Summary: In this paper, the output tracking control of switched systems with time-varying delay under asynchronous switching is investigated. When the augmented system composed of the given switched system and tracking error system is robust exponentially stabilizable, the output tracking control problem under asynchronous switching is solved. A new Lyapunov function dependent on the controllers’ switching signal is constructed, which can effectively counteract the difficulty of controller design to achieve the tracking objective under asynchronous switching. Moreover, the value of the Lyapunov function is allowed to increase during the asynchronous phase. Based on the average dwell time approach, the controllers are designed and the \(H_{\infty }\) performance level is achieved. Finally, a numerical example illustrates the effectiveness of the results.