Further results on delay-dependent stability for continuous system with two additive time-varying delay components

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Further results on delay-dependent stability for continuous system with two additive time-varying delay components$ Xuemei Yu a,n, Xiaomei Wang a, Shouming Zhong a, Kaibo Shi b a b

School of Mathematical Sciences, University of Electronic Science and Technology of China, Sichuan, China School of Information Science and Engineering, Chengdu University, Chengdu, 610106, China

art ic l e i nf o

a b s t r a c t

Article history: Received 10 January 2016 Received in revised form 4 August 2016 Accepted 4 August 2016 This paper was recommended for publication by A.B. Rad

This paper deals with the problem of stability for continuous system with two additive time-varying delay components. By making full use of the information of the marginally delayed state, a novel Lyapunov–Krasovskii functional is constructed. When estimating the derivative of the Lyapunov–Krasovskii functional, we manage to get a fairly tighter upper bound by using the method of reciprocal convex and convex polyhedron. The obtained delay-dependent stability results are less conservative than some existing ones via numerical example comparisons. In addition, this criterion is expressed as a set of linear matrix inequalities, which can be readily tested by using the Matlab LMI toolbox. Finally, four examples are given to illustrate the effectiveness of the proposed method. & 2016 ISA. Published by Elsevier Ltd. All rights reserved.

Keywords: Delay-dependent stability Additive delay components Time-varying delay Lyapunov–Krasovskii functional

1. Introduction

The most commonly used state-space model to represent linear time-delay systems is

During the last several decades, time delay is frequently occur in many practical systems, such as economic systems, biology, neural networks, networked control systems, engineering systems, and so on [1–10]. It is believed to be a major cause of instability and poor performance of the system. Therefore, there has been tremendous interest in developing the stability of time-delay systems. The stability problems for time-delay systems has been widely recognized and studied [11–25]. In addition, according to whether the stability of the system is affected by time-delay, the stability of time-delay system can be divided into delay-dependent stability and delay-independent stability. The former means that the stability of the system is closely related with time delay, but the latter is irrelevant. Delay-independent stability conditions are conservative especially for small size delays. At present, most of the stability of time-delay systems are delay-dependent stability [26–44].

_ ¼ AxðtÞ þ Bxðt  dðtÞÞ; xðtÞ

☆ This work was supported by the National Natural Science Foundation of China (61533006) and the Key Project of Natural Science Research of Anhui Provincial Department of Education (KJ2016A555 and KJ2016A625). n Corresponding author.

ð1Þ

where d(t) is a time delay in the state x(t), which is often assumed to be constant or time-varying satisfying certain conditions, e.g., _ r μ o 1. It can be easily seen that system (1) 0 r dðtÞ r h o 1, dðtÞ has a single time delay d(t) in the state x(t). However, multiple time-delay components in the state should be considered, which can be found in the networked control systems. In some practical situations, signals transmitted from one point to another may experience a few segments of networks, which can possibly induce successive delays with different properties due to the variable network transmission conditions. Fig. 1 shows one simple example of such situation, which can be found in the networked control system. It can be seen that there are two delays in Fig. 1: ds ðtÞ is used to represent the delay from sensor to controller and da ðtÞ is used to represent the delay from controller to actuator. Since the properties of these two delays may not be identical due to the network transmission conditions, it is not reasonable to combine them together. Therefore, when the physical plant and state-feedback controller are, _ ¼ AxðtÞ þ BuðtÞ and uc ðtÞ ¼ Kxc ðtÞ, the respectively, given by xðtÞ closed-loop system is given by _ ¼ AxðtÞ þ BKxðt  ds ðtÞ da ðtÞÞ: xðtÞ

ð2Þ

http://dx.doi.org/10.1016/j.isatra.2016.08.003 0019-0578/& 2016 ISA. Published by Elsevier Ltd. All rights reserved.

Please cite this article as: Yu X, et al. Further results on delay-dependent stability for continuous system with two additive time-varying delay components. ISA Transactions (2016), http://dx.doi.org/10.1016/j.isatra.2016.08.003i

X. Yu et al. / ISA Transactions ∎ (∎∎∎∎) ∎∎∎–∎∎∎

2

 R R _ _ xðsÞdsd θ . We take the state 0 h ttþ θ xðsÞdsd θ as an Rt augmented variable, rather than by hxðtÞ  t  h xðsÞds. The value of the upper bound is estimated to be more strict than previous methods. Therefore, by constructing a new Lyapunov–Krasovskii functional, some new delay-dependent stability criteria are derived to guarantee the stability of time-delay systems. The obtained delay-dependent stability results are less conservative because reciprocally convex approach and convex polyhedron approach are considered. In addition, this criterion is expressed as a set of linear matrix inequalities, which can be readily tested by using the Matlab LMI toolbox. Finally, four examples are given to illustrate the effectiveness of the proposed method. This paper is organized as follows. In Section 2, some preliminaries are briefly given. In Section 3, by choosing a new class of Lyapunov–Krasovskii functional, some less conservative stability criteria are presented to guarantee the systems with two additive time-varying delay components to be asymptotically stable. In Section 4, four examples are given to show the advantage and effectiveness of the proposed result. In last section, conclusion is given. Notations: Throughout this paper, the superscript ‘T’ stands for matrix transposition. Rn denotes the n-dimensional Euclidean space and Rnm refers to the set of all n  m real matrices. The notation X 1 4 X 2 means that the matrix X 1  X 2 is positive definite. And the notation X 1 ZX 2 means that the matrix X 1  X 2 is nonnegative. The symmetric term in the symmetric matrix is denoted by ‘n’, as for example " # " # X1 X2 X1 X2 ¼ : X T2 X 3 n X3 Z

Fig. 1. Networked control system.

