Research on fuzzy symmetrical control of valve controlled asymmetric hydraulic cylinder system

Abstract

In order to solve the problems, such as the asymmetry, complex control methods, inconsistent dynamic characteristics for positive/negative directions and poor stability for asymmetric control system, in this paper, a system symmetric control method based on state feedback was brought out and the state space mathematical model was derived for valve controlled hydraulic cylinder system. With the symmetric control method of the state feedback, the valve controlled asymmetric hydraulic system can be converted into state space symmetric system, then the unified control method can be used for positive and negative directions after conversion, finally the feedback link of the system was educed. Through simulation, the accuracy of the model and the effectiveness of the control method were verified. Through experiments, the input and output characteristics and load response characteristics of the system before and after conversion were compared. It was concluded that the system had symmetric characteristics for inputs and outputs, and better load response characteristics after conversion. This study can be a base for the application of symmetric control theory to large asymmetric hydraulic systems and was of great help to realize symmetric control of large asymmetric hydraulic systems.

Keywords

Valve controlled asymmetric cylinder system state feedback fuzzy symmetrical compensation fuzzy symmetrical

1 Introduction

Valve-controlled asymmetric hydraulic cylinder system has the advantages of compact structure, large power-weight ratio, fast response and good positioning function, which is widely used in many industrial automatic systems and military fields. However, valve-controlled asymmetric hydraulic cylinder is an asymmetric system. “Asymmetric” refers to the difference in content, size, shape and characteristics of a figure or object relative to a point or a straight line [1 –3]. In the valve controlled asymmetric hydraulic cylinder system, from the structure point of view, the left and right structure of the proportional valve zero position is asymmetric, and the structure of the hydraulic cylinder piston is asymmetric, too. Form the load response point of view, the loads to be overcome during piston movements is asymmetric, which leads to the difference of the cylinder motion response in the positive and negative directions, mainly in the aspects of input and output characteristics, overshoot, rising time, load response characteristics and so on [4]. These significant differences generated during the positive and negative motions for valve controlled asymmetric hydraulic cylinder system will easily make the system unstable due to the unreasonable configuration of control parameters.

In order to solve the problem of poor dynamic response quality caused by asymmetry of asymmetric system, many scholars in China and abroad have done a lot of research. As regards the modeling of valve-controlled asymmetric hydraulic system, based on the establishment of a unified nonlinear model of the positive and negative motion of the hydraulic cylinder, Yang Junhong [5, 6] transformred the nonlinear state equation of the system into a global linearization model by an accurate linearization method. In the literature [7], a unified frequency domain model of valve control cylinder is proposed based that asymmetric valve control system is modeled in frequency domain. Ye Xiaohua [8] proposed the method of weighted average to integrate the different mathematical models in two directions into a unified mathematical model. These studies provide ideas for the use of symmetric control theory to achieve high precision position control of asymmetric systems.

As for the control strategy, Xu Dongguang, etc. [9] have carried on the theoretical analysis and the experimental research to position control issue of asymmetric hydraulic cylinder by use of symmetric valve control method, and proposed the best load ratio for the consistency of the positive and negative responses. Bai Yanhong studied the control strategy of electro-hydraulic servo system with of both position and velocity, and then [10] proposed a control strategy combining with speed feedback and position feedback together with load force compensation. Based on the common butterfly characteristic between the piston speed and the hydraulic cylinder oil pressure for asymmetric system, Zhang Fei [11] proposes an adaptive compensation method to reduce the influence of load force on the positive and negative velocity of piston rod and improve the dynamic performance of the system. Zeng Le, etc. [1] put forward the optimal symmetric state model for the proportional valve controlled asymmetric hydraulic cylinder system by calculation, and set it as an invariant structure of the asymmetric system. By using the principle of structural invariance to design the compensation control link, the Fuzzy Symmetrical load flow characteristics of the asymmetric system are obtained, and the unified PID control of the asymmetric system is carried out.

Through the above analysis, it can be seen that although these methods can effectively solve the asymmetric problems in theory, they have these common characteristics: the theories are more complex, more parameters are needed to be adjusted, the design and maintenance are difficult, the requirements for engineers and technicians are high, and universality is poor due to the complex control program or algorithm, so it is difficult to be widely used in industrial field.

