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光滑的非线性系统 x=F(x、u)如果能通过状态变量变换和非线性状态反馈化成能控线性系统,那么找出这样的变换和其线性化系统是很重要的。本文给出一个计算方法,如果系统是能线性化的,则它给出化到线性系统所需的变换;如果系统不能线性化,这个变换将分出其最大的能控线性部分。方法的最重要部分是一连串逐次降一阶的非线性微分方程组的求解,这正是标题中“逐次降价”的意义。这个方法从形式到思想都和定常线性系统化横山标准形的方法一致,因非线性而引起的复杂性只是通过微分方程组的解来构造状态座标变换,以代替原来的线性消去。因我们认为在和线性系统的比较中研究非线性系统是了解非线性本质的一个重要途径,于是本文所提出的方法有进一步理论研究的意义。本文是在秦化淑,程代展二位老师的鼓励和指导下做出的。为了赶投稿日期本文尚有一个例子和一个附录(文中几个引理的证明)未完成,这大概有7、8页的篇幅。
Smoothing Nonlinear Systems x = F (x, u) It is important to find such a transformation and its linearization system if it can be transformed into an energy-controlled linear system through state variable transformation and nonlinear state feedback. This paper presents a computational method that gives the necessary transformations to a linear system if the system is linearizable; if the system can not be linearized, the transformation separates its largest controllable linear part. The most important part of the method is the solution of a series of non-linear differential equations one step after the other, which is the meaning of “successive price cuts” in the title. This method, from form to thought, is consistent with the method of linearizing systematically normalized Yokoyama standard shape. The complexity caused by non-linearity is that the state coordinate transformation is constructed by the solution of differential equations instead of the original linear elimination. Because we think that studying nonlinear systems in comparison with linear systems is an important way to understand the nature of nonlinearity, the method proposed in this paper has the meaning of further theoretical research. This article is made under the encouragement and guidance of two teachers of Qin Hua Shu and Cheng Dai Zhan. There is an example and an appendix (a few proofs of lemma in this article) that have not been completed in order to catch the submission date. This is about 7,8 pages.