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An extended reciprocally convex matrix inequality and its application to stability analysis of systems with additive time-varying delays
An extended reciprocally convex matrix inequality and its application to stability analysis of systems with additive time-varying delays
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An extended reciprocally convex matrix inequality and its application to stability analysis of systems with additive time-varying delays Jianmin Jiao, Rui Zhang PII: DOI: Reference:
S0016-0032(19)30869-5 https://doi.org/10.1016/j.jfranklin.2019.11.065 FI 4300
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Journal of the Franklin Institute
Received date: Revised date: Accepted date:
28 November 2018 31 July 2019 25 November 2019
Please cite this article as: Jianmin Jiao, Rui Zhang, An extended reciprocally convex matrix inequality and its application to stability analysis of systems with additive time-varying delays, Journal of the Franklin Institute (2019), doi: https://doi.org/10.1016/j.jfranklin.2019.11.065
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An extended reciprocally convex matrix inequality and its application to stability analysis of systems with additive time-varying delays Jianmin Jiao ∗ , Rui Zhang Institute of Mathematics and Information Science, Baoji University of Arts and Sciences, Baoji 721013, China *Corresponding author : [email protected] (Jianmin Jiao)
Abstract This paper is concerned with the stability analysis of systems with additive time-varying delays. First, an extended reciprocally convex matrix inequality is presented, which is a generalization of the existing reciprocally convex matrix inequalities. Second, combining the proposed matrix inequality with the Wirtinger-based integral inequality, a new stability criterion of systems with additive time-varying delays is proposed. Meanwhile, an improved stability criterion of systems with a single time-varying in a range is also obtained. Finally, two numerical examples are employed to illustrate the advantage of the obtained theoretical results. Keywords: Time-delay systems; stability; reciprocally convex matrix inequality; Lyapunov-Krasovskii functional(LKF); linear matrix inequality(LMI) 1. Introduction It is well known that time-delay frequently occurs in many practical systems, such as biological systems, chemical systems, electrical systems, network control systems and so on, it is regarded as a major factor causes of the instability, oscillations, and poor performance. So, the study on the stability analysis of time-delay systems is important for theory and practical application. A great number of research and results are dedicated to Preprint submitted to Elsevier
December 3, 2019
the development of time-delay systems over these years [1-7]. For stability analysis of time-delay systems, an efficient method is to construct the delaydependent Lyapunov-Krasovskii functions (LKFs), the main objective is to obtain maximum delay bounds guaranteeing the asymptotically stability of the concerned systems. Since the trial on obtaining tight lower bound for integral quadratic functions which exist in the derivatives of LKFs has been a key step in reducing the conservatism of the stability criteria, various mathematical approaches have been proposed, such as model transformations approach [2], free-weighting-matrix approach [3], delay partition approach [4], integral inequalities approach [5-7], and so on. Actually, a combination of several approaches is used in most stability analysis results. The state model of time-delay systems contain one single delay is investigated commonly and frequently in the past. In [8], a new model for continuous system with additive time-varying delays was proposed. The proposed new model has a strong application background in networked control and long-range control. Take a state-feedback network control for example, because the physical plant, sensor, controller and actuator are located in different places, so in the transmit process, signals are transmitted from one device to another device, time delays will be appear. Among these delays, there are two network-induced ones: one is used to represent the delay from sensor to controller, and the other is used to represent the delay from controller to actuator. Thus, the two additive time delays will appear in the closed-loop system. The properties of these two delays may not be identical due to the network transmission conditions. It is not reasonable to combine them together, hence they should be treated separately. Therefor, it is significant to consider the systems with two additive time-varying delays. In [8,9], Lam et.al. have initially investigated the stability for this kind of system, and some stability criteria were obtained. In [10], by constructing new LKF, a robust stability criterion for uncertain systems with additive time-varying delays was given. In [11], the author gave a stability criterion for singular systems with additive time-varying delays by using free-weighting-matrix approach. By taking more information of the additive time-varying delays in LKFs into account, some new stability criteria were established in [12,13]. In [14], an improved stability criterion for Lurie systems with additive time-varying de2
lays was obtained by taking advantage of integral inequality. Recently, by constructing augmented LKFs, some new less conservative stability criteria are presented in [15,16]. In recent years, estimating integral terms directly via integral inequalities gradually becomes more popular, especially the Jensen integral inequality has been widely used [5]. Very recently, Writinger-based integral inequality that provides a more tighter lower bound than the Jensen inequality has been proposed in [17, 18]. Further, several works have been done to improve the Wirtinger-based integral inequality as more general inequality, such as auxiliary-function-based integral inequality [19], free-matrix-based integral inequality [20,21], Bessel-Legendre integral inequality [22], and polynomialsbased integral inequality [23]. Although the inequalities in above literatures have shown a great interest for constant time-delay systems, however, their application to time-varying delays reveals additional difficulties related to the non-convexity of the resulting terms. To address these terms, the reciprocally convex approach is usually employed. The reciprocally convex combination lemma was proposed in [24]. In [25], an improved version of reciprocally convex combination lemma was provided, and it has been extended in [2632]. However, it is worth stressing that these reciprocally convex matrix inequalities in [24-32] are only applied in the case which the delay interval is divided into two subintervals. In the study of stability for systems with additive time-varying delays, the delay interval often need to be divided into m(m > 2) subintervals, the existing reciprocally convex matrix inequalities cannot be applied directly. Therefore, it is necessary to extend the classical reciprocally convex matrix inequalities to the case of m > 2 subintervals in the delay interval. Motivated by the above of the observation, in this paper, the problem of stability analysis for continuous systems with additive time-varying delays is investigated. The main contributions of this paper are as follow: (1) An extended reciprocally convex matrix inequality is derived. (2) By combining the extended reciprocally convex matrix inequality with the Writinger-based integral inequality, a new stability criterion of continuous systems with additive time-varying delays is proposed. Meanwhile, an improved stability criterion of continuous systems with a single time-varying 3
in a range is also obtained. Notation: Throughout this paper, Rn denotes the n-dimensional Euclidean space, while Rm×n refers to the set of all real matrices with m rows and n columns. The superscripts T and −1 mean the transpose and inverse of a matrix, respectively. P > 0 (P < 0) means P is a real symmetric and positive-definite (negative-definite) matrix. In is the n-dimensional identity matrix. Symmetric term in a symmetric matrix is denoted by ∗, and Sym{X} = X + X T . 2. Problem formulation Consider the following continuous system with additive time-varying delays: ( x(t) ˙ = Ax(t) + Bx(t − d1 (t) − d2 (t)), t > 0, (1) x(t) = ϕ(t), t ∈ [−(d1 + d2 ), 0], where x(t) ∈ Rn is the state vector, ϕ(t) ∈ Rn is the continuous initial function, A, B ∈ Rn×n are constant matrices. The time delays d1 (t) and d2 (t) are time-varying differentiable functions that satisfy 0 6 di (t) 6 di , d˙i (t) 6 µi (i = 1, 2),
(2)
where di and µi (i = 1, 2) are known positive constants. This paper aims to derive a delay-dependent stability criterion of system (1) with time-varying delays d1 (t) and d2 (t) satisfying (2). In order to achieve the objective , the following lemma is reviewed. Lemma 1 ([17] Wirtinger-based integral inequality) For a n × n symmetric matrix R > 0, scalars a and b with a < b, and continuously differentiable function x : [a, b] → Rn , the following integral inequality holds Z
a
b
1 x˙ (s)Rx(s)ds ˙ > b−a T
where ϕ1 = x(b) − x(a), ϕ2 = x(b) + x(a) −
2 b−a
Rb a
ϕ1 ϕ2
x(s)ds. 4
T
R 0 0 3R
ϕ1 ϕ2
,
(3)
3. An extended reciprocally convex matrix inequality In this part, an extended reciprocally convex matrix inequality is proposed, shown as follows. Lemma 2 (extended reciprocally convex matrix inequality) For any real scalars αi > 0(i = 1, 2, · · ·, m), n × n symmetric matrices Ri > 0(i = 1, 2, · · ·, m), and any mn × mn matrix M , the following matrix inequality holds 1 R 0 ··· 0 α1 1 1 0 R ··· 0 α2 2 . . . . . . . . . . . . 0
···
0
1 R αm m
α1 R1−1 0 ··· 0 0 α2 R2−1 · · · 0 T T (4) > −M − M − M M. .. .. .. . . . . . . −1 0 0 · · · αm Rm 1 R 0 ··· 0 α1 1 1 0 R ··· 0 α2 2 Proof Define R = , from Schur comple. . . . . . . . . . . . 0
ment, we have
···
0
R M T −1 ∗ M R M
1 R αm m
> 0.
