Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Kernel-based identification using Lebesgue-sampled data

Rodrigo A. González, Koen Tiels, Tom Oomen

TL;DR

This work tackles the identification of continuous-time LTI systems from Lebesgue-sampled outputs, where measurements occur at threshold crossings and provide intersample bounds. It develops a kernel-based, non-parametric estimator that preserves continuous-time semantics and leverages the bounded in-between behavior via a MAP interpretation, yielding a finite representer expansion with coefficients learned by MAP-EM. Hyperparameters are learned with an Empirical Bayes framework, using an EM scheme that incorporates truncation-based sampling to handle the intractable integrals. A closed-form transfer-function description is obtained and computational efficiency is enhanced through kernel decompositions and QR/Cholesky factorizations. The approach is validated with extensive simulations, including a mass-spring-damper encoder setup and additional benchmarks, demonstrating improved model accuracy and substantially reduced output-sample requirements compared to equidistant (Riemann) sampling.

Abstract

Sampling in control applications is increasingly done non-equidistantly in time. This includes applications in motion control, networked control, resource-aware control, and event-based control. Some of these applications, like the ones where displacement is tracked using incremental encoders, are driven by signals that are only measured when their values cross fixed thresholds in the amplitude domain. This paper introduces a non-parametric estimator of the impulse response and transfer function of continuous-time systems based on such amplitude-equidistant sampling strategy, known as Lebesgue sampling. To this end, kernel methods are developed to formulate an algorithm that adequately takes into account the bounded output uncertainty between the event timestamps, which ultimately leads to more accurate models and more efficient output sampling compared to the equidistantly-sampled kernel-based approach. The efficacy of our proposed method is demonstrated through a mass-spring damper example with encoder measurements and extensive Monte Carlo simulation studies on system benchmarks.

Kernel-based identification using Lebesgue-sampled data

TL;DR

This work tackles the identification of continuous-time LTI systems from Lebesgue-sampled outputs, where measurements occur at threshold crossings and provide intersample bounds. It develops a kernel-based, non-parametric estimator that preserves continuous-time semantics and leverages the bounded in-between behavior via a MAP interpretation, yielding a finite representer expansion with coefficients learned by MAP-EM. Hyperparameters are learned with an Empirical Bayes framework, using an EM scheme that incorporates truncation-based sampling to handle the intractable integrals. A closed-form transfer-function description is obtained and computational efficiency is enhanced through kernel decompositions and QR/Cholesky factorizations. The approach is validated with extensive simulations, including a mass-spring-damper encoder setup and additional benchmarks, demonstrating improved model accuracy and substantially reduced output-sample requirements compared to equidistant (Riemann) sampling.

Abstract

Sampling in control applications is increasingly done non-equidistantly in time. This includes applications in motion control, networked control, resource-aware control, and event-based control. Some of these applications, like the ones where displacement is tracked using incremental encoders, are driven by signals that are only measured when their values cross fixed thresholds in the amplitude domain. This paper introduces a non-parametric estimator of the impulse response and transfer function of continuous-time systems based on such amplitude-equidistant sampling strategy, known as Lebesgue sampling. To this end, kernel methods are developed to formulate an algorithm that adequately takes into account the bounded output uncertainty between the event timestamps, which ultimately leads to more accurate models and more efficient output sampling compared to the equidistantly-sampled kernel-based approach. The efficacy of our proposed method is demonstrated through a mass-spring damper example with encoder measurements and extensive Monte Carlo simulation studies on system benchmarks.
Paper Structure (23 sections, 11 theorems, 70 equations, 8 figures, 2 tables, 2 algorithms)

This paper contains 23 sections, 11 theorems, 70 equations, 8 figures, 2 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Setup and problem formulation
  3. System and setup
  4. Practical framework for Lebesgue-sampled system identification
  5. Problem formulation
  6. Kernel-based continuous-time system identification: Preliminaries
  7. Non-parametric estimation using Lebesgue-sampled data
  8. Representer theorem for Lebesgue-sampled systems
  9. Optimal weights with MAP-EM
  10. Kernel hyper-parameter optimization
  11. Transfer function description
  12. Algorithm
  13. Simulations
  14. Practically relevant example
  15. Effect of the threshold amplitude $h$
  16. ...and 8 more sections

Key Result

Lemma 1

Suppose that Assumptions assumption12 and assumption13 hold, and that $g$ is a zero-mean Gaussian process that is independent of $\{v(i\Delta)\}_{i=0}^N$ and has covariance $\mathbb{E}\{g(t)g(s)\} = k(t,s)/\gamma$. Let $\{t_i\}_{i=1}^{N+M}$ be a finite set of real values such that $t_i=i\Delta$ for Furthermore, define $\breve{g}$ as the solution of the optimization problem where $\|\cdot\|_\math

Figures (8)

  • Figure 1: Lebesgue sampling of a signal $z(t)$ with threshold amplitude $h = 1$. The red dots indicate the sampling instants and thresholds being crossed, and the dashed gray rectangles show the regions where $z(t)$ is known to be located.
  • Figure 2: Block diagram of the Lebesgue sampling scheme. Note that $\mathcal{Q}_h$ delivers a set-valued signal $y$, which is used for identification.
  • Figure 3: Input and output signals of the system \ref{['system1']} corresponding to $8$[s] of one Monte Carlo run.
  • Figure 4: Boxplots of the fit metric for the case study, Section \ref{['sec:practicallyrelevant']}. The Lebesgue-sampling-based estimator $\hat{g}_{\textnormal{leb}}$ achieves a better performance than the Riemann approach $\hat{g}_{\textnormal{rie}}$.
  • Figure 5: Bode magnitude plots of 20 Monte Carlo runs (black), compared to the true frequency response (red). Upper plot: equidistantly-sampled approach pillonetto2010new; middle plot: proposed method; lower plot: oracle method (unattainable). The Bode plots of the Lebesgue-sampling approach, obtained via Lemma \ref{['lemma35']}, show much less variability than the Riemann approach over the Monte Carlo runs, and are comparable to the estimates produced by the oracle method.
  • ...and 3 more figures

Theorems & Definitions (11)

  • Lemma 1
  • Lemma 2
  • Lemma 3
  • Theorem 1
  • Theorem 2
  • Lemma 4
  • Corollary 1
  • Theorem 3
  • Proposition 1
  • Lemma 5
  • ...and 1 more