非线性系统输入状态稳定性分析与设计的不定向量Lyapunov函数导数方法

In the project, we investigate the input-to-state stability of nonlinear systems and discovery new sufficient conditions for them, these conditions are more relaxed than others in that they employ a vector Lyapunov function with an indefinite rather than a negative definite one. While for a nonlinear time-delay system, we describe a vector Lyapunov-Krasovskii functional and a vector Lyapunov-Razumikhin function with an indefinite derivative that can be used to determine the input-to-state stability, respectively. First, an extended vector comparison principle for estimating the solution of a differential inequality with an upper bound that is not (semi)negative definite is presented. Then, that principle of a vector Lyapunov-Krasovskii functional and a vector Lyapunov-Razumikhin function with an indefinite derivative are employed to derive an input-to-state stability criterion. On the other hand, based on the results of the indefinite vector Lyapunov function, we introduce the notion of an indefinite control vector Lyapunov function as a generalization of control Lyapunov functions, and show that input-to-state stabilizability of a nonlinear system is equivalent to the existence of a indefinite control vector Lyapunov function. Moreover, using indefinite control vector Lyapunov functions, we construct a universal decentralized feedback control law for a decentralized nonlinear dynamical system that possesses guaranteed input-to-state stable. In the project, we describe a vector Lyapunov function with indefinite derivative to derive an input-to-state stability criterion, this greatly extends previous work in this field. It generalizes previous results and enlarge the class of systems, which possesses the input-to-state stable property. The results are very interesting and enriches the amount of tools for the analysis and design of nonlinear systems.

本项目针对非线性系统，提出不定向量Lyapunov函数导数方法，即通过一个向量Lyapunov函数，在其每一个分量具有不定导数的情况下，建立系统输入状态稳定性判别方法。针对时滞系统，通过建立一个带时滞的向量值比较引理，估计时滞系统状态，提出不定向量Lyapunov-Krasovskii泛函和Lyapunov-Razumikhin函数导数方法，建立时滞系统输入状态稳定性判别准则。针对非线性控制系统的输入状态稳定控制器设计问题，基于输入状态稳定性分析的研究成果，提出不定控制向量Lyapunov函数这一概念，并利用这个函数建立系统存在输入状态稳定控制器的充要条件。基于提出的不定控制向量Lyapunov函数，设计出输入状态稳定控制器，保证系统的输入状态稳定性。通过本项目的研究，将建立一种不定向量Lyapunov函数导数方法，进一步丰富Lyapunov理论，为非线性系统的分析和设计提供有力的新工具。

本项目针对非线性系统，提出不定向量Lyapunov函数导数方法，判定非线性系统的输入状态稳定性。针对时滞系统，通过建立一个带时滞的比较引理，估计时滞系统状态，提出不定向量Lyapunov-Krasovskii泛函和Lyapunov-Razumikhin函数导数方法，建立时滞系统输入状态稳定性判别准则。针对非线性控制系统的输入状态稳定控制器设计问题，基于输入状态稳定性分析的研究成果，提出不定控制向量Lyapunov函数这一概念，并利用这个函数建立系统存在输入状态稳定控制器的充要条件。基于提出的不定控制向量Lyapunov函数，设计出输入状态稳定控制器，保证系统的输入状态稳定性。通过本项目的研究，建立了一种不定向量Lyapunov函数导数方法，进一步丰富了Lyapunov理论，为非线性系统的分析和设计提供了有力的新工具。