ISA Transactions 53 (2014) 335–340
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Robust master–slave synchronization for general uncertain delayed dynamical model based on adaptive control scheme Tianbo Wang a,n, Wuneng Zhou b, Shouwei Zhao a, Weiqin Yu a a b
College of Fundamental Studies, Shanghai University of Engineering Science, Shanghai 201620, PR China College of Information Sciences and Technology, Donghua University, Shanghai 200051, PR China
art ic l e i nf o
a b s t r a c t
Article history: Received 6 November 2012 Received in revised form 9 September 2013 Accepted 9 November 2013 Available online 21 January 2014 This paper was recommended for publication by Dr. Q.-G. Wang
In this paper, the robust exponential synchronization problem for a class of uncertain delayed master– slave dynamical system is investigated by using the adaptive control method. Different from some existing master–slave models, the considered master–slave system includes bounded unmodeled dynamics. In order to compensate the effect of unmodeled dynamics and effectively achieve synchronization, a novel adaptive controller with simple updated laws is proposed. Moreover, the results are given in terms of LMIs, which can be easily solved by LMI Toolbox in Matlab. A numerical example is given to illustrate the effectiveness of the method. & 2013 ISA. Published by Elsevier Ltd. All rights reserved.
Keywords: Master–slave dynamical model Robust synchronization Time delay Stochastic effect
1. Introduction Synchronization of the dynamical systems has become a very interesting topic in recent years and has attracted many researchers from the fields of science and engineering to study it, which can be found in [1–6] and the references therein. Furthermore, synchronization techniques have been applied in the real world such as under-actuated mechanical systems [7], large scale robotic systems [8] and other fields [9,10]. As seen from the adopted synchronization control methods, they mainly include state feedback control [11], adaptive control [12–14], pinning control [15], impulsive control [16,17] and so on. “Master–slave” synchronization, as an important kind of synchronization types, means that two dynamical systems synchronize with each other in the way that the “slave” system mimics the motion of the “master” system [18,19]. Currently, there exist many papers on the synchronization of the master–slave systems. For example, the author studied the robust synchronization and fault detection for a class of neutral master–slave systems subjected to some nonlinear perturbations with time delays by using Lyapunov– Krasovskii functional and change of variables in [20]. Based on the comparison theorem for the stability of impulsive control systems, the authors considered the adaptive–impulsive synchronization of a master–slave system in [21]. For the discrete case, the authors
n
Corresponding author. Tel.: þ 86 2167791190. E-mail address: [email protected] (T. Wang).
investigated the synchronization control of a master–slave chaotic system based on the discrete-time Lyapunov stability theory and the linear matrix inequality framework in [22]. Besides, more results on this problem can be found in [23–25]. To the best of our knowledge, most of the existing literatures on the master–slave synchronization are mainly associated with two determinant dynamical systems or chaotic systems. However, for the dynamical systems with unmodeled dynamics or stochastic disturbances, the corresponding research results are much less. In this paper, we intend to study the robust master–slave synchronization problem for a general uncertain delayed dynamical system with unmodeled dynamics based on a novel adaptive control method. Our contributions are as follows: (i) The considered master–slave system model is very general, in which the unmodeled dynamics only need to satisfy the bounded condition. (ii) Our method could avoid the passive effect from the time delay for the synchronization of the master–slave systems. (iii) The proposed results are very concise. Moreover, if only the unmodeled dynamics are bounded, the existing method could fail to accomplish the synchronization. The master– slave system can achieve synchronization only through our method. The rest of this paper is organized as follows. In Section 2, the investigated master–slave model and some preliminaries are given. In Section 3, a novel adaptive synchronization scheme is proposed. In Section 4, a numerical example is provided to illustrate the effectiveness of our method. Finally, this paper is ended with a conclusion in Section 5. Notations: Rn and Rnm denote the n-dimensional Euclidean space and the set of all n m real matrices, respectively. For a vector
0019-0578/$ - see front matter & 2013 ISA. Published by Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.isatra.2013.11.009
336
T. Wang et al. / ISA Transactions 53 (2014) 335–340
a ¼ ða1 ; a2 ; …; an ÞT , letting signðaÞ ¼ ðsignða1 Þ; signða2 Þ; …; signðan ÞÞT . AT represents the transpose of matrix A, λmax ðHÞ stands for the maximum eigenvalue of real symmetric matrix H. The notation X ZY (respectively, X 4 Y), where X, Y are symmetric matrices, means that X Y is a symmetric semi-definite matrix (respectively, positive definite matrix). In is the n n identical matrix. ðΩ; F ; fF t gt Z 0 ; PÞ denotes the complete probability space with a filtration fF t gt Z 0 satisfying right continuous and F 0 containing all Pnull sets. Efg denotes the mathematical expectation.
Definition 2. For an n-dimensional vector a ¼ ða1 ; a2 ; …; an ÞT A Rn , we define its norms as sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi n
‖a‖1 ¼ ∑ j ai j ; i¼1
‖a‖2 ¼
n
∑ j ai j 2 ;
i¼1
‖a‖1 ¼ max fj ai j g; 1rirn
respectively. Remark 2. From Definition 2, it is easy to see that aT b r ‖a‖1 ‖b‖1 for any a; b A Rn .
2. Problem statement and preliminaries In this paper, we consider the following uncertain master–slave delayed dynamical model with stochastic disturbance: Master system: ( dxðtÞ ¼ ½AxðtÞ þ Bf ðxðt τÞÞ þ Δg 1 ðt; xðtÞ; xðt τÞÞ dt ð1Þ xðtÞ ¼ ϕðtÞ; t A ½ τ; 0: Slave system: 8 > < dyðtÞ ¼ ½AyðtÞ þ Bf ðyðt τ ÞÞ þ Δg 2 ðt; yðtÞ; yðt τ ÞÞ þ uðtÞ dt þ φðtÞ dwðtÞ yðtÞ ¼ ψ ðtÞ; t A ½ τ; 0; > : ð2Þ where xðtÞ A Rn and yðtÞ A Rn are the state vectors of the master system and the slave system, respectively. A A Rnn and B A Rnn are two known real matrices, f ðÞ : Rn -Rn is a continuous vector function, positive constant τ 40 denotes the transmission time delay and is known. Δg i ðÞ : R þ Rn Rn -Rn ði ¼ 1; 2Þ are continuous vector functions and represent the dynamical uncertainties in the master–slave system. uðtÞ A Rn is the control input, w(t) is 1-dimensional Brownian motion defined on the probability space ðΩ; F ; fF t gt Z 0 ; PÞ with EfwðtÞg ¼ 0;
Efw2 ðtÞg ¼ 1:
The Rn-valued function φðtÞ denotes the noise intensity, where φðtÞ may be dependent on the states xðtÞ; yðtÞ; xðt τÞ or yðt τÞ. Continuous vector functions ϕðtÞ A Rn and ψ ðtÞ A Rn denote the initial conditions of the master–slave system. Let the state error of the master system and the slave system be eðtÞ ¼ yðtÞ xðtÞ. Then we can obtain the following error dynamical system: deðtÞ ¼ fAeðtÞ þ B½f ðyðt τÞÞ f ðxðt τÞÞ þ Δg 2 ðt; yðtÞ; yðt τÞÞ Δg 1 ðt; xðtÞ; xðt τÞÞ þ uðtÞg dt þ φðtÞ dwðtÞ:
ð3Þ
Remark 1. Similar to [26,27], for a given positive scalar s 4 0, writing es ðtÞ ¼ yðtÞ xðt sÞ, we can obtain the following error dynamical system: des ðtÞ ¼ fAes ðtÞ þ B½f ðyðt τÞÞ f ðxðt τ sÞÞ þ Δg 2 ðt; yðtÞ; yðt τÞÞ Δg 1 ðt s; xðt sÞ; xðt τ sÞÞ þ uðtÞg dt þ φðtÞ dwðtÞ:
ð4Þ
Now, some definitions and assumptions which are quite standard and natural in the stability analysis of dynamical systems are given as follows. Definition 1. The master system (1) and the slave system (2) are said to be robustly exponential synchronization in mean square if there exists a positive constant ρ 4 0 such that 1 lim sup log E ‖eðtÞ‖2 r ρ t-1 t
ð5Þ
for any initial conditions ϕðtÞ; ψ ðtÞ and uncertainties Δg i ðtÞ ði ¼ 1; 2Þ.
