Generalization of the Matrix Determinant Lemma and its application to the controllability of single input control systems

TL;DR

This paper generalizes the Matrix Determinant Lemma to sums of outer products and applies it to prove that controllability in single input linear systems remains unchanged under state feedback.

Contribution

It introduces a new generalization of the Matrix Determinant Lemma and provides an alternative proof that controllability is unaffected by state feedback in linear systems.

Findings

Controllability matrices' determinants are invariant under state feedback.

Generalized Matrix Determinant Lemma for sums of outer products.

Alternative proof of controllability invariance in linear control systems.

Abstract

Linear control theory provides a rich source of inspiration and motivation for development in the matrix theory. Accordingly, in this paper, a generalization of Matrix Determinant Lemma to the finite sum of outer products of column vectors is derived and an alternative proof of one of the fundamental results in modern control theory of the linear time--invariant systems $\dot x=Ax+Bu,$ $y=Cx$ is given, namely that the state controllability is unaffected by state feedback, and even more specifically, that for the controllability matrices $\mathcal{C}$ of the single input open and closed loops the equality $\det\left(\mathcal{C}_{(A,B,C)}\right)$ $=\det\left(\mathcal{C}_{(A-BK,B,C)}\right)$ holds.