energies Article
Speed Synchronous Control of Multiple Permanent Magnet Synchronous Motors Based on an Improved Cross-Coupling Structure Wei Chen *
ID
, Jiaojiao Liang
ID
and Tingna Shi
School of Electrical and Information Engineering, Tianjin University, Tianjin 300072, China;
[email protected] (J.L.);
[email protected] (T.S.) * Correspondence:
[email protected]; Tel.: +86-022-2740-2325 Received: 1 January 2018; Accepted: 23 January 2018; Published: 24 January 2018
Abstract: Regarding the shortcomings of the cross-coupling control structure during the start-up of a multi-motor with load—namely, a large synchronization error and a long start-up time—this paper proposes a fuzzy self-adjusting cross-coupling control structure. This structure combines a fuzzy self-adjusting filter and an advanced synchronization compensator. The fuzzy self-adjusting filter adjusts the “softened speed”, a newly established concept, so that each motor follows the trajectory of the softened speed during start-up, thus effectively reducing the synchronization error of the starting process. The advanced synchronization compensator is added to shorten the adjusting time of the motors. In addition, this paper analyzes the synchronization performance of the structure when the steady state is interrupted by a sudden step of load. Finally, this paper establishes an experimental platform for a synchronous speed control system for a permanent magnet synchronous motor, and verifies the effectiveness of the proposed structure and the correctness of the theoretical analysis through performing experiments. Keywords: fuzzy self-adjusting filter model; softened speed; advanced synchronization compensator; cross-coupling control; speed synchronous control
1. Introduction With the development of modern industry, the synchronous control of multi-motors has been widely used in robotics, electric vehicles, steel rolling, papermaking, and other fields [1–8]. Shortening the start-up time and improving the synchronization performance of multi-motors are of both theoretical and practical significance for enhancing the control accuracy and stability of the system [9–15]. A virtual line-shafting control and a cross-coupling control structure are usually adopted in the traditional multi-motor synchronous control [16–20]. Anderson R.G. et al. [21] proposed the concept of “electronic virtual line-shafting”. The sum of electromagnetic torque from the servo motors was taken as the load torque of the main motor, which can reduce the synchronization error caused by the starting delay, but the starting time was longer. Therefore, it’s not applicable to the occasions that require a high starting performance. Then, Koren Y. et al. presented a cross-coupling control structure, where the speed difference between two motors was taken as a feedback. This structure could better reflect the load changes of each motor [22], so, it exhibits good synchronization performance in the steady-state operation. In order to improve the synchronization performance and tracking performance of a multi-motor system under sudden changes of load during steady operation, Gu X., Zhang H. and Liu G. et al. presented [23–25] many advanced control methods, such as a neural network, sliding mode control, and internal model control are applied to the cross-coupling control. However, the
Energies 2018, 11, 282; doi:10.3390/en11020282
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coupling control. However, the above methods still have the shortcomings of a large synchronization above methods havetime the shortcomings of a large errorsystem. and a long starting time error and a longstill starting during the start-up of asynchronization loaded multi-motor during the start-up of a loaded multi-motor system. Aiming at the aforementioned problems, this paper presents a fuzzy self-adjusting cross-coupling Aiming at the that aforementioned this paper presents a fuzzy cross-coupling control structure combines aproblems, fuzzy self-adjusting filter with anself-adjusting advanced synchronization control structure that combines a fuzzy self-adjusting filter with advanced compensator. The concept of “softened speed” is put forward in this newanstructure. Thesynchronization fuzzy controller compensator. The concept of “softened speed” is put according forward intothis fuzzy adjusts the softened coefficient of each motor smoothly the new load structure. torque andThe the actual controller adjusts the softened coefficient of each smoothly according toInthe load torque and can the given speed; therefore, the softened speed can motor be adjusted automatically. this way, motors actual speed; therefore, the softened adjusted In this way,and motors follow given the trajectory of the softened speed speed duringcan thebe start-up of aautomatically. multi-motor with load, thus can follow trajectory of the softened theadvanced start-up of a multi-motor compensator with load, and reduce the the synchronization error. At thespeed same during time, the synchronization is thus reduce the synchronization error. At characteristic the same time,and the obtain advanced synchronization compensator is put forward to utilize the phase advance the phase advance, thus shortening put forward totime utilize themotors. phase advance and platform obtain theofphase advance, thus shortening the adjusting of the Finally, characteristic the experimental the synchronous speed control the adjusting time of the Finally, theofexperimental platform ofcross-coupling the synchronous speed control system is established, andmotors. the effectiveness the fuzzy self-adjusting control structure system is established, the effectiveness of the fuzzy self-adjusting cross-coupling control structure proposed in this paperand is verified through experiments. proposed in this paper is verified through experiments. 