Abstract
Adaptive structures are equipped with sensors and actuators to actively counteract external loads such as wind. This can significantly reduce resource consumption and emissions during the life cycle compared to conventional structures. A common approach for active damping is to derive a port-Hamiltonian model and to employ linear-quadratic control. However, the quadratic control penalization lacks physical interpretation and merely serves as a regularization term. Rather, we propose a controller, which achieves the goal of vibration damping while acting energy-optimal. Leveraging the port-Hamiltonian structure, we show that the optimal control is uniquely determined, even on singular arcs. Further, we prove a stable long-time behavior of optimal trajectories by means of a turnpike property. Last, the proposed controller’s efficiency is evaluated in a numerical study.
Zusammenfassung
Adaptive Strukturen sind mit Sensoren und Aktuatoren ausgestattet, um aktiv äußeren Lasten wie Wind entgegenzuwirken. Dies ermöglicht, den Ressourcenverbrauch und die Emissionen während des Lebenszyklus im Vergleich zu konventionellen Strukturen erheblich zu reduzieren. Ein gängiger Ansatz für aktive Dämpfung ist die Ableitung eines Port-Hamiltonschen-Modells und der Einsatz von linear-quadratischer Steuerung. Die quadratische Kontrollbestrafung jedoch mangelt an physikalischer Interpretation und dient lediglich als Regularisierungsterm. Wir schlagen stattdessen einen Regler vor, der das Ziel der Vibrationsdämpfung erreicht, während er energieoptimal handelt. Unter Ausnutzung der Port-Hamiltonschen-Struktur zeigen wir, dass die optimale Steuerung eindeutig bestimmt ist, sogar auf sogenannten singulären Bögen. Weiterhin beweisen wir ein stabiles Langzeitverhalten optimaler Trajektorien im Sinne einer Turnpike-Eigenschaft. Zuletzt wird die Effizienz des vorgeschlagenen Reglers mittels einer numerischen Studie bewertet.
Funding source: Deutsche Forschungsgemeinschaft (DFG)
Award Identifier / Grant number: Project-ID 27906422
Award Identifier / Grant number: SFB 1244
Award Identifier / Grant number: project B02
Award Identifier / Grant number: Project-ID 507037103
About the authors
Manuel Schaller obtained the M.Sc. and Ph.D. in Applied Mathematics from the University of Bayreuth in 2017 and 2021 respectively. From 2020-2023 he held a PostDoc and Lecturer (tenure track) position in the Optimization-based Control group at Technische Universität Ilmenau, Germany. There, he has been Assistant Professor for differential equations since July 2023. His research focuses on data-driven control with guarantees, port-Hamiltonian systems, and stability in infinite dimensional optimal control. For his research Dr. Schaller has been named junior fellow of the GAMM (Society for Applied Mathematics and Mechanics) and received the Best Poster Award at the workshop on systems theory and PDEs (WOSTAP 2022).
Amelie Zeller received her M.Sc. degree in engineering cybernetics from the University of Stuttgart in 2021 and has been a research assistant at the Institute for System Dynamics at the University of Stuttgart since then. Her research interests include actuator placement and state and disturbance estimation for adaptive structures.
Michael Böhm received a Dipl.-Ing. degree and his PhD in engineering cybernetics from the University of Stuttgart, Stuttgart, Germany, in 2011 and 2017, respectively. Since 2017, he has been the Head of the Construction Systems Engineering Group at the Institute for System Dynamics, University of Stuttgart. His current research interests include dynamic modeling and control of mechanical and hydraulic systems, and distributed parameter systems with applications to civil engineering.
Oliver Sawodny received the Dipl.-Ing. degree in electrical engineering from the University of Karlsruhe, Germany, in 1991, and the Ph.D. degree from the Ulm University, Germany, in 1996. In 2002, he became a Full Professor with the Technical University of Ilmenau, Germany. Since 2005, he has been the Director of the Institute for System Dynamics, University of Stuttgart, Germany. His current research interests include methods of differential geometry, trajectory generation, and applications to mechatronic systems.
Cristina Tarín received the M.Sc. degree in electrical engineering from the Technical University of Valencia, Valencia, Spain, in 1996, and the Ph.D. degree from the University of Ulm, Ulm, Germany, in 2001. She has been teaching at the Universidad Carlos III of Madrid, Madrid, Spain, and was enrolled in several research labs with the Technical University of Valencia. Since 2009, she has been a Full Professor with the Department of System Dynamics, University of Stuttgart, Stuttgart, Germany. Her current research interests include signal processing and filtering as well as estimation and modeling.
Karl Worthmann received his Ph.D. degree in mathematics from the University of Bayreuth, Germany, in 2012. 2014 he become assistant professor for “Differential Equations” at Technische Universität Ilmenau (TU Ilmenau), Germany. 2019 he was promoted to full professor after receiving the Heisenberg-professorship “Optimization-based Control” by the German Research Foundation (DFG). He was recipient of the Ph.D. Award from the City of Bayreuth, Germany, and stipend of the German National Academic Foundation. 2013 he has been appointed Junior Fellow of the Society of Applied Mathematics and Mechanics (GAMM), where he served as speaker in 2014 and 2015.