Pre-multiplying and post-multiplying above inequality by its transpose, respectively, we have
Imn Imn
and
R + M + M T + M T R−1 M > 0, which implies (4). This completes the proof. Remark 1 If taking m = 2, α1 = α ∈ (0, 1), α2 = 1 − α and M = M11 M12 , inequality (4) reduces to the following inequality (5) ∗ M22 # " 1 T T R 0 M11 + M11 M12 + M21 α 1 >− 1 T 0 R ∗ M22 + M22 1−α 2 5
−α
T M11 T M12
R1−1
M11 M12
− (1 − α)
T M21 T M22
R2−1
M21 M22
.
(5) The existing reciprocally convex matrix inequalities can be directly obtained via the proposed matrix inequality (5). For examples, setting M11 = −R1 , M12 = −S1 , M21 = −S2T , M22 = −R2 , we can obtain Lemma 4 of [27]. Furthermore, R S T setting M11 = M22 = −R, M12 = −S, M21 = −S , if > 0, which ∗ R leads to R − SR−1 S T > 0 and R − S T R−1 S > 0, then we can obtain popular reciprocally convex combination lemma given in [24]. Remark 2 Combining the reciprocally convex matrix inequality with integral inequality approach, some stability criteria of linear systems with single delay were established in [24-31]. However, in the study of stability problem for systems with additive delays, the delay interval often need to be divided into multiple subintervals. Thus, the above reciprocally convex matrix inequalities cannot be directly applied. In order to solve this problem, Lemma 2 gives an extended reciprocally convex matrix inequality, which can be applied when the number of subintervals in the delay interval is more than two. Remark 3 Compared with those proofs in [24-27], Lemma 2 is very simple and straightforward in which the Schure complement is simply used. Remark 4 It is worth pointing out that unlike the existing reciprocally convex matrix inequalities, scalars α1 , α2 , · · · , αm in Lemma 2 are not need to P satisfy the constraint condition m i=1 αi = 1. Thus, the new matrix inequality (4) may has a wider application. 4. Stability analysis of systems with additive time-varying delays Before introducing the main result, following notations are defined for simplicity ei = [0n×(i−1)n In 0n×(9−i)n ](i = 1, 2, · · · , 9), e˜i = [0n×(i−1)n In 0n×(8−i)n ](i = 1, 2, · · · , 8), e¯i = [0n×(i−1)n In 0n×(7−i)n ](i = 1, 2, · · · , 7),
6
ξ(t) =[xT (t) xT (t − d1 (t)) xT (t − d1 (t) − d2 (t)) xT (t − d1 (t) − d2 ) Z t Z t−d1 (t) 1 1 T T x (t − d1 − d2 ) x (s)ds xT (s)ds d1 (t) t−d1 (t) d2 (t) t−d1 (t)−d2 (t) Z t−d1 (t)−d2 (t) Z t−d1 (t)−d2 1 1 T xT (s)ds]T , x (s)ds d2 − d2 (t) t−d1 (t)−d2 d1 − d1 (t) t−d1 −d2 " # x(t) − x(t − d1 (t)) Rt η1 (t) = , x(t) + x(t − d1 (t)) − d12(t) t−d1 (t) x(s)ds " # x(t − d1 (t)) − x(t − d1 (t) − d2 (t)) R t−d (t) η2 (t) = , x(t − d1 (t)) + x(t − d1 (t) − d2 (t)) − d22(t) t−d11(t)−d2 (t) x(s)ds " # x(t − d1 (t) − d2 (t)) − x(t − d1 (t) − d2 ) R t−d (t)−d (t) η3 (t) = , x(t − d1 (t) − d2 (t)) + x(t − d1 (t) − d2 ) − d2 −d2 2 (t) t−d11(t)−d22 x(s)ds " # x(t − d1 (t) − d2 ) − x(t − d1 − d2 ) R t−d (t)−d η4 (t) = . x(t − d1 (t) − d2 ) + x(t − d1 − d2 ) − d1 −d2 1 (t) t−d11−d2 2 x(s)ds Now, we provide a stability criterion for the system (1) with time-varying delays d1 (t) and d2 (t) satisfying (2). Theorem 1 For given scalars di > 0, µi > 0(i = 1, 2), the system (1) with time-varying delays d1 (t) and d2 (t) satisfying (2) is asymptotically stable if P1 P2 there exist 2n × 2n matrix > 0, n × n matrices Qi > 0(i = ∗ P3 1, 2, 3, 4) and R > 0, and 8n × 8n matrix M , such that LMIs (6)-(9) hold. Ξ1 + Ξ2 + Ξ3 + d1 Φ1 + d2 Φ2 E T M T E1T E T M T E2T ˜ ∗ − d11 R 0 (6) < 0, ˜ ∗ ∗ −1R d2
Ξ1 + Ξ2 + Ξ3 + d1 Φ1 + d2 Φ3 E T M T E1T E T M T E3T ˜ ∗ − d11 R 0 ˜ ∗ ∗ − d12 R Ξ1 + Ξ2 + Ξ3 + d1 Φ4 + d2 Φ2 E T M T E4T E T M T E2T ˜ ∗ − d11 R 0 ˜ ∗ ∗ − d12 R 7
(7)
(8)
< 0, < 0,
Ξ1 + Ξ2 + Ξ3 + d1 Φ4 + d2 Φ3 E T M T E4T E T M T E3T ˜ ∗ − d11 R 0 < 0, 1 ˜ ∗ ∗ − d2 R
(9)
where T T Ξ1 = Sym{eT 1 (P1 A + P2 )e1 + e1 P1 Be3 − e1 P2 Be5 }, T T Ξ2 = eT 1 (Q1 + Q2 + Q3 + Q4 )e1 − (1 − µ1 )e2 Q1 e2 − (1 − µ1 − µ2 )e3 Q2 e3 − T (1 − µ1 )eT 4 Q3 e4 − e5 Q4 e5 , T T T T T T Ξ3 = (d1 + d2 )(eT 1 A RAe1 + e3 B RBe3 + Sym{e1 A RBe3 + E M E}), T T T T T Φi = Sym{e1 (A P2 +P3 )ei+5 + e3 B P2 ei+5 − e5 P3 ei+5 }(i = 1, 2, 3, 4), e1 − e2 e1 + e2 − 2e6 e2 − e3 e2 + e3 − 2e7 , E= e − e 3 4 e + e − 2e 3 4 8 e4 − e5 e4 + e5 − 2e9 e˜2i−1 Ei = (i = 1, 2, 3, 4), e˜2i R 0 ˜ R= . 0 3R Proof Construct the following LKF candidate V (t) =
3 X
Vi (t),
(10)
i=1
where V1 (t) = V2 (t) = + V3 (t) =
" Z
x(t) Rt x(s)ds t−d1 −d2 t
t−d1 (t) Z t
P1 P2 ∗ P3 Z t T x (s)Q1 x(s)ds +
"
x(t) Rt x(s)ds t−d1 −d2
xT (s)Q4 x(s)ds,
t−d1 −d2
x˙ T (s)Rx(s)dsdθ. ˙
t+θ
8
#
xT (s)Q2 x(s)ds
t−d1 (t)−d2 (t) Z t
xT (s)Q3 x(s)ds +
1 (t)−d2 Zt−d Z t 0
−d1 −d2
#T
,
Then, the time derivatives of functionals Vi (t)(i = 1, 2, 3) along the trajectory of system (1) are given by V˙ 1 (t) =ξ T (t)(Ξ1 + d1 (t)Φ1 + d2 (t)Φ2 + (d2 − d2 (t))Φ3 + (d1 − d1 (t))Φ4 )ξ(t). (11) T ˙ (12) V2 (t) 6 ξ (t)Ξ2 ξ(t). Z t x˙ T (s)Rx(s)ds ˙ V˙ 3 (t) =(d1 + d2 )x˙ T (t)Rx(t) ˙ − t−d1 −d2 t
T
=(d1 + d2 )x˙ (t)Rx(t) ˙ − −
Z
t−d1 (t)−d2 (t)
t−d1 (t)−d2
Z
T
t−d1 (t)
x˙ (s)Rx(s)ds ˙ −
T
x˙ (s)Rx(s)ds ˙ − Z
t−d1 (t)−d2
Z
t−d1 (t)
x˙ T (s)Rx(s)ds ˙
t−d1 (t)−d2 (t)
x˙ T (s)Rx(s)ds, ˙
t−d1 −d2
(13) by using Lemma 1 to estimate the upper bounds of the last four integral terms on the right side of equality (13), one has 1 T ˜ 1 T ˜ η1 (t)Rη1 (t) − η (t)Rη2 (t) d1 (t) d2 (t) 2 1 1 ˜ 3 (t) − ˜ 4 (t) − η3T (t)Rη η T (t)Rη d2 − d2 (t) d1 − d1 (t) 4
V˙ 3 (t) 6 (d1 + d2 )x˙ T (t)Rx(t) ˙ −
= (d1 + d2 )x˙ T (t)Rx(t) ˙ T ˜ − d11(t) R 0 0 0 η1 (t) 1 ˜ 0 − d2 (t) R 0 0 η (t) + 2 1 ˜ 0 0 − d2 −d2 (t) R 0 η3 (t) ˜ η4 (t) 0 0 0 − d1 −d1 1 (t) R
9
η1 (t) η2 (t) η3 (t) η4 (t) (14)
.
