Boundary Energy-Shaping Control of the Shallow Water Equation

Authors

Abstract

The aim of this paper is to apply new results on the boundary stabilisation via energy-shaping of distributed port-Hamiltonian systems to a nonlinear PDE, i.e. a slightly simplified formulation of the shallow water equation. Usually, stabilisation of non-zero equilibria via energy-balancing has been achieved by looking at, or generating, a set of structural invariants (Casimir functions), in closed-loop. This approach is not successful in case of the shallow water equation because at the equilibrium the regulator is supposed to supply an infinite amount of energy (dissipation obstacle). In this paper, it is shown how to construct a controller that behaves as a state-modulated boundary source and that asymptotically stabilises the desired equilibrium. The proposed approach relies on a parametrisation of the dynamics provided by the image representation of the Dirac structure associated to the distributed port-Hamiltonian system. In this way, the effects of the boundary inputs on the state evolution are explicitly shown, and as a consequence the boundary control action that maps the open-loop system into a target one characterised by the desired stability properties, i.e. by a “new” Hamiltonian with an isolated minimum at the equilibrium, is determined.

Keywords

distributed port-Hamiltonian systems; passivity-based control; stability of distributed parameter systems

Citation

Journal: IFAC Proceedings Volumes
Year: 2014
Volume: 47
Issue: 3
Pages: 1586–1591
Publisher: Elsevier BV
DOI: 10.3182/20140824-6-za-1003.00870
Note: 19th IFAC World Congress

BibTeX

@article{Macchelli_2014,
  title={{Boundary Energy-Shaping Control of the Shallow Water Equation}},
  volume={47},
  ISSN={1474-6670},
  DOI={10.3182/20140824-6-za-1003.00870},
  number={3},
  journal={IFAC Proceedings Volumes},
  publisher={Elsevier BV},
  author={Macchelli, Alessandro},
  year={2014},
  pages={1586--1591}
}

Download the bib file

References

Curtain, (1995)
Golo, G., Talasila, V., van der Schaft, A. & Maschke, B. Hamiltonian discretization of boundary control systems. Automatica vol. 40 757–771 (2004) – 10.1016/j.automatica.2003.12.017
Hamroun, B., Dimofte, A., Lefèvre, L. & Mendes, E. Control by Interconnection and Energy-Shaping Methods of Port Hamiltonian Models. Application to the Shallow Water Equations. European Journal of Control vol. 16 545–563 (2010) – 10.3166/ejc.16.545-563
Iftime, O. V. & Sandovici, A. Interconnection of Dirac structures via kernel/image representation. Proceedings of the 2011 American Control Conference 3571–3576 (2011) doi:10.1109/acc.2011.5991091 – 10.1109/acc.2011.5991091
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
Luo, (1999)
Macchelli, A. Energy shaping of distributed parameter port-Hamiltonian systems based on finite element approximation. Systems & Control Letters vol. 60 579–589 (2011) – 10.1016/j.sysconle.2011.04.016
Macchelli, A. Asymptotic stability of forced equilibria for distributed port-Hamiltonian systems. 2012 IEEE 51st IEEE Conference on Decision and Control (CDC) 2934–2939 (2012) doi:10.1109/cdc.2012.6426693 – 10.1109/cdc.2012.6426693
Macchelli, A. On the use of Dirac structures on Hilbert spaces in the synthesis of boundary control laws for port-Hamiltonian systems. 52nd IEEE Conference on Decision and Control 3267–3272 (2013) doi:10.1109/cdc.2013.6760382 – 10.1109/cdc.2013.6760382
Macchelli, Modeling and control of complex physical systems: The port-Hamiltonian approach. (2009)
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
Macchelli, A. & Melchiorri, C. Control by interconnection of mixed port Hamiltonian systems. IEEE Transactions on Automatic Control vol. 50 1839–1844 (2005) – 10.1109/tac.2005.858656
Putting energy back in control. IEEE Control Systems vol. 21 18–33 (2001) – 10.1109/37.915398
Ortega, R., van der Schaft, A., Maschke, B. & Escobar, G. Interconnection and damping assignment passivity-based control of port-controlled Hamiltonian systems. Automatica vol. 38 585–596 (2002) – 10.1016/s0005-1098(01)00278-3
Pasumarthy, R. & van der Schaft, A. J. Achievable Casimirs and its implications on control of port-Hamiltonian systems. International Journal of Control vol. 80 1421–1438 (2007) – 10.1080/00207170701361273
Swaters, (2000)
van der Schaft, (2000)
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