In this study, the state-space mathematical model of valve-controlled asymmetric hydraulic cylinder system is established. By constructing an undetermined feedback matrix and an undetermined feedforward compensation matrix, and assuming that the original asymmetric hydraulic system is transformed into a state-space symmetric system by feedback and compensation. The constraint relationship between the state, control and output matrix is obtained from the unique characteristics of the symmetric system to calculate the feedback matrix and feedforward compensation matrix which meet the symmetric characteristics, and then the feedback link and feedforward compensation link of the symmetric conversion of the system are obtained, and also by using the unified control method for the positive and negative direction of the converted system, the effectiveness of the study is verified by experiments and simulation.

2 State space modeling of valve-controlled asymmetric hydraulic cylinders

2.1 Descriptions

Figure 1 shows the hydraulic schematic diagram of four-way valve controlled asymmetric hydraulic cylinder. The system consists of zero-open four-sided sliding valve and asymmetric hydraulic cylinder. Refers to the displacement of the piston rod of the hydraulic cylinder, B_p refers to the viscous damping coefficient of the piston rod of the hydraulic cylinder, F_L refers to the external load force of the system, x_v refers to the displacement of the valve core of the servo valve, ω refers to the area gradient of the throttling window, P₁, P₂ individually refers to the working pressure of the rodless cavity and the rod cavity of the hydraulic cylinder, A₁, A₂ refers to the effective working area of the rodless cavity and the rod cavity of the hydraulic cylinder, q₁ refers to the effective working area of inflow (or outflow) to the rodless cavity of hydraulic cylinder, q₂ refers to the effective working area of inflow (or outflow) to the rod cavity of hydraulic cylinder, P_S refers to the oil supply pressure of the system.

Fig. 1

Working principle diagram of valve control cylinder.

Supposing that the effective area ratio of rodless cavity and rod cavity of hydraulic cylinder is $n = \frac{A_{2}}{A_{1}}$ , then q₂ = nq₁; Supposing that the effective working area of hydraulic cylinders is, then load flow $q_{L} = \frac{q_{1} + q_{2}}{2}$ .

Definition: When the piston rod is outstretched, at this time x_v > 0, then the flow into the rodless cavity is positive, the flow out of the rod cavity is positive, too; When the piston rod retracts, at this time x_v < 0, then the flow into the rodless cavity is negative, and the flow out of the rod cavity is negative, too.

2.2 Linear equation of servo valve flow

The flow output of the servo valve is nonlinear. The linearized flow equation of the servo valve can be obtained by expanding the Taylor series and omitting the term above the second order.

(1) When the velocity of piston ${\dot{x}}_{p} > 0 (x_{v} > 0)$

The linear flow equation of servo valve is $q_{L} = K_{q 1} x_{v} - K_{c 1} P_{L}$ (1)

In the formula: K_q1 is the flow gain, K_c1 is the pressure gain. Usually during system analysis, the static magnification factor at the original point is used as the performance parameter of the valve. Hence the valve coefficients at the origin of K_q1, K_c1 are considered as zero-position valve coefficients, individually expressed as K_q10, K_c10 [12], then $K_{q 10} = ɛ_{1} c_{d} ω \sqrt{\frac{P_{s}}{ρ}}$ , $K_{c 10} = \frac{π r_{c}^{2} ω_{1}}{32 μ}$ , $ɛ_{1} = \sqrt{\frac{(1 + n)^{2}}{2 (1 + n^{3})}}$ ;

Therein, c_d—flow coefficient of each throttle window; ω₁—area gradient of throttle window 1; x_v—displacement of servo valve; ρ—hydraulic oil density; r_c—radial clearance between servo valve core and valve bush; μ—hydraulic oil dynamic viscosity.

(2) When piston velocity ${\dot{x}}_{p} < 0 (x_{v} < 0)$

The linear flow equation of the servo valve is as follows: $q_{L} = K_{q 2} x_{v} - K_{c 2} P_{L}$ (2)

Therein, K_q2 is the flow gain, K_c2 is the pressure gain. $\begin{matrix} K_{q 2} = K_{q 20} = ɛ_{2} c_{d} ω \sqrt{\frac{P_{s}}{ρ}}, \\ K_{c 2} = K_{c 20} = \frac{π r_{c}^{2} ω_{2}}{32 μ}, ɛ_{2} = \sqrt{\frac{n (1 + n)^{2}}{2 (1 + n^{3})}}, \end{matrix}$

ω₂—Area gradient of throttle window 2.

