Research On Input-To-State Stability And Stabilization Of Hybrid Impulsive Systems

Impulsive systems are a class of discontinuous dynamical systems with instantaneous state jump and continuous-time evolution,which depict the dynamic processes affected by the sudden change of the state.Based on its characteristics,impulsive system is widely applied in various fields,such as robotics,biomedical engineering and so on.In these practical problems,the impulse usually exhibits a hybrid effect,that is,the impulse possesses both stabilizing and destabilizing behavior.Hence,it is of great significance to study the hybrid impulsive system.In addition,the real system is inevitably affected by exogenous disturbances,such as the error in measuring elements.This makes the performance analysis of the system more complicated.In view of this,Sontag proposed the concept of input-to-state stability for the first time,which quantitatively describes the influence of external input on the system,and provides the performance estimation of the system state based on the information of the external input.As a result,the input-to-state stability analysis of impulsive system has followed much interest from numerous scholars.In this paper,the input-to-state stability and input-to-state stabilization of nonlinear impulsive systems are studied respectively.Specifically,it includes the following contents:(1)Based on Lyapunov theory,the input-to-state stability and integral-inputto-state stability of nonlinear delayed impulsive systems are studied.The hybrid effect of delayed impulses is considered,the indefinite derivative of Lyapunov function is dealt with,and the restriction on delay that the intervals between any consecutive impulse instants should be greater than time delay is removed.The relationship between the continuous dynamic,the cumulative strength of hybrid impulses and the time delay existing in impulses is established,so that the inputto-state stability and integral-input-to-state stability of the considered impulsive system are guaranteed.Three numerical simulation examples exhibit the effectiveness and practicability of the theoretical results.(2)With the help of event-triggered impulsive control strategy,the input-tostate stabilization of nonlinear time-delay systems is studied.In the framework of Lyapunov function,an event-triggered mechanism including the information of the state and the external input is proposed.According to the idea of LyapunovRazumikhin method,the results with delay-independent conditions are provided to realize the input-to-state stability of the considered system and avoid the Zeno behavior,simultaneously.By using the method of linear matrix inequality,the design criteria of event-triggered mechanism and impulsive control gain are obtained for a special class of impulsive control systems.Two numerical simulation examples show the validity and practicability of the theoretical results.