2. Traditional Cross-Coupling Control 2. Traditional Cross-Coupling Control The control method, which adopts speed difference as a compensation signal in multi-motor The control method, adopts speed difference as a compensation signal multi-motor synchronous control, holdswhich high reliability [26]. The speed difference between the two in motors is taken synchronous control, holdsinhigh reliability [26]. The speed difference between thecompensation two motors is signal taken as a compensation signal the traditional cross-coupling control method. The as a compensation signal in the traditional cross-coupling control method. The compensation signal can can reduce the speed error of the two motors when the motor is disturbed internally or externally [27,28]. reduce the speed error of the two motors when the motor is disturbed internally or externally [27,28]. The motion equations of a permanent magnet synchronous motor (PMSM) are taken to describe the The motion equations of a permanent motor (PMSM) taken to[29]: describe the dynamic and static performance of twomagnet motorssynchronous in the cross-coupling controlare structure dynamic and static performance of two motors in the cross-coupling control structure [29]: ( J dω Te TL Bω J dω (1) dt d=t Te − TL − Bω (1) TeT= KTKiTqiq e
where JJ is motor; ω isω the rotor angular velocity of the Te is the where is the themoment momentofofinertia inertiaofofthe the motor; is the rotor angular velocity ofmotor; the motor; Te electromagnetic torque; T L is the load torque; K T is the torque coefficient; and B is the damping is the electromagnetic torque; TL is the load torque; KT is the torque coefficient; and B is the coefficient. damping coefficient. For the the convenience convenience of of analysis, analysis, the the motor motor is is usually usually equivalent equivalent to to an For an integral integral part, part, disregarding disregarding the delay of the current loop and speed measurement. The block diagram of the traditional the delay of the current loop and speed measurement. The block diagram of the traditional cross-coupling control structure is shown in Figure 1. cross-coupling control structure is shown in Figure 1.
TL1 e1
ωref
F1
M1 ω1
Te1
β1 K M2
β2 e2
F2
ω2
Te2 TL2
Figure cross-coupling control control structure. structure. Figure 1. 1. Traditional Traditional cross-coupling
In Figure 1, ωref is the given speed of two motors; TLi (i = 1, 2) is the load torque of the ith motor; In Figure 1, ω ref is the given speed of two motors; TLi (i = 1, 2) is the load torque of the ith motor; ωi is the output speed of the ith motor; and the equivalent transfer function of the ith motor Mi is ω i is the output speed of the ith motor; and the equivalent transfer function of the ith motor Mi is given by: given by: 11 Gi G (si)( s= (2) ) (2) JiJsi s
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where FFii isisthe theiith motor speed speed loop loop controller; controller; the the structure structure is is shown shown in in Figure Figure 2. 2. where th motor ωref ωi
Δωi
ei
KPi
Tui
Tei
βi KIi/s
id
PI iq
Speed loop controller Rotary encoder
Uα dq SVPWM αβ Uβ
PI
θ
id
iα
dq αβ iβ
UDC αβ abc
Inverter ia
Ua Ub Uc
ib
PMSM Figure Figure2.2.Control Controlsystem systemof ofdouble doubleclosed-loop closed-looppermanent permanentmagnet magnetsynchronous synchronous motor motor (PMSM). (PMSM).
In Figure 2, Tui is the amplitude-unlimited electromagnetic torque output of the ith motor; Tei is In Figure 2, Tui is the amplitude-unlimited electromagnetic torque output of the ith motor; Tei is the amplitude-limited electromagnetic torque output of the ith motor; and ei is the input of speed the amplitude-limited electromagnetic torque output of the ith motor; and ei is the input of speed loop loop controller of the ith motor, which can be expressed as: controller of the ith motor, which can be expressed as: ei Δωi βi (3) ei = ∆ωi − β i (3) where Δωi is the tracking error of the ith motor; and βi is the compensation value. βi is given by: where ∆ω i is the tracking error of the ith motor; and βi is the compensation value. βi is given by: βi K(ωi ωj ) i j , i 1,2 (4) β i = K (ωcoefficient 6= j, i = 1,two 2 motors. (4) i − ω j ) i between where Kij is the synchronous compensation Taking the ith motor as an example, the speed loop controller Fi can be expressed as: where Kij is the synchronous compensation coefficient between two motors. Taking the ith motor as an example, the speed loopKIcontroller Fi can be expressed as: Fi KPi i (5) s KIi Fi = KPi + (5) In Equation (5), the proportional coefficient KPi and sthe integral coefficient KIi need to be adjusted according to the dynamic and static performance of the speed loop. KPi and KIi are shown in Equation (6). In Equation (5), the proportional coefficient KPi and the integral coefficient KIi need to be adjusted according to the dynamic and static performance KPi fcof Ji the speed loop. KPi and KIi are shown in Equation (6). (6) ( K ( fc )2 J KPiIi = f2c ςJi i (6) f 2 KIi = ( 2ςc ) Ji where fc and ζ are the bandwidth and damping coefficient of the speed loop, respectively. Each motor where andabove ζ are method the bandwidth and damping coefficient of the speed loop, respectively. Each motor is set byf cthe [30], hence: is set by the above method [30], hence: f ( f )2 Fi ( s)·Gi ( s) F( s)·G( s) c c 22 (7) fsc (2(ςsf c)) Fi (s)· Gi (s) = F (s)· G (s) = + (7) s (2ςs)2 Which means that each motor has the same open-loop transfer function, independently of the moment of inertia. Which means that each motor has the same open-loop transfer function, independently of the There are some shortcomings in a traditional cross-coupling control structure during the startmoment of inertia. up ofThere a multi-motor load: are some with shortcomings in a traditional cross-coupling control structure during the start-up of a multi-motor with load: 1. In Figure 2, the tracking error Δωi is much larger than the compensation value βi, according to 1.
