Spatially Aware Dictionary-Free Koopman Eigenfunction Identification for Modeling and Control

David Grasev

cs.LGNov 27, 2025

Spatially Aware Dictionary-Free Koopman Eigenfunction Identification for Modeling and Control

Authors: David Grasev

Organizations: 1*Department of Aviation Technology, University of Defence, Brno, Czech Republic.

Abstract

A spatially aware dictionary-free eigenfunction discovery (SADFED) framework is proposed for identification of low-rank Koopman models from data without prescribing a lifting dictionary, kernel, or neural-network eigenfunction architecture. A reference trajectory is selected and used to determine the Koopman modes by regularized least squares (LS). Then, a transformed temporal basis allows the eigenfunction values at all sampled initial conditions to be obtained by a second regularized LS projection. Consequently, only the real and imaginary parts of the eigenvalues remain as the optimization variables. Interpolation of the identified eigenfunction samples reveals their spatial structure, enabling numerical estimation of their gradients. A joint objective combines trajectory reconstruction error with a normalized Koopman partial differential equation (KPDE) residual, promoting spatial consistency with the KPDE over the sampled region and serving as a physics-informed regularizer. The method is evaluated on a system with analytical Koopman eigenfunctions, the FitzHugh-Nagumo system, the van der Pol oscillator, the Duffing system, and a two-spool turbojet engine. The examples demonstrate recovery of known eigenfunctions, sensitivity to reference trajectory and hyperparameters, limit-cycle harmonics and isochrons, discontinuous indicator eigenfunctions and isostables, symmetry exploitation, and construction of state-dependent lifted input dynamics. For the turbojet example, the identified model is further used for state estimation and design of a gain-scheduled tracking linear quadratic Gaussian controller. The results indicate the applicability of SADFED to Koopman spectral identification and control-oriented modeling of nonlinear dynamical systems.

Figures & tables

Figure 1: Schematic diagram of Koopman eigenfunction lifting and evolution.

Figure 2: Flowchart of the SADFED eigenvalue optimization and the objective function.

Figure 3: SADFED in the context of a broader data-driven modeling workflow.

System	Motivation	SADFED validation
Quadratic manifold	Analytical Koopman representation.	Recovery of analytical eigenfunctions – analytical verification.
FitzHugh–Nagumo with inputs	Nontrivial fixed point and nonlinear input dynamics.	Sensitivity analyses, lifted input dynamics, and comparison with EDMDc using a single controlled trajectory.
Van der Pol	Nonlinear limit cycle and asymptotic phase dynamics.	Spectrum discovery, harmonics, refinement beyond the optimization region, and isochrons.
Duffing	Multiple fixed points, basins of attraction, and discontinuous eigenfunction.	Identification of invariant subsets, indicator eigenfunction and isostables, and symmetry exploitation.
Two-spool turbojet	Practical nonlinear system with multiple states and inputs.	Application to engineering systems, lifted input modeling, state estimation, and nonlinear control.

Table 1: Example systems and their role in the validation of SADFED.

Figure 4: Sparse selection of trajectories from the full $21\times 21$ (blue) grid to a $6\times 6$ (black) subgrid using every second odd row and column.

Figure 5: Ground truth (green solid) vs. reconstructed (black dashed) trajectories. The red line highlights the reference trajectory.

Panel of Figure 6 — Figure 6: Exact (colored surfaces) vs. estimated (black circles) eigenfunctions: (a) $\varphi_{1}$ , (b) $\varphi_{2}$ , and (c) $\varphi_{3}$ . Eigenfunctions were normalized such that $\varphi_{i}(1,1)=1$ .

Figure 7: Reconstructed autonomous trajectories of the FitzHugh-Nagumo system: ground truth (green solid), fit (black dashed), reference trajectory (red solid).

Figure 8: The fitted input-driven dynamics for the first 500 seconds of the validation dataset.

Figure 9: Example of oscillatory behavior for the EDMDc solution with 64 IQ RBFs. Ground truth (green solid) vs. EDMDc prediction (black dashed).

Case	Dimension	L2 reg. par.	Scale, $\epsilon$	MAE	Dominant $\lambda$
SADFED	16	$10^{-6}$	–	0.0347	$-0.3\pm 0.436i$
Poly (3rd)	16	$10^{-2}$	–	0.0949	$-0.19\pm 0.38i$
IQ (16)	16	$10^{-1}$	0.75	0.1063	$-0.2\pm 0.39i$
IQ (64)	64	$10^{-5}$	0.75	0.0270	$-0.23\pm 0.41i$
Gaussian	16	$10^{-1}$	1	0.1270	$-0.2\pm 0.39i$
TPS	16	$10^{-1}$	10	0.1621	$-0.2\pm 0.38i$

Table 2: Comparison of SADFED and EDMDc.

