This paper considers a design method for delayed-inverse systems for time-invariant strictly proper digital systems, which exactly reproduce the input with some delay. The inverse system must be stable in practical applications. It is well known that the inverse system is unstable if the system has invariant zeros outside the unit circle. The purpose of this paper is to give a state space design method for stable delayed-inverse systems such that the series-connected system, which includes the system and the stable delayed-inverse system, is a diagonal rational function. As a preliminary, we construct the delayed-inverse system by the usual method and factorize the delayed-inverse systems into several subsystems. The stable delayed-inverse system is obtained using observer design for subsystems of the delayed-inverse system. A simple numerical example is also shown. Finally, an application of the stable delayed-inverse systems to two-degree-of-freedom control is presented.
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