Impulsive Stabilization Research Articles

Impulsive stabilization for linear neutral‐type time‐delay systems

SummaryThe problem of impulsive stabilization of linear neutral delay systems is investigated by using the impulse‐time‐dependent Lyapunov functions. The construction of impulse‐time‐dependent Lyapunov functions is based on the singular system representation of the impulsively controlled neutral delay system. In order to overcome the difficulty arising from time‐varying state delays, a Razumikhin‐type analysis method is developed and the key technique is to establish the relation between the slow and fast state variables by using a difference inequality. The derived stability criterion weakens the restriction condition on the impulse operators found in the previous result. For constant delay case, an improved stability criterion is derived by applying an impulse‐time‐dependent Lyapunov functional. Next, based on the newly established stability criteria, novel sufficient conditions on the existence of impulsive controllers with range dwell‐time constraint are presented. Finally, two numerical examples are given to demonstrate the effectiveness of the proposed results.

Stabilization of nonlinear time-delay systems: Distributed-delay dependent impulsive control

This technical note studies impulsive stabilization of general nonlinear systems with time-delay. Distributed time-delay is considered in the proposed nonlinear impulsive controller. Using Lyapunov–Razumikhin method, an exponential stability criterion is constructed, which is then applied to investigate stabilization of a linear time-delay system under linear distributed-delay dependent impulsive control. Sufficient conditions on the system parameters, impulsive control gains, impulsive instants and distributed delays are obtained in the form of an inequality for global exponential stability. In these results, it is shown that an unstable time-delay system can be successfully stabilized by distributed-delay dependent impulses. It is worth noting that the proposed impulsive controllers are independent of the system states at each impulsive instant, and the states with distributed delays play the key role in the stabilization process. A numerical example is provided to demonstrate the efficiency of the main results.

Stabilization of Uncertain Fractional-Order Complex Switched Networks via Impulsive Control and Its Application to Blind Source Separation

This paper investigates the impulsive stabilization of fractional-order complex switched networks with parametric uncertainty. Using a fractional-order Lyapunov method and matrix inequality techniques, the dynamical characteristics of the controlled impulsive system are well captured, and a novel impulsive stabilizing criterion is derived in terms of algebraic conditions. The stabilization criterion is dependent on system parameters and on the lengths of impulsive intervals. In addition, a simulation example is given to demonstrate the effectiveness of the newly obtained results. Finally, an application of the obtained control pulse is also presented in the blind source separation.

Exponential Stability and Delayed Impulsive Stabilization of Hybrid Impulsive Stochastic Functional Differential Systems

AbstractThis paper is concerned with the stability and impulsive stabilization of hybrid impulsive stochastic functional differential systems with delayed impulses. Using the Razumikhin techniques and Lyapunov functions, some sufficient conditions for the pth moment exponential stability of the systems under consideration are established. Based on the derived stability results, impulsive controllers are designed to stabilize a given unstable linear or nonlinear hybrid stochastic delayed differential system. Different from the existing stability and impulsive stabilization results in the literature, the results obtained in this paper shown that the delayed part of impulses can make a contribution to the stability of systems. Three examples are provided to present the effectiveness and advantages of the proposed results.

Stability of nonlinear systems with variable-time impulses: B-equivalence method

This paper addresses the stability problem of nonlinear systems with variable-time impulses. By B-equivalence method, we shall show that under the well-selected conditions each solution of the considered systems will intersect each surface of discontinuity exactly once, and that the considered systems can be reduced to the fixed-time impulsive ones, which can be regarded as the comparison systems of the considered variable-time impulsive systems. Based on the stability theory of fixed-time impulsive systems, we propose a set of stability criteria for the variable-time impulsive systems. The theoretical results are illustrated by impulsive stabilization of Chua circuit.

Impulsive stabilization of fractional differential systems

This paper investigates the impulsive stabilization problem of fractional differential systems (FDSs in short). Both the global exponential stability and ultimate boundedness criteria are established using Lyapunov functions, algebraic inequality techniques and boundedness of Mittag-Leffler functions. It is shown that unstable and unbounded FDSs can be stable and bounded respectively under impulsive control. Examples and simulations are also provided to demonstrate the effectiveness of the derived theoretical results.

