Observer based fault detection for two dimensional systems described by Roesser models

  • Zhenheng Wang
  • Helen Shang


Fault detection and isolation for two dimensional (2-D) systems represent a great challenge in both theoretical development and applications. 2-D systems have been commonly represented by the Roesser Model and the Fornasini and Marchesini (F–M) model. Research on fault detection and isolation has been carried out using observer-based methods for the F–M model. In this paper, an observer based fault detection strategy is investigated for systems modelled by the Roesser model. Using the 2-D polynomial matrix technique, a dead-beat observer is developed and the state estimate from the observer is then input to a residual generator to monitor occurrence of faults. An enhanced realization technique is combined to achieve efficient fault detection with reduced computations. Simulation results indicate that the proposed method is effective in detecting faults for systems without disturbances as well as those affected by unknown disturbances.


Fault detection 2-D systems Roesser model Observer  State space realization 



The financial support from Natural Sciences and Engineering Research Council of Canada is gratefully appreciated.


  1. Antoniou, G. E., Paraskevopoulos, P. N., & Varoufakis, S. J. (1988). Minimal state-space realization of factorable 2-D transfer functions. IEEE Transactions on Circuits and Systems, 35(8), 1055–1058.MathSciNetCrossRefGoogle Scholar
  2. Birgit, J. (2002). A review on realization theory for infinite-dimensional systems. Germany: University Dortmund.Google Scholar
  3. Bisiacco, M., & Valcher, M. E. (2006). Observer-based fault detection and isolation for 2D state-space models. Multidimensional Systems and Signal Processing, 17, 219–242.MATHMathSciNetCrossRefGoogle Scholar
  4. Bisiacco, M., & Valcher, M. E. (2008). Dead-beat estimation problems for 2D behaviors. Multidimensional Systems and Signal Processing, 19, 287–306.MATHMathSciNetCrossRefGoogle Scholar
  5. Chen, J., & Patton, R. J. (1999). Robust model-based fault diagnosis for dynamic systems. Massachusetts: Kluwer.MATHCrossRefGoogle Scholar
  6. Criftcibasi, Y., & Yuksel, O. (1983). Sufficient or necessary conditions for model controllability and observability of Roesser’s 2-D system model. IEEE Transactions on Automatic Control, AC–27(4), 527–529.CrossRefGoogle Scholar
  7. Fornasini, E., & Marchesini, G. (1977). Doubly indexed dynamic systems. Mathematics System Theory, 12, 59–72.MathSciNetCrossRefGoogle Scholar
  8. Fornasini, E., & Marchesini, G. (1988). State-space realization theory of two dimensional filters. IEEE Transactions on Automatic Control, AC–21(4), 484–491.MathSciNetGoogle Scholar
  9. Frank, P. M. (1990). Fault diagnosis in dynamic systems using analytical and knowledge-based redundancy-a survey and some new results. Automatica, 26, 459–474.MATHCrossRefGoogle Scholar
  10. Gertler, J. (1991). Analytical redundancy methods in fault detection and isolation. In Proceedings of IFAC/IAMCS symposium on safe process (p. 91), Baden-Baden.Google Scholar
  11. Gertler, J. (1993). Residual generation in model-based fault diagnosis. Control Theory and Advanced Technology, 9(1), 259–285.MathSciNetGoogle Scholar
  12. Gertler, J. (1998). Fault detection and diagnosis in engineering systems. New York: Marcel Dekker.Google Scholar
  13. Hoskins, J. C., & Himmelblau, D. M. (1988). Artificial neural network models of knowledge representation in chemical engineering. Computers and Chemical Engineering, 12, 881–890.CrossRefGoogle Scholar
  14. Isaksson, A. (1993). Analysis of identified 2-D non-causal models. IEEE Transactions on Information Theory, 39(2), 525–534.MATHMathSciNetCrossRefGoogle Scholar
  15. Kaczorek, T. (2001). Perfect observers for singular 2D linear systems. Bulletin of the Polish Academy of Sciences. Technical Sciences, 49(1), 141–147.MATHGoogle Scholar
  16. Kramer, M. A. (1987). Malfunction diagnosis using quantitative models with non-boolean reasoning in expert systems. AIChE Journal, 33(1), 130–140.CrossRefGoogle Scholar
  17. Li, X., & Gao, H. (2012). Robust finite frequency H\(_{\infty }\) filtering for uncertain 2-D Roesser systems. Automatica, 48, 1163–1170.MATHCrossRefGoogle Scholar
  18. Li, X., & Gao, H. (2013). Robust finite frequency H\(_{\infty }\) filtering for uncertain 2-D systems: The FM model case. Automatica, 49, 2446–2452.CrossRefGoogle Scholar
  19. Li, X., Gao, H., & Wang, C. (2012). Generalized Kalman-Yakubovich-Popov Lemma for 2-D FM LSS model. IEEE Transactions on Automatic Control, 57(12), 3090–3103.MathSciNetCrossRefGoogle Scholar
  20. Makoto, O., & Sumihisa, H. (1991). Two dimensional LMS adaptive filters. IEEE Transactions in Consumer Electronics, 37(1), 66–73.CrossRefGoogle Scholar
  21. Maleki, S., Rapisarda, P., Ntogramatzidis, L., & Rogers, E. (2013). A geometric approach to 3D fault identification. In Proceedings of the 8th international workshop on multidimensional systems nDS’13. Germany: Erlagen.Google Scholar
