ISA Transactions 53 (2014) 210–219
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A novel approach to delay-fractional-dependent stability criterion for linear systems with interval delay Jiyao An a,c,n, Zhiyong Li a, Xiaomei Wang b,c a Key Laboratory of Embedded and Network Computing of Hunan Province, College of Information Science and Engineering, Hunan University, Changsha 410082, PR China b School of Mathematics Science, University of Electronic Science and Technology of China, Chengdu 610054, PR China c Department of Applied Mathematics, Faculty of Mathematics, University of Waterloo, Waterloo, Ontario, Canada N2L 3G1
art ic l e i nf o
a b s t r a c t
Article history: Received 14 June 2013 Received in revised form 7 November 2013 Accepted 27 November 2013 Available online 31 December 2013 This paper was recommended for publication by Dr. Q.-G. Wang
This paper considers the problem of delay-fractional-dependent stability analysis of linear systems with interval time-varying state delay. By developing a delay variable decomposition approach, both the information of the variable dividing subinterval delay, and the information of the lower and upper bound of delay can be taken into full consideration. Then a new delay-fractional-dependent stability criterion is derived without involving any direct approximation in the time-derivative of the Lyapunov–Krasovskii (LK) functional via some suitable Jensen integral inequalities and convex combination technique. The merits of the proposed result lie in less conservatism, which are realized by choosing different Lyapunov matrices in the variable delay subintervals and estimating the upper bound of some cross term in LK functional more exactly. At last, two well-known numerical examples are employed to show the effectiveness and less conservatism of the proposed method. Crown Copyright & 2013 Published by Elsevier Ltd. on behalf of ISA. All rights reserved.
Keywords: Lyapunov–Krasovskii (LK) functional Delay-fractional-dependent stability Interval time-varying delay Linear matrix inequality (LMI) Maximum allowable delay bound (MADB)
1. Introduction Stability is a central issue in dynamical system and control theory. A dynamical system is called stable (in the sense of Lyapunov) if starting the system somewhere near its desired operating point implies that it will stay around that point ever after. The stability problem for time-delayed systems [1] has received considerable attention in recent years, see, e.g., [2–9], and the references therein. Recently, Richard [2] summarized current research on time-delay systems and listed some open problems, and argued that the stability analysis should have been focusing on effective reduction of the conservation of the stability conditions and control algorithm. Especially, in real system, the delay is assumed to be an interval time-varying delay [10–17], ha r τðtÞ rhb and ha is not restricted to be 0. The main aim is to derive a maximum allowable delay bound (MADB) of the time-delay such that the concerned system is asymptotically stable for any delay size less than the MADB. Accordingly, the obtained MADB becomes a key performance index to measure the conservatism of a delaydependent stability condition. As well-known, the bounding technology and the model transformation technique are introduced into the stability analysis, although they may lead to some conservation. In order to reduce the conservatism, a free-weighting matrix method was proposed in [3], and hereafter the approach has been extended to two form, that is, the one with a null summing term added to the LK functional derivative [3,7,8,23,24], and the one with free matrices item added to the LK functional combined with the descriptor model transformation [4,6,13]. Recently, some techniques with relation to tighter upper bounds of various functions have been pursued: affine functions of polytopictype uncertain systems, quadratic functions of nonlinear systems, and especially a special type of function combinations in delayed systems [4,5,9,10]. Among the bounding technique, the Jensen inequality lemma [1] takes a key role to be efficiently relaxed some integral and cross terms in the time derivative of LK functional. However, to fully relax the matrix cross-products, it has to introduce slightly
n Corresponding author at: Key Laboratory of Embedded and Network Computing of Hunan Province, College of Information Science and Engineering, Hunan University, Changsha 410082, PR China. Tel.: þ 86 731 88821907; fax: þ 86 731 88822417. E-mail address: [email protected] (J. An).
0019-0578/$ - see front matter Crown Copyright & 2013 Published by Elsevier Ltd. on behalf of ISA. All rights reserved. http://dx.doi.org/10.1016/j.isatra.2013.11.020
J. An et al. / ISA Transactions 53 (2014) 210–219
211
excessive free weighting matrices. As a way of reducing the number of decision variables, at the sacrifice of conservatism, relaxations based on the Jensen inequality lemma [1] have been attracted in [7–14,17–19]. On the other hand, the delay partitioning approach has been attracted in the derivation process. Recently, the method of dividing delay (including the discretized Lyapunov functional approach in [1] and the delay decomposition approach in Gouaisbaut and Peaucelle [20] and Han [21]) is the best one to handle the stability of system (1) with constant delay, by which the result near analytical delay limit can be obtained. Since the augmented matrix S has much more room to be adjusted in the criterion, the functional can achieve the benefit of reducing the conservatism. Furthermore, partitioning of the delay intervals into n 4 2 subintervals may lead to further improvements, for examples, another delay partitioning was introduced in [3,6,20,21,24,29], which corresponded to the partitioning into two subintervals of ½0; ha and of ½ha ; hb . Therefore, it is natural that the less conservative stability results can be obtained when the delay decomposition approach is extended to the case of time-varying delay. So far, by dividing interval ½0; τðtÞ into N non-uniform sub-intervals, the pioneering studies have been provided, see [21,22]. But, the purpose of reducing conservatism is still limited due to the existence of multiple coefficients and the impact of subintervals with uniform size [22,25,26]. Moreover, it is still a quite difficult task to divide interval ½0; τðtÞ in a more reasonable manner [22], so that the functional with the augmented matrix can easily be constructed to obtain less conservative stability results. In practice, the systems almost contain some uncertainties because it is very difficult to obtain an exact mathematical model due to environment noise, uncertain or slowly varying parameters, etc. Most recently, the stability analysis for time-delay systems has been extended to the case of the systems with nonlinear perturbations [25–27] and fuzzy systems [31,32], while the proposed result also has been expanded to disturbance attenuation analysis [33,38] and H1 controller/filter design [31,32,36] and robust fault detection [34,37] for the above systems. Ramakrishnan and Ray [25] studied the problem of stability analysis of the linear systems with nonlinear perturbation by partitioning the delay interval into two segments of equal length. Zhang et al. [26] employed some novel integral inequalities (Lemmas 1 and 2 in [26]) to derive an improved stability criterion for the above systems. He et al. [33] investigated the disturbance attenuation problem and H1 controller design for the linear systems. Lin et al. considered the H1 filter design problem [31] and observer-based H1 control design problem [32] for nonlinear system via T–S fuzzy model approach. Most recently, Liu [35] further studied the stability analysis of linear systems. However, there also exists further room to investigate the upper bound of the time-derivative of the LK functional, and the delay interval may be divided into two unequal subintervals or more subintervals [22,24,29]. This motivates the present research to develop a novel method for stability problem for the concerned systems with less conservatism by making full use of the information of time-delays and constructing a novel LK functional via variable delay dividing technique. Motivated by the above discussions, this paper will focus on the delay-fractional-dependent stability problem of linear systems with interval time-varying delay. Inspired by the reciprocally convex approach in [10], and some integral inequalities in [1,7,13], and then the work of Kim [12], Zhang [22] and Briat [9], we firstly construct a novel delay-fractional-dependent LK functional by developing a delay decomposition approach, in which the integral interval ½t hb ; 0 is decomposed into ½t hb ; t ha ατ, ½t ha ατ; t ha and ½t ha ; t ðha =2Þ, ½t ðha =2Þ; t with 0 o α o1 and τ 9 hb ha . Secondly, we derive a stability criterion for system (1) by suitably using the inequalities such as (4)–(8) in Lemma 1 below. Since a tuning parameter α and different delay partitioning method are introduced, the information about xðt ha ατÞ and xðt ððha =2ÞÞ can be taken into full consideration. Meanwhile, based on the convex combination approach, the cross team in the time derivative of LK functional can be estimated more exactly without any direct approximation of the delay terms in the derivation process. Therefore, using the variable delay decomposition approach, the Lyapunov matrices in LK 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 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 conservatism. Finally, two well-known numerical examples are used to compare with some previous results and demonstrate the effectiveness of the proposed method.
2. Problem formulation Consider the following linear system with a time-varying state delay: (
x_ ðtÞ ¼ AxðtÞ þ Aτ xðt τðtÞÞ
xðtÞ ¼ ϕðtÞ; 8 t A ½ hb ; 0;
ð1Þ
where xðtÞis the state vector. ϕðtÞ is the continuous initial vector function defined on ½ hb ; 0; the matrices A; Aτ are constant matrices with appropriate dimensions. In this paper, the delayτðtÞ is assumed to be time-varying delay as the following two cases: Case 1. τðtÞ is a differentiable function, satisfying for all t Z 0: 0 o ha r τðtÞ rhb ; τ_ ðtÞ r hd
ð2Þ
Case 2. τðtÞ is not differentiable or the upper bound of the derivative of τðtÞ is unknown, and τðtÞ satisfies 0 o ha r τðtÞ rhb
ð3Þ
where ha ; hb and hd are some given values. The purpose of this paper is to find a stability criterion, which is less conservative than the existing results. Such a criterion may be used to compute the tolerable delay bound hb for given ha or vice versa. To end this section, we introduce the following lemmas which are useful in stability analysis for the linear system.
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J. An et al. / ISA Transactions 53 (2014) 210–219
Lemma 1. (Integral inequalities, Gu [1]; Sun et al. [7]; Zhang et al. [13]) Let xðtÞ A ℜn be a vector-valued function with first-order continuous-derivative entries. Then, for any matrices M; N A ℜnn , Z A ℜ2n2n , X ¼ X T A ℜnn , and some given scalars 0 r τ1 o τ2 , the following integral inequalities hold: (1) When X 40 and τ1 ; τ2 are constant values, Z t τ1 Z t τ1 Z xT ðsÞXxðsÞds Z xT ðsÞdsX ðτ2 τ1 Þ t τ2
τ22 τ21
Z
2
τ1
t τ2
Z
τ2
ðτ 2 τ 1 Þ
Z
t t þθ
t τ1 t τ2
xT ðsÞXxðsÞdsdθ Z "
x_ ðsÞX x_ ðsÞds r T
Z
τ1
Z
τ2
xðt τ1 Þ
t τ1 t τ2
t tþθ
#T
xðt τ2 Þ
ð4Þ
xðsÞds
xT ðsÞdsdθX
X
X
n
X
"
Z
τ1
Z
τ2
xðt τ1 Þ
t tþθ
xðsÞdsd θ
ð5Þ
# ð6Þ
xðt τ2 Þ
(2) When X 40 and τ1 ; τ2 are time-varying, h ¼ τ2 τ1 : ¼ hðtÞ Z0, # " #T (" Z t τ1 xðt τ1 Þ M M þ M T M þ N T x_ T ðsÞX x_ ðsÞds r þ h X 1 ½ MT T τ Þ xðt N 2 n N N t τ2
)" NT
xðt τ1 Þ
xðt τ2 Þ
(3) When τ1 ; τ2 are time-varying, h ¼ τ2 τ1 : ¼ hðtÞ Z 0, and X is any symmetric matrix, # )" " #T (" # Z t τ1 xðt τ1 Þ xðt τ1 Þ M þ M T M þ N T x_ T ðsÞX x_ ðsÞds r þ hZ xðt τ2 Þ xðt τ2 Þ n N N T t τ2 X Y Z0, and Y ¼ ½M N. with n Z
# ð7Þ
ð8Þ
Remark 1. Eqs. (4)–(8) are called various integral inequalities presented over the existing literatures. They play a key role in the derivation of a criterion for delay-dependent stability analysis of systems with time-varying delay. In this paper, different from the exiting ones, we will utilize suitably different integral inequality in connection with different cross terms in LK functional or it is time-derivative. Meanwhile, the upper bound of these cross terms in time-derivative of LK functional are estimated and deduced more exactly without involving any direct approximation of uncertain terms in the derivation process.
