite matrix Then for anyx, y∈ Rn

and matrixF , we have

2hF y, xi − hSy, yi ≤ hF S−1FTx, xi

The proof of the above proposition is easily derived from completing square:

hS(y − S−1FTx), y − S−1FTxi ≥ 0

Proposition 2.3 (See, Gu [2]) For any symmetric positive definite matrix W , scalar

ν > 0 and vector function w : [0, ν] −→ Rn

such that the concerned integrals are well defined, then

Z ν

0

w(s)ds

T

W

Z ν

0

w(s)ds



≤ ν

Z ν

0

wT(s)W w(s)ds

3 Main results

Consider system (2.1), where the delay function h(t) satisfies 0 ≤ hm ≤ h(t) ≤

hM, constant τ ≥ 0 and ¯h = max{τ, hM} and the nonlinear perturbation function f(.) sat-isfies the condition (2.2) For given symmetric positive definite matrices P, Q, R, S, T, Z, W

we set

ρ(α) = 2αλmax(P ) +1 − e−2ατ(λmax(R) + λmax(S))

+1 − e−2αhm



λmax(Q) + h2M



e2αhM

− 1λmax(T ) +hM − hm

2

e2αhM

− 1λmax(Z) + τ2e2ατ − 1λmax(W )

Note that, the scalar function ρ(α) is continuous and strictly increasing function in

α ∈ [0, ∞), ρ(0) = 0, ρ(α) → ∞ as α → ∞ Hence, for any λ0 > 0, there is a unique

positive solution α∗ of the equation ρ(α) = λ0, and ρ(α) < λ0 for all α∈ (0, α∗) Let us

set λ1 = λmin(P ), and

λ2 = λmax(P ) + hmλmax(Q) + τλmax(R) + λmax(S)+1

2h

3

Me2α∗ h M

λmax(T ) + 1

2(hM − hm)

2(hM + hm)e2α∗ h M

λmax(Z) +1

2τ

3e2α∗ τ

λmax(W )

The exponential stability of system (2.1) is summarized in the following theorem

Theorem 3.1 Assume that, for system (2.1), there exist matrices Uk,(k = 1, , 7),

symmetric positive definite matricesP, Q, R, S, T, Z, W, and positive number , such that

the following linear matrix inequality hold:

Ξ =





















Ξ11 AT

0U2+ W Ξ13 AT

0U4 −UT

1 + AT

1U4 −UT

3 + AT

1U5 UT

3 D+ AT

1U6 UT

3 + AT

1U7

4

5D− U6 UT

5 − U7

6 + DTU7





















<0,

(3.1) where

Ξ11= AT

0(P + U1) + (P + UT

1 )A0+ a20I+ Q + R − T − W ;

Ξ13= P A1+ UT

1 A1+ AT

0U3+ T ;

Ξ16= P D + UT

1 D+ AT

0U6; Ξ17= AT

0U7+ P + UT

1;

Ξ33= −T − Z + AT

1U3+ UT

3A1+ a2

1I;

Ξ55= −U5− U5T+ S + h2MT + (hM − hm)2Z+ τ2W;

Ξ66= −S + DTU6+ UT

6 D+ a22I;

Ξ77= −I + UT

7 + U7 Then the system (2.1) is globally exponentially stable Moreover, every solution

x(t, φ) of the system satisfies

kx(t, φ)k ≤r λ2

λ1

kφkse−αt, ∀α ∈ (0, α∗], ∀t ≥ 0

Chứng minh Let λ0 = λmin



−Ξ>0 (due to (3.1)) Taking any α > 0 from the interval (0, α∗], we consider the following Lyapunov-Krasovskii functional for the system (2.1)

V(t, xt) =

7

X

i =1

V1 = xT(t)P x(t),

V2 =

Z t

t−h m

e2α(s−t)xT(s)Qx(s)ds

V3 =

Z t

t−τ

e2α(s−t)xT(s)Rx(s)ds,

V4 =

Z t

t−τ

e2α(s−t)˙xT(s)S ˙x(s)ds,

V5 = hM

Z t

t−h M

Z t

s

e2α(θ−t+hM )˙xT(θ)T ˙x(θ)dθds,

V6 = (hM − hm)

Z t−h m

t−h M

Z t

s

e2α(θ−t+hM )˙xT(θ)Z ˙x(θ)dθds,

V7 = τ

Z t

t−τ

Z t

s

e2α(θ+τ −t)˙xT(θ)W ˙x(θ)dθds

Taking the derivative of V1 along the solution of system (2.1) we have

˙

V1 = 2xT(t)P ˙x(t)

