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Novel master–slave synchronization criteria of chaotic Lur’e systems with time delays using sampled-data control
Accepted Manuscript
Novel master-slave synchronization criteria of chaotic Lur’e systems with time delays using sampled-data control Ruimei Zhang, Deqiang Zeng, Shouming Zhong PII: DOI: Reference:
S0016-0032(17)30231-4 10.1016/j.jfranklin.2017.05.008 FI 2982
To appear in:
Journal of the Franklin Institute
Received date: Revised date: Accepted date:
3 October 2016 16 February 2017 1 May 2017
Please cite this article as: Ruimei Zhang, Deqiang Zeng, Shouming Zhong, Novel master-slave synchronization criteria of chaotic Lur’e systems with time delays using sampled-data control, Journal of the Franklin Institute (2017), doi: 10.1016/j.jfranklin.2017.05.008
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Novel master-slave synchronization criteria of chaotic Lur’e systems with time delays using sampled-data control✩ Ruimei Zhanga,∗, Deqiang Zengb , Shouming Zhonga a School
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of Mathematics Sciences, University of Electronic Science and Technology of China, Chengdu Sichuan 611731, PR China b Data Recovery Key Laboratory of Sichuan Province, College of Mathematics and Information Science, Neijiang Normal University, Neijiang 641100, PR China
Abstract
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This paper investigates the problem of master-slave synchronization of chaotic Lur’e systems (CLSs) with time delays by sampled-data control. First, a novel Lyapunov-Krasovskii functional (LKF) is constructed with some new augmented terms, which can fully capture the system characteristics and the available information on the actual sampling pattern. In comparison with existing results, the constraint condition of the positive definition of the LKF is more relax, since it is positive definite only requiring at sampling instants. Second, based on the LKF, a less conservative synchronization criterion is established. Third, the
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desired estimator gain can be designed in terms of the solution to linear matrix inequalities (LMIs). The obtained conditions ensure the master-slave synchronization of CLSs under a longer sampling period than
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remarkable existing works. Finally, three numerical simulations of Chua’s circuit and neural network are provided to show the effectiveness and advantages of the proposed results.
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Keywords:
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Synchronization, Chaotic Lur’e systems, Sampled-data control, Neural networks
1. Introduction
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Since the seminal paper by Pecora and Carroll was published in 1990, chaos synchronization has gained growing research interests as it has been widely applied to many areas, such as chaos generator design, information science, biological systems, chemical reaction, and secure communication [1–6, 8]. Simultaneously, it has been shown that many nonlinear systems, including Chua’s circuit, hyper chaotic attractors, and n-scroll attractors, can be represented as Lur’e-type systems. Thus, the master-slave synchronization of ✩ This
work was supported by the National Natural Science Foundation of China (NO. 61533006) author Email address: [email protected] (Ruimei Zhang)
∗ Corresponding
Preprint submitted to Journal of The Franklin Institute
May 8, 2017
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chaotic Lur’e systems (CLSs) has provoked widespread interests in many research and application fields, and a great number of interesting research results have been reported on this topic [4, 7, 10, 11, 13–16, 19, 27– 34]. Further, because of signal transmissions among neurons and the finite switching speed of amplifiers in the implementation of electrical circuits, time delays are unavoidably encountered CLSs, which may cause divergence, oscillation, or even instability. Accordingly, it is germane to take time delays into account for master-slave synchronization of CLSs.
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From the control strategy point of view, the control is an effective synchronization method for CLSs, and many control approaches have been proposed to synchronize CLSs such as time delay feedback control [10, 13, 16], impulsive control [15], adaptive control [12, 18], sampled-data control [9, 20], and proportional derivative control method [19]. Among these methods, sampled-data control, which uses only the sampled information of the system at its sampling instants to the controller, has garnered wide-scale attention in
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recent years [9, 11, 20–33]. Compared with some continuous control, the sampled-data control has many advantages, such as easy installation, high reliability, maintenance with low cost, and efficiency. Therefore, the investigation of the master-slave synchronization of CLSs with time delays via sampled-data control is important in both theory and application.
During utilizing the sampled-data control to synchronize the CLSs with time delays, how to choose the sampling period is an critical issue to be considered. It is well-known that if the sampling period is large,
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then its operating conditions such as computational burden, limitation of load, and communication capacity can be relaxed [28]. Thus, it is significant to consider the control design problem under a sampling period
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as large as possible. Up to now, many interesting results concerning on finding a longer maximal sampling period for CLSs with time delays have been published in [4, 11, 17, 20–33]. Very recently, by sampled-data control, sufficient conditions were obtained for global asymptotic synchronization of CLSs in [11]. Improved
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synchronization criteria for this problem were proposed in [27–29] by introducing some sawtooth structure terms. Further results on the master-slave synchronization of CLSs with time delays has been proposed in
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[30–33] by introducing some discontinuous LKFs. However, the obtained conditions in [25, 26, 30–32] may be conservative, since the constructed LKFs in these papers are required to be positive definite on the whole sampling interval. In fact, the conservativeness may be further reduced by relaxing this requirement to only
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requiring positive definite at sampling instants. In addition, in [11, 27–33], the main contribution of these work was to construct an appropriate LKF to get a longer sampling period. However, there still exists room for improving the synchronization criteria of [11, 27–33], because some important terms have been ignored in the LKFs proposed by [11, 27–33], and the available information on actual sampling pattern have not been fully adopted, which might lead to some degree of conservatism. Thus, it is necessary to construct an more appropriate LKF to derive less conservative synchronization criteria for the CLSs. Motivated by the above discussion, in this paper, the problem of master-slave synchronization of CLSs with time delays is investigated by using sampled-data control. The main contributions are summarized 2
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below. 1) A novel LKF with some new augmented terms is constructed. Notably, V5 (t) and V6 (t) are the first time to be proposed, which can take full advantage of the available information on the actual sampling pattern. 2) The paper presents relatively relaxed conservative synchronization criteria. The paper differs from the prior results in [25, 26, 30–32, 34] to restrict the constructed LKF to be positive define on the whole
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sampling intervals. In the present paper, the LKF is positive definite only requiring at sampling instants. 3) Compared with the sampling periods in the existing results in [11, 27–34], a longer sampling period is obtained in this paper.
The remainder of this paper is organized as follows. In Section 2, the problem formulation and some preliminaries are introduced. Our main results are presented in Section 3, where the novel sampled-data
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synchronization criterion of CLSs with time delays are obtained. In Section 4, numerical examples are given to demonstrate the effectiveness and the benefit of the proposed results. Finally, the conclusion is drawn in Section 5.
Notations: Throughout this paper,
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X and Y , the notation X > Y means that the matrix X − Y is positive define. diag{· · · } and col{· · · } stand for a block-diagonal matrix and a column vector, respectively. Sym{X} = X + X T . The symmetric term
for algebraic operations.
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in a matrix is denoted by ∗. If not explicitly stated, all matrices are assumed to be compatible dimensions
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2. Problem description and preliminaries Consider the following general master-slave type of delayed Lur’e systems with sampled-data feedback
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control [29–32]: x(t) ˙ = Ax(t) + Bx(t − d) + Wσ(Dx(t)), M : p(t) = Cx(t), z(t) ˙ = Az(t) + Bz(t − d) + Wσ(Dz(t)) + u(t), S : q(t) = Cz(t), C :
u(t) = K(p(tp ) − q(tp )),
(1)
tp ≤ t < tp+1 ,
which consists of the master system M , slave system S , and sampled-data feedback controller C . M and S with u(t) = 0 are identical chaotic time-delay Lur’e systems with state vectors x(t), z(t) ∈
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are known real matrices. K ∈ 0 is the
constant time delay. u(t) ∈
σi (α) − σi (β) ≤ ki+ , α−β
∀α, β, i = 1, 2, . . . , m.
(2)
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For sampled-data feedback synchronization, only discrete measurements of p(t) and q(t) can be used for synchronization purposes, that is, we only have the measurements p(tp ) and q(tp ) at the sampling instant tp . It is assumed that the control signal to be generated by using a zero-order-hold (ZOH) function with a sequence of hold times 0 = t0 < t1 < · · · < tp < · · · . The sampled-data control input may be represented as discrete control signal: tp ≤ t < tp+1 ,
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u(t) = ud (tp ) = ud (t − (t − tp )) = ud (t − r(t)),
where r(t) = t − tp , which is a piecewise linear function, and r(t) ˙ = 1. Sampling interval hp satisfies the following condition: 0 ≤ r(t) ≤ hp = tp+1 − tp ≤ h,
follows:
tp ≤ t < tp+1 ,
(3)
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u(t) = K[p(t − r(t)) − q(t − r(t))],
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where h is the largest sampling interval and bounded. Thus, the state feedback controller is designed as
where K is the sampled-data feedback controller gain matrix to be determined.
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Given the synchronization scheme (1), the synchronization error is defined as e(t) = x(t) − z(t). Then, from (1) and (3), we have the following synchronization error system:
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e(t) ˙ = Ae(t) + Be(t − d) + Wf (De(t), z(t)) − KCe(t − r(t)),
tp ≤ t < tp+1 ,
(4)
where f (De(t), z(t)) = σ(De(t) + Dz(t)) − σ(Dz(t)).
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Since σi (·) belongs to the sector [ki− , ki+ ], from (2), one can obtain that for ∀e(t), z(t) ki− ≤
fi (Di e(t), z(t)) σi (Di (e(t) + z(t))) − σi (Di z(t)) = ≤ ki+ , Di e(t) Di e(t)
Di e(t) 6= 0, i = 1, 2, . . . , m,
(5)
in which Di denotes the ith row vector of D. It is easily found from (5) that [fi (Di e(t), z(t)) − ki− Di e(t)] × [fi (Di e(t), z(t)) − ki+ Di e(t)] ≤ 0,
∀e(t), z(t), i = 1, 2, . . . , m.
(6)
Remark 1. In the condition (5), ki− and ki+ can be positive, negative or zero, which means that the resulting nonlinear function may be non-monotonic. While in [27–30], the class of globally Lipschitz nonlinear function 4
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was described under the condition of ki− = 0 and ki+ > 0. Therefore, the nonlinear description in this paper is quite general that covers the usual sigmoid nonlinear function as a special case, which is more effective to reduce the conservatism of the results. Throughout this paper, we introduce the following lemmas.
holds: Z
a
b
ξ T (s)Rξ(s)ds ≤
4(b − a)2 π2
Z
b
˙ ξ˙T (s)Rξ(s)ds.
a
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Lemma 1 [24]. Let ξ(t) ∈ W [a, b) and ξ(a) = 0. Then, for any n × n matrix R > 0, the following inequality
Lemma 2 [24]. Let Y > 0 and w(s) be an appropriate dimensional vector. Then, we have the following: Z t2 Z t2 w(s)ds, wT (s)Yw(s)ds ≤ (t2 − t1 )ξ T (t)F T Y −1 Fξ(t) + 2ξ T (t)F T − t1
t1
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where matrix F and vector ξ(t) independent on the integral variable are the arbitrary appropriate dimensional ones.
3. Main results
In this section, a new synchronization criterion is proposed for chaotic Lur’e system with time delays based on constructing a novel augmented LKF. The desired sampled-data controller is designed via the
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established conditions to achieve the master-slave synchronization of systems (1). Before presenting the main results, for the sake of representation convenience, The notations for some vectors and matrices are
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defined as follows (See Appendix A).
