$\mathcal{H}_{\infty}$-optimal Interval Observer Synthesis for Uncertain Nonlinear Dynamical Systems via Mixed-Monotone Decompositions

TL;DR

This paper develops an $$-optimal interval observer for uncertain nonlinear systems using mixed-monotone decompositions, ensuring correct state estimation and stability with minimized error gain via SDP and LMI techniques.

Contribution

It introduces a novel $$-optimal interval observer for nonlinear systems that guarantees correctness and stability through mixed-monotone decompositions and convex optimization.

Findings

The proposed observer guarantees correct state estimation without additional constraints.

It provides sufficient conditions for input-to-state stability.

Performance comparisons show improvements over benchmark observers.

Abstract

This paper introduces a novel $\mathcal{H}_{\infty}$-optimal interval observer synthesis for bounded-error/uncertain locally Lipschitz nonlinear continuous-time (CT) and discrete-time (DT) systems with noisy nonlinear observations. Specifically, using mixed-monotone decompositions, the proposed observer is correct by construction, i.e., the interval estimates readily frame the true states without additional constraints or procedures. In addition, we provide sufficient conditions for input-to-state (ISS) stability of the proposed observer and for minimizing the $\mathcal{H}_{\infty}$ gain of the framer error system in the form of semi-definite programs (SDPs) with Linear Matrix Inequalities (LMIs) constraints. Finally, we compare the performance of the proposed $\mathcal{H}_{\infty}$-optimal interval observers with some benchmark CT and DT interval observers.