Using Density Estimation in Comparing Input Signals for Gas Turbine Engine Transient Models

Abstract

Gas turbine engines are already complex and nonlinear systems. Nevertheless, gas turbines must become more complex and nonlinear over time to meet the more challenging requirements in the future. Consequently, the different gas turbine engine model types used in the gas turbine design and development processes must become more complex and nonlinear over time. However, simpler models will still be derived from the complex and nonlinear models for applications such as: controller design. Also, the complex and nonlinear models can be run in transient to generate data for the faster data driven transient models. An input signal is used either to test a derived simpler model or generate transient data for a data driven model. As the models get more complex and nonlinear, typical input signals which test the simpler models or create data for the data driven models may not be enough anymore. Conventionally, the input signals are designed in the frequency domain to excite the frequency range of interest. The designed input signals are composed of small perturbations about different operating points and transitions between these different operating points with relatively larger perturbations. How much perturbation is small or large is determined heuristically. However, comprehensive model testing or data generation requires certainty in covering the feasible state space boundary and interior. Concepts, such as boundary and interior do not have equivalents in the frequency domain. Therefore, a supplementary approach considering the feasible state space geometry is necessary for evaluating the designed input signals in terms of feasible state space coverage. This paper proposes using density estimation to evaluate the designed input signals. Among the density estimation methods, this paper used the K-Nearest-Neighbors (kNN) approach. kNN method calculates the smallest volume which encompasses k number of the closest neighboring points about a density estimation point in a given space. For the demonstration in gas turbines, the sea level static (SLS) condition was chosen. The approach can be extended to different flight conditions. First, the feasible state space boundary was determined at SLS by comparing how much of the feasible state space was enveloped by the steps and chops between idle and maximum thrust in different demand types. The signal which enveloped the most of the feasible state space was chosen as the boundary. Naturally, the signal path points of the boundary defining signal were selected as the boundary density estimation points. Then, a Latin hypercube space filling technique with a thousand points was used to populate the interior of the feasible state space with density estimation points. After determining the density estimation points, a small subset of the input signals in the gas turbine literature were evaluated using the kNN method and the selected density estimation points. Each signal’s average closeness to the density estimation points were calculated. Receiving the expected results verified the proposed approach and observations were made to improve the tested signals further.