引用本文:王　红.非线性控制系统与状态空间的几何结构[J].控制理论与应用,2001,18(5):702~708.[点击复制]
WANG Hong.Nonlinear Control System and Geometrical Structure of State Space[J].Control Theory and Technology,2001,18(5):702~708.[点击复制]
非线性控制系统与状态空间的几何结构
Nonlinear Control System and Geometrical Structure of State Space
摘要点击 1190  全文点击 5061  投稿时间：1999-02-02  修订日期：2000-05-08
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DOI编号  10.7641/j.issn.1000-8152.2001.5.013
  2001,18(5):702-708
中文关键词  非线性系统  黎曼流形  局部坐标表示  对合分布  Kalman分解  状态反馈解耦
英文关键词  nonlinear system  Riemannian manifold  local coordinate representation  involutive distribution  Kalman decomposition  state feedback decoupling
基金项目  国家自然科学基金(19771066); 陕西省自然科学基金(97CS0101); 西北工业大学 “双新计划”资助项目.
作者单位
王　红 西北工业大学 数学与信息科学系, 西安 710072
南开大学 数学科学院, 天津 300071 
中文摘要
      首先从整体化的观点定义了一种建立在黎曼流形上的非线性控制系统, 给出了系统的状态方程在黎曼流形的局部坐标系下的表达式. 讨论了黎曼流形的几何结构对非线性系统的影响, 研究了非线性系统的能控性和能观测性. 其次, 利用对合分布与全测地子流形的性质, 给出了建立在黎曼流形上的非线性系统的局部能控结构分解, 局部能观结构分解和Kalman分解. 第三, 分别利用彼此正交的对合分布族和递增对合分布族与全测地子流形族的性质, 研究了建立在黎曼流形上的非线性控制系统平行解耦问题和级联解耦问题, 以及仿射非线性控制系统的局部干扰解耦问题.
英文摘要
      In this paper, first of all from global viewpoint we define a kind of nonlinear control systems on Riemannian manifold, and give the representation of state equation for the nonlinear systems under a local coordinate system of Riemannian manifold, show that the geometrical structure of Riemannian manifold affect on nonlinear control systems, and discuss the local controllability and observability of nonlinear system on Riemannian manifold. Second, we give the local controllability decomposition of structure, the local observability decomposition of structure and local Kalman decomposition for nonlinear control system on Riemannian manifold by using the involutive distribution and totally geodesic submanifold. Third, we study some decoupling problem of nonlinear control system on Riemannian manifold and describe respectively the parallel decomposition problem and cascade decomposition problem for nonlinear control system on Riemannian manifold in which the characters of a family of mutually orthogonal involutive distributions, a family of increasing involutive distributions and a family of totally geodesic submanifold are used. We also discuss the local disturbance decoupling problem of affine nonlinear control system on Riemannian manifold.