Pulse Control System Stability With Pulse Time Window

The paper mainly focuses on the impulsive control system with impulse time windows.Nowadays,most of the research about impulsive systems are assumed that the occurrence of the impulses is fixed(this system is called fixed time impulsive system),or is decided by the time when the system states hit a certain hyperplane(this system is called state-dependent impulsive system).So we can say:“the occurrence of impulses if fixed or can be calculated.” We know that there is no machine that can put impulses into the system without error.Because there always exists error between the expected input time and the real input one.For example,we plan to add an input of impulse at time t,usually,we set our machine to add the impulse in an impulse time window(t-?,t + ?),where ? is a small positive number and is called time error.This situation is very common in daily life.But there are very few papers about this topic.Compared with the existing impulsive systems,the systems with impulse time windows is of practical importance.So it is very meaningful to study impulse time windowsBased on the impulsive control theorem,we mainly take account of impulse time windows.We try to find the influence of impulse time windows to the system.We call this system impulsive control system with impulse time windows.Specifically,we first fix a control period,in which we confirm the impulse time windows,and every impulse takes place within its impulse time window,and with time goes on,the stabilization of the original system is realized.Generally speaking,our work mainly focuses on the following.Firstly,we formulate the linear impulsive control systems with impulse time windows and study the asymptotic stability in different situations of the system systems.We obtain several stability criteria.Numerical examples are given to verify the effectiveness of the theoretical results.Secondly,By generalizing the traditional impulsive control theory,we then establish the comparison system of the considered system and theoretically prove that the stability of an impulsive control system with impulse time windows can be derived by the corresponding comparison system.By use of the results proposed,the chaotic Lorenz system is stabilized.Thirdly,we list a kind of periodically control system with each of its period contains multiple impulses with impulse time windows.We propose three new mathematical models of of this system,namely single impulse control systems with impulse time windows,alternate control systems with impulse time windows and sandwich control systems with impulse time windows,then by induction,we obtain periodically multiple state-jumps impulsive control systems with impulse time windows.We study the stability of them and obtain their stability criteria by LMIs.The Chua's oscillator is controlled by using three different kinds of methods,respectively.Forthly,we study the impulsive delayed linear system with impulse time window.Moreover,the impulses are delay-dependent.By constructing a Lyaponov function,we obtain some conditions for the uniformly stability of the system.An example is also given to illustrate the efficiency of the results.Lastly,we are trying to combine the T-S fuzzy model and the ideal of impulsive control with impulse time windows together,we set up a T-S fuzzy model of the memristive Chua chaotic system,then we use the periodically single impulse control with impulse time windows to control the T-S fuzzy model,by designing a proper controller,we have finally successfully controlled the system.