# Adaptive Repetitive Control of A Linear Oscillating Motor under Periodic Hydraulic Step Load

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## Abstract

**:**

## 1. Introduction

## 2. Problem Formulation and Dynamic Models

## 3. Nonlinear Adaptive Repetitive Controller Design

#### 3.1. Design Model and Issues to be Addressed

**Assumption**

**1.**

#### 3.2. Projection Mapping and Parameter Adaptation

#### 3.3. Controller Design

#### 3.4. Main Results

**Theorem**

**1.**

**Proof.**

**Remark**

**1.**

## 4. Comparative Experimental Results

#### 4.1. Experiment Setup

#### 4.2. Comparative Experimental Results

- $APC:$ The nonlinear adaptive repetitive controller in the above section. To simplify the controller and save memory, only a small number of unknown parameters are adapted, i.e., m in Equation (12) is set to 3. So ${\phi}_{d}^{T}{\theta}_{d\alpha}$ and its counterpart of motor 2 ${\phi}_{d}^{T}{\theta}_{d\beta}$ all have seven terms. The initial values for the parameters are set to ${\widehat{\theta}}_{\alpha}\left(0\right)={\left(\right)}^{70}T$ and ${\widehat{\theta}}_{\beta}\left(0\right)={\left(\right)}^{70}T$. Viscous damping coefficients ${B}_{1}$ and ${B}_{2}$ as well as the Coulomb friction amplitude ${A}_{f}$ were identified by Wang in the design process of the linear oscillating motor [27]. Even if the wear of linear bears and different working conditions change the true values of parameters slightly, it is reasonable to set identified values as the initial values. ${K}_{s}$ is the elasticity coefficient of two parallel springs. Its initial value was set to its nominal value. The initial value of the load pressure of the pump was set to 2.8 MPa by adjusting the throttle valve. The hydraulic load of linear motor is periodic. Its initial amplitude can be approximated as the product of load pressure and ram area of piston and its phase is synchronous with displacement of the other linear motor. Initial Fourier coefficients were got by applying Fourier transformation to the periodic initial hydraulic load. The bounds of the parameter variations in the two motor systems are the same and are estimated as ${\theta}_{min}={\left(\right)}^{60}T$ and ${\theta}_{max}={\left(\right)}^{100}T$. The magnitude of $\Delta $ is assumed to be less than 500, i.e., $\delta {(x,t)}_{\alpha}\le 500$ and $\delta {(x,t)}_{\beta}\le 500$. The control gains in two motors are the same: ${k}_{1}=20$, ${k}_{s1}=600$. The shape function of Coulomb friction ${S}_{f}\left({x}_{2}\right)=\mathrm{arctan}\left(1000{x}_{2}\right)$ [21]. Adaptation rate matrix $\mathsf{\Gamma}$ in the two systems are identically set: $\mathsf{\Gamma}=\mathrm{diag}[50,5,100,100,100,100,100,100,10,10]$. The control diagram is shown in Figure 3.
- Proportional-integral-derivative (PID): A conventional proportional-integral-derivative controller was built. We optimized the PID parameters in experiments. Derivative section is sensitive to interference, whose coefficient was set to zero. Considering the working condition of the LDP, we paid attention to the steady state response of the linear motor, when we optimized PI coefficients. The hydraulic pressure of 2.8 MPa was applied on the LDP. Because of coupling effect of two motors, the amplitude attenuation and phase lag of the single linear motor cannot reflect the control performance. The output flow rate of LDP was measured and recorded, which was used as the optimization objective. We got the optimal PI coefficients when the flow rate was maximum. The controller parameters are ${k}_{\alpha p}=65,\phantom{\rule{3.33333pt}{0ex}}{k}_{\alpha i}=89,\phantom{\rule{3.33333pt}{0ex}}{k}_{\alpha d}=0$, ${k}_{\beta p}=65,\phantom{\rule{3.33333pt}{0ex}}{k}_{\beta i}=89,\phantom{\rule{3.33333pt}{0ex}}{k}_{\beta d}=0$, which represent the P-gain, I-gain and D-gain of two motors, respectively.

- Maximal absolute value of the tracking errors is defined as$$\begin{array}{c}\hfill {M}_{e}=\underset{i=1,\dots ,N}{max}\mid {z}_{1}\left(i\right)\mid \end{array}$$
- Average tracking error is defined as$$\begin{array}{c}\hfill \kappa =\frac{1}{N}\sum _{I=1}^{n}\mid {z}_{1}\left(i\right)\mid \end{array}$$
- Standard deviation performance index is defined as$$\begin{array}{c}\hfill \sigma =\sqrt{\frac{1}{N}\sum _{I=1}^{n}{\left(\right)}^{|}2}\end{array}$$

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Abbreviations

LDP | Linear drive collaborative rectification structure pump |

ARC | Adaptive robust control |

PID | Proportional-integral-derivative |

RISE | Robust integral of the sign of the error |

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Items | Symbol | Value |
---|---|---|

Stroke of cylinder | S | 4 mm |

Resonant frequency of linear motor | f | 28 Hz |

ram area of piston | ${A}_{p}$ | 48 ${\mathrm{mm}}^{2}$ |

Force coefficient of linear motor | ${K}_{e}$ | 27.5 N/A |

Motor driver amplification coefficient | ${K}_{d}$ | 8 A/V |

Mass of the mover | m | 1.15 kg |

Additive elasticity coefficient of two parallel springs | ${K}_{s}$ | 36,000 N/$\mathrm{m}$ |

Indices | ${\mathit{M}}_{\mathit{e}}$ | $\mathit{\kappa}$ | $\mathit{\sigma}$ |
---|---|---|---|

Motor 1 with PID | 2.3219 | 1.3354 | 0.7263 |

Motor 2 with PID | 4.3039 | 2.7644 | 1.2643 |

Motor 1 with APC | 2.1675 | 0.8506 | 0.6028 |

Motor 2 with APC | 1.5320 | 0.4887 | 0.3950 |

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**MDPI and ACS Style**

Li, X.; Jiao, Z.; Li, Y.; Cao, Y.
Adaptive Repetitive Control of A Linear Oscillating Motor under Periodic Hydraulic Step Load. *Sensors* **2020**, *20*, 1140.
https://doi.org/10.3390/s20041140

**AMA Style**

Li X, Jiao Z, Li Y, Cao Y.
Adaptive Repetitive Control of A Linear Oscillating Motor under Periodic Hydraulic Step Load. *Sensors*. 2020; 20(4):1140.
https://doi.org/10.3390/s20041140

**Chicago/Turabian Style**

Li, Xinglu, Zongxia Jiao, Yang Li, and Yuan Cao.
2020. "Adaptive Repetitive Control of A Linear Oscillating Motor under Periodic Hydraulic Step Load" *Sensors* 20, no. 4: 1140.
https://doi.org/10.3390/s20041140