Keywords

1 Introduction

With the development of industry and engineering, multiple hydraulic cylinders are widely used to drive loads in various fields, such as mechanical manufacturing, marine engineering, national defense, lifting equipment of large stage, synchronous lifting system of large bridge and so on [1, 2]. And the requirement of synchronization accuracy is relatively high.

In the hydraulic system, the large impact load will cause large speed fluctuation of the actuator, especially the change of the comprehensive oil elastic modulus after the gas escaping from the oil caused by the load change, which will cause greater harm to the system. The speed stability of hydraulic actuator is affected by the elastic modulus of oil, which mainly depends on the air content and working pressure of oil [3].

In general, the stiffness of the hydraulic system is often ignored when the load changes suddenly. In some cases, the requirement of the speed stability of the actuator is relatively high, so the influence of the oil elastic modulus should be considered. Through the analysis of the elastic modulus of the hydraulic oil under the pressure change, it is found that in order to make the output speed of the hydraulic cylinder tend to be stable, it is necessary to rely on the control algorithm to reduce the interference of the impact load. Through the mathematical model and simulation verification, it is concluded that the fuzzy PID control weighted algorithm can significantly reduce the impact of load and keep the speed of hydraulic cylinder with high control accuracy [4].

The closed-loop control of electro-hydraulic servo valve or electro-hydraulic proportional valve is often used in multiple cylinders synchronous control system [5]. Hydraulic servo and proportional synchronous control system is composed of various electro-hydraulic proportional valves, electro-hydraulic servo valves or digital valves [6]. The advantage of the synchronous system is that it usually adopts closed-loop control mode, which can continuously detect and feedback the output variables, so that the errors caused by the input signal, system type and system interference can be suppressed in time, and high control accuracy can be obtained while meeting the requirements of rapidity and system stability. At present, the commonly used control algorithms include PID control and related improved control algorithm, adaptive control, sliding mode variable structure control and intelligent control [7, 8].

2 Speed Stability Analysis

2.1 Hydraulic Oil Bulk Modulus of Elasticity

In the work process of hydraulic system, the volume of oil will shrink under a large pressure, which is called the compressibility of oil. The compressibility of hydraulic oil can be expressed by the following equation [9].

$$ E = - \frac{\Delta p}{{\Delta V}} \times V $$
(1)

where \(\Delta {\text{P}}\) is the change amount of the pressure, \(\Delta V\) is the volume change amount of the hydraulic oil, and \(V\) is initial volume of the hydraulic oil.

The change of oil bulk elastic modulus will directly affect the speed rigidity of hydraulic actuator [10, 11]. In general or under low pressure, the relative compression volume of hydraulic oil is very small, which can be ignored and considered as incompressible. But for heavy load system or system with large impact load, the compressibility of oil seriously affects the stability of system speed. The oil bulk elastic modulus is an important parameter for dynamic analysis, modeling and Simulation of hydraulic system. The size of hydraulic oil bulk elastic modulus has a great relationship with the type, air content, working pressure and temperature of hydraulic oil [12].

When the hydraulic oil model is selected, the volume elastic modulus of the hydraulic oil will increase with the increase of pressure according to Eq. (1), and the volume elastic modulus of the hydraulic oil will decrease with the increase of temperature. Assuming that the bulk elastic modulus of the pure hydraulic oil is E0, when the pressure is P, the bulk elastic modulus of oil can be expressed by the following equation [13].

$$ E = \left\{ {\begin{array}{*{20}c} {\frac{{E_{0} }}{{\alpha P_{0} E_{0} (\frac{1}{{P^{2} }} - \frac{1}{{P_{c}^{2} }})}},} & {P_{0} < P < P_{c} } \\ {E_{0} ,} & {P \ge P_{c} } \\ \end{array} } \right. $$
(2)

where a is the air content in the fluid, P0 is the standard atmospheric pressure and Pc is the critical pressure.

