On optimal alarm filter design

Accuracy and efficiency of alarm systems are of paramount importance in safe operations of industrial processes. Accuracy is measured by false and missed alarm rates; while efficiency relates to the detection delay and complexity of the technique used. Moving average filters are often employed in industry for improved alarm accuracy at the expense of some detection delay. Can one do better than moving average filters? The following problem is studied in this paper: Given statistic distributions of both normal and abnormal conditions, and relatively fixed filter complexity, design an optimal alarm filter for best alarm accuracy, minimizing a weighted sum of false and missed alarm rates (probabilities). It turns out that the general form of such optimal alarm filters is the so called log-likelihood ratio (LLR) filters, which can be highly nonlinear and difficult to implement in practice. With fixed filter structures (first or second order), design of optimal linear alarm filters and optimal quadratic alarm filters is studied, and numerical optimization based procedures are proposed. Some key elements in the optimal design include use of characteristic functions from probability theory to facilitate computation of the objective function, and a DE algorithm for optimization (the optimization problem is non-convex and with small gradients in some area). The validity of the proposed methods is illustrated by several design examples in which the optimal filters in the general, linear and quadratic forms are computed, and their relative performance in alarm accuracy is fully discussed.