ISA Transactions ∎ (∎∎∎∎) ∎∎∎–∎∎∎

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New results on delay-range-dependent stability analysis for interval time-varying delay systems with non-linear perturbations Pin-Lin Liu n Department of Automation Engineering Institute of Mechatronoptic System, Chienkuo Technology University, Changhua 500, Taiwan, ROC

art ic l e i nf o

a b s t r a c t

Article history: Received 29 November 2014 Received in revised form 12 February 2015 Accepted 3 March 2015 This paper was recommended for publication by Jeff Pieper

This paper studies the problem of the stability analysis of interval time-varying delay systems with nonlinear perturbations. Based on the Lyapunov–Krasovskii functional (LKF), a sufficient delay-rangedependent criterion for asymptotic stability is derived in terms of linear matrix inequality (LMI) and integral inequality approach (IIA) and delayed decomposition approach (DDA). Further, the delay range is divided into two equal segments for stability analysis. Both theoretical and numerical comparisons have been provided to show the effectiveness and efficiency of the present method. Two well-known examples are given to show less conservatism of our obtained results and the effectiveness of the proposed method. & 2015 ISA. Published by Elsevier Ltd. All rights reserved.

Keywords: Integral inequality approach (IIA) Interval time-varying delay Delayed decomposition approach (DDA) Linear matrix inequality (LMI) Maximum allowable upper bounds (MAUB)

1. Introduction Systems with time delays are universal in many physical and engineering systems. These delays are usually time-varying in nature, and have an adverse impact not only on the system performance but also on its stability; therefore, neglecting the effects of delay in the analysis may lead to instability and incorrect design calculations. Therefore, stability analysis of time-delay systems has been extensively studied by many researchers [1–22]. On the other hand, the range of time-varying delay for systems with nonlinear perturbations considered in [3–5,7,8,10–12,14–17,19–22] is from zero to an upper bound. In the real world, a time-varying interval delay is often encountered, that is, the range of delay varies in an interval for which the lower bound is not restricted to zero. In this case, the criteria in the previous works, such as [3–5,8,10–12,14–16,19,22], are conservative because they do not take into account the information of the lower bound of delay. Systems with interval time-varying delay h1 r hðtÞ r h2 are used to show that the propagated speed of signals is finite and uncertain in systems [6,7,8,13,17,18,20,21], and some delay-range-dependent stability analysis and control synthesis have been carried out by using the Lyapunov–Krasovksii functional approach for interval time-varying delay systems with nonlinear perturbations [6,7,17,18,20,21].

n

Tel.: þ 886 4 7111155; fax: þ 886 4 7111129. E-mail addresses: [email protected], [email protected]

In order to increase the delay bounds, various methods are employed, such as model transformation technique [8], integral inequality matrix method [11], free-weighting matrices technique [15,20,22], Jensen's integral inequality method [16,17] and delayed decomposition approach [12]. Furthermore, some researchers put their effort on reducing the complexity of its calculation. In [3], both delayindependent and delay-dependent stability criteria were proposed for a time-delayed system under nonlinear perturbations. In [5], by employing a descriptor model transformation, delay-dependent robust stability condition was presented for a class of time-delay systems with nonlinear perturbations. An integral inequality approach (IIA) was developed in [11,12] to estimate the derivative of the Lyapunov functional, and the information of delays can be taken into full consideration. In [22], using free-weighting matrices to deal with the cross terms involved in the derivative of the Lyapunov–Krasovskii functional (LKF), a less conservative delay-dependent stability criterion was proposed. All these techniques assume that the delay-range varies from zero to an upper bound. In order to further reduce the conservative, inspired by the discretized Lyapunov method, a delay-partitioning approach was proposed in [19] and shows its improvement of maximum delay bounds [7,17,18]. It is worth pointing out that, different from the delay partitioning approach used in [7,17], the main advantage of the approach which was used in [18] is that when constructing the LKF, not all information of every subinterval is exploited. The terms of LKF result in the present of hðtÞ in the final subinterval, so when delay partitioning number becomes larger, the derived criteria can lead to

http://dx.doi.org/10.1016/j.isatra.2015.03.001 0019-0578/& 2015 ISA. Published by Elsevier Ltd. All rights reserved.

Please cite this article as: Liu P-L. New results on delay-range-dependent stability analysis for interval time-varying delay systems with non-linear perturbations. ISA Transactions (2015), http://dx.doi.org/10.1016/j.isatra.2015.03.001i

P.-L. Liu / ISA Transactions ∎ (∎∎∎∎) ∎∎∎–∎∎∎

2

an improvement. Furthermore, the cost of computation and the complexity of the obtained criteria do not increase. However, it can be found that the Lyapunov functional introduced in [18] only contains some simple integral terms and double integral terms, but does not include triple-integral terms. Adding some triple-Integral terms in the Lyapunov functional may be helpful for the reduction of the conservatism such as that reported in [17]. Motivated by the above discussions, this paper extends recent results [12] for the delay-dependent stability with time-varying delay systems with nonlinear perturbations to the delay-range-dependent robust stability criteria for interval time-varying delay systems with nonlinear perturbations. Based on the linear matrix inequality (LMI) criterion given in [12], an extended delay-range-dependent criterion for time-varying delay systems with nonlinear perturbations is derived by a simple procedure. The advantage of the proposed approach is that the emerging stability criterion can be used efficiently via existing numerical convex optimization algorithms like interiorpoint algorithms for solving LMIs. An important feature of results reported here is that all conditions depend on both lower and upper bounds of the interval time-varying delays. Finally, two well-known examples are given to demonstrate the reduced conservatism and the effectiveness of the proposed method. Notations: Throughout this paper, the superscripts ‘ 1’ and ‘T’ stand for the inverse and transpose of a matrix, respectively; Rnn denotes an n-dimensional Euclidean space; Rmn is the set of all m  n real matrices; P 4 0 means that matrix P is symmetric positive definite; For real symmetric matrices X and Y; the notation X Z Y (respectively, X 4 Y) means that the matrix X  Y is positive semidefinite (respectively, positive definite); I is an appropriately dimensional identity matrix; X ij denotes the element in row i and column j of matrix X; the notation n always denotes the symmetric block in one symmetric matrix. Matrices, if not explicitly stated, are assumed to have compatible dimensions.

