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Control engineering
Critical gain from the Routh–Hurwitz cubic test
Largest gain for which the characteristic polynomial a₃s³ + a₂s² + a₁s + K has all its roots in the left half-plane.
- Ref
- CTL-13
- Rev
- v1
- Field
- Control engineering
- Result
- Kmax
For a₃s³ + a₂s² + a₁s + K = 0 with a₃ > 0, the roots all lie in the left half-plane when a₂ > 0, a₁ > 0, K > 0 and a₂a₁ > a₃K. The last condition sets the largest stable gain Kmax = a₂a₁/a₃. At that gain a pair of roots sits on the imaginary axis at ±j√(a₁/a₃) and the loop oscillates with constant amplitude. The gain K in use is entered to give the gain margin.
Largest gain for which the characteristic polynomial a₃s³ + a₂s² + a₁s + K has all its roots in the left half-plane. At this gain a pair of roots sits on the imaginary axis and the loop oscillates steadily.
Calculator
- Gain margin, Kmax / K3
- Gain margin in decibels9.542 dB
- Oscillation frequency at Kmax, √(a₁/a₃)1.414 rad/s
- Period of that oscillation4.443 s
What drives the result
Each variable is moved across its typical range while the others stay at the values above. The longest bar is the key driver.
- With s³ + 3s² + 2s + K, the loop is stable for K between 0 and 6. At the limit two roots sit at ±j1.414 rad/s and the loop rings with a period of 4.44 s.
- The gain in use is 2, a factor 3 below the limit, or 9.54 dB. Common practice asks for a margin of at least about 6 dB (a factor of 2), so there is some room before the loop becomes uncomfortably oscillatory.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing () | Local step | Change in result |
|---|---|---|---|---|---|---|
| a1 Coefficient of s | 0.5 to 10 | 1.5 | 30 | 28.5 | +10% | +10 % |
| a2 Coefficient of s² | 0.5 to 10 | 1 | 20 | 19 | +10% | +10 % |
| a3 Coefficient of s³ | 0.5 to 5 | 1.2 | 12 | 10.8 | +10% | −9.09 % |
| K Gain in use | 0.1 to 10 | 6 | 6 | 0 | +10% | 0 % |
Sweep of Coefficient of s²
| Coefficient of s² | Critical (maximum stable) gain |
|---|---|
| 0.5 | 1 |
| 1.688 | 3.375 |
| 2.875 | 5.75 |
| 4.063 | 8.125 |
| 5.25 | 10.5 |
| 6.438 | 12.88 |
| 7.625 | 15.25 |
| 8.813 | 17.63 |
| 10 | 20 |
Sweep one variable
Variables and how each one contributes
Critical (maximum stable) gain
What the equation gives you. Everything else feeds it.
Coefficient of s³
The coefficient of s³ in the characteristic polynomial. It must be positive. Inversely proportional: a larger leading coefficient gives a smaller stable gain range. Raising it by 10 % (to 1.1) moves the result by −9.09 %, from 6 to 5.455.
Coefficient of s²
The coefficient of s². It must be positive for stability. Directly proportional: more damping-like content widens the stable range. Raising it by 10 % (to 3.3) moves the result by +10 %, from 6 to 6.6.
Coefficient of s
The coefficient of s. It must be positive for stability. Directly proportional: it widens the stable range and raises the oscillation frequency. Raising it by 10 % (to 2.2) moves the result by +10 %, from 6 to 6.6.
Gain in use
The gain in use, which enters the constant term of the polynomial. No effect on Kmax, which depends only on a₃, a₂ and a₁. It sets how close to the limit the loop is: the gain margin is Kmax/K = 3 at the defaults.
Worked example with the values above
- The conditions a₂ > 0, a₁ > 0 and K > 0 are satisfied by the coefficients 3, 2 and 2.
- The cubic test needs a₂a₁ > a₃K, that is 3 × 2 = 6 > 1 K.
- So K < a₂a₁/a₃ = 6, and K = 2 is stable with a margin of 3.
Assumptions and limits
- A cubic characteristic polynomial with real constant coefficients, a₃ > 0, and the gain K appearing only in the constant term.
- Strict stability is what is tested. Equality a₂a₁ = a₃K is the boundary case of sustained oscillation.
Common mistakes
- Checking only that all coefficients are positive. That is necessary but, from third order up, not sufficient: a₂a₁ > a₃K must hold as well.
- Using it when the gain also changes the other coefficients. The inequality must then be re-derived with the gain in the right places.
Statistics and uncertainty
Kmax = a₂a₁/a₃ is a product-over-quotient, so independent relative errors combine in quadrature: 3 % in each of the three coefficients gives about 5 % in Kmax (√3 × 3). Coefficients that come from identified models are usually uncertain by much more, so a gain margin of only a few per cent should not be trusted.
Where it is used
- Finding the stability limit and gain margin of a third-order loop without computing the roots.
- Predicting the frequency at which a loop will oscillate when its gain is raised too far.
Maths you need
- Characteristic polynomial: the denominator of the closed loop, set to zero
- Left half-plane roots: stability means all roots have negative real part