Stabilization of port-Hamiltonian systems by nonlinear boundary control in the presence of disturbances
Authors
Abstract
In this paper, we are concerned with the stabilization of linear port-Hamiltonian systems of arbitrary orderN∈ ℕ on a bounded 1-dimensional spatial domain (a,b). In order to achieve stabilization, we couple the system to a dynamic boundary controller, that is, a controller that acts on the system only via the boundary pointsa,bof the spatial domain. We use a nonlinear controller in order to capture the nonlinear behavior that realistic actuators often exhibit and, moreover, we allow the output of the controller to be corrupted by actuator disturbances before it is fed back into the system. What we show here is that the resulting nonlinear closed-loop system is input-to-state stable w.r.t. square-integrable disturbance inputs. In particular, we obtain uniform input-to-state stability for systems of orderN= 1 and a special class of nonlinear controllers, and weak input-to-state stability for systems of arbitrary orderN∈ ℕ and a more general class of nonlinear controllers. Also, in both cases, we obtain convergence to 0 of all solutions ast→∞. Applications are given to vibrating strings and beams.
Citation
- Journal: ESAIM: Control, Optimisation and Calculus of Variations
- Year: 2021
- Volume: 27
- Issue:
- Pages: 53
- Publisher: EDP Sciences
- DOI: 10.1051/cocv/2021051
BibTeX
@article{Schmid_2021,
title={{Stabilization of port-Hamiltonian systems by nonlinear boundary control in the presence of disturbances}},
volume={27},
ISSN={1262-3377},
DOI={10.1051/cocv/2021051},
journal={ESAIM: Control, Optimisation and Calculus of Variations},
publisher={EDP Sciences},
author={Schmid, Jochen and Zwart, Hans},
year={2021},
pages={53}
}
References
- Augner, B. Well-Posedness and Stability of Infinite-Dimensional Linear Port-Hamiltonian Systems with Nonlinear Boundary Feedback. SIAM Journal on Control and Optimization vol. 57 1818–1844 (2019) – 10.1137/15m1024901
- Bastin, G. & Coron, J.-M. Stability and Boundary Stabilization of 1-D Hyperbolic Systems. Progress in Nonlinear Differential Equations and Their Applications (Springer International Publishing, 2016). doi:10.1007/978-3-319-32062-5 – 10.1007/978-3-319-32062-5
- Clarke, F. H., Ledyaev, Yu. S. & Stern, R. J. Asymptotic Stability and Smooth Lyapunov Functions. Journal of Differential Equations vol. 149 69–114 (1998) – 10.1006/jdeq.1998.3476
- Curtain, R. & Zwart, H. Introduction to Infinite-Dimensional Systems Theory. Texts in Applied Mathematics (Springer New York, 2020). doi:10.1007/978-1-0716-0590-5 – 10.1007/978-1-0716-0590-5
- Dashkovskiy, S. & Mironchenko, A. Input-to-state stability of infinite-dimensional control systems. Mathematics of Control, Signals, and Systems vol. 25 1–35 (2012) – 10.1007/s00498-012-0090-2
- Dashkovskiy, Conference proceedings of the 11th IFAC Symposium on Nonlinear Control Systems, IFAC-PapersOnLine (2019)
- Dashkovskiy, S., Kapustyan, O. & Schmid, J. A local input-to-state stability result w.r.t. attractors of nonlinear reaction–diffusion equations. Mathematics of Control, Signals, and Systems vol. 32 309–326 (2020) – 10.1007/s00498-020-00256-w
- 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
- Edalatzadeh, M. S. & Morris, K. A. Stability and Well-Posedness of a Nonlinear Railway Track Model. IEEE Control Systems Letters vol. 3 162–167 (2019) – 10.1109/lcsys.2018.2849831
- Fattorini, H. O. Boundary Control Systems. SIAM Journal on Control vol. 6 349–385 (1968) – 10.1137/0306025
- 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
- Jacob, B., Morris, K. & Zwart, H. C 0-semigroups for hyperbolic partial differential equations on a one-dimensional spatial domain. Journal of Evolution Equations vol. 15 493–502 (2015) – 10.1007/s00028-014-0271-1
