Disturbance Decoupling, $(f,g)$-Invariant and Controllability Subspaces of a Class of Homogeneous Polynomial Systems

Abstract

This paper discusses control systems of the type \[\dot x = f(x) + \sum_{j = 1}^m {b_i u_i + \sum_{j = 1}^k {d_j \omega _j ,\quad x \in \mathbb{R}^n } } \] where $f(x)$ is a homogeneous polynomial vector field, $b_i $ and $d_i $ are constant vectors, and the output function is linear. It is shown that the well-known theory of disturbance decoupling of linear systems extends to this class in a very natural way. The resulting theory is far simpler than the general nonlinear theory. More important, all computations needed can be done using very simple algorithms, which require only a finite number of computations and use only methods from linear algebra.