From Observability to Observer Realization: A path via elementary block-diagram manipulations

TL;DR

This paper presents a graphical method for transforming linear systems between observability and observer forms, enhancing pedagogical understanding through elementary block diagram manipulations, and also applies to controllability and controller realization.

Contribution

It introduces a simple, graphical technique for system realization transformations, emphasizing pedagogical utility and leveraging duality in linear systems.

Findings

Graphical transformation method simplifies understanding of system realizations.

Technique applicable to both observability-to-observer and controllability-to-controller transformations.

Enhances teaching methods in linear control courses.

Abstract

Introductory state-space linear control courses focus on linear, time-invariant systems and spend intense efforts by introducing system realizations that allow the student to grasp fundamental concepts, among which controllability, observability, and controller and observer design. This note describes a graphical mechanism to transform a system expressed in observability form into its observer form based on elementary block diagram manipulations, a technique whose pedagogical usefulness tends to be overlooked. Given the well-known duality principle in linear systems with respect to controllability and observability, the proposed graphical mechanism is applicable to transform from controllability to controller realization.