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The performance of digitally controlled systems can be improved by using properly adjusted fractional-order hold (FROH) circuits instead of the common zero-order hold. Nevertheless, some of the previous analytical results formerly presented in the bibliography are exclusively applicable to zero-free second-order continuous-time plants. Consequently, the related FROH tuning methods may only be used in such cases. The mentioned results are extended to first-order plus time-delay systems. In this way, the application of FROH circuits in a wide range of industrial digital control systems is allowed.