状态仿射离散非线性系统可控性与小可控性研究

State-affine nonlinear systems form a fundamental and important class of nonlinear control systems, which involve bilinear systems and many other nonlinear systems of special forms, and can be used to approximate arbitrarily nonlinear systems from the theoretical point of view. In this proposed work, we will investigate fundamental properties of state-affine discrete-time nonlinear systems with focus on controllability and small-controllability. The motivation of the proposed work is based on the following three aspects. Firstly, nonlinear systems are ubiquitous in nature and many fields of science and engineering, and nonlinear systems theory plays an important role in modern science, which is the most important part of control theory. Secondly, controllability is one of the fundamental concepts in control subjects as well as one of the essential properties of control systems, which has played a key role in the development of control theory. Thirdly, thanks to the evolution of computer control technologies, the theory and techniques developed for discrete-time control systems are well implementable and realizable. The analysis and synthesis of discrete-time control systems has become an important part of modern control theory. The innovation of the proposed research is to construct the algebraic criteria for the determination of controllability and small-controllability of state-affine discrete-time nonlinear systems by establishing four study frameworks. Specifically, we will perform a systematic study on various notions of controllability, including single-state-controllability、neighborhood-controllability、null-controllability and near-controllability, as well as on other fundamental properties such as reachability and state feedback stabilization. Successful work in the proposed research direction will substantially advance our understanding of discrete-time nonlinear systems and foster further developments in the fields of the controllability theory and nonlinear systems theory.

状态仿射非线性系统是一类基本的、重要的、包括了双线性等特殊系统的非线性系统，从理论上可逼近任意非线性系统。本项目以状态仿射离散非线性系统为对象，针对可控性和小可控性问题进行攻关。开展此研究是基于以下三方面。1.非线性系统的重要性：非线性系统在现实世界中广泛存在，非线性系统理论在现代科学中扮演关键角色，是控制理论最重要的部分；2.可控性的重要性：可控性是控制学科基本概念和控制系统本质属性，在控制理论中发挥至关重要的作用；3.离散控制系统的重要性：计算机控制的发展使离散控制系统的重要性日益突出，其分析与综合已成为现代控制理论的重要组成部分。本项目首次建立并采用四种研究框架，从不同角度系统的研究状态仿射离散非线性系统的可控性与小可控性，给出代数判据。具体研究内容包括单状态可控性、邻域可控性、零可控性、可达性、临近可控性及状态反馈镇定等。预期研究成果将对可控性和非线性系统理论的发展起重要推动作用。

状态仿射非线性系统是一类重要的、包括了双线性系统的非线性系统，具有广泛的应用背景。本项目以状态仿射离散非线性系统为研究背景，以最具代表性的两类系统，即齐次离散双线性系统和非齐次离散双线性系统为重点研究对象，针对可控性、临近可控性和小可控性等问题进行了研究：1）发展了由项目负责人提出的一种根轨迹方法，将其应用至齐次离散双系统的可控性研究中，并提出了一种利用不变集研究可控性的方法，提供了新思路；2）给出了齐次离散双线性系统的可控性和临近可控性代数判据以及求解实现状态转移控制输入的算法，这些判据可通过代数方法验证，易于应用；3）给出了两类非齐次离散双线性系统的可控性和临近可控性代数判据，均是充要条件，并推导出了可控子空间和临近可控子空间；4）指出了状态仿射离散非线性系统的小可控性可由其近似离散双线性系统的小可控性推导出，并给出了状态仿射离散非线性系统小可控性的判据；5）给出了二维多输入离散双线性系统可控性和临近可控性的充要条件；6）将研究拓展延伸至集群系统，给出了集群可控性和弱集群可控性判据。综上所述，通过本项目的研究，获得了丰富的关于离散双线性系统和状态仿射离散非线性系统的可控性与小可控性成果，这些成果将对非线性系统可控性理论的发展起到重要推动作用。