Controller structures, PID ideas and design trade-offs

Give each action a precise meaning

An ideal parallel PID expression is u=Kp e+Ki z+Kd de/dt with dz/dt=e. The proportional contribution depends on current error. The integral state z accumulates signed error over time and retains a value when error becomes zero. The derivative contribution responds to the error’s rate, not knowledge of its future. Gain units must convert all three contributions into the same input unit.

In a stable, feasible constant-reference regime, integral action can support a nonzero steady actuator input with zero measured error. This helps reject certain constant disturbances. The qualification matters: integral action cannot deliver unavailable actuator capacity, remove sensor bias or guarantee stability. The additional state changes the transient response and must be included in the analysis.

Represent the input that the plant actually receives

An actuator may saturate in magnitude or be limited in its rate of change. If a controller continues integrating an error while its request cannot be applied, its stored contribution may grow far beyond the useful range. This is integral windup. Later recovery can be delayed while that stored contribution unwinds. Logging only the requested input conceals this mismatch.

Anti-windup arrangements can restrict accumulation or use the difference between requested and applied input to adjust the integral state. Their detailed logic depends on direction, bounds and controller structure. Preventing further accumulation that drives saturation is different from preventing useful recovery. A teaching sketch of saturation is not a validated protection function.

Treat noise and reference changes deliberately

Ideal differentiation amplifies high-frequency measurement variation. Practical derivative action is band limited. Filtering reduces that amplification while adding dynamic effects that belong in the model. Differentiating measured output rather than a stepped reference can avoid a reference-change derivative kick, but changes the controller structure and requires consistent signs.

Select a controller by declared requirements, model uncertainty and constraints. Compare command tracking, disturbance rejection, noise sensitivity and actuator duty. Several gains that fit one trace can behave differently under a changed load. A retained integral state also affects restart or mode changes; a smooth transfer requires a deliberate initialization and tracking strategy.

A prescribed error reveals the accumulation problem

Hold e=0.5 m/s as an imposed illustration, set Kp=2, Ki=0.5/s and z(0)=0, with no derivative action. Then z=0.5t metres and requested u=1+0.25t m/s. An illustrative upper applied-input limit of 2 m/s is reached at 4 s. At 10 s the request is 3.5 m/s while the applied value is 2 m/s.

The imposed error is not a solution of the earlier closed-loop plant. It isolates controller accumulation so the two signals can be compared. A complete recovery prediction would need plant dynamics, lower and upper limits, anti-windup law and initial states.

Try the reasoning

1. If error becomes zero, must a PI controller output become zero?

   Show the answer and reasoning

   No. The proportional term vanishes, but the accumulated integral state can retain the input needed to balance the load.

2. If error changes from +.5 to −.5 m/s, does z instantly change sign?

   Show the answer and reasoning

   No. Its slope changes sign. The stored integral state changes continuously under this ideal integration law.

3. Why does a quieter filtered measurement not automatically improve control?

   Show the answer and reasoning

   The filter also changes delay and phase. Noise reduction must be assessed together with the resulting closed-loop dynamics.

What to carry forward

Choose and implement controller actions around the measured plant, constraints and stored states.