A geometric approach to the general autonomous regulator problem in the time-delay framework

The aim of this paper is to show the applicability of geometric techniques to a regulation problem for linear, time-delay systems. Given a plant whose dynamics equations include delays and an exosystem that generates a reference signal, the problem we consider consists in finding a feedback regulator which guarantees asymptotic stability of the regulation loop and asymptotic command following of the reference signal, for any initial condition of the overall system in the presence of disturbances. By associating to the time-delay plant a corresponding abstract system with coefficients in a ring, it is possible to place our investigation in a finite dimensional algebraic context, where intuition and results obtained in the classical case, that is without delays, may be exploited. In particular, using tools and methods of the geometric approach to systems with coefficients in a ring, sufficient conditions for the solvability of the considered problem are found and a constructive procedure, which works under specific hypotheses, is given.