2. Problem description and preliminaries In this paper, consider the interval time-varying delay systems with nonlinear perturbations described by the following linear differential difference equations: x_ ðtÞ ¼ AxðtÞ þBxðt hðtÞÞ þ Ff ðxðtÞ; tÞ þ Ggðxðt  hðtÞÞ; tÞ xðt þ ηÞ ¼ ϕðηÞ;

t 40

8 η A ½  h2 ;  h1 

n

ð6aÞ

the following integral inequality holds 

Rt t  hðtÞ

Z

t t  hðtÞ

x_ T ðsÞX 33 x_ ðsÞds r

h

x ðtÞ T

x ðt  hðtÞÞ T

2 X 11 i 6 T x_ ðsÞ 4 X 12 X T13 T

X 12 X 22 X T23

32 3 X 13 xðtÞ 7 6 X 23 54 xðt  hðtÞÞ 7 5ds: x_ ðsÞ 0

ð6bÞ

Lemma 2. [2]. The following matrix inequality " # Q ðxÞ SðxÞ o0 ST ðxÞ RðxÞ

ð7aÞ

where Q ðxÞ ¼ Q T ðxÞ; RðxÞ ¼ RT ðxÞandSðxÞ depend on affine on x is equivalent to RðxÞ o0

ð7bÞ

Q ðxÞ o 0

ð7cÞ

and Q ðxÞ  SðxÞR  1 ðxÞST ðxÞ o 0

ð7dÞ

Lemma 3. [2]. Given matrices Q ¼ Q T ; D; and E; of appropriate dimensions then Q þ DFðtÞE þ ET F T ðtÞDT o0

ð8aÞ

for all FðtÞ satisfying F T ðtÞFðtÞ rI; if and only if there exists some ε 4 0 such that

ð2Þ

Q þ ε  1 DDT þ εET E o 0

ð8bÞ

nn

f ðxðtÞ; tÞf ðxðtÞ; tÞ r α2 xT ðtÞxðtÞ

ð3Þ

g T ðxðt  hðtÞÞ; tÞgðxðt  hðtÞÞ; tÞ rβ2 xT ðt  hðtÞÞxðt  hðtÞÞ

ð4Þ

where α and β are known positive constants. hðtÞ is a time-varying continuous function satisfies _ rh ; 8t Z0 0 r h1 r hðtÞ r h2 ; hðtÞ d

[11,12]. For any positive semi-definite matrices 3 X 12 X 13 X 22 X 23 7 5 Z0; X T23 X 33

ð1Þ

constant with xðtÞ A R as state vector of the system, A; B; F; G A R matrices, ϕðdÞ continuous vector-valued initial function, f ðxðtÞ; tÞ A Rn ; and gðxðt  hðtÞÞ; tÞ A Rn unknown non-linear perturbations with respect to xðtÞ and xðt  hðtÞÞ; respectively, assumed as T

Lemma 1. 2 X 11 6 T X ¼ 4 X 12 X T13

ð5Þ

where h1 ; h2 and hd are constants. In this paper, our objective is to establish new delay-rangedependent stability criteria for systems (1). To obtain the main results, the following lemmas are indispensable in deriving the proposed stability criteria. In order to make the following clearly, we let δ ¼ h2 h1 :

The main aim is to derive a maximum admissible upper bound (MAUB) of the time-delay such that the concerned system is asymptotically stable for any delay size less than the MAUB. Accordingly, the obtained MAUB becomes a key performance index to measure the conservatism of a delay-dependent stability condition. The delay interval ½h1 ; h2  is divided into two subintervals with an unequal width as ½h1 ; h1 þ ρδ and ½h1 þ ρδ; h2  ð0 o ρ o1; δ ¼ h2  h1 Þ; which is different from the existing method [1,12]. Under case 1, h1 r hðtÞ r h1 þ ρδ, the stability results for case 1 is derived in the following Theorem 1. Theorem 1. If h1 rhðtÞ r h1 þ ρδ; 0 o ρ o1; for given scalars h1 ; h2 ; and hd ; the system (1) subject to (2)–(4) is asymptotically stable if there exist P ¼ P T 4 0; Q i ¼ Q Ti Z 0; Rj ¼ RTj Z 0 ði ¼ 1; 2; 3; 4; 2 3 2 3 Y 11 Y 12 Y 13 X 11 X 12 X 13 T 6 XT 7 6 7 j ¼ 1; 2; 3Þ; X ¼ 4 12 X 22 X 23 5 Z0; Y ¼ 4 Y 12 Y 22 Y 23 5 Z 0; T T T T X 13 X 23 X 33 Y 13 Y 23 Y 33

Please cite this article as: Liu P-L. New results on delay-range-dependent stability analysis for interval time-varying delay systems with non-linear perturbations. ISA Transactions (2015), http://dx.doi.org/10.1016/j.isatra.2015.03.001i

P.-L. Liu / ISA Transactions ∎ (∎∎∎∎) ∎∎∎–∎∎∎

2

Z 11 6 T Z ¼ 4 Z 12 Z T13

3 Z 13 Z 23 7 5 Z 0 and a scalar ε 4 0 such that the following Z 33

Z 12 Z 22 Z T23

LMIs hold: 2 Ω11 Ω12 6 Ω22 6 n 6 6 n n 6 6 n 6 n Ω¼6 6 n n 6 6 n 6 n 6 6 n n 4 n

Z 

Ω14

Ω15

0

0

0

0

Ω25

Ω26

0

Ω33

0

0

0

0

n

Ω44

n

n

0 Ω55

0 0

0 0

n

n

n

Ω66

Ω67

n

n

n

n

Ω77

n

n

n

n

n

n

Ω18

T

þ x ðt  h1 ÞðQ 2  Q 1 Þxðt  h1 Þ

7 Ω28 7 7 Ω38 7 7 7 Ω48 7 7o0 0 7 7 7 0 7 7 0 7 5 Ω88

þ xT ðt  h1  ρδÞðQ 3 Q 2 Þxðt  h1  ρδÞ  xT ðt  h2 ÞQ 3 xðt h2 Þ  xT ðt  hðtÞÞð1  hd ÞQ 4 xðt  hðtÞÞ ð9aÞ

þ x_ T ðtÞ½h1 R1 þ ρδR2 þð1 ρÞδR3 x_ ðtÞ  Z 

ð9bÞ

t  h1  ρδ

t  h2

x_ T ðsÞR1 x_ ðsÞds

x_ T ðsÞR3 x_ ðsÞds

T

þ xT ðt  h1 ÞðQ 2  Q 1 Þxðt  h1 Þ 2

I þ Q 1 þ Q 4 þ h1 X 11 þX 13 þ X T13 ;

þ xT ðt  h1  ρδÞðQ 3 Q 2 Þxðt  h1  ρδÞ  xT ðt  h2 ÞQ 3 xðt h2 Þ  xT ðt  hðtÞÞð1  hd ÞQ 4 xðt  hðtÞÞ þ x_ T ðtÞ½h1 R1 þ ρδR2 þð1 ρÞδR3 x_ ðtÞ Z t x_ T ðsÞðR1  X 33 Þx_ ðsÞds 

ð1 hd ÞQ 4 þ ε2 β I þ ρδY 22 Y 23  Y T23 þ ρδY 11 þY 13 þY T13 ; ρδY T12  Y 13 þ Y 23 ; Ω26 ¼ ρδY 12  Y 13 þ Y T23 ; BT ½h1 R1 þ ρδR2 þ ð1  ρÞδR3 ; 2

Z 

T

Ω33 ¼ ε1 I; Ω38 ¼ F ½h1 R1 þ ρδR2 þ ð1  ρÞδR3 ; Ω44 ¼  ε2 I;

Z

Ω48 ¼ GT ½h1 R1 þ ρδR2 þ ð1  ρÞδR3 ;

