# Real-Time Implementation of the Prescribed Performance Tracking Control for Magnetic Levitation Systems

^{*}

## Abstract

**:**

## 1. Introduction

- A modified TOSMO is applied to quickly estimate the approximate value of the uncertainty and exterior disturbance;
- The novel Prescribed Performance Function (PPF) does not contain a singularity problem and can flexibly adjust lower and upper bounds. Furthermore, it can extend the operation domain at a steady state compared to that of the conventional PPF. With the proposed PPF, the steady-state error boundaries will be symmetric to zero, so, when the transformed error converges to zero, the tracking error also converges to zero;
- A modified function of GFTSMM based on the transformed errors of the PPC is introduced; hence, the error variables quickly converge to the equilibrium point with the prescribed performance;
- The maximum overshoot, convergence index, and steady-state error can be managed within a predefined domain under the proposed controller;
- A novel solution ensures a finite-time stable position of the controlled ball and the possibility of performing different orbit tracking missions with an impressive performance in the terms of tracking accuracy, fast convergence, stabilization, and chattering reduction;
- The effectiveness of the designed control solution was confirmed by simulation and experiment;
- This controller is presented in a way that can be applied to real-time applications. In addition, it can apply not only to MLSs but also to a class of second-order nonlinear systems.

## 2. Problem Statements

**Assumption**

**1.**

## 3. Design of the Proposed Control Method

#### 3.1. Design of the Sliding Mode Surface

#### 3.2. Design of Global Fast Terminal Sliding Mode Control

#### 3.3. Design of a Disturbance Observer for Magnetic Levitation Systems

#### 3.4. Prescribed Performance Control

**Remark**

**1.**

**Remark**

**2.**

- $T\left(\tilde{z}\right)$ is a smooth and strictly increasing function;
- $-1<T\left(\tilde{z}\right)<1$;
- $T\left(\tilde{z}\right)=0$ if $\tilde{z}=0$;
- $\left\{\begin{array}{c}\underset{\tilde{z}\to -\infty}{lim}T\left(\tilde{z}\right)=-1\hfill \\ \underset{\tilde{z}\to +\infty}{lim}T\left(\tilde{z}\right)=1\hfill \end{array}\right.$.

#### 3.5. Proposed Controller Design

**Theorem**

**1.**

**Proof**

**of**

**Theorem**

**1.**

## 4. Simulation and Experimental Results

#### 4.1. Simulation Results

- Step 1: simulates and evaluates the approximation ability of the proposed observer through a comparison between its approximation ability and the conventional TOSMO;
- Step 2: investigates the management of the terms of the proposed PPC including maximum overshoot and steady-state of the controlled errors;
- Step 3: compares the tracking accuracy, maximum overshoot and steady-state of the controlled errors among the four control methods through figures plotted from MATLAB and RMS methods;
- Step 4: considers the chattering behavior that appeared in the control signal of the four methods.

#### 4.2. Experimental Results

**Remark**

**3.**

## 5. Some Remarkable Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 6.**The controlled ball trajectory under four controllers: planned trajectory and actual trajectory.

**Figure 7.**Time evolution of the trajectory errors of the controlled ball using four control methods.

**Figure 11.**Comparison of the controlled ball trajectory with the desired trajectory using four separate controllers; (

**a**) in case of sinusoidal orbit; (

**b**) in case of a rest-to-rest line.

**Figure 12.**The trajectory errors of the controlled ball under four controllers; (

**a**) in case of sinusoidal orbit; (

**b**) in case of a rest-to-rest line.

**Figure 13.**Control voltage of four controllers; (

**a**) in case of sinusoidal orbit; (

**b**) in case of a rest-to-rest line.

**Figure 14.**Time evolution of the MTOSMO’s output; (

**a**) in case of sinusoidal orbit; (

**b**) in case of a rest-to-rest line.

System Parameters | Value | Unit |
---|---|---|

g | 9.81 | m/s${}^{2}$ |

m | 0.02 | kg |

$\lambda $ | $2.48315625\times {10}^{-5}$ | Nm${}^{2}$/A${}^{2}$ |

K | 1.05 | A/V |

$\widehat{\mu}$ | 0.00136487 | (N.m${}^{2}$/kg.V${}^{2}$) |

Controller | Symbol | Value |
---|---|---|

PID | ${K}_{p},\phantom{\rule{0.166667em}{0ex}}{K}_{i},\phantom{\rule{0.166667em}{0ex}}{K}_{d}$ | $300,\phantom{\rule{0.166667em}{0ex}}100,\phantom{\rule{0.166667em}{0ex}}10$ |

SMC | $c,\phantom{\rule{0.166667em}{0ex}}\sigma ,\overline{\Delta}+\kappa $ | $20,50,3.8$ |

GFTSMC | $\lambda ,\omega ,\eta ,\vartheta ,\alpha ,\beta $ | $10,10,1.1,2.2,1.1,0.8$ |

${\sigma}_{1},\overline{\Delta}+{\kappa}_{1}$ | $20,50,3.8$ | |

Proposed Method | ${p}_{0},{p}_{1},{p}_{\infty},r$ | $0.023,0.006,0.0015,3$ |

$\lambda ,\omega ,\eta ,\vartheta ,\alpha ,\beta $ | $10,10,1.1,2.2,1.1,0.8$ | |

${\sigma}_{2},\overline{\delta}+{\kappa}_{2}$ | $0.13,0.1$ | |

${\pi}_{1},{\pi}_{2},{\pi}_{3},\rho $ | $5.45,3.67,6.6,100$ |

Controller | RMSTE |
---|---|

PID | $1.0034\times {10}^{-3}$ |

SMC | $3.1097\times {10}^{-4}$ |

GFTSMC | $1.979\times {10}^{-4}$ |

Proposed Method | $4.2119\times {10}^{-5}$ |

Controller | RMSTE in Case of Sinusoidal Orbit | RMSTE in Case of a Rest-to-Rest Line |
---|---|---|

PID | $1.0411\times {10}^{-3}$ | $1.1342\times {10}^{-3}$ |

SMC | $3.6039\times {10}^{-4}$ | $3.7681\times {10}^{-4}$ |

GFTSMC | $3.3052\times {10}^{-4}$ | $3.4805\times {10}^{-4}$ |

Proposed Method | $1.2203\times {10}^{-4}$ | $1.9601\times {10}^{-4}$ |

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**MDPI and ACS Style**

Truong, T.N.; Vo, A.T.; Kang, H.-J. Real-Time Implementation of the Prescribed Performance Tracking Control for Magnetic Levitation Systems. *Sensors* **2022**, *22*, 9132.
https://doi.org/10.3390/s22239132

**AMA Style**

Truong TN, Vo AT, Kang H-J. Real-Time Implementation of the Prescribed Performance Tracking Control for Magnetic Levitation Systems. *Sensors*. 2022; 22(23):9132.
https://doi.org/10.3390/s22239132

**Chicago/Turabian Style**

Truong, Thanh Nguyen, Anh Tuan Vo, and Hee-Jun Kang. 2022. "Real-Time Implementation of the Prescribed Performance Tracking Control for Magnetic Levitation Systems" *Sensors* 22, no. 23: 9132.
https://doi.org/10.3390/s22239132