Analytical Volume Analysis for the Finite-time Controllable Region of the Linear Discrete-time Systems

TL;DR

This paper introduces a new theorem for analytically computing the finite-time controllability region of linear discrete-time systems, deconstructs control capability factors, and generalizes the results to various cases.

Contribution

It presents a novel analytical theorem for the controllable region volume, extracting control capability factors and extending the analysis to special cases and continuous-time systems.

Findings

New theorem for finite-time controllability zonotope

Analytical factors for control capability derived

Generalization to narrow regions, negative eigenvalues, and continuous systems

Abstract

In this paper, the works on the analytical volume analysis for the controllable regions of the linear discrete-time (LDT) systems in papers \cite{zhaomw202001} and \cite {zhaomw202004} are discussed further and a new theorem on the analytical computing for the finite-time controllability zonotope (controllable region) of LDT systems are proven. And then, three analytical factors describing the control capability of the systems are deconstructed successfully from the analytical volume expression of the controllable region. Finally, the theorem is generalized to three cases: the narrow controllable region, the matrix $A$ with $n$ negative eigenvalues, the linear continuous-time systems.