On the asymptotic stability for linear multi-order fractional-order systems with time delays

The characteristic polynomial-based frequency domain methods are generally used in asymptotic stability analysis of multi-order fractional-order systems. However, the proof for the existing characteristic polynomial-based asymptotic stability conditions are questionable for the fact that the proof utilized the Laplace final value theorem while the Laplace final value theorem holds only when the final value exists. It is of importance to give a rigorously proved characteristic polynomial-based asymptotic stability condition. Under this background, the asymptotic stability problems for multi-order fractional-order systems are revisited in this paper. Necessary and sufficient characteristic polynomial-based conditions for asymptotic stability of multi-order (commensurate and incommensurate) fractional-order systems are derived using inverse Laplace formula. The results fill the gap in the research of the necessary and sufficient conditions for the asymptotic stability of multi-order fractional-order systems.