Panel of Figure 10 — Figure 10: Reference trajectory selection: (a) relationship between $\mathcal{J}$ and $t_{\mathrm{conv}}$ and (b) the selected trajectories colored according to $\mathcal{J}$ (nonlinear color map for better readability).

Panel of Figure 11 — Figure 11: The effect of (a) $\rho$ and (b) $\gamma$ on $\mathcal{J}_{\mathrm{temp}}$ and $\mathcal{J}_{\mathrm{KPDE}}$ .

Panel of Figure 12 — Figure 12: The effect of (a) $n_{\lambda}$ , (b) temporal subgrid size, (c) full grid size, and (d) interpolation grid size on $\mathcal{J}_{\mathrm{temp}}$ and $\mathcal{J}_{\mathrm{KPDE}}$ .

Figure 13: The effect of increasing the variance of additive Gaussian white noise on

\mathcal{J}_{\mathrm{temp}}

and

\mathcal{J}_{\mathrm{KPDE}}

Panel of Figure 14 — Figure 14: (a) The reconstructed trajectories: ground truth (green solid), reconstruction (black dashed), and reference trajectory (red). (b) Level sets of the $\angle\varphi_{1}$ representing approximate isochrons, with selected trajectories propagated for equal time intervals.

Panel of Figure 15 — Figure 15: Comparison of LHS (blue) and RHS (red) of the KPDE for eigenfunctions $\varphi_{2,...,5}$ of the van der Pol oscillator. The corresponding imaginary parts are (a) $\pm 2.48i$ , (b) $\pm 4.12i$ , (c) $\pm 5.78i$ , and (d) $\pm 7.42i$ .

Figure 16: Reconstructed trajectories: ground truth (green solid) vs. reconstructed (black dashed), and the reference trajectory (red).

Figure 17: Refined zero-eigenvalue eigenfunction $\varphi_{0}$ distinguishing the two basins of attraction of the Duffing system. The transition between the approximately constant values identifies the S-shaped separatrix.

Figure 18: Level sets of $|\varphi_{1}|$ giving the isostables of both stable fixed points of the Duffing system. Two trajectories starting from the same level set approached another level set after the same time. The times were 2.5 s (blue and black dashed lines) and 5 s (cyan and violet solid lines). Red squares denote the isostables obtained by EDMD Williams et al. (2015a) .

Figure 19: Reconstructed autonomous trajectories and the temporal cost of the two-spool GTE. Ground truth (green solid) vs. reconstruction (black dashed), and the reference trajectory (red).

Panel of Figure 20 — Figure 20: Comparison of LHS (blue) and RHS (red) of the KPDE terms for the first 4 eigenfunctions of the two-spool GTE. The corresponding eigenvalues are (a,b) $\lambda_{1,2}=-2.835\pm 1.19i$ and (c,d) $\lambda_{3,4}=-2.85\pm 3.29i$ .

Figure 21: Fitted input-driven validation data of the two-spool GTE with the corresponding mean absolute errors.

Figure 22: Fitted example output variables of the two-spool GTE: (top) thrust, (center) turbine inlet temperature, and (bottom) low-pressure compressor pressure ratio.

Panel of Figure 23 — Figure 23: The closed-loop eigenvalues, $\mathbf{\Lambda_{CL}}$ : (a) distribution in the complex plane, and (b) maximum real parts vs. $N_{1}$ and $N_{2}$ .

Case	PI	LQG
$\mathrm{IAE}_{N1}$	1400 RPM s	1200 RPM s
$\mathrm{IAE}_{N2}$	1756 RPM s	821.5 RPM s
$\mathrm{IAE}_{F}$	23077 F s	13684 F s
$t_{\pm 2\,\%\,\Delta F}^{acc}$	1.55 s	0.79 s
$t_{\pm 2\,\%\,\Delta F}^{dec}$	2.02 s	0.70 s
$F$ abs. OS	0.00 $\%$	2.46 $\%$

Table 3: Comparison of PI and Koopman gain-scheduled tracking LQG controllers. The thrust settling times were evaluated using a

\pm 2\,\%

band relative to the thrust step magnitude.

Panel of Figure 24 — Figure 24: Example of tracking LQG (red) performance compared to a PI controller (blue) for (a) acceleration and (b) deceleration of the two-spool GTE. The black dashed lines correspond to the KF estimates for $N_{1}$ and $N_{2}$ , and to the Koopman prediction of thrust.

Panel of Figure 25 — Figure 25: Comparison of (a) the fuel flow and (b) the nozzle area commanded by both controllers. The LQG controller effectively utilized both inputs compared to the PI controller.

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