Abstract : In the holding pressure control section of an injection molding machine, because the polymer melt is a viscoelastic melt, the pressure transmission cannot be as uniform as that of a solid, resulting in a non-uniform distribution of the actual holding pressure. Furthermore, as the holding pressure process progresses, parameters such as melt temperature and density continuously change over time, leading to significant fluctuations in the holding pressure. In addition, changes in material factors, structural factors, injection molding process conditions, and operating conditions all contribute to the strong nonlinear characteristics of the holding pressure process. Therefore, converting this holding pressure process into a nonlinear system model and linearizing it, then applying it to the B&R injection molding machine control system, allows for more accurate control of the holding pressure.
Keywords: Injection molding process; Holding pressure control; Nonlinear system model; B&R control system
1 Introduction
With the increasing progress and development of society, people’s material life has been continuously improved, and the demand for plastics has gradually increased. In today’s society, plastics, concrete, steel and wood are known as the four major industrial materials. Plastic products, as one of the main molded products, are a general term for daily necessities and industrial products made of plastic as the main raw material, including products made of plastic as raw material through all processes such as injection molding and thermoforming [1].
In the injection molding process, the holding pressure control stage may be the most important stage affecting the performance of the final product. If the holding pressure is too high, the plastic product will stick to the mold, which will cause difficulty in demolding. It may also increase the residual stress of the plastic, resulting in burrs or leakage defects. If the holding pressure is too low, the plastic product will shrink, warp or have voids. Therefore, choosing the appropriate holding pressure is crucial to ensuring the performance and quality of the product [3]. The holding pressure stage is a complex stage that is not isothermal and compressible. There are many factors that restrict the improvement of the precision of injection molding equipment, and the holding pressure control in the injection process is one of the core elements. On the one hand, the holding pressure has a very important influence on the shrinkage, warping and residual stress of the product. On the other hand, the injection system is complex, and there are time-varying and nonlinear characteristics in the injection process and serious interference in the operating environment, making it difficult to implement precise control of the holding pressure. Therefore, the holding pressure process of the injection molding machine is a typical nonlinear system.
Based on nonlinear systems, this paper establishes a mathematical model of the holding pressure section of an all-electric injection molding machine and performs linearization processing. Furthermore, based on the application of the model in B&R's control system, the feasibility of the model is verified through experimental data on an injection molding machine of a well-known injection molding machine manufacturer in Shanghai.
2. Introduction to Nonlinear Systems
Suppose the state equation of a nonlinear system is expressed as follows:
(1)
Let f(x) and g(x) be smooth vector fields, and h(x) be a smooth function defined on the point r. If at time t, the control value depends only on the values of the state and the external reference input at the same time, then the control is called static feedback control. Otherwise, if the control also depends on a set of additional state variables, i.e., if the control itself has its own internal state as a proper dynamic system output driven by the external reference input, then a dynamic feedback control is achieved.
In the above system, static feedback control takes the input variable as equal to...
In equation (2), is the external reference input (as shown in Figure 1). In fact, the combination of system (1) will generate a closed-loop control system with the following structure.
The function in equation (2) is defined on this appropriate open set.
Figure 1. Simplified flowchart of the nonlinear system
3. Precise linearization
The exact linearization method of nonlinear systems refers to the linearization process in which no higher-order nonlinear terms are ignored. Therefore, this linearization is not only accurate but also global, that is, it is used for the entire defined region. The exact linearization method of nonlinear systems uses the method of differential geometry and the concept of differential manifolds. It realizes the exact linearization of nonlinear systems by constructing feedback transformation and differential homeomorphism. It reflects the linear characteristics of the integral trajectory of a vector field that satisfies certain conditions in the local or global range, and embodies this characteristic in another coordinate system through the transformation of the coordinate system [5].
Exact linearization reflects the linearity of the integral trajectory of a vector field under certain conditions in a local or global range. It also reflects this characteristic in another coordinate system by changing the coordinate system. Therefore, this linearization is not only accurate but also global, meaning it can be used for the entire defined region.
4. Establishment and processing of the pressure holding section model of the injection molding machine
The injection molding machine used in this article is an all-electric injection molding machine, whose injection system transmission mechanism uses a servo motor , and its mechanical transmission structure is shown in Figure 2:
Figure 2. Actuator diagram of the injection unit
As shown in Figure 2, the servo motor controls the synchronous gear through the synchronous belt. At the other end of the synchronous gear, the servo motor, after deceleration, controls the ball screw to move back and forth, thereby converting the rotational motion of the motor into the linear motion of the injection. Therefore, the process of controlling the injection motion is the process of controlling the point-to-point motion of the servo motor, and the process of controlling the pressure holding section is the process of controlling the torque output of the servo motor. If we find the asymptotically stable equilibrium point of the motor torque in the entire nonlinear system, we can stabilize the pressure holding process by controlling the torque.
