Open AccessArticle

Research on Feedforward-Feedback Composite Anti-Disturbance Control of Electro-Hydraulic Proportional System Based on Dead Zone Compensation

Jianbo Dai

^1,2,*,

Haozhi Xu

^1,2,

Lei Si

^1,2

Dong Wei

^1,2,

Jinheng Gu

^1,2

and

Hang Chen

^3,*

College of Mechanical and Electrical Engineering, China University of Mining and Technology, Xuzhou 221008, China

Jiangsu Collaborative Innovation Center of Intelligent Mining Equipment, China University of Mining and Technology, Xuzhou 221008, China

China Coal Technology & Engineering Group, Chongqing Research Institute, Chongqing 400039, China

Authors to whom correspondence should be addressed.

Machines 2024, 12(12), 855; https://doi.org/10.3390/machines12120855

Submission received: 22 October 2024 / Revised: 5 November 2024 / Accepted: 6 November 2024 / Published: 27 November 2024

(This article belongs to the Special Issue Key Technologies in Intelligent Mining Equipment)

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Browse Figures

Figure 1
Electro-hydraulic proportional control system model. "> Figure 2
Feedforward-feedback composite control system diagram. "> Figure 3
Amesim simulation model. "> Figure 4
Identification model validation. "> Figure 5
Simulation model. "> Figure 6
Simulation model. (a) Trajectory tracking; (b) error curve. "> Figure 7
Small-excavator experimental platform. "> Figure 8
Hydraulic schematic diagram of excavator. "> Figure 9
Hydraulic cylinder reciprocating motion. "> Figure 10
The dead zone compensation process. "> Figure 11
Hydraulic cylinder reciprocating motion after adding dead zone. "> Figure 12
Trajectory tracking experiment. (a) Constant load trajectory tracking; (b) error curve. "> Figure 13
Trajectory tracking experiment. (a) Variable load trajectory tracking; (b) error curve. ">

Versions Notes

Abstract

Considering the complexity and difficulty of obtaining certain parameters in the electro-hydraulic proportional control system, a precise transfer function of the system was derived through parameter identification using experimental data obtained from an Amesim simulation model after establishing a basic mathematical model. This approach reduces the reliance on accurate parameters of individual components. A feedforward-feedback composite controller was designed, and its effectiveness was validated in Simulink using the system’s transfer function. Subsequently, the dead zone range of the proportional valve was determined through experiments, and a dead zone compensation strategy was designed, which reduced the time required for the proportional valve to traverse the dead zone by 89.4%. Based on the dead zone compensation, trajectory tracking experiments were conducted to validate the effectiveness of the feedforward-feedback composite controller. Under fixed disturbances, the trajectory tracking error was reduced by 53.8% compared to feedback control. Under time-varying load disturbances, the trajectory tracking error was reduced by 51.2% compared to feedback control.

Keywords:

electro-hydraulic proportional valve control system; feedforward-feedback composite control; load disturbance; dead zone compensation

1. Introduction

With the increasing demand for automation and intelligence in mining operations, the control precision and responsiveness of heavy-duty robotic arms have become essential. As crucial equipment in mining tasks, these heavy-duty robotic arms undertake high-intensity tasks such as excavation, loading, and unloading, requiring high reliability and operational accuracy. The successful execution of these tasks depends on the arm’s complex internal power system, particularly the hydraulic system at its core.

Within the robotic arm’s power system, the hydraulic cylinder serves as one of the most critical actuators, driving joint movements through changes in hydraulic oil pressure to ensure efficient, precise, and stable operation. Thus, the performance, control precision, and operational reliability of hydraulic cylinders directly affect the overall working efficiency and capacity of the robotic arm. For this reason, research on hydraulic cylinder control is essential. However, due to the influences of hydraulic oil flow, pressure, and variations in external load, the control process for hydraulic cylinders exhibits strong nonlinearity and uncertainty. These challenges particularly limit traditional hydraulic control methods when precise position control is required [1,2,3,4].

