Cumulative Error Regulator for Vehicle Control

• Cumulative Error Regulator (CER) is a control augmentation that integrates cumulative tracking error into deep Koopman models to mitigate long-horizon drift.
• It is implemented within the MPC framework to compensate for model mismatch by accumulating and correcting tracking errors over time.
• Empirical results show CER improves trajectory tracking precision, significantly reducing RMSE in lateral displacement and heading error in vehicle control tests.

Cumulative Error Regulator (CER) denotes an augmentation block introduced for a deep bilinear Koopman model used in real-time vehicle control in the curvilinear Frenet frame. In that setting, CER is a control-side mechanism that augments the learned lifted dynamics with a cumulative tracking-error state, with the stated purpose of compensating for model mismatch, mitigating accumulated prediction errors, improving trajectory tracking precision, and enhancing safety by correcting long-horizon drift (Abtahi et al., 16 Jul 2025).

1. Concept and motivation

The stated motivation for CER is that any finite-dimensional Koopman approximation is imperfect. The learned model has approximation error or model mismatch, this mismatch produces prediction drift over long horizons, and in vehicle tracking—especially for progress along path $S$, lateral deviation $e_y$, and heading error $e_\psi$—these errors accumulate over time because the tracking variables are integrative in nature. CER is introduced specifically to address that accumulation rather than only the instantaneous mismatch between predicted and reference outputs (Abtahi et al., 16 Jul 2025).

The paper characterizes CER as a mechanism designed to predict accumulated error dynamics and force cumulative error to stay near zero, so that the controller can detect and compensate for long-term deviation. In that formulation, CER is meant to compensate for model mismatch, mitigate accumulated prediction errors, improve trajectory tracking precision, and enhance safety by correcting long-horizon drift. A plausible implication is that CER addresses a limitation of predictive controllers whose nominal model is sufficiently accurate for short-horizon prediction but still susceptible to drift under repeated rollout.

2. Position within the Koopman-based control architecture

CER is inserted in the control block, not in the data-training block. The overall architecture is described as three stages: data acquisition, deep bilinear Koopman model learning, and a control block with CER. The control block is “further enhanced with the CER to improve tracking performance,” and, in the CER-MDBK-MPC formulation, the augmented state contains both the lifted Koopman state $\hat Z_k$ and the cumulative tracking error state $\mathbf e_k$ (Abtahi et al., 16 Jul 2025).

This placement is central to the paper’s interpretation of CER. It is not part of the model identification loss and it is not trained as a neural module. Only the Koopman predictor is trained: the encoder learns lifted observables, the matrices $A$, $B$, and $H_i$ are learned from data, and a multi-step prediction loss is used to improve long-horizon fidelity. CER is introduced afterward as a deterministic augmentation used in control time to compensate for residual prediction errors that remain after training.

The underlying learned prediction model is the deep bilinear Koopman network, written as

$$\hat{Z}_{k+1} = AZ_{k} + B U_k + \sum_{i=1}^{m+l} H_i \left[ U^i_{k}\cdot Z_{k} \right].$$

The paper emphasizes that CER does not replace this predictor. Instead, the bilinear Koopman model predicts the system evolution, CER adds a cumulative error state driven by the model’s output mismatch, and the augmented model then feeds MPC.

3. State augmentation and error dynamics

The vehicle states in the Frenet frame are

$$\mathbf{x} = [v_x, v_y, \dot{\psi}, \Delta s, e_y, e_\psi]^\top.$$

CER is tailored to the three tracked outputs

$e_y$0

which are the states singled out as prone to accumulated error because they are tied to path-following over time (Abtahi et al., 16 Jul 2025).

The CER error dynamics are defined by

$e_y$1

where

$e_y$2

Here $e_y$3 is the cumulative tracking error state at time $e_y$4, $e_y$5 is the predicted tracking output, $e_y$6 is the desired reference tracking output, and $e_y$7 is the output-selection matrix mapping predicted lifted state $e_y$8 to the tracking outputs. The paper explicitly describes CER as a discrete-time integrator of output error: it adds the current output tracking error to the previous cumulative error, thereby storing how much tracking deviation has accumulated over time.

The extended lifted state is

$e_y$9

and the augmented input is

$e_\psi$0

The dimension increase is $e_\psi$1 because the cumulative error state is three-dimensional.

The full augmented system is given as

$e_\psi$2

with output

$e_\psi$3

The augmented matrices are

$e_\psi$4

$e_\psi$5

The paper’s interpretation of these blocks is explicit. The top part propagates the original lifted dynamics plus bilinear correction, the bottom part keeps the cumulative error state, the bilinear term affects the original lifted state evolution but not directly the error state, and the error state is driven by output mismatch. The $e_\psi$6 term in $e_\psi$7 is the integrator structure, while the $e_\psi$8 term in $e_\psi$9 injects the reference trajectory into the error dynamics so that error is formed as predicted output minus reference.

4. Role inside model predictive control

CER is embedded directly into the MPC prediction model, producing the CER-MDBK-MPC controller. The finite-horizon optimization problem is written as

$\hat Z_k$0

subject to the augmented CER-Koopman dynamics, the output mapping, input bounds, and the initial augmented state

$\hat Z_k$1

In the paper’s formulation, CER changes MPC by making the optimizer account not only for instantaneous predicted output error but also for the history-integrated error state (Abtahi et al., 16 Jul 2025).

The augmented Koopman system is said to “simultaneously propagate both the observable dynamics and the cumulative tracking error at each time step,” enabling MPC to account for accumulated deviations over the horizon $\hat Z_k$2. This is the mechanism by which CER is claimed to improve closed-loop tracking: the optimizer is exposed to long-term error buildup rather than only one-step prediction discrepancy.

