Nonminimality of the Realizations and Possessing State Matrices With Integer Elements in Linear Discrete-Time Controllers

It is known that discrete-time controllers, whose state matrices have no noninteger element, are beneficial in homomorphic-based encrypted control systems. Nevertheless, it has been recently shown that possessing state matrices with integer elements usually yields unstable discrete-time controllers. In this article, we investigate the problem from a nonminimality perspective. It is shown that nonminimal realizations, in comparison to minimal ones, can theoretically provide a wider framework to obtain controllers having state matrices with integer elements. However, in the case of dealing with bounded-input bounded-output (BIBO) stable controllers, this framework cannot preserve internal stability. But, benefiting from the introduced framework, a class of unstable controllers is introduced, which can be realized by state-space forms having state matrices with integer elements. Numerical examples are presented to verify the usefulness of the introduced framework in the realization of unstable controllers with integer state matrices.