Output-Only System Identification via Generalized Polynomial Chaos
Motivation
The identification of quantities of interest based on sparse observations of a given variable is a challenging problem across many engineering fields. In thermoacoustics, pressure data recorded at a few discrete locations within the system has been used to estimate key parameters of a one-mode Galerkin expansion model through an output-only system identification approach. This strategy is implemented in two main steps:
- Calculation of the Kramers-Moyal (KM) coefficients using time-series data.
- Evaluation of the theoretical KM coefficients, derived by applying a slow-time averaging method to the one-mode Galerkin expansion.
Finally, an optimization procedure is performed to determine the most likely values of the model parameters by minimizing the discrepancy between the theoretical and experimental KM coefficients. This approach has been demonstrated successfully in various studies, including investigations of different types of combustion chambers. The model parameters obtained through this optimization can subsequently be utilized to develop control strategies aimed at mitigating combustion instabilities.
Although successful, the output-only system identification approach described above is not predictive. Changes in operating conditions necessitate a new optimization process because the model is overly simplistic: it typically requires only three parameters, which are not directly tied to distinct physical processes.To address this challenge, more sophisticated and powerful models are essential.
Objectives and Strategy
Our group is currently developing output-only system identification strategies that allow for more complex models than the simple one-mode Galerkin expansion. Instead of relying on the Fokker-Planck equation and its associated Kramers-Moyal (KM) coefficients, our work leverages the Generalized Polynomial Chaos (gPC) expansion. Generalized Polynomial Chaos is an uncertainty quantification technique that models the propagation of probability density functions (PDFs) through systems characterized by partial differential equations (PDEs). We have developed an 'intrusive' gPC expansion of the linearized Euler equations (with source terms), reformulating the entire system in a state-space representation.
In a second step, we optimize the parameters of the expanded model so that the analytical expressions of the gPC coefficients (derived via the intrusive gPC approach) align with their experimental counterparts, which are obtained from experimental data using a variant of the 'non-intrusive' gPC method.
Because the model is general, once the parameters are identified, it becomes possible to vary specific parameters—directly linked to distinct physical processes—to predict the thermoacoustic behavior of the system under different operating conditions.