Thus, a new mathematical model with multiple time delays is proposed [31,36,45–50], which relates to the practical situation in linear time-delay systems has the following form: ! n X _ ¼ AxðtÞ þ Bx t  xðtÞ di ðtÞ ; ð3Þ i¼2

_ r μ o 1. There have been many where 0 r di ðtÞ r hi o1, dðtÞ i studies on the stability of systems with two additive delay components [51–55]. By taking more information of the time-varying delay in Lyapunov–Krasovskii functional into account and by utilizing free matrix variables in approximating certain integral terms, a new stability criterion for the systems with two additive time-varying components was given [34]. An improved stability criterion was proposed by exploiting new Lyapunov–Krasovskii functional [38]. The paper exposed the new model and gave a preliminary result on its stability analysis with two additive timevarying delay components [46]. However, when constructing the Lyapunov–Krasovskii functional, the information about d(t), d1 ðtÞ and d2 ðtÞ have not been fully utilized [34,35,37,38,46], which would be inevitably conservative to some extent. On the other hand, a significant source of conservativeness that could be further reduced lies in the calculation of the derivative of the Lyapunov–Krasovskii functional. For example, in [46], when R0 R0 _ þ αÞdαdβ is bounded with estimating V_ ðtÞ,  d 1 β x_ T ðt þ αÞM 1 xðt R t T T _  t  d ðtÞ x_ ðαÞM 1 xð _ αÞdα, which may lead to cond 1 x_ ðtÞM 1 xðtÞ 1 siderable conservation. In [34,35,37,38,46], since these approaches do not make full use of the information about d(t), d1 ðtÞ and d2 ðtÞ when constructing the Lyapunov–Krasovskii functional, and the conditions given in those papers was still conservative for the calculation of the derivative of a Lyapunov–Krasovskii functional. So there is still room to improve in existing papers. Motivated by the issues discussed above, in this paper, it is our intention to present new stability criteria for systems with two additive time-varying delay components. We still consider the case that two additive time-varying delay components appear in the state, and the idea of this paper can be easily extended to systems with multiple delay components. The advantage of this paper is the establishment of a novel Lyapunov–Krasovskii functional by making full use of the information of the marginally delayed state. We construct a new Lyapunov–Krasovskii functional that employs the information of the marginally delayed state xðt  h1 Þ, xðt  h2 Þ and xðt  hÞ. In addition, when estimating the 2 R0 derivative of the Lyapunov–Krasovskii functional V_ ðtÞ,  h2  h R R T Rt 0 t _T _ _ is bounded with   h t þ θ xðsÞdsd θ t þ θ x ðsÞZ xðsÞdsdθ

R

0 h

Rt

tþθ

2. Problem formulation and some preliminaries Consider the following system with two time-varying delay components in the state: ( _ ¼ AxðtÞ þBxðt d1 ðtÞ  d2 ðtÞÞ; xðtÞ ð4Þ t A ½  h; 0; xðtÞ ¼ ϕðtÞ; where xðtÞ A Rn is the state vector. A, B A Rnn are the known matrices. ϕðtÞ is the initial condition on the segment ½  h; 0. d1 ðtÞ and d2 ðtÞ represent two additive delay components in the state. A nature assumption on the two delays is made 0 r d1 ðtÞ r h1 o 1; 0 r d2 ðtÞ r h2 o 1;

d_ 1 ðtÞ r μ1 o 1; d_ ðtÞ r μ o 1: 2

2

ð5Þ

Our goal is to show that system (4) is asymptotically stable for all delays d1 ðtÞ and d2 ðtÞ satisfying (5). A popular approach to lump d1 ðtÞ and d2 ðtÞ into one delay dðtÞ ¼ d1 ðtÞ þd2 ðtÞ: (

ð6Þ

By substituting (6) into (4), we obtain _ ¼ AxðtÞ þBxðt dðtÞÞ; xðtÞ xðtÞ ¼ ϕðtÞ;

t A ½  h; 0;

ð7Þ

where 0 r dðtÞ r h o1;

_ r μ o1; dðtÞ

with h ¼ h1 þ h2 ;

μ ¼ μ1 þ μ2 :

Remark 2.1. However, it should be recognized that such treatment will be very conservative. On the one hand, the two delay

Please cite this article as: Yu X, et al. Further results on delay-dependent stability for continuous system with two additive time-varying delay components. ISA Transactions (2016), http://dx.doi.org/10.1016/j.isatra.2016.08.003i

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components in the state may have different natures, it is not appropriate to lump them together. On the other hand, d(t) is usually less than h. In such a case, we cannot guarantee that d(t) and h can reach the maximum at the same time. Therefore, it is necessary to study system (4) with the two additive delay components. Lemma 2.1 ([4]). For any symmetric positive definite matrix Σ 4 0, scalars m2 4 m1 4 0, such that the integrations concerned are well defined, then the following inequality holds: Z t  m1 Z t  m1 Z t  m1 xT ðsÞΣ xðsÞds r  xT ðsÞdsΣ xðsÞds; ðm2  m1 Þ t  m2

t  m2

t  m2

ð8Þ 1  ðm22  m21 Þ 2

Z



t

Z

 m1

Z

t t þθ

 m2

xT ðsÞΣ xðsÞds dθ r 

Z

 m1

Z

t t þθ

 m2

xT ðsÞds dθΣ

Z

xðsÞds dθ:

tþθ

 m1

min P

αi j αi 4 0;

 αi ¼ 1

αi

i

f i ðtÞ ¼

X

f i ðtÞ þ max g i;j ðtÞ

i

X g i;j ðtÞ;

ð10Þ

iaj

i

subject to (

" Δ

m

g i;j : R -R; g j;i ðtÞ ¼ g i;j ðtÞ;

f i ðtÞ

g i;j ðtÞ

g j;i ðtÞ

f j ðtÞ

#

) Z0 :

φ11 φ12 φ13 6 n φ22 φ23 6 6 6 n n φ33 6

6 6 6 6 Φ¼6 6 6 6 6 6 6 6 6 6 4

φ18 φ19

φ1;10

0

0

0

0

0 0

0 0

0 0

0 0

0 0

0

0

φ39

φ3;10

0

0

0

0

0

0

0

0

0

0

0

n

n

n

φ44

n

n

n

n

φ55

n

n

n

n

n

φ66

0

0

0

0

n

n

n

n

n

n

φ77

0

0

0

n

n

n

n

n

n

n

n

n

n

n

n

n

n

n

n

n

n

n

n

n

φ88 φ89 n φ99 n

n

φ8;10 0

φ10;10

 þ M 1 þ M 3 þ hM 5

0

0

 M 1 þM 2

 M2

 M3 þ M4

 M4 0  M5  M5   þ M 1 þ M 3 þ hM 5

0

0

 M 1 þM 2

 M2

 M3 þ M4

3 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 5

 M4 0  M5  M5  ; ð9Þ

X1

2

T

 m2

Lemma 2.2 ([29]). Let f 1 ; f 2 ; …; f N : Rm -R have positive values in an open subset D of Rm. Then, the reciprocally convex combination of fi over D satisfies 

