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

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.

💡 Research Summary

The paper addresses a long‑standing gap in vehicle dynamics research: the lack of a rigorous mathematical justification for the common practice of neglecting transient tire dynamics when designing stability controllers. The authors consider a single‑track (bicycle) vehicle model whose rigid‑body dynamics are described by a pair of ordinary differential equations (ODEs) for the sideslip angle β and yaw rate r. In contrast to the traditional static tire models (e.g., Pacejka), the lateral tire forces are generated by distributed friction dynamics that evolve along the contact patch. These dynamics are captured by semilinear hyperbolic partial differential equations (PDEs) for the bristle deformation variables z₁(ξ,t) and z₂(ξ,t), where ξ∈