Koopman Invariant Subspaces and Finite Linear Representations of Nonlinear Dynamical Systems for Control

PloS One
Steven L BruntonJ Nathan Kutz

Abstract

In this wIn this work, we explore finite-dimensional linear representations of nonlinear dynamical systems by restricting the Koopman operator to an invariant subspace spanned by specially chosen observable functions. The Koopman operator is an infinite-dimensional linear operator that evolves functions of the state of a dynamical system. Dominant terms in the Koopman expansion are typically computed using dynamic mode decomposition (DMD). DMD uses linear measurements of the state variables, and it has recently been shown that this may be too restrictive for nonlinear systems. Choosing the right nonlinear observable functions to form an invariant subspace where it is possible to obtain linear reduced-order models, especially those that are useful for control, is an open challenge. Here, we investigate the choice of observable functions for Koopman analysis that enable the use of optimal linear control techniques on nonlinear problems. First, to include a cost on the state of the system, as in linear quadratic regulator (LQR) control, it is helpful to include these states in the observable subspace, as in DMD. However, we find that this is only possible when there is a single isolated fixed point, as systems with multiple fixed ...Continue Reading

References

Apr 23, 1983·The Veterinary Record·P H Holmes
May 1, 1931·Proceedings of the National Academy of Sciences of the United States of America·B O Koopman
Mar 1, 1932·Proceedings of the National Academy of Sciences of the United States of America·B O Koopman, J V Neumann
Jun 8, 2007·Proceedings of the National Academy of Sciences of the United States of America·Josh Bongard, Hod Lipson
Apr 4, 2009·Science·Michael Schmidt, Hod Lipson
Jan 3, 2013·Chaos·Marko BudisićIgor Mezić
Mar 28, 2013·Proceedings of the National Academy of Sciences of the United States of America·Hayden SchaefferStanley Osher
Oct 31, 2013·Proceedings of the National Academy of Sciences of the United States of America·Vidvuds OzolinsStanley Osher
Mar 4, 2015·International Health·Joshua L Proctor, Philip A Eckhoff
Nov 4, 2015·Journal of Neuroscience Methods·Bingni W BruntonJ Nathan Kutz
Apr 2, 2016·Proceedings of the National Academy of Sciences of the United States of America·Steven L BruntonJ Nathan Kutz

❮ Previous
Next ❯

Citations

Jul 25, 2019·Physical Review. E·Gerard McCaulDenys I Bondar
Jun 1, 2017·Nature Communications·Steven L BruntonJ Nathan Kutz
Nov 25, 2018·Nature Communications·Bethany LuschSteven L Brunton
Jan 20, 2018·Physical Review. E·Naoya TakeishiTakehisa Yairi
Jan 21, 2021·Physical Review. E·Jing Hu, Yueheng Lan
Nov 23, 2020·Biophysics Reviews·P Rajendra, V Brahmajirao
Jan 13, 2021·ISA Transactions·Mostafa Al-Gabalawy
Nov 5, 2021·Scientific Reports·Craig R GinJ Nathan Kutz
Jul 12, 2018··Miguel Alfonso MendezJean Marie Buchlin
Jun 2, 2017··Chen LuBalaji Jayaraman

❮ Previous
Next ❯

Software Mentioned

Matlab
LASSO
SINDy

Related Concepts

Trending Feeds

COVID-19

Coronaviruses encompass a large family of viruses that cause the common cold as well as more serious diseases, such as the ongoing outbreak of coronavirus disease 2019 (COVID-19; formally known as 2019-nCoV). Coronaviruses can spread from animals to humans; symptoms include fever, cough, shortness of breath, and breathing difficulties; in more severe cases, infection can lead to death. This feed covers recent research on COVID-19.

Blastomycosis

Blastomycosis fungal infections spread through inhaling Blastomyces dermatitidis spores. Discover the latest research on blastomycosis fungal infections here.

Nuclear Pore Complex in ALS/FTD

Alterations in nucleocytoplasmic transport, controlled by the nuclear pore complex, may be involved in the pathomechanism underlying multiple neurodegenerative diseases including Amyotrophic Lateral Sclerosis and Frontotemporal Dementia. Here is the latest research on the nuclear pore complex in ALS and FTD.

Applications of Molecular Barcoding

The concept of molecular barcoding is that each original DNA or RNA molecule is attached to a unique sequence barcode. Sequence reads having different barcodes represent different original molecules, while sequence reads having the same barcode are results of PCR duplication from one original molecule. Discover the latest research on molecular barcoding here.

Chronic Fatigue Syndrome

Chronic fatigue syndrome is a disease characterized by unexplained disabling fatigue; the pathology of which is incompletely understood. Discover the latest research on chronic fatigue syndrome here.

Evolution of Pluripotency

Pluripotency refers to the ability of a cell to develop into three primary germ cell layers of the embryo. This feed focuses on the mechanisms that underlie the evolution of pluripotency. Here is the latest research.

Position Effect Variegation

Position Effect Variagation occurs when a gene is inactivated due to its positioning near heterochromatic regions within a chromosome. Discover the latest research on Position Effect Variagation here.

STING Receptor Agonists

Stimulator of IFN genes (STING) are a group of transmembrane proteins that are involved in the induction of type I interferon that is important in the innate immune response. The stimulation of STING has been an active area of research in the treatment of cancer and infectious diseases. Here is the latest research on STING receptor agonists.

Microbicide

Microbicides are products that can be applied to vaginal or rectal mucosal surfaces with the goal of preventing, or at least significantly reducing, the transmission of sexually transmitted infections. Here is the latest research on microbicides.