Duality of switched DAEs

  • Ferdinand Küsters
  • Stephan Trenn
Original Article


We present and discuss the definition of the adjoint and dual of a switched differential-algebraic equation (DAE). For a proper duality definition, it is necessary to extend the class of switched DAEs to allow for additional impact terms. For this switched DAE with impacts, we derive controllability/reachability/determinability/observability characterizations for a given switching signal. Based on this characterizations, we prove duality between controllability/reachability and determinability/observability for switched DAEs.


Duality Switched systems Differential-algebraic equations 


  1. 1.
    Balla K, März R (2002) A unified approach to linear differential algebraic equations and their adjoints. Z Anal Anwend 21:783–802MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Barabanov NE (1995) Stability of inclusions of linear type. In: American Control Conference, Proceedings of the 1995, vol 5, pp 3366–3370. doi: 10.1109/ACC.1995.532231
  3. 3.
    Basile G, Marro G (1992) Controlled and conditioned invariants in linear system theory. Prentice-Hall, Englewood CliffszbMATHGoogle Scholar
  4. 4.
    Berger T, Trenn S (2012) The quasi-Kronecker form for matrix pencils. SIAM J Matrix Anal Appl 33(2):336–368. doi: 10.1137/110826278 MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Berger T, Trenn S (2014) Kalman controllability decompositions for differential-algebraic systems. Syst Control Lett 71:54–61. doi: 10.1016/j.sysconle.2014.06.004 MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Berger T, Reis T, Trenn S (2016) Observability of linear differential-algebraic systems. In: Ilchmann A, Reis T (eds) Surveys in differential-algebraic equations IV, Differential-algebraic equations forum. Springer-Verlag, Berlin-Heidelberg (to appear)Google Scholar
  7. 7.
    Campbell SL (1980) Singular systems of differential equations I. Pitman, New YorkzbMATHGoogle Scholar
  8. 8.
    Campbell SL, Nichols NK, Terrell WJ (1991) Duality, observability, and controllability for linear time-varying descriptor systems. Circ Syst Signal Process 10(4):455–470. doi: 10.1007/BF01194883 MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Cobb JD (1984) Controllability, observability and duality in singular systems. IEEE Trans Autom Control AC 29:1076–1082. doi: 10.1109/TAC.1984.1103451
  10. 10.
    Frankowska H (1990) On controllability and observability of implicit systems. Syst Control Lett 14:219–225. doi: 10.1016/0167-6911(90)90016-N MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Kalman RE (1961) On the general theory of control systems. In: Proceedings of the first international congress on automatic control, Moscow 1960. Butterworth’s, London, pp 481–493Google Scholar
  12. 12.
    Knobloch HW, Kappel F (1974) Gewöhnliche Differentialgleichungen. Teubner, StuttgartCrossRefzbMATHGoogle Scholar
  13. 13.
    Küsters F (2015) On duality of switched DAEs. Master’s thesis, TU KaiserslauternGoogle Scholar
  14. 14.
    Küsters F, Trenn S (2015) Duality of switched ODEs with jumps. In: Proc. 54th Conf. on Decision and Control. IEEE, Osaka, Japan, pp 4879–4884. doi: 10.1109/CDC.2015.7402981
  15. 15.
    Küsters F, Ruppert MGM, Trenn S (2015) Controllability of switched differential-algebraic equations. Syst Control Lett 78:32–39. doi: 10.1016/j.sysconle.2015.01.011 MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Lamour R, März R, Tischendorf C (2013) Differential algebraic equations: a projector based analysis, differential-algebraic equations forum, vol 1. Springer-Verlag, Heidelberg-BerlinCrossRefzbMATHGoogle Scholar
  17. 17.
    Lawrence D (2010) Duality properties of linear impulsive systems. In: Proc. 49th IEEE Conf. Decis. Control, Atlanta, pp 6028–6033. doi: 10.1109/CDC.2010.5717765
  18. 18.
    Li Z, Soh CB, Xu X (1999) Controllability and observability of impulsive hybrid dynamic systems. IMA J Math Control Inf 16(4):315–334. doi: 10.1093/imamci/16.4.315 MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Linh V, März R (2015) Adjoint pairs of differential-algebraic equations and their lyapunov exponents. J Dyn Differ Equ, pp 1–30. doi: 10.1007/s10884-015-9474-6
  20. 20.
    Meng B (2006) Observability conditions of switched linear singular systems. In: Proceedings of the 25th Chinese control conference, Harbin, pp 1032–1037Google Scholar
  21. 21.
    Petreczky M, Tanwani A, Trenn S (2015) Observability of switched linear systems. In: Djemai M, Defoort M (eds) Hybrid dynamical systems, Lecture notes in control and information sciences, vol 457. Springer-Verlag, Switzerland, pp 205–240. doi: 10.1007/978-3-319-10795-0_8
  22. 22.
    van der Schaft AJ (1991) Duality for linear systems: external and state space characterization of the adjoint system. In: Bonnard B, Bride B, Gauthier JP, Kupka I (eds) Analysis of controlled dynamical systems, Progress in systems and control theory, vol 8. Birkhäuser, Boston, pp 393–403. doi: 10.1007/978-1-4612-3214-8_35
  23. 23.
    Schwartz L (1957, 1959) Théorie des Distributions. Hermann, ParisGoogle Scholar
  24. 24.
    Sun Z, Ge SS (2005) Switched linear systems. Communications and control engineering. Springer-Verlag, London. doi: 10.1007/1-84628-131-8 CrossRefGoogle Scholar
  25. 25.
    Tanwani A, Trenn S (2010) On observability of switched differential-algebraic equations. In: Proc. 49th IEEE Conf. Decis. Control, Atlanta, pp 5656–5661. doi: 10.1109/CDC.2010.5717685
  26. 26.
    Tanwani A, Trenn S (2012) Observability of switched differential-algebraic equations for general switching signals. In: Proc. 51st IEEE Conf. Decis. Control, Maui, USA, pp 2648–2653, doi: 10.1109/CDC.2012.6427087
  27. 27.
    Tanwani A, Trenn S (2016) Determinability and state estimation for switched differential-algebraic equations, Automatica (under review)Google Scholar
  28. 28.
    Trenn S (2009) Distributional differential algebraic equations. PhD thesis, Institut für Mathematik, Technische Universität Ilmenau, Universitätsverlag Ilmenau, Germany.
  29. 29.
    Trenn S (2009) Regularity of distributional differential algebraic equations. Math Control Signals Syst 21(3):229–264. doi: 10.1007/s00498-009-0045-4 MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Trenn S (2012) Switched differential algebraic equations. In: Vasca F, Iannelli L (eds) Dynamics and control of switched electronic systems—advanced perspectives for modeling, simulation and control of power converters, chap 6. Springer-Verlag, London, pp 189–216. doi: 10.1007/978-1-4471-2885-4_6
  31. 31.
    Trenn S, Willems J (2012) Switched behaviors with impulses—a unifying framework. In: Proc. 51st IEEE Conf. Decis. Control, Maui, pp 3203–3208. doi: 10.1109/CDC.2012.6426883
  32. 32.
    Trumpf J (2003) On the geometry and parametrization of almost invariant subspaces and observer theory. PhD thesis, Universität WürzburgGoogle Scholar

Copyright information

© Springer-Verlag London 2016

Authors and Affiliations

  1. 1.Fraunhofer Institute for Industrial MathematicsKaiserslauternGermany
  2. 2.Technomathematics GroupUniversity of KaiserslauternKaiserslauternGermany

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