Single-rod electrohydraulic system is widely applied in industrial production due to high power-to-weight ratio, but it generally has a low energy efficiency and has many system states which need to be measured. Therefore, an output feedback controller with energy saving is proposed in this paper. The designed controller only needs a displacement sensor to detect the position of the single-rod cylinder; the other states of the system are estimated by extended state observer (ESO). Besides, a nonlinear disturbance observer (NDO) is introduced to estimate the external mismatched disturbance. The output feedback controller based on extended state observer and nonlinear disturbance observer has a better tracking performance compared with other controllers. In addition, a proportional relief valve (PRV) is introduced to control the supply pressure of the system. The variable supply pressure reduces the energy of throttling loss and overflow loss, which achieves energy saving of about 54% according to the simulation results. Meanwhile, the tracking error of the energy saving controller is stable at 0.1 mm. In a word, the proposed controller not only achieves energy saving but also has a satisfactory trajectory tracking performance.

Electrohydraulic systems are widely used in industrial equipment and mobile applications such as excavator, crane, robot manipulator, and hydraulic press [

Generally speaking, the low energy efficiency of the electrohydraulic system is due to the fact that the supply energy produced by hydraulic power is much higher than the required energy by the actuators and the excess energy is lost as heat [

In order to maintain the characteristics of the valve-controlled system with fast response and high control accuracy, many researchers choose to add a proportional relief valve (PRV) on the original hydraulic system to achieve energy saving. PRV has the ability to adjust the supply pressure, which can reduce the throttle loss of the proportional directional valve (PDV). And the PDV is used to improve the position control accuracy of the hydraulic system [

In previous studies, most energy saving control method with variable supply pressure required speed, pressure, and even force sensors of the load for state feedback, but in many cases, only the output of the hydraulic system is measurable. Therefore, it is very meaningful to develop high performance output feedback control strategies [

To further improve the trajectory tracking accuracy of output feedback of the electrohydraulic system, a nonlinear disturbance observer (NDO) was designed to estimate the unknown disturbance [

In this study, the control purpose of the single-rod electrohydraulic servo system is not only to meet the trajectory tracking accuracy requirement but also to reduce energy consumption as possible. Therefore, the complete hydraulic system is introduced, where variable supply pressure control method is to realize the energy saving based on PRV and output feedback control strategy is to realize high performance position tracking based on PDV. ESO is designed to estimate the full states of the hydraulic system and NDO is presented to compensate for external disturbance and internal inaccurate modelling. Finally, the energy efficiency of the robust backstepping controller in variable supply pressure system is calculated.

The rest of the paper is organized as follows. Section

The configuration for position control and energy-saving control purposes of the proposed system is depicted in Figure

Single-rod electrohydraulic servo system.

The dynamics of the single-rod cylinder can be given by Newton’s second law:

The equations of flow into the two chambers flow through the PDV orifices can be written as

Due to the fact that the response of the PDV is much faster than the whole system dynamics, the valve dynamic is often neglected without reducing control performance in the servo system [

In general, the pump supplied flow rate is always higher than the actuator required flow rate, and the supply pressure always exceeds the cracking pressure, so the PRV is always open [

As the current sealing technology is becoming more and more mature, only the internal leakage of the single-rod cylinder is taken into consideration in this paper. In the following, the pressure dynamic equations of asymmetric cylinder are established with the ignorance of the external leakage [

In order to highlight the importance of system supply pressure

Therefore, the pressure of the actuator chambers can be rewritten as

To facilitate subsequent controller design, the state variables are defined as follows:

The state space equations of the whole system can be established through (

In (

In order to improve energy efficiency of the system, the supply pressure of the pump should be equal to the instantaneous pressure required by the actuator. The minimum required supply pressure is used to drive the single-rod cylinder to complete the trajectory tracking, while excessive supply pressure generates large energy consumption through PDV. And the bigger opening of the PDV results in less throttle loss which means energy efficiency is improved [

To track the desired trajectory

If the single-rod cylinder moves forward (

The relationship between the ideal supply pressure

Therefore, the required supply pressure

In (

To observe the unmeasured system states, the integrated disturbance

Then the original system in (

The estimate of

In order to complete the design of ESO,

Combining (

Define the estimate error of the auxiliary variable as

Therefore, the mismatched disturbances can be estimated by NDO in real time, which further improves the observation performance of ESO. Then define an auxiliary variable for ESO as

The dynamics of the auxiliary variable can be transformed as follows:

Define the parameter estimate error as

The characteristic equation of

To test the stability of ESO and NDO, a Lyapunov function is defined as

The time derivative of

The solution of (

From (

In order to achieve precise position tracking while ensuring that the single-rod system has a high energy efficiency, a recursive backstepping controller is proposed and the closed-loop stability of the controller is analysed by Lyapunov technique. The overall control scheme is shown in Figure

The block diagram of control scheme.

