Exponential Stabilization of Port-Hamiltonian Boundary Control Systems via Energy Shaping

Authors

Alessandro Macchelli, Yann Le Gorrec, Hector Ramirez

Abstract

This article is concerned with the exponential stabilization of a class of linear boundary control systems (BCSs) in port-Hamiltonian form through energy shaping. Starting from a first feedback loop that is in charge of modifying the Hamiltonian function of the plant, a second control loop that guarantees exponential convergence to the equilibrium is designed. In this way, a major limitation of standard energy shaping plus damping injection control laws applied to linear port-Hamiltonian BCSs, namely the fact that only asymptotic convergence is assured, has been removed.

Citation

• Journal: IEEE Transactions on Automatic Control
• Year: 2020
• Volume: 65
• Issue: 10
• Pages: 4440–4447
• Publisher: Institute of Electrical and Electronics Engineers (IEEE)
• DOI: 10.1109/tac.2020.3004798

BibTeX

@article{Macchelli_2020,
  title={{Exponential Stabilization of Port-Hamiltonian Boundary Control Systems via Energy Shaping}},
  volume={65},
  ISSN={2334-3303},
  DOI={10.1109/tac.2020.3004798},
  number={10},
  journal={IEEE Transactions on Automatic Control},
  publisher={Institute of Electrical and Electronics Engineers (IEEE)},
  author={Macchelli, Alessandro and Le Gorrec, Yann and Ramirez, Hector},
  year={2020},
  pages={4440--4447}
}

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References

• 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
• 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
• Macchelli, A. & Califano, F. Dissipativity-based boundary control of linear distributed port-Hamiltonian systems. Automatica vol. 95 54–62 (2018) – 10.1016/j.automatica.2018.05.029
• Putting energy back in control. IEEE Control Systems vol. 21 18–33 (2001) – 10.1109/37.915398
• 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
• Olver, P. J. Applications of Lie Groups to Differential Equations. Graduate Texts in Mathematics (Springer New York, 1993). doi:10.1007/978-1-4612-4350-2 – 10.1007/978-1-4612-4350-2
• van der Schaft, A. L2-Gain and Passivity Techniques in Nonlinear Control. Communications and Control Engineering (Springer International Publishing, 2017). doi:10.1007/978-3-319-49992-5 – 10.1007/978-3-319-49992-5
• Macchelli, A. On the control by interconnection and exponential stabilisation of infinite dimensional port-Hamiltonian systems. 2016 IEEE 55th Conference on Decision and Control (CDC) 3137–3142 (2016) doi:10.1109/cdc.2016.7798739 – 10.1109/cdc.2016.7798739
• Fujimoto, K. & Sugie, T. Canonical transformation and stabilization of generalized Hamiltonian systems. Systems & Control Letters vol. 42 217–227 (2001) – 10.1016/s0167-6911(00)00091-8
• macchelli, Energy-based control of a wave equation with boundary anti-damping. Proc 21st IFAC World Congress (0)
• 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
• 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
• Zhang, L., Prieur, C. & Qiao, J. PI boundary control of linear hyperbolic balance laws with stabilization of ARZ traffic flow models. Systems & Control Letters vol. 123 85–91 (2019) – 10.1016/j.sysconle.2018.11.005
• 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
• 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
• Kosaraju, K. C., de Jong, M. C. & Scherpen, J. M. A. A novel passivity based controller for a piezoelectric beam. 2019 18th European Control Conference (ECC) 174–179 (2019) doi:10.23919/ecc.2019.8795988 – 10.23919/ecc.2019.8795988
• 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