Impulsive stabilization of a class of singular systems with time-delays

This paper deals with the impulsive stabilization problem for a class of linear singular systems with time-delays. The stabilization is achieved by only exerting impulsive action on the slow state variables. Two novel Lyapunov methods are presented to determine exponential stability of the impulsively controlled systems. For the case where the time-delay is unknown and may be time-varying, a Lyapunov–Razumikhin method is developed, in which the Razumikhin condition is constructed by exploiting the relation among the fast state variables, the slow state variables, and their initial values. For the case where the delay derivative is strictly less than 1, a descriptor type of impulse-time-dependent Lyapunov functional is introduced, which is discontinuous at impulse times but does not grow along the state trajectories by construction. By using a convex technique, the stability criteria are expressed in terms of linear matrix inequalities (LMIs). Then, the impulsive controllers can be designed in the framework of LMIs. The effectiveness and advantages of the proposed methods are confirmed through simulation results.

Impulsive stabilization of chaos in fractional-order systems

This paper considers a class of nonlinear impulsive Caputo differential equations of fractional order, which models chaotic systems. Computer-assisted proof of chaos suppression by stabilizing the unstable system equilibria is provided. A nonexistence result of periodic solutions is presented, and the commensurate fractional-order Lorenz system is simulated for illustration.

A Memristor-Based Lorenz Circuit and Its Stabilization via Variable-Time Impulsive Control

In this paper, an off-the-shelf memristor emulator of the quadratic memristor is developed and applied to a Lorenz circuit. The impulsive stabilization of the chaotic system by variable moments of impulses is investigated. By B-equivalence method, the considered variable-time impulsive system can be reduced to the fixed-time impulsive one. The simulations of the chaotic behavior verify the effectiveness of the emulator-based memristive system. The stability simulations support the validity of the method.

Impulsive synchronization of drive-response chaotic delayed neural networks

In this paper, we investigate synchronization of drive-response neural networks with time-varying delays via impulsive control. Based on impulsive stability theory, we design proper impulsive controllers and derive some sufficient conditions for achieving synchronization. Noticeably, we adopt adaptive strategy to design unified controllers for different neural networks and relax the restrictions on the impulsive interval. All the obtained results are verified by several numerical examples.

Impulsive stabilization and synchronization of uncertain financial hyperchaotic systems

Impulsive stabilization and synchronization of uncertain financial hyperchaotic systems

Impulsive stabilization and synchronization of Hopfield-type neural networks with impulse time window

This paper studies the problem of global exponential stabilization and synchronization for impulsive Hopfield-type neural networks with impulse time window. By using the stability theory of impulsive dynamical systems, some sufficient conditions guaranteeing the global exponential stabilization and synchronization of Hopfield-type NNs are derived. The main innovation embodies that the impulsive instants are no longer limited at fixed instants, but suggested to be at some certain time intervals, named by impulse time windows. We shall show that impulses occurring randomly in impulse time windows can still stabilize and/or synchronize the considered neural networks under certain suitable assumptions. Two numerical examples are also given to illustrate the effectiveness of theoretical results.

Stability of uncertain impulsive complex-variable chaotic systems with time-varying delays

In this paper, the robust exponential stabilization of uncertain impulsive complex-variable chaotic delayed systems is considered with parameters perturbation and delayed impulses. It is assumed that the considered complex-variable chaotic systems have bounded parametric uncertainties together with the state variables on the impulses related to the time-varying delays. Based on the theories of adaptive control and impulsive control, some less conservative and easily verified stability criteria are established for a class of complex-variable chaotic delayed systems with delayed impulses. Some numerical simulations are given to validate the effectiveness of the proposed criteria of impulsive stabilization for uncertain complex-variable chaotic delayed systems.

Complex hybrid synchronization of complex-variable dynamical network via impulsive control

In this paper, complex hybrid synchronization of network coupled with complex-variable chaotic systems is investigated. The complex-variable dynamical network is synchronized onto a given orbit with respect to a complex matrix using impulsive control. Adaptive strategy is adopted to design adaptive impulsive controllers, which is universal for different dynamical networks. Further, it can relax the restriction on impulsive intervals. Based on impulsive stability theory and Lyapunov function method, several synchronization criteria for achieving complex hybrid synchronization are derived. Numerical examples are provided to show the effectiveness of the theoretical results.

