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Fundamental limitations for error tracking/regulation are obtained for the robust servomechanism problem (RSP) for a sampled system. In studying this problem, the cheap control problem for a multi-input/multi-output discrete time system is considered, and explicit expressions are obtained for the limiting steady state solution of its associated algebraic Riccati equation (ARE), as the weight on the control energy tends to zero. Application of these results is then made to obtain explicit expressions for the limiting performance costs associated with error tracking/regulation in the RSP. These limitations can be characterized by the system order, the dimension of the outputs, the number of the system's transmission zeros and the location of the system's nonminimum phase transmission zeros.