Hybrid predictor-corrector numerical schemes: λ-tracking, sampled-data observers and data assimilation

Abstract

This paper considers hybrid ordinary differential equation (ODE) solvers for process dynamics constructed by combining standard numerical schemes with standard observers. Specifically, we combine the first-order Euler scheme with a Luenberger observer. The key ideas are to take advantage of available process output information and to switch from the numerical scheme to the process output-driven observer when the numerical scheme alone would produce inadequate results. Within this setup, two tasks emerge: How to choose the observer gain? How to choose the step size in the numerical scheme? Underpinning our approach is a λ tracking-based sampled-data observer that invokes a λ dead zone. This λ tracking observer determines the observer gain and the numerical step-size adaptively. The resulting adaptive hybrid algorithm is a time-stepping numerical scheme. Using a sampled-data observer allows for process measurements to be only available at some discrete times, whilst adaptive tuning allows the gains and sampling times to adjust automatically to each other – rather than both being subjected to designer's choice. Results are illustrated with examples of simulation.