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

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.

💡 Research Summary

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This paper tackles the challenging problem of locally stabilizing a semilinear single‑track vehicle model that incorporates distributed tire‑road friction dynamics. Unlike conventional single‑track models that approximate tire forces with static nonlinear functions, the authors adopt a physically‑based distributed friction model (e.g., Dahl, LuGre) in which the internal state of each tire patch is described by a hyperbolic partial differential equation (PDE). The vehicle’s lateral dynamics (lateral velocity and yaw rate) are modeled by ordinary differential equations (ODEs) and are coupled to the PDEs through the slip angles and steering inputs, resulting in an ODE‑PDE interconnection of infinite dimension.

A central observation is that the PDE subsystem is strictly dissipative: the friction mechanisms inherently dissipate energy, which can be captured by a quadratic energy functional. The authors exploit this property in a backstepping design. First, they rewrite the coupled dynamics in a compact state‑space form and construct a Lyapunov functional that combines the ODE energy (quadratic in the lumped states) with the PDE energy (integral of the squared distributed states) and appropriate cross‑terms. By differentiating this functional along system trajectories, they show that the PDE contribution yields a negative definite term thanks to strict dissipativity, while the backstepping gains are chosen to cancel the remaining coupling terms. Consequently, the closed‑loop system satisfies a differential inequality (\dot V \le -\alpha V) for some (\alpha>0), guaranteeing local exponential stability of the equilibrium.

Two controller families are synthesized. The first is a full‑state feedback law that assumes direct measurement of both the lumped vehicle states and the distributed tire states. The second is an output‑feedback scheme suitable for practical implementation. It employs a cascaded observer: a boundary observer estimates the distributed states from available boundary measurements, while a Luenberger‑type observer reconstructs the ODE states. Separate Lyapunov analyses prove exponential convergence of the observer errors, and the combined observer‑controller loop inherits the same exponential stability as the full‑state case.

Numerical simulations validate the theory. A high‑speed (≈150 km/h) oversteer scenario is simulated with realistic parameter uncertainties (±20 %) and slow‑varying wind disturbances. The proposed backstepping controller reduces the peak yaw‑rate and lateral‑velocity overshoot by more than 60 % compared with a conventional lumped‑model ESC, and shortens the settling time by roughly 30 %. The observer‑based output feedback achieves comparable performance despite sensor noise, demonstrating robustness. The simulations also illustrate that the control law uses only bounded functionals of the distributed states, preserving the semilinear structure of the original ODE‑PDE system.

The paper’s contributions are fourfold: (1) introducing a rigorous distributed‑friction PDE model into vehicle dynamics, (2) leveraging the intrinsic passivity of the PDE to design a backstepping controller with provable exponential convergence, (3) providing both state‑feedback and observer‑based output‑feedback implementations, and (4) demonstrating robustness to non‑Lipschitz friction nonlinearities, parameter variations, and external disturbances. The authors suggest future work on global (rather than local) stability guarantees, experimental validation on a test vehicle, and computationally efficient implementations (e.g., model reduction) to enable real‑time deployment.