González, Rodrigo

Oomen, Tom

Tiels, Koen

English

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 Non-parametric Continuous-time System Identification withLebesgue-sampled Output MeasurementsCitation for published version (APA):González, R., Tiels, K., & Oomen, T. (2023). Non-parametric Continuous-time System Identification withLebesgue-sampled Output Measurements. Abstract from 42nd Benelux Meeting on Systems and Control 2023,Elspeet, Netherlands.Document license:CC BY-NC-NDDocument status and date:Published: 21/03/2023Document Version:Accepted manuscript including changes made at the peer-review stagePlease check the document version of this publication:• A submitted manuscript is the version of the article upon submission and before peer-review. There can beimportant differences between the submitted version and the official published version of record. Peopleinterested in the research are advised to contact the author for the final version of the publication, or visit theDOI to the publisher's website.• The final author version and the galley proof are versions of the publication after peer review.• The final published version features the final layout of the paper including the volume, issue and pagenumbers.Link to publicationGeneral rightsCopyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright ownersand it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights.            • Users may download and print one copy of any publication from the public portal for the purpose of private study or research.            • You may not further distribute the material or use it for any profit-making activity or commercial gain            • You may freely distribute the URL identifying the publication in the public portal.If the publication is distributed under the terms of Article 25fa of the Dutch Copyright Act, indicated by the “Taverne” license above, pleasefollow below link for the End User Agreement:www.tue.nl/taverneTake down policyIf you believe that this document breaches copyright please contact us at:openaccess@tue.nlproviding details and we will investigate your claim.Download date: 04. Oct. 2023Non-parametric Continuous-time System Identification withLebesgue-sampled Output MeasurementsRodrigo A. GonzálezTU/er.a.gonzalez@tue.nlKoen TielsTU/ek.tiels@tue.nlTom OomenTU/e and TUDt.a.e.oomen@tue.nl1 BackgroundEvent-based sampling schemes in control design can leadto improvements in overall performance and resource effi-ciency [1], with applications in, e.g., network control andincremental encoders [2]. One of the most popular event-based sampling methods is Lebesgue sampling, which con-sists of sampling the continuous-time signal whenever itcrosses fixed thresholds in the amplitude domain.The problem that is addressed in this work is the estimationof non-parametric continuous-time models from Lebesgue-sampled output data. To this end, we seek an estimator thatcan 1) provide a continuous-time impulse response estimatefrom possibly noisy and short data records, and 2) exploitthe entirety of the information contained in the irregularsampling instants and the bounded intersample behavior.2 Problem formulationConsider a linear and time-invariant, continuous-time sys-tem G (with impulse response g(t)), excited by a known in-put u(t). The noisy output z(t) is Lebesgue-sampled, as inFigure 1. In practice, z(t) is fastly-sampled every ∆ [s] andsamples are retrieved only when z(i∆) crosses a thresholdlevel. Thus, we have access to y(i∆)=[ηi,ηi+h], where ηi isthe lower threshold at time i∆ and h is the threshold ampli-tude. This setup is presented in Figure 2.1 2 3 4 5 6 7−3−2−11234ht[s]z(t)Figure 1:Lebesguesamplingofz(t),with thresholdamplitudeh=1.-u(t) G -x(t) +v(t)k+ z(t)? ∆- Qh -y(i∆)Figure 2: Block diagram of the Lebesgue-sampling scheme. TheQh block delivers a set-valued signal y.The goal is to derive an identification method that estimatesg(t) using {u(t)}t∈[0,∆N] and y1:N = {y(i∆)}Ni=1.3 ApproachBy adopting a kernel-based approach, the goal is to find aregularized estimator of the formĝ = argming∈G(−log p(y1:N |g)+ γ∥g∥2G ),where p(y1:N |g) is the likelihood function, and G is a re-producing kernel Hilbert space (RKHS). The impulse re-sponse ĝ can be obtained via a finite-dimensional opti-mization problem thanks to the Representer Theorem. Thefinite-dimensional problem that is derived is solved withMAP-EM (MAP Expectation-Maximization), which pro-vides closed-form iterations that are shown to converge tothe global optimum if adequately initialized.Any kernel-based estimator will depend on the choice of hy-perparameters. These are computed with Empirical Bayes inthis work, from which MAP-EM iterations are also derived.This time the iterations require computing the second mo-ment of a high-dimensional truncated Gaussian. Such ma-trix can be estimated with Monte Carlo sampling methods.4 ResultsWe consider G(s) = 1/(0.05s2 + 0.2s+ 1). The proposedapproach exploits the output intersample knowledge to de-liver more accurate models than the standard kernel (Rie-mann) approach [3], for all values of h.Figure 3: Fit boxplots for the standard Riemann approach (left),and the proposed method (Lebesgue, right) for differentvalues of threshold amplitude h.AcknowledgmentsThis work is part of the research program VIDI with projectnumber 15698, which is (partly) financed by NWO.References[1] K. J. Åström, and B. Bernhardsson. Systems with Lebesgue sam-pling. In Directions in Mathematical Systems Theory and Optimization, pp.1–13. Springer, 2003.[2] N. Strijbosch and T. Oomen. “Iterative learning control for inter-mittently sampled data: Monotonic convergence, design, and applications”.Automatica, Article 110171, 2022.[3] G. Pillonetto and G. De Nicolao. “A new kernel-based approach forlinear system identification”. Automatica, 46(1), pp. 81–93, 2010.

Non-parametric Continuous-time System Identification with Lebesgue-sampled Output Measurements

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Non-parametric Continuous-time System Identification with Lebesgue-sampled Output Measurements

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