In this paper, a new method to estimate the initial condition set which guarantees the quadratic stability for a class of non-linear systems via dynamic surface control (DSC) in the presence of input saturation has been proposed. The estimated set is enlarged to allow some degree of input saturation, while achieving the simultaneous quadratic stability of the system in the domain. A convex optimization problem to obtain ellipsoidal initial condition sets via a linear matrix inequality (LMI) approach as well as an illustrative example will be presented.