2.
large∆ω during the start-up of a multi-motor with load, thus making the Equation andtracking ei is veryerror In Figure(3), 2, the i is much larger than the compensation value βi , according to much largerwith thanload, the saturation value. amplitude-unlimited output of electromagnetic torque Equation (3), and ei is very large during the start-up of Tauimulti-motor thus making the However, the output of the speed loop controller generally contains limiting part because of amplitude-unlimited output of electromagnetic torque Tui much largera than the saturation value. ei of the speed loop controller will be saturated the system’s safety requirements. The output T However, the output of the speed loop controller generally contains a limiting part because of the for a period of time, during which compensation work, eventually leading for to system’s safety requirements. Thethe output Tei of the value speeddoes loopnot controller will be saturated aalarger periodsynchronization of time, duringerror. which the compensation value does not work, eventually leading to a When the speed fluctuation of the system is large, the fixed gain compensation can’t be adjusted larger synchronization error. according to the disturbance of each motor in real time; this results in a longer adjusting time.
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When the speed fluctuation of the system is large, the fixed gain compensation can’t be adjusted 4 of 16 of each motor in real time; this results in a longer adjusting time.
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3. Fuzzy Self-Adjusting Cross-Coupling Control 3. Fuzzy Self-Adjusting Cross-Coupling Control Aiming at the problem of the cross-coupling control structure in the start-up of a multi-motor Aiming at the problem of the cross-coupling control structure in the of athe multi-motor adds fuzzy self-adjusting control module, andstart-up introduces advanced system with load, this paper adds aa fuzzy self-adjusting control module, and introduces advanced system with load, this paper adds a fuzzy self-adjusting control module, and introduces the advanced synchronous compensator into the traditional cross-coupling control structure. The block diagram of synchronous compensator into the traditional cross-coupling control structure. The block diagram of the two motors is shown in Figure 3. the two motors is shown in Figure 3.
e1 e1 * ωref Fuzzy self-adjusting ωref * ωref Fuzzy ωref self-adjusting control module control module
β1 β1 β2 β2 e2 e2
F1 F1
Te1 Te1
TL1 TL1
M1 M1
ω1 ω1
Advanced synchronization compensator Advanced synchronization compensator F2 F2
M2 M2
Te2 Te2
ω2 ω2
TL2 TL2
Figure Figure 3. 3. Fuzzy Fuzzy self-adjusting self-adjusting cross-coupling cross-coupling control control structure. structure. Figure 3. Fuzzy self-adjusting cross-coupling control structure.
The fuzzy self-adjusting control module and advanced synchronization compensator module The self-adjusting control and advanced synchronization compensator module are The fuzzy fuzzy controlmodule module and advanced synchronization compensator module are discussed as self-adjusting follows. discussed as follows. are discussed as follows. 3.1. Fuzzy Self-Adjusting Control Module 3.1. 3.1. Fuzzy Fuzzy Self-Adjusting Self-Adjusting Control Control Module Module Since the tracking error is much larger than the compensation value in the cross-coupling control Since the error much larger value in control Sinceduring the tracking tracking error is isof much larger than than the the compensation compensation in the the cross-coupling cross-coupling control saturation for a long time, thus structure the start-up a multi-motor with load, Tei is invalue structure during the start-up of a multi-motor with load, T is in saturation for a long time, thus ei ei is in saturation for a long time, thus structure during the start-up of a multi-motor with load, T resulting in a greater synchronization error. In a fuzzy self-adjusting cross-coupling control structure, resulting in synchronization error. fuzzy self-adjusting self-adjusting cross-coupling cross-coupling control control structure, structure, resulting in aa greater greaterω* synchronization error. In In aa fuzzy ref is adjusted by the fuzzy self-adjusting control module, which makes it the softened speed the speed ω* is adjusted by the fuzzy self-adjusting control module, which makes it ref is adjusted by the fuzzy self-adjusting control module, which makes it the softened softened ω*ref ref smoothly. This makes the tracking error close to the output of the synchronous converge to ωspeed converge to ω smoothly. This makes makes the the tracking tracking error error close close to the output the synchronous ref smoothly. This the output of of synchronous converge to ω compensator inrefthe starting process, thus reducing the influence oftothe saturation of the the limiting part compensator in the starting process, thus reducing the influence of the saturation of the limiting part compensator in the starting process, thus reducing the influence of the saturation of the limiting part of the speed loop controller on the synchronization error. The fuzzy self-adjusting control module is of the speed loop controller on the synchronization error. The fuzzy self-adjusting control module of the speed loop4.controller on the synchronization error. The fuzzy self-adjusting control module is is shown in Figure shown shown in in Figure Figure 4. 4.
