The authors have shown that the performance of discrete-time observer-based monitoring systems can be represented by the performance index k$_2$(P) (condition number of the eigensystem P of the observer matrix in terms of L$_2$ norm). The observers with the minimized performance index can be defined as robust observers in the sense that the observer performance can be guaranteed in harsh environments. In this paper, based on the performance index, a design methodology for the robust discrete-time observer is developed. Similar to the continuous-time case, the methodology determines the structure and eigenvalues of the observer matrix simultaneously. A complete design procedure is given for single-output case and is illustrated with a spindle-driver example. The simulation results demonstrate the improved performance compared with a traditional pole-placement observer technique.