According to section 1.2, the linearized flow equation of the servo valve of the valve-controlled asymmetric hydraulic cylinder system is the same in the positive and negative direction, but the control parameters such as flow gain and pressure coefficient are slightly different. Because the existence of two different equations in two different directions will bring difficulties to the design and control of the system, it is necessary to integrate the two Equations [8].

Since the effective working area of asymmetric cylinder rod cavity is always smaller than that of rodless cavity, the retracting speed of a Fuzzy Symmetrical cylinder piston rod is always larger than its outstretched speed, so the equilibrium position of asymmetric cylinder is always shifted to the direction of rodless cavity. That is, the frequency of each parameter when the asymmetric cylinder piston rod retracts in unit time is larger than that of each parameter when the piston rod extends out. Therefore, we adopt the weighted average method to integrate different mathematical models in both positive and negative directions. Then the integrated model is used as the system model of valve controlled asymmetric hydraulic cylinder, and the state space model of the system is integrated into formula (3). $q_{L} = K_{q} x_{v} - K_{c} P_{L}$ (3) $K_{q} = \frac{n}{n + 1} K_{q 1} + \frac{1}{n + 1} K_{q 2}$ $K_{c} = \frac{n}{n + 1} K_{c 1} + \frac{1}{n + 1} K_{c 2}$ $P_{L} = \frac{1}{A_{e}} (A_{1} P_{1} - A_{2} P_{2}) = \frac{A_{1}}{A_{e}} P_{1} - \frac{A_{2}}{A_{e}} P_{2}$

The Equation 3 can also be expressed as follows: $q_{L} = K_{q} x_{v} - K_{c} \frac{A_{1}}{A_{e}} P_{1} + K_{c} \frac{A_{2}}{A_{e}} P_{2}$ (4)

Next, according to the description in Equation 4 and sections 1.1, the linearization expressions of q₁ and q₂ can be expressed as follows: $q_{1} = δ_{1} K_{q} x_{v} - δ_{1} K_{c} \frac{A_{1}}{A_{e}} P_{1} + δ_{1} K_{c} \frac{A_{2}}{A_{e}} P_{2}$ (5) $q_{2} = δ_{2} K_{q} x_{v} - δ_{2} K_{c} \frac{A_{1}}{A_{e}} P_{1} + δ_{2} K_{c} \frac{A_{2}}{A_{e}} P_{2}$ (6)

Therein, $δ_{1} = \frac{2}{n + 1}$ , $δ_{2} = \frac{2 n}{n + 1}$

2.3 Flow continuity equation

(1) When piston velocity ${\dot{x}}_{p} > 0 (x_{v} > 0)$

As shown in Fig. 1, the flow into the left cavity of the hydraulic cylinder (rodless cavity) can be expressed in formula 7: $q_{1} = A_{1} \frac{{dx}_{p}}{dt} + (C_{ip} + C_{ep}) P_{1} - C_{ip} P_{2} + \frac{V_{01}}{β_{e}} \frac{{dP}_{1}}{dt}$ (7)

C_ip—Leakage coefficient inside hydraulic cylinder; C_ep—Leakage coefficient outside hydraulic cylinder; V₀₁—Initial volume of rodless cavity;

β_e—Effective bulk elastic modulus of oil.

The flow from the right cavity of the hydraulic cylinder (rod cavity) can be expressed in formula 8: $q_{2} = A_{2} \frac{{dx}_{p}}{dt} + C_{ip} P_{1} - (C_{ip} + C_{ep}) P_{2} - \frac{V_{02}}{β_{e}} \frac{{dP}_{2}}{dt}$ (8)

V₀₂—Initial volume of rod cavity;

(2) When piston velocity ${\dot{x}}_{p} < 0 (x_{v} < 0)$

At this point, the hydraulic cylinder moves to the left (when the hydraulic rod shrinks), As shown in Fig. 1, according to section 1.1, it is not difficult to deduce that the flow rate into the right cavity of the hydraulic cylinder (rod cavity) can be expressed in formula 8. And Flow rate out of the left cavity of the hydraulic cylinder (rodless cavity) can be expressed in formula 7.

2.4 Balance equation of hydraulic cylinder load

(1) When piston velocity ${\dot{x}}_{p} > 0 (x_{v} > 0)$

As shown in Fig. 1, the force balance equation of piston rod of hydraulic cylinder is established by force analysis: $A_{1} P_{1} = m \frac{d^{2} x_{p}}{{dt}^{2}} + B_{p} \frac{{dx}_{p}}{dt} + A_{2} P_{2} + F_{L}$ (9)

m—Total mass of equivalent mass of load on piston, piston and piston rod;

B_P—Viscoelastic damping coefficient of load, piston.

