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Bias correction and instrumental variables for direct data-driven model-reference control

Manas Mejari, Valentina Breschi, Simone Formentin, Dario Piga

TL;DR

It is demonstrated that using data-based covariance parameterization of the controller enables bias-correction and instrumental variable techniques within the data-driven optimization, thus reducing measurement noise effects as data volume increases.

Abstract

Managing noisy data is a central challenge in direct data-driven control design. We propose an approach for synthesizing model-reference controllers for linear time-invariant (LTI) systems using noisy state-input data, employing novel noise mitigation techniques. Specifically, we demonstrate that using data-based covariance parameterization of the controller enables bias-correction and instrumental variable techniques within the data-driven optimization, thus reducing measurement noise effects as data volume increases. The number of decision variables remains independent of dataset size, making this method scalable to large datasets. The approach's effectiveness is demonstrated with a numerical example.

Bias correction and instrumental variables for direct data-driven model-reference control

TL;DR

It is demonstrated that using data-based covariance parameterization of the controller enables bias-correction and instrumental variable techniques within the data-driven optimization, thus reducing measurement noise effects as data volume increases.

Abstract

Managing noisy data is a central challenge in direct data-driven control design. We propose an approach for synthesizing model-reference controllers for linear time-invariant (LTI) systems using noisy state-input data, employing novel noise mitigation techniques. Specifically, we demonstrate that using data-based covariance parameterization of the controller enables bias-correction and instrumental variable techniques within the data-driven optimization, thus reducing measurement noise effects as data volume increases. The number of decision variables remains independent of dataset size, making this method scalable to large datasets. The approach's effectiveness is demonstrated with a numerical example.

Paper Structure

This paper contains 10 sections, 8 theorems, 55 equations, 3 figures.

Table of Contents

  1. INTRODUCTION
  2. PROBLEM FORMULATION
  3. DATA-BASED CHARACTERIZATION OF THE CLOSED-LOOP DYNAMICS
  4. Model-reference matching with noise-free data
  5. MODEL-REFERENCE MATCHING WITH NOISY DATA
  6. Bias-correction approach
  7. Instrumental variable technique
  8. NUMERICAL EXAMPLE
  9. CONCLUSION
  10. APPENDIX

Key Result

Proposition 1

Let Assumption asm:PE be satisfied. Let us assume that the matrices $G_x, G_r, G_v$ satisfy eq:G. Then, by applying the control law $u(t)= K_x x(t) + K_r r(t)$ to system eq:system, the resulting closed-loop dynamics can be expressed as, where $\bar{W}_0 = A\bar{V}_0 - \bar{V}_1$ is a noise-induced term.

Figures (3)

  • Figure 1: Comparison of bias-correction (BC), instrumental variable (IV) and averaging (Avg.) strategy proposed in bpft21.
  • Figure 2: Reference tracking: Desired state $x_{d,2}(t)$ (red); mean (dashed black) and std. deviation (blue shaded area) of the closed-loop state $x_2(t)$ over $100$ MC runs, avg. SNR= $7$ dB.
  • Figure 3: Effect of data-length $T$ on the performance.

Theorems & Definitions (16)

  • Proposition 1
  • proof
  • Remark 1
  • Theorem 1
  • Proposition 2
  • Remark 2: Computational efficiency
  • Proposition 3
  • proof
  • Proposition 4
  • proof
  • ...and 6 more