This study concerns the observer design for Lipschitz non-linear descriptor systems. It introduces a new observer structure called a generalised dynamic observer, which is more general than the proportional and proportional–integral observers. The originality of the proposed observer is that it provides additional degrees of freedom in the observer design, which can be used to increase steady-state accuracy and to improve robustness in estimation against modelling errors. Conditions for the existence and stability of this observer are given in terms of linear matrix inequalities. Reduced-order and full-order observers can be designed directly by using the proposed unified approach. The effectiveness of the developed method is illustrated by two numerical examples where impulse and impulse-free cases are considered, a comparison between the generalised dynamic observer, the proportional observer, and the proportional–integral observer is given to show the observers performances.

References

1. 1)
  - 34. Yip, E.L., Sincovec, R.F.: ‘Solvability’, controllability, and observability of continuous descriptor systems', IEEE Trans. Autom. Control, 1981, 26, pp. 702–707.
2. 2)
  - 12. Darouach, M.: ‘H∞ unbiased filtering for linear descriptor systems via LMI’, IEEE Trans. Autom. Control, 2009, 54, pp. 1966–1972.
3. 3)
  - 16. Gao, Z., Breikin, T., Wang, H.: ‘Discrete-time proportional and integral observer and observer-based controller for systems with both unknown input and output disturbances’, Opt. Control Appl. Methods, 2008, 29, pp. 171–189.
4. 4)
  - 29. Kou, S.R., Elliot, D.L., Tarn, T.J.: ‘Exponential observers for nonlinear dynamic systems’, J. Inf. Control, 1975, 29, pp. 204–216.
5. 5)
  - 7. Müller, P.C.: ‘Modelling and control of mechatronic systems by the descriptor approach’, J. Theor. Appl. Mech., 2005, 43, pp. 593–607.
6. 6)
  - 24. Zhuang, G., Xia, J., Zhang, B., et al: ‘Robust normalisation and P–D state feedback control for uncertain singular Markovian jump systems with time-varying delays’, IET Control Theory Applic., 2018, 12, pp. 419–427.
7. 7)
  - 31. Raghavar, S., Hedrick, J.K.: ‘Observer-based controllers for nonlinear systems’. Proc. ASME Winter Annual Meeting, Anaheim, California, USA, 1992.
8. 8)
  - 28. Thau, F.E.: ‘Observing the state of nonlinear dynamic systems’, Int. J. Control, 1973, 17, pp. 471–479.
9. 9)
  - 14. Müller, P.C., Hou, M.: ‘On the observer design for descriptor systems’, IEEE Trans. Autom. Control, 1993, 38, (11), pp. 1666–1671.
10. 10)
  - 19. 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, pp. 212–217.
11. 11)
  - 5. Fridman, E., Shaked, U.: ‘A descriptor system approach to H∞ control of linear time-delay systems’, IEEE Trans. Autom. Control, 2002, 47, pp. 253–270.
12. 12)
  - 10. Xu, S., Lam, J.: ‘Robust control and filtering of singular systems’ (Lecture notes in control and information sciences)' (Springer-Verlag, Berlin, 2006).
13. 13)
  - 33. Wu, Z.-G., Su, H., Shi, P., et al: ‘Analysis and synthesis of singular systems with time-delays’, vol. 443 (Springer, Berlin, 2013).
14. 14)
  - 21. Jiang, B., Kao, Y., Gao, C., et al: ‘Passification of uncertain singular semi-Markovian jump systems with actuator failures via sliding mode approach’, IEEE Trans. Autom. Control, 2017, 62, pp. 4138–4143.
15. 15)
  - 20. Feng, Z., Shi, P.: ‘Sliding mode control of singular stochastic Markov jump systems’, IEEE Trans. Autom. Control, 2017, 62, pp. 4266–4273.
16. 16)
  - 26. Kaprielian, S., Turi, J.: ‘An observer for a nonlinear descriptor system’. 1992 Proc. 31st IEEE Conf. on Decision and Control, Tucson, AZ, USA, 1992, pp. 975–976.
17. 17)