Definition 3 (Gu [28]). For an n-dimensional stochastic differential system dxðtÞ ¼ f ðxðtÞ; xðt τÞ; tÞ dt þgðxðtÞ; xðt τÞ; tÞ dυðtÞ
ð6Þ
with the initial condition ξðtÞ A R on t 4 0, where dυðtÞ is an mdimensional Brownian motion, f ðt; xðtÞ; xðt τÞÞ : R þ Rn Rn -Rn and gðt; xðtÞ; xðt τÞÞ : R þ Rn Rn -Rnm are continuous differentiable functions. Letting VðxðtÞ; tÞ A C 2;1 ðRn R þ ; R þ Þ, the operator LVðxðtÞ; tÞ is defined as n
LVðxðtÞ; tÞ ¼ V t ðxðtÞ; tÞ þV x ðxðtÞ; tÞf ðxðtÞ; xðt τÞ; tÞ 1 þ tr g T ðxðtÞ; xðt τÞ; tÞV xx gðxðtÞ; xðt τÞ; tÞ ; 2
ð7Þ
where V t ðxðtÞ; tÞ ¼ ∂VðxðtÞ; tÞ=∂t; V x ðxðtÞ; tÞ ¼ ð∂VðxðtÞ; tÞ=∂x1 ; ∂VðxðtÞ; tÞ=∂x2 ; …; ∂VðxðtÞ; tÞ=∂xn Þ and V xx ¼ ð∂2 V ðxðtÞ; tÞ=∂xi ∂xj Þnn . Assumption 1. Assume that the nonlinear vector function f ðÞ satisfies the Lipshitz condition, i.e., there exits a scalar H 4 0 such that pffiffiffiffi ‖f ðz1 ðtÞÞ f ðz2 ðtÞÞ‖2 r H ‖z1 ðtÞ z2 ðtÞ‖2 for any z1 ðtÞ; z2 ðtÞ A Rn . Assumption 2. Assume that the unmodeled dynamics in the master–slave systems (1) and (2) are bounded and there exist two positive constants L1 4 0 and L2 40 such that ‖Δg 1 ðt; xðtÞ; xðt τÞÞ‖1 r L1 ;
‖Δg 2 ðt; xðtÞ; xðt τÞÞ‖1 r L2
n
for any xðtÞ A R . Assumption 3. Assume that there exist two positive definite matrices M 1 A Rnn and M 2 A Rnn such that
φT ðtÞφðtÞ r 2eT ðtÞM 1 eðtÞ þ2eT ðt τÞM2 eðt τÞ: Remark 3. Dynamical uncertainties often appear in some dynamical models, which closely affects the state trajectories and the stability of the dynamical systems (see [29,30]). However, there exist few results on the synchronization problem of the master–slave systems with unmodeled dynamics. Currently, there also exist many papers such as [31–34] to study the stability problem of uncertain dynamical systems, but most of them demand that the uncertainties have the form of ΔAxðtÞ ¼ DFðtÞExðtÞ with F T ðtÞFðtÞ r I, where D; E; FðtÞ are some matrices with appropriate dimensions. In fact, this is a linear growth condition and only suits for some special models. It is easy to see that the uncertainties in our paper are more general. Our main objective in this paper is to construct the following adaptive controller: 8 uðtÞ ¼ k1 ðtÞeðtÞ k2 ðtÞ signðPeðtÞÞ > < k_ 1 ðtÞ ¼ αeT ðtÞPeðtÞ ð8Þ > :_ k ðtÞ ¼ β‖PeðtÞ‖ 2
1
such that the master system (1) and the slave system (2) robustly exponentially synchronize, where α 4 0 and β 4 0 are two any positive constants. In order to achieve it, we provide two useful lemmas.
T. Wang et al. / ISA Transactions 53 (2014) 335–340
1 þ ðk2 ðtÞ k^ 2 Þ2
Lemma 1 (Han et al. [35]). For any constant positive definite symmetric matrix Ω A Rnn and known positive constants hm and hM, if lðtÞ : ½hm ; hM -Rn is a vector function such that the integrations in the following are well defined, then ! !T Z Z Z h2 ðtÞ
ðh2 ðtÞ h1 ðtÞÞ
h1 ðtÞ
lðϑÞΩl ðϑÞ dϑ Z T
h2 ðtÞ
h1 ðtÞ
h2 ðtÞ
lðϑÞ dϑ Ω
h1 ðtÞ
lðϑÞ dϑ
where h1 ðtÞ and h2 ðtÞ are differentiable functions and satisfy 0 o hm rh1 ðtÞ rh2 ðtÞ rhM .
þ γ3
t0
t
and taking the differential of V(t) along the state trajectories of error system (3), we have ;
dVðtÞ ¼ LVðtÞ dt þ 2eT ðtÞP φðtÞ dwðtÞ:
e cðt sÞ
s 1
t0
þ Δg 2 ðt; yðtÞ; yðt τÞÞ Δg 1 ðt; xðtÞ; xðt τÞÞ þ uðtÞ
mðtÞ r he
ɛðt t 0 Þ
;
R1 0
þ eT ðtÞQeðtÞ eT ðt τÞQeðt τÞ þ τeT ðtÞR1 eðtÞ Z t T Z t eT ðsÞR1 eðsÞ ds þ 2 eðsÞ ds R2 ðeðtÞ eðt τÞÞ tτ
tτ
2 þ ðk1 ðtÞ k^ 1 Þk_ 1 ðtÞ þ ðk2 ðtÞ k^ 2 Þk_ 2 ðtÞ þtr φT ðtÞP φðtÞ 2
α
β
¼ eT ðtÞðPA þ AT PÞeðtÞ þ2eT ðtÞPB½f ðyðt τÞÞ f ðxðt τÞÞ
rðs ξÞmðξÞ dξ ds;
where t 0 Z 0; 0 r τ ðtÞ r τ; r ¼ pair of positive constants, then
ð13Þ
From Definition 3, we get LVðtÞ ¼ 2eT ðtÞP AeðtÞ þ B½f ðyðt τÞÞ f ðxðt τÞÞ
t0
Z
ð12Þ
β
Lemma 2 (Liu et al. [36]). Assume that positive constants c; γ 1 ; γ 2 ; γ 3 satisfy γ 1 þ γ 2 þ rγ 3 o c, m(t) is a nonnegative continuous function on ð 1; þ 1Þ and satisfies the following inequality on an interval ½t 0 ; þ 1Þ: Z t Z t e cðt sÞ mðsÞ ds þ γ 2 e cðt sÞ mðs τðsÞÞ ds mðtÞ r e ct hþ γ 1 Z
337
eɛs rðsÞ ds; rðsÞ 4 0,
þ 2eT ðtÞP½Δg 2 ðt; yðtÞ; yðt τÞÞ Δg 1 ðt; xðtÞ; xðt τÞÞ
τ and h are a
2eT ðtÞP½k1 ðtÞeðtÞ þ k2 ðtÞ signðPeðtÞÞ þ eT ðtÞQeðtÞ Z t eT ðsÞR1 eðsÞ ds eT ðt τÞQeðt τÞ þ τeT ðtÞR1 eðtÞ
t Z t0 ;
tτ
Z
where ɛ is the unique positive solution of the following equation:
þ2
c γ 1 γ 2 eɛτ r γ 3 ¼ ɛ:
t tτ
T eðsÞ ds R2 ðeðtÞ eðt τÞÞ þ 2ðk1 ðtÞ k^ 1 ÞeT ðtÞPeðtÞ
þ 2ðk2 ðtÞ k^ 2 Þ‖PeðtÞ‖1 þtrfφT ðtÞP φðtÞg
Remark 4. The adaptive controller (8) can be divided into two parts. The first part is just the state feedback control, and the second part is to restrain the effect of unmodeled dynamics in the master–slave system inspired by the idea of the bang–bang control and the sliding mode control. On the other hand, in order to attain better control performance, we adjust the control gain according to the weight of the error state. Especially, the second part in controller (8) even relates to the sign of the error state components. So, controller (8) is much more novel.
