Pairs of $k$-step reachability and $m$-step observability matrices

TL;DR

This paper establishes a necessary and sufficient condition for the existence of a system triple $(A,B,C)$ that simultaneously realizes given $k$-step reachability and $m$-step observability matrices, linking these concepts in control theory.

Contribution

It provides a new theoretical criterion connecting $k$-step reachability and $m$-step observability matrices for system realization.

Findings

Derived a necessary and sufficient condition for system realization.

Unified the concepts of reachability and observability matrices.

Contributed to the theoretical understanding of system controllability and observability.

Abstract

Let $V$ and $W$ be matrices of size $ n \times pk$ and $q m \times n $, respectively. A necessary and sufficient condition is given for the existence of a triple $(A,B,C)$ such that $V$ a $k$-step reachability matrix of $(A,B)$ and $W$ an $m$-step observability matrix of $(A,C)$.