= xT(t)hP A0+ AT

0Pix(t) + 2xT(t)PhA1x(t − h(t)) + D ˙x(t − τ ) + f(t)i,

where, for convenient, we denote f(t) =: f(t, x(t), x(t − h(t)), ˙x(t − τ ))

From (2.2) we obtain

ha20xT(t)x(t) + a21xT(t − h(t))x(t − h(t)) + a22˙xT(t − τ ) ˙x(t − τ ) − fT(t)f(t)i ≥ 0,

for any  >0 Therefore, the derivative of V1 satisfies

˙

V1 ≤ xT(t)hP A0+ AT

0P + a2

0Iix(t) + 2xT(t)PhA1x(t − h(t)) + D ˙x(t − τ ) + f(t)i

+ ha21xT(t − h(t))x(t − h(t)) + a22˙xT(t − τ ) ˙x(t − τ ) − fT(t)f(t)i

(3.3)

Next, the derivatives of Vk, k= 2, , 7 give

˙

V2 = xT(t)Qx(t) − e−2αhm

xT(t − hm)Qx(t − hm) − 2αV2;

˙

V3 = xT(t)Rx(t) − e−2ατxT(t − τ )Rx(t − τ ) − 2αV3;

˙

V4 = ˙xT(t)S ˙x(t) − e−2ατ ˙xT(t − τ )S ˙x(t − τ ) − 2αV4;

˙

V5 = h2Me2αhM

˙xT(t)T ˙x(t) − hM

Z t

t−h M

e2α(s−t+hM )˙xT(s)T ˙x(s)ds − 2αV5

≤ h2

Me2αhM ˙xT(t)T ˙x(t) − hM

Z t

t−h M

˙xT(s)T ˙x(s)ds − 2αV5;

(3.4)

and

˙

V6 = (hM − hm)2e2αhm

˙xT(t)Z ˙x(t)

− (hM − hm)

Z t−h m

t−h M

e2α(s−t+hM )˙xT(s)Z ˙x(s)ds − 2αV6

≤ (hM − hm)2e2αhm

˙xT(t)Z ˙x(t)

− (hM − hm)

Z t−h m

t−h M

˙xT(s)Z ˙x(s)ds − 2αV6;

˙

V7 = τ2e2ατ˙xT(t)W ˙x(t) − τ

Z t

t−τ

e2α(s+τ −t)˙xT(s)W ˙x(s)ds − 2αV7

≤ τ2e2ατ˙xT(t)W ˙x(t) − τ

Z t

t−τ

˙xT(s)W ˙x(s)ds − 2αV7;

(3.5)

Applying Proposition 3 and the Leibniz-Newton formula, we have

−hM

Z t

t−h M

˙xT(s)T ˙x(s)ds ≤ −h(t)

Z t

t−h(t)

˙xT(s)T ˙x(s)ds

≤ −

Z t

t−h(t)

˙x(s)ds

T

T

Z t

t−h(t)

˙x(s)ds



≤ −hx(t) − x(t − h(t))iTThx(t) − x(t − h(t))i;

(3.6)

−(hM − hm)

Z t−h m

t−h M

˙xT(s)Z ˙x(s)ds ≤ −(h(t) − hm)

Z t−h m

t−h(t)

˙xT(s)Z ˙x(s)ds

≤ −

Z t−h m

t−h(t)

˙x(s)ds

T

Z

Z t−h m

t−h(t)

˙x(s)ds



≤ −hx(t − hm) − x(t − h(t))iTZhx(t − hm) − x(t − h(t))i;

(3.7)

−τ

Z t

t−τ

˙xT(s)W ˙x(s)ds ≤ −

Z t

t−τ

˙x(s)ds

T

W

Z t

t−τ

˙x(s)ds



≤ −hx(t) − x(t − τ )iTW

h x(t) − x(t − τ )i

(3.8)

By using the following identity relation

− ˙x(t) + D ˙x(t − τ ) + A0x(t) + A1x(t − h(t)) + f(t) = 0,

we obtain

2hxT(t)UT

1 + xT(t − τ )UT

2 + xT(t − h(t))UT

3

+ xT(t − hm)UT

4 + ˙xT(t)UT

5 + ˙xT(t − τ )UT

6 + fT(t)UT

7

i

×h− ˙x(t) + D ˙x(t − τ ) + A0x(t) + A1x(t − h(t)) + f(t)i= 0

(3.9)

