POD-Kalman Observer for Linear Time Invariant Dynamic Systems

  • Saeed Eftekhar Azam
Part of the SpringerBriefs in Applied Sciences and Technology book series (BRIEFSAPPLSCIENCES)


This Chapter investigates the statistical properties of residual errors induced by POD-based reduced order modeling. Such errors enter into the state space equations of the reduced systems in terms of system evolution and observation noise. A fundamental assumption made by recursive Bayesian filters, as exploited in this study, is the whiteness of the aforementioned noises. In this chapter, null hypothesis of the whiteness of the noise signals is tested by making use of the Bartlett’s whiteness test. It is shown that, no matter what the number of POMs retained in the analysis is, the null hypothesis of the whiteness is always to be rejected. However, the spectral power of the embedded periodic signals decreases rapidly by increasing the number of POMs. The speed-up gained by incorporating POD-based reduced models into Kalman observer of linear time invariant systems, is also addressed in this chapter. It is shown that the reduced models incorporated into the Kalman filter dramatically reduce the computing time, leading to speed-up of 300 for a POD model featuring 1 POM, which is able to accurately reconstruct the displacement time history of the structure. Moreover, it is revealed that the coupling of POD and Kalman filter can improve the estimations provided by POD alone.


Kalman Filter Reduce Order Model Tune Mass Damper Acceleration Time History Linear Time Invariant System 
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  1. Bartlett MS (1978) An introduction to stochastic processes with special reference to methods and applications. Cambridge University Press, LondonzbMATHGoogle Scholar
  2. Galvanetto U, Violaris G (2007) Numerical investigation of a new damage detection method based on proper orthogonal decomposition. Mech Sys Signal Process 21:1346–1361CrossRefGoogle Scholar
  3. Goodwin GC, Graebe SF, Salgado ME (2001) Control system design. Pearson, LondonGoogle Scholar
  4. Gustafsson TK, Mäkilä PM (1996) Modelling of uncertain systems via linear programming. Automatica 32:319–334CrossRefzbMATHGoogle Scholar
  5. He J, Sarma P, Durlofsky LJ (2011) Use of reduced-order models for improved data assimilation within an EnKF context. In: Proceedings of SPE reservoir simulation symposium 2011, vol. 2, pp 1181–1195Google Scholar
  6. Ikeda Y (2009) Active and semi-active vibration control of buildings in Japan-practical applications and verification. Struct Control Health Monit 16:703–723CrossRefGoogle Scholar
  7. Kececioglu DB (2002) Reliability engineering handbook, vol 2. DEStech publications Inc, PennsylvaniaGoogle Scholar
  8. Korkmaz S (2011) A review of active structural control: Challenges for engineering informatics. Comput Struct 89:2113–2132CrossRefGoogle Scholar
  9. Lilliefors HW (1967) On the Kolmogorov-Smirnov test for normality with mean and variance. J Am Stat Assoc 62:399–402CrossRefGoogle Scholar
  10. Malmberg A, Holst U, Holst J (2005) Forecasting near-surface ocean winds with Kalman filter techniques. Ocean Eng 32:273–291CrossRefGoogle Scholar
  11. Miller LH (1956) Table of percentage points of Kolmogorov statistics. J Am Stat Assoc 51:111–121CrossRefzbMATHGoogle Scholar
  12. Preumont A (2011) Vibration control of active structures: an introduction. Springer, BerlinCrossRefGoogle Scholar
  13. Reschenhofer E (1989) Adaptive test for white noise. Biometrika 76:629–632CrossRefzbMATHMathSciNetGoogle Scholar
  14. Ruotolo R, Surace C (1999) Using SVD to detect damage in structures with different operational conditions. J Sound Vib 226:425–439CrossRefGoogle Scholar
  15. Shane C, Jha R (2011) Proper orthogonal decomposition based algorithm for detecting damage location and severity in composite beams. Mech Sys Signal Process 25:1062–1072CrossRefGoogle Scholar
  16. Stoica P, Moses RL (1997) Introduction to spectral analysis. Printice Hall Inc, Upper Saddle riverzbMATHGoogle Scholar
  17. Tian X, Xie Z, Sun Q (2011) A POD-based ensemble four-dimensional variational assimilation method. Tellus, Ser A: Dyn Meteorol Oceanogr 63:805–816CrossRefGoogle Scholar
  18. Wikle CK, Cressie N (1999) A dimension-reduced approach to space-time Kalman filtering. Biometrika 86:815–829CrossRefzbMATHMathSciNetGoogle Scholar

Copyright information

© The Author(s) 2014

Authors and Affiliations

  1. 1.Road, Housing & Urban Development Research CenterTehranIran

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