Deep Bilinear Koopman Model for Real-Time Vehicle Control in Frenet Frame

Farhang Motallebi Araghi; Mohammad Abtahi; Navid Mojahed; Shima Nazari

REVIEW 2 major objections 5 minor 42 references

Deep Bilinear Koopman Model for Real-Time Vehicle Control in Frenet Frame

T0 review · 2 major / 5 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read A learned bilinear Koopman model with cumulative error regulation enables a convex MPC that tracks paths up to 97 percent more accurately than EDMD and LTI baselines and solves each control step in 0.58 ms.

desk verdict Useful incremental engineering paper on deep bilinear Koopman MPC with a nice integral-error module; the closed-loop evidence is thin and the MPC derivation drops the curvature bilinear term. read the letter →

arxiv 2507.12578 v1 pith:WUPHU63Y submitted 2025-07-16 eess.SY cs.LGcs.ROcs.SY

Mohammad Abtahi , Farhang Motallebi Araghi , Navid Mojahed , Shima Nazari This is my paper · ORCID

classification eess.SYcs.LGcs.ROcs.SY

keywords KoopmanoperatorbilinearmodelpredictivecontrolautonomousvehiclesFrenetframedeeplearningcumulativeerrorregulatorhardware-in-the-loop

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

Koopman operator theory promises a way to treat nonlinear vehicle dynamics as linear dynamics in a lifted space, but standard lifts are hand-picked and tend to miss input–state coupling. This paper tries to remove that bottleneck by learning the lift and the dynamics together: a deep encoder produces observables, and a final Koopman layer learns a bilinear transition $Z_{k+1}=A Z_k+B U_k+\sum_i H_i(U^i_k Z_k)$ that explicitly couples each control channel to the lifted state. Trained with a multi-step prediction loss, this model is embedded in a convex MPC by freezing the lifted state in the bilinear term over the prediction horizon, and augmented with a cumulative error regulator that propagates and penalizes accumulated tracking error. In hardware-in-the-loop tests against the paper's prior linear deep Koopman model, an Extended Dynamic Mode Decomposition (EDMD) Koopman model, and an LTI model, the controller reduces average tracking RMSE by 33–68 percent and keeps the per-step solve time at 0.58 ms, well inside the 25 ms control period. If the result holds, it shows that learned bilinear Koopman models are a practical route to accurate, convex, real-time vehicle control.

What carries the argument

The central object is the learned bilinear Koopman realization: an encoder $\Phi_\theta$ maps the six vehicle states to learnable observables, the lifted state is $Z_k=[x_k;\Phi_\theta(x_k)]$, and the transition is $Z_{k+1}=A Z_k+B U_k+\sum_i H_i(U^i_k Z_k)$, where $U_k$ includes steering wheel angle, throttle/brake command, and road curvature. This structure makes input–state coupling a tunable part of the dynamics rather than a modeling afterthought. Around it, the Cumulative Error Regulator appends $e_{k+1}=e_k+(y_k-y_k^{\mathrm{ref}})$ to the lifted state, and the MPC preserves convexity by replacing the lifted state inside the bilinear terms with its initial value $Z_t$ over the prediction horizon, producing the linearized update of Eq. 40.

What would settle it

Re-run the same CER-MDBK-MPC against the linear MDK baseline on a more aggressive maneuver, such as a higher-speed double lane change or a slalom with curvature near the training limits, and measure the deviation of the lifted state from its horizon-start value; if the bilinear controller's tracking advantage over the linear model shrinks or vanishes as that deviation grows, the convex linearization assumption is the limiting factor.