Ω77 ¼

Z

t  h1

þ f ðxðtÞ; tÞF T PxðtÞ þ g T ðxðt  hðtÞÞ; tÞGT PxðtÞ

Ω15 ¼ h1 X 11  X 13 þ X T23 ; Ω18 ¼ AT ½h1 R1 þ ρδR2 þ ð1  ρÞδR3 ;

Ω67 ¼

t  h1  ρδ

x_ T ðsÞR2 x_ ðsÞds

t

¼ xT ðtÞðAT P þ PA þQ 1 þ Q 4 ÞxðtÞ þ xT ðtÞPBxðt  hðtÞÞ

Ω11 ¼ A P þ PA þ ε1 α Ω12 ¼ PB; Ω13 ¼ PF; Ω14 ¼ PG;

Ω55 ¼

t  h1

Z

þ xT ðtÞPFf ðxðtÞ; tÞ þ xT ðtÞPGgðxðt  hðtÞÞ; tÞ þ xT ðt  hðtÞÞBT PxðtÞ

T

Ω66 ¼

x_ T ðsÞR3 x_ ðsÞds r xT ðtÞðAT P þ PAþ Q 1 þ Q 4 ÞxðtÞ

þ xT ðt  hðtÞÞBT PxðtÞ þ f ðxðtÞ; tÞF T PxðtÞ þ g T ðxðt  hðtÞÞ; tÞGT PxðtÞ

3

where

Ω28 ¼

t  h2

T

Ω13

R1  X 33 Z 0; R2  Y 33 Z 0; R3  Z 33 Z 0

Ω25 ¼

t  h1  ρδ

þ xT ðtÞPBxðt  hðtÞÞ þ xT ðtÞPFf ðxðtÞ; tÞ þxT ðtÞPGgðxðt  hðtÞÞ; tÞ

and

Ω22 ¼

3



Q 2  Q 1 þh1 X 22  X 23  X T23 þ ρδY 11 þ Y 13 þ Y T13 ; Q 3  Q 2 þρδY 22  Y 23  Y T23 þð1 ρÞδZ 11 þ Z 13 þ Z T13 ; ð1  ρÞδZ 12  Z 13 þ Z T23 ; Q 3 þ ð1  ρÞδZ 22 Z 23  Z T23 ;

Z  Z 

Ω88 ¼ ½h1 R1 þ ρδR2 þ ð1  ρÞδR3 : Proof Construct a Lyapunov–Krasovskii functional candidate as follows: Z t Z t  h1 xT ðsÞQ 1 xðsÞdsþ xT ðsÞQ 2 xðsÞds Vðxt Þ ¼ xT ðtÞPxðtÞ þ Z þ Z þ

t  h1

t  h1  ρδ t  h2 0  h1

Z þ

Z

t

 h2

xT ðsÞQ 3 xðsÞdsþ

Z

t t þθ

t  h1  ρδ

t

t  hðtÞ

x_ ðsÞR1 x_ ðsÞds dθ þ T

tþθ

 h1  ρδ

Z

Z

xT ðsÞQ 4 xðsÞds

 h1  h1  ρδ

Z

t tþθ

x_ T ðsÞR2 x_ ðsÞds dθ

x_ T ðsÞR3 x_ ðsÞds dθ

ð10Þ

t  h1

t  h1 t  h1  ρδ

x_ T ðsÞðR2  Y 33 Þx_ ðsÞds

t  h1  ρδ t  h2 t t  h1

x_ T ðsÞX 33 x_ ðsÞds 

t  h1  ρδ t  h2

x_ T ðsÞðR3  Z 33 Þx_ ðsÞds Z

t  h1

t  h1  ρδ

x_ T ðsÞY 33 x_ ðsÞds

x_ T ðsÞZ 33 x_ ðsÞds

ð11Þ

With the operator for the term x_ T ðtÞ½h1 R1 þ ρδR2 þ ð1  ρÞδR3 x_ ðtÞ as follows: x_ T ðtÞ½h1 R1 þ ρδR2 þ ð1 ρÞδR3 x_ ðtÞ ¼ ½AxðtÞ þ Bxðt  hðtÞÞ þ Ff ðxðtÞ; tÞ þ Ggðxðt  hðtÞÞ; tÞT ½h1 R1 þ ρδR2 þ ð1  ρÞδR3   ½AxðtÞ þ Bxðt  hðtÞÞ þ Ff ðxðtÞ; tÞ þ Ggðxðt  hðtÞÞ; tÞ ¼ xT ðtÞAT ½h1 R1 þ ρδR2 þ ð1  ρÞδR3 AxðtÞ þ xT ðtÞAT ½h1 R1 þρδR2 þ ð1  ρÞδR3 Bxðt  hðtÞÞ þ xT ðtÞAT ½h1 R1 þρδR2 þ ð1  ρÞδR3 Ff ðxðtÞ; tÞ þ xT ðtÞAT ½h1 R1 þρδR2 þ ð1  ρÞδR3 Ggðxðt  hðtÞÞ; tÞ

Calculating the derivative of (10) with respect to t 4 0 along the trajectories of (1) leads to V_ ðxt Þ ¼ xT ðtÞðAT P þ PAÞxðtÞ þ xT ðtÞPBxðt  hðtÞÞ þ xT ðtÞPFf ðxðtÞ; tÞ T

T

T

T

þ xT ðt  hðtÞÞBT ½h1 R1 þρδR2 þ ð1  ρÞδR3 AxðtÞ þ xT ðt  hðtÞÞBT ½h1 R1 þρδR2 þ ð1  ρÞδR3 Bxðt  hðtÞÞ þ xT ðt  hðtÞÞBT ½h1 R1 þρδR2 þ ð1  ρÞδR3 Ff ðxðtÞ; tÞ þ xT ðt  hðtÞÞBT ½h1 R1 þρδR2 þ ð1  ρÞδR3 Ggðxðt  hðtÞÞ; tÞ

T

þ x ðtÞPGgðxðt hðtÞÞ; tÞ þ x ðt  hðtÞÞB PxðtÞ þ f ðxðtÞ; tÞF PxðtÞ

þ f ðxðtÞ; tÞF T ½h1 R1 þ ρδR2 þ ð1  ρÞδR3 AxðtÞ

þ g T ðxðt  hðtÞÞ; tÞGT PxðtÞ þxT ðtÞQ 1 xðtÞ  xT ðt  h1 ÞQ 1 xðt  h1 Þ

þ f ðxðtÞ; tÞF T ½h1 R1 þ ρδR2 þ ð1  ρÞδR3 Bxðt  hðtÞÞ

þ xT ðt  h1 ÞQ 2 xðt h1 Þ xT ðt  h1  ρδÞQ 2 xðt  h1  ρδÞ þ xT ðt  h1  ρδÞQ 3 xðt  h1  ρδÞ  xT ðt  h2 ÞQ 3 xðt  h2 Þ _ þ xT ðtÞQ 4 xðtÞ  xT ðt  hðtÞÞð1  hðtÞÞQ 4 xðt  hðtÞÞ Z t x_ T ðsÞR1 x_ ðsÞds þ x_ T ðtÞh1 R1 x_ ðtÞ  þ x_ T ðtÞρδR2 x_ ðtÞ 

