Lyapunov Stability of Linear Fractional Systems: Part 2 — Derivation of a Stability Condition

This paper, composed of two parts, addresses the stability of linear commensurate order fractional systems, D_n (X) = A X 0 < n < 1, using the infinite state approach. Whereas Part 1 has been dedicated to the definition of fractional systems energy, Part 2 deals with the derivation of a stability condition. When the eigenvalues of A are real, the modal representation shows that system energy is the sum of independent modal energies, so the derivation of a stability condition is straightforward in this case. On the contrary, when the eigenvalues are complex with positive real parts, unusual energy dynamics depending on initial conditions prevent direct derivation of a stability condition. Thus, an indirect method is proposed to formulate a stability condition in the complex eigenvalues case.