max(ωi) max(ωi) max(TLi) max(TLi) ωref ωref Given Given speed speed
Choose rules Fuzzy α Choose rules Fuzzy controller α Mode 1 controller α Mode 1 α Mode 2 ε Mode 2 ε Mode selector Mode selector
Filter Filter model model
ω*ref ω*ref Softened Softened speed speed
Fuzzy self-adjusting control module Fuzzy self-adjusting control module Figure 4. Fuzzy-self-adjusting control module. Figure 4. Fuzzy-self-adjusting control module. Figure 4. Fuzzy-self-adjusting control module.
In Figure 4, the output α of the fuzzy controller is defined as the softened coefficient, and ε is In Figure 4, the output α of the fuzzy controller is defined the softened coefficient, and ε is defined as the steady state control coefficient. In this paper, ε = 1. as Namely: defined as the steady state control coefficient. In this paper, ε = 1. Namely: ωmax max ω1 , ω2 (8) ωmax max ω1 , ω2 (8) TLmax max TL1 , TL2 TLmax max TL1 , TL2 The mode selector is used to switch between modes. It divides the fuzzy controller into two The mode is used to switch between It, divides the fuzzy controller into two operating modesselector based on ωmax. Mode 1: when ωmax modes. < 0.98ωref it is considered that the motor is in a
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In Figure 4, the output α of the fuzzy controller is defined as the softened coefficient, and ε is defined as the steady state control coefficient. In this paper, ε = 1. Namely: (
ωmax = max{ω1 , ω2 } TLmax = max{ TL1 , TL2 }
(8)
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The mode selector is used to switch between modes. It divides the fuzzy controller into two operating modesThe based on controller ω max . Mode 1: the when ω max < 0.98ω motorfollows is in a ref , it is considered fast start stage. fuzzy and filter work simultaneously, so thatthat eachthe motor fast start stage. The fuzzy controller and the filter work simultaneously, so that each motor follows ref, the trajectory of the softened speed to reduce the synchronization error; Mode 2: when ωmax ≥ 0.98ωthe trajectory of the that softened speed gradually to reduce the synchronization error; Mode 2: when ω max the ≥ 0.98ω it is considered the motor enters into the steady state. In this case, only filterref is, it is considered that the motor gradually enters into the steady state. In this case, only the filter is activated, thereby improving the dynamic response of the system. activated, thereby improving the dynamic of the system. In order to make the softened speedresponse transition to the set value at a certain response speed In order make the filter softened transition the set value at a certain response speed smoothly, smoothly, thetofollowing linkspeed is usually takento[31,32]: the following filter link is usually taken [31,32]: * ωref αωref (1 α)ωmax (9) ∗ ωref = αωref + (1 − α)ωmax (9) where 0 < α t3, making each motor the in softened trajectory in a better manner. Then, comparing tfollow 2 working 1 is greater 2 the
, because the given speed
thanspeed that of periodthan in which is on the rise ① ②of , because the given of , ① is greater that ofω* ② period in the conditions ref, the 1 is under ω* condition
longer, t1 > t2 ,① ensuring thatthus each can better the trajectory of ref is on the rise underthus condition is longer, t1 motor > t2, ensuring thatfollow each motor can better which the softened speed. According to Equation since ω max = 0 at the (9), initial starting 0 at the the output initial follow the trajectory of the softened speed. (9), According to Equation since ωmax =time, value α of thethe fuzzy controller different when the given speed is different. (9), it can starting time, output value is α of the fuzzy controller is different when the From givenEquation speed is different. be seen that the(9), values the softened speeds are different at the initial time. From Equation it canofbe seen that the valuesofoftwo the motors softened speeds of two motors arestarting different at Therefore, adjusting theTherefore, softened coefficient according to different operating conditions, fuzzy the initial by starting time. by adjusting the softened coefficient according to the different self-adjusting control the module make each motor follow trajectory of the softened speed. operating conditions, fuzzycan self-adjusting controlbetter module can the make each motor better follow the Furthermore, thesoftened input ofspeed. the speed loop controller is reduced, so that thecontroller time of electromagnetic trajectory of the Furthermore, the input of the speed loop is reduced, so torque the saturation state is shortened. the purpose reducing the synchronization error that theintime of electromagnetic torque in Finally, the saturation state of is shortened. Finally, the purpose of during the start-up is achieved. reducing the synchronization error during the start-up is achieved.
