Fundamental limitations for error tracking/regulation are obtained for the robust servomechanism problem (RSP) for discrete time periodic systems. In studying this problem, the RSP for a multi-input/multi-output discrete time system is considered; application of these results is then made to the "periodic system robust servomechanism problem", and explicit expressions for the limiting costs for error tracking regulation are obtained. These limitations can be characterized completely by the number and location of the non-minimum phase transmission zeros of the system's associated lifted system, together with a term which depends on the order of the system, the number of inputs, the number of transmission zeros and the system's period. (C) 2003 Elsevier Science Ltd. All rights reserved.