It can be seen that the bulk elastic modulus of hydraulic oil is greatly affected by pressure. With the change of pressure, the state of gas in the oil will also change. There are two states of gas in the oil, which are compression and dissolution. The volume change of gas in two states of the hydraulic oil can be expressed by the following equation [14].

$$ \frac{{P_{1} V_{1} }}{{T_{1} }} = \frac{{(P_{1} + \Delta P)(V_{1} + \Delta V_{p} + \Delta V_{d} )^{\gamma } }}{{T_{2} }} $$
(3)

where P1 is the initial state pressure, V1 is the volume of gas in initial state, ΔVp is the volume change of gas under pressure, ΔVd is the volume change of gas dissolution, T1 is the temperature of the hydraulic oil in initial state, T2 is the temperature of the hydraulic oil in working condition, γ is the adiabatic index of gas.

2.2 Hydraulic Cylinder Speed Analysis

The hydraulic cylinder speed mainly depends on the volume of hydraulic oil entering the hydraulic cylinder. Assuming that the pipeline and valve body through which the hydraulic oil flows are rigid bodies without deformation, the continuity equation of the input hydraulic cylinder flow can be expressed as follows [15].

$$ q = Av + \lambda (P + \Delta P) + \frac{{V_{t} }}{E}\frac{d(P + \Delta P)}{{dt}} $$
(4)

where A is the piston area of the hydraulic cylinder, v is the speed of hydraulic cylinde, λ is the Hydraulic cylinder leakage coefficient, P is the hydraulic cylinder input pressure, Vt is the input liquid volume of hydraulic cylinder.

Based on the above theoretical analysis, if an impact load is encountered, it will cause the impact pressure, and the volume elastic modulus of hydraulic oil will change, which will cause the speed of hydraulic cylinder to be unstable and nonlinear. In order to improve the speed stability of the hydraulic cylinder and reduce the influence of the impact pressure and the change of the hydraulic oil bulk elastic modulus, the hydraulic oil bulk elastic modulus and the control algorithm can be used to improve the stability of the speed control. To improve the bulk modulus of elasticity of hydraulic oil, it is necessary to reduce the air content in the oil, which can be achieved by vacuuming [16].

3 Control Algorithm Selection

The displacement control scheme of one single hydraulic cylinder is shown in Fig. 1. The electro-hydraulic proportional directional valve is used to control the displacement of the hydraulic cylinder. The displacement sensor of the hydraulic cylinder collects the displacement signal. The difference between the displacement signal and the set displacement value is used as the control signal of the controller.

Fig. 1
A flow diagram of a hydraulic cylinder’s displacement control process depicts the integration of components including a controller, amplifier, electro-hydraulic proportional directional valve, and load from left to right.

The displacement control scheme of hydraulic cylinder

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3.1 Controller Design

Fuzzy-PID control can effectively improve the control accuracy. Fuzzy-PID control is developed on the basis of ordinary PID control, and it is the product of the combination of fuzzy theory and PID control. Fuzzy-PID control is also a control method often used in industry. Fuzzy-PID control can change the three parameters (proportional coefficient, integral coefficient and differential coefficient) according to the change of error signal, thus improving the control accuracy.

The input objects of Fuzzy-PID controller are e(error) and ec(error change rate). If they change, it will trigger the fuzzy controller to adjust the three coefficients (ΔKp, ΔKi, ΔKd). The control block diagram is shown in Fig. 2.

Fig. 2
A block diagram of the flow of a control system that integrates a fuzzy controller with a P I D controller to manage a controlled object.

Schematic block diagram of fuzzy PID control

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For the hydraulic cylinder speed control system, the fuzzy rule control table can be formulated by summing up the previous experience, and the membership function relationship of parameters can be formulated according to the actual deviation requirements of the project. According to the control accuracy requirements, the 7-segment fuzzy universe is selected, and the triangular membership function shown in Fig. 3. It is selected as the input and output membership function.

Fig. 3
A fuzzy logic membership function graph for decision-making and control systems depicts seven linguistic variables labeled N B, N M, N S, Z, P S, P M, and P S plotted against a numerical scale from negative 6 to 6. The membership values range from 0 to 1.

Membership function

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According to the characteristics of hydraulic cylinder system, the Fuzzy-PID control rules are determined as shown in Table 1.

