Koopman Invariant Subspaces and Finite Linear Representations of Nonlinear Dynamical Systems for Control
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{"title"=>"Koopman invariant subspaces and finite linear representations of nonlinear dynamical systems for control", "type"=>"journal", "authors"=>[{"first_name"=>"Steven L.", "last_name"=>"Brunton", "scopus_author_id"=>"36024889600"}, {"first_name"=>"Bingni W.", "last_name"=>"Brunton", "scopus_author_id"=>"55654236000"}, {"first_name"=>"Joshua L.", "last_name"=>"Proctor", "scopus_author_id"=>"8949981300"}, {"first_name"=>"J. Nathan", "last_name"=>"Kutz", "scopus_author_id"=>"7006290957"}], "year"=>2016, "source"=>"PLoS ONE", "identifiers"=>{"arxiv"=>"1510.03007", "issn"=>"19326203", "doi"=>"10.1371/journal.pone.0150171", "sgr"=>"84960373420", "scopus"=>"2-s2.0-84960373420", "pmid"=>"26919740", "pui"=>"608858495"}, "id"=>"597c8b57-6064-3a14-9ec7-7be62ff0c8be", "abstract"=>"In this work, we explore finite-dimensional linear representations of nonlinear dynamical systems by restricting the Koopman operator to an invariant subspace. The Koopman operator is an infinite-dimensional linear operator that evolves observable functions of the state-space of a dynamical system [Koopman 1931, PNAS]. 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 [Williams et al. 2015, JNLS]. Choosing nonlinear observable functions to form an invariant subspace where it is possible to obtain linear 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 points or more complicated attractors are not globally topologically conjugate to a finite-dimensional linear system, and cannot be represented by a finite-dimensional linear Koopman subspace that includes the state. We then present a data-driven strategy to identify relevant observable functions for Koopman analysis using a new algorithm to determine terms in a dynamical system by sparse regression of the data in a nonlinear function space [Brunton et al. 2015, arxiv]; we show how this algorithm is related to DMD. Finally, we demonstrate how to design optimal control laws for nonlinear systems using techniques from linear optimal control on Koopman invariant subspaces.", "link"=>"http://www.mendeley.com/research/koopman-invariant-subspaces-finite-linear-representations-nonlinear-dynamical-systems-control", "reader_count"=>82, "reader_count_by_academic_status"=>{"Unspecified"=>6, "Student > Doctoral Student"=>7, "Researcher"=>13, "Student > Ph. D. Student"=>32, "Student > Postgraduate"=>5, "Other"=>3, "Student > Master"=>9, "Student > Bachelor"=>6, "Professor"=>1}, "reader_count_by_user_role"=>{"Unspecified"=>6, "Student > Doctoral Student"=>7, "Researcher"=>13, "Student > Ph. D. Student"=>32, "Student > Postgraduate"=>5, "Other"=>3, "Student > Master"=>9, "Student > Bachelor"=>6, "Professor"=>1}, "reader_count_by_subject_area"=>{"Engineering"=>44, "Unspecified"=>9, "Mathematics"=>8, "Agricultural and Biological Sciences"=>4, "Neuroscience"=>2, "Physics and Astronomy"=>6, "Chemical Engineering"=>1, "Chemistry"=>3, "Computer Science"=>5}, "reader_count_by_subdiscipline"=>{"Engineering"=>{"Engineering"=>44}, "Neuroscience"=>{"Neuroscience"=>2}, "Chemistry"=>{"Chemistry"=>3}, "Physics and Astronomy"=>{"Physics and Astronomy"=>6}, "Agricultural and Biological Sciences"=>{"Agricultural and Biological Sciences"=>4}, "Computer Science"=>{"Computer Science"=>5}, "Mathematics"=>{"Mathematics"=>8}, "Unspecified"=>{"Unspecified"=>9}, "Chemical Engineering"=>{"Chemical Engineering"=>1}}, "reader_count_by_country"=>{"Canada"=>1, "South Korea"=>1, "Sweden"=>1, "United States"=>2, "Japan"=>2, "Luxembourg"=>1}, "group_count"=>6}

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Figshare

