Improved delay-dependent exponential stability criteria for time-delay system

Available online at www.sciencedirect.com

Journal of the Franklin Institute 350 (2013) 790–801 www.elsevier.com/locate/jfranklin

Improved delay-dependent exponential stability criteria for time-delay system Jiuwen Cao School of Electrical and Electronic Engineering, Nanyang Technological University, Nanyang Avenue, Singapore 639798, Singapore Received 5 December 2006; received in revised form 4 December 2012; accepted 24 December 2012 Available online 29 January 2013

Abstract This paper considers the problem of time delay-dependent exponential stability criteria for the time-delay linear system. Utilizing the linear inequality matrices (LMIs) and slack matrices, a novel criterion based on the Lyapunov–Krasovskii methodology is derived for the exponential stability of the time-delay system. Based on the criteria condition we concluded that the upper bound of the exponential decay rate for the time-delay system can be easily calculated. In addition, an improved sufficient condition for the robust exponential stability of uncertain time-delay system is also proposed. Numerical examples are provided to show the effectiveness of our results. Comparisons between the results derived by our criteria and the one given in Liu (2004) [1], Mondie and Kharitonov (2005) [2], and Xu et al. (2006) [3] show that our methods are less conservative in general. Furthermore, numerical results also show that our criteria can guarantee larger exponential decay rates than the ones derived by Liu (2004) [1] and Mondie and Kharitonov (2005) [2] in all time delay points we have tested and in some of time delay points obtained by Xu et al. (2006) [3]. & 2013 The Franklin Institute. Published by Elsevier Ltd. All rights reserved.

1. Introduction Time delay phenomenons existed in most of the control systems and may cause unexpected performance, such as the instability of the system. In such a case, analyzing the performance and stability conditions of time delay system plays an important role in system design and control. Over the past many decades, the stability conditions of time delay systems have been thoroughly researched [1–16,19–23]. Among several different E-mail addresses: [email protected], [email protected] 0016-0032/$32.00 & 2013 The Franklin Institute. Published by Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.jfranklin.2012.12.026

J. Cao / Journal of the Franklin Institute 350 (2013) 790–801

791

techniques, the Lyapunov second method [17], which makes use of the Lyapunov function and its first-order derivative instead of requiring the knowledge of the system general solution, has been comprehensively studied. In recent several decades, the stability criteria based on linear inequality matrices (LMIs) became popular due to that when formulated in terms of LMI, the control problem can be efficiently solved with convex optimization algorithms comparing with conventional techniques. The stability of linear time delay system, which is the fundamental but important in system control, has attracted much interest over a half century. Representative results of stability conditions for linear time delay control system based on linear inequality matrices can be found in [1–6] and references therein. Since most of the research papers concerned the asymptotical stability conditions of the linear systems in the past decades, exponential stability conditions of linear systems which guarantees stronger stability constraints than the asymptotical stability are urgent. Some of the pioneer work on the analysis of exponential stability for linear systems can be found in [1–3]. In [1], the author first developed the asymptotical stability condition and then extended to exponential case by introducing the Schur complements [18] into the Lyapunov–Krasovskii function. A modification of standard LMI-type stability which allows to obtain the exponential decay rate of the system solution was shown in [2]. By choosing an appropriate Lyapunov–Krasovskii function, an improved exponential stability condition for time-delay systems has been proposed in [3]. As time delays existed in control systems may lead to the instability of the system, a large number of recent papers have been reported on the time delay dependent stability conditions [1,6,8,10,11,14–16]. Generally, a large time delay condition that can stabilize the system would be less conservative than the one with a small time delay. Recent research has shown that introducing modulatory matrices into the derivative of Lyapunov–Krasovskii function would be helpful to reduce some conservatism in the existing delay-dependent stability conditions and the derivative would remain unaffected. Hence, the flexibility of choosing the slack matrix would result in a less conservative stability condition and a relatively large time delay. In this paper, we study the problem of time delay-dependent exponential stability criteria for the time-delay linear system. Using the recent popular linear inequality matrices (LMIs) and utilizing the Newton–Leibniz formula and slack matrices, a novel criterion based on the Lyapunov–Krasovskii methodology is derived for the exponential stability of the time-delay system. Based on the criteria condition we concluded that the upper bound of the exponential decay rate for the time-delay system can be easily calculated. In addition, an improved sufficient condition for the robust exponential stability of uncertain time-delay system is also proposed. Numerical examples are provided to show the effectiveness of our results. Comparisons between the results derived by our criteria and the one given in [1–3] show that our methods are less conservative in general. Furthermore, numerical results also show that our criteria can guarantee larger exponential decay rates than the ones derived by [1,2] in all the delay points we have tested and in some of the delay points obtained by [3]. Notation: For expression simplicity, throughout this paper, Q40 and Qo0 mean that the matrix Q is a positive definite matrix and a negative definite matrix, respectively. lmin ðAÞ and lmax ðAÞ denote the minimum eigenvalue and the maximum eigenvalue of the matrix A, pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi P respectively. JjJh ¼ supy2½h,0 fJjðyÞJg, JxðtÞJ ¼ ð ni ¼ 1 x2i ðtÞÞ1=2 , and JAJ ¼ lmax ðAT AÞ.

