Module 3/5 · Weeks 7–9 · 27 h

Feedback control

DRT 344 Automation, Robotics and Intelligent Control Systems

About 90 minDraft, awaiting reviewLast updated 26 September 2026

By the end of this module you will be able to

Compare open-loop and closed-loop control
Explain the role of the P, I and D terms and their effect on the response
Calculate the steady-state error of proportional control
Tune a PID controller initially with the Ziegler–Nichols method and state its limits
Explain PX4's cascaded multicopter control and the tuning order

Prerequisites: DRT 111 module 4 (derivatives and integrals) · DRT 341 module 3

Why this matters

A multirotor is not inherently stable. Without a controller adjusting motor speeds hundreds of times per second, it would flip over in a fraction of a second. The controller behind almost every system, from drones and robots to air conditioners, is feedback control, and the most widely used form is PID.

Feedback control is like keeping a car in the middle of its lane. You see how far the car has drifted (the error) and turn the wheel to correct it; the bigger the drift, the more you turn. If a crosswind keeps pushing, you gradually hold the wheel over. If you see the car drifting fast, you correct in advance. These three behaviours are P, I and D.

Open loop versus closed loop

Open loop acts on a plan without measuring the result, such as running motors at fixed power from a table. If wind or load changes, the result drifts and nothing corrects it
Closed loop measures the real output with a sensor, compares it with the setpoint, and adjusts the command using the error $e = r - y$

The PID controller

u (t) = K_{p} e (t) + K_{i} \int_{0}^{t} e (τ) d τ + K_{d} \frac{d e}{d t}

Term	Responds to	Effect of increasing it	Caution
P (proportional)	The present error	Faster response	Too much causes oscillation; alone it usually leaves a residual error
I (integral)	Accumulated past error	Removes residual error	Too much causes oscillation and overshoot; needs anti-windup
D (derivative)	Rate of change of error	Damps oscillation	Amplifies noise; needs filtering

Integrator windup happens when the actuator is saturated, for example motors at full power, while the I term keeps accumulating. When the error reverses, the controller responds very slowly, so the I term must be limited (anti-windup). PX4 does this by clamping in its velocity loop.

Example 1 Why P alone does not reach the target

A first-order plant has steady-state gain $K = 2$ and a P controller with $K_{p} = 3$ . For a step setpoint:

e_{ss} = \frac{1}{1 + K _{p} K} = \frac{1}{1 + 3 \times 2} = \frac{1}{7} \approx 0.143

The system settles about 14.3% short of the target. Raising $K_{p}$ shrinks the error but never removes it, and a real system with delay starts to oscillate. Adding an I term removes the residual error.

Ziegler–Nichols tuning

Ziegler and Nichols (1942) proposed finding initial PID values by experiment. In one method, raise $K_{p}$ gradually (with I and D off) until the system oscillates steadily; record that gain as $K_{u}$ and the oscillation period as $T_{u}$ , then use the table (Åström & Murray, Table 11.1):

Controller	$K_{p}$	$T_{i}$	$T_{d}$
P	$0.5 K_{u}$	–	–
PI	$0.45 K_{u}$	$T_{u} /1.2$	–
PID	$0.6 K_{u}$	$T_{u} /2$	$T_{u} /8$

Example 2 Calculating PID gains from an experiment

On a motor test rig, the system oscillates steadily at $K_{u} = 2.0$ with period $T_{u} = 0.5$ s.

$K_{p} = 0.6 \times 2.0 = 1.2$
$T_{i} = 0.5/2 = 0.25$ s, so $K_{i} = K_{p} / T_{i} = 1.2/0.25 = 4.8$
$T_{d} = 0.5/8 = 0.0625$ s, so $K_{d} = K_{p} T_{d} = 1.2 \times 0.0625 = 0.075$

Åström and Murray warn that the Ziegler–Nichols rules use too little process information and often give closed loops that lack robustness. These values are only a starting point, and never drive a real drone to sustained oscillation; do it only on a test rig or in simulation.

Cascaded control in multirotors

PX4 uses a cascaded control architecture, a mix of P and PID controllers in layers.