3

ð11Þ

φ11 ¼ Q 1 þ Q 4 þ Q 2 þ Q 5 þ Q 3 þ Q 6 þ h2 U 1  U 2 þ D1 A þAT DT1 , φ12 ¼ U 2  S þ D1 B þ AT DT3 , φ13 ¼ S, φ18 ¼ P  D1 þ AT DT2 , φ19 ¼ T 11  T T12 , φ1;10 ¼ T 12  T 22 , φ22 ¼ ðμ  1ÞQ 1 þ S þ ST 2U 2 þ D3 B þ BT DT3 , φ23 ¼ U 2  S, φ28 ¼  D3 þBT DT2 , φ33 ¼  Q 4  U 2  U 3 U 4 , φ35 ¼ U 3 , φ37 ¼ U 4 , φ39 ¼ T T12 T 11 , φ3;10 ¼ T 22  T 12 , φ44 ¼ ðμ1  1ÞQ 2 , φ55 ¼  Q 5  U 3 , φ66 ¼ ðμ2  1ÞQ 3 , φ77 ¼ Q 6  U 4 , φ88 ¼ h1 R1 þh2 4 2 T R2 þh U 2 þ ðh h1 Þ2 U 3 þ ðh  h2 Þ2 U 4 þ h4 Z  D2  DT2 , φ89 ¼ hT 12 , φ8;10 ¼ hT 22 , φ99 ¼ U 1 , φ10;10 ¼  Z.

with

Proof. Construct the following Lyapunov–Krasovskii functional candidate: VðtÞ ¼

7 X

V i ðtÞ;

ð17Þ

i¼1

where

3. Main result

V 1 ðtÞ ¼ xT ðtÞPxðtÞ; In this section, we consider the stability of system (4).

2

Theorem 3.1. Consider system (4) with delays subject to (5). For given scalars hi ði ¼ 1; 2Þ and μi ði ¼ 1; 2Þ, system (4) is asymptotically h i , stable if there exist symmetric positive definite matrices P, Tn11 TT 12 22 Q i ði ¼ 1; …; 6Þ, Rj ðj ¼ 1; 2Þ, U k ðk ¼ 1; …; 4Þ, Z and any matrices M l ðl ¼ 1; …; 5Þ, Dr ðr ¼ 1; 2; 3Þ, S with appropriate dimensions, such that the following LMIs holds: " # U2 S Z 0; ð12Þ n U2 2

Φ  h1 M 1

6n 4 n

2

n

2

0

n

 h2 R2

 h1 R1 n

Φ  h1 M 2

6n 4 n

2

 h1 R1

Φ  h1 M 1

6n 4

 h1 R1 n

Φ  h1 M 2

6n 4 n

where

 h2 M 3

 h2 M 4

7 5 o 0;

 h2 M 3 7 0 5 o 0;

0

n

 h2 R2

Z þ

7 5 o 0;

t  d1 ðtÞ

xT ðsÞQ 4 xðsÞds þ

t t  h1

xT ðsÞQ 2 xðsÞds þ

t  d2 ðtÞ

Z xT ðsÞQ 5 xðsÞds þ

t

t

t  h2

xT ðsÞQ 3 xðsÞds

xT ðsÞQ 6 xðsÞds;

Z

0

t

_ x_ T ðsÞR1 xðsÞds dθ þ

tþθ

 h1

Z

Z

0  h2

t

t þθ

_ x_ T ðsÞR2 xðsÞds dθ ; ð21Þ

Z

0 h

Z

0 h

t tþθ

t tþθ

Z

2

h 2

Z

0 h

xT ðsÞU 1 xðsÞds dθ;

_ x_ T ðsÞU 2 xðsÞds

þ ðh h2 Þ

V 7 ðtÞ ¼ ð16Þ

Z

t

Z

t

t h

V 4 ðtÞ ¼

Z

3

n

3 Rt #2 t  h xðsÞds 4R0 Rt 5; T 22 _  h t þ θ xðsÞdsdθ T 12

ð20Þ

V 6 ðtÞ ¼ h

ð15Þ

xT ðsÞQ 1 xðsÞds þ

t  dðtÞ

ð14Þ

 h2 R2

 h1 R1

V 3 ðtÞ ¼

V 5 ðtÞ ¼ h

3

5

T 11

ð19Þ

Z

3

θ

Z

t

Z ð13Þ

3T "

Rt

xðsÞds 4 R 0 Rt t h _  h t þ θ xðsÞdsd Z

3

7 0 5 o 0;  h2 R2

 h2 M 4

V 2 ðtÞ ¼

ð18Þ

Z θ

 h2

Z

t þθ

h 0

Z

t

t tþλ

dθ þ ðh  h1 Þ

ð22Þ Z

 h1 h

_ x_ T ðsÞU 4 xðsÞds dθ;

_ x_ T ðsÞZ xðsÞds dλdθ;

Z

t tþθ

_ x_ T ðsÞU 3 xðsÞds dθ ð23Þ

ð24Þ

where d(t) is defined in (6). According to the condition of Theorem 3.1, we will show d(t) is positive definite. Then calculating the time derivative of d(t) along the trajectory of (4) yields, we have

Please cite this article as: Yu X, et al. Further results on delay-dependent stability for continuous system with two additive time-varying delay components. ISA Transactions (2016), http://dx.doi.org/10.1016/j.isatra.2016.08.003i

X. Yu et al. / ISA Transactions ∎ (∎∎∎∎) ∎∎∎–∎∎∎

4 7 X

V_ ðtÞ ¼

V_ i ðtÞ;

