# Hybrid Modelling and Sliding Mode Control of Semi-Active Suspension Systems for Both Ride Comfort and Road-Holding

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## Abstract

**:**

## 1. Introduction

## 2. Hybrid Automata

**Definition**

**1.**

- Q is a set of discrete states ${q}_{i}\in Q$;
- X is a set of continuous state vectors $x\in X\subseteq {\mathbb{R}}^{n}$;
- $Init$ is a set of initial hybrid states $({q}_{0},x\left(0\right))\in Init\subset Q\times {\mathbb{R}}^{n}$;
- F is a set of vector fields $f({q}_{i},x)\in F$: $Q\times {\mathbb{R}}^{n}\to {\mathbb{R}}^{n}$;
- I is a set of continuous invariants $Inv\left({q}_{i}\right)\in I$: $Q\to {\mathbb{R}}^{n}$, where $Inv\left({q}_{i}\right)$ restricts the continuous evolution within ${q}_{i}\in Q$;
- E is a set of discrete transitions ${e}_{i}\in E\subseteq Q\times Q$ to switch between discrete states ${q}_{i}\in Q$;
- G is a set of guard conditions $\mathcal{G}\left({e}_{i}\right)\in G$: $E\to {\mathbb{R}}^{n}$;
- R is a set of reset maps $\mathcal{R}({e}_{i},x)\in R$: $E\to {\mathbb{R}}^{n}\times {\mathbb{R}}^{n}$.

## 3. Modelling of Electro-Rheological Damper

## 4. The Quarter Car Suspension Model

## 5. The Sliding Mode Controller Design

## 6. Prototype Implementation and Simulation Results

## 7. Conclusions and Perspectives

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Table 1.**The vector fields and invariants of the discrete states in Figure 4.

Discrete State ${\mathit{q}}_{\mathit{i}}$ | Dynamics $\mathit{f}({\mathit{q}}_{\mathit{i}},\mathit{x})$ | Invariant $\mathbf{Inv}\left({\mathit{q}}_{\mathit{i}}\right)$ |
---|---|---|

${\dot{x}}_{1}={x}_{2}$ | ||

${\dot{x}}_{2}=-\frac{({k}_{s}+{k}_{e})}{{m}_{s}}({x}_{1}-{x}_{3})-\frac{{c}_{e}}{{m}_{s}}({x}_{2}-{x}_{4})-\frac{1}{{m}_{s}}{x}_{5}$ | ||

${q}_{1}$ | ${\dot{x}}_{3}={x}_{4}$ | ${x}_{2}-{x}_{4}<0$ |

${\dot{x}}_{4}=\frac{({k}_{s}+{k}_{e})}{{m}_{u}}({x}_{1}-{x}_{3})+\frac{{c}_{e}}{{m}_{u}}({x}_{2}-{x}_{4})+\frac{1}{{m}_{u}}{x}_{5}-\frac{{k}_{t}}{{m}_{u}}({x}_{3}-w)$ | ||

${\dot{x}}_{5}=-\frac{1}{\tau}({x}_{5}+u)$ | ||

${\dot{x}}_{1}={x}_{2}$ | ||

${\dot{x}}_{2}=-\frac{({k}_{s}+{k}_{e})}{{m}_{s}}({x}_{1}-{x}_{3})-\frac{{c}_{e}}{{m}_{s}}({x}_{2}-{x}_{4})-\frac{1}{{m}_{s}}{x}_{5}$ | ||

${q}_{2}$ | ${\dot{x}}_{3}={x}_{4}$ | ${x}_{2}-{x}_{4}>0$ |

${\dot{x}}_{4}=\frac{({k}_{s}+{k}_{e})}{{m}_{u}}({x}_{1}-{x}_{3})+\frac{{c}_{e}}{{m}_{u}}({x}_{2}-{x}_{4})+\frac{1}{{m}_{u}}{x}_{5}-\frac{{k}_{t}}{{m}_{u}}({x}_{3}-w)$ | ||

${\dot{x}}_{5}=-\frac{1}{\tau}({x}_{5}-u)$ |

**Table 2.**The guard conditions and reset maps of the discrete transitions in Figure 4.

Transition ${\mathit{e}}_{\mathit{i}}$ | Reset $\mathcal{R}({\mathit{e}}_{\mathit{i}},\mathit{x})$ | Guard $\mathcal{G}\left({\mathit{e}}_{\mathit{i}}\right)$ |
---|---|---|

${e}_{1}$ | $x:=x$ | ${x}_{2}-{x}_{4}\ge 0$ |

${e}_{2}$ | $x:=x$ | ${x}_{2}-{x}_{4}\le 0$ |

Parameter | Value | Unit |
---|---|---|

${m}_{s}$ | 2.28 | kg |

${m}_{u}$ | 0.26 | kg |

${k}_{s}$ | 1399 | N/m |

${k}_{e}$ | 186 | N/m |

${c}_{e}$ | 23 | N·s/m |

${k}_{t}$ | 12,270 | N/m |

$\tau $ | 40 | ms |

${c}_{{s}_{max}}$ | 5000 | N·s/m |

${c}_{{g}_{max}}$ | 3000 | N·s/m |

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**MDPI and ACS Style**

Aljarbouh, A.; Fayaz, M.
Hybrid Modelling and Sliding Mode Control of Semi-Active Suspension Systems for Both Ride Comfort and Road-Holding. *Symmetry* **2020**, *12*, 1286.
https://doi.org/10.3390/sym12081286

**AMA Style**

Aljarbouh A, Fayaz M.
Hybrid Modelling and Sliding Mode Control of Semi-Active Suspension Systems for Both Ride Comfort and Road-Holding. *Symmetry*. 2020; 12(8):1286.
https://doi.org/10.3390/sym12081286

**Chicago/Turabian Style**

Aljarbouh, Ayman, and Muhammad Fayaz.
2020. "Hybrid Modelling and Sliding Mode Control of Semi-Active Suspension Systems for Both Ride Comfort and Road-Holding" *Symmetry* 12, no. 8: 1286.
https://doi.org/10.3390/sym12081286