Therefore, from (3.3)-(3.9) we have

˙

V(t, xt) + 2αV (t, xt) ≤ ηT(t)Φη(t), (3.10) where,

ηT(t) =xT(t) xT(t − τ ) xT(t − h(t)) xT(t − hm) ˙xT(t) ˙xT(t − τ ) fT(t) ,

Φ =





















Φ11 AT

0U2+ W Φ13 AT

0U4 −UT

1 + AT

2

1U4 −UT

3 + AT

1U5 UT

3D+ AT

1U6 UT

3 + AT

1U7

4

5D− U6 UT

5 − U7

6 + DTU7



















 ,

and

Φ11 = (A0+ αI)TP + P (A0+ αI) + AT

0U1+ UT

1A0+ a20I + Q + R − W − T ;

Φ13 = P A1+ UT

1A1+ AT

0U3+ T ; Φ16= P D + UT

1D+ AT

0U6;

Φ17 = P + UT

1 + AT

0U7; Φ22= −e−2ατR− W ;

Φ33 = a2

1I− T − Z + AT

1U3+ UT

3A1; Φ44= −e−2αh m

Q− Z;

Φ55 = S + h2Me2αhMT + (hM − hm)2e2αhMZ + τ2e2ατW − U5− UT

5;

Φ66 = −e−2ατS+ UT

6D+ DTU6 + a22I;

Φ77 = −I + U7+ UT

7

Observe thatΦ = Ξ + Ψ, where,

Ψ = diagn2αP, (1 − e−2ατ)R, 0, (1 − e−2αhm

)Q, h2M(e2αhM

− 1)T + (hM − hm)2(e2αhM

− 1)Z + τ2(e2ατ − 1)W, (1 − e−2ατ)S, 0o

hence

˙

V(t, xt) + 2αV (t, xt) ≤ ηT(t)(Ξ + Ψ)η(t) (3.11) Taking (3.11) into account, we finally obtain

˙

V(t, xt) + 2αV (t, xt) ≤ hρ(α) − λ0

i

which implies V(t, xt) ≤ V (0, x0)e−2αt, t ≥ 0 To estimate the exponential stability rate

of the solution, we use (3.2) that

λ1kx(t)k2 ≤ V (t, xt) ≤ λ2kxtk2s, t ∈ R+ and from the differential inequality (3.12), we obtain

kx(t, φ)k ≤r λ2

λ1

kφkse−αt, t≥ 0 which completes the proof of the theorem

Remark 3.1 The exponential convergence rate α in Theorem 1 can be obtained by solv-ing a nonlinear scalar equation ρ(α) = λ0 For this equation, many algorithms and com-putational methods can be used, e.g., iterative or Newton’s method [9] However, for a more explicit condition, we estimate the exponential rate α as follow: From the fact that,

e2α¯h

− 1 ≥ 2α¯h, we have ρ(α) ≤ γe2α¯h

− 1, where, γ = λmax¯(P )

h +hλmax(Q) +

λmax(R) + λmax(S)i + h2

Mλmax(T ) + (hM − hm)2λmax(Z) Therefore, system (2.1) is exponentially stable with the exponential rate0 < α ≤ 1

2¯hln



1 + λ0 γ



Remark 3.2 Theorem 1 gives conditions for the exponential stability of neutral systems with nonlinear uncertainties and interval-time varying state delay These conditions are derived in terms of linear matrix inequalities which can be solved effectively by various computation tools [1] Different from [5, 6, 12, 13], where the α-exponential stability problem is considered, the exponential rate α is given and enters as nonlinear terms in the stability conditions In this paper, the exponential convergence rate is determined in terms

of linear matrix inequalities

4 Numerical examples

In this section, we give some numerical examples to illustrate the effectiveness of our obtained results in comparison with the existing results

Example 4 1 Consider neutral system (2.1), where

A0 =−2 0

 , A1 =0.1 −1

0 −0.1

 , D=0.1 0

0 0.1

 ,

a0 = 0.2, a1 = 0.2, a2 = 0.1, τ = 1, and h(t) = 1 + ψ(t), where, ψ(t) = 0.5 sin(t) if t ∈ I = ∪k≥0[2kπ, (2k + 1)π] and ψ(t) = 0 if t ∈ R+\I

