Calculator · Dynamics & Control
Routh–Hurwitz Stability Calculator
Enter the coefficients of a real characteristic polynomial in descending powers of s. The Routh array is built exactly as it is taught, and the number of right-half-plane roots is read from the sign changes in the first column — without solving for the roots.
Input
s^3 + 2s^2 + 3s + 1
Result
| Right-half-plane roots | 0 |
|---|---|
| First-column sign changes | 0 |
| Strictly Hurwitz | Yes |
| Roots right of Re(s) = −0 | 0 |
| Verdict | Asymptotically stable |
No sign change in the first column, so no root lies in the right half-plane.
Routh array
| Row | First column | c2 |
|---|---|---|
| s^3 | 1 | 3 |
| s^2 | 2 | 1 |
| s^1 | 2.5 | 0 |
| s^0 | 1 | 0 |
- Sign changes
- 0
- RHP roots (Routh)
- 0
- Order
- 3
Each sign change down the first column is one root in the right half-plane. No sign change means no right-half-plane root.
Hurwitz determinants
Strictly Hurwitz — every root in the open left half-plane
- Δ1 > 02
- Δ2 > 05
- Δ3 > 05
With a positive leading coefficient, the polynomial is strictly Hurwitz exactly when every leading principal minor Δ1…Δn of the Hurwitz matrix is positive.
Explore further
Technical reference
Row entries: bᵢ = (row above · row-above-that determinant) / (leading element of the row above)
Right-half-plane root count = number of sign changes down the first column
- P(s)
- characteristic polynomial, descending powers of s
- s^k
- row label; the first two rows are the alternating coefficients of P(s)
Sign changes: each sign change reading down the first column counts one root with Re(s) > 0, so the total sign-change count is reported directly as the right-half-plane root count.
Zero leading element (row not entirely zero): the zero is replaced by ε → 0⁺ (a small positive number scaled to the polynomial's coefficients) and the array continues; the substituted row is flagged rather than hidden.
Entire zero row: the row is replaced by the coefficients of dA/ds, where A(s) is the auxiliary polynomial formed from the row above at that power. This is exactly the classical special case, and it signals roots symmetric about the origin — including possible imaginary-axis pairs — which are also solved for and reported when the auxiliary polynomial is solvable.
Assumptions and limitations
- · Continuous-time systems only: the criterion is about the left half-plane, not the unit circle.
- · Real coefficients, entered in descending powers of s.
- · Zero rows and zero leading elements are handled explicitly, and the special case is reported rather than hidden.
- · Routh counts roots but does not locate them; the root map in the Lab uses the numerical solver for that.
- · Coefficients spanning many orders of magnitude can make the array ill-conditioned; a disagreement between methods is reported instead of a confident verdict.