(2) When piston velocity ${\dot{x}}_{p} < 0 (x_{v} < 0)$

The force balance equation of piston and piston rod of hydraulic cylinder is established by force analysis, see Equation 9.

2.5 State space model of valve-controlled non-symmetric hydraulic cylinder system

Let $x_{1} = {\dot{x}}_{p}$ , x₂ = P₁, x₃ = P₂, u₁ = x_v, u₂ = F_L, then according to equation 5-equation 9, the state equation of valve-controlled asymmetric hydraulic cylinder system can be expressed as: ${\begin{matrix} {\dot{x}}_{1} = - \frac{B_{p}}{m} x_{1} + \frac{A_{1}}{m} x_{2} - \frac{A_{2}}{m} x_{3} - \frac{1}{m} u_{2} \\ {\dot{x}}_{2} = - \frac{β_{e} A_{1}}{V_{01}} x_{1} - (\frac{δ_{1} K_{c} β_{e} A_{1}}{V_{01} A_{e}} + \frac{β_{e} C_{ip} + β_{e} C_{ep}}{V_{01}}) x_{2} + (\frac{δ_{1} K_{c} β_{e} A_{2}}{V_{01} A_{e}} + \frac{β_{e} C_{ip}}{V_{01}}) x_{\begin{matrix} 3 \end{matrix}} + \frac{β_{e} δ_{1} K_{q}}{V_{01}} u_{1} \\ {\dot{x}}_{3} = \frac{β_{e} A_{2}}{V_{02}} x_{1} + (\frac{δ_{2} K_{c} β_{e} A_{1}}{V_{02} A_{e}} + \frac{β_{e} C_{ip}}{V_{02}}) x_{2} - (\frac{δ_{2} K_{c} β_{e} A_{2}}{V_{02} A_{e}} + \frac{β_{e} C_{ip} + β_{e} C_{ep}}{V_{02}}) x_{\begin{matrix} 3 \end{matrix}} - \frac{β_{e} δ_{2} K_{q}}{V_{02}} u_{1} \\ y_{1} = x_{1} \\ y_{2} = \frac{A_{1}}{A_{e}} x_{2} - \frac{A_{2}}{A_{e}} x_{3} \end{matrix}$ (10)

State matrix A: $[\begin{matrix} - \frac{B_{p}}{m} & \frac{A_{1}}{m} & - \frac{A_{2}}{m} \\ - \frac{β_{e} A_{1}}{V_{01}} & - \frac{δ_{1} K_{c} β_{e} A_{1}}{V_{01} A_{e}} + \frac{β_{e} C_{i p} + β_{e} C_{e p}}{V_{01}} & \frac{δ_{1} K_{c} β_{e} A_{2}}{V_{01} A_{e}} + \frac{β_{e} C_{i p}}{V_{01}} \\ \frac{β_{e} A_{2}}{V_{02}} & \frac{δ_{2} K_{c} β_{e} A_{1}}{V_{02} A_{e}} + \frac{β_{e} C_{i p}}{V_{02}} & - (\frac{δ_{2} K_{c} β_{e} A_{2}}{V_{02} A_{e}} P_{2} + \frac{β_{e} C_{i p} + β_{e} C_{e p}}{V_{02}}) \end{matrix}]$ ,

Control matrix B: $[\begin{matrix} \begin{matrix} 0 \\ \frac{β_{e} δ_{1} K_{q}}{V_{01}} \\ - \frac{β_{e} δ_{2} K_{q}}{V_{02}} \end{matrix} & \begin{matrix} - \frac{1}{m} \\ 0 \\ 0 \end{matrix} \end{matrix}]$ , Output matrix C: $[\begin{matrix} 1 & 0 & 0 \\ 0 & \frac{A_{1}}{A_{e}} & - \frac{A_{2}}{A_{e}} \end{matrix}]$

Then Equation 10 can be expressed as: $\begin{matrix} \dot{x} = Ax + Bu \\ y = Cx \end{matrix}$ (11)