  - 6. Liu, P., Zhang, E., Yang, X., et al: ‘Passivity and optimal control of descriptor biological complex systems’, IEEE Trans. Autom. Control, 2008, 53, pp. 122–125.
18. 18)
  - 15. Zhou, L., Lu, G.: ‘Detection and stabilization for discrete-time descriptor systems via a limited capacity communication channel’, Automatica, 2009, 45, (10), pp. 2272–2277.
19. 19)
  - 35. Owens, D.H., Debeljkovic, D.L.: ‘Consistency and Liapunov stability of linear descriptor systems: a geometric analysis’, IMA J. Math. Control Inf., 1985, 2, pp. 139–151.
20. 20)
  - 17. Gao, Z., Ding, S.X.: ‘Fault estimation and fault-tolerant control for descriptor systems via proportional multiple-integral and derivative observer design’, IET Control Theory Applic., 2007, 1, (5), pp. 1208–1218.
21. 21)
  - 9. Osorio-Gordillo, G.L., Darouach, M., Astorga-Zaragoza, C.M., et al: ‘New dynamical observers design for linear descriptor systems’, IET Control Theory Applic., 2016, 10, (17), pp. 2223–2232.
22. 22)
  - 25. Boutayeb, M., Darouach, M.: ‘Observers design for nonlinear descriptor systems’. Proc. IEEE Conf. on Decision and Control, New Orleand, LA, 1995, pp. 2369–2374.
23. 23)
  - 32. Dai, L.: ‘Singular control systems’ (Lecture notes in control and information sciences) (Springer, Berlin, 1989), pp. 29–39.
24. 24)
  - 8. Osorio-Gordillo, G.L., Darouach, M., Astorga-Zaragoza, C.M.: ‘H∞ dynamical observers design for linear descriptor systems, application to state and unknown input estimation’, Eur. J. Control, 2015, 26, pp. 35–43.
25. 25)
  - 36. Skelton, R.E., Iwasaki, T., Grigoriadis, K.: ‘A unified algebraic approach to linear control design’ (CRC Press, Boca Raton, 1998).
26. 26)
  - 4. Boutayeb, M., Darouach, M., Rafaralahy, H.: ‘Generalized state-space observers for chaotic synchronization and secure communication’, IEEE Trans. Circuit Syst., 2002, 49, pp. 345–349.
27. 27)
  - 1. Duan, G.-R.: ‘Analysis and design of descriptor linear systems’, vol. 23 (Springer Science & Business Media, New York, 2010).
28. 28)
  - 30. Zak, S.K.: ‘On the stabilization and observation of nonlinear/uncertain dynamic systems’, IEEE Trans. Autom. Control, 1990, 35, (5), pp. 604–607.
29. 29)
  - 11. Alma, M., Darouach, M.: ‘Adaptive observers design for a class of linear descriptor systems’, Automatica, 2014, 50, pp. 578–583.
30. 30)
  - 13. Darouach, M.: ‘On the functional observers for linear descriptor systems’, Syst. Control Lett., 2012, 61, (3), pp. 427–434.
31. 31)
  - 3. Boulkroune, B., Darouach, M., Zasadzinski, M., et al: ‘A nonlinear observer design for an activated sludge wastewater treatment process’, J. Process Control, 2009, 19, pp. 1558–1565.
32. 32)
  - 27. Kidane, N., Yamashita, Y., Nishitani, H.: ‘Observer based I/O-linearizing control of high index DAE systems’. Proc. American Control Conf., Denver, Colorado, USA, 2003, pp. 3537–3542.
33. 33)
  - 23. Wang, H.S., Yung, C.F., Chang, F.R.: ‘H∞ control for nonlinear descriptor systems’, IEEE Trans. Autom. Control, 2002, 47, (11), pp. 1919–1925.
34. 34)
  - 22. Wang, H.S., Yung, C.F., Chang, F.R.: ‘H∞ control for nonlinear descriptor systems’, IEEE Trans. Autom. Control, 2002, 47, pp. 1919–1925.
35. 35)
  - 2. Araujo, J.M., Barros, P.R., Dorea, C.E.T.: ‘Design of observers with error limitation in discrete-time descriptor systems: a case study of a hydraulic tank system’, IEEE Trans. Control Syst. Technol., 2012, 20, pp. 1041–1047.
36. 36)
  - 18. Gao, Z., Ho, D.W.C.: ‘Proportional multiple-integral observer design for descriptor systems with measurement output disturbances’, IET Control Theory Applic., 2004, 151, (3), pp. 279–288.

Generalised dynamic observer design for Lipschitz non-linear descriptor systems

References

Related content