Periodically driven quantum dynamics has recently attracted much attention in experimental as well as theoretical studies as it offers a promising way for exploring novel quantum phenomena which would be difficult or impossible to observe otherwise. However, analyzing periodically driven systems is in general a hard problem. With a few exceptions, recent studies mostly focus on driving simple (often non-interacting particles) Hamiltonians, whereas such a condition can be satisfied only approximately in realistic experimental conditions. From recent studies, even small integrability-breaking terms are known to be relevant to the long-time dynamics and eventually lead to a state of infinite temperature (a random state) as a final steady state. In the experimental context, this final state is no longer intriguing since one cannot get any information reflected from the system.