Optimal Interval Observers for Bounded Jacobian Nonlinear Dynamical Systems

TL;DR

This paper introduces two novel interval observer designs for discrete-time and continuous-time nonlinear systems with bounded Jacobians, ensuring correct state bounding and stability despite uncertainties.

Contribution

The paper presents new interval observer methods using mixed-monotone decomposition and embedding systems, with optimization techniques for gain computation and stability guarantees.

Findings

The proposed observers guarantee correct state bounds without additional constraints.

They minimize $\\mathcal{H}_{\infty}$ and $L_1$ gains for improved robustness.

Comparative analysis shows improved performance over existing methods.

Abstract

In this chapter, we introduce two interval observer designs for discrete-time (DT) and continuous-time (CT) nonlinear systems with bounded Jacobians that are affected by bounded uncertainties. Our proposed methods utilize the concepts of mixed-monotone decomposition and embedding systems to design correct-by-construction interval framers, i.e., the interval framers inherently bound the true state of the system without needing any additional constraints. Further, our methods leverage techniques for positive/cooperative systems to guarantee global uniform ultimate boundedness of the framer error, i.e., the proposed interval observer is input-to-state stable. Specifically, our two interval observer designs minimize the $\mathcal{H}_{\infty}$ and $L_1$ gains, respectively, of the associated linear comparison system of the framer error dynamics. Moreover, our designs adopt a multiple-gain observer structure, which offers additional degrees of freedom, along with coordinate transformations that may improve the feasibility of the resulting optimization programs. We will also discuss and propose computationally tractable optimization formulations to compute the observer gains. Finally, we compare the efficacy of the proposed designs against existing DT and CT interval observers.