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International Journal of Bifurcation and Chaos, Vol. 18, No. 1 (2008) 187–202 c World Scientific Publishing Company
FURTHER RESULTS ON MASTER-SLAVE SYNCHRONIZATION OF GENERAL LUR’E SYSTEMS WITH TIME-VARYING DELAY FERNANDO O. SOUZA, REINALDO M. PALHARES, ˆ EDUARDO M. A. M. MENDES∗ and LEONARDO A. B. TORRES Department of Electronics Engineering, Federal University of Minas Gerais, Av. Antˆ onio Carlos 6627, Belo Horizonte 31270-901, Brazil ∗[email protected] Received July 13, 2006; Revised January 2, 2007 In this paper, a new approach to analyze the asymptotic, exponential and robust stability of the master-slave synchronization for Lur’e systems using time-varying delay feedback control is proposed. The discussion is motivated by the problem of transmitting information in optical communication systems using chaotic lasers. The approach is based on the Lyapunov–Krasovskii stability theory for functional differential equations and the linear matrix inequality (LMI) technique with the use of a recent Leibniz–Newton model based transformation, without including any additional dynamics. Using the problem of synchronizing coupled Chua’s circuits, three examples are given to illustrate the effectiveness of the proposed methodology. Keywords: Synchronization; Lur’e system; time-delay; robust stability; asymptotic and exponential stability.
1. Introduction
to point by using chaotic carrier signals (e.g. see Sec. 3.4 in [Andrievskii & Fradkov, 2004] and references therein), specific methods are preferred in particular cases, due to practical limitations. One such situation occurs in the secure transmission of information relying on synchronized chaotic lasers, in the context of optical communication systems [Liu et al., 2001; Chen & Liu, 2000]. In this case, the speed constraints related to the processing of optical signals demand simple and robust ways to achieve synchronization [Blakely et al., 2004]. It is important to note two characteristics of chaotic laser systems, namely: (i) Local
Despite the theoretical relevance of synchronizing Lur’e systems with time-varying delay, one of the main focus of the present paper is the analysis of synchronized chaotic oscillators. Since seminal papers were published on theoretical and experimental evidence of chaotic systems synchronization [Fujisaka & Yamada, 1983; Pecora & Carroll, 1990], much effort has been devoted to the analysis of such systems. As pointed out in [Yal¸cin et al., 2001], such oscillators are the key components in chaos based communication systems. Although there are many different approaches to send information from point ∗
Author for correspondence 187
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time-delayed optoelectronic feedback sometimes can be used to produce chaotic oscillations in each laser, at the transmitter, and at the receiver; i.e. the lasers would not exhibit natural chaotic behavior, if this was not done [Liu et al., 2001] and (ii) the coupling structure commonly used to send information is based on the master-slave approach, which implies that one chaotic oscillator — the master system — influences the other chaotic oscillator — the slave system; and not the other way around. The first characteristic indicates that delay lines are already used on some chaotic lasers, and therefore they could be modified to ensure synchronization. The primary role of these internal feedback delay lines is to guarantee chaotic behavior, independently of the coupling with another oscillator. The second characteristic indicates that, in the analysis of the synchronization condition, the propagation delay is immaterial, because the only and harmless effect on the receiver, due to the distance between transmitter and receiver, would be the receiver synchronization with a delayed copy of the transmitter, as depicted in Fig. 1. It is noted that this scenario could be changed if signals from the slave system were sent back to the master system, in a hypothetical bidirectional coupling, but this is not the case. The above facts seem to be relevant to establish, in the authors’ point of view, what are the primary motivations to use artificial delay lines or to consider the impact of time delay signal propagation in chaos based optical communication systems, namely: (i) the “chaotification” of the transmitted signals aimed at alleged system security improvement; (ii) the determination of possible deleterious effects caused by unavoidable internal signal propagation delay, at the controlled receiver; and (iii) the practical feasibility of a control strategy that can be used in the case of ultra-fast chaotic systems synchronization.
Master(t)
τ
Master´(t)
Slave(t)
Fig. 1. Master-slave approach. The existence of propagation delay τ is immaterial, once the slave system synchronizes with a master system delayed copy: master (t) = master(t − τ ). 1
Focusing on the important issue of the so-called secure character of the information transmission system [Alvarez & Li, 2006], some considerations are in order. If one wishes to keep the information transmission system secure, it is desirable to design synchronized systems that are highly sensitive to parameter uncertainty. On the other hand, if the transmission system should be robust, in the sense that the master and slave systems should achieve synchronization despite parameters mismatch, the security would be inevitably lowered down. The main goal in the present work is to theoretically analyze the synchronization stability of systems with time-varying delay, in order to provide less conservative tools that can be used to inspect more carefully the conflicting requirements in the design of chaos based optical communication systems, as stated above.1 The tools here presented follow the approach of constructing linear matrix inequalities (LMI) from the analysis of Lyapunov–Krasovskii functionals, providing sufficient conditions to determine the feasibility of the synchronization between master and slave systems represented as uncertain Lur’e systems with time-varying delay [Yal¸cin et al., 2001; Liao & Chen, 2003; Huang et al., 2006]. The results are also extended to the case of exponential stability analysis. In this paper the notation “∗” is used instead of writing terms in the symmetric matrix. The superscript “T ” represents the transpose. M 0 (≺ 0) means that the matrix M is positive (negative) definite. diag(·) denotes a diagonal matrix. Let R denote the set of real numbers, Rn the n-dimensional Euclidean space and Rn×m the set of all n × m real matrices, 0n×n are n × n matrices of zeros. The paper is organized as follows: in Sec. 2 the main approach is outlined by stating the assumptions and presenting the theoretical developments that lead to less conservative linear matrix inequalities (LMI) for asymptotic stability analysis (Sec. 2.1), exponential stability analysis (Sec. 2.2) and robust stability analysis (Sec. 2.3) of the synchronization error dynamical system. Illustrative examples, through numerical simulations, are provided in Sec. 3. Finally, concluding remarks are presented in Sec. 4.
A very interesting alternative is proposed in [Kim et al., 2004], where the very time delays comprise the “key” to decode the information at the receiver.
Further Results on Master-Slave Synchronization of General Lur’e Systems
2. Further Results Consider the following master-slave synchronization scheme: x(t) ˙ = Ax(t) + Bσ(Cx(t)) M: p(t) = Hx(t) y(t) ˙ = Ay(t) + Bσ(Cy(t)) + u(t) S: (1) q(t) = Hy(t) C : u(t) = −K(x(t) − y(t)) + M (p(t − τ (t)) − q(t − τ (t)))
189
Throughout the paper, the following two hypotheses will be used to derive the conditions for the robust stability. (H1) The time-varying delay, τ (t), is continuous, differentiable and bounded, i.e. 0 ≤ τ (t) ≤ τ,
τ˙ (t) ≤ µ < 1
(3)
where τ and µ are constants. (H2) The nonlinearity η(Ce, y) belongs to the sector [0, κ]: ηi (cTi e, y) σi (cTi e + cTi y) − σi (cTi y) = ≤κ cTi e cTi e
where M is the master system, S is the slave system and C is the time-delay feedback control law with a time-varying delay. The state vectors are given by x, y ∈ Rn , the output of each system are denoted by p, q ∈ Rl , the matrices A ∈ Rn×n , B ∈ Rn×nh , C ∈ Rnh ×n , H ∈ Rl×n , and the function σ(·) satisfies a sector condition, with σi (·) i = 1, 2, . . . , nh belonging to the sector [0, κ], i.e. σi (ν)(σi (ν)− κν) ≤ 0 for i = 1, 2, . . . , nh . Figure 2 shows the scheme of synchronization between the master and the slave systems proposed in [Huang et al., 2006], where the authors applied a time-varying delay feedback to the slave system with control signal u(t) ∈ Rn , feedback matrices K ∈ Rn×n , M ∈ Rn×l and time-varying delay τ (t). Synchronization means agreement or correlation of different processes in time, therefore the design of the controller C aims to make x(t) − y(t) → 0 as t → ∞, where · is a norm defined in Rn . By defining the error signal e(t) = x(t) − y(t), the error system can be represented as follows ˜ ε : e(t) ˙ = Ae(t) + F e(t − τ (t))
then the error system (2) is exponentially stable, where δ is called the rate of exponential stability.
+ Bη(Ce(t), y(t)) (2) ˜ with A = A + K, F = −M H, and η(Ce(t), y(t)) = σ(Ce(t) + Cy(t)) − σ(Cy(t)).
The relationship between the state and the delayed state as well as the delay-dependence can be explicitly taken into account in the stability analysis
Fig. 2.
0≤
where cTi denotes the ith row vector of C. The following inequality holds, ηi (cTi e, y)(ηi (cTi e, y) − κcTi e) ≤ 0, ∀ e, y; i = 1, 2, . . . , nh .
(4)
Notice that the initial condition to error system is e(s) = 0, s ∈ [−τ 0]. The following definition will be also used in the remaining of the paper. Definition 1. Let δ > 0 and (δ) > 0 be given
scalars, such that:
−δt
e(t) ≤ (δ)e
sup e(θ)
−τ ≤θ≤0
+
˙ , sup e(θ)
−τ ≤θ≤0
Master-slave synchronization scheme.
∀t > 0
(5)
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by considering the Newton–Leibniz identity and some slack matrix variables. This strategy is used for reducing conservativeness in the resulting LMIs without introducing any extra dynamics as suggested in [Wu et al., 2004]. Lemma 1. For any matrices Y , W , and X of appropriate dimensions, the null term
2[eT (t)Y + eT (t − τ (t))W + η T (Ce(t), y(t))X] × e(t) −
t
t−τ (t)
e(s)ds ˙ − e(t − τ (t)) = 0
(6)
is equivalent to
Y +YT t ∗ 1 ξ T (t, s) τ (t) t−τ (t) ∗ ∗
−Y + W T −W − W T ∗ ∗
XT −X T 0 ∗
−τ Y −τ W ξ(t, s)ds = 0 −τ X
(7)
0
with ξ(t, s) = [eT (t) eT (t − τ (t)) η T (Ce(t), y(t)) e˙T (s)]T . Using the fact that the Newton–Leibniz identity is given by t e(s)ds ˙ e(t − τ (t)) = e(t) −
Proof.
t−τ (t)
the equality in Eqs. (6) and (7) follows directly.