Impulsive Stabilization and Impulsive Synchronization of Discrete-Time Delayed Neural Networks

This paper investigates the problems of impulsive stabilization and impulsive synchronization of discrete-time delayed neural networks (DDNNs). Two types of DDNNs with stabilizing impulses are studied. By introducing the time-varying Lyapunov functional to capture the dynamical characteristics of discrete-time impulsive delayed neural networks (DIDNNs) and by using a convex combination technique, new exponential stability criteria are derived in terms of linear matrix inequalities. The stability criteria for DIDNNs are independent of the size of time delay but rely on the lengths of impulsive intervals. With the newly obtained stability results, sufficient conditions on the existence of linear-state feedback impulsive controllers are derived. Moreover, a novel impulsive synchronization scheme for two identical DDNNs is proposed. The novel impulsive synchronization scheme allows synchronizing two identical DDNNs with unknown delays. Simulation results are given to validate the effectiveness of the proposed criteria of impulsive stabilization and impulsive synchronization of DDNNs. Finally, an application of the obtained impulsive synchronization result for two identical chaotic DDNNs to a secure communication scheme is presented.

Globally exponential stabilization of neural networks with mixed time delays via impulsive control

The impulsive stabilization problem of neural networks with discrete time-varying delays and unbounded continuously distributed delays is considered. By using impulse-time-dependent Lyapunov function-based techniques to capture the hybrid structure characteristics of the considered impulsive neural networks, two novel global exponential stability criteria are obtained in terms of linear matrix inequalities, which are capable of dealing with the case where both the continuous and discrete dynamics are unstable. When the original continuous-time delayed neural networks are not stable, sufficient conditions are developed for the design of exponentially stable linear impulsive state feedback controllers. Four numerical examples are given to illustrate the less conservatism and effectiveness of the proposed results.

Pinning impulsive directed coupled delayed dynamical network and its applications

The main objective of the present paper is to further investigate pinning synchronisation of a complex delayed dynamical network with directionally coupling by a single impulsive controller. By developing the analysis procedure of pinning impulsive stability for undirected coupled dynamical network previously, some simple yet general criteria of pinning impulsive synchronisation for such directed coupled network are derived analytically. It is shown that a single impulsive controller can always pin a given directed coupled network to a desired homogenous solution, including an equilibrium point, a periodic orbit, or a chaotic orbit. Subsequently, the theoretical results are illustrated by a directed small-world complex network which is a cellular neural network (CNN) and a directed scale-free complex network with the well-known Hodgkin–Huxley neuron oscillators. Numerical simulations are finally given to demonstrate the effectiveness of the proposed control methodology.

Impulsive stabilization of a class of nonlinear system with bounded gain error

Considering mechanical limitation or device restriction in practical application, this paper investigates impulsive stabilization of nonlinear systems with impulsive gain error. Compared with the existing impulsive analytical approaches, the proposed impulsive control method is more practically applicable, which includes control gain error with an acceptable boundary. A sufficient criterion for global exponential stability of an impulsive control system is derived, which relaxes the condition for precise impulsive gain efficiently. The effectiveness of the proposed method is confirmed by theoretical analysis and numerical simulation based on Chua's circuit.

Second-order cluster consensus of multi-agent dynamical systems with impulsive effects

This paper discuss the cluster consensus of multi-agent dynamical systems (MADSs) with impulsive effects and coupling delays. Some sufficient conditions that guarantee cluster consensus in MADS are derived. In each cluster, agents update their position and velocity states according to a leader’s instantaneous information, and interactions among agents are uncertain. Furthermore, switching topology problem in MADS is considered by impulsive stability and adaptive strategy. Finally, numerical simulations are given to verify our theoretical analysis.

Structure identification of an uncertain network coupled with complex-variable chaotic systems via adaptive impulsive control

In this paper, structure identification of an uncertain network coupled with complex-variable chaotic systems is investigated. Both the topological structure and the system parameters can be unknown and need to be identified. Based on impulsive stability theory and the Lyapunov function method, an impulsive control scheme combined with an adaptive strategy is adopted to design effective and universal network estimators. The restriction on the impulsive interval is relaxed by adopting an adaptive strategy. Further, the proposed method can monitor the online switching topology effectively. Several numerical simulations are provided to illustrate the effectiveness of the theoretical results.