On exosubsets and internal models

Discrete-time dynamical systems defined on input, state, and output sets which do not admit binary operations have recently been studied with regard to the regulator problem of feedback control theory. If it is assumed that the regulation condition places no constraint on control states, that the exosystem can be observed from the controller, that all exosystem states are effective in generating output requests, and that feedback is inoperative when the regulation condition is satisfied, then it is known that the controller contains an algebraic model of the exosystem, provided that a suitable exosubset of regulator states exists to represent the action of the exosystem within the context of the overall regulator. This paper examines the possibility of directly constructing the exosubset from elementary dynamical descriptions of plant, exosystem, and controller, with the aid of quotient set representation of exosystem states. The construction indicates that exosubset existence is a non-trivial question, and brings out the importance of product versus coproduct state descriptions for the regulator.