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

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 .

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.