Finite-time Lyapunov 函数和耦合系统的稳定性分析

本课题将利用子系统的Finite-time Lyapunov函数、 Finite-time ISS Lyapunov函数和非保守小收益定理分析耦合系统的稳定性。首先分别利用Finite-time Lyapunov函数，Finite-time ISS Lyapunov函数建立微分系统渐近稳定的充分条件和Lyapunov稳定逆定理，带有输入的微分系统ISS的充分条件和ISS 逆定理。其次提出两种计算Finite-time Lyapunov函数方法: (1)把Finite-time Lyapunov函数的计算问题转化为线性或非线性规划问题, (2)设计算法验证根据状态范数函数构造的函数是所求的Finite-time Lyapunov函数。最后根据Finite-time Lyapunov函数、Finite-time ISS Lyapunov函数建立非保守小收益定理, 分析耦合系统的稳定性。

For this project, we will analyze stability of interconnected systems using Finite-time Lyapunov functions、Finite-time input-to-state stability (ISS) Lyapunov functions and non-conservative small gain theorems. We firstly propose sufficient conditions by Finite-time Lyapunov functions for asymptotic stability of continuous-time systems and Lyapunov stability converse theorems for continuous-time systems, and propose sufficient conditions for ISS of continuous time systems with inputs utilizing Finite-time ISS Lyapunov functions and ISS converse theorems for continuous time systems with inputs. Then we present two novel methods to compute Finite-time Lyapunov functions、Finite-time ISS Lyapunov functions. One way is to solve a linear optimization problem or a mixed integer optimization problem, the solution of which is a Finite-time Lyapunov function、a Finite-time ISS Lyapunov function. The other way is to design an algorithm of verifying the constructed function based on the function of the norm of the state is a Finite-time Lyapunov function、a Finite-time ISS Lyapunov function. Based on the computed Finite-time Lyapunov functions、Finite-time ISS Lyapunov functions of subsystems of interconnected systems, non-conservative small gain theorems are proposed. Furthermore, we construct Finite-time Lyapunov functions for interconnected systems based on non-conservative small gain theorems, and then analyze stability of interconnected systems.