  We consider 1D lattices described by Hubbard or Bose-Hubbard models, in the
presence of periodic high-frequency perturbations, such as uniform ac force or
modulation of hopping coefficients. Effective Hamiltonians for interacting
particles are derived using an averaging method resembling classical canonical
perturbation theory. As is known, a high-frequency force may renormalize
hopping coefficients, causing interesting phenomena such as coherent
destruction of tunnelling and creation of artificial gauge fields. We find
explicitly additional corrections to the effective Hamiltonians due to
interactions, corresponding to non-trivial processes such as single-particle
density-dependent tunnelling, correlated pair hoppings, nearest neighbour
interactions, etc. Some of these processes arise also in multiband lattice
models, and are capable to give rise to a rich variety of quantum phases. The
apparent contradiction with other methods, e.g. Floquet-Magnus expansion, is
explained. The results may be useful for designing effective Hamiltonian models
in experiments with ultracold atoms, as well as in the field of ultrafast
nonequilibrium magnetism. An example of manipulating exchange interaction in a
Mott-Hubbard insulator is considered, where our corrections play an essential
role.