Z

t  h1

t  h1 t  h1  ρδ

x_ T ðsÞR2 x_ ðsÞdsþ x_ T ðtÞð1  ρÞδR3 x_ ðtÞ

T T T

þ f ðxðtÞ; tÞF T ½h1 R1 þ ρδR2 þ ð1  ρÞδR3 Ff ðxðtÞ; tÞ T

þ f ðxðtÞ; tÞF T ½h1 R1 þ ρδR2 þ ð1  ρÞδR3 Ggðxðt  hðtÞÞ; tÞ þ g T ðxðt  hðtÞÞ; tÞGT ½h1 R1 þ ρδR2 þ ð1  ρÞδR3 AxðtÞ þ g T ðxðt  hðtÞÞ; tÞGT ½h1 R1 þ ρδR2 þ ð1  ρÞδR3 Bxðt  hðtÞÞ þ g T ðxðt  hðtÞÞ; tÞGT ½h1 R1 þ ρδR2 þ ð1  ρÞδR3 Ff ðxðtÞ; tÞ þ g T ðxðt  hðtÞÞ; tÞGT ½h1 R1 þ ρδR2 þ ð1  ρÞδR3 Ggðxðt  hðtÞÞ; tÞ ð12Þ

Please cite this article as: Liu P-L. New results on delay-range-dependent stability analysis for interval time-varying delay systems with non-linear perturbations. ISA Transactions (2015), http://dx.doi.org/10.1016/j.isatra.2015.03.001i

P.-L. Liu / ISA Transactions ∎ (∎∎∎∎) ∎∎∎–∎∎∎

4

Alternatively, the following equations are true: R t  h1 _T _ t  h1 x ðsÞX 33 xðsÞds  t  h1  ρδ Z t  h1  ρδ ds  x_ T ðsÞZ 33 x_ ðsÞds t  h2 

Rt

Z

¼ Z 

t

x_ ðsÞX 33 x_ ðsÞds  T

t  h1

t  h1

t  hðtÞ

Z

t  hðtÞ t  h1  ρδ

x_ T ðsÞY 33 x_ ðsÞds

Z

T

ε1 ½α2 xT ðtÞxðtÞ  f ðxðtÞ; tÞf ðxðtÞ; tÞ Z0

x_ T ðsÞY 33 x_ ðsÞ

and ε2 ½β2 xT ðt  hðtÞÞxðt  hðtÞÞ  g T ðxðt hðtÞÞ; tÞgðxðt  hðtÞÞ; tÞ Z 0 x_ ðsÞY 33 x_ ðsÞds T

t  h1  ρδ

t  h2

x_ T ðsÞZ 33 x_ ðsÞds

ð13Þ

Z

Z r

t

t  h1

h

t

t  h1

t  h1  ρδ

Z

2 X 11 i T _xT ðsÞ 6 4 X 12 X T13

xT ðt  h1 Þ

X 12 X 22 X T23

X 13 0

rxT ðtÞh1 X 11 xðtÞ þ xT ðtÞh1 X 12 xðt  h1 Þ þ xT ðtÞX 13 x_ ðsÞds þ xT ðt  h1 Þh1 X T12 xðtÞ

Z

T

T

þ x ðt  h1 Þh1 X 22 xðt h1 Þ þ x ðt  h1 ÞX 23 Z

t

t  h1

x_ T ðsÞdsX T13 xðtÞ þ

Z

t t  h1

32

xðtÞ

3

6 7 X 23 7 54 xðt  h1 Þ 5ds Z

þ

t t  h1

x_ ðsÞ

t t  h1

x_ ðsÞds

þ xT ðt  h1 Þ½h1 X 22  X 23  X T23 xðt  h1 Þ

t  h1  ρδ

ð20Þ

2

xT ðt hðtÞÞ

T

f ðxðtÞ; tÞ

n

Ξ 12 Ξ 22

Ξ 13 Ξ 23

Ξ 14 Ξ 24

Ξ 15 Ξ 25

0 Ξ 26

n

n

Ξ 33

Ξ 34

0

0

n

n

n

Ξ 44

0

0

n

n

n

n

Ξ 55

0

n

n

n

n

n

Ξ 66

n

n

n

n

n

n

Ξ 11

3 0 0 7 7 7 0 7 7 0 7 7 with 7 0 7 7 Ξ 67 7 5 Ξ 77

2

þ X T13 þ AT ½h1 R1 þ ρδR2 þ ð1  ρÞδR3 A; ð14Þ

Ξ 12 ¼ PB þ AT ½h1 R1 þ ρδR2 þ ð1 ρÞδR3 B; Ξ 13 ¼ PF þ AT ½h1 R1 þ ρδR2 þ ð1  ρÞδR3 F; Ξ 14 ¼ PG þAT ½h1 R1 þ ρδR2 þ ð1  ρÞδR3 G; Ξ 15 ¼ h1 X 11 X 13 þ X T23 ; Ξ 22 ¼  ð1  hd ÞQ 4 þ ε2 β2 I þ ρδY 22  Y 23  Y T23 þ ρδY 11 þ Y 13 þ Y T13

x_ T ðsÞY 33 x_ ðsÞds

þ BT ½h1 R1 þ ρδR2 þ ð1  ρÞδR3 B; Ξ 23 ¼ BT ½h1 R1 þ ρδR2 þ ð1  ρÞδR3 F; Ξ 24 ¼ BT ½h1 R1 þ ρδR2 þ ð1  ρÞδR3 G;

þ xT ðt  hðtÞÞ½ρδY 12  Y 13 þY T23 xðt  h1  ρδÞ

Ξ 25 ¼ ρδY T12  Y 13 þ Y 23 ; Ξ 26 ¼ ρδY 12  Y 13 þ Y T23 ;

þ xT ðt  h1  ρδÞ½ρδY T12  Y T13 þ Y 23 xðt hðtÞÞ þ xT ðt  h1  ρδÞ½ρδY 22  Y 23  Y T23 xðt h1  ρδÞ

ð15Þ

Ξ 33 ¼  ε1 I þ F T ½h1 R1 þ ρδR2 þð1 ρÞδR3 F; Ξ 34 ¼ F T ½h1 R1 þ ρδR2 þ ð1  ρÞδR3 G; Ξ 44 ¼  ε2 I þ GT ½h1 R1 þ ρδR2 þ ð1  ρÞδR3 G;

Z

t  h1 t  hðtÞ

Ξ 55 ¼ Q 2 Q 1 þ h1 X 22  X 23 X T23 þ ρδY 11 þ Y 13 þ Y T13 ;

x_ T ðsÞY 33 x_ ðsÞds

Ξ 66 ¼ Q 3 Q 2 þ ρδY 22 Y 23  Y T23 þ ð1  ρÞδZ 11 þZ 13 þ Z T13 ;

rxT ðt  h1 Þ½ρδY 11 þ Y 13 þ Y T13 xðt  h1 Þ þ x ðt  h1 Þ½ρδY 12  Y 13 þ Y T23 xðt  hðtÞÞ þ xT ðt  hðtÞÞ½ρδY T12  Y T13 þY 23 xðt  h1 Þ þ xT ðt  hðtÞÞ½ρδY 22  Y 23 Y T23 xðt  hðtÞÞ

