  Monochromatically driven two-level systems (i.e., Rabi models) are ubiquitous
in various fields of physics. Though they have been exactly solved, the
physical pictures in these exact solutions are not clear. Recently, approximate
analytical solutions with neat physics have been obtained by using the
counterrotating hybridized rotating wave (CHRW) method, which has been proven
to be effective over a wider range of parameters than the previous analytical
solutions. However, the CHRW depends on a parameter {\xi}, which has no
solution in some regimes. Here we combine the double-unitary-transformation
approach with the generalized Van Vleck nearly degenerate perturbation theory,
and present approximate analytical results with clear physics for almost all
parameter regimes, which agree well with the numerical solutions and the
previous experimental results. Moreover, the dynamic frequencies of the Rabi
model are regular, and the frequency with the highest Fourier amplitude changes
from the Rabi frequency to 2n{\omega} with driving frequency {\omega} and
integer n, as the driving intensity increases from weak to deep-strong. In
addition, we further explore the Floquet dynamics of the dissipative open Rabi
model. Remarkably, the dissipations are tunable in the rotating frame, and the
approximate analytical results obtained by our method are in good agreement
with the numerical results in the strong driving regime. These results pave the
way to quantum control using strong and deep-strong driving with applications
in quantum technologies.
