Kalman smoother for detection of heat sources defects

This work is devoted to an application of the Kalman smoother method for the detection of failure times for a system of linear parabolic partial differential equation. For complex processes whose evolution is described by partial differential equations, the detection of failures is often difficult. In thermal systems, for example, when one or more heating sources fail, the sensors placed at a distance perceive the effect with delay. The Kalman smoother is then applied when the classical Kalman filter is used for the estimation of intensity of our sources. From these estimation, it is then necessary to determine the moment of failure and the possible restart of the heating. In the following, the whole problematic and the method of resolution is presented.