Applying observer based FDI techniques to detect faults in dynamic and bounded stochastic distributions

This paper presents novel approach on applying the standard observer based fault detection technique to the change detection of the output probability density functions for dynamic stochastic systems. For such systems, the control inputs of the system appear as a set of variables in the probability density functions of the system output, and these variables affect the shape of the probability density function of the system output. Using the B-splines approximation theory, the measured probability density functions of the system output are represented by a set of weights which are functions of the control inputs to the system. This leads to a unique expression of the dynamic characteristics of the output probability density functions for the system. Using this expression, it has been shown that standard observer based fault detection technique can be used to detect any unexpected changes caused by the additive type of fault in the dynamic part of the system. In specific, two observers are constructed for the fault detection purposes, where the first observer is based on the linear weighted integration of the output probability density functions whilst the second observer uses the non-linear residual signal generated from the integration of the output probability density functions. In both cases, the convergence of the observers have been proved under certain conditions when there is no fault in the system. An applicability study to the detection of unexpected changes of particle size (i.e. flocculation size) in paper-making is included to demonstrate the use of the proposed algorithm and desired results have been obtained.