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Optimal Control Strategies for Energy Harvesting by Regenerative Shock Absorbers in Cars ^{†}

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## Abstract

**:**

## 1. Introduction

## 2. The Model and the Overall Control Architecture

#### 2.1. The Full-Car Regenerative Suspension Model

#### 2.2. Power Flow Analysis

- Ideally one would like to dispose of a control action capable to make both quantities large and sign defined for all time:$${P}_{m}\left(t\right)<<0,\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}{P}_{s}\left(t\right)>>0\phantom{\rule{0.277778em}{0ex}}\mathrm{and}\phantom{\rule{0.277778em}{0ex}}{P}_{m}\left(t\right)=-{P}_{s}\left(t\right),\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}$$If this were possible it would imply $\dot{E}\left(t\right)=0$, which would ensure that the system has a perfect transfer of energy from the road to the electrical batteries.
- Because ${f}_{i}\left(t\right)$ are directly manipulable variables, it is quite easy to make ${P}_{m}\left(t\right)<0$ sign-defined. However,$${\int}_{0}^{T}|{P}_{m}\left(t\right)|dt\le {\int}_{0}^{T}\left|{P}_{s}\left(t\right)\right|dt$$
- Observe first that ${P}_{s}$ is zero when the road profiles are all completely flat (${\dot{z}}_{ri}=0,\phantom{\rule{0.166667em}{0ex}}i=1,\dots ,4$) or when the tire deflections are all zero, $({z}_{ui}\left(t\right)-{z}_{ri}\left(t\right)=0),\phantom{\rule{0.166667em}{0ex}}i=1,\dots ,4$. Thus, high levels of energy harvesting on flat roads are incompatible with good road handling performance. From a control perspective, observe also that ${P}_{s}$ depends on the regulated variables ${z}_{ui}$. Thus, if ${P}_{s}$ were made sign-defined and as large as possible, viz. ${P}_{s}>>0$ via a suitable control law this would increase the harvested energy regardless of the behavior (the sign) of ${P}_{m}$. Roughly speaking, making ${P}_{s}>0$ it would provide a barrier for the internal energy of the system from flowing back towards the road.

#### 2.3. Electromechanical Actuator Model

#### 2.4. The Overall Control Architecture

## 3. RVC Control Design Specifications and LMI Based Designs

#### 3.1. Ride Comfort and Road Handling Control Specifications

#### 3.2. Energy Harvesting Control Specification

#### 3.3. State-Space Realization for the RVC Control Synthesis

#### 3.4. Frequency Shaping of the State-Space Realization

#### 3.5. ${H}_{2}$ and ${H}_{\infty}$ Optimal State Feedback Designs

#### 3.5.1. Optimal ${H}_{\infty}$ Control Synthesis

#### 3.5.2. Optimal ${H}_{2}$ Control Synthesis

## 4. Simulation Results

**C**straight road at 70 Km/h. Figure 5 depicts the corresponding profiles.

#### 4.1. Simulations for Ride Index = 0.25

#### 4.2. Simulations for Ride Index = 0.47

#### 4.3. Simulations for Ride Index = 0.70

#### 4.4. Average Harvested Electrical Power

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 5.**Road profiles ${z}_{r1}\left(t\right)$, ${z}_{r2}\left(t\right)$, ${z}_{r3}\left(t\right)$ and ${z}_{r4}\left(t\right)$ used in the simulations.

**Figure 6.**(

**top**) Accelerations ${\ddot{z}}_{s1}\left(t\right)$, (

**middle**) Actuation current ${i}_{1}\left(t\right)$, (

**down**) Instantaneous harvested electrical power ${P}_{e1}\left(t\right)$.

**Figure 7.**(

**top**) Accelerations ${\ddot{z}}_{s1}\left(t\right)$, (

**middle**) Actuation current ${i}_{1}\left(t\right)$, (

**down**) Instantaneous harvested electrical power ${P}_{e1}\left(t\right)$.

**Figure 8.**(

**top**) Accelerations ${\ddot{z}}_{s1}\left(t\right)$, (

**middle**) Actuation current ${i}_{1}\left(t\right)$, (

**down**) Instantaneous harvested electrical power ${P}_{e1}\left(t\right)$.

RI = 0.25 (Excellent Comfort/Smallest Harvesting) | ||||
---|---|---|---|---|

Control | $\mathbf{\rho}$ | $\mathbf{\beta}$ | $\mathbf{\gamma}$ | $\mathbf{\alpha}$ |

H${}_{\infty ,dec}$ | 0.2067 | 100 | 10 | 30 |

H${}_{\infty ,cen}$ | 0.42 | 3.45 | 1 | 27 |

H${}_{2,cen}$ | 0.5 | 2 | 1 | 29 |

RI = 0.47 (Trade-off between Comfort/Harvesting) | ||||
---|---|---|---|---|

Control | $\mathbf{\rho}$ | $\mathbf{\beta}$ | $\mathbf{\gamma}$ | $\mathbf{\alpha}$ |

H${}_{\infty ,dec}$ | 0.323 | 100 | 10 | 42 |

H${}_{\infty ,cen}$ | 1.67 | 2.25 | 1 | 30 |

H${}_{2,cen}$ | 0.82578 | 3.764 | 1 | 31 |

RI = 0.25 (Perceivable Discomfort/Largest Harvesting) | ||||
---|---|---|---|---|

Control | $\mathbf{\rho}$ | $\mathbf{\beta}$ | $\mathbf{\gamma}$ | $\mathbf{\alpha}$ |

H${}_{\infty ,dec}$ | 0.42 | 100 | 10 | 47 |

H${}_{\infty ,cen}$ | 2.82 | 3 | 1 | 33 |

H${}_{2,cen}$ | 2.167 | 4–69 | 1 | 33 |

**Table 4.**Average harvest electrical power. The percentages express the improvements with respect to the H${}_{\infty ,dec}$ control achievements for the same RI.

Average Harvest Electrical Power | |||
---|---|---|---|

Control | RI = 0.25 | RI = 0.47 | RI = 0.70 |

H${}_{\infty ,dec}$ | 4 × 100 W | 4 × 124 W | 4 × 153 W |

H${}_{\infty ,cen}$ | 4 × 120 W (+20%) | 4 × 141 W (+13%) | 4 × 160 W (+4.5%) |

H${}_{2,cen}$ | 4 × 141 W (+40%) | 4 × 154 W (+24%) | 4 × 168 W (+10%) |

© 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

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**MDPI and ACS Style**

Casavola, A.; Tedesco, F.; Vaglica, P.
^{†}. *Vibration* **2020**, *3*, 99-115.
https://doi.org/10.3390/vibration3020009

**AMA Style**

Casavola A, Tedesco F, Vaglica P.
^{†}. *Vibration*. 2020; 3(2):99-115.
https://doi.org/10.3390/vibration3020009

**Chicago/Turabian Style**

Casavola, Alessandro, Francesco Tedesco, and Pasquale Vaglica.
2020. "^{†}" *Vibration* 3, no. 2: 99-115.
https://doi.org/10.3390/vibration3020009