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Power-Based Methods for Infinite-Dimensional 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

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.

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References

  1. R. Abraham, J. Marsden, T. Ratiu, Manifolds, Tensor Analysis, and Applications, 2nd edn. (Springer, Berlin, 1988)

    Book  MATH  Google Scholar 

  2. G. Blankenstein, Geometric modeling of nonlinear RLC circuits. IEEE Trans. Circuits Syst. I: Regul. Pap. 52(2), 396–404 (2005)

    Article  MathSciNet  Google Scholar 

  3. G. Blankenstein, Power balancing for a new class of non-linear systems and stabilization of RLC circuits. Int. J. Control 78(3), 159–171 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  4. R.K. Brayton, W.L. Miranker, A stability theory for nonlinear mixed initial boundary value problems. Arch. Rat. Mech. Anal. 17(5) (1964)

    Google Scholar 

  5. R.K. Brayton, J.K. Moser, A theory of nonlinear networks i. Q. Appl. Math. 22(1), 1–33 (1964)

    MathSciNet  Google Scholar 

  6. R.K. Brayton, J.K. Moser, A theory of nonlinear networks ii. Q. Appl. Math. 22(2), 81–104 (1964)

    MathSciNet  Google Scholar 

  7. V. Duindam, A. Macchelli, S. Stramigioli, H. Bryuninckx (eds.), Modeling and Control of Complex Physical Systems: The port-Hamiltonian Approach (Springer, Berlin, 2009)

    Google Scholar 

  8. D. Jeltsema, R. Ortega, J.M.A. Scherpen, On passivity and power-balance inequalities of nonlinear RLC circuits, in IEEE Transactions on Circuits and Systems-I: Fundamental Theory and Applications, vol. 50, pp. 1174–1179, September (2003)

    Google Scholar 

  9. D. Jeltsema, J.M.A. Scherpen, Multidomain modeling of nonlinear networks and systems. IEEE Control Syst. Mag. (2009)

    Google Scholar 

  10. D. Jeltsema, A. van der Schaft, Pseudo-gradient and Lagrangian boundary control system formulation of electromagnetic field. J. Phys. A : Math. Theor. 40, 1–17 (2007)

    Article  Google Scholar 

  11. 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)

    Google Scholar 

  12. 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)

    Google Scholar 

  13. R. Ortega, D. Jeltsema, J.M.A. Scherpen, Power shaping: a new paradigm for stabilization of nonlinear RLC circuits. IEEE Trans. Autom. Control 48(10), 1762–1767 (2003)

    Article  MathSciNet  Google Scholar 

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

    Article  MATH  Google Scholar 

  15. A.J. van der Schaft, L2-Gain and Passivity Techniques in Nonlinear Control (Springer, London, 2000)

    Book  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Electrical Engineering, Indian Institute of Technology Madras, Chennai, India

    Krishna Chaitanya Kosaraju & Ramkrishna Pasumarthy

Authors
  1. Krishna Chaitanya Kosaraju
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  2. Ramkrishna Pasumarthy
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Corresponding author

Correspondence to Ramkrishna Pasumarthy .

Editor information

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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Kosaraju, K.C., Pasumarthy, R. (2015). Power-Based Methods for Infinite-Dimensional 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_15

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  • DOI: https://doi.org/10.1007/978-3-319-20988-3_15

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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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