Hydraulic cylinder control of injection molding machine based on differential evolution fractional order PID

LI Ya-qiu，GU Li-chen，YANG Sha，XUE Xu-fei

(School of Mechatronic Engineering, Xi’an University of Architecture and Technology, Xi’an 710055, China)

Abstract： Injection molding machine, hydraulic elevator, speed actuators belong to variable speed pump control cylinder system. Because variable speed pump control cylinder system is a nonlinear hydraulic system, it has some problems such as response lag and poor steady-state accuracy. To solve these problems, for the hydraulic cylinder of injection molding machine driven by the servo motor, a fractional order proportion-integration-diferentiation (FOPID) control strategy is proposed to realize the speed tracking control. Combined with the adaptive differential evolution algorithm, FOPID control strategy is used to determine the parameters of controller on line based on the test on the servo-motor-driven gear-pump-controlled hydraulic cylinder injection molding machine. Then the slef-adaptive differential evolution fractional order PID controller (SADE-FOPID) model of variable speed pump-controlled hydraulic cylinder is established in the test system with simulated loading. The simulation results show that compared with the classical PID control, the FOPID has better steady-state accuracy and fast response when the control parameters are optimized by the adaptive differential evolution algorithm. Experimental results show that SADE-FOPID control strategy is effective and feasible, and has good anti-load disturbance performance.

Key words： variable speed pump-controlled cylinder; fractional order proportion-integration-differentiation (FOPID); self-adaptive differential evolution (SADE); injection molding machine control; anti-load disturbance

0 Introduction

Because the hydraulic system has the characteristics of high power density and strong adaptability, it is widely used in injection molding machines, hydraulic elevators, speed actuators and other equipment. In the control of product forming, the hydraulic system of injection molding machine needs the product to have higher response performance in speed control. According to the characteristics of hydraulic system circuit, it is divided into valve control system and pump control system. The servo valve of the valve control system has throttling effect, which leads to the loss of system energy. The valve control system has a high requirement for oil cleanliness. When the quantitative pump is driven by changing the displacement of the hydraulic pump or by the variable speed servo motor, the hydraulic servo system has the characteristics of simple structure, high energy utilization and strong ability to carry pollution. When the hydraulic system of injection molding machine is driven by servo motor combined with quantitative pump, the hydraulic system has the advantages of wide-range speed regulation and ease of realizing closed-loop control[1-3]. In the process of product stamping and forming by the hydraulic servo system of injection molding machine, there are several working conditions such as mould locking, glue injection, sol cooling and ejection. In each working condition, the working process of the executing cylinder should be precisely controlled. Therefore, it is necessary to ensure the stable output of pressure and flow[4-5].

At the same time, the hydraulic servo system of variable speed pump control cylinder is a strong nonlinear system. It is easily affected by nonlinear and discontinuous factors[6-7]. These factors restrict and interfere the injection molding performance of hydraulic system of injection molding machine. Therefore, it is necessary to propose effective control strategies to balance the influence of uncontrollable factors and improve the tracking control precision and robustness of the system. In the control of the pressure and speed of hydraulic servo system of injection molding machine, Pen et al put forward a fuzzy sliding mode control strategy to achieve accurate control[1]. Chiang et al designed a control strategy for fuzzy synovial control with symbolic distance[6]. This control strategy is based on sliding mode control. Under different target speed and load conditions, this kind of control can achieve fast response of pump control system with variable speed. Peng et al has analyzed the causes of the system speed drop, and carried out speed compensation for each affected parameter[8]. Through the observation results of the extended disturbance observer, the tracking error was obtained[9], and then the control loop was used to compensate the error. For the hydraulic cylinder speed control system, an adaptive fuzzy sliding mode controller was designed in Ref.[10]. This controller is based on pi-sigma fuzzy neural network. It solves the uncertain and nonlinear problems of system parameters. Compared with the classical PID controller and adaptive fuzzy controller, this controller has better control performance and robustness. The electro-hydraulic position servo system is easily affected by parameter time-varying and external load interference. In view of the influence of these uncertainties, Shao et al designed the fractional PID control strategy for tracking accuracy with repetitive control compensation and strong anti-parameter perturbation ability and robustness[11]. These control algorithms have good dynamic performance and robustness, but the implementation of the algorithm is complex and computational intensive. When these algorithms are applied to practical engineering, the performance of hardware will be higher. Therefore, Podlubny[12]proposed a fractional order PID controller (FOPID). FOPID has been successfully applied in engineering[13-15], which indicates that FOPID has a strong inhibitory force on nonlinear systems. Compared with the classical PID controller, FOPID introduces two more adjustable control parameters. It not only improves the accuracy of the controller, but also is more advantageous for the control of nonlinear systems. In addition, the optimization of control parameters has been paid attention to. Huang et al optimized the controller parameters by using differential evolution algorithm[16]. Chen et al optimized the controller parameters based on genetic parameter adaptive adjustment algorithm[17]. Wei et al optimized the controller parameters adopting particle swarm optimization algorithm[18].

