# Balance Adjustment of Power-Line Inspection Robot Using General Type-2 Fractional Order Fuzzy PID Controller

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## Abstract

**:**

## 1. Introduction

## 2. General Type-2 Fuzzy Logic System

#### 2.1. General Type-2 Fuzzy Sets

#### 2.2. Rule Base

#### 2.3. Fuzzy Inference Engine

#### 2.4. Type-Reduction and Defuzzification

## 3. PLI Robot System

#### Mathematical Model of PLI Robot System

## 4. Design of GT2FO-FPID Controller

#### 4.1. Approximations of Fractional Order Operation

#### 4.2. Structure of PI${}^{\lambda}$D${}^{\mu}$ Controller

#### 4.3. Structure of GT2FO-FPID Controller

**Remark**

**1.**

## 5. Simulation

#### 5.1. Case 1: Normal Case

#### 5.2. Case 2: External Disturbance

#### 5.3. Case 3: Uncertainty in Mass

#### 5.4. Case 4: Random Disturbance

## 6. Conclusion and Future Work

#### 6.1. Conclusions

#### 6.2. Future Work

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 7.**The close loop diagram of a general type-2 fractional order fuzzy PID (GT2FO-FPID)/interval type-2 fractional order fuzzy PID (IT2FO-FPID)/type-1 fractional order fuzzy PID (T1FO-FPID) controllers.

**Figure 22.**System output response of the PLI robot with uncertainty in mass ($\Delta {m}_{1}$ = 80 kg, $\Delta {m}_{2}$ = 70 kg).

**Figure 23.**System output response of the PLI robot with uncertainty in mass ($\Delta {m}_{1}$ = 80 kg, $\Delta {m}_{2}$ = 70 kg).

**Figure 24.**System output response of the PLI robot with uncertainty in mass ($\Delta {m}_{1}$ = 90 kg, $\Delta {m}_{2}$ = 80 kg).

**Figure 25.**System output response of the PLI robot with uncertainty in mass ($\Delta {m}_{1}$ = 90 kg, $\Delta {m}_{2}$ = 80 kg).

Symbol | ${\mathit{m}}_{1}$ (kg) | ${\mathit{m}}_{2}$ (kg) | ${\mathit{h}}_{1}$ (m) | ${\mathit{h}}_{20}$ (m) | d (m) | l (m) |
---|---|---|---|---|---|---|

value | 63 | 27 | 0.18 | 0.42 | 0.5 | 0.5 |

${\tilde{\mathit{x}}}_{1}$ | |||||||||
---|---|---|---|---|---|---|---|---|---|

${\tilde{\mathit{x}}}_{\mathbf{2}}$ | ${\mathit{k}}_{\mathit{p}}$ | ${\mathit{k}}_{\mathit{i}}$ | ${\mathit{k}}_{\mathit{d}}$ | ||||||

S | M | B | S | M | B | S | M | B | |

S | M | S | M | B | S | B | M | B | M |

M | B | M | B | B | M | B | S | B | S |

B | M | S | M | M | S | M | M | B | M |

**Table 3.**Optimal parameters for GT2FO-FPID, IT2FO-FPID, T1FO-FPID, and fractional order fuzzy PID (FOPID) controllers in all cases.

Controllers Type | Parameters | |||||
---|---|---|---|---|---|---|

${\mathit{G}}_{\mathit{E}}$ | ${\mathit{G}}_{\mathit{CE}}$ | ${\mathit{G}}_{\mathit{PD}}$ | ${\mathit{G}}_{\mathit{PI}}$ | $\mathbf{\lambda}$ | $\mathbf{\mu}$ | |

GT2FO-FPID | 1.88 | 1.76 | 0.66 | 0.01 | 1.01 | 1.15 |

IT2FO-FPID | 0.01 | 0.01 | 0.50 | 0.01 | 0.80 | 1.25 |

T1FO-FPID | 2.64 | 0.68 | 0.53 | 0.45 | 0.92 | 0.01 |

FOPID | 50.01 | 10.00 | 1.00 | 0.80 | 1.30 | 1.30 |

Performance Index | ${\mathit{x}}_{1}$ | ${\mathit{x}}_{3}$ | ||||||
---|---|---|---|---|---|---|---|---|

FOPID | T1FO-FPID | IT2FO-FPID | GT2FO-FPID | FOPID | T1FO-FPID | IT2FO-FPID | GT2FO-FPID | |

ISE | 0.0425 | 0.0479 | 0.0516 | 0.0443 | 0.0173 | 0.0161 | 0.203 | 0.013 |

IAE | 0.3185 | 0.332 | 0.2837 | 0.2916 | 0.2302 | 0.1877 | 0.1817 | 0.164 |

ITAE | 0.5387 | 0.4973 | 0.255 | 0.3453 | 0.4726 | 0.326 | 0.213 | 0.2323 |

Performance Index | ${\mathit{x}}_{1}$ | ${\mathit{x}}_{3}$ | ||||||
---|---|---|---|---|---|---|---|---|

