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- ⎪⎧ 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 u ⎪ _ s 2π 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� � 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. Eccentric Shaker arm Servo sha Connecting motor B (i) ,e nonlinear links exist in the system: first, there are rod uncertainties such as backlash, initial phase de- viation, and time-varying load disturbance. So the mechanical transmission part N(·) is nonlinear and Reducer unknown. Second, there is an essential nonlinear periodic function relationship between the servo Figure 1: Device structure drawing of the continuous casting mold motor speed n and the mold displacement S in the vibration system driven by the servo motor. forward control channel. (ii) ,e uncertainties need to be compensated by Accel- Accel- adjusting the motor speed. It is a common control erate erate Drive strategy to combine the fuzzy control with the PI Drive Drive side method to improve control performance. However, side side Decel- Decel- the traditional fuzzy PI is mostly suitable for single- Load side erate Load side erate Load side direction speed regulation and will produce large overshoot when the servo motor continuously ac- Figure 2: ,e influence of backlash while the servo motor rotated celerates and decelerates. with continuous acceleration and deceleration. 3. Design of the Nonlinear Compound controller, which usually adopted the PI method with simple structure and easy implementation. ,e mold vibration Controller displacement control is completed by mechanical trans- ,e control purpose is to realize the tracking control of mold mission parts following the servo motor speed. ,e overall vibration displacement. But the elimination of vibration dis- block diagram is shown in Figure 3. placement tracking error is realized by adjusting the motor n∗ In order to realize the mold nonsinusoidal vibration, speed. In this paper, the servo motor is taken as the control can be calculated by expected mold displacement. For ex- object and the controller is mainly of two parts: one is the ample, the mold nonsinusoidal vibration waveform uses the mathematical algorithm, which constructs the functional re- Demark nonsinusoidal equation: lationship between the mold displacement and the motor speed S∗ � h sin(ωt − A sin(ωt)). (3) directly. Based on the algorithm, an observer is designed to estimate the uncertainties and feedforward compensation. ,e ,e corresponding angular velocity of the eccentric shaft is other is the fuzzy control with bidirectional parameter ad- justment. It can improve rapidity and stability in case of V(t) � ω(1 − A cos(ωt)), (4) continuous acceleration and deceleration for the servo motor. where ω � (2π/60)f, in which f is the nonsinusoidal vi- ,e block diagram is shown in Figure 5. bration frequency of the mold. A � πα/(2 sin((π/2) (1 + α))), where α is the wave slope, α ∈ [0, 1). ,en, n∗ can 3.1. Nonlinear Mathematical Algorithm. Let θ represent the be expressed as angular displacement of the eccentric shaft. ,en, t ∗ 60iV(t) θ � (1/(i + Δi)) � ( n ) t + d n � 0 2π /60 d . So it follows that (2π) (5) 2πnp θ_ � ω � , (6) 60iω(1 − A cos(ωt)) p � . 60i (2π) where ωp is the angular velocity of the eccentric shaft ,e expected mold nonsinusoidal vibration velocity corresponding to θ and np is defined as the motor speed waveform and the corresponding servo motor speed are corresponding to θ. In (6), it is a linear correspondence from shown in Figure 4. np to θ. Formula (2) can be expressed as According to Figure 4, the servo motor rotates in single S � h sin(θ). (7) direction (n > 0) and continuously accelerates and de- celerates to realize the mold nonsinusoidal vibration. According to the rotating vector method as shown in However, the following two problems need to be con- Figure 6, every point of the sine or cosine curve is one to one sidered during the controller design: correspondence with the reference circle’s position. So, it is 4 Mathematical Problems in Engineering

40 1000

20 900 800 0 700 –20 600 ( t )/(r/min) ∗

–40 n V ( t )/(mm/s) 500

–60 400

–80 300 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 t/s t/s

(a) (b)

Figure 4: Expected mold nonsinusoidal vibration velocity waveform and the corresponding motor speed (α � 0.24). (a) Mold nonsinusoidal vibration velocity waveform. (b) Servo motor speed waveform.

Fuzzy ,e time derivative of (7) is derived as Speed closed loop control S_ � hθ_ cos(θ). (9) ∆K ∆K P I d n∗ PI u Servo n 2 1 θ S Since the servo motor rotates continuously in one di- h sin(θ) +– controllor motor 60 (i + ∆i) s rection, then according to the rotation vector method, it can + be calculated that – np Mathematical Filter Q algorithm S_ sgn(S_) θ_ � ω � √������. (10) p h2 − S2 Figure 5: Block diagram of the compensation control scheme.

