Design of robust observers is considered in the context of linear discrete-time, time invariant systems. Robustness is achieved with respect to polytopic type uncertainties that affect the dynamics of the plant. At the difference with the uncertainty-free situation, state-feedback / observer separation principle does not hold. Therefore, the observer design has to take into account the state-feedback gain. Results are derived with linear matrix inequality formalism and involve up-to-date slack-variables approach. A numerical example illustrates the results. Limitations of the method are discussed and prospective work for improving these is exposed.