Acknowledgment
The authors would like to thank the referees for the valuable suggestions and their careful reading, which in particular improved the statement of Theorem 4. Further, they are grateful to Benedikt Oppeneiger (TU Ilmenau) for providing valuable feedback.
-
Research ethics: Not applicable.
-
Author contributions: The authors have accepted responsibility for the entire content of this manuscript and approved its submission.
-
Competing interests: The authors state no conflict of interest.
-
Research funding: We appreciate funding by the Deutsche Forschungsgemeinschaft (DFG) through Project-ID 279064222, SFB 1244, project B02 and Project-ID 507037103.
-
Data availability: Not applicable.
References
[1] F. Schlegl, C. Honold, S. Leistner, et al.., “Integration of LCA in the planning phases of adaptive buildings,” Sustainability, vol. 11, no. 16, p. 4299, 2019. https://doi.org/10.3390/su11164299.Search in Google Scholar
[2] W. Sobek and P. Teuffel, “Adaptive systems in architecture and structural engineering,” in Smart Structures and Materials 2001: Smart Systems for Bridges, Structures, and Highways, vol. 4330, SPIE, 2001, pp. 36–45.10.1117/12.434141Search in Google Scholar
[3] B. Jacob and H. J. Zwart, Linear Port-Hamiltonian Systems on Infinite-Dimensional Spaces, vol. 223, Basel, Springer Science & Business Media, 2012.10.1007/978-3-0348-0399-1Search in Google Scholar
[4] R. Ortega, A. Van Der Schaft, B. Maschke, and G. Escobar, “Interconnection and damping assignment passivity-based control of port-controlled Hamiltonian systems,” Automatica, vol. 38, no. 4, pp. 585–596, 2002. https://doi.org/10.1016/s0005-1098(01)00278-3.Search in Google Scholar
[5] A. Venkatraman and A. Van der Schaft, “Full-order observer design for a class of port-Hamiltonian systems,” Automatica, vol. 46, no. 3, pp. 555–561, 2010. https://doi.org/10.1016/j.automatica.2010.01.019.Search in Google Scholar
[6] C. Mehl, V. Mehrmann, and M. Wojtylak, “Linear algebra properties of dissipative Hamiltonian descriptor systems,” SIAM J. Matrix Anal. Appl., vol. 39, no. 3, pp. 1489–1519, 2018. https://doi.org/10.1137/18m1164275.Search in Google Scholar
[7] P. Kotyczka and L. Lefevre, “Discrete-time port-Hamiltonian systems: a definition based on symplectic integration,” Syst. Control Lett., vol. 133, p. 104530, 2019. https://doi.org/10.1016/j.sysconle.2019.104530.Search in Google Scholar
[8] P. Kotyczka, B. Maschke, and L. Lefèvre, “Weak form of Stokes–Dirac structures and geometric discretization of port-Hamiltonian systems,” J. Comput. Phys., vol. 361, pp. 442–476, 2018. https://doi.org/10.1016/j.jcp.2018.02.006.Search in Google Scholar
[9] A. Warsewa, M. Böhm, O. Sawodny, and C. Tarín, “A port-Hamiltonian approach to modeling the structural dynamics of complex systems,” Appl. Math. Model., vol. 89, pp. 1528–1546, 2021. https://doi.org/10.1016/j.apm.2020.07.038.Search in Google Scholar
[10] A. Warsewa, Energy-Based Modeling and Decentralized Observers for Adaptive Structures, Düren, Shaker Verlag, 2021.Search in Google Scholar
[11] A. Warsewa, J. L. Wagner, M. Böhm, O. Sawodny, and C. Tarín, “Networked decentralized control of adaptive structures,” J. Commun., vol. 15, no. 6, pp. 496–502, 2020. https://doi.org/10.12720/jcm.15.6.496-502.Search in Google Scholar
[12] A. Warsewa, J. L. Wagner, M. Böhm, O. Sawodny, and C. Tarín, “Decentralized LQG control for adaptive high-rise structures,” IFAC-PapersOnLine, vol. 53, no. 2, pp. 9131–9137, 2020. https://doi.org/10.1016/j.ifacol.2020.12.2154.Search in Google Scholar
[13] S. Dakova, J. L. Heidingsfeld, M. Böhm, and O. Sawodny, “An optimal control strategy to distribute element wear for adaptive high-rise structures,” in 2022 American Control Conference (ACC), IEEE, 2022, pp. 4614–4619.10.23919/ACC53348.2022.9867396Search in Google Scholar
[14] M. Schaller, F. Philipp, T. Faulwasser, K. Worthmann, and B. Maschke, “Control of port-Hamiltonian systems with minimal energy supply,” Eur. J. Control, vol. 62, pp. 33–40, 2021. https://doi.org/10.1016/j.ejcon.2021.06.017.Search in Google Scholar
