Optimal Control of Port-Hamiltonian Descriptor Systems with Minimal Energy Supply

Authors

Timm Faulwasser, Bernhard Maschke, Friedrich Philipp, Manuel Schaller, Karl Worthmann

Abstract

We consider the singular optimal control problem of minimizing the energy supply of linear dissipative port-Hamiltonian descriptor systems subject to control and terminal state constraints. To this end, after reducing the problem to an ODE with feed-through term, we derive an input-state turnpike towards a subspace for optimal control of generalized port-Hamiltonian ordinary differential equations. We study the reachability properties of the system and prove that optimal states exhibit a turnpike behavior with respect to the conservative subspace. By means of the port-Hamiltonian structure, we show that, despite control constraints, this turnpike property is global in the initial state. Further, we characterize the class of dissipative Hamiltonian matrices and pencils.

Citation

• Journal: SIAM Journal on Control and Optimization
• Year: 2022
• Volume: 60
• Issue: 4
• Pages: 2132–2158
• Publisher: Society for Industrial & Applied Mathematics (SIAM)
• DOI: 10.1137/21m1427723

BibTeX

@article{Faulwasser_2022,
  title={{Optimal Control of Port-Hamiltonian Descriptor Systems with Minimal Energy Supply}},
  volume={60},
  ISSN={1095-7138},
  DOI={10.1137/21m1427723},
  number={4},
  journal={SIAM Journal on Control and Optimization},
  publisher={Society for Industrial & Applied Mathematics (SIAM)},
  author={Faulwasser, Timm and Maschke, Bernhard and Philipp, Friedrich and Schaller, Manuel and Worthmann, Karl},
  year={2022},
  pages={2132--2158}
}