¼ eT ðtÞðPA þ AT PÞeðtÞ þ2eT ðtÞPB½f ðyðt τÞÞ f ðxðt τÞÞ þ 2eT ðtÞP½Δg 2 ðt; yðtÞ; yðt τÞÞ Δg 1 ðt; xðtÞ; xðt τÞÞ þ eT ðtÞQeðtÞ eT ðt τÞQeðt τÞ þ τeT ðtÞR1 eðtÞ Z t T Z t eT ðsÞR1 eðsÞ ds þ 2 eðsÞ ds R2 ðeðtÞ eðt τÞÞ tτ
tτ
2k^ 1 eT ðtÞPeðtÞ 2k^ 2 ‖PeðtÞ‖1 þ trfφT ðtÞP φðtÞg:
ð14Þ
We notice that 3. Adaptive synchronization control criteria
2eT ðtÞP½Δg 2 ðt; yðtÞ; yðt τÞÞ Δg 1 ðt; xðtÞÞ; xðt τÞÞ
In this section, we will use the adaptive controller (8) to guarantee the master-salve system (1) and (2) to be the robust exponential synchronization in mean square. Theorem 1. Let Assumptions 1–3 hold. If there exist positive definite symmetric matrices P; Q ; R1 ; R2 A Rnn , scalars k^ 1 4 0; k^ 2 4 0; μ 4 0 and δ 4 0 such that P o μ In ; 2 6
0 δ H I n Q þ μM 2
0 4 R2
Φ1 ¼ 6 6
R2 R2
PB 0
R2
τ 1 R1
0
0
0
BT P
δ In
7 7 7 5
2eT ðtÞPB½f ðyðt τÞÞ f ðxðt τÞÞ rδ
4n4n
ð11Þ
where Φ1;11 ¼ PA þ A P þ Q þ τR1 þ μM 1 2k^ 1 P, L ¼ maxfL1 ; L2 g, then the master–slave system (1) and (2) is robust exponential synchronization in mean square under the action of adaptive controller (8). T
Proof. Choosing the following Lyapunov function: Z t Z 0 Z t VðtÞ ¼ eT ðtÞPeðtÞ þ eT ðsÞQeðsÞ ds þ eT ðϑÞR1 eðϑÞ dϑ ds t τ
t t τ
T Z eðsÞ ds R2
τ
t tτ
tþs
1 eðsÞ ds þ ðk1 ðtÞ k^ 1 Þ2
α
1 T
e ðtÞPBBT PeðtÞ
þ δ½f ðyðt τÞÞ f ðxðt τÞÞT ½f ðyðt τÞÞ f ðxðt τÞÞ 1 T
e ðtÞPBBT PeðtÞ þ δHeT ðt τÞeðt τÞ:
By Lemma 1, one gets Z Z t τ eT ðsÞR1 eðsÞ ds Z
o 0;
k^ 2 4 2L;
ð15Þ
and
rδ
3
ð10Þ
þ
r 4L‖eT ðtÞP‖1
ð9Þ
Φ1;11
Z
r 2‖eT ðtÞP‖1 ‖Δg 2 ðt; yðtÞ; yðt τÞÞ Δg 1 ðt; xðtÞ; xðt τÞÞ‖1
tτ
t t τ
T Z eðsÞ ds R1
ð16Þ
t tτ
eðsÞ ds :
ð17Þ
Thus, according to (15)–(17) and Assumption 3, we have LVðtÞ r eT ðtÞðPA þAT PÞeðtÞ þ δ
1 T
e ðtÞPBBT PeðtÞ
þ δHe ðt τÞeðt τÞ þ 4L‖eT ðtÞP‖1 T
þ eT ðtÞQeðtÞ eT ðt τÞQeðt τÞ Z t T Z 1 þ τeT ðtÞR1 eðtÞ eðsÞ ds R1 Z þ2
τ
t tτ
tτ
t tτ
eðsÞ ds
T eðsÞ ds R2 ðeðtÞ eðt τÞÞ 2k^ 1 eT ðtÞPeðtÞ
2k^ 2 ‖PeðtÞ‖1 þ μ½eT ðtÞM 1 eðtÞ þ eT ðt τÞM 2 eðt τÞ ^ χ ðtÞ þð4L 2k^ Þj eT ðtÞP‖ ; ¼ χ T ðtÞΦ 1 1 1
ð18Þ
338
T. Wang et al. / ISA Transactions 53 (2014) 335–340
Rt where χ ðtÞ ¼ ðeT ðtÞ; eT ðt τÞ; ð t τ eðsÞ dsÞT ÞT , 2 ^ 3 Φ 1;11 0 R2 7 ^ ¼6 Φ δH In Q þ μM2 R2 5; 4 0 1
P o μ In ; ð19Þ
2
τ 1 R1
R2
R2
δ 40 such that
1 T ^ PBBT P þ Q þ τR1 þ μM 1 2k^ 1 P. While and Φ 1;11 ¼ PA þA P þ δ the inequalities (9)–(11) have feasible solutions, by the Schur complement lemma, we yield
LV ðtÞ r 0
6
ð27Þ
Φ3;11
0
δH In Q þ μM2
0 4 R2
Φ3 ¼ 6 6
R2
PB
R2
0
R2
τ 1 R1
0
0
0
BT P
3 7 7 7 5
δ In
o 0; 4n4n
ð28Þ
ð20Þ
for all t Z0. On the other hand, by Itô formula, we have dðeɛs VðsÞÞ ¼ eɛs ½ɛVðsÞ þLVðsÞ ds þ 2eɛs eT ðsÞP φðsÞ dwðsÞ:
ð21Þ
Hence, for any t 4 0, we obtain Z t Efeɛt VðtÞg EfVð0Þg ¼ E eɛs ½ɛVðsÞ þ LVðsÞ ds 0 t
Z r ɛ1
where Φ3;11 ¼ PAþ A P þQ þ τR1 þ μM 1 2k^ 1 P, L ¼ maxfL1 ; L2 g, then the master–slave system (1) and (2) without uncertainties is exponential synchronization in mean square under the action of the following adaptive controller: ( uðtÞ ¼ k1 ðtÞeðtÞ ð29Þ k_ ðtÞ ¼ αeT ðtÞPeðtÞ: T
1
Efeɛs VðsÞg ds:
ð22Þ
0
Proof. Similar to the proof of Theorem 1, choosing a Lyapunov function as
That is, EfVðtÞg r e ɛt EfVð0Þg þ ɛ1
Z
t
Z
e ɛðt sÞ EfVðsÞg ds;
VðtÞ ¼ eT ðtÞPeðtÞ þ
0
where ɛ 1 o ɛ. Then by Lemma 2, we have EfVðtÞg r EfV ð0Þge ρt ;
Z
t Z 0;
þ
where ρ ¼ ɛ ɛ 1 . Due to
t
tτ
t tτ
Z eT ðsÞQeðsÞ dsþ
T Z eðsÞ ds R2
t t τ
0 τ
Z
t t þs
eT ðϑÞR1 eðϑÞ dϑ ds
1 eðsÞ ds þ ðk1 ðtÞ k^ 1 Þ2 ;
α
with similar deductions, we can obtain Corollary 2.
VðtÞ Z λmax ðPÞeT ðtÞeðtÞ;
ð30Þ
□
This implies
Remark 5. The lag synchronization problem is also an interesting topic for dynamical systems, which can be found in [26,27]. The master system (1) and the slave system (2) are said to be robust exponential lag synchronization in mean square if there exists a positive constant ρ 4 0 such that
1 lim sup log E ‖eðtÞ‖2 r ρ: t-1 t
1 lim sup log E ‖es ðtÞ‖2 r ρ t-1 t
Therefore, the master–slave system (1) and (2) robustly exponentially synchronize well. This completes the proof. □
for any initial conditions ϕðtÞ; ψ ðtÞ and uncertainties Δg i ðtÞ ði ¼ 1; 2Þ. It should be pointed out, under the action of adaptive controller (8), if the conditions in Theorem 1 are satisfied, then the master–slave system (1) and (2) is also robust exponential lag synchronization in mean square. However, to the best of our knowledge, for the master–slave system model, there still does not exist similar results until now.
then EfVð0Þg ρt E ‖eðtÞ‖2 r e : λmax ðPÞ
Next, we provide some useful corollaries associated with two special cases of master–slave system (1) and (2). Corollary 1. Let Assumptions 1–3 hold. If there exist positive definite symmetric matrix P A Rnn , scalars k^ 1 4 0; k^ 2 4 0; μ 4 0 and δ 4 0 such that P o μ In ; "
Φ2 ¼
ð23Þ
Φ2;11
PB
BT P
δ In
# o 0;
ð24Þ
ð31Þ
Remark 6. By means of the idea in [37,38], we will study the H1 synchronization, bilateral synchronization and state estimation problems on the master–slave systems with unmodeled dynamics and mixed time delays in our future work.
2n2n
k^ 2 4 2L;
ð25Þ
where Φ2;11 ¼ PA þ A P þ δH I n þ μM 1 2k^ 1 P, L ¼ maxfL1 ; L2 g, then the master–slave system (1) and (2) without time delay (τ ¼0) is robustly exponential synchronization in mean square under the action of the adaptive controller (8).
