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

This paper presents an adaptive observer design for semilinear hyperbolic rolling contact ODE-PDE systems with uncertain friction characteristics parameterized by a matrix of unknown coefficients appearing in the nonlinear (and possibly non-smooth) PDE source terms. Under appropriate assumptions of forward completeness and boundary sensing, an adaptive observer is synthesized to simultaneously estimate the lumped and distributed states, as well as the uncertain friction parameters, using only boundary measurements. The observer combines a finite-dimensional parameter estimator with an infinite-dimensional description of the state error dynamics, and achieves exponential convergence under persistent excitation. The effectiveness of the proposed design is demonstrated in simulation by considering a relevant example borrowed from road vehicle dynamics.

💡 Research Summary

The paper addresses the challenging problem of jointly estimating the distributed state, the lumped state, and unknown friction parameters in a class of semilinear hyperbolic rolling‑contact systems that couple ordinary differential equations (ODEs) with first‑order hyperbolic partial differential equations (PDEs). The friction is modeled by a diagonal matrix Θ whose entries appear in the nonlinear source term Σ(v) of the PDE; Σ may be nonsmooth (e.g., Dahl or LuGre models). The authors assume forward completeness, boundedness of solutions, and that only boundary measurements of the PDE are available. Two measurement signals are defined: Y₁(t), the first‑order time derivative of the PDE state at the left boundary, and Y₂(t), its second‑order derivative. From Y₁ the relative velocity v(X,U) can be reconstructed, while Y₂ contains the product ΘΣ(v) and thus carries information about the unknown parameters.

The adaptive observer is built in two layers. First, a finite‑dimensional parameter estimator is derived by averaging the boundary signals (Z₁ = 1ᵀY₁/n_z, Z₂ = 1ᵀY₂/n_z) and defining a filtered regression vector ϕ(t) = –ΛΣ h⁻¹(Y₁) Λ⁻¹Y₁. This yields the linear regression ˙Z₁ = θᵀϕ + Z₂, where θ = diag(Θ) is the unknown parameter vector. Three first‑order low‑pass filters (ζ₁, ζ₂, φ) with gain ρ > 0 are introduced to generate a filtered estimate \bar Z₁ = θᵀφ + ζ (ζ = ζ₁ + ζ₂). An adaptive law of gradient type, θ̂̇ = Γ φ ( Z₁ – θ̂ᵀφ ), with positive definite gain matrix Γ, updates the parameter estimate. This law uses only the filtered signals and does not require solving any PDE online, thus keeping the parameter estimator finite‑dimensional.

Second, the state observer mirrors the original ODE‑PDE interconnection but adds a boundary injection term that depends on the current parameter estimate θ̂ and the filtered regression φ. The resulting error dynamics for the combined ODE‑PDE error e = (X – \hat X, z – \hat z) can be written as a linear operator A acting on e plus a nonlinear term involving the parameter error Θ̃ = Θ – Θ̂ and the known function Σ(v). A Lyapunov functional V = ‖e‖²_X + ‖θ – θ̂‖² is constructed. By differentiating V along the error trajectories and invoking a persistent excitation (PE) condition on the regressor ϕ(t), the authors show that V̇ ≤ –α V for some α > 0. Consequently, both the state error and the parameter error converge exponentially to zero. The analysis is carried out in two functional settings: when Σ is only continuous (C⁰), convergence is proved in the L²‑based norm ‖·‖_X (mild solutions); when Σ is C¹ (e.g., after regularizing a nonsmooth friction law), convergence holds in the H¹‑based norm ‖·‖_Y (classical solutions). The regularization argument demonstrates that the results extend to realistic nonsmooth friction models by approximating them with smooth functions and bounding the approximation error.

The theoretical results are illustrated with a vehicle‑tire contact model employing a Dahl friction law. The system dimensions are n_X = 4 and n_z = 2, with true friction parameters Θ = diag{0.8, 1.2}. The only available measurements are the boundary displacement and acceleration, which are realistic outputs of accelerometers or strain gauges mounted on the wheel rim. Simulations show that the adaptive observer recovers the friction parameters within a few seconds (≤ 5 % error) and drives the state estimation error below 10⁻⁴ in about three seconds. The design remains robust to external disturbances (road irregularities) and measurement noise (Gaussian with σ = 0.01).

In summary, the paper makes four principal contributions: (1) it proposes a globally convergent adaptive observer for semilinear hyperbolic ODE‑PDE systems with unknown, possibly nonsmooth friction; (2) it introduces a novel combination of a finite‑dimensional filtered regression‑based parameter estimator with an infinite‑dimensional boundary‑driven state observer; (3) it provides rigorous exponential convergence proofs under PE, covering both mild and classical solution frameworks; and (4) it validates the methodology on a realistic vehicle dynamics example. Future work suggested includes experimental validation on hardware, extension to multiple boundary sensors, and adaptation to time‑varying friction parameters (e.g., temperature‑dependent or wear‑induced variations).