Figure 3 shows a simplified schematic diagram of a servo motor. The model will be built around it. The rotor voltage is constant, and the stator voltage is selected as a control variable. In DC motors in engineering, the excitation voltage is usually called the stator voltage, and the armature voltage is called the rotor voltage.
Figure 3 Schematic diagram of servo motor
The above system can be represented by a set of three first-order differential equations. The first equation describes the voltage balance in the stator winding.
In the formula, represents the stator current, represents the resistance of the stator winding, represents the inductance of the stator winding, and represents the stator voltage. The second equation describes the voltage balance in the rotor winding.
In the formula, represents the rotor current, represents the resistance of the rotor winding, represents the inductance of the rotor winding, represents the rotor voltage (which is assumed to be constant), and E represents the back electromotive force. The third equation describes the mechanical balance of the load, assuming only viscous friction (i.e., the frictional torque depends only on the rotor angular velocity).
In the formula, F represents the coefficient of viscous friction, ω represents the angular velocity of the motor shaft, ω represents the moment of inertia of the load, and T represents the torque generated on the motor shaft. The coupling between the three equations is described by the following relationship:
In the formula, represents the magnetic flux related to the stator winding, k represents a constant, assuming the energy conversion efficiency is 100%, and the constants of back electromotive force and torque are taken to have the same value. The state variables are selected as follows (8):
Using the stator voltage as the input variable, substituting it into equation (1), where,
To ensure that this system can achieve precise linearization through state feedback and coordinate changes, it is necessary to verify the two conditions of Theorem 1.
Theorem 1: Suppose a system is given, where and are smooth vector fields. The problem of exact linearization of the state space is solvable in the vicinity of the point (i.e., there exists an "output" function for which the relative order of the system at the point is ) if and only if the following conditions are satisfied:
i) The matrix has a rank of n.
ii) Distributed in the vicinity is consistent.
like
Each point in the distribution has a dimension of 2 in the following dense distribution:
Therefore, it is integrable on U, and thus we can conclude that condition ii of Theorem 1 is also satisfied near any point on U. To transform this system into a linear and controllable system, we formulate partial differential equations.
The following describes the zero dynamics of this system. First, we define an output mapping h(x). In this system, the angular velocity of the motor shaft is the natural output variable. Because we keep the rotor voltage constant during modeling and use the stator voltage as the input, it's suitable to control the angular velocity near a rated non-zero value. Therefore, we take it as an output, i.e., the offset of the angular velocity from a fixed reference value. For this system, the zero-scale output corresponds to all the initial states and inputs we seek, and these initial states and inputs produce an angular velocity that is always equal to a certain value. Therefore, we have...
As can be seen from equation (17), at each point where the relative order of this system is 2, having zero output means that there is a state existing in the set, that is, there exists a manifold (as shown in Figure 4).
By using input, zero output can be achieved.
Figure 4. Schematic diagram of the system manifold
When the input is set to and initial conditions are chosen on the manifold, the zero dynamics of the system describe its internal behavior, and the constraints of this system are .
5. Implementation of the holding pressure section in the injection molding machine system by B&R
Figure 5 shows the B&R injection molding pressure control parameters.
Figure 5. B&R Pressure Holding Control Parameter Configuration Diagram
The initial stage of the pressure holding control phase in an injection molding machine is the injection stage. Based on the injection settings (as shown in Figure 5), the control system calculates the speed and pressure planning curves. Once the target injection position and speed are set, the host CPU downloads the speed curve to the servo driver via the POWERLINK bus. The servo driver then controls the servo motor based on this path planning. When the injection process ends and the pressure holding process begins (i.e., when the pressure holding switching condition is triggered – this article uses time switching, which is checked in Figure 5), the pressure curve is still calculated using the pressure planning function. Since the curve based on the pressure holding switching condition and the position-planned path has uncertainty, the servo driver executes the set speed output by the pressure controller. At this time, the pressure controller and speed controller work simultaneously. The host CPU downloads the pressure holding setting curve to the servo driver via the POWERLINK curve. The pressure controller controls the servo motor to output the set speed based on this setting, and the speed controller receives this set speed and controls the servo motor to run.
