This paper is concerned with a generalization of the concept of an inverse system far a linear time-invariant dynamical system. Most of the previous studies on inverse systems have treated only the case where all components of the input (or their α-th integrals) of the original system can be recovered from the output signal. Even when not all components are recoverable, there is still a possibility of reproducing some of the components of the input or their linear functions. As a generalization of the concept of an inverse system to include such cases, the “α-integral F-inverse” is proposed in this paper. This inverse is a system which reproduces the α-th integral of a linear function Fu of the input u.
The definition of the α-integral F-inverse is given first. Then a necessary and sufficient condition for the existence of an α-integral F-inverse is derived. Finally, a construction procedure of such an inverse is given and a numerical example is presented.