Theorem 1. Under condition (5), for given scalars d > 0, h > 0, υi (i = 1, 2, . . . , 6), the master sys-
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tem M and the slave system S in (1) are globally asymptotically synchronous, if there exist symmetric positive define matrices Q2 , R11 , U, positive diagonal matrix Θ, ∆, Λ, symmetric matrices P, Q1 , X , X¯ , 1 ¯T ¯ I P I1 + I¯2T Q1 I¯2 + (I¯1T − I¯2T )Q2 (I¯1 − I¯2 ) > 0, d 1 Φ(0, 0, 0) I3T S < 0, ∗ − π42 U33 Φ(h, 0, 0) I3T S < 0, ∗ − π42 U33 ¯ Φ(h, h, 0) I3T S YY < 0. ∗ − π42 U33 0 ∗ ∗ − h1 R11
Ψ=
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¯ such that and any appropriate dimensional matrices S, Y, F, F,
5
(7)
(8)
(9)
(10)
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Furthermore, the desired controller gain is given as ¯ K = F −1 F. Proof. For the detailed proofs, see Appendix B. Remark 2. It should be mentioned that, compared with the existing results in [25–34], a less conservative synchronization criterion is obtained in this work. There are two reasons for that: first, a newly augmented
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LKF is constructed in (21). Three (tp , tp+1 )-dependent terms V4 (t), V5 (t), and V6 (t) are constructed, which can take full advantage of the available information on the actual sampling pattern. Note that V5 (t) and V6 (t) are the first time to be proposed for synchronization of chaotic Lur’e system. However, the two (tp , tp+1 )-dependent terms V5 (t) and V6 (t) are neglected in [27–34]. Second, according to (35), (38) and (43), the positive definition of the LKF in (21) is ensured only requiring positive definite at the sampling
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instants. But in [25, 26, 30–32, 34], the proposed LKFs are positive definite on the whole sampling interval by restricting all the symmetric matrices to be positive or the sum of several terms of the LKF to be positive. Thus, an improved synchronization criterion is obtained here.
Z
t
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M
Remark 3. Note that when R12 = 0 and R22 = 0, V4 (t) in (21) reduces to (tp+1 − t) e˙ T (s)R11 e(s)ds, ˙ tp Z t which has been applied in [27, 29, 32]. That is, the terms (tp+1 − t) e˙ T (s)R12 e(tp )ds and (tp+1 − tp Z t T t) e (tp )R22 e(tp )ds are ignored in [27, 29, 32], which can make good use of the available information tp R11 R12 is restricted to be about the actual sampling pattern. On the other hand, in [30, 31], R = ∗ R22 positive. However, only R11 is needed to be positive in this paper. Therefore, the constraint condition of the matrices in V4 (t) is more relaxed than that in [30, 31].
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Remark 4. Inspired by the techniques introduced in [20], a completely new term V5 (t) is introduced in h i h iT X+X T −X+X e(t) e(t) 1 2 this paper. The term (tp+1 − t) e(tp ) T e(tp ) used in [25, 26] is constrained by ∗ −X1 −X1T + X+X 2 T X+X −X+X1 2 the structure of to keep the continuity of the LKF. But such a requirement is X+X T T ∗
−X1 −X1 +
2
not necessary any more for V5 (t). Thus, V5 (t) is more flexible and effective than that [25, 26] in terms of
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conservatism reduction.
Remark 5. In the LKF (21), V6 (t) contains the sampling instants tp , which could make full use of the sawtooth structure characteristic of the sampling Zinput delay. On the other Z hand, when U11 = 0, U12 = 0, U13 = t π2 t T 2 0, U23 = 0, and U33 = 0, V6 (t) reduces to h e˙ (s)U22 e(s)ds ˙ − (e(s) − e(tp ))T U22 (e(s) − e(tp ))ds, 4 tp tp which is similar to the term in the LKF in [20, 34]. That is to say, more harmonic matrices such as U11 , U12 , U13 , U23 , and U33 are introduced in V6 (t). Thus, V6 (t) has advantages to reduce the conservatism. Remark 6. When υ4 = υ5 = υ6 = 0, the equation in (32) can be reduced to 0 = 2(eT (t)υ1 + e˙ T (t)υ2 + eT (tp )υ3 )F[−e(t) ˙ + Ae(t) + Be(t − d) + Wf (De(t), z(t)) − KCe(tp )], 6
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which has been adopted in [29–32]. Compared with the existing equation in [29–32], more adjustable parameters are introduced to harmonize the relation among terms, which can provide great freedom in conservatism reducing. Remark 7. When hp = h, p = 0, 1, . . ., from (34), it is noted that the inequality V˙ (t) < 0 holds if Φ(h, 0, 1) < 0 and Φ(h, h, 1) < 0. Thus, the inequality (8) in Theorem 1 can be removed when the sampling is periodic.
C :
u(t) = K(p(tp ) − q(tp )),
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Remark 8. When B = 0, then system (1) is transformed into the following model without time delay: x(t) ˙ = Ax(t) + Wσ(Dx(t)), M : p(t) = Cx(t), z(t) (11) ˙ = Az(t) + Wσ(Dz(t)) + u(t), S : q(t) = Cz(t), tp ≤ t < tp+1 ,
which has been studied in [27–34]. Correspondingly, the synchronization error system (4) is changed to e(t) ˙ = Ae(t) + Wf (De(t), z(t)) − KCe(t − r(t)), and the LKF in (21) reduces to 6 X
k=1
Vk (t), k 6= 3, tp ≤ t < tp+1 ,
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V (t) =
tp ≤ t < tp+1 ,
(12)
(13)
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where Vk (t), k = 1, 2, 4, 5, 6 are given in (21). Similar to the proof of Theorem 1, One can obtain asymptotically synchronous criterion as follows.
Corollary 1. Under condition (5), for given scalars h > 0, υi (i = 1, 2, . . . , 5), the master system M and
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the slave system S in (11) are globally asymptotically synchronous, if there exist symmetric positive define matrices P, R11 , U, positive diagonal matrix Θ, ∆, Λ, symmetric matrices X , X¯ , and any appropriate
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¯ such that dimensional matrices S, Y, F, F, ˆ 0, 0) Iˆ T S Φ(0, 3 < 0, ∗ − π42 U33 ˆ Φ(h, 0, 0) Iˆ3T S < 0, ∗ − π42 U33 T ˆ ˆ ˆ Φ(h, h, 0) I3 S YY 4 ∗ < 0. − π2 U33 0 ∗ ∗ − h1 R11
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(14)
(15)
(16)
Furthermore, the desired controller gain is given as ¯ K = F −1 F. 7
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4. Numerical examples
Fig. 1. Standard Chua’s circuit and v − i characteristic of Chua’s diode [35].
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In this section, three examples are used to demonstrate the validity and less conservativeness of the proposed new synchronization scheme.
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Example 1. Consider the following standard Chua’s circuit (See Fig. 1), which can be expressed as [35]: dνc1 1 C1 = (νc2 − νc1 ) − gNR (νc1 ), dt R 1 dνc2 (17) = (νc1 − νc2 ) − iL , C2 dt R diL L = −νc2 , dt
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where νc1 and νc2 denote the voltages across the capacitors C1 and C2 , respectively. iL is the current through the inductor L, and the nonlinear function gNR (νc1 ), which defines the ν − i characteristic of Chua’s diode,
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is described by the piecewise-linear function (See Fig. 1) 1 gNR (νc1 ) = Gb νc1 + (Ga − Gb )(|νc1 + E| − |νc1 − E|). 2
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The slopes in the inner and outer regions are Ga , Gb < 0, respectively, which ±E denotes the breakpoints. By recalling the variable: x1 =
νc1 E
, x2 =
νc2 E
, x3 =
iL EG ,
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version of standard Chua’s circuit is rewritten by x˙ 1 = a(x2 − h(x1 )), x˙ 2 = x1 − x2 + x3 , x˙ = −bx , 3 2
where x˙ =
dx dτ ,
a=
C2 C1
> 0 and b =
C2 G2 L
τ =
Gt C2
with G =
1 R.
Therefore, the normalized
(18)
> 0 are the main bifurcation parameters. The nonlinear function is
given by 1 h(x1 ) = m1 x1 + (m0 − m1 )(|x1 + c| − |x1 − c|), 2 8
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with m1 =
Gb G ,
m0 =
Ga G .
With the parameter values: m0 = −1/7, m1 = 2/7, a = 9, b = 14.28, c = 1, the
Chua’s system will obtain the double scroll attractor [36]. Moreover, in the implementation of Lurie network systems, it is often inevitable to introduce time delay in the signals transmitted among neurons because it is an inherent feature of many physical processes.
Chua’s circuit with constant time delay can be expressed as: x˙ 1 (t) = a(x2 (t) − m1 x1 (t) − σ(x1 (t))) − cx1 (t − d), x˙ 2 (t) = x1 (t) − x2 (t) + x3 (t) − cx1 (t − d), x˙ (t) = −bx (t) + c(2x (t − d) − x (t − d)). 3 2 1 3
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Therefore, consider the following delayed Chua’s circuit via sampled-data control [29–32]. The equation of
(19)
It can be found that the Chua’s circuit can be rewritten as the Lur’e form with the following parameters: −am1
A= 1 0
a −1 −b
0
1 , 0
−c 0
B = −c 0 2c 0
0
−a(m0 − m1 )
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,
0
−c
W=
0 0
,
1
T
C=D= 0 , 0
with σ(x1 (t)) = 21 (|x1 (t) + 1| − |x1 (t) − 1|) belonging to sector [0, 1], c = 0.1 and d = 1. In this example, we choose υ1 = 0.0, υ2 = 1.0, υ3 = 1.15, υ4 = 0.1, υ5 = 0.5, and υ6 = 0.1.
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The initial conditions of the master and slave systems are chosen as x(0) = [0.2 0.3 0.2]T and z(0) = [−0.3 − 0.1 0.4]T , t ∈ [−1, 0]. Figs. 2 and 3 show the master system and the slave system states with
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u(t) = 0, respectively. In addition, the response curves of error system (4) with u(t) = 0 is given in Fig. 4. From Fig. 4, we can found that the master system and slave system in (1) are non-synchronization under
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the condition u(t) = 0.
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0 x3(t)
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2
−2
−4 1
0.5
0 x2(t)
−0.5
−2
−1
0
1
2
3
x (t) 1
Fig. 2. State trajectory of master system M in (1) in Example 1.
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2
z3(t)
0
−2
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−4 1
0.5
0 z2(t)
−0.5
−1
−2
2
1
0 z1(t)
3
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Fig. 3. State trajectory of slave system S with u(t) = 0 in (1) in Example 1.
8
6
4
e(t)
2
0
M
−2
−4
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−6
−8
0
10
20
30
40
50 time t
60
70
80
90
100
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Fig. 4. State responses of error system (4) without u(t) in Example 1.
Table 1 provides some comparisons of the maximum sampling intervals involved in the results of this
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paper and those in [29], [30], [31], [32], and [4]. From Table 1, it can be seen that our results provide a larger sampling interval than others, which show the advantage of the results of this paper.
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In order to demonstrate ulteriorly the superiority of our results, by applying Theorem 1, the following state feedback controller is obtained under the maximum sampling interval h = 0.4789: K=
h
3.5324
0.2411
−2.9187
iT
.
Under the above controller gain, the error signal e(t) and control inputs u(t) are given in Figs. 5 and 6, respectively. It can be seen from Fig. 5 that the synchronization error is tending to zero, which shows the master and slave systems in (1) can achieve synchronization successfully by the proposed sampled-data controller. Moreover, when t = 20, h = 0.4789, the simulation results in Fig. 5 imply that only 41 sampled 10
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Table 1: Comparison of maximum allowable sampling interval in Example 1.
Method
[29]
[30]
[31]
[32]
[4]
Theorem 1
h
0.3582
0.4355
0.4357
0.4369
0.4371
0.4789
Table 2: Comparison of maximum allowable sampling interval for c = 0 in Example 1.
[27]
[29]
[28]
[30]
[31]
Corollary 1
h
0.3914
0.3981
0.4800
0.5144
0.5147
0.5185
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Method
signals need to be sent out to the controller, which shows that the sampled-data control uses less information of the system than that by feedback control [8]. 2
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1.5
1
e(t)
0.5
0
−0.5
−1.5
0
2
4
6
8
10 time t
12
14
16
18
20
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−2
M
−1
Fig. 5. State response of error system (4) with u(t) in Example 1.
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When c = 0, the time-delay Chua’s circuit system (19) becomes a Lur’e system without time delay, which has been investigated in [27–31]. The maximum sampling intervals by different methods are listed in
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Table 2. From this table, we can find that by using the method proposed in [27], [29], [28], [30], and [31], the maximum sampling intervals are 0.3914, 0.3981, 0.4800, 0.5144 and 0.5147, respectively. When υ1 = 0.09, υ2 = 1.0, υ3 = 0.75, υ4 = 0.0, υ5 = 0.15, the maximum sampling interval h by Corollary 1 is 0.5185. It can
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be seen that the synchronization criterion in this paper is less conservative than the existing ones. Moreover, the corresponding state feedback controller gain is K=
h
3.1756
0.1406
−2.8992
iT
.