To address this issue, modern hydraulic control technology has gradually incorporated electro-hydraulic proportional control systems, which can continuously adjust the flow and pressure of hydraulic cylinders based on external control signals, thereby achieving higher control accuracy and responsiveness. Particularly in position control, electro-hydraulic proportional systems effectively overcome issues of delay and oscillation common in traditional control methods, significantly enhancing control flexibility and stability. Therefore, investigating the application of electro-hydraulic proportional position control technology in hydraulic cylinders holds substantial theoretical and practical value [5,6,7,8].

At present, the classical PID control algorithm is widely used in electro-hydraulic proportional control system. However, it is prone to overshoot and oscillations under external disturbances. Advanced strategies, such as sliding mode control, adaptive robust control, and optimal control, can achieve better control outcomes but typically require high accuracy of the system’s mathematical model and offer limited robustness against changes in system parameters [9,10,11]. Feedforward control, as an open-loop control, offers advantages such as simple structure, fast response, and no delay. Yet, due to its inability to sense and correct errors, it has limited accuracy and disturbance rejection. The integration of feedforward and feedback control can leverage the strengths of both approaches, significantly improving the control accuracy, responsiveness, and reliability of electro-hydraulic systems [12,13,14].

To address the high precision and responsiveness demands in the control of heavy-duty robotic arms, especially given the challenges posed by the nonlinearity and uncertainty inherent in hydraulic systems, researchers worldwide have conducted extensive studies in the field of electro-hydraulic control, proposing various improved methods and control strategies. For example, Yao et al. developed feedforward compensation for the system based on the invariance principle and pole zero theory, effectively improving the tracking performance of the system and greatly expanding the system bandwidth [15]. Lee et al. utilized feedforward control to accelerate tracking speed, enabling precise tracking [16]. Ibrahim et al. proposed a real-time tracking control method based on pulse width modulation for electro-hydraulic proportional systems, which achieves position tracking with minimal error and low transient time by controlling the speed of the actuator [17]. Kong et al. proposed a multi-switching-mode intelligent hybrid control method for electro-hydraulic proportional systems, which integrates PID control law, neural fuzzy control law, and expert-based control law [18]. Wang et al. proposed a model free linear active disturbance suppression controller with integrated dead zone inverse compensation for electro-hydraulic proportional systems with unknown dead zones [19]. Liu et al. designed a linear extended state observer for an electro-hydraulic system controlled by a proportional valve. They also designed nonlinear functions to compensate for dead zones and improve trajectory tracking accuracy under external disturbances [20].

To enhance the response characteristics of the electro-hydraulic proportional control system, this paper proposes a feedforward-feedback composite control strategy based on dead zone compensation. By compensating for the proportional valve’s dead zone, the response speed is improved. The feedforward-feedback controller is also designed to increase control accuracy under external disturbances. A mathematical and simulation model is established for theoretical analysis, and an experimental platform is set up to validate the effectiveness of the control strategy.

2. Electro-Hydraulic Proportional Control System Model

2.1. The Components of the Electro-Hydraulic Proportional Control System

The electro-hydraulic proportional control system studied in this paper is shown in Figure 1. The system is built around a hydraulic cylinder, which serves as the driving component of a small excavator. The stable control of the hydraulic cylinder determines the overall stability control of the excavator. The system mainly consists of a load, sensors, a controller, a proportional directional valve, a hydraulic cylinder, a motor, a gear pump, external disturbances, and other components. The electro-hydraulic proportional control system collects signals through sensors and feeds them back to the controller. After processing, the controller calculates the control signal and transmits it to the proportional valve to drive the extension and retraction movement of the hydraulic cylinder.

2.2. Mathematical Model of Electro-Hydraulic Proportional Control System

2.2.1. Mathematical Model of a Hydraulic Cylinder

The flow equation for a proportional valve can be expressed as follows:

Q_{L} = K_{sv} x_{v} - K_{c} P_{L}

(1)

In the equation,

K_{s v}

is the flow gain of the servo valve;

x_{v}

is the displacement of the valve spool;

K_{c}

is the flow-pressure coefficient; and

P_{L}

is the load pressure.