Ξ 67 ¼ ð1  ρÞδZ 12  Z 13 þ Z T23 ; Ξ 77 ¼  Q 3 þ ð1  ρÞδZ 22  Z 23  Z T23 :

T

ð16Þ

and Z 

x_ T ðsÞðR3  Z 33 Þx_ ðsÞds

Ξ 11 ¼ A P þPA þ ε1 α I þQ 1 þ Q 4 þ h1 X 11 þ X 13

rxT ðt  hðtÞÞ½ρδY 11 þ Y 13 þ Y T13 xðt  hðtÞÞ



and

T

Similarly, we obtain



t  h2

g T ðxðt  hðtÞÞ; tÞ  xT ðt  h1 Þ xT ðt  h1  ρδÞ xT ðt  h2 Þ

x_ T ðsÞdsX T23 xðt  h1 Þ

þ xT ðt  h1 Þ½h1 X T12  X T13 þ X 23 xðtÞ

t  hðtÞ

t  h1  ρδ

h ξT ðtÞ ¼ xT ðtÞ

6 6 6 6 6 6 Ξ¼6 6 6 6 6 4

¼ xT ðtÞ½h1 X 11 þ X 13 þ X T13 xðtÞ þ xT ðtÞ½h1 X 12  X 13 þX T23 xðt h1 Þ

Z

 where

x_ T ðsÞX 33 x_ ðsÞds

xT ðtÞ

ð19Þ

Substituting (12)–(19) into (11), we obtain Rt _ Vðxt Þ r ξT ðtÞΞξðtÞ  t  h1 x_ T ðsÞðR1 X 33 Þx_ ðsÞds Z t  h1 x_ T ðsÞðR2  Y 33 Þx_ ðsÞds 

By utilizing Lemma 1 and the Leibniz–Newton formula, we have



ð18Þ

t  h1  ρδ

t  h2

x_ T ðsÞZ 33 x_ ðsÞds

Case 2. When h1 þ ρδ r hðtÞ r h2

rxT ðt  h1 ρδÞ½ð1 ρÞδZ 11 þ Z 13 þ Z T13 xðt h1 ρδÞ þ x ðt  h1  ρδÞ½ð1  ρÞδZ 12  Z 13 þZ T23 xðt  h2 Þ þ xT ðt  h2 Þ½ð1  ρÞδZ T12  Z T13 þ Z 23 xðt  h1  ρδÞ þ xT ðt  h2 Þ½ð1  ρÞδZ 22  Z 23  Z T23 xðt  h2 Þ

When R1 X 33 Z 0; R2  Y 33 Z 0; R3  Z 33 Z 0; and h1 r hðtÞ r h1 þ ρδ; the last three terms in (20) are all less than 0. Thus by _ t Þ rξT ðtÞΞξðtÞ o 0: It Schur complements of Lemma 2, we have Vðx follows from Lyapunov–Krasovskii stability theorem that timevarying delay systems with nonlinear perturbations (1) are asymptotically stable.

T

ð17Þ

Note that for any ε1 Z 0; ε2 Z0; it follows from (3) and (4) that

Alternatively, the following equations are true: Rt R th  t  h1 x_ T ðsÞX 33 x_ ðsÞds  t  h11 ρδ x_ T ðsÞY 33 x_ ðsÞds Z t  h1  ρδ x_ T ðsÞZ 33 x_ ðsÞds  t  h2

¼

Z

t t  h1

x_ T ðsÞX 33 x_ ðsÞds 

Z

t  h1 t  h1  ρδ

x_ T ðsÞY 33 x_ ðsÞds

Please cite this article as: Liu P-L. New results on delay-range-dependent stability analysis for interval time-varying delay systems with non-linear perturbations. ISA Transactions (2015), http://dx.doi.org/10.1016/j.isatra.2015.03.001i

P.-L. Liu / ISA Transactions ∎ (∎∎∎∎) ∎∎∎–∎∎∎

Z 

t  hðtÞ t  h2

x_ T ðsÞZ 33 x_ ðsÞds 

Z

t  h1  ρδ t  hðtÞ

x_ T ðsÞZ 33 x_ ðsÞds

2

ð21Þ

Theorem 2. If h1 þ ρδ rhðtÞ r h2 ; 0 o ρo 1; for given scalars h1 ; h2 ; and hd ; the system (1) subject to (2)–(4) is asymptotically stable if P ¼ P T 4 0; Q i ¼ Q Ti Z 0; Rj ¼ RTj Z 0 ði ¼ 1; 2; 3; 4; 2 3 2 3 Y 11 Y 12 Y 13 X 11 X 12 X 13 T 6 XT 7 6 7 j ¼ 1; 2; 3Þ; X ¼ 4 12 X 22 X 23 5 Z 0; Y ¼ 4 Y 12 Y 22 Y 23 5 Z 0; T T T T X 13 X 23 X 33 Y 13 Y 23 Y 33 2 3 Z 11 Z 12 Z 13 6 T 7 Z ¼ 4 Z 12 Z 22 Z 23 5 Z 0 and a scalar ε 4 0 such that the following Z T13 Z T23 Z 33

there

exist

LMIs hold: 2 Ω11 Ω12 6 Ω 22 6 n 6 6 n n 6 6 n 6 n Ω¼6 6 n n 6 6 n 6 n 6 6 n n 4 n

n

Ω13 0

Ω14 0

Ω15 0

0 Ω 26

0 Ω 27

Ω33

0

0

0

0

n

Ω44

0

0

0

n

n

Ω 55

Ω 56

0

n

n

n

Ω 66

0

n

n

n

n

Ω77

n

n

n

n

n

3 Ω18 7 Ω28 7 7 Ω38 7 7 7 Ω48 7 7o0 0 7 7 7 0 7 7 0 7 5 Ω88

Z 12 Z 22 Z T23

following LMIs hold: 2 ^ 11 Ω12 Ω13 Ω 6 ^ 22 6 n Ω 0 6 6 n n Ω33 6 6 6 n n n ^ ¼6 Ω 6 n n n 6 6 6 n n n 6 6 n n n 4 n

n

n

3 Z 13 Z 23 7 5 Z0 and a scalar ε 4 0 such that the Z 33 Ω14

Ω15

0

0

0

Ω25

Ω26

0

0 Ω44

0 0

0 0

0 0

n

Ω55

0

0

n

n

Ω66

Ω67

n

n

n

Ω77

n

n

n

n

Ω18

3

7 Ω28 7 7 Ω38 7 7 7 Ω48 7 7o0 0 7 7 7 0 7 7 0 7 5 Ω88

ð23aÞ

and R1  X 33 Z0; R2 Y 33 Z0; R3  Z 33 Z 0

ð23bÞ

where ^ 11 ¼ AT P þ PA þ ε1 α2 I þQ 4 þ h1 X 11 þ X 13 þX T13 ; Ω ð22aÞ

^ 22 ¼ ε2 β2 I þρδY 22  Y 23  Y T23 þρδY 11 þ Y 13 þ Y T13 ; Ω and Ωij ; ði; j ¼ 1; 2; :::; 8; i o j r8Þ are defined in (9). Proof. If the matrix Q 4 ¼ 0 is selected in (10).This proof can be completed in a similar formulation to Theorem 1. Corollary 2. If h1 þ ρδ r hðtÞ r h2 ; 0 oρ o 1; for given scalars h1 andh2 ; the system (1) subject to (2)–(4) is asymptotically stable

