# Differential Evolution with Shadowed and General Type-2 Fuzzy Systems for Dynamic Parameter Adaptation in Optimal Design of Fuzzy Controllers

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Type-2 Fuzzy Systems and Shadowed Sets

- -
- The elevated region for the membership degrees with a value of 1.
- -
- The reduced region for the membership degrees with a value of 0.
- -
- The shaded region with degree of membership in [0, 1].

## 3. General Type-2 Fuzzy Systems

_{x}⊆ [0, 1], x represents a primary membership function partition, and u represents a secondary membership function partition.

_{x}(u) is used for general type-2 fuzzy logic systems.

_{α,}and it is the union of all primary membership functions of Ã, which secondary membership degrees are higher or equal to α (0 ≤ α ≤ 1) [48,49]. The visual representation of an alpha plane can be found in Figure 5, in the same way the expression of the alpha plane is given by Equation (9).

## 4. Differential Evolution Algorithm

**Structure of the Population**

**Initialization**

**Mutation**

**Crossover**

**Selection**

## 5. Differential Evolution Algorithm with Dynamic Parameter Adaptation

- ➢
**Shadowed Type 2 fuzzy systems**

- ➢
**General Type 2 fuzzy systems**

## 6. Experiments Whit the D.C. Motor Speed Controller

_{1}represents the variants using GT2FDE and µ

_{2}represents the variants using ST2FDE.

- Ho:
- The results of the GT2FDE methodology without noise and with noise are higher than the methodology ST2FDE without noise and with noise.
- Ha:
- The results of the GT2FDE methodology without noise and with noise are lower than the methodology ST2FDE without noise and with noise.

## 7. Discussion of Results

## 8. Conclusions

^{−01}and 9.73 × 10

^{−01}, respectively, which are very similar. The same can be observed with the average of the e × periments where the results were 9.84$\text{}\times \text{}{10}^{-01}$ and 9.85 $\times {10}^{-01}$, respectively. We can say that the difference between the two is small and the statistical test shows that ST2FDE is better than GT2FDE.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Ethical Approval

## References

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**Figure 16.**Graphical representation of the error using ST2FDE without noise and with different noise levels.

**Figure 17.**Simulation of the best result obtained in the experimentation with GT2FDE without noise in the FLC.

**Figure 18.**Simulation of the best result obtained in the experimentation with GT2FDE with noise of 0.5 in the FLC.

**Figure 19.**Simulation of the best result obtained in the experimentation with GT2FDE with a noise of 0.7 in the FLC.

**Figure 20.**Simulation of the best result obtained in the experimentation with GT2FDE with a noise of 0.9 in the FLC.

**Figure 21.**Graphical representation of the error using GT2FDE without noise and with different noise levels.

**Figure 22.**Comparison of the best results between ST2FDE and GT2FDE without noise and with different noise levels.

Generation | F | ||
---|---|---|---|

Low | Medium | High | |

Low | − | − | Low |

Medium | − | Medium | − |

High | High | − | − |

Generalized Type-2 Fuzzy Logic Sets | |
---|---|

Low | ${\mu}_{1}\left(x\right)=\mathrm{max}\left(\mathrm{min}\left(\frac{x-0.5}{-0.08+0.5},\frac{0.4-x}{0.4+0.08}\right),0\right)and$ ${\mu}_{2}\left(x\right)=\mathrm{max}\left(\mathrm{min}\left(\frac{x+0.4}{0.08+0.4},\frac{0.5-x}{0.5-0.08}\right),0\right)$ $\overline{\mu}\left(x\right)=\{\begin{array}{c}\mathrm{max}\left({\mu}_{1}\left(\mathrm{x}\right),\text{}{\mu}_{2}\text{}\left(\mathrm{x}\right)\right){\forall}_{x}\notin \left(-0.08,0.08\right)\\ 1{\forall}_{x}\in \left(-0.08,0.08\right)\end{array}$ $\underset{\_}{\mu}\left(x\right)=\mathrm{min}\left({\mu}_{1}\left(\mathrm{x}\right),\text{}{\mu}_{2}\text{}\left(\mathrm{x}\right)\right)$ ${\rho}_{x}=\mathrm{max}\left(\mathrm{min}\left(\frac{x-{a}_{x}}{{b}_{x}-{a}_{x}}\right),\left(\frac{{c}_{x}-x}{{c}_{x}-{b}_{x}}\right),0\right),where$ ${a}_{x}=\frac{-0.5-0.4}{2},{b}_{x}=\frac{-0.8-0.08}{2},{c}_{x}=\frac{-0.4-0.5}{2},$ $\delta =\overline{\mu}\left(x\right)-\underset{\_}{\mu}\left(x\right)$ ${\sigma}_{u}=\frac{1+\rho}{2\surd 3}\delta +\epsilon $ $Where\rho =0.5$ |

