Topological equivalence canonical forms for linear multivariable systems without control

TL;DR

This paper investigates the classification of uncontrolled linear multivariable systems, revealing invariants like observability and stability under topological equivalence, and provides explicit canonical forms for specific cases.

Contribution

It introduces a new classification framework for uncontrolled systems based on topological equivalence, including explicit canonical forms and invariance properties.

Findings

Observability and stability are invariant under topological equivalence.

Explicit canonical forms are derived for three-dimensional systems with scalar observation.

System decomposition based on eigenvalues and observability is established.

Abstract

In this paper, we discuss the classification problem for linear time-invariant multivariable systems without control. It turns out that the observability and stability are invariant for topological equivalent systems. Abstract results concerning system decomposition according to eigenvalues and observability are obtained. Finally, as concrete examples, the topological equivalence canonical forms for a three dimensional system equipped with a scalar observation are presented explicitly.