The problem of designing a globally optimal robust output-feedback controller for time-varying polytopic uncertain systems is a well-known non-convex optimization problem. In this paper, new sufficient conditions for robust H∞outputfeedback control synthesis are proposed in terms of a special type of bilinear matrix inequalities (BMIs), which can be solved effectively using linear matrix inequality (LMI) optimization plus a line search. In order to reduce the conservatism of robust output-feedback control methods based on single quadratic Lyapunov function, we utilize multiple Lyapunov functions. The associated robust output-feedback controller is constructed as a switching-type full-order dynamic output-feedback controller, consisting of a family of linear subcontrollers and a min-switching logic. The proposed approach features the important property of computational efficiency with stringent performance. Its effectiveness and advantages have been demonstrated through numerical studies.