For matrix M , it follows Lemma 2 that ˜ − d11(t) R 0 0 0 1 ˜ 0 − d2 (t) R 0 0 1 ˜ 0 0 0 − d2 −d2 (t) R ˜ 0 0 0 − d1 −d1 1 (t) R ˜ −1 d1 (t)R 0 0 −1 ˜ 0 d (t) R 0 2 +M T ˜ −1 0 0 (d2 − d2 (t))R 0
0
9
6 M + MT
0 0 0 ˜ −1 (d1 − d1 (t))R
M.
(15)
From (14) and (15), one can obtain ˜ −1 E1 M E + d2 (t)E T M T E T R ˜ −1 E2 M E V˙ 3 (t) 6 ξ T (t)(Ξ3 + d1 (t)E T M T E1T R 2 ˜ −1 E3 M E + (d2 − d2 (t))E T M T E3T R ˜ −1 E4 M E)ξ(t). + (d1 − d1 (t))E T M T E4T R
(16)
Then, combing (11),(12) with (16), we can get ˜ −1 E1 M E) V˙ (t) 6 ξ T (t)(Ξ1 + Ξ2 + Ξ3 + d1 (t)(Φ1 + E T M T E1T R ˜ −1 E2 M E + d2 (t)(Φ2 + E T M T E T R 2 T
˜ −1 E3 M E) + (d2 − d2 (t))(Φ3 + E M T E3T R ˜ −1 E4 M E))ξ(t). + (d1 − d1 (t))(Φ4 + E T M T E4T R Based on convex combination technique, V˙ (t) < 0 holds if the following four inequalities hold ˜ −1 E1 M E)+d2 (Φ2 +E T M T E T R ˜ −1 E2 M E) < 0, Ξ1 +Ξ2 +Ξ3 +d1 (Φ1 +E T M T E1T R 2 ˜ −1 E1 M E)+d2 (Φ3 +E T M T E3T R ˜ −1 E3 M E) < 0, Ξ1 +Ξ2 +Ξ3 +d1 (Φ1 +E T M T E1T R ˜ −1 E2 M E) < 0, ˜ −1 E4 M E)+d2 (Φ2 +E T M T E T R Ξ1 +Ξ2 +Ξ3 +d1 (Φ4 +E T M T E4T R 2 ˜ −1 E4 M E)+d2 (Φ3 +E T M T E T R ˜ −1 E3 M E) < 0, Ξ1 +Ξ2 +Ξ3 +d1 (Φ4 +E T M T E4T R 3 which are respectively guaranteed by (6)-(9) based on Schur complement. Thus, if (6)-(9) hold, then V˙ (t) < 0 holds for any ξ(t) 6= 0, which implies 10
that system (1) with the delays satisfying (2) is asymptotically stable. The proof is completed. Remark 5 By splitting the whole delay interval [−(d1 + d2 ), 0] into four subintervals [−(d1 +d2 ), −(d1 (t)+d2 )], [−(d1 (t)+d2 ), −(d1 (t)+d2 (t))], [−(d1 (t)+ d2 (t)), −d1 (t)] and [−d1 (t), 0], we construct the LKF (10) which depend on the subintervals. Thus, the interactions among the vectors of x(t), x(t − d1 (t)), x(t − d1 (t) − d2 (t)), x(t − d1 (t) − d2 ) and x(t − d1 − d2 ) were sufficiently Rt R t−d (t) considered. Furthermore, by taking the states t−d1 (t) x(s)ds, t−d11(t)−d2 (t) x(s)ds, R t−d (t)−d R t−d1 (t)−d2 (t) x(s)ds and t−d11−d2 2 x(s)ds as augmented variables, the stabilt−d1 (t)−d2 ity criterion in Theorem 1 utilizes more information on state variables, and these terms are not employed in previous literatures. Therefore, Theorem 1 is expected to less conservative than some results in the literatures, which will be illustrated through numerical examples in the next section. Remark 6 When dealing with the derivative of the LKF (10), the general approach is to use Jensen’s or Wirtinger-based integral inequalities to scale the integral terms firstly, then employed convex optimization approach to handle time-varying delays existing in denominator of the formula. It should be pointed out that the existing reciprocally convex matrix inequalities in [24-32] are only in the case when the delay interval is divided into two subintervals. In this paper, the delay interval is divided into four subintervals, so the existing reciprocally convex matrix inequalities can not be applied directly. We introduced an extended reciprocally convex matrix inequality to deal with the time derivative on double integral terms directly. When d1 (t) or d2 (t) is a constant, the system (1) reduces to linear system with a single time-varying delay in a range. Without loss generality, suppose that ( x(t) ˙ = Ax(t) + Bx(t − d(t)), t > 0, (17) x(t) = ϕ(t), t ∈ [−h2 , 0], time-varying delay d(t) satisfying ˙ 6 µ, h1 6 d(t) 6 h2 , d(t) where h1 , h2 and µ are constants. We choose the following LKF candidate [27] 11
(18)
T
V (t) = ζ (t)P ζ(t) + + h1 Z
Z
0
Z
Z
t T
x (s)Q1 x(s)ds +
t−h1
xT (s)Q2 x(s)ds
t−h2
t−h1
t
Z
x˙ T (s)Z x(s)dsdθ ˙ + (h2 − h1 )
−h1 t+θ −h1 Z −h1 Z t
Z
−h1
−h2
Z
t
x˙ T (s)Rx(s)dsdθ ˙
t+θ
x˙ T (u)U x(u)dudθds, ˙ t+θ s −h2 R t−h1 T Rt x (s)ds]T , P > 0, Qi > 0(i = xT (s)ds where ζ(t) = [xT (t) t−h2 t−h1 1, 2, 3), Z > 0, R > 0 and U > 0. Then, along a similar line as in the derivation of Theorem 1 we can derive the following stability criterion for system (17). Theorem 2 For given scalars h1 , h2 and µ, system (17) with time-varying delay d(t) satisfying (18) is asymptotically stable if there exist 3n×3n matrix P11 P12 P13 ∗ P22 P23 > 0, n×n matrices Qi > 0(i = 1, 2, 3), Z > 0, R > 0, U > ∗ ∗ P33 0, and 4n × 4n matrix M , such that LMIs (19)-(20) hold. +
Γ + (h2 − h1 )Ψ1 E¯ T M T E¯1T ¯ − U¯ ∗ −R Γ + (h2 − h1 )Ψ2 E¯ T M T E¯2T ¯ ∗ −R
< 0,
(19)
< 0,
(20)
where 1 2 T 2 T 2 T Γ = e¯T e1 + 1 (Q1 +Q3 +h1 A ZA+(h2 −h1 ) A RA+ 2 (h2 −h1 ) A U A−4Z)¯ T T 2 T 2 T e¯2 (−Q1 + Q2 − 4Z + 2U )¯ e2 + e¯3 (−(1 − µ)Q3 + h1 B ZB + (h2 − h1 ) B RB + 1 2 T (h2 −h1 ) B U B+2U )¯ e3 +¯ eT e4 +¯ eT e5 +¯ eT e6 +¯ eT e7 + 4 (−Q2 )¯ 5 (−12Z)¯ 6 (10U )¯ 7 (−2U )¯ 2 T T T 2 T Sym{¯ e1 (P11 A + P12 )¯ e1 + e¯1 (−P12 + P13 − 2Z)¯ e2 + e¯1 (P11 B + h1 A ZB + (h2 − 1 2 T 2 T T T h1 ) A RB + 2 (h2 − h1 ) A U B)¯ e3 + e¯1 (−P13 )¯ e4 + e¯T 1 (h1 A P12 + h1 P22 + T T 6Z)¯ e5 +¯ eT e3 +¯ eT e5 +¯ eT e6 +¯ eT e5 + 2 (2U )¯ 2 (−h1 P22 +h1 P23 +6Z)¯ 2 (−4U )¯ 3 (h1 B P12 )¯ T T T T T ¯ ¯ e¯3 (−6U )¯ e6 + e¯3 (2U )¯ e7 + e¯4 (−h1 P23 )¯ e5 + E M E}, T T Ψi = Sym{¯ eT (A P + P )¯ e + e¯T ei+5 + e¯T ¯i+5 − 13 23 i+5 1 2 (−P23 + P33 )¯ 3 B P13 e T e¯4 P33 e¯i+5 }(i = 1, 2),
12
