ROBUST SYNCHRONIZATION OF CHAOTIC LUR'E SYSTEMS VIA REPLACING VARIABLES CONTROL

Sampled-data feedback control for master-slave synchronization schemes that consist of identical chaotic Lur'e systems is studied. Sufficient conditions for global asymptotic synchronization of such chaotic Lur'e systems are obtained using the free-weighting matrix approach and expressed in terms of linear matrix inequalities (LMIs). With the help of the LMI solvers, the sampled-data feedback control law can easily be obtained to globally asymptotically synchronize Lur'e chaotic systems. The effectiveness of the proposed method is finally illustrated via numerical simulations of chaotic Chua's circuits.

In this paper, we develop a systematic Lyapunov approach to the regional stability and performance analysis of saturated systems in a general feedback configuration. The only assumptions we make about the system are well-posedness of the algebraic loop and local stability. Problems to be considered include the estimation of the domain of attraction, the reachable set under a class of bounded energy disturbances and the nonlinear L2 gain. The regional analysis is established through an effective treatment of the algebraic loop and the saturation/deadzone function. This treatment yields two forms of differential inclusions, a polytopic differential inclusion (PDI) and a norm-bounded differential inclusion (NDI) that contain the original system. Adjustable parameters are incorporated into the differential inclusions to reflect the regional property. The main idea behind the regional analysis is to ensure that the state remain inside the level set of a certain Lyapunov function where the PDI or the NDI is valid. With quadratic Lyapunov functions, conditions for stability and performances are derived as linear matrix inequalities (LMIs). To obtain less conservative conditions, we use a pair of conjugate non-quadratic Lyapunov functions, the convex hull quadratic function and the max quadratic function. These functions yield bilinear matrix inequalities (BMIs) as conditions for stability and guaranteed performance level. The BMI conditions cover the corresponding LMI conditions as special cases, hence the BMI results are guaranteed to be as good as the LMI results. In most examples, the BMI results are significantly better than the LMI results

We study the dynamical regimes that emerge from the strong coupling between two Chua's circuits with parameters mismatch. For the region around the perfect synchronous state we show how to combine parameter diversity and coupling in order to robustly and precisely target a desired regime. This target process allows us to obtain regimes that may lie outside parameter ranges accessible for any isolated circuit. The results are obtained by following a recently developed theoretical technique, the order parameter expansion, and are verified both by numerical simulations and on electronic circuits. The theoretical results indicate that the same predictable change in the collective dynamics can be obtained for large populations of strongly coupled circuits with parameter mismatches.

This paper discusses the problem of finding a common quadratic Lyapunov function (CQLF) for planar systems in linear matrix inequality (LMI) regions. The main contribution of this paper is to give a complete characterization of quadratic Lyapunov functions (QLFs) for a given planar system, which is stable in an LMI region, and obtain a necessary and sufficient condition for such a class of planar systems to share a CQLF in the LMI region. The results are not only verifiable but also constructive, which can be realized by computer easily.

In this paper we present a synchronization method for chaotic Lur'e systems by constructing a new piecewise Lyapunov function. Using a delayed feedback control scheme, a delay-dependent stability criterion is derived for the synchronization of chaotic systems that are represented by Lur'e systems with deadzone nonlinearity. Based on the Lyapunov—Krasovskii functional and by using some properties of the nonlinearity, a new delay-dependent stabilization condition for synchronization is obtained via linear matrix inequality (LMI) formulation. The criterion is less conservative than existing ones, and it will be verified through a numerical example.

Dichotomy, or monostability, is one of the most important properties of nonlinear dynamic systems. For a dichotomous system, the solution of the system is either unbounded or convergent to a certain equilibrium, thus periodic or chaotic states cannot exist in the system. In this paper, a new methodology for the analysis of dichotomy of a class of nonlinear systems is proposed, and a linear matrix inequality (LMI)-based criterion is derived. The results are then extended to uncertain systems with real convex polytopic uncertainties in the linear part, and the LMI representation for robust dichotomy allows the use of parameter-dependent Lyapunov function. Based on the results, a dynamic output feedback controller guaranteeing robust dichotomy is designed, and the controller parameters are explicitly expressed by a set of feasible solutions of corresponding linear matrix inequalities. An extended Chua's circuit with two nonlinear resistors is given at the end of the paper to demonstrate the validity and applicability of the proposed approach. It is shown that by investigating the convergence of the bounded oscillating solutions of the system, our results suggests a viable and effective way for chaos control in nonlinear circuits.

For a large class of relay feedback systems (RFS) there will be limit cycle oscillations. Conditions to check existence and local stability of limit cycles for these systems are well known. Global stability conditions, however, are practically nonexistent. The paper presents conditions in the form of linear matrix inequalities (LMIs) that, when satisfied, guarantee global asymptotic stability of limit cycles induced by relays with hysteresis in feedback with linear time-invariant (LTI) stable systems. The analysis consists in finding quadratic surface Lyapunov functions for Poincare maps associated with RFS. These results are based on the discovery that a typical Poincare map induced by an LTI flow between two hyperplanes can be represented as a linear transformation analytically parametrized by a scalar function of the state. Moreover, level sets of this function are convex subsets of linear manifolds. The search for quadratic Lyapunov functions on switching surfaces is done by solving a set of LMIs. Although this analysis methodology yields only a sufficient criterion of stability, it has proved very successful in globally analyzing a large number of examples with a unique locally stable symmetric unimodal limit cycle. In fact, it is still an open problem whether there exists an example with a globally stable symmetric unimodal limit cycle that could not be successfully analyzed with this new methodology. Examples analyzed include minimum-phase systems, systems of relative degree larger than one, and of high dimension. Such results lead us to believe that globally stable limit cycles of RFS frequently have quadratic surface Lyapunov functions.

