access icon free Reduced-order dynamic observer error linearisation for discrete-time systems: an extension of the classical observer error linearisation

This study addresses reduced-order dynamic observer error linearisation (RDOEL) for discrete-time systems, which is a modified version of dynamic observer error linearisation (DOEL). The authors define the concept of RDOEL and provide a necessary and sufficient condition for RDOEL. Moreover, the relation between RDOEL and DOEL is discussed when the auxiliary dynamics is an arbitrary linear controllable system. Using the relationship and an existing algorithm for DOEL, an algorithm for RDOEL also proposed, which offers lower-dimensional observers than DOEL. Finally, the authors illustrate the theory by applying it to the discrete-time hyperchaotic system due to Wang, which was derived from the generalised Rössler system via a first-order difference algorithm.

Inspec keywords: linear systems; reduced order systems; linearisation techniques; nonlinear systems; discrete time systems; controllability; chaos; observers

Other keywords: discrete-time hyperchaotic system; RDOEL; reduced-order dynamic observer error linearisation; first-order difference algorithm; linear controllable system; generalised Rössler system; classical observer error linearisation; auxiliary dynamics; lower-dimensional observers

Subjects: Simulation, modelling and identification; Nonlinear control systems; Discrete control systems; Control system analysis and synthesis methods

References

    1. 1)
      • 30. Lee, W., Nam, K.: ‘Observer design for autonomous discrete-time nonlinear systems’, Syst. Control Lett., 1991, 17, pp. 4958 (doi: 10.1016/0167-6911(91)90098-Y).
    2. 2)
      • 39. Wang, X.Y.: ‘Chaos in complex nonlinear systems’ (Publishing House of Electronics Industry, Beijing, 2003) (in Chinese).
    3. 3)
      • 31. Lin, W., Byrnes, C.I.: ‘Remarks on linearization of discrete-time autonomous systems and nonlinear observer design’, Syst. Control Lett., 1995, 25, pp. 3140 (doi: 10.1016/0167-6911(94)00054-Y).
    4. 4)
      • 12. Respondek, W., Pogromsky, A., Nijmeijer, H.: ‘Time scaling for observer design with linearizable error dynamics’, Automatica, 2004, 40, (2), pp. 277285 (doi: 10.1016/j.automatica.2003.09.012).
    5. 5)
      • 25. Boutat, D., Busawon, K.: ‘On the transformation of nonlinear dynamical systems into the extended nonlinear observable canonical form’, Int. J. Control, 2011, 84, (1), pp. 94106 (doi: 10.1080/00207179.2010.541285).
    6. 6)
      • 29. Chung, S.-T., Grizzle, J.W.: ‘Sampled-data observer error linearization’, Automatica, 1990, 26, (6), pp. 9971007 (doi: 10.1016/0005-1098(90)90084-U).
    7. 7)
      • 14. Keller, H., ‘Non-linear observer design by transformation into a generalized observer canonical form’, Int. J. Control, 1987, 46, (6), pp. 19151930 (doi: 10.1080/00207178708934024).
    8. 8)
      • 16. C.Li, W., Tao, L.W.: ‘Observing non-linear time-variable systems through a canonical form observer’, Int. J. Control, 1986, 44, (6), pp. 17031713 (doi: 10.1080/00207178608933695).
    9. 9)
      • 22. Back, J., Seo, J.H.: ‘An algorithm for system immersion into nonlinear observer form: SISO case’, Automatica, 2006, 42, (2), pp. 321328 (doi: 10.1016/j.automatica.2005.09.014).
    10. 10)