e¯2 − e¯3 e6 e¯ + e¯3 − 2¯ E¯ = 2 , e¯3 − e¯4 e¯3 + e¯4 − 2¯ e7 e¯2i−1 E¯i = (i = 1, 2), e¯2i R 0 ¯ R= , 0 3R U 0 ¯ U= . 0 3U Remark 7 When µ is unknown, setting Q3 = 0 in Theorem 2 we can obtain a delay-rate-independent stability criterion for system (17). In addition, ¯ − U¯ −S2T −R it is easy to derive that setting Q3 = 0 and M = ¯ in The−S1 −R orem 2 yield precisely the Theorem 8 in [27]. Hence, compared with Theorem 8 in [27], the Theorem 2 in this paper overcomes the conservativeness. Remark 8 The obtained results in this paper are formulated in terms of LMIs, they can be easily solved by using any LMI toolbox, e.g. Matlab or Scilab. Remark 9 More recently, generalized integral inequalities, tighter than Wirtinger-based integral inequality, have been developed. For example, auxiliary function-based inequality [19], free-matrix-based inequality [20,21], and Bessel-Legendre inequality [22]. The further improved stability criteria can be obtained by combing those inequalities and the proposed extended reciprocally convex matrix inequality, while it will need more computational burden. 5. Numerical examples To illustrate the effectiveness and the less conservatism of obtained results in this paper, two numerical examples are given in this section. Example 1 Consider the system (1) with the following parameters: A=
−2 0 0 −0.9
,
13
B=
−1 0 −1 −1
.
The delay derivatives µ1 and µ2 are assumed as 0.1 and 0.8, respectively. Our purpose is to find the upper bounded d1 of delay d1 (t), or d2 of delay d2 (t), when the other is known, below which the system is asymptotically stable. The calculation results obtained by the methods in [8-16] and Theorem 1 in this paper are listed in Table 1 and Table 2 under different cases. From Table 1 and Table 2, we can see that by using Theorem 1, the results we obtained are less conservative than those of [8-16]. Table 1. Comparison of delay upper bound d2 for various d1
d1 [8] [9] [10] [11] [12] [13] [14] [15] [16] Theorem 1
Example 2
1.0 0.415 0.512 0.595 0.873 0.982 0.983 0.988 0.999 1.163 1.233
1.2 0.376 0.406 0.462 0.673 0.782 0.849 0.854 0.972 0.965 1.035
1.5 0.248 0.283 0.312 0.452 0.482 0.671 0.675 0.680 0.669 0.752
Consider the system (17) with the following parameters: −2 0 −1 0 A= , B= . 0 −0.9 −1 −1
Assuming that the delay derivative µ is unknown. The maximal upper bounds of h2 with respect to various h1 calculated by [17, 24, 27] and Theorem 2 (Q3 = 0) in this paper are listed in Table 3. We can see that stability condition presented in this paper is less conservative than those in [17, 24, 27]. The obtained results by using Theorem 2 are better than compared ones since in which the extended reciprocally convex inequality is used. Remark 10 As a matter of fact in example 1 and example 2, the number of decision variables is 68.5n2 + 3.5n and 23n2 + 4n, respectively. To get higher bounds with less decision variables is our next research aim. 14
Table 2. Comparison of delay upper bound d1 for various d2
d2 [8] [9] [10] [11] [12] [13] [14] [15] [16] Theorem 1
0.3 1.324 1.453 1.531 1.808 1.682 2.329 2.337 1.880 1.875 2.441
0.4 1.039 1.214 1.313 1.593 1.582 2.065 2.069 1.779 1.773 2.145
0.5 0.806 1.021 1.140 1.424 1.482 1.836 1.842 1.675 1.671 1.912
Table 3. Comparison of delay upper bound h2 for various h1
h1 [24] [17] [27] Theorem 2
0 1.868 2.113 2.213 2.233
0.4 1.882 2.179 2.256 2.264
0.7 1.953 2.237 2.286 2.289
6. Conclusion In this paper, an extended reciprocally convex matrix inequality has been introduced, which is a generalization of the existing reciprocally convex matrix inequality. Further, a stability criterion with less conservatism for systems with additive time-varying delays has been established by combining the proposed matrix inequality and Writinger-based integral inequality. The advantages of the proposed matrix inequality and the corresponding criterion have been shown via a numerical example. In the future, our main work include two aspects: (1) how to reduce the complexity of the proposed reciprocally convex matrix inequality; (2) trying to apply proposed method in this paper to study other control problem of systems with additive time-varying delays, such as stabilization controller design[33], robust filtering analysis 15
[34], H∞ control problem [35], and so on. 7. Acknowledgment This work was supported by the National Natural Science Foundation of China grants 31600299 and 11626035, and Research Foundation of Baoji University of Arts and Sciences grant ZK16010. References [1] E. Fridman. Introduction to Time-Delay Systems: Analysis and Control. Birkhouser, Boston, 2014 . [2] C. Briat. Linear Parameter-Varying and Time-Delay Systems: Analysis, Observation, Filtering & Control. Springer-Verlag, London, 2015. [3] M. Wu, Y. He, J.H. She, G.P. Liu. Delay-dependent criteria for robust stability of time-varying delay systems. Automatica 40(8) (2004) 14351439. [4] K. Gu. A further refinement of discretized Lyapunov functional method for the stability of time-delay systems. International Journal of Control 74(10) (2001) 967-976. [5] K. Gu, J. Chen, V.L. Kharitonov. Stability of time-delay systems. Springer Science & Business Media, 2003. [6] O.M. Kwon, M.J Park, J.H. Park, S.M. Lee, E.J. Cha. Improved results on stability of linear systems with time-varying delays via Wirtingerbased integral inequality. Journal of the Franklin Institute 351(12) (2014) 5386-5398. [7] T.H. Lee, J.H. Park. Improved stability conditions of time-varying delay systems based on new Lyapunov functionals. Journal of the Franklin Institute 355(3) (2018) 1176-1191.
