Computationally Efficient L₁ and ℌ_∞ Optimal Interval Observer Design

This paper extends existing ${{\mathcal{H}}_\infty }$ and L<inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</inf> interval observer designs for discrete-time (DT) and continuous-time (CT) nonlinear systems with bounded Jacobians and uncertainties by significantly reducing their computational complexity from mixed-integer programs to (generally) linear or semidefinite programs, without adding any conservatism. Our methods similarly employ a multiple-gain observer structure and are based on mixed-monotone decomposition and embedding systems to design correct-by-construction interval framers that inherently bound the true system state and whose framer error system is input-to-state stable. without needing additional positivity constraints. Moreover, we propose a novel CT mixed-monotone decomposition function that is less conservative than previous nonlinear bounding approaches, resulting in smaller framer errors. Finally, we demonstrate the effectiveness of the proposed interval observer on several DT and CT examples.