and R1  X 33 Z 0; R2  Y 33 Z 0; R3  Z 33 Z 0

ð22bÞ

where Ω 22 ¼  ð1  hd ÞQ 4 þ ε2 β2 I þ ð1  ρÞδZ 11 þZ 13 þ Z T13 þ ð1  ρÞδZ 22  Z 23 Z T23 ; Ω 26 ¼ ð1  ρÞδZ T12  Z T13 þ Z 23 ; Ω 27 ¼ ð1  ρÞδZ 12 Z 13 þ Z T23 ; Ω 55 ¼

Z 11 6 T Z 0; Z ¼ 4 Z 12 Z T13

5

Q 2  Q 1 þ h1 X 22 X 23  X T23 þ ρδY 11 þY 13 þY T13 ; Ω 56 ¼ ρδY 12  Y 13 þ Y T23 ;

Ω 66 ¼ Q 3  Q 2 þ ρδY 22  Y 23  Y T23 þ ð1  ρÞδZ 11 þ Z 13 þZ T13 :

if there exist P ¼ P T 4 0; Q i ¼ Q Ti Z 0; Rj ¼ RTj Z 0 ði ¼ 1; 2; 3; 4; 2 3 2 3 Y 11 Y 12 Y 13 X 11 X 12 X 13 T 6 XT 6 7 7 j ¼ 1; 2; 3Þ; X ¼ 4 12 X 22 X 23 5 Z0; Y ¼ 4 Y 12 Y 22 Y 23 5 Z 0; T T T T X 13 X 23 X 33 Y 13 Y 23 Y 33 2 3 Z 11 Z 12 Z 13 6 T 7 Z ¼ 4 Z 12 Z 22 Z 23 5 Z 0 and a scalar ε 4 0 such that the following T T Z 13 Z 23 Z 33 LMIs hold: 2 ^ 11 Ω12 Ω 6 ^ 22 6 n Ω 6 6 n n 6 6 6 n n ^ ¼6 Ω 6 n n 6 6 6 n n 6 6 n n 4 n

and Ωij ; ði; j ¼ 1; 2; :::; 8; io j r 8Þ are defined in (9).

n

Ω13

Ω14

Ω15

0

0

0

0

0

Ω 26

Ω 27

Ω33

0

0

0

0

n

Ω44

0

0

0

n

n

Ω 55

Ω 56

0

n

n

n

Ω 66

n

n

n

n

0 Ω77

n

n

n

n

n

Ω18

3

7 Ω28 7 7 Ω38 7 7 7 Ω48 7 7o0 0 7 7 7 0 7 7 0 7 5

ð24aÞ

Ω88

and Proof. This proof can be completed in a similar formulation to Theorem 1. When the time-varying delay hðtÞ is not differentiable or the information of the time derivative of delay is unknown, by eliminating Q 4 in Theorems 1 and 2, one can easily get the following Corollaries 1 and 2. Corollary 1. If h1 r hðtÞ r h1 þ ρδ; 0 o ρo 1; for given scalars h1 and h2 ; the system (1) subject to (2)–(4) is asymptotically stable if there exist P ¼ P T 4 0; Q i ¼ Q Ti Z0; Rj ¼ RTj Z 0 ði ¼ 1; 2; 3; 4; 2 3 2 3 Y 11 Y 12 Y 13 X 11 X 12 X 13 T 6 XT 7 6 7 j ¼ 1; 2; 3Þ; X ¼ 4 12 X 22 X 23 5 Z0; Y ¼ 4 Y 12 Y 22 Y 23 5 T T T T X 13 X 23 X 33 Y 13 Y 23 Y 33

R1  X 33 Z0; R2 Y 33 Z0; R3  Z 33 Z 0

ð24bÞ

^ 22 ¼ ε2 β2 I þ ð1  ρÞδZ 11 þ Z 13 þZ T þ ð1  ρÞδZ 22  Z 23  where Ω 13 Z T23 ; and Ωij ðΩ ij Þ; ði; j ¼ 1; 2; :::; 8; i o j r8Þ are defined in (9) and (22) andΩij ; ði; j ¼ 1; 2; :::; 8; io j r 8Þ are defined in (9). Proof. If the matrix Q 4 ¼ 0 is selected in (10). This proof can be completed in a similar formulation to Theorem 2. Remark 1. Since the interval ½h1 ; h2  is divided into two variable subintervals ½h1 ; h1 þρδ and ½h1 þ ρδ; h2  ð0 o ρo 1; δ ¼ h2  h1 Þ in whichρ is a tunable parameter, it is clear that the Lyapunov– Krasovskii functional defined in Theorem 1 is more general than the one in [1,11]. Therefore, using the delayed decomposition approach, the Lyapunov matrices in Lyapunov–Krasovskii functional of this paper may be different in the delay intervals and the LMIs also may be different in the stability conditions, and thus compared with

Please cite this article as: Liu P-L. New results on delay-range-dependent stability analysis for interval time-varying delay systems with non-linear perturbations. ISA Transactions (2015), http://dx.doi.org/10.1016/j.isatra.2015.03.001i

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6

the methods using the same Lyapunov matrices and the uniformly dividing delay subintervals, the variable and different Lyapunov matrices-based method may lead to less conservativeness. It is worth mentioning that the delayed decomposition is different, and the stability criterion of proposed approach is also different, and then the calculated maximum allowable upper bounds (MAUB) on h2 may be different. Examples below can demonstrate the merit and reduced conservatism of the proposed approach. Remark 2. Theorems 1 and 2 provide delay-range-dependent asymptotic stability criteria for interval time-varying delay systems with nonlinear perturbations (1) in terms of solvability of LMIs [2]. We thereby obtain maximum allowable upper bounds (MAUB) 0 r h1 r hðtÞ r h2 , such that (1) is asymptotically stable by solving this convex optimization problem ( Maximize h2 ð25Þ Subjectto ð9Þ; ð22Þ Inequality (25) is a convex optimization problem and can be obtained efficiently with the MATLAB LMI Toolbox. Furthermore, such variable decomposition method may lead to reduction of conservativeness if being able to set a suitable dividing Table 1 MAUB h2 for different α and unknown hd for Example 1 (h1 ¼ 0). α and β

