# A Generic WebLab Control Tuning Experience Using the Ball and Beam Process and Multiobjective Optimization Approach

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## Abstract

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## 1. Introduction

#### 1.1. Control Engineering Learning Involving the Ball and Beam Process

#### 1.1.1. Related Works Associated with the Ball and Beam Modeling and Simulation

#### 1.1.2. Related Works Associated with the Construction of the Ball and Beam Plant

## 2. Ball and Beam Process Description and Modelling

#### 2.1. The Ball and Beam Apparatus

#### 2.2. Physical Modeling

#### 2.2.1. Simplified Model

#### 2.2.2. Full Model

## 3. PID Control and Performance Measures

## 4. Multiobjective Optimization Applied to Control Engineering

#### 4.1. Multiobjective Problem Statement

#### 4.2. Multiobjective Optimization Process

#### 4.3. Multicriteria Decision Making

## 5. Remote Experiment Description

#### 5.1. Experiment Structure

#### 5.2. Experimental Procedures

## 6. Results and Discussion

#### 6.1. Multiobjective Optimization Procedures

#### 6.1.1. Problem Definition

#### 6.1.2. Multiobjective Optimization through NSGA-II

Algorithm 1 NSGA-II procedures |

1: Initialize the population |

2: Evaluate the objective functions for the individuals |

3: Rank the individual based on non-dominated sorting |

4: Calculate the crowding distance |

5: While (Stopping Criteria is not satisfied) |

6: Select the individuals by using a binary tournament for the mating pool |

7: Apply the genetic operators, crossover, and mutation, to the mating pool |

8: Evaluate the objective functions of the offspring population |

9: Combine the offspring population with the current generation |

10: Rank the individual based on non-dominated sorting |

11: Calculate the crowding distance |

12: Select better solutions until complete the size of the population |

13: End While |

14: Output the non-dominated solutions |

#### 6.1.3. Multicriteria Decision Making Strategy

## 7. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**The ball and beam process used as a case study in this research [25].

**Figure 2.**The movement of the sphere is affected by gravity when the beam is not on the horizontal position.

**Figure 7.**Block diagrams of Proportional-Integral-Derivative (PID) control variations: (

**a**) Proportional Integral-Derivative (PI-D); and, (

**b**) Integral-Proportional Derivative (I-PD).

**Figure 8.**Multiobjective optimization procedures for control systems (adapted from [44]).

**Figure 9.**Representation of both non-dominated and dominated solutions in the cost functions domain.

**Figure 10.**ELSA-SP structure (adapted from [62]).

Model | ${\mathit{K}}_{\mathit{p}}$ | ${\mathit{T}}_{\mathit{i}}$ | ${\mathit{T}}_{\mathit{d}}$ |
---|---|---|---|

${G}_{m}\left(s\right)=\frac{{K}_{m}{e}^{-s{t}_{m}}}{{s}^{2}}$ | $\frac{0.0214}{{K}_{m}{t}_{m}^{2}}$ | $17.570{t}_{m}$ | $14.019{t}_{m}$ |

${\mathit{K}}_{\mathit{p}}$ | ${\mathit{K}}_{\mathit{d}}$ | ${\mathit{K}}_{\mathit{h}}$ | ${\mathit{K}}_{\mathit{p}1}$ | ${\mathit{J}}_{1}\left(\mathit{\phi}\right)$ | ${\mathit{J}}_{2}\left(\mathit{\phi}\right)$ | |

Reference Controller | 0.5 | 0.8 | 8 | 8 | 1931.2 | 240.4 |

Optimized Controller | 50 | 11.3588 | 40.6189 | 0.1376 | 1064.7 | 191.9 |

${\mathit{K}}_{\mathit{p}}$ | ${\mathit{T}}_{\mathit{i}}$ | ${\mathit{T}}_{\mathit{d}}$ | ${\mathit{J}}_{1}\left(\mathit{\phi}\right)$ | ${\mathit{J}}_{2}\left(\mathit{\phi}\right)$ | ||

Åström Controller | 21.8814 | 0.1757 | 0.1402 | - | 1952.0 | 233.5 |

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**MDPI and ACS Style**

Kagami, R.M.; da Costa, G.K.; Uhlmann, T.S.; Mendes, L.A.; Freire, R.Z. A Generic WebLab Control Tuning Experience Using the Ball and Beam Process and Multiobjective Optimization Approach. *Information* **2020**, *11*, 132.
https://doi.org/10.3390/info11030132

**AMA Style**

Kagami RM, da Costa GK, Uhlmann TS, Mendes LA, Freire RZ. A Generic WebLab Control Tuning Experience Using the Ball and Beam Process and Multiobjective Optimization Approach. *Information*. 2020; 11(3):132.
https://doi.org/10.3390/info11030132

**Chicago/Turabian Style**

Kagami, Ricardo Massao, Guinther Kovalski da Costa, Thiago Schaedler Uhlmann, Luciano Antônio Mendes, and Roberto Zanetti Freire. 2020. "A Generic WebLab Control Tuning Experience Using the Ball and Beam Process and Multiobjective Optimization Approach" *Information* 11, no. 3: 132.
https://doi.org/10.3390/info11030132