- Jacob, B. & Kaiser, J. T. Well-posedness of systems of 1-D hyperbolic partial differential equations. Journal of Evolution Equations vol. 19 91–109 (2018) – 10.1007/s00028-018-0470-2
- Jacob, B., Nabiullin, R., Partington, J. R. & Schwenninger, F. L. Infinite-Dimensional Input-to-State Stability and Orlicz Spaces. SIAM Journal on Control and Optimization vol. 56 868–889 (2018) – 10.1137/16m1099467
- Jacob, B., Schwenninger, F. L. & Zwart, H. On continuity of solutions for parabolic control systems and input-to-state stability. Journal of Differential Equations vol. 266 6284–6306 (2019) – 10.1016/j.jde.2018.11.004
- Jacob, B., Mironchenko, A., Partington, J. R. & Wirth, F. Noncoercive Lyapunov Functions for Input-to-State Stability of Infinite-Dimensional Systems. SIAM Journal on Control and Optimization vol. 58 2952–2978 (2020) – 10.1137/19m1297506
- Jacob, B., Schwenninger, F. L. & Vorberg, L. A. Remarks on input-to-state stability of collocated systems with saturated feedback. Mathematics of Control, Signals, and Systems vol. 32 293–307 (2020) – 10.1007/s00498-020-00264-w
- Jayawardhana, B. & Weiss, G. State Convergence of Passive Nonlinear Systems With an $L^{2}$ Input. IEEE Transactions on Automatic Control vol. 54 1723–1727 (2009) – 10.1109/tac.2009.2020661
- Kankanamalage, H. G., Lin, Y. & Wang, Y. On Lyapunov-Krasovskii Characterizations of Input-to-Output Stability. IFAC-PapersOnLine vol. 50 14362–14367 (2017) – 10.1016/j.ifacol.2017.08.2015
- Karafyllis, I. & Krstic, M. ISS with Respect to Boundary Disturbances for 1-D Parabolic PDEs. IEEE Transactions on Automatic Control vol. 61 3712–3724 (2016) – 10.1109/tac.2016.2519762
- Karafyllis, I. & Krstic, M. ISS In Different Norms For 1-D Parabolic Pdes With Boundary Disturbances. SIAM Journal on Control and Optimization vol. 55 1716–1751 (2017) – 10.1137/16m1073753
- Karafyllis, I. & Krstic, M. Input-to-State Stability for PDEs. Communications and Control Engineering (Springer International Publishing, 2019). doi:10.1007/978-3-319-91011-6 – 10.1007/978-3-319-91011-6
- Kawan, C., Mironchenko, A., Swikir, A., Noroozi, N. & Zamani, M. A Lyapunov-Based Small-Gain Theorem for Infinite Networks. IEEE Transactions on Automatic Control vol. 66 5830–5844 (2021) – 10.1109/tac.2020.3042410
- Le Gorrec, Y., Zwart, H. & Maschke, B. Dirac structures and Boundary Control Systems associated with Skew-Symmetric Differential Operators. SIAM Journal on Control and Optimization vol. 44 1864–1892 (2005) – 10.1137/040611677
- Mazenc, F. & Prieur, C. Strict Lyapunov functions for semilinear parabolic partial differential equations. Mathematical Control & Related Fields vol. 1 231–250 (2011) – 10.3934/mcrf.2011.1.231
- Miletic, M., Sturzer, D., Arnold, A. & Kugi, A. Stability of an Euler-Bernoulli Beam With a Nonlinear Dynamic Feedback System. IEEE Transactions on Automatic Control vol. 61 2782–2795 (2016) – 10.1109/tac.2015.2499604
- Mironchenko, A. Local input-to-state stability: Characterizations and counterexamples. Systems & Control Letters vol. 87 23–28 (2016) – 10.1016/j.sysconle.2015.10.014
- Mironchenko, A. & Wirth, F. Characterizations of Input-to-State Stability for Infinite-Dimensional Systems. IEEE Transactions on Automatic Control vol. 63 1692–1707 (2018) – 10.1109/tac.2017.2756341
- Mironchenko, A. & Wirth, F. Lyapunov characterization of input-to-state stability for semilinear control systems over Banach spaces. Systems & Control Letters vol. 119 64–70 (2018) – 10.1016/j.sysconle.2018.07.007
- Mironchenko, A., Karafyllis, I. & Krstic, M. Monotonicity Methods for Input-to-State Stability of Nonlinear Parabolic PDEs with Boundary Disturbances. SIAM Journal on Control and Optimization vol. 57 510–532 (2019) – 10.1137/17m1161877
- Mironchenko, A. Criteria for Input-to-State Practical Stability. IEEE Transactions on Automatic Control vol. 64 298–304 (2019) – 10.1109/tac.2018.2824983
