Notes on the State Space Realizations of Rational Order Transfer Functions

In this paper, the concept of minimal state space realization for a fractional order system is defined from the inner dimension point of view. Some basic differences of the minimal realization concept in the fractional and integer order systems are discussed. Five lower bounds are obtained for the inner dimension of a minimal state space realization of a fractional order transfer function. Also, the concept of optimal realization, which can be a helpful concept in practice, is introduced for transfer functions having rational orders. An algorithm is suggested to obtain the optimal realizations of rational order transfer functions. The introduced concept might be used to get minimal realizations of rational order transfer functions. This point is illustrated by presenting some examples.