  • {"files"=>["https://ndownloader.figshare.com/files/4725034"], "description"=>"<div><p>In 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 <i>nonlinear</i> 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 points or more complicated attractors are not globally topologically conjugate to a finite-dimensional linear system, and cannot be represented by a finite-dimensional linear Koopman subspace that includes the state. We then present a data-driven strategy to identify relevant observable functions for Koopman analysis by leveraging a new algorithm to determine relevant terms in a dynamical system by ℓ<sub>1</sub>-regularized regression of the data in a nonlinear function space; we also show how this algorithm is related to DMD. Finally, we demonstrate the usefulness of nonlinear observable subspaces in the design of Koopman operator optimal control laws for fully nonlinear systems using techniques from linear optimal control.</p></div>", "links"=>[], "tags"=>["DMD", "Finite Linear Representations", "LQR", "Nonlinear Dynamical Systems", "subspace", "Koopman operator", "Koopman analysis", "Koopman Invariant Subspaces", "nonlinear systems", "nonlinear function space"], "article_id"=>3023200, "categories"=>["Physical Sciences not elsewhere classified", "Cell Biology", "Neuroscience", "Biotechnology", "Mathematical Sciences not elsewhere classified", "Developmental Biology", "Science Policy", "Mental Health"], "users"=>["Steven L. Brunton", "Bingni W. Brunton", "Joshua L. Proctor", "J. Nathan Kutz"], "doi"=>"https://dx.doi.org/10.1371/journal.pone.0150171", "stats"=>{"downloads"=>13, "page_views"=>16, "likes"=>0}, "figshare_url"=>"https://figshare.com/articles/Koopman_Invariant_Subspaces_and_Finite_Linear_Representations_of_Nonlinear_Dynamical_Systems_for_Control/3023200", "title"=>"Koopman Invariant Subspaces and Finite Linear Representations of Nonlinear Dynamical Systems for Control", "pos_in_sequence"=>0, "defined_type"=>6, "published_date"=>"2016-02-26 09:54:45"}
  • {"files"=>["https://ndownloader.figshare.com/files/4725415"], "description"=>"<p>The Koopman optimal controller achieves a much smaller overall cost, <i>J</i>, approximately 1/3 of the cost of the standard LQR solution.</p>", "links"=>[], "tags"=>["DMD", "Finite Linear Representations", "LQR", "Nonlinear Dynamical Systems", "subspace", "Koopman operator", "Koopman analysis", "Koopman Invariant Subspaces", "nonlinear systems", "nonlinear function space"], "article_id"=>3023512, "categories"=>["Physical Sciences not elsewhere classified", "Cell Biology", "Neuroscience", "Biotechnology", "Mathematical Sciences not elsewhere classified", "Developmental Biology", "Science Policy", "Mental Health"], "users"=>["Steven L. Brunton", "Bingni W. Brunton", "Joshua L. Proctor", "J. Nathan Kutz"], "doi"=>"https://dx.doi.org/10.1371/journal.pone.0150171.g004", "stats"=>{"downloads"=>0, "page_views"=>13, "likes"=>0}, "figshare_url"=>"https://figshare.com/articles/Illustration_of_LQR_control_around_a_nonlinear_fixed_point_using_standard_linearization_black_and_truncated_Koopman_red_/3023512", "title"=>"Illustration of LQR control around a nonlinear fixed point using standard linearization (black) and truncated Koopman (red).", "pos_in_sequence"=>0, "defined_type"=>1, "published_date"=>"2016-02-26 09:54:45"}
  • {"files"=>["https://ndownloader.figshare.com/files/4725325"], "description"=>"<p>The attracting slow manifold is shown in red, the constraint is shown in blue, and the slow unstable subspace of <a href=\"http://www.plosone.org/article/info:doi/10.1371/journal.pone.0150171#pone.0150171.e046\" target=\"_blank\">Eq (25)</a> is shown in green. Black trajectories of the linear Koopman system in <b>y</b> project onto trajectories of the full nonlinear system in <b>x</b> in the <i>y</i><sub>1</sub>-<i>y</i><sub>2</sub> plane. Here, <i>μ</i> = −0.05 and <i>λ</i> = 1. Figure is reproduced with Code 1 in <a href=\"http://www.plosone.org/article/info:doi/10.1371/journal.pone.0150171#pone.0150171.s001\" target=\"_blank\">S1 Appendix</a>.