4. A numerical example
T
Proof. Similar to the proof of Theorem 1, choosing a Lyapunov function as 1 1 VðtÞ ¼ eT ðtÞPeðtÞ þ ðk1 ðtÞ k^ 1 Þ2 þ ðk2 ðtÞ k^ 2 Þ2 ;
α
β
ð26Þ
then we can obtain the results in Corollary 1 after some direct computation. □ Corollary 2. Let Assumptions 1–3 hold, if there exist positive definite symmetric matrices P; Q ; R1 ; R2 A Rnn , scalars k^ 1 4 0; μ 4 0 and
In order to show the effectiveness of our proposed method, we provide an illustration example in the section. It is well known that Chua's dynamical system is 8 _ > < x 1 ðtÞ ¼ 10½x2 ðtÞ x1 ðtÞ f 1 ðx1 ðtÞÞ x_ 2 ðtÞ ¼ x1 ðtÞ x2 ðtÞ þ x3 ðtÞ ð32Þ > : x_ ðtÞ ¼ 15x ðtÞ 0:0385x ðtÞ; 3 2 3 where f 1 ðx1 ðtÞÞ ¼ bx1 ðtÞ þ 0:5ða bÞðjx1 ðtÞ þ1j jx1 ðtÞ 1jÞ, a, b are two constants. Furthermore, it has been shown that the system (32) possesses a chaotic behavior if a and b are appropriately chosen. Obviously, the system can be written as the vector form _ ¼ AxðtÞ þ Bf ðxðtÞÞ; xðtÞ
ð33Þ
T. Wang et al. / ISA Transactions 53 (2014) 335–340
where xðtÞ ¼ ðx1 ðtÞ; x2 ðtÞ; x3 ðtÞÞT , 2 3 10 10 0 6 1 7 1 1 A¼4 5; 0 15 0:0385
60
f ðxðtÞÞ ¼ ðf 1 ðx1 ðtÞÞ; 0; 0ÞT , B ¼ diagf 10; 1; 1g. Due to the fact that
20
339
x1(t) x (t)
40
2
x (t)
x(t),y(t)
^ ^ 1 ðtÞÞj ‖f ðxðtÞÞ f ðxðtÞÞ‖ 2 ¼ j f 1 ðx1 ðtÞÞ f 1 ðx r ðj bj þ j a bj Þ j x1 ðtÞ x^ 1 ðtÞj ^ r ðj bj þ j a bj Þ ‖xðtÞ xðtÞ‖ 2
3
y (t) 3
y (t) 2
0
y (t) 1
−20 ^ A R3 , f ðxðtÞÞ satisfies Assumption 1. for any xðtÞ; xðtÞ We take the master system as
−40
_ ¼ AxðtÞ þ Bf ðxðt τÞÞ þ Δg 1 ðt; xðtÞ; xðt τÞÞ xðtÞ
−60
and the slave system as dyðtÞ ¼ ½AyðtÞ þ Bf ðyðt τÞÞ þ Δg 2 ðt; yðtÞ; yðt τÞÞ þ uðtÞ dt þ φðtÞ dwðtÞ;
Δg1 ðt; xðtÞ; xðt τÞÞ ¼ ð sin ðx1 ðtÞÞ; cos ðx2 ðt τÞÞ; 1Þ ; Δg2 ðt; yðtÞ; yðt τÞÞ ¼ ð cos ðx1 ðt τÞÞ; 0:5; sin ðx3 ðt τÞÞÞT ; φðtÞ ¼ ðx1 ðtÞ y1 ðtÞ; 0; x3 ðt τÞ y3 ðt τÞÞT ; T
a ¼6, b ¼ 0.75, τ ¼2. It is easy to verify that Assumptions 1–3 hold while setting H ¼ 7:52 ; L1 ¼ L2 ¼ 1; M 1 ¼ diagf0:5; 0; 0g and M 2 ¼ diagf0; 0; 0:5g. By the LMI toolbox in Matlab, we obtain the feasible solutions of the inequalities (9)–(11) as follows: 2 3 1:6743 0:0314 0:0221 6 7 2:1212 2:1213 5; P ¼ 4 0:0314 0:0221 2:1213 2:2719 2 3 129:6021 2:3307 0:0369 6 7 Q ¼ 4 2:3307 150:9423 0:02619 5; 0:0369 0:02619 163:3586 9:7733
6 R1 ¼ 4 6:6116 0:0957 2 2:6489 6 R2 ¼ 4 1:1102 0:0156
6:6116
0:0957
4
k^ 1 ¼ 80; k^ 2 ¼ 3; μ ¼ 26:5676; δ ¼ 2:1616. According to Theorem 1, we know that the master–slave system in this example robustly synchronizes well. Taking α ¼ β ¼ 0:1 and the initial conditions xð0Þ ¼ ð 30; 10; 30ÞT and yð0Þ ¼ ð50; 20; 50ÞT , we can draw some figures as follows. In particular, Fig. 1 shows the state trajectories of the master–slave system without control input. Figs. 2 and 3 present the state trajectories and the state error trajectories of the master–slave system under the action of the adaptive controller (8) for the given initial conditions, respectively, which shows that our designed controller is effective. Fig. 4 is the state trajectories of the updated laws in the adaptive controller, which shows that the control gains almost are constants. Remark 7. It is noted that Ref. [38] also studied the synchronization of dynamical systems with uncertain nonlinear terms by using a similar discontinuous controller as that in our paper. Furthermore, the nonlinear terms also satisfy the bounded condition. However, they did not consider the factors as the time delay and stochastic disturbances. In addition, the methods in the two papers are different. Compared with them, our proposed method need not do some complex computation and discussions. Furthermore, we may adjust the convergence rate by tuning the control gains α and β. Besides them, the method in [38] only could be used to solve low-dimensional dynamical systems such as
8
10
50 x3(t),y3(t) x2(t),y2(t)
0
3
7 70:6853 0:1531 5; 0:1531 71:2835 3 1:1102 0:0156 12:8618 0:0184 7 5; 0:0184 12:8662
6
Fig. 1. State trajectories of the master–slave system without control input for the given initial conditions.
x (t),y (t) 1
−50
0
0.5
1
1
1.5
2
2.5
3
t/second Fig. 2. State trajectories of the master–slave system under the action of the adaptive controller (8) for the given initial conditions.
80 e (t) 1
60
e2(t) e (t)
40
3
20
e(t)
2
2
t/second
x(t),y(t)
respectively, where
0
0 −20 −40 −60 −80
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
t Fig. 3. The state error trajectories of the master–slave system under the action of the adaptive controller (8) for the given initial conditions.
340
T. Wang et al. / ISA Transactions 53 (2014) 335–340
300 [8]
250 k (t)
[9]
1
k (t)
k1(t),k2(t)
200
2
[10]
150
[11]
100
[12]
50
[13]
0
[14]
0
0.5
1
1.5
2
2.5
3
t/second Fig. 4. State trajectories of the updated laws in the adaptive controller (8).
[15] [16] [17]
1-dimensional or 2-dimensional ones, for the high-dimensional case, which is invalidate.
[18] [19]
5. Conclusion In this paper, we have investigated the robust master–slave synchronization for a class of uncertain delayed dynamical systems based on the adaptive control method. Different from some existing results, the unmodeled dynamics only need to satisfy the bounded condition. In order to achieve synchronization, a new controller has been constructed by using the idea of the bang–bang control and the sliding mode control. Numerical example has shown that our proposed method is effective. Acknowledgments
[20]
[21]
[22]
[23]
[24] [25]
[26]
This work was supported by the National Natural Science Foundation of China (61203128), Tian Yuan Special Foundation (NO 11326132), the Chen Guang Project of Shanghai Municipal Education Commission and Shanghai Education Development Foundation (12CG65), the Domestic Visiting Project of Shanghai University Teachers, Connotative Construction Project of Shanghai University of Engineering Science (nhky-2012-13).