Medium | ${\mu}_{1}\left(x\right)=\mathrm{max}\left(\mathrm{min}\left(\frac{x+0.084}{0.4+0.084},\frac{0.92-x}{0.92-0.4}\right),0\right)and$ ${\mu}_{2}\left(x\right)=\mathrm{max}\left(\mathrm{min}\left(\frac{x-0.084}{0.5-0.084},\frac{1.07-x}{1.07-0.5}\right),0\right)$ $\overline{\mu}\left(x\right)=\{\begin{array}{c}\mathrm{max}\left({\mu}_{1}\left(\mathrm{x}\right),\text{}{\mu}_{2}\text{}\left(\mathrm{x}\right)\right){\forall}_{x}\notin \left(0.4,0.5\right)\\ 1{\forall}_{x}\in \left(0.4,0.5\right)\end{array}$ $\underset{\_}{\mu}\left(x\right)=\mathrm{min}\left({\mu}_{1}\left(\mathrm{x}\right),\text{}{\mu}_{2}\text{}\left(\mathrm{x}\right)\right)$ ${\rho}_{x}=\mathrm{max}\left(\mathrm{min}\left(\frac{x-{a}_{x}}{{b}_{x}-{a}_{x}}\right),\left(\frac{{c}_{x}-x}{{c}_{x}-{b}_{x}}\right),0\right),where$ ${a}_{x}=\frac{-0.084+0.084}{2},{b}_{x}=\frac{0.4-0.5}{2},{c}_{x}=\frac{0.92-1.09}{2},$ $\delta =\overline{\mu}\left(x\right)-\underset{\_}{\mu}\left(x\right)$ ${\sigma}_{u}=\frac{1+\rho}{2\surd 3}\delta +\epsilon $ $Where\rho =0.5$ |

High | ${\mu}_{1}\left(x\right)=\mathrm{max}\left(\mathrm{min}\left(\frac{x-0.4}{0.92-0.4},\frac{1.4-x}{1.4-0.92}\right),0\right)and$ ${\mu}_{2}\left(x\right)=\mathrm{max}\left(\mathrm{min}\left(\frac{x-0.5}{1.07-0.5},\frac{1.5-x}{1.5-1.07}\right),0\right)$ $\overline{\mu}\left(x\right)=\{\begin{array}{c}\mathrm{max}\left({\mu}_{1}\left(\mathrm{x}\right),\text{}{\mu}_{2}\text{}\left(\mathrm{x}\right)\right){\forall}_{x}\notin \left(0.92,1.07\right)\\ 1{\forall}_{x}\in \left(0.92,1.07\right)\end{array}$ $\underset{\_}{\mu}\left(x\right)=\mathrm{min}\left({\mu}_{1}\left(\mathrm{x}\right),\text{}{\mu}_{2}\text{}\left(\mathrm{x}\right)\right)$ ${\rho}_{x}=\mathrm{max}\left(\mathrm{min}\left(\frac{x-{a}_{x}}{{b}_{x}-{a}_{x}}\right),\left(\frac{{c}_{x}-x}{{c}_{x}-{b}_{x}}\right),0\right),where$ ${a}_{x}=\frac{0.4+0.5}{2},{b}_{x}=\frac{0.92-1.07}{2},{c}_{x}=\frac{1.4-1.5}{2},$ $\delta =\overline{\mu}\left(x\right)-\underset{\_}{\mu}\left(x\right)$ ${\sigma}_{u}=\frac{1+\rho}{2\surd 3}\delta +\epsilon $ $Where\rho =0.5$ |

No. | Inputs | Output | |
---|---|---|---|

Error | Change in Error | Voltage | |

1 | NegV | ErrNeg | Dis |

2 | NegV | SinErr | Dis |

3 | NegV | ErrMax | Dis_m |

4 | ZeroV | ErrNeg | Aum_m |

5 | ZeroV | ErrMax | Dis_m |

6 | PosV | ErrNeg | Aum_m |

7 | PosV | SinErr | Aum |

8 | PosV | ErrMax | Aum |

9 | ZeroV | SinErr | Man |

10 | NegV | ErrNeg_M | Dis |

11 | ZeroV | ErrNeg_M | Aum_m |

12 | PosV | ErrNeg_M | Aum |

13 | PosV | ErrMax_M | Aum |

14 | ZeroV | ErrMax_M | Dis_m |

15 | NegV | ErrMax_M | Dis |

Parameters | ST2FDE and GT2FDE |
---|---|

Population | 50 |

Dimensions | 45 |

Generations | 30 |

Number of experiments | 30 |

F | Dynamic |

Cr | 0.3 |

ST2FDE | ||||
---|---|---|---|---|

Method | ST2FDE without Noise FLC | ST2FDE with Noise 0.5 FLC | ST2FDE with Noise 0.7 FLC | ST2FDE with Noise 0.9 FLC |