Under the above given state feedback controller gain, the state responses of error system (4) and the control inputs u(t) are given in Figs. 7 and 8, where the initial conditions of the master and slave systems are chosen as x(0) = [0.7 − 0.3 0.4]T and z(0) = [−0.7 − 0.5 0.5]T . It can be seen from Fig. 7 that the 11
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6
4
ui(t)
2
0
−4
−6
0
2
4
6
8
10 time t
12
14
16
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−2
18
20
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Fig. 6. Control input u(t) in Example 1.
1.5
1
0.5
e(t)
0
M
−0.5
−1
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−1.5
−2
0
2
4
6
8
10 time t
12
14
16
18
20
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Fig. 7. State response of error system (4) with u(t) for c = 0 in Example 1.
synchronization error is tending to zero, which show the effectiveness of our control scheme.
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Example 2. Consider the master-slave synchronization of two unidirectionally coupled Chua’s circuits via
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sampled-data feedback control [32–34]. The system can be presented as: x˙ 1 (t) = a(x2 (t) − g(x1 (t))), x˙ 2 (t) = x1 (t) − x2 (t) + x3 (t), x˙ 3 (t) = −bx2 (t), x˙ 4 (t) = a(x5 (t) − g(x4 (t))), x˙ 5 (t) = x4 (t) − x5 (t) + x6 (t) + m(x5 (t) − x2 (t)), x˙ 6 (t) = −bx5 (t), 12
(20)
ACCEPTED MANUSCRIPT
6
4
ui(t)
2
0
−4
−6
0
2
4
6
8
10 time t
12
14
16
CR IP T
−2
18
20
with the nonlinear characteristic satisfying
AN US
Fig. 8. Control input u(t) for c = 0 in Example 1.
1 g(xi (t)) = l1 xi (t) + (l0 − l1 )(|xi (t) + c| − |xi (t) − c|), i = 1, 4, 2
and parameters l0 = −1/7, l1 = 2/7, a = 9, b = 14.28, c = 1, m = 0.01. Obviously, the system can be represented in the Lur’e system form with −al1
a
0
0
0
1
−1
1
0
0
M
0
−b
0
0
0
0
0
0
0
al1
a
0
0
−m
0
1
−1 + m
1
0
0
0
−b
0
0
0
0
−a(l0 − l1 ) 0 0 , W = 0 0 0
ED
A=
PT
1 0 0 0 0 , C = D = 0 −a(l0 − l1 ) 0 0 0 0 0
0
T
0 0 , 1 0 0
CE
with σ1 (x1 (t)) = 21 (|x1 (t)+c|−|x1 (t)−c|) and σ2 (x4 (t)) = 21 (|x4 (t)+c|−|x4 (t)−c|) belonging to sector [0, 1]. Choose υ1 = 0.1, υ2 = 1.0, υ3 = 0.75, υ4 = 0.0, υ5 = 0.2, and assume the initial values of the the master
and slave systems in (11) to be x(0) = [0.1 0.2 0.3 0.4 0.5 0.6]T and z(0) = [−0.8 − 0.6 − 0.4 − 0.2 0 0.2]T ,
AC
respectively. The response of error system (12) with u(t) = 0 is shown in Fig. 9. Applying Corollary 1, our obtained maximum sampling period h and the detailed comparison with
those in [32–34] are given in Table 3. From Table 3, it can be clearly seen that when compared with the existing results, our newly derived criterion gives a much larger maximum sampling period. For example, the maximum sampling period h is 0.25 in [33], 0.26 in [32], 0.28 in [34], and 0.51 in this paper, which shows
that our criterion provides a much less conservative result. Moreover, by Corollary 1, the corresponding
13
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6 4 2
e(t)
0
−4 −6 −8
0
10
20
30
40
time t
CR IP T
−2
50
AN US
Fig. 9. State responses of error system (12) without u(t) in Example 2.
Table 3: Comparison of maximum allowable sampling interval in Example 2.
Method
Corollary 1
h
0.51
Improvement rates (%)
K=
3.2151
0.1545
0.0004
0.0003
[34]
0.25
0.26
0.28
104.00
96.15
82.14
−2.9211
−0.0016
−0.0007
0.0054
0.0014
3.2165
0.1548
−2.9284
ED
[32]
M
state feedback controller gain is obtained as
[33]
T
.
Under the above given state feedback controller gain, the state responses of error system (12) and the
PT
control inputs u(t) are given in Figs. 10 and 11. It can be seen from Fig. 10 that the synchronization error is tending to zero, which demonstrates that the control law can guarantee the asymptotically synchronization
CE
of the master-slave system (11).
Example 3. Consider the master and slave systems in (11) with the following parameters [25, 26, 29, 32]:
−1
AC
A= 0 0
0
0
−1
0
0
−1
,
1.2
W = 1.24 0
−1.6 1 2.2
0
0.9 , 1.5
1
C=D= 0 0
0 1 0
0
T
0 , 1
which implies that the Lur’e system reduces to a neural network with three neurons. Moreover, the neuron activation functions σ(xi (t)) =
k1+ = k2+ = k3+ = 1.
1 2 (|xi (t)
+ 1| − |xi (t) − 1|), i = 1, 2, 3, and thus k1− = k2− = k3− = 0 and
The initial states of the master and slave systems are chosen as x(0) = [0.4 0.3 0.8]T and z(0) = [0.2 0.4 0.9]T , respectively. The trajectories of the master system and the slave system with u(t) = 0 are 14
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3
2
1
e(t)
0
−2
−3
−4
0
2
4
6
8
10 time t
12
14
16
CR IP T
−1
18
20
AN US
Fig. 10. State response of error system (12) with u(t) in Example 2.
8
6
4
ui(t)
2
0
M
−2
−4
ED
−6
−8
0
2
4
6
8
10 time t
12
14
16
18
20
PT
Fig. 11. Control input u(t) in Example 2.
shown in Figs. 12 and 13, respectively. Moreover, the response of error system (12) with u(t) = 0 is given in
CE
Fig. 14. From Fig. 14, we can found that the master system and slave system in (11) are non-synchronization under the condition u(t) = 0.
AC
The maximum sampling intervals by different methods are listed in Table 4. From this table, we can find that by applying the design methods proposed in [25], [26], [29], and [32], maximum sampling intervals are 0.1732, 0.3212, 0.3687, and 0.3818, respectively. While based on Corollary 1 in this paper, when υ1 = 0.855,
υ2 = 1.0, υ3 = 0.45, υ4 = −0.5, υ5 = 0.07, it can be calculated that maximum sampling interval h = 0.9140,
which is much larger than the ones provided in [25, 26, 29, 32]. It means our results is less conservative than the ones given in [25, 26, 29, 32]. Under the maximum allowable sampling interval h = 0.9140, the following state feedback controller is
15
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3 2
x3(t)
1 0 −1
−3 1.5 1
CR IP T
−2
2
0.5
1
0
0 −0.5
−1 −1
x2(t)
−2
x (t) 1
AN US
Fig. 12. State trajectory of master system M in (11) in Example 3.
3 2
z3(t)
1 0 −1
−3 1 0.5 0
M
−2
0
ED
−0.5
z2(t)
2 1 −1
−1
−2
z (t) 1
0.9290
−0.9602
−0.0225
0.7543
0.6297
1.3892
1.0455
CE
obtained:
PT
Fig. 13. State trajectory of slave system S with u(t) = 0 in (11) in Example 3.
AC
K = 0.5895 −0.3886
.
Under the above state feedback controller gain, and initial condition e(0) = [0.2 − 0.1 − 0.1]T , Figs. 15
and 16 depict the state trajectories of the error system (12) and the controller u(t), respectively. Fig. 15 shows that the synchronization error is tending to zero. Thus, we can synchronize successfully the master and slave systems by the proposed sampled-data controller. Remark 9. In this section, three numerical examples show the advantages of our obtained results. The first one is based on the standard Chua’s circuit. The second one shows the usefulness of our obtained results for higher dimension Chua’s system, since the number of the states is always large in real application. Then, 16
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5 4 3 2
e(t)
1 0
−2 −3 −4 −5
0
50
100 time t
150
CR IP T
−1
200
AN US
Fig. 14. State responses of error system (12) without u(t) in Example 3.
Table 4: Comparison of maximum allowable sampling interval in Example 3.
Method
Corollary 1
h
0.9140
Improvement rates (%)
[25]
[26]
[29]
[32]
0.1732
0.3212
0.3687
0.3818
427.71
184.55
147.89
139.39
ED
established synchronization criteria.
M
the third one is based on neural network. From the three examples, it shows the wide applicability of our
5. Conclusion
PT
In this paper, a sampled-data control scheme was proposed to investigate the master-slave synchronization problem for CLSs with time delays. In comparison with existing results on synchronization of CLSs with time delays, there are two advantages: first, a novel LKF with some new augmented terms is con-
CE
structed, which can fully capture the system characteristics and the available information on the actual sampling pattern. Second, the LKF is positive definite only requiring at sampling instants, while slackening
AC
restrictive conditions used in some existing results. As a result, the synchronization criteria improved the published results in the literature. In practical implementation, due to the limitation of the communication and computation resources, it
is necessary to use effective control strategies to reduce the utilization of the network bandwidth. Eventtriggered control [37, 38] can greatly reduce the utilization of the limited network bandwidth, since it only sends necessary sampling signal through the network. Therefore, how to design event-triggered sampling control to investigate synchronization for chaotic Lur’e system is our future research topic.
17
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0.2
0.15
e(t)
0.1
0.05
−0.05
−0.1 0
5
10 time t
CR IP T
0
15
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Fig. 15. State trajectory of the error system (12) with u(t) in Example 3.
0.3
0.2
0.1
ui(t)
0
M
−0.1
−0.2
ED
−0.3
−0.4
0
5
10
15
time t
PT
Fig. 16. Control input u(t) in Example 3.
CE
6. Acknowledgment
This research was particularly supported by the National Natural Science Foundation of China (NO.61533006).
