Observability of Switched Linear Systems

Chapter
Part of the Lecture Notes in Control and Information Sciences book series (LNCIS, volume 457)

Abstract

Observability of switched linear systems has been well studied during the past decade and depending on the notion of observability, several criteria have appeared in the literature. The main difference in these approaches is how the switching signal is viewed: Is it a fixed and known function of time, is it an unknown external signal, is it the result of a discrete dynamical system (an automaton) or is it controlled and is therefore an input? We will focus on the recently introduced geometric characterization of observability, which assumes knowledge of the switching signal. These geometric conditions depend on computing the exponential of the matrix and require the exact knowledge of switching times. To relieve the computational burden, some relaxed conditions that do not rely on the switching times are given; this also allows for a direct comparison of the different observability notions. Furthermore, the generalization of the geometric approach to linear switched differential algebraic systems is possible and presented as well.

References

  1. 1.
    Armentano, V.A.: The pencil (sE-A) and controllability-observability for generalized linear systems: a geometric approach. SIAM J. Control Optim. 24, 616–638 (1986)Google Scholar
  2. 2.
    Babaali, M., Egerstedt, M.: Observability of switched linear systems. In: Hybrid Systems: Computation and Control, Lecture Notes in Computer Science, vol. 2993, pp. 48–63. Springer (2004)Google Scholar
  3. 3.
    Babaali, M., Pappas, G.J.: Observability of switched linear systems in continuous time. In: Hybrid Systems: Computation and Control. Lecture Notes in Computer Science, vol. 3414, pp. 103–117. Springer, Berlin (2005)Google Scholar
  4. 4.
    Baglietto, M., Battistelli, G., Scardovi, L.: Active mode observation of switching systems based on set-valued estimation of the continuous state. Int. J. Robust Nonlinear Control 19(14), 1521–1540 (2009)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Balluchi, A., Benvenuti, L., Di Benedetto, M., Sangiovanni-Vincentelli, A.: Observability for hybrid systems. In: Proceedings of 42nd IEEE Conference Decision and Control, Hawaii, USA, vol. 2, pp. 1159–1164 (2003). doi:10.1109/CDC.2003.1272764
  6. 6.
    Berger, T., Ilchmann, A., Trenn, S.: The quasi-Weierstraß form for regular matrix pencils. Lin. Alg. Appl. 436(10), 4052–4069 (2012). doi:10.1016/j.laa.2009.12.036 MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Berger, T., Trenn, S.: The quasi-Kronecker form for matrix pencils. SIAM J. Matrix Anal. Appl. 33(2), 336–368 (2012). doi:10.1137/110826278
  8. 8.
    Berger, T., Trenn, S.: Addition to “The quasi-Kronecker form for matrix pencils”. SIAM J. Matrix Anal. Appl. 34(1), 94–101 (2013). doi:10.1137/120883244 MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Boukhobza, T., Hamelin, F.: Observability of switching structured linear systems with unknown input. A graph-theoretic approach. Automatica 47(2), 395–402 (2011)Google Scholar
  10. 10.
    Chaib, S., Boutat, D., Benali, A., Barbot, J.P.: Observability of the discrete state for dynamical piecewise hybrid systems. Nonlin. Anal. Th. Meth. Appl. 63(3), 423–438 (2005)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Collins, P., van Schuppen, J.H.: Observability of piecewise-affine hybrid systems. In: Alur, R., Pappas G.J. (eds.) Hybrid Systems: Computation and Control, Lecture Notes in Computer Science, vol. 2993, pp. 265–279. Springer-Verlag (2004)Google Scholar
  12. 12.
    De Santis, E.: On location observability notions for switching systems. Syst. Control Lett. 60(10), 807–814 (2011). doi: http://dx.doi.org/10.1016/j.sysconle.2011.06.004
  13. 13.
    Fliess, M., Join, C., Perruquetti, W.: Real-time estimation for switched linear systems. In: Proceedings of 47th IEEE Conference on Decision and Control, Cancun, Mexico, pp. 941–946 (2008). doi: 10.1109/CDC.2008.4739053
