Output-feedback controlled synchronization problems for a class of nonlinear unstable systems under information constraints imposed by limited capacity of the communication channel are analyzed. A binary time-varying coder-decoder scheme is described, and a theoretical analysis for multidimensional master-slave systems represented in Lurie form (linear part plus nonlinearity depending only on measurable outputs) is provided. An output-feedback control law is proposed based on the passification theorem. It is shown that the synchronization error exponentially tends to zero for sufficiently high transmission rate (channel capacity). The results obtained for the synchronization problem can be extended to tracking problems in a straightforward manner if the reference signal is described by an external (exogenous) state space model. The results are illustrated by the controlled synchronization of two chaotic Chua systems via a communication channel with limited capacity.