This paper proposes a tube-based method for the asynchronous observation problem of discrete-Time switched linear systems in the presence of amplitude-bounded disturbances. Sufficient stability conditions of the nominal observer error system under mode-dependent persistent dwell-Time MPDT switching are first established. Taking the disturbances into account, a novel asynchronous MPDT robust positive invariant RPI set and an asynchronous MPDT generalized RPI GRPI set are determined for the difference system between the nominal and disturbed observer error systems. Further, the global uniform asymptotical stability of the observer error system is established in the sense of converging to the asynchronous MPDT GRPI set, i.e., the cross section of the tube of the observer error system. Finally, the proposed results are validated on a space robot manipulator example.