</p>", "links"=>[], "tags"=>["DMD", "Finite Linear Representations", "LQR", "Nonlinear Dynamical Systems", "subspace", "Koopman operator", "Koopman analysis", "Koopman Invariant Subspaces", "nonlinear systems", "nonlinear function space"], "article_id"=>3023410, "categories"=>["Physical Sciences not elsewhere classified", "Cell Biology", "Neuroscience", "Biotechnology", "Mathematical Sciences not elsewhere classified", "Developmental Biology", "Science Policy", "Mental Health"], "users"=>["Steven L. Brunton", "Bingni W. Brunton", "Joshua L. Proctor", "J. Nathan Kutz"], "doi"=>"https://dx.doi.org/10.1371/journal.pone.0150171.g003", "stats"=>{"downloads"=>0, "page_views"=>17, "likes"=>0}, "figshare_url"=>"https://figshare.com/articles/Visualization_of_three_dimensional_linear_Koopman_system_from_Eq_25_along_with_projection_of_dynamics_onto_the_i_x_i_sub_1_sub_i_x_i_sub_2_sub_plane_/3023410", "title"=>"Visualization of three-dimensional linear Koopman system from Eq (25) along with projection of dynamics onto the <i>x</i><sub>1</sub>-<i>x</i><sub>2</sub> plane.", "pos_in_sequence"=>0, "defined_type"=>1, "published_date"=>"2016-02-26 09:54:45"}
  • {"files"=>["https://ndownloader.figshare.com/files/4725226"], "description"=>"<p>In both cases, <i>μ</i> = −0.05 and <i>λ</i> = −1.</p>", "links"=>[], "tags"=>["DMD", "Finite Linear Representations", "LQR", "Nonlinear Dynamical Systems", "subspace", "Koopman operator", "Koopman analysis", "Koopman Invariant Subspaces", "nonlinear systems", "nonlinear function space"], "article_id"=>3023347, "categories"=>["Physical Sciences not elsewhere classified", "Cell Biology", "Neuroscience", "Biotechnology", "Mathematical Sciences not elsewhere classified", "Developmental Biology", "Science Policy", "Mental Health"], "users"=>["Steven L. Brunton", "Bingni W. Brunton", "Joshua L. Proctor", "J. Nathan Kutz"], "doi"=>"https://dx.doi.org/10.1371/journal.pone.0150171.g002", "stats"=>{"downloads"=>1, "page_views"=>13, "likes"=>0}, "figshare_url"=>"https://figshare.com/articles/Illustration_of_two_examples_with_a_slow_manifold_/3023347", "title"=>"Illustration of two examples with a slow manifold.", "pos_in_sequence"=>0, "defined_type"=>1, "published_date"=>"2016-02-26 09:54:45"}
  • {"files"=>["https://ndownloader.figshare.com/files/4725130"], "description"=>"<p>The dashed lines from <i>y</i><sub><i>k</i></sub> → <i>x</i><sub><i>k</i></sub> indicate that we would like to be able to recover the original state.</p>", "links"=>[], "tags"=>["DMD", "Finite Linear Representations", "LQR", "Nonlinear Dynamical Systems", "subspace", "Koopman operator", "Koopman analysis", "Koopman Invariant Subspaces", "nonlinear systems", "nonlinear function space"], "article_id"=>3023263, "categories"=>["Physical Sciences not elsewhere classified", "Cell Biology", "Neuroscience", "Biotechnology", "Mathematical Sciences not elsewhere classified", "Developmental Biology", "Science Policy", "Mental Health"], "users"=>["Steven L. Brunton", "Bingni W. Brunton", "Joshua L. Proctor", "J. Nathan Kutz"], "doi"=>"https://dx.doi.org/10.1371/journal.pone.0150171.g001", "stats"=>{"downloads"=>1, "page_views"=>14, "likes"=>0}, "figshare_url"=>"https://figshare.com/articles/Schematic_illustrating_the_Koopman_operator_for_nonlinear_dynamical_systems_/3023263", "title"=>"Schematic illustrating the Koopman operator for nonlinear dynamical systems.", "pos_in_sequence"=>0, "defined_type"=>1, "published_date"=>"2016-02-26 09:54:45"}

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Relative Metric

{"start_date"=>"2016-01-01T00:00:00Z", "end_date"=>"2016-12-31T00:00:00Z", "subject_areas"=>[]}
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