Borja, Luis Pablo

Scherpen, Jacquelien M. A.

Wesselink, T. C.

English

Proceedings - University of Groningen

   University of GroningenStabilization of a class of port-Hamiltonian systems using saturated controllersBorja, Luis Pablo; Wesselink, T. C. ; Scherpen, Jacquelien M. A.Published in:38th Benelux Meeting on Systems and ControlIMPORTANT NOTE: You are advised to consult the publisher's version (publisher's PDF) if you wish to cite fromit. Please check the document version below.Document VersionFinal author's version (accepted by publisher, after peer review)Publication date:2019Link to publication in University of Groningen/UMCG research databaseCitation for published version (APA):Borja, L. P., Wesselink, T. C., & Scherpen, J. M. A. (2019). Stabilization of a class of port-Hamiltoniansystems using saturated controllers. In 38th Benelux Meeting on Systems and Control (pp. 130-130). KULeuven.CopyrightOther than for strictly personal use, it is not permitted to download or to forward/distribute the text or part of it without the consent of theauthor(s) and/or copyright holder(s), unless the work is under an open content license (like Creative Commons).The publication may also be distributed here under the terms of Article 25fa of the Dutch Copyright Act, indicated by the “Taverne” license.More information can be found on the University of Groningen website: https://www.rug.nl/library/open-access/self-archiving-pure/taverne-amendment.Take-down policyIf you believe that this document breaches copyright please contact us providing details, and we will remove access to the work immediatelyand investigate your claim.Downloaded from the University of Groningen/UMCG research database (Pure): http://www.rug.nl/research/portal. For technical reasons thenumber of authors shown on this cover page is limited to 10 maximum.Download date: 04-06-2022Stabilization of a class of port-Hamiltonian systems using saturatedcontrollers.L.P. Borja, T.C. Wesselink and J.M.A. Scherpen.Nijenborgh 4, 9747 AG Groningen, The Netherlands.e-mail: l.p.borja.rosales[j.m.a.scherpen]@rug.nl,t.c.wesselink@student.rug.nlAbstractThe port-Hamiltonian (PH) framework has been extensivelyused to model physical systems. One of the main advan-tages of this modeling approach is that it provides a system-atic procedure to obtain mathematical models that capturethe nonlinear phenomena and preserve conservation lawspresent in physical systems [1]. Additionally, in this frame-work, the roles of the interconnection pattern, the dissipa-tion, and the energy are underscored. Thus, a natural way tocontrol passive systems is to design controllers that give thedesired shape to the energy of the closed-loop system. Thisprocess is known as energy-shaping and it is the main ideaof several passivity-based control (PBC) approaches.While PBC techniques have been proved to be effective tosolve the stabilization problem of physical systems [2], [3],some important