Finite Time Stability Analysis Of Linear Time - Delay Systems

System stability problem is one of the most basic and important problems of control theory research. Finite time stability(FTS) divides into three elements: a specific time interval, the boundedness of the initial condition and the boundedness of the system state.So in order to judge whether system is finite time stability, first of all, we should according to the actual requirement to give a period of time interval, the boundedness of the initial conditions and the boundedness of the system state. Then we should inspect the system state in the time interval whether in the boundaries of given in the advance. It should be noted that, finite time stabilization and asymptotic stability are two different concepts, a finite-time stable system may not satisfy the asymptotic stability condition, on the other hand, a asymptotic stable system may not satisfy the finite-time stability condition either. In this paper, we combine with the method of looking for a Lyapunov Function with differential item to research finite time stability problems of linear time delay system. The paper involves a line of several aspects: linear time delay system, linear time varying delay system,finite time boundedness, input-output finite time stability. Main works of this paper are as follows:First, we research on the input-output finite time stabilization of a class of linear time delay system, by constructing Lyapunov function containing differential, we eliminate the influence of time delay phenomenon on the system stability and then according Linear Matrix Inequality(LMIs)to turn the system stability problem into Linear Matrix Inequality problem, next we through the scaling of inequality, acquire the sufficient conditions for the finite time boundedness and the input-output finite-time stabilization of linear Time-delay system. It should be pointed out that FTB consider the system state in a finite time inerrval while IO-FTS only consider the output in a finite time interval. However, in some control research, system state boundedness and output boundenness in a finite time interval are both concerned, Hence, by combining FTB and IO- FTS,the behaviors of system state and output can be considered comprehensively. Finally design state feedback controller and prove the condition is correct by simulation results.Second, on the basis of the linear time delay system, we promote the research of the inputoutput finite time stability to the linear time-varying delay system, also use the same method of constructing Lyapunov function containing differential,then using linear matrix inequality make the linear time-varying delay system steer stable in finite time and design state feedbacke controller, the simulation results are presented showing correct of the proposed methods.