1100
ωref*/(r/min)
900
①
③ ④
700
②
⑥
500
⑤ 300 t3
t2
t1
100 0
0.01
0.02
0.03
0.04
0.05
t/(s) Figure 8. Softened speed under different operating conditions. Figure 8. Softened speed under different operating conditions.
3.2. The Advanced Synchronization Compensator When a load of the motor changes greatly, the fixed gain synchronous compensator will give rise to an excessive fluctuation of speed, and it takes a long time for the fluctuation to be smoothed out. For each motor, the speed fluctuation of any other motor can be regarded as a time-varying interference. In order to eliminate the effect of interference on the output characteristic of the system, corrective control is employed to further improve the adjusting time. The synchronous compensator
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3.2. The Advanced Synchronization Compensator When a load of the motor changes greatly, the fixed gain synchronous compensator will give rise to an excessive fluctuation of speed, and it takes a long time for the fluctuation to be smoothed out. For each motor, the speed fluctuation of any other motor can be regarded as a time-varying interference. In order to eliminate the effect of interference on the output characteristic of the system, corrective control is employed to further improve the adjusting time. The synchronous compensator with anticipatory control shown in Figure 9. Energies 2018, 11, x FOR PEER is REVIEW 8 of 16
ω1 ω2
K
Fg Correction
βi
Advanced synchronization compensator Figure9.9.Advanced Advancedsynchronous synchronouscompensator. compensator. Figure
In Figure 9, K is the synchronization compensation coefficient; and Fg is the correction controller. In Figure 9, K is the synchronization compensation coefficient; and Fg is the correction controller. In order to accelerate the system’s dynamic response [37], this paper introduces the classic advanced In order to accelerate the system’s dynamic response [37], this paper introduces the classic advanced correction, namely: correction, namely: η · Ts +11 η·Ts Fg (sF) = (10) (10) g ( s) Ts + 1 Ts 1 where T is advanced time constant; and η is attenuation factor and η > 1. where T is advanced time constant; and η is attenuation factor and η > 1. The advanced compensation controller compensates the output speed of its own according to the The advanced compensation controller compensates the output speed of its own according to disturbance level of the other motor. The amount of advancement needed by the system is obtained by the disturbance level of the other motor. The amount of advancement needed by the system is using the phase advance characteristic of Fg , thus increasing the cut-off frequency, and finally reducing obtained by using the phase advance characteristic of Fg, thus increasing the cut-off frequency, and the influence of the disturbance. finally reducing the influence of the disturbance. The following analysis indicates the influence of the speed fluctuation of the second motor on The following analysis indicates the influence of the speed fluctuation of the second motor on the first motor in the fuzzy self-adjusting cross-coupling control structure. As shown in Figure 3, the the first motor in the fuzzy self-adjusting cross-coupling control structure. As shown in Figure 3, the output speed ω 2 of the second motor is taken as the input, and the output speed ω 1 of the first motor output speed ω2 of the second motor is taken as the input, and the output speed ω1 of the first motor is taken as the output. The transfer function in this case is: is taken as the output. The transfer function in this case is: KFKF (s) G (s) ω1 ( s ) g ( s )(F ω=( s) g s) F( s)G( s) ω2 (s) 1 1 + [1 + KFg (s)] F (s) G (s) ω2 ( s) 1 [1 KFg ( s)]F( s)G( s)
(11) (11)
A Bode diagram of phase-frequency characteristics and amplitude-frequency characteristics when A Bode diagram of phase-frequency characteristics and amplitude-frequency characteristics K = 2 is shown in Figure 10. In order to facilitate the comparison with the traditional cross-coupling when K = 2 is shown in Figure 10. In order to facilitate the comparison with the traditional control structure, the transfer function of the traditional cross-coupling structure is deduced as: cross-coupling control structure, the transfer function of the traditional cross-coupling structure is deduced as: ω1tra (s) KF (s) G (s) = (12) ω2tra (ωs) ( s)1 + (1 + K ) F (s) G (s) KF ( s ) G ( s ) 1tra (12) K )amplitude-frequency 1 F( s)G( s) Figure 10 also shows the Bode ωdiagram of(1the characteristics and 2tra ( s) phase-frequency characteristics of the traditional structure. As can be seen from the figure, the Figure 10 also shows the Bode diagram of the amplitude-frequency characteristics and fuzzy self-adjusting cross-coupling control structure increases the cut-off frequency and improves the phase-frequency characteristics of the traditional structure. As can be seen from the figure, the fuzzy system’s dynamic response compared with the traditional cross-coupling control structure. The phase self-adjusting cross-coupling control structure increases the cut-off frequency and improves the margin increases, and the stability of the system is improved. system’s dynamic response compared with the traditional cross-coupling control structure. The phase margin increases, and the stability of the system is improved.