The continuity equation for the flow rate in a hydraulic cylinder can be expressed as follows:

Q_{L} = A_{p} s x_{p} + C_{t} P_{L} + \frac{V_{t}}{4 l_{e}} s P_{L}

(2)

In the equation,

A_{p}

is the effective area of the hydraulic cylinder piston;

C_{t}

is the total leakage coefficient of the hydraulic cylinder;

V_{t}

is the total compressible volume;

l_{e}

is the effective bulk modulus; and

x_{p}

is the displacement of the piston rod.

The force balance equation for a hydraulic cylinder and its load can be expressed as follows:

A_{p} P_{L} = m_{t} s^{2} x_{p} + B_{p} s x_{p} + K_{s} x_{p}

(3)

In the equation,

m_{t}

is the load mass;

B_{p}

is the load damping coefficient; and

K_{s}

is the load stiffness coefficient.

By combining Equations (1)–(3) and simplifying, the transfer function of the hydraulic cylinder can be obtained:

G_{1} (s) = \frac{1 / A_{p}}{s (\frac{s^{2}}{ω_{h}^{2}} + \frac{2 δ_{h} s}{ω_{h}} + 1)}

(4)

In the equation,

ω_{h} = \sqrt{\frac{4 l_{e} A_{p}^{2}}{V_{t} m_{t}}}

is the natural frequency of the hydraulic cylinder;

δ_{h} = \frac{K_{c e}}{A_{p}} \sqrt{\frac{l_{e} m_{t}}{V_{t}}}

is the damping ratio of the hydraulic cylinder; and

K_{c e}

is the total flow-pressure coefficient.

2.2.2. Mathematical Model of the Proportional Directional Valve

As the core control component in an electro-hydraulic position proportional system, the role of the proportional directional valve is to achieve electro-hydraulic signal conversion output. It controls the movement of the valve spool according to the input electrical signal, thereby precisely adjusting the flow rate of hydraulic oil to control the position and speed of the actuator, the hydraulic cylinder.

In engineering, the proportional valve is regarded as a typical second-order element, with its transfer function as follows:

G_{2} (s) = x (s) / Δ U (s) = \frac{K_{a} K_{v}}{\frac{1}{ω_{v}^{2}} s^{2} + \frac{2 ζ_{v}}{ω_{v}} s + 1}

(5)

In the equation,

Δ U

is the input voltage signal of the proportional amplifier,

Δ U (s) = U_{g} (s) - x_{p} (s) K_{m}

;

U_{g}

is the input voltage signal of the proportional valve;

K_{m}

is the feedback gain of the position sensor;

K_{a}

is the conversion gain of the proportional amplifier;

K_{v}

is the gain of the proportional directional valve;

ω_{v}

is the phase bandwidth of the proportional directional valve; and

ζ_{v}

is the damping ratio of the proportional directional valve.

2.2.3. Mathematical Model of Transfer Function for the Impact of Load Disturbance on the System

When high-precision control of electro-hydraulic proportional systems is required, traditional control methods face various challenges, with external disturbances being a particularly significant factor. If the control process accounts for external disturbances and incorporates a compensatory measure, the stability of the electro-hydraulic proportional system under disturbance can be greatly improved. Therefore, it is essential to consider a mathematical model of the system under the influence of external disturbances. According to the system’s operating principles, the impact of load disturbances is transmitted through hydraulic oil to the proportional valve via the hydraulic cylinder, affecting the movement of the valve spool. The formula is as follows:

x_{v} (s) = F_{d} (s) G_{1} (s) G_{2} (s) = F_{d} (s) (\frac{1 / A_{p}}{s (\frac{s^{2}}{ω_{h}^{2}} + \frac{2 δ_{h} s}{ω_{h}} + 1)}) (\frac{K_{a} K_{v}}{\frac{1}{ω_{v}^{2}} s^{2} + \frac{2 ζ_{v}}{ω_{v}} s + 1})

(6)

In the equation,

x_{v} (s)

is the spool movement of the proportional valve; and

F_{d} (s)

is the external load disturbance.

