Necessary and sufficient conditions for relative controllability of discrete linear delay systems

TL;DR

This paper establishes fundamental controllability conditions for discrete linear delay systems, including a Kalman-type criterion and Gramian matrix analysis, supported by numerical validation.

Contribution

It introduces necessary and sufficient conditions for relative controllability of discrete linear delay systems, extending classical control theory to delayed systems.

Findings

Kalman-type rank condition for delayed systems

Non-singularity of the discrete Gramian matrix

Control function formulation validated by numerical examples

Abstract

In this paper, we investigate delayed linear difference systems and establish several fundamental results. We first provide a Kalman-type rank condition tailored for delayed linear difference systems. Furthermore, we construct the discrete Gramian matrix and prove its non-singularity, which is essential for analyzing system properties. Additionally, we obtain necessary and sufficient conditions ensuring the relative controllability of the system. Finally, we formulate the control function for effective system control and validate our theoretical findings with numerical examples. These examples illustrate the practical behavior of discrete linear delay systems.