# Single-Neuron Adaptive Hysteresis Compensation of Piezoelectric Actuator Based on Hebb Learning Rules

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## Abstract

**:**

## 1. Introduction

## 2. The Hysteretic Nonlinearity of the PEA

#### 2.1. Experimental Setup

#### 2.2. Characteristics of the PEA’s Hysteresis

## 3. Single-Neuron Adaptive Controller Design

#### 3.1. Single-Neuron Adaptive Control Algorithm

_{i}, y, and δ are the state variable, output, and threshold, respectively; and w

_{i}represents the weight of x

_{i}that can be adjusted by the learning rules.

_{i}(t) during learning is proportional to the signal p

_{i}(t) and decays slowly, the learning rule of the neuron can be expressed as

_{i}(t) is the learning rules. To further improve the adaptability of neurons, the following learning rules are employed:

#### 3.2. Controller Design for the PEA

_{1}(t), x

_{2}(t), and x

_{3}(t) are adopted as the state variables to the neuron system.

_{i}(t) will converge to a stable value if c is small enough. According to the common experience of single-neuron adaptive control, d is typically less than 0.5. In this paper, d = 0.4 is adopted.

_{i}(t) is calculated using Equation (4). Three state variables correspond to three control outputs produced by the neuron, which are the proportional feedback u

_{1}(t), first-order differential feedback u

_{2}(t), and second-order differential feedback u

_{3}(t), respectively. The proportional feedback can quickly reduce the tracking error. The first-order differential feedback can improve the system’s transient state performance, i.e., the response speed and overshoot. The second-order differential feedback ensures that the system remains stable during a fast response. The change in the weight reflects the dynamic characteristics of the controlled plant and the response process. The neuron continuously adjusts the weight through its own learning rules, and quickly eliminates the error and enters the steady-state under the correlation of the three feedbacks.

## 4. Experimental Verifications

_{max}and Y

_{max}are the maximum allowable control input and maximum displacement output, respectively. The open-loop control represents the basic characteristics of the system as no controller is utilized.

_{p}is tuned with the other gains set to 0. Subsequently, the other gains are adjusted after the K

_{p}is specified. For the PEA, PI control is found to be adequate to achieve satisfactory performance. In fact, a trial and error process is inevitable to finely tune the PID gains to achieve satisfactory results.

#### 4.1. Step Response

_{p}= 0.8, K

_{i}= 1000, and K

_{d}= 0 following the critical ratio method. Step response experiments are carried out and the experimental results are shown in Figure 4. The step response of the open-loop system is also provided for the purpose of comparison.

#### 4.2. Tracking of Sinusoidal Trajectories

_{p}, K

_{i}, and K

_{d}are tuned to 1.11, 100, and 0, respectively. Figure 5 shows the tracking performance of the 1 Hz sinusoidal trajectory. Compared to the open-loop system, both the proposed method and the PID controller can successfully compensate the hysteresis of the PEA. The PEA can follow the desired trajectory well. It can be observed that the steady-state tracking error of the proposed method can be reduced to the noise level. The steady-state tracking error of the PID controller is slightly higher than the noise level but is still comparable to the proposed method. In the following experiments, sinusoidal and triangular trajectories with higher frequencies are utilized while the parameters of the two controllers are fixed to the above values. This helps to test the robustness and adaptability of the proposed method against the rate-dependence of the PEA’s hysteresis.

_{i}and r

_{i}represents the i th values of the actual and desired trajectories, respectively, and N is the length of sampling data.

#### 4.3. Tracking of Triangular Trajectories

#### 4.4. Hysteresis Compensation Efficiency

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

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Frequency (Hz) | The Proposed Method | PID | ||
---|---|---|---|---|

RMSE (nm) | RRMSE (%) | RMSE (nm) | RRMSE (%) | |

1 | 64.5 | 0.76 | 101.9 | 1.20 |

5 | 90.2 | 1.06 | 349.3 | 4.11 |

10 | 108.6 | 1.28 | 626.1 | 7.37 |

20 | 133.0 | 1.56 | 941.5 | 11.08 |

50 | 170.2 | 2.02 | 1158.8 | 13.63 |

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**MDPI and ACS Style**

Qin, Y.; Duan, H.
Single-Neuron Adaptive Hysteresis Compensation of Piezoelectric Actuator Based on Hebb Learning Rules. *Micromachines* **2020**, *11*, 84.
https://doi.org/10.3390/mi11010084

**AMA Style**

Qin Y, Duan H.
Single-Neuron Adaptive Hysteresis Compensation of Piezoelectric Actuator Based on Hebb Learning Rules. *Micromachines*. 2020; 11(1):84.
https://doi.org/10.3390/mi11010084

**Chicago/Turabian Style**

Qin, Yanding, and Heng Duan.
2020. "Single-Neuron Adaptive Hysteresis Compensation of Piezoelectric Actuator Based on Hebb Learning Rules" *Micromachines* 11, no. 1: 84.
https://doi.org/10.3390/mi11010084