Source-linked AI summary
Model predictive traction control system based on the Koopman operator
Josip Kir Hromatko, Šandor Ileš
TL;DR
Traction control requires handling difficult nonlinear tire models while maintaining practical computation. The paper uses EDMD-based Koopman lifting to build a linear MPC controller and compares it with NMPC. In simulation, the controllers show comparable tracking, while the Koopman-based approach reduces execution time.
Problem
Nonlinear tire dynamics are difficult to model accurately and make advanced traction-control optimization computationally challenging.
Method
The paper uses a Koopman operator approximated with EDMD and polynomial lifting to formulate a linear MPC controller for single-wheel traction control.
Results
Comparable slip-tracking performance was observed between the Koopman-based MPC and NMPC, while linear MPC solved controller steps at least an order of magnitude faster.
Takeaways & Limitations
The Koopman-based algorithm provides comparable control performance with reduced execution time in the reported simulation and hardware evaluations.
Abstract
from arXiv · showhide
Due to their importance, traction control and anti-lock braking systems have become standard equipment in modern vehicles. However, accurate models of tire dynamics are often difficult to obtain and usually include nonlinearities, making their use in control systems challenging. This paper describes a traction control system based on model predictive control and Koopman operator theory, which aims to approximate nonlinear systems with linear ones through a state space transformation. A linear model predictive controller based on the Koopman predictor is compared to a standard nonlinear model predictive controller. Experiments in a high-fidelity vehicle dynamics simulation environment show a comparable reference tracking performance of the two controllers, with a reduced execution time for the proposed Koopman operator-based algorithm, both on a standard PC and embedded hardware.
I. INTRODUCTION
Traction control improves vehicle safety and performance by regulating tire forces, while MPC and Koopman-based linearization address nonlinear modeling and online optimization costs.
- Motivation: Traction forces drive vehicle behavior, so controlling tire slip can improve safety and performance.Modern vehicles commonly use traction control and ABS for this purpose.
- Model predictive control: MPC repeatedly solves a constrained optimization problem over a prediction horizon and applies its first control input.Its benefits include explicit constraints and intuitive handling of multiple inputs or outputs.
- Model predictive control: MPC requires a plant model and incurs online optimization costs, motivating methods that simplify modeling and reduce execution time.Deriving an accurate plant model can also be time-consuming.
- Koopman-based control: Linearizing nonlinear models can make MPC optimization convex and substantially faster when constraints and costs are convex.Koopman operator theory performs this approximation through a nonlinear transformation of the state space.
- Contribution: The paper develops a single-wheel traction-control MPC using an EDMD-approximated Koopman operator and compares it with NMPC.The comparison uses a high-fidelity simulator and embedded hardware, targeting faster real-time execution.
- Wheel dynamics: The wheel model uses longitudinal and rotational wheel speeds as states, with motor torque as input and a simplified Pacejka tire-force model.The setup assumes quarter-vehicle vertical load when acceleration load transfer is ignored, and neglects aerodynamic, friction, and rolling-resistance effects.
A. Koopman operator theory for controlled systems
Koopman theory represents controlled nonlinear dynamics in a lifted space where an infinite-dimensional linear operator predicts transformed states.
- Controlled-system formulation: Controlled-system Koopman modeling introduces an extended state containing the current state and the future input sequence.This converts the controlled dynamics into an autonomous formulation using the left-shift operator.
- Linear prediction: The Koopman operator acts linearly on lifting functions, or observables, to approximate the nonlinear dynamics.Including the original states among the observables allows predicted states of interest to be recovered with a selection matrix.
B. Extended dynamic mode decomposition
EDMD identifies a finite-dimensional linear approximation of nonlinear dynamics from state-transition data and lifting functions, while its untransformed inputs limit applicability.
- EDMD formulation: EDMD approximates the Koopman operator from a dataset of state, input, and successor-state triples using a dictionary of basis functions.The method is formulated as an optimization problem over the collected data.
- Linear predictor: The identified linear predictor uses lifted states, matrices  and B̂, and a selection matrix Ĉ to recover system outputs.The selection matrix may be manually specified or obtained by least-squares minimization.
- Identification properties: The EDMD least-squares problems can be solved analytically, and the resulting predictor minimizes one-step prediction error.Datasets may contain multiple trajectories and can require as little as one sample per trajectory.
- Limitation: Because inputs enter EDMD untransformed, prediction accuracy can be limited for nonlinear input-to-state mappings.The method is therefore more suitable for systems with a linear input-to-state mapping.
C. Approximating the wheel dynamics
The traction-control model uses data-driven polynomial lifting and a state representation selected to support Koopman identification and control.
- Data generation: The identification dataset contains 1000 trajectories of 250 samples at a 2 ms sampling time.Vehicle speed, slip ratio, and motor torque were randomized within specified ranges.
- State selection: The chosen state representation is x = [s ω]⊤, where s = ωR − v, rather than alternative speed- and slip-based formulations.The choice reflects faster rotational dynamics and a linear input-output relationship desirable for EDMD.
- Choosing the lifting function: Polynomial basis functions lift the original state into a higher-dimensional space for Koopman approximation.Polynomial functions achieved the highest approximation accuracy among the tested basis choices.
- Choosing the lifting function: Adding the slip-ratio function to the lifted state enables slip tracking within a linear MPC framework.This also supports representing nonlinear constraints and costs through the lifted state.
4) Approximation results:
The identified Koopman predictor is evaluated for traction-control conditions and supports prediction over horizons commonly used in traction-control design. The section also situates traction control around slip regulation, driver-request preservation, and front-wheel-drive state estimation.
- The Koopman predictor was identified using EDMD with fourth-order polynomial basis functions.
