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

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.