Phase/(˚/(˚) ) Phase
Amplitude/dB /dB Amplitude
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0 0 20 20 40 40 60 60 0 0
Before correction Before correction After correction After correction
-30 -30 -60 -60 -90 -90
Before correction Before correction After correction After correction 1011 10
1022 10
1033 1044 10 10 f (rad/s) f (rad/s)
1055 10
1066 10
Figure 10. 10. Bode plots plots characteristic of of the controlled controlled object before before and after after correction. Figure Figure 10.Bode Bode plotscharacteristic characteristic ofthe the controlledobject object beforeand and aftercorrection. correction.
4. Performance Performance Analysis of of Fuzzy Self-adjusting Self-adjusting Cross-Coupling Control Control 4. 4. PerformanceAnalysis Analysis ofFuzzy Fuzzy Self-adjustingCross-Coupling Cross-Coupling Control
ei(r/min) ei(r/min)
4.1. Synchronization Synchronization Performance Analysis Analysis during the the Start-Up of of Motors with with Load 4.1. 4.1. SynchronizationPerformance Performance Analysisduring during theStart-Up Start-Up ofMotors Motors withLoad Load To simplify simplify the analysis, analysis, it is assumed that that theload load torqueof of thefirst first motorisisTTL1L1 >> 0, 0, and the the To To simplifythe the analysis,ititisisassumed assumed thatthe the loadtorque torque ofthe the firstmotor motor is TL1 > 0,and and the second motor starts without load. Therefore, for the output speeds of the first motor and the second, second speeds ofof thethe first motor and thethe second, it secondmotor motorstarts startswithout withoutload. load.Therefore, Therefore,for forthe theoutput output speeds first motor and second, 1 < ω 2.In In Figure 11, e iis is the input of the ith (i = 1,2) motor’s speed loop controller; and it holds that ω holds that ω < ω . Figure 11, e the input of the ith (i = 1,2) motor’s speed loop controller; and it holds that1ω1 < 2ω2. In Figure 11,i ei is the input of the ith (i = 1,2) motor’s speed loop controller; and is the tracking tracking error of of the ith ith motor. Setting Setting the rated rated torque as as TN, then, the the saturation value value Δωii is ∆ω the tracking error error ofthe the ithmotor. motor. Settingthe the ratedtorque torque asTTNN,,then, then, thesaturation saturation value Δωi is the N . equals 1.2T equals equals1.2T 1.2TNN..
e1 e1 Δω1 Δω1
900 900 600 600 300 300 0 0 0 0
0.01 0.02 0.03 0.04 0.05 0 0.01 0.02 0.03 0.04 0.05 0 t/s t/s
ei(r/min) ei(r/min)
(a) (a)
300 300 200 200 100 100 0 0 0 0
e1 e1 Δω1 Δω1
0.01 0.02 0.03 0.04 0.05 0 0.01 0.02 0.03 0.04 0.05 0 t/s t/s (b) (b)
e2 e2 Δω2 Δω2
0.01 0.02 0.03 0.04 0.05 0.01 0.02 0.03 0.04 0.05 t/s t/s
e2 e2 Δω2 Δω2
0.01 0.02 0.03 0.04 0.05 0.01 0.02 0.03 0.04 0.05 t/s t/s
Figure 11. Input e1 of the speed loop controller of the two structures: (a) e1 in the traditional crossFigure 11. Input e1 of the speed loop controller of the two structures: (a) e1 in the traditional crossFigure 11.control Inputstructure; e1 of the(b) speed loopfuzzy controller of the two structures: control (a) e1 structure. in the traditional self-adjusting cross-coupling coupling e1 in the coupling control structure; (b) e1 in the fuzzy self-adjusting cross-coupling control structure. cross-coupling control structure; (b) e1 in the fuzzy self-adjusting cross-coupling control structure.