ð25Þ

i¼1

where _ V_ 1 ðtÞ ¼ 2xT ðtÞP xðtÞ; 2

V_ 2 ðtÞ ¼

3T "

Rt

t  h xðsÞds 24 R 0 R t _  h t þ θ xðsÞdsd

" ¼ 2ηT ðtÞ where

ð26Þ

eT9 eT10

#T "

θ

5

T 11

T 12

n

T 22

T 11

T 12

n

T 22

#"

#"

xðtÞ  xðt  hÞ _  xðtÞ þ xðt  hÞ hxðsÞ

#

#

eT1  eT3 T

eT3  eT1 þ he8

ηðtÞ;

ð27Þ

h

i

xT ðt  h2 Þ x_ T ðtÞ π T1 π T2 ;

π2 ¼

 d1 ðtÞvT1 ðtÞR1 v1 ðtÞ  ðh1  d1 ðtÞÞvT2 ðtÞR1 v2 ðtÞ

2 V_ 5 ðtÞ ¼ xT ðtÞðh U 1 ÞxðtÞ  h

t h

tþθ

h

I nn

0nð10  iÞn

iT

t

t h

 xðsÞds :

ð34Þ

T

t h

þx ðt  hÞð Q 4 Þxðt  hÞ þ x ðt h1 Þð  Q 5 Þxðt  h1 Þ þxT ðt  h2 Þð  Q 6 Þxðt  h2 Þ; _  V_ 4 ðtÞ ¼ h1 x_ ðtÞR1 xðtÞ T

_  þ h2 x_ ðtÞR2 xðtÞ T

_ x_ ðsÞR1 xðsÞds T

t  h1 t

Z

t  h2

_ x_ ðsÞR2 xðsÞds:

Z

t t h

_ ¼ h x_ T ðsÞU 2 xðsÞds



t  d1 ðtÞ t  h1

 d1 ðtÞvT1 ðtÞR1 v1 ðtÞ  ðh1  d1 ðtÞÞvT2 ðtÞR1 v2 ðtÞ;

v1 ðtÞ ¼ v2 ðtÞ ¼ t

t  h2

1 d1 ðtÞ

t t  d1 ðtÞ

1 h1  d1 ðtÞ

Z

v4 ðtÞ ¼

Z

t  d1 ðtÞ t  h1

ð30Þ

 ðh  h1 Þ

_ xðsÞds:

Z

t t  d2 ðtÞ

_  x_ T ðsÞR2 xðsÞds

t  d2 ðtÞ

Z

_ xðsÞds;

t  d2 ðtÞ t  h2

_ xðsÞds;

1

d1 ðtÞ-0d1 ðtÞ

t  dðtÞ

th

_ x_ T ðsÞU 2 xðsÞds

h ðxðtÞ xðt  dðtÞÞÞT U 2 ðxðtÞ  xðt dðtÞÞÞ dðtÞ

h ðxðt  dðtÞÞ  xðt  hÞÞT U 2 ðxðt dðtÞÞ xðt hÞÞ h  dðtÞ

h h ðe1  e2 ÞU 2 ðe1  e2 ÞT þ ðe2  e3 Þ dðtÞ h  dðtÞ

S U2

#"

eT1  eT2 eT2  eT3

#

ηðtÞ;

Z

t  h1 th

_ x_ T ðsÞU 3 xðsÞdsr  ηT ðtÞðeT5  eT3 ÞT U 3 ðeT5  eT3 ÞηðtÞ; ð38Þ

t  d2 ðtÞ t  h2

_ x_ T ðsÞR2 xðsÞds ð31Þ

Z  ðh  h2 Þ

t  h2 th

_ x_ T ðsÞU 4 xðsÞdsr  ηT ðtÞðeT7  eT3 ÞT U 4 ðeT7  eT3 ÞηðtÞ; ð39Þ

2 _ V_ 6 ðtÞ r x_ T ðtÞ½h U 2 þ ðh  h1 Þ2 U 3 þ ðh h2 Þ2 U 4 xðtÞ " T # " #" T # T T T e1 e2 e1  e2 U2 S ηðtÞ  ηT ðtÞ T e2  eT3 eT2 eT3 n U2

 ηT ðtÞðeT5 eT3 ÞT U 3 ðeT5  eT3 ÞηðtÞ  ηT ðtÞðeT7  eT3 ÞT U 4 ðeT7  eT3 ÞηðtÞ;

and Z

Z

From (36)–(39), we have

t

1 h2  d2 ðtÞ

_ h x_ T ðsÞU 2 xðsÞds

ð37Þ

 d2 ðtÞvT3 ðtÞR2 v3 ðtÞ  ðh2  d2 ðtÞÞvT4 ðtÞR2 v4 ðtÞ;

1 d2 ðtÞ

t t  dðtÞ

U 2 ðe2  e3 ÞT ηðtÞ " T #T " e1  eT2 U2 T r  η ðtÞ T e2  eT3 n Z

where v3 ðtÞ ¼

ð36Þ

_ xðsÞds;

_ ¼ x_ T ðsÞR2 xðsÞds r

Z

¼  ηT ðtÞ½

_ x_ T ðsÞR1 xðsÞds

where Z

_ x_ T ðsÞU 4 xðsÞds;

ð29Þ

t  d1 ðtÞ

r

th

r

T

According to Lemma 2.1, we have Z t Z t _ _ ¼ x_ T ðsÞR1 xðsÞds  x_ T ðsÞR1 xðsÞds

t  h1

t h

t  h2

Using Lemma 2.1 and 2.2, we have

ð28Þ h

t

Z

 ðh  h2 Þ

T

Z

ð35Þ

h i 2 _ V_ 6 ðtÞ ¼ x_ T ðtÞ h U 2 þ ðh  h1 Þ2 U 3 þ ðh  h2 Þ2 U 4 xðtÞ Z t  h1 Z t _ _ x_ T ðsÞU 2 xðsÞds  ðh  h1 Þ x_ T ðsÞU 3 xðsÞds h

ði ¼ 1; …; 10Þ:

