The FCC Stability Criterion for Fractional-Order Linear Time-Invariant Systems with Commensurate or Incommensurate Orders

In this paper, a new stability criterion is proposed for the stability analysis of fractional-order linear time-invariant (FO-LTI) systems with incommensurate or commensurate orders. Although Matignon's theorem is well-established for studying the stability of FO-LTI systems with commensurate orders, there is no advanced and reliable criterion to check the stability of FO-LTI systems with incommensurate (not commensurate) orders. Therefore, developing a new criterion to check the stability of this type of FO-LTI systems is a necessity. The proposed criterion is based on constructing a monodromy operator for the solution of FO-LTI systems along with the use of short memory principle. Moreover, the monodromy operator is approximated by the use of the fractional Chebyshev collocation method, and necessary and sufficient conditions of the stability are obtained. The advantages of the proposed criterion are illustrated in three practical examples.