Based on the previous analysis, we have designed an FOPID controller. It is based on fractional calculus theory and adaptive differential evolution algorithm. By using the adaptive differential evolution algorithm (SADE), the controller can adjust the five control parameters of the FOPID. Since the parameters of FOPID algorithm are fixed, it may lead to overshoot, integration saturation and local optimization. The controller designed in our work can solve these problems. Based on the simulation loading technology of actual working conditions of injection molding machine developed by Zheng et al[19], simulation and experiment on different target tracking speeds and loading conditions are carried out. The results show that the SADE-FOPID control strategy is effective in the hydraulic system of injection molding machine.

1 Structure and model of hydraulic system

1.1 Structure of hydraulic system

The hydraulic system structure adopted in this test bed is shown in Fig.1. The system has a driving system and a loading system. The driving system is driven by the permanent magnet synchronous motor (PMSM), which is directly controlled by the controller servo driver. The output flow of the quantitative pump is controlled by the speed of the PMSM. The loading system consists of ordinary motor and Atos proportional overflow valve. The loading system is an experimental system developed in the early stage to simulate the actual working conditions of injection molding machine based on digital loading technology. The output force of the loading cylinder is controlled by the electro-hydraulic loading device, and the linear motion of the hydraulic cylinder is simulated. The load transformation of driving hydraulic cylinder can be realized through simulation. The displacement sensor measures the speed at which the hydraulic cylinder is driven, which is processed online and transmitted to the controller. The system pressure can be detected by the pressure sensor and transmitted to the controller. The system drives the hydraulic cylinder required by the pressure and flow by the driving system to exactly match the output. The sensor responds with pressure and speed signals. The controller generates a control signal by comparing the desired speed signal of the driving cylinder with the feedback signal. At this point, according to the input voltage of the control signal, the servo driver adjusts the speed of the servo motor, thus realizing direct control of the driving hydraulic cylinder. The reversing action of the hydraulic cylinder can be accomplished by the electromagnetic reversing valve.

1—Permanent magnet synchronous motor; 2—Gear pump; 3—Safety valve; 4—Pressure transmitter; 5—Electromagnetic directional valve; 6—Hydraulic cylinder; 7—Displacement sensor; 8—Safety valve; 9—Gear pump; 10—Three-phase induction motor; 11—Electromagnetic directional valve；12—Check calve；13—Proportional relief valve; 14—Hydraulic cylinder; 15—Signal measurement and control system

1.2 Mathematical model of variable speed pump control cylinder system

1) Speed control model of permanent magnet synchronous motor

Servo driver and PMSM driving system has a small time constant, the inertia of the servo motor is small, and the acceleration response time and deceleration response time are short. Practice has proven that most industrial processes have aperiodic and overdamping characteristics, which can often be expressed as first-order or second-order inertial links. Therefore, in this paper, the PMSM is simplified as a first-order inertial link as[1]

(1)

and in the time domain, the servo motor model can be expressed as

(2)

whereKvis the speed gain, which is determined by the characteristics of the servo driver;Uis the input voltage signal of the servo driver;nis the speed value of the motor; andTdis the time constant.

2) The output flow equation of the quantitative pump is expressed as

(3)

whereDpis the displacement of the quantitative pump;ppis the output pressure of the pump;βeis the elastic modulus of oil volume;ηis the oil viscosity; andCipis the leakage coefficient of the quantitative pump.

