In this paper, we enhance the quality of our preceding results on periodically time-varying memory state-feedback controller (PTVMSFC) synthesis for discrete-time linear periodic/time-invariant systems. We firstly revisit PTVMSFC synthesis for certain systems and derive a necessary and sufficient LMI condition for the existence of the desired H∞-PTVMSFCs. Based on these LMIs, we next consider robust H∞-PTVMSFC synthesis for polytopic-type uncertain systems and demonstrate that we can obtain less conservative results. We finally derive a test to verify that the designed PTVMSFC is "exact" in the sense that it attains the best achievable robust H∞ performance. This exactness verification test works fine in practice, and we show via numerical examples that exact robust control is indeed possible via PTVMSFCs, even for those problems where the standard static state-feedback fails.