This paper studies a new approach to linear quadratic control for linear systems with input saturation. Our work presents an optimal sector bound to model the mismatch between the unsaturated controller and saturated one and an optimised control design associated with this sector bound. This leads to a new characterisation of invariant sets and new switching controllers. The main outcome of this paper is that better performance can be guaranteed for the same region of attraction, or equivalently, a larger region of attraction is given for the same level of guaranteed performance.