Bayesian Estimation & its Application in System Fault Diagnosis and Instrumentation Calibration

Estimation of unmeasured states and monitoring of changes in the statistical parameters of the residues/innovations, form an important approach towards model-based fault detection & diagnosis (FDD). This requires the formulation of system dynamics in the state-space framework, wherein the conditional probability density function (pdf) of the state-vector xk, conditioned on the measurement zk, is propagated through a predictor-corrector process to obtain the optimum estimate of the state, while minimizing its error covariance. The Bayesian formulation yields the conditional pdf of the kth state, p{Xk|zk}, which is equated to the likelihood function, p{zk|Xk} & the prior pdf p{Xk|zk- 1} and it is this formulation which governs the Bayesian estimation methodology. Here an overview of the Bayesian estimation problem is presented, which discusses the formulation of the Kalman filter as a Bayesian estimator resulting in a closed form solution, provided the dynamics are linear and the uncertainties are Gaussian. The sequential Monte-Carlo filters (SMC), or particle filters, which addresses both non-linear & non-Gaussian problems, but do not offer a closed form solution, are also introduced. The unscented Kalman filter (UKF), which overcomes some of the disadvantages of the particle filter in terms of being computationally intensive & not guaranteeing convergence for all initial sample sets, are also introduced. The model-based diagnostic problem, by study of the behavior of the estimated states, Xk & the residues [rk = (zk – HXk)], along with the convergence of the error covariance matrix Pk, and by use of multiple-model filtering, GLR (generalized likelihood ratio) methods, sequential probability ratio tests (SPRT) on the residues, etc. are also explained, along with typical applications in process & electrical equipment.