This brief studies the synchronization problem of nonlinear multi-agent systems (MASs) with the quadratic condition (QUAD-condition) via prescribed performance control (PPC), where a novel performance function is employed that can configure the convergence rate arbitrarily. With the designed protocol, all relative states’ evolution is within the preset performance envelopes, and the system achieves state synchronization asymptotically. In comparison with the existing studies, we first investigate the coordinated control of QUAD nonlinear MASs with prescribed trajectory behaviors. Moreover, the convergence rate of the proposed performance constraint function can be selected arbitrarily, which has better versatility. Simulations on coupled chaotic circuit networks verify the theoretical results.