[论文解读] LaSalle Invariance Principle for Discrete-time Dynamical Systems: A Concise and Self-contained Tutorial

Wenjun Mei, Francesco Bullo|arXiv (Cornell University)|Oct 10, 2017

Distributed Control Multi-Agent Systems参考文献 14被引用 35

一句话总结

本文为离散时间非线性动力系统的LaSalle不变性原理提供了一套自包含的教程，严格证明了LaSalle原始著作中对建立渐近稳定性至关重要的引理。通过使用李雅普诺夫函数和不变集，本文建立了差分方程的该原理，表明即使在有限时间后李雅普诺夫函数在定义域边界未定义，轨迹仍会收敛到李雅普诺夫函数不递减的最大的不变集。

ABSTRACT

LaSalle invariance principle was originally proposed in the 1950's and has become a fundamental mathematical tool in the area of dynamical systems and control. In both theoretical research and engineering practice, discrete-time dynamical systems have been at least as extensively studied as continuous-time systems. For example, model predictive control is typically studied in discrete-time via Lyapunov methods. However, there is a peculiar absence in the standard literature of standard treatments of Lyapunov functions and LaSalle invariance principle for discrete-time nonlinear systems. Most of the textbooks on nonlinear dynamical systems focus only on continuous-time systems. In Chapter 1 of the book by LaSalle [11], the author establishes the LaSalle invariance principle for difference equation systems. However, all the useful lemmas in [11] are given in the form of exercises with no proof provided. In this document, we provide the proofs of all the lemmas proposed in [11] that are needed to derive the main theorem on the LaSalle invariance principle for discrete-time dynamical systems. We organize all the materials in a self-contained manner. We first introduce some basic concepts and definitions in Section 1, such as dynamical systems, invariant sets, and limit sets. In Section 2 we present and prove some useful lemmas on the properties of invariant sets and limit sets. Finally, we establish the original LaSalle invariance principle for discrete-time dynamical systems and a simple extension in Section~3. In Section 4, we provide some references on extensions of LaSalle invariance principles for further reading. This document is intended for educational and tutorial purposes and contains lemmas that might be useful as a reference for researchers.

研究动机与目标

• 为标准教科书中缺乏对离散时间非线性系统的LaSalle不变性原理的全面处理提供补足。
• 提供LaSalle原始著作（JPL:76）中所有推导主要定理所必需的引理的完整、自包含证明。
• 以严谨且易于理解的方式，为研究人员和学生建立离散时间LaSalle不变性原理。
• 将经典原理扩展到李雅普诺夫函数和系统映射在定义域边界未定义，但解在有限时间后仍保持在紧子集内的情况。

提出的方法

• 使用连续映射T和迭代x(n+1) = T(x(n))的形式化定义离散半动力系统。
• 引入关键概念：极限集、不变集及其使用闭包和序列收敛的拓扑性质。
• 证明在连续映射和有界轨道下，极限集的紧致性和不变性相关引理。
• 推导主要LaSalle不变性原理：若李雅普诺夫函数V沿轨迹非增且系统有界，则轨迹收敛到V的零差集内最大的不变集。
• 将该原理扩展至T和V在集合G的边界未定义的情况，只要解在有限时间N后保持在紧子集Gc内。
• 利用序列紧致性和连续性证明极限集非空、紧致且不变，从而支持收敛性分析。

实验结果

研究问题

• RQ1 鉴于标准文献中缺乏完整证明，如何严谨地建立离散时间非线性系统的LaSalle不变性原理？
• RQ2 基于李雅普诺夫函数的递减性，离散时间系统轨迹收敛到不变集的必要与充分条件是什么？
• RQ3 经典LaSalle原理能否扩展到李雅普诺夫函数和动力系统在定义域边界未定义的情况？
• RQ4 极限集和不变集的哪些拓扑性质可确保离散时间系统中的收敛性？
• RQ5 映射T和李雅普诺夫函数V的紧致性与连续性属性如何保证渐近稳定性？

主要发现

• 任意有界轨迹的极限集Ω(x₀)非空、紧致，且在映射T下不变。
• 若李雅普诺夫函数V沿轨迹非增且系统有界，则轨迹收敛到V(T(x)) - V(x) = 0集合内最大的不变集M。
• 即使T和V在G的边界未定义，只要解在有限时间N后保持在紧集Gc内，扩展的LaSalle原理依然成立。
• 对于某个c ∈ ℝ，x(n)收敛到M ∩ V⁻¹(c)是确定的，其中M是V的零差集内最大的不变集。
• 证明依赖于序列紧致性和连续性，表明轨迹的极限点同时属于V⁻¹(c)和不变集M。
• 本文通过提供离散时间LaSalle不变性原理所需的所有引理的完整、自包含证明，填补了文献中的关键空白。