Authors

Krishna Chaitanya Kosaraju, Ramkrishna Pasumarthy

Abstract

In this chapter we aim to extend the Brayton Moser (BM) framework for modeling infinite-dimensional systems. Starting with an infinite-dimensional port-Hamiltonian system we derive a BM equivalent which can be defined with respect to a non-canonical Dirac structure. Based on this model we derive stability and new passivity properties for the system. The state variables in this case are the “effort” variables and the storage function is a “power-like” function called the mixed potential. The new property is derived by “differentiating” one of the port variables. We present our results with the Maxwell’s equations, and the transmission line with non-zero boundary conditions as examples.

Keywords

Infinite Dimensional Systems; port-Hamiltonian Systems; Dirac Structure; Storage Function; Passive Properties

Citation

BibTeX

@inbook{Kosaraju_2015,
  title={{Power-Based Methods for Infinite-Dimensional Systems}},
  ISBN={9783319209883},
  ISSN={1610-7411},
  DOI={10.1007/978-3-319-20988-3_15},
  booktitle={{Mathematical Control Theory I}},
  publisher={Springer International Publishing},
  author={Kosaraju, Krishna Chaitanya and Pasumarthy, Ramkrishna},
  year={2015},
  pages={277--301}
}

Download the bib file

References

  • Abraham, R., Marsden, J. E. & Ratiu, T. Manifolds, Tensor Analysis, and Applications. Applied Mathematical Sciences (Springer New York, 1988). doi:10.1007/978-1-4612-1029-0 – 10.1007/978-1-4612-1029-0
  • Blankenstein, G. Geometric modeling of nonlinear RLC circuits. IEEE Transactions on Circuits and Systems I: Regular Papers vol. 52 396–404 (2005) – 10.1109/tcsi.2004.840481
  • Blankenstein, G. Power balancing for a new class of non-linear systems and stabilization of RLC circuits. International Journal of Control vol. 78 159–171 (2005) – 10.1080/00207170500036191
  • Brayton, R. K. & Miranker, W. L. A stability theory for nonlinear mixed initial boundary value problems. Archive for Rational Mechanics and Analysis vol. 17 358–376 (1964) – 10.1007/bf00250472
  • Brayton, R. K. & Moser, J. K. A theory of nonlinear networks. I. Quarterly of Applied Mathematics vol. 22 1–33 (1964) – 10.1090/qam/169746
  • Brayton, R. K. & Moser, J. K. A theory of nonlinear networks. II. Quarterly of Applied Mathematics vol. 22 81–104 (1964) – 10.1090/qam/169747
  • V. Duindam, A. Macchelli, S. Stramigioli, H. Bryuninckx (eds.), Modeling and Control of Complex Physical Systems: The port-Hamiltonian Approach (Springer, Berlin, 2009) (2009)
  • Jeltsema, D., Ortega, R. & Scherpen, J. M. A. On passivity and power-balance inequalities of nonlinear rlc circuits. IEEE Transactions on Circuits and Systems I: Fundamental Theory and Applications vol. 50 1174–1179 (2003) – 10.1109/tcsi.2003.816332
  • D. Jeltsema, J.M.A. Scherpen, Multidomain modeling of nonlinear networks and systems. IEEE Control Syst. Mag. (2009)
  • Jeltsema, D. & Schaft, A. van der. Pseudo-gradient and Lagrangian boundary control system formulation of electromagnetic fields. Journal of Physics A: Mathematical and Theoretical vol. 40 11627–11643 (2007) – 10.1088/1751-8113/40/38/013
  • K.C. Kosaraju, R. Pasumarthy, D. Jeltsema, Alternate Passive Maps and Stability of Infinite-dimensional Systems via Mixed Potential Functions, in Proceedings of the 5th IFAC Workshop on Lagrangian and Hamiltonian Methods for Non Linear Control, Lyon, France, 4–7 July (2015)
  • R. Pasumarthy, K.C. Kosaraju, A. Chandrasekar, On Power Balancing and Stabilization for A Class of Infinite-dimensional Systems, in Proceedings of the Mathematical Theory of Networks and Systems, Groningen, The Netherlands, July, pp. 7–11 (2014)
  • Ortega, R., Jeltsema, D. & Scherpen, J. M. A. Power shaping: A new paradigm for stabilization of nonlinear RLC circuits. IEEE Transactions on Automatic Control vol. 48 1762–1767 (2003) – 10.1109/tac.2003.817918
  • 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
  • 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. 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