On Homogeneous Polynomial Approximations of Nolinear Control Systems

In this paper we are going to study the chain of some distributions required by nonlinear control problems associated to homogeneous polynomial nonlinear systems. The idea is due to Ruiz and Nijmeijer and the basic facts are strongly inspiered by their paper. The distribution studied here is the controlled invariant distribution used in the problems of local disturbance decoupling and input-output decoupling of nonlinear systems. 1 Construction of the non-standard splitting of modules We shall give some details about the construction of a graduation of the Lie algebra of the homogeneous polynomial vector eld induced by a dilation. For a more extensive treatement of this subject, the reader is referred to [Bacc92] (Appendix B) or [Good76]. 1. Let feig1 i n denote the canonical basis ofR . For any n-tuple of natural numbers (w1; w2; :::; wn), ordered nondecreasing (1 w1 ::: wn def = r) we de ne a family of maps : R n 7! R by: