流形上的状态反馈控制策略

State feedback control strategies on manifolds

在流形上研究非线性系统的反馈镇定问题 ,针对线性化系统存在不可控不稳定子空间和不可控中心子空间几种情形 ,提出通过构造中心流形的控制策略 ,使线性化系统变为完全可控系统 .给出的系列定理表明 :①在线性化系统完全可控条件下 ,线性多输入反馈控制足可以使非线性系统镇定于原点 ;若原点为双曲的 ,则单输入线性控制是足够的 ;②线性化系统部份可控时 ,若不可控子空间是不稳定子空间 ,则存在中心流形控制器 ,使系统在原点邻域的平衡点上变为完全可控系统 ;若不可控子空间是中心子空间 ,则既可以通过中心流形将系统反馈镇定于原点 ,又可以重新构造中心流形使系统在原点的邻域内变为完全可控系统 ;③将存在不可控单零特征根的系统镇定于原点 ,构成了控制器的设计算法 .

Feedback stabilization for nonlinear systems is researched on manifolds, which establishes a kind of nonlinear control strategies on the center manifolds to enable the linearized systems completely controllable. It addresses such cases that there exist uncontrollable and unstable subspaces or center subspaces. The developed theorems show: (a) the linear multiple inputs are adequate to control or stabilize nonlinear systems to the equilibrium if their linearized systems are completely controllable, furthermore, if the equilibriums are hyperbolic then a single input control is adequate; (b) if the linearized systems are not completely controllable and the uncontrollable subspaces are unstable subspaces,then there exists the center manifolds controller to construct a center manifold and transform the systems completely controllable at a vicinity of the equilibrium; if the linearized systems are not completely controllable and the uncontrollable subspaces are center subspaces, then the nonlinear system can be stabilized to the equilibrium through a nonlinear controller designed on its center manifolds or on the center manifolds constructed by the center manifolds controller; (c) the nonlinear control is finally realized through the stabilizer for stabilizing a reduced nonlinear system with a single zero eigenvalue.