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This paper presents a new method for designing a gain-scheduled missile autopilot. The nonlinear missile dynamics are modeled as a switched linear parameter-varying (SLPV) system, as is also common for many hybrid system models. A gain-scheduled autopilot for the SLPV system is then designed using a new synthesis technique that is based on a nonsmooth dissipation framework. The proposed algorithm extends previously published Lyapunov-based LPV approaches in three important ways: the SLPV system can provide a more accurate model of the missile dynamics; the nonsmooth Lyapunov function used in the synthesis is a more general class of Lyapunov functions; and the gridding technique typically used to solve the synthesis can be eliminated from the problem. As a result, the new approach yields a reliable gain-scheduled autopilot with better performance than the LPV controllers available in the literature.