Source-linked AI summary
Learning-Based Predictive Control Method for Vehicle Lateral Control with a Multi-Step Gaussian Process Regression Prediction
Hasan Zakeri, Baisravan HomChaudhuri
TL;DR
Safety-constrained control needs accurate multi-step state-distribution forecasts when uncertainty depends on state and control. The paper uses multi-step Gaussian Process Regression in MPC and reports reduced excessive conservatism in vehicle lane-change simulations while retaining safety assurances.
Problem
Stochastic MPC requires accurate state-distribution forecasts across the control horizon, which is challenging when uncertainty depends on state and control inputs.
Source: §1 · p. 1Method
Results
Vehicle lateral-control simulations report advantages over traditional uncertainty approximations, particularly in avoiding excessive conservatism over extended horizons while minimizing performance compromises.
Source: §6 · p. 9Takeaways & Limitations
The lane-change case study demonstrates the practical utility of multi-step uncertainty prediction for balancing safety assurances and performance.
Source: §6 · pp. 7, 9The setup allows physical parameters to be unknown and assumes process noises are i.i.d. and spatially uncorrelated.
Source: §2 · p. 2
Abstract
from arXiv · showhide
We introduce a novel approach to model predictive control that incorporates multi-step uncertainty prediction for safely controlling systems characterized by uncertainties dependent on both state and control variables. The discrepancy between real-world systems and their control-oriented representations arises from inherent uncertainties, which frequently correlate with state and control variables, a common occurrence in modeling errors. As these uncertainties accumulate and propagate over time, they can produce substantial deviations over extended horizons, potentially compromising the integrity of safety-critical applications. Although existing stochastic control frameworks can maintain system operation within safety boundaries at specified confidence levels, they necessitate accurate prediction of state distributions throughout the control horizon. This prediction represents a significant challenge for systems where uncertainties vary with state and control inputs. Our contribution addresses this challenge through a Model Predictive Controller leveraging multi-step Gaussian Process Regression to capture and anticipate uncertainties that are state- and control-dependent. We further propose an iterative solution to the optimization problem in our MPC framework and discuss the convergence of the algorithm. To demonstrate the method in a practical application, we conduct an in-depth analysis of vehicle lateral control, particularly during lane-changing maneuvers, examining how errors propagate through the system model. The effectiveness of our proposed methodology is validated through comprehensive simulations.
1 Introduction
Safety-constrained MPC needs accurate forecasts of state distributions, but vehicle uncertainties can depend on state and control and accumulate over prediction horizons. The paper proposes multi-step GPR and a successive MPC solution, analyzed for convergence and applied to lane changes.
Motivation
- Modeling mismatch can propagate across prediction steps and produce substantial long-horizon errors that challenge safety analysis.The paper focuses on characterizing error distributions across horizons under state- and control-dependent uncertainty and incorporating them into MPC.Source: §1 · p. 1
Research gap
- Existing GPR methods mainly quantify uncertainty at a single step, while stochastic MPC needs multi-step error distributions over finite horizons.Prior alternatives include immediate-discrepancy models, black-box vehicle dynamics, linearized propagation, and restrictive or conservative approximations.Source: §1 · p. 1
Lane-change setting
- Lane changes combine coupled nonlinear lateral and longitudinal dynamics, non-convex collision constraints, and state- and control-dependent uncertainty.The introduction notes that existing approaches may rely on predefined trajectories or simple vehicle models, while some coordination methods use instantaneous positions rather than predicted trajectories.Source: §1 · pp. 1–2
Core contributions
- The paper combines multi-step GPR for state- and control-dependent uncertainty with safety-constrained MPC and a successive optimization method whose convergence is analyzed.The contributions also include a more comprehensive data-gathering strategy, a general MPC formulation, and a scenario simulation.Source: §1 · p. 2
Prior work and extension
- The present work extends the authors’ earlier multi-step GPR strategy with theoretical improvements and more detailed analysis.The stated additions include comprehensive data gathering, a general MPC formulation, convergence analysis, and scenario simulation.Source: §1 · p. 2
2 Problem Description
The section defines mismatch between an actual system and a control-oriented model, then sets up vehicle dynamics and a chance-constrained MPC formulation for lane changes.
2 Problem Description
- Mismatch ε(k) is the difference between actual and control-model states and is added at each prediction step; its distribution over horizon H can depend on state and input.The control-oriented model may be nonlinear or derived from first-principles modeling, data, or simplification; physical parameters may be unknown, and process noises are i.i.d. and spatially uncorrelated.Source: §2 · p. 2
Lateral Vehicle Control System Dynamics.
- The vehicle case study uses a two-degree-of-freedom bicycle model, with lateral position, velocity, yaw, and yaw rate as states; its kinematic model supports lane-change trajectories.Steering and acceleration are inputs to the kinematic model, while velocity is external or time-varying.Source: Lateral Vehicle Control System Dynamics · p. 2
2.2 Safety Constrained Model Predictive Control.
- The finite-horizon MPC tracks a reference trajectory and input while constraining the vehicle to collision-free regions at confidence 1−ε_safe.The future distributions come from multiple learned models, and the safety constraint depends on the input being optimized, making the optimization nontrivial.Source: §2.2 · p. 3
3 Uncertainty Propagation Model
The paper models mismatch at each prediction step with multi-step Gaussian processes, then uses realistic lane-change inputs to generate training data for uncertainty predictions across the horizon. These predictions support tightening the safety region to account for model mismatch.
