Synchronization criteria of Lur’e systems with time-delay feedback control

doi:10.1016/j.chaos.2004.06.025

Chaos, Solitons & Fractals

Volume 23, Issue 4, February 2005, Pages 1285-1298

https://doi.org/10.1016/j.chaos.2004.06.025 Get rights and content

Abstract

In this paper time-delay effects on the master–slave synchronization scheme are studied by a time-delay feedback control technique. Several new delay-independent and delay-dependent sufficient conditions are presented for master–slave synchronization of Lur’e systems based upon Lyapunov method and linear matrix inequalities (LMI’s) approaches. These new synchronization criteria are easily verifiable and offer some fairly adjustable real parameters, which are of important significance in the design and applications of such chaos synchronization systems, and the proposed results improve and generalize the earlier works.

Introduction

In recent times, the subject of chaotic synchronization has received considerable attentions, e.g., [2], [3], [14], [16], [19], [21], [22], [23], [24], [25] and references cited therein. One unified approach to chaos synchronization is to reformulate it as a (generalized) Lur’e system and then discuss the absolute stability of its error dynamics [5], [6], [12], [16], [17], [20]. A typical approach in the Lur’e system approach to chaos synchronization problems is to derive a sufficient condition in the form of a special Lyapunov functional or a linear matrix inequality. In [18], several sufficient conditions were obtained for synchronization of identical or nonidentical systems. Synchronization schemes were investigated in the literature [10], [11], [12], [17], [20]. In [3], an overview of synchronization methods has been recently presented. Synchronization has opened the way to investigate an engineering application of chaos such as chaotic communications. In [4], it was reported for propagation delay in master–slave synchronization schemes, the authors introduced the possibility of applying chaotic synchronization to optical communication, called this problem a phase sensitivity due to the distance between two remote chaotic systems and reported that the existence of a time-delay may destroy synchronization, however, theoretical studies of this problem are few, and delays often result in instabilities. Therefore, stability analysis of time-delay systems is an important topic in control theory [9], [13].

In this paper, we will continue the study of the master–slave type of chaos synchronization problems, for a general form of Lur’e systems by a time-delay feedback control technique, initiated in [12], [20]. Several delay-independent and delay-dependent criteria for global asymptotic stability of the error system are given which is expressed as matrix inequalities and are derived from several new extended Lyapunov–Krasovskii functionals and inequalities techniques. It shows that the obtained results also improve and generalize the earlier works. In addition, the obtained criteria offer several adjustable real parameters, and are easy to verify, which are of important significance in the design and application of such chaos synchronization systems.

The organization of this paper is as follows. In Section 2, problem formulation and the master–slave synchronization scheme are presented for Lur’e systems with time-delay. In Section 3, based on several new Lyapunov–Krasovskii functional and linear matrix inequalities (LMI), several new delay-independent and delay-dependent criteria are given to ascertain the global asymptotic stability of the error system such that the master–slave system synchronizes. In Section 4, we provide an example for Lur’e systems and demonstrate the effectiveness of the proposed results. In Section 5, concluding remarks are given.

Section snippets

Problem formulation

Consider a general master–slave synchronization scheme with static error feedback and time-delay τ $M : \{\begin{matrix} \dot{x} (t) = Ax (t) + B σ (Cx (t)), \\ p (t) = Hx (t), \end{matrix}$ $S : \{\begin{matrix} \dot{y} (t) = Ay (t) + B σ (Cy (t)) + u (t), \\ q (t) = Hy (t), \end{matrix}$ $C : u (t) = - K (x (t) - y (t)) + M (p (t - τ) - q (t - τ)),$ with master system $M$ , slave system $S$ and controller $C$ , see Fig. 1, where the time-delay τ > 0. The master and slave systems are Lur’e systems with state vectors $x, y \in R^{n}$ , output of subsystems $p, q \in R^{l}$ respectively, and matrices $H \in R^{l \times n}, K, A \in R^{n \times n}, B \in R^{n \times n_{h}}, C \in R^{n_{h} \times n}$ . σ_i(·) satisfies a sector

Synchronization criteria of Lur’e systems

To give our main results, we need the following lemmas which could be found in [1], [8], [15].

