Control Systems - Level 4
The I Term
Concept
The plant in this level has gravity pulling the mass down. With only a proportional controller, the mass settles at some point below the setpoint: once you’re there, the residual error generates just enough upward force to balance gravity. The P controller can’t eliminate this steady-state error without infinite gain.
Enter the integral term. It accumulates error over time:
u(t) = Kp · e(t) + Ki · ∫ e(t) dt
As long as there’s any nonzero error, the integral keeps growing and pumping
up the command. The only steady state is one where e = 0, which is exactly what we want!
Try it:
- Set Kp = 4, Ki = 0. Watch the mass settle below the setpoint.
- Slowly raise Ki. The steady-state error vanishes.
- Raise Ki too far: oscillation and sluggish overshoot (the integral accumulates too much before the error flips).
Aside, integral windup: if the actuator is saturated for a long time, the
integral can grow huge and cause massive overshoot later. Real implementations
clamp the integral (we do, via iMax).
Goal
Bring the mass to the setpoint (±0.05) and hold it there for 2 seconds despite gravity pulling it down. Try Kp around 4 and Ki around 0.8, and let the sim run for a few seconds so the integrator has time to work.