A new approach to the zero-dynamics assignment problem for nonlinear discrete-time systems using functional equations

The present paper work aims at the development of a systematic method to arbitrarily assign the zero dynamics of a nonlinear discrete-time system by constructing the requisite synthetic output maps. The problem under consideration is motivated by the need to address the control problem of nonminimum-phase nonlinear discrete-time systems, since the latter represent a rather broad class of systems due to the well-known effect of sampling on the stability of zero-dynamics. In the proposed approach, the above objective can be attained through: a systematic computation of synthetic output maps that induce minimum-phase behavior, and the subsequent integration into the methodological framework of currently available nonminimum-phase compensation schemes that rely on output redefinition. The mathematical formulation of the zero-dynamics assignment problem is realized via a system of nonlinear functional equations (NFEs), a rather general set of necessary and sufficient conditions for solvability is derived. The solution to the above system of NFEs can be proven to be locally analytic, and this enables the development of a solution method that is easily programmable with the aid of a symbolic software package. The synthetic output maps that induce the prescribed zero dynamics for the original nonlinear discrete-time systems can be explicitly computed on the basis of the solution to the aforementioned system of NFEs.