ð1  μ2 ÞxT ðt  d2 ðtÞÞQ 3 xðt  d2 ðtÞÞ

lim

ð33Þ

2 V_ 5 ðtÞ rxT ðtÞðh U 1 ÞxðtÞ  π T1 U 1 π 1 ;

ð1  μ1 ÞxT ðt  d1 ðtÞÞQ 2 xðt  d1 ðtÞÞ



xT ðsÞU 1 xðsÞds:

By substituting (34) into (33), we obtain

_ xðsÞds dθ ;

ð1  μÞxT ðt dðtÞÞQ 1 xðt  dðtÞÞ

Z

th

ð32Þ

xðsÞds; th 0 Z t

Z

V_ 3 ðtÞ r xT ðtÞ½Q 1 þ Q 4 þ Q 2 þ Q 5 þ Q 3 þ Q 6 xðtÞ



t

t h

t

h ei ¼ 0nði  1Þn

Z

Z

According to Lemma 2.1, we have Z t T Z Z t xT ðsÞU 1 xðsÞds r  xðsÞds U 1 h

with

π1 ¼

By substituting (30) and (31) into (29), we obtain _ þ h2 x_ T ðtÞR2 xðtÞ _ V_ 4 ðtÞ rh1 x_ T ðtÞR1 xðtÞ  d2 ðtÞvT3 ðtÞR2 v3 ðtÞ  ðh2  d2 ðtÞÞvT4 ðtÞR2 v4 ðtÞ;

ηT ðtÞ ¼ xT ðtÞ xT ðt  dðtÞÞ xT ðt  hÞ xT ðt  d1 ðtÞÞ xT ðt h1 Þ xT ðt  d2 ðtÞÞ

Z

Z t  d1 ðtÞ 1 _ _  h1 Þ; xðsÞds ¼ xðt d1 ðtÞ-h1 h1  d1 ðtÞ t  h1 Z t 1 _ _ lim xðsÞds ¼ xðtÞ; d2 ðtÞ-0d2 ðtÞ t  d2 ðtÞ Z t  d2 ðtÞ 1 _ _  h2 Þ: lim xðsÞds ¼ xðt d2 ðtÞ-h2 h2  d2 ðtÞ t  h2 lim

t t  d1 ðtÞ

_ _ xðsÞds ¼ xðtÞ;

ð40Þ

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Using Lemma 2.1, we have ! 4 2Z 0 Z t h h _  _ Z xðsÞ V_ 7 ðtÞ ¼ x_ T ðsÞ θ x_ T ðsÞZ xðsÞdsd 4 2 h t þθ ! !T ! Z 0 Z t Z 0 Z t 4 h T _  _ _ Z xðsÞ xðsÞdsd xðsÞdsd r x_ ðsÞ θ Z θ : 4 h tþθ h t þθ

where

ξT11 ðtÞ ¼ ½ηT ðtÞvT1 vT3 ; ξT12 ðtÞ ¼ ½ηT ðtÞvT1 vT4 ; 2 Φ  h1 M 1 ~ ¼6 n  h1 R1 Φ 4 11 n

ð41Þ From the Newton–Leibniz formula, the following equations are true for any matrices M l ðl ¼ 1; …; 5Þ and Dr ðr ¼ 1; 2; 3Þ with appropriate dimensions: 2ηT ðtÞM 1 ½xðtÞ  xðt  d1 ðtÞÞ  d1 ðtÞv1  ¼ 0;

ð42Þ

2ηT ðtÞM 2 ½xðt  d1 ðtÞÞ  xðt h1 Þ  ðh1  d1 ðtÞÞv2  ¼ 0;

ð43Þ

2ηT ðtÞM 3 ½xðtÞ  xðt  d2 ðtÞÞ  d2 ðtÞv3  ¼ 0;

ð44Þ

2ηT ðtÞM 4 ½xðt  d2 ðtÞÞ  xðt h2 Þ  ðh2  d2 ðtÞÞv4  ¼ 0;

ð45Þ

2ηT ðtÞM 5 ½hxðtÞ  π 1  π 2  ¼ 0;

ð46Þ

_ þ AxðtÞ þ Bxðt  dðtÞÞ ¼ 0; 2½xT ðtÞD1 þ x_ T ðtÞD2 þ xT ðt  dðtÞÞD3 ½  xðtÞ ð47Þ

where ξ ðtÞ ¼ ηT ðtÞvT1 vT2 vT3 vT4 . According to (17)–(47), we have T



~ ðtÞ; VðtÞ r ξ ðtÞΦξ

ð48Þ

Φ  d1 ðtÞM1

6n 6 ~ ¼6 6n Φ 6 6n 4 n

 ðh1  d1 ðtÞÞM 2

 d2 ðtÞM 3

 ðh2  d2 ðtÞÞM 4

 d1 ðtÞR1

0

0

0

n

 ðh1  d1 ðtÞÞR1

0

0

n

n

 d2 ðtÞR2

n

n

n

0  ðh2  d2 ðtÞÞR2

3 7 7 7 7: 7 7 5

~ o 0, then there exists a scalar ε 4 0, such that If Φ ~ ðtÞ o  εxT ðtÞxðtÞ; 8 xðtÞ a 0: VðtÞ r ξ ðtÞΦξ T

ð49Þ

Suppose d1 ðtÞ-h1 or d1 ðtÞ-0, we have d1 ðtÞ T ~ ξ ðtÞ þ h1  d1 ðtÞξT ðtÞΦ ~ ξ ðtÞ ¼ ξT ðtÞΦξ ~ ðtÞ o  εxT ðtÞxðtÞ; 8 xðtÞ a 0; ξ ðtÞΦ 1 1 2 2 2 h1 1 h1

where 



ξT1 ðtÞ ¼ ηT ðtÞvT1 vT3 vT4 ; ξ

T 2 ðtÞ ¼

2



ηT ðtÞvT2 vT3 vT4 ;