It can be seen from Figure 11a that in the traditional cross-coupling control structure, for the load It can be seen from Figure 11a that in the traditional cross-coupling control structure, for the load 1, indicating the traditional compensation value β1 < 0 and e1 during the start-up motor, it holds thatfrom e1 > Δω It be seen Figure 11a thatthat in the cross-coupling control structure, for the 1, indicating that the compensation value β1 < 0 and e1 during the start-up motor,can it holds that e1 > Δω u1 of the controller corresponding to the input of the motors with load is large. As a result, the output T load motor, it with holdsload thatise1large. > ∆ωAs the compensation valuecorresponding β1 < 0 and e1 during the 1 , indicating to the input of the motors a result, thethat output Tu1 of the controller e1 is far greater than the saturation value, so Te1 is equal to 1.2 TN. The compensation value basically e1 is far greater than the saturation value, so Te1 is equal to 1.2 TN. The compensation value basically does not work. For the initial value of the no-load motor during starting, it holds that e2 > Δω2; does not work. For the initial value of the no-load motor during starting, it holds that e2 > Δω2;
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start-up of the motors with load is large. As a result, the output Tu1 of the controller corresponding to the input e1 is far greater than the saturation value, so Te1 is equal to 1.2 TN . The compensation value basically does not work. For the initial value of the no-load motor during starting, it holds that e2 > ∆ω 2 ; the compensation value still does not work. Only when β2 increases to a certain value does the compensation value begin to play a compensatory role. In Figure 11b, the load motor satisfies e1 < ∆ω 1 , indicating that the compensation value β1 > 0, and the advanced synchronous compensator compensates for the speed. The advanced synchronous compensator of the no-load motor also compensates for the speed. Regarding the early stages during the start-up, the controller of each motor is working in the nonlinear region. Ignoring the integral action of the controller, the input ei (i = 1, 2) of the speed loop controller with load in the starting process is: ∗ − ω − β = [ αω + (1 − α )(ω − ∆ω ) − ω ] − K ( ω − ω ) F ei = ∆ωi − β i = ωref i i i i i j g ref = α(ωref − ωi ) − (1 − α)∆ω − KFg ∆ω = α∆ωi − [KFg + (1 − α)]∆ω
(13)
Similarly, the input of the speed loop controller in the traditional cross-coupling control structure is: ei(tra) = ∆ωi − K∆ω (14) Comparing Equation (13) with (14), it should be noted that in the starting process of the fuzzy self-adjusting cross-coupling control structure, because 0 < α 1, the coefficient before the first term ∆ω i of Equation (13) is reduced; that is, the tracking proportional coefficient decreases. Further, the coefficient before the second ∆ω increases; that is, the synchronization proportional coefficient increases. In summary, in the multi-motor starting process, compared with the traditional cross-coupling control structure, the tracking error of the fuzzy self-adjusting cross-coupling control structure decreases, and the synchronous compensator output increases simultaneously. Hence, the input of the speed loop controller decreases. Therefore, the time for Tu1 to be saturated is shortened, and the synchronization error is thus reduced. When the controller of each motor works in the linear region, the difference of accelerations between two motors in the fuzzy self-adjusting cross-coupling control structure is: a1 − a2 = fc J1 [αωref +(1J−1 α)ωmax −ω1 ] − = f c (1 + K )(ω2 − ω1 )
K f c J1 (ω1 −ω2 ) J1
+
K f c J2 (ω2 −ω1 ) J2
−
f c J2 [αωref +(1−α)ωmax −ω2 ] J2
(15)
From Equation (15), it should be noted that the acceleration difference between two motors is independent of α; that is, changing the softened coefficient α will not influence the synchronization performance of the system in the working area of the linear zone. 4.2. Synchronization Performance Analysis of Steady State with Sudden Changes of Load In this section, the synchronization performance of the steady state with sudden changes of loads in a fuzzy self-adjusting cross-coupling control structure is analyzed. The ith motor output speed is: ωi =
ωref FG − TLi Gi + KFg FG (ω1 + ω2 ) 1 + (1 + 2KFg ) FG
(16)
Derived from Equation (16), the synchronization error between two motors can be expressed as: ∆ω = ω1 − ω2 =
TL2 G2 − TL1 G1 1 + (1 + 2KFg ) FG
(17)
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(18)
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From the above analysis, it should be noted that η > 1, therefore, 1 − Fg < 0. Compared with the traditional cross-coupling control structure, Δωmotors of a fuzzy self-adjusting Similarly, the synchronization error ofthe two in the traditional cross-coupling cross-coupling control control structure obviously; that is, the synchronization error between two motors in steady state structure decreases is: T G2 − TL1 G1 with sudden changes of load decreases. (18) ∆ωtra = ω1 − ω2 = L2 1 + (1 + 2K ) FG
5. Experiments and Analysis From the above analysis, it should be noted that η > 1, therefore, 1 − Fg < 0. Compared with the traditional cross-coupling control structure, the ∆ω of a fuzzy self-adjusting cross-coupling control 5.1. Experimental System and Parameter structure decreases obviously; that is,Selection the synchronization error between two motors in steady state withTo sudden changes of load decreases. verify the effectiveness of the fuzzy self-adjusting cross-coupling control structure, the PMSM speed control system experimental platform is constructed, as shown in Figure 12. The hardware 5. Experiments and Analysis system is mainly composed of two PMSMs. The PMSMs used in this paper are surface-mounted. One motor is connected to theand magnetic powder brake, and the other is connected to the load motor, which 5.1. Experimental System Parameter Selection is interlinked over a resistance box. The controlling circuit features mainly include a main control verifycircuit, the effectiveness of the fuzzy current self-adjusting cross-coupling control structure, the PMSM panel,Topower drive circuit, voltage, sensors, and so on. The DSP chip TMS320F28335 speed control system experimental platform is constructed, as shown in Figure 12. The hardware produced by TI is used as the core in the main control panel. The power devices use Mitsubishi’s IPM system is mainly composed of two PMSMs. The PMSMs used in this paper are surface-mounted. module PS21867, and the switching frequency is 5 kHz. The parameters of two motors are shown in One motor is connected to the magnetic powder brake, and the other is connected to the load motor, Table 3. which is interlinked over a resistance box. The controlling circuit features mainly include a main control panel, power circuit, drive circuit, voltage, current sensors, so on. The DSP(PMSMs). chip TMS320F28335 Table 3. Experiment parameters of permanent magnetand synchronous motors produced by TI is used as the core in the main control panel. The power devices use Mitsubishi’s IPM Motors TN (N·m) J (kg·m²) nNThe (r/min) p PNof(kW) module PS21867, and the switching frequency is 5 kHz. parameters two motors are shown in −3 M 15 2.72 × 10 1500 2 2.3 Table 3.
