  We consider construction of effective Hamiltonians for periodically driven
interacting systems in the case of resonant driving. The standard
high-frequency expansion is not expected to converge due to the resonant
creation of collective excitations, and one option is to resort to the
application of degenerate perturbation theory (DPT) in the extended
Floquet-Hilbert space. We propose an extension of DPT whereby the degenerate
subspace includes not only the degenerate levels of interest but rather all the
states of the system. The resulting approach, which we call extended DPT
(EDPT), is shown to resemble a high-frequency expansion, provided the
quasienergy matrix is constructed such that each $m$th diagonal block contains
energies reduced to the $m$th Floquet zone. The proposed theory is applied to a
driven Bose-Hubbard model and is shown to yield more accurate quasienergy
spectra than the conventional DPT. The computational complexity of EDPT is
intermediate between DPT and the numerically exact approach, thus providing a
practical compromise between accuracy and efficiency.