      • 26. Yang, J., Back, J., Seo, J.H., Shim, H.: ‘Reduced-order dynamic observer error linearization’. IFAC Symp. NOLCOS, Bologna, Italy, 2010.
    11. 11)
      • 13. Zheng, G., Boutat, D., Barbot, J.P.: ‘Single output-dependent observability normal form’, SIAM J. Control Optim., 2007, 46, (6), pp. 22422255 (doi: 10.1137/050627137).
    12. 12)
      • 35. Kazantzis, N., Kravaris, C.: ‘Discrete-time nonlinear observer design using functional equations’, Syst. Control Lett., 2001, 42, pp. 8194 (doi: 10.1016/S0167-6911(00)00071-2).
    13. 13)
      • 18. Phelps, R.A., Krener, A.J.: ‘Computation of observer normal form using Macsyma’, in Byrnes, C.I., Martin, C.F., Saeks, R.E.: (Eds.): ‘Analysis and control of nonlinear systems’ (North-Holland, 1988), pp. 475482.
    14. 14)
      • 20. Jouan, P.: ‘Immersion of nonlinear systems into linear systems modulo output injection’, SIAM J. Control Optim., 2003, 41, (6), pp. 17561778 (doi: 10.1137/S0363012901391706).
    15. 15)
      • 1. Krener, J.A., Isidori, A.: ‘Linearization by output injection and nonlinear observers’, Syst. Control Lett., 1983, 3, (1), pp. 4752 (doi: 10.1016/0167-6911(83)90037-3).
    16. 16)
      • 21. Back, J., Seo, J.H.: ‘Immersion of non-linear systems into linear systems up to output injection: characteristic equation approach’, Int. J. Control, 2004, 77, (8), pp. 723734 (doi: 10.1080/00207170410001715040).
    17. 17)
      • 28. Yang, J.: ‘Nonlinear observer design by reduced-order dynamic observer error linearization’. PhD dissertation, Seoul National University, 2011.
    18. 18)
      • 41. Ding, Z.: ‘Differential stability and design of reduced-order observers for non-linear systems’, IET Control Theory Applic., 2011, 5, (2), pp. 315322 (doi: 10.1049/iet-cta.2009.0523).
    19. 19)
      • 17. Zheng, G., Boutat, D.: ‘Synchronisation of chaotic systems via reduced observers’, IET Control Theory Applic., 2011, 5, (2), pp. 308314 (doi: 10.1049/iet-cta.2010.0078).
    20. 20)
      • 6. Noh, D., Jo, N.H., Seo, J.H.: ‘Nonlinear observer design by dynamic observer error linearization’, IEEE Trans. Autom. Control, 2004, 49, (10), pp. 17461750 (doi: 10.1109/TAC.2004.835397).
    21. 21)
      • 36. Xiao, M., Kazantzis, N., Kravaris, C., Krener, A.J.: ‘Nonlinear discrete-time observer design with linearizable error dynamics’, IEEE Trans. Autom. Control, 2003, 48, (4), pp. 622626 (doi: 10.1109/TAC.2003.809793).
    22. 22)
      • 34. Boutat, D., Boutat-Baddas, L., Darouach, M.: ‘A new reduced-order observer normal form for nonlinear discrete time systems’, Syst. Control Lett., 2012, 61, pp. 10031008 (doi: 10.1016/j.sysconle.2012.07.007).
    23. 23)
      • 37. Lin, W., Wei, J.: ‘Observer design for a class of discrete-time nonlinear systems’, Eur. J. Control, 2009, 15, pp. 184193 (doi: 10.3166/ejc.15.184-193).
    24. 24)
      • 7. Boutat, D., Benali, A., Hammouri, H., Busawon, K.: ‘New algorithm for observer error linearization with a diffeomorphism on the outputs’, Automatica, 2009, 45, (10), pp. 21872193 (doi: 10.1016/j.automatica.2009.05.030).
    25. 25)
      • 4. Levine, J., Marino, R.: ‘Nonlinear system immersion, observers and finite-dimensional filters’, Syst. Control Lett., 1986, 7, (2), pp. 133142 (doi: 10.1016/0167-6911(86)90019-8).
    26. 26)
      • 2. Bestle, D., Zeitz, M.: ‘Canonical form observer design for non-linear time-variable systems’, Int. J. Control, 1983, 38, (2), pp. 419431 (doi: 10.1080/00207178308933084).
