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Koopman Eigenfunction-Based Identification and Optimal Nonlinear Control of Turbojet Engine

David Grasev

TL;DR

This work addresses the challenge of modeling and controlling nonlinear gas turbine engines by learning a compact, interpretable governing equation from data using sparse identification (SINDy) and transforming the autonomous dynamics into Koopman eigenfunctions. By optimizing eigenvalues (via PSO) and computing analytic eigenfunctions from KPDE, the authors construct a Koopman eigenfunction model (KEM) and pair it with a Kalman observer to realize a globally optimal nonlinear controller (Koopman LQGI). Compared to classical PI, LPV-PI, and SINDy-IMC, the KEM-based controller demonstrates superior reference tracking and disturbance rejection across sea-level and varying flight conditions, validating the global nature and spectral-targeting capabilities of the approach. This framework enables mode-targeted optimization and robust performance, with potential extensions to multi-spool engines and model-predictive control in future work.

Abstract

Gas turbine engines are complex and highly nonlinear dynamical systems. Deriving their physics-based models can be challenging because it requires performance characteristics that are not always available, often leading to many simplifying assumptions. This paper discusses the limitations of conventional experimental methods used to derive component-level and locally linear parameter-varying models, and addresses these issues by employing identification techniques based on data collected from standard engine operation under closed-loop control. The rotor dynamics are estimated using the sparse identification of nonlinear dynamics. Subsequently, the autonomous part of the dynamics is mapped into an optimally constructed Koopman eigenfunction space. This process involves eigenvalue optimization using metaheuristic algorithms and temporal projection, followed by gradient-based eigenfunction identification. The resulting Koopman model is validated against an in-house reference component-level model. A globally optimal nonlinear feedback controller and a Kalman estimator are then designed within the eigenfunction space and compared to traditional and gain-scheduled proportional-integral controllers, as well as a proposed internal model control approach. The eigenmode structure enables targeting individual modes during optimization, leading to improved performance tuning. Results demonstrate that the Koopman-based controller surpasses other benchmark controllers in both reference tracking and disturbance rejection under sea-level and varying flight conditions, due to its global nature.

Koopman Eigenfunction-Based Identification and Optimal Nonlinear Control of Turbojet Engine

TL;DR

This work addresses the challenge of modeling and controlling nonlinear gas turbine engines by learning a compact, interpretable governing equation from data using sparse identification (SINDy) and transforming the autonomous dynamics into Koopman eigenfunctions. By optimizing eigenvalues (via PSO) and computing analytic eigenfunctions from KPDE, the authors construct a Koopman eigenfunction model (KEM) and pair it with a Kalman observer to realize a globally optimal nonlinear controller (Koopman LQGI). Compared to classical PI, LPV-PI, and SINDy-IMC, the KEM-based controller demonstrates superior reference tracking and disturbance rejection across sea-level and varying flight conditions, validating the global nature and spectral-targeting capabilities of the approach. This framework enables mode-targeted optimization and robust performance, with potential extensions to multi-spool engines and model-predictive control in future work.

Abstract

Gas turbine engines are complex and highly nonlinear dynamical systems. Deriving their physics-based models can be challenging because it requires performance characteristics that are not always available, often leading to many simplifying assumptions. This paper discusses the limitations of conventional experimental methods used to derive component-level and locally linear parameter-varying models, and addresses these issues by employing identification techniques based on data collected from standard engine operation under closed-loop control. The rotor dynamics are estimated using the sparse identification of nonlinear dynamics. Subsequently, the autonomous part of the dynamics is mapped into an optimally constructed Koopman eigenfunction space. This process involves eigenvalue optimization using metaheuristic algorithms and temporal projection, followed by gradient-based eigenfunction identification. The resulting Koopman model is validated against an in-house reference component-level model. A globally optimal nonlinear feedback controller and a Kalman estimator are then designed within the eigenfunction space and compared to traditional and gain-scheduled proportional-integral controllers, as well as a proposed internal model control approach. The eigenmode structure enables targeting individual modes during optimization, leading to improved performance tuning. Results demonstrate that the Koopman-based controller surpasses other benchmark controllers in both reference tracking and disturbance rejection under sea-level and varying flight conditions, due to its global nature.
Paper Structure (39 sections, 89 equations, 29 figures, 1 table, 2 algorithms)

This paper contains 39 sections, 89 equations, 29 figures, 1 table, 2 algorithms.

Table of Contents

  1. Introduction
  2. Component-level model of a turbojet engine
  3. Models of GTE components
  4. Atmospheric conditions and losses
  5. Compressor
  6. Combustion chamber
  7. Turbine
  8. Nozzle
  9. Shaft dynamics and control system
  10. Transient solver
  11. Koopman operator theory
  12. Proposed approach - SINDy-KEM
  13. Identification of the control-affine governing equation
  14. Optimization of the eigenvalues via temporal projection
  15. Identification of analytical eigenfunctions from KPDE
  16. ...and 24 more sections

Figures (29)

  • Figure 1: Block diagram of the system
  • Figure 2: A workflow diagram of the entire identification and control design process
  • Figure 3: Examples of nonlinear sampling approaches
  • Figure 4: The IMC block diagram
  • Figure 5: The relative spool speed command profile utilized for the control law optimization
  • ...and 24 more figures