# A Novel Iterative Linear Matrix Inequality Design Procedure for Passive Inter-Substructure Vibration Control

^{1}

^{2}

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## Abstract

**:**

## 1. Introduction

## 2. Connections between Interstory FVDs and Decentralized Velocity-Feedback Controllers

## 3. Design of Interstory FVD Systems Using a Decentralized SOF ${\mathit{H}}_{\infty}$ Approach

#### 3.1. Decentralized SOF ${H}_{\infty}$ Controllers

**Remark**

**1.**

**Remark**

**2.**

#### 3.2. ILMI Design Procedure

#### 3.2.1. Initialization

#### 3.2.2. Iterations

**Step****i.a**- Solve the auxiliary LMI optimization problem ${\mathcal{P}}_{\phantom{\rule{-0.166667em}{0ex}}\mathrm{aux}}({\mathit{X}}_{R},{\mathit{Y}}_{R})$ with fixed matrices ${\mathit{X}}_{R}={\mathit{X}}_{R}^{(i-1)}$ and ${\mathit{Y}}_{R}={\mathit{Y}}_{R}^{(i-1)}$ to obtain an optimal triplet $\left(\right)$.
**Step****i.b**- Solve the structured LMI optimization problem ${\mathcal{P}}_{\phantom{\rule{-0.166667em}{0ex}}\mathrm{sof}\phantom{\rule{0.166667em}{0ex}}}(\mathit{L},\mathbb{D})$ with fixed matrix $\mathit{L}={\mathit{L}}^{\left(i\right)}$ to obtain an optimal quartet $\left(\right)$ and a diagonal gain matrix ${\mathit{K}}^{\left(i\right)}={\mathit{Y}}_{R}^{\left(i\right)}{\left\{{\mathit{X}}_{R}^{\left(i\right)}\right\}}^{-1}$ with associated $\gamma $-value ${\gamma}_{{K}^{\left(i\right)}}\le {\left\{{\eta}_{i}\right\}}^{1/2}.$

**Theorem**

**1.**

**Proof.**

## 4. Numerical Results

#### 4.1. FVD System Design

`null()`, has the following form

#### 4.2. Seismic Response

**Remark**

**3.**

**Remark**

**4.**

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Abbreviations

BI | base isolation |

BMI | bilinear matrix inequality |

BRL | bounded real lemma |

FRF | frequency response function |

FVD | fluid viscous damper |

ILMI | iterative linear matrix inequality |

ISSD | inter-substructure damper |

LMI | linear matrix inequaltiy |

SOF | static output feedback |

SSF | static state feedback |

SVF | static velocity feedback |

TD | tuned damper |

TMD | tuned mass damper |

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**Figure 2.**Schematic model of a multistory building equipped with a complete set of interstory fluid viscous dampers.

**Figure 3.**Multistory building equipped with a complete set of ideal interstory force-actuation devices and collocated interstory-velocity sensors.

**Figure 4.**Maximum singular values of the FRFs corresponding to the uncontrolled building (black dotted line), the optimal SSF controller (blue dash-dotted line) and the computed FVD system (red solid line). (

**a**) Overall view. (

**b**) Close-up view of the main resonant peaks.

**Figure 5.**El Centro 1940 ground-acceleration seismic record, full-scale 180-component with an absolute acceleration-peak of 3.417 m/s${}^{2}$.

**Figure 6.**Time-response peak-values. (

**a**) Maximum absolute interstory drifts ($\times {10}^{-2}$ m). (

**b**) Maximum absolute total accelerations (m/s${}^{2}$). (

**c**) Maximum absolute actuation forces ($\times {10}^{6}$ N).

**Figure 7.**Close-up view of the interstory-drift time-response ${r}_{3}\left(t\right)$ for the FVD system and the optimal SSF ${H}_{\infty}$ controller. The black dotted line in the background represents the clipped response of the uncontrolled building.

**Figure 8.**Close-up view of the total-acceleration time-response ${a}_{3}\left(t\right)$ for the FVD system and the optimal SSF ${H}_{\infty}$ controller. The black dotted line in the background represents the clipped response of the uncontrolled building.

**Figure 9.**Actuation-force time-history ${u}_{3}\left(t\right)$ for the FVD system and the optimal SSF ${H}_{\infty}$ controller.

**Figure 10.**Plots of actuation force versus interstory drift in the 3rd story level corresponding to the FVD system and the optimal SSF ${H}_{\infty}$ controller.

**Figure 11.**Plots of actuation force versus interstory velocity in the 3rd story level corresponding to the FVD system and the optimal SSF ${H}_{\infty}$ controller.

Mode | 1 | 2 | 3 | 4 | 5 |
---|---|---|---|---|---|

natural frequency (Hz) | 1.008 | 2.825 | 4.493 | 5.797 | 6.773 |

Step | 0 | 1 | 2 | 3 | 4 | 5 | 6 |
---|---|---|---|---|---|---|---|

Upper bound ${\eta}_{i}^{1/2}$ | 0.0933 | 0.0911 | 0.0902 | 0.0900 | 0.0899 | 0.0898 | 0.0898 |

Building Level | 1 | 2 | 3 | 4 | 5 |
---|---|---|---|---|---|

Damping coefficient ${\widehat{c}}_{i}$ ($\times {10}^{6}$ Ns/m) | 8.4144 | 6.6570 | 5.7092 | 5.1799 | 4.9493 |

Building Level | 1 | 2 | 3 | 4 | 5 |
---|---|---|---|---|---|

FVD system | 68.89 | 68.01 | 66.93 | 66.65 | 69.81 |

SSF controller | 70.58 | 76.76 | 72.00 | 63.89 | 58.61 |

Building Level | 1 | 2 | 3 | 4 | 5 |
---|---|---|---|---|---|

FVD system | 38.09 | 58.32 | 75.70 | 64.78 | 65.65 |

SSF controller | 27.43 | 23.48 | 60.28 | 62.54 | 62.85 |

© 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**

Rubió-Massegú, J.; Palacios-Quiñonero, F.; Rossell, J.M.; Karimi, H.R.
A Novel Iterative Linear Matrix Inequality Design Procedure for Passive Inter-Substructure Vibration Control. *Appl. Sci.* **2020**, *10*, 5859.
https://doi.org/10.3390/app10175859

**AMA Style**

Rubió-Massegú J, Palacios-Quiñonero F, Rossell JM, Karimi HR.
A Novel Iterative Linear Matrix Inequality Design Procedure for Passive Inter-Substructure Vibration Control. *Applied Sciences*. 2020; 10(17):5859.
https://doi.org/10.3390/app10175859

**Chicago/Turabian Style**

Rubió-Massegú, Josep, Francisco Palacios-Quiñonero, Josep M. Rossell, and Hamid Reza Karimi.
2020. "A Novel Iterative Linear Matrix Inequality Design Procedure for Passive Inter-Substructure Vibration Control" *Applied Sciences* 10, no. 17: 5859.
https://doi.org/10.3390/app10175859