Bundle Reduction and the Alignment Distance on Spaces of State-Space LTI Systems

Abstract

This paper introduces a large class of differential-geometric distances between finite-dimensional linear dynamical systems, collectively called the alignment distance. Contrary to the existing distances, the alignment distance is based on the state-space description of dynamical systems, and is defined on the manifolds of systems of fixed order and fixed input–output dimension under a matrix rank constraint (e.g., minimality, controllability, or observability). While the quotient topology and principal fiber bundle structure associated with such manifolds have been known since the early days of modern control theory, distances natural to this structure have not been studied. The starting point for defining such a distance is to identify a linear system of order $n$ with its equivalence class of state-space realizations, all related by the so-called similarity action, i.e., state-space change of basis under $GL(n)$ , the Lie group of nonsingular $n \times n$ matrices. The main idea of the alignment distance is to first find the best state-space change of basis that brings a realization of a system “as close as possible” to a realization of another system (the alignment step), and then compare the aligned realizations. A direct implementation of this idea, due to noncompactness of $GL(n)$ , is complicated. However, using the notion of “reduction of the structure group” of a principal bundle, we show that the change of basis can be restricted to an orthogonal change of basis, provided one uses realizations in a reduced subbundle. This key observation brings about significant computational benefits. As a technical contribution (possibly of independent interest), we show that several forms of realization balancing available in the control literature have differential-geometric significance, and are, indeed, examples of reducing the structure group from $GL(n)$ to its subgroup of orthogonal matrices $O(n)$ . The alignment distance can be defined for stable and unstable systems, discrete or continuous-time, and stochastic systems.