Case 2. |S| � h and sin θ � ±1.

S S · · · S S > 0, S < 0 S > 0, S > 0 According to formula (4), the corresponding angular B A A B A B velocity can be obtained as follows: h θp ω � ω 1 − A cosωt ��, (11) x 0 t 0 t p0 0 · · t S < 0, S < 0 S < 0, S > 0 where 0 is the time when the nonsinusoidal vibration displacement curve reaches its peak and t0 is not unique. Figure 6: Corresponding relation between the sine (or cosine) Considering the motor single-direction rotation, ωp0 is a function and the reference circle. positive constant. Only considering the time when the vi- bration displacement reaches the peak upward in this paper, concluded that a point of the reference circle corresponds to it follows that a position of the mold, which can avoid the multisolution of π sine or cosine function in the single cycle. ωt − A sinωt � − � 0. (12) 0 0 2 Theorem 1. As to (6) and (7), according to the rotating ,e wave slope α is defined as vector method, the functional relationship between the ec- 4t 4t − (T/4) � 4t 2ωt centric shaft angular velocity ωp and the mold displacement S α � m � 0 � 0 − 1 � 0 − 1, (13) can be expressed as T T T π _ θp � ωp where T is the vibration cycle and tm is the lagging time, − . . which is the time difference to the peak displacement be- � ω +|S_|z 0 5�sgnz � + z0 51 − sgnz � �� − sgnz �ω , p0 r r r r r p0 tween nonsinusoidal waveform and sinusoidal waveform. (8) ,en, it follows that 2 2 πα πα where zr � h − S and ωp0 � ω0(1 + (πα/2)tan(πα/2)) is A � � . (14) constant, α ∈ [0, 1). 2 sin((π/2)(1 + α)) 2 cos(πα/2) From (13) and (14), one has Proof. □ πα π πα A cos� � � A cos�ωt − � � A sinωt � � . (15) Case 1. |S| ≠ h and sin θ ≠ ±1. 2 0 2 0 2 Mathematical Problems in Engineering 5

( t ) � ( A) ( t ) � Speed closed loop ������������From (15), sin ω 0 πα/2 and cos ω 0 d 2 n∗+ PI u Servo n 2 1 θ S − 1 − (πα/2A) . ,en, h sin(θ) ����������� + – controllor motor 60 (i + ∆i) s πα 2 2 ( ) A cosωt0 � � − A − � � . 16 + 2 – np Mathematical Filter Q algorithm Substituting (14) and (16) into (11), it follows that e disturbance observer ���������� (πα)2 Figure 7: Block diagram of the disturbance observer. ω � ω 1 − A cosωt �� � ω⎛⎝1 + A2 − ⎞⎠ p0 0 4 ��������������� Kp (n – 1) πα 1 (17) ⎛⎝ ⎞⎠ ∆K � ω 1 + e P Kp (n) 2 ()cos2(πα/2) − 1 Ke Kpu Fuzzy πα πα rules � ω�1 + tan� ��. ec ∆KI K (n) 2 2 Kec Kiu i

,erefore, when S reaches the peak, ωp0 is related to ω Gain Ki (n – 1) and α, which are the same to the given signal. coefficient From (10) and (17), it follows that