J. Cao / Journal of the Franklin Institute 350 (2013) 790–801

792

2. Main results In this section, we first show the general expression of the linear time delay system and the concept of the exponential stability of the system solution. Then, we describe a general theorem to show the exponential stability condition of the linear system. We show that in terms of the linear inequality matrix (LMI), the upper bound of the exponential decay rate for the time-delay system can be easily calculated. A numerical example is given to test the proposed theorem and followed by the comparisons with three related results. 2.1. Time delay-dependent exponential stability condition Consider the following time-delay system: ( _ ¼ AxðtÞ þ BxðthÞ, ðaÞ xðtÞ xðt0 þ yÞ ¼ jðyÞ,

ðbÞ

ð1Þ

_ denotes the first-order derivative of x(t) with respect where x(t) is the state vector and xðtÞ to the variable of t. y 2 ½h,0 and j 2 Cð½h,0,Rn Þ is a continuous vector valued initial function of system (1). A,B 2 Rnn are the system coefficient matrices and h40 is the time delay. Generally, the system is said to be exponentially stable if the solution of system (1) satisfies the following definition. Definition 2.1. System (1) is said to be exponentially stable if there exist scalars a40 and gZ0 such that for every solution x(t) of Eq. (1), the following inequality holds: JxðtÞJrgeat JjJh ,

tZ0:

ð2Þ

Fact 1 (Schur complement, Zhang [18]). Given constant symmetric matrices S1 , S2 , and S3 where ST1 ¼ S1 and ST2 ¼ S2 40, then S1 þ ST3 S1 2 S3 o0 holds if and only if " # " # T S2 S3 S1 S3 o0 or o0: ST3 S1 S3 S2

According to the Lyapunov stability theory and by using the linear inequality and the slack matrices, we can derived the following theorem. Theorem 2.1. The time-delay system described by Eq. (1) is exponentially stable for any delays h satisfying hmax Zh40, if there exist real n  n positive definite matrices P, Q, M, a positive constant b40, and two slack matrices Y, W, satisfy the following inequality: 2 3 o11 PBY þ W T hmax Y mhmax AT Q 6 7 6 D W W T M hmax W mhmax BT Q 7 7o0, O¼6 ð3Þ 6 D 7 D hmax Q 0 4 5 D D D mhmax Q where o11 ¼ PA þ AT P þ Y þ Y T þ e2bhmax M þ 2bP and m ¼ ðe2bh 1Þ=2b. D denotes the elements below the main diagonal of a symmetric block matrix.