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Extended reading notes

Core claim

The central claim is that modeling the lifted vehicle dynamics as $Z_{k+1}=A Z_k+B U_k+\sum_i H_i(U^i_k Z_k)$, where $Z_k$ concatenates the original Frenet-frame states with an encoder-learned observable vector and the matrices $H_i$ are learned end-to-end, captures input–state interactions that linear Koopman models miss. The paper supports this by showing that the learned $H_i$ matrices have sparsity patterns aligned with the nonlinear bicycle-model equations, by reporting lower open-loop prediction RMSE than the linear MDK-Net and EDMDK baselines, and by closing the loop with a convex MPC that freezes the lifted state in the bilinear terms to its horizon-start value and augments the state with a cumulative tracking-error integrator. On a double-lane-change maneuver with a high-fidelity vehicle simulator as the plant, the CER-MDBK-MPC cuts tracking error of path progress, lateral deviation, and heading error by 5–97 percent relative to EDMDK-MPC and by 35–77 percent relative to the linear MDK baseline, with average reductions of 33 percent, 38 percent, and 68 percent against the linear Koopman, EDMD, and LTI controllers.

Load-bearing premise

The load-bearing premise is that holding the lifted state fixed at its starting value for the 20-step MPC horizon keeps the linearized bilinear model close to the true learned dynamics; if the vehicle maneuvers aggressively enough for the lifted state to leave that neighborhood, the model the controller optimizes no longer represents the bilinear dynamics, and the accuracy advantage would weaken—and the encoder is only as reliable as the simulation training distribution it was learned from.

Editorial extensions

If this is right

The convex freezing trick turns the bilinear terms into a state-scheduled input matrix, so the same MPC formulation can be reused for any learned bilinear Koopman model without losing real-time solvability.
Open-loop, MDBK-Net reports 2-second prediction RMSE for lateral velocity and yaw rate reduced by 43 percent and 36 percent relative to MDK-Net, and by 66 percent and 50 percent relative to EDMDK, with the largest gains in states where input–state coupling matters.
Closed-loop, CER-MDBK-MPC reduces path progress, lateral deviation, and heading-error RMSE by 5 percent, 97 percent, and 88 percent versus EDMDK-MPC, and by 35 percent, 77 percent, and 59 percent versus the linear MDK baseline.
At 0.58 ms average solve time per step, the controller leaves roughly 24 ms of the sampling interval for sensing, communication, and safety checks, directly supporting the paper's real-time implementation claim.
The cumulative error regulator keeps pushing accumulated tracking errors toward zero over the whole horizon, which is why the largest visible improvements appear during and after the aggressive second lane change rather than only at steady state.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

A testable interpretation of the CER module is that it acts as output integral action on the tracking errors: at steady state the cumulative error must remain constant for the penalty to stay bounded, which cancels constant model-mismatch bias; the paper does not draw this equivalence explicitly.
Because the bilinear term is frozen at the horizon-start lifted state, the resulting controller is effectively a gain-scheduled LTV MPC whose input matrix changes with the current lifted state; comparing against an explicit LTV MPC built from Jacobian linearization would isolate what the learned bilinear schedule adds.
The same encoder-plus-bilinear-Koopman-layer-plus-error-integrator architecture should transfer to other actuated nonlinear systems such as manipulators or small aircraft, but that transfer is untested and the model's validity is bounded by the simulation distribution it was trained on.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Short version: a serious, incremental engineering paper. The genuinely new part is the deep bilinear Koopman model combined with a cumulative error regulator, applied to Frenet-frame vehicle control with driver inputs. The open-loop prediction results over 1,425 test trajectories are credible and show MDBK-Net beating the linear MDK and EDMDK baselines, especially in lateral states. The HIL setup on dSPACE with CarSim RT is real work, and the reported 0.58 ms per-step compute time is a solid real-time result.

The soft spots are real but not fatal. The biggest is in the MPC derivation. Equation (40) freezes the lifted state to its initial value Z_t for the whole horizon, which is the known quasi-LPV trick from Ref. [37]. But it also drops the curvature bilinear term H3(Z_t * kappa) entirely. That is not an approximation; it is an omission. Since curvature is an exogenous input and Z_t is known, H3(Z_t * kappa) is a known affine term that could be added for free. The result is that the controller's internal model is not even the local linearization of the learned bilinear model, and the paper's claim that it captures curvature nonlinearities in the control loop is unsupported. The improvement over CER-MDK still demonstrates the value of the H1 and H2 state-dependent input scheduling, so the bilinear benefit is not entirely an artifact, but the authors should include H3 or justify its absence.

