Existence and stability of limit cycles in control of anti-lock braking systems with two boundaries via perturbation theory

ABSTRACT

This paper presents a two-phase control logic for anti-lock braking systems (ABS). ABS are by now a standard component in every modern car, preventing the wheels from going into a lock situation where the wheels are fixed by the brake and the stopping distances are greatly prolonged. There are different approaches to such control logics. An ABS design proposed in recent literature controls the wheel's slip by creating stable limit cycles in the corresponding phase space. This design is modified via an analytical approach that is derived from perturbation theory. Simulation results document shorter braking distance compared to available tests in the literature.

KEYWORDS:

• Vehicle dynamics
• closed-loop switching control
• ABS
• quarter car model
• hybrid systems
• limit cycles
• non-smooth dynamics
• perturbation theory
• stability

Acknowledgements

This research started during the stay of the third author in the University of Cologne under a Humboldt Postdoctoral Fellowship. The authors thank anonymous referees for constructive remarks that helped to improve the paper.

Disclosure statement

No potential conflict of interest was reported by the authors.

Notes

1. We omit here any rolling friction.

2. As mentioned in Olson, Shaw, and tephan (2005), in the special case of packed snow, we find μ(1; θ_r) = max μ(λ; θ_r) rendering the concept of an ABS useless. In practice, ABS are here manually or automatically switched off.

3. Nevertheless, the control logic is needed to work at any occurring velocity.

4. Since it is physically not possible that the braking pressure becomes negativem we find T_b ≥ 0. Thus, to be more precise, ${\dot{T}}_b=u$ is restricted to u ∈ {0, k₁} at T_b = 0.

5. By this restriction, we will stay within the manifold when adding two vectors u, v ∈ Σ₀, simplifying the calculations. Since every Poincaré section is as a hypersurface locally equivalent to ${\mathbb{R}}^{n-1}$ and our analysis is operating locally, we do not lose any generality by this confinement.

6. The system is a special case of more general, cylindrical phase spaces such as $\mathcal{V}:=\{V_0+\left[(v-V_0)\text{mod}{V}_2\right]|v\in \mathbb{R}\}$, where V₀ < V₁ < V₂ and Σ₀ = {(x, v) ∈ Ω∣v = V₀≅V₂} and Σ₁ = {(x, v) ∈ Ω∣v = V₁} that can also be analysed by this method.

Additional information

Funding

The research is partially supported by National Science Foundation [grant number CMMI-1436856];Russian Foundation for Basic Research [grant number 13-01-00347].