Koopman form of nonlinear systems with inputs

Lucian Cristian Iacob (Corresponding author), Roland Tóth (Corresponding author), Maarten Schoukens (Corresponding author)

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3 Citaten (Scopus)
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Samenvatting

The Koopman framework proposes a linear representation of finite-dimensional nonlinear systems through a generally infinite-dimensional globally linear embedding. Originally, the Koopman formalism has been derived for autonomous systems. In applications for systems with inputs, generally a linear time invariant (LTI) form of the Koopman model is assumed, as it facilitates the use of control techniques such as linear quadratic regulation and model predictive control. However, it can be easily shown that this assumption is insufficient to capture the dynamics of the underlying nonlinear system. Proper theoretical extension for actuated continuous-time systems with a linear or a control-affine input has been worked out only recently, however extensions to discrete-time systems and general continuous-time systems have not been developed yet. In the present paper, we systematically investigate and analytically derive lifted forms under inputs for a rather wide class of nonlinear systems in both continuous and discrete time. We prove that the resulting lifted representations give Koopman models where the state transition is linear, but the input matrix becomes state-dependent (state and input-dependent in the discrete-time case), giving rise to a specially structured linear parameter-varying (LPV) description of the underlying system. We also provide error bounds on how much the dependency of the input matrix contributes to the resulting representation and how well the system behavior can be approximated by an LTI Koopman representation. The introduced theoretical insight greatly helps for performing proper model structure selection in system identification with Koopman models as well as making a proper choice for LTI or LPV techniques for the control of nonlinear systems through the Koopman approach.

Originele taal-2Engels
Artikelnummer111525
Aantal pagina's10
TijdschriftAutomatica
Volume162
DOI's
StatusGepubliceerd - apr. 2024

Bibliografische nota

Publisher Copyright:
© 2024 The Authors

Financiering

This work has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement nr. 714663 ). This research has also been supported by the Eötvös Loránd Research Network (Grant Number: SA-77/2021 ). The material in this paper was not presented at any conference. This paper was recommended for publication in revised form by Associate Editor Gianluigi Pillonetto under the direction of Editor Alessandro Chiuso. This work has received funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (grant agreement nr. 714663). This research has also been supported by the Eötvös Loránd Research Network (Grant Number: SA-77/2021). The material in this paper was not presented at any conference. This paper was recommended for publication in revised form by Associate Editor Gianluigi Pillonetto under the direction of Editor Alessandro Chiuso.

FinanciersFinanciernummer
Horizon 2020 Framework Programme
European Research Council
Horizon 2020714663, SA-77/2021

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