2.1. Asymptotic stability analysis Consider now the asymptotic stability of the error system with time-varying delay τ (t). The main idea is to add the null term in Lemma 1
PA˜ + A˜TP + Q + Y T + Y ∗ ∗ Ξ = ∗ ∗
to the derivative of the Lyapunov functional to obtain a delay-dependent condition. This simple strategy results in a less conservative approach to analyze the problem of stability. The first main result is stated in the following theorem. Theorem 1. Let τ > 0 and µ < 1 be the upper
bounds for the time-delays and their rate of variation respectively. Suppose that hypothesis (H1) and (H2) hold. If there exist P = P T > 0, Q = QT > 0, Z = Z T > 0, Λ = diag(λ1 , λ2 , . . . , λnh ) > 0, and any matrices X, Y and W of appropriate dimensions such that the following LMI is satisfied:
PF + W T − Y −(1 − µ)Q − W T − W ∗ ∗ ∗
then the master-slave synchronization scheme (1) synchronizes with error system (2) having an unique and globally asymptotic stable equilibrium point, e = 0, where A˜ = A + K and F = −M H. Select the following Lyapunov–Krasovskii functional candidate as in [Palhares et al., 2005a, 2005b]:
Proof.
V (et ) = V1 (et ) + V2 (et ) + V3 (et ) where V1 (et ) eT (t)P e(t)
(9)
P B + X T + κC TΛ −τ Y −X T −τ W −2Λ −τ X ∗ −τ Z ∗ ∗ V2 (et ) V3 (et )
t t−τ (t)
0 −τ
t
t+θ
τ A˜T Z τFT Z τ BT Z ≺ 0 0 −τ Z
(8)
eT (s)Qe(s)ds e˙ T (s)Z e(s)dsdθ ˙
with P = P T 0, Q = QT 0, Z = Z T 0. Taking the time derivative of V (et ) in (9) along with the error state trajectories, e(t), in (2) yields V˙ (et ) = V˙ 1 (et ) + V˙ 2 (et ) + V˙ 3 (et )
(10)
Further Results on Master-Slave Synchronization of General Lur’e Systems
191
with ˙ + e˙ T (t)P e(t) V˙ 1 (et ) = eT (t)P e(t) ˜ ˜ = eT (t)P [Ae(t) + F e(t − τ (t)) + Bη(Ce(t), y(t))] + [Ae(t) + F e(t − τ (t)) + Bη(Ce(t), y(t))]T P e(t) ˜ + 2eT (t)P Bη(Ce(t), y(t)) + 2eT (t)P F e(t − τ (t)) = 2eT (t)PAe(t) t 1 ˜ 2eT (t)PAe(t) + 2eT (t)P Bη(Ce(t), y(t)) + 2eT (t)P F e(t − τ (t))ds (11) ≤ τ (t) t−τ (t) T ˙ (t − τ (t))Qe(t − τ (t)) + eT (t)Qe(t) V˙ 2 (et ) = −(1 − d(t))e
≤ −(1 − µ)eT (t − τ (t))Qe(t − τ (t)) + eT (t)Qe(t) t
1 −(1 − µ)eT (t − τ (t))Qe(t − τ (t)) + eT (t)Qe(t) ds = τ (t) t−τ (t) t T ˙ ˙ − e˙T (s)Z e(s)ds ˙ V3 (et ) = τ e˙ (t)Z e(t) ˙ − ≤ τ e˙ T (t)Z e(t) 1 ≤ τ (t) ≤
1 τ (t)
t t−τ (t)
t t−τ (t)
(12)
t−τ t
t−τ (t)
e˙ T (s)Z e(s)ds ˙
[τ e˙T (t)Z e(t) ˙ − e˙ T (s)τ Z e(s)]ds ˙ ˜ [eT (t)τ A˜TZ Ae(t) + eT (t)τ A˜TZF e(t − τ (t)) + eT (t)τ A˜TZBη(Ce(t), y(t))
T
˜ + eT (t − τ (t))τ F TZF e(t − τ (t)) + e (t − τ (t))τ F TZAe(t) ˜ + η T (Ce(t), y(t)) + eT (t − τ (t))τ F TZBη(Ce(t), y(t)) + η T (Ce(t), y(t))τ B TZAe(t) ˙ × τ B TZF e(t − τ (t)) + η T (Ce(t), y(t))τ B TZBη(Ce(t), y(t)) − e˙T (s)τ Z e(s)]ds
where, again, A˜ = A + K and F = −M H. Using hypothesis (H2) that deals with the nonlinearities and adding in (10) the null term given in Lemma 1, it follows that: V˙ (et ) ≤ V˙ 1 (et ) + V˙ 2 (et ) + V˙ 3 (et ) −2
nh i=1
+ 2[eT(t)Y + eT(t − τ (t))W
× e(t) −
t
t−τ (t)
e(s)ds ˙ − e(t − τ (t)) (14)
that is equivalent to t 1 ˙ ξ T (t, s)Ωξ(t, s)ds V (et ) ≤ τ (t) t−τ (t)
Ω12 Ω22 ∗ ∗
Ω13 Ω23 Ω33 ∗
Ω14 Ω24 Ω34 Ω44
(16)
with
λi ni (CiTe, y)(ηi (cTie, y) − κcTie)
− η T(Ce(t), y(t))X]
Ω11 ∗ Ω= ∗ ∗
(13)
(15)
ξ(t, s) = [eT(t) eT(t − τ (t)) η T(Ce(t), y(t)) e˙T (s)]T Ω11 Ω12 Ω13 Ω14 Ω22 Ω23 Ω24 Ω33 Ω34 Ω44
PA˜ + A˜TP + Q + τ A˜TZA˜ + Y + Y T PF + τ A˜TZF − Y + W T P B + X T + τ A˜TZB + κC TΛ −τ Y −(1 − µ)Q + τ F TZF − W − W T −X T + τ F TZB −τ W −2Λ + τ B TZB −τ X −τ Z
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Applying Schur’s complement in (16), Ξ defined in (8) is obtained. If Ξ ≺ 0, then V˙ (et ) < 0 for any ξ(t, s) = 0. As a result the master-slave synchronization scheme (1) synchronizes with the error system in Eq. (2) yielding a unique and globally asymptotic stable equilibrium point, e = 0, in the context of the Lyapunov–Krasovskii theory.
controller C design, and the time-delay τ (t), can be analyzed as tuning parameters to adjust the speed synchronization. Then the following Theorem proposes a sufficient condition for the exponential stability analysis of error synchronization system (2) in the master-slave synchronization given by (1).
2.2. Exponential stability analysis
upper bounds for the time-delay, the rate of variation and the exponential convergence rate of error system (2), respectively. Suppose that hypotheses (H1) and (H2) hold. If there exist P = P T > 0, Q = QT > 0, Z = Z T > 0, Λ = diag(λ1 , λ2 , . . . , λnh ) > 0, and any matrices X, Y and W of appropriate dimensions such that the following LMI is satisfied:
Theorem 2. Let τ > 0, µ < 1, and δ > 0 be the
The exponential stability analysis is important since it is often desired that the master-slave synchronization scheme converges in an exponential rate to ensure fast response for the synchronization. In this context, the gain matrices K and M , in the
Ξexp
=
PA˜ + A˜TP + 2δP + Q + Y T + Y
PF + W T − Y
PB + X T + κC TΛ
−τ Y
∗
−e2δτ (1 − µ)Q − W T − W
−X T
−τ W
∗
∗
−2Λ
−τ X
∗
∗
∗
−τ Z
∗
∗
∗
∗
τ A˜TZ
τ F TZ τ B TZ ≺ 0 0 −τ Z (17)
then the master-slave synchronization scheme in (2) synchronizes and the error system in (2) has a unique and globally exponential stable equilibrium point at e = 0, with the exponential convergence rate, δ, A˜ = A + K and F = −M H.
V3 exp (et )
Select the Lyapunov–Krasovskii functional candidate as:
PT
Proof.
