  We analyze a family of Runge-Kutta based quadrature algorithms for the
approximation of the gramians of linear time invariant dynamical systems. The
approximated gramians are used to obtain an approximate balancing
transformation similar to the approach used in balanced POD. It is shown that
hereby rational interpolation is performed, as the approximants span certain
Krylov subspaces. The expansion points are mainly determined by the time step
sizes and the eigenvalues of the matrices given by the Butcher tableaus.
