Control Engineering: Fourth-Order Motion Setup

Overview

In this project, my partner and I designed and implemented a feedback and feedforward control system for a fourth-order PATO motion setup — a two-mass spring-damper system representative of the dynamics found in semiconductor lithography and high-speed printing. The primary objective was to achieve the fastest possible scanning motion over a 120-radian stroke while maintaining the lowest tracking error on the non-collocated (end-effector) side.

The project was conducted as part of the Control Engineering course at TU Eindhoven (Group 67, with Kayden Knapik) and written in IEEE conference format.

Full Report — Control Engineering: Fourth-Order Motion Setup

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System Identification

Before designing a controller, we performed frequency-domain system identification using a 3-point measurement technique. Broadband noise was injected into the plant, and three signals were recorded simultaneously: the disturbance, the plant input, and the plant output. This allowed us to compute an accurate open-loop FRF even under closed-loop conditions.

The Frequency Response Function was estimated using Welch's method with a Hanning window and 50% overlap:

H(f) = \frac{S_{dy}(f)}{S_{du}(f)}

where $S_{dy}$ is the cross-power spectral density between the disturbance and output, and $S_{du}$ is the cross-power spectral density between the disturbance and input.

The plant exhibits two distinct transfer functions:

Collocated (motor-side encoder): contains an anti-resonance/resonance pair
Non-collocated (end-effector): fourth-order roll-off with resonance only

Controller Design

Feedforward

Feedforward was designed to handle the bulk of reference tracking, with three components tuned sequentially:

Coulomb friction gain ( $K_{fc}$ ) — matched deceleration/acceleration errors
Viscous friction gain ( $K_{fv}$ ) — minimized constant-velocity error
Acceleration feedforward ( $K_{fa}$ ) — eliminated remaining error peaks

This reduced peak tracking error from 1.4 rad to 0.1 rad.

Feedback (Loop Shaping)

The feedback controller was designed using classical loop-shaping:

| Component | Parameters | |-----------|-----------| | Proportional Gain | 2.5 | | Notch Filter (60 Hz) | Zero damping: 0.01, Pole damping: 0.7 | | Notch Filter (3.45 Hz) | Zero damping: 0.1, Pole damping: 0.001 | | Lead-Lag Filter | Zero: 14 Hz, Pole: 60 Hz | | 2nd Order Low-Pass | Pole: 200 Hz, Damping: 0.2 |

The 60 Hz notch targeted the main resonance frequency, while the 3.45 Hz notch addressed a low-frequency mode caused by non-ideal mechanical coupling. The lead filter provided phase margin around the crossover frequency.

Results

The final controller achieved Region I performance:

RMS Tracking Error: 2.252 mrad (requirement: < 6 mrad)
Peak Tracking Error: 9.425 mrad (requirement: < 12 mrad)
Sensitivity margin ( $\|S(j\omega)\|$ ): below 6 dB

The system was validated on real hardware (Raspberry Pi-based real-time controller connected to the physical PATO setup), with Bode plots, Nyquist diagrams, and power spectral density analysis confirming stability and performance.

What I Learned

This project deepened my understanding of frequency-domain control design — from the subtleties of FRF measurement (coherence, spectral leakage) to the practical trade-offs of loop shaping (bandwidth vs. noise amplification). Tuning feedforward before feedback proved to be a powerful design philosophy that I've since applied in other projects.

Technologies Used

MATLAB, Simulink, Raspberry Pi (real-time control), physical PATO motion setup, IEEE LaTeX template