Vexp (et ) = V1 exp (et ) + V2 exp (et ) + V3 exp (et ) (18) where V1 exp (et ) e2δt eT (t)P e(t)
V2 exp (et )
t
t−τ (t) 0
−τ
t t+θ
e2δs eT (s)Qe(s)ds e2δs e˙ T (s)Z e(s)dsdθ ˙
0, Q = QT 0, Z = Z T 0. with P = Taking the time derivative of V (et ) in (18) along with the error state trajectories, e(t), in (2) yields V˙ exp (et ) = V˙ 1 exp (et ) + V˙ 2 exp (et ) + V˙ 3 exp (et ) (19) with
˙ + e2δt e˙ T (t)P e(t) V˙ 1 exp (et ) = 2δe2δt eT (t)P e(t) + e2δt eT (t)P e(t) ˙ + e˙T (t)P e(t)} = e2δt {2δeT (t)P e(t) + eT (t)P e(t) e2δt t ˜ 2eT (t)[δP + PA]e(t) + 2eT (t)PBη(Ce(t), y(t)) + 2eT (t)PF e(t − τ (t))ds ≤ τ (t) t−τ (t) T ˙ (t − τ (t))Qe(t − τ (t)) + e2δt eT (t)Qe(t) V˙ 2 exp (et ) = −e2δ(t−τ (t)) (1 − d(t))e ≤ −e2δ(t−τ ) (1 − µ)eT (t − τ (t))Qe(t − τ (t)) + e2δt eT (t)Qe(t) = e2δt {−e2δτ (1 − µ)eT (t − τ (t))Qe(t − τ (t)) + eT (t)Qe(t)} e2δt t −e2δτ (1 − µ)eT (t − τ (t))Qe(t − τ (t)) + eT (t)Qe(t)ds = τ (t) t−τ (t)
(20)
(21)
Further Results on Master-Slave Synchronization of General Lur’e Systems
˙ − V˙ 3 exp (et ) = τ e2δt e˙T (t)Z e(t) ˙ − ≤ τ e2δt e˙T (t)Z e(t) ≤ =
e2δt τ (t) e2δt τ (t)
t
t−τ (t)
t
t−τ (t)
t t−τ
193
e2δt e˙T (s)Z e(s)ds ˙
t t−τ (t)
e2δt e˙ T (s)Z e(s)ds ˙
[τ e˙T (t)Z e(t) ˙ − e˙ T (s)τ Z e(s)]ds ˙ ˜ [eT (t)τ A˜TZAe(t) + eT (t)τ A˜TZF e(t − τ (t)) + eT (t)τ A˜TZBη(Ce(t), y(t))
˜ + eT (t − τ (t))τ F TZF e(t − τ (t)) + eT (t − τ (t))τ F TZAe(t) ˜ + eT (t − τ (t))τ F TZBη(Ce(t), y(t)) + η T (Ce(t), y(t))τ B TZAe(t) + η T (Ce(t), y(t)) ˙ × τ B TZF e(t − τ (t)) + η T (Ce(t), y(t))τ B TZBη(Ce(t), y(t)) − e˙T (s)τ Z e(s)]ds with A˜ = A + K and F = −M H. Using hypothesis (H2) (nonlinearities) and adding in (19) the null term in Lemma 1 results in V˙ exp (et ) ≤ V˙ 1 exp (et ) + V˙ 2 exp (et ) + V˙ 3 exp (et ) −2
nh i=1 T
λi ni (CiTe, y)(ηi (cTie, y) − κcTie)
+ 2[e (t)Y + eT (t − τ (t))W − η T (Ce(t), y(t))X] t e(s)ds ˙ − e(t − τ (t)) × e(t) − t−τ (t)
(23) that is equivalent to: e2δt t ˙ ξ T (t, s)Ωexp ξ(t, s)ds (24) Vexp (et ) ≤ τ (t) t−τ (t) Ωexp(11) Ωexp(12) Ωexp(13) Ωexp(14) ∗ Ωexp(22) Ωexp(23) Ωexp(24) Ωexp = ∗ ∗ Ωexp(33) Ωexp(34) ∗ ∗ ∗ Ωexp(44)
Ωexp(14) −τ Y Ωexp(22) −e2δτ (1 − µ)Q + τ F TZF − W − W T Ωexp(23) −X T + τ F TZB Ωexp(24) −τ W Ωexp(33) −2Λ + τ B TZB Ωexp(34) −τ X Ωexp(44) −τ Z Applying Schur complements in (25), Ξexp defined in (17) is obtained. If Ξexp ≺ 0, then V˙ exp (et ) < 0 for any ξ(t, s) = 0. Therefore, it follows that Vexp (e(t)) ≤ Vexp (e(0)),
Vexp (e(t)) ≤ e2δt λmax (P )e(t)2 t +λmax (Q) e2δs e(s)2 ds t−τ (t) 0 t
+ λmax (Z)
with
and ˜T
˜T
Ωexp(11) 2δP + PA˜ + A P + Q + τ A ZA˜ Ωexp(12)
+Y +Y T PF + τ A˜TZF − Y + W T
Ωexp(13) PB + X T + τ A˜TZB + κC TΛ
for t ≥ 0
From (18), it can be observed that:
(25) ξ(t, s) = [eT (t) eT (t−τ (t)) η T (Ce(t), y(t)) e˙ T (s)]T
(22)
−τ
t+θ
2 e2δs e(s) ˙ dsdθ
Furthermore, notice that [Xu et al., 2005]:
0
−τ
t
t+θ
2 e2δs e(s) ˙ dsdθ t
s−t
= ≤τ
t−τ −τ t 2δs t−τ
e
2 e2δs e(s) ˙ dθds 2 e(s) ˙ ds
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Therefore, it follows that
+λmax (Q)φ2
Vexp (e(t)) ≤ e2δt λmax (P )e(t)2 t + λmax (Q) e2δs e(s)2 ds
˙ 2 + τ λmax (Z)φ
t−τ (t) t
+ τ λmax (Z)
t−τ
≤ λmax (P )φ2
2
and for t = 0
−τ (0) 0
+ τ λmax (Z) ≤ λmax (P )φ2
−τ
2 e2δs e(s) ˙ ds
0
−τ (0) 0 −τ
e2δs ds
e2δs ds
1 − e−2δτ + λmax (Q)φ 2δ −2δτ ˙ 2 1−e + τ λmax (Z)φ 2δ
2 e2δs e(s) ˙ ds
Vexp (e(0)) ≤ λmax (P )e(0)2 0 + λmax (Q) e2δs e(s)2 ds
with φ
sup e(θ),
−τ ≤θ≤0
˙ φ
sup e(θ) ˙
−τ ≤θ≤0
On the other hand, it is clear that Vexp (e(t)) ≥ e2δt λmin (P )e(t)2 Since Vexp (e(t)) ≤ Vexp (e(0)), e(t) is bounded above by
1 − e−2δτ 1 − e−2δτ λmax (Z) λmax (P ) + λmax (Q) 2δ 2δ −δt ˙ −δt φe φe + e(t) ≤ λmin (P ) λmin (P ) ˙ ≤ ρ(δ)e−δt {φ + φ} with 1 − e−2δτ λ (P ) + λmax (Q) max 2δ , ρ(δ) max λ (P ) min
Then, from Definition 1, the master-slave synchronization scheme in (2) synchronizes with error system (2) having a unique and globally exponential stable equilibrium point at e = 0 with the exponential convergence rate, δ.
2.3. Robust stability analysis Notice that Theorems 1 and 2 can be extended to the case of robust stability analysis when the system matrices are not exactly known. Suppose that the matrices in the scheme of master-slave synchronization (1) are unknown but belonging to a polytope type uncertain domain P defined as the set of all matrices taken from the
−2δτ 1−e λmax (Z) 2δ λmin (P )
convex combination of the vertices: v γi (Ai , Bi ); P (A, B) : (A, B) = i=1
γi ≥ 0,
v
γi = 1 ,
(26)
i=1
where Ai and Bi , i = 1, . . . , v, are the polytope vertices. The robust stability analysis problem is to verify if system (2) is stable for all (A, B) ∈ P. In this case, the easiest sufficient condition for robust stability is based on the concept of quadratic stability, where a single Lyapunov–Krasovskii functional
Further Results on Master-Slave Synchronization of General Lur’e Systems
is used to prove the stability for the entire polytope. As a consequence, the stability of a polytope of matrices can be checked by a feasibility test of a set of LMIs involving only the vertices of the uncertainty domain. The following Theorem states a sufficient condition for the robust stability analysis of error system (2), which extends the result in Theorem 1 for uncertain systems in terms of LMIs, and in doing so it guarantees the master-slave synchronization.
PA˜i + A˜TiP + Q + Y T + Y ∗ Ξi = ∗ ∗ ∗
Theorem 3. Consider the uncertain error system given in (2) with (A, B) ∈ P as in (26). Let τ > 0, and µ < 1 be the upper bounds for the time-delay and its rate of variation, respectively. Suppose that hypotheses (H1) and (H2) hold. The uncertain error system (2) is quadratically stable, if there exist matrices P = P T > 0, Q = QT > 0, Z T > 0, Λ = diag(λ1 , λ2 , . . . , λnh ) > 0, and any matrices X, Y and W with appropriate dimensions satisfying the following LMIs:
PF + WT − Y −(1 − µ)Q − W T − W ∗ ∗ ∗
∀ i = 1, . . . , v, and A˜i = Ai + K. As was done for Theorem 1, Theorem 2 can be easily extended to the problem of robust exponential stability for synchronization by considering the LMI presented in (17), where A˜ (where A˜ = A + K) and B are taken with the subscript i, ∀ i = 1, . . . , v, where i denotes the vertices of the polytope.
3. Illustrative Examples In this section three examples using the same Chua’s Circuit as in [Yal¸cin et al., 2001] and [Huang et al., 2006] are given to demonstrate the efficiency of the proposed method. In particular, the first example is used for comparison purposes with the methods proposed in those papers. Example 1. Consider the Chua Circuit governed by the following equations x˙ = a(y − h(x)) y˙ = x − y − z (28) z˙ = −by
where h(x), the nonlinear characteristic of the system, is given by 1 h(x) = m1 x + (m0 − m1 )(|x + c| − |x − c|) 2 and the parameters are a = 9, b = 14.28, c = 1, m0 = −1/7, m1 = 2/7. With these set of parameters, the system settles on the well-known double scroll attractor.
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P Bi + X T + κC TΛ −τ Y −X T −τ W −2Λ −τ X ∗ −τ Z ∗ ∗
τ A˜TiZ τ F TZ τ BiTZ ≺0 0 −τ Z (27)
System (28) can be represented in the Lur’e form as follows −a(m0 − m1 ) −am1 a 0 −1 1 , B = 0 A= 1 , 0 −b 0 0
C= 1 0 0 and σ(ν) = 1/2(|ν + c| − |ν − c|) belongs to sector [0 κ] with κ = 1. As considered in [Yal¸cin et al., 2001], the output matrix is selected as H = [1 0 0], which means that the master system is connected to the slave system by the first state variable. By choosing K = 0n×n and M = [6.0229
1.3367
− 2.1264]T
the system in [Yal¸cin et al., 2001] is now investigated. For comparison purposes, consider µ = 0, i.e. the time-delay is lumped and constant. Applying Theorem 1 proposed in this paper, the largest time-delay is τ = 0.141. Whereas applying Theorem 2 in [Yal¸cin et al., 2001] the largest time-delay is τ = 0.039 and applying Theorem 1 in [Huang et al., 2006] τ = 0.083. It is clear that the proposed method is less conservative than the methods in [Yal¸cin et al., 2001] and [Huang et al., 2006] for this first example. To further demonstrate this, τ = 0.141 is now considered. Figures 3 and 4 show the error signal and the double scroll attractors generated for
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e(t) = x(t)−y(t)
0.4 0.2 0 −0.2 −0.4 −0.6 −0.8
0
10
20
30
40
50
time
Fig. 3.
Error signal, e(t) = x(t) − y(t), with τ = 0.141 and µ = 0.
the master and slave systems, respectively. From simulations, it can be noticed that the coupled Chua’s Circuit do not synchronize for τ = 0.21, as depicted in Figs. 5 and 6. The initial conditions of the master and slave systems in this case were x(s) = [0.4 −0.53 0.17]T and y(s) = [0.28 −0.34 0.81]T ∀ s ∈ [−τ 0]. On the other hand, when a set of non-null matrix gains, K, with M = [6.0229 1.3367 − 2.1264]T in (1) is considered, the largest timedelays obtained by applying Theorem 1 proposed in this paper and Theorem 1 in [Huang et al., 2006] are
shown in Table 1. Note that the proposed method is less conservative. The reader should be reminded that the approach proposed in [Yal¸cin et al., 2001] cannot be applied in cases where the gain matrix K is non-null. Unlike the main purpose of the approach in [Huang et al., 2006], which is also developed to obtain a synchronization criteria dependent on the size of the time-delay, the matrices K and M chosen in [Huang et al., 2006] for this same example make the error system in (2)
Remark.
