Stability Analysis Of Time-delay Hybrid Systems Under Average Framework

Hybrid system is a class of unified dynamical systems formed by the interaction of a continuous variable dynamical system and a discrete variable dynamical system.Hybrid systems are widely used in many fields,such as aerospace technology,control engineering,and biological science,and their theoretical research is of great significance.The impul-sive systems and switched systems are two typical hybrid systems.Meanwhile,time delay is inevitable in practical engineering applications,it has a certain impact on the stability of the system.Delayed systems have received a lot of attention.This paper mainly stud-ies the stability analysis of time-delay systems with delayed impulses effect and switching systems under average framework.It mainly uses Lyapunov function and average dwell time method to obtain the sufficiency criterion for system stability.The main research contents are as follows.Firstly,the exponential stability problem of switched systems with hybrid delayed impulses is investigated,where the hybrid delayed impulses depend on both current states and historical states.Based on Lyapunov function,Razumikhin technique and average dwell time method,the adequacy criteria for exponential stability of switched system are obtained.In particular,the potential stabilizing and destabilizing effects of time delays in impulses are investigated.In other words,the time delays may destabilize an origi-nally stable system or,conversely,stabilize an originally unstable system.Interestingly,under certain conditions,such delayed impulses cannot destroy the stability of the whole system for all impulsive time sequences with arbitrarily bounded impulsive intervals,and the sizes of the delays in the impulses can theoretically be arbitrarily finite.Secondly,the stability of a class of mixed delayed systems with delayed impulse effect is studied.The time delay considered here is the mixed delay including both constant delay and distributed delay.Based on the Lyapunov function method,the ideas of aver-age impulsive interval and average impulsive delay,and the proposed concepts of average positive impulsive estimation and average impulsive estimation,some sufficient criteria for the stability of nonlinear impulsive systems are established from the perspectives of impulsive perturbation and impulsive control,respectively.The concept of average impul-sive estimation drops the restriction on the common threshold of impulsive estimation at every impulsive point.Here the impulsive estimation can be time varying.Especially,the derived conditions do not impose any restriction on the magnitude relationship between the delay in continuous flow and impulsive delay in the case of impulsive perturbation.It also shows that the delay in continuous flow might have a potential effect on system stability.It also shows that the delay in continuous flow might have a potential effect on system stability.The results of stability analysis are applied to the synchronization of complex networks with mixed delays and impulses.Thirdly,the stability of a class of nonlinear systems under flexible delay pulse con-trol is studied.A sufficient condition for the global exponential stability of the system is obtained based on the Lyapunov-Razumikhin approach.Utilizing the proposed method of average impulsive estimation,the rate coefficients are flexible,and the impulsive delay can be integrated to guarantee the effect of stabilization of impulses.It is shown that the size of delay in continuous dynamics can be flexible.Specially,it can be smaller or larger than the impulsive intervals,and there is no magnitude relationship between the delay in continuous flow and impulsive delay.As an application,the theoretical results are applied to the synchronization of a chaotic neural network,and the impulsive control input is formalized in terms of linear matrix inequalities.Finally,the stability and l₂/L₂-gain analysis for asynchronously switched system-s in both discrete-time and continuous-time contexts are studied with considering the asynchronous switching between the subsystems and controllers.Based on the con-structed Lyapunov-like functions with average dwell time method and inequality analysis technique,the stability of switched systems without disturbance and a non-weighted l₂/L₂-gain under external disturbance can be guaranteed.The constructed piecewise Lyapunov-like functions are dependent on controller mode rather than system mode and are continuous at the switching instants of subsystems and discontinuous when the modes are matched.This is in keeping with the key feature of asynchronous switching behavior.