A graphic stability criterion for non-commensurate fractional-order time-delay systems

This study focuses on a graphical approach to determine the stability of non-commensurate fractional-order systems with time-delay. By the iterative decomposition technique, the fractional-order systems described by the transfer functions are represented as a series of closed loop systems. The innermost closed loop system is straightforward to test the stability by its coefficient and the fractional-order. A function with respect to each open loop system is defined, and the stability of a non-commensurate fractional-order system is determined by the unstable poles of the innermost system and the times of the curve depending on the defined function encircling the origin in the clockwise direction. Finally, three illustrative examples are provided to demonstrate the effectiveness of the proposed criterion.