3) Cylinder flow continuity equation

Considering that the pressure of the oil return chamber is ignored, because the hydraulic pipeline adopts stainless steel pipeline, the pressure drop of the high-pressure pipeline is ignored too.The change of flow in the rodless chamber of the hydraulic cylinder is expresed as

(4)

whereA1is the effective area of the rodless chamber of the hydraulic cylinder;Vgis the total volume of the high pressure chamber; andCepis the total leakage coefficient of the hydraulic cylinder.

4) Force balance equation of hydraulic cylinder movement is expressed as

(5)

wheremtis the equivalent mass of piston rod and load;Bpis the viscous damping coefficient of driving cylinder;Kis the elastic coefficient of load; andFLis the external load.

(6)

2 Design of FOPID controller

2.1 Definition of fractional calculus

(7)

whereaandtare the upper and lower limits of calculus, respectively;ρis the order of the FOC operator;Dis the fractional operator; and Re(ρ) is the real part of the order.

Fractional calculus is defined as

(8)

where [(t-a)/h] is the integer part; Γ(·) is the Euler gamma function;jis the sub-interval segment divided by equal intervals of FOC interval;his the length of the sub-interval of the FOC interval (calculated step size).

Under the condition of zero initial value, the Laplace transform of the definition of G-L is expressed as

(9)

2.2 FOPID controller

The transfer function of the FOPID controller can be defined as

Gc(s)=Kp+Kis-λ+Kdsμ,

(10)

whereKp,KiandKdare proportional, integral, and differential gains, respectively;λandμare fractional order integral order and fractional order differential order, respectively.

Because the hydraulic system of injection molding machine is generally nonlinear system, the use of classical PID control will cause a large control tracking error. Therefore, parametersλandμare introduced into the controller to increase the flexibility of the controller. In the time domain, FOPID controller can be designed as

u(t)=kpe(t)+kiD-λe(t)+kdDμe(t).

(11)

2.3 FOPID parameter optimization of adaptive differential evolution algorithm

The principle of the controller is shown in Fig.2. FOPID controller introduces two more control parameters, which makes the adjustment performance more flexible while leads to difficulties in adjusting multiple control parameters and maintaining the system dynamic control quality.

Fig.2 Schematic diagram of SADE-FOPID controller

Differential evolution algorithm (DE) put forward by Storn et al is a kind of competitive global search optimization algorithm based on population[21]. To obtain the optimal parameters of the standard solution, the algorithm includes the processes of population initialization, mutation and crossover and choice according to the objective function. The core process of the algorithm is as follows.

The initial individuals of the population is given as

whereMis the population size;Gis the population algebra; wheni=0 andG=0, the initial population individual is determined.

In the current population algebra, three vectors are randomly selected and the weighted difference vector between the two vectors is added to the third vector to produce mutants to complete the mutation operation, which is expressed as

(12)

In order to enrich the diversity of the population, the variation value obtained through the mutation operation and the target value were discreetly crossed, and the crossover rule is

(13)

whereRjis the random number between [0,1];pcris the cross probability,pcr∈[0,1].

(14)

wheref(·) is the fitness function.

In the process of parameter optimization, since the variation factorFand the crossover probability factorpcrare fixed values, the DE algorithm cannot meet the performance requirements of the algorithm for the control parameters. According to the fitness value of the evolutionary individual, the adaptive differential evolutionary algorithm can adjust the adaptive control parameters reasonably to reduce the dependence of the DE algorithm on the structural parameters of the adaptive control strategy, as shown in Fig.3.

Fig.3 Parameter optimization process

The adjustment strategies for variantion factorFand crossover factorpcrare expressed as

(15)

(16)

The five control parameters of FOPID to be optimized are proportional gainKp, integral gainKi, differential gainKd, fractional integral orderλand fractional differential orderμ.These parameters are taken as the initial population individuals of the adaptive differential evolutionary algorithm as

Xi={Kp,Ki,Kd,λ,μ).

2.4 Selection of fitness function

In the selection of fitness function, error and control signal should be considered as weighted values based on integrated time and absolute error (ITAE) criterion. Then the fitness function of SADE algorithm is expressed as

(17)

whereω1,ω2andω3are the weighted coefficients, and they are random numbers within [0,1].

2.5 Parameter optimization of SADE-FOPID

1) Parameter setting. Set the parameters of SADE and FOPID.

