New geometric results in eigenstructure assignment

TL;DR

This paper explores the intrinsic geometric properties of reachability and Rosenbrock matrices in LTI systems, revealing eigenvalue-independent relationships that enhance understanding of eigenstructure assignment and control design.

Contribution

It uncovers eigenvalue-independent interactions among subspace chains in geometric control, extending analysis beyond stationarity to system structural properties.

Findings

Subspace chains are linked to system structure independently of eigenvalues.

Partial chains relate to the system's ability to assign spectra without complex Jordan forms.

These geometric properties hold even before reaching stationarity.

Abstract

The focus of this paper is the connection between two foundational areas of LTI systems theory: geometric control and eigenstructure assignment. In particular, we study the properties of the null-spaces of the reachability matrix pencil and of the Rosenbrock system matrix, which have been extensively used as two computational building blocks for the calculation of pole placing state feedback matrices and pole placing friends of output-nulling subspaces. Our objective is to show that the subspaces in the chains of kernels obtained in the construction of these feedback matrices interact with each other in ways that are entirely independent from the choice of eigenvalues. So far, these chains of subspaces have only been studied in the case of stationarity. In this case, it is known that these chains converge to the classic Kalman reachable subspace for the reachability matrix pencil and to the largest reachability subspace in the case of the Rosenbrock matrix, respectively. Here we are interested in showing that even before stationarity has been reached, the partial chains are linked to structural properties of the system, and are therefore independent of the closed-loop eigenvalues that we wish to assign. We further characterize these subspaces by investigating the notion of largest subspace on which it is possible to assign the closed-loop spectrum (possibly maintaining the output at zero) without resorting to non-trivial Jordan forms.