A class of controlled synchronization systems under information constraints imposed by limited information capacity of the coupling channel is analyzed. It is shown that the framework proposed by Fradkov, [Phys. Rev. E 73, 066209 (2006)] is suitable not only for observer-based synchronization but also for controlled master-slave synchronization via a communication channel with limited information capacity. A simple first-order coder-decoder scheme is proposed and a theoretical analysis for multidimensional master-slave systems represented in the Lurie form (linear part plus nonlinearity depending only on measurable outputs) is provided. An output feedback control law is proposed based on the passification method. It is shown that for systems with passifiable linear part (satisfying the hyperminimum phase condition) the upper bound of the limiting synchronization error is proportional to the upper bound of the transmission error. As a consequence, both upper and lower bounds of the limiting synchronization error are proportional to the maximum rate of the coupling signal and inversely proportional to the information transmission rate (channel capacity). The results are applied to controlled synchronization of two chaotic Chua systems coupled via a controller and a channel with limited capacity. It is shown by computer simulation that, unlike for the case of observer-based synchronization, the hyperminimum phase property cannot be violated for controlled synchronization.