Abstract
In this paper, a fuzzy PI control method based on nonlinear feedforward compensation is proposed for the nonsinusoidal vibration system of mold driven by servo motor, rotated in single direction with variable speed. During controller design, there are mainly two issues to consider: (i) nonlinear relationship (approximate periodic function) between mold displacement and servo motor speed and (ii) uncertainties caused by backlash due to motor variable speed. So, firstly, the relationship between mold displacement and motor rotation speed is built directly based on the rotation vector method. Then, an observer is designed to estimate the uncertainties and feedforward compensation. Secondly, as the motor rotates in single direction with variable speed, a fuzzy control with bidirectional parameter adjustment is adopted to improve rapidity and stability based on the traditional PI method. Finally, some simulation results show the effectiveness of the proposed control method.
1. Introduction
Nonsinusoidal vibration of continuous casting mold is one of the key technologies to develop efficient continuous casting [1, 2]. It is a new drive mode that servo motor, rotated in single direction with variable speed, drives continuous casting mold vibrated in nonsinusoidal waveform [3]. Compared with the existing modes, it has a simplified and compact structure, long service life, and energy-saving property and is of convenient maintenance. In this system, there are mainly three parts: a servo motor as an actuator, transmission mechanism (including a gear reducer, an eccentric shaft, and a connector), and a mold. Through the transmission mechanism, when the servo motor rotates at a constant speed, the mold vibrates in sinusoidal form. When the servo motor rotates at variable speed, the mold vibrates in a nonsinusoidal manner. This paper mainly considers the controller design in case of mold nonsinusoidal vibration. The tracking control of mold vibration displacement is realized by the control of servo motor speed. To develop the corresponding controller, two major issues need to be addressed: The first issue is nonlinear relationship [4] (approximate sine periodic function and the inverse mapping is not unique) between servo motor speed and mold displacement. Second, the transmission mechanism is realized by gear meshing, and there is backlash between the gears to avoid being stuck due to the heating and expansion caused by friction. It is necessary to compensate the uncertainties caused by backlash, while the servo motor rotated in single direction with variable speed.
Since the mold displacement is in the periodic form, if the expected mold displacement is taken as the given signal directly and the closed-loop control is carried out, the servo motor rotation speed will change alternately positive and negative, which cannot meet the technological requirements for the motor. As to that problem, the piecewise mapping function [5, 6] is used to convert the mold displacement to motor angular displacement by using the arcsine function to keep the mapping unique. In fact, the tracking error of mold displacement is mitigated by adjusting the motor speed. If the relationship between motor speed and mold displacement can be established directly, the tracking control of mold vibration displacement can be converted to servo motor speed control. In this paper, a new mathematical algorithm based on the rotation vector method is proposed to build the relationship directly.
On the other hand, under the integral action of the eccentric shaft, the backlash may cause the accumulation disturbance to mold displacement. Generally, when the backlash’s width is known, the optimal control [7] or adaptive control method [8, 9] could be adopted. Actually, the backlash is nonlinear and difficult to establish a model. In [10–12], the backlash is seen as a black box or bounded disturbance and compensated by robust terms. In [13, 14], the adaptive fuzzy and neural estimated inverse control method are proposed to mitigate the hysteresis nonlinearities for actuator and effective to promote tracking performance. Although the backlash exists in transmission mechanism (not the actuator), the inverse model of the transmission mechanism could be established based on the mathematical algorithm as an observer to estimate the effect of backlash on mold vibration displacement and compensated by adjusting the motor speed.
For the motor speed closed-loop control, the PI control method has advantages of simple structure and easy implementation to the motor control. As to the fixed parameters in traditional PI, the combination of fuzzy control and PI has strong robustness to system parameter perturbation and disturbance [15–17]. In view of the mold nonsinusoidal vibration, the servo motor rotates in single direction and variable speed, which means the motor needs to continuously accelerate and decelerate. However, in traditional fuzzy rules, the fuzzy gain adjustment direction is only one way when the speed error is positive or negative, which makes it only applicable to one-way speed regulation (or only applicable to the accelerating, the decelerating will produce large overshoot). So, it needs the fuzzy gain has two-way speed regulation to improve rapidity and stability.
