# Modeling and Control of a Linear Piezoelectric Actuator

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Design Model of a Piezoelectric Actuator

_{1}point. The amplification mechanisms AM2–AM4 are composed of levers, and the pointers are O

_{1}, O

_{2}, and O

_{3}, respectively. When the vertical piezoelectric stack is operated, the force point positions (F

_{2}and F

_{3}) of AM2 and AM3 increase. Under the action of levers AM2 and AM3, the position of the pointer O

_{3}of AM4 is lowered, and the position of the force point F

_{4}increases, thus realizing the multi-stage amplification of the output displacement of the piezoelectric stack. In turn, the multi-stage amplification produces a displacement δ.

_{p}is the stiffness of the piezoelectric stack, m, c, and K are the equivalent mass of the moving mechanism, damping of the amplifying mechanism, and stiffness, respectively, x

_{p}is the telescopic displacement of the piezoelectric stack, x is the output displacement of the mechanism, and F is the output force of the piezoelectric stack.

_{n}is the natural frequency, and ξ is the damping ratio. Their expression can be written as follows:

## 3. Hysteretic Nonlinear Model

_{i}and r

_{i}are the weights and thresholds of each operator, respectively, and n is the total number of operators.

_{in}and n

_{out}are the numbers of neurons in the input and output layers, respectively, and k is a constant, usually between one and five. Finally, 12 neuron nodes were selected after several attempts.

_{ij}, and threshold, r

_{j}, in Equation (8) and the weight, w

_{j}

_{1}, threshold, θ, and constant coefficients, k and b, in Equation (9) were constantly modified to ensure that the error reached the expected value to complete the fitting of the hysteresis model.

_{i}, is the mean square error between the fitting and experimental results, which is the evaluation standard for the quality of individuals in the group.

_{i}of each individual being selected was

## 4. Compound Control

_{min}, e

_{max}] and [ec

_{min}, ec

_{max}], respectively, and the required domains are [e*

_{min}, e*

_{max}] and [ec*

_{min}, ec*

_{max}], respectively, the change formula is

_{e}and k

_{ec}are the scale factors of e and ec, respectively, as follows:

_{p}, ΔT

_{i}, and ΔT

_{d}was similarly defined as (−3,3).

_{p}, can speed up the response, but if it is too large, it leads to an overshoot. The integration link, T

_{i}, can eliminate static errors; however, it can easily cause integral saturation and overshoot in the early stages. The differential link, T

_{d}, can anticipate and act before the deviation becomes large; however, this increases the adjustment time. Based on the above experience, the general ideas for establishing fuzzy rules are as follows:

- (1)
- When e is large, K
_{p}should be set to a large value to ensure system response speed. Meanwhile, to avoid a large overshoot in the system and limit the integration function, T_{i}should be set to zero and T_{d}should be set to a smaller value. - (2)
- When e is medium, the K
_{p}value should be reduced. Simultaneously, to prevent the system from becoming unstable owing to noise, the value of T_{i}should be increased appropriately to reduce the time to reach a steady state. - (3)
- When e is small, to reduce the steady-state time of the system and avoid a decrease in adjustment accuracy caused by steady-state errors, K
_{p}and T_{i}should be set to larger values. The T_{d}value is mainly adjusted based on the absolute value of ec. When the absolute value of ec is large, the T_{d}value is small to prevent multiple shocks in the system. In other cases, the T_{d}value is medium.

## 5. Simulation and Experimental Test

#### 5.1. Simulation and Verification of the Model

^{4}, the training precision target was 10

^{−7}, the training speed was 0.05, and the rest were default values. A performance comparison is shown in Figure 12. The BP neural network improved by the GA exhibited the same operational results every time. Although the maximum absolute error (MAE) and running time were slightly worse than those of the BP neural network, the relative error (RE) and root mean square error (RMSE) were smaller than those of the BP neural network. This shows that the GA-BP neural network can determine the global optimal solution, which has a higher fitting effect. Therefore, the superiority of the GA-BP neural network was verified.

_{p}was 0.2, the integral coefficient T

_{i}was 80, and the differential coefficient T

_{d}was 0.055.

#### 5.2. Experiment of Compound Control

## 6. Conclusions

- (1)
- The maximum output displacement of the proposed piezoelectric actuator was 558.3 μm under a driving voltage of 150 V and a driving frequency of 1 Hz.
- (2)
- The proposed GA-BP neural network model solved problems, such as strong randomness, falling easily into the local optimal solution, and slow convergence of the BP neural network, while ensuring modeling accuracy.
- (3)
- The stability adjustment time of the feedforward fuzzy PID control was 11% and 22% of that of the feedforward fuzzy PID control. When interference occurred, the overshoot was reduced by 69.74% compared to the feedforward control.
- (4)
- The feedforward fuzzy PID control system has the advantages of a fast response speed and strong anti-interference ability.

