A Comparison of Extended Kalman Filter Approaches Using Non-linear Temperature and Ultrasound Time-of-flight Measurement Models for Heating Source Localization of a Transient Heat Transfer Problem

ABSTRACT State estimation procedures using the extended Kalman filter are investigated for a transient heat transfer problem in which a heat source is applied on one side of a thin plate and ultrasonic time of flight or temperature is measured on the other side of the plate. A forward conduction solution is developed using finite element methods and verified using closed-form analytic solutions and validated using thermal conduction experiments. Parameter identification of experiment heating source magnitude and convection coefficient is conducted with least squares, extended Kalman filter, and extended information filter. Comparisons of six heating source localization measurement models are conducted where temperature or ultrasonic time of flight readings provide the measurement update to the extended Kalman filter. For each quantity of interest being investigated, two different measurement models are implemented: 1) directly using the quantity of interest at the sensor locations as the measurement vector and 2) indirectly obtaining distance from the quantity of interest and then using these obtained distances 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 localization results and convergence behavior are compared for the six measurement models investigated.