Magnitude–frequency responses of fractional order systems: properties and subsequent results

Abstract

This study deals with the properties of magnitude–frequency responses in fractional order systems. Using Phragmén–Lindelöf theorem in complex analysis, it is shown that the supremum of the magnitude–frequency response of a fractional system with a commensurate order less than one cannot be greater than that of its integer order bounded-input, bounded-output stable counterpart. Further results are also obtained on magnitude–frequency response of stable/unstable fractional order systems. Moreover, it is found that the supremum (infimum) of the magnitude-scaling frequency of the family of fractional order systems having a fixed structure and different orders in the range (0, 2) is a piecewise logarithmically convex (concave) function of the scaling frequency. On the basis of the properties of magnitude–frequency responses in fractional order systems, some sample consequent results which are useful in analysis of fractional order circuits and systems are derived. These results are validated by various numerical examples.