  A hybrid dynamical system switches between dynamic regimes at time- or
state-triggered events. We propose an offline algorithm that simultaneously
estimates discrete and continuous components of a hybrid system's state. We
formulate state estimation as a continuous optimization problem by relaxing the
discrete component and use a robust loss function to accommodate large changes
in the continuous component during switching events. Subsequently, we develop a
novel nonsmooth variable projection algorithm with Gauss-Newton updates to
solve the state estimation problem and prove the algorithm's global convergence
to stationary points. We demonstrate the effectiveness of our approach on
simple piecewise-linear and -nonlinear mechanical systems undergoing
intermittent impact.