Note that, the delay function h(t) is continuous, but non-differentiable on R+ Therefore, the stability results obtained in [3, 10-12, 16, 18-21] are not applicable By using LMI toolbox of Matlab, we can verify that, the LMI (3.1) is satisfied with hm =

1, hM = 1.5,  = 10 and

P = 11.1343 −8.6199

−8.6199 33.9554

 , Q= 6.4391 −2.7561

−2.7561 23.0890

 , R= 5.9522 −1.4762

−1.4762 10.8871

 ,

S= 1.1647 −0.1417

−0.1417 2.0581

 , T = 0.4364 −0.0872

−0.0872 0.8021

 , W = 0.5411 −0.0846

−0.0846 1.1756

 ,

Z = 2.6249 −1.2851

−1.2851 12.4711

 , U1 =−14.3260 41.8769

−7.9793 −28.4459

 , U2 = 0.1456 0.0452

−0.0317 0.2562

 ,

U3 =0.3302 −0.7932

0.7994 −8.1438

 , U4 =−0.2285 1.8629

−0.0296 0.2517

 , U5 =

 3.3512 7.8831

−10.1070 7.1532

 ,

U6 =−0.0126 −0.8276

0.7984 0.0933

 , U7 =1.1847 −8.1379

8.1228 1.3701



We have λ0 = 0.3635 and

ρ(α) = 73.6906α + 36.90841 − e−2α+ 1.1867e2α− 1+ 5.0081e3α− 1

The unique positive solution of equation ρ(α) = λ0 is α∗ = 0.0022057 Then all solution x(t, φ) of the system satisfies the following inequality

kx(t, φ)k ≤ 3.1803kφkse−0.0022t, ∀t ≥ 0

Example 4 2 Consider the system studied in ([15, 20]):

d

dt[x(t) − Dx(t − τ )] = A0x(t) + A1x(t − τ ) + f(t, x(t), x(t − τ )), (4.1) where,

A0 =−2 0.5

 , A1 = 1 0.4

0.4 −1

 , D=0.2 1

0 0.2

 , a0 = 0.2, a1 = 0.1

Applying Corollary 1 for hm = 0, hM = τ and a2 = 0 we obtain the allowable value of the delay for the asymptotic stability of system (4.1) is τ = 1.8106, while the upper bound of value τ given in [15] and [20] is0.583 and 1.7043, respectively

5 Conclusion

In this paper, we have proposed new delay-dependent exponential stability condi-tions for a class of nonlinear neutral systems with non-differentiable interval time-varying delay Based on the improved Lyapunov-Krasovskii functionals and linear matrix inequal-ity technique, new delay-dependent sufficient conditions for the exponential stabilinequal-ity of the systems have been established in terms of LMIs Numerical examples are given to show the effectiveness of our results

Acknowledgments

This work was partially supported by Hanoi National University of Education and the Min-istry of Education and Training, Vietnam

REFERENCES

[1] S Boyd, L.E Ghaoui, E Feron, & V Balakrishnan (1994) Linear Matrix Inequalities

in System and Control Theory, Philadelphia: SIAM

[2] K Gu (2000) An integral inequality in the stability problem of time delay systems Proc IEEE Conf on Decision and Control, New York

[3] Q.L Han & L Yu (2004) Robust stability of of linear neutral systems with nonlinear parameter perturbations IEE Proceeding, Control Theory and Application 151(5), 539-546

[4] Y He, Q Wang, C Lin & M Wu (2007) Delay-range-dependent stability for systems with time-varying delay Automatica, 43, 371-376

[5] L.V Hien & V.N Phat (2009) Exponential stability and stabilization of a class of uncertain linear time-delay systems Journal of the Franklin Institute, 346, 611-625 [6] L.V Hien & V.N Phat (2009) Stability and stabilization of switched linear dynamic systems with time delay and uncertainties Applied Mathematics and Computation,

210, 223-231

[7] X Jiang & Q.L Han (2006) Delay-dependent robust stability for uncertain linear systems with interval time-varying delay Automatica, 42, 1059-1065

[8] X Jiang & Q Han (2008) New stability criteria uncertain linear systems with interval time-varying delay Automatica, 44, 2680-2685

[9] C.T Kelley (2003) Solving nonlinear equations with Newton’s method, Philadelphia: SIAM

[10] O.M Kwon, J.H Park & S.M Lee (2009) Augmented Lyapunov functional ap-proach to stability of uncertain neutral systems with time-varying delays Applied Mathematics and Computation, 207, 202-212