An Algebraic State Observer for a Class of Physical Systems

TL;DR

This paper introduces a novel algebraic approach to designing state observers for nonlinear physical systems, providing explicit relations between unmeasurable states and measurable signals without requiring observability conditions.

Contribution

The paper presents a new algebraic observer design method that applies to nonlinear physical systems, avoiding traditional asymptotic or excitation conditions.

Findings

Provides explicit algebraic relations for state estimation

Works for a broad class of physical systems without observability constraints

Offers a major departure from classical asymptotic observer designs

Abstract

In this paper we present a radically new approach to design state observers for nonlinear systems, with particular emphasis on physical ones. Our objective is to obtain an algebraic relation between the unmeasurable part of the state and filtered versions of the systems inputs and outputs, which holds true for all $t \geq 0$. The latter qualifier should be contrasted with the usual asymptotic (or fixed/finite time) objective. The standing assumption for our design is the availability -- or possibility of constructing, via coordinate change -- state components with measurable derivatives. In the physical systems studied in the paper this condition is naturally satisfied. The next step in the design is the application of the Swapping Lemma to pull out from the dynamics the derivative of one of these signals. The design is completed replacing the latter by the measurable signals and arranging the remaining terms. The algebraic observer constitutes a refreshing major departure from classical asymptotic observer designs, even in the case of electrical motors and mechanical systems that have been exhaustively studied. Particularly notable is the fact that no observability or excitation condition is imposed for the construction of the algebraic observer.