Final exam — Altitude control of a drone

Thu, 11 Jun 2026 00:00:00 +0000

Context

This is the final exam of the Sampled Control Systems course I taught at ECE Paris to the full year group (500 students) from April to June 2026. Rather than a set of disconnected questions, it is built as a single running case study: all four parts work on the same system — the altitude control of an agricultural observation drone — and each part picks up where the previous one left off.

Duration: 1h30 — calculator and documents not allowed. 20 points, plus 2 bonus points.

The system under study: an observation drone holding a constant altitude over a field

What the paper covers

Warm-up — quiz and two exercises. Five multiple-choice questions on PID gain effects, the mapping between the Laplace plane and the $z$-plane, first-order dynamics and discrete stability; then an inverse Laplace transform by partial fractions, and the discretisation of a first-order plant behind a zero-order hold.

Part 1 — Modelling. From Newton’s second law on the vertical axis to the transfer functions of the drone: four brushless motors, fluid friction, and gravity treated as a constant disturbance. The motor itself is identified graphically from a measured step response ($K_m$, $\tau$), then the open-loop transfer function is factorised and its poles and zeros placed in the Laplace plane.

Part 2 — Closed-loop control. Black’s formula on the closed loop, then the final-value theorem applied by superposition to both the setpoint and the gravity disturbance — which exposes the steady-state error a proportional controller cannot remove. The part closes on integral action, a feedforward alternative (bonus), and the risk derivative action creates when the drone flies in gusty wind.

Part 3 — Discretisation and implementation. The controller moves onto an ESP32 behind a zero-order hold: backward-Euler substitution, the recurrence equations for the P, I and D terms, the low-pass filter the derivative branch needs in practice, a Python implementation to complete, and a comparison of continuous versus sampled step responses.

Part 4 — Filtering. Anti-aliasing ahead of the accelerometer’s ADC, then sensor fusion: which filter merges a 10 Hz GPS with a 100 Hz accelerometer, at what rate it runs, and how the two sensors’ standard deviations drive the trade-off between short-term dynamics and long-term drift.

The paper

The exam is written in French, as the course is taught in French. A full transcription with all figures and formulas is available on the French side of this site:

📄 Read the full transcription (French)

Solutions are not published.

Related documents: exam paper · answer sheet · all course material

PID | Etienne Chassaing

Final exam — Altitude control of a drone

Context

What the paper covers

The paper