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5Cfefcfd1682f70bb4a20064c6b7 Hindawi Mathematical Problems in Engineering Volume 2019, Article ID 1728768, 9 pages https://doi.org/10.1155/2019/1728768 Research Article Nonlinear Compound Control for Nonsinusoidal Vibration of the Mold Driven by Servo Motor with Variable Speed Qiang Li ,1 Yi-ming Fang ,1,2 Jian-xiong Li ,1 and Zhuang Ma1 1Key Lab of Industrial Computer Control Engineering of Hebei Province, Yanshan University, Qinhuangdao, Hebei Province 066004, China 2National Engineering Research Center for Equipment and Technology of Cold Strip Rolling, Qinhuangdao, Hebei Province 066004, China Correspondence should be addressed to Yi-ming Fang; [email protected] Received 24 April 2019; Revised 3 September 2019; Accepted 10 September 2019; Published 22 October 2019 Academic Editor: Francesc Pozo Copyright © 2019 Qiang Li et al. ,is is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 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. ,en, 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 addressed: ,e first issue is nonlinear relationship [4] (ap- proximate sine periodic function and the inverse mapping is Nonsinusoidal vibration of continuous casting mold is one not unique) between servo motor speed and mold dis- of the key technologies to develop efficient continuous placement. Second, the transmission mechanism is realized casting [1, 2]. It is a new drive mode that servo motor, by gear meshing, and there is backlash between the gears to rotated in single direction with variable speed, drives con- avoid being stuck due to the heating and expansion caused tinuous casting mold vibrated in nonsinusoidal waveform by friction. It is necessary to compensate the uncertainties [3]. Compared with the existing modes, it has a simplified caused by backlash, while the servo motor rotated in single and compact structure, long service life, and energy-saving direction with variable speed. property and is of convenient maintenance. In this system, Since the mold displacement is in the periodic form, if there are mainly three parts: a servo motor as an actuator, the expected mold displacement is taken as the given signal transmission mechanism (including a gear reducer, an ec- directly and the closed-loop control is carried out, the servo centric shaft, and a connector), and a mold. ,rough the motor rotation speed will change alternately positive and transmission mechanism, when the servo motor rotates at a negative, which cannot meet the technological requirements constant speed, the mold vibrates in sinusoidal form. When for the motor. As to that problem, the piecewise mapping the servo motor rotates at variable speed, the mold vibrates function [5, 6] is used to convert the mold displacement to in a nonsinusoidal manner. ,is paper mainly considers the motor angular displacement by using the arcsine function to controller design in case of mold nonsinusoidal vibration. keep the mapping unique. In fact, the tracking error of mold ,e tracking control of mold vibration displacement is displacement is mitigated by adjusting the motor speed. If realized by the control of servo motor speed. To develop the the relationship between motor speed and mold displace- corresponding controller, two major issues need to be ment can be established directly, the tracking control of 2 Mathematical Problems in Engineering mold vibration displacement can be converted to servo analyzed in Section 2. ,e main results and theoretical motor speed control. In this paper, a new mathematical analysis are given in Section 3. Simulation results are given algorithm based on the rotation vector method is proposed in Section 4, followed by Section 5 that concludes the work. to build the relationship directly. On the other hand, under the integral action of the 2. Mathematical Model of the Mold Vibration eccentric shaft, the backlash may cause the accumulation disturbance to mold displacement. Generally, when the System and Problem Statement backlash’s width is known, the optimal control [7] or ,e continuous casting mold is driven by the servo motor adaptive control method [8, 9] could be adopted. Actually, through the coupling, reducer, eccentric shaft, and con- the backlash is nonlinear and difficult to establish a model. In necting rod. ,e device structure drawing is shown in [10–12], the backlash is seen as a black box or bounded Figure 1. disturbance and compensated by robust terms. In [13, 14], ,e model of servo motor is expressed as follows: the adaptive fuzzy and neural estimated inverse control : p B T method are proposed to mitigate