Detectability Conditions and State Estimation for Linear Time-Varying and Nonlinear Systems

TL;DR

This paper introduces a new detectability condition for linear time-varying systems using the exponential dichotomy spectrum, enabling low-complexity observer design, and extends this approach to certain nonlinear systems with proven local convergence.

Contribution

It proposes a novel detectability condition based on the exponential dichotomy spectrum and extends observer design to nonlinear systems with convergence guarantees.

Findings

Observer gain depends only on unstable modes.

Numerical results confirm the effectiveness of the proposed observer.

Extension to nonlinear systems with local convergence proven.

Abstract

This work proposes a detectability condition for linear time-varying systems based on the exponential dichotomy spectrum. The condition guarantees the existence of an observer, whose gain is determined only by the unstable modes of the system. This allows for an observer design with low computational complexity compared to classical estimation approaches. An extension of this observer design to a class of nonlinear systems is proposed and local convergence of the corresponding estimation error dynamics is proven. Numerical results show the efficacy of the proposed observer design technique.