Energy-optimal control of adaptive structures
Authors
Manuel Schaller, Amelie Zeller, Michael Böhm, Oliver Sawodny, Cristina Tarín, Karl Worthmann
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.
Citation
- Journal: at - Automatisierungstechnik
- Year: 2024
- Volume: 72
- Issue: 2
- Pages: 107–119
- Publisher: Walter de Gruyter GmbH
- DOI: 10.1515/auto-2023-0090
BibTeX
@article{Schaller_2024,
title={{Energy-optimal control of adaptive structures}},
volume={72},
ISSN={2196-677X},
DOI={10.1515/auto-2023-0090},
number={2},
journal={at - Automatisierungstechnik},
publisher={Walter de Gruyter GmbH},
author={Schaller, Manuel and Zeller, Amelie and Böhm, Michael and Sawodny, Oliver and Tarín, Cristina and Worthmann, Karl},
year={2024},
pages={107--119}
}
References
- Schlegl, F. et al. Integration of LCA in the Planning Phases of Adaptive Buildings. Sustainability vol. 11 4299 (2019) – 10.3390/su11164299
- Sobek, W. & Teuffel, P. <title>Adaptive systems in architecture and structural engineering</title> SPIE Proceedings vol. 4330 36–45 (2001) – 10.1117/12.434141
- Jacob, B. & Zwart, H. J. Linear Port-Hamiltonian Systems on Infinite-Dimensional Spaces. (Springer Basel, 2012). doi:10.1007/978-3-0348-0399-1 – 10.1007/978-3-0348-0399-1
- Ortega, R., van der Schaft, A., Maschke, B. & Escobar, G. Interconnection and damping assignment passivity-based control of port-controlled Hamiltonian systems. Automatica vol. 38 585–596 (2002) – 10.1016/s0005-1098(01)00278-3
- Venkatraman, A. & van der Schaft, A. J. Full-order observer design for a class of port-Hamiltonian systems. Automatica vol. 46 555–561 (2010) – 10.1016/j.automatica.2010.01.019
- Mehl, C., Mehrmann, V. & Wojtylak, M. Linear Algebra Properties of Dissipative Hamiltonian Descriptor Systems. SIAM Journal on Matrix Analysis and Applications vol. 39 1489–1519 (2018) – 10.1137/18m1164275
- Kotyczka, P. & Lefèvre, L. Discrete-time port-Hamiltonian systems: A definition based on symplectic integration. Systems & Control Letters vol. 133 104530 (2019) – 10.1016/j.sysconle.2019.104530
- Kotyczka, P., Maschke, B. & Lefèvre, L. Weak form of Stokes–Dirac structures and geometric discretization of port-Hamiltonian systems. Journal of Computational Physics vol. 361 442–476 (2018) – 10.1016/j.jcp.2018.02.006
- Warsewa, A., Böhm, M., Sawodny, O. & Tarín, C. A port-Hamiltonian approach to modeling the structural dynamics of complex systems. Applied Mathematical Modelling vol. 89 1528–1546 (2021) – 10.1016/j.apm.2020.07.038
- A. Warsewa, Energy-Based Modeling and Decentralized Observers for Adaptive Structures, Düren, Shaker Verlag, 2021.
