Poly-quadratic stability and global chaos synchronization of discrete time hybrid systems

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Abstract

This paper considers global chaos synchronization as an observer design problem. Chaos generators are ascribed the form of piecewise linear maps, a particular class of discrete time hybrid systems. Sufficient conditions of global synchronization are formulated using a poly-quadratic stability concept. Such a concept is based on the use of parameter-dependent Lyapunov functions to check stability and leads to a reduction of the restricitions relative to conditions involving a unique Lyapunov function.

Introduction

Since the pioneering works of Carroll and Pecora [1], [2], the past decade has witnessed a tremendous surge of interest concerning chaos synchronization. Indeed, it was pointed out that the reproducibility of chaotic behaviors through synchronization might be interesting for secure communications and this has spurred numerous advancements in encoding, masking or encrypting of information.

Chaos synchronization has been particularly investigated from the control theory point of view. To date, various techniques have been suggested using a wide variety of schemes. For a unidirectional coupling between two systems, chaos synchronization can be formulated as a reconstruction problem and based on the notion of observer borrowed from the field of automatic control [3], [4], [5], [6]. It involves an instantaneous difference between an estimated signal derived from the state of a system called receiver and the transmitted signal derived from the state of a chaos generator called transmitter. The reconstruction is based on a suitable gain matrix computation of the observer.

One of the challenging problems consists in the necessity for the state vector of the receiver to converge towards that of the transmitter for arbitrarily initial conditions. As nonlinearity is fundamental in order to exhibit chaotic motions, synchronization can be considered as a global stability problem of nonlinear dynamical systems. Yet, the proof of global convergence, and so the achievement of global synchronization is difficult to attain in the general case.

Motivated by this reason, the problem in this paper has been restricted to transmitters and receivers described by piecewise linear maps, a particular class of discrete time hybrid systems. For this class, global synchronization can be achieved under conditions involving quadratic Lyapunov functions and derived from robust control theoretic results [7]. Sufficient conditions of global stability involving a unique Lyapunov function for all the local linear dynamics of the map can be stated but suffer from conservatism. The term “conservative” is used in a sense different from that of dynamical systems theory. Here, it is derived from the automatic control field and means “restrictive”. And yet, conservatism is an important point to be studied. Indeed, the more conservative it is, the more the conditions might cause the problem to be not feasible even thought there could exist a solution. As a matter of fact, it is more suited to choose a Lyapunov function related to the each local linear dynamics. Nevertheless, ensuring independently stability for each local linear dynamics does not necessarily lead to global stability. Here, to face this situation, we introduce a poly-quadratic stability concept. Such a concept is based on using parameter-dependent Lyapunov functions to check stability. The Lyapunov function depends on parameters related to each local linear dynamics.

The layout of this paper is as follows. Section 2 is devoted to the statement of chaos synchronization as an observer problem. In Section 3, necessary and sufficient conditions of poly-quadratic stability are established for general dynamical systems where the dynamical matrix evolves in a polytope defined by its vertices. In Section 4 two proposition are given. The first gives a sufficient condition for global synchronization involving a unique Lyapunov function. The second proposition is based on poly-quadratic stability and gives a new sufficient condition of global synchronization. It involves a parameter-dependent Lyapunov function and leads to less conservative result. The conditions are given in terms of linear matrix inequalities (LMIs). Finally, in Section 5, an illustrative example is presented showing the conservatism reduction obtained by the poly-quadratic stability-based conditions.

Section snippets

Synchronization scheme: the observer configuration

The synchronization scheme to be considered in this paper is described by the n-dimensional state space representation of the respective transmitter and receiver:Xk+1t=AiXkt+BiYkt=CiXkt;Xk+1r=AiXkr+Bi+Li(Ykt−Ykr)Ykr=CiXkr

  • Xkt and Xk+1t (respectively, Xkr and Xk+1r) are the n-dimensional state vectors of the transmitter (respectively, receiver) at the discrete times k and k+1. Superscript ‘t’ (respectively, ‘r’) stands for transmitter (respectively, receiver).

  • The output vector Ykt=CiXkt of

Poly-quadratic stability

This section is devoted to a general definition of poly-quadratic stability. Consider a dynamical discrete time systemxk+1=Ak)xkwhere x∈Rn is the state vector, ρk∈Ξ⊂RM is an unknown but bounded time-varying parameter. The structure of the dynamical matrix A is assumed to depend in a polytopic way on the parameter ρk:Ak)=l=1MρklÃl,ρkl≥0,l=1Mρkl=1where Ãl are given constant matrices known as vertices. When we are concerned with checking the stability of system (4), the following theorem

Conditions of global synchronization

In this section, an LMI-based approach is developed to establish the conditions of poly-quadratic stability and hence global convergence of Eq. (2) and global synchronization of Eq. (1).

Consider Eq. (2) governing the dynamics of the error of synchronization. Global convergence of Eq. (2) is equivalent to global convergence of the dual systemηk+1=Akk=l=1Nξkl(Al−LlCl)Tηkwhich is a particular form of Eq. (4) with

  • M=N representing the number of regions;

  • ρk replaced by ξk;

  • Ã=(Al−LlCl)T.

Illustrative example

In this section, a conservatism assessment of Proposition 1, Proposition 2 is performed by comparing the feasibility of the respective problems with distinct choices of output matrices.

Consider a map T of which state space representation of the transmitter and the receiver is of the form (1) with

  • Xkt=[ xktyktzkt]T and Xkr=[ xkrykrzkr]T;

  • Ai=00.890.5hi0.890−0.100.9 with h1=−a and h2=λ;

  • B1=[ 000]T and B2=[ 0−6(a+λ)0]T;

  • the two regions associated, respectively, with A1 and A2 are R1 and R2. R1 is the

Conclusion

A procedure to compute the matrix gain of an observer ensuring global chaos synchronization for piecewise linear maps as a particular class of discrete time hybrid systems was proposed. By introducing poly-quadratic stability and PDLF, two propositions expressed in terms of linear matrix inequalities were established. The first involves a unique quadratic Lyapunov function. Then, the conservatism has been reduced by using a parameter-dependent Lyapunov function. The parameterization is related

Acknowledgements

The authors would like to thank Professor Mosekilde for his valuable suggestions which improved this paper.

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