Implementation of fractional-order transfer functions in the viewpoint of the required fractional-order capacitors

In this paper, based on the concept of restricted difference basis (RDB) for a natural number, the general structures of the pseudo state-space realisations of some classes of commensurate fractional-order transfer functions are obtained. It is shown that the number of needed fractional-order capacitors for implementing a transfer function belonging to each of these classes depends on the commensurate order and also the RDB based on which the pseudo state-space realisation of the considered transfer function is obtained. In this regard, three general classes of commensurate fractional-order transfer functions are introduced which all can be implemented by the same number of fractional-order capacitors. Also, four classes of fractional-order transfer functions of commensurate order 1/122${1 \mathord{/ {\vphantom {1 2}} \kern-\nulldelimiterspace} 2}$ are introduced which all can be implemented by only one fractional-order capacitor of order 1/122${1 \mathord{/ {\vphantom {1 2}} \kern-\nulldelimiterspace} 2}$. Finally, as a more general case, four classes of fractional-order transfer functions of commensurate order 1/1rr(r∈N)${1 \mathord{/ {\vphantom {1 r}} \kern-\nulldelimiterspace} r}(r \in \mathbb{N})$ are studied in aspect of the number and the order of the fractional-order capacitors needed for their implementations. It is proved that these classes of commensurate fractional-order transfer functions can be implemented by r − 1 number of fractional-order capacitors of order 1/1rr${1 \mathord{/ {\vphantom {1 r}} \kern-\nulldelimiterspace} r}$.