α ¼ 0; β ¼ 0:1

α ¼ 0:1; β ¼ 0:1

Cao and Lam [3] Han [5] Zuo and Wang [22] Qiu et al. [14] Chen et al. [4] Qiu et al. [15] Kwon et al. [8] Kwon and Park [9] Liu [11] Rakkiyappan et al. [16] Lakshmanan et al. [10] Liu [12] ðρ ¼ 0:15Þ Hui [5] ðρ ¼ 0:4Þ Wang et al. [19] ðρ ¼ 0:15Þ Corollary 1 ðρ ¼ 0:15Þ Wang et al. [19] ðρ ¼ 0:17Þ Corollary 1 ðρ ¼ 0:17Þ Wang et al. [19] ðρ ¼ 0:2Þ Corollary 1 ðρ ¼ 0:2Þ Wang et al. [19] ðρ ¼ 0:25Þ Corollary 1 ðρ ¼ 0:25Þ Wang et al. [19] ðρ ¼ 0:3Þ Corollary 1 ðρ ¼ 0:3Þ Wang et al. [19] ðρ ¼ 0:4Þ Corollary 1 ðρ ¼ 0:4Þ

0.6811 1.3279 2.7422 2.7423 2.7423 2.7757 2.7758 2.7753 2.7429 2.9816 3.0853 3.4863 3.5326 9.4195 10.4159 8.1330 9.2077 7.0658 7.8303 5.6545 6.2668 4.7107 5.2249 3.5326 3.9223

0.6129 1.2503 1.8753 1.8753 1.8753 1.8959 1.8959 1.8959 1.8895 1.9805 2.0974 2.6144 3.3907 9.0418 9.3986 7.9781 8.2949 6.7814 7.0529 5.4215 5.6435 4.5209 4.7036 3.3907 3.5286

point with relation to ρ: For seeking an appropriate ρ satisfying 0 o ρ o1; such that the maximum allowable upper bounds (MAUB) on h2 for fixed lower bound h1 of delay hðtÞ subjecting to (9a) and (9b) are maximal, we give an algorithm as follows, the Nelder–Mead simplex algorithm such as [1,12]. Algorithm 1. (Maximizing h2 for a fixed h1 ) Step 1: For given hd ; choose upper bound on δðδ ¼ h2  h1 Þ satisfying (9a) and (9b); select this upper bound as initial value δ0 of δ: Step 2: Set appropriate step lengths, δstep and ρstep ; for δ and ρ; respectively. Set k as a counter, and choose k ¼ 1: Meanwhile, let δ ¼ δ0 þ δstep and the initial value ρ0 of ρ equals ρstep : Step 3: Let ρ ¼ kρstep ; if inequalities (9a) and (9b) are feasible, go to Step 4; otherwise, go to Step 5. Step 4: Let δ0 ¼ δ; ρ0 ¼ ρ; k ¼ 1 and δ ¼ δ0 þ δstep ; go to Step 3. Step 5: Let k ¼ k þ 1: If kρstep o 1; then go to Step 3; otherwise, stop.

Remark 3. Similar to Algorithm 1, an algorithm for seeking an appropriate ρ such that the upper bound of delay h1 þρδ r hðtÞ r h2 ; subjecting to (22a) and (22b) is maximal can be easily obtained. Remark 4. The Nelder–Mead simplex algorithm is used for the optimization of a function with some dividing variables, and the simplex adapts itself to the local and scrape, and contracts to the final optimal value. As one of the feasible algorithms, it has been proven effective and computationally compact for the optimization problem with several variables. 3. Illustrative examples In this section, two well-known examples are provided to illustrate the advantages of the proposed stability results. Example 1. In order to demonstrate effectiveness of the method, we present the same example used in [3–22]. It is given in this section to compare with the results of the previous methods. The system is described as follows: x_ ðtÞ ¼ AxðtÞ þ Bxðt  hðtÞÞ þFf ðxðtÞ; tÞ þ Ggðxðt  hðtÞÞ; tÞ ð25Þ 

    0:6 0:7 ; B¼ ;  0:1  1 1  0:8   1 0 T F ¼G¼ ; f ðxðtÞ; tÞf ðxðtÞ; tÞ rα2 xT ðtÞxðtÞ 0 1 A¼

 1:2

0:1

and g T ðxðt  hðtÞÞ; tÞgðxðt hðtÞÞ; tÞ r β2 xT ðt  hðtÞÞxðt  hðtÞÞ:

Table 2 MAUB h2 for different hd ; α and β for Example 1 (h1 ¼ 0). α and β

α ¼ 0; β ¼ 0:1

hd Cao and Lam [3] Han [5] Zuo and Wang [22] Zhang et al. [20] Wang and Shen [18] Liu [12] Theorem 1 ðρ ¼ 0:5Þ Liu [12] Theorem 2 ðρ ¼ 0:18Þ Wang et al. [19] ðρ ¼ 0:5Þ Liu et al. [13] Theorem 1 ðρ ¼ 0:5Þ Theorem 2 ðρ ¼ 0:95Þ

0.5 0.5467 0.6743 1.1424 1.442 1.4430 2.3503 2.6772 2.8249 3.0678 3.1695 2.4247

α ¼ 0:1; β ¼ 0:1 0.9 0.279 – 0.738 1.280 1.408 1.8477 1.7785 2.8261 1.6396 3.0755 2.4235

1.1 – – 0.735 1.280 1.408 1.8459 1.7785 2.8257 1.6396 3.0755 2.4235

0.5 0.4950 0.5716 1.0097 1.284 1.3896 2.0930 2.3789 2.7125 2.0559 2.5970 1.7614

0.9 0.255 – 0.714 1.209 1.3677 1.7684 1.7244 2.7125 1.3647 2.4570 1.7614

1.1 – – 0.714 1.209 1.3677 1.7684 1.7244 2.7125 1.3647 2.4570 1.7614

Please cite this article as: Liu P-L. New results on delay-range-dependent stability analysis for interval time-varying delay systems with non-linear perturbations. ISA Transactions (2015), http://dx.doi.org/10.1016/j.isatra.2015.03.001i

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Solution: Consider the nonlinear system (25) with constant time-delay. Table 1 shows the maximum upper bound of timedelay under different values of α and β: The table shows the proposed method presented in this paper providing less conservative results than the previous results [3–6,8–12,14–16,19,22]. In Table 1 ‘unknown hd ’ means that hd can be arbitrary values, even if hd is very large or hðtÞ is not differentiable. Assume that nonlinear perturbations f ðxðtÞ; tÞ and gðxðt hðtÞÞ; tÞ satisfy (3) and (4) respectively and delay hðtÞ satisfies (5), we calculate the allowable upper bound of h2 ðh1 ¼ 0Þ that guarantees robust stability of system (25) under different hd ; α and β listed in Table 2. On the basis of stability criteria given in [3,5,12,13,18–20,22] and Theorem 1 in this paper, computational results appear in Table 2. It can be seen that Theorems 1 and 2 provides much less conservative results than others. When hd 4 1, the stability criteria proposed in [3,5] cannot be applied to check robust stability of the system (1). For given values of α; β; h1 ; and hd ; we apply Theorems 1 and 2 to calculate the maximal allowable value h2 that guarantees the asymptotical stability of the system as listed in Table 3. From the table, it is easy to see that our proposed stability criterion gives a much less conservative results than those in [6,7,17,20,21] since the proposed analysis uses a delay-central point method as well as tighter bounding on the time-derivative of LKF. Example 2. In order to demonstrate effectiveness of the method we have presented, the same example used in [12,19] is given in this section to compare with results of the previous methods. The system is described as x_ ðtÞ ¼ AxðtÞ þBxðt hðtÞÞ þ Ff ðxðtÞ; tÞ þ Ggðxðt  hðtÞÞ; tÞ ð26Þ

