On state representations of nonlinear implicit systems

Abstract

This work considers a semi-implicit system Δ, that is, a pair (S, y), where S is an explicit system described by a state representation , where x(t) ∈ ℝ n and u(t) ∈ ℝ m , which is subject to a set of algebraic constraints y(t) = h(t, x(t), u(t)) = 0, where y(t) ∈ ℝ l . An input candidate is a set of functions v = (v 1, …, v s ), which may depend on time t, on x, and on u and its derivatives up to a finite order. The problem of finding a (local) proper state representation ż = g(t, z, v) with input v for the implicit system Δ is studied in this article. The main result shows necessary and sufficient conditions for the solution of this problem, under mild assumptions on the class of admissible state representations of Δ. These solvability conditions rely on an integrability test that is computed from the explicit system S. The approach of this article is the infinite-dimensional differential geometric setting of Fliess, Lévine, Martin, and Rouchon (1999) (‘A Lie-Bäcklund Approach to Equivalence and Flatness of Nonlinear Systems’, IEEE Transactions on Automatic Control, 44(5), (922–937)).