Non-linear dynamic modelling of a switching valve driven by pulse width modulation in the hydraulic braking system of a vehicle

Abstract

Fast-switching valves driven by puse width modulation have been widely used in the hydraulic braking systems (such as anti-lock braking systems or electric stability control systems) of vehicles, because of their lower cost and their functions which are similar to those of proportional valves. However, proportional valves have been widely investigated, whereas there is significantly less literature on the switching valves of the hydraulic control units in the braking systems of vehicles. In order to investigate the controllability of pressure, construction of an accurate theoretical model is the key to attaining this. Based on theoretical analysis and finite element analysis, this paper presents a non-linear dynamics model of a typical fast-switching valve. The non-linear model is composed of three submodels: a mechanical submodel, an electromagnetic submodel and a fluid dynamics submodel. The fluid momentum theorem combined with the simulation curve-fitting method is used to model accurately the fluid dynamics which are due to the varying flow field. The hydraulic force acting on the spool can be obtained directly from this model. Finally, the feasibility of the response time and the pressurization performance of the non-linear dynamics model is proved by comparing the simulation results from MATLAB/Simulink with the experimental results obtained on a test bench. The study shows that the model can be regarded as a predictive tool for future investigation.

An autonomous emergency braking strategy based on non‐linear model predictive deceleration control

“…The structure of IV is shown in Figure 4. Ignore the viscous force and friction force, the dynamics equations of spool can be obtained as follows [15,23,26]:…”

“…For system (26), there is a well-known lemma obtained as follows, which has been proved in [34]: Lemma 2. For system (26), the feasibility of the optimal control problem (30) at time t = 0 implies its feasibility for all t > 0. Then, for any t≥0 and 𝜏 ∈ (t , t + 𝛿] the optimal value function satisfies…”

“…Theorem 1. For system (26), suppose that assumption 1 holds, and the optimal control problem ( 30) is feasible at time t = 0. Then, the state trajectory will converge to the set Ω(ε) in finite time t 0 .…”

“…And we have the following lemma [34]: Lemma 1. For system (26), suppose that assumption 1 holds. Given Q > 0 and R > 0, there exist P > 0 and ε > 0 such that: 1) Kx ∈ U ;…”

“…The structure of IV is shown in Figure 4. Ignore the viscous force and friction force, the dynamics equations of spool can be obtained as follows [15, 23, 26]: $$$$\begin{equation}{m_s}\ddot z = {F_m} - {F_s} - {F_k}\end{equation}$$$$where m s and z are the mass and displacement of the spool, respectively, z = 0 mm when the valve is fully closed, z = 0.2 mm when the valve is fully open. F m is the electromagnetic force, F s is the hydraulic force, and F k is the spring force.…”