In this paper, the robust synchronization problem is addressed for recurrent neural networks with time-varying delay by linear feedback control. Robustness in the present paper is referred to as the allowance of parameters mismatch between the drive system and the response system. Sufficient conditions for robust synchronization with a synchronization error bound, expressed as linear matrix inequality (LMI), are derived based on Lyapunov–Krasovskii functionals. Both delay-dependent and delay-independent conditions are obtained. Two examples are given to illustrate the results.

The paper presents stabilization of a Networked Control System (NCS) with uncertain, time-varying delays using two stabilization approaches namely, Parameter Dependant Lyapunov Function (PDLF) and Quadratic Lyapunov Function (QLF). The PDLF approach depends upon the time varying delay parameters. Thus, for larger delays the system's stability is ensured. In NCS, the uncertainties arising out from the variations in time-delay is modeled using polytopic approach and the stability region of the system is presented in terms of controller gain and maximum tolerable delay. The overall NCS is represented in discrete domain. The proposed PDLF stabilization results have been compared with that of the QLF approach. Lyapunov-based stability criterion is presented in Linear Matrix inequalities (LMIs). The stabilized state feedback controller is designed using LMIs and considering the bounded time-varying delays. The proposed approach is illustrated by two numerical examples. From the results obtained, it is observed that the PDLF approach improves the stability region of the system.

This paper increases the upper bound of sampling interval for synchronization of chaotic Lur'e systems. First, the characteristics of a sampling pattern and nonlinearities are fully utilized in a new piecewise-differentiable Lyapunov functional, which is continuous at sampling instant. Second, appropriate slack variables are used for handling a zero constraint. Using the slack variables gives the relaxation of a synchronization criterion. Based on the Lyapunov functional and the zero constraint, the synchronization criterion is derived by applying free-matrix-based integral inequalities and Finsler's lemma. The free-matrix-based integral inequalities are used for yielding tight bounds on integral terms related to the sampling pattern. Finsler's lemma is applied for transforming a bilinear matrix inequality that is generated by slack variables into a linear matrix inequality. This paper applies the proposed criterion to the application to the secure communication. Simulation results show the effectiveness of the proposed criterion.

The onset of jump resonance in the driven Chua's circuit is deeply described in this paper, further confirming the role of Chua's circuit in nonlinear circuits and systems theory. The study of the conditions under which this peculiar behavior occurs is based on the possibility of obtaining a Lur'e representation of the driven Chua's circuit and applying the describing function approach. Design considerations to select the Chua's circuit parameters in order to shape the jump resonance features are given, providing the nonlinear core for highly selective frequency drift sensors.

This paper investigates anticipating synchronization of a general form of chaotic lur'e master-slave systems. Through the analysis of the set-valued nonlinearities, linear matrix inequalities (LMI)-based output feedback is firstly proposed to achieve globally asymptotic prediction of the given system's future states. Then, based on the ideology of active control, sliding mode controllers are adopted to ensure accurate anticipation of Lur'e systems, which are provided with high robustness against possible disturbances. The control techniques are both verified through Lyapunov stability analysis, and illustrated by numerical simulation of Chua's circuit. Simulation results show that active sliding mode control scheme has better performance.

This study presents a sum-of-squares (SOS) approach to stable control and guaranteed cost control of discrete polynomial fuzzy systems. The SOS framework presented in this paper offers a new paradigm over the existing linear matrix inequality (LMI) approaches to discrete Takagi-Sugeno (T-S) fuzzy models. At first, this study, newly introduces a discrete polynomial fuzzy model that is more general and effective than the well-known discrete T-S fuzzy model. With considering the operating domain, stable control design conditions are then derived based on state-dependent Lyapunov functions that contain quadratic Lyapunov functions as a special case. Hence, the design approach discussed in this study could be more general than the LMI design approaches based on quadratic Lyapunov functions. Moreover, this study also discusses a guaranteed cost control design which is carried out by minimising the upper bound of a given performance function. All the design conditions derived in this study can be represented in terms of SOS and are symbolically and numerically solved via the SOSOPT and the SeDuMi, respectively. Finally, the ball-and-beam system is provided as an example to illustrate the utility of the proposed SOS-based design approach.

In recent years synchronization of chaotic dynamics has received ever increasing attention. Herein, cellular neural networks (CNNs) are considered as a tool for generating hyperchaotic behaviors. By exploiting a system theory approach, a technique for synchronizing a large class of CNNs is developed. In particular, a necessary and sufficient condition for hyperchaos synchronization is given, which is based on the controllability property of linear systems. Finally, in order to show the effectiveness of the proposed technique, the synchronization of a CNN constituted by Chua's circuits is illustrated.

Global synchronization of coupled map lattices are investigated in this paper. A sufficient condition for global synchronization is derived. Unlike other approaches, the coupled lattices is not linearized in this paper. Instead, the sufficient condition for the synchronization is obtained here by introducing a reference state along with Lyapunov function method, and this condition is simply given in terms of the maximum absolute value of eigenvalues of the coupling matrices those are not equal to one. Moreover, we give how to construct the coupling matrix to guarantee global synchronization of coupled map lattices based on two comparing matrices, which is very convenient to use. A typical example of coupled map lattices with chaotic nodes is finally used to verify the theoretical results and the effectiveness of the proposed synchronization scheme.