J. Cao / Journal of the Franklin Institute 350 (2013) 790–801

793

Proof. With the Lyapunov second stability method, it is straightforward to construct the following Lyapunov–Krasovskii functionals: V ðxt Þ ¼ V1 ðxt Þ þ V2 ðxt Þ þ V3 ðxt Þ,

ð4Þ

where V1 ðxt Þ ¼ xT ðtÞPxðtÞ, Z

0 Z

t

_ ds dx, x_ T ðsÞQe2bðstxÞ xðsÞ

V2 ðxt Þ ¼

80ZxZh,

ð6Þ

tþx

h

Z

ð5Þ

t

V3 ðxt Þ ¼

xT ðsÞMe2bðstþhÞ xðsÞ ds:

ð7Þ

th

Here we defined g1 and g2 as ( g1 ¼ lmin ðPÞ, g2 ¼ lmax ðPÞ þ hmax e2bhmax lmax ðQÞ þ hmax e2bhmax lmax ðMÞ, and

ðaÞ ðbÞ

ð8Þ

( Jxt Jmax ¼ max

_ sup fJjðyÞJg, sup fJjðyÞJgg: y2½h,0

y2½h,0

From Eq. (8) and the Lyapunov–Krasovskii functionals we have constructed, we can obtain the following inequalities: g1 JxðtÞJ2 rV ðxt Þrg2 Jxt J2max :

ð9Þ

Utilizing the Newton–Leibniz formula, it is straightforward to get the following transform equation: Z t _ ds: xðthÞ ¼ xðtÞ xðsÞ ð10Þ th

The time derivative of V ðxt Þ along the trajectories of the dynamic system (1) can be easily obtained as d d V ðxt Þ ¼ ðV1 ðxt Þ þ V2 ðxt Þ þ V3 ðxt ÞÞ, ð11Þ dt dt where for each function, it is given in the following equations correspondingly: V_ 1 ðxt Þ ¼ 2xT ðtÞP½AxðtÞ þ BxðthÞ Z t _ ds ¼ 2xT ðtÞPðA þ BÞxðtÞ2xT ðtÞPB xðsÞ th Z t _ ds ¼ 2xT ðtÞPðA þ BÞxðtÞ þ 2xT ðtÞðY PBÞ xðsÞ Z t Z t th Z t T T T _ _ _ ds þ2x ðthÞW xðsÞ ds2x ðtÞY xðsÞ ds2x ðthÞW xðsÞ th th th Z 1 t ¼ 2½xT ðtÞðPA þ Y ÞxðtÞ þ 2xT ðtÞðPBY þ W T ÞxðthÞ h th T _ _ 2xT ðthÞWxðthÞ2xT ðtÞhY xðsÞ2x ðthÞhW xðsÞ ds,

J. Cao / Journal of the Franklin Institute 350 (2013) 790–801

794

V_ 2 ðxt Þ ¼

Z

0

_ þ xÞ dx _ ½x_ T ðtÞQe2bx xðtÞ x_ T ðt þ xÞQxðt Z 0Z t _ ds dx 2b x_ T ðsÞQe2bðstxÞ xðsÞ h tþx Z Z t e2bh 1 t T _ ds _ ds2bV2 ðxt Þ x_ ðtÞQxðtÞ x_ T ðsÞQxðsÞ ¼ 2b th th Z t 1 _ _ ¼ ½x_ T ðtÞmhQxðtÞ x_ T ðsÞhQxðsÞ ds2bV2 ðxt Þ h th Z 1 t ¼ f½AxðtÞ þ BxðthÞT mhQ½AxðtÞ þ BxðthÞ h th _ x_ T ðsÞhQxðsÞg ds2bV2 ðxt Þ Z 1 t T ¼ ½x ðtÞmhAT QAxðtÞ þ 2xT ðtÞmhAT QBxðthÞ h th _ xT ðthÞmhBT QBxðthÞx_ T ðsÞhQxðsÞ ds2bV2 ðxt Þ, Z t V_ 3 ðxt Þ ¼ xT ðtÞMe2bh xðtÞxT ðthÞMxðthÞ2b xT ðsÞMe2bðstþhÞ xðsÞ ds th Z 1 t T ¼ x ðtÞMe2bh xðtÞxT ðthÞMxðthÞ ds2bV3 ðxt Þ: h th Combining V_ 1 ðxt Þ, V_ 2 ðxt Þ, and V_ 3 ðxt Þ, we have Z 1 t T V_ ðxt Þ ¼ w ðt,sÞO1 wðt,sÞ ds2b½V2 ðxt Þ þ V3 ðxt Þ, h th h

where wðt,sÞ ¼ ½xT ðtÞ xT ðthÞ x_ T ðsÞT and ! ! 2 PBY þ W T PA þ AT P þ Y þ Y T 6 þmhAT QA þ e2bh M þmhAT QB 6 6 ! 6 O1 ¼ 6 W W T M 6 D 6 þmhBT QB 4 D D Hence, we have 1 V_ ðxt Þ þ 2bV ðxt Þ ¼ h where

