Exponential Stabilization of Boundary Controlled Port-Hamiltonian Systems With Dynamic Feedback

Authors

Hector Ramirez, Yann Le Gorrec, Alessandro Macchelli, Hans Zwart

Abstract

It is shown that a strictly-input passive linear finite dimensional controller exponentially stabilizes a large class of partial differential equations actuated at the boundary of a one dimensional spatial domain. This follows since the controller imposes exponential dissipation of the total energy. The result can by use for control synthesis and for the stability analysis of complex systems modeled by sets of coupled PDE’s and ODE’s. The result is specialized to port-Hamiltonian control systems and a simplified DNA-manipulation process is used to illustrate the result.

Citation

• Journal: IEEE Transactions on Automatic Control
• Year: 2014
• Volume: 59
• Issue: 10
• Pages: 2849–2855
• Publisher: Institute of Electrical and Electronics Engineers (IEEE)
• DOI: 10.1109/tac.2014.2315754

BibTeX

@article{Ramirez_2014,
  title={{Exponential Stabilization of Boundary Controlled Port-Hamiltonian Systems With Dynamic Feedback}},
  volume={59},
  ISSN={1558-2523},
  DOI={10.1109/tac.2014.2315754},
  number={10},
  journal={IEEE Transactions on Automatic Control},
  publisher={Institute of Electrical and Electronics Engineers (IEEE)},
  author={Ramirez, Hector and Le Gorrec, Yann and Macchelli, Alessandro and Zwart, Hans},
  year={2014},
  pages={2849--2855}
}

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References

• yosida, Functional Analysis (1995)
• Wen, J. T. Time domain and frequency domain conditions for strict positive realness. IEEE Transactions on Automatic Control vol. 33 988–992 (1988) – 10.1109/9.7263
• 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
• Willems, J. C. Dissipative dynamical systems part I: General theory. Archive for Rational Mechanics and Analysis vol. 45 321–351 (1972) – 10.1007/bf00276493
• van der Schaft, A. L2 - Gain and Passivity Techniques in Nonlinear Control. Communications and Control Engineering (Springer London, 2000). doi:10.1007/978-1-4471-0507-7 – 10.1007/978-1-4471-0507-7
• Boudaoud, M., Haddab, Y. & Le Gorrec, Y. Modeling and Optimal Force Control of a Nonlinear Electrostatic Microgripper. IEEE/ASME Transactions on Mechatronics vol. 18 1130–1139 (2013) – 10.1109/tmech.2012.2197216
• Macchelli, A. & Melchiorri, C. Modeling and Control of the Timoshenko Beam. The Distributed Port Hamiltonian Approach. SIAM Journal on Control and Optimization vol. 43 743–767 (2004) – 10.1137/s0363012903429530
• Tucsnak, M. & Weiss, G. Observation and Control for Operator Semigroups. (Birkhäuser Basel, 2009). doi:10.1007/978-3-7643-8994-9 – 10.1007/978-3-7643-8994-9
• Luo, Z.-H., Guo, B.-Z. & Morgul, O. Stability and Stabilization of Infinite Dimensional Systems with Applications. Communications and Control Engineering (Springer London, 1999). doi:10.1007/978-1-4471-0419-3 – 10.1007/978-1-4471-0419-3
• 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
• 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
• villegas, A Port-Hamiltonian Approach to Distributed Parameter Systems (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
• 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
• 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