Second, the closed-loop evaluation is a single double-lane-change maneuver at constant curvature, with no error bars or significance tests. 'Significant reductions' is too strong for a single run. The open-loop numbers are better supported because they average over many trajectories, but they also lack confidence intervals.

Third, no code or data is released. For a data-driven method, that limits reproducibility, though it is not a correctness flaw.

The paper is clearly written and the authors engage honestly with prior literature. The self-citation to MDK-Net is fair because it is the direct predecessor. The bilinear matrices in Fig. 5 align with known physics, which is a nice check.

Bottom line: this is a useful incremental contribution for the Koopman-MPC community. It deserves serious peer review, not a desk reject. A referee should ask for the H3 term to be addressed, and for at least one more maneuver plus statistical reporting before acceptance.

Referee Report

2 major / 5 minor

Summary. The paper proposes a Multi-Step Deep Bilinear Koopman Network (MDBK-Net) that jointly learns lifting functions and bilinear Koopman dynamics for vehicle state evolution in the Frenet frame, and integrates the learned model with a Cumulative Error Regulator (CER) into a model predictive controller (CER-MDBK-MPC). The model is evaluated in open-loop prediction against EDMDK and the authors' previous linear MDK-Net on 1,425 CarSim test trajectories, and in closed-loop hardware-in-the-loop experiments on a dSPACE SCALEXIO system with a CarSim RT vehicle model. The paper reports lower prediction RMSE than the baselines and closed-loop tracking-error reductions of 33% versus CER-MDK-MPC, 38% versus EDMDK-MPC, and 68% versus LTI-MPC, with an average controller compute time of 0.58 ms per step.

Significance. If the claims hold, the work is a useful demonstration of data-driven Koopman modeling for vehicle control: it shows that learned bilinear couplings between inputs and lifted states can be trained with a multi-step loss and deployed in real-time MPC on embedded hardware. The open-loop evaluation over 1,425 trajectories is a strength, as is the HIL validation on a commercial real-time simulator. The comparison against a linear Koopman baseline with the same CER module is a reasonable attempt to isolate the effect of bilinear modeling. However, the closed-loop evidence is limited to a single scenario, and the MPC internal model is not exactly the learned bilinear model, which weakens the attribution of the reported improvements to bilinear modeling.

major comments (2)

[III-H, Eqs. (39b) and (40)] The predictive model actually implemented in the MPC is not the learned bilinear dynamics in Eq. (24). First, the equality constraint in (39b) sums only over i = 1..m, omitting the curvature-channel bilinear term H_3(Z_k * κ) that appears in the learned dynamics; even when κ is constant, H_3(Z_t κ) is an affine offset that is discarded. Second, Eq. (40) fixes the lifted state to its initial value Z_t throughout the entire prediction horizon, so the resulting recurrence is linear in the input with a state-scheduled input matrix. The paper attributes the Table III tracking improvements to the 'bilinear terms introduced in the proposed framework,' but the MPC is not solving the learned bilinear model even at the first prediction step. The comparison against CER-MDK-MPC therefore contrasts two different learned models plus a gain-scheduling approximation, not a controlled test of bilinear modeling. Please either include the H_3 term in the MPC formulation, report the drift ||Z_k - Z_t|| over the 20-step horizon for the test scenario to justify the freezing, or explicitly discuss why this approximation is valid and what its error magnitude is.
[IV, Table III and Fig. 7] The closed-loop validation rests on a single double-lane-change scenario, and the text reports a constant curvature profile of κ = 0.001 m⁻¹, which is not representative of an actual lane-change maneuver (a real DLC has varying curvature). No repeated runs, variations in initial conditions, speeds, or road geometries are provided, so the reported RMSE reductions (e.g., 97% for e_y versus EDMDK) are based on one trajectory. The abstract's claim of 'significant reductions' is therefore not statistically substantiated. Please evaluate the controllers on multiple scenarios (varying curvature profiles, speeds, and maneuvers), report mean and standard deviation over repeated runs, and, if possible, include a significance test. This is especially important because the closed-loop controller uses the frozen-state approximation, and its validity may depend on the maneuver.

