Synchronization Of Nonlinear Systems Research Articles

Manifold Calculus in System Theory and Control—Fundamentals and First-Order Systems

The aim of the present tutorial paper is to recall notions from manifold calculus and to illustrate how these tools prove useful in describing system-theoretic properties. Special emphasis is put on embedded manifold calculus (which is coordinate-free and relies on the embedding of a manifold into a larger ambient space). In addition, we also consider the control of non-linear systems whose states belong to curved manifolds. As a case study, synchronization of non-linear systems by feedback control on smooth manifolds (including Lie groups) is surveyed. Special emphasis is also put on numerical methods to simulate non-linear control systems on curved manifolds. The present tutorial is meant to cover a portion of the mentioned topics, such as first-order systems, but it does not cover topics such as covariant derivation and second-order dynamical systems, which will be covered in a subsequent tutorial paper.

Fuzzy remodeling and synchronization of nonlinear systems

Synchronization is one of the crucial control problems in coordinated behavior systems. A wide range of applications necessitates the development and improvement of approaches to solving the linearization problem for non-linear systems with complex behavior. Hence the paper offers an approach to solving the problem of chaotic system coordinate synchronization based on the use of fuzzy remodeling and superstability conditions. It is shown that transition from general (non-linear) synchronization problem statement to fuzzy description with fuzzy Takagi-Sugeno models allows transforming the initial task to a well understood problem of fuzzy T-S system stabilization that describes the dynamics synchronization error. As a candidate solution for the problem we offer a way of implementing superstability conditions that reduce the task of finding fuzzy regulator to solving a linear programming task. It is shown that implementation of superstability conditions not only presents a simple way of solving the problem, but also has practical value. Superstability provides the monotonesness of the transient process of synchronization without sharp spikes of solution norm. The efficiency of the offered approach is demonstrated by the example of synchronization of two hyperchaotic systems. To prove the efficiency of superstability conditions the solution of the robust synchronization problem with parametric uncertainty in system matrices is given. The presented results can be applied for synchronization of various nonlinear systems demonstrating chaotic dynamics.

Adaptive Synchronization of Time Delay Chaotic Systems with Uncertain and Unknown Parameters via Aperiodically Intermittent Control

Time delay and uncertainty are the two common phenomena in nonlinear systems, which seriously influence the control and synchronization of nonlinear systems. In this paper, an adaptive intermittent control strategy is proposed to realize the synchronization of time delay chaotic systems with uncertain and unknown parameters. For the uncertain and unknown parameters, some sufficient conditions for complete synchronization are derived based on the stability theory. In the part of numerical simulation, two cases of linear and non-linear delay are discussed, which both show the effectiveness of the theoretical results.

Generalized PID Synchronization of Higher Order Nonlinear Systems With a Recursive Lyapunov Approach

This paper investigates the problem of synchronization for nonlinear systems. Following a Lyapunov approach, we first study the global synchronization of nonlinear systems in the canonical control form with both distributed proportional-derivative and proportional-integral-derivative control actions of any order. To do so, we develop a constructive methodology and generate in an iterative way inequality constraints on the coupling matrices that guarantee the solvability of the problem or, in a dual form, provide the nonlinear weights on the coupling links between the agents such that the network synchronizes. The same methodology allows us to include a possible distributed integral action of any order to enhance the rejection of heterogeneous disturbances. The considered approach does not require any dynamic cancellation, thus preserving the original nonlinear dynamics of the agents. The results are then extended to linear and nonlinear systems admitting a canonical control transformation. Numerical simulations validate the theoretical results.

Cluster Synchronization of Diffusively Coupled Nonlinear Systems: A Contraction-Based Approach

Finding the conditions that foster synchronization in networked oscillatory systems is critical to understanding a wide range of biological and mechanical systems. However, the conditions proved in the literature for synchronization in nonlinear systems with linear coupling, such as has been used to model neuronal networks, are in general not strict enough to accurately determine the system behavior. We leverage contraction theory to derive new sufficient conditions for cluster synchronization in terms of the network structure, for a network where the intrinsic nonlinear dynamics of each node may differ. Our result requires that network connections satisfy a cluster-input-equivalence condition, and we explore the influence of this requirement on network dynamics. For application to networks of nodes with neuronal spiking dynamics, we show that our new sufficient condition is tighter than those found in previous analyses which used nonsmooth Lyapunov functions. Improving the analytical conditions for when cluster synchronization will occur based on network configuration is a significant step toward facilitating understanding and control of complex oscillatory systems.

