Optimal Construction of Koopman Eigenfunctions for Prediction and Control
| Autor: | Igor Mezic, Milan Korda |
|---|---|
| Přispěvatelé: | Équipe Méthodes et Algorithmes en Commande (LAAS-MAC), Laboratoire d'analyse et d'architecture des systèmes (LAAS), Université Toulouse Capitole (UT Capitole), Université de Toulouse (UT)-Université de Toulouse (UT)-Institut National des Sciences Appliquées - Toulouse (INSA Toulouse), Institut National des Sciences Appliquées (INSA)-Université de Toulouse (UT)-Institut National des Sciences Appliquées (INSA)-Université Toulouse - Jean Jaurès (UT2J), Université de Toulouse (UT)-Université Toulouse III - Paul Sabatier (UT3), Université de Toulouse (UT)-Centre National de la Recherche Scientifique (CNRS)-Institut National Polytechnique (Toulouse) (Toulouse INP), Université de Toulouse (UT)-Université Toulouse Capitole (UT Capitole), Université de Toulouse (UT), University of California [Santa Barbara] (UC Santa Barbara), University of California (UC), Université Toulouse - Jean Jaurès (UT2J)-Université Toulouse 1 Capitole (UT1), Université Fédérale Toulouse Midi-Pyrénées-Université Fédérale Toulouse Midi-Pyrénées-Centre National de la Recherche Scientifique (CNRS)-Université Toulouse III - Paul Sabatier (UT3), Université Fédérale Toulouse Midi-Pyrénées-Institut National des Sciences Appliquées - Toulouse (INSA Toulouse), Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Institut National Polytechnique (Toulouse) (Toulouse INP), Université Fédérale Toulouse Midi-Pyrénées-Université Toulouse - Jean Jaurès (UT2J)-Université Toulouse 1 Capitole (UT1), Université Fédérale Toulouse Midi-Pyrénées, University of California [Santa Barbara] (UCSB), University of California |
| Rok vydání: | 2020 |
| Předmět: |
0209 industrial biotechnology
Data-driven methods [MATH.MATH-DS]Mathematics [math]/Dynamical Systems [math.DS] Linear prediction Dynamical Systems (math.DS) 02 engineering and technology Dynamical system [SPI.AUTO]Engineering Sciences [physics]/Automatic Mathematics - Spectral Theory 020901 industrial engineering & automation Operator (computer programming) FOS: Mathematics Applied mathematics Model predictive control Mathematics - Dynamical Systems Electrical and Electronic Engineering Invariant (mathematics) Koopman operator Mathematics - Optimization and Control Spectral Theory (math.SP) Mathematics Eigenfunctions Eigenfunction Linear subspace Computer Science Applications Optimization and Control (math.OC) Control and Systems Engineering Convex optimization [MATH.MATH-OC]Mathematics [math]/Optimization and Control [math.OC] |
| Zdroj: | IEEE Transactions on Automatic Control IEEE Transactions on Automatic Control, 2020, 65 (12), pp.5114-5129. ⟨10.1109/TAC.2020.2978039⟩ IEEE Transactions on Automatic Control, Institute of Electrical and Electronics Engineers, 2020, 65 (12), pp.5114-5129. ⟨10.1109/TAC.2020.2978039⟩ |
| ISSN: | 2334-3303 0018-9286 |
| DOI: | 10.1109/tac.2020.2978039 |
| Popis: | This article presents a novel data-driven framework for constructing eigenfunctions of the Koopman operator geared toward prediction and control. The method leverages the richness of the spectrum of the Koopman operator away from attractors to construct a set of eigenfunctions such that the state (or any other observable quantity of interest) is in the span of these eigenfunctions and hence predictable in a linear fashion. The eigenfunction construction is optimization-based with no dictionary selection required. Once a predictor for the uncontrolled part of the system is obtained in this way, the incorporation of control is done through a multistep prediction error minimization, carried out by a simple linear least-squares regression. The predictor so obtained is in the form of a linear controlled dynamical system and can be readily applied within the Koopman model predictive control (MPC) framework of (M. Korda and I. Mezic, 2018) to control nonlinear dynamical systems using linear MPC tools. The method is entirely data-driven and based predominantly on convex optimization. The novel eigenfunction construction method is also analyzed theoretically, proving rigorously that the family of eigenfunctions obtained is rich enough to span the space of all continuous functions. In addition, the method is extended to construct generalized eigenfunctions that also give rise Koopman invariant subspaces and hence can be used for linear prediction. Detailed numerical examples demonstrate the approach, both for prediction and feedback control. * * Code for the numerical examples is available from https://homepages.laas.fr/mkorda/Eigfuns.zip . |
| Databáze: | OpenAIRE |
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