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Dirac Structures and Control by Interconnection for Distributed Port-Hamiltonian Systems

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Mathematical Control Theory I

Part of the book series: Lecture Notes in Control and Information Sciences ((LNCIS,volume 461))

Abstract

The aim of this work is to show how the Dirac structure properties can be exploited in the development of energy-based boundary control laws for distributed port-Hamiltonian systems. Stabilisation of non-zero equilibria has been achieved by looking at, or generating, a set of structural invariants, namely Casimir functions, in closed-loop, and geometric conditions for the problem to be solved are determined. However, it is well known that this method fails when an infinite amount of energy is required at the equilibrium (dissipation obstacle). So, a novel approach that enlarges the class of stabilising controllers within the control by interconnection paradigm is also discussed. In this respect, it is shown how to determine a different control port that is instrumental for removing the intrinsic constraints imposed by the dissipative structure of the system. The general theory is illustrated with the help of two related examples, namely the boundary stabilisation of the shallow water equation with and without distributed dissipation.

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References

  1. J. Cervera, A. van der Schaft, A. Baños, Interconnection of port-Hamiltonian systems and composition of Dirac structures. Automatica 43(2), 212–225 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  2. R. Curtain, H. Zwart, An Introduction to Infinite Dimensional Linear Systems Theory (Springer, New York, 1995)

    Book  MATH  Google Scholar 

  3. M. Dalsmo, A. van der Schaft, On representation and integrability of mathematical structures in energy-conserving physical systems. SIAM J. Control Optim. 37, 54–91 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  4. V. Duindam, A. Macchelli, S. Stramigioli, H. Bruyninckx, Modeling and Control of Complex Physical Systems: The Port-Hamiltonian Approach (Springer, Berlin, 2009)

    Book  Google Scholar 

  5. B. Hamroun, A. Dimofte, L. Lefèvre, E. Mendes, Control by interconnection and energy-shaping methods of port Hamiltonian models. Application to the shallow water equations. Eur. J. Control 16(5), 545–563 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  6. O. Iftime, A. Sandovici, Interconnection of Dirac Structures via Kernel/Image Representation, in Proceedings of the American Control Conference (ACC 2011), CA, San Francisco, USA, 2011, pp. 3571–3576

    Google Scholar 

  7. O. Iftime, A. Sandovici, G. Golo, Tools for Analysis of Dirac Structures on Banach Spaces, in Proceedings of the 44th IEEE Conference on Decision and Control and European Control Conference (CDC-ECC 2005), 2005, pp. 3856–3861

    Google Scholar 

  8. D. Jeltsema, R. Ortega, J. Scherpen, An energy-balancing perspective of interconnection and damping assignment control of nonlinear systems. Automatica 40(9), 1643–1646 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  9. Y. Le Gorrec, H. Zwart, B. Maschke, Dirac structures and boundary control systems associated with skew-symmetric differential operators. SIAM J. Control Optim. 44(5), 1864–1892 (2005)

    Article  MathSciNet  Google Scholar 

  10. A. Macchelli, Passivity-Based Control of Implicit Port-Hamiltonian Systems, in 2013 European Control Conference (ECC), Zürich, Switzerland, 2013, pp. 2098–2103

    Google Scholar 

  11. A. Macchelli, Dirac structures on Hilbert spaces and boundary control of distributed port-Hamiltonian systems. Syst. Control Lett. 68, 43–50 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  12. A. Macchelli, Passivity-based control of implicit port-Hamiltonian systems. SIAM J. Control Optim. 52(4), 2422–2448 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  13. A. Macchelli, B. Maschke, Modeling and Control of Complex Physical Systems: The Port-Hamiltonian Approach, Chapter Infinite-Dimensional Port-Hamiltonian Systems, pp. 211–271. In: Duindam et al. [4] (2009)

    Google Scholar 

  14. A. Macchelli, C. Melchiorri, Modeling and control of the Timoshenko beam. The distributed port Hamiltonian approach. SIAM J. Control Optim. 43(2), 743–767 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  15. A. Macchelli, C. Melchiorri, Control by interconnection of mixed port Hamiltonian systems. IEEE Trans. Autom. Control 50(11), 1839–1844 (2005)

    Article  MathSciNet  Google Scholar 

  16. A. Macchelli, Y. Le Gorrec, H. Ramírez, H. Zwart, On the synthesis of boundary control laws for distributed port-Hamiltonian systems. IEEE Trans. Autom. Control (2014) (submitted)