Φ h1 M1

d2 ðtÞM 3

 ðh2  d2 ðtÞÞM 4

6 6n

 h1 R1

0

0

4n

n

 d2 ðtÞR2

0

n

n

 ðh2  d2 ðtÞÞR2

d2 ðtÞM 3

 ðh2  d2 ðtÞÞM 4

 h1 R1

0

0

n

 d2 ðtÞR2

0

n

n

 ðh2  d2 ðtÞÞR2

~ ¼6 Φ 1 6

n

2

n

Φ h1 M2

6 n ~ ¼6 6 Φ 2 6n 4 n

3 7 7 7; 7 5

ð50Þ

3 7 7 7: 7 5

 h2 M 3 0

3 7 5;

ð53Þ

 h2 R2

n

Φ  h1 M 1

~ ¼6 Φ 4n 12

 h2 M 4

 h1 R1

0

n

 h2 R2

3 7 5:

ð54Þ

~ ~ ~ is convex in d ðtÞ A ½0; h . Φ And Φ 1 2 2 11 and Φ 12 are defined in Theorem 3.1. Similarly, LMI (51) heads for d2 ðtÞ-0 to the following equation: d2 ðtÞ T ~ ξ ðtÞ þ h2  d2 ðtÞξT ðtÞΦ ~ ξ ðtÞ ¼ ξT ðtÞΦ ~ ξ ðtÞ o 0; ξ ðtÞΦ 21 21 22 22 2 2 22 2 h2 21 h2 ð55Þ where 



ξT21 ðtÞ ¼ ηT ðtÞvT2 vT3 ;  ξT22 ðtÞ ¼ ηT ðtÞvT2 vT4 ; 2 Φ  h1 M 2 ~ ¼6 n  h1 R1 Φ 4 21 n

n

Φ  h1 M 2

~ ¼6 Φ 4n 22 n

where 2

2

2

T

5

 h1 R1 n

 h2 M 3

3 7 5;

ð56Þ

3  h2 M 4 7 0 5:

ð57Þ

0  h2 R2

 h2 R2

~ ~ ~ is convex in d ðtÞ A ½0; h . Φ And Φ 2 2 2 21 and Φ 22 are defined in Theorem 3.1. Motivated by the issues discussed above, system (4) subject to (5) is asymptotically stable if the LMIs (12)–(16) hold. The proof is completed. □ Remark 3.1. In fact, when constructing the Lyapunov–Krasovskii function, many approaches do not make full use of the information about d(t), d1 ðtÞ and d2 ðtÞ. Therefore, it would be inevitably conservative to some extent. And the purpose of reducing conservatism is still remains challenging. Then, when constructing the Lyapunov– Krasovskii function, we make full use of the information about d(t), d1 ðtÞ and d2 ðtÞ. In addition, a novel Lyapunov–Krasovskii function with the triple-integral of terms is constructed. Therefore, the stability criteria in this paper may be more applicable. Remark 3.2. In this paper, a term V 2 ðtÞ is included in the Lyapunov–Krasovskii function V(t). It is noticeable that V 2 ðtÞ plays a key role in reducing conservativeness of our results. In addition, we Rt R0 Rt _ take the states t  h xðsÞds and  h t þ θ xðsÞdsd θ as augmented variables. Thus, the information of the state variables is fully used in Theorem 3.1. To reduce the conservatism, when estimating V_ ðtÞ, Rt R t  d ðtÞ Rt _ _ _  t  d1 ðtÞ x_ T ðsÞR1 xðsÞds,  t  h11 x_ T ðsÞR1 xðsÞds,  t  d2 ðtÞ x_ T ðsÞR2 xðsÞds, R t  d2 ðtÞ T T _ are bounded with  d1 ðtÞv1 ðtÞ R1 v1 ðtÞ,  t  h2 x_ ðsÞR2 xðsÞds  ðh1 d1 ðtÞÞvT2 ðtÞR1 v2 ðtÞ,  d2 ðtÞvT3 ðtÞR2 v3 ðtÞ,  ðh2  d2 ðtÞÞvT4 ðtÞR2 v4 ðtÞ,

ð51Þ

~ is convex in d ðtÞ A ½0; h . And Φ 1 1 LMI (50) heads for d2 ðtÞ-h2 to the following equation: d2 ðtÞ T ~ ξ ðtÞ þ h2  d2 ðtÞξT ðtÞΦ ~ ξ ðtÞ ¼ ξT ðtÞΦ ~ ξ ðtÞ o 0; ξ ðtÞΦ 11 11 12 12 1 1 12 1 h2 11 h2 ð52Þ

Rt 2 R0 _ θ respectively. According to Lemma 2.1,  h2  h t þ θ x_ T ðsÞZ xðsÞdsd R R T R R  0 t 0 t _ _ is bounded with   h t þ θ xðsÞdsd θ Z  h t þ θ xðsÞds dθ . R0 Rt _ Moreover, we take the state  h t þ θ xðsÞdsd θ as an augmented Rt variable, rather than by hxðtÞ  t  h xðsÞds. The value of the upper bound is estimated to be more strict than previous methods. ~ is negative define in the Remark 3.3. It is easy to show that Φ rectangle 0 r d1 ðtÞ rh1 o1, 0 r d2 ðtÞ r h2 o 1, only if it is

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6

negative definite at all vertices. We define that this method to be the convex polyhedron method. This method plays a key role in reducing conservativeness of derived results. Remark 3.4. Although we only consider system (4) with two additive time-varying delay components in the state, the results of this paper can be extended to the following systems with multiple additive delay components: ! n X _xðtÞ ¼ AxðtÞ þ Bx t  di ðtÞ : i¼2

And a nature assumption on the delay di ðtÞ is made 0 r di ðtÞ r hi o 1; d_ i ðtÞ r μi o 1: Then, we will construct the Lyapunov–Krasovskii function as follows: V~ ðtÞ ¼ V 1 ðtÞ þV 2 ðtÞ þ V~ 3 ðtÞ þ V~ 4 ðtÞ þ V 5 ðtÞ þ V 6 ðtÞ þ V 7 ðtÞ; where V 1 ðtÞ, V 2 ðtÞ, V 5 ðtÞ, V 6 ðtÞ, V 7 ðtÞ are given in (18), (19), (22), (23) and (24), respectively. And Z t n Z t X V~ 3 ðtÞ ¼ xT ðsÞQ 1 xðsÞds þ xT ðsÞQ 2i xðsÞds t  dðtÞ