[30]
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Adaptive synchronization and parameter identification of chaotic system with unknown parameters and mixed delays based on a special matrix structure. ISA Trans, 2013;52:738–43. Zhu QX, Cao JD. Adaptive synchronization under almost every initial data for stochastic neural networks with time-varying delays and distributed delays. Commun Nonlinear Sci Numer Simul 2011;16:2139–59. Lu WL, Li X, Rong ZH. Global stabilization of complex networks with digraph topologies via a local pinning algorithm. Automatica 2010;46:116–21. Zheng S, Dong GG, Bi QS. Impulsive synchronization of complex networks with non-delayed and delayed coupling. Phys Lett A 2009;373:4255–9. Lu JQ, Ho Daniel WC, Cao JD. A unified synchronization criterion for impulsive dynamical networks. Automatica 2010;46:1215–21. Antonio L. Master–slave synchronization of fourth-order Lü chaotic oscillators via linear output feedback. IEEE Trans Circuits Syst II: Express Pap 2010;57:213–7. Wu CW, Hsu CF, Hwang CK. Master–slave chaos synchronization using adaptive TSK-type CMAC neural control. J Frankl Inst 2011;348:1847–68. Karimi HR. Robust synchronization and fault detection of uncertain master– slave systems with mixed time-varying delays and nonlinear perturbations. Int J Control Autom Syst 2011;9:671–80. Sun M, Zeng CY, Tao YW, Tian LX. Adaptive–impulsive synchronization in drive-response networks of continuous systems and its application. Phys Lett A 2009;373:3041–6. Lee S, Kapila V, Porfiri M, Panda A. Master–slave synchronization of continuously and intermittently coupled sampled-data chaotic oscillators. Commun Nonlinear Sci Numer Simul 2010;15:4100–13. Karimi HR. A sliding mode approach to H1 synchronization of master–slave time-delay systems with Markovian jumping parameters and nonlinear uncertainties. J Frankl Inst 2012;349:1480–96. Du HY. Function projective synchronization in drive-response dynamical networks with non-identical nodes. Chaos Solitons Fractals 2011;44:510–4. Karimi HR, Zapateiro M, Luo NS. Adaptive synchronization of master–slave systems with mixed neutral and discrete time-delays and nonlinear perturbations. Asian J Control 2012;14:251–7. Yang XS, Zhu QX, Huang CX. Lag stochastic synchronization of chaotic mixed time-delayed neural networks with uncertain parameters or perturbations. Neurocomputing 2011;74:1617–25. Guo WL. Lag synchronization of complex networks via pinning control. Nonlinear Anal: Real World Appl 2011;12:2579–85. Gu HB. Adaptive synchronization for competitive neural networks with different time scales and stochastic perturbation. Neurocomputing 2009;73:350–6. Dong WJ, Farrell JA. Decentralized cooperative control of multiple nonholonomic dynamic systems with uncertainty. Automatica 2009;45:706–10. Liu YH. Robust adaptive observer for nonlinear systems with unmodeled dynamics. Automatica 2009;45:1891–5. Qian W, Cong S, Sun YX, Fei SM. Novel robust stability criteria for uncertain systems with time-varying delay. 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Contents lists available at ScienceDirect
ISA Transactions journal homepage: www.elsevier.com/locate/isatrans
Robust master–slave synchronization for general uncertain delayed dynamical model based on adaptive control scheme Tianbo Wang a,n, Wuneng Zhou b, Shouwei Zhao a, Weiqin Yu a a b
College of Fundamental Studies, Shanghai University of Engineering Science, Shanghai 201620, PR China College of Information Sciences and Technology, Donghua University, Shanghai 200051, PR China
art ic l e i nf o
a b s t r a c t
Article history: Received 6 November 2012 Received in revised form 9 September 2013 Accepted 9 November 2013 Available online 21 January 2014 This paper was recommended for publication by Dr. Q.-G. Wang
In this paper, the robust exponential synchronization problem for a class of uncertain delayed master– slave dynamical system is investigated by using the adaptive control method. Different from some existing master–slave models, the considered master–slave system includes bounded unmodeled dynamics. In order to compensate the effect of unmodeled dynamics and effectively achieve synchronization, a novel adaptive controller with simple updated laws is proposed. Moreover, the results are given in terms of LMIs, which can be easily solved by LMI Toolbox in Matlab. A numerical example is given to illustrate the effectiveness of the method. & 2013 ISA. Published by Elsevier Ltd. All rights reserved.
Keywords: Master–slave dynamical model Robust synchronization Time delay Stochastic effect
1. Introduction Synchronization of the dynamical systems has become a very interesting topic in recent years and has attracted many researchers from the fields of science and engineering to study it, which can be found in [1–6] and the references therein. Furthermore, synchronization techniques have been applied in the real world such as under-actuated mechanical systems [7], large scale robotic systems [8] and other fields [9,10]. As seen from the adopted synchronization control methods, they mainly include state feedback control [11], adaptive control [12–14], pinning control [15], impulsive control [16,17] and so on. “Master–slave” synchronization, as an important kind of synchronization types, means that two dynamical systems synchronize with each other in the way that the “slave” system mimics the motion of the “master” system [18,19]. Currently, there exist many papers on the synchronization of the master–slave systems. For example, the author studied the robust synchronization and fault detection for a class of neutral master–slave systems subjected to some nonlinear perturbations with time delays by using Lyapunov– Krasovskii functional and change of variables in [20]. Based on the comparison theorem for the stability of impulsive control systems, the authors considered the adaptive–impulsive synchronization of a master–slave system in [21]. For the discrete case, the authors
n
Corresponding author. Tel.: þ 86 2167791190. E-mail address: [email protected] (T. Wang).
investigated the synchronization control of a master–slave chaotic system based on the discrete-time Lyapunov stability theory and the linear matrix inequality framework in [22]. Besides, more results on this problem can be found in [23–25]. To the best of our knowledge, most of the existing literatures on the master–slave synchronization are mainly associated with two determinant dynamical systems or chaotic systems. However, for the dynamical systems with unmodeled dynamics or stochastic disturbances, the corresponding research results are much less. In this paper, we intend to study the robust master–slave synchronization problem for a general uncertain delayed dynamical system with unmodeled dynamics based on a novel adaptive control method. Our contributions are as follows: (i) The considered master–slave system model is very general, in which the unmodeled dynamics only need to satisfy the bounded condition. (ii) Our method could avoid the passive effect from the time delay for the synchronization of the master–slave systems. (iii) The proposed results are very concise. Moreover, if only the unmodeled dynamics are bounded, the existing method could fail to accomplish the synchronization. The master– slave system can achieve synchronization only through our method. The rest of this paper is organized as follows. In Section 2, the investigated master–slave model and some preliminaries are given. In Section 3, a novel adaptive synchronization scheme is proposed. In Section 4, a numerical example is provided to illustrate the effectiveness of our method. Finally, this paper is ended with a conclusion in Section 5. Notations: Rn and Rnm denote the n-dimensional Euclidean space and the set of all n m real matrices, respectively. For a vector
0019-0578/$ - see front matter & 2013 ISA. Published by Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.isatra.2013.11.009
336
T. Wang et al. / ISA Transactions 53 (2014) 335–340
a ¼ ða1 ; a2 ; …; an ÞT , letting signðaÞ ¼ ðsignða1 Þ; signða2 Þ; …; signðan ÞÞT . AT represents the transpose of matrix A, λmax ðHÞ stands for the maximum eigenvalue of real symmetric matrix H. The notation X ZY (respectively, X 4 Y), where X, Y are symmetric matrices, means that X Y is a symmetric semi-definite matrix (respectively, positive definite matrix). In is the n n identical matrix. ðΩ; F ; fF t gt Z 0 ; PÞ denotes the complete probability space with a filtration fF t gt Z 0 satisfying right continuous and F 0 containing all Pnull sets. Efg denotes the mathematical expectation.
Definition 2. For an n-dimensional vector a ¼ ða1 ; a2 ; …; an ÞT A Rn , we define its norms as sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi n
‖a‖1 ¼ ∑ j ai j ; i¼1
‖a‖2 ¼
n
∑ j ai j 2 ;
i¼1
‖a‖1 ¼ max fj ai j g; 1rirn
respectively. Remark 2. From Definition 2, it is easy to see that aT b r ‖a‖1 ‖b‖1 for any a; b A Rn .
2. Problem statement and preliminaries In this paper, we consider the following uncertain master–slave delayed dynamical model with stochastic disturbance: Master system: ( dxðtÞ ¼ ½AxðtÞ þ Bf ðxðt τÞÞ þ Δg 1 ðt; xðtÞ; xðt τÞÞ dt ð1Þ xðtÞ ¼ ϕðtÞ; t A ½ τ; 0: Slave system: 8 > < dyðtÞ ¼ ½AyðtÞ þ Bf ðyðt τ ÞÞ þ Δg 2 ðt; yðtÞ; yðt τ ÞÞ þ uðtÞ dt þ φðtÞ dwðtÞ yðtÞ ¼ ψ ðtÞ; t A ½ τ; 0; > : ð2Þ where xðtÞ A Rn and yðtÞ A Rn are the state vectors of the master system and the slave system, respectively. A A Rnn and B A Rnn are two known real matrices, f ðÞ : Rn -Rn is a continuous vector function, positive constant τ 40 denotes the transmission time delay and is known. Δg i ðÞ : R þ Rn Rn -Rn ði ¼ 1; 2Þ are continuous vector functions and represent the dynamical uncertainties in the master–slave system. uðtÞ A Rn is the control input, w(t) is 1-dimensional Brownian motion defined on the probability space ðΩ; F ; fF t gt Z 0 ; PÞ with EfwðtÞg ¼ 0;
Efw2 ðtÞg ¼ 1:
The Rn-valued function φðtÞ denotes the noise intensity, where φðtÞ may be dependent on the states xðtÞ; yðtÞ; xðt τÞ or yðt τÞ. Continuous vector functions ϕðtÞ A Rn and ψ ðtÞ A Rn denote the initial conditions of the master–slave system. Let the state error of the master system and the slave system be eðtÞ ¼ yðtÞ xðtÞ. Then we can obtain the following error dynamical system: deðtÞ ¼ fAeðtÞ þ B½f ðyðt τÞÞ f ðxðt τÞÞ þ Δg 2 ðt; yðtÞ; yðt τÞÞ Δg 1 ðt; xðtÞ; xðt τÞÞ þ uðtÞg dt þ φðtÞ dwðtÞ:
ð3Þ
Remark 1. Similar to [26,27], for a given positive scalar s 4 0, writing es ðtÞ ¼ yðtÞ xðt sÞ, we can obtain the following error dynamical system: des ðtÞ ¼ fAes ðtÞ þ B½f ðyðt τÞÞ f ðxðt τ sÞÞ þ Δg 2 ðt; yðtÞ; yðt τÞÞ Δg 1 ðt s; xðt sÞ; xðt τ sÞÞ þ uðtÞg dt þ φðtÞ dwðtÞ:
ð4Þ
Now, some definitions and assumptions which are quite standard and natural in the stability analysis of dynamical systems are given as follows. Definition 1. The master system (1) and the slave system (2) are said to be robustly exponential synchronization in mean square if there exists a positive constant ρ 4 0 such that 1 lim sup log E ‖eðtÞ‖2 r ρ t-1 t
ð5Þ
for any initial conditions ϕðtÞ; ψ ðtÞ and uncertainties Δg i ðtÞ ði ¼ 1; 2Þ.
