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

Luigi Romano, Ole Morten Aamo, Miroslav Krstić, Jan Åslund, Erik Frisk

TL;DR

This work addresses stability and stabilization of semilinear single-track vehicle models with distributed tire friction dynamics, formulated as interconnections between slow ODEs and fast hyperbolic PDEs. By introducing the perturbation parameter $\epsilon = \bar{L}/v_x$, it proves that, for sufficiently small $\epsilon$, the system can be approximated by a reduced-order ODE plus boundary-layer PDE, enabling standard finite-dimensional stability and control techniques. The authors derive local stability results via Lyapunov analysis for the reduced and boundary-layer subsystems and extend them to small $\epsilon$ for both state-feedback and output-feedback controllers (with observers). This provides the first rigorous justification for using conventional finite-dimensional vehicle designs when accounting for distributed tire dynamics, bridging empirical tire-transient observations with a formal ODE–PDE framework and illustrating the practical effectiveness through simulations.

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.

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

TL;DR

This work addresses stability and stabilization of semilinear single-track vehicle models with distributed tire friction dynamics, formulated as interconnections between slow ODEs and fast hyperbolic PDEs. By introducing the perturbation parameter , it proves that, for sufficiently small , the system can be approximated by a reduced-order ODE plus boundary-layer PDE, enabling standard finite-dimensional stability and control techniques. The authors derive local stability results via Lyapunov analysis for the reduced and boundary-layer subsystems and extend them to small for both state-feedback and output-feedback controllers (with observers). This provides the first rigorous justification for using conventional finite-dimensional vehicle designs when accounting for distributed tire dynamics, bridging empirical tire-transient observations with a formal ODE–PDE framework and illustrating the practical effectiveness through simulations.

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.
Paper Structure (22 sections, 10 theorems, 79 equations, 9 figures, 1 table)

This paper contains 22 sections, 10 theorems, 79 equations, 9 figures, 1 table.

Table of Contents

  1. Introduction
  2. Model description and preliminaries
  3. Model description
  4. Lateral vehicle dynamics with distributed tire friction dynamics
  5. State-space representation
  6. Assumptions and preliminaries
  7. Structural assumptions
  8. Assumptions for controller and observer design
  9. Stability
  10. Analysis via singular perturbation theory
  11. Reduced ODE subsystem
  12. Boundary layer PDE subsystem
  13. Stability analysis for sufficiently small $\epsilon$
  14. Stabilization
  15. State-feedback stabilization
  16. ...and 7 more sections

Key Result

Theorem 2.1

Suppose that $\Sigma \in C^0(\mathbb{R}^{2};\mathbf{M}_{2}(\mathbb{R}))$ and $h_1,h_2 \in C^0(\mathbb{R}^2;\mathbb{R}^2)$ are locally Lipschitz continuous, and $U \in C^0([0,T];\mathbb{R}^{2})$. Then, for all initial conditions (ICs) $(X_0,z_0) \triangleq (X(0),z(\cdot,0)) \in \altmathcal{X}$, there

Figures (9)

  • Figure 1: Single-track vehicle model.
  • Figure 2: Schematic representation of the ODE-PDE interconnection \ref{['eq:originalSystems']}.
  • Figure 3: Typical trends of the function $\Phi(\cdot)$ obtained for constant and exponentially decreasing pressure distributions $\bar{p}_i(\xi)$, $i \in \{1,2\}$.
  • Figure 4: Local stability chart for a semilinear single-track vehicle model \ref{['eq:originalSystems']} with constant pressure distribution and flexible tire carcass (Parametrization \ref{['param:2']}) with $b = 0$, linearized around the zero equilibrium $(X^\star, z^\star(\xi), U^\star) = (0,0,0)$, for different values of the understeer index $\frac{\tilde{C}_{1}(0)l_1}{\tilde{C}_{2}(0)l_2}$ and longitudinal speed $v_x$. The unstable regions (in white) correspond to combinations of parameters associated with micro-shimmy oscillations that are not detected by reduced order representations. Model parameters as in SemilinearV.
  • Figure 5: Open loop behavior of the lumped states and steering inputs, for $L$ as in \ref{['Lnum']}: (a) kinematic variables; (b) axle forces; (c) steering inputs.
  • ...and 4 more figures

Theorems & Definitions (22)

  • Theorem 2.1: Local existence and uniqueness of mild solutions
  • proof
  • Proposition 2.1: Strict dissipativity
  • proof
  • Lemma 2.1
  • proof
  • Remark 1
  • Proposition 3.1
  • Remark 2
  • Remark 3
  • ...and 12 more