minor comments (5)

[IV, Table II] The RMSE values are described as being computed after normalizing each state using the mean and standard deviation of the test dataset, yet the table headers show physical units (km/h, deg/s, m). This is potentially misleading; please report physical units with normalized values clearly distinguished, or provide both normalized and unnormalized errors.
[IV, text after Table III] The sentence claiming reductions 'by 5%, 97%, and 88% compared to the EDMDK-MPC' is numerically inconsistent with Table III: for S the reduction is approximately 2.4% (0.206 vs 0.211), not 5%. Please correct this value.
[IV, experimental setup] The statement 'The same optimal control problem formulation is employed for all controllers' is imprecise: the dynamic constraints differ between the baselines and the proposed method. Please clarify that the cost weights, constraints, and prediction horizon are identical, while the internal models differ.
[III-H, Eq. (40)] The notation H_i = [h_{i1}, ..., h_{ip}] and \hat{H}_j = [h_{1j}, ..., h_{mj}] is not fully defined; if h_{ij} are columns of H_i (each of dimension p), then \hat{H}_j is p × m, but this should be stated explicitly to avoid confusion.
[IV and Conclusion] The paper alternates between 'MDK-MPC' and 'CER-MDK-MPC' when referring to the linear Koopman baseline; please use the full name 'CER-MDK-MPC' consistently, since the CER module is part of that baseline controller.

Circularity Check

0 steps flagged · score 2.0 of 10

No significant circularity: the tracking-error claims are external closed-loop benchmarks; the only self-citation is the MDK-Net baseline, which is not load-bearing.

full rationale

The derivation chain is not circular under the seven defined patterns. The central claims are empirical: open-loop prediction RMSE in Table II is evaluated on a held-out 10% test split against CarSim ground truth, and closed-loop tracking RMSE in Table III is measured on a dSPACE SCALEXIO hardware-in-the-loop CarSim plant. None of these benchmark RMSE values appear in the training losses of Eqs. (27)-(32), so the results are not fitted inputs renamed as predictions. The comparison CER-MDBK-MPC versus CER-MDK-MPC is a fair controlled comparison: both share the same CER block, the same encoder-based Koopman architecture family, the same training data, and identical MPC weighting matrices, so the difference isolates the learned bilinear terms. Eq. (40) fixes the lifted state in the bilinear term to its initial value to preserve convexity; this is an approximation in the MPC's internal model, not an identity that forces the measured tracking outcomes. The full bilinear recurrence is used during training and open-loop evaluation via Eqs. (24) and (29), and the closed-loop advantage is still verified against an external plant rather than against the paper's own training objective. The only author self-citation is the prior MDK-Net work, Ref. [31], which is used as a comparison baseline and as a lineage statement, not as evidence that the central correctness claim holds or as a means of excluding alternative models. Under the review rule that self-citation is not itself circularity, there are no circular steps; the score of 2 reflects only the presence of a minor, non-load-bearing self-citation.

Assumptions & free parameters 4 free parameters · 4 assumptions · 0 invented entities

The central claim rests on the learned model parameters (the encoder and Koopman matrices) fitted to CarSim data, plus several hand-chosen hyperparameters and the linearization approximation. No new physical entities are introduced.

free parameters (4)

Lifted state dimension p = 66
Chosen by validation as the best trade-off between model fidelity and computational cost; affects the capacity of the learned Koopman subspace and the MPC problem size.
Loss weights alpha1..alpha4 = [0.1, 1.0, 1.6, 1e-4]
Tuned on validation performance; directly shape the learned encoder and Koopman matrices, and therefore the downstream control results.
Regularization factors lambda_theta, lambda_AB, lambda_H = 10, 1, 100
Set by hand to balance fitting and overfitting; influence the learned model and its stability.
Forgetting factor beta in multi-step loss = 0.9
Discounts recursive prediction errors in the multi-step loss; chosen by hand and affects long-horizon prediction accuracy.