Slave
Master
0.4
0.4 0.2
0.2
x2
y2
0 0
−0.2 −0.2
−0.4
−0.4 4
−0.6 4 2
4 2
0
0
−2 x
3
−4
−4
2
4 0
−2
−2 x1
2
0
y3
−4
−4
−2 y1
Fig. 4. Three-dimensional view on the double scroll attractors generated for, master and slave systems, with τ = 0.141 and µ = 0.
Further Results on Master-Slave Synchronization of General Lur’e Systems
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Error 4 3
e(t) = x(t)−y(t)
2 1 0 −1 −2 −3 −4
0
20
40
60
80
100
time
Fig. 5.
Error signal, e(t) = x(t) − y(t), with τ = 0.21 and µ = 0.
Slave
Master
0.4
1.5
0.2
1 0.5
x
2
y2
0 −0.2
0 −0.5
−0.4
−1
−0.6 4
−1.5 10 2
5
4 2
0
0
−2 x3
−4
−4
5 0
0
−5
−2
−10
y3
x1
−5
y1
Fig. 6. Three-dimensional view on the double scroll attractors generated for, master and slave systems, with τ = 0.21 and µ = 0.
Table 1. K K1 K2 K3 K4 K5
= diag(−0.52, −1.642, −2.623) = diag(−1.52, −2.642, −3.623) = diag(−2.52, −3.642, −4.623) = diag(−3.52, −4.642, −5.623) = diag(−4.52, −5.642, −6.623)
The largest time-delay, τ , with µ = 0. Theorem 1 in [Huang et al., 2006] τ τ τ τ τ
= 0.112 = 0.131 = 0.154 = 0.180 = 0.213
Theorem 1, Proposed τ τ τ τ τ
= 0.159 = 0.173 = 0.189 = 0.208 = 0.233
Largest Allowable Time-Delay, by Simulation τ τ τ τ τ
= 0.231 = 0.261 = 0.297 = 0.350 = 0.447
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become delay-independent, i.e. the synchronization occurs regardless of the size of time-delay. In [Huang et al., 2006] the control matrices are chosen as K = diag(−8.52, −9.642, −10.623) and M = [4.0229, 1.3367, −2.1267]T which, as illustrated in Fig. 7, make the error dynamics stable even for a large time-delay as, e.g. τ = 100. If one chooses K = diag(−8.52, −9.642, −10.623) and
M = [6.0229, 1.3367, −2.1267]T the same characteristic can be again noticed as seen in Fig. 8. For this last case, the maximum time-delay found by applying Theorem 1 proposed in this paper or Theorem 1 in [Huang et al., 2006] is τ = 0.828. At first glance this could mean that this kind of delay-independence synchronization may be desired. However, from the authors’ point of
Error 0.2 0.1 0
e(t) = x(t)−y(t)
−0.1 −0.2 −0.3 −0.4 −0.5 −0.6 −0.7
0
Fig. 7.
500
1000 time
1500
2000
Error signal, e(t) = x(t) − y(t), with τ = 100 and µ = 0.
Error 0.2 0.1 0
e(t) = x(t)−y(t)
−0.1 −0.2 −0.3 −0.4 −0.5 −0.6 −0.7
0
Fig. 8.
500
1000
1500 time
2000
2500
Error signal, e(t) = x(t) − y(t), with τ = 100 and µ = 0.
3000
Further Results on Master-Slave Synchronization of General Lur’e Systems
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Example 2. Consider the same Chua’s Circuit with the same parameters as in Example 1, K = 0n×n , and M = [6.0229 1.3367 −2.1264]T . Applying Theorem 2 and considering constant time-delay, that is, µ = 0, the exponential stability is obtained as can be seen in Table 2. The efficiency of the proposed method is demonstrated in Figs. 9 to 11, in the context that if one desires to design appropriate gains with specific exponential stability characteristics. In this scenario, Theorem 2 may be useful.
view, this kind of problem had been issued in the context of delay-dependence as supported by the Introduction of this paper concerning optical communication. To highlight this discussion, suppose that the error system (2) can be represented by a linear description: ˜ ε : e(t) ˙ = Ae(t) + F e(t − τ (t)) with A˜ = A + K and F = −M H. As presented in [Niculescu, 1996], a necessary condition for the above system to be delay-independent is that A˜ + F and A˜ − F be Hurwitz stable. Considering K = diag(−8.52, −9.642, −10.623) and M = [6.0229, 1.3367, −2.1267]T it is easy to verify that A˜ + F and A˜ − F are Hurwitz stable. Notice that this does not hold for the set of matrices K in Table 1 and the last matrix M . Notice also that in terms of sharpness of results it is important to note that u(t) in [Yal¸cin et al., 2001] is different; it contains no part with the K matrix which makes it harder to synchronize since an explicit dependence of time-delay size is now imposed, i.e. it turns a delay-dependent condition test.
Example 3. Consider the same Chua’s Circuit as in
Example 1, but with the parameters a and b varying in the following intervals: 8 ≤ a ≤ 10 13.28 ≤ b ≤ 15.28 Considering that the uncertainty belongs to a polytope type uncertain domain P as in (26), then the Table 2. Rates of exponential stability, δ, for the time-delays, τ (t), with µ = 0. τ δ
0.050 0.813
0.100 0.792
0.140 0.223
Error 1 0.8 0.6
e(t) = x(t)−y(t)
0.4 0.2 0 −0.2 −0.4 −0.6 0
1
2
3
4
5
6
7
time
Fig. 9. Error signal, e(t) = x(t) − y(t), with τ = 0.05 and µ = 0 (solid) and the rate exponential decay e−2δt with δ = 0.813 (dotted).
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Error 1 0.8 0.6
e(t) = x(t)−y(t)
0.4 0.2 0 −0.2 −0.4 −0.6 0
1
2
3
4
5
6
7
time
Fig. 10. Error signal, e(t) = x(t) − y(t), with τ = 0.10 and µ = 0 (solid) and the rate exponential decay e−2δt with δ = 0.792 (dotted).
Error 1 0.8 0.6
e(t) = x(t)−y(t)
0.4 0.2 0 −0.2 −0.4 −0.6 0
5
10 time
15
20
Fig. 11. Error signal, e(t) = x(t) − y(t), with τ = 0.14 and µ = 0 (solid) and the rate exponential decay e−2δt with δ = 0223 (dotted).
Further Results on Master-Slave Synchronization of General Lur’e Systems Table 3.
Largest allowable time-delay, τ , with µ = 0.
K K0 K1 K2 K3 K4 K5
= diag(0, 0, 0) = diag(−0.52, −1.642, −2.623) = diag(−1.52, −2.642, −3.623) = diag(−2.52, −3.642, −4.623) = diag(−3.52, −4.642, −5.623) = diag(−4.52, −5.642, −6.623)
Largest Allowable Time-Delay, τ , by Theorem 3 τ τ τ τ τ τ
= 0.125 = 0.149 = 0.162 = 0.177 = 0.194 = 0.215
following matrices denotes the vertices of the uncertain scheme master-slave synchronization (1): −2.2857 8.00 0 −1.00 1 , A1 = 1.0000 0 −13.28 0 −2.2857 8.00 0 −1.00 1 , A2 = 1.0000 0 −15.28 0 −2.8571 10.00 0 −1.00 1 , A3 = 1.0000 0 −13.28 0 −2.8571 10.00 0 −1.00 1 , A4 = 1.0000 0 −15.28 0 3.4286 4.2857 B1 = 0 , B2 = 0 . 0 0 Using Theorem 3 the largest time-delays related to each K are shown in Table 3 with M = [6.0229 1.3367 −2.1264]T .
4. Concluding Remarks A new delay-dependent approach to analyze the asymptotic, exponential and robust stability of the master-slave synchronization for Lur’e systems using time-varying delay feedback control has been presented in this paper. This approach follows from the Lyapunov–Krasovskii stability theory, the linear matrix inequality (LMI) machinery and free weighting matrices based on the Leibniz–Newton formula. The paper has also pointed out some facts that seem to be relevant to clarify the significance of delay in the analysis of synchronized lasers used in the design of chaos based optical communication
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systems, by shedding some light on the associated conflicting requirements. Finally, numerical examples illustrate the efficiency of the proposed method and show that the results obtained in this paper are less conservative than others previously published.
Acknowledgments This work has been supported in part by the Brazilian agencies CNPq, CAPES and FAPEMIG.
References Alvarez, G. & Li, S. [2006] “Some basic cryptographic requirements for chaos-based cryptosystems,” Int. J. Bifurcation and Chaos 16, 2129–2151. Andrievskii, B. R. & Fradkov, A. L. [2004] “Control of chaos: Methods and applications. II. Applications,” Autom. Rem. Contr. 65, 505–533. Blakely, J. N., Illing, L. & Gauthier, D. J. [2004] “Controlling fast chaos in delay dynamical systems,” Phys. Rev. Lett. 92, 193901-1–193901-4. Chen, H. F. & Liu, J. M. [2000] “Open-loop chaotic synchronization of injection-locked semiconductor lasers with gigahertz range modulation,” IEEE J. Quant. Electron. 36, 27–34. Fujisaka, H. & Yamada, T. [1983] “Stability theory of synchronized motion in coupled oscillator systems,” Progr. Theoret. Phys. 69, 32–47. Huang, H., Li, H.-X. & Zhong, J. [2006] “Master-slave synchronization of general Lur’e systems with timevarying delay and parameter uncertainty,” Int. J. Bifurcation and Chaos 16, 281–294. Kim, C.-M., Kye, W.-H., Rim, S. & Lee, R.-Y. [2004] “Communication key using delay times in timedelayed chaos synchronization,” Phys. Lett. A 333, 235–240. Liao, X. & Chen, G. [2003] “Chaos synchronization of general Lur’e systems via time-delay feedback control,” Int. J. Bifurcation and Chaos 13, 207–213. Liu, J. M., Chen, H. F. & Tang, S. [2001] “Opticalcommunication systems based chaos in semiconductor lasers,” IEEE Trans. Circuits Syst.-I 48, 1475–1483. Niculescu, S. I. [1996] Sur la Stabilit´e et la Stabilisation ´ des Syst`emes Lin´eaires a ` Etats Retard´es. PhD thesis, Institut National Polytechnique de Grenoble. Palhares, R. M., Campos, C. D., Ekel, P. Y., Leles, M. C. R. & D’Angelo, M. F. S. V. [2005a] “Delaydependent robust H∞ control of uncertain linear systems with lumped delays,” IEE Proc. Contr. Th. Appl. 152, 27–33. Palhares, R. M., Campos, C. D., Ekel, P. Y., Leles, M. C. R. & D’Angelo, M. F. S. V. [2005b]
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“Delay-dependent robust H∞ control of uncertain linear systems with time-varying delays,” Comput. Math. Appl. 50, 13–32. Pecora, L. M. & Carroll, T. L. [1990] “Synchronization in chaotic systems,” Phys. Rev. Lett. 64, 821–824. Wu, M., He, Y., She, J.-H. & Liu, G.-P. [2004] “Delaydependent criteria for robust stability of time-varying delay systems,” Automatica 40, 1435–1439.
