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Observer Design for Switched Linear Systems with State Jumps

  • Aneel TanwaniEmail author
  • Hyungbo Shim
  • Daniel Liberzon
Chapter
Part of the Lecture Notes in Control and Information Sciences book series (LNCIS, volume 457)

Abstract

An observer design for switched linear systems with state resets is proposed based on the geometric conditions for large-time observability from our recent work. Without assuming the observability of individual subsystems, the basic idea is to combine the maximal information available from each mode to obtain a good estimate of the state after a certain time interval (over which the switched system is observable) has passed. We first study systems where state reset maps at switching instants are invertible, in which case it is possible to collect all the observable and unobservable information separately at one time instant. One can then annihilate the unobservable component of all the modes and obtain an estimate of the state by introducing an error correction map at that time instant. However, for the systems with non-invertible jump maps, this approach needs to be modified and a recursion-based error correction scheme is proposed. In both approaches, the criterion for choosing the output injection matrices is given, which leads to the asymptotic recovery of the system state.

Keywords

Observer Design Error Correction Term Switching Instant Switch Linear System State Estimation Error 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

References

  1. 1.
    Alessandri, A., Coletta, P.: Switching observers for continuous-time and discrete-time linear systems. Proc. Am. Control Conf. 2001, 2516–2521 (2001)CrossRefGoogle Scholar
  2. 2.
    Babaali, M., Pappas, G.J.: Observability of switched linear systems in continuous time. Hybrid Systems: Computation and Control. Lecture Notes in Computer Science, vol. 3414, pp. 103–117. Springer, Berlin (2005)Google Scholar
  3. 3.
    Balluchi, A., Benvenuti, L., Di Benedetto, M., Sangiovanni-Vincentelli, A.: Design of observers for hybrid systems. Hybrid Systems: Computation and Control. Lecture Notes in Computer Science, vol. 2289, pp. 76–89. Springer, Berlin (2002)Google Scholar
  4. 4.
    Balluchi, A., Benvenuti, L., Di Benedetto, M., Sangiovanni-Vincentelli, A.: Observability for hybrid systems. In: Proceedings of the 42nd IEEE Conference on Decision and Control, Hawaii, USA, vol. 2, pp. 1159–1164 (2003). doi: 10.1109/CDC.2003.1272764
  5. 5.
    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, Heidelberg (2004)Google Scholar
  6. 6.
    De Santis, E.: On location observability notions for switching systems. Syst. Control Lett. 60(10), 807–814 (2011). http://dx.doi.org/10.1016/j.sysconle.2011.06.004
  7. 7.
    Hespanha, J.P., Morse, A.S.: Stability of switched systems with average dwell-time. In: Proceedings of the 38th IEEE Conference on Decision and Control, pp. 2655–2660 (1999)Google Scholar
  8. 8.
    Lee, C., Ping, Z., Shim, H.: Real time switching signal estimation of switched linear systems with measurement noise. In: Proceedings of the 12th European Control Conference 2013, Zurich, Switzerland, pp. 2180–2185 (2013)Google Scholar
  9. 9.
    Medina, E.A., Lawrence, D.A.: Reachability and observability of linear impulsive systems. Automatica 44, 1304–1309 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Medina, E.A., Lawrence, D.A.: State estimation for linear impulsive systems. Proc. Am. Control Conf. 2009, 1183–1188 (2009)Google Scholar
  11. 11.
    Meghnous, A.R., Pham, M.T., Lin-Shi, X.: A hybrid observer for a class of DC-DC power converters. Proc. Am. Control Conf. 2013, 6225–6230 (2013)Google Scholar
  12. 12.
    Pait, F.M., Morse, A.S.: A cyclic switching strategy for parameter-adaptive control. IEEE Trans. Autom. Control 6(39), 1172–1183 (1994)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Pettersson, S.: Designing switched observers for switched systems using multiple Lyapunov functions and dwell-time switching. In: IFAC Conference on Analysis and Design of Hybrid Systems, pp. 18–23 (2006)Google Scholar
  14. 14.
    Shim, H., Tanwani, A.: On a sufficient condition for observability of switched nonlinear systems and observer design strategy. Proc. Am. Control Conf. 2011, 1206–1211 (2011)Google Scholar
  15. 15.
    Shim, H., Tanwani, A.: Hybrid-type observer design based on a sufficient condition for observability in switched nonlinear systems. Int. J. Robust Nonlinear Control 24(6), 1064–1089 (2014). doi: 10.1002/rnc.2901
  16. 16.
    Sontag, E.D.: Mathematical Control Theory: Deterministic Finite Dimensional Systems, 2nd edn. Springer, New York (1998)CrossRefzbMATHGoogle Scholar
  17. 17.
    Sun, Z., Ge, S.S.: Switched linear systems. Communications and Control Engineering. Springer, London (2005). doi: 10.1007/1-84628-131-8 Google Scholar
  18. 18.
    Tanwani, A., Liberzon, D.: Robust invertibility of switched linear systems. In: Proceedings of the 50th IEEE Conference on Decision and Control and European Control Conference ECC 2011, Orlando, USA, pp. 441–446 (2011)Google Scholar
  19. 19.
    Tanwani, A., Shim, H., Liberzon, D.: Observability implies observer design for switched linear systems. In: Proceedings of the ACM Conference on Hybrid Systems: Computation and Control, pp. 3–12 (2011)Google Scholar
  20. 20.
    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
  21. 21.
    Tanwani, A., Shim, H., Liberzon, D.: Comments on “Observability of switched linear systems: characterization and observer design” (2014). Submitted for publication; Preprint available on authors’ websitesGoogle Scholar
  22. 22.
    Tanwani, A., Trenn, S.: An observer for switched differential-algebraic equations based on geometric characterization of observability. In: Proceedings of the 52nd IEEE Conference on Decision and Control, Florence, Italy, pp. 5981–5986 (2013)Google Scholar
  23. 23.
    Trenn, S.: Regularity of distributional differential algebraic equations. Math. Control Signals Syst. 21(3), 229–264 (2009). doi: 10.1007/s00498-009-0045-4 MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Van Gorp, J., Defoort, M., Djemai, M., Manamanni, N.: Hybrid observer for the multicellular converter. In: 4th IFAC Conference on Analysis and Design of Hybrid Systems (2012)Google Scholar
  25. 25.
    Vidal, R., Chiuso, A., Soatto, S., Sastry, S.: Observability of linear hybrid systems. Hybrid Systems: Computation and Control. Lecture Notes in Computer Science, vol. 2623, pp. 526–539. Springer, Berlin (2003)Google Scholar
  26. 26.
    Vu, L., Liberzon, D.: Invertibility of switched linear systems. Automatica 44(4), 949–958 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    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

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of KaiserslauternKaiserslauternGermany
  2. 2.ASRI School of Electrical EngineeringSeoul National UniversitySeoulSouth Korea
  3. 3.Coordinated Science LaboratoryUniversity of Illinois at Urbana-ChampaignUrbanaUSA

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