Invariant sets for a class of nonlinear control systems tractable by symbolic computation

Abstract

A set S is said to be controlled invariant with respect to a control system if a state feedback law exists such that the closed loop system has S as an invariant set. In the present paper we generalise results on input-affine polynomial control systems and algebraic varieties (i.e. sets described by the zeros of polynomial equations) considered in Zerz and Walcher (2012) to an extended class of vector fields. More precisely, we consider vector fields of the form f = F ° h, where F is a polynomial vector and h is a continuously differentiable function with certain (algebraic) properties, as well as sets Vh as the preimages of varieties under h. We will see that for example polynomial expressions in sine and cosine satisfy the mentioned properties. The main advantage of the considered function class is that it is accessible to symbolic computation. We give computational methods (based on the theory of Gröbner bases) to decide the controlled invariance of Vh.