Cascade control

Example 1 What happens when the outer-loop gain is too high?

A simple roll model: a PI rate loop, motors with a 30 ms time constant and a 10 ms delay. Command a 20° tilt and change only the angle gain (the model’s values are not ArduPilot units, except that the angle gain uses the same range).

import math
import numpy as np

def step(k_angle, k_rp=0.15, k_ri=0.1, T=2.0, dt=0.0025, tau=0.03, J=0.02, delay=0.01):
    n = int(T / dt)
    ang = rate = u_act = integ = 0.0
    buf = [0.0] * int(delay / dt)
    out = []
    for _ in range(n):
        rate_cmd = k_angle * (math.radians(20) - ang)      # outer loop: angle -> rate
        e = rate_cmd - rate
        integ += e * dt
        buf.append(k_rp * e + k_ri * integ)                # inner loop: PI on rate
        u_act += (buf.pop(0) - u_act) * dt / tau           # motor lag
        rate += u_act / J * dt
        ang += rate * dt
        out.append(math.degrees(ang))
    out = np.array(out)
    overshoot = (out.max() - 20) / 20
    outside = np.where(abs(out - 20) > 0.4)[0]
    settle = (outside[-1] + 1) * dt
    rise = np.argmax(out >= 18) * dt
    return overshoot, settle, rise

for k in (3, 4.5, 9, 12):
    os_, ts, tr = step(k)
    print(f"angle P {k:>4}: rise {tr:.2f} s, overshoot {os_:5.1%}, settle (2%) {ts:.2f} s")

angle P    3: rise 0.53 s, overshoot  1.8%, settle (2%) 0.64 s
angle P  4.5: rise 0.37 s, overshoot 12.3%, settle (2%) 1.25 s
angle P    9: rise 0.23 s, overshoot 40.0%, settle (2%) 1.55 s
angle P   12: rise 0.20 s, overshoot 54.4%, settle (2%) 1.96 s

The higher the outer-loop gain, the faster the approach, but the larger the overshoot and the longer the settling, because the outer loop commands rates faster than the inner loop can follow. The inner loop must be clearly faster than the outer loop, so tune the inner loop first and then raise the outer loop. These numbers belong to the model; a real drone must be checked with logs.

Example 2 Tilting 20° at different accelerations

import math

presets = {"default 110000": 110000, "Medium 108000": 108000, "Slow 72000": 72000, "VerySlow 30000": 30000}   # cdeg/s²
ANGLE = 20                                      # degrees
for name, cdeg in presets.items():
    a = cdeg / 100                              # deg/s²
    t = 2 * math.sqrt(ANGLE / a)
    peak = math.sqrt(ANGLE * a)
    print(f"{name:<15} {a:6.0f} deg/s2: {t:.2f} s to {ANGLE} deg, peak rate {peak:5.1f} deg/s")

default 110000    1100 deg/s2: 0.27 s to 20 deg, peak rate 148.3 deg/s
Medium 108000     1080 deg/s2: 0.27 s to 20 deg, peak rate 147.0 deg/s
Slow 72000         720 deg/s2: 0.33 s to 20 deg, peak rate 120.0 deg/s
VerySlow 30000     300 deg/s2: 0.52 s to 20 deg, peak rate  77.5 deg/s

Slow settings suit camera drones or large drones with slow motors, while fast settings suit small, agile drones. Setting it faster than the motors can deliver saturates them and makes control worse, so match it to the aircraft rather than setting it as fast as possible.