Definition 3 (Gu [28]). For an n-dimensional stochastic differential system dxðtÞ ¼ f ðxðtÞ; xðt τÞ; tÞ dt þgðxðtÞ; xðt τÞ; tÞ dυðtÞ
ð6Þ
with the initial condition ξðtÞ A R on t 4 0, where dυðtÞ is an mdimensional Brownian motion, f ðt; xðtÞ; xðt τÞÞ : R þ Rn Rn -Rn and gðt; xðtÞ; xðt τÞÞ : R þ Rn Rn -Rnm are continuous differentiable functions. Letting VðxðtÞ; tÞ A C 2;1 ðRn R þ ; R þ Þ, the operator LVðxðtÞ; tÞ is defined as n
LVðxðtÞ; tÞ ¼ V t ðxðtÞ; tÞ þV x ðxðtÞ; tÞf ðxðtÞ; xðt τÞ; tÞ 1 þ tr g T ðxðtÞ; xðt τÞ; tÞV xx gðxðtÞ; xðt τÞ; tÞ ; 2
ð7Þ
where V t ðxðtÞ; tÞ ¼ ∂VðxðtÞ; tÞ=∂t; V x ðxðtÞ; tÞ ¼ ð∂VðxðtÞ; tÞ=∂x1 ; ∂VðxðtÞ; tÞ=∂x2 ; …; ∂VðxðtÞ; tÞ=∂xn Þ and V xx ¼ ð∂2 V ðxðtÞ; tÞ=∂xi ∂xj Þnn . Assumption 1. Assume that the nonlinear vector function f ðÞ satisfies the Lipshitz condition, i.e., there exits a scalar H 4 0 such that pffiffiffiffi ‖f ðz1 ðtÞÞ f ðz2 ðtÞÞ‖2 r H ‖z1 ðtÞ z2 ðtÞ‖2 for any z1 ðtÞ; z2 ðtÞ A Rn . Assumption 2. Assume that the unmodeled dynamics in the master–slave systems (1) and (2) are bounded and there exist two positive constants L1 4 0 and L2 40 such that ‖Δg 1 ðt; xðtÞ; xðt τÞÞ‖1 r L1 ;
‖Δg 2 ðt; xðtÞ; xðt τÞÞ‖1 r L2
n
for any xðtÞ A R . Assumption 3. Assume that there exist two positive definite matrices M 1 A Rnn and M 2 A Rnn such that
φT ðtÞφðtÞ r 2eT ðtÞM 1 eðtÞ þ2eT ðt τÞM2 eðt τÞ: Remark 3. Dynamical uncertainties often appear in some dynamical models, which closely affects the state trajectories and the stability of the dynamical systems (see [29,30]). However, there exist few results on the synchronization problem of the master–slave systems with unmodeled dynamics. Currently, there also exist many papers such as [31–34] to study the stability problem of uncertain dynamical systems, but most of them demand that the uncertainties have the form of ΔAxðtÞ ¼ DFðtÞExðtÞ with F T ðtÞFðtÞ r I, where D; E; FðtÞ are some matrices with appropriate dimensions. In fact, this is a linear growth condition and only suits for some special models. It is easy to see that the uncertainties in our paper are more general. Our main objective in this paper is to construct the following adaptive controller: 8 uðtÞ ¼ k1 ðtÞeðtÞ k2 ðtÞ signðPeðtÞÞ > < k_ 1 ðtÞ ¼ αeT ðtÞPeðtÞ ð8Þ > :_ k ðtÞ ¼ β‖PeðtÞ‖ 2
1
such that the master system (1) and the slave system (2) robustly exponentially synchronize, where α 4 0 and β 4 0 are two any positive constants. In order to achieve it, we provide two useful lemmas.
T. Wang et al. / ISA Transactions 53 (2014) 335–340
1 þ ðk2 ðtÞ k^ 2 Þ2
Lemma 1 (Han et al. [35]). For any constant positive definite symmetric matrix Ω A Rnn and known positive constants hm and hM, if lðtÞ : ½hm ; hM -Rn is a vector function such that the integrations in the following are well defined, then ! !T Z Z Z h2 ðtÞ
ðh2 ðtÞ h1 ðtÞÞ
h1 ðtÞ
lðϑÞΩl ðϑÞ dϑ Z T
h2 ðtÞ
h1 ðtÞ
h2 ðtÞ
lðϑÞ dϑ Ω
h1 ðtÞ
lðϑÞ dϑ
where h1 ðtÞ and h2 ðtÞ are differentiable functions and satisfy 0 o hm rh1 ðtÞ rh2 ðtÞ rhM .
þ γ3
t0
t
and taking the differential of V(t) along the state trajectories of error system (3), we have ;
dVðtÞ ¼ LVðtÞ dt þ 2eT ðtÞP φðtÞ dwðtÞ:
e cðt sÞ
s 1
t0
þ Δg 2 ðt; yðtÞ; yðt τÞÞ Δg 1 ðt; xðtÞ; xðt τÞÞ þ uðtÞ
mðtÞ r he
ɛðt t 0 Þ
;
R1 0
þ eT ðtÞQeðtÞ eT ðt τÞQeðt τÞ þ τeT ðtÞR1 eðtÞ Z t T Z t eT ðsÞR1 eðsÞ ds þ 2 eðsÞ ds R2 ðeðtÞ eðt τÞÞ tτ
tτ
2 þ ðk1 ðtÞ k^ 1 Þk_ 1 ðtÞ þ ðk2 ðtÞ k^ 2 Þk_ 2 ðtÞ þtr φT ðtÞP φðtÞ 2
α
β
¼ eT ðtÞðPA þ AT PÞeðtÞ þ2eT ðtÞPB½f ðyðt τÞÞ f ðxðt τÞÞ
rðs ξÞmðξÞ dξ ds;
where t 0 Z 0; 0 r τ ðtÞ r τ; r ¼ pair of positive constants, then
ð13Þ
From Definition 3, we get LVðtÞ ¼ 2eT ðtÞP AeðtÞ þ B½f ðyðt τÞÞ f ðxðt τÞÞ
t0
Z
ð12Þ
β
Lemma 2 (Liu et al. [36]). Assume that positive constants c; γ 1 ; γ 2 ; γ 3 satisfy γ 1 þ γ 2 þ rγ 3 o c, m(t) is a nonnegative continuous function on ð 1; þ 1Þ and satisfies the following inequality on an interval ½t 0 ; þ 1Þ: Z t Z t e cðt sÞ mðsÞ ds þ γ 2 e cðt sÞ mðs τðsÞÞ ds mðtÞ r e ct hþ γ 1 Z
337
eɛs rðsÞ ds; rðsÞ 4 0,
þ 2eT ðtÞP½Δg 2 ðt; yðtÞ; yðt τÞÞ Δg 1 ðt; xðtÞ; xðt τÞÞ
τ and h are a
2eT ðtÞP½k1 ðtÞeðtÞ þ k2 ðtÞ signðPeðtÞÞ þ eT ðtÞQeðtÞ Z t eT ðsÞR1 eðsÞ ds eT ðt τÞQeðt τÞ þ τeT ðtÞR1 eðtÞ
t Z t0 ;
tτ
Z
where ɛ is the unique positive solution of the following equation:
þ2
c γ 1 γ 2 eɛτ r γ 3 ¼ ɛ:
t tτ
T eðsÞ ds R2 ðeðtÞ eðt τÞÞ þ 2ðk1 ðtÞ k^ 1 ÞeT ðtÞPeðtÞ
þ 2ðk2 ðtÞ k^ 2 Þ‖PeðtÞ‖1 þtrfφT ðtÞP φðtÞg
Remark 4. The adaptive controller (8) can be divided into two parts. The first part is just the state feedback control, and the second part is to restrain the effect of unmodeled dynamics in the master–slave system inspired by the idea of the bang–bang control and the sliding mode control. On the other hand, in order to attain better control performance, we adjust the control gain according to the weight of the error state. Especially, the second part in controller (8) even relates to the sign of the error state components. So, controller (8) is much more novel.
