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The evolution of the error caused by the amplitude quantization in digital control systems can be studied by introducing bounded disturbances at the points of the quantization. Such a formulation provides an upper bound for the actual quantization error. This paper introduces the problem of the quantization error in the light of the discrete-time optimum control theory. The determination of the worst possible effect of the error due to quantization is reduced to solving a common optimum control problem; the "worst effect" is defined as the maximum of a chosen performance criterion. For a design, a method is proposed for the minimization of the worst effect of the quantization error. Computational solutions illustrate the presentation.