- Networked Decentralized Control of Adaptive Structures. Journal of Communications 496–502 (2020) doi:10.12720/jcm.15.6.496-502 – 10.12720/jcm.15.6.496-502
- Warsewa, A., Wagner, J. L., Böhm, M., Sawodny, O. & Tarín, C. Decentralized LQG Control for Adaptive High-Rise Structures. IFAC-PapersOnLine vol. 53 9131–9137 (2020) – 10.1016/j.ifacol.2020.12.2154
- Dakova, S., Heidingsfeld, J. L., Bohm, M. & Sawodny, O. An Optimal Control Strategy to Distribute Element Wear for Adaptive High-Rise Structures. 2022 American Control Conference (ACC) 4614–4619 (2022) doi:10.23919/acc53348.2022.9867396 – 10.23919/acc53348.2022.9867396
- Schaller, M., Philipp, F., Faulwasser, T., Worthmann, K. & Maschke, B. Control of port-Hamiltonian systems with minimal energy supply. European Journal of Control vol. 62 33–40 (2021) – 10.1016/j.ejcon.2021.06.017
- Willems, J. Least squares stationary optimal control and the algebraic Riccati equation. IEEE Transactions on Automatic Control vol. 16 621–634 (1971) – 10.1109/tac.1971.1099831
- Faulwasser, T., Maschke, B., Philipp, F., Schaller, M. & Worthmann, K. Optimal Control of Port-Hamiltonian Descriptor Systems with Minimal Energy Supply. SIAM Journal on Control and Optimization vol. 60 2132–2158 (2022) – 10.1137/21m1427723
- Philipp, F., Schaller, M., Faulwasser, T., Maschke, B. & Worthmann, K. Minimizing the energy supply of infinite-dimensional linear port-Hamiltonian systems. IFAC-PapersOnLine vol. 54 155–160 (2021) – 10.1016/j.ifacol.2021.11.071
- 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.
- van der Schaft, A. & Jeltsema, D. Port-Hamiltonian Systems Theory: An Introductory Overview. Foundations and Trends® in Systems and Control vol. 1 173–378 (2014) – 10.1561/2600000002
- Cardoso-Ribeiro, F. L., Matignon, D. & Lefèvre, L. A partitioned finite element method for power-preserving discretization of open systems of conservation laws. IMA Journal of Mathematical Control and Information vol. 38 493–533 (2020) – 10.1093/imamci/dnaa038
- J. W. Strutt and B. Rayleigh, The Theory of Sound, London, Macmillan, 1877.
- Grüne, L. Economic receding horizon control without terminal constraints. Automatica vol. 49 725–734 (2013) – 10.1016/j.automatica.2012.12.003
- Grüne, L. & Pannek, J. Nonlinear Model Predictive Control. Communications and Control Engineering (Springer International Publishing, 2017). doi:10.1007/978-3-319-46024-6 – 10.1007/978-3-319-46024-6
- Trélat, E., Zhang, C. & Zuazua, E. Steady-State and Periodic Exponential Turnpike Property for Optimal Control Problems in Hilbert Spaces. SIAM Journal on Control and Optimization vol. 56 1222–1252 (2018) – 10.1137/16m1097638
- K. Zhou, J. Doyle, and K. Glover, Robust and Optimal Control, Upper Saddle River, NJ, Prentice Hall, 1996.
- W. S. Levine, The Control Handbook: Control System Advanced Methods, Boca Raton, CRC Press, 2011.
- 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.
- 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.
- Locatelli, A. Optimal Control. (Birkhäuser Basel, 2001). doi:10.1007/978-3-0348-8328-3 – 10.1007/978-3-0348-8328-3
- Andersson, J. A. E., Gillis, J., Horn, G., Rawlings, J. B. & Diehl, M. CasADi: a software framework for nonlinear optimization and optimal control. Mathematical Programming Computation vol. 11 1–36 (2018) – 10.1007/s12532-018-0139-4
- P. Schwerdtner and M. Schaller, “Structured optimization-based model order reduction for parametric systems,” Preprint arXiv:2209.05101, 2022.
- Reis, T. & Voigt, M. Linear-Quadratic Optimal Control of Differential-Algebraic Systems: The Infinite Time Horizon Problem with Zero Terminal State. SIAM Journal on Control and Optimization vol. 57 1567–1596 (2019) – 10.1137/18m1189609
- Faulwasser, T., Flaßkamp, K., Ober-Blöbaum, S., Schaller, M. & Worthmann, K. Manifold turnpikes, trims, and symmetries. Mathematics of Control, Signals, and Systems vol. 34 759–788 (2022) – 10.1007/s00498-022-00321-6