Search

This technical note addresses the problem of transforming a single-input–single-output discrete-time system into the extended observer form, which comprise a linear time-invariant observable component, and a nonlinear injection term, which depends on the input, output, and their forward shifts up to a finite order. Intrinsic necessary and sufficient conditions are provided for obtaining the extended observer form via a parametrized state transformation. The conditions are formulated directly in terms of the state equations and do not rely on input–output equations as in the earlier papers. Further, an algorithm for obtaining the required transformation is presented. Unlike the existing results on observer forms, the results are not restricted to reversible systems but to more general submersive systems, i.e., to systems, which are reversible via static state feedback.

The problem of finding a minimal realization of a single-input–single-output nonlinear retarded time-delay input–output equation with constant commensurable delays is addressed. First, an algorithm is given to find an observable realization of dimension $n$ of a $n$th order input–output equation. Then, this algorithm is modified under two technical assumptions to find a realization of lower dimension, which is observable and accessible. It is shown that under the same technical assumptions an observable and accessible realization has minimal possible state-space dimension among all possible realizations. An algebraic approach based on polynomial tools and modules of differential 1-forms is used to derive the results.