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

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

TL;DR

This work tackles local stabilization of a semilinear single-track vehicle model with distributed tire friction, formulating the dynamics as an ODE-PDE interconnection between a 2D rigid-body system and two semilinear bristle PDEs. A passivity-exploiting backstepping design leverages the PDE subsystem's dissipativity to achieve exponential stabilization around a prescribed equilibrium $(X^\star,z^\star)$, yielding both state-feedback and output-feedback controllers via a cascaded observer. The authors provide constructive proofs based on Lyapunov functionals that couple lumped and distributed states, establish local well-posedness and exponential convergence, and validate the approach through simulations under disturbances and uncertainties. Results indicate effective stabilization of oversteer at high speeds and robustness to measurement noise and delays, highlighting the method's potential for enhancing automotive safety and performance in real-world tire-road interactions.

Abstract

This paper addresses the local stabilization problem for semilinear single-track vehicle models with distributed tire friction dynamics, represented as interconnections of ordinary differential equations (ODEs) and hyperbolic partial differential equations (PDEs). A passivity-exploiting backstepping design is presented, which leverages the strict dissipativity properties of the PDE subsystem to achieve exponential stabilization of the considered ODE-PDE interconnection around a prescribed equilibrium. Sufficient conditions for local well-posedness and exponential convergence are derived by constructing a Lyapunov functional combining the lumped and distributed states. Both state-feedback and output-feedback controllers are synthesized, the latter relying on a cascaded observer. The theoretical results are corroborated with numerical simulations, considering non-ideal scenarios and accounting for external disturbances and uncertainties. Simulation results confirm that the proposed control strategy can effectively and robustly stabilize oversteer vehicles at high speeds, demonstrating the relevance of the approach for improving the safety and performance in automotive applications.

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

TL;DR

This work tackles local stabilization of a semilinear single-track vehicle model with distributed tire friction, formulating the dynamics as an ODE-PDE interconnection between a 2D rigid-body system and two semilinear bristle PDEs. A passivity-exploiting backstepping design leverages the PDE subsystem's dissipativity to achieve exponential stabilization around a prescribed equilibrium , yielding both state-feedback and output-feedback controllers via a cascaded observer. The authors provide constructive proofs based on Lyapunov functionals that couple lumped and distributed states, establish local well-posedness and exponential convergence, and validate the approach through simulations under disturbances and uncertainties. Results indicate effective stabilization of oversteer at high speeds and robustness to measurement noise and delays, highlighting the method's potential for enhancing automotive safety and performance in real-world tire-road interactions.

Abstract

This paper addresses the local stabilization problem for semilinear single-track vehicle models with distributed tire friction dynamics, represented as interconnections of ordinary differential equations (ODEs) and hyperbolic partial differential equations (PDEs). A passivity-exploiting backstepping design is presented, which leverages the strict dissipativity properties of the PDE subsystem to achieve exponential stabilization of the considered ODE-PDE interconnection around a prescribed equilibrium. Sufficient conditions for local well-posedness and exponential convergence are derived by constructing a Lyapunov functional combining the lumped and distributed states. Both state-feedback and output-feedback controllers are synthesized, the latter relying on a cascaded observer. The theoretical results are corroborated with numerical simulations, considering non-ideal scenarios and accounting for external disturbances and uncertainties. Simulation results confirm that the proposed control strategy can effectively and robustly stabilize oversteer vehicles at high speeds, demonstrating the relevance of the approach for improving the safety and performance in automotive applications.
Paper Structure (14 sections, 8 theorems, 75 equations, 10 figures, 1 table)

This paper contains 14 sections, 8 theorems, 75 equations, 10 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. Controller design
  8. State-feedback controller design
  9. Step 1
  10. Step 2
  11. Step 3
  12. Output-feedback controller design
  13. Simulation results
  14. Conclusions

Key Result

Theorem 2.1

Suppose that $\Sigma \in C^0(\mathbb{R}^{2};\mathbf{M}_{2}(\mathbb{R}))$ is 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 exists $t_\textnormal{max} \leq \infty$ such that

Figures (10)

  • Figure 1: Single-track vehicle model.
  • Figure 2: Schematic representation of the ODE-PDE interconnection \ref{['eq:originalSystems']}.
  • Figure 3: Normalized steady-state lateral tire forces $\bar{F}_{yi} \triangleq \frac{F_{yi}}{2F_{zi}}$, $i \in \{1,2\}$. Model parameters as in Table \ref{['tab:parameters']}.
  • Figure 4: True states (solid tick blue line) and observer estimates (dashed blue line) for the open loop system \ref{['eq:originalSystems']}-\ref{['eq:MatricesOOO']}, with $p=2$.
  • Figure 5: Open-loop convergence of the observer error estimate for three different values of observer gains $p=2$, 6, and 10.
  • ...and 5 more figures

Theorems & Definitions (21)

  • Theorem 2.1: Local existence and uniqueness of mild solutions
  • proof
  • Remark 1
  • Remark 2
  • Proposition 2.1: Strict dissipativity
  • proof
  • Proposition 2.2: Passivity
  • proof
  • Lemma 2.1
  • proof
  • ...and 11 more