Polynomial Control Systems: Invariant Sets Given By Algebraic Equations/Inequations

Consider a nonlinear input-affine control system <(x)over dot>(t) = f (x(t)) g(x(t))u(t), y(t) = h(x(t)), where f, g, h are polynomial functions. Let S be a set given by algebraic equations and inequations (in the sense of not equal). Such sets appear, for instance, in the theory of the Thomas decomposition, which is used to write a variety as a disjoint union of simpler subsets. The set S is called controlled invariant if there exists a polynomial state feedback law u(t) = alpha(x(t)) such that S is an invariant set of the closed loop system <(x)over dot> = (f + g alpha)(x). If it is possible to achieve this goal with a polynomial output feedback law u(t) = beta(y(t)), then S is called controlled and conditioned invariant. These properties are discussed and algebraically characterized, and algorithms are provided for checking them with symbolic computation methods. (C) 2017, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved.