A comparison of extended Kalman filter, ultrasound time-of-flight measurement models for heating source localization

Comparisons of six heating source localisation measurement models are conducted where temperature or ultrasonic time of flight readings provide the measurement update to the extended Kalman filter for estimating the location of a high heat flux spot source on a flat plate. For a particular measurement model, one of two processes are used: (1) directly using the measurements as the measurement vector in the extended Kalman filter or (2) indirectly obtaining the distance from the sensor to the heating source based on the measurement and then using the obtained distance as the measurement vector in the extended Kalman filter. For the direct models, the Jacobian required by the extended Kalman filter is obtained numerically using finite differences from the finite element forward conduction solution. For the indirect models, the derivatives of the distances with respect to the state variables are obtained in closed form. Heating source localisation results and convergence behaviour are compared for the six measurement models investigated. The ellipse from ultrasonic pulse one-way time of flight measurement model produces the best results when considering accuracy of converged solution, ability to converge to the correct solution given different initial guesses, and smoothness of convergence behaviour. Additionally, extended Kalman filter, extended information filter, and least squares inverse methods are compared for a parameter identification to quantify the heat flux and convection coefficient on the plate. All three inversion methods produce similar results which is significant as future work will consider only the extended Kalman filter.