时变时滞系统的L-K泛函

L-K functional for time-varying delay systems

对时变时滞系统稳定性条件的研究一直是控制理论中的热点问题。目前,还没有统一的稳定性条件。原因在于对于时滞系统而言,选取的Lyapunov-Krasovskii泛函(简称L-K泛函)不同,对泛函求导后积分项处理的方法不同,那么稳定性条件就不同。针对这一问题,研究了时变时滞系统L-K泛函的构造和对泛函求导后对积分项的处理方法。论文主要工作分2方面展开:首先,构造的L-K泛函有双重积分型、多重积分型、增广型,以及时滞分割法构造L-K泛函。在研究L-K泛函的构造时,充分考虑了区间时滞上、下界的信息,得到系统时滞依赖的稳定性条件。其次,L-K泛函求导后积分项的处理方法,通常采用自由权矩阵法,Jensen积分不等式和Wirtinger积分不等式方法。较Jensen积分不等式,Wirtinger积分不等式有更小的保守性。运用不等式的目的在于以线性矩阵不等式的形式(LMI)给出稳定性条件,这样方便利用MATLAB求解。

The study of stability condition of time-varying systems is a focus in control theory.Till now,there isn’t an identical stability condition.The reason is that for the time-delay system,if the selected Lyapunov-Krasovskii functional(L-K functional)is different and processing of the integration after derivation on functional is different,then the stability condition is also different.Consideration on the issues,the problems of the structure of L-K functional for time-varying delay system and the processing of the integral term after the functional derivation are studied in this paper.The main work is divided into two aspects:Firstly,the L-K functional is respectively constructed by double integral,multi-integral,augmented dimension and time-delay partition method.The information of the upper and lower bounds of the interval time-varying delay is fully considered in constructing of L-K functional to obtain stability condition of time-delay system.Secondly,the problem of processing the integral term after derivation on L-K functional is addressed.Generally,there are free weights matrix method,Jensen integral inequality and Wirtinger integral inequality method.Compared with Jensen integral inequality,Wirtinger integral inequality is with less conservatism.The aim of applying the inequalities is to obtain stability condition in the form of linear matrix inequality(LMI),which is feasible via MATLAB.