非线性系统精确线性化方法探究

【作者】 钟宁帆；

【导师】 邹云；

【作者基本信息】 南京理工大学 ， 控制理论与控制工程， 2004， 硕士

【摘要】 非线性系统是一种最为普遍的系统，许多非线性系统都能在数学上用非线性常微分方程加以描述，仿射非线性系统是其中的一类。针对不同类型的非线性系统已有了很多种控制方法，其中针对仿射非线性系统的非线性系统微分几何理论是一种非常系统的方法。非线性系统微分几何理论已经发展了二十多年，然而由于该理论自身的特点，在理论上的使用和实际工程应用中都有很大的问题。本文针对其中的比较典型的问题，以几何理论中的精确线性化方法为背景在如下两个主要方面进行了较为深入的研究，获得了一些有意义的进展。 (1)减弱了已有的非线性系统精确线性化的条件，由此得到了更为一般的精确线性化后的系统形式。在这部分的讨论中，本文提出了光滑向量场的线性不变集和生成集的概念，讨论了概念之间的关系。在此基础上讨论了光滑向茸场在不同坐标系下的表示，提出了在一定坐标系下具有线性化形式的光滑向量场应满足的条件。进一步，本文对单输入单输出非线性系统的精确线性化问题进行了讨论，在更弱的意义下给出了系统精确线性化的条件，并且得到更为一般的线性化后的形式。 (2)讨论了利用精确线性化方法对受外界干扰的非线性系统进行H_∞控制器设计的问题。在这里本文讨论了如何克服使用精确线性化方法所带来的鲁棒性问题。通过设计状态反馈控制器，克服了一类系统参数不确定性所带来的影响；同时，使的闭环系统对外界干扰有一定的抑制功能。更多还原

【Abstract】 In this dissertation, the problems of exact linearization techniques for affine systems are investigated. Affine systems are a class of the most important nonlinear systems, which appear in many practical fields. Nonlinearity is one of the most common properties of the system behaviors in the world. It is very complex and hence is hard to cope with. Usually, nonlinear systems can be modeled by a group of nonlinear differential equations. In mathematics, it is very difficult to find an approach available for analyzing the dynamic behaviors of all these nonlinear equations. Hence, in nonlinear analysis, different approaches are developed to study different class of nonlinear systems, which has become an inherent essence in nonlinear analysis.In recent decades, nonlinear geometric approach is proven to be one the most efficient methods for the analysis and synthesis of affine control systems in both theoretic and engineering fields. However, some serious defects also appear in the implementations of it. Hence, it is of much interest to improve the performance of the existing nonlinear geometric approach. To this end, this thesis studies the following two topics on improving the existing skills of geometric approach in exact linearization as well as H-infinity controller designs for affine systems.(i) The existing conditions for exactly-linearizing an affine system are improved, by which, a much more general representation of exact linearization of affine systems is obtained, which is of certain potential impacts in the controller design for affine systems. In order to obtain this result, some new concepts called the linear invariant set of smooth vector field, and the generation set of smooth vector are proposed for affine systems. The relationships between these two concepts and the different forms of smooth vector field in different local coordinates are discussed, based on which, some improved conditions for the exact linearizaion of SISO affine nonlinear system are obtained.(ii) A design technique of H-infinity controllers is proposed for SISO affine nonlinear system Via exact linearization method. Inherently, it is well-known that the process of exact linearization is not robust at all. It relies on the preciosity of the system models very much. To overcome this shortcoming, a nonlinear state feedback controller is designed to ensure the behaviors of close-loop system satisfy a certain H-infinity performance specification. The simulations show that the proposed controller is very effective to solve such a problem.更多还原