Stability and stabilization of semilinear single-track vehicle models with distributed tire friction dynamics via singular perturbation analysis

TL;DR

This paper provides a rigorous mathematical framework connecting distributed tire friction models with traditional vehicle dynamics, using singular perturbation theory to analyze stability and design controllers for semilinear single-track vehicle models.

Contribution

It introduces a novel singular perturbation approach to justify neglecting tire dynamics in vehicle models and designs stabilizing controllers within this framework.

Findings

Validates the use of simplified models for small perturbation parameters

Designs stabilizing controllers based on the new theoretical framework

Bridges empirical observations with rigorous mathematical analysis

Abstract

This paper investigates the stability and stabilization of semilinear single-track vehicle models with distributed tire friction dynamics, modeled as interconnections of ordinary differential equations (ODEs) and hyperbolic partial differential equations (PDEs). Motivated by the long-standing practice of neglecting transient tire dynamics in vehicle modeling and control, a rigorous justification is provided for such simplifications using singular perturbation theory. A perturbation parameter, defined as the ratio between a characteristic rolling contact length and the vehicle's longitudinal speed, is introduced to formalize the time-scale separation between rigid-body motion and tire dynamics. For sufficiently small values of this parameter, it is demonstrated that standard finite-dimensional techniques can be applied to analyze the local stability of equilibria and to design stabilizing controllers. Both state-feedback and output-feedback designs are considered, under standard stabilizability and detectability assumptions. Whilst the proposed controllers follow classical approaches, the novelty of the work lies in establishing the first mathematical framework that rigorously connects distributed tire models with conventional vehicle dynamics. The results reconcile decades of empirical findings with a formal theoretical foundation and open new perspectives for the analysis and control of ODE-PDE systems with distributed friction in automotive applications.