2

3

P

0

0

6 O2 ¼ 4 0 0

0 0

7 05 0

and O3 ¼ O1 þ 2bO2

Z

3 hY 7 7 7 7 7: hW 7 7 5 hQ

ð12Þ

t

th

wT ðt,sÞfO1 þ 2bO2 gwðt,sÞ ds,

ð13Þ

J. Cao / Journal of the Franklin Institute 350 (2013) 790–801

2 6 6 6 6 ¼6 6 6 4

PA þ AT P þ Y þ Y T

!

PBY þ W T

þmhAT QA þ e2bh M þ 2bP

!

þmhAT QB

D

W W T M þmhBT QB

D

D

!

795

3 hY 7 7 7 7 7: hW 7 7 5 hQ

ð14Þ

Now applying the Schur complement equivalent to Eq. (14) and for all h satisfying hmax ZhZ0, we carried out that O3 o0 is equivalent to the following inequality by following a similar line as described in Theorem 1 of [6], that is Oo0, where O was defined in Eq. (3). Combining the inequalities (3) and (13), we can get the following inequality: V_ ðxt Þ þ 2bV ðxt Þo0: The above inequality leads to the following one: V ðxt Þoe2bt V ðjÞ,

8tZ0:

ð15Þ

Therefore, combining the left inequality of Eqs. (9) and (15), it is easy to obtain the following inequalities: g1 Jxðt,jÞJ2 rV ðxt Þoe2bt V ðjÞrg2 e2bt Jxt J2max ,

8tZ0:

With the above expression, it is straightforward to get the following result: rffiffiffiffiffi g Jxðt,jÞJo 2 ebt Jxt Jmax , 8tZ0: g1 Hence, the exponential stability of system described by Eqs. (1) and (2) is proved. That finishes our proof. & Remark 2.1. In the proof of Theorem 2.1, two slack variable matrices Y and W are introduced. As studied by many previous research work, introducing modulatory matrices into the derivative of Lyapunov–Krasovskii function would be helpful to reduce some conservatism in the existing delay-dependent stability conditions and the derivative would remain unaffected. As evident by the proof Theorem 2.1, one can easily find that while introducing the slack matrices Y and W, the V_ 1 ðxt Þ remains unaffected. The feasibility of choosing the modulatory matrices can reduce the conservatism in stability results for time delay systems. Moreover, the advantages of introduced slack matrices is also verified by the following numerical example. 2.2. Numerical example In this subsection, a numerical example introduced in [1–3] is used to evaluated our proposed theorem. Utilizing the LMI control tool-box in Matlab, the maximum exponential decay rates with respect to different time delays are obtained and comparisons

J. Cao / Journal of the Franklin Institute 350 (2013) 790–801

796

with the results given in [1–3] are shown. Consider the following linear time delay system:     3 2 0:5 0:1 _ ¼ xðtÞ xðtÞ þ xðthÞ: 1 0 0:3 0 For testing, the time delays are varying from 0.8 to 2 with the interval 0.2. The maximum decay rates for each time delay are obtained by the LMI control tool-box and the corresponding slack matrices Y and W are derived. Table 1 shows the simulation results and Table 2 describes the corresponding slack matrices, respectively. From Table 1, one can easily find that using the slack matrices Y and W, only when h¼ 1.2 and h ¼ 1.4, the exponential decay rates obtained by our proposed theorem are a little smaller than the ones in [3]. While for other delay points, the max exponential decay rates obtained by our theorem are larger than the ones in [1–3]. Hence, the condition given by Theorem 2.1 is less conservative than [1–3] in general. 3. The uncertain time-delay systems In this section, we study the robust exponential stability condition of the uncertain timedelay system with the time-varying norm-bounded parameter uncertainties. We first give a brief description of uncertain time-delay system. Then, we show the robust exponential stability condition in terms of LMI. Numerical simulation and comparisons are given subsequently. 3.1. Robust time delay-dependent exponential stability condition Consider the following system: 8 _ > < xðtÞ ¼ ½A þ DAðtÞxðtÞ þ ½B þ DBðtÞxðthÞ, xðt0 þ yÞ ¼ jðyÞ, > : ½DAðtÞ DBðtÞ ¼ DF ðtÞ½E E , A B