3. Feedforward-Feedback Composite Controller Design

3.1. Feedforward-Feedback Control System Design

The basic principle of feedback control is to continuously monitor the system output, compare the actual output with the desired output, and adjust the system input based on the deviation, so that the system output approaches the desired value. Industrial control is often subject to many disturbances, and conventional feedback control alone may struggle to achieve good control quality. In such cases, feedforward and feedback control are often combined. The basic principle of feedforward control is to adjust the control variables directly by understanding the relationship between system input and the desired output in advance, thereby counteracting or reducing the impact of disturbances on the output. Unlike feedback control, feedforward control does not rely on output information for adjustments; instead, it pre-calculates the required control value through mathematical models or empirical formulas, improving the system’s response speed and stability. A feedforward-feedback composite control system combines the advantages of both feedforward and feedback control methods. The block diagram of the feedforward-feedback composite control system is shown in Figure 2. Assuming that the disturbances encountered by the control system are measurable.

In the figure,

G_{f} (s)

is the transfer function of the disturbance channel,

G_{d} (s)

is the transfer function of the feedforward controller for disturbances,

G_{c} (s)

is the transfer function of the feedback controller,

G_{1} (s)

is the transfer function of the control channel,

G_{2} (s)

is the common transfer function, and

H (s)

is the transfer function of the feedback channel. The transfer function of the feedforward-feedback control system for disturbances is as follows:

\frac{Y (s)}{M (s)} = \frac{(G_{f} (s) + G_{d} (s) G_{1} (s)) G_{2} (s)}{1 + G_{c} (s) G_{1} (s) G_{2} (s) H (s)}

(7)

Under pure feedforward control, the system’s transfer function for disturbances is as follows:

\frac{Y (s)}{M (s)} = (G_{f} (s) + G_{d} (s) G_{1} (s)) G_{2} (s)

(8)

It can be seen that, compared to a purely feedforward control system, the influence of disturbances on the controlled quantity in the feedforward-feedback control system is reduced to

{1} / {1 + G_{c} (s) G_{1} (s) G_{2} (s) H (s)}

of its original value, thereby decreasing the impact of disturbances on the system. At the same time, feedback control is incorporated, enhancing the system’s feedback regulation capability.

It can be derived that the condition for the feedforward-feedback control system to achieve complete compensation for disturbances is as follows:

G_{d} (s) = - \frac{G_{f} (s)}{G_{1} (s)}

(9)

It can be seen that using a feedforward-feedback control system allows for the advance compensation of one or several significant disturbances affecting the system output before they impact it. For the remaining disturbances, feedback control can be utilized to compensate for deviations through changes in the output.

For the feedback control portion, this paper employs the PID control, which is the most widely used in industrial control. Proportional control adjusts based on the current error, where the control output is directly proportional to the error, represented by the term

P = K_{p} \times e (t)

, with

K_{p}

as the proportional gain and

e_{t}

as the current error. While increasing

K_{p}

can speed up the system response, an excessively high

K_{p}

may lead to oscillations or even instability. Integral control adjusts based on the accumulated error over time and is primarily used to eliminate steady-state error, represented by the integral term

I = K_{I} \int e (t) d t

, where

K_{i}

is the integral gain. This action ensures that even small errors affect the output over time, thereby reducing steady-state error. However, a high integral gain may slow the response and lead to overshoot and oscillations. Derivative control adjusts based on the rate of change of the error, predicting its trend so the system can respond preemptively. The derivative term is

D = K_{d} \frac{d e (t)}{d t}

, where

K_{d}

is the derivative gain. Derivative action helps improve the dynamic response, reducing overshoot and oscillations. However, it is sensitive to noise, and too much derivative gain can destabilize the system. The PID controller output is the sum of the following three parts:

u (t) = K_{p} e (t) + K_{i} \int e (t) d t + K_{d} \frac{d e (t)}{d t}

(10)

By adjusting appropriately, the PID controller allows the system to achieve the desired target while balancing response speed and stability.

3.2. Identification of Control System Parameters

According to the theoretical derivation above, the hydraulic valve-controlled cylinder system can be considered a third-order system. Due to system nonlinearity, the parameter values in the mathematical model need to be derived based on the current position’s parameters at different working positions, which complicates calculations and increases workload. Using a model derived from a single position to represent the entire motion process fails to account for the system’s dynamic characteristics. Therefore, this paper establishes an Amesim simulation model to identify the system’s transfer function by collecting parameters throughout the motion process, ensuring that the identified model aligns as closely as possible with the system’s dynamic response characteristics. The Amesim simulation model developed in this paper is shown in Figure 3, and some parameter settings for the simulation system are listed in Table 1.