A study of synchronization processes of nonlinear systems in the difference space of phase variables

The analysis of trajectories in the phase space of the systems of ordinary differential equations has been made. Classification of phase trajectories has been developed. Synchronization in Rossler systems, coupled by the scheme “main–controlled” system, has been studied. In the controlled system, variables in the right –hand side are replaced by functions of time, which are solutions to the main system. The analysis of processes in nonlinear systems was made by means of replacement with the help of synchronization matrix and transfer to the linearized system of variables equal to the difference of phase variables of the main and controlled systems. As a result of this analysis, there have been set the values of the synchronization matrix elements in which there occur different types of synchronization: complete, phase and topological. It is shown that even in the absence of communication between Rossler systems in the difference space of phase variables of the main and controlled systems with nonlinear dynamics, there occurs topological synchronization and there is formed an attractor with low spatial complexity that is an open trajectory of limited values. The criterion for the absence of synchronization of nonlinear systems is the unlimited growth of the difference of phase variables

Composite nonlinear feedback control technique for master/slave synchronization of nonlinear systems

In this paper, we propose a design approach of composite nonlinear feedback control technique for the synchronization of master/slave nonlinear systems with time-varying delays, Lipschitz nonlinear functions and parametric uncertainties. Based on the Lyapunov–Krasovskii stabilization theory and linear matrix inequalities, a new sufficient condition is generated for the synchronization of chaotic systems with nonlinearities and perturbations on the master and slave systems. By using the Barbalat’s lemma, the proposed control method guarantees that the states of the master and slave systems are synchronized with an asymptotic convergence rate. Simulation results are demonstrated on two forms of Chua’s chaotic system, which illustrate that the suggested design technique yields satisfactory transient performance.

Master-slave synchronization for nonlinear systems via reliable control with gaussian stochastic process

In this paper, master-slave synchronization methods for nonlinear system are proposed under a reliable control scheme. In order to consider a more realistic situation in designing a reliable controller, Gaussian stochastic process is introduced to the concept of reliable control system. To confirm an effect of reliable control system, a new synchronization criterion of nonlinear systems under time-varying delay feedback and reliable control with Gaussian transition probabilities is derived within the framework of linear matrix inequalities (LMIs) based on Lyapunov method. Finally, two numerical examples are provided to show the effectiveness and applicability of the of the proposed results.

Noise-induced binary synchronization in nonlinear systems

The phenomenon of noise-induced binary synchronization has been discovered in two independent dynamical systems generating aperiodic binary signals under the action of a common noise source. The presence of a synchronous regime was confirmed by the calculation of Lyapunov exponents for the two systems. The mechanism of development of the noise-induced binary synchronization regime has been found. A relation of the observed regime to binary generalized synchronization is established.

Hybrid stabilization and synchronization of nonlinear systems with unbounded delays

The stabilization of nonlinear systems with bounded and unbounded time-delays via hybrid control is studied. We investigate and identify switching rules for which stabilization can be verified a priori. When this approach is inadequate, stabilizing state-dependent switching rules are constructed. This method is based on partitioning the state-space into switching regions. Unwanted physical behavior, such as chattering and Zeno behavior, is avoided. Sufficient conditions are established using Razumikhin-like theorems. The theoretical results provide insight into how hybrid control strategies can be constructed to synchronize a class of nonlinear systems with unbounded delay. The findings are illustrated through numerical simulations.

Synchronization of nonlinear systems with communication delays and intermittent information exchange

This paper studies the synchronization problem of second-order nonlinear multi-agent systems with intermittent communication in the presence of irregular communication delays and possible information losses. The control objective is to steer all agents’ positions to a common position with a prescribed desired velocity available only to some leaders. Based on the small-gain framework, we propose a synchronization scheme relying on an intermittent information exchange protocol in the presence of time delays and possible packet dropouts. We show that our control objectives are achieved with a simple selection of the control gains provided that the directed graph, describing the interconnection between agents, contains a spanning tree. An example of Euler–Lagrange systems is considered to illustrate the application and effectiveness of the proposed approach.

Impulsive control and synchronization of nonlinear system with impulse time window

In this paper, the issue on impulsive control and synchronization of nonlinear system with impulse time window are investigated. The considered impulsive effects can be stochastically occurred at a determined time window. Hence, the impulses here can be more general and more applicable than the fixed impulses. Some novel and easy-to-check criteria with impulse time window are obtained to guarantee the impulsive control and synchronization global asymptotical stable. Especially, on the basis of the our analysis, we only choose an efficient impulse time window instead of choosing fixed-impulse sequences. Finally, the simulation results further demonstrate the effectiveness of our theoretical results.