ðaÞ ðbÞ ðcÞ

ð16Þ

where F ðtÞ 2 Rkl is an unknown time-varying matrix function with Lebesgue measurable elements bounded by the following inequality: F T ðtÞF ðtÞrI:

ð17Þ

Before stating our results, we give the following fact which will be used in the proof of the following result. Table 1 Comparisons of maximum exponential decay rates b on different time delays. Time delay h¼

0.8

1

1.2

1.4

1.6

1.8

2

max b [1] max b [2] max b [3] Theorem 1

0.7344 0.9367 0.9366 0.9429

0.6715 0.5903 0.9192 0.9220

0.6145 0.3400 0.8991 0.8894

0.5642 0.1813 0.8115 0.7951

0.5202 0.0752 0.6990 0.7043

0.4818 0.0014 0.6148 0.6313

0.4481 0 0.5494 0.5719

J. Cao / Journal of the Franklin Institute 350 (2013) 790–801

797

Table 2 Slack matrices for different time delays. Time delays

Y

h¼ 0.8



3:5790 10:7162

h¼ 1



h¼ 1.2



h¼ 1.4 h¼ 1.6 h¼ 1.8 h¼ 2

W 



32:2855 23:0370



10:7162 12:8302  123:3203 109:8176



23:0370 24:4876  168:1059 171:2564

109:8176 139:6253  6:3800 14:8464



171:2564 219:9938  6:3857 14:7104

14:8464 36:4709   0:0169 0:1335 103  0:1335 1:0262   0:1286 0:8270 3 10  0:8270 5:9665   2:5203 11:7921

14:7104 36:8187   0:0172 0:1366 103  0:1366 1:0364   0:1221 0:8386 3 10  0:8386 6:0238   2:2320 11:7608

11:7921 70:2668   0:0733 0:4232 103  0:4232 2:5392

11:7608 71:0371   0:0706 0:4321 103  0:4321 2:5783

Fact 2. For any F(t) satisfying Eq. (17) and any scalar s40, we have the following results: ! 2 3 " # DAðtÞPT   T DB ðtÞP 7 PE TA 6 D T T 6 þPDAT ðtÞ 7¼ F ðtÞ½EA P EB P  þ F T ðtÞ½DT 0 4 5 PE TB 0 PT DBðtÞ 0 " # " # T sDT D 0 1 PE A þ r ½EA PT EB PT : s PE TB 0 0 With the similar method in Theorem 2.1, we extended to uncertain system described by Eq. (16), we have the following results. Theorem 3.1. The uncertain system described by Eq. (16) is robust exponentially stable for any delay h satisfying hmax Zh40 if there exist real n  n positive definite matrices P, Q, M, positive constants b40, s40, and two slack matrices Y, W satisfy the following inequality: ! 3 2 PBY þ W T T mPD 7 hmax Y mhmax A Q 6 o11 þsEAT EB 7 6 7 6 ! 7 6 T W W M 7 6 T 7 6 D 0 hmax W mhmax B Q T 7o0, 6 ð18Þ O¼6 þsEB EB 7 7 6 7 6 D D hmax Q 0 0 7 6 7 6 D D mhmax Q mhmax QD 5 4 D D D D D msI where o11 ¼ PA þ AT P þ Y þ Y T þ e2bhmax M þ sEAT EA þ 2bP and m ¼ ðe2bh 1Þ=2b. D denotes the elements below the main diagonal of a symmetric block matrix.

