Proper orthogonal decomposition (POD) is the mathematical basis of a method of constructing low-order mathematical models for the "gray-box" fault- detection algorithm that is a component of a diagnostic system known as beaconbased exception analysis for multimissions (BEAM). POD has been successfully applied in reducing computational complexity by generating simple models that can be used for control and simulation for complex systems such as fluid flows. In the present application to BEAM, POD brings the same benefits to automated diagnosis.
Selected aspects of BEAM have been described in numerous prior NASA Tech Briefs articles. To summarize briefly: BEAM is a method of real-time or offline, automated diagnosis of a complex dynamic system.The gray-box approach makes it possible to utilize incomplete or approximate knowledge of the dynamics of the system that one seeks to diagnose. In the gray-box approach, a deterministic model of the system is used to filter a time series of system sensor data to remove the deterministic components of the time series from further examination. What is left after the filtering operation is a time series of residual quantities that represent the unknown (or at least unmodeled) aspects of the behavior of the system. Stochastic modeling techniques are then applied to the residual time series (see figure). The procedure for detecting abnormal behavior of the system then becomes one of looking for statistical differences between the residual time series and the predictions of the stochastic model.
The need for POD or another method to construct simple approximate models for use in the gray-box approach arises because in a typical case, a detailed deterministic model of the system to be diagnosed may not exist, or, if it exists, may be too complex for real-time computations. One or more simplified deterministic model(s) that describe the system to acceptable degrees of accuracy are therefore desired. The simplified deterministic models can be created from computational simulations of the system and/or empirical data on the operation of the system.
POD modeling requires two steps. The first step is to extract the "mode shapes" or basis functions from experimental data or detailed simulations of the system. This step can involve principal- component analysis, i.e., singularvalue decomposition. In the second step, the basis functions are projected to a low-order or few-dimensional approximate dynamical model by use of the Galerkin method.
The present POD-based method has been verified by creating a low-order dynamical model of a system represented by Burgers' equation, which is a partial differential equation that describes a diverse set of wave phenomena such as flowing gases, flood waters, glaciers, and automobile traffic. It was demonstrated that a loworder (7 POD modes) dynamical model that exhibited high fidelity could be created, even in the presence of noise. In the absence of noise, the model was found to simulate the system with 1 percent error. In the presence of 10-percent uncorrelated Gaussian noise, the model was found to simulate the system with 5 percent error.