issues, that might arise during the imple-mentation of these controllers, are not addressed by thesetechniques, e.g., the lack of sensors to measure some vari-ables or constraints in the actuators. Indeed, these are perva-sive problems in nonlinear control, therefore, finding a solu-tion to them is paramount from a practical perspective.In this work, we propose a PBC approach that deals with theproblem of set-point regulation for a class of physical sys-tems. Moreover, this approach has the following appealingfeatures• It obviates the necessity of solving partial differentialequations, which, in general, are one of the main con-straints of the PBC techniques.• The controllers derived from this approach are sat-urated, and the saturation limits can be straightfor-wardly adjusted.• The control design only requires a partial measure-ment of the state. Particularly for mechanical sys-tems, it is not necessary to measure the velocities (mo-menta).• It is possible to include an integral-like term that en-sures the convergence of the steady-state error to zero.The proposed PBC approach follows the ideas previouslyreported in [4] and [5]. As proposed in the latter, the controldesign requires the definition of a “virtual state” whose dy-namics are designed in such a way that it is possible to injectdamping in coordinates that cannot be measured. Moreover,a direct extension of the results reported in [4] ensures thesaturation of the control signal. This is sometimes requiredto avoid malfunctions in the actuators of physical systems.With the aim of presenting our results, we, first, provide acharacterization of the PH systems for which the proposedPBC approach is suitable. Towards this goal, we provide as-sumptions that can be verified in a systematic way. Then, weproceed with the control design, and we provide the stabil-ity analysis of the closed-loop system, which is carried outusing passivity arguments and Lyapunov theory. Finally, weillustrate the applicability of the technique through the ex-perimental results obtained for a planar robot with two linksand flexible joints.References[1] Duindam, V., Macchelli, A., Stramigioli, S., &Bruyninckx, H. (Eds.). (2009). Modeling and control ofcomplex physical systems: the port-Hamiltonian approach.Springer Science & Business Media.[2] Ortega, R., Perez, J. A. L., Nicklasson, P. J., &Sira-Ramirez, H. J. (2013). Passivity-based control ofEuler-Lagrange systems: mechanical, electrical and elec-tromechanical applications. Springer Science & BusinessMedia.[3] van der Schaft, A. J. (2000). L2-gain and passivitytechniques in nonlinear control. London: Springer.[4] Borja, P., Cisneros, R., & Ortega, R. (2016). A con-structive procedure for energy shaping of portHamiltoniansystems. Automatica, 72, 230-234.[5] Dirksz, D. A., & Scherpen, J. M. A. (2013). On track-ing control of rigid-joint robots with only position measure-ments. IEEE Transactions on Control Systems Technology,21(4), 1510-1513.

