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

David Grasev

cs.LGApr 2, 2026

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

Authors: David Grasev

Organizations: Department of Aviation Technology, University of Defence, Brno, Czech Republic.

Abstract

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.

Figures & tables

Symbol	Dimension / unit	Description
$N_{1},N_{2}$	RPM	Low- and high-pressure spool speeds
$W_{\mathrm{f}}$	kg/s	Fuel flow
$A_{\mathrm{n}}$	m 2	Nozzle area
$\Pi_{\mathrm{EPR}}$	–	Engine pressure ratio
$F$	N	Net thrust
$\mathbf{x}$	$\mathbb{R}^{2}$	State vector, $\mathbf{x}=[N_{1},N_{2}]^{\top}$

Table 1: Summary of the main notation used in the manuscript.

Figure 1: Turbofan GTE diagram with the main components considered in this paper.

Figure 2: Schematic diagram of Koopman eigenfunction lifting and evolution.

Figure 3: The AKMPC diagram showing the interaction between the Koopman system and MPC optimization.

Panel of Figure 4 — Figure 4: (a) Thrust and (b) fuel flow steady-state validation against the GasTurb 15 simulation software in $H=0$ km, $M_{0}=0$ and $H=10$ km, $M_{0}=0.8$ .

Steady-state	$H=0$ km, $M_{0}=0$	$H=10$ km, $M_{0}=0.8$
$\mathrm{MAE_{F}}$	1010.4 N	403.75 N
$\mathrm{MAPE_{F}}$	4.49 $\%$	2.37 $\%$
$\mathrm{MAE_{WF}}$	0.0183 kg/s	0.0031 kg/s
$\mathrm{MAPE_{WF}}$	2.86 $\%$	1.42 $\%$

Table 2: MAE and MAPE values for the steady-state validation.

Figure 5: Transient performance validation against the GasTurb 15 simulation software for the sea-level conditions. The fuel flow is rescaled for better visualization.

Transient	$H=0$ km, $M_{0}=0$
$\mathrm{MAE_{N1}}$	83.7 RPM
$\mathrm{MAPE_{N1}}$	0.744 $\%$
$\mathrm{MAE_{N2}}$	51.2 RPM
$\mathrm{MAPE_{N2}}$	0.274 $\%$

Table 3: MAE and MAPE values for the transient validation.

Figure 6: Dataset used for identification of the 2-spool GTE (without noise for clarity).

Function	LF	GAU	IQ
$\mathrm{MAE_{N1}}$ (RPM)	31.92	19.14	20.3
$\mathrm{MAE_{N2}}$ (RPM)	30.58	17.05	15.17
$\mathrm{MAE_{EPR}}$	0.0182	0.0116	0.0114
$\mathrm{MAPE_{N1}}$ $\%$	1.52	0.656	0.68
$\mathrm{MAPE_{N2}}$ $\%$	1.76	0.55	0.6
$\mathrm{MAPE_{EPR}}$ $\%$	2.2	1.21	1.12

Table 4: Comparison of the observable functions for MH-EDMD.

Figure 7: Prediction results of the identified Koopman system after numerical integration.

Metric	EDMDc, GAU	MH-EDMD, IQ
$\mathrm{MAE}_{N_{1}}$ (RPM)	372.3	20.3
$\mathrm{MAE}_{N_{2}}$ (RPM)	433.7	15.17
$\mathrm{MAE}_{\mathrm{EPR}}$	0.0523	0.0114

Table 5: Comparison of the MH-EDMD with the EDMDc with Gaussian RBF.

Figure 8: Spool speed response in sea-level conditions.

Panel of Figure 9 — Figure 9: (a) Fuel flow and (b) nozzle area comparison for $N_{1}-N_{2}$ in sea-level conditions.

Controller	$\mathrm{IAE}\ N_{1}$	$\mathrm{IAE}\ N_{2}$
Unit	$\mathrm{RPM\,s}$	$\mathrm{RPM\,s}$
PI	17381	14600
K-FBLC	16763	13526
K-FBLC-I	15740	12660
AKMPC	14869	11744

Table 6: The IAE criterion values for the controllers in sea-level conditions.

Figure 10: Estimated disturbances for the

N_{1}-N_{2}

case in sea-level conditions.