0 1

 ;

T

f ðxðtÞ; tÞf ðxðtÞ; tÞ r α2 xT ðtÞxðtÞ; g T ðxðt  hðtÞÞ; tÞgðxðt  hðtÞÞ; tÞ rβ2 xT ðt  hðtÞÞ xðt  hðtÞÞ and h1 r hðtÞ r h2 :

1.0

1 ðρ ¼ 0:3Þ 2 ðρ ¼ 0:9Þ

1 ðρ ¼ 0:3Þ 2 ðρ ¼ 0:9Þ

1.442 1.558 1.5636 1.824 1.8599 2.1789 2.1012 1.543 1.760 1.7897 1.9930 2.0650 2.2709 2.2759

1.338 1.558 1.5636 1.824 1.8599 2.1714 2.1008 1.543 1.760 1.7897 1.9930 2.0650 2.2705 2.2749

1.338 1.558 1.5636 1.824 1.8599 2.1714 2.1008 1.543 1.760 1.7897 1.9930 2.0650 2.2705 2.2749

25:3932



4:6140  0:4063



0:2210 0:2642

Y 12 ¼ Y 22 ¼

 0:0469 2:1517 Y 33 ¼  0:0800   0:0093 Z 12 ¼ 0:0012  0:2370 Z 22 ¼  0:0228  0:0479 Z 33 ¼ 0:0061 

   14:9529 0:1853 ; Y 11 ¼ ; 8:0435 0:1853 0:2941     0:1104  5:2605  0:3351 ; Y 13 ¼ ;  0:0534 0:1943  0:2154     0:0469 0:3737  0:1585 ; Y 23 ¼ ; 0:0124  0:0127 0:0410     0:0800 0:1614  0:0155 ; Z 11 ¼ ; 0:1632  0:0155 0:0026    0:0012  0:0171 0:0022 ; Z 13 ¼ ; 0:0002 0:0022  0:0003    0:0228 0:0171  0:0022 ; Z 23 ¼ ; 0:0033  0:0022 0:0003   0:0061 ; ε1 ¼ 63:5640; ε2 ¼ 15:2841: 0:0018

4:6140

Remark 5. Some comparisons have been made with the same examples that appear in many recent papers. Our results show them less conservative than those reports.

Table 3 MAUB h2 with different values of hd and h1 ¼ 0:5; 1. [20] [17] [6] [7] [21] Theorem Theorem [20] [17] [6] [7] [21] Theorem Theorem

 X 33 ¼

The maximum allowable time delay upper bounds for h2 ðh1 ¼ 0Þ for different ρ are shown in Table 4. From Table 4, it is easy to see that the stability conditions derived in this study are more effective. Fig. 1 shows simulation  of theT above system (26) for h¼ 4.57 with the initial state  1 1 : As the diagram indicates, the system (26) would be asymptotically stable if the delay time h2 ðh1 ¼ 0Þ is less than 4.57.

It is assumed that non-linear perturbations satisfy

0.5

Solution: By taking h1 ¼ 0; and ρ ¼ 0:4; we get Corollary 1 remains feasible for any delay time h2 r4:5785: In case of h2 ¼ 4:5785; Corollary 1 yields the following set of feasible solutions:     11:7461  0:2176 21:3272  0:0347 P¼ ; Q1 ¼ ;  0:2176 0:4710  0:0347 4:4372     10:9743  0:3720 1:2500  0:1185 Q2 ¼ ; Q3 ¼ ;  0:3720 0:4974  0:1185 0:0157     27:8965 4:0036 2:1959  0:0855 R1 ¼ ; R2 ¼ ; 4:0036 14:1357  0:0855 0:1649     0:0902  0:0114 16:6486 3:2977 R3 ¼ ; X 11 ¼ ;  0:0114 0:0034 3:2977 11:2651      16:6415  2:4003  17:6764  3:7718 X 12 ¼ ; X 13 ¼ ; 4:4457  2:0049  4:5613  6:9002     28:3694 3:4234 25:5804 4:8068 X 22 ¼ ; X 23 ¼ ; 3:4234 8:3420 3:6470 2:8215

1.284 1.384 1.3858 1.524 1.6622 1.9573 1.6908 1.408 1.532 1.5647 1.6380 1.8188 1.9629 1.8336

1.245 1.384 1.3858 1.524 1.6622 1.9573 1.6908 1.408 1.532 1.5647 1.6380 1.8188 1.9629 1.8334

1.245 1.384 1.3858 1.524 1.6622 1.9573 1.6907 1.408 1.532 1.5647 1.6380 1.8188 1.9629 1.8333

Time delay 4.57 sec 1 x1 x2

0.8 0.6 0.4

Output x1,x2

where      2 0 1 0 1 A¼ ; B¼ ; F¼ 0 1 1 1 0   1 0 G¼ ; α ¼ 0:05; β ¼ 0:1: 0 1

7

0.2 0 -0.2 -0.4 -0.6

Table 4 MAUB h2 ðh1 ¼ 0Þ for different ρ. ρ Liu [12] Wang et al. [19] Corollary 1

0.15 3.6654 9.5416 12.7256

-0.8 0.2 3.1908 7.1562 9.5632

0.25 2.7079 5.7250 7.6677

0.35 2.1155 4.0892 5.4900

-1

0

5

10

15

20

25

Time(sec) Fig. 1. The simulation of the Example 2 for h2 ¼ 4:57 ðh1 ¼ 0Þ s.

Please cite this article as: Liu P-L. New results on delay-range-dependent stability analysis for interval time-varying delay systems with non-linear perturbations. ISA Transactions (2015), http://dx.doi.org/10.1016/j.isatra.2015.03.001i

30

P.-L. Liu / ISA Transactions ∎ (∎∎∎∎) ∎∎∎–∎∎∎

8

4. Conclusion This paper has investigated the delay-range-dependent stability analysis of linear systems with interval time-varying delay and nonlinear perturbations. By combining integral inequality approach (IIA) and delayed decomposition approach (DDA), some improved delay-range-dependent criteria with less conservatism have been obtained. An important feature of results reported here is that all conditions depend on both lower and upper bounds of the interval time-varying delays. Finally, two well-known examples are given to show the effectiveness of the proposed approach.

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Please cite this article as: Liu P-L. New results on delay-range-dependent stability analysis for interval time-varying delay systems with non-linear perturbations. ISA Transactions (2015), http://dx.doi.org/10.1016/j.isatra.2015.03.001i

New results on delay-range-dependent stability analysis for interval time-varying delay systems with non-linear perturbations.

This paper studies the problem of the stability analysis of interval time-varying delay systems with nonlinear perturbations. Based on the Lyapunov-Kr...
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