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Model-free Dynamic Mode Adaptive Control using Matrix RLS

Parham Oveissi, Ankit Goel

TL;DR

The paper tackles control of complex, high-dimensional, time-varying systems for which explicit models are unavailable. It proposes Dynamic Mode Adaptive Control (DMAC), a model-free data-driven framework that combines a dynamics-approximation module based on a matrix recursive least squares formulation with a sparse-measurement, reference-tracking controller. DMAC links to dynamic mode decomposition, showing equivalence of its identification objective to DMD, and uses a forgetting factor to adapt to changing dynamics. Through four numerical examples and sensitivity analyses, it demonstrates robust performance with respect to learning hyperparameters and system parameters, while offering computational advantages over reinforcement learning approaches.

Abstract

This paper presents a novel, model-free, data-driven control synthesis technique known as dynamic mode adaptive control (DMAC) for synthesizing controllers for complex systems whose mathematical models are not suitable for classical control design. DMAC consists of a dynamics approximation module and a controller module. The dynamics approximation module is motivated by data-driven reduced-order modeling techniques and directly approximates the system's dynamics in state-space form using a matrix version of the recursive least squares algorithm. The controller module includes an output tracking controller that utilizes sparse measurements from the system to generate the control signal. The DMAC controller design technique is demonstrated through various dynamic systems commonly found in engineering applications. A systematic sensitivity study demonstrates the robustness of DMAC with respect to its own hyperparameters and the system's parameters.

Model-free Dynamic Mode Adaptive Control using Matrix RLS

TL;DR

The paper tackles control of complex, high-dimensional, time-varying systems for which explicit models are unavailable. It proposes Dynamic Mode Adaptive Control (DMAC), a model-free data-driven framework that combines a dynamics-approximation module based on a matrix recursive least squares formulation with a sparse-measurement, reference-tracking controller. DMAC links to dynamic mode decomposition, showing equivalence of its identification objective to DMD, and uses a forgetting factor to adapt to changing dynamics. Through four numerical examples and sensitivity analyses, it demonstrates robust performance with respect to learning hyperparameters and system parameters, while offering computational advantages over reinforcement learning approaches.

Abstract

This paper presents a novel, model-free, data-driven control synthesis technique known as dynamic mode adaptive control (DMAC) for synthesizing controllers for complex systems whose mathematical models are not suitable for classical control design. DMAC consists of a dynamics approximation module and a controller module. The dynamics approximation module is motivated by data-driven reduced-order modeling techniques and directly approximates the system's dynamics in state-space form using a matrix version of the recursive least squares algorithm. The controller module includes an output tracking controller that utilizes sparse measurements from the system to generate the control signal. The DMAC controller design technique is demonstrated through various dynamic systems commonly found in engineering applications. A systematic sensitivity study demonstrates the robustness of DMAC with respect to its own hyperparameters and the system's parameters.
Paper Structure (14 sections, 4 theorems, 42 equations, 15 figures)

This paper contains 14 sections, 4 theorems, 42 equations, 15 figures.

Table of Contents

  1. INTRODUCTION
  2. Dynamic Mode Adaptive Control
  3. Dynamic Mode Approximation
  4. Connection with Dynamic Mode Decomposition
  5. Control Law Update
  6. Numerical Examples
  7. Mass-Damper-Spring System
  8. 3-Mass system
  9. Van Der Pol
  10. Burgers' Equation
  11. Conclusions
  12. Appendix
  13. Full-State Feedback Control with Integral Action
  14. Matrix RLS

Key Result

Proposition 2.1

Consider the function eq:J_k_def. Then, the minimizer $\Theta_k$ satisfies where, for all $k \geq 0,$$\gamma_k \stackrel{\triangle}{=} \lambda + \phi_{k-1}^{\rm T} {\mathcal{P}}_{k-1} \phi_{k-1},$ and $\Theta_0 = 0,$${\mathcal{P}}_0 \stackrel{\triangle}{=} R_\Theta^{-1}.$

Figures (15)

  • Figure 1: Dynamic Mode Adaptive Control (DMAC) architecture for model-free, data-driven, and learning-based control of sampled-data systems.
  • Figure 2: Closed-loop response of \ref{['eq:MCK']} with DMAC. a) shows the output $y_k$ and the reference signal $r,$ b) shows the control signal $u_k,$ c) shows the absolute value of the tracking error $z_k$ on a logarithmic scale, and d) shows the estimate matrix $\Theta_k$ computed by DMAC.
  • Figure 3: Effect of DMAC hyperparameters on the closed-loop performance.
  • Figure 4: Effect of varying the system's physical parameters on the closed-loop performance.
  • Figure 5: Three masses connected in series.
  • ...and 10 more figures

Theorems & Definitions (9)

  • Definition 2.1
  • Proposition 2.1
  • proof
  • Proposition 5.1
  • proof
  • Proposition 5.2
  • proof
  • Proposition 5.3
  • proof