  14. 14.
    Gantmacher, F.R.: The Theory of Matrices (Vol. I & II). Chelsea, New York (1959)Google Scholar
  15. 15.
    Ji, Z., Wang, L., Guo, X.: On controllability of switched linear systems. IEEE Trans. Autom. Control 53(3), 796–801 (2008). doi:10.1109/TAC.2008.917659 MathSciNetCrossRefGoogle Scholar
  16. 16.
    Lee, C., Ping, Z., Shim, H.: Real time switching signal estimation of switched linear systems with measurement noise. In: Proceedings of 12th European Control Conference 2013, Zurich, Switzerland, pp. 2180–2185 (2013)Google Scholar
  17. 17.
    Liu, B., Marquez, H.: Controllability and observability for a class of controlled switching impulsive systems. IEEE Trans. Autom. Control 53(10), 2360–2366 (2008)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Medina, E.A., Lawrence, D.A.: State estimation for linear impulsive systems. In: Proceedings of American Control Conference 2009, pp. 1183–1188 (2009)Google Scholar
  19. 19.
    Medina, E.A., Lawrence, D.A.: Reachability and observability of linear impulsive systems. Automatica 44, 1304–1309 (2008)MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    Mincarelli, D., Floquet, T.: On second-order sliding mode observers with residuals’ projection for switched systems. In: Proceedings of 51st IEEE Conference on Decision and Control, Maui, USA, pp. 1929–1934 (2012). doi: 10.1109/CDC.2012.6426719
  21. 21.
    Orani, N., Pisano, A., Franceschelli, M., Giua, A., Usai, E.: Robust reconstruction of the discrete state for a class of nonlinear uncertain switched systems. Nonlinear Anal. Hybrid Syst. 5(2), 220–232 (2011). doi:10.1016/j.nahs.2010.10.011 MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    Petreczky, M.: Realization theory of hybrid systems. Ph.D. thesis, Vrije Universiteit, Amsterdam (2006)Google Scholar
  23. 23.
    Petreczky, M.: Realization theory of linear and bilinear switched systems: A formal power series approach—part I: realization theory of linear switched systems. ESAIM Control Optim. Calc. Var., pp. 410–445 (2011)Google Scholar
  24. 24.
    Petreczky, M.: Realization theory of linear and bilinear switched systems: A formal power series approach—part II: bilinear switched systems. ESAIM Control Optim. Calc. Var., pp. 446–471 (2011)Google Scholar
  25. 25.
    Petreczky, M., van Schuppen, J.H.: Realization theory for linear hybrid systems. IEEE Trans. Autom. Control 55(10), 2282–2297 (2010)CrossRefGoogle Scholar
  26. 26.
    Petreczky, M., van Schuppen, J.H.: Span-reachability and observability of bilinear hybrid systems. Automatica 46(3), 501–509 (2010). doi:10.1016/j.automatica.2010.01.008 MathSciNetCrossRefMATHGoogle Scholar
  27. 27.
    Ríos, H., Davila, J., Fridman, L.: High-order sliding mode observers for nonlinear autonomous switched systems with unknown inputs. J. Franklin Inst. 349(10), 2975–3002 (2012)MathSciNetCrossRefMATHGoogle Scholar
  28. 28.
    De Santis, E., Di Benedetto, M.D., Pola, G.: A structural approach to detectability for a class of hybrid systems. Automatica 45(5), 1202–1206 (2009). doi:10.1016/j.automatica.2008.12.014 MathSciNetCrossRefMATHGoogle Scholar
  29. 29.
    Schwartz, L.: Théorie des Distributions I, II. No. IX, X in Publications de l’institut de mathématique de l’Universite de Strasbourg. Hermann, Paris (1950, 1951)Google Scholar
  30. 30.
    Shim, H., Tanwani, A.: Hybrid-type observer design based on a sufficient condition for observability in switched nonlinear systems. Int. J. Robust Nonlinear Control (Special Issue on High Gain Observers and Nonlinear Output Feedback Control 24(6), 1064–1089, 2014). doi:10.1002/rnc.2901
  31. 31.
    Sun, Z., Ge, S.S.: Switched linear systems. Communications and Control Engineering. Springer-Verlag, London (2005). doi: 10.1007/1-84628-131-8
  32. 32.
    Tanwani, A., Liberzon, D.: Invertibility of switched nonlinear systems. Automatica 46(12), 1962–1973 (2010). doi:10.1016/j.automatica.2010.08.002