The state errors

Define

Since

In accordance with (

The resulting virtual control input

Similar to state error

Substituting the actual control input

In order to verify the stability of the proposed controller based on ESO and NDO in the single-rod electrohydraulic system, a positive definite function

Combining (

According to (

According to Lasalle’s invariance principle, there will be

In order to verify the trajectory tracking effect and energy saving effect of the proposed controller, the entire model of single-rod electrohydraulic system is established and parameters are listed in Table

Single-rod electrohydraulic system parameters.

Parameters | Description | Value | Units |
---|---|---|---|

_{1} | Effective area of the piston chamber | 1.9625 × 10^{−3} | m^{2} |

Α_{2} | Effective area of the rod chamber | 9.4514 × 10^{−4} | m^{2} |

_{s} | Supply pressure | 5 × 10^{6} | Pa |

_{r} | Tank pressure | 0 | Pa |

_{01} | Initial volumes of piston chamber | 3.9 × 10^{−4} | m^{−3} |

_{02} | Initial volumes of rod chamber | 1.9 × 10^{−4} | m^{−3} |

_{e} | Effective hydraulic fluid bulk modulus | 10^{9} | Pa |

_{t} | Total leakage coefficient of chambers | 9.2 × 10^{−13} | m^{5}/(N·s) |

Equivalent mass of the load | 80 | kg | |

Damping coefficient | 3000 | N·s/m | |

_{sv} | Voltage flow gain of PDV | 7.245 × 10^{−8} | m^{3}/(V·s·Pa^{1/2}) |

_{rv} | Pressure gain of PRV | 2.1 | MPa/V |

Contrast controller parameters.

Controllers | Parameters |
---|---|

PID | |

SMC | |

OFC | |

OFDC | |

OFDESC |

The motion trajectory is designed as

The desired trajectory of the system.

Comparison of tracking effect between different controllers.

Estimated disturbance of NDO for the system.

Figure

Tracking errors of OFDC and OFDESC.

Estimated pressures of OFDESC for the system.

Energy provided by the pump of OFDC and OFDESC.

In order to further study the energy utilization of the single-rod electrohydraulic system, the energy of throttling loss and overflow loss is calculated. Figure

Flow rate of the pump and the actuator.

Energy distribution of OFDC and OFDESC for the system.

In this paper, an output feedback controller based on extended state observer and nonlinear disturbance observer with energy saving is proposed. For a single-rod electrohydraulic system with only displacement signal output, the ESO is used to estimate all the states of system and the NDO is used to estimate the external disturbance. The simulation results show that the observers have a satisfactory observation effect in this system. Compared with PID, SMC, and OFC, the proposed controller has a better performance of trajectory tracking with about 0.1 mm steady-state error. Additionally, the variable supply pressure method controlled by PRV greatly reduces the supply energy. Simulation results show that the supply energy of OFDC is 4.16 kJ, while the supply energy of OFDESC is only 1.91 kJ, whose energy saving of the system is about 54% with parameters and desired trajectory in this paper. Meanwhile, the tracking performance of OFDESC is as good as OFDC with a stable error of 0.1 mm. So the proposed variable supply pressure method is adaptable, which not only achieves energy saving but also does not affect the trajectory tracking effect. However, it can be found from the energy distribution diagram that the overflow loss energy still accounts for most of the total energy, which is mainly because of the fixed flow rate of the constant pump. The actuator requires different supply flow rate with different expected trajectories, if the desired trajectory is so slow that the more supply flow will be directly back to the tank through PRV, which makes the energy of overflow loss increased. Therefore, more research will focus on improving the tracking performance of different desired trajectories, reducing the energy of overflow loss, and further improving energy efficiency. Furthermore, relevant experimental data will be supplemented in future work.

The data used to support the findings of this study are included within the article.

The authors declare that there are no conflicts of interest regarding the publication of this paper.

This work was supported by the National Natural Science Foundation of China (Grants U1910212 and 51675519) and Fundamental Research Funds for the Central Universities (2019XKQYMS37).