Finite-time Lyapunov 函数和耦合系统的稳定性分析

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.

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。

生物、物理、机械、自动化、化学等领域的许多现象可以用动力系统来描述。这些现象的稳定性分析在各自领域是非常重要的研究课题。Lyapunov 函数对于动力系统的稳定性分析具有重要的作用。对于所考虑的动力系统特别是非线性动力系统,我们可以应用 Lyapunov 第二方法分析其稳定性并估计其吸引域。该方法的优点是无需计算描述所考虑系统的微分或差分方程的解。因此，如何构造和计算 Lyapunov 函数成了稳定性分析的研究重点。学者们提出了许多Lyapunov 函数的计算方法，比如 Zubov 方法、SOS (Semidefinite optimization for sum of squares polynomials)方法、CPA (Continuous and piecewise affine (CPA) method)方法、 分配法等。在此基础上一些学者研究条件件减弱的 Lyapunov 函数，并得到有关稳定性的相关结论。根据已有的参考文献，我们研究了时变和时不变微分系统的 Finite-time Lyapunov 函数，根据其结果研究了脉冲微分微分系统的控制稳定性问题；并研究了带有输入的时变和时不变微分系统的 Finite-time ISS Lyapunov 函数及其计算方法；在此基础上，将提出非保守小收益定理，并利用其分析耦合系统的稳定性与估计耦合系统的吸引域。关于该课题的研究，我们已取得了一定的成果，在国际SCI期刊上，项目负责人已发表四篇论文。该课题的结论一方面将丰富 Finite-time Lyapunov 函数、非保守小收益定理、耦合系统稳定性分析的研究内容；另一方面将为实际现象的稳定性分析比如生物系统的稳定性分析、优化控制领域的控制 Lyapunov 函数的设计等提供一定的理论依据和分析方法。