Xu, S., Lam, J., Ho, D. W. C. & Zou, Y. [2005] “Delaydependent exponential stability for a class of neural networks with time delays,” J. Comput. Appl. Math. 183, 16–28. Yal¸cin, M. E., Suykens, J. A. K. & Vandewalle, J. [2001] “Master-slave synchronization of Lur’e systems with time-delay,” Int. J. Bifurcation and Chaos 11, 1707–1722.
FURTHER RESULTS ON MASTER-SLAVE SYNCHRONIZATION OF GENERAL LUR’E SYSTEMS WITH TIME-VARYING DELAY FERNANDO O. SOUZA, REINALDO M. PALHARES, ˆ EDUARDO M. A. M. MENDES∗ and LEONARDO A. B. TORRES Department of Electronics Engineering, Federal University of Minas Gerais, Av. Antˆ onio Carlos 6627, Belo Horizonte 31270-901, Brazil ∗[email protected] Received July 13, 2006; Revised January 2, 2007 In this paper, a new approach to analyze the asymptotic, exponential and robust stability of the master-slave synchronization for Lur’e systems using time-varying delay feedback control is proposed. The discussion is motivated by the problem of transmitting information in optical communication systems using chaotic lasers. The approach is based on the Lyapunov–Krasovskii stability theory for functional differential equations and the linear matrix inequality (LMI) technique with the use of a recent Leibniz–Newton model based transformation, without including any additional dynamics. Using the problem of synchronizing coupled Chua’s circuits, three examples are given to illustrate the effectiveness of the proposed methodology. Keywords: Synchronization; Lur’e system; time-delay; robust stability; asymptotic and exponential stability.
1. Introduction
to point by using chaotic carrier signals (e.g. see Sec. 3.4 in [Andrievskii & Fradkov, 2004] and references therein), specific methods are preferred in particular cases, due to practical limitations. One such situation occurs in the secure transmission of information relying on synchronized chaotic lasers, in the context of optical communication systems [Liu et al., 2001; Chen & Liu, 2000]. In this case, the speed constraints related to the processing of optical signals demand simple and robust ways to achieve synchronization [Blakely et al., 2004]. It is important to note two characteristics of chaotic laser systems, namely: (i) Local
Despite the theoretical relevance of synchronizing Lur’e systems with time-varying delay, one of the main focus of the present paper is the analysis of synchronized chaotic oscillators. Since seminal papers were published on theoretical and experimental evidence of chaotic systems synchronization [Fujisaka & Yamada, 1983; Pecora & Carroll, 1990], much effort has been devoted to the analysis of such systems. As pointed out in [Yal¸cin et al., 2001], such oscillators are the key components in chaos based communication systems. Although there are many different approaches to send information from point ∗
Author for correspondence 187
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time-delayed optoelectronic feedback sometimes can be used to produce chaotic oscillations in each laser, at the transmitter, and at the receiver; i.e. the lasers would not exhibit natural chaotic behavior, if this was not done [Liu et al., 2001] and (ii) the coupling structure commonly used to send information is based on the master-slave approach, which implies that one chaotic oscillator — the master system — influences the other chaotic oscillator — the slave system; and not the other way around. The first characteristic indicates that delay lines are already used on some chaotic lasers, and therefore they could be modified to ensure synchronization. The primary role of these internal feedback delay lines is to guarantee chaotic behavior, independently of the coupling with another oscillator. The second characteristic indicates that, in the analysis of the synchronization condition, the propagation delay is immaterial, because the only and harmless effect on the receiver, due to the distance between transmitter and receiver, would be the receiver synchronization with a delayed copy of the transmitter, as depicted in Fig. 1. It is noted that this scenario could be changed if signals from the slave system were sent back to the master system, in a hypothetical bidirectional coupling, but this is not the case. The above facts seem to be relevant to establish, in the authors’ point of view, what are the primary motivations to use artificial delay lines or to consider the impact of time delay signal propagation in chaos based optical communication systems, namely: (i) the “chaotification” of the transmitted signals aimed at alleged system security improvement; (ii) the determination of possible deleterious effects caused by unavoidable internal signal propagation delay, at the controlled receiver; and (iii) the practical feasibility of a control strategy that can be used in the case of ultra-fast chaotic systems synchronization.
Master(t)
τ
Master´(t)
Slave(t)
Fig. 1. Master-slave approach. The existence of propagation delay τ is immaterial, once the slave system synchronizes with a master system delayed copy: master (t) = master(t − τ ). 1
Focusing on the important issue of the so-called secure character of the information transmission system [Alvarez & Li, 2006], some considerations are in order. If one wishes to keep the information transmission system secure, it is desirable to design synchronized systems that are highly sensitive to parameter uncertainty. On the other hand, if the transmission system should be robust, in the sense that the master and slave systems should achieve synchronization despite parameters mismatch, the security would be inevitably lowered down. The main goal in the present work is to theoretically analyze the synchronization stability of systems with time-varying delay, in order to provide less conservative tools that can be used to inspect more carefully the conflicting requirements in the design of chaos based optical communication systems, as stated above.1 The tools here presented follow the approach of constructing linear matrix inequalities (LMI) from the analysis of Lyapunov–Krasovskii functionals, providing sufficient conditions to determine the feasibility of the synchronization between master and slave systems represented as uncertain Lur’e systems with time-varying delay [Yal¸cin et al., 2001; Liao & Chen, 2003; Huang et al., 2006]. The results are also extended to the case of exponential stability analysis. In this paper the notation “∗” is used instead of writing terms in the symmetric matrix. The superscript “T ” represents the transpose. M 0 (≺ 0) means that the matrix M is positive (negative) definite. diag(·) denotes a diagonal matrix. Let R denote the set of real numbers, Rn the n-dimensional Euclidean space and Rn×m the set of all n × m real matrices, 0n×n are n × n matrices of zeros. The paper is organized as follows: in Sec. 2 the main approach is outlined by stating the assumptions and presenting the theoretical developments that lead to less conservative linear matrix inequalities (LMI) for asymptotic stability analysis (Sec. 2.1), exponential stability analysis (Sec. 2.2) and robust stability analysis (Sec. 2.3) of the synchronization error dynamical system. Illustrative examples, through numerical simulations, are provided in Sec. 3. Finally, concluding remarks are presented in Sec. 4.
A very interesting alternative is proposed in [Kim et al., 2004], where the very time delays comprise the “key” to decode the information at the receiver.
Further Results on Master-Slave Synchronization of General Lur’e Systems
2. Further Results Consider the following master-slave synchronization scheme: x(t) ˙ = Ax(t) + Bσ(Cx(t)) M: p(t) = Hx(t) y(t) ˙ = Ay(t) + Bσ(Cy(t)) + u(t) S: (1) q(t) = Hy(t) C : u(t) = −K(x(t) − y(t)) + M (p(t − τ (t)) − q(t − τ (t)))
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Throughout the paper, the following two hypotheses will be used to derive the conditions for the robust stability. (H1) The time-varying delay, τ (t), is continuous, differentiable and bounded, i.e. 0 ≤ τ (t) ≤ τ,
τ˙ (t) ≤ µ < 1
(3)
where τ and µ are constants. (H2) The nonlinearity η(Ce, y) belongs to the sector [0, κ]: ηi (cTi e, y) σi (cTi e + cTi y) − σi (cTi y) = ≤κ cTi e cTi e
where M is the master system, S is the slave system and C is the time-delay feedback control law with a time-varying delay. The state vectors are given by x, y ∈ Rn , the output of each system are denoted by p, q ∈ Rl , the matrices A ∈ Rn×n , B ∈ Rn×nh , C ∈ Rnh ×n , H ∈ Rl×n , and the function σ(·) satisfies a sector condition, with σi (·) i = 1, 2, . . . , nh belonging to the sector [0, κ], i.e. σi (ν)(σi (ν)− κν) ≤ 0 for i = 1, 2, . . . , nh . Figure 2 shows the scheme of synchronization between the master and the slave systems proposed in [Huang et al., 2006], where the authors applied a time-varying delay feedback to the slave system with control signal u(t) ∈ Rn , feedback matrices K ∈ Rn×n , M ∈ Rn×l and time-varying delay τ (t). Synchronization means agreement or correlation of different processes in time, therefore the design of the controller C aims to make x(t) − y(t) → 0 as t → ∞, where · is a norm defined in Rn . By defining the error signal e(t) = x(t) − y(t), the error system can be represented as follows ˜ ε : e(t) ˙ = Ae(t) + F e(t − τ (t))
then the error system (2) is exponentially stable, where δ is called the rate of exponential stability.
+ Bη(Ce(t), y(t)) (2) ˜ with A = A + K, F = −M H, and η(Ce(t), y(t)) = σ(Ce(t) + Cy(t)) − σ(Cy(t)).
The relationship between the state and the delayed state as well as the delay-dependence can be explicitly taken into account in the stability analysis
Fig. 2.
0≤
where cTi denotes the ith row vector of C. The following inequality holds, ηi (cTi e, y)(ηi (cTi e, y) − κcTi e) ≤ 0, ∀ e, y; i = 1, 2, . . . , nh .
