Koopman-Based Nonlinear Identification and Model Predictive Control of a Turbofan Engine

The gist The proposed approach uses an adaptive Koopman-based model predictive controller (AKMPC) to achieve robust multivariable control of a two-spool turbofan engine by identifying a time-varying low-order Koopman system using MH-EDMD, which enables the handling of nonlinear dynamics and constraints across varying flight conditions.

How it works

The core methodology involves developing an in-house physics-based component-level model to generate training data for the identification process The identification procedure is tailored to ensure an accurate prediction of both spool speeds and EPR for control purposes A meta-heuristic extended dynamic mode decomposition (MH-EDMD) method was employed to obtain the Koopman system, which incorporates L2 regularization and optimizes the nonlinear parameters of observable functions using a metaheuristic method, such as a genetic algorithm or particle swarm optimization The objective function minimizes prediction error across the entire training and validation time series, computed by numerically integrating the Koopman system, rather than relying solely on the one-step prediction error This accumulated error is difficult to address using gradient-based optimization; hence, the metaheuristic approach The final optimization problem is thus min p 1 Nt X Nt k=1 s.t. Re(Λ)min ≤ Re(Λ) ≤ Re(Λ)max, rank [Bc ABc · · · AnΨ−1Bc] = nΨ, rank [C⊺ (CA) A −1· · · (CAnΨ−1) A −1] = nΨ, inf D(p) ≥ dmin, where p ∈ P is a vector of parameters of the lifting functions and D is an upper-triangular matrix, measuring the Euclidean distance of observables in the parameter space P, with elements dij = ∥pi − pj∥2, i = 1, …, nΨ−3, j = i+ 1, …, nΨ −2, and the remaining elements replaced by large numbers

Model Development and Identification

A physics-based component-level model was developed to generate training data and validate the controllers The model accounts for atmospheric and flight conditions using the International Standard Atmosphere and adiabatic relations between the inlet velocity and the static and total pressures and temperatures Component characteristics of compressors and turbines were extracted from the mixed-flow turbofan example from the GasTurb 15 simulation software [54] and scaled to the design point of the target GTE The training dataset was obtained via closed-loop simulation of the dynamics, including quasi-amplitudemodulated pseudo-random binary sequences and sinusoidal changes in the setpoint The corrected parameters were utilized [14], and the data were normalized as follows The IQ observables were selected for the Koopman model, as they achieved an accurate prediction of both spool speeds and EPR

Control Strategies

Three control approaches were developed using the identified Koopman model:

1. An adaptive Koopman MPC (AKMPC) with a disturbance observer (DO) The AKMPC evaluates the time-varying Koopman model at the current time step k and holds it constant over the prediction horizon It includes a Kalman filter-based DO to compensate for mismatch caused by variations in flight conditions and modeling uncertainty The AKMPC can be interpreted as a computationally efficient alternative between the offset-free KMPC with a fixed model and online adaptive Koopman identification

2. A Koopman-based feedback linearization controller (K-FBLC) This controller is benchmarked against the AKMPC and its integrator-augmented version (K-FBLC-I)

3. A Koopman-based feedback linearization controller with integrators (K-FBLC-I) The integrators in K-FBLC-I reduced steady-state offsets, but they led to occasional overshoots and retained the structural dependence on the selected output relative degree

Performance Evaluation

The results demonstrate that the proposed identification approach enables accurate predictions of both spool speeds and EPR, allowing the Koopman model to be reused flexibly across different control formulations While all strategies achieve comparable performance in sea-level conditions, the AKMPC demonstrates improved performance under varying flight conditions due to its ability to capture nonlinear dynamics, handle constraints, and compensate for model mismatch The EPR control strategy improves the thrust response The AKMPC provides the best tracking performance in terms of IAE, overshoots, and noise in the control inputs under sea-level conditions Under varying flight conditions, the AKMPC again exhibits the best performance in terms of IAE and adherence to the setpoint The computational-time analysis indicated that the online AKMPC update, including Koopman prediction and QP, is computationally tractable in the tested MATLAB implementation

Conclusion

The MH-EDMD was compared with a classical EDMDc approach, showing the benefit of using state-dependent input dynamics and optimizing the nonlinear observable parameters The proposed framework further demonstrates that traditional nonlinear GTE output limiters can be handled directly through linear inequality constraints thanks to the Koopman framework The simulation results showed that the AKMPC provides the most consistent performance across both sea-level and varying flight conditions The present study focuses on practical closed-loop behavior over the investigated operating envelope rather than on a theorem-based stability analysis The computational tractability of the online AKMPC update suggests its suitability for real-time application in turbofan engine control

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Koopman-Based Nonlinear Identification and Model Predictive Control of a Turbofan Engine David Grasev1∗ Department of Aviation Technology, University of Defence, Brno, Czech Republic. Corresponding author(s). E-mail(s): david.grasev@unob.

1. Bold header: Adaptive Koopman MPC (AKMPC) implementation

This improved system can handle constraints, and compensate for model mismatch under varying flight conditions by using a time-varying low-order Koopman system identified offline with MH-EDMD, allowing it to preserve the quadratic-programming formulation while being robust to operating condition variations.

2. Bold header: Multi-criteria objective function for identification

The identification process is enhanced by a modified multi-criteria objective function that penalizes prediction error accumulated over the training time series and explicitly includes penalties for lightly unstable eigenvalues, ensuring both prediction accuracy and system stability during model construction.

3. Bold header: Explicit nonlinear output constraint handling

The AKMPC framework can now handle nonlinear GTE output limiters directly through linear inequality constraints thanks to the Koopman framework, allowing the optimizer to automatically restrict the control action according to the active constraints for outputs like rotor acceleration rates and TIT.

This paper investigates Koopman operator-based approaches for multivariable control of a two-spool turbofan engine. A physics-based component-level model is developed to generate training data and validate the controllers. A meta-heuristic extended dynamic mode decomposition is adapted, with a cost function designed to accurately capture both spool-speed dynamics and the engine pressure ratio (EPR), enabling the construction of a single Koopman model that can be reused across multiple control strategies. Using the identified time-varying Koopman model, an adaptive Koopman-based model predictive controller (AKMPC) with a disturbance observer is developed and compared with a Koopman-based feedback linearization controller (K-FBLC) and its integrator-augmented version (K-FBLC-I). The Koopman representation further enables nonlinear GTE output limiters, such as rotor-acceleration and turbine-inlet-temperature limits, to be expressed as linear constraints in the AKMPC. The controllers are evaluated for two control configurations of spool speeds and EPR, under both sea-level and varying flight conditions. The results demonstrate that the proposed identification approach enables accurate predictions of both spool speeds and EPR, allowing the Koopman model to be reused flexibly across different control formulations. While all strategies achieve comparable performance in sea-level conditions, the AKMPC demonstrates improved performance under varying flight conditions due to its ability to capture nonlinear dynamics, handle constraints, and compensate for model mismatch. Moreover, the EPR control strategy improves the thrust response. The study highlights the applicability of Koopman-based control and the advantages of the AKMPC framework for robust turbofan engine control.