Routh–Hurwitz Criterion: Stability Without Finding Roots
This animation explains how engineers determine closed-loop system stability without solving for polynomial roots. Starting from the characteristic equation 1+G(s)H(s)=0, it maps pole locations in the s-plane to stability outcomes, then introduces the Routh array. A side-by-side Routh table and s-plane show how sign changes in the first column directly count right-half-plane roots. Useful for control systems students learning quick, root-free stability analysis for higher-order polynomials.
16:9 · every frame verified for overlaps, spacing and edges before rendering
The prompt that made it
Create a 90-second animated educational video titled: “Routh–Hurwitz Criterion: Stability Without Finding Roots” Audience: Undergraduate engineering students studying Control Systems. Format: 16:9, Full HD, clean light background, professional engineering animation, dark-blue headings, large readable equations, clear English narration. Start with the closed-loop characteristic equation: 1 + G(s)H(s) = 0 Then show a characteristic polynomial: P(s) = aₙsⁿ + aₙ₋₁sⁿ⁻¹ + ... + a₁s + a₀ Animate an s-plane. Show: Left-half-plane poles → Stable Right-half-plane poles → Unstable Imaginary-axis poles → Special stability condition Then show a high-order polynomial such as: s⁶ + 3s⁵ + 7s⁴ + 12s³ + 15s² + 8s + 4 = 0 Ask visually: “Do we really need to calculate all six roots?” Introduce the Routh–Hurwitz criterion. Narration: “The Routh–Hurwitz criterion determines the number of roots of the characteristic equation that lie in the right-half of the s-plane, without explicitly calculating the roots.” Show the central rule: NUMBER OF SIGN CHANGES IN THE FIRST COLUMN NUMBER OF RIGHT-HALF-PLANE ROOTS Animate a Routh table on the left and an s-plane on the right. End with: 0 sign changes → No RHP roots 1 sign change → 1 RHP root 2 sign changes → 2 RHP roots Final question on screen: “Can we determine stability without solving the polynomial?” Keep the mathematical notation accurate and readable.
Make the next one in this series
- Show how to handle a zero in the first column of the Routh table with the epsilon method.
- Animate the special case of an entire row of zeros and its link to symmetric root pairs.
- Demonstrate Routh–Hurwitz applied to a real feedback control system with a variable gain K.
