# Generalised Proportional Integral Control for Magnetic Levitation Systems Using a Tangent Linearisation Approach

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## Abstract

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## 1. Introduction

## 2. The Magnetic Levitation System and Problem Formulation

#### 2.1. The Nonlinear Model

#### 2.2. Tangent Linearisation

#### 2.3. Flatness of the Linearised Magnetic Levitation System

#### 2.4. Problem Formulation

## 3. Generalised Proportional Integral Controller Design

#### 3.1. A Flatness-Based Pole Placement Approach for Stabilisation

#### 3.2. Using GPI to Locally Control the Magnetic Levitation System

## 4. Computer Simulations

#### 4.1. Tracking a Smooth Rest-to-Rest Trajectory

#### 4.2. Robustness with Regard to Measurement Noises

#### 4.3. Robustness with Regard to Controller Gain Mismatches

#### 4.4. Comparison of the Controllers Based on Integral Criteria

## 5. Experimental Results on the Magnetic Levitation Platform

#### 5.1. Schematics of the Experimental Implementation of the Controller

#### 5.2. Results of the Experiments

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

ALS | Air Levitation System |

FO | Fractional order |

GPI | Generalised Proportional Integral |

IAE | Integral Absolute Tracking Error |

ISE | Integral Squared Tracking Error |

ITAE | Integral Time Absolute Tracking Error |

LQ | Linear Quadratic |

LQR | Linear Quadratic Regulator |

MISO | Multiple Input Single Output |

MLS | Magnetic Levitation Systems |

PI | Proportional Integral |

PID | Proportional Integral Derivative |

RTW | Real Time Workshop |

RTWT | Real Time Windows Target |

SISO | Single Input Single Output |

## Appendix A. Closed-Loop Stability Analysis of the System against Errors in the Estimation of the β Parameter

## References

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**Figure 3.**Evolution of the tracking trajectory of the magnetic ball, ${y}^{*}\left(t\right)\left[m\right]$ vs. $y\left(t\right)\left[m\right]$, nominal case (Test 1).

**Figure 4.**Evolution of tracking error trajectory of the magnetic ball, $e\left(t\right)=y\left(t\right)-{y}^{*}\left(t\right)\left[m\right]$, nominal case (Test 1).

**Figure 5.**Control input of the magnetic levitation system, $u\left(t\right)\left[V\right]$, nominal case (Test 1).

**Figure 6.**Evolution of the tracking trajectory of the magnetic ball, ${y}^{*}\left(t\right)\left[m\right]$ vs. $y\left(t\right)\left[m\right]$, with measurement noises at the control input and the output of the MLS (Test 2).

**Figure 7.**Evolution of tracking error trajectory of the magnetic ball, $e\left(t\right)=y\left(t\right)-{y}^{*}\left(t\right)\left[m\right]$, with measurement noises at the control input and the output of the MLS (Test 2).

**Figure 8.**Control input of the magnetic levitation system, $u\left(t\right)\left[V\right]$, with measurement noise at the control input and the output of the MLS (Test 2).

**Figure 9.**Evolution of the tracking trajectory of the magnetic ball, ${y}^{*}\left(t\right)\left[m\right]$ vs. $y\left(t\right)\left[m\right]$, with measurement noises and controller gain mismatches (Test 3).

**Figure 10.**Evolution of tracking error trajectory of the magnetic ball, $e\left(t\right)=y\left(t\right)-{y}^{*}\left(t\right)\left[m\right]$, with measurement noises and controller gain mismatches (Test 3).

**Figure 11.**Control input of the magnetic levitation system, $u\left(t\right)\left[V\right]$, with measurement noises and controller gain mismatches (Test 3).

**Figure 15.**Evolution of the tracking trajectory of the magnetic ball, ${y}^{*}\left(t\right)\left[m\right]$ vs. $y\left(t\right)\left[m\right]$, (Experimental Test).

**Figure 16.**Evolution of tracking error trajectory of the magnetic ball, $e\left(t\right)=y\left(t\right)-{y}^{*}\left(t\right)\left[m\right]$, (Experimental Test).

**Figure 17.**Control input of the magnetic levitation system, $u\left(t\right)\left[V\right]$, (Experimental Test).

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**MDPI and ACS Style**

Belmonte, L.M.; Segura, E.; Fernández-Caballero, A.; Somolinos, J.A.; Morales, R.
Generalised Proportional Integral Control for Magnetic Levitation Systems Using a Tangent Linearisation Approach. *Mathematics* **2021**, *9*, 1424.
https://doi.org/10.3390/math9121424

**AMA Style**

Belmonte LM, Segura E, Fernández-Caballero A, Somolinos JA, Morales R.
Generalised Proportional Integral Control for Magnetic Levitation Systems Using a Tangent Linearisation Approach. *Mathematics*. 2021; 9(12):1424.
https://doi.org/10.3390/math9121424

**Chicago/Turabian Style**

Belmonte, Lidia M., Eva Segura, Antonio Fernández-Caballero, José A. Somolinos, and Rafael Morales.
2021. "Generalised Proportional Integral Control for Magnetic Levitation Systems Using a Tangent Linearisation Approach" *Mathematics* 9, no. 12: 1424.
https://doi.org/10.3390/math9121424