J. Cao / Journal of the Franklin Institute 350 (2013) 790–801

798

Proof. With the same Lyapunov–Krasovskii functions in Theorem 2.1, the time derivative of V ðxt Þ along the trajectories of the dynamic system is d d V ðxt Þ ¼ ðV1 ðxt Þ þ V2 ðxt Þ þ V3 ðxt ÞÞ, dt dt

ð19Þ

where 1 V_ 1 ðxt Þ ¼ h

Z

t

f2xT ðtÞ½PðA þ DAðtÞÞ þ Y xðtÞ

th

þ2xT ðtÞ½PðB þ DBðtÞÞY þ W T xðthÞ T _ _ ðthÞhW xðsÞg ds, 2xT ðthÞWxðthÞ2xT ðtÞhY xðsÞ2x

Z 1 t T fx ðtÞ½A þ DAðtÞT mhQ½A þ DAðtÞxðtÞ V_ 2 ðxt Þ ¼ h th þ2xT ðtÞ½A þ DAðtÞT mhQ½B þ DBðtÞ  xðthÞ _ þxT ðthÞ½B þ DBðtÞT mhQ½B þ DBðtÞxðthÞx_ T ðsÞhQxðsÞg ds2bV2 ðxt Þ, Z t 1 xT ðtÞe2bh MxðtÞxT ðthÞMxðthÞ ds2bV3 ðxt Þ: V_ 3 ðxt Þ ¼ h th Combining the time derivatives of V1 ðxt Þ, V2 ðxt Þ and V3 ðxt Þ, then we have Z 1 t T V_ ðxt Þ ¼ w ðt,sÞO1 wðt,sÞ ds2b½V2 ðxt Þ þ V3 ðxt Þ, h th where wT ðt,sÞ ¼ ½xT ðtÞ xT ðthÞ x_ T ðsÞ and 2 $11 $12 6 D W W T þ ½B þ DBðtÞT mhQ½B þ DBðtÞM O1 ¼ 4 D D

hY

3

hW 7 5 hQ

with $11 ¼ PA þ AT P þ Y þ Y T þ PDAðtÞ þ DAT ðtÞP þ mhAT QA þmhDAT ðtÞQDAðtÞ þ mhDAT ðtÞQA þ mhAT QDAðtÞ þ e2bh M, $12 ¼ PðB þ DBðtÞÞY þ W T þ mhAT QB þ mhAT QDBðtÞ þmhDAT ðtÞQB þ mhDAT ðtÞQDBðtÞ: Thus, we have 1 V_ ðxt Þ þ 2bV ðxt Þ ¼ h

Z

t

wT ðt,sÞO2 wðt,sÞ ds,

ð20Þ

th

where the items of the matrix O2 is the same to O1 except for the following changed items and the relations between them is that $211 ¼ $11 þ 2bP

J. Cao / Journal of the Franklin Institute 350 (2013) 790–801

799

Since hmax ZhZ0, using the Fact 2 and the Schur complement, the matrix O2 o0 is equivalence to the following linear matrix inequality: ! 3 2 PBY þ W T T mPD 7 hmax Y mhmax A Q 6 o11 þsEAT EB 7 6 7 6 ! 7 6 T W W M 7 6 T 7 6 D W mh B Q 0 h max max T 7o0 6 ð21Þ O¼6 þsEB EB 7 7 6 7 6 D D hmax Q 0 0 7 6 7 6 D D mhmax Q mhmax QD 5 4 D D D D D msI with o11 ¼ PA þ AT P þ Y þ Y T þ e2bhmax M þ sEAT EA þ 2bP. Then, if the inequality (21) holds, Eq. (20) changes to V_ ðxt Þ þ 2bV ðxt Þo0: The above inequality leads to the following one: V ðxt Þoe2bt V ðjÞ,

8tZ0:

ð22Þ

Similarly, we arrive at the following conclusion: rffiffiffiffiffi g Jxðt,jÞJo 2 ebt Jxt Jmax , 8tZ0 g1 with g1 and g2 defined by Eq. (8). This finishes our proof.