Z

þ V~ 4 ðtÞ ¼

t t h

n Z X i¼1

xT ðsÞQ 4 xðsÞds þ

0  hi

Z

i ¼ 1 t  di ðtÞ n Z t X i¼1

t tþθ

t  hi

xT ðsÞQ 3i xðsÞds;

_ x_ T ðsÞRi xðsÞds dθ :

According to the proof of Theorem 3.1, we can obtain the similar results. Therefore, the results obtained in this paper can be extended to multiple delay case.

have the upper bound h1 of d1 ðtÞ tabulated in Table 1. In which ‘–’ means that the results are not applicable to the corresponding cases. It can easily be seen from Table 1 that the computed admissible upper bound of the time-varying delay is larger than the stability criteria in [31,36,46,38,34], because reciprocally convex approach and convex polyhedron approach are considered. For h1 ¼ 1 and h2 ¼ 0:928, the asymptotic stability with the initial state ½2;  2T is given in Fig. 2. For h2 ¼ 0:3 and h1 ¼ 1:689, the asymptotic stability with the initial state ½  0:5; 0:5T is given in Fig. 3. From Figs. 2 and 3 we can see the effectiveness and superiority of the method proposed in this paper. This paper has the potential to enable us to obtain less conservative results than those obtained by [31,36,46,38,34]. Example 2. To demonstrate the superiority of our method, we consider system (4) with the following parameters:



 2 0 1 0 A¼ ; B¼ : 0  0:9 1 1 Suppose we know that d_ 1 ðtÞ r 0:1, d_ 2 ðtÞ r 0:8. In this paper, h1 and h2 represent the delay upper bound of d1 ðtÞ and d2 ðtÞ, respectively. We calculated delay bounds for different cases by Theorem 3.1 and the results in [50,31,19,23,36,46,38,34,52] are listed in Table 2, in which ‘–’ means that the results are not applicable to the corresponding cases. It can easily be seen from Table 2 that the computed admissible upper bound of the time-varying delay is larger than the stability criteria in [50,31,19,23,36,46,38,34,52], because reciprocally convex approach and convex polyhedron approach are considered. For h1 ¼ 1 and h2 ¼ 0:982, the asymptotic stability with the initial state ½  3; 3T is given in Fig. 4. For h2 ¼ 0:3 and h1 ¼ 1:682, the asymptotic stability with the initial state ½ 5; 5T is given in Fig. 5. From Figs. 4 and 5 we can see the effectiveness and 2 x (t) 1

1.5

4. Illustrative examples

Example 1. To demonstrate consider system (4) with the



2 0 1 A¼ ; B¼ 0 9 1

the superiority of our method, we following parameters:  0 : 1

2

1

Amplitude

In this section, four examples are given to show the effectiveness of the proposed method.

x (t)

0.5 0 −0.5 −1 −1.5

Suppose we know that d_ 1 ðtÞ r 0:1, d_ 2 ðtÞ r 0:8. Our goal is to find the upper bound h1 of delay d1 ðtÞ, or h2 of d2 ðtÞ, when the other is known. By employing Theorem 3.1 with h1 ¼ 1, 1.1, 1.2 and 1.5, we obtain the upper bound h2 of d2 ðtÞ tabulated in Table 1. Similarly, when h2 ¼0.3, 0.4 and 0.5, we also

−2

0

1

2

3

4

5

6

7

8

9

10

t

Fig. 2. For h1 ¼ 1 and h2 ¼0.928, the dynamical behavior of system (4) in Example 1.

Table 1 Calculated delay bounds for different cases in Example 1. Method

[31] [36] [46] [38] [34] Theorem 3.1

Delay bound h2 for given h1

Delay bound h1 for given h2

h1 ¼ 1

h1 ¼ 1:1

h1 ¼ 1:2

h1 ¼ 1:5

h2 ¼ 0:3

h2 ¼ 0:4

h2 ¼ 0:5

0.180 0.378 0.415 0.512 0.872 0.928

0.080 0.278 0.376 0.457 0.772 0.889

– 0.178 0.340 0.406 0.672 0.789

– – 0.248 0.283 0.371 0.459

0.880 1.078 1.324 1.453 1.572 1.689

0.780 0.978 1.039 1.214 1.472 1.528

0.680 0.878 0.806 1.021 1.372 1.446

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Example 3. To demonstrate the superiority of our method, we consider system (4) with the following parameters:



 2 0 1 0 A¼ ; B¼ : 0 0:09 1  1 Suppose we know that d_ 1 ðtÞ r 0:1, d_ 2 ðtÞ r 0:8. The purpose is to find the upper bound h1 of delay d1 ðtÞ, or h2 of d2 ðtÞ, when the other is known. We calculated delay bounds for different cases by Theorem 3.1 and the results in [50,19,31] are listed in Table 3, in which ‘–’ means that the results are not applicable to the corresponding cases. It can easily be seen from Table 3 that the computed admissible upper bound of the timevarying delay is larger than the stability criteria in [50,19,31], because reciprocally convex approach and convex polyhedron approach are considered. For h1 ¼ 1 and h2 ¼ 0:580, the asymptotic stability with the initial state ½  1; 1T is given in Fig. 6. For h2 ¼ 0:3 and h1 ¼ 1:546, the asymptotic stability with the initial state ½  1:5; 1:5T is given in Fig. 7. From Figs. 6 and 7 we can see the effectiveness and superiority of the method proposed in this paper. This paper has the potential to enable us to obtain less conservative results than those obtained by [50,19,31].

distillation column upon the change in the relative compositions of the reactants in Fig. 10. _ ¼ A0 xðtÞ þ A1 xðt  1Þ with the following Consider the system xðtÞ parameters: 2

4:93 6 3:20 6 A0 ¼ 6 4 6:40 0

 1:01

0

 5:30 0:347 0:833

 12:8  32:5 11:0

0

3

2

0 7 7 7;  1:04 5  3:96

1:92 6 0 6 A1 ¼ 6 4 0 0

0

0

1:92 0 0

0 1:87 0

3

0

0 7 7 7: 0 5 0:724

Consider the linear constant system with time lag as follows: _ ¼ A0 xðtÞ þA1 xðtÞ þ BuðtÞ; xðtÞ

ð58Þ

3

2

1 Amplitude

superiority of the method proposed in this paper. This paper has the potential to enable us to obtain less conservative results than those obtained by [50,31,19,23,36,46,38,34,52].

