  We extend the concepts of quantum complete synchronization and phase
synchronization, which are proposed firstly in [Phys. Rev. Lett, 111 103605
(2013)], to more widespread quantum generalized synchronization. The
generalized synchronization can be considered as a necessary condition or a
more flexible derivative of complete synchronization, and its criterion and
synchronization measurement are further proposed and analyzed in this paper. As
an example, we consider two typical generalized synchronizations in a designed
optomechanical system. Unlike the effort to construct a special coupling
synchronization system, we purposefully design extra control fields based on
Lyapunov control theory. We find that the Lyapunov function can adapt to more
flexible control objectives, which is more suitable for generalized
synchronization control, and the control fields can be achieved simply with a
time-variant voltage. Finally, the existence of quantum entanglement in
different generalized synchronizations is also discussed.