Stabilization of a class of port-Hamiltonian systems using saturated controllers

ARTS repository - University of Groningen

   University of GroningenStabilization of a class of port-Hamiltonian systems using saturated controllersBorja, Luis Pablo; Wesselink, T. C. ; Scherpen, Jacquelien M. A.Published in:38th Benelux Meeting on Systems and ControlIMPORTANT NOTE: You are advised to consult the publisher's version (publisher's PDF) if you wish to cite fromit. Please check the document version below.Document VersionFinal author's version (accepted by publisher, after peer review)Publication date:2019Link to publication in University of Groningen/UMCG research databaseCitation for published version (APA):Borja, L. P., Wesselink, T. C., & Scherpen, J. M. A. (2019). Stabilization of a class of port-Hamiltoniansystems using saturated controllers. In 38th Benelux Meeting on Systems and Control (pp. 130-130). KULeuven.CopyrightOther than for strictly personal use, it is not permitted to download or to forward/distribute the text or part of it without the consent of theauthor(s) and/or copyright holder(s), unless the work is under an open content license (like Creative Commons).The publication may also be distributed here under the terms of Article 25fa of the Dutch Copyright Act, indicated by the “Taverne” license.More information can be found on the University of Groningen website: https://www.rug.nl/library/open-access/self-archiving-pure/taverne-amendment.Take-down policyIf you believe that this document breaches copyright please contact us providing details, and we will remove access to the work immediatelyand investigate your claim.Downloaded from the University of Groningen/UMCG research database (Pure): http://www.rug.nl/research/portal. For technical reasons thenumber of authors shown on this cover page is limited to 10 maximum.Download date: 05-12-2023Stabilization of a class of port-Hamiltonian systems using saturatedcontrollers.L.P. Borja, T.C. Wesselink and J.M.A. Scherpen.Nijenborgh 4, 9747 AG Groningen, The Netherlands.e-mail: l.p.borja.rosales[j.m.a.scherpen]@rug.nl,t.c.wesselink@student.rug.nlAbstractThe port-Hamiltonian (PH) framework has been extensivelyused to model physical systems. One of the main advan-tages of this modeling approach is that it provides a system-atic procedure to obtain mathematical models that capturethe nonlinear phenomena and preserve conservation lawspresent in physical systems [1]. Additionally, in this frame-work, the roles of the interconnection pattern, the dissipa-tion, and the energy are underscored. Thus, a natural way tocontrol passive systems is to design controllers that give thedesired shape to the energy of the closed-loop system. Thisprocess is known as energy-shaping and it is the main ideaof several passivity-based control (PBC) approaches.While PBC techniques have been proved to be effective tosolve the stabilization problem of physical systems [2], [3],some important