3. Main results Firstly, we divide the delay interval 0; ha and ha ; hb into four segments: hi 1 ; hi ; i ¼ 1; 2; 3; 4 where h0 ¼ 0; h1 ¼ ha =2, 2 2 h2 ¼ ha ; h3 ¼ ha þ ατ; h4 ¼ hb ; 0 o α o 1. For the sake of convenience, we denote τ ¼ hb ha , and τi ¼ hi hi 1 , r i ¼ hi hi 1 , ði ¼ 1; 2; 3; 4Þ and τ0 ¼ ha 0. For the system (1), we derive a stability condition by using the variable delay decomposition approach as follows: Theorem 1. In Case 1, for given scalars 0 o ha r hb ; 0 o α o 1; hd , the system (1) is asymptotically stable if there exist real symmetric matrices " # " # Z i1 Z i2 S11 S12 4 0, Z i ¼ , and any matrices M i ; N i ; ði ¼ 1; 2; 3; 4Þ with appropriate dimensions such P,Q i 4 0; Ri ; ði ¼ 1; 2; 3; 4Þ, Q τ Z 0; Rτ , n Z i3 n S22 that the LMIs in (9) and (10) are feasible. 2 4 2τ 3 2τ 2 2τ 2 i r11 R1 r22 R2 6Pþ ∑ r 6 i¼1 i 6 2τ 1 1 2 2 6 n 6 τ1 Q 1 þ r 1 R1 þ τ0 S11 τ0 S12 6 Vp ¼ 6 2 τ2 1 2 n n 6 τ2 Q 2 þ r 2 R2 þ τ0 S22 6 6 n n n 6 4 n
n
2τ23 r 3 R3
0
0
0
0
2τ 3 1 τ 3 Q 3 þ r 3 R3
n
0 2τ 4 τ4 Q 4 þ r 4 R4 1
n
Ωði; kÞ : ¼ Ω0 þ Ωk1 þ Ωk2 þ Ξ ki o 0; ði ¼ 1; 2; k ¼ 3; 4Þ with " τ3 R3 þ ð1 hd ÞRτ n
½ M1 Z1
N1
#
" Z 0;
3 7 7 7 7 7 7 740 7 7 7 7 5
ð9Þ
ð10Þ
τ4 R4 þ ð1 hd ÞRτ ½M 4 n
2τ24 r 4 R4
N4 Z2
# Z 0:
J. An et al. / ISA Transactions 53 (2014) 210–219
213
where 2 6 6 6 6 Ω0 : ¼ 6 6 6 6 6 4 2
PAþ AT P þ Q 1 þ S11 R1
PAτ
S12 þ R1
0
0
0
n
ð1 hd ÞQ τ
0
0
0
0
n
n
Q 2 Q 1 þ S22 S11 R1 R2
n
n
n
S12 þR2 Q 3 Q 2 þ Q τ S22 R2
0 0
0 0
n
n
n
n
Q4 Q3
0
n
n
n
n
n
Q4
0
6n 6 6 6n 3 Ω1 : ¼ 6 6n 6 6 6n 4
0
0
0
0
N 1 NT1 þ τ3 M 2 þ τ3 M T2
0
M T1 þ N 1
τ3 M 2 þ τ3 N T2
n
0
n
n
0 M 1 þ M T1
0 0
n
n
n
R4 τ3 N 2 τ3 N T2
n
n
n
n
n
2
0 6n 6 6 6n 4 Ω1 : ¼ 6 6n 6 6 6n 4
0
0 7 7 7 0 7 7 ; 0 7 7 7 R4 7 5 R4
0
0
0
M T4 þ N4
3
0
τ
0
n
0
0
0
0
n
n
M 1 þ M T1 þ τ3 Z 11
M 1 þ NT1 þ τ3 Z 12
0
n
n
n
N 1 NT1 þM 4 þ M T4 þ τ3 Z 13
0
n
n
n
n
τ4 ðN 3 þ N T3 Þ
!
4
∑ τ2i Ri þ τ3 Rτ Γ ;
i¼1
8 Z 13 > > > T > > Z > < 12 Ξ k1 : ¼ τk Z k ; Z 3 : ¼ ½Z 3ij 66 ; Z 3ij 9 Z 11 > > > Z 12 > > > : 0
Ξ k2 : ¼ τ2k Φk Rk 1 ðΦk ÞT ;
! 4 ∑ τ2i Ri þ τRτ Γ ; Γ : ¼ A
Ω42 : ¼ Γ T
Aτ
i¼1
i ¼ 2; j ¼ 4 i ¼ 4; j ¼ 4 ; i ¼ 4; j ¼ 2 other
Φ3 : ¼ colf0; M 2 ; 0; 0; N 2 ; 0g;
τ4 M 3 þ τ
0
8 Z 23 > > > T > > Z > < 22 Z 4 : ¼ ½Z 4ij 66 ; Z 4ij 9 Z 21 > > > Z 22 > > > : 0
i ¼ 2; j ¼ 2
7 7 7 7 7 7; 7 7 7 5
3
0
N 4 NT4 þ 4 ðM 3 þ M T3 Þ
n
Ω32 : ¼ Γ T
0
3
0
0
T 4 N3
7 7 7 7 7 7; 7 7 7 5
0 ;
i ¼ 2; j ¼ 2 i ¼ 2; j ¼ 5 i ¼ 5; j ¼ 5 ; i ¼ 5; j ¼ 2 other
Φ4 : ¼ colf0; M3 ; 0; 0; 0; N 3 g:
ð11Þ
Proof. Choose the following delay-fractional-dependent LK functional: Vðt; xt Þ ¼ V 1 ðt; xt Þ þ V 2 ðt; xt Þ þ V 3 ðt; xt Þ þ V 4 ðt; xt Þ
ð12Þ
where xt denotes the function xðsÞdefined on the interval ½t hb ; t, and V 1 ðt; xt Þ ¼ xT ðtÞPxðtÞ; Z t hi 1 Z 4 xT ðsÞQ i xðsÞds þ V 2 ðt; xt Þ ¼ ∑ t hi
i¼1
Z V 3 ðt; xt Þ ¼
"
t
#T "
xðsÞ xðs τ20 Þ
t h1
S11
S12
n
S22
t h2 t τ ðtÞ
#"
xT ðsÞQ τ xðsÞds #
xðsÞ
ds
xðs τ20 Þ
Z h2 Z t x_ T ðsÞRi x_ ðsÞdsdθ þ x_ T ðsÞRτ x_ ðsÞdsdθ t þθ hi τðtÞ t þ θ i¼1 " # S11 S12 with P,Q i 4 0ði ¼ 1; 2; 3; 4Þ,Q τ Z 0, 4 0,Ri ði ¼ 1; 2; 3; 4Þ,Rτ being real symmetric matrices. n S22 First, we show that the condition Vðt; xt Þ Z ε1 ‖xðtÞ‖; ðε1 4 0Þ is satisfied. Assume that Q i 40; Ri 40; ði ¼ 1; 2; 3; 4Þ, then, suitably applying Lemma 1 it yields 4
V 4 ðt; xt Þ ¼ ∑ τi
Z
hi 1
Z Vðt; xt Þ Z xT ðtÞPxðtÞ þ Z þ
t
t h1
"
Z
t
t h2
t τðtÞ
xðsÞ
Z xT ðsÞQ τ xðsÞds þ #T
τ0
xðs 2 Þ
" ds
n We denote ζ ðtÞ : ¼ col xðtÞ Vðt; xt Þ Z ζ ðtÞV p ζ ðtÞ þ T
Z
t h2 t τðtÞ
S11
S12
n
S22
Rt t h1
h2 τðtÞ
#Z
t tþθ
"
t t h1
τ0
xðs 2 Þ
t h2
Z
h2 τ ðtÞ
Z
4
1
i¼1
τi
x_ T ðsÞRτ x_ ðsÞdsdθ þ ∑
xðsÞ
R t h1
xðsÞds
xT ðsÞQ τ xðsÞdsþ
Z
xðsÞds t t þθ
#
2τ i i ¼ 1 ri 4
ds þ ∑ R t h2 t h3
xðsÞds
x_ T ðsÞRτ x_ ðsÞdsdθ;
Z
Z
t hi 1 t hi
hi 1 hi
R t h3 t h4
Z
t tþθ
Z xT ðsÞdsQ i
x_ T ðsÞdsdθRi
t hi 1
xðsÞds t hi
Z
hi 1 hi
Z
t t þθ
x_ ðsÞdsdθ
o xðsÞds , after the simple arrangement, we have
ð13Þ
214
J. An et al. / ISA Transactions 53 (2014) 210–219
2
4 2τ 3 i 6Pþ ∑ r 6 i¼1 i 6 6 n 6 6 Vp 96 n 6 6 6 n 6 4
2τ 21 r1
2τ 1 2 τ1 Q 1 þ r1 R1 þ τ0 S11 1
n
2τ 22 r2
2
τ0 S12
2τ23 r3
0
0
0
0
n
2τ 2 2 τ2 Q 2 þ r2 R2 þ τ0 S22
n
n
2τ 3 1 τ3 Q 3 þ r3 R3
n
n
n
1
3
2τ24 r4
0 2τ 4 τ4 Q 2 þ r 4 R4 1
7 7 7 7 7 7 7 7 7 7 7 5
ð14Þ
Therefore, if the right side of (14) is positive, it yields a sufficient condition to guarantee that Vðt; xt Þ Z ε1 ‖xðtÞ‖; ðε1 4 0Þ. Moreover, the LMIs in (10) imply that Ri 4 0; ði ¼ 1; 2; 3; 4Þ; Rτ 4 0. Consequently, if the LMIs in (9) are satisfied, Q i 40; ði ¼ 1; 2; 3; 4Þ, and Q τ Z 0, then Vðt; xt Þ Z ε1 J xðtÞ J ; ðε1 4 0Þ. _ xt Þ r ε2 J xðtÞ J ; ðε2 40Þ is also guaranteed if the LMIs in (10) and Q τ Z0, Secondly, we show that the condition Vðt; " # S11 S12 4 0 hold. Taking the derivative of (12) with respect to t along the trajectory of system (1), we have Q i 4 0; ði ¼ 1; 2; 3; 4Þ, n S22 T ð15Þ V_ 1 ðt; xt Þ ¼ x_ ðtÞPxðtÞ þ xT ðtÞP x_ ðtÞ ¼ ½AxðtÞ þ Aτ xðt τðtÞÞT PxðtÞ þxT ðtÞP½AxðtÞ þ Aτ xðt τðtÞÞ 4
V_ 2 ðt; xt Þ ¼ ∑ ½xT ðt hi 1 ÞQ i xðt hi 1 Þ xT ðt hi ÞQ i xðt hi Þ i¼1
þ xT ðt h2 ÞQ τ xðt h2 Þ ð1 τ_ ðtÞÞxT ðt τðtÞÞQ τ xðt τðtÞÞ 4
r ∑ ½xT ðt hi 1 ÞQ i xðt hi 1 Þ xT ðt hi ÞQ i xðt hi Þ i¼1
þ xT ðt h2 ÞQ τ xðt h2 Þ ð1 hd ÞxT ðt τðtÞÞQ τ xðt τðtÞÞ " V_ 3 ðt; xt Þ ¼
#T "
xðtÞ xðt h1 Þ
S11
S12
n
S22
#"
#
xðtÞ xðt h1 Þ
"
xðt h1 Þ xðt h2 Þ
#T "
ð16Þ
S11
S12
n
S22
#"
xðt h1 Þ xðt h2 Þ
# ð17Þ
4
V_ 4 ðt; xt Þ ¼ ∑ τ2i x_ T ðtÞRi x_ ðtÞ þ ðτðtÞ h2 Þx_ T ðtÞRτ x_ ðtÞ i¼1
4
∑ τi i¼1
Z
t hi 1
Z
x_ T ðsÞRi x_ ðsÞds ð1 τ_ ðtÞÞ
t hi
t h2
x_ T ðsÞRτ x_ ðsÞds
t τ ðtÞ
ð18Þ
For any t Z0, it is the fact that ha r τðtÞ r ha þ ατ or ha þ ατ r τðtÞ r hb , (0 o α o 1). In the case of ha r τðtÞ r ha þ ατ, i.e., τðtÞ A ½h2 ; h3 , k ¼ 3, suitably using the integral inequalities in Lemma 1, the following inequalities are true: ðτðtÞ h2 Þx_ T ðtÞRτ x_ ðtÞ r ατx_ T ðtÞRτ x_ ðtÞ ¼ τ3 x_ T ðtÞRτ x_ ðtÞ
τi
Z
τ3
t hi 1 t hi
Z
t h2 t h3
" x_ T ðsÞRi x_ ðsÞdsr
xðt hi 1 Þ
#T "
xðt hi Þ
x_ T ðsÞR3 x_ ðsÞds ð1 τ_ ðtÞÞ
Z
t h2 t τ ðtÞ
ð19Þ
Ri
Ri
n
Ri
#"
xðt hi 1 Þ
# ; ði ¼ 1; 2; 4Þ
xðt hi Þ
x_ T ðsÞRτ x_ ðsÞds r τ3 Z ¼ " r
Z
t h2 t h3
t h2
t τ ðtÞ
x_ T ðsÞR3 x_ ðsÞds ð1 hd Þ
#T ("
xðt τðtÞÞ "
xðt τðtÞÞ
with
M1
n
N1
It follows from (15) to (21) that " _ xt Þ r ρξT ðtÞ Ω0 þ Ω3 þ Γ T Vðt; 1
4
xðt h3 Þ
N 1 NT1
M2 N2
Z 0, and ρ ¼ ðτðtÞ h2 =ατÞ; 0 r ρ r 1.