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19
Journal Pre-proof
An extended reciprocally convex matrix inequality and its application to stability analysis of systems with additive time-varying delays Jianmin Jiao, Rui Zhang PII: DOI: Reference:
S0016-0032(19)30869-5 https://doi.org/10.1016/j.jfranklin.2019.11.065 FI 4300
To appear in:
Journal of the Franklin Institute
Received date: Revised date: Accepted date:
28 November 2018 31 July 2019 25 November 2019
Please cite this article as: Jianmin Jiao, Rui Zhang, An extended reciprocally convex matrix inequality and its application to stability analysis of systems with additive time-varying delays, Journal of the Franklin Institute (2019), doi: https://doi.org/10.1016/j.jfranklin.2019.11.065
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An extended reciprocally convex matrix inequality and its application to stability analysis of systems with additive time-varying delays Jianmin Jiao ∗ , Rui Zhang Institute of Mathematics and Information Science, Baoji University of Arts and Sciences, Baoji 721013, China *Corresponding author : [email protected] (Jianmin Jiao)
Abstract This paper is concerned with the stability analysis of systems with additive time-varying delays. First, an extended reciprocally convex matrix inequality is presented, which is a generalization of the existing reciprocally convex matrix inequalities. Second, combining the proposed matrix inequality with the Wirtinger-based integral inequality, a new stability criterion of systems with additive time-varying delays is proposed. Meanwhile, an improved stability criterion of systems with a single time-varying in a range is also obtained. Finally, two numerical examples are employed to illustrate the advantage of the obtained theoretical results. Keywords: Time-delay systems; stability; reciprocally convex matrix inequality; Lyapunov-Krasovskii functional(LKF); linear matrix inequality(LMI) 1. Introduction It is well known that time-delay frequently occurs in many practical systems, such as biological systems, chemical systems, electrical systems, network control systems and so on, it is regarded as a major factor causes of the instability, oscillations, and poor performance. So, the study on the stability analysis of time-delay systems is important for theory and practical application. A great number of research and results are dedicated to Preprint submitted to Elsevier
December 3, 2019
the development of time-delay systems over these years [1-7]. For stability analysis of time-delay systems, an efficient method is to construct the delaydependent Lyapunov-Krasovskii functions (LKFs), the main objective is to obtain maximum delay bounds guaranteeing the asymptotically stability of the concerned systems. Since the trial on obtaining tight lower bound for integral quadratic functions which exist in the derivatives of LKFs has been a key step in reducing the conservatism of the stability criteria, various mathematical approaches have been proposed, such as model transformations approach [2], free-weighting-matrix approach [3], delay partition approach [4], integral inequalities approach [5-7], and so on. Actually, a combination of several approaches is used in most stability analysis results. The state model of time-delay systems contain one single delay is investigated commonly and frequently in the past. In [8], a new model for continuous system with additive time-varying delays was proposed. The proposed new model has a strong application background in networked control and long-range control. Take a state-feedback network control for example, because the physical plant, sensor, controller and actuator are located in different places, so in the transmit process, signals are transmitted from one device to another device, time delays will be appear. Among these delays, there are two network-induced ones: one is used to represent the delay from sensor to controller, and the other is used to represent the delay from controller to actuator. Thus, the two additive time delays will appear in the closed-loop system. The properties of these two delays may not be identical due to the network transmission conditions. It is not reasonable to combine them together, hence they should be treated separately. Therefor, it is significant to consider the systems with two additive time-varying delays. In [8,9], Lam et.al. have initially investigated the stability for this kind of system, and some stability criteria were obtained. In [10], by constructing new LKF, a robust stability criterion for uncertain systems with additive time-varying delays was given. In [11], the author gave a stability criterion for singular systems with additive time-varying delays by using free-weighting-matrix approach. By taking more information of the additive time-varying delays in LKFs into account, some new stability criteria were established in [12,13]. In [14], an improved stability criterion for Lurie systems with additive time-varying de2
lays was obtained by taking advantage of integral inequality. Recently, by constructing augmented LKFs, some new less conservative stability criteria are presented in [15,16]. In recent years, estimating integral terms directly via integral inequalities gradually becomes more popular, especially the Jensen integral inequality has been widely used [5]. Very recently, Writinger-based integral inequality that provides a more tighter lower bound than the Jensen inequality has been proposed in [17, 18]. Further, several works have been done to improve the Wirtinger-based integral inequality as more general inequality, such as auxiliary-function-based integral inequality [19], free-matrix-based integral inequality [20,21], Bessel-Legendre integral inequality [22], and polynomialsbased integral inequality [23]. Although the inequalities in above literatures have shown a great interest for constant time-delay systems, however, their application to time-varying delays reveals additional difficulties related to the non-convexity of the resulting terms. To address these terms, the reciprocally convex approach is usually employed. The reciprocally convex combination lemma was proposed in [24]. In [25], an improved version of reciprocally convex combination lemma was provided, and it has been extended in [2632]. However, it is worth stressing that these reciprocally convex matrix inequalities in [24-32] are only applied in the case which the delay interval is divided into two subintervals. In the study of stability for systems with additive time-varying delays, the delay interval often need to be divided into m(m > 2) subintervals, the existing reciprocally convex matrix inequalities cannot be applied directly. Therefore, it is necessary to extend the classical reciprocally convex matrix inequalities to the case of m > 2 subintervals in the delay interval. Motivated by the above of the observation, in this paper, the problem of stability analysis for continuous systems with additive time-varying delays is investigated. The main contributions of this paper are as follow: (1) An extended reciprocally convex matrix inequality is derived. (2) By combining the extended reciprocally convex matrix inequality with the Writinger-based integral inequality, a new stability criterion of continuous systems with additive time-varying delays is proposed. Meanwhile, an improved stability criterion of continuous systems with a single time-varying 3
in a range is also obtained. Notation: Throughout this paper, Rn denotes the n-dimensional Euclidean space, while Rm×n refers to the set of all real matrices with m rows and n columns. The superscripts T and −1 mean the transpose and inverse of a matrix, respectively. P > 0 (P < 0) means P is a real symmetric and positive-definite (negative-definite) matrix. In is the n-dimensional identity matrix. Symmetric term in a symmetric matrix is denoted by ∗, and Sym{X} = X + X T . 2. Problem formulation Consider the following continuous system with additive time-varying delays: ( x(t) ˙ = Ax(t) + Bx(t − d1 (t) − d2 (t)), t > 0, (1) x(t) = ϕ(t), t ∈ [−(d1 + d2 ), 0], where x(t) ∈ Rn is the state vector, ϕ(t) ∈ Rn is the continuous initial function, A, B ∈ Rn×n are constant matrices. The time delays d1 (t) and d2 (t) are time-varying differentiable functions that satisfy 0 6 di (t) 6 di , d˙i (t) 6 µi (i = 1, 2),
(2)
where di and µi (i = 1, 2) are known positive constants. This paper aims to derive a delay-dependent stability criterion of system (1) with time-varying delays d1 (t) and d2 (t) satisfying (2). In order to achieve the objective , the following lemma is reviewed. Lemma 1 ([17] Wirtinger-based integral inequality) For a n × n symmetric matrix R > 0, scalars a and b with a < b, and continuously differentiable function x : [a, b] → Rn , the following integral inequality holds Z
a
b
1 x˙ (s)Rx(s)ds ˙ > b−a T
where ϕ1 = x(b) − x(a), ϕ2 = x(b) + x(a) −
2 b−a
Rb a
ϕ1 ϕ2
x(s)ds. 4
T
R 0 0 3R
ϕ1 ϕ2
,
(3)
3. An extended reciprocally convex matrix inequality In this part, an extended reciprocally convex matrix inequality is proposed, shown as follows. Lemma 2 (extended reciprocally convex matrix inequality) For any real scalars αi > 0(i = 1, 2, · · ·, m), n × n symmetric matrices Ri > 0(i = 1, 2, · · ·, m), and any mn × mn matrix M , the following matrix inequality holds 1 R 0 ··· 0 α1 1 1 0 R ··· 0 α2 2 . . . . . . . . . . . . 0
···
0
1 R αm m
α1 R1−1 0 ··· 0 0 α2 R2−1 · · · 0 T T (4) > −M − M − M M. .. .. .. . . . . . . −1 0 0 · · · αm Rm 1 R 0 ··· 0 α1 1 1 0 R ··· 0 α2 2 Proof Define R = , from Schur comple. . . . . . . . . . . . 0
ment, we have
···
0
R M T −1 ∗ M R M
1 R αm m
> 0.
