# Rate-Dependent Hysteresis Model of a Giant Magnetostrictive Actuator Based on an Improved Cuckoo Algorithm

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## Abstract

**:**

## 1. Introduction

## 2. Rate-Dependent Asymmetric PI Model

#### 2.1. PI Model

_{m}[0,t

_{E}] is assumed to be the set of all continuous piecewise functions in the definitional domain of [0,t

_{E}], in which 0 = t

_{0}< t

_{1}< t

_{2}… < t

_{E}, u(t)∈C

_{m}, and the output of the Play operator is:

_{r}[u(0)] is the output of a single Play operator at the initial moment; F

_{r}[u(t)] denotes the output of a single Play operator at time t, in which u(t) stands for the input signal at time t; F

_{r}[u(t − 1)] represents the output of the Play operator at the time before t. r is the threshold of the Play operator.

_{ri}stands for the weight of each Play operator.

#### 2.2. Asymmetric PI (API) Model

_{r},

_{β}[u(t)] is the output of the improved Play operator at the current moment; F

_{r},

_{β}[u(t − 1)] stands for the output of the operator at the previous moment. The value of β could change the output amplitude and output gap of the Play operator, thus affecting the output of the asymmetric PI model. With the above improvements combined, the asymmetric PI model is expressed as shown in Formula (7).

#### 2.3. RAPI Model

_{0}(f)… a

_{n}(f) denote frequency-related parameters. The parameter values were identified by combining the output under different frequencies, and the unknown parameters in Formula (9) were acquired through curve fitting. Then, each frequency function was substituted into Formula (8) so as to obtain the rate-dependent RAPI model.

## 3. Improved CS Algorithm

#### 3.1. CS Algorithm

- There was only one bird egg in each nest, and the eggs were randomly distributed.
- In each evolution, the bird egg with the highest fitness was reserved.
- Under a fixed number of bird’s nests, the host bird abandoned the cuckoo eggs at Pa, and the eliminated nests would be replaced by new nests.

#### 3.2. CS Algorithm Optimization

#### 3.2.1. Adaptive Step (AS) Strategy

_{min}= 0.8 is the minimum value of the inertia factor; w

_{max}= 2 is the maximum value of the inertia factor; K is the number of iterations; and K

_{max}is the maximum number of iterations, which was set to K

_{max}= 1000. The change in the w value is displayed in Figure 1:

#### 3.2.2. Bird’s Nest Disturbance Strategy

#### 3.2.3. ICS Algorithm Flow

_{max}, the number of nests N, w

_{max}, and w

_{min}.

#### 3.3. Performance Test of the ICS Algorithm

^{−16}in the later stage of iterations, but its optimization accuracy remained higher than that of other algorithms. For F7 (a simple low-dimension function), the ICS algorithm and other algorithms converged to the global optimum, indicating its excellent optimization ability on low-dimension functions. According to the Std value, the standard deviation of the ICS algorithm was 0 in many tests under different functions, manifesting its favorable stability in optimization.

## 4. Parameter Identification and Experimental Verification of the RAPI Model

#### 4.1. Parameter Optimization

#### 4.1.1. Optimization of the Number of Hysteresis Factors

#### 4.1.2. Order Optimization of the Auxiliary Function

#### 4.1.3. Optimization of the Number of Play Operators

#### 4.2. Parameter Identification of the RAPI Model

#### 4.2.1. Parameter Identification of the API Model

#### 4.2.2. Parameter Identification of the RAPI Model

#### 4.3. Verification of the RAPI Model

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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Function | Expression | Definitional Domain | Optimal Value |
---|---|---|---|

F1 | $f\left(x\right)={\displaystyle \sum _{i=1}^{n}{x}_{i}^{2}}$ | [−100, 100] | 0 |

F2 | $f\left(x\right)={\displaystyle \sum _{i=1}^{n}\left|{x}_{i}\right|}+{\displaystyle \prod _{i=1}^{n}\left|{x}_{i}\right|}$ | [−10, 10] | 0 |