issues, that might arise during the imple-mentation of these controllers, are not addressed by thesetechniques, e.g., the lack of sensors to measure some vari-ables or constraints in the actuators. Indeed, these are perva-sive problems in nonlinear control, therefore, finding a solu-tion to them is paramount from a practical perspective.In this work, we propose a PBC approach that deals with theproblem of set-point regulation for a class of physical sys-tems. Moreover, this approach has the following appealingfeatures• It obviates the necessity of solving partial differentialequations, which, in general, are one of the main con-straints of the PBC techniques.• The controllers derived from this approach are sat-urated, and the saturation limits can be straightfor-wardly adjusted.• The control design only requires a partial measure-ment of the state. Particularly for mechanical sys-tems, it is not necessary to measure the velocities (mo-menta).• It is possible to include an integral-like term that en-sures the convergence of the steady-state error to zero.The proposed PBC approach follows the ideas previouslyreported in [4] and [5]. As proposed in the latter, the controldesign requires the definition of a “virtual state” whose dy-namics are designed in such a way that it is possible to injectdamping in coordinates that cannot be measured. Moreover,a direct extension of the results reported in [4] ensures thesaturation of the control signal. This is sometimes requiredto avoid malfunctions in the actuators of physical systems.With the aim of presenting our results, we, first, provide acharacterization of the PH systems for which the proposedPBC approach is suitable. Towards this goal, we provide as-sumptions that can be verified in a systematic way. Then, weproceed with the control design, and we provide the stabil-ity analysis of the closed-loop system, which is carried outusing passivity arguments and Lyapunov theory. Finally, weillustrate the applicability of the technique through the ex-perimental results obtained for a planar robot with two linksand flexible joints.References[1] Duindam, V., Macchelli, A., Stramigioli, S., &Bruyninckx, H. (Eds.). (2009). Modeling and control ofcomplex physical systems: the port-Hamiltonian approach.Springer Science & Business Media.[2] Ortega, R., Perez, J. A. L., Nicklasson, P. J., &Sira-Ramirez, H. J. (2013). Passivity-based control ofEuler-Lagrange systems: mechanical, electrical and elec-tromechanical applications. Springer Science & BusinessMedia.[3] van der Schaft, A. J. (2000). L2-gain and passivitytechniques in nonlinear control. London: Springer.[4] Borja, P., Cisneros, R., & Ortega, R. (2016). A con-structive procedure for energy shaping of portHamiltoniansystems. Automatica, 72, 230-234.[5] Dirksz, D. A., & Scherpen, J. M. A. (2013). On track-ing control of rigid-joint robots with only position measure-ments. IEEE Transactions on Control Systems Technology,21(4), 1510-1513.

