# A Sliding Mode Control Strategy for an ElectroHydrostatic Actuator with Damping Variable Sliding Surface

^{1}

^{2}

^{3}

^{4}

^{*}

## Abstract

**:**

## 1. Introduction

- (1)
- To alleviate the conflict between the overshoot and rapidity of the EHA system, a control strategy based on SMC is proposed. Compared to the classical SMC method, the overshoot is suppressed without undermining the speed, which is in line with both simulative and experimental results.
- (2)
- For parameter adjustment, a parameter-tuning method for SMC is established. For the controller, damping-ratio-based parameter tuning is optimized, which further improves the industrial applications of our controller.

## 2. Prerequisite

#### 2.1. Brushless DC Motor (BLDCM) Model

_{e}is the back-electromotive force (EMF) coefficient; T

_{e}is the corresponding electromagnetic torque of the motor; J

_{m}is the moment of inertia of the motor; T

_{l}is the equivalent external load; and B

_{m}is the viscous friction coefficient.

#### 2.2. Pump-Controlled Cylinder Model

_{i}and Q

_{o}be the inlet and outlet flow of plunger pump. The working flow of the EHA can be modeled as follows:

_{p}is the pump displacement; L

_{i}and L

_{o}represent the internal and external leakage coefficients; p

_{i}, p

_{o}, and p

_{a}are the inlet pressure, outlet pressure, and back pressure of the oil tank pump; V

_{in}and V

_{out}are the equivalent inlet and outlet volume; and β

_{e}is the elastic modulus of the fluid.

_{c}stands for the internal leakage.

_{o}represents the effective volume of the chamber; L

_{a}is proportional to the pressure difference Δp is the total leakage coefficient of the pump and the cylinder; Q

_{a}is the unconsidered flow loss; M stands for the total equivalent mass of the cylinder and the load; B

_{c}is the viscous friction coefficient of the cylinder; K

_{s}is the elastic load coefficient; F

_{f}stands for the static friction; and F

_{L}is the load.

_{a}stands for the total rotational inertia of the motor and the pump.

#### 2.3. Problem Formulation

_{4}is a virtual control input with respect to the EHA model, based on which a three-order subsystem is formed by the first three equations. It can be observed that this high-order system contains both matched disturbances and mismatched disturbances. SMC is used to guarantee the robustness but is insensitive to mismatched disturbances [30,31]. Specifically, the control with mismatched disturbances is more challenging than that with only matched disturbances, and only a few related results have been proposed [32].

_{1}z

_{2}and z

_{3}represent the position, velocity, and acceleration of the cylinder, respectively. Hence, the first three terms from Equation (6) can be elaborated on by Equation (7):

_{un}stands for the flow loss that has not been considered.

## 3. Methodology

#### 3.1. Sliding Mode Controller with Damping Variable Sliding Surface

_{1}and c

_{2}are positive, which meets the requirements of the Routh‒Hurwitz stability criterion [37]. The error e, as such, approaches 0 during the control process.

_{1}and c

_{2}with respect to the controlling is restricted to c

_{1}, c

_{2}> 0. Despite the range of c

_{1}and c

_{2}within a two-dimensional plane, the values of these two parameters will inevitably affect the dynamic performance.

_{d}, which is:

_{d}is nonderivable at the start time, ${\dot{X}}_{d}$ and ${\ddot{X}}_{d}$ do not exist in theory. However, in reality, ${\dot{X}}_{d}$ and ${\ddot{X}}_{d}$ will converge to the largest number instead of diverging to infinity. Furthermore, the enormous number can cause a saturation of control output (u* in Equation (14)) and a plunge of output following the step moment, which results in an impact. However, this impact is the source of a large overshoot or an oscillation during the system adjustment process. For this reason, the overshoot is limited by an artificial ceiling of ${\dot{X}}_{d}$ and ${\ddot{X}}_{d}$, as mentioned in Section 1.

_{1}and c

_{2}is facilitated by introducing the undamped natural frequency ω

_{n}and the damping ratio ξ, which are:

#### 3.2. Extended State Observer (ESO)

_{1}is measured by a displacement sensor, from which z

_{2}and z

_{3}are derived by using a differentiator. The noise generated by the differentiator will cause the distortion of z

_{2}and z

_{3}. Moreover, let ${f}_{d}\left(t\right)$ stand for the system disturbance, which is compensated for by the robust term in Equation (13). Nevertheless, this compensation is such an overcompensation that it can bring cause chattering in the steady phase of the servo system. For this reason, a fourth-order ESO, not only for estimating the system state, but also for revising the compensation via observing the system disturbance, is established in Equation (22):

#### 3.3. Stability Analysis

## 4. Numerical Simulations

#### 4.1. Model Establishing

_{d}from the host computer, DV-SMC collects the state variables, i.e., z

_{i}(i = 1,2,3,4), from ESO to generate the output u*. Consequently, the final output, which is exactly represented by u

_{c}

^{d}, is calculated by dual-PID.

