# Proposal for Mediative Fuzzy Control: From Type-1 to Type-3

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Mediative Fuzzy Logic for Type-1 Controller

_{A}(x)〉 | x ∈ X},

_{A}(x) has values in [0, 1]. In this expression, µ

_{A}(x) is the membership function (MF) that assigns a membership to each x. This fuzzy set is now called type-1.

_{B}(x), ν

_{B}(x)〉 | x ∈ X},

_{B}(x), ν

_{B}(x) are the membership and non-membership degrees of x ∈ X to a set B in X, 0 ≤ µ

_{B}(x), ν

_{B}(x) ≤ 1 and

_{B}(x) + ν

_{B}(x) ≤ 1.

_{B}(x) = 1 − μ

_{B}(x) − ν

_{B}(x),

_{B}defines the uncertainty degree. The key idea is that uncertainty in real problems is captured by π.

_{C}(x) = min(μ

_{C}(x), ν

_{C}(x))

_{C}(x) represents the agreement MF, and ν

_{C}(x) is the non-agreement MF [4].

## 3. Proposed Mediative Type-1 Fuzzy Controller

## 4. Mediative Type-2 Fuzzy Controller

## 5. Mediative Type-3 Fuzzy Controller

## 6. Illustrative Example

_{1}and ϴ

_{2}are the angles which are changed by the main and tail motors. TRMS uses the DC motors to generate the torques which have nonlinear characteristics and are expressed as

_{1}and τ

_{2}are dependent on the DC motor voltage, and are expressed as

_{1}and u

_{2}are, respectively, the main motor and tail rotor voltages, and k

_{1}, k

_{2}, T

_{11}, T

_{10}and T

_{20}are motor parameters.

_{GY}, T

_{0}and T

_{p}are system parameters in the TRMS model.

## 7. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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Main Rotor | |||||||

MF | a | b | c | d | $\mathit{\lambda}$ | ${\mathbf{\ell}}_{\mathbf{1}}$ | ${\mathbf{\ell}}_{\mathbf{2}}$ |

NB | −5 | −5 | −3.5 | −0.5 | 0.3 | 0.2 | 0.2 |

N | −3.5 | −1.5 | 0 | 0.3 | 0.2 | 0.2 | |

Z | −1.5 | 0 | 1.5 | - | 0.3 | 0.2 | 0.2 |

P | 0 | 1.5 | 3 | 0.3 | 0.2 | 0.2 | |

PB | 0.5 | 3 | 5 | 5 | 0.3 | 0.2 | 0.2 |

Tail Rotor | |||||||

MF | a | b | c | d | $\mathit{\lambda}$ | ${\mathbf{\ell}}_{\mathbf{1}}$ | ${\mathbf{\ell}}_{\mathbf{2}}$ |

NB | −5 | −5 | −3 | −1 | 0.3 | 0.2 | 0.2 |

N | −3 | −1 | −0.5 | 0.3 | 0.2 | 0.2 | |

Z | −1 | −0.5 | 1 | - | 0.3 | 0.2 | 0.2 |

P | −0.5 | 1 | 2.5 | 0.3 | 0.2 | 0.2 | |

PB | 1 | 2.5 | 5 | 5 | 0.3 | 0.2 | 0.2 |

Main Rotor | |||||||

MF | a | b | c | d | $\mathit{\lambda}$ | ${\mathbf{\ell}}_{\mathbf{1}}$ | ${\mathbf{\ell}}_{\mathbf{2}}$ |

NB | −5 | −5 | −2.5 | −1 | 0.3 | 0.2 | 0.2 |

N | −2.5 | −1 | 0 | 0.3 | 0.2 | 0.2 | |

Z | −1 | 0 | 1.5 | - | 0.3 | 0.2 | 0.2 |

P | 0 | 1.5 | 3 | 0.3 | 0.2 | 0.2 | |

PB | 1.5 | 3 | 5 | 5 | 0.3 | 0.2 | 0.2 |

Tail Rotor | |||||||

MF | a | b | c | d | $\mathit{\lambda}$ | ${\mathbf{\ell}}_{\mathbf{1}}$ | ${\mathbf{\ell}}_{\mathbf{2}}$ |

NB | −5 | −5 | −3 | −1 | 0.3 | 0.2 | 0.2 |

N | −3 | −1.5 | −0.5 | 0.3 | 0.2 | 0.2 | |

Z | −1.5 | −0.5 | 1.5 | - | 0.3 | 0.2 | 0.2 |

P | −0.5 | 1.5 | 2.5 | 0.3 | 0.2 | 0.2 | |

PB | 1.5 | 2.5 | 5 | 5 | 0.3 | 0.2 | 0.2 |

Main and Tail Rotors | |||||||
---|---|---|---|---|---|---|---|

MF | a | b | c | d | $\mathit{\lambda}$ | ${\mathbf{\ell}}_{1}$ | ${\mathbf{\ell}}_{2}$ |

NB | −10 | −10 | −7.5 | −0.5 | 0.3 | 0.2 | 0.2 |

N | −7.5 | −5 | 0 | 0.3 | 0.2 | 0.2 | |

Z | −5 | 0 | 5 | - | 0.3 | 0.2 | 0.2 |

P | 0 | 5 | 7.5 | 0.3 | 0.2 | 0.2 | |

PB | 5 | 7.5 | 10 | 10 | 0.3 | 0.2 | 0.2 |

Reference Signal | Rotor | PID [28] | Intuitionistic [28] | Mediative Type-1 (This Work) |
---|---|---|---|---|

Step | Main | 1.465 | 1.128 | 1.060 |

Step | Tail | 3.963 | 2.713 | 2.355 |

Saw | Main | 1.549 | 1.170 | 1.085 |

Saw | Tail | 7.743 | 6.664 | 6.344 |

Sinus | Main | 0.811 | 0.226 | 0.155 |

Sinus | Tail | 8.380 | 6.486 | 6.222 |

**Table 5.**Comparison of results of the mediative type-3 fuzzy controller with respect to the type-2 and type-1 fuzzy controllers.

Reference Signal | Rotor | Mediative Type-1 | Mediative Type-2 | Mediative Type-3 |
---|---|---|---|---|

Step | Main | 1.060 | 1.031 | 0.871 |

Step | Tail | 2.355 | 2.299 | 2.132 |

Saw | Main | 1.085 | 1.037 | 0.925 |

Saw | Tail | 6.344 | 6.289 | 6.021 |

Sinus | Main | 0.155 | 0.127 | 0.094 |

Sinus | Tail | 6.222 | 6.056 | 5.812 |

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**MDPI and ACS Style**

Castillo, O.; Melin, P.
Proposal for Mediative Fuzzy Control: From Type-1 to Type-3. *Symmetry* **2023**, *15*, 1941.
https://doi.org/10.3390/sym15101941

**AMA Style**

Castillo O, Melin P.
Proposal for Mediative Fuzzy Control: From Type-1 to Type-3. *Symmetry*. 2023; 15(10):1941.
https://doi.org/10.3390/sym15101941

**Chicago/Turabian Style**

Castillo, Oscar, and Patricia Melin.
2023. "Proposal for Mediative Fuzzy Control: From Type-1 to Type-3" *Symmetry* 15, no. 10: 1941.
https://doi.org/10.3390/sym15101941