# Data-Driven Adaptive Iterative Learning Method for Active Vibration Control Based on Imprecise Probability

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## Abstract

**:**

## 1. Introduction

## 2. State–Space Model and P-Type IL Method

## 3. Dynamic Linearization and MFA Controller Design

**Lemma**

**1:**

## 4. The Stopping Criteria Design

#### 4.1. Preliminary Notion of Imprecise Probability

#### 4.2. The Diagnosis Method Design

#### 4.2.1. Fault Reliability

#### 4.2.2. Establish Fault Probability Interval

#### 4.2.3. Diagnosis Cost Functions and Decision-Making

## 5. The Summary of the Proposed Method

## 6. Numerical Simulations

#### 6.1. FE Modeling and Setting of Controller Parameters

#### 6.2. Harmonic Excitation

**/**b and sensor c are shown in Figure 4c,d. In comparison with the P-type IL method, smaller amplitudes were obtained as long as the piezoelectric cantilevered plate was controlled by the robust MFA-IL control and the proposed method. The root mean square (RMS) values of the dynamic responses and measurement signals were used to quantitatively analyze the performance of the P-type IL method, the robust MFA-IL control, and the proposed method, which are listed in Table 3. From Table 3, both the robust MFA-IL control and the proposed method had better control performance by comparing the P-type IL method. The vibration amplitude was reduced 41.22% under the control of the proposed method, and the vibration amplitudes reduced 40.36% under the control of the robust MFA-IL control. The proposed method and the robust MFA-IL had similar control precision.

**/**b had faster convergence speed than that connected with actuator c. Based on the theory of imprecise probability, all controllers could learn sufficiently, and satisfactory control performance could be achieved.

**/**b and sensor c are shown in Figure 9b. When noise signals begin to excite, the dynamic responses of the plate and measurement signals from sensors change greatly; however, the divergence phenomenon was not found. After stopping the excitation of the noise signals, the vibration control system was restored to the stability state by the proposed method.

#### 6.3. Random Excitation

**/**b and actuator c are presented in Figure 12a,b. The measurement signals from sensor a

**/**b and sensor c are displayed in Figure 12c,d. The feedback gains are depicted in Figure 13.

## 7. Experiments

#### 7.1. ExperimentSetup

#### 7.2. Modal Analysis

#### 7.3. ExperimentResults

**/**b and sensor c are presented in Figure 17a,b, and the control voltages of actuator a

**/**b and actuator c are presented in Figure 17c,d. The data used to calculate the RMSs are recorded after learning termination, and the RMSs are given in Table 3. From Table 3, the proposed method reached the comparatively ideal control performance: the vibration amplitudes were reduced 33.9% under the control of the proposed method, and the vibration amplitudes were reduced 31.8% under the control of the P-type IL method. This excellent performance was obtained by integrating the MFA method into time-varying P-type IL method. The learning processes of the feedback gains are depicted in Figure 18. The proposed method is feasible to simultaneous maintain the control performance and damp down quickly for structural vibration. Under the control of different methods, the feedback gains obtained from the same controllers result in distinct values. The real-time diagnosis curves for the fused information and single information sources are given in Figure 19. By the theory of imprecise probability, the learning processes of feedback gains can be diagnosed in real time. The decisions made based on the designed stopping criteria causes all controllers to learn sufficiently, and excellent control performance was obtained.

## 8. Conclusions and Outlooks

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 4.**The time-domain measurement signals of the actuators/sensors: (

