<p>In this paper, we propose a novel criterion for practical synchronization of discrete-time Lur’e-type complex dynamic networks (CDNs) with event-triggered control. In particular, we consider that each node is a Lur’e system subject to control saturation and persistent disturbances, operating with an asynchronous relaxed event-triggering mechanism (ETM). Based on the Lyapunov Theory, we derive constructive conditions expressed in terms of quasi-linear matrix inequalities to guarantee that the synchronization error of the CDN is ultimately bounded considering prescribed sets of admissible initial errors and disturbances. These conditions are cast into a semidefinite programming problem to codesign the decentralized controllers and the parameters of the ETMs aiming at reducing the frequency of control updates compared to the time-triggered strategy. The synchronization error norm in the ultimate bounded regime is upper bounded by a positive constant specified by the designer. A numerical example is presented to illustrate the proposed method.</p>