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Is Data All That Matters? The Role of Control Frequency for Learning-Based Sampled-Data Control of Uncertain Systems

Ralf Römer, Lukas Brunke, Siqi Zhou, Angela P. Schoellig

TL;DR

This work uses Gaussian processes to learn a continuous-time model and compute a corresponding discrete-time controller, for which the result is an uncertain sampled-data control system, for which it derives robust stability conditions.

Abstract

Learning models or control policies from data has become a powerful tool to improve the performance of uncertain systems. While a strong focus has been placed on increasing the amount and quality of data to improve performance, data can never fully eliminate uncertainty, making feedback necessary to ensure stability and performance. We show that the control frequency at which the input is recalculated is a crucial design parameter, yet it has hardly been considered before. We address this gap by combining probabilistic model learning and sampled-data control. We use Gaussian processes (GPs) to learn a continuous-time model and compute a corresponding discrete-time controller. The result is an uncertain sampled-data control system, for which we derive robust stability conditions. We formulate semidefinite programs to compute the minimum control frequency required for stability and to optimize performance. As a result, our approach enables us to study the effect of both control frequency and data on stability and closed-loop performance. We show in numerical simulations of a quadrotor that performance can be improved by increasing either the amount of data or the control frequency, and that we can trade off one for the other. For example, by increasing the control frequency by 33%, we can reduce the number of data points by half while still achieving similar performance.

Is Data All That Matters? The Role of Control Frequency for Learning-Based Sampled-Data Control of Uncertain Systems

TL;DR

This work uses Gaussian processes to learn a continuous-time model and compute a corresponding discrete-time controller, for which the result is an uncertain sampled-data control system, for which it derives robust stability conditions.

Abstract

Learning models or control policies from data has become a powerful tool to improve the performance of uncertain systems. While a strong focus has been placed on increasing the amount and quality of data to improve performance, data can never fully eliminate uncertainty, making feedback necessary to ensure stability and performance. We show that the control frequency at which the input is recalculated is a crucial design parameter, yet it has hardly been considered before. We address this gap by combining probabilistic model learning and sampled-data control. We use Gaussian processes (GPs) to learn a continuous-time model and compute a corresponding discrete-time controller. The result is an uncertain sampled-data control system, for which we derive robust stability conditions. We formulate semidefinite programs to compute the minimum control frequency required for stability and to optimize performance. As a result, our approach enables us to study the effect of both control frequency and data on stability and closed-loop performance. We show in numerical simulations of a quadrotor that performance can be improved by increasing either the amount of data or the control frequency, and that we can trade off one for the other. For example, by increasing the control frequency by 33%, we can reduce the number of data points by half while still achieving similar performance.
Paper Structure (12 sections, 7 theorems, 37 equations, 3 figures)

This paper contains 12 sections, 7 theorems, 37 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Problem Statement
  3. Background
  4. Gaussian Process Regression
  5. Sampled-Data Systems
  6. Methodology
  7. Model Learning and Linearization
  8. Robust Sampled-Data Control of the Uncertain System
  9. Optimizing the Control Performance
  10. Evaluation
  11. Discussion
  12. Conclusion

Key Result

Lemma 1

The control eq:sd_control_law_lin asympto-tically stabilizes system eq:sd_nominal_open_loop_system for all samplings satisfying Assumption ass:sd_sampling_interval if there exist matrices $\bm{Q}_1 = \bm{Q}_1^\mathsf{T} \succ \bm{0}$, $\bm{Q}_2$, $\bm{Q}_3$, $\bm{Z}_1$, $\bm{Z}_2$, $\bm{Z}_3$, $\bm{ where $*$ denotes symmetry, ${\bm{\Xi} = \bm{Q}_2 + \bm{Q}_2^\mathsf{T} + T_\mathrm{s} \bm{Z}_1}$ a

Figures (3)

  • Figure 1: In digital control systems (blue shaded box), the sampler, controller, and zero-order-hold (ZOH) operate at a certain control frequency $f_\mathrm{c}$. We propose a framework (dashed-line boxes) to simultaneously compute the minimum control frequency and design a controller using an uncertain model learned from data using Gaussian process (GP) regression.
  • Figure 2: Quadrotor trajectories for different control frequencies ${f_\mathrm{c}=\xi f_\mathrm{c,min}}$, where ${\xi\in \{{\color{mycolor1} 1},{\color{mycolor2} 1.25},{\color{mycolor3} 1.5},{\color{mycolor4} 2}\}}$, and different amounts of training data. The shaded areas represent $\pm$ one standard deviation. Convergence to the setpoint significantly improves, and variance reduces if the control frequency is increased from its minimum value $f_\mathrm{c}=f_\mathrm{c,min}$.
  • Figure 3: Tradeoff between the control frequency and the amount of training data in terms of closed-loop performance. The white area indicates $(N,f_\mathrm{c})$ pairs for which a stabilizing controller is found for less than 50% of the randomly drawn datasets. For an increasing amount of data, similar performance can be achieved at a lower control frequency. Vice versa, increasing the frequency can compensate for a significant lack of data.

Theorems & Definitions (15)

  • Definition 1
  • Lemma 1: Fridman.2004, Lemma 2.3
  • proof
  • Lemma 2
  • proof
  • Lemma 3
  • Lemma 4: Xie.1996
  • Theorem 1
  • proof
  • Remark 1
  • ...and 5 more