Parameter Estimation in Adaptively Coupled Kuramoto Oscillators

The problem of joint state and parameter estimation for adaptively coupled Kuramoto oscillators consisting of the oscillator phases and coupling strengths is addressed, focusing on the analysis of underlying identifiability and observability properties required for convergence of the estimates. In particular, we focus on a solution provided by an Extended Kalman Filter, and characterize the state space in terms of regions for which local observability can be guaranteed - and thus the estimator should be able to reconstruct the actual state and parameter values - and those, in which local observability cannot be ensured. It turns out that trajectories typically cross through these regions several times. Furthermore, it is shown that for the case of oscillator phase locking local observability can not be guaranteed and an observer can at most estimate a ratio of parameters that characterizes the dynamics of the coupling strengths. The theoretical findings are illustrated for different scenarios using numerical simulation.