Accessibility and orbits for discrete-time nonlinear control systems

Abstract

The paper develops an algebraic formalism that is based on the language of ideals and modules, associated with the analytic control system given by a set of difference equations. Using this language we show how the orbits of the system can be determined by the generators of some ideal of the ring of germs of analytic functions. The ideal is invariant with respect to shift operators that are defined by the system. Since the orbits are related to the accessibility property, the conditions for accessibility of the system will be given using the germs of analytic functions and the germs of one-forms, associated with the modules in the ring of germs of analytic functions.