University of Groningen

   University of GroningenStabilization of a class of port-Hamiltonian systems using saturated controllersBorja, Luis Pablo; Wesselink, T. C. ; Scherpen, Jacquelien M. A.Published in:38th Benelux Meeting on Systems and ControlIMPORTANT NOTE: You are advised to consult the publisher's version (publisher's PDF) if you wish to cite fromit. Please check the document version below.Document VersionFinal author's version (accepted by publisher, after peer review)Publication date:2019Link to publication in University of Groningen/UMCG research databaseCitation for published version (APA):Borja, L. P., Wesselink, T. C., & Scherpen, J. M. A. (2019). Stabilization of a class of port-Hamiltoniansystems using saturated controllers. In 38th Benelux Meeting on Systems and Control (pp. 130-130). KULeuven.CopyrightOther than for strictly personal use, it is not permitted to download or to forward/distribute the text or part of it without the consent of theauthor(s) and/or copyright holder(s), unless the work is under an open content license (like Creative Commons).The publication may also be distributed here under the terms of Article 25fa of the Dutch Copyright Act, indicated by the “Taverne” license.More information can be found on the University of Groningen website: https://www.rug.nl/library/open-access/self-archiving-pure/taverne-amendment.Take-down policyIf you believe that this document breaches copyright please contact us providing details, and we will remove access to the work immediatelyand investigate your claim.Downloaded from the University of Groningen/UMCG research database (Pure): http://www.rug.nl/research/portal. For technical reasons thenumber of authors shown on this cover page is limited to 10 maximum.Download date: 19-11-2022Stabilization of a class of port-Hamiltonian systems using saturatedcontrollers.L.P. Borja, T.C. Wesselink and J.M.A. Scherpen.Nijenborgh 4, 9747 AG Groningen, The Netherlands.e-mail: l.p.borja.rosales[j.m.a.scherpen]@rug.nl,t.c.wesselink@student.rug.nlAbstractThe port-Hamiltonian (PH) framework has been extensivelyused to model physical systems. One of the main advan-tages of this modeling approach is that it provides a system-atic procedure to obtain mathematical models that capturethe nonlinear phenomena and preserve conservation lawspresent in physical systems [1]. Additionally, in this frame-work, the roles of the interconnection pattern, the dissipa-tion, and the energy are underscored. Thus, a natural way tocontrol passive systems is to design controllers that give thedesired shape to the energy of the closed-loop system. Thisprocess is known as energy-shaping and it is the main ideaof several passivity-based control (PBC) approaches.While PBC techniques have been proved to be effective tosolve the stabilization problem of physical systems [2], [3],some important issues, that might arise during the imple-mentation of these controllers, are not addressed by thesetechniques, e.g., the lack of sensors to measure some vari-ables or constraints in the actuators. Indeed, these are perva-sive problems in nonlinear control, therefore, finding a solu-tion to them is paramount from a practical perspective.In this work, we propose a PBC approach that deals with theproblem of set-point regulation for a class of physical sys-tems. Moreover, this approach has the following appealingfeatures• It obviates the necessity of solving partial differentialequations, which, in general, are one of the main con-straints of the PBC techniques.• The controllers derived from this approach are sat-urated, and the saturation limits can be straightfor-wardly adjusted.• The control design only requires a partial measure-ment of the state. Particularly for mechanical sys-tems, it is not necessary to measure the velocities (mo-menta).• It is possible to include an integral-like term that en-sures the convergence of the steady-state error to zero.The proposed PBC approach follows the ideas previouslyreported in [4] and [5]. As proposed in the latter, the controldesign requires the definition of a “virtual state” whose dy-namics are designed in such a way that it is possible to injectdamping in coordinates that cannot be measured. Moreover,a direct extension of the results reported in [4] ensures thesaturation of the control signal. This is sometimes requiredto avoid malfunctions in the actuators of physical systems.With the aim of presenting our results, we, first, provide acharacterization of the PH systems for which the proposedPBC approach is suitable. Towards this goal, we provide as-sumptions that can be verified in a systematic way. Then, weproceed with the control design, and we provide the stabil-ity analysis of the closed-loop system, which is carried outusing passivity arguments and Lyapunov theory. Finally, weillustrate the applicability of the technique through the ex-perimental results obtained for a planar robot with two linksand flexible joints.References[1] Duindam, V., Macchelli, A., Stramigioli, S., &Bruyninckx, H. (Eds.). (2009). Modeling and control ofcomplex physical systems: the port-Hamiltonian approach.Springer Science & Business Media.[2] Ortega, R., Perez, J. A. L., Nicklasson, P. J., &Sira-Ramirez, H. J. (2013). Passivity-based control ofEuler-Lagrange systems: mechanical, electrical and elec-tromechanical applications. Springer Science & BusinessMedia.[3] van der Schaft, A. J. (2000). L2-gain and passivitytechniques in nonlinear control. London: Springer.[4] Borja, P., Cisneros, R., & Ortega, R. (2016). A con-structive procedure for energy shaping of portHamiltoniansystems. Automatica, 72, 230-234.[5] Dirksz, D. A., & Scherpen, J. M. A. (2013). On track-ing control of rigid-joint robots with only position measure-ments. IEEE Transactions on Control Systems Technology,21(4), 1510-1513.

