Stabilization of infinite dimensional port-Hamiltonian systems by nonlinear dynamic boundary control
Authors
Hector Ramirez, Hans Zwart, Yann Le Gorrec
Abstract
The conditions for existence of solutions and stability, asymptotic and exponential, of a large class of boundary controlled systems on a 1D spatial domain subject to nonlinear dynamic boundary actuation are given. The consideration of such class of control systems is motivated by the use of actuators and sensors with nonlinear behavior in many engineering applications. These nonlinearities are usually associated to large deformations or the use of smart materials such as piezo actuators and memory shape alloys. Including them in the controller model results in passive dynamic controllers with nonlinear potential energy function and/or nonlinear damping forces. First it is shown that under very natural assumptions the solutions of the partial differential equation with the nonlinear dynamic boundary conditions exist globally. Secondly, when energy dissipation is present in the controller, then it globally asymptotically stabilizes the partial differential equation. Finally, it is shown that assuming some additional conditions on the interconnection and on the passivity properties of the controller (consistent with physical applications) global exponential stability of the closed-loop system is achieved.
Keywords
Boundary control systems; Port-Hamiltonian systems; Nonlinear control; Existence of solutions; Stabilization
Citation
- Journal: Automatica
- Year: 2017
- Volume: 85
- Issue:
- Pages: 61–69
- Publisher: Elsevier BV
- DOI: 10.1016/j.automatica.2017.07.045
BibTeX
@article{Ramirez_2017,
title={{Stabilization of infinite dimensional port-Hamiltonian systems by nonlinear dynamic boundary control}},
volume={85},
ISSN={0005-1098},
DOI={10.1016/j.automatica.2017.07.045},
journal={Automatica},
publisher={Elsevier BV},
author={Ramirez, Hector and Zwart, Hans and Le Gorrec, Yann},
year={2017},
pages={61--69}
}
References
- Augner, Well-posedness and stability of linear port-Hamiltonian systems with nonlinear boundary feedback. SIAM Journal on Control and Optimization (2016)
- Augner, B. & Jacob, B. Stability and stabilization of infinite-dimensional linear port-Hamiltonian systems. Evolution Equations & Control Theory vol. 3 207–229 (2014) – 10.3934/eect.2014.3.207
- Borazjani, I. Fluid–structure interaction, immersed boundary-finite element method simulations of bio-prosthetic heart valves. Computer Methods in Applied Mechanics and Engineering vol. 257 103–116 (2013) – 10.1016/j.cma.2013.01.010
- Boudaoud, Modeling and optimal force control of a nonlinear electrostatic microgripper. IEEE/ASME Transactions on Mechatronics (2012)
- Collet, M., David, P. & Berthillier, M. Active acoustical impedance using distributed electrodynamical transducers. The Journal of the Acoustical Society of America vol. 125 882–894 (2009) – 10.1121/1.3026329
- Curtain, (1995)
- (2009)
- Ishizaka, K. & Flanagan, J. L. Synthesis of Voiced Sounds From a Two-Mass Model of the Vocal Cords. Bell System Technical Journal vol. 51 1233–1268 (1972) – 10.1002/j.1538-7305.1972.tb02651.x
- 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, Linear port-Hamiltonian systems on infinite-dimensional spaces. (2012)
- 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
- Macchelli, A., Le Gorrec, Y., Ramirez, H. & Zwart, H. On the Synthesis of Boundary Control Laws for Distributed Port-Hamiltonian Systems. IEEE Transactions on Automatic Control vol. 62 1700–1713 (2017) – 10.1109/tac.2016.2595263
- 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
- Oostveen, Strongly stabilizable distributed parameter systems. (2000)
- Pazy, Semigroups of linear operators and applications to partial differential equations. (1983)
- 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
- van der Schaft, A. J. & Maschke, B. M. Hamiltonian formulation of distributed-parameter systems with boundary energy flow. Journal of Geometry and Physics vol. 42 166–194 (2002) – 10.1016/s0393-0440(01)00083-3
- Villegas, (2007)
- 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
- Villegas, J. A., Zwart, H., Le Gorrec, Y., Maschke, B. & van der Schaft, A. J. Stability and Stabilization of a Class of Boundary Control Systems. Proceedings of the 44th IEEE Conference on Decision and Control 3850–3855 doi:10.1109/cdc.2005.1582762 – 10.1109/cdc.2005.1582762
- Zheng, Nonlinear evolution equations. (2004)
- Zwart, H., Le Gorrec, Y., Maschke, B. & Villegas, J. Well-posedness and regularity of hyperbolic boundary control systems on a one-dimensional spatial domain. ESAIM: Control, Optimisation and Calculus of Variations vol. 16 1077–1093 (2009) – 10.1051/cocv/2009036