Nonlinear System Identification of Variable-Pitch Propellers Using a Wiener Model

1. Problem Statement

The paper addresses the challenge of modeling and controlling Variable-Pitch Propellers (VPPs) for Unmanned Aerial Vehicles (UAVs). While VPPs offer significant advantages over fixed-pitch propellers (FPPs)—such as improved agility, efficiency, and fault tolerance through the dual control of motor speed ($\omega$) and blade pitch ($\beta$)—they introduce complex nonlinear dynamics.

• Complexity: The coupling between RPM and pitch actuation, combined with unsteady aerodynamic effects (e.g., wake-induced lags, vortex shedding), makes traditional quasi-steady models insufficient for fast transients.
• Goal: The authors aim to develop a compact, interpretable, and computationally efficient model suitable for real-time applications, specifically for Reinforcement Twinning (RT) frameworks. RT requires an adaptive digital twin that can be continuously updated via system identification to support model-free reinforcement learning agents.

2. Methodology

The study employs a hybrid approach combining first-principles physics with data-driven system identification.

A. Theoretical Framework: The Wiener Model

The authors propose a Wiener-like architecture to simplify the full electromechanical-aerodynamic system.

• Structure: The model separates the system into two blocks:
  1. Linear Dynamic Block: Processes the input commands (PWM signals for RPM and pitch) through linear first-order dynamics to estimate the internal states (rotor speed $\omega$ and pitch angle $\beta$). This assumes electrical dynamics are much faster than mechanical ones and that inflow dynamics are negligible or quasi-steady.
  2. Static Nonlinear Block: Maps the estimated states ($\omega, \beta$) to the output thrust ($T$) using a static nonlinear function.
• Key Innovation: Unlike traditional Wiener models where the static map is fixed by aerodynamic theory, this study treats the nonlinear mapping as data-driven. It is refined using experimental data to capture deviations from ideal aerodynamic laws and unsteady effects.

B. Experimental Setup

• Hardware: A custom VPP mechanism was built at the von Karman Institute, featuring a hollow shaft with a linear actuator driving a lever system to adjust blade pitch.
• Sensors:
  • Thrust: 10 kg load cell (amplified and filtered).
  • RPM: Infrared LED sensor with reflective tape.
  • Control: Closed-loop PI control for RPM and PD control for pitch.
• Data Collection:
  • Static Characterization: A 10x10 grid of RPM ($2000-6000$) and pitch ($-10^\circ$ to $10^\circ$) points to generate a static thrust map.
  • Dynamic Characterization: Step responses (5 up, 5 down) for both RPM and pitch across various operating points to capture transient dynamics.

C. Identification and Optimization Process

1. Initial Estimation: Time constants for the linear dynamic blocks were estimated via least-squares minimization on step responses.
2. Model Structure: The final model uses a 4th-order realization (two time constants for RPM, two for pitch) and a 3rd-order polynomial for the static thrust map:
   $$\hat{T} = c_{20}\omega^2 + c_{11}\omega\beta + c_{21}\omega^2\beta + c_{12}\omega\beta^2 + c_{30}\omega^3$$
3. Fine-Tuning: A forward sensitivity gradient descent method was used to optimize all parameters (time constants and polynomial coefficients) simultaneously. The cost function minimized the error between predicted and measured thrust over open-loop step sequences.

D. Control Validation

A preliminary model-based control system was implemented using two PID controllers (one for $\omega_{ref}$, one for $\beta_{ref}$) to track a thrust setpoint. The gains were optimized using the Nelder-Mead simplex algorithm to minimize thrust tracking error.

3. Key Contributions

• Novel Wiener Architecture for VPPs: The paper demonstrates that a Wiener model, where the static nonlinearity is identified from dynamic data rather than fixed theory, effectively captures the complex coupling of VPPs.
• Parsimonious Representation: The model reduces a complex 13-parameter physical system to a manageable set of 9 parameters (4 time constants + 5 polynomial coefficients) while maintaining high accuracy.
• Integration with Digital Twins: The work provides a practical foundation for the Reinforcement Twinning (RT) framework, offering a model that is both physically interpretable and computationally light enough for real-time updates.
• Experimental Validation: Comprehensive testing of both static and dynamic responses, including the identification of actuator limitations (e.g., delayed pitch response).

4. Results

• Static Accuracy: The fitted static thrust map achieved an adjusted $R^2$ of 0.996, confirming the propeller operates largely in a linear lift regime where thrust scales quadratically with speed.
• Dynamic Performance:
  • Initial time constants were identified (e.g., $T_{\omega,1} \approx 0.15s$, $T_{\beta,2} \approx 0.28s$).
  • After fine-tuning, the model accuracy improved by 38.3% (cost function reduction from $5.19 \times 10^{-4}$ to $3.2 \times 10^{-4}$).
  • The model successfully reproduced measured dynamics for step changes in RPM, pitch, and combined inputs.
• Control Performance:
  • The optimized dual-input controller (RPM + Pitch) achieved the thrust setpoint in approximately 0.5 seconds with no overshoot.
  • Comparison with an RPM-only controller showed that the VPP with dual control offers superior response, particularly for thrust reduction (step-down) scenarios.

5. Significance and Future Work

• Significance: This study bridges the gap between high-fidelity physics-based models (which are too slow for real-time control) and pure black-box machine learning (which lacks interpretability). The Wiener model offers a "sweet spot" for UAV control applications requiring rapid adaptation.
• Limitations: The current setup suffered from noise in the load cell and a slow, non-linear response from the linear actuator (LAC), which limited the fidelity of the pitch dynamics identification.
• Future Directions:
  • Upgrading hardware: Replacing the LAC with a custom controller, using high-resolution encoders for RPM, and employing a tri-axial thrust/torque sensor.
  • Reinforcement Twinning: The identified model will be integrated into the full RT framework to test adaptive control policies where the digital twin evolves alongside the learning agent.

In conclusion, the paper successfully validates a data-driven Wiener model as a robust, real-time capable digital twin for variable-pitch propellers, paving the way for advanced adaptive control strategies in UAVs.