# An ARX Model-Based Predictive Control of a Semi-Active Vehicle Suspension to Improve Passenger Comfort and Road-Holding

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## Abstract

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## 1. Introduction

## 2. One-Quarter Semi-Active Suspension, Including the Actuator’s Dynamics

## 3. Predictive Control Based on an ARX Model

#### 3.1. Driver Block

#### 3.2. Predictive Model

## 4. Performance Criteria

- Passenger comfort in low frequencies.Below 5 Hz, keep the relation gain between the sprung mass displacement and the road profile, i.e., (${z}_{s}/{z}_{p}$) less than 2. The aim is to decrease the maximum peak (around 1 Hz for an average city vehicle). For this criterion, ${z}_{p}$ is a sinusoidal signal defined by ${z}_{p}=0.015sin\left(wt\right)$ m.
- Vehicle stability between 0 and 15 Hz.Measured through the division of the relation (${z}_{us}/{z}_{p}$), i.e., unsprung mass displacement over the road profile. The goal is to cut down the maximum peak observed in the range [10–13] Hz, for an average city vehicle. For this test, ${z}_{p}$ is represented by $0.001sin\left(wt\right)$ m.
- Passenger Comfort; the acceleration criterion.From 4 to 30 Hz, keep the root mean square acceleration (rms) of the sprung mass (one quarter of the chassis), below the maximum rms vertical acceleration limit developed by the International Standard ISO 2631 as explained in [2], to assure passenger comfort for up to 8 h. To run this test, apply the ${z}_{p}$ as in vehicle stability.
- Suspension Deflection.From 0 to 4 Hz, keep (${z}_{s}-{z}_{us}$) within the physical limits of shock absorber to avoid unmodeled dynamics and a premature suspension wear-off. The herein employed MR damper has $\pm 2.5$ cm as displacement physical limits, as explained in [47]. For this test, ${z}_{p}$ is the same defined for passenger comfort in low frequencies.
- Performance in time domain.With ${z}_{p}$ representing a road bump profile; the objective is to decrease, as much as possible, overshoot, undershoot and settling time for: (${z}_{s}/{z}_{p}$), (${z}_{us}/{z}_{p}$), rms of ${z}_{us}$, and (${z}_{s}-{z}_{us}$). The degree of improvement is measured with respect to passive suspension.

## 5. Results and Discussion

#### 5.1. Simulation Work

#### 5.2. Predictive Controller Design

#### 5.3. Results in the Frequency Domain

#### 5.4. Results in Time Domain

## 6. Conclusions and Future Work

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

ARX | AutoRegressive with eXogenous input |

DOF | Degrees of Freedom |

LPV | Linear Parameter-Varying control |

MPC | Model Predictive Control |

MR | Magneto-Rheological |

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**Figure 1.**Semi-active One-quarter-vehicle suspension model with a magneto-rheological (MR) damper. The actuator is represented by means of the Bouc-Wen model. Figure taken and modified from [46].

**Figure 4.**Unsprung mass displacement gain $({z}_{us}/{z}_{p})$ for the passive and predictive controller.

**Figure 5.**Chassis acceleration relation ${\ddot{z}}_{s}$ for the passive and predictive controller. Black line indicates the limit of the acceleration comfort.

**Figure 6.**Suspension deflection (${z}_{s}-{z}_{us}$) for the passive and predictive controlled suspensions.

**Figure 7.**Chassis or sprung mass displacement for the predictive controlled and passive suspensions.

Step | Description |
---|---|

1 | Get reference ${z}_{s}^{sp}\left(k\right)$ |

2 | Get process output ${z}_{s}\left(k\right)$ |

3 | Compute ${{\phi}_{i}}^{\left(\lambda \right)}$, ${{\delta}_{i}}^{\left(\lambda \right)}$, ${\mu}^{\left(\lambda \right)}$ |

4 | Compute ${z}_{s}^{d}\left(\right)open="("\; close=")">k+\lambda \mid k$ |

5 | Compute ${{\widehat{e}}_{i}}^{\left(\lambda \right)}$, ${{\widehat{g}}_{i}}^{\left(\lambda \right)}$, ${\widehat{h}}^{\left(\lambda \right)}$ |

6 | Compute control signal ${i}_{damp}\left(k\right)$ |

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

${m}_{s}$ | 450 kg |

${m}_{us}$ | 45 kg |

${k}_{s}$ | 16,000 N/m |

${k}_{t}$ | 210,000 N/m |

c | 1000 Ns/m |

Suspension | (${\mathit{z}}_{\mathit{s}}/{\mathit{z}}_{\mathit{p}}$) Gain | (${\mathit{z}}_{\mathbf{us}}/{\mathit{z}}_{\mathit{p}}$) Gain | ${\ddot{z}}_{\mathit{s}}$ | Max (${\mathit{z}}_{\mathit{s}}-{\mathit{z}}_{\mathbf{us}}$) in cm. |
---|---|---|---|---|

Passive | 3.23 | 2.97 | ✓ | 2.77 × |

Semi-active Predictive | 1.64 | 1.95 | ✓ | 1.63 ✓ |

**Table 4.**Percentage of improvement of reported work and proposed predictive control strategy against passive case. Result are reported in frequency domain.

LPV/${\mathit{H}}_{\mathit{\infty}}$ [51] | LPV/${\mathit{H}}_{\mathit{\infty}}$ [52] | ${\mathit{H}}_{\mathit{\infty}}$ [18] | ${\mathit{H}}_{2}$ [21] | Predictive Control | |
---|---|---|---|---|---|

(${z}_{s}/{z}_{p}$) | Not Reported | 12.1 | 46 | 40.8 | 49.2 |

(${z}_{us}/{z}_{p}$) | 8 | 7 | 39.7 | 40 | 68.9 |

$\left({z}_{s}-{z}_{us}\right)$ | 16.2 | 8 | 65 | 61 | 41 |

${\ddot{z}}_{s}$ | 18 | 25.4 | Negative | Negative | 14 |

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**MDPI and ACS Style**

Piñón, A.; Favela-Contreras, A.; Félix-Herrán, L.C.; Beltran-Carbajal, F.; Lozoya, C.
An ARX Model-Based Predictive Control of a Semi-Active Vehicle Suspension to Improve Passenger Comfort and Road-Holding. *Actuators* **2021**, *10*, 47.
https://doi.org/10.3390/act10030047

**AMA Style**

Piñón A, Favela-Contreras A, Félix-Herrán LC, Beltran-Carbajal F, Lozoya C.
An ARX Model-Based Predictive Control of a Semi-Active Vehicle Suspension to Improve Passenger Comfort and Road-Holding. *Actuators*. 2021; 10(3):47.
https://doi.org/10.3390/act10030047

**Chicago/Turabian Style**

Piñón, Alejandro, Antonio Favela-Contreras, Luis C. Félix-Herrán, Francisco Beltran-Carbajal, and Camilo Lozoya.
2021. "An ARX Model-Based Predictive Control of a Semi-Active Vehicle Suspension to Improve Passenger Comfort and Road-Holding" *Actuators* 10, no. 3: 47.
https://doi.org/10.3390/act10030047