Singularly perturbed zero dynamics of nonlinear systems

Stability properties of zero dynamics are among the crucial input-output properties of both linear and nonlinear systems. Unstable, or nonminimum phase, zero dynamics are a major obstacle to input-output linearization and high-gain designs. An analysis of the effects of regular perturbations in system equations on zero dynamics shows that whenever a perturbation decreases the system´s relative degree, it manifests itself as a singular perturbation of zero dynamics. Conditions are given under which the zero dynamics evolve in two time scales characteristic of a standard singular perturbation form that allows a separate analysis of slow and fast parts of the zero dynamics. The slow part is shown to be identical to the zero dynamics of the unperturbed system, while the fast part, represented by the so-called boundary layer system, describes the effects of perturbation