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References

• 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
• Beattie, C., Mehrmann, V., Xu, H. & Zwart, H. Linear port-Hamiltonian descriptor systems. Mathematics of Control, Signals, and Systems vol. 30 (2018) – 10.1007/s00498-018-0223-3
• Berger, T., Ilchmann, A. & Trenn, S. The quasi-Weierstraß form for regular matrix pencils. Linear Algebra and its Applications vol. 436 4052–4069 (2012) – 10.1016/j.laa.2009.12.036
• Berger, T. & Reis, T. Controllability of Linear Differential-Algebraic Systems—A Survey. Surveys in Differential-Algebraic Equations I 1–61 (2013) doi:10.1007/978-3-642-34928-7_1 – 10.1007/978-3-642-34928-7_1
• Brogliato, B., Maschke, B., Lozano, R. & Egeland, O. Dissipative Systems Analysis and Control. Communications and Control Engineering (Springer London, 2007). doi:10.1007/978-1-84628-517-2 – 10.1007/978-1-84628-517-2
• Carlson, D. & Schneider, H. Inertia theorems for matrices: The semidefinite case. Journal of Mathematical Analysis and Applications vol. 6 430–446 (1963) – 10.1016/0022-247x(63)90023-4
• Curtain, R. F. & Zwart, H. An Introduction to Infinite-Dimensional Linear Systems Theory. Texts in Applied Mathematics (Springer New York, 1995). doi:10.1007/978-1-4612-4224-6 – 10.1007/978-1-4612-4224-6
• Damm, T., Grüne, L., Stieler, M. & Worthmann, K. An Exponential Turnpike Theorem for Dissipative Discrete Time Optimal Control Problems. SIAM Journal on Control and Optimization vol. 52 1935–1957 (2014) – 10.1137/120888934
• Duindam, V., Macchelli, A., Stramigioli, S. & Bruyninckx, H. Modeling and Control of Complex Physical Systems. (Springer Berlin Heidelberg, 2009). doi:10.1007/978-3-642-03196-0 – 10.1007/978-3-642-03196-0
• Faulwasser T.. Turnpikes, Trims and Symmetries, arXiv:2104.03039 (2021)
• Faulwasser, T., Flaßkamp, K., Ober-Blöbaum, S. & Worthmann, K. A Dissipativity Characterization of Velocity Turnpikes in Optimal Control Problems for Mechanical Systems. IFAC-PapersOnLine vol. 54 624–629 (2021) – 10.1016/j.ifacol.2021.06.125
• Faulwasser T.. Turnpike Properties in Optimal Control: An Overview of Discrete-Time and Continuous-Time Results, arXiv:2011.13670 (2021)
• Faulwasser T.. The Interval Turnpike Property for Adjoints, arXiv:2005.12120 (2020)
• Faulwasser, T., Korda, M., Jones, C. N. & Bonvin, D. On turnpike and dissipativity properties of continuous-time optimal control problems. Automatica vol. 81 297–304 (2017) – 10.1016/j.automatica.2017.03.012
• Faulwasser, T., Korda, M., Jones, C. N. & Bonvin, D. On turnpike and dissipativity properties of continuous-time optimal control problems. Automatica vol. 81 297–304 (2017) – 10.1016/j.automatica.2017.03.012
• Faulwasser T.. Optimal Control of Port-Hamiltonian Descriptor Systems with Minimal Energy Supply, arXiv:2106.06571 (2021)
• Flaßkamp, K., Ober-Blöbaum, S. & Worthmann, K. Symmetry and motion primitives in model predictive control. Mathematics of Control, Signals, and Systems vol. 31 455–485 (2019) – 10.1007/s00498-019-00246-7
• Grüne, L. & Guglielmi, R. On the relation between turnpike properties and dissipativity for continuous time linear quadratic optimal control problems. Mathematical Control & Related Fields vol. 11 169–188 (2021) – 10.3934/mcrf.2020032
• Grüne, L. & Müller, M. A. On the relation between strict dissipativity and turnpike properties. Systems & Control Letters vol. 90 45–53 (2016) – 10.1016/j.sysconle.2016.01.003
• Grüne, L., Schaller, M. & Schiela, A. Exponential sensitivity and turnpike analysis for linear quadratic optimal control of general evolution equations. Journal of Differential Equations vol. 268 7311–7341 (2020) – 10.1016/j.jde.2019.11.064
• Heiland J.. Classical System Theory Revisited for Turnpike in Standard State Space Systems and Impulse Controllable Descriptor Systems, arXiv:2007.13621 (2020)
• Hermes H.. Functional Analysis and Time Optimal Control (1969)
• Ilchmann, A., Leben, L., Witschel, J. & Worthmann, K. Optimal control of differential‐algebraic equations from an ordinary differential equation perspective. Optimal Control Applications and Methods vol. 40 351–366 (2019) – 10.1002/oca.2481
• 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
• Lee E. B.. Foundations of Optimal Control Theory (1986)
• Liberzon, D. Calculus of Variations and Optimal Control Theory. (2012) doi:10.1515/9781400842643 – 10.1515/9781400842643
• Macki J.. Introduction to Optimal Control Theory (2012)
• 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
• Mehl, C., Mehrmann, V. & Wojtylak, M. Distance problems for dissipative Hamiltonian systems and related matrix polynomials. Linear Algebra and its Applications vol. 623 335–366 (2021) – 10.1016/j.laa.2020.05.026
• Ortega, R., van der Schaft, A., Castanos, F. & Astolfi, A. Control by Interconnection and Standard Passivity-Based Control of Port-Hamiltonian Systems. IEEE Transactions on Automatic Control vol. 53 2527–2542 (2008) – 10.1109/tac.2008.2006930
• Philipp F.. Minimizing the Energy Supply of Infinite-Dimensional Linear Port-Hamiltonian Systems, arXiv:2105.03873 (2021)
• Pighin D.. The Turnpike with Lack of Observability, arXiv:2007.14081 (2020)
• Porretta, A. & Zuazua, E. Long Time versus Steady State Optimal Control. SIAM Journal on Control and Optimization vol. 51 4242–4273 (2013) – 10.1137/130907239
• 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
• Sontag, E. D. Mathematical Control Theory. Texts in Applied Mathematics (Springer New York, 1998). doi:10.1007/978-1-4612-0577-7 – 10.1007/978-1-4612-0577-7
• Trélat, E. & Zhang, C. Integral and measure-turnpike properties for infinite-dimensional optimal control systems. Mathematics of Control, Signals, and Systems vol. 30 (2018) – 10.1007/s00498-018-0209-1
• 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
• Trélat, E. & Zuazua, E. The turnpike property in finite-dimensional nonlinear optimal control. Journal of Differential Equations vol. 258 81–114 (2015) – 10.1016/j.jde.2014.09.005
• van der Schaft, A. J. Port-Hamiltonian Differential-Algebraic Systems. Surveys in Differential-Algebraic Equations I 173–226 (2013) doi:10.1007/978-3-642-34928-7_5 – 10.1007/978-3-642-34928-7_5
• 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
• van der Schaft, A. & Maschke, B. Generalized port-Hamiltonian DAE systems. Systems & Control Letters vol. 121 31–37 (2018) – 10.1016/j.sysconle.2018.09.008
• van der Schaft, A. & Maschke, B. Dirac and Lagrange Algebraic Constraints in Nonlinear Port-Hamiltonian Systems. Vietnam Journal of Mathematics vol. 48 929–939 (2020) – 10.1007/s10013-020-00419-x
• Volpe, R. & Khosla, P. Analysis and experimental verification of a forth order plant model for manipulator force control. IEEE Robotics & Automation Magazine vol. 1 4–13 (1994) – 10.1109/100.298481
• Willems, J. C. Dissipative dynamical systems part I: General theory. Archive for Rational Mechanics and Analysis vol. 45 321–351 (1972) – 10.1007/bf00276493
• Yip, E. & Sincovec, R. Solvability, controllability, and observability of continuous descriptor systems. IEEE Transactions on Automatic Control vol. 26 702–707 (1981) – 10.1109/tac.1981.1102699