The simulation input signal is a step signal with a step value of 15 cm. The simulation duration is set to 10 s, with a sampling frequency of 1000 Hz. A standard integrator type is selected, and linear analysis mode is used for the simulation. The model identification program written in MATLAB-2021a reads the Jacobian matrix from the Amesim simulation results for parameter identification. The transfer function from the proportional directional valve signal to the valve spool displacement is as follows:

G_{1} (s) = \frac{2.527 \times 10^{5}}{s^{2} + 804.2 s + 2.527 \times 10^{5}}

(11)

The transfer function from valve spool displacement to hydraulic cylinder displacement is as follows:

G_{2} (s) = \frac{380.6}{s^{3} + 6.149 s^{2} + (9.359 \times 10^{4}) s}

(12)

The identified parameters of the proportional valve indicate a natural frequency of 502.6548 rad/s and a damping ratio of 0.8. The natural frequency of the valve-controlled cylinder system is 305.9184 rad/s with a damping ratio of 0.1. To verify the accuracy of the parameter identification results, the displacement curve calculated from the transfer function is compared with the displacement curve from the Amesim simulation model. The step signal value is set to 0.15 m. Before 4.5 s, the hydraulic cylinder displacement rises steadily, and at 4.5 s, the hydraulic cylinder displacement stabilizes, ultimately reaching 0.15 m.

Figure 4 shows that the absolute error of the displacement tracking curve between the simulation and parameter identification reaches its maximum of approximately 0.00152 m at around 1 s, with a relative deviation of 1.5%. The average error during the entire process is 1.40 × 10⁻⁸ m. Therefore, it can be concluded that the transfer function model derived from parameter identification is essentially consistent with the simulation model in Amesim, allowing for the validation of the control effects of different control methods using the transfer function model.

4. Simulink Simulation Analysis

Based on the transfer functions of each component of the electro-hydraulic position proportional system identified in the previous text, a mathematical simulation model is established in the Simulink module of MATLAB-2021a, as shown in Figure 5.

In the simulation, a sine signal is taken as the desired displacement for the electro-hydraulic position proportional system. The amplitude of the sine signal is 30 cm, with a vertical offset of 30 cm and a period of 2 s. The simulation time is set to 4 s, with a sampling frequency of 1000 Hz. Apply a load disturbance of 70 kg at the end of the first sine cycle, which is the second second, and continue until the end of the simulation. The simulation results and error curves of two different control methods, PID feedback control and feedforward-feedback composite control, are shown in Figure 6. The advantages and disadvantages of different control methods are measured by calculating the average error and maximum error during the tracking process. The calculation of the average error is shown in Formula (13).

e_{a v r} = \frac{\sum_{i = 1}^{n} |a_{i} - b_{i}|}{n}

(13)

Among them,

e_{a v r}

is the average error,

a_{i}

is the desired displacement at the i-th point,

b_{i}

is the trajectory tracking displacement at the i-th point, and

n

is the number of sampling points.

The sinusoidal trajectory tracking simulation results in Figure 6 indicate that both the feedforward-feedback composite control and the feedback control methods can effectively track the desired displacement. However, after adding a 70 kg load disturbance in the second, the error of the feedback control method fluctuated. In contrast, feedforward-feedback control effectively compensates for disturbances through the feedforward controller before they affect the entire system, resulting in a stable motion trajectory without fluctuations caused by disturbances. The average tracking error of the feedforward-feedback composite control is 0.0057 cm, which is better than the 0.0064 cm of the standalone feedback control. Based on the results obtained from the simulation, the feedforward-feedback composite control proves to be more effective than standalone feedback control in the electro-hydraulic position proportional system when facing load disturbances.

5. Experimental Verification

In order to verify the effectiveness of the control method proposed in this article, an electro-hydraulic proportional control system experimental platform was built based on the boom cylinder of a small excavator, as shown in Figure 7. The hydraulic schematic of the experimental equipment is shown in Figure 8. The three hydraulic cylinders in the figure control the movement of the excavator’s boom, bucket, and boom, respectively.