The paper also addresses tractability. Because the bilinear dynamics make the MPC non-convex, it applies a standard approximation that fixes the lifted state in the bilinear term to its initial value over the horizon,

$\hat Z_k$3

yielding the linearized approximation

$\hat Z_k$4

CER remains active in this linearized formulation. The stated point is that the bilinear Koopman dynamics are approximated for tractability, while CER remains the mechanism that counters accumulated tracking drift in the predictive loop.

5. Empirical behavior and reported performance

Closed-loop performance is evaluated through hardware-in-the-loop experiments using a CarSim RT model as the target plant, with real-time validation conducted on a dSPACE SCALEXIO system. The reported controller comparisons are CER-MDBK-MPC, CER-MDK-MPC, EDMDK-MPC, and LTI-MPC. The paper reports that CER-MDBK-MPC more closely follows the reference trajectory across almost all tracking states, performs especially well during the second aggressive lane change, and reduces accumulated tracking deviation in the final stages of the simulation (Abtahi et al., 16 Jul 2025).

The closed-loop RMSE values reported for CER-MDBK-MPC are $\hat Z_k$5 for $\hat Z_k$6 (km/h), $\hat Z_k$7 for $\hat Z_k$8 (km/h), $\hat Z_k$9 for $\mathbf e_k$0 (deg/s), $\mathbf e_k$1 for $\mathbf e_k$2 (m), $\mathbf e_k$3 for $\mathbf e_k$4 (m), and $\mathbf e_k$5 for $\mathbf e_k$6 (deg). The corresponding CER-MDK-MPC values are $\mathbf e_k$7, $\mathbf e_k$8, $\mathbf e_k$9, $A$0, $A$1, and $A$2; the LTI-MPC values are $A$3, $A$4, $A$5, $A$6, $A$7, and $A$8; and the EDMDK-MPC values are $A$9, $B$0, $B$1, $B$2, $B$3, and $B$4.

The relative improvements emphasized in the paper concern the path-following variables most closely tied to accumulated drift. Compared to EDMDK-MPC, CER-MDBK-MPC is reported to reduce RMSE by $B$5 in $B$6, $B$7 in $B$8, and $B$9 in $H_i$0. Compared to CER-MDK-MPC, the reported reductions are $H_i$1 in $H_i$2, $H_i$3 in $H_i$4, and $H_i$5 in $H_i$6. The paper also states an average relative reduction across all states of $H_i$7 better than CER-MDK-MPC, $H_i$8 better than EDMDK-MPC, and $H_i$9 better than LTI-MPC.

These results are presented as evidence that CER matters most in lateral displacement $$\hat{Z}_{k+1} = AZ_{k} + B U_k + \sum_{i=1}^{m+l} H_i \left[ U^i_{k}\cdot Z_{k} \right].$$0, heading error $$\hat{Z}_{k+1} = AZ_{k} + B U_k + \sum_{i=1}^{m+l} H_i \left[ U^i_{k}\cdot Z_{k} \right].$$1, and the final stages of the maneuver, which is consistent with the stated purpose of suppressing accumulated drift. Real-time feasibility is also reported explicitly: the average computation time is $$\hat{Z}_{k+1} = AZ_{k} + B U_k + \sum_{i=1}^{m+l} H_i \left[ U^i_{k}\cdot Z_{k} \right].$$2 ms for a sampling interval of $$\hat{Z}_{k+1} = AZ_{k} + B U_k + \sum_{i=1}^{m+l} H_i \left[ U^i_{k}\cdot Z_{k} \right].$$3 ms.

6. Scope, ambiguity, and unrelated uses of the acronym

Within the materials considered here, “Cumulative Error Regulator” is a specific term from real-time vehicle control rather than a general-purpose designation across fields. The acronym “CER” is used elsewhere for unrelated constructs. In OCR post-correction and handwritten text recognition, CER means Character Error Rate, defined as the number of character substitutions, deletions, and insertions divided by the ground-truth string length, or, in the HTR formulation, normalized Levenshtein or edit distance; neither usage denotes a regulator (Kanerva et al., 3 Feb 2025, Al-Azzawi et al., 23 Jan 2026).

In LLM reasoning, CER denotes Cumulative Entropy Regulation, a training-time mechanism associated with the “Explore Briefly, Then Decide” paradigm and the Token Entropy Cumulative Average, again unrelated to the vehicle-control augmentation (Jiang et al., 2 Oct 2025). In AI insurance, CER denotes Control boundary, Evidence reconstruction, and Insurance response, and is defined as a use-case-level diagnostic for AI residual risk transfer rather than a control or estimation device (Leung et al., 2 Jun 2026).

Two further contrasts sharpen the boundary of the term. The quantum-computing paper on Error Mitigation by Restricted Evolution does not define a method called CER and instead introduces EMRE and HEMRE, with the text explicitly treating any CER reference there as a naming mismatch (Saxena et al., 2024). In arithmetic geometry, “the regulator” refers to the Mordell–Weil regulator of an elliptic curve over a global function field, not to a cumulative-error mechanism (Pazuki, 2022).

Taken together, these usages show that “CER” is strongly context-dependent. In the literature represented here, “Cumulative Error Regulator” refers specifically to the error-integrating augmentation applied to deep bilinear Koopman MPC for Frenet-frame vehicle tracking, whereas the same acronym elsewhere names metrics, reward-shaping mechanisms, and diagnostic frameworks in entirely different research domains.