Pre-multiplying and post-multiplying above inequality by its transpose, respectively, we have
Imn Imn
and
R + M + M T + M T R−1 M > 0, which implies (4). This completes the proof. Remark 1 If taking m = 2, α1 = α ∈ (0, 1), α2 = 1 − α and M = M11 M12 , inequality (4) reduces to the following inequality (5) ∗ M22 # " 1 T T R 0 M11 + M11 M12 + M21 α 1 >− 1 T 0 R ∗ M22 + M22 1−α 2 5
−α
T M11 T M12
R1−1
M11 M12
− (1 − α)
T M21 T M22
R2−1
M21 M22
.
(5) The existing reciprocally convex matrix inequalities can be directly obtained via the proposed matrix inequality (5). For examples, setting M11 = −R1 , M12 = −S1 , M21 = −S2T , M22 = −R2 , we can obtain Lemma 4 of [27]. Furthermore, R S T setting M11 = M22 = −R, M12 = −S, M21 = −S , if > 0, which ∗ R leads to R − SR−1 S T > 0 and R − S T R−1 S > 0, then we can obtain popular reciprocally convex combination lemma given in [24]. Remark 2 Combining the reciprocally convex matrix inequality with integral inequality approach, some stability criteria of linear systems with single delay were established in [24-31]. However, in the study of stability problem for systems with additive delays, the delay interval often need to be divided into multiple subintervals. Thus, the above reciprocally convex matrix inequalities cannot be directly applied. In order to solve this problem, Lemma 2 gives an extended reciprocally convex matrix inequality, which can be applied when the number of subintervals in the delay interval is more than two. Remark 3 Compared with those proofs in [24-27], Lemma 2 is very simple and straightforward in which the Schure complement is simply used. Remark 4 It is worth pointing out that unlike the existing reciprocally convex matrix inequalities, scalars α1 , α2 , · · · , αm in Lemma 2 are not need to P satisfy the constraint condition m i=1 αi = 1. Thus, the new matrix inequality (4) may has a wider application. 4. Stability analysis of systems with additive time-varying delays Before introducing the main result, following notations are defined for simplicity ei = [0n×(i−1)n In 0n×(9−i)n ](i = 1, 2, · · · , 9), e˜i = [0n×(i−1)n In 0n×(8−i)n ](i = 1, 2, · · · , 8), e¯i = [0n×(i−1)n In 0n×(7−i)n ](i = 1, 2, · · · , 7),
6
ξ(t) =[xT (t) xT (t − d1 (t)) xT (t − d1 (t) − d2 (t)) xT (t − d1 (t) − d2 ) Z t Z t−d1 (t) 1 1 T T x (t − d1 − d2 ) x (s)ds xT (s)ds d1 (t) t−d1 (t) d2 (t) t−d1 (t)−d2 (t) Z t−d1 (t)−d2 (t) Z t−d1 (t)−d2 1 1 T xT (s)ds]T , x (s)ds d2 − d2 (t) t−d1 (t)−d2 d1 − d1 (t) t−d1 −d2 " # x(t) − x(t − d1 (t)) Rt η1 (t) = , x(t) + x(t − d1 (t)) − d12(t) t−d1 (t) x(s)ds " # x(t − d1 (t)) − x(t − d1 (t) − d2 (t)) R t−d (t) η2 (t) = , x(t − d1 (t)) + x(t − d1 (t) − d2 (t)) − d22(t) t−d11(t)−d2 (t) x(s)ds " # x(t − d1 (t) − d2 (t)) − x(t − d1 (t) − d2 ) R t−d (t)−d (t) η3 (t) = , x(t − d1 (t) − d2 (t)) + x(t − d1 (t) − d2 ) − d2 −d2 2 (t) t−d11(t)−d22 x(s)ds " # x(t − d1 (t) − d2 ) − x(t − d1 − d2 ) R t−d (t)−d η4 (t) = . x(t − d1 (t) − d2 ) + x(t − d1 − d2 ) − d1 −d2 1 (t) t−d11−d2 2 x(s)ds Now, we provide a stability criterion for the system (1) with time-varying delays d1 (t) and d2 (t) satisfying (2). Theorem 1 For given scalars di > 0, µi > 0(i = 1, 2), the system (1) with time-varying delays d1 (t) and d2 (t) satisfying (2) is asymptotically stable if P1 P2 there exist 2n × 2n matrix > 0, n × n matrices Qi > 0(i = ∗ P3 1, 2, 3, 4) and R > 0, and 8n × 8n matrix M , such that LMIs (6)-(9) hold. Ξ1 + Ξ2 + Ξ3 + d1 Φ1 + d2 Φ2 E T M T E1T E T M T E2T ˜ ∗ − d11 R 0 (6) < 0, ˜ ∗ ∗ −1R d2
Ξ1 + Ξ2 + Ξ3 + d1 Φ1 + d2 Φ3 E T M T E1T E T M T E3T ˜ ∗ − d11 R 0 ˜ ∗ ∗ − d12 R Ξ1 + Ξ2 + Ξ3 + d1 Φ4 + d2 Φ2 E T M T E4T E T M T E2T ˜ ∗ − d11 R 0 ˜ ∗ ∗ − d12 R 7
(7)
(8)
< 0, < 0,
Ξ1 + Ξ2 + Ξ3 + d1 Φ4 + d2 Φ3 E T M T E4T E T M T E3T ˜ ∗ − d11 R 0 < 0, 1 ˜ ∗ ∗ − d2 R
(9)
where T T Ξ1 = Sym{eT 1 (P1 A + P2 )e1 + e1 P1 Be3 − e1 P2 Be5 }, T T Ξ2 = eT 1 (Q1 + Q2 + Q3 + Q4 )e1 − (1 − µ1 )e2 Q1 e2 − (1 − µ1 − µ2 )e3 Q2 e3 − T (1 − µ1 )eT 4 Q3 e4 − e5 Q4 e5 , T T T T T T Ξ3 = (d1 + d2 )(eT 1 A RAe1 + e3 B RBe3 + Sym{e1 A RBe3 + E M E}), T T T T T Φi = Sym{e1 (A P2 +P3 )ei+5 + e3 B P2 ei+5 − e5 P3 ei+5 }(i = 1, 2, 3, 4), e1 − e2 e1 + e2 − 2e6 e2 − e3 e2 + e3 − 2e7 , E= e − e 3 4 e + e − 2e 3 4 8 e4 − e5 e4 + e5 − 2e9 e˜2i−1 Ei = (i = 1, 2, 3, 4), e˜2i R 0 ˜ R= . 0 3R Proof Construct the following LKF candidate V (t) =
3 X
Vi (t),
(10)
i=1
where V1 (t) = V2 (t) = + V3 (t) =
" Z
x(t) Rt x(s)ds t−d1 −d2 t
t−d1 (t) Z t
P1 P2 ∗ P3 Z t T x (s)Q1 x(s)ds +
"
x(t) Rt x(s)ds t−d1 −d2
xT (s)Q4 x(s)ds,
t−d1 −d2
x˙ T (s)Rx(s)dsdθ. ˙
t+θ
8
#
xT (s)Q2 x(s)ds
t−d1 (t)−d2 (t) Z t
xT (s)Q3 x(s)ds +
1 (t)−d2 Zt−d Z t 0
−d1 −d2
#T
,
Then, the time derivatives of functionals Vi (t)(i = 1, 2, 3) along the trajectory of system (1) are given by V˙ 1 (t) =ξ T (t)(Ξ1 + d1 (t)Φ1 + d2 (t)Φ2 + (d2 − d2 (t))Φ3 + (d1 − d1 (t))Φ4 )ξ(t). (11) T ˙ (12) V2 (t) 6 ξ (t)Ξ2 ξ(t). Z t x˙ T (s)Rx(s)ds ˙ V˙ 3 (t) =(d1 + d2 )x˙ T (t)Rx(t) ˙ − t−d1 −d2 t
T
=(d1 + d2 )x˙ (t)Rx(t) ˙ − −
Z
t−d1 (t)−d2 (t)
t−d1 (t)−d2
Z
T
t−d1 (t)
x˙ (s)Rx(s)ds ˙ −
