Introductory feedback control

Why this matters

Lithium batteries charged in heat wear quickly and risk becoming dangerous, so the swap station’s charging cabinet must hold its temperature with a cooling fan. Ways of commanding the fan range from the simplest, on when hot and off when cool, to a PID controller that adjusts fan power continuously. The same principle is used in a drone’s flight controller, holding attitude hundreds of times a second.

The feedback block diagram

A simple cabinet model: charging heat $Q$ warms it, the fan removes up to $P_{\max }$, and the walls exchange heat with the surrounding air $T_a$ at rate $h$. With no fan at all, the temperature settles at $T_a+Q/h$, here $32+300/10=62$ °C, which is far too hot.

ON/OFF, P, I and D

• ON/OFF runs the fan at full when too hot and off when cool enough. It needs hysteresis (a dead band), such as on above 30.5 °C and off below 29.5 °C; otherwise the fan switches so often it wears out. The temperature therefore always oscillates within that band
• P (proportional) commands in proportion to the present error: the hotter, the harder the fan runs. It usually leaves a steady-state error, because some error must remain to keep the fan running
• I (integral) accumulates the error over time. As long as an error remains, the command keeps rising until the steady-state error disappears
• D (derivative) looks at the rate of change of the error, helping reduce overshoot, but is sensitive to noise; slow temperature processes often omit D

def cabinet(mode, kp=0.3, ki=0.002, t_end=3600, dt=1.0):
    C, h, Ta, Q, Pmax, sp = 20000.0, 10.0, 32.0, 300.0, 600.0, 30.0   # J/K, W/K, °C, W, W, °C
    temp, integ, on, switches, hist = 32.0, 0.0, False, 0, []
    for _ in range(int(t_end / dt) + 1):
        e = temp - sp                                    # positive when too hot
        if mode == "ON/OFF":
            new_on = True if temp > sp + 0.5 else False if temp < sp - 0.5 else on
            switches += new_on != on
            on = new_on
            u = 1.0 if on else 0.0
        elif mode == "P":
            u = min(max(kp * e, 0.0), 1.0)
        else:                                            # PI, not integrating while saturated
            integ += e * dt
            raw = kp * e + ki * integ
            u = min(max(raw, 0.0), 1.0)
            if u != raw:
                integ -= e * dt
        temp += dt * (Q - Pmax * u - h * (temp - Ta)) / C
        hist.append(temp)
    last = hist[-1200:]                                  # the last 20 minutes
    return min(last), max(last), switches


for mode in ("ON/OFF", "P", "PI"):
    lo, hi, sw = cabinet(mode)
    print(f"{mode:<6} last 20 min: {lo:.2f}–{hi:.2f} °C" + (f", fan switched {sw} times" if mode == "ON/OFF" else ""))

ON/OFF last 20 min: 29.50–30.51 °C, fan switched 52 times
P      last 20 min: 31.68–31.68 °C
PI     last 20 min: 30.00–30.00 °C

Integral windup

If the command saturates, for example the fan is already at full but it is still too hot, the I term keeps accumulating with no effect. When the temperature reaches the target, the excess accumulation keeps the fan on until it is too cold. This is integral windup. The example prevents it the simplest way, by pausing accumulation while the command is saturated. PX4’s flight controller also limits the I term in its rate loop.

Module lab

Common mistakes

Summary

• ON/OFF is simple but always oscillates, and needs hysteresis
• P reduces error but usually leaves a steady-state error; I removes it; D reduces overshoot but is noise-sensitive
• PI suits slow processes such as temperature
• Integral windup must be prevented when the command saturates

Check your understanding

1. A cabinet has $T_a=30$ °C, $Q=200$ W and $h=8$ W/K. Without a fan, where does its temperature settle?
2. A P controller has $K_p=0.3$ and an error of 2 °C. What is the command?
3. Which PID term removes steady-state error?
4. Why does ON/OFF need hysteresis?
5. When does integral windup occur?