assumptions (4)

domain assumption A finite-dimensional bilinear model (Eq. 16) can approximate the controlled vehicle dynamics (Eq. 18) in the lifted space.
Section III-C and II-C assume the learned encoder plus matrices A, B, Hi span a nearly invariant subspace; in general such a subspace does not exist for nonlinear systems, so the model is an approximation whose quality is measured empirically.
domain assumption The CarSim RT model is a sufficiently faithful surrogate for a real vehicle to support the claim of suitability for embedded autonomous vehicle systems.
Section IV uses CarSim RT as the target plant; no physical vehicle tests are performed, so the real-world transferability is assumed.
ad hoc to paper Fixing the lifted state to its initial value in the MPC (Eq. 40) is a valid linearization over the prediction horizon.
Section III-H invokes this to retain convexity; its accuracy depends on the lifted state not deviating far from Z_t within N=20 steps, and this is not verified in the paper.
domain assumption Training data from CarSim with steering plus or minus 40 degrees and curvature plus or minus 0.004 per meter covers the deployment scenario.
Section III-B generates the dataset; the closed-loop test uses a double lane change at curvature 0.001, inside this range, but generalization outside the range is untested.

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Cite this review

Pith. "Pith review of Deep Bilinear Koopman Model for Real-Time Vehicle Control in Frenet Frame." pith.science (2026). https://pith.science/paper/WUPHU63Y

@misc{pith2026250712578,
  author       = {Pith},
  title        = {Pith review of: Deep Bilinear Koopman Model for Real-Time Vehicle Control in Frenet Frame},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/WUPHU63Y}},
  note         = {Machine review of arXiv:2507.12578}
}

read the original abstract

Accurate modeling and control of autonomous vehicles remain a fundamental challenge due to the nonlinear and coupled nature of vehicle dynamics. While Koopman operator theory offers a framework for deploying powerful linear control techniques, learning a finite-dimensional invariant subspace for high-fidelity modeling continues to be an open problem. This paper presents a deep Koopman approach for modeling and control of vehicle dynamics within the curvilinear Frenet frame. The proposed framework uses a deep neural network architecture to simultaneously learn the Koopman operator and its associated invariant subspace from the data. Input-state bilinear interactions are captured by the algorithm while preserving convexity, which makes it suitable for real-time model predictive control (MPC) application. A multi-step prediction loss is utilized during training to ensure long-horizon prediction capability. To further enhance real-time trajectory tracking performance, the model is integrated with a cumulative error regulator (CER) module, which compensates for model mismatch by mitigating accumulated prediction errors. Closed-loop performance is evaluated through hardware-in-the-loop (HIL) experiments using a CarSim RT model as the target plant, with real-time validation conducted on a dSPACE SCALEXIO system. The proposed controller achieved significant reductions in tracking error relative to baseline controllers, confirming its suitability for real-time implementation in embedded autonomous vehicle systems.

Figures

Figures reproduced from arXiv: 2507.12578 by the authors.

**Figure 2.** Vehicle states in the curvilinear Frenet frame: longitudinal velocity ˙ [PITH_FULL_IMAGE:figures/full_fig_p002_2.png] view at source ↗

**Figure 3.** Architecture of the proposed MDBK-Net framework. The system consists of three main components: (i) a data acquisition block for preparing training [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗

Figures from the paper (5 more)

**Figure 4.** Figure 4: Long-term prediction results of vehicle dynamics for proposed [PITH_FULL_IMAGE:figures/full_fig_p009_4.png]

**Figure 5.** Figure 5: Visualization of bilinear Koopman matrices [PITH_FULL_IMAGE:figures/full_fig_p010_5.png]

**Figure 6.** Figure 6: Hardware-in-the-loop (HIL) setup used to evaluate the real-time perfor [PITH_FULL_IMAGE:figures/full_fig_p012_6.png]

**Figure 7.** Figure 7: Comparative performance of CER MDK MPC, EDMDK MPC, CER [PITH_FULL_IMAGE:figures/full_fig_p012_7.png]

**Figure 8.** Figure 8: Task turnaround time comparison of all controllers during HIL [PITH_FULL_IMAGE:figures/full_fig_p013_8.png]

Discussion (0). Continue with ORCID to comment.

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