3 Uncertainty Propagation Model
- Multi-step GPR models mismatch at every prediction step, directly modeling horizon uncertainty rather than propagating a one-step error through an assumed linear-Gaussian model.The approach is intended to provide higher-confidence predictions with tighter bounds.Source: §3 · p. 3
3.1 Input Sequence Generation.
- Lane-change input sequences retain physically meaningful maneuvers, using kinematic-model trajectories with specified lane, speed, and heading conditions.Erratic alternations between steering directions or input magnitudes are discarded.Source: §3.1 · p. 3
3.2 Data Generation and Collection.
- Training data record the deviation between the actual and simple models from a shared initial condition for each prediction step across the horizon.The resulting error samples are available at each prediction step.Source: §3.2 · pp. 3–4
3.3 Multi-Step Gaussian Process Regression.
- The method trains H Gaussian process models, one per prediction step, with means and covariances depending on the current state and horizon input.Each model represents uncertainty at its corresponding time step.Source: §3.3 · p. 4, Figure 2 · p. 4
4 Successive Controller Design
The controller handles control-dependent uncertainty by successively solving a convexified MPC problem using the previous iteration’s control solution. Under stated feasibility conditions, its cost is non-increasing and bounded below, so the approach converges to a local minimum.
4.1 Formulation of the Optimization Problem.
- Because future state distributions and safety constraints depend on the unknown control, the MPC problem is non-convex; the controller updates predictions and constraint tightening from the previous control iterate to solve a convex program.The safety constraint uses a back-off computed from the predicted uncertainty mean and covariance to satisfy the specified confidence level.Source: §4.1 · p. 4
- The controller begins with a reference trajectory and input, then iterates from the current vehicle state until changes in the objective or control meet a stopping criterion.The preliminary control estimate comes from the path planner.Source: §4.1 · pp. 4–5
4.2 Convergence Analysis.
- The feasibility argument relies on bounded changes in GP-based constraint back-offs, which preserve feasibility of the previous trajectory in the updated constraint set.The paper states that these bounds limit changes in the contracted safety set between iterations.Source: §4.2 · p. 5
- The successive approach converges toward a local minimum because feasible iterations have non-increasing, non-negative cost.The argument assumes an initially feasible control trajectory and that each previous solution remains feasible under updated constraints.Source: §4.2 · p. 5
5 Numerical Simulations
The simulations evaluate uncertainty prediction on an inverted-pendulum benchmark and a realistic vehicle-control case. The proposed approach stays calibrated on the benchmark and supports safe lane changes where propagation-based constraints can become infeasible.
5.1 Benchmark Problem: Inverted Pendulum.
- Pendulum training compares nonlinear and linearized model trajectories under identical inputs, with ±10% length error and N(0, 0.02) measurement noise.The resulting trajectory differences are observations used to train the GP model of discrepancy.Source: §5.1 · pp. 5–6
- A separate GP at each prediction step learns accumulated discrepancies, unlike the one-step GP, which iteratively propagates uncertainty; evaluation measures coverage, mean accuracy, and over-approximation.Test trajectories are separated by initial condition, and repeated trajectories per training condition help estimate the spread of discrepancy.Source: §5.1 · p. 6
- 0.01 vs 0.27 NMSE for θ and 0.02 vs 0.33 for θ̇: horizon-averaged scores for the proposed method and propagation method, respectively.The propagation method’s error grows by more than an order of magnitude beyond the first two or three steps, while the proposed method remains between 0.01 and 0.02 across the horizon.Source: Figure 4 · p. 6, §5.1 · p. 6
- 1.06× vs 14.9× for θ and 1.23× vs 16.4× for θ̇: proposed and propagation methods’ average over-approximation ratios, with 95.9% coverage for both states.The proposed method stays close to the ideal ratio of unity; the propagation method’s over-approximation is more than an order of magnitude.Source: Figure 5 · p. 6, §5.1 · p. 6
5.2 Lateral Control and Lane Change Scenario.
- The vehicle case has the ego car leave the middle lane to pass slower traffic, then return to the middle lane by 35 s as the road clears.It moves to the upper lane, holds it while passing traffic, then crosses toward the lower lane before settling back into the middle lane.Source: §5.2 · p. 7
- The propagation method’s uncertainty margin empties the feasible set in tight collision-free regions, while the proposed controller safely completes the lane-change maneuver.In the driving scenario, the propagation controller has no feasible input at the three-step prediction, whereas the proposed method handles the lane changes under tighter constraints.Source: §5.2 · p. 7
6 Conclusion
The paper proposes multi-step Gaussian Process Regression to track state- and control-dependent uncertainty across prediction horizons. In vehicle lane-change simulations, it reduces excessive conservatism compared with traditional uncertainty approximations while maintaining safety assurances and minimizing performance compromises.
6 Conclusion
- The multi-step Gaussian Process Regression approach models state- and control-dependent uncertainty and tracks its propagation across prediction horizons.The authors also discuss convergence of the algorithm.Source: §6 · pp. 7, 9
- The lane-change visualization marks vehicle centers and lane boundaries, with collision-free zones defined relative to vehicle centers rather than physical bounds.It shows the ego vehicle relative to other vehicles at seven instants.Source: Figure 9 · p. 8
- In the illustrated one-step estimation example, overestimated uncertainty makes the third predicted collision-free zone a null set at 3 s.The depicted closed-loop trajectory ends at 1.5 s.Source: Figure 10 · p. 9
- On the same trajectory and with the same parameters, the one-step method becomes infeasible while the proposed method navigates successfully.The comparison illustrates the practical result during vehicle lateral control.Source: Figure 11 · p. 9, §6 · p. 9
- The simulations show advantages over traditional uncertainty approximations, particularly by preventing excessive conservatism over extended prediction horizons.The authors report this balance as improving safety assurances while minimizing performance compromises.Source: §6 · p. 9