7

0

−1

−2

Example 4. An example feedback controller design. Consider an example of a typical control problem occurring in the chemical and petroleum industries as described in [56]. A block diagram of the control system with this feedback controller is shown in Fig. 8. The block diagram of Fig. 9 depicts a refining plant. The block in the recycle loop represents the effects of the cooler, decanter, and

0

4

0.3

3

0.2

2 Amplitude

0.4

−0.2

−2

−0.3

−3

−0.4

−4 −5

−0.5 2

3

4

5

6

7

8

9

4

5

6

7

8

9

10

0

−0.1

1

3

1

−1

0

2

Fig. 4. For h1 ¼1 and h2 ¼0.982, the dynamical behavior of system (4) in Example 2.

5

0

1

t

0.5

0.1 Amplitude

−3

10

0

1

2

3

4

5

6

7

8

9

10

t

t

Fig. 3. For h2 ¼0.3 and h1 ¼ 1.689, the dynamical behavior of system (4) in Example 1.

Fig. 5. For h2 ¼ 0.3 and h1 ¼ 1.682, the dynamical behavior of system (4) in Example 2.

Table 2 Calculated delay bounds for different cases in Example 2. Method

[50] [31,19,23] [36] [46] [38] [34] [52] Theorem 3.1

Delay bound h2 for given h1

Delay bound h1 for given h2

h1 ¼ 1

h1 ¼ 1:2

h1 ¼ 1:5

h2 ¼ 0:3

h2 ¼ 0:4

h2 ¼ 0:5

– 0.180 0.378 0.415 0.512 0.872 0.873 0.982

– – 0.178 0.340 0.406 0.672 0.673 0.782

– – – 0.248 0.283 0.371 0.373 0.482

– 0.880 1.078 1.324 1.453 1.572 1.573 1.682

– 0.780 0.978 1.039 1.214 1.472 1.473 1.582

– 0.680 0.878 0.806 1.021 1.372 1.373 1.482

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Table 3 Calculated delay bounds for different cases in Example 3. Method

Delay bound h2 for given h1

[50] [19] [31] Theorem 3.1

Delay bound h1 for given h2

h1 ¼ 1

h1 ¼ 1:1

h1 ¼ 1:5

h2 ¼ 0:3

h2 ¼ 0:4

h2 ¼ 0:5

– 0.180 0.180 0.580

– 0.080 0.080 0.468

– – – 0.342

0.098 1.080 1.080 1.546

– 0.980 0.980 1.432

– 0.080 0.880 1.346

1 0.8 0.6

Amplitude

0.4 0.2 0 −0.2 −0.4 −0.6 −0.8 −1

0

1

2

3

4

5

6

7

8

9

10

t

Fig. 6. For h1 ¼1 and h2 ¼ 0.580, the dynamical behavior of system (4) in Example 3. 1.5

1

Amplitude

0.5

0

−0.5

−1

−1.5

0

1

2

3

4

5

6

7

8

9

10

t

Fig. 7. For h2 ¼0.3 and h1 ¼1.546, the dynamical behavior of system (4) in Example 3. Fig. 8. Block of system with (optimal) feedback controller in Example 4.

with the following parameters: 2

4:93 6 3:20 6 A0 ¼ 6 4 6:40 0

2

1

0

0

0

60 6 B¼6 40

 1:01  5:30 0:347 0:833

0  12:8  32:5 11:0

3 0 0 7 7 7; 1:04 5 3:96

2

1:92 6 0 6 A1 ¼ 6 4 0 0

0 1:92 0 0

0 0 1:87 0

3 0 0 7 7 7; 0 5 0:724

3

17 7 7: 05

where uðtÞ ¼ KxðtÞ, based on Matlab LMI toolbox, we have

  5:3855 4:1750 6:2336 7:1194 K¼ :  1:6726 0:5893 1:0495 0:5580 Fig. 11 shows a typical time response of this system. The response is sluggish; that is, a long time elapses before an initial disturbance is significantly damped. The system time response was markedly improved with the addition of the feedback control, as shown in Fig. 12. Therefore, a good system performance has

been obtained for system (58), and the design of the controller has been established.

5. Conclusion In this paper, the problem of stability for continuous system with two additive time-varying delay components is investigated. A novel Lyapunov–Krasovskii functional is constructed by making full use of the information of the marginally delayed state. Based on reciprocally convex approach and convex polyhedron approach, combined with Jensen integral inequality and linear matrix inequality (LMI), a delay-dependent stability criterion has been obtained. In addition, the obtained delay-dependent stability results are less conservative because reciprocally convex approach and convex polyhedron approach are considered. The stability results are expressed as a set of linear matrix inequalities. Finally, four examples have been provided to show the effectiveness of the proposed results.

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Fig. 9. Refining plant in Example 4.

References

Fig. 10. Chemical reactor with recycle loop in Example 4.

0.1 0.08 0.06

Amplitude

0.04 0.02 0 −0.02 −0.04 −0.06

0

1

2

3

4

5

6

7

t

Fig. 11. Response of system before addition of feedback controller in Example 4.

0.1

0.08

Amplitude

0.06

0.04

0.02

0

−0.02

−0.04

0

1

2

3

4

5

6

7

t

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Fig. 12. Response of system after addition of feedback controller in Example 4.

Please cite this article as: Yu X, et al. Further results on delay-dependent stability for continuous system with two additive time-varying delay components. ISA Transactions (2016), http://dx.doi.org/10.1016/j.isatra.2016.08.003i

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Please cite this article as: Yu X, et al. Further results on delay-dependent stability for continuous system with two additive time-varying delay components. ISA Transactions (2016), http://dx.doi.org/10.1016/j.isatra.2016.08.003i