This study considers the functional observers design for a class of discrete-time switched singular systems simultaneously subject to state delays, unknown inputs (UIs) and arbitrary switching sequences. The singular matrix E is assumed to be switching-mode-dependent, and the UIs are present in both the state and the measurement channels. A mode-dependent, delay-type observer is first constructed, then, based on the unknown input decoupling and the switched Lyapunov theory, a method is proposed to design such observer which ensures both: the total elimination of the UIs and exponential estimate of the given function of the state vector. The conditions for the existence of the proposed observer are given through algebraic matrix inequalities, and exponential stability of the observation error dynamics is derived by using a properly constructed decay-rate-dependent switched Lyapunov function and the linear matrix inequality technique. Finally, an illustrative example is given to show the effectiveness of the obtained results.

References

1. 1)
  - 1. Lungu, M., Lungu, R.: ‘Full-order observer design for linear systems with unknown inputs’, Int. J. Control, 2012, 85, (10), pp. 1602–1615 (doi: 10.1080/00207179.2012.695397).
2. 2)
  - 2. Fernando, T., MacDougall, S., Sreeram, V., Trinh, H.: ‘Existence conditions for unknown input functional observers’, Int. J. Control, 2013, 86, (1), pp. 22–28 (doi: 10.1080/00207179.2012.715802).
3. 3)
  - 3. Bhattacharyya, S.P.: ‘Observer design for linear systems with unknown inputs’, IEEE Trans. Autom. Control, 1978, 23, (3), pp. 483–484 (doi: 10.1109/TAC.1978.1101758).
4. 4)
  - 4. Walcott, B., Żak, S.H.: ‘State observation of nonlinear uncertain dynamical systems’, IEEE Trans. Autom. Control, 1987, 32, (2), pp. 166–170 (doi: 10.1109/TAC.1987.1104530).
5. 5)
  - 5. Kudva, P., Viswanadham, N., Ramakrishna, A.: ‘Observers for linear systems with unknown inputs’, IEEE Trans. Autom. Control, 1980, 25, (1), pp. 113–115 (doi: 10.1109/TAC.1980.1102245).
6. 6)
  - 6. Darouach, M., Zasadzinski, M., Xu, S.J.: ‘Full-order observers for linear systems with unknown inputs’, IEEE Trans. Autom. Control, 1994, 39, (3), pp. 606–609 (doi: 10.1109/9.280770).
7. 7)
  - 7. Liu, H.-Y., Duan, Z.-S.: ‘Unknown input observer design for systems with monotone non-linearities’, IET Control Theory Appl., 2012, 6, (12), pp. 1941–1947 (doi: 10.1049/iet-cta.2011.0611).
8. 8)
  - 8. Trinh, H., Fernando, T.: ‘Functional observers for dynamical systems’ (Springer, Berlin, 2012).
9. 9)
  - 9. Darouach, M.: ‘Existence and design of functional observers for linear systems’, IEEE Trans. Autom. Control, 2000, 45, (5), pp. 940–943 (doi: 10.1109/9.855556).
10. 10)
  - 10. Darouach, M.: ‘Linear functional observers for systems with delays in state variables: the discrete-time case’, IEEE Trans. Autom. Control, 2005, 50, (2), pp. 228–233 (doi: 10.1109/TAC.2004.841932).
11. 11)
  - 11. Fiacchini, M., Millerioux, G.: ‘Dead-beat functional observers for discrete-time LPV systems with unknown inputs’, IEEE Trans. Autom. Control, 2013, 58, (12), pp. 3230–3235 (doi: 10.1109/TAC.2013.2261712).
12. 12)
  - 12. Dai, L.: ‘Singular control systems’ (Springer-Verlag, Berlin-Heidelberg, 1989).
13. 13)
  - 13. Darouach, M., Zasadzinski, M., Hayar, M.: ‘Reduced-order observer design for descriptor systems with unknown inputs’, IEEE Trans. Autom. Control, 1996, 41, (7), pp. 1068–1072 (doi: 10.1109/9.508918).
14. 14)
  - 14. Ma, S., Cheng, Z.: ‘Observer design for discrete time-delay singular systems with unknown inputs’. American Control Conf., 2005, pp. 4215–4219.
15. 15)
  - 15. Koenig, D.: ‘Unknown input proportional multiple-integral observer design for linear descriptor systems: application to state and fault estimation’, IEEE Trans. Autom. Control, 2005, 50, (2), pp. 212–217 (doi: 10.1109/TAC.2004.841889).
