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

Saber Omidi, Rene Akupan Ebunle, Se Young Yoon

Abstract

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.

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

Abstract

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.
Paper Structure (13 sections, 2 theorems, 42 equations, 7 figures, 2 tables, 1 algorithm)

This paper contains 13 sections, 2 theorems, 42 equations, 7 figures, 2 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Data-Driven Derivative Feedback Control Through Policy Iteration
  3. Derivative Feedback LQR Control for LTI System
  4. Model-Based Policy Iteration (PI)
  5. Model-free Policy Iteration
  6. Data-Driven Derivative Feedback Control Through System Identification
  7. Dynamic Mode Decomposition with Control
  8. Prediction Error Minimization
  9. Implementation and Testing on a Magnetic Levitation System
  10. Active Magnetic Levitation (AML) System
  11. Simulation Result
  12. Experimental Result
  13. Conclusion

Key Result

Theorem 2.1

Let $\boldsymbol{K}_1$ to be any stabilizing state DFC gain for the system LTI system and let $\boldsymbol{P}_i>\boldsymbol{0}$ represent the unique positive definite solution to the equation where $\boldsymbol{A}_{i}^{-1}=\boldsymbol{A}^{-1}(\boldsymbol{I}+\boldsymbol{BK}_{i})$. For $\boldsymbol{K}_{i+1}$ defined as for $i=0,1,2,\ldots,$ the following properties hold:

Figures (7)

  • Figure 1: Schematic of the magnetic levitation system.
  • Figure 2: Top: The evolution of cost function values computed via \ref{['cost_iteration']} and \ref{['finite value DFC cost functiont']} over three epochs. Middle: The Frobenius norm of the error between the iterative $\boldsymbol{P}_i$ and the model-based solution \ref{['DFC ARE']} over the PI iterations of the first epoch. Bottom: The convergence of the control gain error norm $||\boldsymbol{K}_i - \boldsymbol{K}_\text{ARE}||_F$ over the PI iterations of the first epoch.
  • Figure 3: The experimental setup for the two-disk magnetic levitation (MagLev Model 730).
  • Figure 4: Frequency response of a candidate identified model vs uncertainty range of measured data
  • Figure 5: Convergence of the Algorithm \ref{['algorithm1']} observed for the experimental test. The top graph shows the cost function value over four epochs. A dashed line at $0.0647$ is the nominal cost achieved by $\boldsymbol{K}_\text{ARE}$, which serves as the reference point. The graph also shows the cost values for the solutions of two identified models: $\boldsymbol{K}_{id,1}$ (solid) and $\boldsymbol{K}_{id,2}$ (dotted). The middle and bottom graphs illustrates the convergence of the PI iteration during the first epoch ($\kappa=1$), showing the Frobenius norm of iterative updates for the value matrix ($\|\boldsymbol{P}_i - \boldsymbol{P}_{i-1}\|_F$) and the control gain ($\|\boldsymbol{K}_i - \boldsymbol{K}_{i-1}\|_F$), respectively.
  • ...and 2 more figures

Theorems & Definitions (5)

  • Theorem 2.1
  • proof
  • Theorem 2.2
  • proof
  • Remark 1