2) Initialization. According to population size, population algebra and set control parameters, the population is initialized.

3) Individual evaluation. According to Eqs.(12)-(14), the mutation and crossover operations are completed, and the fitness value of individual population of fitness function is set according to Eq.(17).

4) UpdatepcrandF. By comparing the adaptive value of the current population with the average adaptive value, the update of the crossover probability factorpcrand variation factorFis completed by Eqs.(15) and (16) or the next step is taken.

5) Optimal parameter output. Judging whether the convergence standard or the number of population iterations is reached, output the optimal control parameters to the FOPID controller or enter step 3.

3 Simulation

3.1 Principle of semi-physical simulation test platform

In order to verify the effectiveness of the control strategy proposed, according to the mathematical model of the hydraulic system and the FOPID control principle, the simulation model of the control system was established in Matlab/Simulink, and the adaptive differential evolution algorithm was written using the Matlab Function module. The relevant parameters of the hydraulic system are shown in Table 1.

Through a large number of experiments, the controller parameters were set as follows. The population size was set to be 20. The FOPID parameters included in the population wereKp,Ki,Kdandλ,μ. Variantion factorFand crossover factorpcrwere adjusted dynamically according to the adaptive adjustment strategy.

Table 1 Main parameters of hydraulic system

In order to reduce the iteration times of parameter optimization of the adaptive differential evolution algorithm, the step response experiment was conducted to ensure the control algorithm the reasonable parameter values. The sampling frequency was 100 Hz. The control parameter of SADE-FOPID control algorithm wereM=20,G=20,F=0.5,pcr=0.6,Kp=12.3,Ki=0.36,Kd=0.85,λ=0.6,μ=0.78 andKv=100. Based on the above parameter settings,pcrandFwere dynamically adjusted with the adaptive differential evolution algorithm. The simulation results are shown in Figs.4 and 5.

Fig.4 Dynamic speed response curves of step input with load disturbance

Fig.4 shows the speed step response curve of the system, which is compared with classical PID and SADE-FOPID, in the case that the speed step of the system reaches 40 mm/s at 0.4 s and 10 kN is loaded at 3 s and then maintained for 2 s. It can be seen that neither controller has overshoot. The response time of the step response of the SADE-FOPID controller is 0.52 s, and the response time under classical PID control is 0.96 s； The steady-state error of the SADE-FOPID controller is 0.04 mm/s，and the steady-state error of the classical PID controller is 0.13 mm/s. When the load disturbance of 10 kN is loaded at the time of 3 s, the SADE-FOPID adjustment time is 0.3 s and then returns to the desired speed value. The classical PID adjustment time is 0.51 s. After the load disturbance is maintained for 2 s, the SADE-FOPID controller recovers to the target velocity value by 0.29 s less than the classical PID controller.

Fig.5 Dynamic speed response curves of combined signal

Fig.5 shows the dynamic response curve of the combined signal under load of 5 kN. The combined signal includes step signal, ramp signal and sinusoidal response signal. At 0.02 s, the target speed is entered at a step of 25 mm/s. At 2 s, the target velocity drops to 5 mm/s with a slope of -10 mm/s. After the output of the system is stable, the sinusoidal input is taken as the expected speed signal at 5 s. The cycle is 5 s, and the speed drops to 5 mm/s after half a cycle. At this point, under the action of the SADE-FOPID controller, the dynamic response error of the combined input signal is 0.18 mm/s-0.49 mm/s. Under the classical PID control, the dynamic response error is 0.22 mm/s-1.42 mm/s. Compared with the classical PID control, the dynamic error of the SADE-FOPID controller with parameter optimization is reduced by about 65% and the response time is shortened by about 48%.

The simulation results show that the FOPID control strategy based on adaptive differential evolution algorithm has the capability of online fine tuning under constant conditions. It can suppress load disturbance effectively, and its shorter adjustment time can reduce the effect on the dynamic response due to the uncertainty of system parameters as well as the dynamic error when the system speed tracks the dynamic changes. Compared with the conventional controller, the SADE-FOPID controller has better control effect.

3.2 Result analysis

In this paper, a FOPID controller based on adaptive differential evolution algorithm is proposed. In order to verify the feasibility of the controller, the test platform was built according to the hydraulic system schematic diagram shown in Fig.1. The experimental platform is shown in Fig.6. Under constant load and step load, the desired velocity of the proposed SADE-FOPID control strategy was experimentally analyzed.