Table 1 Fuzzy control rules
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3.2 Simulation Analysis

The displacement control system of single hydraulic cylinder is established by using AMESim software, as shown in Fig. 4.

Fig. 4
A A M E Sim software-generated model of a single hydraulic cylinder displacement control system with various components and connections essential for its operation.

AMESim model of hydraulic cylinder displacement control system

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The fuzzy controller is built in Simulink, and the final combined simulation model is shown in Fig. 5.

Fig. 5
A block diagram depicts the flow of a complex control system simulation model, highlighting the integration of various components like P I D and fuzzy logic controller for efficient system performance.

The combined simulation model

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The hydraulic cylinder will encounter impact load during its movement, which will cause hydraulic impact on the system and affect the stability of the hydraulic cylinder speed. In the simulation model, the load pressure is set to 5 MPa, and the air content of the hydraulic oil is set to 10 and 20%. When the system runs to the 10th second, the load will suddenly change to 2 times of the original. The simulation results are shown in Fig. 6.

Fig. 6
A multi-line graph plots distance per meter versus time in seconds for different air contents labeled 0%, 10%, and 20%. The lines are marked from the bottom left to the top right indicating an increasing trend.

The hydraulic cylinder displacement simulation results

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From the simulation results, it can be seen that the greater the air content, the worse the speed stability of the hydraulic cylinder. Due to the sudden change of the load, the system began to oscillate at the 10th second, but then the system returned to the equilibrium state in a short time, indicating that the Fuzzy-PID control has a good ability to resist external interference.

4 Synchronization Control Strategy

Theoretically, synchronization can be achieved with the same input flow of the same hydraulic cylinder, but in practice, different manufacturing errors of the hydraulic cylinder, different loads, and different characteristics of the control elements will lead to non-synchronization of multiple hydraulic cylinders.

The general hydraulic synchronization circuit limits the synchronization accuracy of multiple hydraulic cylinders. It is necessary to use advanced controllers to realize the synchronization of hydraulic cylinders with uneven load. The control of each hydraulic cylinder needs to be controlled separately [17].

There are many control strategies for hydraulic synchronization. Open loop control adopts synchronous loop, and closed loop control adopts “parallel synchronization” and “leader–follower” control strategies. Closed loop control often can achieve high control accuracy. In order to achieve high-precision synchronization, many researchers have done a lot of work and achieved relevant research results [18,19,20]. In order to obtain better synchronization accuracy and dynamic response, referring to a large number of relevant literature and similar research experience, the combination of “leader–follower” control strategy and Fuzzy-PID is planned to be adopted.

The “leader–follower” control strategy is to select one of the leader cylinders and the other cylinders as the follower cylinders, and take the output displacement of the leader cylinder as the input of other hydraulic cylinders, so as to track and achieve synchronization. The control scheme is shown in Fig. 7.

Fig. 7
A block diagram of the leader-follower control scheme depicts a complex system with multiple components and pathways, illustrating the interaction between a leader and follower in a control system at three levels.

“Leader–follower” control scheme

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4.1 Design of Fuzzy-PID Controller

Fuzzy-PID control has excellent robustness and good dynamic and static control performance in principle, and can meet the requirements of online adjustment of various parameters in the PID controller when the structural characteristics of the system change [21, 22], which can achieve the best control effect of synchronization. A two-dimensional input and a three-dimensional output fuzzy controller are selected, with the deviation e and the deviation change rate ec as the input values, and the increment of the three parameters of the PID controller as the output values of the fuzzy controller. The flow quantity is controlled by controlling the spool displacement of the electro-hydraulic proportional valve, so the displacement of the hydraulic cylinder can be controlled.

According to the synchronization characteristics of multiple hydraulic cylinders, a Fuzzy-PID control rule is determined. In the actual synchronization process, according to the control quantity u, the common weighted average method is used to solve ambiguity [23]. For example \(\Delta Kp\), the following equation is established.

$$ \Delta K_{p} = \frac{{\sum\limits_{i}^{n} {u_{i} \mu (i)} }}{{\sum\limits_{i = 1}^{n} {\mu (i)} }} $$
(5)

where \(u_{i}\) is the number of control quantity, \(\Delta Kp\) is the output values and \(\mu (i)\) is the membership of control quantity.