Figure 2 Cascaded multicopter control loops in PX4

Loop	Type	Output	Example PX4 parameters
Position	P	Velocity setpoint	`MPC_XY_P`, `MPC_Z_P`
Velocity	PID	Acceleration setpoint	`MPC_XY_VEL_P_ACC`, `MPC_XY_VEL_I_ACC`, `MPC_XY_VEL_D_ACC`
Attitude	P on quaternion	Rate setpoint	`MC_ROLL_P`, `MC_PITCH_P`, `MC_YAW_P`
Angular rate	PID	Torque to the mixer	`MC_ROLLRATE_P/I/D`, `MC_PITCHRATE_P/I/D`

Inner loops run faster than outer loops. The rate loop takes filtered gyro data, and the gyro publication rate is set by IMU_GYRO_RATEMAX. The PX4 tuning guide stresses tuning the rate controller first, because it affects all flight modes; the attitude loop is then much easier. It suggests raising gains by 20–30% per iteration, then 5–10% for fine tuning, testing with fast step inputs in hover.

Class activity

Activity: Simulating an altitude controller

Write a Python simulation of a discrete first-order plant with P, PI and PID controllers.
Compare rise time, overshoot and residual error for each.
Add an actuator limit and observe integrator windup, then add anti-windup.
If available, read a PX4 SITL tuning log and identify which loop was tuned first.

Common mistakes

Watch out

Tuning outer loops before inner loops, fixing the wrong problem
Raising D without filtering, so motors vibrate and heat up
No anti-windup when actuators saturate
Treating Ziegler–Nichols values as final without checking robustness
Driving a real drone to its stability limit during tests

Summary

A closed loop measures the output and corrects using the error $e = r - y$
P reacts to the present error, I removes residual error, D damps oscillation
P control of a first-order plant leaves $e_{ss} = 1/ (1 + K_{p} K)$
Ziegler–Nichols PID: $K_{p} = 0.6 K_{u}$ , $T_{i} = T_{u} /2$ , $T_{d} = T_{u} /8$ , only a starting point
PX4 cascades P → PID → P → PID; tune the rate loop first

Check your understanding

Which PID term removes residual error?
A first-order plant with $K = 4$ uses P control with $K_{p} = 2$ . What is the steady-state error?
An experiment gives $K_{u} = 4$ and $T_{u} = 0.8$ s. Calculate PID $K_{p}$ , $K_{i}$ and $K_{d}$ by Ziegler–Nichols.
In PX4, which loop sends the rate setpoint to the rate loop?
Why tune the rate loop before the others?

Answers

The I (integral) term
$1/ (1 + 2 \times 4) = 1/9 \approx 0.111$ , about 11%
$K_{p} = 2.4$ ; $T_{i} = 0.4$ s so $K_{i} = 6$ ; $T_{d} = 0.1$ s so $K_{d} = 0.24$
The attitude loop, a P controller on the quaternion
Every outer loop acts through the rate loop; if the inner loop is poor, the outer loops cannot be tuned well, and it affects all flight modes

Key formulas

PID controller	$u (t) = K_{p} e (t) + K_{i} \int_{0}^{t} e (τ) d τ + K_{d} \frac{d e}{d t}$
Steady-state error (first-order plant, P control)	$e_{ss} = \frac{1}{1 + K _{p} K}$
Ziegler–Nichols ultimate-gain PID	$K_{p} = 0.6 K_{u}, T_{i} = \frac{T _{u}}{2}, T_{d} = \frac{T _{u}}{8}$
Converting to gains	$K_{i} = \frac{K _{p}}{T _{i}}, K_{d} = K_{p} T_{d}$

Key references

Åström, K. J., & Murray, R. M. (2021). Feedback systems: An introduction for scientists and engineers (2nd ed.). Princeton University Press. link
Nise, N. S. (2019). Control systems engineering (8th ed.). Wiley.
Ziegler, J. G., & Nichols, N. B. (1942). Optimum settings for automatic controllers. Transactions of the ASME, 64(8), 759–765. link
PX4 Autopilot. Controller diagrams; Multicopter PID tuning guide. PX4 user guide (main). link
Beard, R. W., & McLain, T. W. (2012). Small unmanned aircraft: Theory and practice. Princeton University Press.
Quan, Q. (2017). Introduction to multicopter design and control. Springer.

In class / field

Lecture, case discussion and in-class problem solving

Learning evidence: Quiz results and submitted exercises

Module quiz

This is a formative self-check, not a graded exam

Knowledge domain: Control, autopilot and navigation