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What is Routh–Hurwitz stability criterion?

A method that decides, from the coefficients of the characteristic equation alone and without solving it, whether all its roots (the closed-loop poles) lie in the left half of the s-plane. A necessary condition is that all coefficients have the same sign with no missing term; in addition, the system is stable if the first column of the Routh array all has the same sign, and the number of sign changes equals the number of roots in the right half-plane. A cubic s³ + a₂s² + a₁s + a₀ = 0 is stable when all coefficients are positive and a₂a₁ > a₀. It is used to find the range of a gain for which a system is stable. Routh's method and Hurwitz's method give the same conditions.

Japanese name: ラウス・フルビッツの安定判別法

Routh–Hurwitz stability criterion appears in 2 practice questions across 1 topic, including Control Engineering Basics.

Field: Control Engineering Basics

Try it: questions about Routh–Hurwitz stability criterion

A unity-feedback system has the loop transfer function K/(s(s + 1)(s + 2)). Which of the following is the range of K for which the closed-loop system is stable?

  1. A0 < K < 2
  2. B0 < K < 3
  3. C0 < K < 6
Show answer and explanation

Answer: C. 0 < K < 6

The characteristic equation is s(s + 1)(s + 2) + K = s³ + 3s² + 2s + K = 0. By the Routh–Hurwitz stability criterion, a cubic s³ + a₂s² + a₁s + a₀ = 0 is stable when all coefficients are positive and a₂a₁ > a₀, so 3 × 2 > K, that is, 0 < K < 6. At K = 6 the poles move to s = ±j√2, giving a sustained oscillation at an angular frequency of about 1.41 rad/s (a period of about 4.4 s). For larger K, the oscillation keeps growing.

Which of the following is closest to the gain margin of the loop transfer function L(s) = 2/(s(s + 1)(s + 2))?

  1. AAbout 4.8 dB
  2. BAbout 9.5 dB
  3. CAbout 15.6 dB
Show answer and explanation

Answer: B. About 9.5 dB

The phase is −90° − tan⁻¹ω − tan⁻¹(ω/2); the two angles add up to 90° when ω × (ω/2) = 1, so the phase reaches −180° at ω = √2 rad/s. The gain there is |L(j√2)| = 2/(√2 × √3 × √6) = 1/3, so the gain margin is 20 log₁₀3 ≈ 9.5 dB. Multiplying the gain by 3 (K = 6) would reach the limit, which agrees with the stable range 0 < K < 6 found with the Routh–Hurwitz stability criterion.

Routh–Hurwitz stability criterion in the explanations

Questions that mention Routh–Hurwitz stability criterion (2)

  1. A unity-feedback system has the loop transfer function K/(s(s + 1)(s + 2)). Which of the following is the range of K for which the closed-loop system is stable?
  2. Which of the following is closest to the gain margin of the loop transfer function L(s) = 2/(s(s + 1)(s + 2))?

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Related terms

Transfer function Gain margin Block diagram Bode plot Feedback control Feedforward control First-order system Integral windup Laplace transform Overshoot

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