The design of state feedback gain-scheduled controllers for linear parameter-varying systems with saturating actuators is addressed in the paper. The parameters can vary arbitrarily fast inside a polytope with known vertices. Sufficient conditions for the existence of gain-scheduled controllers assuring asymptotic stability for initial conditions inside a region of the state space are provided in terms of parameter-dependent linear matrix inequalities. A complete characterization of the solutions of these inequalities is given in terms of homogeneous polynomially parameter-dependent matrices of arbitrary degree that can be obtained from finite linear matrix conditions, written in terms of the vertices of the polytope. A procedure based on an extension of P贸lya''s Theorem produces a sequence of sufficient conditions which tend to the necessity as the level of relaxation increases. The scheduled controller is a homogeneous polynomially parameter-dependent state feedback gain of arbitrary degree that quadratically stabilizes the closed-loop system and provides an estimate of the domain of attraction of the origin, as illustrated by numerical examples.
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