Design of coupling for synchronization in time-delayed systems

Abstract

We study the problem of controlled network synchronization of coupled semipassive systems in the case when the outputs (the coupling variables) and the inputs are subject to constant time-delay (as it is often the case in a networked context). Predictor-based dynamic output feedback controllers are proposed to interconnect the systems on a given network. Using Lyapunov-Krasovskii functional and the notion of semipassivity, we prove that under some mild assumptions, the solutions of the interconnected systems are globally ultimately bounded. Sufficient conditions on the systems to be interconnected, on the network topology, on the coupling dynamics, and on the time-delays that guarantee global state synchronization are derived. A local analysis is provided in which we compare the performance of our predictor-based control scheme against the existing static diffusive couplings available in the literature. We show (locally) that the time-delay that can be induced to the network may be increased by including the predictors in the loop. The results are illustrated by computer simulations of coupled Hindmarsh-Rose neurons. V C 2015 AIP Publishing LLC. [http://dx.doi.org/10.1063/1.4906820] This manuscript focuses on controlled synchronization of identical nonlinear systems interacting on networks with general topologies and interconnected through predictor-based diffusive dynamic couplings. The systems are said to be diffusively coupled, if they interact through weighted differences of the form c(y j À y i) with some positive constant c called the coupling strength and y i , y j denoting the outputs of the ith and jth systems. An important element of our control scheme is the use of a communication network. Network communication is necessary in the study of synchronization to transmit and receive measurement and control data among the systems. Because of the time needed to transmit data over the network, the use of net-worked communication to exchange information results in unavoidable time-delays. This networked-induced delays are undesirable because they may lead to the loss of synchrony. Hence, when studying synchronization among dynamical systems with networked communication , it is important to design control algorithms which are robust with respect to time-delays. The results presented here follow the same research line as Refs. 1 and 2, where sufficient conditions for synchronization of diffusively interconnected nonlinear systems with and without time-delays are derived. In order to derive their results, the authors assume that the individual systems are semipassive 3 with respect to the coupling variable y i , and their corresponding internal dynamics have some desired stability properties (convergent internal dynamics 4). In particular, in Ref. 2, the authors study the problem of network synchronization of diffusively time-delayed coupled semipassive systems. They prove that under some mild assumptions, there always exists a region S in the parameter space (coupling strength c versus time-delay s), such that ...

General hybrid projective complete dislocated synchronization between a class of chaotic real nonlinear systems and a class of chaotic complex nonlinear systems

“…Many different types of synchronization phenomenon have been intensively investigated and a lot of theoretical results have been obtained in the past 20 years, such as complete synchronization [1], anti-synchronization [2], generalized synchronization [3,4], phase synchronization [5], anti-phase synchronization [6], lag synchronization [7], partial synchronization [8], projective synchronization [9,10,11,12,13], time scale synchronization [14], combination synchronization [15,16], compound synchronization [17] etc. During the past decades, there exist the following methods to realize chaos synchronization such as OGY method [19], feedback control method [20,21,22], H ∞ control method [23], optimal control method [24], PID control method [25], active control method [26], passive control method [27], backstepping method [28], adaptive control method [29,30], sliding mode control method [31], impulsive control method [32], coupling control method [33,34,35] etc.…”

Finite-time synchronization between two complex-variable chaotic systems with unknown parameters via nonsingular terminal sliding mode control

“…A fundamental scheme of chaos synchronization is the drive-response configuration, and the output of the response system should track the drive system asymptotically. Several control techniques have been investigated to realize chaos synchronization such as OGY method [1], feedback control method [2,3], active control method [4], backstepping method [5], adaptive control method [6][7][8][9][10][11][12][13], impulsive control method [14,15], coupling control method [16][17][18], sliding control method [19] and switching control method [20][21][22][23].…”

Compound-combination synchronization of chaos in identical and different orders chaotic systems

“…This shows that compound-combination synchronization between the drive systems (6)-(8) and the response systems (9)- (11) is achieved. Finally, the full 1 2 3 ( , , ) q q q is 2 2 2 2 2 2 2 1 3 2 2 1 3 1 3 2 3 2 2 2 2 2 2 3 3 3 3 3 3 3 3 3 2 3 2 3 3 3 3 3 3 3 2 3 2 3 1 1 1 1 1 2 2 2 2 2 2 2 2 2 3 3 3 3 3 3 3 3 1 1 1 1 1 2 2 2 2 2 2 2 2 2 3 3 3 3 3 3 3 3 Solving the drive system (6)-(8) and the response systems (9)-(11) with the controllers defined in (26) using the following initial conditions (15) 1 3 2 3 2 2 2 2 2 2 3 3 3 3 3 3 3 3 3 2 3 2 3 3 3 3 3 3 3 2 3 2 3 n ))) Solving the drive system (6)-(8) and the response system (9) with the controllers defined in (27) using the following initial conditions using the initial conditions of the drive systems and response systems as …”

“…This interest has led to the discovery of different synchronization types and schemes such as complete synchronization [24], phase synchronization, anti-synchronization [25], projective synchronization [26], time delay synchronization [27], generalized synchronization [28], function projective synchronization [29], increased order synchronization [30], reduced order synchronization [31] and others [23,32]. Most of the previous discoveries on synchronization focus on synchronization between one drive and one response oscillator only.…”