LMI criteria for robust chaos synchronization of a class of chaotic systems

Based on the Lyapunov stability theory and LMI technique, a new sufficient criterion, formulated in the LMI form, is established in this paper for chaos robust synchronization by linear-state-feedback approach for a class of uncertain chaotic systems with different parameters perturbation and different external disturbances on both master system and slave system. The new sufficient criterion can guarantee that the slave system will robustly synchronize to the master system at an exponential convergence rate. Meanwhile, we also provide a criterion to find out proper feedback gain matrix K$K$ that is still a pending problem in literature [H. Zhang, X.K. Ma, Synchronization of uncertain chaotic systems with parameters perturbation via active control, Chaos, Solitons and Fractals 21 (2004) 39–47]. Finally, the effectiveness of the two criteria proposed herein is verified and illustrated by the chaotic Murali–Lakshmanan–Chua system and Lorenz systems, respectively.     

State feedback and output feedback control of a class of nonlinear systems with delayed measurements

This paper is concerned with the problem of state feedback and output feedback control of a class of nonlinear systems with delayed measurements. This class of nonlinear systems is made up of continuous-time linear systems with nonlinear perturbations. The nonlinearity is assumed to satisfy a global Lipschitz condition and the time delay is assumed to be time-varying and have no restriction on its derivative. On the basis of the Lyapunov–Krasovskii approach, sufficient conditions for the existence of the state feedback controller and the output feedback controller are derived in terms of linear matrix inequalities. Methods of calculating the controller gain matrices are also presented. Two numerical examples are given to illustrate the effectiveness of the proposed methods.     

Absolute stability criteria for a class of nonlinear singular systems with time delay

This paper deals with the problem of absolute stability for a class of time-delay singular systems with sector-bounded nonlinearity. Both delay-independent and delay-dependent criteria are presented and formulated in the form of linear matrix inequalities (LMIs). Neither model transformation nor a bounding technique for cross-terms, nor a slack variable method is involved in obtaining the stability criteria. Numerical examples are given to show the effectiveness and improvements over some existing results.     

A new augmented Lyapunov-Krasovskii functional approach for stability of linear systems with time-varying delays

This paper proposes improved delay-dependent conditions for asymptotic stability of linear systems with time-varying delays. The proposed method employs a suitable Lyapunov-Krasovskii’s functional for new augmented system. Based on Lyapunov method, delay-dependent stability criteria for the systems are established in terms of linear matrix inequalities (LMIs) which can be easily solved by various optimization algorithms. Three numerical examples are included to show that the proposed method is effective and can provide less conservative results.     

On robustly exponential stability of uncertain neutral systems with time-varying delays and nonlinear perturbations

The issue of robustly exponential stability for uncertain neutral-type systems is considered in this paper. The uncertainties are nonlinear and the delays are time-varying. In terms of a linear matrix inequality (LMI), the new sufficient stability condition with delay dependence is presented. The model transformation and bounding techniques for cross terms are avoided based on an integral inequality. Two illustrative examples are proposed to show the effectiveness of our method.     

A semi-free weighting matrices approach for neutral-type delayed neural networks

In this paper, a new approach is proposed for stability issues of neutral-type neural networks (DNNs) with constant delay. First, the semi-free weighting matrices are proposed and used instead of the known free weighting matrices to express the relationship between the terms in the Leibniz–Newton formula to simplify the system synthesis and to obtain less computation demand. Second, global exponential stability conditions which are less conservative and restrictive than the known results are derived. At the same time, based on the above approach, fewer variable matrices are introduced in the construction of the Lyapunov functional and augmented Lyapunov functional. Two examples are given to show their effectiveness and advantages over others.     

Asymptotic pinning synchronization of nonlinear multi-agent systems: Its application to tunnel diode circuit

This work proposes a pinning state-feedback control technique for synchronizing non-linear multi-agent systems (MASs) with time delays. A collection of switching-directed graphs describes the communication exchanges between all of the agents. The challenge of asymptotic stability analysis for some error systems is translated into the construction of a leader-following synchronization of the relevant MASs. The closed-loop system could be acquired by building a convenient Lyapunov–Krasovskii functional (LKF) that has two integral terms, and by using Kronecker product qualities combined with matrix inequality techniques. When these conditions are met, a state-feedback pinning controller can be built with linear matrix inequalities (LMIs), which can be derived easily from a number of efficient optimization algorithms. Further, the performance of the proposed control design system is verified based on a tunnel diode circuit (TDC) by numerical simulations.     

Synchronization of coupled chaotic FitzHugh-Nagumo systems

This paper addresses dynamic synchronization of two FitzHugh-Nagumo (FHN) systems coupled with gap junctions. All the states of the coupled chaotic system, treating either as single-input or two-input control system, are synchronized by stabilizing their error dynamics, using simplest and locally robust control laws. The local asymptotic stability, chosen by utilizing the local Lipschitz nonlinear property of the model to address additionally the non-failure of the achieved synchronization, is ensured by formulating the matrix inequalities on the basis of Lyapunov stability theory. In the presence of disturbances, it ensures the local uniform ultimate boundedness. Furthermore, the robustness of the proposed methods is ensured against bounded disturbances besides providing the upper bound on disturbances. To the best of our knowledge, this is the computationally simplest solution for synchronization of coupled FHN modeled systems along with unique advantages of less conservative local asymptotic stability of synchronization errors with robustness. Numerical simulations are carried out to successfully validate the proposed control strategies.     

Global asymptotic stability for a class of nonlinear neural networks with multiple delays

This paper investigates the global asymptotic stability (GAS) for a class of nonlinear neural networks with multiple delays. Based on Lyapunov stability theory and the linear matrix inequality (LMI) technique, a less conservative delay-dependent stability criterion is derived. The present result is shown to be less conservative than those given in the literature.     