16. 16)
  - 16. Koenig, D.: ‘Observer design for unknown input nonlinear descriptor systems via convex optimization’, IEEE Trans. Autom. Control, 2006, 51, (6), pp. 1047–1052 (doi: 10.1109/TAC.2006.876807).
17. 17)
  - 17. Marx, B., Koenig, D., Ragot, J.: ‘Design of observers for Takagia–Sugeno descriptor systems with unknown inputs and application to fault diagnosis’, IET Control Theory Appl., 2007, 1, (5), pp. 1487–1495 (doi: 10.1049/iet-cta:20060412).
18. 18)
  - 18. Feng, J.-E., Lam, J., Xu, S.: ‘Finite-time functional observers for descriptor systems’, Int. J. Control Autom., 2009, 7, (3), pp. 341–347 (doi: 10.1007/s12555-009-0302-9).
19. 19)
  - 19. Feng, J.-e.: ‘Finite time functional observers for discrete-time singular systems with unknown inputs’. Proc. 29th Chinese Control Conf., Beijing, China, July 2010, pp. 65–70.
20. 20)
  - 20. Darouach, M.: ‘On the functional observers for linear descriptor systems’, Syst. Control Lett., 2012, 61, (3), pp. 427–434 (doi: 10.1016/j.sysconle.2012.01.006).
21. 21)
  - 21. Ezzine, M., Darouach, M., Ali, H.S., Messaoud, H.: ‘Unknown inputs functional observers designs for delay descriptor systems’, Int. J. Control, 2013, 86, (10), pp. 1850–1858 (doi: 10.1080/00207179.2013.797607).
22. 22)
  - 22. DeCarlo, R.A., Branicky, M.S., Pettersson, S., Lennartson, B.: ‘Perspectives and results on the stability and stabilizability of hybrid systems’, Proc. IEEE, 2000, 38, (7), pp. 1069–1082 (doi: 10.1109/5.871309).
23. 23)
  - 23. Sun, Z., Ge, S.S.: ‘Stability theory of switched dynamical systems’ (Springer, London, 2011).
24. 24)
  - 24. Zhao, S., Sun, J.: ‘Geometric approach for observability and accessibility of discrete-time non-linear switched impulsive systems’, IET Control Theory Appl., 2013, 7, (7), pp. 1014–1021 (doi: 10.1049/iet-cta.2012.0919).
25. 25)
  - 25. Tanwani, A., Shim, H., Liberzon, D.: ‘Observability for switched linear systems: characterization and observer design’, IEEE Trans. Autom. Control, 2013, 58, (4), pp. 891–904 (doi: 10.1109/TAC.2012.2224257).
26. 26)
  - 26. Morales-Morales, C., Adam-Medina, M., Cervantes, I., Vela-Valdés, L.G., Beltran, C.D.G.: ‘Virtual estimator for piecewise linear systems based on observability analysis’, Sensors, 2013, 13, (3), pp. 2735–2749 (doi: 10.3390/s130302735).
27. 27)
  - 27. Zhao, X., Liu, H., Zhang, J., Li, H.: ‘Multiple-mode observer design for a class of switched linear systems’, IEEE Trans. Autom. Sci. Eng., 2015, 12, (1), pp. 272–280 (doi: 10.1109/TASE.2013.2281466).
28. 28)
  - 28. Djemai, M., Defoort, M. (Eds.): ‘Hybrid dynamical systems observation and control’ (Springer International Publishing, 2015).
29. 29)
  - 29. Millerioux, G., Daafouz, J.: ‘Unknown input observers for switched linear discrete time systems’. American Control Conf., 2004, pp. 5802–5805.
30. 30)
  - 30. Sundaram, S., Hadjicostis, C.N.: ‘Designing stable inverters and state observers for switched linear systems with unknown inputs’. Proc. 45th IEEE Conf. Decision and Control, San Diego, USA, December 2006, pp. 4105–4110.
31. 31)
  - 31. Chen, J., Lagoa, C.M.: ‘Robust observer design for a class of switched systems’. Proc. 45th IEEE Conf. Decision and Control, San Diego, USA, December 2006, pp. 1659–1664.
32. 32)
  - 32. Bejarano, F.J., Pisano, A., Usai, E.: ‘Finite-time converging jump observer for switched linear systems with unknown inputs’, Nonlinear Anal., Hybrid Syst., 2011, 5, (2), pp. 174–188 (doi: 10.1016/j.nahs.2010.04.010).
33. 33)
  - 33. Bejarano, F.J., Pisano, A.: ‘Switched observers for switched linear systems with unknown inputs’, IEEE Trans. Autom. Control, 2011, 56, (3), pp. 681–686 (doi: 10.1109/TAC.2010.2095990).
34. 34)
  - 34. Van Gorp, J., Defoort, M., Veluvolu, K., Djemai, M.: ‘Hybrid sliding mode observer for switched linear systems with unknown inputs’, J. Franklin Inst., 2014, 351, (7), pp. 3987–4008 (doi: 10.1016/j.jfranklin.2014.04.002).
