David Grasev

David Grasev

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 developed, 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 suitable for multiple control objectives. Using the identified time-varying Koopman model, two controllers are developed: an adaptive Koopman-based model predictive controller (AKMPC) with a disturbance observer and a Koopman-based feedback linearization controller (K-FBLC), which serves as a benchmark. The controllers are evaluated for two control strategies, namely 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 both control strategies achieve comparable performance in steady conditions, the AKMPC exhibits superior robustness compared with the K-FBLC under varying flight conditions due to its ability to compensate for model mismatch. Moreover, the EPR control strategy improves the thrust response. The study highlights the applicability of Koopman-based control and demonstrates the advantages of the AKMPC-based framework for robust turbofan engine control.

David Grasev, Miguel A. Mendez

This work presents the system identification of a variable-pitch propeller (VPP) powertrain, encompassing the full actuation chain from PWM signals to thrust generation, with the aim of developing compact models suitable for real-time digital twinning and control applications. The identification is grounded in experimental data covering both static and dynamic responses of the system. The proposed model takes the form of a Wiener-like architecture, where the PWM inputs are first processed through linear first-order dynamics describing the motor and pitch actuation, and the resulting states are then mapped via a static nonlinear relation to the generated thrust. This structure naturally arises under the assumptions that the electronic actuation operates on a much faster time scale than the mechanical response, and that the contribution of the aerodynamically induced torque is negligible in the tested regime. The resulting parsimonious representation is shown to reproduce the measured dynamics with good accuracy while remaining interpretable and computationally light, thereby providing a practical basis for integration in control-oriented digital twin frameworks.

David Grasev

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.

David Grasev

A new approach to data-driven discovery of Koopman eigenfunctions without a pre-defined set of basis functions is proposed. The approach is based on a reference trajectory, for which the Koopman mode amplitudes are first identified, and the Koopman mode decomposition is transformed to a new basis, which contains fundamental functions of eigenvalues and time. The initial values of the eigenfunctions are obtained by projecting trajectories onto this basis via a regularized least-squares fit. A global optimizer was employed to optimize the eigenvalues. Mapping initial-state values to eigenfunction values reveals their spatial structure, enabling the numerical computation of their gradients. Thus, deviations from the Koopman partial differential equation are penalized, leading to more robust solutions. The approach was successfully tested on several benchmark nonlinear dynamical systems, including the FitzHugh-Nagumo system with inputs, van der Pol and Duffing oscillators, and a 2-spool turbojet engine with control. The study demonstrates that incorporating principal eigenvalues and spatial structure integrity promotion significantly improves the accuracy of Koopman predictors. The approach effectively discovers Koopman spectral components even with sparse state-space sampling and reveals geometric features of the state space, such as invariant partitions. Finally, the numerical approximation of the eigenfunction gradient can be used for input dynamics modeling and control design. The results support the practicality of the approach for use with various dynamical systems.