POD-Kalman Observer for Linear Time Invariant Dynamic Systems

Abstract

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.