In this paper, a fuzzy PI control strategy based on nonlinear feedforward compensation is proposed for the mold vibration system. The main contributions are summarized as follows:(1)A mathematical algorithm is proposed to build the relationship between mold displacement and servo motor speed directly. Based on the algorithm, an observer is designed to estimate the effect of the backlash and other factors on mold displacement and feedforward compensate.(2)In comparison to [6], the tracking control of mold vibration displacement could be converted to servo motor speed control.(3)As to the mold nonsinusoidal vibration, the servo motor rotates in single direction and continuously accelerates and decelerates. Combined with the PI control method, a fuzzy control method with bidirectional parameter adjustment is adopted to improve rapidity and stability for the motor speed control.
The rest of this paper is organized as follows. The mathematical model of the mold vibration system is analyzed in Section 2. The main results and theoretical analysis are given in Section 3. Simulation results are given in Section 4, followed by Section 5 that concludes the work.
2. Mathematical Model of the Mold Vibration System and Problem Statement
The continuous casting mold is driven by the servo motor through the coupling, reducer, eccentric shaft, and connecting rod. The device structure drawing is shown in Figure 1.
The model of servo motor is expressed as follows:where is the rotate speed of the motor, and are stator d- and q-axes currents, and are the stator d- and q-axes voltages, is the pole pair numbers, is the stator resistance, is the stator inductance, is the flux linkage, is the rotor inertia, is the viscous friction coefficient, and is the load torque.
In order to decouple the speed and currents, the vector control strategy of is used. Here, two PI controllers, which are used to stabilize the d-q axes current errors, are adopted in the two current loops, respectively. In this paper, the speed loop controller is designed mainly.
The transmission mechanism mainly comprises a reducer, an eccentric shaft, and a connecting rod. The reducer realizes transmission through gear meshing. The backlash is essential to avoid being stuck caused by tooth friction, heating, and expansion. When the servo motor rotates at constant speed or accelerates in one direction, the backlash can be ignored and the reduction ratio is fixed. When the servo motor rotates in single direction with continuous acceleration and deceleration, the backlash may affect the speed regulation system just as a hysteresis disturbance. The impact is shown in Figure 2.
Under the integral action of the eccentric shaft, the backlash may cause the accumulation disturbance to the angle of the eccentric shaft. Meanwhile, the initial mechanical zero deviation will also cause the initial phase difference to the angle. So the equation of the mold vibration displacement can be expressed as [6]where is the mold displacement, is the actual motor speed, is the transmission ratio, is the uncertainty caused by the backlash and other factors, and is the initial phase offset of the eccentric shaft as a constant.
Through the analysis of the mathematical model, the expected motor speed is transmitted to the speed controller, which usually adopted the PI method with simple structure and easy implementation. The mold vibration displacement control is completed by mechanical transmission parts following the servo motor speed. The overall block diagram is shown in Figure 3.
In order to realize the mold nonsinusoidal vibration, can be calculated by expected mold displacement. For example, the mold nonsinusoidal vibration waveform uses the Demark nonsinusoidal equation:
The corresponding angular velocity of the eccentric shaft iswhere , in which is the nonsinusoidal vibration frequency of the mold. , where is the wave slope, . Then, can be expressed as
The expected mold nonsinusoidal vibration velocity waveform and the corresponding servo motor speed are shown in Figure 4.
(a)
(b)
According to Figure 4, the servo motor rotates in single direction () and continuously accelerates and decelerates to realize the mold nonsinusoidal vibration.
However, the following two problems need to be considered during the controller design:(i)The nonlinear links exist in the system: first, there are uncertainties such as backlash, initial phase deviation, and time-varying load disturbance. So the mechanical transmission part is nonlinear and unknown. Second, there is an essential nonlinear periodic function relationship between the servo motor speed and the mold displacement in the forward control channel.(ii)The uncertainties need to be compensated by adjusting the motor speed. It is a common control strategy to combine the fuzzy control with the PI method to improve control performance. However, the traditional fuzzy PI is mostly suitable for single-direction speed regulation and will produce large overshoot when the servo motor continuously accelerates and decelerates.
3. Design of the Nonlinear Compound Controller
The control purpose is to realize the tracking control of mold vibration displacement. But the elimination of vibration displacement tracking error is realized by adjusting the motor speed. In this paper, the servo motor is taken as the control object and the controller is mainly of two parts: one is the mathematical algorithm, which constructs the functional relationship between the mold displacement and the motor speed directly. Based on the algorithm, an observer is designed to estimate the uncertainties and feedforward compensation. The other is the fuzzy control with bidirectional parameter adjustment. It can improve rapidity and stability in case of continuous acceleration and deceleration for the servo motor. The block diagram is shown in Figure 5.