AC
Appendix A
I¯1 = [In 0n,n ], I¯2 = [0n,n In ], Ii = [0n,(i−1)n In 0n,(5−i)n+m ] (i = 1, 2, 3), I4 = [0m,3n Im 0m,2n ], Ij = [0n,(j−2)n+m In 0n,(6−j)n ] (j = 5, 6), η1 (t) = col e(t), ˙ e(tp ) , η2 (t) = col e(t), e(t) − e(tp ) , Z t η3 (t) = col e(s)ds, e(t) − e(tp ) , η4 (t) = col e(t), e(t), ˙ e(tp ) , tp
η5 (t) = col
Z
t
tp
e(s)ds, e(t) − e(tp ), r(t)e(tp ) , ς(t, s) = col e(t), e(t + s) , 18
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Z t ςˆ(t) = col e(t), e(t), ˙ e(tp ), f (De(t), z(t)), e(s)ds , ς(t) = col ςˆ(t), e(t − d) , tp
R=
R11
R12
∗
R22
U¯12 =
U13 U23
,
X =
X11
X12
∗
X22
,
U11
U = ∗ ∗
U12
U13
U22
U23
∗
U33
U11 ¯ , U11 = ∗
U12 U22
,
− , ∆ = diag{δ1 , δ2 , . . . , δm }, Λ = diag{λ1 , λ2 , . . . , λm }, K − = diag{k1− , k2− , . . . , km },
+ K + = diag{k1+ , k2+ , . . . , km }, Y¯ = [I1T I2T · · · I6T ], Υ =
6 X
k=1
CR IP T
υk IkT , Yˆ = [Iˆ1T Iˆ2T · · · Iˆ5T ],
ˆ = Iˆi = [0n,(i−1)n In 0n,(4−i)n+m ] (i = 1, 2, 3), Iˆ4 = [0m,3n Im 0m,n ], Iˆ5 = [0n,3n+m In ], Υ
5 X
k=1
υk IˆkT ,
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Φ(hp , r(t), ρ) =Sym{I1T P I2 } + Sym{(I4 − K − DI1 )T ∆DI2 } + Sym{(K + DI1 − I4 )T ΛDI2 } + I1T Q1 I1 − I6T Q1 I6 + d2 I2T Q2 I2 − (I1 − I6 )T Q2 (I1 − I6 ) + (hp − r(t))[I2T I3T ]R[I2T I3T ]T − Sym{(I1 − I3 )T R12 I3 } − r(t)I3T R22 I3 −1 T ¯ T ¯ ¯ + ρr(t)YYR 11 Y Y + Sym{YY(I1 − I3 )}
+ Sym{(hp − r(t)) [I1T I1T − I3T ]X [I2T I2T ]T + [I5T I1T − I3T ]X¯ [I1T I2T ]T }
M
− [I1T I1T − I3T ]X [I1T I1T − I3T ]T + I3T X11 I3 − [I5T I1T − I3T ]X¯ [I5T I1T − I3T ]T π2 T T [I I1 − I3T ]U¯11 [I5T I1T − I3T ]T 4 5 π2 T −1 T I1T − I3T ]U¯12 I3 } + I (r(t)S + r(t)S T + ρSU33 S )I3 4 3
−
ED
+ h2 [I1T I2T I3T ]U[I1T I2T I3T ]T − π2 r(t)Sym{[I5T 4
PT
− Sym{(I4 − K − DI1 )T Θ(I4 − K + DI1 )} + Sym{ΥF(−I2 + AI1 + BI6 + WI4 )} ¯ 3 }, − Sym{ΥFCI
CE
ˆ p , r(t), ρ) =Sym{Iˆ1T P Iˆ2 } + Sym{(Iˆ4 − K − DIˆ1 )T ∆DIˆ2 } + Sym{(K + DIˆ1 − Iˆ4 )T ΛDIˆ2 } Φ(h
AC
+ (hp − r(t))[Iˆ2T Iˆ3T ]R[Iˆ2T Iˆ3T ]T − Sym{(Iˆ1 − Iˆ3 )T R12 Iˆ3 } − r(t)Iˆ3T R22 Iˆ3 −1 T ˆ T ˆ ˆ ˆ ˆ + ρr(t)YYR 11 Y Y + Sym{YY(I1 − I3 )} + Sym{(hp − r(t)) [Iˆ1T Iˆ1T − Iˆ3T ]X [Iˆ2T Iˆ2T ]T + [Iˆ5T Iˆ1T − Iˆ3T ]X¯ [Iˆ1T Iˆ2T ]T }
− [Iˆ1T Iˆ1T − Iˆ3T ]X [Iˆ1T Iˆ1T − Iˆ3T ]T + Iˆ3T X11 Iˆ3 − [Iˆ5T Iˆ1T − Iˆ3T ]X¯ [Iˆ5T Iˆ1T − Iˆ3T ]T
π2 ˆT ˆT ˆT ¯ ˆT ˆT ˆT T + h2 [Iˆ1T Iˆ2T Iˆ3T ]U[Iˆ1T Iˆ2T Iˆ3T ]T − [I I1 − I3 ]U11 [I5 I1 − I3 ] 4 5 π2 π2 ˆT −1 T ˆ r(t)Sym{[Iˆ5T Iˆ1T − Iˆ3T ]U¯12 Iˆ3 } + I (r(t)S + r(t)S T + ρSU33 S )I3 − 4 4 3 ˆ − Sym{(Iˆ4 − K − DIˆ1 )T Θ(Iˆ4 − K + DIˆ1 )} + Sym{ΥF(− Iˆ2 + AIˆ1 + W Iˆ4 )} ¯ Iˆ3 }. ˆ FC − Sym{Υ 19
ACCEPTED MANUSCRIPT
Appendix B Proof of Theorem 1. Consider the following LKF for the synchronization error system (4): V (t) =
6 X
tp ≤ t < tp+1 ,
Vk (t),
k=1
(21)
V1 (t) = eT (t)Pe(t), Z Di e(t) Z m m X X − V2 (t) = 2 δi (fi (s) − ki s)ds + 2 λi V3 (t) =
i=1 t
eT (s)Q1 e(s)ds + d
t−d
V4 (t) = (tp+1 − t)
Z
Z
0
−d
t
i=1
Z
(ki+ s − fi (s))ds,
t
e˙ T (s)Q2 e(s)dsdθ, ˙
t+θ
η1T (s)Rη1 (s)ds,
tp
0
AN US
Z
0
Di e(t)
CR IP T
where
T T T ¯ V5 (t) = (tp+1 − t) η2 (t)X η2 (t) − e (tp )X11 e(tp ) + η3 (t)X η3 (t) , V6 (t) = h2
Z
t
tp
η4T (s)Uη4 (s)ds −
π2 4
Z
t
tp
η5T (s)Uη5 (s)ds.
M
Now, let us calculate the time derivative of V (t) along the trajectories of the synchronization error system (4) yields the following result: 6 X
V˙ k (t),
(22)
ED
V˙ (t) =
k=1
where V˙ k (t) (k = 1, 2, . . . , 6) are calculated as follows:
m X i=1
(23)
δi (fi (Di e(t), z(t)) − ki− Di e(t))Di e(t) ˙ +2
CE
V˙ 2 (t) =2
PT
V˙ 1 (t) = 2eT (t)P e(t), ˙
−
T
+
m X i=1
λi (ki+ Di e(t) − fi (Di e(t), z(t)))Di e(t) ˙
(24)
T
AC
=2(f (De(t), z(t)) − K De(t)) ∆De(t) ˙ + 2(K De(t) − f (De(t), z(t))) ΛDe(t), ˙
V˙ 3 (t) =eT (t)Q1 e(t) − eT (t − d)Q1 e(t − d) + d2 e˙ T (t)Q2 e(t) ˙ −d T
T
2 T
Z
t
e˙ T (s)Q2 e(s)ds ˙
t−d
(25)
T
≤e (t)Q1 e(t) − e (t − d)Q1 e(t − d) + d e˙ (t)Q2 e(t) ˙ − (e(t) − e(t − d)) Q2 (e(t) − e(t − d)),
V˙ 4 (t) =(tp+1 − t)η1T (t)Rη1 (t) − =(tp+1 − t)η1T (t)Rη1 (t) −
Z
t
tp Z t tp
η1T (s)Rη1 (s)ds e˙ T1 (s)R11 e˙ 1 (s)ds − 2(e(t) − e(tp ))T R12 e(tp )
− r(t)eT (tp )R22 e(tp ), 20
(26)
ACCEPTED MANUSCRIPT
V˙ 5 (t) =2(tp+1 − t) η2T (t)X η˙ 2 (t) + η3T (t)X¯ η˙ 3 (t) − η2T (t)X η2 (t) − eT (tp )X11 e(tp ) + η3T (t)X¯ η3 (t) , (27)
π2 T η (t)Uη5 (t) V˙ 6 (t) =h2 η4T (t)Uη4 (t) − 4 5 2 π η3T (t)U¯11 η3 (t) + 2r(t)η3T (t)U¯12 e(tp ) + r2 (t)eT (tp )U33 e(tp ) . =h2 η4T (t)Uη4 (t) − 4 Note that U > 0, which implies U33 > 0. Thus, for any matrix S ∈
CR IP T
−1 T r(t)U33 + S U33 r(t)U33 + S ≥ 0,
which is equivalent to
−1 T −r(t)2 U33 ≤ r(t)S + r(t)S T + SU33 S .
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From Lemma 2, for any appropriate dimensional matrix Y, we have Z t T T e˙ T1 (s)R11 e˙ 1 (s)ds ≤ r(t)ς T (t)YR−1 − 11 Y ς(t) + 2ς (t)Y(e(t) − e(tp )). tp
(28)
(29)
(30)
According to (6), for any positive diagonal matrix Θ ∈
(31)
M
On the other hand, based on the system (4), for any appropriate dimensional matrix F , the following equation holds: Z t 0 = 2 υ1 eT (t) + υ2 e˙ T (t) + υ3 eT (tp ) + υ4 f T (De(t), z(t)) + υ5 eT (s)ds + υ6 eT (t − d) F tp
ED
− e(t) ˙ + Ae(t) + Be(t − d) + Wf (De(t), z(t)) − KCe(tp ) ,
(32)
PT
which can be transformed into
¯ 0 = 2ς T (t)ΥF[−e(t) ˙ + Ae(t) + Be(t − d) + Wf (De(t), z(t))] − 2ς T (t)ΥFCe(t p ),
CE
¯ = FK. where F
(33)
Combining (22)–(31) and (33), we get
AC
V˙ (t) ≤ς T (t)Φ(hp , r(t), 1)ς(t) hp − r(t) r(t) h − hp T Φ(0, 0, 1) + Φ(h, 0, 1) + Φ(h, h, 1) ς(t). =ς (t) h h h
(34)
According to Schur complement, it can be obtained from (8)–(10) that Φ(0, 0, 1) < 0, Φ(h, 0, 1) < 0, and
Φ(h, h, 1) < 0, which mean from (34) that V˙ (t) < 0, tp ≤ t < tp+1 , ς(t) 6= 0.
(35)
Next, under the condition of Theorem 1, we show the LKF (21) is positive definite. 21
ACCEPTED MANUSCRIPT
According to Jensen inequality in [29], we can get that Z 0 Z 0 V3 (t) ≥ eT (t + s)Q1 e(t + s)ds + (eT (t) − eT (t + s))Q2 (e(t) − e(t + s))ds. −d
(36)
−d
From (7) and (36), when ς(t, s) 6= 0, one has 1 d
Z
0
−d
eT (t)Pe(t)ds + V3 (t) ≥
Z
0
ς T (t, s)Ψς(t, s)ds > 0.
(37)
−d
CR IP T
V1 (t) + V3 (t) =
Noting V2 (t) ≥ 0 and Vi (tp ) = 0 (i = 4, 5, 6), we can find from (21) and (37) that V (tp ) ≥ V1 (tp ) + V3 (tp ) > 0,
(38)
which implies the LKF (21) is positive definite at t = tp , p = 0, 1, 2 . . ..
On the other hand, noting V1 (t), V2 (t) and V3 (t) are continuous functionals, we have
AN US
lim Vk (t) = Vk (tp ), k = 1, 2, 3.
t→t− p
Moreover, it can be found that Z tp lim− V4 (t) = (tp − tp ) η1T (s)Rη1 (s)ds = 0, t→tp
tp−1
tp
tp
η1T (s)Rη1 (s)ds
and
(40)
= 0,
M
lim V4 (t) = (tp+1 − tp )
t→t+ p
Z
(39)
AC
CE
PT
ED
h i X11 X12 e(tp ) − eT (tp−1 )X11 e(tp−1 ) lim V5 (t) =(tp − tp ) eT (tp ) eT (tp ) − eT (tp−1 ) t→t− p ∗ X22 e(tp ) − e(tp−1 ) R tp h R i e(s)ds tp t p−1 T T T ¯ + = 0, e (s)ds e (tp ) − e (tp−1 ) X tp−1 e(tp ) − e(tp−1 ) h i X11 X12 e(tp ) − eT (tp )X11 e(tp ) lim+ V5 (t) =(tp+1 − tp ) eT (tp ) eT (tp ) − eT (tp ) t→tp ∗ X22 e(tp ) − e(tp ) R tp h R i e(s)ds tp T tp T T ¯ = 0, + X e (s)ds e (tp ) − e (tp ) tp e(tp ) − e(tp ) (41)
which imply that V4 (t) and V5 (t) vanish before tp and after tp . According to Lemma 1, It can be found that lim V6 (t) ≥ 0,
(42)
t→t− p
which implies limt→t− V6 (t) ≥ V6 (tp ) = 0. p 22
ACCEPTED MANUSCRIPT
Therefore, from (39)–(42), we can obtain that V (t− p ) ≥ V (tp ), p = 0, 1, . . . .
(43)
From (35) and (43), we can find V (t) is monotonically decreasing for t ∈ [t0 , +∞).
V (t) > V (t− p+1 ) ≥ V (tp+1 ) ≥ 0, t ∈ [tp , tp+1 ), p = 0, 1, . . . . Thus, V (t) is positive definite.
CR IP T
By (35), (38) and (43), for ς(t) 6= 0, we have (44)
By (35), (43), and (44), we can obtain that the synchronization error system (4) is globally asymptotically stable. Therefore, the master system M and the slave system S in (1) are globally asymptotically
AN US
synchronous. This completes the proof of Theorem 1.
