Optimal Experimental Design for Uncertain Systems Based on Coupled Differential Equations

We consider the optimal experimental design (OED) problem for an uncertain system described by coupled ordinary differential equations (ODEs), whose parameters are not completely known. The primary objective of this work is to develop a general experimental design strategy that is applicable to any ODE-based model in the presence of uncertainty. For this purpose, we focus on non-homogeneous Kuramoto models in this study as a vehicle to develop the OED strategy. A Kuramoto model consists of $N$ interacting oscillators described by coupled ODEs, and they have been widely studied in various domains to investigate the synchronization phenomena in biological and chemical oscillators. Here we assume that the pairwise coupling strengths between the oscillators are non-uniform and unknown. This gives rise to an uncertainty class of possible Kuramoto models, which includes the true unknown model. Given an uncertainty class of Kuramoto models, we focus on the problem of achieving robust synchronization of the uncertain model through external control. Should experimental budget be available for performing experiments to reduce model uncertainty, an important practical question is how the experiments can be prioritized so that one can select the sequence of experiments within the budget that can most effectively reduce the uncertainty. In this paper, we present an OED strategy that quantifies the objective uncertainty of the model via the mean objective cost of uncertainty (MOCU), based on which we identify the optimal experiment that is expected to maximally reduce the MOCU. We demonstrate the importance of quantifying the operational impact of the potential experiments in designing optimal experiments and show that the MOCU-based OED scheme enables us to minimize the cost of robust control of the uncertain Kuramoto model with the fewest experiments compared to other alternatives. The proposed scheme is fairly general and can be applied to any uncertain complex system represented by coupled ODEs.