Panel of Figure 11 — Figure 11: (a) Altitude and (b) Mach number of flight for the evaluation.

Figure 12: Spool speed response in varying flight conditions.

Figure 13: Comparison of fuel flow with limits in varying flight conditions.

Controller	$\mathrm{IAE}\ N_{1}$	$\mathrm{IAE}\ N_{2}$
Unit	$\mathrm{RPM\,s}$	$\mathrm{RPM\,s}$
PI	24632	22621
K-FBLC	27971	21807
K-FBLC-I	25760	20710
AKMPC	23835	19548

Table 7: The IAE criterion values for the controllers in varying flight conditions.

Panel of Figure 14 — Figure 14: (a) Spool speeds, (b) fuel flow, and (c) nozzle area comparison for AKMPC with and without the DO in varying flight conditions.

Figure 15: Estimated disturbances for the

N_{1}-N_{2}

case in varying flight conditions.

Item	Specification
CPU	AMD Ryzen 7 8845HS 3.8 GHz
RAM	32 GB DDR5
Operating System	Microsoft Windows 11 10.0.26200
MATLAB Version	R2025b Update 4

Table 8: Hardware and software specifications.

Case	Mean (ms)	STD (ms)	Median (ms)
SL	0.94	0.27	0.86
VFCs	0.98	0.30	0.88

Table 9: The per-step computational times for the AKMPC and Koopman predictor.

Panel of Figure 16 — Figure 16: Effect of the TIT and $\dot{N}_{1}$ limiters on the limited outputs - (a) the $\dot{N}_{1}$ and (b) the TIT. The limiters successfully prevented the system from exceeding the set operational limits.

Figure 17: The relation between the corrected thrust and engine pressure ratio across varying Mach numbers.

Figure 18: EPR and

N_{1}

response in sea-level conditions. Black circles mark K-FBLC and K-FBLC-I overshoots.

Panel of Figure 19 — Figure 19: (a) Fuel flow and (b) nozzle area comparison for $\Pi_{\mathrm{EPR}}-N_{1}$ in sea-level conditions.

Controller	$\mathrm{IAE}\ N_{1}$	$\mathrm{IAE}\ \Pi_{\mathrm{EPR}}$
Unit	$\mathrm{RPM\,s}$	(-)
K-FBLC	21380	3.88
K-FBLC-I	21400	3.785
AKMPC	20922	3.738

Table 10: The IAE criterion values for the controllers in sea-level conditions.

Figure 20: Estimated disturbances for the

\Pi_{\mathrm{EPR}}-N_{1}

case in sea-level conditions.

Figure 21: Thrust comparison for the two control strategies. Black circles mark regions with excessive oscillations for the

N_{1}-N_{2}

controller.

Figure 22: EPR and

N_{1}

response in varying flight conditions. Black circles mark the regions with offset and overshoots.

Panel of Figure 23 — Figure 23: (a) Fuel flow and (b) nozzle area comparison for $\Pi_{\mathrm{EPR}}-N_{1}$ in varying flight conditions.

Controller	$\mathrm{IAE}\ N_{1}$	$\mathrm{IAE}\ \Pi_{\mathrm{EPR}}$
Unit	$\mathrm{RPM\,s}$	(-)
K-FBLC	25366	4.703
K-FBLC-I	24500	4.55
AKMPC	23807	4.331

Table 11: The IAE criterion values for the controllers in varying flight conditions.

Figure 24: Estimated disturbances for the

\Pi_{\mathrm{EPR}}-N_{1}

case in varying flight conditions.

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The Koopman operator has gained considerable attention due to its ability to provide a global linear representation of highly complex dynamical systems. The operator describes nonlinear dynamics in a linear way through the lens of real- or complex-valued observable functions. Data-driven techniques, like extended dynamic mode decomposition (EDMD), kernel EDMD, and machine-learning methods, can be used to generate finite-dimensional approximations accompanied by finite-data error bounds. In this tutorial paper, we provide a concise introduction into Koopman operator theory and its use in systems and control. A particular focus is put on data-driven surrogate models, their extension to systems with inputs, and controller design using Koopman operator theory. Moreover, we demonstrate the key techniques, i.e., EDMD and Koopman MPC. To this end, we provide simulation studies including source code on GitHub to enable the interested reader to experience the Koopman operator in systems and control step by step.

Abstract

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