In this work, interval time-delay switched systems having completely unstable and mixed stable matrices of the state vector are considered. An observer-based controller is designed for finite-time boundedness and H-infinity-control of these systems. New sufficient conditions on the existence of a desired observer are developed and new average dwell-time bounds are introduced separately in case of unstable and mixed stable subsystems. An algorithm is presented for the calculation of unknown constants in the average dwell-time bounds which depend on nonlinear matrices in terms of the cone complementarity linearization method. Finally, numerical examples are given for the effectiveness and validity of the proposed solutions.