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This study focuses on the problem of regional stability analysis of rational control systems with saturating actuators. Estimates of the region of attraction are computed by means of invariant domains associated to rational Lyapunov functions. Conditions for computing these invariant domains (regions of stability) are proposed in the form of linear matrix inequalities (LMIs). These conditions are derived considering a differential-algebraic representation of the rational system dynamics. The saturation effects are taken into account by means of a generalised sector condition for deadzone non-linearities. The obtained conditions are cast in convex optimisation schemes in order to compute a Lyapunov function which leads to a maximised estimate of the region of attraction.