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Chaotic behavior due to the round-off effect in digital control systems is called the chaotic rounding error. First, we model digital control systems with finite-wordlength digital compensators by mixed mappings. A mixed mapping system is described jointly by a point mapping and a cell mapping, that is, its state consists of real numbers and integers. After presenting some definitions and fundamental results of mixed mapping systems, we prove certain sufficient conditions for the existence of the chaotic rounding error. Finally, as an illustration, we discuss the chaotic rounding error of a system with a second-order plant and first-order digital compensator.