A Behavioral Approach to the Control of Discrete Linear Repetitive Processes

This paper formulates the theory of linear discrete time repetitive processes in the setting of behavioral systems theory. A behavioral, latent variable model for repetitive processes is developed and for the physically defined inputs and outputs as manifest variables, a kernel representation of their behavior is determined. Conditions for external stability and controllability of the behavior are then obtained. A sufficient condition for stabilizability is also developed for the behavior and it is shown under a mild restriction that, whenever the repetitive system is stabilizable, a regular constant output feedback stabilizing controller exists. Next a notion of eigenvalues is defined for the repetitive process under an action of a closed loop controller. It is then shown how under controllability of the original repetitive process, an arbitrary assignment of eigenvalues for the closed loop response can be achieved by a constant gain output feedback controller under the above restriction. These results on the existence of constant gain output feedback controllers are among the most striking properties enjoyed by repetitive systems, discovered in this paper. Results of this paper utilize the behavioral model of the repetitive process which is an analogue of the 1D equivalent model of the dynamics studied in earlier work on repetitive processes