⎪⎧ S_ sgn(S_) Figure 8: Structure diagram of the fuzzy controller. ⎨⎪ √������ , |S| ≠ h, sin θ ≠ ±1, _ h2 − S2 p θp � ωp � ⎪ (18) ⎪ 3.2. Fuzzy PI Controller. As the servo motor rotates in a ⎩ , |S| � h, � . ωp0 sin θp ±1 single direction and continuously accelerates and decelerates to realize the mold nonsinusoidal vibration, new fuzzy rules Define z � h2 − S2 and convert equation (18) into an r with bidirectional speed regulation are adopted for the PI equation expressed as follows: self-tuning to improve rapidity and stability of the servo _ θp � ωp motor speed control ,e structure diagram of the fuzzy − . . controller is shown in Figure 8. � ω +|S_|z 0 5�sgnz � + z0 51 − sgnz � �� − sgnz �ω , p0 r r r r r p0 ,e inputs of fuzzy controller are speed error e and error (19) rate ec, and the outputs are the adjustment of PI control parameters ΔKP and ΔKI. ,e final PI parameters are where ωp0 � ω(1 + (πα/2)tan(πα/2)) is constant and only related to ω and α, α ∈ [0, 1). ,is completes the proof. ⎧⎨ K � K′P + ΔK , P P (22) Based on ,eorem 1 and formula (6), np can be calculated as ⎩ KI � KI′ + ΔKI, − 1 np � N (S) where K′P and KI′ are initial setting values of proportion and 60i integral coefficient, respectively. � �ω +|S_|z− 0.5�sgnz � + z0.51 − sgnz � �� − sgnz �ω �. 2π p0 r r r r r p0 As to input variables and output values, seven linguistic (20) 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. ,e Remark 1. If S and S are known, the corresponding servo normalized domain corresponds to {− 3, − 2, − 1, 0, 1, 2, and 3}. motor speed can be calculated based on equation (20). ,e amplitude of the mold displacement is 3 mm, and the error e is mainly in [− 3, 3]. So the range of e is set as To mitigate the influence of disturbance in the trans- [− 3, 3] (if the actual value exceeds the range, take the mission mechanism part, a disturbance observer is designed limiting value). According to the mold vibration model, the based on the mathematical algorithm as shown in Figure 7. ec range can be measured as [− 80, 80]. ,e quantitative np is the motor speed corresponding to the actual mold factors of e and ec are 1 and 0.0375, respectively. displacement S. ,e difference between np and n can be According to the experience of PI parameter setting, it regarded as the effect of the backlash and other factors on the should be adjusted quickly when the error is large and it speed regulation system and can be compensated through requires fine adjustment when the error is small. ,erefore, the feedforward. Q is a low-pass filter. ,e forward channel is a membership function should be close to the equilibrium point high gain to suppress the influence of disturbance on the and sparse on both sides of the fuzzy domain, which will output. Q is adopted as increase the amount of program implemented [18, 19]. In this 3(λs + 1) paper, the gain coefficient [20] is introduced to adjust the Q(s) � , (21) parameters based on the symmetric membership function. (λs)3 + 3(λs)2 + 3λs + 1 ,e gain coefficients of ΔKP and ΔKI are β1 and β2, where λ is the time constant and s is the complex parameter. respectively, and the setting principle is as follows: 6 Mathematical Problems in Engineering

Table (1) When |e| is large, take larger β1 and β2 to speed up 1: Fuzzy control rules of ΔKP. the response speed. e ec (2) When |e| is medium large, reduce β1 and β2 to make NB NM NS ZO PS PM PB the change of proportion and integral coefficient not NB PB NS ZO PS NM NB PS be dramatic. NM PM NM ZO PS NM NB PS (3) When |e| is small, in the case of ec larger, appro- NS PM NM ZO PM NS NM PM priately increase β1 and reduce β2 to prevent KI from ZO PS NB NS PB NS NS PM increasing too fast to produce overshoot. In the case PS PS NB NS PM ZO ZO PB PM PS NB NM PS ZO PS PB of ec being medium or small, increase β2 to improve