!
#
∑ τ2i Ri þ ατRτ Γ þ ατZ 3 ξðtÞ
i¼1
n
#
Z1
t τðtÞ
M 1 þ N T1
"
τ3 "
τ3 R3 þ ð1 hd ÞRτ
t h2
M 1 þ M T1
#T (
þ ð1 ρÞ U ατ U "
Z
x_ T ðsÞRτ x_ ðsÞds
x_ T ðsÞðτ3 R3 þ ð1 hd ÞRτ Þx_ ðsÞds τ3
xðt h2 Þ
þ
ð20Þ
M 2 þ M T2
#
Z
t τ ðtÞ t h3
x_ T ðsÞR3 x_ ðsÞds
þ ρ U ατ U Z 1
M 2 þN T2
)"
xðt h2 Þ
#
xðt τðtÞÞ
#
N 2 NT2 9 " #T " # M 2 = xðt τðtÞÞ n
# R3 1
N2
;
xðt h3 Þ
ð21Þ
J. An et al. / ISA Transactions 53 (2014) 210–219
" þ ð1 ρÞξ ðtÞ Ω0 þ Ω T
" with
3 1þ
τ3 R3 þ ð1 hd ÞRτ n
M1
Γ
N1
Z1
T
4
∑ τ
i¼1
! 2 i Ri þ
ατRτ Γ þ ðατÞ2 Φ3 R3 1 ðΦ3 ÞT ξðtÞ
Ω0 þ Ω
3 1þ
Γ
T
ð22Þ
# Z0
n 3 3 where Ω0 ; Ω1 ; Γ ; Z 3 ; Φ are defined in (11), and ξðtÞ : ¼ col xðtÞ xðt τðtÞÞ xðt h1 Þ xðt h2 Þ xðt h3 Þ Since 0 r ρ r 1, applying the convex combination method, we conclude that if the following LMIs: !
Ω0 þ Ω31 þ Γ T
215
o xðt h4 Þ .
4
∑ τ2i Ri þ ατRτ Γ þ ατZ 3 o0
i¼1 4
∑ τ
i¼1
! 2 i Ri þ
ατRτ Γ þ ðατÞ2 Φ3 R3 1 ðΦ3 ÞT o0
ð23Þ
are feasible, then x is true for a sufficiently small ε240. Meanwhile, if ha þ ατ r τðtÞ r hb , i.e., τðtÞ A h3 ; h4 ,k ¼ 4, similar to the above deduction process, we also can obtain the similar stability condition. This completes the proof. □ Remark 2. By constructing delay-fractional-dependent LK Functional (12) and using suitably integral inequalities in Lemma 1, Theorem 1 is obtained without involving any approximation of the delay term when exploiting a convex combination of the uncertain terms involved, and the value of the upper bound of some cross term is estimated more exactly than the previous methods, So, such a feature may lead to less conservative results compared to the existing ones. Moreover, by proving the positive definiteness of the constructed LK functional, the constraints on some functional parameters were relaxed and the conservatism of the stability results may be further reduced. Remark 3. Since the interval ha ; hb is divided into two variable subintervals ha ; ha þ ατ and ha þ ατ; hb in which α is a tunable h i h i parameter, the lower bound of the delay 0; ha is also divided into two equal subintervals 0; h2a and h2a ; ha for the simplification, it is clear that the LK functional defined in Theorem 1 are more general than the ones in [3–5,8,15–18,25–28], etc. Therefore, both the information of delayed state xðt h2a Þ and xðt ha ατÞ can be taken into account. It is worthy mentioning that the delay decomposition is different, and the stability criterion of proposed approach is also different, and then the calculated MADB on hb may be different. Examples below can demonstrate the merit and reduced conservatism of the proposed approach. So, such variable decomposition method may lead to reduction of conservatism if being able to set a suitable dividing point with relation to α. About how to seek an appropriate α satisfying 0 o α o 1, such that we can obtain the MADB on hb for fixed lower bound ha of time delay. We give a simple algorithm as follows, such as [15,16,23]. Algorithm 1. (Maximizing hb for a fixed ha ) Step 1: For given hd , choose an upper bound on hb in the existing literatures, and then select this upper bound as the initial value hb ð0Þ of hb . Step 2: Set appropriate step lengths, hb;step and αstep for hb and α, respectively. Set k as a counter, and k ¼ 1. Let hb ¼ hb ð0Þ þ hb;step and the initial value α0 ¼ αstep . Step 3: Let α ¼ kαstep , if the LMIs in (9) and (10) are feasible, go to step 4; otherwise, go to step 5. Step 4: Let hb ð0Þ ¼ hb , α0 ¼ αstep , k ¼ 1, and hb ¼ hb ð0Þ þ hb;step , go to step 3. Step 5: Let k ¼ k þ 1, if kαstep o 1, then go to step 3; otherwise, stop. Here, we only provide an alternative way to maximize hb by using the simple iterative optimization procedure (optimally dividing scheme). Generally speaking, when less conservative stability results can be found for a certain group of α than other groups, the corresponding dividing scheme can be called the optimally dividing delay interval [22]. Moreover, how to find the MADB on hb and the corresponding optimal dividing parameter α, we can apply the Nelder–Mead simplex algorithm to obtain the above parameters, the reader is referred to the algorithm in [30]. 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 landscape, 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. However, what is the detailed relationship between hb and α for a fixed ha ?, or what function is the hb ¼ f ðha ; αÞ? Furthermore, how to solve the proposed criteria in a more effective way? They are still some challenging problems. Remark 4. In the proof of Theorem 1, Lemma 1 plays a key effect on the present results, especially, if the lower and upper term in integral is uncertain and required to be approximated with its lower or upper bound then use of (7) and (8) would be beneficial since the free variables M; N; Z are introduced. By employing some convex combination technologies with regard to different uncertain terms, a stability criterion is derived. Unlike the free-weighting matrix method, the slack matrix variables M; N; Z in Theorem 1 are not redundant. In the literatures, to achieve the delay-dependence of the proposed results, the Leibniz–Newton formula was employed and many free-weighting matrix variables were introduced. In fact, these free-weighting matrix variables are all redundant and can be eliminated, which was theoretically proved in [15]. Generally speaking, the integral inequality based results are carried out more efficiently as compared with the general free-weighting matrix based on the Newton–Leibniz formula [3,6,8,11,14,16,23]. For the case of unknown hd , that is, the delay is not differential or the time-derivative of delay is unknown, one may set Q τ ¼ 0; Rτ ¼ 0 in (12). Then Theorem 1 reduces to the following Corollary 1.
216
J. An et al. / ISA Transactions 53 (2014) 210–219
Corollary 1. In Case 2, for given scalars 0 o ha r hb , 0 o α o1, the system (1) is asymptotically stable if there exist real symmetric matrices " # " # Z i1 Z i2 S11 S12 P,Q i 4 0; Ri ; ði ¼ 1; 2; 3; 4Þ, Q τ Z 0; Rτ , 4 0, Z i ¼ , and any matrices M i ; N i ; ði ¼ 1; 2; 3; 4Þ with appropriate dimensions such n Z i3 n S22 that the LMIs in (9) and (10) with Q τ ¼ 0; Rτ ¼ 0 are feasible. Moreover, for comparison with the existing ones, we reduce to the following results when the lower bound of the delay is 0, that is, ha ¼ 0. In case that the interval team ½0; ha is missing, one may set Q 1 ¼ 0; Q 2 ¼ 0; Sij ¼ 0; R1 ¼ 0; R2 ¼ 0 in (12). According to the line of Theorem 1, we reduce to the following Corollary 2. Corollary 2. In Case 1, for given scalars 0 ¼ ha r hb ; 0 o α o 1; hd , the system (1) is asymptotically stable if there exist real symmetric matrices " # Z i1 Z i2 P,Q i 4 0; Ri ; ði ¼ 3; 4Þ, Q τ Z 0; Rτ , Z i ¼ , and any matrices M i ; Ni ; ði ¼ 1; 2; 3; 4Þ with appropriate dimensions such that the LMIs in (24) n Z i3 and (25) are feasible. Meanwhile, in Case 2, the system (1) is asymptotically stable if the LMIs in (24) and (25) with Q τ ¼ 0; Rτ ¼ 0 are feasible. 2 3 4 2τ 3 2τ 2 2τ 2 i r33 R3 r44 R4 7 6 P þ ∑ r Ri 6 7 i¼3 i 6 7 V^ p ¼ 6 2τ 3 740 1 n Q þ R 0 6 7 τ3 3 r3 3 4 5 2τ 4 1 n n Q þ R 4 4 τ4 r4 k
k
k
^ ði; kÞ : ¼ Ω ^ þΩ ^ þΩ ^ þΞ ^ o 0; ði ¼ 1; 2; k ¼ 3; 4Þ Ω 0 i 1 2 with " τ3 R3 þ ð1 hd ÞRτ "
½ M1
n
N1
Z1
τ4 R4 þ ð1 hd ÞRτ ½ M4 N 4 n
ð24Þ
ð25Þ
# Z 0; # Z 0:
Z2
where 2 6 ^ : ¼6 6 Ω 0 6 4 2 6 ^ 3 : ¼6 6 Ω 1 6 4 2 6 6
PA þ AT P þ Q τ þ Q 3
PAτ
0
0
n
ð1 hd ÞQ τ
0
0
n
n
Q4 Q3
0
n
n
n
Q4
M 1 þ N T1
0
n
N 1 N T1 þ τ3 M 2 þ τ3 M T2
τ3 M 2 þ τ3 NT2
n
n
R4 τ3 N 2 τ3 N T2
n
n
n
τ
3
k
0
7 0 7 7; 7 R4 5 R4
0
M 1 þ NT1 þ τ3 Z 12
n
N 4 NT4 þ τ 4 ðM 3 þ M T3 Þ
M T4 þ N4
n
n
N 1 NT1 þM 4 þ M T4 þ τ3 Z 13
n 4
T
n
!
^; ∑ τ2i Ri þ τ3 Rτ Γ
i¼3
4
^ : ¼Γ ^ Ω 2
T
4
n
!