Fig.6 Mechanical & electrical and hydraulic remote measurement and control experiment platform

The GK6073-6AF31 PMSM with a power of 11.6 kW was used in our experiment, and the rated speed of the motor is 3 000 r/min. PGP505 gear pump manufactured by parker company was used in the test. The displacement of the pump is 6 mL/r, and the speed range of the pump is 500 r/min-3 000 r/min. The hydraulic cylinder is UG21D63/36-300TYCR hydraulic cylinder. Pressure and displacement sensors were used to detect system performance. The measured value of the sensor was used as the feedback signal of pressure and speed. The load variation of loading cylinder was simulated by digital loading technology. The system pressure varies with the load, and the driving speed of the hydraulic cylinder depends on the speed of the PMSM. Pressure and displacement signals were fed back from the sensor. These signals were transmitted to the analog input port of the high-speed acquisition card. Lab-VIEW8.6 software was used to write the control program and the adaptive differential evolution algorithm to optimize the controller parameters. After calculating the error, the output control quantity of the controller control servo driver were obtained. By controlling the speed of the motor, the execution speed of the hydraulic cylinder was be controlled.

Fig.7 shows the experimental results of the combined signal as velocity tracking signal.

Fig.7 Dynamic speed response curves of combined signal from experiment

The combined signal is step signal, ramp signal and sinusoidal signal. The test was conducted according to the servo performance requirements of hydraulic system of injection molding machine and the loading loop was set at a constant load of 5 kN. The adjustment time of step response was 0.68 s, and the maximum steady-state error was 1.73 mm/s. The steady-state error of the slope response was 2.42 mm/s at most, at the steady-state error of the sinusoidal response was 2.13 mm/s at most. The experimental results show that the ramp descent and sinusoidal tracking of the system are delayed for a certain time under constant load. There exists a velocity decline process under an inertial load. The specific speed response tracking performance is shown in Table 2.

Tabal 2 Performance indicators of speed tracking response

Fig.8 shows the dynamic speed response curve of the system when the expected speed step tracking response is added with a step load of 10 kN at 3 s and a load reduction at 5 s. Under the step loading condition, the increase of hydraulic oil pressure leads to the increase of system leakage and oil compression while the decrease of system output flow and speed. SADE-FOPID controller quickly compensates for the disturbance change by 0.62 s and then recovers to the desired velocity value.

Fig.8 Dynamic speed response curve of step input with load disturbance from experiment

Experimental results show that when the loading is constantly exerted on the system under the action of the SADE-FOPID controller, the target speed tracking performance is better, and the dynamic error meets the system output requirements. The variable speed pump control servo system has good response performance and keeps good robustness for step load.

4 Conclusion

In this study, the speed servo system of variable speed pump controlled hydraulic cylinder is simulated by digital loading technology, and the mathematical model is established. The hydraulic system of injection molding machine is a hydraulic servo system which requires higher execution speed. This paper presents a FOPID speed controller for such systems. Compared with the three control parameters of the classical PID controller, the SADE-FOPID controller introduces two additional control parameters λ and μ, which is convenient to obtain better control performance and use adaptive differential evolution to optimize FOPID controller parameters online.

Numerical simulation results show that the response time of the SADE-FOPID controller at the step tracking speed is 0.52 s, which is 54% of that of the classical PID controller. The steady-state error of the SADE-FOPID controller is 0.73 mm/s, which is 30% of that of the classical PID controller. Under the disturbance of step load, the maximum deviation of the SADE-FOPID controller is 1.74 mm/s, which is only 47.8% of that of the PID control. The adjustment time is 0.3 s, which is only 58.8% of that of PID control.

The experimental results show that the tracking performance of the control system is good at step, slope and sinusoidal target velocity. Its maximum tracking error is only 2.42 mm/s, which meets the target accuracy requirement of the system.

SADE-FOPID controller has the ability to control parameters adaptively. Compared with PID controller with fixed parameters, its dynamic response characteristics and steady-state control accuracy are better, and its anti-load disturbance performance is improved. The SADE-FOPID controller is suitable for the hydraulic system of injection molding machine with strong nonlinearity.