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This paper presents optimal dynamic quantizers for controlling linear time-invariant systems with the discrete- valued input. The quantizers considered here are in the form of a difference equation, for which we find a quantizer such that the system composed of a given linear plant and the quantizer is an optimal approximation of the given linear plant in the sense of the input-output relation. First, we derive a closed form expression for the performance (the degree of the approximation) of a class of dynamic quantizers. Next, based on this, an optimal dynamic quantizer and its performance, corresponding to the performance limitation of the dynamic quantizers, are provided. Finally, the validity of the proposed quantizer is shown by numerical simulations.