(4)
Notice that the initial condition to error system is e(s) = 0, s ∈ [−τ 0]. The following definition will be also used in the remaining of the paper. Definition 1. Let δ > 0 and (δ) > 0 be given
scalars, such that:
−δt
e(t) ≤ (δ)e
sup e(θ)
−τ ≤θ≤0
+
˙ , sup e(θ)
−τ ≤θ≤0
Master-slave synchronization scheme.
∀t > 0
(5)
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by considering the Newton–Leibniz identity and some slack matrix variables. This strategy is used for reducing conservativeness in the resulting LMIs without introducing any extra dynamics as suggested in [Wu et al., 2004]. Lemma 1. For any matrices Y , W , and X of appropriate dimensions, the null term
2[eT (t)Y + eT (t − τ (t))W + η T (Ce(t), y(t))X] × e(t) −
t
t−τ (t)
e(s)ds ˙ − e(t − τ (t)) = 0
(6)
is equivalent to
Y +YT t ∗ 1 ξ T (t, s) τ (t) t−τ (t) ∗ ∗
−Y + W T −W − W T ∗ ∗
XT −X T 0 ∗
−τ Y −τ W ξ(t, s)ds = 0 −τ X
(7)
0
with ξ(t, s) = [eT (t) eT (t − τ (t)) η T (Ce(t), y(t)) e˙T (s)]T . Using the fact that the Newton–Leibniz identity is given by t e(s)ds ˙ e(t − τ (t)) = e(t) −
Proof.
t−τ (t)
the equality in Eqs. (6) and (7) follows directly.
2.1. Asymptotic stability analysis Consider now the asymptotic stability of the error system with time-varying delay τ (t). The main idea is to add the null term in Lemma 1
PA˜ + A˜TP + Q + Y T + Y ∗ ∗ Ξ = ∗ ∗
to the derivative of the Lyapunov functional to obtain a delay-dependent condition. This simple strategy results in a less conservative approach to analyze the problem of stability. The first main result is stated in the following theorem. Theorem 1. Let τ > 0 and µ < 1 be the upper
bounds for the time-delays and their rate of variation respectively. Suppose that hypothesis (H1) and (H2) hold. If there exist P = P T > 0, Q = QT > 0, Z = Z T > 0, Λ = diag(λ1 , λ2 , . . . , λnh ) > 0, and any matrices X, Y and W of appropriate dimensions such that the following LMI is satisfied:
PF + W T − Y −(1 − µ)Q − W T − W ∗ ∗ ∗
then the master-slave synchronization scheme (1) synchronizes with error system (2) having an unique and globally asymptotic stable equilibrium point, e = 0, where A˜ = A + K and F = −M H. Select the following Lyapunov–Krasovskii functional candidate as in [Palhares et al., 2005a, 2005b]:
Proof.
V (et ) = V1 (et ) + V2 (et ) + V3 (et ) where V1 (et ) eT (t)P e(t)
(9)
P B + X T + κC TΛ −τ Y −X T −τ W −2Λ −τ X ∗ −τ Z ∗ ∗ V2 (et ) V3 (et )
t t−τ (t)
0 −τ
t
t+θ
τ A˜T Z τFT Z τ BT Z ≺ 0 0 −τ Z
(8)
eT (s)Qe(s)ds e˙ T (s)Z e(s)dsdθ ˙
with P = P T 0, Q = QT 0, Z = Z T 0. Taking the time derivative of V (et ) in (9) along with the error state trajectories, e(t), in (2) yields V˙ (et ) = V˙ 1 (et ) + V˙ 2 (et ) + V˙ 3 (et )
(10)
Further Results on Master-Slave Synchronization of General Lur’e Systems
191
with ˙ + e˙ T (t)P e(t) V˙ 1 (et ) = eT (t)P e(t) ˜ ˜ = eT (t)P [Ae(t) + F e(t − τ (t)) + Bη(Ce(t), y(t))] + [Ae(t) + F e(t − τ (t)) + Bη(Ce(t), y(t))]T P e(t) ˜ + 2eT (t)P Bη(Ce(t), y(t)) + 2eT (t)P F e(t − τ (t)) = 2eT (t)PAe(t) t 1 ˜ 2eT (t)PAe(t) + 2eT (t)P Bη(Ce(t), y(t)) + 2eT (t)P F e(t − τ (t))ds (11) ≤ τ (t) t−τ (t) T ˙ (t − τ (t))Qe(t − τ (t)) + eT (t)Qe(t) V˙ 2 (et ) = −(1 − d(t))e
≤ −(1 − µ)eT (t − τ (t))Qe(t − τ (t)) + eT (t)Qe(t) t
1 −(1 − µ)eT (t − τ (t))Qe(t − τ (t)) + eT (t)Qe(t) ds = τ (t) t−τ (t) t T ˙ ˙ − e˙T (s)Z e(s)ds ˙ V3 (et ) = τ e˙ (t)Z e(t) ˙ − ≤ τ e˙ T (t)Z e(t) 1 ≤ τ (t) ≤
1 τ (t)
t t−τ (t)
t t−τ (t)
(12)
t−τ t
t−τ (t)
e˙ T (s)Z e(s)ds ˙
[τ e˙T (t)Z e(t) ˙ − e˙ T (s)τ Z e(s)]ds ˙ ˜ [eT (t)τ A˜TZ Ae(t) + eT (t)τ A˜TZF e(t − τ (t)) + eT (t)τ A˜TZBη(Ce(t), y(t))
T
˜ + eT (t − τ (t))τ F TZF e(t − τ (t)) + e (t − τ (t))τ F TZAe(t) ˜ + η T (Ce(t), y(t)) + eT (t − τ (t))τ F TZBη(Ce(t), y(t)) + η T (Ce(t), y(t))τ B TZAe(t) ˙ × τ B TZF e(t − τ (t)) + η T (Ce(t), y(t))τ B TZBη(Ce(t), y(t)) − e˙T (s)τ Z e(s)]ds
where, again, A˜ = A + K and F = −M H. Using hypothesis (H2) that deals with the nonlinearities and adding in (10) the null term given in Lemma 1, it follows that: V˙ (et ) ≤ V˙ 1 (et ) + V˙ 2 (et ) + V˙ 3 (et ) −2
nh i=1
+ 2[eT(t)Y + eT(t − τ (t))W
× e(t) −
t
t−τ (t)
e(s)ds ˙ − e(t − τ (t)) (14)
that is equivalent to t 1 ˙ ξ T (t, s)Ωξ(t, s)ds V (et ) ≤ τ (t) t−τ (t)
Ω12 Ω22 ∗ ∗
Ω13 Ω23 Ω33 ∗
Ω14 Ω24 Ω34 Ω44
(16)
with
λi ni (CiTe, y)(ηi (cTie, y) − κcTie)
− η T(Ce(t), y(t))X]
Ω11 ∗ Ω= ∗ ∗
(13)
(15)
ξ(t, s) = [eT(t) eT(t − τ (t)) η T(Ce(t), y(t)) e˙T (s)]T Ω11 Ω12 Ω13 Ω14 Ω22 Ω23 Ω24 Ω33 Ω34 Ω44
PA˜ + A˜TP + Q + τ A˜TZA˜ + Y + Y T PF + τ A˜TZF − Y + W T P B + X T + τ A˜TZB + κC TΛ −τ Y −(1 − µ)Q + τ F TZF − W − W T −X T + τ F TZB −τ W −2Λ + τ B TZB −τ X −τ Z
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Applying Schur’s complement in (16), Ξ defined in (8) is obtained. If Ξ ≺ 0, then V˙ (et ) < 0 for any ξ(t, s) = 0. As a result the master-slave synchronization scheme (1) synchronizes with the error system in Eq. (2) yielding a unique and globally asymptotic stable equilibrium point, e = 0, in the context of the Lyapunov–Krasovskii theory.
controller C design, and the time-delay τ (t), can be analyzed as tuning parameters to adjust the speed synchronization. Then the following Theorem proposes a sufficient condition for the exponential stability analysis of error synchronization system (2) in the master-slave synchronization given by (1).
2.2. Exponential stability analysis
upper bounds for the time-delay, the rate of variation and the exponential convergence rate of error system (2), respectively. Suppose that hypotheses (H1) and (H2) hold. If there exist P = P T > 0, Q = QT > 0, Z = Z T > 0, Λ = diag(λ1 , λ2 , . . . , λnh ) > 0, and any matrices X, Y and W of appropriate dimensions such that the following LMI is satisfied:
Theorem 2. Let τ > 0, µ < 1, and δ > 0 be the
The exponential stability analysis is important since it is often desired that the master-slave synchronization scheme converges in an exponential rate to ensure fast response for the synchronization. In this context, the gain matrices K and M , in the
Ξexp
=
PA˜ + A˜TP + 2δP + Q + Y T + Y
PF + W T − Y
PB + X T + κC TΛ
−τ Y
∗
−e2δτ (1 − µ)Q − W T − W
−X T
−τ W
∗
∗
−2Λ
−τ X
∗
∗
∗
−τ Z
∗
∗
∗
∗
τ A˜TZ
τ F TZ τ B TZ ≺ 0 0 −τ Z (17)
then the master-slave synchronization scheme in (2) synchronizes and the error system in (2) has a unique and globally exponential stable equilibrium point at e = 0, with the exponential convergence rate, δ, A˜ = A + K and F = −M H.
V3 exp (et )
Select the Lyapunov–Krasovskii functional candidate as:
PT
Proof.