the integral effect. PB PS NB NM PS ZO PS PB In this paper, the new fuzzy control rules with bi- directional output adjustment are adopted to improve the Table K responsiveness. With the rotational speed error as the main 2: Fuzzy control rules of Δ I. criterion, KP and KI can be further refined as follows: e ec (1) When |e| is large, the tracking requirement can be NB NM NS ZO PS PM PB achieved only through the proportional component NB NB NM ZO PB PB ZO NM due to the large error. Even if the integral is in- NM NB NM ZO PB PM ZO NM troduced, the response cannot be further improved, NS NB NS PS PB PM ZO NB ZO NB NS PS PB PS ZO NB while the overshoot may appear. So KP can be larger and K can be zero. PS NB ZO PM PB PS ZO NB I PM NB ZO PM PB ZO NS NB (2) When |e| is a medium value, take smaller KP to PB NB ZO PB PB ZO NS NB reduce overshoot. At the same time, add KI slowly to converge the static difference that cannot be elimi- nated by the proportional and avoid the integral Table saturation. 3: Parameters of the servo motor. Parameters of the servo motor Value (3) When |e| is small, increasing KP and KI is beneficial to improve the system steady state performance and Rated power (kW) 20.4 enhance the antidisturbance ability. Rated current (A) 45 Rated speed (r/min) 1500 ,e effect of the time-varying load and the friction should Stator inductance (mH) 4.6 be considered in the fuzzy control rules. ,e direction of the Stator resistance (Ω) 0.14 load and friction is always opposite to the motor. In the Number of pole pairs 3 acceleration phase, the friction force and load torque will Rotor flux linkage (Wb) 0.704 decrease |ec|, while in the deceleration phase it will increase Rotor inertia (kg·m2) 0.1110 |ec|. ,e larger |ec| is, the easier it is to cause overshoot. ,erefore, the fuzzy control rules are adjusted as follows: 4. Simulation Research (1) Reducing the lower limit of KP in the deceleration section could decrease the increasing rate of KP. It Some simulations are used to illustrate the effectiveness of could also reduce the torque of the motor when |e| is the proposed control method. a medium or small value, so as to reduce overshoot ,e mechanical transmission part parameters are as and make the control effect close to the acceleration follows: reduction ratio: i � 5; amplitude of the mold: section. h � 3mm. ,e nominal values of the servo motor parameters (2) ,e rules for K are basically the same as that in the (Table 3) are selected as follows [4]. I ,e servo motor’s speed loop and current loop use the acceleration section. In the deceleration section, KI increases from zero and the value after the system same set of parameters: Kpn � 3.894 A·s/rad, τn � 43.2 ms, deceleration overshoot is much larger than that Kpiq � 12.982 V/A, τiq � 2 ms, Kpid � 12.982 V/A, and before the overshoot. So the integral output can τid � 2 ms. In filter Q, λ � 0.002. quickly change from negative to positive and reach ,e control purpose is to realize the tracking control of the required value when the speed is stable, thus mold vibration displacement, and the servo motor is taken as reducing the overshoot. the controller object. ,e expected signal for the mold is taken as Demark nonsinusoidal function: S∗ � h sin According to the above rules and combined with the (ωt − A sin(ωt)). ,e corresponding given servo motor speed ∗ actual situation of the control system, ΔKP and ΔKI fuzzy is n (t) � 60iω(1 − A cos(ωt))/(2π), where ω � 2π f/60. In rules are shown in Tables 1 and 2. this paper, h � 3 mm, α � 0.24, and f � 130 times/min. ,e weighted average method is often used in ambiguity ,e load disturbance is adopted as follows [6]: resolution to obtain the modified parameter values ΔKP and T � (5.1335 + 6.4985 sin(ωt − A sin ωt)) Nm. (23) ΔKI. L Mathematical Problems in Engineering 7

