Control by Interconnection of Distributed Port-Hamiltonian Systems Beyond the Dissipation Obstacle

Authors

Alessandro Macchelli, Luis Pablo Borja, Romeo Ortega

Abstract

The main contribution of this paper is a general methodology for the definition of a new passive output that is instrumental for the stabilisation of a large class of distributed port-Hamiltonian systems defined on a one-dimensional spatial domain. This new output is in fact employed within the control by interconnection via Casimir generation paradigm for the synthesis of boundary stabilising control laws. It is well-known that this control technique is limited by the so-called “dissipation obstacle” when the passive controller is interconnected to the natural input/output port of the plant. When it is the case, it is impossible to shape the energy of the system along the directions in which dissipation is present. In this paper, it is shown how these limitations can be removed by interconnecting the boundary controller to the new passive output of the system, and then how the control by interconnection can easily deal with the dissipation obstacle. The general theory is illustrated with the help of a concluding example, the boundary stabilization of the shallow water equation.

Keywords

distributed port-Hamiltonian systems; control by interconnection; boundary control; passivity-based control

Citation

• Journal: IFAC-PapersOnLine
• Year: 2015
• Volume: 48
• Issue: 13
• Pages: 99–104
• Publisher: Elsevier BV
• DOI: 10.1016/j.ifacol.2015.10.221
• Note: 5th IFAC Workshop on Lagrangian and Hamiltonian Methods for Nonlinear Control LHMNC 2015- Lyon, France, 4–7 July 2015

BibTeX

@article{Macchelli_2015,
  title={{Control by Interconnection of Distributed Port-Hamiltonian Systems Beyond the Dissipation Obstacle}},
  volume={48},
  ISSN={2405-8963},
  DOI={10.1016/j.ifacol.2015.10.221},
  number={13},
  journal={IFAC-PapersOnLine},
  publisher={Elsevier BV},
  author={Macchelli, Alessandro and Borja, Luis Pablo and Ortega, Romeo},
  year={2015},
  pages={99--104}
}

Download the bib file

References

• Curtain, (1995)
• Duindam, (2009)
• 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
• Jacob, (2012)
• Jeltsema, D., Ortega, R. & M.A. Scherpen, J. An energy-balancing perspective of interconnection and damping assignment control of nonlinear systems. Automatica vol. 40 1643–1646 (2004) – 10.1016/j.automatica.2004.04.007
• 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. Boundary energy shaping of linear distributed port-Hamiltonian systems. European Journal of Control vol. 19 521–528 (2013) – 10.1016/j.ejcon.2013.10.002
• Macchelli, A. Towards a port-based formulation of macro-economic systems. Journal of the Franklin Institute vol. 351 5235–5249 (2014) – 10.1016/j.jfranklin.2014.09.009
• Macchelli, Modeling and Control of Complex Physical Systems: The Port-Hamiltonian Approach. chapter Infinite-Dimensional Port-Hamiltonian Systems (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
• Macchelli, On the synthesis of boundary control laws for distributed port-Hamiltonian systems.. Automatic Control, IEEE Transactions on (2014)
• Macchelli, Asymptotic stabilisation of distributed port-Hamiltonian systems by boundary energy-shaping control. (2015)
• Ortega, New results on control by interconnection and energy-balancing passivity-based control of port-Hamiltonian systems. (2014)
• Putting energy back in control. IEEE Control Systems vol. 21 18–33 (2001) – 10.1109/37.915398
• 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
• Pasumarthy, Port-Hamiltonian formulation of shallow water equations with Coriolis force and topography. (2008)
• Rodriguez, On stabilization of nonlinear distributed parameter port-controlled Hamiltonian systems via energy shaping. (2001)
• Schoberl, M. & Siuka, A. On Casimir Functionals for Infinite-Dimensional Port-Hamiltonian Control Systems. IEEE Transactions on Automatic Control vol. 58 1823–1828 (2013) – 10.1109/tac.2012.2235739
• 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
• van der Schaft, A. & Jeltsema, D. Port-Hamiltonian Systems Theory: An Introductory Overview. Foundations and Trends® in Systems and Control vol. 1 173–378 (2014) – 10.1561/2600000002
• 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
• Venkatraman, A. & van der Schaft, A. Energy Shaping of Port-Hamiltonian Systems by Using Alternate Passive Input-Output Pairs. European Journal of Control vol. 16 665–677 (2010) – 10.3166/ejc.16.665-677