Routh-Hurwitz criterion of stability and robust stability for fractional-order systems with order α ∈ [1, 2)

The paper proves that for D^α x = A x with α ∈ [1,2), asymptotic stability is equivalent to positivity of all even-order leading principal minors of a 2n×2n block-Toeplitz matrix H_α built from cos((n−j)απ/2) and sin((n−j)απ/2) weighted coefficients of the characteristic polynomial f(λ) = λ^n + a_1 λ^{n-1} + … + a_n. The proof rotates the complex plane by θ = (α−1)π/2, evaluates g(iλ) = f(λ e^{iαπ/2}), and invokes the generalized Routh–Hurwitz (Hermite–Biehler) criterion to match those minors with the 2k-th leading principal minors of H_α (Theorem 1 and Definition 1) . The candidate solution follows exactly this structure, differing only by notational choices for θ and by giving a slightly more explicit expansion of g(iy) into real and imaginary parts that produce the entries of H_α. One minor caveat in the candidate’s remarks is an overstatement that for α=1 the matrix “collapses to the usual Hurwitz matrix”; while the conditions reduce to the classical Hurwitz stability test via the Hermite–Biehler framework, the displayed H_α is a generalized Hurwitz matrix built from the real/imaginary parts on the imaginary axis, not the standard Hurwitz matrix per se. Aside from this nuance, both proofs are equivalent in substance and correct, relying on the same rotation argument, sector condition (Matignon), and generalized Routh–Hurwitz lemma .