On state representations of nonlinear implicit systems

This work considers a semi-implicit system ∆, that is, a pair (S, y), where S is a explicit system described by a state representation ẋ(t) = f(t, x(t), u(t)), where x(t) ∈ Rn and u(t) ∈ Rm, which is subject to a set of algebraic constraints y(t) = h(t, x(t), u(t)) = 0, where y(t) ∈ Rl. An input candidate is a set of functions v = (v1, . . . , vs), 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 ∆ is studied in this paper. 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 ∆. These solvability conditions rely on an integrability test that is computed from the explicit system S. The approach of this paper is the infinitedimensional differential geometric setting of Fliess et al. (1999).