#### 4.2. Model Simulation Results

#### 4.3. Damping Ratio Selection

## 5. Experiments

#### 5.1. Experimental Settings

#### 5.2. Results

## 6. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 9.**Settling time of different control strategies. 1+ in Figure 9 represents that the settling time of SMC and PID on the 5 mm step input is longer than the signal collecting time (1s).

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

Piston effective area $\left({\mathrm{m}}^{2}\right)$ | $1.134\times {10}^{-3}$ |

Effective stroke $\left(\mathrm{m}\right)$ | $0.1$ |

Leakage coefficient $\left({\mathrm{m}}^{3}/\left(\mathrm{s}/\mathrm{Pa}\right)\right)$ | $2.5\times {10}^{-11}$ |

Fluid elastic modulus $\left(\mathrm{N}/{\mathrm{m}}^{2}\right)$ | $6.86\times {10}^{8}$ |

Hydraulic cylinder volume $\left({\mathrm{m}}^{3}\right)$ | $4\times {10}^{-4}$ |

Cylinder viscous friction $\left(\mathrm{N}/\left(\mathrm{m}/\mathrm{s}\right)\right)$ | $1000$ |

Mass of cylinder and load $\left(\mathrm{kg}\right)$ | $243$ |

Pump displacement $\left({\mathrm{m}}^{3}/\mathrm{rad}\right)$ | $3.98\times {10}^{-7}$ |

Motor viscous friction $\left(\mathrm{N}\cdot \mathrm{m}/\left(\mathrm{rad}/\mathrm{s}\right)\right)$ | $6\times {10}^{-4}$ |

Phase resistance $\left(\mathsf{\Omega}\right)$ | $0.2$ |

Phase inductance $\left(\mathrm{mH}\right)$ | $1.33$ |

Motor spindle moment of inertia $\left(\mathrm{kg}\cdot {\mathrm{m}}^{2}\right)$ | $4\times {10}^{-4}$ |

Torque coefficient $\left(\mathrm{N}\cdot \mathrm{m}/\mathrm{A}\right)$ | $0.351$ |

Back EMF coefficient $\left(\mathrm{V}/\left(\mathrm{rad}/\mathrm{s}\right)\right)$ | $0.234$ |

Elastic load coefficient $\left(\mathrm{N}/\mathrm{m}\right)$ | $8\times {10}^{8}$ |

Bus voltage $\left(\mathrm{VDC}\right)$ | $270$ |

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

Rated pressure $\left(\mathrm{MPa}\right)$ | $11$ |

Rated speed $\left(\mathrm{mm}/\mathrm{s}\right)$ | $300$ |

Rated force $\left(\mathrm{kN}\right)$ | $12$ |

Effective displacement $\left(\mathrm{mm}\right)$ | $0~110$ |

Rated power supply $\left(\mathrm{VDC}\right)$ | $270$ |

Bandwidth $\left(\mathrm{Hz}\right)$ | $5$ |

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**MDPI and ACS Style**

Wang, M.; Wang, Y.; Yang, R.; Fu, Y.; Zhu, D.
A Sliding Mode Control Strategy for an ElectroHydrostatic Actuator with Damping Variable Sliding Surface. *Actuators* **2021**, *10*, 3.
https://doi.org/10.3390/act10010003

**AMA Style**

Wang M, Wang Y, Yang R, Fu Y, Zhu D.
A Sliding Mode Control Strategy for an ElectroHydrostatic Actuator with Damping Variable Sliding Surface. *Actuators*. 2021; 10(1):3.
https://doi.org/10.3390/act10010003

**Chicago/Turabian Style**

Wang, Mingkang, Yan Wang, Rongrong Yang, Yongling Fu, and Deming Zhu.
2021. "A Sliding Mode Control Strategy for an ElectroHydrostatic Actuator with Damping Variable Sliding Surface" *Actuators* 10, no. 1: 3.
https://doi.org/10.3390/act10010003