The problems of Extended Strict Positive Realness (ESPR) analysis and synthesis for a class of discrete-time systems with polytopic uncertainty are considered. An equivalent condition of ESPR for discrete-time systems is first given in terms of linear matrix inequalities (LMIs). Moreover, a sufficient condition for the uncertain systems to be robustly stable with ESPR is established by stability with ESPR of vertex systems. Furthermore, a condition on robust stability with ESPR for the uncertain systems is obtained in terms of linear matrix inequalities. Based on this result, a simple approach to the design of a linear state-feedback controller which stabilizes robustly the uncertain systems and achieves ESPR is proposed. A numerical example is worked out to illustrate the efficiency of the proposed approach.