Routh-Hurwitz Calculator
For s⁴ + 2s³ + 3s² + 4s + 5 = 0 the Routh array has two sign changes in its first column. Unstable: 2 roots in the right half-plane.
For s⁴ + 2s³ + 3s² + 4s + 5 enter 1, 2, 3, 4, 5. Include zeros for missing powers.
Unstable: 2 roots in the right half-plane
- Sign changes in the first column
- 2
- Special case
- none
- Roots (numerical check)
- 0.287815 + j1.41609, 0.287815 − j1.41609, -1.28782 + j0.857897, -1.28782 − j0.857897
| Row | col 1 | col 2 | col 3 |
|---|---|---|---|
| s^4 | 1 | 3 | 5 |
| s^3 | 2 | 4 | 0 |
| s^2 | 1 | 5 | 0 |
| s^1 | -6 | 0 | 0 |
| s^0 | 5 | 0 | 0 |
Step-by-step solution
- Rows s^4 and s^3 take alternate coefficients
- Each further entry: (b₁a₂ − a₁b₂) ÷ b₁ from the two rows above
- First column sign changes = 2 → 2 roots in the right half-plane
Routh-Hurwitz Formula
bᵢ = (a₁a₂ᵢ − a₀a₂ᵢ₊₁)/a₁
number of sign changes in the first column = number of roots in the right half-plane
Worked Example
The characteristic equation s⁴ + 2s³ + 3s² + 4s + 5 = 0:
- Rows s^4 and s^3 take alternate coefficients
- Each further entry: (b₁a₂ − a₁b₂) ÷ b₁ from the two rows above
- First column sign changes = 2 → 2 roots in the right half-plane
Special case: none · Roots (numerical check): 0.287815 + j1.41609, 0.287815 − j1.41609, -1.28782 + j0.857897, -1.28782 − j0.857897
| Row | col 1 | col 2 | col 3 |
|---|---|---|---|
| s^4 | 1 | 3 | 5 |
| s^3 | 2 | 4 | 0 |
| s^2 | 1 | 5 | 0 |
| s^1 | -6 | 0 | 0 |
| s^0 | 5 | 0 | 0 |
GATE tip
For a range of K that keeps a system stable, keep K symbolic in the array mentally and require every first-column entry to be positive. The K that makes a row zero gives the frequency of oscillation from the auxiliary equation.
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Second-Order System Calculator
Mₚ = e^(−ζπ/√(1−ζ²)), tₚ = π/ω_d
ω_d = ωₙ√(1 − ζ²), t_r = (π − arccos ζ)/ω_d, t_s ≈ 4/(ζωₙ) (2 %)
Example: A system with ζ = 0.5 and ωₙ = 4 rad/s (s² + 4s + 16): Peak overshoot Mₚ = 16.3034 %.
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Steady-State Error Calculator
e_ss = A/(1 + Kₚ), A/K_v, A/Kₐ
Kₚ = lim G(s), K_v = lim sG(s), Kₐ = lim s²G(s) as s → 0
Example: G(s) = 10 / [s(s + 1)(s + 5)] with a unit ramp input: Steady-state error (ramp) e_ss = 0.5.
Frequently asked questions
What if a first-column entry is zero?
Replace it with a small positive ε and continue; sign changes as ε → 0⁺ give the count.
What does a row of zeros mean?
The polynomial has roots symmetric about the origin, often a pair on the jω axis. Replace the row with the derivative of the auxiliary polynomial from the row above.
Sources
- The International System of Units (SI), 9th edition (SI Brochure), Bureau International des Poids et Mesures .
Last reviewed: . Factors are stored exactly and computed with 50-digit decimal arithmetic. See how we verify conversion factors.
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APA
Scientific Converters. (2026, October 15). Routh-Hurwitz Calculator. Retrieved October 25, 2026, from https://scientificconverters.com/calculators/routh-hurwitz-calculator/
IEEE
Scientific Converters, "Routh-Hurwitz Calculator," scientificconverters.com. Accessed: Oct. 25, 2026. [Online]. Available: https://scientificconverters.com/calculators/routh-hurwitz-calculator/