3.1. Nonlinear Mathematical Algorithm
Let represent the angular displacement of the eccentric shaft. Then, . So it follows thatwhere is the angular velocity of the eccentric shaft corresponding to and is defined as the motor speed corresponding to . In (6), it is a linear correspondence from to . Formula (2) can be expressed as
According to the rotating vector method as shown in Figure 6, every point of the sine or cosine curve is one to one correspondence with the reference circle’s position. So, it is concluded that a point of the reference circle corresponds to a position of the mold, which can avoid the multisolution of sine or cosine function in the single cycle.
Theorem 1. As to (6) and (7), according to the rotating vector method, the functional relationship between the eccentric shaft angular velocity and the mold displacement can be expressed aswhere and is constant, .
Proof.
Case 1. and .
The time derivative of (7) is derived asSince the servo motor rotates continuously in one direction, then according to the rotation vector method, it can be calculated that
Case 2. and .
According to formula (4), the corresponding angular velocity can be obtained as follows:where is the time when the nonsinusoidal vibration displacement curve reaches its peak and is not unique.
Considering the motor single-direction rotation, is a positive constant. Only considering the time when the vibration displacement reaches the peak upward in this paper, it follows that
The wave slope is defined aswhere is the vibration cycle and is the lagging time, which is the time difference to the peak displacement between nonsinusoidal waveform and sinusoidal waveform. Then, it follows that
From (15), and . Then,
Substituting (14) and (16) into (11), it follows that
Therefore, when reaches the peak, is related to and , which are the same to the given signal.
From (10) and (17), it follows that
Define and convert equation (18) into an equation expressed as follows:where is constant and only related to and , . This completes the proof.
Based on Theorem 1 and formula (6), can be calculated as
Remark 1. If and are known, the corresponding servo motor speed can be calculated based on equation (20).
To mitigate the influence of disturbance in the transmission mechanism part, a disturbance observer is designed based on the mathematical algorithm as shown in Figure 7. is the motor speed corresponding to the actual mold displacement . The difference between and can be regarded as the effect of the backlash and other factors on the speed regulation system and can be compensated through feedforward. is a low-pass filter. The forward channel is a high gain to suppress the influence of disturbance on the output. is adopted as where is the time constant and is the complex parameter.
3.2. Fuzzy PI Controller
As the servo motor rotates in a single direction and continuously accelerates and decelerates to realize the mold nonsinusoidal vibration, new fuzzy rules with bidirectional speed regulation are adopted for the PI self-tuning to improve rapidity and stability of the servo motor speed control The structure diagram of the fuzzy controller is shown in Figure 8.
The inputs of fuzzy controller are speed error and error rate , and the outputs are the adjustment of PI control parameters and . The final PI parameters arewhere and are initial setting values of proportion and integral coefficient, respectively.
As to input variables and output values, seven linguistic variables in its domain are taken: NB, NM, NS, Z, PS, PM, and PB, which mean negative big, negative medium, negative small, zero, positive small, positive medium, and positive big. The normalized domain corresponds to {−3, −2, −1, 0, 1, 2, and 3}.
The amplitude of the mold displacement is 3 mm, and the error is mainly in . So the range of is set as (if the actual value exceeds the range, take the limiting value). According to the mold vibration model, the range can be measured as . The quantitative factors of and are 1 and 0.0375, respectively.
According to the experience of PI parameter setting, it should be adjusted quickly when the error is large and it requires fine adjustment when the error is small. Therefore, the membership function should be close to the equilibrium point and sparse on both sides of the fuzzy domain, which will increase the amount of program implemented [18, 19]. In this paper, the gain coefficient [20] is introduced to adjust the parameters based on the symmetric membership function.
The gain coefficients of and are and , respectively, and the setting principle is as follows:(1)When is large, take larger and to speed up the response speed.(2)When is medium large, reduce and to make the change of proportion and integral coefficient not be dramatic.(3)When is small, in the case of larger, appropriately increase and reduce to prevent from increasing too fast to produce overshoot. In the case of being medium or small, increase to improve the integral effect.
