This article focuses on distributed consensus control for nonlinear multiagent systems under directed graphs of dynamic frequency switches. The agents exchange information with each other through directed graphs that are switched based on a new general switching law. Not all graphs contain a spanning tree with the leader as the root. Under this condition, a novel distributed switched observer is constructed for each follower, and it can accurately estimate the state information of the leader by means of the switching law. Subsequently, a unified distributed controller is designed for each follower with the observer. Based on the Lyapunov stability theory, it is strictly proved that all agents can achieve a consensus with the controller. More importantly, our proposed algorithm can render the number of switches of the graphs to grow faster than an affine function of the length of a time interval, which greatly reduces the conservativeness on graphs switches. Moreover, the consensus algorithm has a wide application prospect in cooperative control. Finally, the effectiveness of theoretical results is illustrated by making numerical simulations.