¼ eT ðtÞðPA þ AT PÞeðtÞ þ2eT ðtÞPB½f ðyðt τÞÞ f ðxðt τÞÞ þ 2eT ðtÞP½Δg 2 ðt; yðtÞ; yðt τÞÞ Δg 1 ðt; xðtÞ; xðt τÞÞ þ eT ðtÞQeðtÞ eT ðt τÞQeðt τÞ þ τeT ðtÞR1 eðtÞ Z t T Z t eT ðsÞR1 eðsÞ ds þ 2 eðsÞ ds R2 ðeðtÞ eðt τÞÞ tτ
tτ
2k^ 1 eT ðtÞPeðtÞ 2k^ 2 ‖PeðtÞ‖1 þ trfφT ðtÞP φðtÞg:
ð14Þ
We notice that 3. Adaptive synchronization control criteria
2eT ðtÞP½Δg 2 ðt; yðtÞ; yðt τÞÞ Δg 1 ðt; xðtÞÞ; xðt τÞÞ
In this section, we will use the adaptive controller (8) to guarantee the master-salve system (1) and (2) to be the robust exponential synchronization in mean square. Theorem 1. Let Assumptions 1–3 hold. If there exist positive definite symmetric matrices P; Q ; R1 ; R2 A Rnn , scalars k^ 1 4 0; k^ 2 4 0; μ 4 0 and δ 4 0 such that P o μ In ; 2 6
0 δ H I n Q þ μM 2
0 4 R2
Φ1 ¼ 6 6
R2 R2
PB 0
R2
τ 1 R1
0
0
0
BT P
δ In
7 7 7 5
2eT ðtÞPB½f ðyðt τÞÞ f ðxðt τÞÞ rδ
4n4n
ð11Þ
where Φ1;11 ¼ PA þ A P þ Q þ τR1 þ μM 1 2k^ 1 P, L ¼ maxfL1 ; L2 g, then the master–slave system (1) and (2) is robust exponential synchronization in mean square under the action of adaptive controller (8). T
Proof. Choosing the following Lyapunov function: Z t Z 0 Z t VðtÞ ¼ eT ðtÞPeðtÞ þ eT ðsÞQeðsÞ ds þ eT ðϑÞR1 eðϑÞ dϑ ds t τ
t t τ
T Z eðsÞ ds R2
τ
t tτ
tþs
1 eðsÞ ds þ ðk1 ðtÞ k^ 1 Þ2
α
1 T
e ðtÞPBBT PeðtÞ
þ δ½f ðyðt τÞÞ f ðxðt τÞÞT ½f ðyðt τÞÞ f ðxðt τÞÞ 1 T
e ðtÞPBBT PeðtÞ þ δHeT ðt τÞeðt τÞ:
By Lemma 1, one gets Z Z t τ eT ðsÞR1 eðsÞ ds Z
o 0;
k^ 2 4 2L;
ð15Þ
and
rδ
3
ð10Þ
þ
r 4L‖eT ðtÞP‖1
ð9Þ
Φ1;11
Z
r 2‖eT ðtÞP‖1 ‖Δg 2 ðt; yðtÞ; yðt τÞÞ Δg 1 ðt; xðtÞ; xðt τÞÞ‖1
tτ
t t τ
T Z eðsÞ ds R1
ð16Þ
t tτ
eðsÞ ds :
ð17Þ
Thus, according to (15)–(17) and Assumption 3, we have LVðtÞ r eT ðtÞðPA þAT PÞeðtÞ þ δ
1 T
e ðtÞPBBT PeðtÞ
þ δHe ðt τÞeðt τÞ þ 4L‖eT ðtÞP‖1 T
þ eT ðtÞQeðtÞ eT ðt τÞQeðt τÞ Z t T Z 1 þ τeT ðtÞR1 eðtÞ eðsÞ ds R1 Z þ2
τ
t tτ
tτ
t tτ
eðsÞ ds
T eðsÞ ds R2 ðeðtÞ eðt τÞÞ 2k^ 1 eT ðtÞPeðtÞ
2k^ 2 ‖PeðtÞ‖1 þ μ½eT ðtÞM 1 eðtÞ þ eT ðt τÞM 2 eðt τÞ ^ χ ðtÞ þð4L 2k^ Þj eT ðtÞP‖ ; ¼ χ T ðtÞΦ 1 1 1
ð18Þ
338
T. Wang et al. / ISA Transactions 53 (2014) 335–340
Rt where χ ðtÞ ¼ ðeT ðtÞ; eT ðt τÞ; ð t τ eðsÞ dsÞT ÞT , 2 ^ 3 Φ 1;11 0 R2 7 ^ ¼6 Φ δH In Q þ μM2 R2 5; 4 0 1
P o μ In ; ð19Þ
2
τ 1 R1
R2
R2
δ 40 such that
1 T ^ PBBT P þ Q þ τR1 þ μM 1 2k^ 1 P. While and Φ 1;11 ¼ PA þA P þ δ the inequalities (9)–(11) have feasible solutions, by the Schur complement lemma, we yield
LV ðtÞ r 0
6
ð27Þ
Φ3;11
0
δH In Q þ μM2
0 4 R2
Φ3 ¼ 6 6
R2
PB
R2
0
R2
τ 1 R1
0
0
0
BT P
3 7 7 7 5
δ In
o 0; 4n4n
ð28Þ
ð20Þ
for all t Z0. On the other hand, by Itô formula, we have dðeɛs VðsÞÞ ¼ eɛs ½ɛVðsÞ þLVðsÞ ds þ 2eɛs eT ðsÞP φðsÞ dwðsÞ:
ð21Þ
Hence, for any t 4 0, we obtain Z t Efeɛt VðtÞg EfVð0Þg ¼ E eɛs ½ɛVðsÞ þ LVðsÞ ds 0 t
Z r ɛ1
where Φ3;11 ¼ PAþ A P þQ þ τR1 þ μM 1 2k^ 1 P, L ¼ maxfL1 ; L2 g, then the master–slave system (1) and (2) without uncertainties is exponential synchronization in mean square under the action of the following adaptive controller: ( uðtÞ ¼ k1 ðtÞeðtÞ ð29Þ k_ ðtÞ ¼ αeT ðtÞPeðtÞ: T
1
Efeɛs VðsÞg ds:
ð22Þ
0
Proof. Similar to the proof of Theorem 1, choosing a Lyapunov function as
That is, EfVðtÞg r e ɛt EfVð0Þg þ ɛ1
Z
t
Z
e ɛðt sÞ EfVðsÞg ds;
VðtÞ ¼ eT ðtÞPeðtÞ þ
0
where ɛ 1 o ɛ. Then by Lemma 2, we have EfVðtÞg r EfV ð0Þge ρt ;
Z
t Z 0;
þ
where ρ ¼ ɛ ɛ 1 . Due to
t
tτ
t tτ
Z eT ðsÞQeðsÞ dsþ
T Z eðsÞ ds R2
t t τ
0 τ
Z
t t þs
eT ðϑÞR1 eðϑÞ dϑ ds
1 eðsÞ ds þ ðk1 ðtÞ k^ 1 Þ2 ;
α
with similar deductions, we can obtain Corollary 2.
VðtÞ Z λmax ðPÞeT ðtÞeðtÞ;
ð30Þ
□
This implies
Remark 5. The lag synchronization problem is also an interesting topic for dynamical systems, which can be found in [26,27]. The master system (1) and the slave system (2) are said to be robust exponential lag synchronization in mean square if there exists a positive constant ρ 4 0 such that
1 lim sup log E ‖eðtÞ‖2 r ρ: t-1 t
1 lim sup log E ‖es ðtÞ‖2 r ρ t-1 t
Therefore, the master–slave system (1) and (2) robustly exponentially synchronize well. This completes the proof. □
for any initial conditions ϕðtÞ; ψ ðtÞ and uncertainties Δg i ðtÞ ði ¼ 1; 2Þ. It should be pointed out, under the action of adaptive controller (8), if the conditions in Theorem 1 are satisfied, then the master–slave system (1) and (2) is also robust exponential lag synchronization in mean square. However, to the best of our knowledge, for the master–slave system model, there still does not exist similar results until now.
then EfVð0Þg ρt E ‖eðtÞ‖2 r e : λmax ðPÞ
Next, we provide some useful corollaries associated with two special cases of master–slave system (1) and (2). Corollary 1. Let Assumptions 1–3 hold. If there exist positive definite symmetric matrix P A Rnn , scalars k^ 1 4 0; k^ 2 4 0; μ 4 0 and δ 4 0 such that P o μ In ; "
Φ2 ¼
ð23Þ
Φ2;11
PB
BT P
δ In
# o 0;
ð24Þ
ð31Þ
Remark 6. By means of the idea in [37,38], we will study the H1 synchronization, bilateral synchronization and state estimation problems on the master–slave systems with unmodeled dynamics and mixed time delays in our future work.
