A stability criterion for linear fractional order systems in frequency domain

A stability theorem for linear fractional order systems is proposed by analyzing the relationship between the phase angle increment of the denominator of the transfer function and the stability in the frequency domain. Two functions about the denominator coefficients are defined, the stability conditions are presented by analyzing the relationship of the positive real solutions of these two functions and the relationship between the number of solutions and the highest order of the denominator. This stability theorem generalizes the Hermite-Biehler theorem for integer order linear systems and extend it to fractional order systems in the frequency domain. Finally, the results of two numerical examples are analyzed to illustrate the validity of the proposed stability theorem.