^; ∑ τ2i Ri þ τRτ Γ
0
3
7 τ4 M 3 þ τ4 N T3 7 7 7; 7 0 5 T τ4 ðN 3 þ N 3 Þ
Γ^ : ¼ ½ A Aτ 0 0 ;
i¼3
Ξ^ 1 : ¼ τk Z^ ; ðk ¼ 3; 4Þ; k
3 3 3 Z^ : ¼ ½Z^ ij 44 ; Z^ ij 9
k
3
M 1 þ M T1 þ 3 Z 11
6 4
^ : ¼Γ ^ Ω 2
7 7 7; 7 5
M 1 þ M T1
^ : ¼6 Ω 6 1 4
3
k
8 Z 11 > > > > > > < Z 12 Z T12
> > > Z > > > 13 : 0 k
i ¼ 1; j ¼ 1 i ¼ 1; j ¼ 2 i ¼ 2; j ¼ 1 ;
4 4 4 Z^ : ¼ ½Z^ ij 44 ; Z^ ij 9
i ¼ 2; j ¼ 2 other
^ R 1 ðΦ ^ ÞT ; ðk ¼ 3; 4Þ; Ξ^ 2 : ¼ τ2k Φ k
3
^ : ¼ colf0; M ; N ; 0g; Φ 2 2
8 Z 23 > > > > ZT > > < 22 Z 21
> > > Z > > > 22 : 0 4
i ¼ 2; j ¼ 2 i ¼ 2; j ¼ 3 i ¼ 3; j ¼ 3 ; i ¼ 3; j ¼ 2 other
^ : ¼ colf0; M ; 0; N g: Φ 3 3
ð26Þ
Remark 5. It is worthy mentioning that the variable delay decomposition approach proposed in this paper can be applied to the stabilization and H1 control synthesis, and the corresponding stabilization and control criteria can be obtained. Moreover, we would like to point out that the proposed results can be extended to more general/practical systems such as time-delay systems with nonlinear perturbations as [25–27], fuzzy systems as [24,29,31,32,36], fault detection as [34,37] and disturbance system as [33], and the corresponding results will appear in the near future.
J. An et al. / ISA Transactions 53 (2014) 210–219
217
4. Numerical examples In this section, two well-known numerical examples are given to show the merit and effectiveness of the proposed method. Example 1. Consider the system (1) with the following system parameters: 0 0 0 1 A¼ ; Aτ ¼ 1 1 1 2 analytical
which has the analytical bound hmax
¼ π for the constant delay case (that is, τ_ ðtÞ 0).
In order to make a comparison, we employ the stability criteria given in [4,7,10,14,16,17] and in the results of this paper. When the derivative of delay is available, for the lower bound ha ¼ 0, the admissible upper bounds hb of the time-varying delay are obtained for different maximum delay derivative hd and the computational results are shown in Table 1. When the derivative of delay is unknown or does not exist, for varies lower bound ha , the MADB on hb are obtained and shown in Table 2. Moreover, when hd ¼ 0:3 and ha ¼ 1:0, the MADB computed by Shao and Han [11] and Qian [14] are 2.35 and 2.493, respectively, while the result in Theorem 1 is computed as 3.3873; When ha ¼ 5:0, using the stability criteria in Sun et al. [7], Liu [35] and Theorem 1 of this paper, the MADB on hb are hb ¼ 5:2970, hb ¼ 5:4661 and hb ¼ 5:5710, respectively. So, it can be seen from Tables 1 and 2 that the stability results obtained in this paper are less conservative that those in [4,7] and [10,11,14,16,17,35]. Table 1 MADB on hb with varying hd and ha ¼ 0 for Example 1. Method
hd
Shao [4] Sun et al. [7] Zhu et al.[16] Tang et al. [17] Corollary 2 (α ¼ 0:65) Corollary 2 (α ¼ 0:7)
0.1
0.3
0.5
0.8
1
5.4630 5.4764 5.4780 5.4940 7.8011 7.8401
2.2160 2.2160 2.2850 2.3070 3.5932 3.6962
1.1270 1.1272 1.2080 1.2330 1.7334 1.7655
0.8710 0.8714 1.0200 1.0440 1.2999 1.3123
0.8710 0.8714 1.0200 1.0440 1.2999 1.3123
Table 2 MADB on hb with varying ha and unknown hd for Example 1. Method
ha
Shao [4] Sun et al. [7] Zhu et al.[16] Tang et al. [17] Park et al. [10] Qian et al. [14] Liu [35] Corollary 1 (α ¼ 0:55) Corollary 1 (α ¼ 0:6)
0.3
0.5
0.8
1
2
1.0715 1.0716 1.2043 1.2246 1.2400 1.3500 1.4250 1.6351 1.6837
1.2191 1.2196 1.3429 1.3619 1.3800 1.4700 1.5694 1.7751 1.8120
1.4539 1.4552 1.5663 1.5838 1.6000 1.6800 1.7945 1.9978 2.0209
1.6169 1.6189 1.7228 1.739 1.7500 1.8100 1.9493 2.1519 2.1672
2.4798 2.4884 2.5600 2.5700 2.5800 2.6100 2.7674 2.9669 2.9552
Table 3 MADB on hb with given ha for different hd for Example 2. ha
Methods
hd ¼ 0:1
hd ¼ 0:3
hd ¼ 0:5
hd ¼ 0:9
1
Sun et al. [7] Liu et al. [18] Theorem 1 (α ¼ 0:65)
4.1935 4.4045 5.0950
3.0538 3.1208 3.6840
2.3058 2.3513 2.8418
1.9008 2.0921 2.5389
2
Sun et al. [7] Liu et al. [18] Theorem 1 (α ¼ 0:65)
4.4932 4.5729 5.0458
3.0129 3.1092 3.4397
2.5663 2.6987 3.0240
2.5663 2.6987 3.0240
3
Sun et al. [7] Liu et al. [18] Theorem 1 (α ¼ 0:65)
4.3979 4.5406 4.8247
3.3408 3.4186 3.6616
3.3408 3.4186 3.6616
3.3408 3.4186 3.6616
4
Sun et al. [7] Liu et al. [18] Theorem 1 (α ¼ 0:65)
4.1978 4.2367 4.5762
4.1690 4.2097 4.3788
4.1690 4.2097 4.3788
4.1690 4.2097 4.3788
5
Sun et al. [7] Liu et al. [18] Theorem 1 (α ¼ 0:65)
5.0275 5.0440 5.1453
5.0275 5.0440 5.1453
5.0275 5.0440 5.1453
5.0275 5.0440 5.1453
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J. An et al. / ISA Transactions 53 (2014) 210–219
Table 4 MADB on hb with given hd for different ha for Example 2. hd
Methods
ha ¼ 0
ha ¼ 1
ha ¼ 2
ha ¼ 3
ha ¼ 4
ha ¼ 5
0.5
Ramakrishnan and Ray [28] Hui et al. [26] Our results
2.2022 2.1471 2.3223
2.3912 2.5652 2.8418
2.9578 3.1124 3.0240
3.6384 3.7448 3.6616
4.3736 4.4369 4.3788
5.1463 5.1713 5.1453
0.9
Ramakrishnan and Ray [28] Hui et al. [26] Our results
1.8828 2.1377 2.2416
2.3585 2.5627 2.5389
2.9578 3.1085 3.0240
3.6384 3.7408 3.6616
4.3736 4.4340 4.3788
5.1463 5.1703 5.1453
Any hd
Ramakrishnan and Ray [28] Hui et al. [26] Our results
1.6654 2.1377 2.2416
2.1251 2.5627 2.4552
2.7113 3.1085 3.0312
3.3839 3.7408 3.7035
4.1136 4.4340 4.4203
5.1463 5.1703 5.1665
Table 5 MADB on hb with varying ha and unknown hd for Example 2. Method
ha
Shao [4] Sun et al. [7] Zhu et al. [16] Tang et al. [17] Park et al. [10] Shao and Han [11] Zhang et al. [27] Qian et al. [14] Liu et al. [18] Guo et al. [19] Ramakrishnan and Ray [25] Corollary 1 (α ¼ 0:55) Corollary 1 (α ¼ 0:6)
1
2
3
4
5
1.8737 1.9008 2.0273 2.0448 2.0600 2.0665 2.0665 2.0100 2.0921 2.1202 2.4454 2.4552 2.5449
2.5048 2.5663 2.5915 2.6051 2.6100 2.6181 2.6181 2.5680 2.6987 2.7199 2.9312 3.0312 3.0545
3.2591 3.3408 3.3010 3.3098 3.3100 3.3173 3.3173 3.3420 3.4186 3.4569 3.5087 3.7035 3.6939
4.0744 4.1690 4.0855 4.0877 4.0900 4.0905 4.0905 4.1730 4.2097 4.2574 4.1509 4.4203 4.4026
– 5.0275 – – – – – 5.0360 5.0440 5.0976 – 5.1665 5.1557
Table 6 Comparison with MADB on hb with varying hd (ha ¼ 0). Method
hd
Park and Ko [5] J.-H. Kim [12] Corollary 2 (α ¼ 0:65) Corollary 2 (α ¼ 0:7)
0.0
0.05
0.10
0.5
3.0
4.47 4.97 5.11 5.18
4.01 4.35 4.46 4.54
3.66 3.86 3.96 4.08
2.33 2.33 2.48 2.58
1.86 1.86 2.33 2.34
Example 2. Consider system (1) with A¼
2
0
0
0:9
;
Aτ ¼
1
0
1
1
analytical
which has the analytical bound hmax
¼ 6:17258 for the constant delay (when τ_ ðtÞ 0).
As with Example 1, here we calculate the MADB that guarantees the asymptotic stability of system (1) using the methods proposed in [7,18], and Theorem 1 in this paper. For unknown maximum derivative hd of the time delay and varying lower bound ha of the time delay, the MADB on hb are obtained and shown in Table 3. For the Table, it can be seen that our method yields less conservative than [7,18]. Moreover, compare to the existing ones in [26,28], computational results in this paper are obtained and these are listed in Table 4 for various case. Note that our results in Table 4 use Theorem 1 (α ¼ 0:65) when hd are known and ha 4 0; Corollary 1 (α ¼ 0:55) when hd are unknown and ha 4 0; Corollary 2 (α ¼ 0:6) when ha ¼ 0. It is seen that the stability criteria in [26] and Theorem 1 in this paper are not covered by each other, although they are less conservative than those in [28]. However, the criteria in [26] are complex and involve more matrix variables than ours. Meanwhile, for the case of ha ¼ 0 and unknown hd (any hd ), from this example, it seems that our results (Corollaries 1 and 2) are less conservative than those in [26,28] when the lower bound of the varying delay, i.e., ha is small. For unknown hd , the MADB on hb , which guarantee the asymptotic stability of system (1) for given lower bounds ha , are listed in Table 5 for Corollary 1 and the exiting ones in [4,7,10,11,14,16–19,25,27], respectively. Moreover, comparing to the existing ones in [5,12], we assume that ha ¼ 0, and can calculate the result by Corollary 2 of this paper and those in [5,12], which is shown in Table 6. From Tables 5 and 6, it also can be seen that the stability results obtained in the paper are less conservative than those in [4,5,7,10–12,14,16–19,25,27]. It should be noted that hd ¼ 0 does not mean a constant delay (that is, τ_ ðtÞ 0), but it means a time-varying delay with τ_ ðtÞ r 0; 8 t Z 0. Also, our result gives less conservativeness for a relatively small value of hd .
