线性时变系统有限时间稳定性分析

Finite time stability analysis of the linear time-varying systems

针对一般非线性时变系统的有限时间稳定性分析问题,考虑到系统初始时刻在有限时间区间内的变化,本文分别提出了一般非线性时变系统的一致有限时间稳定性,一致收缩稳定性和固定调节时间一致收缩稳定性定义.针对一类线性时变系统,基于计算系统的所有轨线的包线的思想,本文分别给出了判定该类系统的收缩稳定性、固定调节时间收缩稳定性、一致有限时间稳定性、一致收缩稳定性、固定调节时间一致收缩稳定性的充分必要条件,同时给出了判定该类系统一致收缩稳定性及固定调节时间一致收缩稳定性的3个充分条件.进一步,本文将所得定理结果推广到了周期线性时变系统,所得结论为判定周期线性时变系统关于任意初始时刻的一致有限时间稳定性,一致收缩稳定性及固定调节时间一致收缩稳定性提供了理论依据.最后,以4个数值算例和两航天器相对运动过程为例验证了本文结果的正确性.

Considering the problem of the finite time stability analysis of the general nonlinear time-varying system with its initial time varying in a finite interval of time, the paper proposes the concepts of uniform finite time stability, uniform contractive stability and uniform contractive stability with fixed settling time, respectively. For a class of linear time-vaxying （LTV） systems, by computing the envelope of all of its trajectories, the corresponding sufficient and necessary decision theorems for its contractive stability, contractive stability with fixed settling time, uniform finite time stability, uniform contractive stability and uniform contractive stability with fixed settling time are derived respectively. Moreover, three sufficient decision theorems for the uniform contractive stability and uniform contractive stability with fixed settling time of the LTV systems are also proposed. Fnrther, the obtained theorems axe generalized to the periodic LTV system, and the generalized results provide theoretical basis for deciding the uniform finite time stability, uniform contractive stability and uniform contractive stability with fixed settling time of the periodic LTV system with respect to arbitrary initial time. Finally, four numerical examples and an example about relative motion process of two spacecraft axe given to verify the correctness of the obtained results.