Based on the control characteristics of servo motors, the current control function is usually executed internally by the servo driver. Therefore, the speed control function becomes the final execution unit in the entire pressure holding control stage. Furthermore, as verified in the previous section, this system can find the asymptotically stable equilibrium point of the servo motor torque. During the pressure holding process, the motor needs frequent pressure increases and decreases. To achieve better results, we use an S-shaped acceleration/deceleration control method to reach the system's equilibrium point.
The S-shaped acceleration/deceleration control method controls the rate of change of acceleration during servo motor start-up, shutdown, and sudden speed changes. This ensures a smooth transition in motor speed during sudden speed changes, thereby maintaining stable pressure changes. The S-shaped acceleration/deceleration curve refers to minimizing the sudden impact on the machinery and motor through precise control during acceleration and deceleration. The curve is divided into seven stages: acceleration phase, uniform acceleration phase, deceleration phase, uniform speed phase, acceleration/deceleration phase, uniform deceleration phase, and deceleration/deceleration phase. (This article only uses the most basic path planning curve; other more complex planning methods may exist depending on the process.) The basic path planning is shown in Figure 6.
Figure 6. B&R Pressure Holding Control Parameters
Based on this planning function, the motion curve during the injection process can be planned. During the pressure holding phase, if a sudden speed change is detected, it can be limited by parameters. The execution function can be divided into four execution objects: PressureController, SwitchSelector, RateLimiter, and PressureComparator. A flowchart is shown in Figure 7.
Figure 7 Flowchart of B&R Pressure Holding Control Model
The PressureComparator detects abnormal pressure during operation. When the actual pressure sampled by the pressure sensor is abnormal, the injection process stops immediately, regardless of the stage the equipment is in. The SwitchSelector detects the holding pressure switching condition and determines whether the set holding pressure condition has been met. During the injection stage, only the SpeedController is executed. When switching to the holding pressure stage, the PressureController is executed first, outputting the set speed to the SpeedController, which then controls the servo motor. The RateLimiter controls the acceleration change rate Jerk. The SpeedController's output is compared with Jerk; if it exceeds the range, the driver limits the output according to the aforementioned limits. The expression for the PID controller composed of the PressureController and SpeedController is shown in Equation 24.
All the above objects are written as C language programs, then encapsulated into the function library pQCont, called in the main program, and then compiled and downloaded to the B&R control system CPU. Its hardware topology is shown in Figure 8.
Figure 8 Hardware topology diagram of B&R control system
In the actual test, the parameters were set as shown in Figure 9:
Figure 9. B&R Pressure Holding Control Parameter Settings
The experimentally measured curves of the set pressure and the actual pressure are shown in Figure 10.
Figure 10 Pressure curve during the pressure holding process
As shown in the curve above, when transitioning from the injection stage to the holding pressure stage, the holding pressure experiences a slight overshoot and stabilizes at the set pressure. In the following period, the holding pressure remains stable without overshoot or oscillation, which is a relatively ideal result.
6 Conclusion
This paper introduces the basic concept of nonlinear system models, establishes a corresponding mathematical model based on the characteristics of the holding pressure of an all-electric injection molding machine, and linearizes it using a precise linearization method. The asymptotically stable equilibrium point of the output is found and applied to the B&R control system. Through path planning, actual experiments and data analysis are conducted to verify the feasibility of the model, and the control results achieve an ideal effect.
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About the author:
Sui Penghao (1993-), male, from Chifeng, Inner Mongolia. He is a 2016 graduate of the Department of Automation, School of Electronic Information and Electrical Engineering, Shanghai Jiao Tong University, and currently works at B&R Industrial Automation (China) Co., Ltd., mainly engaged in the design and development of control solutions for the plastics industry.
Yang Yupu (1957-), male, from Shaanxi Province. Professor of Automation, School of Electronic Information and Electrical Engineering, Shanghai Jiao Tong University, and member of the Application Professional Committee of the Chinese Association of Automation. His main research areas include intelligent measurement and control and automation devices, industrial measurement and control computer networks and intelligent information processing, and intelligent control theory and applications.
Chen Zhiping (1970-), male, from Xinning, Hunan Province. Technical Manager and Senior Engineer of the Application Technology Department at B&R Industrial Automation (China) Co., Ltd., primarily engaged in technical solution consulting and review. He is skilled in application algorithm design for machine control, synchronous control of multi-axis systems, CNC and robot systems, and has accumulated extensive application development experience in OEM and PA fields.
Wang Yishuai (1987-), male, from Luoyang, Henan Province. He is a senior application technology engineer at B&R Industrial Automation (China) Co., Ltd., mainly engaged in software development and technical support for the plastics industry.
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