0.3 3 4 2.8 2.6 0.2 3 0.55 0.6 0.65 2 0.1

1 0 0 –0.1

–1 (mm) error Tracking Tracking curves (mm) Tracking –0.2 –2

–3 0 0.5 1 1.5 2 2.5 3 0 0.5 1 1.5 2 2.5 3 t/s t/s Expected signal Tracking without compensation Tracking without compensation Tracking with compensation Tracking with compensation (a) (b)

Figure 9: (a) Tracking performance with and without compensation. (b) Tracking errors with and without compensation.

1000 ×10–3 6

800 5 4 600 3

400 2 1 200 0 Mapping motor speed (rad/s) motor Mapping

0 –1 0 0.5 1 1.5 2 2.5 3 (rad/s) error speed tracking Motor –2 t/s 0 0.5 1 1.5 2 2.5 3 3.5 4 Figure 10: Mapping function between motor speed and mold t/s vibration displacement. Fuzzy PI control PI control

Figure 12: Comparison of servo motor speed tracking errors of the proposed fuzzy PI control and the traditional PI control. 1000 900 According to common requirements of machining 800 precision ±3%, it takes the worst situation: Δi � 3%i. ,e initial zero offset of the eccentric shaft is d � − 0.2 rad. 700 To show the effectiveness of the proposed controller, the 600 simulation is conducted under two cases: with and without nonlinear feedforward compensation. ,e simulation results 500 Motor speed (rad/s) Motor are illustrated in Figures 9–12. Figure 9 clearly shows that the 400 tracking error when the nonlinear feedforward compensa- tion applied is much better than the case without the 300 0 0.5 1 1.5 2 2.5 3 compensation. It should be emphasized that Figure 9 suf- t/s ficiently illustrates the validity of the proposed control scheme because of the precisely tracking performance and Given motor speed small tracking error. Motor speed with feedforward compensation Based on the nonlinear mathematical algorithm shown Figure 11: Curves of servo motor speed with feedforward as equation (20), the mapping between the mold displace- compensation. ment and the servo motor speed is constructed to meet the 8 Mathematical Problems in Engineering technological requirements for the motor rotated in single servomotors,” Journal of Iron and Steel Research In- direction and unique mapping. Figure 10 is the mapping ternational, vol. 24, no. 3, pp. 251–257, 2017. curve and illustrates the algorithm effective. [3] L. Liu, Y. R. Dun, Y. M. Fang, and G. Y. Li, “Modeling and Figure 11 shows the given motor speed and the actual verification of the nonlinear system of oscillation platform of motor speed with feedforward compensation, respectively. continuous casting mold driven by servo motor,” Advances in ,e difference between the actual motor speed and the given Mechanical Engineering, vol. 8, no. 7, pp. 1–9, 2016. speed is used to compensate the uncertainties caused by the [4] Y. M. Fang, G. Y. Li, J. X. Li, and L. Liu, “,e model and analysis for displacement system of the continuous casting backlash and other factors in mechanical transmission parts. mold driven by servo motor,” Chinese Journal of Scientific Figure 12 illustrates the servo motor speed tracking Instrument, vol. 35, no. 11, pp. 2615–2632, 2014. errors of the proposed fuzzy PI control scheme and the [5] H. C. Zheng, L. Liu, Y. M. Fang, and Q. Li, “Nonlinear auto traditional PI control. It should be noted that the traditional disturbance rejection control for vibration displacement PI method has a large overshoot than the proposed control system of continuous cast mold driven by servo motor,” in scheme when the motor changes direction. Proceedings of the 35th Chinese Control Conference, CCC2016, pp. 879–884, Chengdu, China, July 2016. 5. Conclusion [6] K. S. Kang, L. Liu, Y. M. Fang, and H. C. Zheng, “Backstepping sliding mode control for continuous cast mold oscillation In this paper, a fuzzy PI control strategy based on nonlinear displacement system driven by servo motor,” Control Ceory feedforward compensation is proposed for the mold dis- & Applications, vol. 33, no. 11, pp. 1442–1448, 2016. placement system. Firstly, a mathematical algorithm is [7] G. Tao, X. Ma, and Y. Ling, “Optimal and nonlinear proposed to build the relationship between mold displace- decoupling control of systems with sandwiched backlash,” ment and servo motor speed. Based on the algorithm, an Automatica, vol. 37, no. 2, pp. 165–176, 2001. observer is designed to estimate the disturbance and feed- [8] A. Taware and G. Tao, “An adaptive dead-zone inverse forward compensation. Secondly, a fuzzy control method controller for systems with sandwiched dead-zones,” In- ternational Journal of Control, vol. 76, no. 8, pp. 755–769, with bidirectional parameter adjustment is adopted com- 2003. pound with the PI control method to improve rapidity and [9] X. Sun and L. F. Zhou, “Discrete-time adaptive control of stability of the servo motor speed control. Finally, the linear system with backlash nonlinearity,” Control Ceory & simulation results show the proposed method is effective. Applications, vol. 8, no. 1, pp. 96–100, 1991. ,e mathematical algorithm is mainly based on the [10] J. Guo, B. Yao, Q. W. Chen, and J. Jiang, “Adaptive robust Demark nonsinusoidal function in this paper. Future works control for nonlinear system with input backlash or backlash- will focus on the algorithm applying to other nonsinusoidal like hysteresis,” in Proceedings of the IEEE International vibration displacement waveforms, such as piecewise Conference on Control and Automation, pp. 1962–1967, IEEE, waveform and composite waveform. Christchurch, New Zealand, December 2009. [11] C. Y. Su, M. Oya, and H. Hong, “Stable adaptive fuzzy control Data Availability of nonlinear systems preceded by unknown backlash-like hysteresis,” IEEE Transactions on Fuzzy Systems, vol. 11, no. 1, ,e data used to support the findings of this study are in- pp. 1–8, 2003. cluded within the article. [12] C. Wen and J. Zhou, “Decentralized adaptive stabilization in the presence of unknown backlash-like hysteresis,” Auto- matica, vol. 43, no. 3, pp. 426–440, 2007. Conflicts of Interest [13] X. Y. Zhang, Y. Wang, X. K. 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[19] S. B. Cai, S. L. Wu, and F. J. Bao, “Position tracking control of PMSM based on fuzzy PID- variable structure adaptive control,” Mathematical Problems in Engineering, vol. 2013, Article ID 375483, 2013. [20] H. Peng, J. Z. Wang, W. Shen, and D. Y. Li, “Double fuzzy control with compensating factor for electronic-hydraulic servovalve-controlled system,” Journal of Mechanical Engi- neering, vol. 53, no. 24, pp. 184–192, 2017. Advances in Advances in Journal of The Scientifc Journal of Operations Research Decision Sciences Applied Mathematics World Journal Probability and Statistics Hindawi Hindawi Hindawi Hindawi Publishing Corporation Hindawi www.hindawi.com Volume 2018 www.hindawi.com Volume 2018 www.hindawi.com Volume 2018 http://www.hindawi.comwww.hindawi.com Volume 20182013 www.hindawi.com Volume 2018

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