Stability: Routh-Hurwitz Criterion With easy examples.

The Routh-Hurwitz criterion decides stability algebraically, without finding poles. From the characteristic polynomial you build the Routh array; if every first-column entry shares the same sign, all poles are left-half-plane and the system is stable. Each sign change counts one unstable pole.

Factoring high-order polynomials by hand is impractical, yet stability is the first design question. Routh-Hurwitz answers it in minutes and, better, reveals the stable range of a parameter like controller gain K: exactly what a designer needs before tuning.

Key Concepts

• Characteristic equation: denominator polynomial a_n x s^n + … + a_0 = 0.
• Routh array: rows from polynomial coefficients; each new row derives from the two above.
• Stability rule: stable iff all first-column entries are nonzero with the same sign; sign changes count right-half-plane poles.
• Zero row: signals roots symmetric about the origin; build the auxiliary equation from the row above, differentiate, and continue. Means marginal stability.
• Zero leading element: replace with small epsilon and take the limit.
• Necessary condition: all coefficients present and same-signed; otherwise unstable with no array needed.

Formulas

• Construction: rows s^n down to s^0; each entry = (diagonal products' difference) / first element of the row above.
• Cubic shortcut: s^3 + a s^2 + b s + c: s^1 entry = (a x b - c)/a; stability needs a,b,c > 0 and a x b > c.
• Assumptions: linear time-invariant, polynomial characteristic equation (approximate delays first).

Worked Example

Stability of s^3 + 6s^2 + 11s + 6 = 0:

1. s^3 row: 1, 11. s^2 row: 6, 6.
2. s^1: (6 x 11 - 1 x 6)/6 = 60/6 = 10. s^0: 6.
3. First column 1, 6, 10, 6: all positive, zero sign changes → stable (roots -1, -2, -3).
4. Had the constant been 66, the s^1 entry would hit zero: the edge of instability. Keep K symbolic and the first-column inequalities give the exact stable gain range.

Common Misconceptions

• "It finds pole locations." It only counts right-half-plane poles, never their positions.
• "A zero row means unstable." It means symmetric roots, often an imaginary-axis pair: marginally stable. The auxiliary equation identifies them.
• "It handles time delays." e^(-tau x s) is not polynomial; approximate it (e.g. Pade) before building the array.

Real-World Applications

• Gain design: maximum stable controller gain K.
• Parameter ranges: stable spring/mass/circuit values.
• Filter checks: active-filter stability before building.
• Interviews/exams: the fastest hand stability check in classical control.

Routh-Hurwitz settles stability from the polynomial alone: build the array, check first-column signs, count sign changes for unstable poles. Zero rows and zero leading elements have standard fixes. Leave gain symbolic and solve the inequalities for the exact stable range.