Orders of Input/Output Differential Equations and State-Space Dimensions

Abstract

This paper deals with the orders of input/output equations satisfied by nonlinear systems. Such equations represent differential (or difference, in the discrete-time case) relations between high-order derivatives (or shifts, respectively) of input and output signals. It is shown that, under analyticity assumptions, there cannot exist equations of order less than the minimal dimension of any observable realization; this generalizes the known situation in the classical linear case. The results depend on new facts, themselves of considerable interest in control theory, regarding universal inputs for observability in the discrete case, and observation spaces in both the discrete and continuous cases. Included in the paper is also a new and simple self-contained proof of Sussmann’s universal input theorem for continuous-time analytic systems.