Therefore, the modified PID parameters can be obtain by adding the calculated output values and the initial parameters. The calculation equation is as follows.

$$ K_{p} = K_{p0} + \{ e,ec\} K_{p} = K_{p0} + \Delta K_{p} $$
(6)
$$ K_{i} = K_{i0} + \{ e,ec\} K_{i} = K_{i0} + \Delta K_{i} $$
(7)
$$ K_{d} = K_{d0} + \{ e,ec\} K_{d} = K_{d0} + \Delta K_{d} $$
(8)

4.2 Simulation and Analysis of Synchronous Control

The Simulink simulation model of fuzzy PID control is established for the synchronous loop of four hydraulic cylinders, as shown in Fig. 8.

Fig. 8
A block diagram of a simulation model for synchronizing four hydraulic cylinders. The diagram illustrates the integration of various components including P I D controllers, fuzzy logic controllers, and subsystems for precise control and synchronization.

Four hydraulic cylinders synchronization simulation model

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The displacement of the hydraulic cylinder is set to 280 mm. After simulation, the results are shown in Fig. 9. The maximum synchronization error of the piston during movement is about 0.4 mm.

Fig. 9
A multi-line graph plots distance per meter versus time in seconds for the displacement of four cylinders. All cylinders have a similar trend, initially stable (0, 0) to (15, 0), then sharply increasing (30, 275), and finally plateaus. Values are estimated.

Synchronous simulation results of the four hydraulic cylinders

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5 Synchronization Control Experiment

The designed four cylinders synchronous circuit is shown in Fig. 10. Each of the four hydraulic cylinders is equipped with position sensors. The data acquisition card collects the displacement signals of the hydraulic cylinder, and the operator console sends instructions to control the speed and displacement of the first hydraulic cylinder. The output displacement signals are used as input instructions for other hydraulic cylinders. The displacement sensor continuously detects the displacement signal, and the controller continuously corrects the displacement error, until the synchronization of each hydraulic cylinder is achieved. An experimental platform was built based on the hydraulic circuit, as shown in Fig. 11.

Fig. 10
A block diagram of a four-cylinder synchronous hydraulic circuit depicts the interaction between the controller, input, and displacement sensors crucial for hydraulic systems operation in machinery or industrial applications.

Four cylinder synchronous hydraulic circuit

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Fig. 11
A photograph of a technical setup with multiple hydraulic hoses connected to a series of four cylinders mounted on a panel, above a control unit with various switches and indicators.

The experimental platform

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In the circuit, a closed-loop control system consisting of an electro-hydraulic proportional directional valve and a hydraulic cylinder is used. The displacement signal of one hydraulic cylinder is used as input to the other three hydraulic cylinders, and the proportional directional valve current is controlled through displacement feedback to keep the four cylinders synchronized. The instantaneous pressure change of the electromagnetic overflow valve is used as the impact load. The displacement curves of the four hydraulic cylinders obtained through the data acquisition card are shown in Fig. 12. It can be seen that the displacement curves are basically coincident, with a synchronization error of about 0.5 mm.

Fig. 12
A multi-line graph plots the result from an experiment with a rapid increase in a variable up to 280 millimeters over a short period, then it plateaus and ends on the top right.

The experimental results

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6 Conclusions

  1. 1.

    Through theoretical analysis of the bulk elastic modulus of hydraulic oil, a mathematical model is established to obtain the factors and measures that affect the speed stability of the hydraulic cylinder. On the basis of theoretical analysis, a simulation analysis of a single hydraulic cylinder system with different air content was conducted. It was shown that an increase in air content would cause a decrease in speed stability, and the control algorithm used could improve the speed stability of the hydraulic cylinder, which was verified by experiments.

  2. 2.

    Based on the analysis of various synchronization control methods, a hydraulic cylinder synchronization control scheme based on “leader–follower” control strategy is proposed for multi hydraulic cylinder synchronization. A fuzzy PID controller is designed, and simulation and experiments verify that this control method can still achieve high synchronization accuracy under impact loads.