**a**) actuator a

**/**b; (

**b**) actuator c; (

**c**) sensor a

**/**b; (

**d**) sensor c

**.**

**Figure 9.**Verification curves with the stability of the proposed method: (

**a**) displacement responses and (

**b**) measurement signals of sensors.

**Figure 12.**The time-domain measurement signals of actuators/sensors: (

**a**) actuator a

**/**b; (

**b**) actuator c; (

**c**) sensor a

**/**b; (

**d**) sensor c.

**Figure 16.**Vibration responses excited by actuator a: (

**a**) the time-domain measurement signals of the sensor a; (

**b**) frequency response of the sensor a

**.**

**Figure 17.**The time-domain measurement signals of actuators/sensors: (

**a**) sensor a/b; (

**b**) sensor c; (

**c**) actuator a/b; (

**d**) actuator c.

Graphite-Epoxy (GE) | Piezoelectric Material |
---|---|

Yong’s modulus ($\mathrm{GPa}$) | Elastic stiffness ($\mathrm{GPa}$) |

${E}_{11}=40.51$ | ${C}_{11}=126$ ${C}_{12}=79.5$ |

${E}_{22}={E}_{33}=13.96$ | ${C}_{13}=84.1$ ${C}_{33}=117$ |

Shear modulus ($\mathrm{GPa}$) | ${C}_{44}=23.3$ ${C}_{66}=23$ |

${G}_{12}={G}_{13}=3.1$ | Piezoelectric stain ($\mathrm{C}/{\mathrm{m}}^{2}$) |

${G}_{23}=1.55$ | ${e}_{16}={e}_{25}=17$ |

Poisson’s ratio | ${e}_{31}={e}_{32}=-6.5$ |

${v}_{12}={v}_{13}=0.22$ | ${e}_{33}=23.3$ |

${v}_{23}=0.11$ | Permittivity ($\mathrm{F}/\mathrm{m}$) |

Density ($\mathrm{kg}/{\mathrm{m}}^{3}$) | ${\epsilon}_{11}={\epsilon}_{22}=1.503\times {10}^{-8}$ |

$\rho =1830$ | ${\epsilon}_{33}=1.3\times {10}^{-8}$ |

Density ($\mathrm{kg}/{\mathrm{m}}^{3}$) | |

$\rho =7500$ |

Mode | Numerical (Hz) | Experimental (Hz) | Error Percentage |
---|---|---|---|

1 | 5.4377 | 5.326 | 2.1% |

2 | 24.217 | 21.259 | 13.9% |

3 | 28.683 | 31.593 | −9.2% |

Algorithm | Case 1 | Case 2 | Experiment | |||||||
---|---|---|---|---|---|---|---|---|---|---|

Point A | Point B | Sensor a/b | Sensor c | Point A | Point B | Sensor a/b | Sensor c | Sensor a/b | Sensor c | |

Uncontrolled | 2.262 × 10^{−3} | 8.805 × 10^{−3} | 7.420 | 4.314 | 3.430 × 10^{−3} | 12.793 × 10^{−3} | 9.934 | 5.249 | 3.764 | 2.167 |

P-type IL | 1.526 × 10^{−3} | 5.909 × 10^{−3} | 4.590 | 2.706 | 2.238 × 10^{−3} | 8.169 × 10^{−3} | 6.323 | 3.356 | 2.563 | 1.488 |

Robust MFA-IL | 1.479 × 10^{−3} | 5.719 × 10^{−3} | 4.348 | 2.561 | - | - | - | - | - | - |

Proposed method | 1.488 × 10^{−3} | 5.763 × 10^{−3} | 4.413 | 2.598 | 2.080 × 10^{−3} | 7.590 × 10^{−3} | 5.800 | 3.065 | 2.480 | 1.444 |

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**MDPI and ACS Style**

Bai, L.; Feng, Y.-W.; Li, N.; Xue, X.-F.; Cao, Y.
Data-Driven Adaptive Iterative Learning Method for Active Vibration Control Based on Imprecise Probability. *Symmetry* **2019**, *11*, 746.
https://doi.org/10.3390/sym11060746

**AMA Style**

Bai L, Feng Y-W, Li N, Xue X-F, Cao Y.
Data-Driven Adaptive Iterative Learning Method for Active Vibration Control Based on Imprecise Probability. *Symmetry*. 2019; 11(6):746.
https://doi.org/10.3390/sym11060746

**Chicago/Turabian Style**

Bai, Liang, Yun-Wen Feng, Ning Li, Xiao-Feng Xue, and Yong Cao.
2019. "Data-Driven Adaptive Iterative Learning Method for Active Vibration Control Based on Imprecise Probability" *Symmetry* 11, no. 6: 746.
https://doi.org/10.3390/sym11060746