35. 35)
  - 35. Koenig, D., Marx, B., Jacquet, D.: ‘Unknown input observers for switched nonlinear discrete time descriptor systems’, IEEE Trans. Autom. Control, 2008, 53, (1), pp. 373–379 (doi: 10.1109/TAC.2007.914226).
36. 36)
  - 36. Niculescu, S.-I.: ‘Delay effects on stability: a robust control approach’ (Springer, London, 2001).
37. 37)
  - 37. Karimi, H.R., Maass, P.: ‘Delay-range-dependent exponential H∞ synchronization of a class of delayed neural networks’, Chaos Solitons Fractals, 2009, 41, (3), pp. 1125–1135 (doi: 10.1016/j.chaos.2008.04.051).
38. 38)
  - 38. Karimi, H.R., Gao, H.: ‘New delay-dependent exponential synchronization for uncertain neural networks with mixed time delays’, IEEE Trans. Syst., Man Cybern. B, Cybern., 2010, 40, (1), pp. 173–185 (doi: 10.1109/TSMCB.2009.2024408).
39. 39)
  - 39. Boukas, E.-K.: ‘Control of singular systems with random abrupt changes’ (Springer, Berlin, 2007).
40. 40)
  - 40. Wang, Z., Rodrigues, M., Theilliol, D., Shen, Y.: ‘Actuator fault estimation observer design for discrete-time linear parameter-varying descriptor systems’, Int. J. Adapt. Control Signal Process.2015, 29, (2), pp. 242–258 (doi: 10.1002/acs.2469).
41. 41)
  - 41. Du, Z., Zhang, Q., Chang, G.: ‘State feedback stabilization for switched singular networked control systems with time-delay’. Proc. 2009 Chinese Control and Decision, Guilin, China, July 2009, pp. 5587–5591.
42. 42)
  - 42. Krishnasamy, R., Balasubramaniam, P.: ‘A descriptor system approach to the delay-dependent exponential stability analysis for switched neutral systems with nonlinear perturbations’, Nonlinear Anal., Hybrid Syst., 2015, 15, pp. 23–36 (doi: 10.1016/j.nahs.2014.07.001).
43. 43)
  - 43. Raouf, J., Michalska, H.: ‘Exponential stabilization of singular systems by controlled switching’. Proc. 49th IEEE Conf. Decision and Control, Atlanta, GA, USA, December 2010, pp. 414–419.
44. 44)
  - 44. Zhang, W.-A., Yu, L.: ‘Stability analysis for discrete-time switched time-delay systems’, Automatica, 2009, 45, (10), pp. 2265–2271 (doi: 10.1016/j.automatica.2009.05.027).
45. 45)
  - 45. de Oliveira, M.C., Skelton, R.E.: ‘Stability tests for constrained linear systems’, in Reza, Moheimani S.O., (Ed.): ‘Perspectives in robust control’ (Springer, London, 2001), pp. 241–257.
46. 46)
  - 46. Rao, C.R., Mitra, S.K.: ‘Generalized inverse of matrices and its applications’ (Wiley, New York, 1971).
47. 47)
  - 47. Pipeleers, G., Demeulenaere, B., Swevers, J., Vandenberghe, L.: ‘Extended LMI characterizations for stability and performance of linear systems’, Syst. Control Lett., 2009, 58, (17), pp. 510–518 (doi: 10.1016/j.sysconle.2009.03.001).
48. 48)
  - 18. Wang, D., Wang, W., Shi, P.: ‘Robust fault detection for switched linear systems with state delays’, IEEE Trans. Syst. Man Cybern. B, 2009, 39, (3), pp. 800–805 (doi: 10.1109/TSMCB.2008.2007498).
49. 49)
  - 49. Zhang, H., Chen, Q., Yan, H., Liu, J.: ‘Robust H∞ filtering for switched stochastic system with missing measurements’, IEEE Trans. Signal Process., 2009, 57, (9), pp. 3466–3474 (doi: 10.1109/TSP.2009.2020376).
50. 50)
  - 50. Karimi, H.R.: ‘Robust H∞ filter design for uncertain linear systems over network with network-induced delays and output quantization’, Model. Ident. Control, 2009, 30, (1), pp. 27–37 (doi: 10.4173/mic.2009.1.3).
51. 51)
  - 51. Chadli, M., Karimi, H.R.: ‘Robust observer design for unknown inputs Takagi–Sugeno models’, IEEE Trans. Fuzzy Syst., 2013, 21, (1), pp. 158–164 (doi: 10.1109/TFUZZ.2012.2197215).

Functional observer for switched discrete-time singular systems with time delays and unknown inputs

Functional observer for switched discrete-time singular systems with time delays and unknown inputs

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