This paper addresses the stability characterisation problem for a class of nonlinear discrete-time control systems subject to actuator saturation and energy bounded disturbance inputs. The considered class of systems covers all nonlinear discrete-time systems that can be modelled by rational difference equations. Based on quadratic and piecewise quadratic Lyapunov functions, conditions based on linear matrix inequalities are proposed to analyse the asymptotic stability (internal stability) and the (2) input-to-state stability (external stability) of the closed-loop system. The proposed conditions are then incorporated into convex optimisation problems to either maximise an estimate of the region of attraction or a bound on the admissible (2) disturbances, and also to obtain an estimate of the system (2)-gain for an admissible set of disturbances.