| In order to facilitate the evaluation of directional residual based fault isolation tests, three performance indexes are studied in this work for linear static and linear dynamic systems. The analysis of the generalized likelihood ratio test (GLRT) has resulted in the probability density function and several invariant properties of the GLRT statistics. The conditional probability of correct fault isolation is also derived to help establish a performance index, which can only be optimized when the fault-to-noise ratio is large. A simplified version of this criterion is used to develop a numerical optimization algorithm. The min-max performance index is defined as the best possible fault isolation performance in the most difficult situation. As a function of the Cramer-Rao bound, this index is shown to be solely determined by the system structure and cannot be optimized under any linear transformation. The Information Matrix based optimization criterion focuses on the robustness and simplicity of the fault isolation test. It yields an algorithm that simultaneously diagonalizes the fault response matrix and the residual covariance matrix. This optimization algorithm can be applied to both static and dynamic systems. Solution existence conditions and the noise rank defect situations are discussed as well. |
