Dead-beat control in the behavioral approach

In this paper, the concepts of controllability and zero-controllability of a variable ${bf w}$ , appearing either in a standard or in a latent variable description (as manifest variable), are introduced and characterized. By assuming this perspective, the dead-beat control (DBC) problem is posed as the problem of designing a controller, involving both ${bf w}$ and the latent variable ${bf c}$, such that, for the resulting controlled behavior, the variable ${bf w}$ goes to zero in a finite number of steps in every trajectory. Zero-controllability of ${bf w}$ turns out to be a necessary and sufficient condition for the existence of “admissible” DBCs as well as for the existence of regular DBCs. The class of minimal DBCs, namely DBCs with the least possible number of rows, is singled-out and a parametrization of such controllers is provided. Finally, a necessary and sufficient condition for the existence of DBCs that can be implemented via a feedback law, for which ${bf w}$ is the input and the latent variable ${bf c}$ the corresponding output, is provided.