NARCIS 

Dissertations of the University of Groningen

   University of GroningenStabilization of a class of port-Hamiltonian systems using saturated controllersBorja, Luis Pablo; Wesselink, T. C. ; Scherpen, Jacquelien M. A.Published in:38th Benelux Meeting on Systems and ControlIMPORTANT NOTE: You are advised to consult the publisher's version (publisher's PDF) if you wish to cite fromit. Please check the document version below.Document VersionFinal author's version (accepted by publisher, after peer review)Publication date:2019Link to publication in University of Groningen/UMCG research databaseCitation for published version (APA):Borja, L. P., Wesselink, T. C., & Scherpen, J. M. A. (2019). Stabilization of a class of port-Hamiltoniansystems using saturated controllers. In 38th Benelux Meeting on Systems and Control (pp. 130-130). KULeuven.CopyrightOther than for strictly personal use, it is not permitted to download or to forward/distribute the text or part of it without the consent of theauthor(s) and/or copyright holder(s), unless the work is under an open content license (like Creative Commons).The publication may also be distributed here under the terms of Article 25fa of the Dutch Copyright Act, indicated by the “Taverne” license.More information can be found on the University of Groningen website: https://www.rug.nl/library/open-access/self-archiving-pure/taverne-amendment.Take-down policyIf you believe that this document breaches copyright please contact us providing details, and we will remove access to the work immediatelyand investigate your claim.Downloaded from the University of Groningen/UMCG research database (Pure): http://www.rug.nl/research/portal. For technical reasons thenumber of authors shown on this cover page is limited to 10 maximum.Download date: 10-09-2023Stabilization of a class of port-Hamiltonian systems using saturatedcontrollers.L.P. Borja, T.C. Wesselink and J.M.A. Scherpen.Nijenborgh 4, 9747 AG Groningen, The Netherlands.e-mail: l.p.borja.rosales[j.m.a.scherpen]@rug.nl,t.c.wesselink@student.rug.nlAbstractThe port-Hamiltonian (PH) framework has been extensivelyused to model physical systems. One of the main advan-tages of this modeling approach is that it provides a system-atic procedure to obtain mathematical models that capturethe nonlinear phenomena and preserve conservation lawspresent in physical systems [1]. Additionally, in this frame-work, the roles of the interconnection pattern, the dissipa-tion, and the energy are underscored. Thus, a natural way tocontrol passive systems is to design controllers that give thedesired shape to the energy of the closed-loop system. Thisprocess is known as energy-shaping and it is the main ideaof several passivity-based control (PBC) approaches.While PBC techniques have been proved to be effective tosolve the stabilization problem of physical systems [2], [3],some important issues, that might arise during the imple-mentation of these controllers, are not addressed by thesetechniques, e.g., the lack of sensors to measure some vari-ables or constraints in the actuators. Indeed, these are perva-sive problems in nonlinear control, therefore, finding a solu-tion to them is paramount from a practical perspective.In this work, we propose a PBC approach that deals with theproblem of set-point regulation for a class of physical sys-tems. Moreover, this approach has the following appealingfeatures• It obviates the necessity of solving partial differentialequations, which, in general, are one of the main con-straints of the PBC techniques.• The controllers derived from this approach are sat-urated, and the saturation limits can be straightfor-wardly adjusted.• The control design only requires a partial measure-ment of the state. Particularly for mechanical sys-tems, it is not necessary to measure the velocities (mo-menta).• It is possible to include an integral-like term that en-sures the convergence of the steady-state error to zero.The proposed PBC approach follows the ideas previouslyreported in [4] and [5]. As proposed in the latter, the controldesign requires the definition of a “virtual state” whose dy-namics are designed in such a way that it is possible to injectdamping in coordinates that cannot be measured. Moreover,a direct extension of the results reported in [4] ensures thesaturation of the control signal. This is sometimes requiredto avoid malfunctions in the actuators of physical systems.With the aim of presenting our results, we, first, provide acharacterization of the PH systems for which the proposedPBC approach is suitable. Towards this goal, we provide as-sumptions that can be verified in a systematic way. Then, weproceed with the control design, and we provide the stabil-ity analysis of the closed-loop system, which is carried outusing passivity arguments and Lyapunov theory. Finally, weillustrate the applicability of the technique through the ex-perimental results obtained for a planar robot with two linksand flexible joints.References[1] Duindam, V., Macchelli, A., Stramigioli, S., &Bruyninckx, H. (Eds.). (2009). Modeling and control ofcomplex physical systems: the port-Hamiltonian approach.Springer Science & Business Media.[2] Ortega, R., Perez, J. A. L., Nicklasson, P. J., &Sira-Ramirez, H. J. (2013). Passivity-based control ofEuler-Lagrange systems: mechanical, electrical and elec-tromechanical applications. Springer Science & BusinessMedia.[3] van der Schaft, A. J. (2000). L2-gain and passivitytechniques in nonlinear control. London: Springer.[4] Borja, P., Cisneros, R., & Ortega, R. (2016). A con-structive procedure for energy shaping of portHamiltoniansystems. Automatica, 72, 230-234.[5] Dirksz, D. A., & Scherpen, J. M. A. (2013). On track-ing control of rigid-joint robots with only position measure-ments. IEEE Transactions on Control Systems Technology,21(4), 1510-1513.