Stability analysis for continuous system with additive time-varying delays: A less conservative result

This paper presents a less conservative result for stability analysis of continuous-time systems with additive delays by constructing a new Lyapunov-Krasovskii functional and utilizing free matrix variables in approximating certain integral quadratic terms in obtaining the stability condition in terms of linear matrix inequalities. Numerical example is provided to show the effectiveness of the proposed method compared to some recent results.     

Robust fault tolerant control of continuous-time switched systems: An LMI approach

The present paper proposes a new robust fault tolerant control (RFTC) design for continuous-time switched systems. The main objective is to design in an integrated way the couple (controller, observer) that allows to stabilize switched systems even in the presence of actuator faults. A state/fault estimation observer is designed to simultaneously estimate system state and actuator faults. Based on this observer, a fault tolerant controller is developed to stabilize the system and accommodate the actuator faults automatically. The RFTC problem is formalized in the form of linear matrix inequalities (LMI) rather than bilinear matrix inequalities (BMI), to avoid the difficulty of solving BMIs. Finally, a numerical example composed of unstable subsystems is studied to show the applicability and efficiency of the obtained results.     

Synchronization stability of general complex dynamical networks with time-varying delays: A piecewise analysis method

The synchronization problem of some general complex dynamical networks with time-varying delays is investigated. Both time-varying delays in the network couplings and time-varying delays in the dynamical nodes are considered. The delays considered in this paper are assumed to vary in an interval, where the lower and upper bounds are known. Based on a piecewise analysis method, the variation interval of the time delay is firstly divided into several subintervals, by checking the variation of the derivative of a Lyapunov function in every subinterval, then the convexity of matrix function method and the free weighting matrix method are fully used in this paper. Some new delay-dependent synchronization stability criteria are derived in the form of linear matrix inequalities. Two numerical examples show that our method can lead to much less conservative results than those in the existing references.     

Adaptive backstepping sliding mode control for chaos synchronization of two coupled neurons in the external electrical stimulation

In this paper, a robust control system combining backstepping and sliding mode control techniques is used to realize the synchronization of two gap junction coupled chaotic FitzHugh-Nagumo (FHN) neurons in the external electrical stimulation. A backstepping sliding mode approach is applied firstly to compensate the uncertainty which occur in the control system. However, the bound of uncertainty is necessary in the design of the backstepping sliding mode controller. To relax the requirement for the bound of uncertainty, an adaptive backstepping sliding mode controller with a simple adaptive law to adapt the uncertainty in real time is designed. The adaptive backstepping sliding mode control system is robust for time-varying external disturbances. The simulation results demonstrate the effectiveness of the control scheme.     

Robust chaos suppression for the family of nonlinear chaotic systems with noise perturbation

This paper investigates the robust chaos suppression problem for some classical Rössler systems using the sliding mode controller (SMC). Based on the proportional-integral (PI) switching surface, a SMC is derived to not only guarantee asymptotical stability of the equilibrium points of the Rössler systems but also reduce the effect of noise perturbation to an _H∞$H_{\infty }$-norm performance. The parameter matrix necessary for constructing both PI switching surface and the SMC can be easily solved by the linear matrix inequality (LMI) optimization technique. Finally, two illustrative examples are provided to demonstrate the efficacy of the proposed control methodology.     

Improved stability criteria for time-varying delayed T-S fuzzy systems via delay partitioning approach

This paper focuses on the stability analysis for uncertain Takagi-Sugeno (T-S) fuzzy systems with interval time-varying delay. The uncertainties of system parameter matrices are assumed to be time-varying and norm-bounded. Some new Lyapunov-Krasovskii functionals (LKFs) are constructed by nonuniformly dividing the whole delay interval into multiple segments and choosing different Lyapunov functionals to different segments in the LKFs. By employing these LKFs, some new delay-derivative-dependent stability criteria are established for the nominal and uncertain T-S fuzzy systems in a convex way. These stability criteria are derived that depend on both the upper and lower bounds of the time derivative of the delay. By employing the new delay partitioning approach, the obtained stability criteria are stated in terms of linear matrix inequality (LMI). They are equivalent or less conservative while involving less decision variables than the existing results. Finally, numerical examples are given to illustrate the effectiveness and reduced conservatism of the proposed results.     

A semi-global finite-time convergent observer for a class of nonlinear systems with bounded trajectories

This paper considers the problem of designing finite-time convergent observers for a class of lower-triangular nonlinear systems with bounded solution trajectories. Using the homogeneous domination approach, we construct an observer with homogeneous structure and saturation design, whose states will converge to the real states in a finite time by adjusting the observer gain. Several application examples of this finite-time convergent observer are discussed in this paper.     

Dissipativity analysis for singular systems with time-varying delays

This paper is concerned with the dissipativity analysis problem for singular systems with time-varying delays. A delay-dependent criterion is established to guarantee the dissipativity of the underlying systems using the delay partitioning technique. All the results given in this paper are not only dependent upon the time delay, but also dependent upon the number of delay partitions. The effectiveness and the reduced conservatism of the derived results are demonstrated by two illustrative examples.     

Improved delay-dependent stability criterion for neural networks with time-varying delay

In this paper, the problem of delay-dependent asymptotic stability criterion for neural networks with time-varying delay has been considered. A new class of Lyapunov functional which contains a triple-integral term is constructed to derive some new delay-dependent stability criteria. The obtained criteria are less conservative because free-weighting matrices method and a convex optimization approach are considered. Finally, numerical examples are given to illustrate the effectiveness of the proposed method.