J. An et al. / ISA Transactions 53 (2014) 210–219
219
5. Conclusion In this paper, delay-fractional-dependent stability problem for linear systems with interval time-varying delay is investigated. Through employing a novel delay-dependent LK functional via variable delay interval dividing technique, the proposed criterion provides better results since approximation of the uncertain delay factors in the time-derivative of LK functional has been avoided, and making full use of the delay information and the relationship among xðtÞ; xðt τðtÞÞ; xðt ðha =2ÞÞ; xðt ha Þ, xðt ha ατÞ and xðt hb Þ with τ 9hb ha ,0 o α o 1, and thus some less conservative stability conditions are obtained in the form of LMIs. Finally, by two well-known examples, we showed the usefulness and less conservatism of the proposed approach. Moreover, the proposed approach is simple and may easily be extended to robust stability, stabilization problems for linear time-delay systems and the systems with nonlinear perturbations. Acknowledgments This work is supported in part by the Natural Science Foundation of China under Grants 61173107 and 61370097, and in part by the State Scholarship Fund of China Scholarship Council (CSC) under Grant 201206135001, and in part by the Planned Science and Technology Project of Hunan Province under Grant 2012FJ4130. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34] [35] [36] [37] [38]
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A novel approach to delay-fractional-dependent stability criterion for linear systems with interval delay Jiyao An a,c,n, Zhiyong Li a, Xiaomei Wang b,c a Key Laboratory of Embedded and Network Computing of Hunan Province, College of Information Science and Engineering, Hunan University, Changsha 410082, PR China b School of Mathematics Science, University of Electronic Science and Technology of China, Chengdu 610054, PR China c Department of Applied Mathematics, Faculty of Mathematics, University of Waterloo, Waterloo, Ontario, Canada N2L 3G1
art ic l e i nf o
a b s t r a c t
Article history: Received 14 June 2013 Received in revised form 7 November 2013 Accepted 27 November 2013 Available online 31 December 2013 This paper was recommended for publication by Dr. Q.-G. Wang
This paper considers the problem of delay-fractional-dependent stability analysis of linear systems with interval time-varying state delay. By developing a delay variable decomposition approach, both the information of the variable dividing subinterval delay, and the information of the lower and upper bound of delay can be taken into full consideration. Then a new delay-fractional-dependent stability criterion is derived without involving any direct approximation in the time-derivative of the Lyapunov–Krasovskii (LK) functional via some suitable Jensen integral inequalities and convex combination technique. The merits of the proposed result lie in less conservatism, which are realized by choosing different Lyapunov matrices in the variable delay subintervals and estimating the upper bound of some cross term in LK functional more exactly. At last, two well-known numerical examples are employed to show the effectiveness and less conservatism of the proposed method. Crown Copyright & 2013 Published by Elsevier Ltd. on behalf of ISA. All rights reserved.
Keywords: Lyapunov–Krasovskii (LK) functional Delay-fractional-dependent stability Interval time-varying delay Linear matrix inequality (LMI) Maximum allowable delay bound (MADB)
1. Introduction Stability is a central issue in dynamical system and control theory. A dynamical system is called stable (in the sense of Lyapunov) if starting the system somewhere near its desired operating point implies that it will stay around that point ever after. The stability problem for time-delayed systems [1] has received considerable attention in recent years, see, e.g., [2–9], and the references therein. Recently, Richard [2] summarized current research on time-delay systems and listed some open problems, and argued that the stability analysis should have been focusing on effective reduction of the conservation of the stability conditions and control algorithm. Especially, in real system, the delay is assumed to be an interval time-varying delay [10–17], ha r τðtÞ rhb and ha is not restricted to be 0. The main aim is to derive a maximum allowable delay bound (MADB) of the time-delay such that the concerned system is asymptotically stable for any delay size less than the MADB. Accordingly, the obtained MADB becomes a key performance index to measure the conservatism of a delaydependent stability condition. As well-known, the bounding technology and the model transformation technique are introduced into the stability analysis, although they may lead to some conservation. In order to reduce the conservatism, a free-weighting matrix method was proposed in [3], and hereafter the approach has been extended to two form, that is, the one with a null summing term added to the LK functional derivative [3,7,8,23,24], and the one with free matrices item added to the LK functional combined with the descriptor model transformation [4,6,13]. Recently, some techniques with relation to tighter upper bounds of various functions have been pursued: affine functions of polytopictype uncertain systems, quadratic functions of nonlinear systems, and especially a special type of function combinations in delayed systems [4,5,9,10]. Among the bounding technique, the Jensen inequality lemma [1] takes a key role to be efficiently relaxed some integral and cross terms in the time derivative of LK functional. However, to fully relax the matrix cross-products, it has to introduce slightly
n Corresponding author at: Key Laboratory of Embedded and Network Computing of Hunan Province, College of Information Science and Engineering, Hunan University, Changsha 410082, PR China. Tel.: þ 86 731 88821907; fax: þ 86 731 88822417. E-mail address: [email protected] (J. An).
0019-0578/$ - see front matter Crown Copyright & 2013 Published by Elsevier Ltd. on behalf of ISA. All rights reserved. http://dx.doi.org/10.1016/j.isatra.2013.11.020
J. An et al. / ISA Transactions 53 (2014) 210–219
211
excessive free weighting matrices. As a way of reducing the number of decision variables, at the sacrifice of conservatism, relaxations based on the Jensen inequality lemma [1] have been attracted in [7–14,17–19]. On the other hand, the delay partitioning approach has been attracted in the derivation process. Recently, the method of dividing delay (including the discretized Lyapunov functional approach in [1] and the delay decomposition approach in Gouaisbaut and Peaucelle [20] and Han [21]) is the best one to handle the stability of system (1) with constant delay, by which the result near analytical delay limit can be obtained. Since the augmented matrix S has much more room to be adjusted in the criterion, the functional can achieve the benefit of reducing the conservatism. Furthermore, partitioning of the delay intervals into n 4 2 subintervals may lead to further improvements, for examples, another delay partitioning was introduced in [3,6,20,21,24,29], which corresponded to the partitioning into two subintervals of ½0; ha and of ½ha ; hb . Therefore, it is natural that the less conservative stability results can be obtained when the delay decomposition approach is extended to the case of time-varying delay. So far, by dividing interval ½0; τðtÞ into N non-uniform sub-intervals, the pioneering studies have been provided, see [21,22]. But, the purpose of reducing conservatism is still limited due to the existence of multiple coefficients and the impact of subintervals with uniform size [22,25,26]. Moreover, it is still a quite difficult task to divide interval ½0; τðtÞ in a more reasonable manner [22], so that the functional with the augmented matrix can easily be constructed to obtain less conservative stability results. In practice, the systems almost contain some uncertainties because it is very difficult to obtain an exact mathematical model due to environment noise, uncertain or slowly varying parameters, etc. Most recently, the stability analysis for time-delay systems has been extended to the case of the systems with nonlinear perturbations [25–27] and fuzzy systems [31,32], while the proposed result also has been expanded to disturbance attenuation analysis [33,38] and H1 controller/filter design [31,32,36] and robust fault detection [34,37] for the above systems. Ramakrishnan and Ray [25] studied the problem of stability analysis of the linear systems with nonlinear perturbation by partitioning the delay interval into two segments of equal length. Zhang et al. [26] employed some novel integral inequalities (Lemmas 1 and 2 in [26]) to derive an improved stability criterion for the above systems. He et al. [33] investigated the disturbance attenuation problem and H1 controller design for the linear systems. Lin et al. considered the H1 filter design problem [31] and observer-based H1 control design problem [32] for nonlinear system via T–S fuzzy model approach. Most recently, Liu [35] further studied the stability analysis of linear systems. However, there also exists further room to investigate the upper bound of the time-derivative of the LK functional, and the delay interval may be divided into two unequal subintervals or more subintervals [22,24,29]. This motivates the present research to develop a novel method for stability problem for the concerned systems with less conservatism by making full use of the information of time-delays and constructing a novel LK functional via variable delay dividing technique. Motivated by the above discussions, this paper will focus on the delay-fractional-dependent stability problem of linear systems with interval time-varying delay. Inspired by the reciprocally convex approach in [10], and some integral inequalities in [1,7,13], and then the work of Kim [12], Zhang [22] and Briat [9], we firstly construct a novel delay-fractional-dependent LK functional by developing a delay decomposition approach, in which the integral interval ½t hb ; 0 is decomposed into ½t hb ; t ha ατ, ½t ha ατ; t ha and ½t ha ; t ðha =2Þ, ½t ðha =2Þ; t with 0 o α o1 and τ 9 hb ha . Secondly, we derive a stability criterion for system (1) by suitably using the inequalities such as (4)–(8) in Lemma 1 below. Since a tuning parameter α and different delay partitioning method are introduced, the information about xðt ha ατÞ and xðt ððha =2ÞÞ can be taken into full consideration. Meanwhile, based on the convex combination approach, the cross team in the time derivative of LK functional can be estimated more exactly without any direct approximation of the delay terms in the derivation process. Therefore, using the variable delay decomposition approach, the Lyapunov matrices in LK 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 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 conservatism. Finally, two well-known numerical examples are used to compare with some previous results and demonstrate the effectiveness of the proposed method.
2. Problem formulation Consider the following linear system with a time-varying state delay: (
x_ ðtÞ ¼ AxðtÞ þ Aτ xðt τðtÞÞ
xðtÞ ¼ ϕðtÞ; 8 t A ½ hb ; 0;
ð1Þ
where xðtÞis the state vector. ϕðtÞ is the continuous initial vector function defined on ½ hb ; 0; the matrices A; Aτ are constant matrices with appropriate dimensions. In this paper, the delayτðtÞ is assumed to be time-varying delay as the following two cases: Case 1. τðtÞ is a differentiable function, satisfying for all t Z 0: 0 o ha r τðtÞ rhb ; τ_ ðtÞ r hd
ð2Þ
Case 2. τðtÞ is not differentiable or the upper bound of the derivative of τðtÞ is unknown, and τðtÞ satisfies 0 o ha r τðtÞ rhb
ð3Þ
where ha ; hb and hd are some given values. The purpose of this paper is to find a stability criterion, which is less conservative than the existing results. Such a criterion may be used to compute the tolerable delay bound hb for given ha or vice versa. To end this section, we introduce the following lemmas which are useful in stability analysis for the linear system.
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J. An et al. / ISA Transactions 53 (2014) 210–219
Lemma 1. (Integral inequalities, Gu [1]; Sun et al. [7]; Zhang et al. [13]) Let xðtÞ A ℜn be a vector-valued function with first-order continuous-derivative entries. Then, for any matrices M; N A ℜnn , Z A ℜ2n2n , X ¼ X T A ℜnn , and some given scalars 0 r τ1 o τ2 , the following integral inequalities hold: (1) When X 40 and τ1 ; τ2 are constant values, Z t τ1 Z t τ1 Z xT ðsÞXxðsÞds Z xT ðsÞdsX ðτ2 τ1 Þ t τ2
τ22 τ21
Z
2
τ1
t τ2
Z
τ2
ðτ 2 τ 1 Þ
Z
t t þθ
t τ1 t τ2
xT ðsÞXxðsÞdsdθ Z "
x_ ðsÞX x_ ðsÞds r T
Z
τ1
Z
τ2
xðt τ1 Þ
t τ1 t τ2
t tþθ
#T
xðt τ2 Þ
ð4Þ
xðsÞds
xT ðsÞdsdθX
X
X
n
X
"
Z
τ1
Z
τ2
xðt τ1 Þ
t tþθ
xðsÞdsd θ
ð5Þ
# ð6Þ
xðt τ2 Þ
(2) When X 40 and τ1 ; τ2 are time-varying, h ¼ τ2 τ1 : ¼ hðtÞ Z0, # " #T (" Z t τ1 xðt τ1 Þ M M þ M T M þ N T x_ T ðsÞX x_ ðsÞds r þ h X 1 ½ MT T τ Þ xðt N 2 n N N t τ2
)" NT
xðt τ1 Þ
xðt τ2 Þ
(3) When τ1 ; τ2 are time-varying, h ¼ τ2 τ1 : ¼ hðtÞ Z 0, and X is any symmetric matrix, # )" " #T (" # Z t τ1 xðt τ1 Þ xðt τ1 Þ M þ M T M þ N T x_ T ðsÞX x_ ðsÞds r þ hZ xðt τ2 Þ xðt τ2 Þ n N N T t τ2 X Y Z0, and Y ¼ ½M N. with n Z
# ð7Þ
ð8Þ
Remark 1. Eqs. (4)–(8) are called various integral inequalities presented over the existing literatures. They play a key role in the derivation of a criterion for delay-dependent stability analysis of systems with time-varying delay. In this paper, different from the exiting ones, we will utilize suitably different integral inequality in connection with different cross terms in LK functional or it is time-derivative. Meanwhile, the upper bound of these cross terms in time-derivative of LK functional are estimated and deduced more exactly without involving any direct approximation of uncertain terms in the derivation process.