Vexp (et ) = V1 exp (et ) + V2 exp (et ) + V3 exp (et ) (18) where V1 exp (et ) e2δt eT (t)P e(t)
V2 exp (et )
t
t−τ (t) 0
−τ
t t+θ
e2δs eT (s)Qe(s)ds e2δs e˙ T (s)Z e(s)dsdθ ˙
0, Q = QT 0, Z = Z T 0. with P = Taking the time derivative of V (et ) in (18) along with the error state trajectories, e(t), in (2) yields V˙ exp (et ) = V˙ 1 exp (et ) + V˙ 2 exp (et ) + V˙ 3 exp (et ) (19) with
˙ + e2δt e˙ T (t)P e(t) V˙ 1 exp (et ) = 2δe2δt eT (t)P e(t) + e2δt eT (t)P e(t) ˙ + e˙T (t)P e(t)} = e2δt {2δeT (t)P e(t) + eT (t)P e(t) e2δt t ˜ 2eT (t)[δP + PA]e(t) + 2eT (t)PBη(Ce(t), y(t)) + 2eT (t)PF e(t − τ (t))ds ≤ τ (t) t−τ (t) T ˙ (t − τ (t))Qe(t − τ (t)) + e2δt eT (t)Qe(t) V˙ 2 exp (et ) = −e2δ(t−τ (t)) (1 − d(t))e ≤ −e2δ(t−τ ) (1 − µ)eT (t − τ (t))Qe(t − τ (t)) + e2δt eT (t)Qe(t) = e2δt {−e2δτ (1 − µ)eT (t − τ (t))Qe(t − τ (t)) + eT (t)Qe(t)} e2δt t −e2δτ (1 − µ)eT (t − τ (t))Qe(t − τ (t)) + eT (t)Qe(t)ds = τ (t) t−τ (t)
(20)
(21)
Further Results on Master-Slave Synchronization of General Lur’e Systems
˙ − V˙ 3 exp (et ) = τ e2δt e˙T (t)Z e(t) ˙ − ≤ τ e2δt e˙T (t)Z e(t) ≤ =
e2δt τ (t) e2δt τ (t)
t
t−τ (t)
t
t−τ (t)
t t−τ
193
e2δt e˙T (s)Z e(s)ds ˙
t t−τ (t)
e2δt e˙ T (s)Z e(s)ds ˙
[τ e˙T (t)Z e(t) ˙ − e˙ T (s)τ Z e(s)]ds ˙ ˜ [eT (t)τ A˜TZAe(t) + eT (t)τ A˜TZF e(t − τ (t)) + eT (t)τ A˜TZBη(Ce(t), y(t))
˜ + eT (t − τ (t))τ F TZF e(t − τ (t)) + eT (t − τ (t))τ F TZAe(t) ˜ + eT (t − τ (t))τ F TZBη(Ce(t), y(t)) + η T (Ce(t), y(t))τ B TZAe(t) + η T (Ce(t), y(t)) ˙ × τ B TZF e(t − τ (t)) + η T (Ce(t), y(t))τ B TZBη(Ce(t), y(t)) − e˙T (s)τ Z e(s)]ds with A˜ = A + K and F = −M H. Using hypothesis (H2) (nonlinearities) and adding in (19) the null term in Lemma 1 results in V˙ exp (et ) ≤ V˙ 1 exp (et ) + V˙ 2 exp (et ) + V˙ 3 exp (et ) −2
nh i=1 T
λi ni (CiTe, y)(ηi (cTie, y) − κcTie)
+ 2[e (t)Y + eT (t − τ (t))W − η T (Ce(t), y(t))X] t e(s)ds ˙ − e(t − τ (t)) × e(t) − t−τ (t)
(23) that is equivalent to: e2δt t ˙ ξ T (t, s)Ωexp ξ(t, s)ds (24) Vexp (et ) ≤ τ (t) t−τ (t) Ωexp(11) Ωexp(12) Ωexp(13) Ωexp(14) ∗ Ωexp(22) Ωexp(23) Ωexp(24) Ωexp = ∗ ∗ Ωexp(33) Ωexp(34) ∗ ∗ ∗ Ωexp(44)
Ωexp(14) −τ Y Ωexp(22) −e2δτ (1 − µ)Q + τ F TZF − W − W T Ωexp(23) −X T + τ F TZB Ωexp(24) −τ W Ωexp(33) −2Λ + τ B TZB Ωexp(34) −τ X Ωexp(44) −τ Z Applying Schur complements in (25), Ξexp defined in (17) is obtained. If Ξexp ≺ 0, then V˙ exp (et ) < 0 for any ξ(t, s) = 0. Therefore, it follows that Vexp (e(t)) ≤ Vexp (e(0)),
Vexp (e(t)) ≤ e2δt λmax (P )e(t)2 t +λmax (Q) e2δs e(s)2 ds t−τ (t) 0 t
+ λmax (Z)
with
and ˜T
˜T
Ωexp(11) 2δP + PA˜ + A P + Q + τ A ZA˜ Ωexp(12)
+Y +Y T PF + τ A˜TZF − Y + W T
Ωexp(13) PB + X T + τ A˜TZB + κC TΛ
for t ≥ 0
From (18), it can be observed that:
(25) ξ(t, s) = [eT (t) eT (t−τ (t)) η T (Ce(t), y(t)) e˙ T (s)]T
(22)
−τ
t+θ
2 e2δs e(s) ˙ dsdθ
Furthermore, notice that [Xu et al., 2005]:
0
−τ
t
t+θ
2 e2δs e(s) ˙ dsdθ t
s−t
= ≤τ
t−τ −τ t 2δs t−τ
e
2 e2δs e(s) ˙ dθds 2 e(s) ˙ ds
F. O. Souza et al.
194
Therefore, it follows that
+λmax (Q)φ2
Vexp (e(t)) ≤ e2δt λmax (P )e(t)2 t + λmax (Q) e2δs e(s)2 ds
˙ 2 + τ λmax (Z)φ
t−τ (t) t
+ τ λmax (Z)
t−τ
≤ λmax (P )φ2
2
and for t = 0
−τ (0) 0
+ τ λmax (Z) ≤ λmax (P )φ2
−τ
2 e2δs e(s) ˙ ds
0
−τ (0) 0 −τ
e2δs ds
e2δs ds
1 − e−2δτ + λmax (Q)φ 2δ −2δτ ˙ 2 1−e + τ λmax (Z)φ 2δ
2 e2δs e(s) ˙ ds
Vexp (e(0)) ≤ λmax (P )e(0)2 0 + λmax (Q) e2δs e(s)2 ds
with φ
sup e(θ),
−τ ≤θ≤0
˙ φ
sup e(θ) ˙
−τ ≤θ≤0
On the other hand, it is clear that Vexp (e(t)) ≥ e2δt λmin (P )e(t)2 Since Vexp (e(t)) ≤ Vexp (e(0)), e(t) is bounded above by
1 − e−2δτ 1 − e−2δτ λmax (Z) λmax (P ) + λmax (Q) 2δ 2δ −δt ˙ −δt φe φe + e(t) ≤ λmin (P ) λmin (P ) ˙ ≤ ρ(δ)e−δt {φ + φ} with 1 − e−2δτ λ (P ) + λmax (Q) max 2δ , ρ(δ) max λ (P ) min
Then, from Definition 1, the master-slave synchronization scheme in (2) synchronizes with error system (2) having a unique and globally exponential stable equilibrium point at e = 0 with the exponential convergence rate, δ.
2.3. Robust stability analysis Notice that Theorems 1 and 2 can be extended to the case of robust stability analysis when the system matrices are not exactly known. Suppose that the matrices in the scheme of master-slave synchronization (1) are unknown but belonging to a polytope type uncertain domain P defined as the set of all matrices taken from the
−2δτ 1−e λmax (Z) 2δ λmin (P )
convex combination of the vertices: v γi (Ai , Bi ); P (A, B) : (A, B) = i=1
γi ≥ 0,
v
γi = 1 ,
(26)
i=1
where Ai and Bi , i = 1, . . . , v, are the polytope vertices. The robust stability analysis problem is to verify if system (2) is stable for all (A, B) ∈ P. In this case, the easiest sufficient condition for robust stability is based on the concept of quadratic stability, where a single Lyapunov–Krasovskii functional
Further Results on Master-Slave Synchronization of General Lur’e Systems
is used to prove the stability for the entire polytope. As a consequence, the stability of a polytope of matrices can be checked by a feasibility test of a set of LMIs involving only the vertices of the uncertainty domain. The following Theorem states a sufficient condition for the robust stability analysis of error system (2), which extends the result in Theorem 1 for uncertain systems in terms of LMIs, and in doing so it guarantees the master-slave synchronization.
PA˜i + A˜TiP + Q + Y T + Y ∗ Ξi = ∗ ∗ ∗
Theorem 3. Consider the uncertain error system given in (2) with (A, B) ∈ P as in (26). Let τ > 0, and µ < 1 be the upper bounds for the time-delay and its rate of variation, respectively. Suppose that hypotheses (H1) and (H2) hold. The uncertain error system (2) is quadratically stable, if there exist matrices P = P T > 0, Q = QT > 0, Z T > 0, Λ = diag(λ1 , λ2 , . . . , λnh ) > 0, and any matrices X, Y and W with appropriate dimensions satisfying the following LMIs:
PF + WT − Y −(1 − µ)Q − W T − W ∗ ∗ ∗
∀ i = 1, . . . , v, and A˜i = Ai + K. As was done for Theorem 1, Theorem 2 can be easily extended to the problem of robust exponential stability for synchronization by considering the LMI presented in (17), where A˜ (where A˜ = A + K) and B are taken with the subscript i, ∀ i = 1, . . . , v, where i denotes the vertices of the polytope.
3. Illustrative Examples In this section three examples using the same Chua’s Circuit as in [Yal¸cin et al., 2001] and [Huang et al., 2006] are given to demonstrate the efficiency of the proposed method. In particular, the first example is used for comparison purposes with the methods proposed in those papers. Example 1. Consider the Chua Circuit governed by the following equations x˙ = a(y − h(x)) y˙ = x − y − z (28) z˙ = −by
where h(x), the nonlinear characteristic of the system, is given by 1 h(x) = m1 x + (m0 − m1 )(|x + c| − |x − c|) 2 and the parameters are a = 9, b = 14.28, c = 1, m0 = −1/7, m1 = 2/7. With these set of parameters, the system settles on the well-known double scroll attractor.
195
P Bi + X T + κC TΛ −τ Y −X T −τ W −2Λ −τ X ∗ −τ Z ∗ ∗
τ A˜TiZ τ F TZ τ BiTZ ≺0 0 −τ Z (27)
System (28) can be represented in the Lur’e form as follows −a(m0 − m1 ) −am1 a 0 −1 1 , B = 0 A= 1 , 0 −b 0 0
C= 1 0 0 and σ(ν) = 1/2(|ν + c| − |ν − c|) belongs to sector [0 κ] with κ = 1. As considered in [Yal¸cin et al., 2001], the output matrix is selected as H = [1 0 0], which means that the master system is connected to the slave system by the first state variable. By choosing K = 0n×n and M = [6.0229
1.3367
− 2.1264]T
the system in [Yal¸cin et al., 2001] is now investigated. For comparison purposes, consider µ = 0, i.e. the time-delay is lumped and constant. Applying Theorem 1 proposed in this paper, the largest time-delay is τ = 0.141. Whereas applying Theorem 2 in [Yal¸cin et al., 2001] the largest time-delay is τ = 0.039 and applying Theorem 1 in [Huang et al., 2006] τ = 0.083. It is clear that the proposed method is less conservative than the methods in [Yal¸cin et al., 2001] and [Huang et al., 2006] for this first example. To further demonstrate this, τ = 0.141 is now considered. Figures 3 and 4 show the error signal and the double scroll attractors generated for
196
F. O. Souza et al. Error 0.8 0.6
e(t) = x(t)−y(t)
0.4 0.2 0 −0.2 −0.4 −0.6 −0.8
0
10
20
30
40
50
time
Fig. 3.
