一类齐次离散双线性系统可控性分析

Controllability of a class of homogeneous discrete-time bilinear systems

在系统分析中,可控性是系统的一个重要特性.在工程实际操作中,往往需要对一个连续系统进行离散化处理,人们希望系统在离散化后能保留原系统的重要系统特征,比如可控性.对于线性系统,我们有成熟的判断方法.然而,对于非线性系统则无统一的判别方法.Elliott在2005年给出了一个二阶双线性系统经过离散化后,可控性发生变化的例子.它表明一个系统在离散化前后,它的可控性可能会发生改变.本文旨在给出一类二阶离散化双线性系统可控性充分条件,并和已有结果作比较,表明本文结果更具有一般性.另外,本文对于3阶及以上的这类系统可控性做出了不可控的判断.

In the system analysis, controllability is an important feature of the system. In engineering practice, it is often required to discretize a continuous system. It is hoped the system after discretization to preserve important characteristics of the original system, such as controllability. For linear systems, we have a mature judgment method. However, for nonlinear systems there is no general determination method. Elliott, in 2005, gave a second-order bilinear system for which the controllability is changed after discretization. It shows that a system before and after discretization may have different controllability. We give a sufficient condition for the controllability of a class of second-order discrete bilinear system, and prove that it is more general than other existing results. Besides, we affirm that this kind of systems with orders higher than two is not controllable.