3 State-space symmetry model of valve-controlled asymmetric hydraulic cylinder system

3.1 Solve the mathematical model of state space symmetric system

Supposing that there is a feedback control matrix K and a feedforward control matrix L, which can transform system 11 into a state-space symmetric system, then the transformed new input matrix: $v = Kx + Lu$

The feedback system can be expressed as: $\dot{x} = Ax + Bv \Rightarrow \dot{x} = Ax + B (Kx + Lu)$ that is: $\begin{matrix} \dot{x} = (A + BK) x + BLu \\ y = Cx \end{matrix}$ (12)

According to the definition of state space symmetric system, if system 12 is state space symmetric system, then: $\begin{matrix} (A + BK)^{T} = A + BK \\ BL = C^{T} \end{matrix}$ (13)

To solve the above matrix equations, we obtain: $\begin{matrix} K = V [\frac{1}{2} S^{- 1} U_{1}^{T} (A^{T} - A) U_{1} + ZS S^{- 1} U_{1}^{T} (A^{T} - A) U_{2}] U^{T} \\ L = B^{-} C^{T} . \end{matrix}$

Therein, singular values of B^T are decomposed into $B^{T} = V [S O] [\begin{matrix} U_{1}^{T} \\ U_{2}^{T} \end{matrix}]$ , and Z is an arbitrary symmetric matrix of the corresponding dimensions, According to the theory of control system, the necessary and sufficient condition of system stability is $U_{2}^{T} {AU}_{2} < 0$ .

3.2 Symmetry control strategy of valve-controlled asymmetric hydraulic cylinder system

Through constructing a undetermined matrix K based on state feedback (hereinafter referred to feedback matrix) and a matrix L based on input compensation (hereinafter referred to feedforward matrix), and assuming that the original asymmetric hydraulic system is transformed into a state-space symmetric system by state feedback and input compensation, the constraint relationship between state, control and output matrix is obtained from the unique characteristics of symmetric system, and then the feedback matrix K and feedforward matrix L satisfying symmetry characteristics are obtained; Obviously, applying the obtained feedback matrix and compensation matrix to the original asymmetric system, then the original asymmetric system can be transformed into a state space symmetric system.

The block diagram of the simulated structure of the valve-controlled asymmetric hydraulic cylinder system (formula 11) is shown in Fig. 2, and the block diagram of the simulated structure of the system (formula 12) after symmetry transformation is shown in Fig. 3.

Fig. 2

The block diagram of the simulated structure of the system before transformation.

Fig. 3

The block diagram of the simulated structure of the system after transformation.

4 Simulation study of state space symmetric model

In order to verify the effectiveness of Fuzzy Symmetrical transformation of valve-controlled asymmetric hydraulic cylinder system, the stability and better load characteristics after system transformation, the method shown in Fig. 3 is adopted. Taking the original (Fig. 2) and transformed (Fig. 3) valve-controlled asymmetric hydraulic cylinder system as the research objects, two simulation methods are adopted: 1-System Symmetry Simulation Study: For the external load forces with the same value but different directions - positive and negative, compare the loads of the system before and after transformation; 2-For different valve core displacements, compare the load response characteristics of the system before and after transformation; Matlab simulation is used to verify the accuracy of the established mathematical model of valve control asymmetric hydraulic cylinder system and the effectiveness of the model symmetric control method.

The simulation and test parameters are shown in Table 1, the initial value of x₁ is 0 m/s, and the initial value of x₂ and x₃ is 0 MPa, in the matlab, the state space model before and after symmetry transformation of valve-controlled asymmetric hydraulic cylinder system is established.

Table 1
Basic parameters of system

Parameters Spec Parameters Spec

Mass of piston and load on piston-m 300 kg Oil density-r 890 Kg/m3

Area of rodless cavity of hydraulic cylinder-A₁ 0.001963m² Oil supply pressure-Ps 10 MPa

Area of rod cavity of hydraulic cylinder-A₂ 0.001347m² Internal leakage factor-C_ip 3 × 10^-11 m³/sPa

Equivalent radius of proportional valve-r_c 5 × 10^-5m Flow coefficient-C_d 0.61

Viscous damping coefficient of pistons and loads-B_p 800 Area gradient-w 2.5 × 10^-3m

Initial volume of rodless cavity-V₀₁ 4.9 × 10^-4m³ External leakage factor-C_ep 1.5 × 10^-12m³/sPa