References
[1] T.L. Carroll, L.M. Pecora, Synchronizing chaotic systems, IEEE Trans. Circuits Syst. I, Fundam. Theory Appl. 38 (4) (1991) 453–456.
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Novel master-slave synchronization criteria of chaotic Lur’e systems with time delays using sampled-data control Ruimei Zhang, Deqiang Zeng, Shouming Zhong PII: DOI: Reference:
S0016-0032(17)30231-4 10.1016/j.jfranklin.2017.05.008 FI 2982
To appear in:
Journal of the Franklin Institute
Received date: Revised date: Accepted date:
3 October 2016 16 February 2017 1 May 2017
Please cite this article as: Ruimei Zhang, Deqiang Zeng, Shouming Zhong, Novel master-slave synchronization criteria of chaotic Lur’e systems with time delays using sampled-data control, Journal of the Franklin Institute (2017), doi: 10.1016/j.jfranklin.2017.05.008
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Novel master-slave synchronization criteria of chaotic Lur’e systems with time delays using sampled-data control✩ Ruimei Zhanga,∗, Deqiang Zengb , Shouming Zhonga a School
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of Mathematics Sciences, University of Electronic Science and Technology of China, Chengdu Sichuan 611731, PR China b Data Recovery Key Laboratory of Sichuan Province, College of Mathematics and Information Science, Neijiang Normal University, Neijiang 641100, PR China
Abstract
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This paper investigates the problem of master-slave synchronization of chaotic Lur’e systems (CLSs) with time delays by sampled-data control. First, a novel Lyapunov-Krasovskii functional (LKF) is constructed with some new augmented terms, which can fully capture the system characteristics and the available information on the actual sampling pattern. In comparison with existing results, the constraint condition of the positive definition of the LKF is more relax, since it is positive definite only requiring at sampling instants. Second, based on the LKF, a less conservative synchronization criterion is established. Third, the
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desired estimator gain can be designed in terms of the solution to linear matrix inequalities (LMIs). The obtained conditions ensure the master-slave synchronization of CLSs under a longer sampling period than
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remarkable existing works. Finally, three numerical simulations of Chua’s circuit and neural network are provided to show the effectiveness and advantages of the proposed results.
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Keywords:
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Synchronization, Chaotic Lur’e systems, Sampled-data control, Neural networks
1. Introduction
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Since the seminal paper by Pecora and Carroll was published in 1990, chaos synchronization has gained growing research interests as it has been widely applied to many areas, such as chaos generator design, information science, biological systems, chemical reaction, and secure communication [1–6, 8]. Simultaneously, it has been shown that many nonlinear systems, including Chua’s circuit, hyper chaotic attractors, and n-scroll attractors, can be represented as Lur’e-type systems. Thus, the master-slave synchronization of ✩ This
work was supported by the National Natural Science Foundation of China (NO. 61533006) author Email address: [email protected] (Ruimei Zhang)
∗ Corresponding
Preprint submitted to Journal of The Franklin Institute
May 8, 2017
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chaotic Lur’e systems (CLSs) has provoked widespread interests in many research and application fields, and a great number of interesting research results have been reported on this topic [4, 7, 10, 11, 13–16, 19, 27– 34]. Further, because of signal transmissions among neurons and the finite switching speed of amplifiers in the implementation of electrical circuits, time delays are unavoidably encountered CLSs, which may cause divergence, oscillation, or even instability. Accordingly, it is germane to take time delays into account for master-slave synchronization of CLSs.
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From the control strategy point of view, the control is an effective synchronization method for CLSs, and many control approaches have been proposed to synchronize CLSs such as time delay feedback control [10, 13, 16], impulsive control [15], adaptive control [12, 18], sampled-data control [9, 20], and proportional derivative control method [19]. Among these methods, sampled-data control, which uses only the sampled information of the system at its sampling instants to the controller, has garnered wide-scale attention in
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recent years [9, 11, 20–33]. Compared with some continuous control, the sampled-data control has many advantages, such as easy installation, high reliability, maintenance with low cost, and efficiency. Therefore, the investigation of the master-slave synchronization of CLSs with time delays via sampled-data control is important in both theory and application.
During utilizing the sampled-data control to synchronize the CLSs with time delays, how to choose the sampling period is an critical issue to be considered. It is well-known that if the sampling period is large,
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then its operating conditions such as computational burden, limitation of load, and communication capacity can be relaxed [28]. Thus, it is significant to consider the control design problem under a sampling period
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as large as possible. Up to now, many interesting results concerning on finding a longer maximal sampling period for CLSs with time delays have been published in [4, 11, 17, 20–33]. Very recently, by sampled-data control, sufficient conditions were obtained for global asymptotic synchronization of CLSs in [11]. Improved
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synchronization criteria for this problem were proposed in [27–29] by introducing some sawtooth structure terms. Further results on the master-slave synchronization of CLSs with time delays has been proposed in
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[30–33] by introducing some discontinuous LKFs. However, the obtained conditions in [25, 26, 30–32] may be conservative, since the constructed LKFs in these papers are required to be positive definite on the whole sampling interval. In fact, the conservativeness may be further reduced by relaxing this requirement to only
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requiring positive definite at sampling instants. In addition, in [11, 27–33], the main contribution of these work was to construct an appropriate LKF to get a longer sampling period. However, there still exists room for improving the synchronization criteria of [11, 27–33], because some important terms have been ignored in the LKFs proposed by [11, 27–33], and the available information on actual sampling pattern have not been fully adopted, which might lead to some degree of conservatism. Thus, it is necessary to construct an more appropriate LKF to derive less conservative synchronization criteria for the CLSs. Motivated by the above discussion, in this paper, the problem of master-slave synchronization of CLSs with time delays is investigated by using sampled-data control. The main contributions are summarized 2
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below. 1) A novel LKF with some new augmented terms is constructed. Notably, V5 (t) and V6 (t) are the first time to be proposed, which can take full advantage of the available information on the actual sampling pattern. 2) The paper presents relatively relaxed conservative synchronization criteria. The paper differs from the prior results in [25, 26, 30–32, 34] to restrict the constructed LKF to be positive define on the whole
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sampling intervals. In the present paper, the LKF is positive definite only requiring at sampling instants. 3) Compared with the sampling periods in the existing results in [11, 27–34], a longer sampling period is obtained in this paper.
The remainder of this paper is organized as follows. In Section 2, the problem formulation and some preliminaries are introduced. Our main results are presented in Section 3, where the novel sampled-data
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synchronization criterion of CLSs with time delays are obtained. In Section 4, numerical examples are given to demonstrate the effectiveness and the benefit of the proposed results. Finally, the conclusion is drawn in Section 5.
Notations: Throughout this paper,
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X and Y , the notation X > Y means that the matrix X − Y is positive define. diag{· · · } and col{· · · } stand for a block-diagonal matrix and a column vector, respectively. Sym{X} = X + X T . The symmetric term
for algebraic operations.
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in a matrix is denoted by ∗. If not explicitly stated, all matrices are assumed to be compatible dimensions
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2. Problem description and preliminaries Consider the following general master-slave type of delayed Lur’e systems with sampled-data feedback
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control [29–32]: x(t) ˙ = Ax(t) + Bx(t − d) + Wσ(Dx(t)), M : p(t) = Cx(t), z(t) ˙ = Az(t) + Bz(t − d) + Wσ(Dz(t)) + u(t), S : q(t) = Cz(t), C :
u(t) = K(p(tp ) − q(tp )),
(1)
tp ≤ t < tp+1 ,
which consists of the master system M , slave system S , and sampled-data feedback controller C . M and S with u(t) = 0 are identical chaotic time-delay Lur’e systems with state vectors x(t), z(t) ∈
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are known real matrices. K ∈
constant time delay. u(t) ∈
σi (α) − σi (β) ≤ ki+ , α−β
∀α, β, i = 1, 2, . . . , m.
(2)
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For sampled-data feedback synchronization, only discrete measurements of p(t) and q(t) can be used for synchronization purposes, that is, we only have the measurements p(tp ) and q(tp ) at the sampling instant tp . It is assumed that the control signal to be generated by using a zero-order-hold (ZOH) function with a sequence of hold times 0 = t0 < t1 < · · · < tp < · · · . The sampled-data control input may be represented as discrete control signal: tp ≤ t < tp+1 ,
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u(t) = ud (tp ) = ud (t − (t − tp )) = ud (t − r(t)),
where r(t) = t − tp , which is a piecewise linear function, and r(t) ˙ = 1. Sampling interval hp satisfies the following condition: 0 ≤ r(t) ≤ hp = tp+1 − tp ≤ h,
follows:
tp ≤ t < tp+1 ,
(3)
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u(t) = K[p(t − r(t)) − q(t − r(t))],
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where h is the largest sampling interval and bounded. Thus, the state feedback controller is designed as
where K is the sampled-data feedback controller gain matrix to be determined.
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Given the synchronization scheme (1), the synchronization error is defined as e(t) = x(t) − z(t). Then, from (1) and (3), we have the following synchronization error system:
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e(t) ˙ = Ae(t) + Be(t − d) + Wf (De(t), z(t)) − KCe(t − r(t)),
tp ≤ t < tp+1 ,
(4)
where f (De(t), z(t)) = σ(De(t) + Dz(t)) − σ(Dz(t)).
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Since σi (·) belongs to the sector [ki− , ki+ ], from (2), one can obtain that for ∀e(t), z(t) ki− ≤
fi (Di e(t), z(t)) σi (Di (e(t) + z(t))) − σi (Di z(t)) = ≤ ki+ , Di e(t) Di e(t)
Di e(t) 6= 0, i = 1, 2, . . . , m,
(5)
in which Di denotes the ith row vector of D. It is easily found from (5) that [fi (Di e(t), z(t)) − ki− Di e(t)] × [fi (Di e(t), z(t)) − ki+ Di e(t)] ≤ 0,
∀e(t), z(t), i = 1, 2, . . . , m.
(6)
Remark 1. In the condition (5), ki− and ki+ can be positive, negative or zero, which means that the resulting nonlinear function may be non-monotonic. While in [27–30], the class of globally Lipschitz nonlinear function 4
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was described under the condition of ki− = 0 and ki+ > 0. Therefore, the nonlinear description in this paper is quite general that covers the usual sigmoid nonlinear function as a special case, which is more effective to reduce the conservatism of the results. Throughout this paper, we introduce the following lemmas.
holds: Z
a
b
ξ T (s)Rξ(s)ds ≤
4(b − a)2 π2
Z
b
˙ ξ˙T (s)Rξ(s)ds.
a
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Lemma 1 [24]. Let ξ(t) ∈ W [a, b) and ξ(a) = 0. Then, for any n × n matrix R > 0, the following inequality
Lemma 2 [24]. Let Y > 0 and w(s) be an appropriate dimensional vector. Then, we have the following: Z t2 Z t2 w(s)ds, wT (s)Yw(s)ds ≤ (t2 − t1 )ξ T (t)F T Y −1 Fξ(t) + 2ξ T (t)F T − t1
t1
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where matrix F and vector ξ(t) independent on the integral variable are the arbitrary appropriate dimensional ones.
3. Main results
In this section, a new synchronization criterion is proposed for chaotic Lur’e system with time delays based on constructing a novel augmented LKF. The desired sampled-data controller is designed via the
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established conditions to achieve the master-slave synchronization of systems (1). Before presenting the main results, for the sake of representation convenience, The notations for some vectors and matrices are
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defined as follows (See Appendix A).