- Mironchenko, A. Small gain theorems for networks of heterogeneous systems. IFAC-PapersOnLine vol. 52 538–543 (2019) – 10.1016/j.ifacol.2019.12.017
- Mironchenko, A., Kawan, C. & Glück, J. Nonlinear small-gain theorems for input-to-state stability of infinite interconnections. Mathematics of Control, Signals, and Systems vol. 33 573–615 (2021) – 10.1007/s00498-021-00303-0
- Mironchenko, A. & Prieur, C. Input-to-State Stability of Infinite-Dimensional Systems: Recent Results and Open Questions. SIAM Review vol. 62 529–614 (2020) – 10.1137/19m1291248
- Nabiullin, R. & Schwenninger, F. L. Strong input-to-state stability for infinite-dimensional linear systems. Mathematics of Control, Signals, and Systems vol. 30 (2018) – 10.1007/s00498-018-0210-8
- Oostveen, J. Strongly Stabilizable Distributed Parameter Systems. (2000) doi:10.1137/1.9780898719864 – 10.1137/1.9780898719864
- Pazy, A. Semigroups of Linear Operators and Applications to Partial Differential Equations. Applied Mathematical Sciences (Springer New York, 1983). doi:10.1007/978-1-4612-5561-1 – 10.1007/978-1-4612-5561-1
- Pepe, P. On Liapunov–Krasovskii functionals under Carathéodory conditions. Automatica vol. 43 701–706 (2007) – 10.1016/j.automatica.2006.10.024
- Ramirez, H., Le Gorrec, Y., Macchelli, A. & Zwart, H. Exponential Stabilization of Boundary Controlled Port-Hamiltonian Systems With Dynamic Feedback. IEEE Transactions on Automatic Control vol. 59 2849–2855 (2014) – 10.1109/tac.2014.2315754
- Ramirez, H., Zwart, H. & Le Gorrec, Y. Stabilization of infinite dimensional port-Hamiltonian systems by nonlinear dynamic boundary control. Automatica vol. 85 61–69 (2017) – 10.1016/j.automatica.2017.07.045
- Schmid, J. Weak input-to-state stability: characterizations and counterexamples. Mathematics of Control, Signals, and Systems vol. 31 433–454 (2019) – 10.1007/s00498-019-00248-5
- Schmid, Conference proceedings of the 11th Symposium on Nonlinear Control Systems, IFAC-PapersOnLine (2019)
- Schmid, J., Dashkovskiy, S., Jacob, B. & Laasri, H. Well-posedness of non-autonomous semilinear systems. IFAC-PapersOnLine vol. 52 216–220 (2019) – 10.1016/j.ifacol.2019.11.781
- Schmid, J. Infinite-time admissibility under compact perturbations. Operator Theory: Advances and Applications 73–82 (2020) doi:10.1007/978-3-030-35898-3_3 – 10.1007/978-3-030-35898-3_3
- Schwenninger, F. L. Input-to-state stability for parabolic boundary control:linear and semilinear systems. Operator Theory: Advances and Applications 83–116 (2020) doi:10.1007/978-3-030-35898-3_4 – 10.1007/978-3-030-35898-3_4
- Tucsnak, M. & Weiss, G. Well-posed systems—The LTI case and beyond. Automatica vol. 50 1757–1779 (2014) – 10.1016/j.automatica.2014.04.016
- Villegas, J. A., Zwart, H., Le Gorrec, Y. & Maschke, B. Exponential Stability of a Class of Boundary Control Systems. IEEE Transactions on Automatic Control vol. 54 142–147 (2009) – 10.1109/tac.2008.2007176
- Weiss, G. Admissibility of Unbounded Control Operators. SIAM Journal on Control and Optimization vol. 27 527–545 (1989) – 10.1137/0327028
- Zheng, J. & Zhu, G. Input-to-state stability with respect to boundary disturbances for a class of semi-linear parabolic equations. Automatica vol. 97 271–277 (2018) – 10.1016/j.automatica.2018.08.007
- Zheng, J. & Zhu, G. A De Giorgi Iteration-Based Approach for the Establishment of ISS Properties for Burgers’ Equation With Boundary and In-domain Disturbances. IEEE Transactions on Automatic Control vol. 64 3476–3483 (2019) – 10.1109/tac.2018.2880160
- Zheng, J. & Zhu, G. A weak maximum principle-based approach for input-to-state stability analysis of nonlinear parabolic PDEs with boundary disturbances. Mathematics of Control, Signals, and Systems vol. 32 157–176 (2020) – 10.1007/s00498-020-00258-8