In this paper, the new fuzzy control rules with bidirectional output adjustment are adopted to improve the responsiveness. With the rotational speed error as the main criterion, and can be further refined as follows:(1)When is large, the tracking requirement can be achieved only through the proportional component due to the large error. Even if the integral is introduced, the response cannot be further improved, while the overshoot may appear. So can be larger and can be zero.(2)When is a medium value, take smaller to reduce overshoot. At the same time, add slowly to converge the static difference that cannot be eliminated by the proportional and avoid the integral saturation.(3)When is small, increasing and is beneficial to improve the system steady state performance and enhance the antidisturbance ability.
The effect of the time-varying load and the friction should be considered in the fuzzy control rules. The direction of the load and friction is always opposite to the motor. In the acceleration phase, the friction force and load torque will decrease , while in the deceleration phase it will increase . The larger is, the easier it is to cause overshoot. Therefore, the fuzzy control rules are adjusted as follows:(1)Reducing the lower limit of in the deceleration section could decrease the increasing rate of . It could also reduce the torque of the motor when is a medium or small value, so as to reduce overshoot and make the control effect close to the acceleration section.(2)The rules for are basically the same as that in the acceleration section. In the deceleration section, increases from zero and the value after the system deceleration overshoot is much larger than that before the overshoot. So the integral output can quickly change from negative to positive and reach the required value when the speed is stable, thus reducing the overshoot.
According to the above rules and combined with the actual situation of the control system, and fuzzy rules are shown in Tables 1 and 2.
The weighted average method is often used in ambiguity resolution to obtain the modified parameter values and .
4. Simulation Research
Some simulations are used to illustrate the effectiveness of the proposed control method.
The mechanical transmission part parameters are as follows: reduction ratio: ; amplitude of the mold: . The nominal values of the servo motor parameters (Table 3) are selected as follows [4].
The servo motor’s speed loop and current loop use the same set of parameters: , , , , , and . In filter , .
The control purpose is to realize the tracking control of mold vibration displacement, and the servo motor is taken as the controller object. The expected signal for the mold is taken as Demark nonsinusoidal function: . The corresponding given servo motor speed is , where . In this paper, , , and times/min.
The load disturbance is adopted as follows [6]:
According to common requirements of machining precision , it takes the worst situation: . The initial zero offset of the eccentric shaft is .
To show the effectiveness of the proposed controller, the simulation is conducted under two cases: with and without nonlinear feedforward compensation. The simulation results are illustrated in Figures 9–12. Figure 9 clearly shows that the tracking error when the nonlinear feedforward compensation applied is much better than the case without the compensation. It should be emphasized that Figure 9 sufficiently illustrates the validity of the proposed control scheme because of the precisely tracking performance and small tracking error.
(a)
(b)
Based on the nonlinear mathematical algorithm shown as equation (20), the mapping between the mold displacement and the servo motor speed is constructed to meet the technological requirements for the motor rotated in single direction and unique mapping. Figure 10 is the mapping curve and illustrates the algorithm effective.
Figure 11 shows the given motor speed and the actual motor speed with feedforward compensation, respectively. The difference between the actual motor speed and the given speed is used to compensate the uncertainties caused by the backlash and other factors in mechanical transmission parts.
Figure 12 illustrates the servo motor speed tracking errors of the proposed fuzzy PI control scheme and the traditional PI control. It should be noted that the traditional PI method has a large overshoot than the proposed control scheme when the motor changes direction.
5. Conclusion
In this paper, a fuzzy PI control strategy based on nonlinear feedforward compensation is proposed for the mold displacement system. Firstly, a mathematical algorithm is proposed to build the relationship between mold displacement and servo motor speed. Based on the algorithm, an observer is designed to estimate the disturbance and feed-forward compensation. Secondly, a fuzzy control method with bidirectional parameter adjustment is adopted compound with the PI control method to improve rapidity and stability of the servo motor speed control. Finally, the simulation results show the proposed method is effective.
The mathematical algorithm is mainly based on the Demark nonsinusoidal function in this paper. Future works will focus on the algorithm applying to other nonsinusoidal vibration displacement waveforms, such as piecewise waveform and composite waveform.
Data Availability
The data used to support the findings of this study are included within the article.
Conflicts of Interest
The authors declare that there are no competing interests regarding the publication of this paper.
Acknowledgments
This work was supported by the National Natural Science Foundation of China (no. 61873226), the National Science Foundation of Hebei Province (no. F2017203304), and the Hebei province Talent Project Training Funds of Hebei Province (no. A2016015002).