In the experiment, only the motion control of the boom cylinder was carried out, and the oil circuits of the bucket hydraulic cylinder and the boom hydraulic cylinder were in a closed state. The experimental system uses a motor as the power source, drives a gear pump through a coupling, and delivers hydraulic oil to the hydraulic cylinder through a proportional directional valve to provide power. The signal control part consists of an upper computer, PLC, signal amplification board, proportional directional valve, displacement sensor, and other components. The PLC is the S7-1200 manufactured by Siemens, Germany. The displacement sensor transmits the real-time displacement of the hydraulic cylinder to the upper computer through the signal amplification board and PLC. After the upper computer collects the displacement signal, it calculates the displacement control signal based on the feedforward-feedback composite control algorithm and transmits it to the PLC. The control signal output by the PLC is amplified by the signal amplification board to control the movement of the proportional valve core, change the flow direction and speed of the hydraulic oil, and thus control the expansion and contraction movement of the hydraulic cylinder.

5.1. Proportional Valve Dead Zone Compensation

In an electro-hydraulic proportional system, the control signal for the proportional directional valve is a ±10 V voltage signal. A reciprocating motion control experiment was conducted, causing the proportional valve to switch directions. In the motion control experiment, the hydraulic cylinder first extends and then retracts. The impact of the dead zone on the system is observed during the directional switching of the proportional valve. Displacement data of the hydraulic cylinder and the control signals of the proportional valve were collected and are plotted in Figure 9. Based on the figure, the impact of the proportional valve’s dead zone on the response characteristics of the electro-hydraulic proportional system was observed. In the figure, the black curve represents the displacement of the hydraulic cylinder, and the red curve represents the control voltage signal of the proportional valve during the motion. The green horizontal dashed line representing a voltage of −2 V intersects the red curve at points A and B, corresponding to displacement points E and F, respectively. Similarly, the green horizontal dashed line representing a voltage of 2V intersects the red curve at points C and D, corresponding to displacement points G and H, respectively. The control voltage signal in the AB segment of the red curve corresponds to the displacement of the hydraulic cylinder in the EF segment of the black curve. It can be observed that, prior to the AB segment, when the control signal is between 0 and −2 V, the proportional valve is in the dead zone, the hydraulic circuit is not yet open, and the hydraulic cylinder does not move. When the voltage signal is less than −2 V, the proportional valve crosses the dead zone, the hydraulic circuit opens, and the hydraulic cylinder begins to extend. When the proportional valve control signal is in the BC segment, the voltage signal remains between −2 V and +2 V, the proportional valve is in the dead zone, the hydraulic circuit remains closed, and the hydraulic cylinder stops moving. When the voltage signal is in the CD segment, the proportional valve control voltage exceeds +2 V, the proportional valve crosses the dead zone, and the hydraulic cylinder begins to retract. After point D, when the control voltage is less than 2 V, the hydraulic cylinder stops moving. The experiment shows that the dead zone of this proportional valve is ±20%. The existence of the dead zone in proportional valves significantly restricts the response speed of electro-hydraulic proportional systems.

When the proportional valve control signal

U_{1}

, output by the controller, is transmitted to TIA Portal via the upper computer, TIA Portal determines whether the control signal is within the dead zone of the proportional valve. If the control signal is in the dead zone, a voltage step signal

U_{2}

is added as compensation to the controller’s output voltage signal through programming in TIA Portal. The controller’s output signal

U_{1}

, together with the dead zone compensation signal

U_{2}

, forms the drive signal for the proportional valve, enabling it to quickly pass through the dead zone. The dead zone compensation process is shown in Figure 10. Before the dead zone compensation signal is added, the drive signal of the proportional valve varies between −10 V and +10 V. After adding the dead zone compensation signal, the drive signal of the proportional valve falls within the ranges of −10 V to −2 V and +2 V to +10 V. As shown in Figure 10, after adding the dead zone compensation signal, the drive signal of the proportional valve will no longer appear within the dead zone range of −2 V to 2 V during directional switching. A repetitive motion control experiment is conducted again to observe the control effect after adding the dead zone compensation signal. The displacement data of the hydraulic cylinder and the control signal of the proportional valve from the experiment were collected and plotted in Figure 11.