ð23Þ &

3.2. Numerical example In this subsection, a numerical example used in [1,3] is introduced to test our proposed result. Comparisons on the maximum exponential decay rate with [1,3] are also shown in this subsection. Consider the following uncertain linear time delay system:       4 1 0:1 0 _ ¼ xðtÞ þ DAðtÞ xðtÞ þ þ DBðtÞ xðthÞ 0 4 4 0:1 with JDAðtÞJr0:2

and

JDBðtÞJr0:2:

Similar to the one in [1,3], we consider D ¼ 0:2I,EA ¼ EB ¼ I, and set s ¼ 0:540. To verify the performance of the proposed theorem, the time delays are varying from 0.3 to 1.5 with interval 0.2. The maximum decay rates of the robust exponential stability for each time delay are obtained by the LMI control tool-box and the corresponding slack matrices Y and W are derived. Table 3 shows the simulation results and Table 4 describes the corresponding slack matrices, respectively. From Table 3, it is apparent that the max exponential decay rate b obtained by our proposed Theorem 3.1 is larger than the results given by [1] for each delay point. In the mean time, we can also find that when h ¼ 0:321:1 the exponential decay rate derived by Theorem 3.1 is larger than [3]. However, when the time delay increase to h¼ 1.3 and h¼ 1.5, the exponential decay rate obtained by Theorem 3.1 is a little smaller than [3]. Hence, we can conclude that the proposed robust exponential

800

J. Cao / Journal of the Franklin Institute 350 (2013) 790–801

Table 3 Comparisons of maximum exponential decay rates b on different time delays. Time delay h¼

0.3

0.5

0.7

0.9

1.1

1.3

1.5

max b [1] max b [3] Theorem 2

0.6255 1.0108 1.1813

0.4760 0.8366 0.9457

0.3825 0.7103 0.7510

0.3191 0.6165 0.6291

0.2735 0.5425 0.5448

0.2392 0.4845 0.4823

0.2125 0.4375 0.4335

Table 4 Slack matrices for different time delays. Time delays

Y

W

h¼0.3





h¼0.5

 78:9903 14:0128 14:0128 2:5038   27:3728 8:2838

h¼0.7



8:2838 2:5484  15:3497 2:4009



8:2814 2:5489  15:3497 2:4009

h¼0.9



2:4009 0:3811  9:1002 0:9479



2:4009 0:3812  9:1012 0:9475

h¼1.1



0:9479 0:1198  3:6971 0:8916



0:9475 0:1201  3:6976 0:8913

h¼1.3



0:8916 0:2283  1:5005 0:7007



0:8913 0:2285  1:5004 0:7007

h¼1.5



0:7007 0:3282  0:7143 0:4973



0:7007 0:3281  0:7143 0:4971

0:4973 0:3577

 78:9846 14:0117 14:0117 2:5035   27:3744 8:2814

0:4971 0:3571

stability condition for uncertain linear time delay system is generally better than the ones from [1,3].

4. Conclusion The exponential stability condition for linear time delay system has been considered in this paper. A new delay-dependent exponential stability criteria in terms of the linear inequality matrix and introducing slack matrices has been derived. Based on this criteria, the upper bound of the exponential decay rate for the time-delay system can be easily obtained by utilizing a convex optimization scheme of LMI. The general result has been extended to the uncertain linear time delay control system and a robust exponential stability condition has been presented in this paper. To illustrate the effectiveness of the developed theorems, two numerical examples have been tested and the maximum exponential decay rates with respect to different time delays have been shown for comparison. The simulations have shown that the stability criteria given in this paper is generally less conservative than several existed results.