Initial volume of rod cavity-V₀₂ 3.4 × 10^-4m³ Viscosity-μ 1.37 × 10^-4P_a • s

Effective bulk elastic modulus-b_e 7 × 10⁸Pa Hydraulic cylinder stroke-l 0.5 m

Parameters	Spec	Parameters	Spec
Mass of piston and load on piston-m	300 kg	Oil density-r	890 Kg/m3
Area of rodless cavity of hydraulic cylinder-A₁	0.001963m²	Oil supply pressure-Ps	10 MPa
Area of rod cavity of hydraulic cylinder-A₂	0.001347m²	Internal leakage factor-C_ip	3 × 10^-11 m³/sPa
Equivalent radius of proportional valve-r_c	5 × 10^-5m	Flow coefficient-C_d	0.61
Viscous damping coefficient of pistons and loads-B_p	800	Area gradient-w	2.5 × 10^-3m
Initial volume of rodless cavity-V₀₁	4.9 × 10^-4m³	External leakage factor-C_ep	1.5 × 10^-12m³/sPa
Initial volume of rod cavity-V₀₂	3.4 × 10^-4m³	Viscosity-μ	1.37 × 10^-4P_a • s
Effective bulk elastic modulus-b_e	7 × 10⁸Pa	Hydraulic cylinder stroke-l	0.5 m

4.1 Analysis of controllability for valve-control asymmetric hydraulic cylinder system

Controllability and observability are essential to a control system. When designing the control system, we must know whether the system is controllable and observable, and what we need to control are the output variables, so we only need to study the output controllability of the system. The complete output controllability of the system is as follows:

For system: $\begin{matrix} \dot{x} = A^{'} x + B^{'} u \\ y = C^{'} x + D^{'} u \end{matrix}$ (therein, A′, B′, C′, D′ are respectively constant matrices of n × n, n × r, m × n and m × r), the necessary and sufficient condition for controllable output is: rank (Q) = q, Therein, $Q = [C^{'} B^{'} C^{'} A^{'} B^{'} C' {A^{'}}^{2} B^{'} \dots C^{'} {A^{'}}^{n-1} B^{'} D^{'}]$ , which is the output controllable matrix with dimension of m × (n + 1) r, and q is the dimension of the output variable.

In order to analyze the controllability of the system, the state matrix of the system is obtained by using the basic parameters of the system (see Table 1) and the derived state space mathematical model 12 of the valve-control asymmetric hydraulic cylinder system after symmetry conversion:

A′ = A + B × K, Input matrix of the system: B′ = C^T

Output matrix of the system: C′ = C

And the direct matrix of the system is $D^{'} = [\begin{matrix} 0 & 0 \\ 0 & 0 \end{matrix}]$ . Per the calculation, rank [Q] = 2 = q, so the system is output controllable.

4.2 Analysis of observability for valve-control asymmetric hydraulic cylinder system

System observability means that the change of system state can be reflected by measuring system outputs, and its precise definition can be found in reference. The state observability of the system is as follows:

For system: $\begin{matrix} \dot{x} = A^{'} x + B^{'} u \\ y = C^{'} x + D^{'} u \end{matrix}$ , its necessary and sufficient condition of state observability is: rank (N) = n. therein, $N = [\begin{matrix} C^{'} \\ C^{'} A^{'} \\ C^{'} {A^{'}}^{2} \\ \begin{matrix} \dots \\ C^{'} {A^{'}}^{n-1} \end{matrix} \end{matrix}]$ , which is the state observable matrix with dimension of m × (n + 1) r.

According to C′ and A′ in section 3.1, it can be calculated: rank (N) =3 = n, so the system is observable.

4.3 System symmetry simulation study

When the displacement of proportional valve core is 0.003 m, and the external load force is 20000 N and – 20000 N, the load pressure outputs of the system before and after Fuzzy Symmetrical transformation of valve-controlled asymmetric hydraulic cylinder system are shown in Figs. 4 and 5, respectively.

Fig. 4

Load pressure output response of system input symmetric load force before transformation.

Fig. 5

Load pressure output response of system input symmetric load force after transformation.

It can be seen from Figs. 4 and 5 that before symmetric control for the valve-controlled asymmetric hydraulic cylinder system, applying the same value of the external loads at positive and negative directions, the output load pressure at both directions are unequal due to the asymmetry of its structure and external characteristics. After the state space symmetry control based on state feedback and feedforward compensation, the valve-controlled asymmetric hydraulic cylinder system outputs the same load pressure value in both positive and negative directions for the same value of the external loads, that is, it is symmetric in both directions.

