Control systems and models

Why this matters

A flight controller decides hundreds of times per second how hard to drive each motor. Without understanding how the drone responds to commands, tuning becomes guesswork. UAT 314 covered feedback and PID; this course goes deeper into models, response, practical PID implementation and the structure of the PX4 and ArduPilot autopilots.

The whole course uses one hypothetical case: a lab tuning a 1.5 kg training quadcopter in SITL before real flights. All numbers are made up for practice. The Python code for every module can be downloaded from /downloads/uat-206/.

The closed-loop block diagram

A closed-loop system measures the output $y$ and compares it with the desired value $r$ to get the error $e=r-y$. The controller uses $e$ to compute a command $u$ for the aircraft, which is also pushed by disturbances such as wind. A sensor measures the result and feeds it back for the next comparison, like a driver watching the road and correcting the steering all the time (Åström and Murray, 2021).

A first-order motor model

When the throttle steps up, the propeller does not speed up instantly because of its inertia. Thrust rises as a first-order response $y(t)=K(1-e^{-t/\tau })$, where $\tau$ is the time constant, the time for the response to reach 63.2% of its final value. As a transfer function, a way of writing the input–output relationship in the Laplace domain, it is $G(s)=K/(\tau s+1)$ (Nise, 2019).

import math

DT = 0.01
thrust = [0.0, 1.09, 1.98, 2.71, 3.3, 3.79, 4.19, 4.52, 4.79, 5.01, 5.19, 5.34, 5.46, 5.55, 5.64,
          5.7, 5.76, 5.8, 5.84, 5.87, 5.89, 5.91, 5.93, 5.94, 5.95, 5.96, 5.97, 5.97, 5.98, 5.98, 5.99]
final = thrust[-1]
target = 0.632 * final
i = next(k for k, v in enumerate(thrust) if v >= target)
t63 = (i - 1) * DT + (target - thrust[i - 1]) / (thrust[i] - thrust[i - 1]) * DT
print(f"final thrust {final:.2f} N, 63.2% = {target:.2f} N, tau from the curve = {t63 * 1000:.1f} ms")

pts = [(k * DT, math.log(1 - v / final)) for k, v in enumerate(thrust[:15])]
slope = sum(t * y for t, y in pts) / sum(t * t for t, _ in pts)
print(f"tau from a log fit of the first 0.14 s = {-1 / slope * 1000:.1f} ms")

final thrust 5.99 N, 63.2% = 3.79 N, tau from the curve = 49.9 ms
tau from a log fit of the first 0.14 s = 49.6 ms

Vertical dynamics and linearisation

Vertically, the total thrust $T$ works against the weight $mg$, giving $m\ddot{z}=T-mg$. Propeller thrust is roughly proportional to the square of throttle, $T=k{u}^2$, which is nonlinear. But we mostly control around hover, so a straight line around that point is a good approximation: $\Delta T\approx 2k{u}_h\Delta u$. Beard and McLain (2012) use the same approach for small unmanned aircraft.

import math

M, G, N_MOTORS, K_MAX = 1.5, 9.80665, 4, 8.0      # kg, m/s², number of motors, N at full throttle
t_hover = M * G / N_MOTORS
u_hover = math.sqrt(t_hover / K_MAX)
gain = N_MOTORS * 2 * K_MAX * u_hover                # N per 1.0 throttle (all motors together)
print(f"hover thrust per motor {t_hover:.2f} N, hover throttle {u_hover:.3f}")
print(f"local gain {gain:.1f} N per unit throttle -> +0.01 throttle gives {gain * 0.01 / M:.2f} m/s^2")
print(f"thrust-to-weight ratio {N_MOTORS * K_MAX / (M * G):.2f}")

hover thrust per motor 3.68 N, hover throttle 0.678
local gain 43.4 N per unit throttle -> +0.01 throttle gives 0.29 m/s^2
thrust-to-weight ratio 2.18

Second-order transfer functions

Many systems combine two first-order parts, or have mass and spring behaviour, so they respond as second-order systems. The standard form is $G(s)={\omega }_0^2/(s^2+2\zeta {\omega }_{0}s+{\omega }_0^2)$, where ${\omega }_0$ is the natural frequency and $\zeta$ is the damping ratio, which sets how much the response oscillates. Module 2 uses this form to study response and stability.

Module lab

Common mistakes

Summary

• A closed loop measures the output, compares it with the reference and keeps correcting, while disturbances act on it.
• A motor can be approximated by a first-order model; $\tau$ is the time to reach 63.2% of the final value.
• Vertical dynamics $m\ddot{z}=T-mg$ with thrust $T=k{u}^2$ are linearised around hover.
• A second-order system is defined by ${\omega }_0$ and $\zeta$.

Check your understanding

1. A first-order system has $\tau =0.1$ s. When does the response reach 63.2% of its final value?
2. At $3\tau$, what percentage of the final value has been reached?
3. A 2 kg drone has four motors. What is the hover thrust per motor?
4. If $T=8u^2$ and hover thrust per motor is 2 N, what is the hover throttle?
5. Why linearise the model around hover?