Stability: why raising the gain makes it diverge
When a pole crosses into the right half-plane, the system diverges. The relationship between gain, delay and stability margins.
Poles and stability
The pole positions of the closed-loop transfer function set the response.
| Pole position | Response |
|---|---|
| On the real axis, left side | Converges without ringing |
| Left side, with an imaginary part | Converges with ringing |
| On the imaginary axis | Oscillates forever |
| Right side | Diverges |
As you raise the gain the poles move, and at some point they cross the imaginary axis. This is what is behind "keep raising P and it suddenly starts to shake" (the root locus).
Delay eats stability
Sensor filters, ADC conversion, calculation time and the PWM update are all delay. Delay makes the signal come back late, so feedback acts to increase the error instead of reducing it. So with the same gain, a system diverges more easily when the control period is slow or the filter is heavy.
Practical measures
- Do not filter the sensor harder than necessary
- Make the control period comfortably faster than the plant (10 to 20 times the bandwidth)
- Limit the output and the integrator so the system does not run away even with a large error
- Check the stability margins as numbers (the phase and gain margins in the next note)