T
x˙ (s)Rx(s)ds ˙ − Z
t−d1 (t)−d2
Z
t−d1 (t)
x˙ T (s)Rx(s)ds ˙
t−d1 (t)−d2 (t)
x˙ T (s)Rx(s)ds, ˙
t−d1 −d2
(13) by using Lemma 1 to estimate the upper bounds of the last four integral terms on the right side of equality (13), one has 1 T ˜ 1 T ˜ η1 (t)Rη1 (t) − η (t)Rη2 (t) d1 (t) d2 (t) 2 1 1 ˜ 3 (t) − ˜ 4 (t) − η3T (t)Rη η T (t)Rη d2 − d2 (t) d1 − d1 (t) 4
V˙ 3 (t) 6 (d1 + d2 )x˙ T (t)Rx(t) ˙ −
= (d1 + d2 )x˙ T (t)Rx(t) ˙ T ˜ − d11(t) R 0 0 0 η1 (t) 1 ˜ 0 − d2 (t) R 0 0 η (t) + 2 1 ˜ 0 0 − d2 −d2 (t) R 0 η3 (t) ˜ η4 (t) 0 0 0 − d1 −d1 1 (t) R
9
η1 (t) η2 (t) η3 (t) η4 (t) (14)
.
For matrix M , it follows Lemma 2 that ˜ − d11(t) R 0 0 0 1 ˜ 0 − d2 (t) R 0 0 1 ˜ 0 0 0 − d2 −d2 (t) R ˜ 0 0 0 − d1 −d1 1 (t) R ˜ −1 d1 (t)R 0 0 −1 ˜ 0 d (t) R 0 2 +M T ˜ −1 0 0 (d2 − d2 (t))R 0
0
9
6 M + MT
0 0 0 ˜ −1 (d1 − d1 (t))R
M.
(15)
From (14) and (15), one can obtain ˜ −1 E1 M E + d2 (t)E T M T E T R ˜ −1 E2 M E V˙ 3 (t) 6 ξ T (t)(Ξ3 + d1 (t)E T M T E1T R 2 ˜ −1 E3 M E + (d2 − d2 (t))E T M T E3T R ˜ −1 E4 M E)ξ(t). + (d1 − d1 (t))E T M T E4T R
(16)
Then, combing (11),(12) with (16), we can get ˜ −1 E1 M E) V˙ (t) 6 ξ T (t)(Ξ1 + Ξ2 + Ξ3 + d1 (t)(Φ1 + E T M T E1T R ˜ −1 E2 M E + d2 (t)(Φ2 + E T M T E T R 2 T
˜ −1 E3 M E) + (d2 − d2 (t))(Φ3 + E M T E3T R ˜ −1 E4 M E))ξ(t). + (d1 − d1 (t))(Φ4 + E T M T E4T R Based on convex combination technique, V˙ (t) < 0 holds if the following four inequalities hold ˜ −1 E1 M E)+d2 (Φ2 +E T M T E T R ˜ −1 E2 M E) < 0, Ξ1 +Ξ2 +Ξ3 +d1 (Φ1 +E T M T E1T R 2 ˜ −1 E1 M E)+d2 (Φ3 +E T M T E3T R ˜ −1 E3 M E) < 0, Ξ1 +Ξ2 +Ξ3 +d1 (Φ1 +E T M T E1T R ˜ −1 E2 M E) < 0, ˜ −1 E4 M E)+d2 (Φ2 +E T M T E T R Ξ1 +Ξ2 +Ξ3 +d1 (Φ4 +E T M T E4T R 2 ˜ −1 E4 M E)+d2 (Φ3 +E T M T E T R ˜ −1 E3 M E) < 0, Ξ1 +Ξ2 +Ξ3 +d1 (Φ4 +E T M T E4T R 3 which are respectively guaranteed by (6)-(9) based on Schur complement. Thus, if (6)-(9) hold, then V˙ (t) < 0 holds for any ξ(t) 6= 0, which implies 10
that system (1) with the delays satisfying (2) is asymptotically stable. The proof is completed. Remark 5 By splitting the whole delay interval [−(d1 + d2 ), 0] into four subintervals [−(d1 +d2 ), −(d1 (t)+d2 )], [−(d1 (t)+d2 ), −(d1 (t)+d2 (t))], [−(d1 (t)+ d2 (t)), −d1 (t)] and [−d1 (t), 0], we construct the LKF (10) which depend on the subintervals. Thus, the interactions among the vectors of x(t), x(t − d1 (t)), x(t − d1 (t) − d2 (t)), x(t − d1 (t) − d2 ) and x(t − d1 − d2 ) were sufficiently Rt R t−d (t) considered. Furthermore, by taking the states t−d1 (t) x(s)ds, t−d11(t)−d2 (t) x(s)ds, R t−d (t)−d R t−d1 (t)−d2 (t) x(s)ds and t−d11−d2 2 x(s)ds as augmented variables, the stabilt−d1 (t)−d2 ity criterion in Theorem 1 utilizes more information on state variables, and these terms are not employed in previous literatures. Therefore, Theorem 1 is expected to less conservative than some results in the literatures, which will be illustrated through numerical examples in the next section. Remark 6 When dealing with the derivative of the LKF (10), the general approach is to use Jensen’s or Wirtinger-based integral inequalities to scale the integral terms firstly, then employed convex optimization approach to handle time-varying delays existing in denominator of the formula. It should be pointed out that the existing reciprocally convex matrix inequalities in [24-32] are only in the case when the delay interval is divided into two subintervals. In this paper, the delay interval is divided into four subintervals, so the existing reciprocally convex matrix inequalities can not be applied directly. We introduced an extended reciprocally convex matrix inequality to deal with the time derivative on double integral terms directly. When d1 (t) or d2 (t) is a constant, the system (1) reduces to linear system with a single time-varying delay in a range. Without loss generality, suppose that ( x(t) ˙ = Ax(t) + Bx(t − d(t)), t > 0, (17) x(t) = ϕ(t), t ∈ [−h2 , 0], time-varying delay d(t) satisfying ˙ 6 µ, h1 6 d(t) 6 h2 , d(t) where h1 , h2 and µ are constants. We choose the following LKF candidate [27] 11
(18)
T
V (t) = ζ (t)P ζ(t) + + h1 Z
Z
0
Z
Z
t T
x (s)Q1 x(s)ds +
t−h1
xT (s)Q2 x(s)ds
t−h2
t−h1
t
Z
x˙ T (s)Z x(s)dsdθ ˙ + (h2 − h1 )
−h1 t+θ −h1 Z −h1 Z t
Z
−h1
−h2
Z
t
x˙ T (s)Rx(s)dsdθ ˙
t+θ
x˙ T (u)U x(u)dudθds, ˙ t+θ s −h2 R t−h1 T Rt x (s)ds]T , P > 0, Qi > 0(i = xT (s)ds where ζ(t) = [xT (t) t−h2 t−h1 1, 2, 3), Z > 0, R > 0 and U > 0. Then, along a similar line as in the derivation of Theorem 1 we can derive the following stability criterion for system (17). Theorem 2 For given scalars h1 , h2 and µ, system (17) with time-varying delay d(t) satisfying (18) is asymptotically stable if there exist 3n×3n matrix P11 P12 P13 ∗ P22 P23 > 0, n×n matrices Qi > 0(i = 1, 2, 3), Z > 0, R > 0, U > ∗ ∗ P33 0, and 4n × 4n matrix M , such that LMIs (19)-(20) hold. +
Γ + (h2 − h1 )Ψ1 E¯ T M T E¯1T ¯ − U¯ ∗ −R Γ + (h2 − h1 )Ψ2 E¯ T M T E¯2T ¯ ∗ −R
< 0,
(19)
< 0,
(20)
