Applied Kalman filter theory

The objective of this study is to examine three problems that arise in experimental mechanics where Kalman filter (KF) theory is used. The first is estimating the steady state KF gain from measurements in the absence of process and measurement noise statistics. In an off-line setting the estimation of noise covariance matrices, and the associated filter gain from measurements is theoretically feasible but lead to an ill-conditioned linear least square problem. In this work the merit of Tikhonov’s regularization is examined in order to improve the poor estimates of the noise covariance matrices and steady state Kalman gain. The second problem is on state estimation using a nominal model that represents the actual system. In this work the errors in the nominal model are approximated by fictitious noise and covariance of the fictitious noise is calculated using stored data on the premise that the norm of discrepancy between correlation functions of the measurements and their estimates from the nominal model is minimum. Additionally, the problem of state estimation using a nominal model in on-line operating conditions is addressed and feasibility of extended KF (EKF) based combined state and parameter estimation method is examined. This method takes the uncertain parameters as part of the state vector and a combined parameter and state estimation problem is solved as a nonlinear estimation using EKF. The last problem is the issue of using the filter as a damage detector when the process and measurement noise statistics vary during the monitoring. The basic idea used to implement the filter as a detector is the fact that the innovation process is white. When the system changes due to damage the innovations are no longer white and correlation can be used to detect it. A difficulty arises, however, when the process and/or measurement noise covariance fluctuate because the filter detects these changes also and it becomes necessary to differentiate what comes from damage and what does not. In this work a modified whiteness test for innovations process is examined. The test uses correlation functions of the innovations evaluated at higher lags in order to increase the relative sensitivity of damage over noise fluctuations.