Theorem 1. Under condition (5), for given scalars d > 0, h > 0, υi (i = 1, 2, . . . , 6), the master sys-
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tem M and the slave system S in (1) are globally asymptotically synchronous, if there exist symmetric positive define matrices Q2 , R11 , U, positive diagonal matrix Θ, ∆, Λ, symmetric matrices P, Q1 , X , X¯ , 1 ¯T ¯ I P I1 + I¯2T Q1 I¯2 + (I¯1T − I¯2T )Q2 (I¯1 − I¯2 ) > 0, d 1 Φ(0, 0, 0) I3T S < 0, ∗ − π42 U33 Φ(h, 0, 0) I3T S < 0, ∗ − π42 U33 ¯ Φ(h, h, 0) I3T S YY < 0. ∗ − π42 U33 0 ∗ ∗ − h1 R11
Ψ=
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¯ such that and any appropriate dimensional matrices S, Y, F, F,
5
(7)
(8)
(9)
(10)
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Furthermore, the desired controller gain is given as ¯ K = F −1 F. Proof. For the detailed proofs, see Appendix B. Remark 2. It should be mentioned that, compared with the existing results in [25–34], a less conservative synchronization criterion is obtained in this work. There are two reasons for that: first, a newly augmented
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LKF is constructed in (21). Three (tp , tp+1 )-dependent terms V4 (t), V5 (t), and V6 (t) are constructed, which can take full advantage of the available information on the actual sampling pattern. Note that V5 (t) and V6 (t) are the first time to be proposed for synchronization of chaotic Lur’e system. However, the two (tp , tp+1 )-dependent terms V5 (t) and V6 (t) are neglected in [27–34]. Second, according to (35), (38) and (43), the positive definition of the LKF in (21) is ensured only requiring positive definite at the sampling
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instants. But in [25, 26, 30–32, 34], the proposed LKFs are positive definite on the whole sampling interval by restricting all the symmetric matrices to be positive or the sum of several terms of the LKF to be positive. Thus, an improved synchronization criterion is obtained here.
Z
t
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Remark 3. Note that when R12 = 0 and R22 = 0, V4 (t) in (21) reduces to (tp+1 − t) e˙ T (s)R11 e(s)ds, ˙ tp Z t which has been applied in [27, 29, 32]. That is, the terms (tp+1 − t) e˙ T (s)R12 e(tp )ds and (tp+1 − tp Z t T t) e (tp )R22 e(tp )ds are ignored in [27, 29, 32], which can make good use of the available information tp R11 R12 is restricted to be about the actual sampling pattern. On the other hand, in [30, 31], R = ∗ R22 positive. However, only R11 is needed to be positive in this paper. Therefore, the constraint condition of the matrices in V4 (t) is more relaxed than that in [30, 31].
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Remark 4. Inspired by the techniques introduced in [20], a completely new term V5 (t) is introduced in h i h iT X+X T −X+X e(t) e(t) 1 2 this paper. The term (tp+1 − t) e(tp ) T e(tp ) used in [25, 26] is constrained by ∗ −X1 −X1T + X+X 2 T X+X −X+X1 2 the structure of to keep the continuity of the LKF. But such a requirement is X+X T T ∗
−X1 −X1 +
2
not necessary any more for V5 (t). Thus, V5 (t) is more flexible and effective than that [25, 26] in terms of
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conservatism reduction.
Remark 5. In the LKF (21), V6 (t) contains the sampling instants tp , which could make full use of the sawtooth structure characteristic of the sampling Zinput delay. On the other Z hand, when U11 = 0, U12 = 0, U13 = t π2 t T 2 0, U23 = 0, and U33 = 0, V6 (t) reduces to h e˙ (s)U22 e(s)ds ˙ − (e(s) − e(tp ))T U22 (e(s) − e(tp ))ds, 4 tp tp which is similar to the term in the LKF in [20, 34]. That is to say, more harmonic matrices such as U11 , U12 , U13 , U23 , and U33 are introduced in V6 (t). Thus, V6 (t) has advantages to reduce the conservatism. Remark 6. When υ4 = υ5 = υ6 = 0, the equation in (32) can be reduced to 0 = 2(eT (t)υ1 + e˙ T (t)υ2 + eT (tp )υ3 )F[−e(t) ˙ + Ae(t) + Be(t − d) + Wf (De(t), z(t)) − KCe(tp )], 6
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which has been adopted in [29–32]. Compared with the existing equation in [29–32], more adjustable parameters are introduced to harmonize the relation among terms, which can provide great freedom in conservatism reducing. Remark 7. When hp = h, p = 0, 1, . . ., from (34), it is noted that the inequality V˙ (t) < 0 holds if Φ(h, 0, 1) < 0 and Φ(h, h, 1) < 0. Thus, the inequality (8) in Theorem 1 can be removed when the sampling is periodic.
C :
u(t) = K(p(tp ) − q(tp )),
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Remark 8. When B = 0, then system (1) is transformed into the following model without time delay: x(t) ˙ = Ax(t) + Wσ(Dx(t)), M : p(t) = Cx(t), z(t) (11) ˙ = Az(t) + Wσ(Dz(t)) + u(t), S : q(t) = Cz(t), tp ≤ t < tp+1 ,
which has been studied in [27–34]. Correspondingly, the synchronization error system (4) is changed to e(t) ˙ = Ae(t) + Wf (De(t), z(t)) − KCe(t − r(t)), and the LKF in (21) reduces to 6 X
k=1
Vk (t), k 6= 3, tp ≤ t < tp+1 ,
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V (t) =
tp ≤ t < tp+1 ,
(12)
(13)
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where Vk (t), k = 1, 2, 4, 5, 6 are given in (21). Similar to the proof of Theorem 1, One can obtain asymptotically synchronous criterion as follows.
Corollary 1. Under condition (5), for given scalars h > 0, υi (i = 1, 2, . . . , 5), the master system M and
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the slave system S in (11) are globally asymptotically synchronous, if there exist symmetric positive define matrices P, R11 , U, positive diagonal matrix Θ, ∆, Λ, symmetric matrices X , X¯ , and any appropriate
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¯ such that dimensional matrices S, Y, F, F, ˆ 0, 0) Iˆ T S Φ(0, 3 < 0, ∗ − π42 U33 ˆ Φ(h, 0, 0) Iˆ3T S < 0, ∗ − π42 U33 T ˆ ˆ ˆ Φ(h, h, 0) I3 S YY 4 ∗ < 0. − π2 U33 0 ∗ ∗ − h1 R11
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(14)
(15)
(16)
Furthermore, the desired controller gain is given as ¯ K = F −1 F. 7
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4. Numerical examples
Fig. 1. Standard Chua’s circuit and v − i characteristic of Chua’s diode [35].
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In this section, three examples are used to demonstrate the validity and less conservativeness of the proposed new synchronization scheme.
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Example 1. Consider the following standard Chua’s circuit (See Fig. 1), which can be expressed as [35]: dνc1 1 C1 = (νc2 − νc1 ) − gNR (νc1 ), dt R 1 dνc2 (17) = (νc1 − νc2 ) − iL , C2 dt R diL L = −νc2 , dt
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where νc1 and νc2 denote the voltages across the capacitors C1 and C2 , respectively. iL is the current through the inductor L, and the nonlinear function gNR (νc1 ), which defines the ν − i characteristic of Chua’s diode,
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is described by the piecewise-linear function (See Fig. 1) 1 gNR (νc1 ) = Gb νc1 + (Ga − Gb )(|νc1 + E| − |νc1 − E|). 2
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The slopes in the inner and outer regions are Ga , Gb < 0, respectively, which ±E denotes the breakpoints. By recalling the variable: x1 =
νc1 E
, x2 =
νc2 E
, x3 =
iL EG ,
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version of standard Chua’s circuit is rewritten by x˙ 1 = a(x2 − h(x1 )), x˙ 2 = x1 − x2 + x3 , x˙ = −bx , 3 2
where x˙ =
dx dτ ,
a=
C2 C1
> 0 and b =
C2 G2 L
τ =
Gt C2
with G =
1 R.
Therefore, the normalized
(18)
> 0 are the main bifurcation parameters. The nonlinear function is
given by 1 h(x1 ) = m1 x1 + (m0 − m1 )(|x1 + c| − |x1 − c|), 2 8
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with m1 =
Gb G ,
m0 =
Ga G .
With the parameter values: m0 = −1/7, m1 = 2/7, a = 9, b = 14.28, c = 1, the
Chua’s system will obtain the double scroll attractor [36]. Moreover, in the implementation of Lurie network systems, it is often inevitable to introduce time delay in the signals transmitted among neurons because it is an inherent feature of many physical processes.
Chua’s circuit with constant time delay can be expressed as: x˙ 1 (t) = a(x2 (t) − m1 x1 (t) − σ(x1 (t))) − cx1 (t − d), x˙ 2 (t) = x1 (t) − x2 (t) + x3 (t) − cx1 (t − d), x˙ (t) = −bx (t) + c(2x (t − d) − x (t − d)). 3 2 1 3
CR IP T
Therefore, consider the following delayed Chua’s circuit via sampled-data control [29–32]. The equation of
(19)
It can be found that the Chua’s circuit can be rewritten as the Lur’e form with the following parameters: −am1
A= 1 0
a −1 −b
0
1 , 0
−c 0
B = −c 0 2c 0
0
−a(m0 − m1 )
AN US
,
0
−c
W=
0 0
,
1
T
C=D= 0 , 0
with σ(x1 (t)) = 21 (|x1 (t) + 1| − |x1 (t) − 1|) belonging to sector [0, 1], c = 0.1 and d = 1. In this example, we choose υ1 = 0.0, υ2 = 1.0, υ3 = 1.15, υ4 = 0.1, υ5 = 0.5, and υ6 = 0.1.
M
The initial conditions of the master and slave systems are chosen as x(0) = [0.2 0.3 0.2]T and z(0) = [−0.3 − 0.1 0.4]T , t ∈ [−1, 0]. Figs. 2 and 3 show the master system and the slave system states with
ED
u(t) = 0, respectively. In addition, the response curves of error system (4) with u(t) = 0 is given in Fig. 4. From Fig. 4, we can found that the master system and slave system in (1) are non-synchronization under
PT
the condition u(t) = 0.
AC
0 x3(t)
CE
2
−2
−4 1
0.5
0 x2(t)
−0.5
−2
−1
0
1
2
3
x (t) 1
Fig. 2. State trajectory of master system M in (1) in Example 1.
9
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2
z3(t)
0
−2
CR IP T
−4 1
0.5
0 z2(t)
−0.5
−1
−2
2
1
0 z1(t)
3
AN US
Fig. 3. State trajectory of slave system S with u(t) = 0 in (1) in Example 1.
8
6
4
e(t)
2
0
M
−2
−4
ED
−6
−8
0
10
20
30
40
50 time t
60
70
80
90
100
PT
Fig. 4. State responses of error system (4) without u(t) in Example 1.
Table 1 provides some comparisons of the maximum sampling intervals involved in the results of this
CE
paper and those in [29], [30], [31], [32], and [4]. From Table 1, it can be seen that our results provide a larger sampling interval than others, which show the advantage of the results of this paper.
AC
In order to demonstrate ulteriorly the superiority of our results, by applying Theorem 1, the following state feedback controller is obtained under the maximum sampling interval h = 0.4789: K=
h
3.5324
0.2411
−2.9187
iT
.
Under the above controller gain, the error signal e(t) and control inputs u(t) are given in Figs. 5 and 6, respectively. It can be seen from Fig. 5 that the synchronization error is tending to zero, which shows the master and slave systems in (1) can achieve synchronization successfully by the proposed sampled-data controller. Moreover, when t = 20, h = 0.4789, the simulation results in Fig. 5 imply that only 41 sampled 10
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Table 1: Comparison of maximum allowable sampling interval in Example 1.
Method
[29]
[30]
[31]
[32]
[4]
Theorem 1
h
0.3582
0.4355
0.4357
0.4369
0.4371
0.4789
Table 2: Comparison of maximum allowable sampling interval for c = 0 in Example 1.
[27]
[29]
[28]
[30]
[31]
Corollary 1
h
0.3914
0.3981
0.4800
0.5144
0.5147
0.5185
CR IP T
Method
signals need to be sent out to the controller, which shows that the sampled-data control uses less information of the system than that by feedback control [8]. 2
AN US
1.5
1
e(t)
0.5
0
−0.5
−1.5
0
2
4
6
8
10 time t
12
14
16
18
20
ED
−2
M
−1
Fig. 5. State response of error system (4) with u(t) in Example 1.
PT
When c = 0, the time-delay Chua’s circuit system (19) becomes a Lur’e system without time delay, which has been investigated in [27–31]. The maximum sampling intervals by different methods are listed in
CE
Table 2. From this table, we can find that by using the method proposed in [27], [29], [28], [30], and [31], the maximum sampling intervals are 0.3914, 0.3981, 0.4800, 0.5144 and 0.5147, respectively. When υ1 = 0.09, υ2 = 1.0, υ3 = 0.75, υ4 = 0.0, υ5 = 0.15, the maximum sampling interval h by Corollary 1 is 0.5185. It can
AC
be seen that the synchronization criterion in this paper is less conservative than the existing ones. Moreover, the corresponding state feedback controller gain is K=
h
3.1756
0.1406
−2.8992
iT
.
