Boundary adaptive observer design for semilinear hyperbolic rolling contact ODE-PDE systems with uncertain friction

Technical Summary: Boundary Adaptive Observer for Semilinear Hyperbolic Rolling Contact ODE-PDE Systems

Problem Statement
This paper addresses the joint state and parameter estimation problem for a class of semilinear, homodirectional Ordinary Differential Equation-Partial Differential Equation (ODE-PDE) systems. These systems model rolling contact dynamics, such as those found in vehicle tire-road interactions, where rigid body dynamics (ODEs) are coupled with distributed friction effects described by hyperbolic PDEs. The primary challenge lies in the presence of uncertain friction characteristics, parameterized by a diagonal matrix of unknown coefficients ($\Theta$) appearing in nonlinear, and potentially non-smooth, PDE source terms. The objective is to simultaneously estimate the lumped states (rigid body), distributed states (contact interface deformation), and the uncertain friction parameters using only boundary measurements from the PDE subsystem. Unlike classical adaptive control where estimation serves stabilization, the focus here is on accurate reconstruction per se for monitoring and diagnostics.

Methodology
The proposed solution synthesizes a boundary adaptive observer that combines a finite-dimensional parameter estimator with an infinite-dimensional state observer.

1. System Formulation: The system is modeled as a coupled ODE-PDE where the PDE subsystem features transport velocities and source terms governed by unknown parameters. The coupling occurs through integral functionals of the distributed state and boundary terms.
2. Measurement Strategy: The design relies on two sets of boundary measurements:
   • $Y_1(t)$: The time derivative of the distributed state at the boundary, allowing the reconstruction of the rigid relative velocity.
   • $Y_2(t)$: The second time derivative of the distributed state at the boundary, which provides a linear parametric model for the unknown friction parameters.
3. Parameter Estimation: A finite-dimensional adaptive law is derived based on filtered regression equations. A gradient law with an integral cost function is employed to estimate the unknown parameter vector $\theta$ (derived from $\Theta$). The design utilizes standard filters to generate regressor signals and ensures boundedness of the estimation error.
4. State Estimation: An infinite-dimensional observer is constructed to estimate the lumped and distributed states. The observer dynamics incorporate the estimated parameters and boundary measurements. The error dynamics are analyzed using Lyapunov stability theory.
5. Convergence Analysis:
   • For systems with continuous but non-smooth source terms (typical of friction models like Dahl), convergence is analyzed in the $L^2$-based norm ($\|\cdot\|_X$). The analysis involves a regularization technique where the non-smooth problem is approximated by a sequence of smooth problems, allowing the application of standard Lyapunov arguments.
   • For systems with continuously differentiable source terms, convergence is established in the stronger $H^1$-based norm ($\|\cdot\|_Y$).
   • Exponential convergence of both state and parameter estimation errors is guaranteed under the assumption of forward completeness and the Persistent Excitation (PE) condition of the filtered regressor signal.

Key Contributions

• Novel Observer Architecture: The paper proposes a boundary-sensing-based adaptive observer specifically for semilinear, homodirectional ODE-PDE systems with unknown parameters in the PDE source terms. This extends previous work which was largely limited to linear or heterodirectional systems.
• Handling Non-Smoothness: The methodology addresses the mathematical challenge of non-smooth friction terms (e.g., absolute values in source terms) by utilizing regularization techniques to prove convergence for mild solutions, while also covering classical solutions for smoother cases.
• Joint Estimation: The design achieves simultaneous estimation of lumped states, distributed states, and uncertain friction parameters using only boundary data, avoiding the need for internal state sensing.
• Theoretical Guarantees: The paper provides rigorous proofs for the existence of unique solutions and establishes exponential convergence rates for the estimation errors under PE conditions.

Results
The effectiveness of the proposed design is validated through numerical simulations using a single-track vehicle dynamics model with distributed Dahl-type tire friction.

• Simulation Setup: The model includes a rigid vehicle body coupled with two distributed PDEs representing tire bristle deflection. The simulation involves a step change in friction parameters to mimic a $\mu$-split maneuver (sudden change in road friction).
• Performance:
  • Parameter Convergence: The estimated friction parameters ($\hat{\theta}$) converged rapidly to the true values (within 2 seconds for the initial phase and 8 seconds after a step change), demonstrating robustness to abrupt parameter variations.
  • State Convergence: The observer successfully reconstructed both the lumped states (lateral velocity, yaw rate) and the distributed states. The error norms decayed exponentially, reaching near-zero values within 0.25 seconds.
  • Robustness: Additional simulations indicated that the observer maintains satisfactory performance even when the transport velocity (longitudinal speed) used in the observer is mismatched from the true value, suggesting robustness to parameter uncertainty in the transport term.

Significance and Claims
The authors claim that this work fills a gap in the literature regarding adaptive observers for coupled ODE-PDE systems with nonlinear, non-smooth source terms and unknown parameters. The significance lies in the ability to perform accurate state and parameter reconstruction for rolling contact systems using only boundary measurements, which is critical for applications like Advanced Driver Assistance Systems (ADAS) and fleet coordination where direct measurement of friction coefficients is impossible. The paper emphasizes that the proposed framework is distinct from control-oriented adaptive designs, focusing instead on the reliability of the estimation itself. Theoretical results are limited to forward-complete systems with specific structural conditions on the nonlinearities, and the convergence guarantees are contingent upon the persistent excitation of the input signals. Future work, as noted by the authors, is directed toward experimental validation and extension to bidirectional PDEs and multi-contact configurations.