3. Main results Firstly, we divide the delay interval 0; ha and ha ; hb into four segments: hi 1 ; hi ; i ¼ 1; 2; 3; 4 where h0 ¼ 0; h1 ¼ ha =2, 2 2 h2 ¼ ha ; h3 ¼ ha þ ατ; h4 ¼ hb ; 0 o α o 1. For the sake of convenience, we denote τ ¼ hb ha , and τi ¼ hi hi 1 , r i ¼ hi hi 1 , ði ¼ 1; 2; 3; 4Þ and τ0 ¼ ha 0. For the system (1), we derive a stability condition by using the variable delay decomposition approach as follows: Theorem 1. In Case 1, for given scalars 0 o ha r hb ; 0 o α o 1; hd , the system (1) is asymptotically stable if there exist real symmetric matrices " # " # Z i1 Z i2 S11 S12 4 0, Z i ¼ , and any matrices M i ; N i ; ði ¼ 1; 2; 3; 4Þ with appropriate dimensions such P,Q i 4 0; Ri ; ði ¼ 1; 2; 3; 4Þ, Q τ Z 0; Rτ , n Z i3 n S22 that the LMIs in (9) and (10) are feasible. 2 4 2τ 3 2τ 2 2τ 2 i r11 R1 r22 R2 6Pþ ∑ r 6 i¼1 i 6 2τ 1 1 2 2 6 n 6 τ1 Q 1 þ r 1 R1 þ τ0 S11 τ0 S12 6 Vp ¼ 6 2 τ2 1 2 n n 6 τ2 Q 2 þ r 2 R2 þ τ0 S22 6 6 n n n 6 4 n
n
2τ23 r 3 R3
0
0
0
0
2τ 3 1 τ 3 Q 3 þ r 3 R3
n
0 2τ 4 τ4 Q 4 þ r 4 R4 1
n
Ωði; kÞ : ¼ Ω0 þ Ωk1 þ Ωk2 þ Ξ ki o 0; ði ¼ 1; 2; k ¼ 3; 4Þ with " τ3 R3 þ ð1 hd ÞRτ n
½ M1 Z1
N1
#
" Z 0;
3 7 7 7 7 7 7 740 7 7 7 7 5
ð9Þ
ð10Þ
τ4 R4 þ ð1 hd ÞRτ ½M 4 n
2τ24 r 4 R4
N4 Z2
# Z 0:
J. An et al. / ISA Transactions 53 (2014) 210–219
213
where 2 6 6 6 6 Ω0 : ¼ 6 6 6 6 6 4 2
PAþ AT P þ Q 1 þ S11 R1
PAτ
S12 þ R1
0
0
0
n
ð1 hd ÞQ τ
0
0
0
0
n
n
Q 2 Q 1 þ S22 S11 R1 R2
n
n
n
S12 þR2 Q 3 Q 2 þ Q τ S22 R2
0 0
0 0
n
n
n
n
Q4 Q3
0
n
n
n
n
n
Q4
0
6n 6 6 6n 3 Ω1 : ¼ 6 6n 6 6 6n 4
0
0
0
0
N 1 NT1 þ τ3 M 2 þ τ3 M T2
0
M T1 þ N 1
τ3 M 2 þ τ3 N T2
n
0
n
n
0 M 1 þ M T1
0 0
n
n
n
R4 τ3 N 2 τ3 N T2
n
n
n
n
n
2
0 6n 6 6 6n 4 Ω1 : ¼ 6 6n 6 6 6n 4
0
0 7 7 7 0 7 7 ; 0 7 7 7 R4 7 5 R4
0
0
0
M T4 þ N4
3
0
τ
0
n
0
0
0
0
n
n
M 1 þ M T1 þ τ3 Z 11
M 1 þ NT1 þ τ3 Z 12
0
n
n
n
N 1 NT1 þM 4 þ M T4 þ τ3 Z 13
0
n
n
n
n
τ4 ðN 3 þ N T3 Þ
!
4
∑ τ2i Ri þ τ3 Rτ Γ ;
i¼1
8 Z 13 > > > T > > Z > < 12 Ξ k1 : ¼ τk Z k ; Z 3 : ¼ ½Z 3ij 66 ; Z 3ij 9 Z 11 > > > Z 12 > > > : 0
Ξ k2 : ¼ τ2k Φk Rk 1 ðΦk ÞT ;
! 4 ∑ τ2i Ri þ τRτ Γ ; Γ : ¼ A
Ω42 : ¼ Γ T
Aτ
i¼1
i ¼ 2; j ¼ 4 i ¼ 4; j ¼ 4 ; i ¼ 4; j ¼ 2 other
Φ3 : ¼ colf0; M 2 ; 0; 0; N 2 ; 0g;
τ4 M 3 þ τ
0
8 Z 23 > > > T > > Z > < 22 Z 4 : ¼ ½Z 4ij 66 ; Z 4ij 9 Z 21 > > > Z 22 > > > : 0
i ¼ 2; j ¼ 2
7 7 7 7 7 7; 7 7 7 5
3
0
N 4 NT4 þ 4 ðM 3 þ M T3 Þ
n
Ω32 : ¼ Γ T
0
3
0
0
T 4 N3
7 7 7 7 7 7; 7 7 7 5
0 ;
i ¼ 2; j ¼ 2 i ¼ 2; j ¼ 5 i ¼ 5; j ¼ 5 ; i ¼ 5; j ¼ 2 other
Φ4 : ¼ colf0; M3 ; 0; 0; 0; N 3 g:
ð11Þ
Proof. Choose the following delay-fractional-dependent LK functional: Vðt; xt Þ ¼ V 1 ðt; xt Þ þ V 2 ðt; xt Þ þ V 3 ðt; xt Þ þ V 4 ðt; xt Þ
ð12Þ
where xt denotes the function xðsÞdefined on the interval ½t hb ; t, and V 1 ðt; xt Þ ¼ xT ðtÞPxðtÞ; Z t hi 1 Z 4 xT ðsÞQ i xðsÞds þ V 2 ðt; xt Þ ¼ ∑ t hi
i¼1
Z V 3 ðt; xt Þ ¼
"
t
#T "
xðsÞ xðs τ20 Þ
t h1
S11
S12
n
S22
t h2 t τ ðtÞ
#"
xT ðsÞQ τ xðsÞds #
xðsÞ
ds
xðs τ20 Þ
Z h2 Z t x_ T ðsÞRi x_ ðsÞdsdθ þ x_ T ðsÞRτ x_ ðsÞdsdθ t þθ hi τðtÞ t þ θ i¼1 " # S11 S12 with P,Q i 4 0ði ¼ 1; 2; 3; 4Þ,Q τ Z 0, 4 0,Ri ði ¼ 1; 2; 3; 4Þ,Rτ being real symmetric matrices. n S22 First, we show that the condition Vðt; xt Þ Z ε1 ‖xðtÞ‖; ðε1 4 0Þ is satisfied. Assume that Q i 40; Ri 40; ði ¼ 1; 2; 3; 4Þ, then, suitably applying Lemma 1 it yields 4
V 4 ðt; xt Þ ¼ ∑ τi
Z
hi 1
Z Vðt; xt Þ Z xT ðtÞPxðtÞ þ Z þ
t
t h1
"
Z
t
t h2
t τðtÞ
xðsÞ
Z xT ðsÞQ τ xðsÞds þ #T
τ0
xðs 2 Þ
" ds
n We denote ζ ðtÞ : ¼ col xðtÞ Vðt; xt Þ Z ζ ðtÞV p ζ ðtÞ þ T
Z
t h2 t τðtÞ
S11
S12
n
S22
Rt t h1
h2 τðtÞ
#Z
t tþθ
"
t t h1
τ0
xðs 2 Þ
t h2
Z
h2 τ ðtÞ
Z
4
1
i¼1
τi
x_ T ðsÞRτ x_ ðsÞdsdθ þ ∑
xðsÞ
R t h1
xðsÞds
xT ðsÞQ τ xðsÞdsþ
Z
xðsÞds t t þθ
#
2τ i i ¼ 1 ri 4
ds þ ∑ R t h2 t h3
xðsÞds
x_ T ðsÞRτ x_ ðsÞdsdθ;
Z
Z
t hi 1 t hi
hi 1 hi
R t h3 t h4
Z
t tþθ
Z xT ðsÞdsQ i
x_ T ðsÞdsdθRi
t hi 1
xðsÞds t hi
Z
hi 1 hi
Z
t t þθ
x_ ðsÞdsdθ
o xðsÞds , after the simple arrangement, we have
ð13Þ
214
J. An et al. / ISA Transactions 53 (2014) 210–219
2
4 2τ 3 i 6Pþ ∑ r 6 i¼1 i 6 6 n 6 6 Vp 96 n 6 6 6 n 6 4
2τ 21 r1
2τ 1 2 τ1 Q 1 þ r1 R1 þ τ0 S11 1
n
2τ 22 r2
2
τ0 S12
2τ23 r3
0
0
0
0
n
2τ 2 2 τ2 Q 2 þ r2 R2 þ τ0 S22
n
n
2τ 3 1 τ3 Q 3 þ r3 R3
n
n
n
1
3
2τ24 r4
0 2τ 4 τ4 Q 2 þ r 4 R4 1
7 7 7 7 7 7 7 7 7 7 7 5
ð14Þ
Therefore, if the right side of (14) is positive, it yields a sufficient condition to guarantee that Vðt; xt Þ Z ε1 ‖xðtÞ‖; ðε1 4 0Þ. Moreover, the LMIs in (10) imply that Ri 4 0; ði ¼ 1; 2; 3; 4Þ; Rτ 4 0. Consequently, if the LMIs in (9) are satisfied, Q i 40; ði ¼ 1; 2; 3; 4Þ, and Q τ Z 0, then Vðt; xt Þ Z ε1 J xðtÞ J ; ðε1 4 0Þ. _ xt Þ r ε2 J xðtÞ J ; ðε2 40Þ is also guaranteed if the LMIs in (10) and Q τ Z0, Secondly, we show that the condition Vðt; " # S11 S12 4 0 hold. Taking the derivative of (12) with respect to t along the trajectory of system (1), we have Q i 4 0; ði ¼ 1; 2; 3; 4Þ, n S22 T ð15Þ V_ 1 ðt; xt Þ ¼ x_ ðtÞPxðtÞ þ xT ðtÞP x_ ðtÞ ¼ ½AxðtÞ þ Aτ xðt τðtÞÞT PxðtÞ þxT ðtÞP½AxðtÞ þ Aτ xðt τðtÞÞ 4
V_ 2 ðt; xt Þ ¼ ∑ ½xT ðt hi 1 ÞQ i xðt hi 1 Þ xT ðt hi ÞQ i xðt hi Þ i¼1
þ xT ðt h2 ÞQ τ xðt h2 Þ ð1 τ_ ðtÞÞxT ðt τðtÞÞQ τ xðt τðtÞÞ 4
r ∑ ½xT ðt hi 1 ÞQ i xðt hi 1 Þ xT ðt hi ÞQ i xðt hi Þ i¼1
þ xT ðt h2 ÞQ τ xðt h2 Þ ð1 hd ÞxT ðt τðtÞÞQ τ xðt τðtÞÞ " V_ 3 ðt; xt Þ ¼
#T "
xðtÞ xðt h1 Þ
S11
S12
n
S22
#"
#
xðtÞ xðt h1 Þ
"
xðt h1 Þ xðt h2 Þ
#T "
ð16Þ
S11
S12
n
S22
#"
xðt h1 Þ xðt h2 Þ
# ð17Þ
4
V_ 4 ðt; xt Þ ¼ ∑ τ2i x_ T ðtÞRi x_ ðtÞ þ ðτðtÞ h2 Þx_ T ðtÞRτ x_ ðtÞ i¼1
4
∑ τi i¼1
Z
t hi 1
Z
x_ T ðsÞRi x_ ðsÞds ð1 τ_ ðtÞÞ
t hi
t h2
x_ T ðsÞRτ x_ ðsÞds
t τ ðtÞ
ð18Þ
For any t Z0, it is the fact that ha r τðtÞ r ha þ ατ or ha þ ατ r τðtÞ r hb , (0 o α o 1). In the case of ha r τðtÞ r ha þ ατ, i.e., τðtÞ A ½h2 ; h3 , k ¼ 3, suitably using the integral inequalities in Lemma 1, the following inequalities are true: ðτðtÞ h2 Þx_ T ðtÞRτ x_ ðtÞ r ατx_ T ðtÞRτ x_ ðtÞ ¼ τ3 x_ T ðtÞRτ x_ ðtÞ
τi
Z
τ3
t hi 1 t hi
Z
t h2 t h3
" x_ T ðsÞRi x_ ðsÞdsr
xðt hi 1 Þ
#T "
xðt hi Þ
x_ T ðsÞR3 x_ ðsÞds ð1 τ_ ðtÞÞ
Z
t h2 t τ ðtÞ
ð19Þ
Ri
Ri
n
Ri
#"
xðt hi 1 Þ
# ; ði ¼ 1; 2; 4Þ
xðt hi Þ
x_ T ðsÞRτ x_ ðsÞds r τ3 Z ¼ " r
Z
t h2 t h3
t h2
t τ ðtÞ
x_ T ðsÞR3 x_ ðsÞds ð1 hd Þ
#T ("
xðt τðtÞÞ "
xðt τðtÞÞ
with
M1
n
N1
It follows from (15) to (21) that " _ xt Þ r ρξT ðtÞ Ω0 þ Ω3 þ Γ T Vðt; 1
4
xðt h3 Þ
N 1 NT1
M2 N2
Z 0, and ρ ¼ ðτðtÞ h2 =ατÞ; 0 r ρ r 1.