  33. 33.
    Tanwani, A., Trenn, S.: An observer for switched differential-algebraic equations based on geometric characterization of observability. In: Proceedings of 52nd IEEE Conference on Decision and Control, Florence, Italy, pp. 5981–5986 (2013). doi:10.1109/CDC.2013.6760833
  34. 34.
    Tanwani, A., Trenn, S.: Observability of switched differential-algebraic equations for general switching signals. In: Proceedings of 51st IEEE Conference on Decision and Control, Maui, USA, pp. 2648–2653 (2012). doi: 10.1109/CDC.2012.6427087
  35. 35.
    Tanwani, A., Trenn, S.: On observability of switched differential-algebraic equations. In: Proceedings of 49th IEEE Conference on Decision and Control, Atlanta, USA, pp. 5656–5661 (2010). doi:10.1109/CDC.2010.5717685
  36. 36.
    Tanwani, A., Shim, H., Liberzon, D.: Observability for switched linear systems: Characterization and observer design. IEEE Trans. Autom. Control 58(4), 891–904 (2013). doi:10.1109/TAC.2012.2224257 MathSciNetCrossRefGoogle Scholar
  37. 37.
    Trenn, S.: Distributional differential algebraic equations. Ph.D. thesis, Institut für Mathematik, Technische Universität Ilmenau, Universitätsverlag Ilmenau, Ilmenau, Germany (2009). URL http://www.db-thueringen.de/servlets/DocumentServlet?id=13581
  38. 38.
    Trenn, S.: Regularity of distributional differential algebraic equations. Math. Control Signals Syst. 21(3), 229–264 (2009). doi:10.1007/s00498-009-0045-4 MathSciNetCrossRefMATHGoogle Scholar
  39. 39.
    Trenn, S.: 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, pp. 189–216. Springer, London (2012). doi:10.1007/978-1-4471-2885-4_6
  40. 40.
    Trenn, S.: Stability of switched DAEs. In: Daafouz, J., Tarbouriech, S., Sigalotti, M. (eds.) Hybrid Systems with Constraints, Automation—Control and Industrial Engineering Series, pp. 57–83. Wiley, London (2013). doi:10.1002/9781118639856.ch3
  41. 41.
    Trenn, S., Wirth, F.R.: Linear switched DAEs: Lyapunov exponent, converse Lyapunov theorem, and Barabanov norm. In: Proceedings of 51st IEEE Conference on Decision and Control, Maui, USA, pp. 2666–2671 (2012). doi: 10.1109/CDC.2012.6426245
  42. 42.
    Vidal, R., Chiuso, A., Sastry, S.: Observability and identifiability of jump linear systems. In: Proceedings of 41st IEEE Conference on Decision and Control, pp. 3614–3619 (2002)Google Scholar
  43. 43.
    Vidal, R., Chiuso, A., Soatto, S., Sastry, S.: Observability of linear hybrid systems. In: Hybrid Systems: Computation and Control. Lecture Notes in Computer Science, vol. 2623, pp. 526–539. Springer, Berlin (2003)Google Scholar
  44. 44.
    Vu, L., Liberzon, D.: Invertibility of switched linear systems. Automatica 44(4), 949–958 (2008)MathSciNetCrossRefMATHGoogle Scholar
  45. 45.
    Weierstraß, K.: Zur Theorie der bilinearen und quadratischen Formen. Berl. Monatsb. pp. 310–338 (1868)Google Scholar
  46. 46.
    Wong, K.T.: The eigenvalue problem \(\lambda Tx + Sx \). J. Diff. Eqns. 16, 270–280 (1974)CrossRefMATHGoogle Scholar
  47. 47.
    Wonham, W.M.: Linear Multivariable Control: A Geometric Approach, 3rd edn. Springer, New York (1985)CrossRefMATHGoogle Scholar
  48. 48.
    Xie, G., Wang, L.: Necessary and sufficient conditions for controllability and observability of switched impulsive control systems. IEEE Trans. Autom. Control 49(6), 960–966 (2004). doi:10.1109/TAC.2004.829656 MathSciNetCrossRefGoogle Scholar
  49. 49.
    Yu, L., Zheng, G., Boutat, D., Barbot, J.P.: Observability and observer design for a class of switched systems. IET Control Theory Appl. 5(9), 1113–1119 (2011). doi:10.1049/iet-cta.2010.0475 MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  • Mihaly Petreczky
    • 1
  • Aneel Tanwani
    • 2
  • Stephan Trenn
    • 2
  1. 1.Department of Automatic Control and Computer ScienceEcole des Mines de DouaiDouaiFrance
  2. 2.Department of MathematicsUniversity of KaiserslauternKaiserslauternGermany

Personalised recommendations