where 1 2 T 2 T 2 T Γ = e¯T e1 + 1 (Q1 +Q3 +h1 A ZA+(h2 −h1 ) A RA+ 2 (h2 −h1 ) A U A−4Z)¯ T T 2 T 2 T e¯2 (−Q1 + Q2 − 4Z + 2U )¯ e2 + e¯3 (−(1 − µ)Q3 + h1 B ZB + (h2 − h1 ) B RB + 1 2 T (h2 −h1 ) B U B+2U )¯ e3 +¯ eT e4 +¯ eT e5 +¯ eT e6 +¯ eT e7 + 4 (−Q2 )¯ 5 (−12Z)¯ 6 (10U )¯ 7 (−2U )¯ 2 T T T 2 T Sym{¯ e1 (P11 A + P12 )¯ e1 + e¯1 (−P12 + P13 − 2Z)¯ e2 + e¯1 (P11 B + h1 A ZB + (h2 − 1 2 T 2 T T T h1 ) A RB + 2 (h2 − h1 ) A U B)¯ e3 + e¯1 (−P13 )¯ e4 + e¯T 1 (h1 A P12 + h1 P22 + T T 6Z)¯ e5 +¯ eT e3 +¯ eT e5 +¯ eT e6 +¯ eT e5 + 2 (2U )¯ 2 (−h1 P22 +h1 P23 +6Z)¯ 2 (−4U )¯ 3 (h1 B P12 )¯ T T T T T ¯ ¯ e¯3 (−6U )¯ e6 + e¯3 (2U )¯ e7 + e¯4 (−h1 P23 )¯ e5 + E M E}, T T Ψi = Sym{¯ eT (A P + P )¯ e + e¯T ei+5 + e¯T ¯i+5 − 13 23 i+5 1 2 (−P23 + P33 )¯ 3 B P13 e T e¯4 P33 e¯i+5 }(i = 1, 2),
12
e¯2 − e¯3 e6 e¯ + e¯3 − 2¯ E¯ = 2 , e¯3 − e¯4 e¯3 + e¯4 − 2¯ e7 e¯2i−1 E¯i = (i = 1, 2), e¯2i R 0 ¯ R= , 0 3R U 0 ¯ U= . 0 3U Remark 7 When µ is unknown, setting Q3 = 0 in Theorem 2 we can obtain a delay-rate-independent stability criterion for system (17). In addition, ¯ − U¯ −S2T −R it is easy to derive that setting Q3 = 0 and M = ¯ in The−S1 −R orem 2 yield precisely the Theorem 8 in [27]. Hence, compared with Theorem 8 in [27], the Theorem 2 in this paper overcomes the conservativeness. Remark 8 The obtained results in this paper are formulated in terms of LMIs, they can be easily solved by using any LMI toolbox, e.g. Matlab or Scilab. Remark 9 More recently, generalized integral inequalities, tighter than Wirtinger-based integral inequality, have been developed. For example, auxiliary function-based inequality [19], free-matrix-based inequality [20,21], and Bessel-Legendre inequality [22]. The further improved stability criteria can be obtained by combing those inequalities and the proposed extended reciprocally convex matrix inequality, while it will need more computational burden. 5. Numerical examples To illustrate the effectiveness and the less conservatism of obtained results in this paper, two numerical examples are given in this section. Example 1 Consider the system (1) with the following parameters: A=
−2 0 0 −0.9
,
13
B=
−1 0 −1 −1
.
The delay derivatives µ1 and µ2 are assumed as 0.1 and 0.8, respectively. Our purpose is to find the upper bounded d1 of delay d1 (t), or d2 of delay d2 (t), when the other is known, below which the system is asymptotically stable. The calculation results obtained by the methods in [8-16] and Theorem 1 in this paper are listed in Table 1 and Table 2 under different cases. From Table 1 and Table 2, we can see that by using Theorem 1, the results we obtained are less conservative than those of [8-16]. Table 1. Comparison of delay upper bound d2 for various d1
d1 [8] [9] [10] [11] [12] [13] [14] [15] [16] Theorem 1
Example 2
1.0 0.415 0.512 0.595 0.873 0.982 0.983 0.988 0.999 1.163 1.233
1.2 0.376 0.406 0.462 0.673 0.782 0.849 0.854 0.972 0.965 1.035
1.5 0.248 0.283 0.312 0.452 0.482 0.671 0.675 0.680 0.669 0.752
Consider the system (17) with the following parameters: −2 0 −1 0 A= , B= . 0 −0.9 −1 −1
Assuming that the delay derivative µ is unknown. The maximal upper bounds of h2 with respect to various h1 calculated by [17, 24, 27] and Theorem 2 (Q3 = 0) in this paper are listed in Table 3. We can see that stability condition presented in this paper is less conservative than those in [17, 24, 27]. The obtained results by using Theorem 2 are better than compared ones since in which the extended reciprocally convex inequality is used. Remark 10 As a matter of fact in example 1 and example 2, the number of decision variables is 68.5n2 + 3.5n and 23n2 + 4n, respectively. To get higher bounds with less decision variables is our next research aim. 14
Table 2. Comparison of delay upper bound d1 for various d2
d2 [8] [9] [10] [11] [12] [13] [14] [15] [16] Theorem 1
0.3 1.324 1.453 1.531 1.808 1.682 2.329 2.337 1.880 1.875 2.441
0.4 1.039 1.214 1.313 1.593 1.582 2.065 2.069 1.779 1.773 2.145
0.5 0.806 1.021 1.140 1.424 1.482 1.836 1.842 1.675 1.671 1.912
Table 3. Comparison of delay upper bound h2 for various h1
h1 [24] [17] [27] Theorem 2
0 1.868 2.113 2.213 2.233
0.4 1.882 2.179 2.256 2.264
0.7 1.953 2.237 2.286 2.289
6. Conclusion In this paper, an extended reciprocally convex matrix inequality has been introduced, which is a generalization of the existing reciprocally convex matrix inequality. Further, a stability criterion with less conservatism for systems with additive time-varying delays has been established by combining the proposed matrix inequality and Writinger-based integral inequality. The advantages of the proposed matrix inequality and the corresponding criterion have been shown via a numerical example. In the future, our main work include two aspects: (1) how to reduce the complexity of the proposed reciprocally convex matrix inequality; (2) trying to apply proposed method in this paper to study other control problem of systems with additive time-varying delays, such as stabilization controller design[33], robust filtering analysis 15
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