   University of GroningenStabilization of a class of port-Hamiltonian systems using saturated controllersBorja, Luis Pablo; Wesselink, T. C. ; Scherpen, Jacquelien M. A.Published in:38th Benelux Meeting on Systems and ControlIMPORTANT NOTE: You are advised to consult the publisher's version (publisher's PDF) if you wish to cite fromit. Please check the document version below.Document VersionFinal author's version (accepted by publisher, after peer review)Publication date:2019Link to publication in University of Groningen/UMCG research databaseCitation for published version (APA):Borja, L. P., Wesselink, T. C., & Scherpen, J. M. A. (2019). Stabilization of a class of port-Hamiltoniansystems using saturated controllers. In 38th Benelux Meeting on Systems and Control (pp. 130-130). KULeuven.CopyrightOther than for strictly personal use, it is not permitted to download or to forward/distribute the text or part of it without the consent of theauthor(s) and/or copyright holder(s), unless the work is under an open content license (like Creative Commons).The publication may also be distributed here under the terms of Article 25fa of the Dutch Copyright Act, indicated by the “Taverne” license.More information can be found on the University of Groningen website: https://www.rug.nl/library/open-access/self-archiving-pure/taverne-amendment.Take-down policyIf you believe that this document breaches copyright please contact us providing details, and we will remove access to the work immediatelyand investigate your claim.Downloaded from the University of Groningen/UMCG research database (Pure): http://www.rug.nl/research/portal. For technical reasons thenumber of authors shown on this cover page is limited to 10 maximum.Download date: 05-06-2022Stabilization of a class of port-Hamiltonian systems using saturatedcontrollers.L.P. Borja, T.C. Wesselink and J.M.A. Scherpen.Nijenborgh 4, 9747 AG Groningen, The Netherlands.e-mail: l.p.borja.rosales[j.m.a.scherpen]@rug.nl,t.c.wesselink@student.rug.nlAbstractThe port-Hamiltonian (PH) framework has been extensivelyused to model physical systems. One of the main advan-tages of this modeling approach is that it provides a system-atic procedure to obtain mathematical models that capturethe nonlinear phenomena and preserve conservation lawspresent in physical systems [1]. Additionally, in this frame-work, the roles of the interconnection pattern, the dissipa-tion, and the energy are underscored. Thus, a natural way tocontrol passive systems is to design controllers that give thedesired shape to the energy of the closed-loop system. Thisprocess is known as energy-shaping and it is the main ideaof several passivity-based control (PBC) approaches.While PBC techniques have been proved to be effective tosolve the stabilization problem of physical systems [2], [3],some important issues, that might arise during the imple-mentation of these controllers, are not addressed by thesetechniques, e.g., the lack of sensors to measure some vari-ables or constraints in the actuators. Indeed, these are perva-sive problems in nonlinear control, therefore, finding a solu-tion to them is paramount from a practical perspective.In this work, we propose a PBC approach that deals with theproblem of set-point regulation for a class of physical sys-tems. Moreover, this approach has the following appealingfeatures• It obviates the necessity of solving partial differentialequations, which, in general, are one of the main con-straints of the PBC techniques.• The controllers derived from this approach are sat-urated, and the saturation limits can be straightfor-wardly adjusted.• The control design only requires a partial measure-ment of the state. Particularly for mechanical sys-tems, it is not necessary to measure the velocities (mo-menta).• It is possible to include an integral-like term that en-sures the convergence of the steady-state error to zero.The proposed PBC approach follows the ideas previouslyreported in [4] and [5]. As proposed in the latter, the controldesign requires the definition of a “virtual state” whose dy-namics are designed in such a way that it is possible to injectdamping in coordinates that cannot be measured. Moreover,a direct extension of the results reported in [4] ensures thesaturation of the control signal. This is sometimes requiredto avoid malfunctions in the actuators of physical systems.With the aim of presenting our results, we, first, provide acharacterization of the PH systems for which the proposedPBC approach is suitable. Towards this goal, we provide as-sumptions that can be verified in a systematic way. Then, weproceed with the control design, and we provide the stabil-ity analysis of the closed-loop system, which is carried outusing passivity arguments and Lyapunov theory. Finally, weillustrate the applicability of the technique through the ex-perimental results obtained for a planar robot with two linksand flexible joints.References[1] Duindam, V., Macchelli, A., Stramigioli, S., &Bruyninckx, H. (Eds.). (2009). Modeling and control ofcomplex physical systems: the port-Hamiltonian approach.Springer Science & Business Media.[2] Ortega, R., Perez, J. A. L., Nicklasson, P. J., &Sira-Ramirez, H. J. (2013). Passivity-based control ofEuler-Lagrange systems: mechanical, electrical and elec-tromechanical applications. Springer Science & BusinessMedia.[3] van der Schaft, A. J. (2000). L2-gain and passivitytechniques in nonlinear control. London: Springer.[4] Borja, P., Cisneros, R., & Ortega, R. (2016). A con-structive procedure for energy shaping of portHamiltoniansystems. Automatica, 72, 230-234.[5] Dirksz, D. A., & Scherpen, J. M. A. (2013). On track-ing control of rigid-joint robots with only position measure-ments. IEEE Transactions on Control Systems Technology,21(4), 1510-1513.