In Figure 11, the green horizontal dashed line representing a voltage of −2 V intersects the red curve at points B and C, while the green horizontal dashed line representing a voltage of 2V intersects the red curve at points F and G. Points A, D, E, and H are the turning points of the voltage signal. It can be observed that the voltage signal curve jumps from 0 V to −2 V in segment AB, from −2 V to 0 V in segment CD, from 0 V to 2 V in segment EF, and from 2 V to 0 V in segment GH. The designed dead zone compensation strategy enables the proportional valve to quickly overcome the dead zone. Under the same controller parameter conditions, the time required for the hydraulic cylinder to transition from extension to retraction decreased from 340 s to 36 s, reducing the time for the proportional valve to cross the dead zone by 89.4%. Therefore, it can be concluded that this dead zone compensation method is effective. Subsequent trajectory control experiments for the hydraulic cylinder will be conducted based on this dead zone compensation strategy.

5.2. Feedforward-Feedback Combined Control Experiment Based on Dead Zone Compensation

During the experiment, load disturbances were applied to the boom cylinder. There are two types of load disturbances. The first type is a fixed load disturbance, where an external load of 70 kg is applied at the start of the experiment and remains until its conclusion. The second type is a variable load disturbance, where a 70 kg load disturbance is applied at the 10 s mark and removed at the 120 s mark. The effects of the fixed and variable load disturbances on the motion of the hydraulic cylinder during the application and removal of the load are observed, as well as the actual control performance of the feedforward-feedback combined controller based on dead zone compensation under external load disturbances.

The experimental results of fixed load disturbance trajectory tracking are shown in Figure 12. After applying a fixed load, the peak error of the feedback control reached a maximum of 0.61 cm at 21 s, with an average error of 0.39 cm. In contrast, the peak error for the feedforward-feedback composite control was around 0.3 cm, with an average error of 0.18 cm. The average error in the trajectory tracking experiment under the feedforward-feedback combined control with dead zone compensation was reduced by 53.8% compared to feedback control.

The experimental results of trajectory tracking under time-varying load disturbances are shown in Figure 13. A load is applied after 10 s of system motion and removed at the end of the first sine wave signal. Under the influence of varying loads, the feedback control exhibits a peak error of 0.81 cm and an average error of 0.43 cm. In contrast, the feedforward-feedback composite control achieves a peak error of 0.31 cm and an average error of 0.21 cm. Compared to feedback control, the feedforward-feedback composite control with dead zone compensation reduces the average error by 51.2% in the trajectory tracking experiment. The experimental results of the two scenarios are summarized in Table 2.

6. Conclusions

To improve the response characteristics of the electro-hydraulic proportional control system, this paper conducts an in-depth study on the motion control of the boom cylinder of a small excavator. The research findings are summarized as follows:

(1): After establishing the mathematical model of the electro-hydraulic proportional control system, a parameter identification method was used to identify the system’s transfer function, reducing the control system’s reliance on an accurate mathematical model.
(2): A dead zone compensation strategy was designed to address the dead zone of the proportional directional valve. Under the same controller parameter conditions, the time required for the proportional valve to traverse the dead zone was reduced by 89.4%.
(3): Under fixed load disturbances, the average error of trajectory tracking experiments using the feedforward-feedback composite control with dead zone compensation was reduced by 53.8% compared to feedback control. Under time-varying load disturbances, the average error was reduced by 51.2% compared to feedback control.

Author Contributions

Conceptualization, J.D.; methodology, J.D. and L.S.; software, H.X.; validation, L.S.; formal analysis, H.C.; investigation, H.X.; data curation, D.W.; writing—original draft preparation, J.D.; writing—review and editing, H.X.; supervision, D.W. and J.G.; project administration, J.G. and H.C.; funding acquisition, J.D. All authors have read and agreed to the published version of the manuscript.

Funding

This research was funded by the National Natural Science Foundation of China, grant number 52404179, Jiangsu Province Natural Science Fund, grant number BK20210495, Chinese Postdoctoral Science Foundation, grant number 2020M681761.

Data Availability Statement

Data are contained within the article.