Under the above given state feedback controller gain, the state responses of error system (4) and the control inputs u(t) are given in Figs. 7 and 8, where the initial conditions of the master and slave systems are chosen as x(0) = [0.7 − 0.3 0.4]T and z(0) = [−0.7 − 0.5 0.5]T . It can be seen from Fig. 7 that the 11
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6
4
ui(t)
2
0
−4
−6
0
2
4
6
8
10 time t
12
14
16
CR IP T
−2
18
20
AN US
Fig. 6. Control input u(t) in Example 1.
1.5
1
0.5
e(t)
0
M
−0.5
−1
ED
−1.5
−2
0
2
4
6
8
10 time t
12
14
16
18
20
PT
Fig. 7. State response of error system (4) with u(t) for c = 0 in Example 1.
synchronization error is tending to zero, which show the effectiveness of our control scheme.
CE
Example 2. Consider the master-slave synchronization of two unidirectionally coupled Chua’s circuits via
AC
sampled-data feedback control [32–34]. The system can be presented as: x˙ 1 (t) = a(x2 (t) − g(x1 (t))), x˙ 2 (t) = x1 (t) − x2 (t) + x3 (t), x˙ 3 (t) = −bx2 (t), x˙ 4 (t) = a(x5 (t) − g(x4 (t))), x˙ 5 (t) = x4 (t) − x5 (t) + x6 (t) + m(x5 (t) − x2 (t)), x˙ 6 (t) = −bx5 (t), 12
(20)
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6
4
ui(t)
2
0
−4
−6
0
2
4
6
8
10 time t
12
14
16
CR IP T
−2
18
20
with the nonlinear characteristic satisfying
AN US
Fig. 8. Control input u(t) for c = 0 in Example 1.
1 g(xi (t)) = l1 xi (t) + (l0 − l1 )(|xi (t) + c| − |xi (t) − c|), i = 1, 4, 2
and parameters l0 = −1/7, l1 = 2/7, a = 9, b = 14.28, c = 1, m = 0.01. Obviously, the system can be represented in the Lur’e system form with −al1
a
0
0
0
1
−1
1
0
0
M
0
−b
0
0
0
0
0
0
0
al1
a
0
0
−m
0
1
−1 + m
1
0
0
0
−b
0
0
0
0
−a(l0 − l1 ) 0 0 , W = 0 0 0
ED
A=
PT
1 0 0 0 0 , C = D = 0 −a(l0 − l1 ) 0 0 0 0 0
0
T
0 0 , 1 0 0
CE
with σ1 (x1 (t)) = 21 (|x1 (t)+c|−|x1 (t)−c|) and σ2 (x4 (t)) = 21 (|x4 (t)+c|−|x4 (t)−c|) belonging to sector [0, 1]. Choose υ1 = 0.1, υ2 = 1.0, υ3 = 0.75, υ4 = 0.0, υ5 = 0.2, and assume the initial values of the the master
and slave systems in (11) to be x(0) = [0.1 0.2 0.3 0.4 0.5 0.6]T and z(0) = [−0.8 − 0.6 − 0.4 − 0.2 0 0.2]T ,
AC
respectively. The response of error system (12) with u(t) = 0 is shown in Fig. 9. Applying Corollary 1, our obtained maximum sampling period h and the detailed comparison with
those in [32–34] are given in Table 3. From Table 3, it can be clearly seen that when compared with the existing results, our newly derived criterion gives a much larger maximum sampling period. For example, the maximum sampling period h is 0.25 in [33], 0.26 in [32], 0.28 in [34], and 0.51 in this paper, which shows
that our criterion provides a much less conservative result. Moreover, by Corollary 1, the corresponding
13
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6 4 2
e(t)
0
−4 −6 −8
0
10
20
30
40
time t
CR IP T
−2
50
AN US
Fig. 9. State responses of error system (12) without u(t) in Example 2.
Table 3: Comparison of maximum allowable sampling interval in Example 2.
Method
Corollary 1
h
0.51
Improvement rates (%)
K=
3.2151
0.1545
0.0004
0.0003
[34]
0.25
0.26
0.28
104.00
96.15
82.14
−2.9211
−0.0016
−0.0007
0.0054
0.0014
3.2165
0.1548
−2.9284
ED
[32]
M
state feedback controller gain is obtained as
[33]
T
.
Under the above given state feedback controller gain, the state responses of error system (12) and the
PT
control inputs u(t) are given in Figs. 10 and 11. It can be seen from Fig. 10 that the synchronization error is tending to zero, which demonstrates that the control law can guarantee the asymptotically synchronization
CE
of the master-slave system (11).
Example 3. Consider the master and slave systems in (11) with the following parameters [25, 26, 29, 32]:
−1
AC
A= 0 0
0
0
−1
0
0
−1
,
1.2
W = 1.24 0
−1.6 1 2.2
0
0.9 , 1.5
1
C=D= 0 0
0 1 0
0
T
0 , 1
which implies that the Lur’e system reduces to a neural network with three neurons. Moreover, the neuron activation functions σ(xi (t)) =
k1+ = k2+ = k3+ = 1.
1 2 (|xi (t)
+ 1| − |xi (t) − 1|), i = 1, 2, 3, and thus k1− = k2− = k3− = 0 and
The initial states of the master and slave systems are chosen as x(0) = [0.4 0.3 0.8]T and z(0) = [0.2 0.4 0.9]T , respectively. The trajectories of the master system and the slave system with u(t) = 0 are 14
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3
2
1
e(t)
0
−2
−3
−4
0
2
4
6
8
10 time t
12
14
16
CR IP T
−1
18
20
AN US
Fig. 10. State response of error system (12) with u(t) in Example 2.
8
6
4
ui(t)
2
0
M
−2
−4
ED
−6
−8
0
2
4
6
8
10 time t
12
14
16
18
20
PT
Fig. 11. Control input u(t) in Example 2.
shown in Figs. 12 and 13, respectively. Moreover, the response of error system (12) with u(t) = 0 is given in
CE
Fig. 14. From Fig. 14, we can found that the master system and slave system in (11) are non-synchronization under the condition u(t) = 0.
AC
The maximum sampling intervals by different methods are listed in Table 4. From this table, we can find that by applying the design methods proposed in [25], [26], [29], and [32], maximum sampling intervals are 0.1732, 0.3212, 0.3687, and 0.3818, respectively. While based on Corollary 1 in this paper, when υ1 = 0.855,
υ2 = 1.0, υ3 = 0.45, υ4 = −0.5, υ5 = 0.07, it can be calculated that maximum sampling interval h = 0.9140,
which is much larger than the ones provided in [25, 26, 29, 32]. It means our results is less conservative than the ones given in [25, 26, 29, 32]. Under the maximum allowable sampling interval h = 0.9140, the following state feedback controller is
15
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3 2
x3(t)
1 0 −1
−3 1.5 1
CR IP T
−2
2
0.5
1
0
0 −0.5
−1 −1
x2(t)
−2
x (t) 1
AN US
Fig. 12. State trajectory of master system M in (11) in Example 3.
3 2
z3(t)
1 0 −1
−3 1 0.5 0
M
−2
0
ED
−0.5
z2(t)
2 1 −1
−1
−2
z (t) 1
0.9290
−0.9602
−0.0225
0.7543
0.6297
1.3892
1.0455
CE
obtained:
PT
Fig. 13. State trajectory of slave system S with u(t) = 0 in (11) in Example 3.
AC
K = 0.5895 −0.3886
.
Under the above state feedback controller gain, and initial condition e(0) = [0.2 − 0.1 − 0.1]T , Figs. 15
and 16 depict the state trajectories of the error system (12) and the controller u(t), respectively. Fig. 15 shows that the synchronization error is tending to zero. Thus, we can synchronize successfully the master and slave systems by the proposed sampled-data controller. Remark 9. In this section, three numerical examples show the advantages of our obtained results. The first one is based on the standard Chua’s circuit. The second one shows the usefulness of our obtained results for higher dimension Chua’s system, since the number of the states is always large in real application. Then, 16
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5 4 3 2
e(t)
1 0
−2 −3 −4 −5
0
50
100 time t
150
CR IP T
−1
200
AN US
Fig. 14. State responses of error system (12) without u(t) in Example 3.
Table 4: Comparison of maximum allowable sampling interval in Example 3.
Method
Corollary 1
h
0.9140
Improvement rates (%)
[25]
[26]
[29]
[32]
0.1732
0.3212
0.3687
0.3818
427.71
184.55
147.89
139.39
ED
established synchronization criteria.
M
the third one is based on neural network. From the three examples, it shows the wide applicability of our
5. Conclusion
PT
In this paper, a sampled-data control scheme was proposed to investigate the master-slave synchronization problem for CLSs with time delays. In comparison with existing results on synchronization of CLSs with time delays, there are two advantages: first, a novel LKF with some new augmented terms is con-
CE
structed, which can fully capture the system characteristics and the available information on the actual sampling pattern. Second, the LKF is positive definite only requiring at sampling instants, while slackening
AC
restrictive conditions used in some existing results. As a result, the synchronization criteria improved the published results in the literature. In practical implementation, due to the limitation of the communication and computation resources, it
is necessary to use effective control strategies to reduce the utilization of the network bandwidth. Eventtriggered control [37, 38] can greatly reduce the utilization of the limited network bandwidth, since it only sends necessary sampling signal through the network. Therefore, how to design event-triggered sampling control to investigate synchronization for chaotic Lur’e system is our future research topic.
17
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0.2
0.15
e(t)
0.1
0.05
−0.05
−0.1 0
5
10 time t
CR IP T
0
15
AN US
Fig. 15. State trajectory of the error system (12) with u(t) in Example 3.
0.3
0.2
0.1
ui(t)
0
M
−0.1
−0.2
ED
−0.3
−0.4
0
5
10
15
time t
PT
Fig. 16. Control input u(t) in Example 3.
CE
6. Acknowledgment
This research was particularly supported by the National Natural Science Foundation of China (NO.61533006).
