Time-Invariant Polytopic and Interval Observers for Uncertain Linear Systems via Non-Square Transformation

TL;DR

This paper introduces new polytopic and interval observers for uncertain linear systems that ensure state enclosure and stability, applicable to all detectable systems stabilized by optimal observers, using a non-square transformation.

Contribution

It proposes a novel observer design method employing non-square transformations and polyhedral Lyapunov functions, extending applicability beyond previous approaches.

Findings

Guarantees enclosure of true states and ISS stability.

Applicable to all detectable systems stabilized by optimal observers.

Demonstrated effectiveness through multiple uncertain system examples.

Abstract

This paper presents novel polytopic and interval observer designs for uncertain linear continuous-time (CT) and discrete-time (DT) systems subjected to bounded disturbances and noise. Our approach guarantees enclosure of the true state and input-to-state stability (ISS) of the polytopic and interval set estimates. Notably, our approach applies to all detectable systems that are stabilized by any optimal observer design, utilizing a potentially non-square (lifted) time-invariant coordinate transformation based on polyhedral Lyapunov functions and mixed-monotone embedding systems that do not impose any positivity constraints, enabling feasible and optimal observer designs, even in cases where previous methods fail. The effectiveness of our approach is demonstrated through several examples of uncertain linear CT and DT systems.