For discrete-time nonlinear systems with uncertainty, this paper presents an interval analysis approach to design controller and compute the estimate of the closed-loop robust domain of attraction (RDOA). The dynamics of the system is modelled using difference inclusions. A robust negative-definite and invariant set (RNIS) in the state-control space is proposed. An RNIS is defined by the combination of a robust negativedefinite set (RNS) and a robust controlled invariant set (RCIS), which leads to sufficient conditions for Lyapunov stability of the system. The estimate of RDOA can be obtained by projecting an RNIS along the state space. However, the RNIS is hard to obtain by its definition. Drawing inspiration from the RCIScomputation approach, we define a mapping that utilizes the predecessor operator in the state-control space to compute a set limit. Then, the RNIS can be obtained by finding the limit set for an RNS. The computations of RNS and the limit set are based on interval analysis. An algorithm to estimate the RNIS is introduced with rigorous convergence analysis. Finally, we formulate an optimization problem that is solvable, and enlarges the RNIS and the estimate of RDOA. The method is validated on examples of nonlinear systems subject to actuator saturation.