How to use the Routh-Hurwitz Stability Calculator
- 1. Type the characteristic equationFor example s^4+2s^3+3s^2+4s+5, or a coefficient list.
- 2. Add K if neededUse K in the equation to scan for the stable range.
- 3. Check stabilitySee the table, sign changes, verdict and the real roots as a check.
How the Routh array works
Write the coefficients of a(s) = a₀sⁿ + a₁sⁿ⁻¹ + ... in two rows: a₀, a₂, a₄ ... and a₁, a₃, a₅ .... Each new entry uses the two rows above it: (b₁ × a(k+1) − a₁ × b(k+1)) / b₁, where b₁ is the first element of the row just above. The number of sign changes in the first column equals the number of roots in the right half of the s-plane. No sign changes means stable.
Special case 1, zero in the first column: replace the 0 with a small positive ε, finish the table, then let ε → 0⁺. For s⁵ + 2s⁴ + 2s³ + 4s² + 11s + 10, the s³ row starts with 0. With ε, the first column becomes 1, 2, ε, −12/ε, 6, 10: two sign changes, so two roots in the right half-plane.
Special case 2, row of zeros: form the auxiliary polynomial A(s) from the row above, and use dA/ds as the new row. The roots of A(s) are symmetric about the origin, often on the jω axis. For s³ + 2s² + s + 2, A(s) = 2s² + 2 gives roots ±j1: marginally stable.
Range of K and exam questions
For s³ + 3s² + 2s + K, the s¹ row is (6 − K)/3 and the s⁰ row is K, so the system is stable for 0 < K < 6. At K = 6 the s¹ row is zero and A(s) = 3s² + 6 gives sustained oscillation at ω = √2 rad/s. Type K in the equation and the tool scans the range you set.
- Find the number of right half-plane roots of s⁴ + 2s³ + 3s² + 4s + 5 (two).
- Find K for marginal stability and the frequency of oscillation.
- Use the ε method when a first-column element is zero.
Frequently asked questions
What is the ε method?
When the first entry of a row is zero but the row is not all zeros, you cannot divide by it. Replace it with a tiny positive number ε, finish the table, and read the signs as ε → 0⁺.
What does a row of zeros mean?
The polynomial has roots placed symmetrically about the origin, such as ±jω or ±σ. Use the auxiliary polynomial from the row above to continue and to find those roots.
Does the tool also give the actual roots?
Yes. It lists the numerical roots of the equation so you can check the Routh result.
Is this tool free?
Yes. You enter your name and WhatsApp number once to see your result, and all other Uncodemy tools then open without the form.