Conflicts of Interest

Author Hang Chen was employed by the company China Coal Technology & Engineering Group. The remaining authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

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Figure 1. Electro-hydraulic proportional control system model.

Figure 2. Feedforward-feedback composite control system diagram.

Figure 3. Amesim simulation model.

Figure 4. Identification model validation.

Figure 5. Simulation model.

Figure 6. Simulation model. (a) Trajectory tracking; (b) error curve.

Figure 7. Small-excavator experimental platform.

Figure 8. Hydraulic schematic diagram of excavator.

Figure 9. Hydraulic cylinder reciprocating motion.

Figure 10. The dead zone compensation process.

Figure 11. Hydraulic cylinder reciprocating motion after adding dead zone.

Figure 12. Trajectory tracking experiment. (a) Constant load trajectory tracking; (b) error curve.

Figure 13. Trajectory tracking experiment. (a) Variable load trajectory tracking; (b) error curve.

Table 1. Simulation Parameter Table.

Parameter Name	Value
Piston Diameter/mm	50
Piston Rod Diameter/mm	32
Hydraulic Cylinder Stroke/m	0.6
Viscous Friction Coefficient/(N·s·m⁻¹)	1000
Proportional Valve Frequency/Hz	80
Proportional Valve Maximum Flow Rate/(L/min)	80
Proportional Valve Rated Current/mA	200

Table 2. Experimental Results Table.

	Fixed Load	Time-Varying Load
feedback control peak error/cm	0.61	0.81
feedback control average error/cm	0.39	0.43
feedforward-feedback composite control peak error/cm	0.3	0.31
feedforward-feedback composite control average error/cm	0.18	0.21

Through comparative experiments under two load conditions, it can be concluded that, with dead zone compensation, the performance of the feedforward-feedback composite control method is superior to that of feedback control alone.

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Dai, J.; Xu, H.; Si, L.; Wei, D.; Gu, J.; Chen, H. Research on Feedforward-Feedback Composite Anti-Disturbance Control of Electro-Hydraulic Proportional System Based on Dead Zone Compensation. Machines 2024, 12, 855. https://doi.org/10.3390/machines12120855

AMA Style

Dai J, Xu H, Si L, Wei D, Gu J, Chen H. Research on Feedforward-Feedback Composite Anti-Disturbance Control of Electro-Hydraulic Proportional System Based on Dead Zone Compensation. Machines. 2024; 12(12):855. https://doi.org/10.3390/machines12120855

Chicago/Turabian Style

Dai, Jianbo, Haozhi Xu, Lei Si, Dong Wei, Jinheng Gu, and Hang Chen. 2024. "Research on Feedforward-Feedback Composite Anti-Disturbance Control of Electro-Hydraulic Proportional System Based on Dead Zone Compensation" Machines 12, no. 12: 855. https://doi.org/10.3390/machines12120855

APA Style

Dai, J., Xu, H., Si, L., Wei, D., Gu, J., & Chen, H. (2024). Research on Feedforward-Feedback Composite Anti-Disturbance Control of Electro-Hydraulic Proportional System Based on Dead Zone Compensation. Machines, 12(12), 855. https://doi.org/10.3390/machines12120855

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Research on Feedforward-Feedback Composite Anti-Disturbance Control of Electro-Hydraulic Proportional System Based on Dead Zone Compensation

Abstract

1. Introduction

2. Electro-Hydraulic Proportional Control System Model

2.1. The Components of the Electro-Hydraulic Proportional Control System

2.2. Mathematical Model of Electro-Hydraulic Proportional Control System

2.2.1. Mathematical Model of a Hydraulic Cylinder

2.2.2. Mathematical Model of the Proportional Directional Valve

2.2.3. Mathematical Model of Transfer Function for the Impact of Load Disturbance on the System

3. Feedforward-Feedback Composite Controller Design

3.1. Feedforward-Feedback Control System Design

3.2. Identification of Control System Parameters

4. Simulink Simulation Analysis

5. Experimental Verification

5.1. Proportional Valve Dead Zone Compensation

5.2. Feedforward-Feedback Combined Control Experiment Based on Dead Zone Compensation

6. Conclusions

Author Contributions

Funding

Data Availability Statement

Conflicts of Interest

References

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