  We consider the problem of finding an event-based sampling scheme that
optimizes the trade-off between average sampling rate and control performance
in a linear-quadratic-Gaussian (LQG) control problem setting with output
feedback. Our analysis is based on a recently presented sampled-data controller
structure, which remains LQG-optimal for any choice of sampling scheme. We show
that optimization of the sampling scheme is related to an elliptic
convection-diffusion type partial differential equation over a domain with free
boundary, a so called Stefan problem. A numerical method is presented to solve
this problem for second order systems, and thus obtain an optimal sampling
scheme. The method also directly generalizes to higher order systems, although
with a higher computational cost. For the special case of multidimensional
integrator systems, we present the optimal sampling scheme on closed form, and
prove that it will always outperform its periodic counterpart. Tight bounds on
the improvement are presented. The improved performance is also demonstrated in
numerical examples, both for an integrator system and a more general case.
