# An Improved Passivity-based Control of Electrostatic MEMS Device

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. The Mathematical Model of the Electrostatic MEMS Actuator

^{T}=$\left[{x}_{1},{x}_{2},{x}_{3}\right]$

^{T}where p is the momentum. Thus, the equations can be rewritten as [24]:

**Remark**

**2.1.**

## 3. The Pull-in Stability Analysis of the System

**Remark**

**3.1.**

## 4. Design of the Controller and the Observer for the MEMS Actuator

#### 4.1. The Port-Hamiltonian Model of the System

#### 4.2. The Controller Design Based on IDA-PBC Method

**Proposition**

**5.1.**

**Proof of Proposition**

**5.1.**

**Condition 1:**

**Remark**

**5.1.**

**Remark**

**5.2.**

#### 4.3. The Design of the Speed Observer

## 5. Numerical Simulations

#### 5.1. Simulations Using MATLAB/Simulink environment

#### 5.2. Simulations Using Multiphysics Modelling Software COMSOL

## 6. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 3.**The normalized displacement with and without the coupling constant at a gap distance of 0.3.

**Figure 4.**The normalized displacement with and without the coupling constant at a gap distance of 0.5.

**Figure 5.**The normalized displacement with and without the coupling constant at a gap distance of 0.8.

**Figure 10.**The normalized displacement with and without the coupling constant at a gap distance of 0.8 and damping ratio ζ = 0.01.

**Figure 11.**The normalized estimated velocity with the coupling constant at a gap distance of 0.8 and damping ratio ζ = 0.01.

**Figure 12.**The normalized displacement with different coupling terms at a gap reference of 0.8 and damping ratio ζ = 0.01.

**Figure 17.**Control input (voltage) at two positions: top graph (0.046 $\mathsf{\mu}\mathrm{m}$), bottom graph (0.9 $\mathsf{\mu}\mathrm{m}$ ).

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

Plate length | L = 50 μm |

Plate thickness | t = 2 μm |

Gap between the plates | ${q}_{z}=1.1\mathsf{\mu}\mathrm{m}=1.1\mathsf{\mu}\mathrm{m}$ |

Permittivity | $\u03f5=4.5={4.5}^{\xb0}$ |

Density | $\rho =2320\mathrm{kg}/{\mathrm{m}}^{3}/{\mathrm{m}}^{3}$ |

Damping ratio | ζ = 1 × ${10}^{-6}$ |

Spring stiffness | k = 128 N/m |

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**MDPI and ACS Style**

Ryalat, M.; Salim Damiri, H.; ElMoaqet, H.; AlRabadi, I.
An Improved Passivity-based Control of Electrostatic MEMS Device. *Micromachines* **2020**, *11*, 688.
https://doi.org/10.3390/mi11070688

**AMA Style**

Ryalat M, Salim Damiri H, ElMoaqet H, AlRabadi I.
An Improved Passivity-based Control of Electrostatic MEMS Device. *Micromachines*. 2020; 11(7):688.
https://doi.org/10.3390/mi11070688

**Chicago/Turabian Style**

Ryalat, Mutaz, Hazem Salim Damiri, Hisham ElMoaqet, and Imad AlRabadi.
2020. "An Improved Passivity-based Control of Electrostatic MEMS Device" *Micromachines* 11, no. 7: 688.
https://doi.org/10.3390/mi11070688