[15] J. Willems, “Least squares stationary optimal control and the algebraic Riccati equation,” IEEE Trans. Automat. Control, vol. 16, no. 6, pp. 621–634, 1971. https://doi.org/10.1109/tac.1971.1099831.Search in Google Scholar
[16] T. Faulwasser, B. Maschke, F. Philipp, M. Schaller, and K. Worthmann, “Optimal control of port-Hamiltonian descriptor systems with minimal energy supply,” SIAM J. Control Optim., vol. 60, no. 4, pp. 2132–2158, 2022. https://doi.org/10.1137/21m1427723.Search in Google Scholar
[17] F. Philipp, M. Schaller, T. Faulwasser, B. Maschke, and K. Worthmann, “Minimizing the energy supply of infinite-dimensional linear port-Hamiltonian systems,” IFAC-PapersOnLine, vol. 54, no. 19, pp. 155–160, 2021. https://doi.org/10.1016/j.ifacol.2021.11.071.Search in Google Scholar
[18] T. Faulwasser, J. Kirchhoff, V. Mehrmann, F. Philipp, M. Schaller, and K. Worthmann, “Hidden regularity in singular optimal control of port-Hamiltonian systems,” Preprint arXiv:2305.03790, 2023.Search in Google Scholar
[19] A. Van Der Schaft and D. Jeltsema, “Port-Hamiltonian systems theory: an introductory overview,” Found. Trends Syst. Control, vol. 1, nos. 2–3, pp. 173–378, 2014. https://doi.org/10.1561/2600000002.Search in Google Scholar
[20] F. L. Cardoso-Ribeiro, D. Matignon, and L. Lefèvre, “A partitioned finite element method for power-preserving discretization of open systems of conservation laws,” IMA J. Math. Control Inform., vol. 38, no. 2, pp. 493–533, 2021. https://doi.org/10.1093/imamci/dnaa038.Search in Google Scholar
[21] J. W. Strutt and B. Rayleigh, The Theory of Sound, London, Macmillan, 1877.Search in Google Scholar
[22] L. Grüne, “Economic receding horizon control without terminal constraints,” Automatica, vol. 49, no. 3, pp. 725–734, 2013. https://doi.org/10.1016/j.automatica.2012.12.003.Search in Google Scholar
[23] L. Grüne, J. Pannek, L. Grüne, and J. Pannek, Nonlinear Model Predictive Control, Cham, Springer, 2017.10.1007/978-3-319-46024-6Search in Google Scholar
[24] E. Trélat, C. Zhang, and E. Zuazua, “Steady-state and periodic exponential turnpike property for optimal control problems in Hilbert spaces,” SIAM J. Control Optim., vol. 56, no. 2, pp. 1222–1252, 2018. https://doi.org/10.1137/16m1097638.Search in Google Scholar
[25] K. Zhou, J. Doyle, and K. Glover, Robust and Optimal Control, Upper Saddle River, NJ, Prentice Hall, 1996.Search in Google Scholar
[26] W. S. Levine, The Control Handbook: Control System Advanced Methods, Boca Raton, CRC Press, 2011.Search in Google Scholar
[27] F. Philipp, M. Schaller, K. Worthmann, T. Faulwasser, and B. Maschke, “Optimal control of port-Hamiltonian systems: energy, entropy, and exergy,” Preprint arXiv:2306.08914, 2023.Search in Google Scholar
[28] E. B. Lee and L. Markus, Foundations of Optimal Control Theory. The SIAM Series in Applied Mathematics, London, Sydney, John Wiley & Sons New York, 1967.Search in Google Scholar
[29] A. Locatelli, Optimal Control: An Introduction, Basel, Boston, Berlin, Birkhäuser, 2001.10.1007/978-3-0348-8328-3Search in Google Scholar
[30] J. A. E. Andersson, J. Gillis, G. Horn, J. B. Rawlings, and M. Diehl, “CasADi – a software framework for nonlinear optimization and optimal control,” Math. Program. Comput., vol. 11, no. 1, pp. 1–36, 2019. https://doi.org/10.1007/s12532-018-0139-4.Search in Google Scholar
[31] P. Schwerdtner and M. Schaller, “Structured optimization-based model order reduction for parametric systems,” Preprint arXiv:2209.05101, 2022.Search in Google Scholar
[32] T. Reis and M. Voigt, “Linear-quadratic optimal control of differential-algebraic systems: the infinite time horizon problem with zero terminal state,” SIAM J. Control Optim., vol. 57, no. 3, pp. 1567–1596, 2019. https://doi.org/10.1137/18m1189609.Search in Google Scholar
[33] T. Faulwasser, K. Flaßkamp, S. Ober-Blöbaum, M. Schaller, and K. Worthmann, “Manifold turnpikes, trims, and symmetries,” Math. Control, Signals, Syst., vol. 34, no. 4, pp. 759–788, 2022. https://doi.org/10.1007/s00498-022-00321-6.Search in Google Scholar
© 2023 Walter de Gruyter GmbH, Berlin/Boston