4.4 Simulation study of system load response characteristics

When the external load force of the system u₂ is 20000 sin(0.5πt), and proportional valve core displacement is 0.001 m,0.003 m,0.005 m, the load pressures of the system output before and after the Fuzzy Symmetrical transformation of the valve control asymmetric hydraulic cylinder system is shown in Figs. 6 and 7, respectively.

Fig. 6

Simulation of System Load Response Characteristics with Different Valve Core Displacement before Transformation.

Fig. 7

Simulation of System Load Response Characteristics with Different Valve Core Displacement after Transformation.

By comparing Figs. 6 and 7, it can be seen that before the Fuzzy Symmetrical transformation control, with the inputs of the same external loads, the load pressure of the valve-controlled asymmetric hydraulic cylinder system will change with the change of the valve core displacement; For the same external loads, the bigger valve core displacement, the higher load pressure of the system. Here the valve core displacement can be regarded as the interference of the load pressure. After the symmetry transformation of the system, for different valve core displacements and same external loads, the load pressure curve of the valve control asymmetric hydraulic cylinder system is nearly no change, that is, for different valve core displacements, the system has the stable load pressure curve.

5 Experimental research

5.1 Basic test conditions

The schematic diagram of the valve controlled asymmetric hydraulic cylinder system simulation test table is shown in Fig. 8. The test system consists of a proportional valve controlled power system and a proportional relief valve controlled load system. The load system has two hydraulic pumps, which supply oil for the two cavities of the load cylinders independently in order to establish the external load force conveniently and quickly. The parameters of the experimental platform are shown in Table 1.

Fig. 8

The schematic diagram of the system simulation test Table 1 Drive pump; 2 Load pump; 3 relief valve; 4 Load hydraulic cylinder; 5 Proportional relief valve; 6 Proportional servo valve; 7 Drive hydraulic cylinder; 8 Displacement transducer.

5.2 Test method

The power system controls the maximum pressure of the system through a quantitative pump plus an electromagnetic relief valve, adjusts the valve core displacement by adjusting the current of the proportional valve (system input 1-x_v), provides external load force by bi-opposite cylinders, and controls the pressure of two cavities of the load cylinder by the proportional relief valve in order to establish the external load forces (system input 2-F_l). The displacement and velocity of the load (state 1- ${\dot{x}}_{p}$ ) are measured by a displacement sensor which can move with the piston rod at the connection of the piston rod of the hydraulic cylinder. The pressure of the two cavities (state 2-P₁, state 3-P₂) can be measured by the installed pressure sensors in both rodless cavity and rod cavity of the drive cylinder. Finally, the data curve of the system can be obtained and the state feedback can be carried out in real time through the data collecting system.

5.3 Test results and analysis

5.3.1 Valve-controlled a Fuzzy Symmetrical hydraulic cylinder system before symmetry

The maximum pressure of the system is set to 8 MPa, adjust the input displacement of the valve core is 0.002 m, 0.003 m, 0.004 m, respectively, adjust the pressure of two cavities of the load system to make the external load forces vary between – 2000 N∼20000 N, record the valve core displacements, external load forces, and load pressure with the interval of 2000 N, to obtain the curve of the input and output characteristics of the load as shown in Fig. 9.

Fig. 9

Effect of different valve core displacement on system load response characteristics before transformation.

5.3.2 Valve-controlled asymmetric hydraulic cylinder system after symmetry

Add the control strategy shown in Fig. 10 to the valve-controlled asymmetric hydraulic system, and then set the input parameters and collect data according to the method shown in Section 4.3.1, the load response characteristic curve shown in Fig. 11 is obtained.

Fig. 10

Control Strategy of Test System.

Fig. 11

Effect of different valve core displacement on system load response characteristics after transformation.

5.3.3 Analysis of test results

Figs. 9 and 11 show that the pressure stability and the input & output symmetry of the system are different before and after the symmetry transformation control for valve control asymmetric hydraulic cylinder system. Figure 9 show that before the symmetry transformation, the load pressure of the system will change with the change of the valve core displacement, the bigger the valve core displacement, the greater the load pressure needed by the system, and the input & output of the system show obvious asymmetry. Figure 11 show that after symmetry transformation, with the change of valve core displacement, the load pressure of the system is almost only related to the external load force. Moreover, the external load force-load pressure input & output characteristics of the system are almost Fuzzy Symmetrical in the positive and negative directions.

