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This paper addresses the robust stabilization problem for a class of switched linear systems affected by time-varying uncertainties with saturating actuators. The objective is to design a switching law and a state feedback control law such that the closed-loop system is asymptotically stable at the origin with a large domain of attraction. Via the multiple Lyapunov functions method, sufficient conditions for robust stabilization are derived. If some scalars parameters are selected in advance, the state feedback control law and the estimation of domain of attraction are presented by solving a convex optimization problem with LMI constraints. A numerical example is given to show the effectiveness of the proposed method.