A Note on Observability Canonical Forms for Nonlinear Systems

For nonlinear systems affine in the input with state x ∈ R, input u ∈ R and output y ∈ R, it is a well-known fact that, if the function mapping (x, u, . . . , u(n−1)) into (u, . . . , u(n−1), y, . . . , y(n−1)) is an injective immersion, then the system can be locally transformed into an observability normal form with a triangular structure appropriate for a high-gain observer. In this technical note we extend this result to the case of systems not necessarily affine in the input and such that the injectivity condition holds for the function mapping (x, u, . . . , u(p−1)) into (u, . . . , u(p−1), y, . . . , y(p−1)) with p ≥ n. The forced uncertain harmonic oscillator is taken as elementary example to illustrate the theory.