A preprint study of vehicle traction control reports that a controller using a linear Koopman model showed similar slip tracking and torque reduction to nonlinear model-predictive control, alongside lower reported solve times on a standard PC. The pattern held when road friction changed, although the Koopman controller produced somewhat larger overshoots. At a friction coefficient of 0.6, neither controller reached the slip reference because motor torque was already at maximum.
Preliminary timing on a dSPACE MicroLabBox pointed in the same direction: the reported mean solve time was 1.26 milliseconds for NMPC and 0.023 milliseconds for KMPC. The hardware figures are timing results, while the closed-loop comparison was conducted in simulation.
How the comparison was set up
The researchers asked whether Koopman-based linear model-predictive control (MPC) could match a nonlinear MPC (NMPC) controller's reference-tracking performance while reducing execution time on a standard PC and embedded hardware. The approach represents the vehicle's nonlinear behavior through a linear predictor. The study used a single-wheel traction-control model and compared it with NMPC in the high-fidelity IPG CarMaker simulator and on embedded hardware.
To identify the Koopman predictor, the researchers used a dataset of 1,000 simulated trajectories. Each trajectory contained 250 samples taken every 2 milliseconds. They used extended dynamic mode decomposition, or EDMD, with fourth-order polynomial basis functions.
The closed-loop evaluation consisted of two maneuvers in IPG CarMaker. Both began at 2 km/h with free-rolling wheels, ramped torque references and a requested slip ratio of 0.1. The controllers used a 2-millisecond sampling time and a five-step prediction horizon.
Both controllers imposed the same motor-torque bound, from zero to the reference torque. NMPC used explicit fourth-order Runge-Kutta discretization and a nonlinear primal-dual interior-point solver. KMPC used a primal-dual interior-point solver, with both implementations connected through FORCESPRO interfaces.
What happened when the surface changed
With a constant road-friction coefficient of 0.3, NMPC and KMPC showed similar slip tracking and torque reduction. In the changing-friction sequence, the actual coefficient moved from 0.3 to 0.15, then 0.6 and back to 0.3, while both controllers continued to assume a constant value of 0.3 and were unaware of the changes.
Under that changing condition, the two controllers again performed similarly overall, but KMPC had slightly larger overshoots. When the coefficient reached 0.6, neither controller reached the slip reference because motor torque was already at maximum.
The numbers behind the speed gap
On a standard PC, the difference appeared in both mean and maximum solve times. With constant friction, NMPC took 0.197 milliseconds on average and 0.600 milliseconds at most, compared with 0.004 milliseconds and 0.0432 milliseconds for KMPC. With varying friction, the corresponding figures were 0.210 milliseconds and 0.507 milliseconds for NMPC, versus 0.0037 milliseconds and 0.0627 milliseconds for KMPC.
The study's tracking results were qualitative: it did not report numerical tracking-error, overshoot or variability estimates. The timing section gives mean and maximum solve times for the PC tests, without reporting run counts or dispersion. It also did not report the computation time required to evaluate the lifting functions used by the Koopman model.
What the result shows
The authors conclude that the linear Koopman model provided a good approximation, comparable simulated controller performance and lower execution time than NMPC. The reported evidence comprises a single-wheel model, two closed-loop maneuvers in IPG CarMaker and preliminary mean timing on dSPACE MicroLabBox, so the findings are about computational performance and simulated tracking.
The manuscript is marked arXiv:2608.25753v1 and dated 26 Aug 2026. The work received partial support from the Croatian Science Foundation under project UIP-2019-04-6487.
Paper data and sources
Original title: Model predictive traction control system based on the Koopman operator
Authors: Josip Kir Hromatko, Šandor Ileš
Journal/Repository: Proc. 2024 28th Int. Conf. Syst. Theory Control Comput. (ICSTCC), 2024, pp. 26-31
Status: Preprint, not yet peer-reviewed
First online: 2026-08-26
DOI: 10.1109/icstcc62912.2024.10744655
Original paper · Full text