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Checking dissipativity of linear behavior systems given in kernel representation

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Abstract

The behavior approach and the problem of dissipativity have both been introduced and studied extensively by Willems and others. Current methods to check dissipativity will either check the solvability of a linear matrix inequality or rely on symbolic computations with image representations via computer algebra packages. We will discuss a new characterization for linear behavior systems in kernel representation that allows to check dissipativity via the solution of structured eigenvalue problems. The complexity and efficiency of our new method will be compared to the existing methods.

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References

  1. Antoulas AC (2005) Approximation of large-scale dynamical systems. SIAM Publications, Philadelphia

    Book  MATH  Google Scholar 

  2. Balas G, Chiang R, Packard A, Safonov M (2010) Robust control toolboxTM3. MATLAB user’s guide, The MathWorks, Inc

  3. Boyd S, Balakrishnan V, Kabamba P (1989) A bisection method for computing the H ∞ norm of a transfer matrix and related problems. Math. Control Signals Syst 2: 207–219

    Article  MathSciNet  MATH  Google Scholar 

  4. Boyd S, Ghaoui LE, Feron E, Balakrishnan V (1994) Linear matrix inequalities in system and control theory. SIAM Publications, Philadelphia

    Book  MATH  Google Scholar 

  5. Brüll T (2010) Checking dissipativity of behavioral systems using para-hermitian matrix polynomials. Preprint 683, MATHEON Preprint. Extended version of the present paper. http://www.matheon.de/research/list_preprints.asp

  6. Brüll T, Mehrmann V (2007) STCSSP: A FORTRAN 77 routine to compute a structured staircase form for a (skew-) symmetric / (skew-) symmetric matrix pencil. Preprint 31–2007, Institut für Mathematik, TU Berlin. http://www.math.tu-berlin.de/preprints/

  7. Byers R, Mehrmann V, Xu H (2007) A structured staircase algorithm for skew-symmetric/ symmetric pencils. Electron Trans Numer Anal 26: 1–33

    MathSciNet  MATH  Google Scholar 

  8. Chu C, Wong N (2006) A fast passivity test for descriptor systems via structure-preserving transformations of skew-Hamiltonian/Hamiltonian matrix pencils. In: Design automation conference, 2006 43rd ACM/IEEE, pp 261–266. doi:10.1109/DAC.2006.229221

  9. Demmel J, Kågström B (1993) The generalized Schur decomposition of an arbitrary pencil A—zB: robust software with error bounds and applications. Part I and II. ACM Trans Math Softw 19(2): 160–201

    Article  MATH  Google Scholar 

  10. van der Geest R, Trentelman H (1997) The Kalman–Yakubovich–Popov lemma in a behavioral framework. Syst Control Lett 32(5): 283–290

    Article  MATH  Google Scholar 

  11. Golub GH, Loan CFV (1996) Matrix computations, 3rd edn. The Johns Hopkins University Press, Baltimore

    MATH  Google Scholar 

  12. Löfberg J (2011) YALMIP release R20110314. Downloaded from http://users.isy.liu.se/johanl/yalmip/

  13. MATLAB (2010) Version 7.11.0 (R2010b)

  14. Mehrmann V, Schröder C, Simoncini V (2009) An implicitly-restarted Krylov method for real symmetric/skew-symmetric eigenproblems. Preprint 591, MATHEON Preprint. http://www.matheon.de/research/list_preprints.asp

  15. Polderman JW, Willems JC (1998) Introduction to mathematical systems theory: a behavioral approach. Springer, Berlin

    Google Scholar 

  16. Polik I, Romanko O (2010) SeDuMi version 1.3. Downloaded from http://sedumi.ie.lehigh.edu/

  17. Poppe L, Schröder C, Thies I (2009) PEPACK: a software package for computing the numerical solution of palindromic and even eigenvalue problems using the pencil laub trick. Preprint 22–2009, Institut für Mathematik, TU Berlin. http://www.math.tu-berlin.de/preprints/

  18. Rapisarda P, Willems JC (1997) State maps for linear systems. SIAM J Control Optim 35(3): 1053–1091

    Article  MathSciNet  MATH  Google Scholar 

  19. Shankar S (1999) The Nullstellensatz for systems of PDE. Adv Appl Math 23(4): 360–374

    Article  MathSciNet  MATH  Google Scholar 

  20. Toh KC, Todd MJ, Tutuncu RH (2009) SDPT3 version 4.0—a MATLAB software for semidefinite-quadratic-linear programming. Downloaded from http://www.math.nus.edu.sg/~mattohkc/sdpt3.html

  21. Trentelman HL, Willems JC (1997) Every storage function is a state function. Syst Control Lett 32: 249–259

    Article  MathSciNet  MATH  Google Scholar 

  22. Vandenberghe L, Boyd S (1995) A primal-dual potential reduction method for problems involving matrix inequalities. Math Program 69: 205–236

    MathSciNet  MATH  Google Scholar 

  23. Willems JC (1972a) Dissipative dynamical systems, part I: general theory. Arch Ration Mech Anal 45: 321–351

    Article  MathSciNet  MATH  Google Scholar 

  24. Willems JC (1972b) Dissipative dynamical systems, part II: Linear systems with quadratic supply rates. Arch Ration Mech Anal 45: 352–393

    Article  MathSciNet  MATH  Google Scholar 

  25. Willems JC (1991) Paradigams and puzzles in the theory of dynamical systems. IEEE Trans Autom Control 36(3): 259–294

    Article  MathSciNet  MATH  Google Scholar 

  26. Willems JC, Trentelman HL (1998) On quadratic differential forms. SIAM J Control Optim 36(5): 1703–1749

    Article  MathSciNet  MATH  Google Scholar 

  27. Youla DC (1961) On the factorization of rational matrices. IRE Trans Inf Theory 7(3): 172–189

    Article  Google Scholar 

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Authors and Affiliations

  1. Technische Universität Berlin, Sekretariat MA 4-5, Straße des 17. Juni 136, 10623, Berlin, Germany

    Tobias Brüll

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  1. Tobias Brüll
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Correspondence to Tobias Brüll.

Additional information

This research was supported by Deutsche Forschungsgemeinschaft, through the DFG Research Center Matheon Mathematics for Key Technologies in Berlin. An extended version of the present paper can be found in [5].

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Brüll, T. Checking dissipativity of linear behavior systems given in kernel representation. Math. Control Signals Syst. 23, 159–175 (2011). https://doi.org/10.1007/s00498-011-0065-8

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  • DOI: https://doi.org/10.1007/s00498-011-0065-8

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