- The prediction experiment used an initial vehicle speed of 20 km/h and PRBS control inputs ranging from 50 to 175 Nm.
- Traction control aims to prevent excessive wheel slip while generating optimal traction force, commonly by tracking a wheel-slip setpoint near the tire-force peak.
- The controller may only reduce requested motor torque, preserving expected vehicle behavior while preventing wheel slip.
- For front-wheel-drive electric vehicles, longitudinal speed and wheel slip can be estimated from rear-wheel speeds.
B. Nonlinear MPC
The nonlinear MPC baseline uses the nonlinear wheel dynamics directly in a constrained optimization problem with integral action and torque-related objectives. Its continuous-time formulation is discretized into a nonlinear program solved repeatedly online.
- B. Nonlinear MPC: NMPC uses nonlinear system dynamics directly, while commonly employing polytopic constraints and a quadratic cost to improve convergence.
- B. Nonlinear MPC: The NMPC state includes slip ratio, wheel speed, and an integral state to improve reference tracking under disturbances.
- B. Nonlinear MPC: Motor torque is constrained according to the driver-set or maneuver-specific torque reference.
- B. Nonlinear MPC: The nonlinear cost penalizes slip-tracking error, the integral state, and motor-torque reduction through a weighted sum of squares.
- B. Nonlinear MPC: The continuous-time NMPC problem is discretized and converted into a nonlinear program with prediction horizon T and tuning weights wp, wi, and wu.
- B. Nonlinear MPC: The nonlinear program is solved at every time step, and only the first optimal control input is applied.
- B. Nonlinear MPC: Terminal cost and constraint ingredients were omitted because invariant sets and control Lyapunov functions are difficult to compute for nonlinear systems.
C. Koopman operator-based MPC
The Koopman MPC formulation uses a lifted linear predictor with added integral and reference states, yielding a quadratic program under constant torque-reference assumptions. Dense reformulation can reduce computational dependence on lifted-state dimension but may become numerically inaccurate in some regimes.
- C. Koopman operator-based MPC: Once identified, the Koopman predictor serves as the prediction model in a linear MPC framework and can be extended with additional states.
- C. Koopman operator-based MPC: The KMPC model adds a discrete-time integral state and a zero-dynamics slip-reference state to support tracking and retain linear state-space form.
- C. Koopman operator-based MPC: The full KMPC state is ξ = [z eint κref]⊤, with outputs selected to match the NMPC cost formulation.
- C. Koopman operator-based MPC: Dense reformulation makes system outputs depend only on the initial state and input sequence, while keeping transition-matrix size independent of lifted-state dimension.
- C. Koopman operator-based MPC: The Koopman operator-based MPC algorithm is represented schematically in Fig. 2.
- C. Koopman operator-based MPC: The dense reformulation can be numerically inaccurate for marginally stable systems and longer prediction horizons.
- C. Koopman operator-based MPC: With constant torque reference, linear approximated dynamics, and simple input constraints, the KMPC cost becomes quadratic and produces a quadratic program.
- C. Koopman operator-based MPC: The KMPC algorithm uses stacked output and input vectors and, like the NMPC baseline, omits terminal ingredients.
A. Test environment
The controllers were evaluated in high-fidelity closed-loop vehicle simulations using constant and varying road-friction conditions. They showed similar tracking performance, while the linear MPC formulation reduced controller-step execution time relative to nonlinear MPC.
- Test setup: Tests used IPG CarMaker, a high-fidelity vehicle-dynamics simulator, with two acceleration maneuvers starting at 2 km/h and a slip reference of 0.1.The torque reference ramped to the motor's maximum torque, and the slip reference was estimated from the tire model.
- Controller implementation: Both controllers used FORCESPRO, with NMPC solved as a nonlinear program and KMPC solved using a primal-dual interior-point method.The NMPC dynamics used fourth-order explicit Runge–Kutta discretization, while KMPC used the YALMIP-based formulation.
- Constant road friction: Under constant road friction of 0.3, the controllers delivered similar slip-tracking and torque-reduction performance.After an initial torque ramp and slight slip-reference overshoot, torque reduction decreased slip and improved traction.
- Execution time: Linear MPC solved each controller step at least an order of magnitude faster than nonlinear MPC, including in maximum execution time.Maximum execution time determines the hardware requirements for practical deployment.
- Varying road friction: With friction varying as 0.3-0.15-0.6-0.3, both controllers again performed similarly, although KMPC produced slightly larger overshoots.Neither controller knew the friction changes; at coefficient 0.6, neither reached the slip reference because motor torque was already maximal.
D. Execution time
The Koopman-based linear MPC achieved comparable performance to NMPC while substantially reducing controller execution time on both PC and embedded hardware. This speed advantage can lower hardware implementation costs, although predictor identification and model deployment remain practical considerations.
- Execution-time comparison: At least an order of magnitude faster, linear MPC reduced both mean and maximum controller-step execution times relative to nonlinear MPC.The maximum time matters because it helps determine the hardware required in practice.
- Embedded-hardware results: 1.26 ms for NMPC versus 0.023 ms for KMPC on embedded hardware confirmed the reported reduction in solve time.These preliminary measurements were obtained using a dSPACE MicroLabBox.
- Timing scope: Evaluating the nonlinear lifting functions Φ(·) also contributes to optimal-input calculation time, although this cost was not reported because the expressions were low-complexity.The reported execution values therefore focus on optimization solve time.
- Implementation considerations: Identifying the Koopman predictor is not straightforward, and data-driven identification methods have their own benefits and caveats.The paper notes that future software and computing developments could simplify this modelling process.
- Practical implication: Comparable controller performance with lower execution times can reduce the cost of hardware implementation.The paper presents this as a practical advantage of the Koopman operator-based approach.