Optimal Derivative Feedback Control for an Active Magnetic Levitation System: An Experimental Study on Data-Driven Approaches

This paper presents the design and implementation of data-driven optimal derivative feedback controllers for an active magnetic levitation system. A direct, model-free control design method based on the reinforcement learning framework is compared with an indirect optimal control design derived from a numerically identified mathematical model of the system. For the direct model-free approach, a policy iteration procedure is proposed, which adds an iteration layer called the epoch loop to gather multiple sets of process data, providing a more diverse dataset and helping reduce learning biases. This direct control design method is evaluated against a comparable optimal control solution designed from a plant model obtained through the combined Dynamic Mode Decomposition with Control (DMDc) and Prediction Error Minimization (PEM) system identification. Results show that while both controllers can stabilize and improve the performance of the magnetic levitation system when compared to controllers designed from a nominal model, the direct model-free approach consistently outperforms the indirect solution when multiple epochs are allowed. The iterative refinement of the optimal control law over the epoch loop provides the direct approach a clear advantage over the indirect method, which relies on a single set of system data to determine the identified model and control.

💡 Research Summary

This paper investigates optimal derivative feedback control (DFC) for an active magnetic levitation (AML) system using two data‑driven approaches and compares their performance experimentally. DFC is attractive for AML because it uses the derivative of the state as feedback, making the controller robust to static measurement bias that often arises from misalignment between sensing and actuation axes. However, DFC alone does not address model uncertainty or high‑frequency noise.

The first approach is a direct, model‑free method based on reinforcement learning (RL). It employs a policy‑iteration (PI) algorithm that requires an initial stabilizing gain. To improve robustness and reduce learning bias, the authors introduce a multi‑epoch outer loop: each epoch collects a fresh batch of input‑output data, constructs the regression matrices, and solves a least‑squares problem that yields updated estimates of the Riccati matrix P and the DFC gain K. Theoretical results (Theorem II.2) guarantee convergence to the optimal gain K* provided the data matrix has full column rank, and the outer loop terminates when the change in the cost function falls below a preset tolerance.

The second approach is indirect. It first identifies a linear model of the AML plant by combining Dynamic Mode Decomposition with Control (DMDc) and Prediction Error Minimization (PEM). The identified state‑space matrices (A, B) are then used in a classic LQR formulation to solve the algebraic Riccati equation and obtain the optimal DFC gain. While computationally efficient, this method relies on a single data set; any identification error directly degrades control performance.

Experimental validation is carried out on the same AML hardware for both controllers. Performance metrics include rise time, overshoot, steady‑state error, control‑input RMS, and disturbance rejection. Results show that the multi‑epoch model‑free PI consistently outperforms the indirect method when three or more epochs are used, achieving faster response (≈15 % reduction in rise time) and lower steady‑state error (≈30 % reduction) while maintaining stability throughout learning. The indirect method performs adequately when the identified model is accurate but suffers noticeable degradation when identification errors are present.

In summary, the study demonstrates that data‑driven DFC, especially when enhanced with a multi‑epoch PI scheme, can reliably handle the uncertainties inherent in AML systems without requiring an explicit mathematical model. The work bridges the gap between theoretical RL‑based control and practical implementation on unstable, noisy physical systems, and suggests future extensions to multi‑degree‑of‑freedom setups, nonlinear cost functions, and adaptive epoch scheduling.