    Google Scholar 

  17. A. Macchelli, Y. Le Gorrec, H. Ramírez, Asymptotic Stabilisation of Distributed Port-Hamiltonian Systems by Boundary Energy-Shaping Control, in Proceedings of the 8th International Conference on Mathematical Modelling (MATHMOD 2015), Vienna, 2015

    Google Scholar 

  18. R. Ortega, L. Borja, New Results on Control By Interconnection and Energy-balancing Passivity-based Control of Port-hamiltonian Systems, in 2014 IEEE 53rd Annual Conference on Decision and Control (CDC), Los Angeles, California, USA, 2014, pp. 2346–2351

    Google Scholar 

  19. R. Ortega, A. van der Schaft, I. Mareels, B. Maschke, Putting energy back in control. IEEE Control Syst. Mag. 21(2), 18–33 (2001)

    Google Scholar 

  20. R. Pasumarthy, J. van der Schaft, Achievable Casimirs and its implications on control by interconnection of port-Hamiltonian systems. Int. J. Control 80(9), 1421–1438 (2007)

    Article  MATH  Google Scholar 

  21. R. Pasumarthy, V. Ambati, A. van der Schaft, Port-Hamiltonian Formulation of Shallow Water Equations with Coriolis Force and Topography, in Proceedings of the 18th International Symposium on Mathematical Theory of Networks and Systems (MTNS 2008), Blacksburg, VA, USA, 2008

    Google Scholar 

  22. H. Ramírez, Y. Le Gorrec, A. Macchelli, H. Zwart, Exponential stabilization of boundary controlled port-Hamiltonian systems with dynamic feedback. IEEE Trans. Autom. Control 59(10), 2849–2855 (2014)

    Article  Google Scholar 

  23. H. Rodriguez, A. van der Schaft, R. Ortega, On Stabilization of Nonlinear Distributed Parameter Port-Controlled Hamiltonian Systems via Energy Shaping, in Proceedings of the 40th IEEE Conference on Decision and Control (CDC 2001), vol. 1, 2001, pp. 131–136

    Google Scholar 

  24. M. Schöberl, A. Siuka, On Casimir functionals for infinite-dimensional port-Hamiltonian control systems. IEEE Trans. Autom. Control 58(7), 1823–1828 (2013)

    Article  Google Scholar 

  25. A. van der Schaft, \(L_2\) -Gain and Passivity Techniques in Nonlinear Control, Communication and Control Engineering (Springer, New York, 2000)

    Google Scholar 

  26. A. van der Schaft, D. Jeltsema, Port-Hamiltonian systems theory: an introductory overview. Found. Trends\(^{\textregistered }\) Syst. Control 1(2–3), 173–378 (2014)

    Google Scholar 

  27. A. van der Schaft, B. Maschke, Hamiltonian formulation of distributed parameter systems with boundary energy flow. J. Geom. Phys. 42(1–2), 166–194 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  28. A. Venkatraman, A. van der Schaft, Energy shaping of port-Hamiltonian systems by using alternate passive input-output pairs. Eur. J. Control 16(6), 665–677 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  29. J. Villegas, H. Zwart, Y. Le Gorrec, B. Maschke, Exponential stability of a class of boundary control systems. IEEE Trans. Autom. Control 54(1), 142–147 (2009)

    Article  Google Scholar 

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Authors and Affiliations

  1. Department of Electrical, Electronic, and Information Engineering, University of Bologna, Viale del Risorgimento 2, 40136, Bologna, Italy

    Alessandro Macchelli

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  1. Alessandro Macchelli
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Correspondence to Alessandro Macchelli .

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Editors and Affiliations

  1. Johann Bernoulli Institute for Mathematics and Computer Science, University of Groningen, Groningen, The Netherlands

    M. Kanat Camlibel

  2. Department of Electrical, Computer and Systems Engineering, Rensselaer Polytechnic Institute, Troy, New York, USA

    A. Agung Julius

  3. Department of Electrical Engineering, Indian Institute of Technology, Chennai, Tamil Nadu, India

    Ramkrishna Pasumarthy

  4. Engineering and Technology Institute Groningen, University of Groningen, Groningen, The Netherlands

    Jacquelien M.A. Scherpen

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Macchelli, A. (2015). Dirac Structures and Control by Interconnection for Distributed Port-Hamiltonian Systems. In: Camlibel, M., Julius, A., Pasumarthy, R., Scherpen, J. (eds) Mathematical Control Theory I. Lecture Notes in Control and Information Sciences, vol 461. Springer, Cham. https://doi.org/10.1007/978-3-319-20988-3_2

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  • DOI: https://doi.org/10.1007/978-3-319-20988-3_2

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-20987-6

  • Online ISBN: 978-3-319-20988-3

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