!
#
∑ τ2i Ri þ ατRτ Γ þ ατZ 3 ξðtÞ
i¼1
n
#
Z1
t τðtÞ
M 1 þ N T1
"
τ3 "
τ3 R3 þ ð1 hd ÞRτ
t h2
M 1 þ M T1
#T (
þ ð1 ρÞ U ατ U "
Z
x_ T ðsÞRτ x_ ðsÞds
x_ T ðsÞðτ3 R3 þ ð1 hd ÞRτ Þx_ ðsÞds τ3
xðt h2 Þ
þ
ð20Þ
M 2 þ M T2
#
Z
t τ ðtÞ t h3
x_ T ðsÞR3 x_ ðsÞds
þ ρ U ατ U Z 1
M 2 þN T2
)"
xðt h2 Þ
#
xðt τðtÞÞ
#
N 2 NT2 9 " #T " # M 2 = xðt τðtÞÞ n
# R3 1
N2
;
xðt h3 Þ
ð21Þ
J. An et al. / ISA Transactions 53 (2014) 210–219
" þ ð1 ρÞξ ðtÞ Ω0 þ Ω T
" with
3 1þ
τ3 R3 þ ð1 hd ÞRτ n
M1
Γ
N1
Z1
T
4
∑ τ
i¼1
! 2 i Ri þ
ατRτ Γ þ ðατÞ2 Φ3 R3 1 ðΦ3 ÞT ξðtÞ
Ω0 þ Ω
3 1þ
Γ
T
ð22Þ
# Z0
n 3 3 where Ω0 ; Ω1 ; Γ ; Z 3 ; Φ are defined in (11), and ξðtÞ : ¼ col xðtÞ xðt τðtÞÞ xðt h1 Þ xðt h2 Þ xðt h3 Þ Since 0 r ρ r 1, applying the convex combination method, we conclude that if the following LMIs: !
Ω0 þ Ω31 þ Γ T
215
o xðt h4 Þ .
4
∑ τ2i Ri þ ατRτ Γ þ ατZ 3 o0
i¼1 4
∑ τ
i¼1
! 2 i Ri þ
ατRτ Γ þ ðατÞ2 Φ3 R3 1 ðΦ3 ÞT o0
ð23Þ
are feasible, then x is true for a sufficiently small ε240. Meanwhile, if ha þ ατ r τðtÞ r hb , i.e., τðtÞ A h3 ; h4 ,k ¼ 4, similar to the above deduction process, we also can obtain the similar stability condition. This completes the proof. □ Remark 2. By constructing delay-fractional-dependent LK Functional (12) and using suitably integral inequalities in Lemma 1, Theorem 1 is obtained without involving any approximation of the delay term when exploiting a convex combination of the uncertain terms involved, and the value of the upper bound of some cross term is estimated more exactly than the previous methods, So, such a feature may lead to less conservative results compared to the existing ones. Moreover, by proving the positive definiteness of the constructed LK functional, the constraints on some functional parameters were relaxed and the conservatism of the stability results may be further reduced. Remark 3. Since the interval ha ; hb is divided into two variable subintervals ha ; ha þ ατ and ha þ ατ; hb in which α is a tunable h i h i parameter, the lower bound of the delay 0; ha is also divided into two equal subintervals 0; h2a and h2a ; ha for the simplification, it is clear that the LK functional defined in Theorem 1 are more general than the ones in [3–5,8,15–18,25–28], etc. Therefore, both the information of delayed state xðt h2a Þ and xðt ha ατÞ can be taken into account. It is worthy mentioning that the delay decomposition is different, and the stability criterion of proposed approach is also different, and then the calculated MADB on hb may be different. Examples below can demonstrate the merit and reduced conservatism of the proposed approach. So, such variable decomposition method may lead to reduction of conservatism if being able to set a suitable dividing point with relation to α. About how to seek an appropriate α satisfying 0 o α o 1, such that we can obtain the MADB on hb for fixed lower bound ha of time delay. We give a simple algorithm as follows, such as [15,16,23]. Algorithm 1. (Maximizing hb for a fixed ha ) Step 1: For given hd , choose an upper bound on hb in the existing literatures, and then select this upper bound as the initial value hb ð0Þ of hb . Step 2: Set appropriate step lengths, hb;step and αstep for hb and α, respectively. Set k as a counter, and k ¼ 1. Let hb ¼ hb ð0Þ þ hb;step and the initial value α0 ¼ αstep . Step 3: Let α ¼ kαstep , if the LMIs in (9) and (10) are feasible, go to step 4; otherwise, go to step 5. Step 4: Let hb ð0Þ ¼ hb , α0 ¼ αstep , k ¼ 1, and hb ¼ hb ð0Þ þ hb;step , go to step 3. Step 5: Let k ¼ k þ 1, if kαstep o 1, then go to step 3; otherwise, stop. Here, we only provide an alternative way to maximize hb by using the simple iterative optimization procedure (optimally dividing scheme). Generally speaking, when less conservative stability results can be found for a certain group of α than other groups, the corresponding dividing scheme can be called the optimally dividing delay interval [22]. Moreover, how to find the MADB on hb and the corresponding optimal dividing parameter α, we can apply the Nelder–Mead simplex algorithm to obtain the above parameters, the reader is referred to the algorithm in [30]. 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 landscape, 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. However, what is the detailed relationship between hb and α for a fixed ha ?, or what function is the hb ¼ f ðha ; αÞ? Furthermore, how to solve the proposed criteria in a more effective way? They are still some challenging problems. Remark 4. In the proof of Theorem 1, Lemma 1 plays a key effect on the present results, especially, if the lower and upper term in integral is uncertain and required to be approximated with its lower or upper bound then use of (7) and (8) would be beneficial since the free variables M; N; Z are introduced. By employing some convex combination technologies with regard to different uncertain terms, a stability criterion is derived. Unlike the free-weighting matrix method, the slack matrix variables M; N; Z in Theorem 1 are not redundant. In the literatures, to achieve the delay-dependence of the proposed results, the Leibniz–Newton formula was employed and many free-weighting matrix variables were introduced. In fact, these free-weighting matrix variables are all redundant and can be eliminated, which was theoretically proved in [15]. Generally speaking, the integral inequality based results are carried out more efficiently as compared with the general free-weighting matrix based on the Newton–Leibniz formula [3,6,8,11,14,16,23]. For the case of unknown hd , that is, the delay is not differential or the time-derivative of delay is unknown, one may set Q τ ¼ 0; Rτ ¼ 0 in (12). Then Theorem 1 reduces to the following Corollary 1.
216
J. An et al. / ISA Transactions 53 (2014) 210–219
Corollary 1. In Case 2, for given scalars 0 o ha r hb , 0 o α o1, the system (1) is asymptotically stable if there exist real symmetric matrices " # " # Z i1 Z i2 S11 S12 P,Q i 4 0; Ri ; ði ¼ 1; 2; 3; 4Þ, Q τ Z 0; Rτ , 4 0, Z i ¼ , and any matrices M i ; N i ; ði ¼ 1; 2; 3; 4Þ with appropriate dimensions such n Z i3 n S22 that the LMIs in (9) and (10) with Q τ ¼ 0; Rτ ¼ 0 are feasible. Moreover, for comparison with the existing ones, we reduce to the following results when the lower bound of the delay is 0, that is, ha ¼ 0. In case that the interval team ½0; ha is missing, one may set Q 1 ¼ 0; Q 2 ¼ 0; Sij ¼ 0; R1 ¼ 0; R2 ¼ 0 in (12). According to the line of Theorem 1, we reduce to the following Corollary 2. Corollary 2. In Case 1, for given scalars 0 ¼ ha r hb ; 0 o α o 1; hd , the system (1) is asymptotically stable if there exist real symmetric matrices " # Z i1 Z i2 P,Q i 4 0; Ri ; ði ¼ 3; 4Þ, Q τ Z 0; Rτ , Z i ¼ , and any matrices M i ; Ni ; ði ¼ 1; 2; 3; 4Þ with appropriate dimensions such that the LMIs in (24) n Z i3 and (25) are feasible. Meanwhile, in Case 2, the system (1) is asymptotically stable if the LMIs in (24) and (25) with Q τ ¼ 0; Rτ ¼ 0 are feasible. 2 3 4 2τ 3 2τ 2 2τ 2 i r33 R3 r44 R4 7 6 P þ ∑ r Ri 6 7 i¼3 i 6 7 V^ p ¼ 6 2τ 3 740 1 n Q þ R 0 6 7 τ3 3 r3 3 4 5 2τ 4 1 n n Q þ R 4 4 τ4 r4 k
k
k
^ ði; kÞ : ¼ Ω ^ þΩ ^ þΩ ^ þΞ ^ o 0; ði ¼ 1; 2; k ¼ 3; 4Þ Ω 0 i 1 2 with " τ3 R3 þ ð1 hd ÞRτ "
½ M1
n
N1
Z1
τ4 R4 þ ð1 hd ÞRτ ½ M4 N 4 n
ð24Þ
ð25Þ
# Z 0; # Z 0:
Z2
where 2 6 ^ : ¼6 6 Ω 0 6 4 2 6 ^ 3 : ¼6 6 Ω 1 6 4 2 6 6
PA þ AT P þ Q τ þ Q 3
PAτ
0
0
n
ð1 hd ÞQ τ
0
0
n
n
Q4 Q3
0
n
n
n
Q4
M 1 þ N T1
0
n
N 1 N T1 þ τ3 M 2 þ τ3 M T2
τ3 M 2 þ τ3 NT2
n
n
R4 τ3 N 2 τ3 N T2
n
n
n
τ
3
k
0
7 0 7 7; 7 R4 5 R4
0
M 1 þ NT1 þ τ3 Z 12
n
N 4 NT4 þ τ 4 ðM 3 þ M T3 Þ
M T4 þ N4
n
n
N 1 NT1 þM 4 þ M T4 þ τ3 Z 13
n 4
T
n
!
^; ∑ τ2i Ri þ τ3 Rτ Γ
i¼3
4
^ : ¼Γ ^ Ω 2
T
4
n
!