    27. 27)
      • 8. Wang, Y., Lynch, A.F.: ‘Multiple time scalings of a multiple-output observer form’, IEEE Trans. Autom. Control, 2010, 55, (4), pp. 966971 (doi: 10.1109/TAC.2010.2041616).
    28. 28)
      • 10. Glumineau, A., Moog, C.H., Diop, S., Plestan, F.: ‘New algebro-geometric conditions for the linearization by input-output injection’, IEEE Trans. Autom. Control, 1996, 41, (4), pp. 598603 (doi: 10.1109/9.489283).
    29. 29)
      • 32. Lilge, T.: ‘On observer design for nonlinear discrete-time systems’, Eur. J. Control, 1998, 4, pp. 306319 (doi: 10.1016/S0947-3580(98)70124-4).
    30. 30)
      • 9. Leon, E.A., Solsona, J.A.: ‘Design of reduced-order nonlinear observers for energy conversion applications’, IET Control Theory Applic., 2010, 4, (5), pp. 724734 (doi: 10.1049/iet-cta.2009.0095).
    31. 31)
      • 15. Marino, R., Tomei, P.: ‘Adaptive observers with arbitrary exponential rate of convergence for nonlinear systems’, IEEE Trans. Autom. Control, 1995, 40, (7), pp. 13001304 (doi: 10.1109/9.400471).
    32. 32)
      • 40. Yan, Z.: ‘Q-S (complete or anticipated) synchronization backsteeping scheme in a class of discrete-time chaotic (hyperchaotic) systems: a symbolic-numeric computation approach’, Chaos, 2006, 16, (1), p. 013119 (doi: 10.1063/1.1930727).
    33. 33)
      • 33. Huijberts, H.J.C.: ‘On existence of extended observers for nonlinear discrete-time systems’, in Nijmeijer, H., Fossen, T.I. (Eds.): ‘New directions in nonlinear observer design’ (Springer-Verlag, London, 1999), pp. 7392.
    34. 34)
      • 24. Yu, K.T., Back, J., Seo, J.H.: ‘Constructive algorithm for dynamic observer error linearization via integrators: single output case’, Int. J. Robust Nonlinear Control, 2007, 17, (1), pp. 2549 (doi: 10.1002/rnc.1117).
    35. 35)
      • 11. Guay, M., ‘Observer linearization by output-dependent time-scale transformation’, IEEE Trans. Autom. Control, 2002, 47, (10), pp. 17301735 (doi: 10.1109/TAC.2002.803547).
    36. 36)
      • 38. Zhang, J., Feng, G., Xu, H.: ‘Observer design for nonlinear discrete-time systems: immersion and dynamic observer error linearization techniques’, Int. J. Robust Nonlinear Control, 2010, 20, (5), pp. 504514.
    37. 37)
      • 19. Röbenack, K., Lynch, A.F.: ‘An efficient method for observer design with approximately linear error dynamics’, Int. J. Control, 2004, 77, (7), pp. 607612 (doi: 10.1080/00207170410001682515).
    38. 38)
      • 3. Krener, J.A., Respondek, W.: ‘Nonlinear observers with linearizable error dynamics’, SIAM J. Control Optim., 1985, 23, (2), pp. 197216 (doi: 10.1137/0323016).
    39. 39)
      • 5. Xia, H.X., Gao, W.B.: ‘Nonlinear observer design by observer error linearization’, SIAM J. Control Optim., 1989, 27, (1), pp. 199216 (doi: 10.1137/0327011).
    40. 40)
      • 23. Back, J., Yu, K.T., Seo, J.H.: ‘Dynamic observer error linearization’, Automatica, 2006, 42, (12), pp. 21952200 (doi: 10.1016/j.automatica.2006.07.009).
    41. 41)
      • 27. Yang, J., Back, J., Seo, J.H.: ‘A complete solution to a simple case of dynamic observer error linearization: new approach to observer error linearization’. IEICE Trans. Fundamentals, 2011, E94-A, (1), pp. 424429 (doi: 10.1587/transfun.E94.A.424).
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