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 pulse time of flight is measured between spatially separated transducers on the other side of the plate. This work is an integral part of an effort to develop a system capable of locating the boundary layer transition region on a hypersonic vehicle aeroshell. Results from thermal conduction experiments involving one-way ultrasonic pulse time of flight measurements are presented. Uncertainties in the experiments and sensitivity to heating source location are discussed. Comparisons of heating source localization measurement models are conducted where ultrasonic pulse time of flight readings provide the measurement update to the extended Kalman filter. Two different measurement models are compared: 1) directly using the one-way ultrasonic pulse time of flight as the measurement vector and 2) indirectly obtaining distance from the one-way ultrasonic pulse time of flight and then using these obtained distances as the measurement vector in the extended Kalman filter. For the direct model, the Jacobian required by the extended Kalman filter is obtained numerically using finite differences from the finite element forward conduction solution. For the indirect model, 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 two measurement models. Two areas of sensitivity analyses are presented: 1) heat source location relative to sensor array position, and 2) sensor noise. The direct measurement model produced the best results when considering accuracy of converged solution, ability to converge to the correct solution given different initial guesses, and smoothness of convergence behavior.
Convergence analysis of the extended Kalman filter used in the ultrasonic time-of-flight estimation