FOPID | T1FO-FPID | IT2FO-FPID | GT2FO-FPID | FOPID | T1FO-FPID | IT2FO-FPID | GT2FO-FPID | |

ISE | 0.0674 | 0.0677 | 0.0791 | 0.0624 | 0.0293 | 0.0223 | 0.0301 | 0.0170 |

IAE | 0.5884 | 0.5479 | 0.5188 | 0.4759 | 0.4034 | 0.3211 | 0.3240 | 0.2506 |

ITAE | 3.4889 | 2.8400 | 2.7343 | 2.2592 | 2.4462 | 1.8743 | 1.7791 | 1.1752 |

Performance Index | ${\mathit{x}}_{1}$ | ${\mathit{x}}_{3}$ | ||||||
---|---|---|---|---|---|---|---|---|

FOPID | T1FO-FPID | IT2FO-FPID | GT2FO-FPID | FOPID | T1FO-FPID | IT2FO-FPID | GT2FO-FPID | |

ISE | 0.6962 | 0.6483 | 0.7800 | 0.5791 | 0.2877 | 0.1360 | 0.2323 | 0.1218 |

IAE | 1.6255 | 1.5699 | 1.5518 | 1.3700 | 1.1442 | 0.7570 | 0.9225 | 0.6576 |

ITAE | 14.7873 | 13.8458 | 13.6874 | 11.7064 | 10.9172 | 6.7755 | 8.4116 | 5.7021 |

**Table 7.**Performance index results for uncertainty in mass ($\Delta {m}_{1}$ = 80 kg, $\Delta {m}_{2}$ = 70 kg).

Performance Index | ${\mathit{x}}_{1}$ | ${\mathit{x}}_{3}$ | ||||||
---|---|---|---|---|---|---|---|---|

FOPID | T1FO-FPID | IT2FO-FPID | GT2FO-FPID | FOPID | T1FO-FPID | IT2FO-FPID | GT2FO-FPID | |

ISE | 0.0458 | 0.0524 | 0.0518 | 0.0444 | 0.0198 | 0.0196 | 0.0205 | 0.0130 |

IAE | 0.4174 | 0.4559 | 0.3045 | 0.3051 | 0.3265 | 0.3113 | 0.2016 | 0.1758 |

ITAE | 1.9357 | 2.2453 | 0.5391 | 0.5272 | 1.8253 | 2.0714 | 0.4839 | 0.3895 |

**Table 8.**Performance index results for uncertainty in mass ($\Delta {m}_{1}$ = 90 kg, $\Delta {m}_{2}$ = 80 kg).

Performance Index | ${\mathit{x}}_{1}$ | ${\mathit{x}}_{3}$ | ||||||
---|---|---|---|---|---|---|---|---|

FOPID | T1FO-FPID | IT2FO-FPID | GT2FO-FPID | FOPID | T1FO-FPID | IT2FO-FPID | GT2FO-FPID | |

ISE | 0.0494 | 0.0566 | 0.0520 | 0.0445 | 0.0224 | 0.0228 | 0.0206 | 0.0131 |

IAE | 0.4599 | 0.5021 | 0.3120 | 0.3114 | 0.3678 | 0.3575 | 0.2088 | 0.1812 |

ITAE | 2.5408 | 2.9033 | 0.6438 | 0.6131 | 2.4092 | 2.7289 | 0.5831 | 0.4624 |

Performance Index | ${\mathit{x}}_{1}$ | ${\mathit{x}}_{3}$ | ||||||
---|---|---|---|---|---|---|---|---|

FOPID | T1FO-FPID | IT2FO-FPID | GT2FO-FPID | FOPID | T1FO-FPID | IT2FO-FPID | GT2FO-FPID | |

ISE | 0.0450 | 0.0506 | 0.0521 | 0.0445 | 0.0181 | 0.0170 | 0.0205 | 0.0130 |

IAE | 0.4648 | 0.4819 | 0.3488 | 0.3257 | 0.3096 | 0.2711 | 0.2200 | 0.1768 |

ITAE | 2.8277 | 2.8882 | 1.2784 | 0.8732 | 1.7259 | 1.6870 | 0.8235 | 0.4333 |

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**MDPI and ACS Style**

Chen, Y.; Zhao, T.; Dian, S.; Zeng, X.; Wang, H.
Balance Adjustment of Power-Line Inspection Robot Using General Type-2 Fractional Order Fuzzy PID Controller. *Symmetry* **2020**, *12*, 479.
https://doi.org/10.3390/sym12030479

**AMA Style**

Chen Y, Zhao T, Dian S, Zeng X, Wang H.
Balance Adjustment of Power-Line Inspection Robot Using General Type-2 Fractional Order Fuzzy PID Controller. *Symmetry*. 2020; 12(3):479.
https://doi.org/10.3390/sym12030479

**Chicago/Turabian Style**

Chen, Yao, Tao Zhao, Songyi Dian, Xiaodong Zeng, and Haipeng Wang.
2020. "Balance Adjustment of Power-Line Inspection Robot Using General Type-2 Fractional Order Fuzzy PID Controller" *Symmetry* 12, no. 3: 479.
https://doi.org/10.3390/sym12030479