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Reduction of conservatism is one of the key and difficult problems in missile robust gain scheduling autopilot design based on multipliers. This article presents a scheme of adopting linear parameter-varying (LPV) control approach with full block multipliers to design a missile robust gain scheduling autopilot in order to eliminate conservatism. A model matching design structure with a high demand on matching precision is constructed based on the missile linear fractional transformation (LFT) model. By applying full block S-procedure and elimination lemma, a convex feasibility problem with an infinite number of constraints is formulated to satisfy robust quadratic performance specifications. Then a grid method is adopted to transform the infinite-dimensional convex feasibility problem into a solvable finite-dimensional convex feasibility problem, based on which a gain scheduling controller with linear fractional dependence on the flight Mach number and altitude is derived. Static and dynamic simulation results show the effectiveness and feasibility of the proposed scheme.