F3 | $f\left(x\right)={\displaystyle \sum _{i=1}^{n}i{x}^{4}}$ | [−1.28, 0.64] | 0 |

F4 | $f\left(x\right)=\frac{1}{4000}{\displaystyle \sum _{i=1}^{n}{x}_{i}^{2}-{\displaystyle \prod _{i=1}^{n}\mathrm{cos}\left(\frac{{x}_{i}}{\sqrt{i}}\right)}}+1$ | [−600, 600] | 0 |

F5 | $f\left(x\right)=-20\mathrm{exp}\left[-0.2\sqrt{\frac{1}{n}{\displaystyle \sum _{i=1}^{n}{x}_{i}^{2}}}\right]-\mathrm{exp}\left[\frac{1}{n}{\displaystyle \sum _{i=1}^{n}\mathrm{cos}\left(2\pi {x}_{i}\right)}\right]$ | [−10, 10] | 0 |

F6 | $f\left(x\right)={\displaystyle \sum _{i=1}^{n}\left|{x}_{i}\mathrm{sin}\left({x}_{i}\right)+0.1{x}_{i}\right|}$ | [−10, 10] | 0 |

F7 | $f\left(x\right)=0.5+\frac{{\mathrm{sin}}^{2}\left({x}_{1}^{2}-{x}_{2}^{2}\right)-0.5}{{\left[1+0.001\left({x}_{1}^{2}+{x}_{2}^{2}\right)\right]}^{2}}$ | [−100, 100] | 0 |

Algorithm | Parameter |
---|---|

PSO | c_{1} = 1.5, c_{2} = 1.5, w = 0.5 |

CS | P_{a} = 0.25, a = 0.01 |

ASCS | Step_{min} = 0, Step_{max} = 0.01, P_{a} = 0.25 |

Function | Mean/Std/Best | PSO | CS | ASCS | ICS |
---|---|---|---|---|---|

F1 | Mean | 8.92 | 3.46 × 10^{−5} | 5.21 × 10^{−9} | 0.00 |

Std | 3.18 | 7.20 × 10^{−10} | 2.85 × 10^{−8} | 0.00 | |

Best | 4.42 | 5.94 × 10^{−6} | 4.52 × 10^{−267} | 0.00 | |

F2 | Mean | 3.51 × 10 | 3.48 × 10^{−2} | 2.07 × 10^{−2} | 0.00 |

Std | 1.07 × 10 | 8.58 × 10^{−4} | 7.40 × 10^{−2} | 0.00 | |

Best | 1.96 × 10 | 2.12 × 10^{−2} | 8.53 × 10^{−3} | 0.00 | |

F3 | Mean | 3.18 | 1.06 × 10^{2} | 7.13 × 10^{−16} | 0.00 |

Std | 2.29 | 3.68 × 10 | 3.90 × 10^{−15} | 0.00 | |

Best | 3.17 × 10^{−1} | 3.98 × 10 | 0.00 | 0.00 | |

F4 | Mean | 2.26 × 10 | 8.79 × 10^{−2} | 8.30 × 10^{−4} | 0.00 |

Std | 6.55 | 4.00 × 10^{−2} | 4.55 × 10^{−3} | 0.00 | |

Best | 1.13 × 10 | 2.20 × 10^{−2} | 0.00 | 0.00 | |

F5 | Mean | 1.18 × 10 | 1.90 | 8.60 × 10^{−4} | 8.88 × 10^{−16} |

Std | 1.12 | 3.93 × 10^{−1} | 4.71 × 10^{−3} | 0.00 | |

Best | 9.41 | 1.19 | 4.44 × 10^{−15} | 8.88 × 10^{−16} | |

F6 | Mean | 1.52 × 10 | 7.17 | 6.93 × 10^{−1} | 0.00 |

Std | 3.46 | 9.83×10^{−1} | 2.29 | 0.00 | |

Best | 8.46 | 5.16 | 3.88 × 10^{−11} | 0.00 | |

F7 | Mean | 1.56 × 10^{−14} | 1.57 × 10^{−6} | 7.10 × 10^{−9} | 0.00 |

Std | 5.79 × 10^{−14} | 1.83 × 10^{−6} | 1.96 × 10^{−8} | 0.00 | |

Best | 0.00 | 0.00 | 0.00 | 0.00 |

Evaluation Index | Expression |
---|---|

RMSE | $RMSE=\sqrt{\frac{{{\displaystyle \sum _{i=1}^{N}\left({c}_{\mathrm{exp}}^{i}-{c}_{mx}^{i}\right)}}^{2}}{N}}$ |