Error signal, e(t) = x(t) − y(t), with τ = 0.141 and µ = 0.
the master and slave systems, respectively. From simulations, it can be noticed that the coupled Chua’s Circuit do not synchronize for τ = 0.21, as depicted in Figs. 5 and 6. The initial conditions of the master and slave systems in this case were x(s) = [0.4 −0.53 0.17]T and y(s) = [0.28 −0.34 0.81]T ∀ s ∈ [−τ 0]. On the other hand, when a set of non-null matrix gains, K, with M = [6.0229 1.3367 − 2.1264]T in (1) is considered, the largest timedelays obtained by applying Theorem 1 proposed in this paper and Theorem 1 in [Huang et al., 2006] are
shown in Table 1. Note that the proposed method is less conservative. The reader should be reminded that the approach proposed in [Yal¸cin et al., 2001] cannot be applied in cases where the gain matrix K is non-null. Unlike the main purpose of the approach in [Huang et al., 2006], which is also developed to obtain a synchronization criteria dependent on the size of the time-delay, the matrices K and M chosen in [Huang et al., 2006] for this same example make the error system in (2)
Remark.
Slave
Master
0.4
0.4 0.2
0.2
x2
y2
0 0
−0.2 −0.2
−0.4
−0.4 4
−0.6 4 2
4 2
0
0
−2 x
3
−4
−4
2
4 0
−2
−2 x1
2
0
y3
−4
−4
−2 y1
Fig. 4. Three-dimensional view on the double scroll attractors generated for, master and slave systems, with τ = 0.141 and µ = 0.
Further Results on Master-Slave Synchronization of General Lur’e Systems
197
Error 4 3
e(t) = x(t)−y(t)
2 1 0 −1 −2 −3 −4
0
20
40
60
80
100
time
Fig. 5.
Error signal, e(t) = x(t) − y(t), with τ = 0.21 and µ = 0.
Slave
Master
0.4
1.5
0.2
1 0.5
x
2
y2
0 −0.2
0 −0.5
−0.4
−1
−0.6 4
−1.5 10 2
5
4 2
0
0
−2 x3
−4
−4
5 0
0
−5
−2
−10
y3
x1
−5
y1
Fig. 6. Three-dimensional view on the double scroll attractors generated for, master and slave systems, with τ = 0.21 and µ = 0.
Table 1. K K1 K2 K3 K4 K5
= diag(−0.52, −1.642, −2.623) = diag(−1.52, −2.642, −3.623) = diag(−2.52, −3.642, −4.623) = diag(−3.52, −4.642, −5.623) = diag(−4.52, −5.642, −6.623)
The largest time-delay, τ , with µ = 0. Theorem 1 in [Huang et al., 2006] τ τ τ τ τ
= 0.112 = 0.131 = 0.154 = 0.180 = 0.213
Theorem 1, Proposed τ τ τ τ τ
= 0.159 = 0.173 = 0.189 = 0.208 = 0.233
Largest Allowable Time-Delay, by Simulation τ τ τ τ τ
= 0.231 = 0.261 = 0.297 = 0.350 = 0.447
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F. O. Souza et al.
become delay-independent, i.e. the synchronization occurs regardless of the size of time-delay. In [Huang et al., 2006] the control matrices are chosen as K = diag(−8.52, −9.642, −10.623) and M = [4.0229, 1.3367, −2.1267]T which, as illustrated in Fig. 7, make the error dynamics stable even for a large time-delay as, e.g. τ = 100. If one chooses K = diag(−8.52, −9.642, −10.623) and
M = [6.0229, 1.3367, −2.1267]T the same characteristic can be again noticed as seen in Fig. 8. For this last case, the maximum time-delay found by applying Theorem 1 proposed in this paper or Theorem 1 in [Huang et al., 2006] is τ = 0.828. At first glance this could mean that this kind of delay-independence synchronization may be desired. However, from the authors’ point of
Error 0.2 0.1 0
e(t) = x(t)−y(t)
−0.1 −0.2 −0.3 −0.4 −0.5 −0.6 −0.7
0
Fig. 7.
500
1000 time
1500
2000
Error signal, e(t) = x(t) − y(t), with τ = 100 and µ = 0.
Error 0.2 0.1 0
e(t) = x(t)−y(t)
−0.1 −0.2 −0.3 −0.4 −0.5 −0.6 −0.7
0
Fig. 8.
500
1000
1500 time
2000
2500
Error signal, e(t) = x(t) − y(t), with τ = 100 and µ = 0.
3000
Further Results on Master-Slave Synchronization of General Lur’e Systems
199
Example 2. Consider the same Chua’s Circuit with the same parameters as in Example 1, K = 0n×n , and M = [6.0229 1.3367 −2.1264]T . Applying Theorem 2 and considering constant time-delay, that is, µ = 0, the exponential stability is obtained as can be seen in Table 2. The efficiency of the proposed method is demonstrated in Figs. 9 to 11, in the context that if one desires to design appropriate gains with specific exponential stability characteristics. In this scenario, Theorem 2 may be useful.
view, this kind of problem had been issued in the context of delay-dependence as supported by the Introduction of this paper concerning optical communication. To highlight this discussion, suppose that the error system (2) can be represented by a linear description: ˜ ε : e(t) ˙ = Ae(t) + F e(t − τ (t)) with A˜ = A + K and F = −M H. As presented in [Niculescu, 1996], a necessary condition for the above system to be delay-independent is that A˜ + F and A˜ − F be Hurwitz stable. Considering K = diag(−8.52, −9.642, −10.623) and M = [6.0229, 1.3367, −2.1267]T it is easy to verify that A˜ + F and A˜ − F are Hurwitz stable. Notice that this does not hold for the set of matrices K in Table 1 and the last matrix M . Notice also that in terms of sharpness of results it is important to note that u(t) in [Yal¸cin et al., 2001] is different; it contains no part with the K matrix which makes it harder to synchronize since an explicit dependence of time-delay size is now imposed, i.e. it turns a delay-dependent condition test.
Example 3. Consider the same Chua’s Circuit as in
Example 1, but with the parameters a and b varying in the following intervals: 8 ≤ a ≤ 10 13.28 ≤ b ≤ 15.28 Considering that the uncertainty belongs to a polytope type uncertain domain P as in (26), then the Table 2. Rates of exponential stability, δ, for the time-delays, τ (t), with µ = 0. τ δ
0.050 0.813
0.100 0.792
0.140 0.223
Error 1 0.8 0.6
e(t) = x(t)−y(t)
0.4 0.2 0 −0.2 −0.4 −0.6 0
1
2
3
4
5
6
7
time
Fig. 9. Error signal, e(t) = x(t) − y(t), with τ = 0.05 and µ = 0 (solid) and the rate exponential decay e−2δt with δ = 0.813 (dotted).
200
F. O. Souza et al.
Error 1 0.8 0.6
e(t) = x(t)−y(t)
0.4 0.2 0 −0.2 −0.4 −0.6 0
1
2
3
4
5
6
7
time
Fig. 10. Error signal, e(t) = x(t) − y(t), with τ = 0.10 and µ = 0 (solid) and the rate exponential decay e−2δt with δ = 0.792 (dotted).
Error 1 0.8 0.6
e(t) = x(t)−y(t)
0.4 0.2 0 −0.2 −0.4 −0.6 0
5
10 time
15
20
Fig. 11. Error signal, e(t) = x(t) − y(t), with τ = 0.14 and µ = 0 (solid) and the rate exponential decay e−2δt with δ = 0223 (dotted).
Further Results on Master-Slave Synchronization of General Lur’e Systems Table 3.
Largest allowable time-delay, τ , with µ = 0.
K K0 K1 K2 K3 K4 K5
= diag(0, 0, 0) = diag(−0.52, −1.642, −2.623) = diag(−1.52, −2.642, −3.623) = diag(−2.52, −3.642, −4.623) = diag(−3.52, −4.642, −5.623) = diag(−4.52, −5.642, −6.623)
Largest Allowable Time-Delay, τ , by Theorem 3 τ τ τ τ τ τ
= 0.125 = 0.149 = 0.162 = 0.177 = 0.194 = 0.215
following matrices denotes the vertices of the uncertain scheme master-slave synchronization (1): −2.2857 8.00 0 −1.00 1 , A1 = 1.0000 0 −13.28 0 −2.2857 8.00 0 −1.00 1 , A2 = 1.0000 0 −15.28 0 −2.8571 10.00 0 −1.00 1 , A3 = 1.0000 0 −13.28 0 −2.8571 10.00 0 −1.00 1 , A4 = 1.0000 0 −15.28 0 3.4286 4.2857 B1 = 0 , B2 = 0 . 0 0 Using Theorem 3 the largest time-delays related to each K are shown in Table 3 with M = [6.0229 1.3367 −2.1264]T .
4. Concluding Remarks A new delay-dependent approach to analyze the asymptotic, exponential and robust stability of the master-slave synchronization for Lur’e systems using time-varying delay feedback control has been presented in this paper. This approach follows from the Lyapunov–Krasovskii stability theory, the linear matrix inequality (LMI) machinery and free weighting matrices based on the Leibniz–Newton formula. The paper has also pointed out some facts that seem to be relevant to clarify the significance of delay in the analysis of synchronized lasers used in the design of chaos based optical communication
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systems, by shedding some light on the associated conflicting requirements. Finally, numerical examples illustrate the efficiency of the proposed method and show that the results obtained in this paper are less conservative than others previously published.
Acknowledgments This work has been supported in part by the Brazilian agencies CNPq, CAPES and FAPEMIG.
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