AC
Appendix A
I¯1 = [In 0n,n ], I¯2 = [0n,n In ], Ii = [0n,(i−1)n In 0n,(5−i)n+m ] (i = 1, 2, 3), I4 = [0m,3n Im 0m,2n ], Ij = [0n,(j−2)n+m In 0n,(6−j)n ] (j = 5, 6), η1 (t) = col e(t), ˙ e(tp ) , η2 (t) = col e(t), e(t) − e(tp ) , Z t η3 (t) = col e(s)ds, e(t) − e(tp ) , η4 (t) = col e(t), e(t), ˙ e(tp ) , tp
η5 (t) = col
Z
t
tp
e(s)ds, e(t) − e(tp ), r(t)e(tp ) , ς(t, s) = col e(t), e(t + s) , 18
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Z t ςˆ(t) = col e(t), e(t), ˙ e(tp ), f (De(t), z(t)), e(s)ds , ς(t) = col ςˆ(t), e(t − d) , tp
R=
R11
R12
∗
R22
U¯12 =
U13 U23
,
X =
X11
X12
∗
X22
,
U11
U = ∗ ∗
U12
U13
U22
U23
∗
U33
U11 ¯ , U11 = ∗
U12 U22
,
− , ∆ = diag{δ1 , δ2 , . . . , δm }, Λ = diag{λ1 , λ2 , . . . , λm }, K − = diag{k1− , k2− , . . . , km },
+ K + = diag{k1+ , k2+ , . . . , km }, Y¯ = [I1T I2T · · · I6T ], Υ =
6 X
k=1
CR IP T
υk IkT , Yˆ = [Iˆ1T Iˆ2T · · · Iˆ5T ],
ˆ = Iˆi = [0n,(i−1)n In 0n,(4−i)n+m ] (i = 1, 2, 3), Iˆ4 = [0m,3n Im 0m,n ], Iˆ5 = [0n,3n+m In ], Υ
5 X
k=1
υk IˆkT ,
AN US
Φ(hp , r(t), ρ) =Sym{I1T P I2 } + Sym{(I4 − K − DI1 )T ∆DI2 } + Sym{(K + DI1 − I4 )T ΛDI2 } + I1T Q1 I1 − I6T Q1 I6 + d2 I2T Q2 I2 − (I1 − I6 )T Q2 (I1 − I6 ) + (hp − r(t))[I2T I3T ]R[I2T I3T ]T − Sym{(I1 − I3 )T R12 I3 } − r(t)I3T R22 I3 −1 T ¯ T ¯ ¯ + ρr(t)YYR 11 Y Y + Sym{YY(I1 − I3 )}
+ Sym{(hp − r(t)) [I1T I1T − I3T ]X [I2T I2T ]T + [I5T I1T − I3T ]X¯ [I1T I2T ]T }
M
− [I1T I1T − I3T ]X [I1T I1T − I3T ]T + I3T X11 I3 − [I5T I1T − I3T ]X¯ [I5T I1T − I3T ]T π2 T T [I I1 − I3T ]U¯11 [I5T I1T − I3T ]T 4 5 π2 T −1 T I1T − I3T ]U¯12 I3 } + I (r(t)S + r(t)S T + ρSU33 S )I3 4 3
−
ED
+ h2 [I1T I2T I3T ]U[I1T I2T I3T ]T − π2 r(t)Sym{[I5T 4
PT
− Sym{(I4 − K − DI1 )T Θ(I4 − K + DI1 )} + Sym{ΥF(−I2 + AI1 + BI6 + WI4 )} ¯ 3 }, − Sym{ΥFCI
CE
ˆ p , r(t), ρ) =Sym{Iˆ1T P Iˆ2 } + Sym{(Iˆ4 − K − DIˆ1 )T ∆DIˆ2 } + Sym{(K + DIˆ1 − Iˆ4 )T ΛDIˆ2 } Φ(h
AC
+ (hp − r(t))[Iˆ2T Iˆ3T ]R[Iˆ2T Iˆ3T ]T − Sym{(Iˆ1 − Iˆ3 )T R12 Iˆ3 } − r(t)Iˆ3T R22 Iˆ3 −1 T ˆ T ˆ ˆ ˆ ˆ + ρr(t)YYR 11 Y Y + Sym{YY(I1 − I3 )} + Sym{(hp − r(t)) [Iˆ1T Iˆ1T − Iˆ3T ]X [Iˆ2T Iˆ2T ]T + [Iˆ5T Iˆ1T − Iˆ3T ]X¯ [Iˆ1T Iˆ2T ]T }
− [Iˆ1T Iˆ1T − Iˆ3T ]X [Iˆ1T Iˆ1T − Iˆ3T ]T + Iˆ3T X11 Iˆ3 − [Iˆ5T Iˆ1T − Iˆ3T ]X¯ [Iˆ5T Iˆ1T − Iˆ3T ]T
π2 ˆT ˆT ˆT ¯ ˆT ˆT ˆT T + h2 [Iˆ1T Iˆ2T Iˆ3T ]U[Iˆ1T Iˆ2T Iˆ3T ]T − [I I1 − I3 ]U11 [I5 I1 − I3 ] 4 5 π2 π2 ˆT −1 T ˆ r(t)Sym{[Iˆ5T Iˆ1T − Iˆ3T ]U¯12 Iˆ3 } + I (r(t)S + r(t)S T + ρSU33 S )I3 − 4 4 3 ˆ − Sym{(Iˆ4 − K − DIˆ1 )T Θ(Iˆ4 − K + DIˆ1 )} + Sym{ΥF(− Iˆ2 + AIˆ1 + W Iˆ4 )} ¯ Iˆ3 }. ˆ FC − Sym{Υ 19
ACCEPTED MANUSCRIPT
Appendix B Proof of Theorem 1. Consider the following LKF for the synchronization error system (4): V (t) =
6 X
tp ≤ t < tp+1 ,
Vk (t),
k=1
(21)
V1 (t) = eT (t)Pe(t), Z Di e(t) Z m m X X − V2 (t) = 2 δi (fi (s) − ki s)ds + 2 λi V3 (t) =
i=1 t
eT (s)Q1 e(s)ds + d
t−d
V4 (t) = (tp+1 − t)
Z
Z
0
−d
t
i=1
Z
(ki+ s − fi (s))ds,
t
e˙ T (s)Q2 e(s)dsdθ, ˙
t+θ
η1T (s)Rη1 (s)ds,
tp
0
AN US
Z
0
Di e(t)
CR IP T
where
T T T ¯ V5 (t) = (tp+1 − t) η2 (t)X η2 (t) − e (tp )X11 e(tp ) + η3 (t)X η3 (t) , V6 (t) = h2
Z
t
tp
η4T (s)Uη4 (s)ds −
π2 4
Z
t
tp
η5T (s)Uη5 (s)ds.
M
Now, let us calculate the time derivative of V (t) along the trajectories of the synchronization error system (4) yields the following result: 6 X
V˙ k (t),
(22)
ED
V˙ (t) =
k=1
where V˙ k (t) (k = 1, 2, . . . , 6) are calculated as follows:
m X i=1
(23)
δi (fi (Di e(t), z(t)) − ki− Di e(t))Di e(t) ˙ +2
CE
V˙ 2 (t) =2
PT
V˙ 1 (t) = 2eT (t)P e(t), ˙
−
T
+
m X i=1
λi (ki+ Di e(t) − fi (Di e(t), z(t)))Di e(t) ˙
(24)
T
AC
=2(f (De(t), z(t)) − K De(t)) ∆De(t) ˙ + 2(K De(t) − f (De(t), z(t))) ΛDe(t), ˙
V˙ 3 (t) =eT (t)Q1 e(t) − eT (t − d)Q1 e(t − d) + d2 e˙ T (t)Q2 e(t) ˙ −d T
T
2 T
Z
t
e˙ T (s)Q2 e(s)ds ˙
t−d
(25)
T
≤e (t)Q1 e(t) − e (t − d)Q1 e(t − d) + d e˙ (t)Q2 e(t) ˙ − (e(t) − e(t − d)) Q2 (e(t) − e(t − d)),
V˙ 4 (t) =(tp+1 − t)η1T (t)Rη1 (t) − =(tp+1 − t)η1T (t)Rη1 (t) −
Z
t
tp Z t tp
η1T (s)Rη1 (s)ds e˙ T1 (s)R11 e˙ 1 (s)ds − 2(e(t) − e(tp ))T R12 e(tp )
− r(t)eT (tp )R22 e(tp ), 20
(26)
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V˙ 5 (t) =2(tp+1 − t) η2T (t)X η˙ 2 (t) + η3T (t)X¯ η˙ 3 (t) − η2T (t)X η2 (t) − eT (tp )X11 e(tp ) + η3T (t)X¯ η3 (t) , (27)
π2 T η (t)Uη5 (t) V˙ 6 (t) =h2 η4T (t)Uη4 (t) − 4 5 2 π η3T (t)U¯11 η3 (t) + 2r(t)η3T (t)U¯12 e(tp ) + r2 (t)eT (tp )U33 e(tp ) . =h2 η4T (t)Uη4 (t) − 4 Note that U > 0, which implies U33 > 0. Thus, for any matrix S ∈
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−1 T r(t)U33 + S U33 r(t)U33 + S ≥ 0,
which is equivalent to
−1 T −r(t)2 U33 ≤ r(t)S + r(t)S T + SU33 S .
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From Lemma 2, for any appropriate dimensional matrix Y, we have Z t T T e˙ T1 (s)R11 e˙ 1 (s)ds ≤ r(t)ς T (t)YR−1 − 11 Y ς(t) + 2ς (t)Y(e(t) − e(tp )). tp
(28)
(29)
(30)
According to (6), for any positive diagonal matrix Θ ∈
(31)
M
On the other hand, based on the system (4), for any appropriate dimensional matrix F , the following equation holds: Z t 0 = 2 υ1 eT (t) + υ2 e˙ T (t) + υ3 eT (tp ) + υ4 f T (De(t), z(t)) + υ5 eT (s)ds + υ6 eT (t − d) F tp
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− e(t) ˙ + Ae(t) + Be(t − d) + Wf (De(t), z(t)) − KCe(tp ) ,
(32)
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which can be transformed into
¯ 0 = 2ς T (t)ΥF[−e(t) ˙ + Ae(t) + Be(t − d) + Wf (De(t), z(t))] − 2ς T (t)ΥFCe(t p ),
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¯ = FK. where F
(33)
Combining (22)–(31) and (33), we get
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V˙ (t) ≤ς T (t)Φ(hp , r(t), 1)ς(t) hp − r(t) r(t) h − hp T Φ(0, 0, 1) + Φ(h, 0, 1) + Φ(h, h, 1) ς(t). =ς (t) h h h
(34)
According to Schur complement, it can be obtained from (8)–(10) that Φ(0, 0, 1) < 0, Φ(h, 0, 1) < 0, and
Φ(h, h, 1) < 0, which mean from (34) that V˙ (t) < 0, tp ≤ t < tp+1 , ς(t) 6= 0.
(35)
Next, under the condition of Theorem 1, we show the LKF (21) is positive definite. 21
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According to Jensen inequality in [29], we can get that Z 0 Z 0 V3 (t) ≥ eT (t + s)Q1 e(t + s)ds + (eT (t) − eT (t + s))Q2 (e(t) − e(t + s))ds. −d
(36)
−d
From (7) and (36), when ς(t, s) 6= 0, one has 1 d
Z
0
−d
eT (t)Pe(t)ds + V3 (t) ≥
Z
0
ς T (t, s)Ψς(t, s)ds > 0.
(37)
−d
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V1 (t) + V3 (t) =
Noting V2 (t) ≥ 0 and Vi (tp ) = 0 (i = 4, 5, 6), we can find from (21) and (37) that V (tp ) ≥ V1 (tp ) + V3 (tp ) > 0,
(38)
which implies the LKF (21) is positive definite at t = tp , p = 0, 1, 2 . . ..
On the other hand, noting V1 (t), V2 (t) and V3 (t) are continuous functionals, we have
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lim Vk (t) = Vk (tp ), k = 1, 2, 3.
t→t− p
Moreover, it can be found that Z tp lim− V4 (t) = (tp − tp ) η1T (s)Rη1 (s)ds = 0, t→tp
tp−1
tp
tp
η1T (s)Rη1 (s)ds
and
(40)
= 0,
M
lim V4 (t) = (tp+1 − tp )
t→t+ p
Z
(39)
AC
CE
PT
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h i X11 X12 e(tp ) − eT (tp−1 )X11 e(tp−1 ) lim V5 (t) =(tp − tp ) eT (tp ) eT (tp ) − eT (tp−1 ) t→t− p ∗ X22 e(tp ) − e(tp−1 ) R tp h R i e(s)ds tp t p−1 T T T ¯ + = 0, e (s)ds e (tp ) − e (tp−1 ) X tp−1 e(tp ) − e(tp−1 ) h i X11 X12 e(tp ) − eT (tp )X11 e(tp ) lim+ V5 (t) =(tp+1 − tp ) eT (tp ) eT (tp ) − eT (tp ) t→tp ∗ X22 e(tp ) − e(tp ) R tp h R i e(s)ds tp T tp T T ¯ = 0, + X e (s)ds e (tp ) − e (tp ) tp e(tp ) − e(tp ) (41)
which imply that V4 (t) and V5 (t) vanish before tp and after tp . According to Lemma 1, It can be found that lim V6 (t) ≥ 0,
(42)
t→t− p
which implies limt→t− V6 (t) ≥ V6 (tp ) = 0. p 22
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Therefore, from (39)–(42), we can obtain that V (t− p ) ≥ V (tp ), p = 0, 1, . . . .
(43)
From (35) and (43), we can find V (t) is monotonically decreasing for t ∈ [t0 , +∞).
V (t) > V (t− p+1 ) ≥ V (tp+1 ) ≥ 0, t ∈ [tp , tp+1 ), p = 0, 1, . . . . Thus, V (t) is positive definite.
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By (35), (38) and (43), for ς(t) 6= 0, we have (44)
By (35), (43), and (44), we can obtain that the synchronization error system (4) is globally asymptotically stable. Therefore, the master system M and the slave system S in (1) are globally asymptotically
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synchronous. This completes the proof of Theorem 1.
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