the hysteresis non- >8 1 5 ψf 60 60 L > n_ � iq − n − ; linearities for actuator and effective to promote tracking > J 2π J 2π J > performance. Although the backlash exists in transmission <> 2π R pψ 2π uq mechanism (not the actuator), the inverse model of the _ si f ; (1) > iq � − pnid − q − n + transmission mechanism could be established based on the > 60 L L 60 L > mathematical algorithm as an observer to estimate the effect > R 2π u > _ s d of backlash on mold vibration displacement and compen- : id � − id + pniq + ; L 60 L sated by adjusting the motor speed. For the motor speed closed-loop control, the PI control where n is the rotate speed of the motor, id and iq are stator method has advantages of simple structure and easy imple- d- and q-axes currents, ud and uq are the stator d- and q-axes mentation to the motor control. As to the fixed parameters in voltages, p is the pole pair numbers, Rs is the stator re- traditional PI, the combination of fuzzy control and PI has sistance, L is the stator inductance, ψf is the flux linkage, J is strong robustness to system parameter perturbation and the rotor inertia, B is the viscous friction coefficient, and TL disturbance [15–17]. In view of the mold nonsinusoidal vi- is the load torque. bration, the servo motor rotates in single direction and var- In order to decouple the speed and currents, the vector iable speed, which means the motor needs to continuously ∗ control strategy of id � 0 is used. Here, two PI controllers, accelerate and decelerate. However, in traditional fuzzy rules, which are used to stabilize the d-q axes current errors, are the fuzzy gain adjustment direction is only one way when the adopted in the two current loops, respectively. In this paper, speed error is positive or negative, which makes it only ap- the speed loop controller is designed mainly. plicable to one-way speed regulation (or only applicable to the ,e transmission mechanism mainly comprises a re- accelerating, the decelerating will produce large overshoot). So, ducer, an eccentric shaft, and a connecting rod. ,e reducer it needs the fuzzy gain has two-way speed regulation to im- realizes transmission through gear meshing. ,e backlash is prove rapidity and stability. essential to avoid being stuck caused by tooth friction, In this paper, a fuzzy PI control strategy based on heating, and expansion. When the servo motor rotates at nonlinear feedforward compensation is proposed for the constant speed or accelerates in one direction, the backlash mold vibration system. ,e main contributions are sum- can be ignored and the reduction ratio is fixed. When the marized as follows: servo motor rotates in single direction with continuous (1) A mathematical algorithm is proposed to build the acceleration and deceleration, the backlash may affect the relationship between mold displacement and servo speed regulation system just as a hysteresis disturbance. ,e motor speed directly. Based on the algorithm, an impact is shown in Figure 2. observer is designed to estimate the effect of the Under the integral action of the eccentric shaft, the backlash and other factors on mold displacement backlash may cause the accumulation disturbance to the and feedforward compensate. angle of the eccentric shaft. Meanwhile, the initial me- chanical zero deviation will also cause the initial phase (2) In comparison to [6], the tracking control of mold difference to the angle. So the equation of the mold vibration vibration displacement could be converted to servo displacement can be expressed as [6] motor speed control. 1 t 2πn (3) As to the mold nonsinusoidal vibration, the servo motor S � h sin Z dt + d!; (2) rotates in single direction and continuously accelerates i + Δi 0 60 and decelerates. Combined with the PI control method, where S is the mold displacement, n is the actual motor a fuzzy control method with bidirectional parameter speed, i is the transmission ratio, Δi is the uncertainty caused adjustment is adopted to improve rapidity and stability by the backlash and other factors, and d is the initial phase for the motor speed control. offset of the eccentric shaft as a constant. ,e rest of this paper is organized as follows. ,e ,rough the analysis of the mathematical model, the mathematical model of the mold vibration system is expected motor speed n∗ is transmitted to the speed Mathematical Problems in Engineering 3 d ∗ Servo motor speed n n 2 1 θ S closed-loop control h sin(θ) 60 (i + ∆i) s Mold system D Mechanical C N (·) transmission Coupling Figure 3: Diagram of the mold driven by servo motor.
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