Definition and Analytical Expression on State Observe Ability for Linear Discrete-time Systems with the Bounded Noise Energy

TL;DR

This paper systematically defines and analyzes the observe ability of linear discrete-time systems with bounded noise energy, establishing theoretical foundations, dual relations, and practical computation methods to optimize signal detection performance.

Contribution

It introduces a novel concept of observe ability for practical systems with noise, providing analytical expressions and methods for its computation and optimization.

Findings

Defined observe ability for systems with bounded noise energy.

Proved dual relation between observability and controllability ellipsoids.

Validated methods through numerical experiments.

Abstract

In this article, the definition on the observe ability and its relation to the signal detecting performance are studied systematically for the linear discrete-time(LDT) systems. Firstly, to define and analyze the observe ability for the practical systems with the measured noise, six kinds of bounded noise models are classified. For the noise energy bounded case, the observability ellipsoid and the image observability ellipsoid are defined by the state observed error and then a novel concept on the LDT systems, called as the observe ability, is proposed. Based on that, some theorems and properties about the observe ability and the signal detecting performances are given and proven, and then the reason that to maximize the observe ability is to optimize the signal detecting performances is established. Secondly, a dual relation between the observability ellipsoid and the controllability ellipsoid, which volumes and radii are respectively with some inverse relations, is stated and proven. Accordingly, the analytical computing equations for the volume of the two observability ellipsoids are got and some analytical shape factors of these ellipsoids are deconstructed. Based on these effective compting for the volumes, radii, and shape factors, analyzing and optimizing for the observe ability can be carried out. Thirdly, to compare rationally the state observe ability between the different systems or different system parameters, the normalization of the output variables, the state variables, and the system models are discussed. Finally, some numerical experiments and their results show the effectiveness of the computing and comparing methods for the observe ability.