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The "direct method" of Liapunov is utilized to obtain an upper bound on dynamic (transient and steady-state) quantization error in digital control systems. If a linear digital control system is designed such that its motions (solutions) are uniformly asymptotically stable in-the-large, it is shown that its motions are uniformly bounded, in the sense of Liapunov, with the addition of quantization in the presence of any finite excitation. A quadratic form Liapunov function is used to evaluate an upper bound, on the domain of uniform boundedness of solutions, in the presence of quantization. The technique is unique, as compared to previous publications, in that an explicit analytical solution is available in general for a dynamic upperbound regardless of whether the system impulse response matrix is overdamped or underdamped.