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

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.

💡 Research Summary

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The paper presents a novel data‑driven framework for identifying and optimally controlling a turbojet engine, whose dynamics are highly nonlinear and coupled. Traditional physics‑based component‑level models (CLMs) rely on performance maps that are often unavailable or costly to obtain, while linear parameter‑varying (LPV) models provide only locally linear approximations. To overcome these limitations, the authors exploit data collected during normal closed‑loop engine operation, avoiding any modification of the existing control system.

The methodology proceeds in three main stages. First, the Sparse Identification of Nonlinear Dynamics (SINDy) algorithm is applied to the measured input‑output data. By constructing a library of candidate nonlinear functions and solving an L1‑regularized least‑squares problem (LASSO), SINDy yields a parsimonious set of governing equations that accurately capture rotor dynamics. Importantly, the autonomous part of the identified model—i.e., the dynamics that evolve without external inputs—is isolated for subsequent processing.

Second, the autonomous dynamics are lifted into a Koopman eigenfunction space. The Koopman operator provides a linear representation of nonlinear dynamics in an infinite‑dimensional space; practical implementation requires a finite set of observables. Rather than relying on trial‑and‑error or generic dictionaries, the authors adopt a spectral approach that directly seeks Koopman eigenfunctions. Eigenvalue candidates are optimized globally using a genetic algorithm (GA). The GA evaluates the projection of autonomous trajectories onto a span of exponential functions e^{λt} and minimizes the projection error, thereby identifying a set of eigenvalues that best describe the observed decay/growth rates. Once eigenvalues are fixed, a gradient‑based optimization refines the corresponding eigenfunctions by minimizing the residual of the Koopman eigenvalue equation. This two‑step procedure yields a compact, low‑order Koopman model (KEM) that faithfully reproduces the original nonlinear behavior while remaining linear in the eigenfunction coordinates.

Third, control and estimation are performed entirely in the eigenfunction space. A globally optimal linear‑quadratic‑Gaussian (LQG) controller with integral action (K‑LQGI) is synthesized. The design follows the classic steps: (i) define state‑ and input‑weighting matrices in the eigenfunction coordinates, (ii) solve the continuous‑time algebraic Riccati equation to obtain the optimal feedback gain, (iii) augment the state with an integral of the tracking error to guarantee zero steady‑state error, and (iv) construct a Kalman filter that estimates the eigenfunction states from measured outputs. Because the Koopman representation is linear, the LQG synthesis is straightforward, and the integral term ensures robustness to constant disturbances.

The authors validate the approach on an in‑house MATLAB/Simulink component‑level model of a generic single‑spool turbojet, which has been cross‑validated against the commercial GasTurb simulation suite. Four controllers are compared: (a) a conventional nonlinear PI controller, (b) a gain‑scheduled LPV‑PI controller, (c) an internal‑model‑control (IMC) scheme based on the SINDy model, and (d) the proposed K‑LQGI. Performance metrics include reference‑tracking error, disturbance rejection, settling time, and fuel consumption under sea‑level, high‑altitude, and rapid‑disturbance scenarios. The Koopman‑based controller consistently outperforms the benchmarks, achieving 15–30 % reductions in tracking error and settling time, and demonstrating superior robustness to external perturbations. The eigenmode structure of the Koopman model enables targeted tuning of individual modes (e.g., compressor‑torque oscillations), which is not feasible with conventional LPV or PI designs.

Key contributions of the paper are:

1. Integration of SINDy and Koopman eigenfunction methods to obtain an interpretable, low‑order, globally linear representation of a highly nonlinear engine.
2. Global optimization of eigenvalues via a genetic algorithm combined with a gradient‑based eigenfunction identification, addressing the longstanding challenge of selecting appropriate observables.
3. Design of a globally optimal LQG controller with integral action directly in the eigenfunction space, yielding superior tracking and disturbance rejection without the computational burden of high‑order LPV or nonlinear MPC schemes.
4. Demonstration of practical feasibility using only closed‑loop operational data, eliminating the need for dedicated test‑stand experiments or extensive map generation.

In conclusion, the study shows that a data‑driven Koopman eigenfunction framework, seeded by sparse nonlinear identification, can deliver high‑fidelity models and globally optimal controllers for turbojet engines. This approach bridges the gap between black‑box machine‑learning techniques and physics‑based modeling, offering a scalable solution for modern aerospace propulsion systems where sensor limitations, computational constraints, and the need for robust performance across a wide flight envelope are critical. Future work is suggested on real‑flight data validation, online adaptation of eigenfunctions, and extension to multi‑engine or multi‑mode propulsion architectures.