   University of GroningenStabilization of a class of port-Hamiltonian systems using saturated controllersBorja, Luis Pablo; Wesselink, T. C. ; Scherpen, Jacquelien M. A.Published in:38th Benelux Meeting on Systems and ControlIMPORTANT NOTE: You are advised to consult the publisher's version (publisher's PDF) if you wish to cite fromit. Please check the document version below.Document VersionFinal author's version (accepted by publisher, after peer review)Publication date:2019Link to publication in University of Groningen/UMCG research databaseCitation for published version (APA):Borja, L. P., Wesselink, T. C., & Scherpen, J. M. A. (2019). Stabilization of a class of port-Hamiltoniansystems using saturated controllers. In 38th Benelux Meeting on Systems and Control (pp. 130-130). KULeuven.CopyrightOther than for strictly personal use, it is not permitted to download or to forward/distribute the text or part of it without the consent of theauthor(s) and/or copyright holder(s), unless the work is under an open content license (like Creative Commons).The publication may also be distributed here under the terms of Article 25fa of the Dutch Copyright Act, indicated by the “Taverne” license.More information can be found on the University of Groningen website: https://www.rug.nl/library/open-access/self-archiving-pure/taverne-amendment.Take-down policyIf you believe that this document breaches copyright please contact us providing details, and we will remove access to the work immediatelyand investigate your claim.Downloaded from the University of Groningen/UMCG research database (Pure): http://www.rug.nl/research/portal. For technical reasons thenumber of authors shown on this cover page is limited to 10 maximum.Download date: 11-09-2023Stabilization of a class of port-Hamiltonian systems using saturatedcontrollers.L.P. Borja, T.C. Wesselink and J.M.A. Scherpen.Nijenborgh 4, 9747 AG Groningen, The Netherlands.e-mail: l.p.borja.rosales[j.m.a.scherpen]@rug.nl,t.c.wesselink@student.rug.nlAbstractThe port-Hamiltonian (PH) framework has been extensivelyused to model physical systems. One of the main advan-tages of this modeling approach is that it provides a system-atic procedure to obtain mathematical models that capturethe nonlinear phenomena and preserve conservation lawspresent in physical systems [1]. Additionally, in this frame-work, the roles of the interconnection pattern, the dissipa-tion, and the energy are underscored. Thus, a natural way tocontrol passive systems is to design controllers that give thedesired shape to the energy of the closed-loop system. Thisprocess is known as energy-shaping and it is the main ideaof several passivity-based control (PBC) approaches.While PBC techniques have been proved to be effective tosolve the stabilization problem of physical systems [2], [3],some important issues, that might arise during the imple-mentation of these controllers, are not addressed by thesetechniques, e.g., the lack of sensors to measure some vari-ables or constraints in the actuators. Indeed, these are perva-sive problems in nonlinear control, therefore, finding a solu-tion to them is paramount from a practical perspective.In this work, we propose a PBC approach that deals with theproblem of set-point regulation for a class of physical sys-tems. Moreover, this approach has the following appealingfeatures• It obviates the necessity of solving partial differentialequations, which, in general, are one of the main con-straints of the PBC techniques.• The controllers derived from this approach are sat-urated, and the saturation limits can be straightfor-wardly adjusted.• The control design only requires a partial measure-ment of the state. Particularly for mechanical sys-tems, it is not necessary to measure the velocities (mo-menta).• It is possible to include an integral-like term that en-sures the convergence of the steady-state error to zero.The proposed PBC approach follows the ideas previouslyreported in [4] and [5]. As proposed in the latter, the controldesign requires the definition of a “virtual state” whose dy-namics are designed in such a way that it is possible to injectdamping in coordinates that cannot be measured. Moreover,a direct extension of the results reported in [4] ensures thesaturation of the control signal. This is sometimes requiredto avoid malfunctions in the actuators of physical systems.With the aim of presenting our results, we, first, provide acharacterization of the PH systems for which the proposedPBC approach is suitable. Towards this goal, we provide as-sumptions that can be verified in a systematic way. Then, weproceed with the control design, and we provide the stabil-ity analysis of the closed-loop system, which is carried outusing passivity arguments and Lyapunov theory. Finally, weillustrate the applicability of the technique through the ex-perimental results obtained for a planar robot with two linksand flexible joints.References[1] Duindam, V., Macchelli, A., Stramigioli, S., &Bruyninckx, H. (Eds.). (2009). Modeling and control ofcomplex physical systems: the port-Hamiltonian approach.Springer Science & Business Media.[2] Ortega, R., Perez, J. A. L., Nicklasson, P. J., &Sira-Ramirez, H. J. (2013). Passivity-based control ofEuler-Lagrange systems: mechanical, electrical and elec-tromechanical applications. Springer Science & BusinessMedia.[3] van der Schaft, A. J. (2000). L2-gain and passivitytechniques in nonlinear control. London: Springer.[4] Borja, P., Cisneros, R., & Ortega, R. (2016). A con-structive procedure for energy shaping of portHamiltoniansystems. Automatica, 72, 230-234.[5] Dirksz, D. A., & Scherpen, J. M. A. (2013). On track-ing control of rigid-joint robots with only position measure-ments. IEEE Transactions on Control Systems Technology,21(4), 1510-1513.

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Stabilization of a class of port-Hamiltonian systems using saturated controllers

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