Passivity-exploiting stabilization of semilinear single-track vehicle models with distributed tire friction dynamics

Problem Statement
This paper addresses the local stabilization problem for semilinear single-track vehicle models that incorporate distributed tire friction dynamics. Traditional vehicle control strategies often rely on lumped tire models, which approximate tire forces as static nonlinearities. However, these simplifications fail to capture the inherently distributed nature of tire-road interactions, which introduce delays, memory effects, and complex nonlinear behaviors critical for predicting vehicle stability under transient conditions (e.g., sudden steering inputs or road disturbances). When coupled with rigid-body lateral vehicle dynamics, distributed friction models (such as Dahl or LuGre formulations) result in an infinite-dimensional system described by an interconnection of Ordinary Differential Equations (ODEs) and Hyperbolic Partial Differential Equations (PDEs). The challenge lies in designing rigorous stabilization algorithms for these nonlinear, infinite-dimensional ODE-PDE systems, particularly for oversteer vehicles operating at high speeds where lateral instabilities are a significant risk.

Methodology
The authors propose a passivity-exploiting backstepping design to stabilize the ODE-PDE interconnection around a prescribed equilibrium. The methodology is built upon the following structural insights and steps:

1. System Modeling: The vehicle is modeled using a single-track representation where the lateral velocity and yaw rate (ODE states) are coupled with the distributed deflection of bristle-like elements within the tire contact patch (PDE states). The PDE subsystem describes the evolution of these deflections across the contact patch length.
2. Exploitation of Passivity: A central insight of the approach is that the PDE subsystem, governed by frictional mechanisms, possesses strict dissipativity properties. The tire-road rolling contact inherently dissipates energy through slip losses. The authors construct a Lyapunov functional that combines the lumped (ODE) and distributed (PDE) states, leveraging this strict dissipativity to achieve exponential stabilization.
3. Controller Synthesis:
   • State-Feedback: A backstepping controller is synthesized using a coordinate transformation that maps the original system into a target system with desired stability properties. This involves defining a virtual control law for the ODE subsystem and designing a control input for the PDE subsystem to cancel coupling terms and ensure the decay of the Lyapunov function.
   • Output-Feedback: Since full state measurement of the distributed PDE states is often impractical, a cascaded observer is designed. The observer estimates the rigid-body states and the distributed states based on measurements of the rigid relative velocity (which can be obtained via intelligent tire sensors or combined yaw rate/accelerometer data). The separation principle is utilized, proving that the observer error dynamics and the closed-loop system dynamics can be stabilized simultaneously.
4. Stability Analysis: Sufficient conditions for local well-posedness and exponential convergence are derived. The analysis relies on the construction of a composite Lyapunov functional and the verification of strict dissipativity and Lipschitz continuity assumptions for the friction models.

Key Contributions

• Novel Control Framework: The paper presents a rigorous stabilization algorithm for nonlinear vehicle models with distributed tire friction, a class of systems that has remained largely unexplored compared to linear or lumped-parameter models.
• Passivity-Based Design: Unlike previous approaches that may require analyzing closed-loop behavior a posteriori or rely on boundary term injections limited to linear models, this work synthesizes a controller that uses bounded functionals of the distributed states. This preserves the semilinear structure of the ODE-PDE interconnection.
• Observer Integration: The development of an output-feedback strategy utilizing a cascaded observer for infinite-dimensional vehicle models, enabling stabilization without direct access to the full distributed state.
• Robustness Validation: Theoretical results are corroborated by numerical simulations in MATLAB/Simulink, accounting for non-ideal scenarios including parametric uncertainties, external wind disturbances, measurement noise, and control input delays.

Results
Numerical simulations were conducted on an oversteer vehicle model traveling at a high longitudinal speed (50 m/s), a regime where the vehicle is inherently unstable.

• Stabilization: The proposed output-feedback controller successfully stabilized the vehicle around the desired equilibrium (zero lateral velocity and yaw rate) within approximately 2 seconds, preventing the system norm from exceeding safety thresholds.
• Observer Performance: The observer accurately predicted the true states, with convergence rates dependent on the observer gain. Higher gains yielded faster convergence but reduced filtering effectiveness against noise.
• Comparison: When compared to a controller based on a reduced-order (lumped) model, the infinite-dimensional controller demonstrated superior handling of the distributed dynamics. Notably, the infinite-dimensional controller commanded steering actions in the opposite direction during the initial transient compared to the reduced-order approach, highlighting the importance of distributed modeling for accurate control.
• Robustness Limits: The controller demonstrated robustness to input delays up to 0.6 seconds. However, delays of 1 second or initial conditions significantly far from the equilibrium (e.g., large sideslip angles) led to instability, consistent with the local nature of the derived stability conditions.

Significance and Claims
The paper claims that the proposed control strategy offers a principled way to stabilize vehicles subject to complex contact dynamics, particularly under oversteer conditions at high speeds and in the presence of micro-shimmy oscillations. By explicitly accounting for the distributed nature of tire-road interaction, the method represents a step beyond conventional strategies relying on lumped tire models.

The authors position this work as a foundational step for improving safety and performance in automotive applications, such as Electronic Stability Control (ESC) and Advanced Driver-Assistance Systems (ADAS). They acknowledge that the derived stability conditions are locally valid and somewhat conservative, particularly regarding the size of the initial conditions and equilibrium points required for convergence. The paper concludes that while the current results are specific to single-track models, they provide a rigorous basis for future extensions to more refined vehicle representations (including roll, pitch, and heave) and alternative actuation strategies like combined braking and steering.