J. Cao / Journal of the Franklin Institute 350 (2013) 790–801

801

References [1] P.L. Liu, Exponential stability for linear time-delay systems with delay dependence, Journal of the Franklin Institute 340 (2004) 481–488. [2] S. Mondie, V.L. Kharitonov, Exponential estimates for retarded time-delay system: a LMI approach, IEEE Transactions on Automatic Control 50 (February (2)) (2005) 268–273. [3] S. Xu, J. Lam, M. Zhou, New exponential estimates for time-delay systems, IEEE Transactions on Automatic Control 51 (September (9)) (2006) 1501–1505. [4] V.L. Kharitonov, Robust stability analysis of time delay systems: a survey, Annual Reviews in Control 23 (1) (1999) 185–196. [5] S.I. Niculescu, E.I. Verriest, L. Dugard, J.M.Dion, Stability and robust stability of time-delay system: a guided tour, in: Lecture Notes in Control and Information Sciences, vol. 228, Springer-Verlag, New York, 1997, pp. 1–71. [6] S. Xu, J. Lam, Improved delay-dependent stability criteria for time-delay systems, IEEE Transactions on Automatic Control 50 (Mar. (3)) (2005) 384–387. [7] J.W. Cao, S.M. Zhong, Y.Y. Hu, Global stability analysis for a class of neural networks with varying delays and control input, Applied Mathematics and Computation 189 (2) (2007) 1480–1490. [8] J.W. Cao, S.M. Zhong, New delay-dependent condition for absolute stability of Lurie control system with multiple time delays and nonlinearities, Applied Mathematics and Computation 194 (1) (2007) 250–258. [9] J.W. Cao, S.M. Zhong, Y.Y. Hu, A descriptor system approach to robust stability of uncertain degenerate system with discrete and distribute delays, Journal of Control Theory and Applications 5 (4) (2007) 357–364. [10] J.W. Cao, S.M. Zhong, Y.Y. Hu, Delay-dependent condition for absolute stability of Lurie control system with multiple time delays and nonlinearities, Journal of Mathematical Analysis and Applications 338 (1) (2008) 497–504. [11] J.W. Cao, S.M. Zhong, Y.Y. Hu, Novel delay-dependent stability conditions for a class of MIMO networked control systems with nonlinear perturbation, Applied Mathematics and Computation 197 (2) (2008) 797–809. [12] J.Z. Cui, J.W. Cao, S.M. Zhong, Y.Y. Hu, Robust stability for uncertain neutral systems with mixed timevarying delays, Journal of Control Theory and Applications 7 (2) (2009) 192–196. [13] C. Peng, D. Yue, E. Tian, Z. Gu, A delay distribution based stability analysis and synthesis approach for networked control systems, Journal of the Franklin Institute 346 (2009) 349–365. [14] M. Li, L. Liu, A delay-dependent stability criterion for linear neutral delay systems, Journal of the Franklin Institute 346 (2009) 33–37. [15] X.D. Zhao, Q.S. Zeng, New robust delay-dependent stability and H1 analysis for uncertain Markovian jump systems with time-varying delays, Journal of the Franklin Institute 347 (2010) 863–874. [16] K. Ramakrishnan, G. Ray, Improved delay-range-dependent robust stability criteria for a class of Lure systems with sector-bounded nonlinearity, Journal of the Franklin Institute 348 (2011) 1769–1786. [17] A.M. Lyapunov, The General Problem of the Stability of Motion, Doctoral Dissertation, University of Kharkiv, 1892 (English translations, Taylor & Francis, London 1992). [18] F.Z. Zhang, The Schur Complement and its Applications, Springer, 2005. [19] S. Xu, J. Lam, Y. Zou, A simplified descriptor system approach to delay-dependent stability and performance analyses for time-delay systems, IEE Proceedings—Control Theory and Applications 152 (2) (2005) 147–151. [20] S. Xu, J. Lam, Y. Zou, Further results on delay-dependent robust stability conditions of uncertain neutral systems, International Journal of Robust and Nonlinear Control 15 (5) (2005) 233–246. [21] J.K. Hale, Theory of Functional Differential Equations, Spring-Verlag, New York, 1977. [22] V.B. Kolmanovskii, A.D. Myshkis, Introduction to the Theory and Application of Functional Differential Equations, Kluwer, Dordrecht, the Netherlands, 1999. [23] S.I. Niculescu, Delay Effects on Stability: A Robust Control Approach, Springer-Verlag, Berlin, Germany, 2001.