On the application of Karhunen–Loève transform to transient dynamic systems

Abstract

The Karhunen–Loève transform (KLT) has become a popular method in various fields of engineering science. Due to its ability to identify the most prominent features in the underlying system dynamics the KLT is a favorable method for such tasks as process monitoring, model order reduction or optimum control. However, it is a well-known fact that the KLT is ‘case sensitive’. That is that changes in the dynamic system behavior can decisively affect the KLT results. As much as this property is desired for monitoring problems, it limits the performance of KLT in model order reduction or optimum control problems, if systems are subject to structural changes. Recent research interest focuses on extending applications of KLT to systems with transient dynamic behavior or changing boundary conditions. Approaches have been published that circumvent the limitations of KLT by either assuming reasonable comparability of system dynamics or by measuring the representative performance of KLT-bases a posteriori. However, such methods require additional simulations of the full size system and thus jeopardize the idea of model order reduction. In this paper, we introduce a novel a priori measure to evaluate the performance of the current KLT-basis. This procedure can be of great help in either monitoring or adaptive control of systems that show intermittent transient and (quasi-)stationary dynamic behavior. This a priori measure prepares the path for adaptive model order reduction schemes. Moreover, it can be used to measure the stationarity of multidimensional dynamic processes.