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

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**Figure 1.**Structure of the multi-range linear piezoelectric actuator (Ref. [21]): (

**a**) the assembly drawing; (

**b**) 2D drawing.

**Figure 3.**Finite element analysis of the displacement amplifying mechanism: (

**a**) mesh model; (

**b**) deformation results.

**Figure 5.**Model identification based on frequency response method: (

**a**) experimental results; (

**b**) system identification.

**Figure 6.**Hysteresis curve of the piezoelectric actuator: (

**a**) input voltage and output displacement comparison diagram; (

**b**) hysteresis curve.

**Figure 12.**Comparison of GA-BP and BP neural network algorithms: (

**a**) relative errors; (

**b**) root mean square errors; (

**c**) maximum absolute errors; (

**d**) computation times.

**Figure 16.**Experimental test system for the piezoelectric actuator (Ref. [21]).

**Figure 17.**Output characteristics of the piezoelectric actuator at 1 Hz: (

**a**) displacement varies with time; (

**b**) displacement varies with voltage.

**Figure 18.**Output characteristics of the proposed piezoelectric actuator: (

**a**) displacement varies with time at 2 Hz; (

**b**) displacement varies with voltage at 2 Hz; (

**c**) displacement varies with time at 3 Hz; (

**d**) displacement varies with voltage at 3 Hz.

Load (g) | Maximum Output Displacement (μm) | Magnifying Factor |
---|---|---|

0 | 683.0 | 22.77 |

50 | 671.4 | 22.38 |

100 | 665.7 | 22.19 |

150 | 660.3 | 22.01 |

200 | 654.6 | 21.82 |

e | ec | ||||||
---|---|---|---|---|---|---|---|

NB | NM | NS | ZO | PS | PM | PB | |

NB | PB | PB | PM | PM | PS | ZO | ZO |

NM | PB | PB | PM | PS | PS | ZO | NS |

NS | PM | PM | PM | PS | ZO | NS | NS |

ZO | PM | PM | PS | ZO | NS | NM | NM |

PS | PS | PS | ZO | NS | NS | NM | NM |

PM | PS | ZO | NS | NM | NM | NM | NB |

PB | ZO | ZO | NM | NM | NM | NB | NB |

e | ec | ||||||
---|---|---|---|---|---|---|---|

NB | NM | NS | ZO | PS | PM | PB | |

NB | NB | NB | NM | NM | NS | ZO | ZO |

NM | NB | NB | NM | NS | NS | ZO | ZO |

NS | NB | NM | NS | NS | ZO | PS | PS |

ZO | NM | NM | NS | ZO | PS | PM | PM |

PS | NM | NS | ZO | PS | PS | PM | PB |

PM | ZO | ZO | PS | PS | PM | PB | PB |

PB | ZO | ZO | PS | PM | PM | PB | PB |

e | ec | ||||||
---|---|---|---|---|---|---|---|

NB | NM | NS | ZO | PS | PM | PB | |

NB | PS | NS | NB | NB | NB | NM | PS |

NM | PS | NS | NB | NM | NM | NS | ZO |

NS | ZO | NS | NM | NM | NS | NS | ZO |

ZO | ZO | NS | NS | NS | NS | NS | ZO |

PS | ZO | ZO | ZO | ZO | ZO | ZO | ZO |

PM | PB | NS | PS | PS | PS | PS | PB |

PB | PB | PM | PM | PM | PS | PS | PB |

Fitting Algorithm | RE (%) | RMSE (μm) | MAE (μm) |
---|---|---|---|

PI model | 1.1505 | 4.1574 | 12.6382 |

Polynomial model | 0.3010 | 1.0881 | 8.3264 |

MPI model | 0.2082 | 0.9141 | 3.8659 |

BP neural network model | 0.0480 | 0.1736 | 0.6938 |

GA-BP neural network model | 0.0467 | 0.1698 | 0.6314 |

Driving Frequencies | PID (μm) | BP (μm) | GA-BP Fuzzy PID (μm) | Uncontrolled (μm) |
---|---|---|---|---|

1 Hz | 2.3780 | 2.5048 | 1.2734 | 62.3220 |

2 Hz | 4.9461 | 5.0520 | 2.2116 | 95.5618 |

3 Hz | 15.9856 | 14.6087 | 11.8361 | 127.8361 |

Items | This Work | Ling et al. [34] | Li et al. [35] | Wang et al. [36] | Yu et al. [37] |
---|---|---|---|---|---|

Operation frequency (Hz) | 1–5 | 650 | 5000 | 1600 | 213 |

Maximum output displacement (μm) | 558.3 | / | 3.17 | / | / |

Maximum load (N) | 49.33 | 0.6 | 3.5 | 5 | / |

Resolution (μm) | 0.1 | / | 0.17 | 17.38 | 1.1 |

Drive type | compliant mechanism | inchworm mechanism | Stick-slip type | Stick-slip type | compliant mechanism |

Magnification factor | 21.55 | 3.75 | / | / | 8.99 |

Control method | GA-BP fuzzy PID control | / | / | / | Sliding-mode control |

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**MDPI and ACS Style**

Li, H.; Tong, Y.; Li, C.
Modeling and Control of a Linear Piezoelectric Actuator. *Actuators* **2024**, *13*, 55.
https://doi.org/10.3390/act13020055

**AMA Style**

Li H, Tong Y, Li C.
Modeling and Control of a Linear Piezoelectric Actuator. *Actuators*. 2024; 13(2):55.
https://doi.org/10.3390/act13020055

**Chicago/Turabian Style**

Li, Huaiyong, Yujian Tong, and Chong Li.
2024. "Modeling and Control of a Linear Piezoelectric Actuator" *Actuators* 13, no. 2: 55.
https://doi.org/10.3390/act13020055