This paper studies the asynchronous control problem for a class of switched systems with nonlinear perturbation. Asynchrony means that the switching of the controllers experien-ces a time delay with respect to that of the subsystems. The analytical solution to the state equation of the asynchronously switched systems is utilized to study the dynamics of systems directly without constructing any Lyapunov function. Sufficient conditions that ensure the exponential stability of the closed-loop system under the mode-dependent average dwell time (MDADT) scheme are proposed. MDADT means that each subsystem has its own average dwell time (ADT), which is more general than ADT. Moreover, asynchronously switched controllers are designed in terms of a set of solvable linear matrix inequalities. Finally, a numerical example is provided to illustrate the effectiveness of the proposed method.