6 Conclusions

In this paper, the valve controlled asymmetric hydraulic system is taken as the research object. Aim at solving the problems caused by the system asymmetry including the different control methods and inconsisten dynamic characteristics between postitive direction and negative direction, a symmetry method for asymmetric system based on state feedback is proposed. The parameterized expression of feedback matrix and feed forward matrix are given. The effectiveness of the research method is verified with MATLAB. The fiasibility of the system is verified by the simulation platform of hydraulic servo system.

The controllability and observability of the system were studied by MATLAB, and it was proved that the system was controllable and observable after fuzzy symmetry transformation.

When input the same amount of external load as force in positive and negative directions, the output of both directions are almost the same, which means the system has the same input and output characteristic in the postive and negtive directions after transformation

The load (in form of pressure) of the system is almost only related to the external force after symmetrical transformation, and varies very few while the spool displacement changes. This means the system is much more stable load characteristic after transformation.

This research lays a foundation that the symmetrical control theory can be applied after the asymmetric system is converted into symmetric system, and it is of great help to the realization of high-precision position tracking control of hydraulic servo system; this research is expected to solve the problems of huge debugging work and narrow control range of asymmetric system controller due to the difficulty of balancing the inconsistent response characteristics in positive and negative directions at the same time for a long time, to make the reliability and response characteristics of system can be greatly improved.

Conflicts of interest

The author(s) declare(s) that there is no conflict of interest regarding the publication of this paper.

Footnotes

Acknowledgments

Project (2014CB049405) supported by the National Basic Research Program (973 Program) of China; Project (2009ZX04005-031) supported by the National Science and Technology Major Special; Project (zzyjkt2013) supported by the State Key Laboratory of High Performance Complex Manufacturing.

Hunan Provincial Education Department scientific research project 20C0652.

References

Zeng

, Yang

and Tan

, Compensation control of a non-zero open valve controlled asymmetric cylinder system based on model invariance, Journal of Huazhong University of Science and Technology45(3) (2017), 29–34.

Lazić

D.V.

and Ristanović

M.R.

, Electrohydraulic thrust vector control of twin rocket engines with Position feedback via angular transducers, Control Engineering Practice15(5) (2007), 583–594.

van Oijen

, Komsta

and Antoszkiewicz

, Integral sliding mode compensator for load pressure control of die-cushion cylinder drive, Control Engineering Practice21(5) (2013), 708–718.

Xiao

L.F.

, Bin

B.L.

, Yu

, et al., Cascaded sliding mode force control for a single-rod electrohydraulic actuato, NeurocomPuting156(0) (2015), 117–120.

Yang

J.H.

, Yin

Z.Q.

and Li

S.Y.

, Nonlinear modeling and feedback linearization of valve controlled asymmetric hydraulic cylinder, Chinese Journal of Mechanical Engineering (05) (2006), 203–207.

Yang

J.H.

, Li

S.Y.

and Dai

Y.F.

, Research on robust control strategy of valve controlled asymmetric hydraulic cylinder position servo system, Machine Tool & Hydraulics (07) (2007), 115–118.

Y.S.

, Frequency domain modeling of valve controlled asymmetric hydraulic cylinder, Chinese Journal of Mechanical Engineering (09) (2007), 122–126.

Sun

, Li

C.C.

and Yan

, Nonlinear modeling of valve controlled asymmetric cylinder position system, Journal of Beijing Jiaotong University (04) (2012), 164–168+177.

D.G.

, Wu

S.L.

and Zhao

K.D.

, Theoretical analysis and experimental study on forward and reverse velocity characteristics of valve controlled asymmetric hydraulic cylinder position servo system, Chinese Hydraulics & Pneumatics (06) (2005), 35–39.

10.

Bai

Y.H.

and Quan

, Control strategy of electro hydraulic position and velocity compound servo system, Chinese Journal of Mechanical Engineering 12, 46(24) (2010), 150–155.

11.

Zhang

Fei

, Chen

J.Z.

and Pen

K.X.

, Adaptive compensation of hydraulic position control system, Chinese Journal of Mechanical Engineering 5, 41(5) (2005), 94–97.

12.

Qing

, Tian

and Dan

, Robust H∞ Positional control of 2-DOF robotic arm driven by electro-hydraulic servo system, ISA Transactions59 (2015), 55–64.