Lemma 1

[15]

Given any real matrices W₁, W₂, Q of appropriate dimensions and a number ν > 0 such that 0 < Q = Q^T. Then the following inequality holds: $W_{1}^{T} W_{2} + W_{2}^{T} W_{1} ⩽ ν W_{1}^{T} {QW}_{1} + ν^{- 1} W_{2}^{T} Q^{- 1} W_{2} .$

Lemma 2

Schur complement [1]

For a given matrix $S = [\begin{matrix} S_{11} & S_{12} \\ S_{12}^{T} & S_{22} \end{matrix}] > 0,$ where $S_{11} = S_{11}^{T}, S_{12} = S_{12}^{T}$ , is equivalent to $S_{22} > 0, S_{11} - S_{12} S_{22}^{- 1} S_{12}^{T} > 0 .$

Proof

Note that $[\begin{matrix} I & - S_{12} S_{22}^{- 1} \\ 0 & I \end{matrix}] [\begin{matrix} S_{11} & S_{12} \\ S_{12}^{T} & S_{22} \end{matrix}] {[\begin{matrix} I & - S_{12} S_{22}^{- 1} \\ 0 & I \end{matrix}]}^{T} = [\begin{matrix} S_{11} - S_{12} S_{22}^{- 1} S_{12}^{T} & 0 \\ 0 & S_{22} \end{matrix}] .$ This completes the proof. □

Lemma 3

Jensen inequality [8]

For any constant matrix $W \in R^{m \times m}$ , W = W^T > 0, scalar

An example

Example

Consider the following scroll circuit given by $\{\begin{matrix} \dot{x} = a (y - h (x)), \\ \dot{y} = x - y + z, \\ \dot{z} = - by, \end{matrix}$ with nonlinear characteristic $h (x) = m_{1} x + \frac{1}{2} (m_{0} - m_{1}) (| x + c | - | x - c |),$ and parameters a = 9, b = 14.28, c = 1, m₀ = − (1/7), m₁ = 2/7. The system can be represented in Lur’e form by $A = [\begin{matrix} - {am}_{1} & a & 0 \\ 1 & - 1 & 1 \\ 0 & - b & 0 \end{matrix}], B = [\begin{matrix} - a (m_{0} - m_{1}) \\ 0 \\ 0 \end{matrix}],$ C = H = [1, 0, 0], and $σ (ξ) = \frac{1}{2} (| ξ + c | - | ξ - c |)$ belonging to sector [0, k] with k = 1. This representation results in n_h = 1. Let M = [4.0229, 1.3367, −2.1264]^T, α = 0.8, K = −8I and F = −MH. By direct computation with Matlab tools, when τ = 0.01, there exist Λ

Conclusion

A master–slave synchronization scheme has been addressed for Lur’e systems with a known delay existing between master and slave systems. Several new delay-independent and delay-dependent synchronization criteria, which have been expressed as matrix inequalities, have been given for Lur’e systems by introducing several new Lyapunov–Krasovskii functionals and LMI approach. These synchronization criteria are easily verifiable and offer some fairly adjustable real parameters α, β and γ, and the

Acknowledgments

This work was jointly supported by the National Natural Science Foundation of China under Grant 60373067 and S0390064, the National Basic Research Program of China under Grant 2003CB716206, and the Natural Science Foundation of Jiangsu Province, China under Grants BK2003053.

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Synchronization criteria of Lur’e systems with time-delay feedback control

Abstract

Introduction

Section snippets

Problem formulation

Synchronization criteria of Lur’e systems

[15]

Schur complement [1]

Jensen inequality [8]

An example

Conclusion

Acknowledgments

Chaos, Solitons & Fractals

Chaos, Solitons & Fractals

Chaos, Solitons & Fractals

Linear matrix inequalities in system and control theory

A unified definition of synchronization for dynamical systems

Chaos

From chaos to order-perspectives, methodologies, and applications

Open-loop chaotic synchronization of injection-locked semiconductor lasers with Gigahertz range modulation

IEEE J. Quant. Electron.

Absolute stability theory and the synchronization problem

Int. J. Bifurcat. Chaos

Absolute stability theory and master–slave synchronization

Int. J. Bifurcat. Chaos

Special matrices and their applications in numerical mathematics

Stability of time-delay systems

Introduction to functional differential equations