^; ∑ τ2i Ri þ τRτ Γ
0
3
7 τ4 M 3 þ τ4 N T3 7 7 7; 7 0 5 T τ4 ðN 3 þ N 3 Þ
Γ^ : ¼ ½ A Aτ 0 0 ;
i¼3
Ξ^ 1 : ¼ τk Z^ ; ðk ¼ 3; 4Þ; k
3 3 3 Z^ : ¼ ½Z^ ij 44 ; Z^ ij 9
k
3
M 1 þ M T1 þ 3 Z 11
6 4
^ : ¼Γ ^ Ω 2
7 7 7; 7 5
M 1 þ M T1
^ : ¼6 Ω 6 1 4
3
k
8 Z 11 > > > > > > < Z 12 Z T12
> > > Z > > > 13 : 0 k
i ¼ 1; j ¼ 1 i ¼ 1; j ¼ 2 i ¼ 2; j ¼ 1 ;
4 4 4 Z^ : ¼ ½Z^ ij 44 ; Z^ ij 9
i ¼ 2; j ¼ 2 other
^ R 1 ðΦ ^ ÞT ; ðk ¼ 3; 4Þ; Ξ^ 2 : ¼ τ2k Φ k
3
^ : ¼ colf0; M ; N ; 0g; Φ 2 2
8 Z 23 > > > > ZT > > < 22 Z 21
> > > Z > > > 22 : 0 4
i ¼ 2; j ¼ 2 i ¼ 2; j ¼ 3 i ¼ 3; j ¼ 3 ; i ¼ 3; j ¼ 2 other
^ : ¼ colf0; M ; 0; N g: Φ 3 3
ð26Þ
Remark 5. It is worthy mentioning that the variable delay decomposition approach proposed in this paper can be applied to the stabilization and H1 control synthesis, and the corresponding stabilization and control criteria can be obtained. Moreover, we would like to point out that the proposed results can be extended to more general/practical systems such as time-delay systems with nonlinear perturbations as [25–27], fuzzy systems as [24,29,31,32,36], fault detection as [34,37] and disturbance system as [33], and the corresponding results will appear in the near future.
J. An et al. / ISA Transactions 53 (2014) 210–219
217
4. Numerical examples In this section, two well-known numerical examples are given to show the merit and effectiveness of the proposed method. Example 1. Consider the system (1) with the following system parameters: 0 0 0 1 A¼ ; Aτ ¼ 1 1 1 2 analytical
which has the analytical bound hmax
¼ π for the constant delay case (that is, τ_ ðtÞ 0).
In order to make a comparison, we employ the stability criteria given in [4,7,10,14,16,17] and in the results of this paper. When the derivative of delay is available, for the lower bound ha ¼ 0, the admissible upper bounds hb of the time-varying delay are obtained for different maximum delay derivative hd and the computational results are shown in Table 1. When the derivative of delay is unknown or does not exist, for varies lower bound ha , the MADB on hb are obtained and shown in Table 2. Moreover, when hd ¼ 0:3 and ha ¼ 1:0, the MADB computed by Shao and Han [11] and Qian [14] are 2.35 and 2.493, respectively, while the result in Theorem 1 is computed as 3.3873; When ha ¼ 5:0, using the stability criteria in Sun et al. [7], Liu [35] and Theorem 1 of this paper, the MADB on hb are hb ¼ 5:2970, hb ¼ 5:4661 and hb ¼ 5:5710, respectively. So, it can be seen from Tables 1 and 2 that the stability results obtained in this paper are less conservative that those in [4,7] and [10,11,14,16,17,35]. Table 1 MADB on hb with varying hd and ha ¼ 0 for Example 1. Method
hd
Shao [4] Sun et al. [7] Zhu et al.[16] Tang et al. [17] Corollary 2 (α ¼ 0:65) Corollary 2 (α ¼ 0:7)
0.1
0.3
0.5
0.8
1
5.4630 5.4764 5.4780 5.4940 7.8011 7.8401
2.2160 2.2160 2.2850 2.3070 3.5932 3.6962
1.1270 1.1272 1.2080 1.2330 1.7334 1.7655
0.8710 0.8714 1.0200 1.0440 1.2999 1.3123
0.8710 0.8714 1.0200 1.0440 1.2999 1.3123
Table 2 MADB on hb with varying ha and unknown hd for Example 1. Method
ha
Shao [4] Sun et al. [7] Zhu et al.[16] Tang et al. [17] Park et al. [10] Qian et al. [14] Liu [35] Corollary 1 (α ¼ 0:55) Corollary 1 (α ¼ 0:6)
0.3
0.5
0.8
1
2
1.0715 1.0716 1.2043 1.2246 1.2400 1.3500 1.4250 1.6351 1.6837
1.2191 1.2196 1.3429 1.3619 1.3800 1.4700 1.5694 1.7751 1.8120
1.4539 1.4552 1.5663 1.5838 1.6000 1.6800 1.7945 1.9978 2.0209
1.6169 1.6189 1.7228 1.739 1.7500 1.8100 1.9493 2.1519 2.1672
2.4798 2.4884 2.5600 2.5700 2.5800 2.6100 2.7674 2.9669 2.9552
Table 3 MADB on hb with given ha for different hd for Example 2. ha
Methods
hd ¼ 0:1
hd ¼ 0:3
hd ¼ 0:5
hd ¼ 0:9
1
Sun et al. [7] Liu et al. [18] Theorem 1 (α ¼ 0:65)
4.1935 4.4045 5.0950
3.0538 3.1208 3.6840
2.3058 2.3513 2.8418
1.9008 2.0921 2.5389
2
Sun et al. [7] Liu et al. [18] Theorem 1 (α ¼ 0:65)
4.4932 4.5729 5.0458
3.0129 3.1092 3.4397
2.5663 2.6987 3.0240
2.5663 2.6987 3.0240
3
Sun et al. [7] Liu et al. [18] Theorem 1 (α ¼ 0:65)
4.3979 4.5406 4.8247
3.3408 3.4186 3.6616
3.3408 3.4186 3.6616
3.3408 3.4186 3.6616
4
Sun et al. [7] Liu et al. [18] Theorem 1 (α ¼ 0:65)
4.1978 4.2367 4.5762
4.1690 4.2097 4.3788
4.1690 4.2097 4.3788
4.1690 4.2097 4.3788
5
Sun et al. [7] Liu et al. [18] Theorem 1 (α ¼ 0:65)
5.0275 5.0440 5.1453
5.0275 5.0440 5.1453
5.0275 5.0440 5.1453
5.0275 5.0440 5.1453
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J. An et al. / ISA Transactions 53 (2014) 210–219
Table 4 MADB on hb with given hd for different ha for Example 2. hd
Methods
ha ¼ 0
ha ¼ 1
ha ¼ 2
ha ¼ 3
ha ¼ 4
ha ¼ 5
0.5
Ramakrishnan and Ray [28] Hui et al. [26] Our results
2.2022 2.1471 2.3223
2.3912 2.5652 2.8418
2.9578 3.1124 3.0240
3.6384 3.7448 3.6616
4.3736 4.4369 4.3788
5.1463 5.1713 5.1453
0.9
Ramakrishnan and Ray [28] Hui et al. [26] Our results
1.8828 2.1377 2.2416
2.3585 2.5627 2.5389
2.9578 3.1085 3.0240
3.6384 3.7408 3.6616
4.3736 4.4340 4.3788
5.1463 5.1703 5.1453
Any hd
Ramakrishnan and Ray [28] Hui et al. [26] Our results
1.6654 2.1377 2.2416
2.1251 2.5627 2.4552
2.7113 3.1085 3.0312
3.3839 3.7408 3.7035
4.1136 4.4340 4.4203
5.1463 5.1703 5.1665
Table 5 MADB on hb with varying ha and unknown hd for Example 2. Method
ha
Shao [4] Sun et al. [7] Zhu et al. [16] Tang et al. [17] Park et al. [10] Shao and Han [11] Zhang et al. [27] Qian et al. [14] Liu et al. [18] Guo et al. [19] Ramakrishnan and Ray [25] Corollary 1 (α ¼ 0:55) Corollary 1 (α ¼ 0:6)
1
2
3
4
5
1.8737 1.9008 2.0273 2.0448 2.0600 2.0665 2.0665 2.0100 2.0921 2.1202 2.4454 2.4552 2.5449
2.5048 2.5663 2.5915 2.6051 2.6100 2.6181 2.6181 2.5680 2.6987 2.7199 2.9312 3.0312 3.0545
3.2591 3.3408 3.3010 3.3098 3.3100 3.3173 3.3173 3.3420 3.4186 3.4569 3.5087 3.7035 3.6939
4.0744 4.1690 4.0855 4.0877 4.0900 4.0905 4.0905 4.1730 4.2097 4.2574 4.1509 4.4203 4.4026
– 5.0275 – – – – – 5.0360 5.0440 5.0976 – 5.1665 5.1557
Table 6 Comparison with MADB on hb with varying hd (ha ¼ 0). Method
hd
Park and Ko [5] J.-H. Kim [12] Corollary 2 (α ¼ 0:65) Corollary 2 (α ¼ 0:7)
0.0
0.05
0.10
0.5
3.0
4.47 4.97 5.11 5.18
4.01 4.35 4.46 4.54
3.66 3.86 3.96 4.08
2.33 2.33 2.48 2.58
1.86 1.86 2.33 2.34
Example 2. Consider system (1) with A¼
2
0
0
0:9
;
Aτ ¼
1
0
1
1
analytical
which has the analytical bound hmax
¼ 6:17258 for the constant delay (when τ_ ðtÞ 0).
As with Example 1, here we calculate the MADB that guarantees the asymptotic stability of system (1) using the methods proposed in [7,18], and Theorem 1 in this paper. For unknown maximum derivative hd of the time delay and varying lower bound ha of the time delay, the MADB on hb are obtained and shown in Table 3. For the Table, it can be seen that our method yields less conservative than [7,18]. Moreover, compare to the existing ones in [26,28], computational results in this paper are obtained and these are listed in Table 4 for various case. Note that our results in Table 4 use Theorem 1 (α ¼ 0:65) when hd are known and ha 4 0; Corollary 1 (α ¼ 0:55) when hd are unknown and ha 4 0; Corollary 2 (α ¼ 0:6) when ha ¼ 0. It is seen that the stability criteria in [26] and Theorem 1 in this paper are not covered by each other, although they are less conservative than those in [28]. However, the criteria in [26] are complex and involve more matrix variables than ours. Meanwhile, for the case of ha ¼ 0 and unknown hd (any hd ), from this example, it seems that our results (Corollaries 1 and 2) are less conservative than those in [26,28] when the lower bound of the varying delay, i.e., ha is small. For unknown hd , the MADB on hb , which guarantee the asymptotic stability of system (1) for given lower bounds ha , are listed in Table 5 for Corollary 1 and the exiting ones in [4,7,10,11,14,16–19,25,27], respectively. Moreover, comparing to the existing ones in [5,12], we assume that ha ¼ 0, and can calculate the result by Corollary 2 of this paper and those in [5,12], which is shown in Table 6. From Tables 5 and 6, it also can be seen that the stability results obtained in the paper are less conservative than those in [4,5,7,10–12,14,16–19,25,27]. It should be noted that hd ¼ 0 does not mean a constant delay (that is, τ_ ðtÞ 0), but it means a time-varying delay with τ_ ðtÞ r 0; 8 t Z 0. Also, our result gives less conservativeness for a relatively small value of hd .
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5. Conclusion In this paper, delay-fractional-dependent stability problem for linear systems with interval time-varying delay is investigated. Through employing a novel delay-dependent LK functional via variable delay interval dividing technique, the proposed criterion provides better results since approximation of the uncertain delay factors in the time-derivative of LK functional has been avoided, and making full use of the delay information and the relationship among xðtÞ; xðt τðtÞÞ; xðt ðha =2ÞÞ; xðt ha Þ, xðt ha ατÞ and xðt hb Þ with τ 9hb ha ,0 o α o 1, and thus some less conservative stability conditions are obtained in the form of LMIs. Finally, by two well-known examples, we showed the usefulness and less conservatism of the proposed approach. Moreover, the proposed approach is simple and may easily be extended to robust stability, stabilization problems for linear time-delay systems and the systems with nonlinear perturbations. Acknowledgments This work is supported in part by the Natural Science Foundation of China under Grants 61173107 and 61370097, and in part by the State Scholarship Fund of China Scholarship Council (CSC) under Grant 201206135001, and in part by the Planned Science and Technology Project of Hunan Province under Grant 2012FJ4130. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34] [35] [36] [37] [38]
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