Synchronization induced by disorder of phase directions

We study the influence of randomly distributed phase directions of external force in an array of coupled pendula, instead of studying the influence of continuous phase. We find that with the increase of the absolute value of the phase, the chaotic behaviors of the coupled arrays may be controlled and different synchronized patterns can be induced. These results demonstrate that by introducing the randomness of the phase directions, rather than the continuous value of the phase, it can lead to a synchronization in nonlinear systems. This finding may provide a new insight for understanding the mechanism of disorder induced synchronization.

Complete synchronization induced by disorder in coupled chaotic lattices

Abstract The effect of phase disorder in external forces introduced into two-dimensional lattices of coupled chaotic pendulums is investigated. As the increase of the disorder, we find complete synchronization between the pendulums in each chain and different periodic synchronized patterns, while the chain remains asynchronous if all driving forces have the same phase. Applying the master stability function method, an analytic solution is given to support the numerical results. All these findings may provide further insight into chaos control and synchronization in nonlinear systems.

Synchronization of nonlinear Lurie systems on the basis of passification and backstepping

The problems of passification and synchronization of nonlinear systems are solved with the aid of the backstepping method for cascade systems of the Lurie type with functional uncertainty. Estimates of the errors caused by bounded disturbances and controller discreteness are given. The stated theorem is extended to the case of net systems. The obtained results are applied to the synchronization problem of movement of two mobile wheeled robots and are illustrated by way of computational modeling.

Synchronization of nonlinear systems under information constraints

A brief survey of control and synchronization under information constraints (limited information capacity of the coupling channel) is given. Limit possibilities of nonlinear observer-based synchronization systems with first-order coders or full-order coders are considered in more detail. The existing and new theoretical results for multidimensional drive-response Lurie systems (linear part plus nonlinearity depending only on measurable outputs) are presented. It is shown that the upper bound of the limit synchronization error (LSE) is proportional to the upper bound of the transmission error. As a consequence, the upper and lower bounds of LSE are proportional to the maximum coupling signal rate and inversely proportional to the information transmission rate (channel capacity). The analysis is extended to networks having a “chain,” “star,” or “star-chain” topology. Adaptive chaotic synchronization under information constraints is analyzed. The results are illustrated by example: master-slave synchronization of two chaotic Chua systems coupled via a channel with limited capacity.

The existence of two types of generalized synchronization of nonlinear systems

The existence of two types of generalized synchronization of chaotic nonlinear systems is studied. When the modified system collapses to a stable equilibrium or periodic oscillation, the existence of generalized synchronization can be converted to the problem of compression fixed point under certain conditions. Strict theoretical proofs are given to the exponential attractive property of generalized synchronization manifold. Numerical simulations illustrate the correctness of the present theory.

Synchronisation of nonlinear systems by bidirectional coupling with time-varying delay

This paper considers the synchronization problem for coupled nonlinear systems with time-varying delay. In previous work, we have already derived a sufficient condition for synchronisation and boundedness of two identical strictly semi-passive systems coupled using state feedback with time-delay. This method, however, requires that all states of each system are coupled to the other systems and the coupling delay is constant. In this paper we extend the conditions to identical systems coupled using output feedback with time-varying delay where a bound on the length of the delay and an upper bound of the time-derivative is known. Firstly, we show, using the small-gain theorem, that the trajectories of coupled strictly semi-dissipative systems converge to a bounded region. Then we derive a sufficient condition for synchronisation of the systems coupled with time-varying delay by using a delay range dependent on the stability criterion.

Preservation of stability and synchronization in nonlinear systems

Preservation of stability in the presence of structural and/or parametric changes is an important issue in the study of dynamical systems. A specific case is the synchronization of chaos in complex networks where synchronization should be preserved in spite of changes in the network parameters and connectivity. In this work, a methodology to establish conditions for preservation of stability in a class of dynamical system is given in terms of Lyapunov methods. The idea is to construct a group of dynamical transformations under which stability is retained along certain manifolds. Some synchronization examples illustrate the results.

Using synchronization to obtain dynamic logic gates

We introduce a scheme to obtain key logic-gate structures, using synchronization of nonlinear systems. We demonstrate the idea explicitly by numerics and experiments on nonlinear circuits. A significant feature of this scheme is that a single nonlinear drive-response circuit can be used to flexibly yield the different logic gates, and switch logic behavior by small changes in the parameter of the response system; so the response system can act as a "logic output controller." Thus this scheme may help to construct dynamic general-purpose computational hardware with reconfigurable abilities.