Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

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

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

TL;DR

This work tackles the joint estimation of states and friction parameters in a semilinear, boundary‑sensing ODE–PDE interconnection for rolling contact. It introduces a boundary adaptive observer that combines a finite‑dimensional parameter estimator with an infinite‑dimensional state observer, achieving exponential convergence under persistent excitation. The friction parameters are captured by a diagonal matrix $\Theta$, and the method accommodates non‑smooth nonlinear sources in the PDE. Validation on a road‑vehicle lateral dynamics model with distributed friction demonstrates accurate state reconstruction and rapid, robust parameter convergence, highlighting practical relevance for vehicle safety and control applications.

Abstract

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.

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

TL;DR

This work tackles the joint estimation of states and friction parameters in a semilinear, boundary‑sensing ODE–PDE interconnection for rolling contact. It introduces a boundary adaptive observer that combines a finite‑dimensional parameter estimator with an infinite‑dimensional state observer, achieving exponential convergence under persistent excitation. The friction parameters are captured by a diagonal matrix , and the method accommodates non‑smooth nonlinear sources in the PDE. Validation on a road‑vehicle lateral dynamics model with distributed friction demonstrates accurate state reconstruction and rapid, robust parameter convergence, highlighting practical relevance for vehicle safety and control applications.

Abstract

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.
Paper Structure (12 sections, 5 theorems, 55 equations, 5 figures, 1 table)

This paper contains 12 sections, 5 theorems, 55 equations, 5 figures, 1 table.

Table of Contents

  1. Introduction
  2. Problem statement
  3. Structure of the considered ODE-PDE system
  4. Assumptions and considerations on the available measurements
  5. Adaptive observer design
  6. Parameter identifiers and adaptive laws
  7. Adaptive observer
  8. An example from road vehicle dynamics
  9. Lateral vehicle dynamics
  10. Simulation results
  11. Conclusion
  12. Proof of Lemma \ref{['lemma:convergence']}

Key Result

Theorem 2.1

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

Figures (5)

  • Figure 1: Single-track vehicle model. The kinematic variables are depicted in blue, the dynamic ones are in red.
  • Figure 2: Convergence of the parameter estimate $\hat{\theta}(t)$ to the true value $\theta$.
  • Figure 3: True states (solid tick blue line), observer estimates (dotted blue line), and observer error (solid orange line).
  • Figure 4: Convergence of the observer error estimate.
  • Figure 5: Evolution of the PDE error state $\tilde{z}_1(\xi,t)$, along with its IC (blue line) and BC (orange line).

Theorems & Definitions (12)

  • Theorem 2.1: Local existence and uniqueness of solutions
  • proof
  • Remark 1
  • Theorem 3.1
  • proof
  • Lemma 3.1
  • proof
  • Theorem 3.2
  • proof
  • Theorem 3.3
  • ...and 2 more