Controlled Invariant Hypersurfaces of Polynomial Control Systems

Abstract

We study input-affine control systems with polynomial nonlinearity. A variety V is said to be controlled invariant if there exists a feedback law of polynomial type that causes the closed loop system to have V as an invariant variety. Using the theory of Grobner bases, we show how to constructively decide whether a given variety is controlled invariant for a given system, and if so, how to determine all feedback laws achieving the task. We also describe a set of “trivial” vector fields for which V is invariant. If V is a smooth hypersurface, then V is only invariant for its trivial vector fields. We discuss conditions under which the converse is also true.