Best | 9.66$\times {10}^{-01}$ | 9.41$\times {10}^{-01}$ | 5.59$\times {10}^{-01}$ | 4.52$\times {10}^{-01}$ |

Worst | 9.98$\times {10}^{-01}$ | 9.96$\times {10}^{-01}$ | 6.11$\times {10}^{-01}$ | 6.56$\times {10}^{-01}$ |

Average | 9.84$\times {10}^{-01}$ | 9.73$\times {10}^{-01}$ | 5.86$\times {10}^{-01}$ | 5.81$\times {10}^{-01}$ |

Std. | 8.45$\times {10}^{-03}$ | 1.17$\times {10}^{-02}$ | 1.40$\times {10}^{-02}$ | 6.13$\times {10}^{-02}$ |

GT2FDE | ||||
---|---|---|---|---|

Method | GT2FDE without Noise FLC | GT2FDE with Noise 0.5 FLC | GT2FDE with Noise 0.7 FLC | GT2FDE with Noise 0.9 FLC |

Best | 9.73$\times {10}^{-01}$ | 9.38$\times {10}^{-01}$ | 5.48$\times {10}^{-01}$ | 4.35$\times {10}^{-02}$ |

Worst | 9.95$\times {10}^{-01}$ | 9.91$\times {10}^{-01}$ | 6.08$\times {10}^{-01}$ | 6.53$\times {10}^{-01}$ |

Average | 9.85$\times {10}^{-01}$ | 9.75$\times {10}^{-01}$ | 5.79$\times {10}^{-01}$ | 5.51$\times {10}^{-01}$ |

Std. | 5.88$\times {10}^{-03}$ | 1.25$\times {10}^{-02}$ | 1.70$\times {10}^{-02}$ | 7.46$\times {10}^{-02}$ |

Parameter | Value |
---|---|

Level of Confidence | 95% |

Alpha | 0.05% |

H_{a} | µ_{1} < µ_{2} |

H_{0} | µ_{1} ≥ µ_{2} |

Critical Value | −1.645 |

Statistical Tests | ||||
---|---|---|---|---|

Case Study | ${\mathit{\mu}}_{1}$ | ${\mathit{\mu}}_{2}$ | Z Value | Evidence |

Speed control in a D.C. Motor | GT2FDE without FCL noise | ST2FDE without FCL noise | 0.5321 | Not Significant |

GT2FDE with FCL 0.5 noise | ST2FDE without FCL 0.5 noise | 0.6398 | Not Significant | |

GT2FDE with FCL 0.7 noise | ST2FDE without FCL 0.7 noise | −1.7410 | Significant | |

GT2FDE with FCL 0.9 noise | ST2FDE without FCL 0.9 noise | −1.7018 | Significant |

D.C. Motor Speed Controller | RMSE | Method | Best |

Original DE | 4.72$\times {10}^{-01}$ | ||

DEFIS 1 | 4.57$\times {10}^{-01}$ | ||

DEFIS 2 | 4.80$\times {10}^{-01}$ | ||

DEFIS 3 | 2.36$\times {10}^{-01}$ | ||

Original HS | 4.72$\times {10}^{-01}$ | ||

HSFIS 1 | 4.57$\times {10}^{-01}$ | ||

HSFIS 2 | 4.80$\times {10}^{-01}$ | ||

HSFIS 3 | 2.36$\times {10}^{-01}$ | ||

GT2FDE with noise 0.9 FLC | 4.35 × 10^{−02} |

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**MDPI and ACS Style**

Ochoa, P.; Castillo, O.; Melin, P.; Soria, J.
Differential Evolution with Shadowed and General Type-2 Fuzzy Systems for Dynamic Parameter Adaptation in Optimal Design of Fuzzy Controllers. *Axioms* **2021**, *10*, 194.
https://doi.org/10.3390/axioms10030194

**AMA Style**

Ochoa P, Castillo O, Melin P, Soria J.
Differential Evolution with Shadowed and General Type-2 Fuzzy Systems for Dynamic Parameter Adaptation in Optimal Design of Fuzzy Controllers. *Axioms*. 2021; 10(3):194.
https://doi.org/10.3390/axioms10030194

**Chicago/Turabian Style**

Ochoa, Patricia, Oscar Castillo, Patricia Melin, and José Soria.
2021. "Differential Evolution with Shadowed and General Type-2 Fuzzy Systems for Dynamic Parameter Adaptation in Optimal Design of Fuzzy Controllers" *Axioms* 10, no. 3: 194.
https://doi.org/10.3390/axioms10030194