基于隐函数方法的离散双线性系统临近可控性研究

A nonlinear system is called a bilinear one if it is linear in state and linear in control, but not jointly linear in both. This work investigates near-controllability of discrete-time bilinear systems. The purpose of this work is based on the following three aspects. Firstly, bilinear systems are a special class of nonlinear systems, which exist widely in the real world such as engineering、economic、biology, etc. Among nonlinear systems, bilinear systems are the most close to linear systems. Furthermore, bilinear systems have well approximation properties. It is in general more accurate to use a bilinear model to represent the dynamics of a nonlinear system than to use a linear model. Secondly, due to the wide applications of computer technology, the importance of discrete-time control systems is well realized nowadays. The analysis and synthesis of discrete-time control systems has become an important part of modern control theory. Thirdly, controllability is one of the fundamental concepts in control subjects and has played an essential role in the development of mathematical control theory. The notion of near-controllability was introduced recently, which not only contains the notion of controllability, but also can better characterize the properties of control systems. This work will apply an implicit function approach to investigate, and solve, the near-controllability problems of discrete-time bilinear systems. Specific contents of this work include the study on self-controllability、local controllability、connectedness、near-controllability、stabilizability, etc. Expected research results will play an important significance of promoting the development of bilinear system theory.

一非线性系统，若关于状态和控制分别、但不同时是线性的，则称为双线性系统。本课题以离散双线性系统为对象，针对临近可控性问题展开研究。开展此项研究是基于以下三方面。第一，双线性系统是一类特殊的非线性系统，广泛存在于现实世界中，如工程、经济、生物等领域，是最接近于线性系统的非线性系统。用双线性系统逼近非线性系统具有比线性逼近更准确、精确的效果。第二，随着计算机技术的普及，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。第三，可控性是控制科学的基本概念，并在数学控制理论中发挥重要作用。临近可控性则是近年新提出、包含了可控性的更广泛的概念，更全面地描述了系统的性质。本课题采用一种隐函数方法，重点研究并解决离散双线性系统的临近可控性问题，给出判据。具体研究内容包括自身可控性、局部可控性、连通性、临近可控性、镇定性等问题。预期研究成果将对双线性系统理论的发展具有重要推动意义。

双线性系统是一类重要的非线性系统，可以广泛描述现实世界中许多动态过程，从理论上可以逼近任意非线性系统，是国际上非线性系统理论的研究热点之一。临近可控性是由项目负责人近年提出的包含了可控性的更广泛的新概念，可以更全面地描述控制系统的性质。本项目是第一个为临近可控性立项的课题，在本项目中我们针对离散双线性系统的临近可控性等关键问题进行了研究：1）我们发展了由项目负责人提出的针对离散系统可控性问题的隐函数方法，并进一步完善了由项目负责人提出的一种可以求解出控制输入的根轨迹方法，为离散系统可控性的研究提供了新思路；2）我们给出了离散双线性系统的临近可控性判据，均为易检验的代数判据，同时给出了计算实现状态转移的控制输入的算法；3）我们利用临近可控性研究并解决离散双线性系统的可控性问题，揭示了临近可控性的重要性和价值；4）我们提出了临近可控子空间的概念，是对线性系统可控子空间概念的推广，对非线性系统的几何控制理论具有重要研究意义。关于离散双线性系统可控性与临近可控性的工作目前已发表论文20篇，其中包括SCI检索期刊论文11篇。综上所述，通过本项目的研究，取得了关于离散双线性系统可控性与临近可控性研究的丰硕成果，将对双线性系统理论、离散系统的分析与综合以及根轨迹理论的发展起到重要推动作用。