2n2n
k^ 2 4 2L;
ð25Þ
where Φ2;11 ¼ PA þ A P þ δH I n þ μM 1 2k^ 1 P, L ¼ maxfL1 ; L2 g, then the master–slave system (1) and (2) without time delay (τ ¼0) is robustly exponential synchronization in mean square under the action of the adaptive controller (8).
4. A numerical example
T
Proof. Similar to the proof of Theorem 1, choosing a Lyapunov function as 1 1 VðtÞ ¼ eT ðtÞPeðtÞ þ ðk1 ðtÞ k^ 1 Þ2 þ ðk2 ðtÞ k^ 2 Þ2 ;
α
β
ð26Þ
then we can obtain the results in Corollary 1 after some direct computation. □ Corollary 2. Let Assumptions 1–3 hold, if there exist positive definite symmetric matrices P; Q ; R1 ; R2 A Rnn , scalars k^ 1 4 0; μ 4 0 and
In order to show the effectiveness of our proposed method, we provide an illustration example in the section. It is well known that Chua's dynamical system is 8 _ > < x 1 ðtÞ ¼ 10½x2 ðtÞ x1 ðtÞ f 1 ðx1 ðtÞÞ x_ 2 ðtÞ ¼ x1 ðtÞ x2 ðtÞ þ x3 ðtÞ ð32Þ > : x_ ðtÞ ¼ 15x ðtÞ 0:0385x ðtÞ; 3 2 3 where f 1 ðx1 ðtÞÞ ¼ bx1 ðtÞ þ 0:5ða bÞðjx1 ðtÞ þ1j jx1 ðtÞ 1jÞ, a, b are two constants. Furthermore, it has been shown that the system (32) possesses a chaotic behavior if a and b are appropriately chosen. Obviously, the system can be written as the vector form _ ¼ AxðtÞ þ Bf ðxðtÞÞ; xðtÞ
ð33Þ
T. Wang et al. / ISA Transactions 53 (2014) 335–340
where xðtÞ ¼ ðx1 ðtÞ; x2 ðtÞ; x3 ðtÞÞT , 2 3 10 10 0 6 1 7 1 1 A¼4 5; 0 15 0:0385
60
f ðxðtÞÞ ¼ ðf 1 ðx1 ðtÞÞ; 0; 0ÞT , B ¼ diagf 10; 1; 1g. Due to the fact that
20
339
x1(t) x (t)
40
2
x (t)
x(t),y(t)
^ ^ 1 ðtÞÞj ‖f ðxðtÞÞ f ðxðtÞÞ‖ 2 ¼ j f 1 ðx1 ðtÞÞ f 1 ðx r ðj bj þ j a bj Þ j x1 ðtÞ x^ 1 ðtÞj ^ r ðj bj þ j a bj Þ ‖xðtÞ xðtÞ‖ 2
3
y (t) 3
y (t) 2
0
y (t) 1
−20 ^ A R3 , f ðxðtÞÞ satisfies Assumption 1. for any xðtÞ; xðtÞ We take the master system as
−40
_ ¼ AxðtÞ þ Bf ðxðt τÞÞ þ Δg 1 ðt; xðtÞ; xðt τÞÞ xðtÞ
−60
and the slave system as dyðtÞ ¼ ½AyðtÞ þ Bf ðyðt τÞÞ þ Δg 2 ðt; yðtÞ; yðt τÞÞ þ uðtÞ dt þ φðtÞ dwðtÞ;
Δg1 ðt; xðtÞ; xðt τÞÞ ¼ ð sin ðx1 ðtÞÞ; cos ðx2 ðt τÞÞ; 1Þ ; Δg2 ðt; yðtÞ; yðt τÞÞ ¼ ð cos ðx1 ðt τÞÞ; 0:5; sin ðx3 ðt τÞÞÞT ; φðtÞ ¼ ðx1 ðtÞ y1 ðtÞ; 0; x3 ðt τÞ y3 ðt τÞÞT ; T
a ¼6, b ¼ 0.75, τ ¼2. It is easy to verify that Assumptions 1–3 hold while setting H ¼ 7:52 ; L1 ¼ L2 ¼ 1; M 1 ¼ diagf0:5; 0; 0g and M 2 ¼ diagf0; 0; 0:5g. By the LMI toolbox in Matlab, we obtain the feasible solutions of the inequalities (9)–(11) as follows: 2 3 1:6743 0:0314 0:0221 6 7 2:1212 2:1213 5; P ¼ 4 0:0314 0:0221 2:1213 2:2719 2 3 129:6021 2:3307 0:0369 6 7 Q ¼ 4 2:3307 150:9423 0:02619 5; 0:0369 0:02619 163:3586 9:7733
6 R1 ¼ 4 6:6116 0:0957 2 2:6489 6 R2 ¼ 4 1:1102 0:0156
6:6116
0:0957
4
k^ 1 ¼ 80; k^ 2 ¼ 3; μ ¼ 26:5676; δ ¼ 2:1616. According to Theorem 1, we know that the master–slave system in this example robustly synchronizes well. Taking α ¼ β ¼ 0:1 and the initial conditions xð0Þ ¼ ð 30; 10; 30ÞT and yð0Þ ¼ ð50; 20; 50ÞT , we can draw some figures as follows. In particular, Fig. 1 shows the state trajectories of the master–slave system without control input. Figs. 2 and 3 present the state trajectories and the state error trajectories of the master–slave system under the action of the adaptive controller (8) for the given initial conditions, respectively, which shows that our designed controller is effective. Fig. 4 is the state trajectories of the updated laws in the adaptive controller, which shows that the control gains almost are constants. Remark 7. It is noted that Ref. [38] also studied the synchronization of dynamical systems with uncertain nonlinear terms by using a similar discontinuous controller as that in our paper. Furthermore, the nonlinear terms also satisfy the bounded condition. However, they did not consider the factors as the time delay and stochastic disturbances. In addition, the methods in the two papers are different. Compared with them, our proposed method need not do some complex computation and discussions. Furthermore, we may adjust the convergence rate by tuning the control gains α and β. Besides them, the method in [38] only could be used to solve low-dimensional dynamical systems such as
8
10
50 x3(t),y3(t) x2(t),y2(t)
0
3
7 70:6853 0:1531 5; 0:1531 71:2835 3 1:1102 0:0156 12:8618 0:0184 7 5; 0:0184 12:8662
6
Fig. 1. State trajectories of the master–slave system without control input for the given initial conditions.
x (t),y (t) 1
−50
0
0.5
1
1
1.5
2
2.5
3
t/second Fig. 2. State trajectories of the master–slave system under the action of the adaptive controller (8) for the given initial conditions.
80 e (t) 1
60
e2(t) e (t)
40
3
20
e(t)
2
2
t/second
x(t),y(t)
respectively, where
0
0 −20 −40 −60 −80
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
t Fig. 3. The state error trajectories of the master–slave system under the action of the adaptive controller (8) for the given initial conditions.
340
T. Wang et al. / ISA Transactions 53 (2014) 335–340
300 [8]
250 k (t)
[9]
1
k (t)
k1(t),k2(t)
200
2
[10]
150
[11]
100
[12]
50
[13]
0
[14]
0
0.5
1
1.5
2
2.5
3
t/second Fig. 4. State trajectories of the updated laws in the adaptive controller (8).
[15] [16] [17]
1-dimensional or 2-dimensional ones, for the high-dimensional case, which is invalidate.
[18] [19]
5. Conclusion In this paper, we have investigated the robust master–slave synchronization for a class of uncertain delayed dynamical systems based on the adaptive control method. Different from some existing results, the unmodeled dynamics only need to satisfy the bounded condition. In order to achieve synchronization, a new controller has been constructed by using the idea of the bang–bang control and the sliding mode control. Numerical example has shown that our proposed method is effective. Acknowledgments
[20]
[21]
[22]
[23]
[24] [25]
[26]
This work was supported by the National Natural Science Foundation of China (61203128), Tian Yuan Special Foundation (NO 11326132), the Chen Guang Project of Shanghai Municipal Education Commission and Shanghai Education Development Foundation (12CG65), the Domestic Visiting Project of Shanghai University Teachers, Connotative Construction Project of Shanghai University of Engineering Science (nhky-2012-13).
[30]
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