  22. Mehra, R. K., & Peschon, J. (1971). An innovations approach to fault detection and diagnosis in dynamic systems. Automatica, 7, 637–640.CrossRefGoogle Scholar
  23. Ntogramatzidis, L., & Cantoni, M. (2012). Detectability subspaces and observer synthesis for two-dimensional systems. Multidimensional Systems and Signal Processing, 23(1–2), 79–96.MATHMathSciNetCrossRefGoogle Scholar
  24. Ramos, J. (1993). A subspace algorithm for identifying 2-D separable in denominator filters. IEEE Transactions on Circuits and Systems: Analog and Digital Signal Processing, 41(1), 63–67.CrossRefGoogle Scholar
  25. Ramos, J. R., Alenany, A., Shang, H., & Santos, P. J. L. (2011). Subspace algorithms for identifying separable in denominator 2-D systems with deterministic inputs. IET Control Theory & Applications, 5(15), 1748–1765.MathSciNetCrossRefGoogle Scholar
  26. Rikus, E. (1979). Controllability and observability of 2-D system. IEEE Transactions on Automatic Control, AC–24(1), 121–133.Google Scholar
  27. Roesser, R. P. (1975). A discrete state space model for linear image processing. IEEE Transactions on Automatic Control, AC–20, 1–10.MathSciNetCrossRefGoogle Scholar
  28. Russell, E. L., Chiang, L. H., & Braatz, R. D. (2000). Data-driven techniques for fault detection and diagnosis in chemical processes. London: Springer.CrossRefGoogle Scholar
  29. Wang, D. (1998). Identification and approximation of 1-D and 2-D digital filters. Ph.D. thesis, Florida Atlantic University, United States.Google Scholar
  30. Willsky, A. S., & Jones, H. L. (1976). A generalized likelihood ratio approach to the detection and estimation of jumps in linear systems. IEEE Transactions on Automatic Control, AC–21, 108–112.MathSciNetCrossRefGoogle Scholar
  31. Woods, J., & Radewan, C. (1977). Kalman filtering in two dimensions. IEEE Transactions on Information Theory, IT–23(4), 473–482.MathSciNetCrossRefGoogle Scholar
  32. Wu, L., Shi, P., Gao, H., & Wang, C. (2008). \(\text{ H }_{\infty }\) filtering for 2D Markovian jump systems. Automatica, 44, 1849–1858.MATHMathSciNetCrossRefGoogle Scholar
  33. Wu, L., & Ho, D. W. C. (2009). Fuzzy filter design for Ito stochastic systems with application to sensor fault detection. IEEE Transactions on Fuzzy Systems, 17, 233–242.CrossRefGoogle Scholar
  34. Wu, L., Su, X., & Shi, P. (2011). Mixed \(\text{ H }_2/\text{ H }_{\infty }\) approach to fault detection of discrete linear repetitive processes. Journal of The Franklin Institute, 348, 393–414.MATHMathSciNetCrossRefGoogle Scholar
  35. Wu, L., Yao, X., & Zheng, W. X. (2012). Generalized \(\text{ H }_2\) fault detection for two-dimensional Markovian jump systems. Automatica, 48, 1741–1750.MATHMathSciNetCrossRefGoogle Scholar
  36. Xu, L., Fan, H., Lin, Z., & Bose, N. K. (2008). A direct-construction approach to multidimensional realization and LFR uncertainty modeling. Multidimensional System and Signal Processing, 19, 323–359.MATHMathSciNetCrossRefGoogle Scholar
  37. Xu, H., & Zou, Y. (2011). \(\text{ H }_{\infty }\) control for 2-D singular delayed systems. International Journal of Systems Science, 42(4), 609–619.MATHMathSciNetCrossRefGoogle Scholar
  38. Xu, H., & Zou, Y. (2012). Robust \(\text{ H }_{\infty }\) filtering for uncertain two-dimensional discrete systems with state-varying delays. International Journal of Control Automation and Systems, 8(4), 720–726.CrossRefGoogle Scholar
  39. Xu, H., Lin, Z., & Makur, A. (2012). The existence and design of functional observers for two-dimensional systems. Systems & Control Letters, 61(2), 362–368.MATHMathSciNetCrossRefGoogle Scholar
  40. Yang, R., Xie, L., & Zhang, C. (2006). \(\text{ H }_2\) and mixed \(\text{ H }_{2}/\text{ H }_\infty \) control of two dimensional systems in Roesser model. Automatica, 42, 1507–1514.MATHMathSciNetCrossRefGoogle Scholar
  41. Yao, X., Wu, L., Zheng, W. X., & Wang, C. (2011). Fault detection filter design for Markovian jump singular systems with intermittent measurements. IEEE Transactions on Signal Processing, 59, 3099–3109.MathSciNetCrossRefGoogle Scholar
  42. Ye, S., Wang, W., Zou, Y., & Xu, H. (2011). Non-Fragile robust guaranteed cost control of 2-D discrete uncertain systems described by the general models. Circuits, Systems and Signal Processing, 30, 899–914.MATHMathSciNetCrossRefGoogle Scholar
  43. Zampieri, S. (1991). 2D residual generation and dead beat observers. Systems & Control Letters, 17, 483–492.MATHMathSciNetCrossRefGoogle Scholar
  44. Zhao, P., & Yu, D. (1993). An unbiased and computationally efficient LS estimation method for identifying parameters of 2D noncausal SAR models. IEEE Transactions on Signal Processing, 41(2), 849–857.MATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  1. 1.School of EngineeringLaurentian UniversitySudburyCanada

Personalised recommendations