This paper proposes a two-step interval estimation method for discrete-time switched linear systems with unknown but bounded uncertainties. Based on the ellipsoidal analysis, the proposed estimator provides upper and lower bounds of the system state with high computational analysis. The size of the obtained ellipsoids is minimized using the trace criterion. The idea of introducing an $L_\infty$ performance is also employed in order to improve the estimation accuracy. Its design conditions are given in terms of Linear Matrix Inequalities (LMIs). Finally, a numerical example emphasizes the effectiveness of the contribution.