eesti teaduste
akadeemia kirjastus
SINCE 1952
Proceeding cover
of the estonian academy of sciences
ISSN 1736-7530 (Electronic)
ISSN 1736-6046 (Print)
Impact Factor (2020): 1.045

Observer-based residual generation for nonlinear discrete-time systems; pp. 325–336

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Arvo Kaldmäe, Ülle Kotta



The paper studies the possibility of constructing observer-based residuals to detect faults in a nonlinear discrete-time system. The residuals are generated in such a manner that they detect one specific fault and are not affected by other faults and disturbances. Thus, a bank of residuals has been found to detect and isolate different faults in the system. An algebraic method called functions’ algebra is used to construct an algorithm which computes the residuals. The key fact in residual generation is that any discrete-time observable system can be taken into the extended observer form. This form is used to construct the observer to estimate the system states under the assumption that there are no faults in the system. The state estimates are then compared to the measured values of the states. An example is added to illustrate the theoretical results. In the example it is also demonstrated how to combine the fault detection with the plant reconfiguration step of fault tolerant control.




1. Berdjag, D., Christophe, C., Cocquempot, V., and Jiang, B. Nonlinear model decomposition for robust fault detection and isolation using algebraic tools. IJICIC, 2006, 2(6), 1337–1354.

2. Blanke, M., Kinnaert, M., Lunze, J., and Staroswiecki, M. Diagnosis and Fault-Tolerant Control. Springer, Berlin, 2016.

3. Califano, C., Monaco, S., and Normand-Cyrot, D. Canonical observer forms for multi-output systems up to coordinate and output transformations in discrete time. Automatica, 2009, 45(11), 2483–2490.

4. Huijberts, H. J. C. On existence of extended observers for nonlinear discrete-time systems. In New Directions in Nonlinear Observer Design (Nijmeijer, H. and Fossen, T. I., eds). Springer, London, 1999, 73–92.

5. Huijberts, H. J. C., Lilge, T., and Nijmeijer, H. Synchronization and observers for nonlinear discrete time systems. In Proceedings of the European Control Conference. Karlsruhe, Germany, 1999, 1–9.

6. Huijberts, H. J. C., Nijmeijer, H., and Pogromsky, A. Y. Discrete-time observers and synchronization. In Controlling Chaos and Bifurcations in Engineering Systems. CRC Press, Boca Raton, FL, USA, 1999, 439–456.

7. Isermann, R. Fault diagnosis of machines via parameter estimation and knowledge processing. Automatica, 1993, 29(4), 815–835.

8. Kaldmäe, A., Kotta, Ü., Jiang, B., Shumsky, A., and Zhirabok, A. Faulty plant reconfiguration based on disturbance decoupling methods. Asian J. Control, 2016, 18(3), 858–867.

9. Kaparin, V. and Kotta, Ü. Extended observer form for discrete-time nonlinear control systems. In The 9th IEEE International Conference on Control and Automation. Santiago, Chile, 2011, 507–512.

10. Kaparin, V. and Kotta, Ü. Transformation of nonlinear MIMO discrete-time systems into the extended observer form. Asian J. Control, 2018, DOI 10.1002/asjc.1824.

11. Kaparin, V., Kotta, Ü., and Mullari, T. Extended observer form: simple existence conditions. Int. J. Control, 2013, 86(5), 794–803.

12. Kaparin, V., Kotta, Ü., Shumsky, A. Y., and Zhirabok, A. N. A note on the relationship between single- and multi-experiment observability for discrete-time nonlinear control systems. Proc. Estonian Acad. Sci., 2011, 60(3), 174–178.

13. Krener, A. J. and Respondek, W. Nonlinear observers with linearizable error dynamics. SIAM J. Control Optim., 1985, 23(2), 197–216.

14. Lee, W. and Nam, K. Observer design for autonomous discrete-time nonlinear systems. Syst. Control Lett., 1991, 17(1), 49–58.

15. Li, L. Fault Detection and Fault-Tolerant Control for Nonlinear Systems. Springer, 2016.

16. Patton, R., Clark, R., and Frank, P. Fault Diagnosis in Dynamic Systems: Theory and Applications. Prentice-Hall, NJ, USA, 1989.

17. Persis, C. D. and Isidori, A. A geometric approach to nonlinear fault detection and isolation. IEEE Trans. Autom. Control, 2001, 46(6), 853–865.

18. Plestan, F. and Glumineau, A. Linearization by generalized input-output injection. Syst. Control Lett., 1997, 31(2), 115–128.

19. Shumsky, A. and Zhirabok, A. Unified approach to the problem of full decoupling via output feedback. European J. Control, 2010, 16(4), 313–325.

20. Staroswiecki, M. and Comtet-Varga, G. Analytical redundancy relations for fault detection and isolation in algebraic dynamic systems. Automatica, 2001, 37(5), 687–699.

21. Wang, Y. and Sontag, E. D. Orders of input/output differential equations and state-space dimensions. SIAM J. Control Optim., 1995, 33(4), 1102–1126.

22. Witczak, M. Fault Diagnosis and Fault-Tolerant Control Strategies for Non-Linear Systems. Springer, 2014.

23. Zhang, Y. and Jiang, J. Bibliographical review on reconfigurable fault-tolerant control systems. Ann. Rev. Control, 2008, 32(2), 229–252.

24. Zhirabok, A. and Shumsky, A. The Algebraic Methods for Analysis of Nonlinear Dynamic Systems. Dalnauka, Vladivostok, Russia, 2008 (in Russian).


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