MAE | $MAE=\frac{1}{N}{\displaystyle \sum _{i=1}^{N}\left|{c}_{\mathrm{exp}}^{i}-{c}_{mx}^{i}\right|}$ |

MAD | $MAD=\underset{1\le i\le N}{\mathrm{max}}\left|{c}_{\mathrm{exp}}^{i}-{c}_{mx}^{i}\right|$ |

Number of β | RMSE | MAE | MAD |
---|---|---|---|

1 | 0.48 | 0.29 | 0.6 |

2 | 0.52 | 0.36 | 0.7 |

5 | 0.68 | 0.81 | 2.2 |

10 | 0.84 | 1.02 | 2.7 |

Operator No. | Threshold | Weight |
---|---|---|

0 | 0 | 2.2526 |

1 | 0.5 | 2.2068 |

2 | 1 | 1.7295 |

3 | 1.5 | 2.4370 |

4 | 2 | 1.8501 |

5 | 2.5 | 0.6715 |

6 | 3 | −0.0693 |

7 | 3.5 | 0.3086 |

8 | 4 | 0.2692 |

9 | 4.5 | 0.2779 |

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

β | 1.0056 |

a_{0} | 0.2699 |

a_{1} | 2.7287 |

a_{2} | −0.2615 |

a_{3} | −0.0547 |

c | 1 |

Frequency | β | α_{3} | α_{2} | α_{1} | α_{0} | c |
---|---|---|---|---|---|---|

1 | 0.9739 | −0.0681 | −0.0249 | 2.0342 | −0.1091 | 1 |

25 | 0.9872 | −0.0616 | −0.0214 | 1.2331 | 0.0429 | 1.2192 |

40 | 1.0198 | −0.0557 | −0.0169 | −0.0524 | 0.3016 | 1.3033 |

55 | 1.0321 | −0.0429 | −0.0141 | −0.0524 | 0.3016 | 1.3615 |

70 | 1.0472 | −0.0384 | −0.0115 | −1.261 | 0.6109 | 1.4099 |

85 | 1.048 | −0.0309 | −0.0095 | −2.9429 | 0.7222 | 1.4637 |

100 | 1.0904 | −0.0308 | −0.0064 | −3.1027 | 0.8558 | 1.4659 |

120 | 1.1193 | −0.0185 | −0.0009 | −4.543 | 1.2016 | 1.6500 |

Index | RMSE | MAE | MAD |
---|---|---|---|

RAPI | 1.37 | 1.12 | 5.1 |

PI | 4.98 | 4.62 | 12.6 |

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**MDPI and ACS Style**

Liu, Y.; Meng, J.; Cao, J.
Rate-Dependent Hysteresis Model of a Giant Magnetostrictive Actuator Based on an Improved Cuckoo Algorithm. *Actuators* **2023**, *12*, 400.
https://doi.org/10.3390/act12110400

**AMA Style**

Liu Y, Meng J, Cao J.
Rate-Dependent Hysteresis Model of a Giant Magnetostrictive Actuator Based on an Improved Cuckoo Algorithm. *Actuators*. 2023; 12(11):400.
https://doi.org/10.3390/act12110400

**Chicago/Turabian Style**

Liu, Yang, Jianjun Meng, and Jingnian Cao.
2023. "Rate-Dependent Hysteresis Model of a Giant Magnetostrictive Actuator Based on an Improved Cuckoo Algorithm" *Actuators* 12, no. 11: 400.
https://doi.org/10.3390/act12110400