  We propose a `Floquet engineering' formalism to systematically design a
periodic driving protocol in order to stroboscopically realize the desired
system starting from a given static Hamiltonian. The formalism is applicable to
quantum systems which have an underlying closed Lie-algebraic structure, for
example, solid-state systems with noninteracting particles moving on a lattice
or its variant described by the ultra-cold atoms moving on an optical lattice.
Unlike previous attempts at Floquet engineering, our method produces the
desired Floquet Hamiltonian at any driving frequency and is not restricted to
the fast or slow driving regimes. The approach is based on Wei-Norman ansatz,
which was originally proposed to construct a time-evolution operator for any
arbitrary driving. Here, we apply this ansatz to the micro-motion dynamics,
defined within one period of the driving, and obtain the driving protocol by
fixing the gauge of the micro-motion. To illustrate our idea, we use a two-band
system or the systems consisting of two sub-lattices as a testbed.
Particularly, we focus on engineering the cross-stitched lattice model that has
been a paradigmatic flat-band model.