(a)
(b)
Figure Figure 12. 12. Experiment Experimentsystem: system:(a) (a)The Theexperimental experimentalplatform platformfor forthe the synchronous synchronous speed speed control control system; (b) System of PMSMs and the magnetic powder brake. system; (b) System of PMSMs and the magnetic powder brake.
5.2. Performance Comparison of Two Structures During themagnet Start-Up of Motorsmotors with Load Table 3. Experiment parameters of permanent synchronous (PMSMs). The experimental parameters are set as follows: initially, the two motors start at a given speed Motors T N (N·m) nN and (r/min) PN (kW) J (kg·m2 ) brake, of 500 r/min. Motor 1 is connected to the magnetic Motor 2pis connected to a load motor. − 3 M 1500 is no-load. 2 The rated 2.3 torque is TN, and 2.72 × the 10 load motor The torque of the magnetic brake15is 8 N·m, and the saturation value of the speed loop controller is set as 1.2 TN. A performance comparison of the synchronization error and output speed between the structures during start-up of motors 5.2. Performance Comparison of Two Structures During thetwo Start-Up of Motors withthe Load with load are shown in Figures 13 and 14. In the figure, ts is the adjusting time, and tp is the peak time. The experimental parameters are set as follows: initially, the two motors start at a given speed The electromagnetic torque outputs of the first and second motor during the start-up are shown in of 500 r/min. Motor 1 is connected to the magnetic brake, and Motor 2 is connected to a load motor. Figures 15 and 16, respectively. Performances of the traditional cross-coupling control structure The torque of the magnetic brake is 8 N·m, and the load motor is no-load. The rated torque is TN , and the saturation value of the speed loop controller is set as 1.2 TN . A performance comparison of the synchronization error and output speed between the two structures during the start-up of motors
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∆n [50r/min/div] [50r/min/div] ∆n∆n [50r/min/div]
(abbreviated as traditional structure hereafter) are shown in Figure 13a of every figures below, and (abbreviated asoftraditional structure hereafter) are shown in Figure 13a of(abbreviated every figuresasbelow, and performances the fuzzy self-adjusting cross-coupling control structure improved (abbreviated as traditional structure hereafter) are shown in Figure 13a of every figures below, and performances of the fuzzy self-adjusting cross-coupling control structure (abbreviated as improved with loadhereafter) are shown inshown Figures and 14. In figure, ts is the adjusting time, and tp is the peak structure in13 Figure 13b of the every figures below. performances of theare fuzzy self-adjusting cross-coupling control structure (abbreviated as improved structure hereafter) shown in 13, Figure 13b of every figures below. time.As The electromagnetic torque outputs of the firstsynchronization and second motor during the start-up are shown can be seen are from Figure the maximum structure hereafter) are shown in Figure 13b of every figures below. error of the improved structure As can be seen from Figure 13, the maximum synchronization error of thewith improved structure in Figures 15 and 16, respectively. Performances the traditional control structure reduces r/min 70 r/min, which is a of reduction of 55%cross-coupling compared the traditional As from can be157 seen fromtoFigure 13, the maximum synchronization error of the improved structure reduces from 157 r/min to 70 r/min, which is a reduction of 55% compared with the traditional (abbreviated as traditional structure hereafter) are shown in Figure 13a of every figures below, and structure. Meanwhile, the dynamic response of synchronization error increases by 43%. In Figure 14, reduces from 157 r/min to 70 r/min, which is a reduction of 55% compared with the traditional structure. Meanwhile, the dynamic response of synchronization error increases by 43%. In Figure 14, performances of the fuzzy self-adjusting cross-coupling control structure (abbreviated as improved the adjusting time reduces from 360 ms to 275 ms, which is a 23% reduction; and the peak time structure. Meanwhile, the dynamic response of synchronization error increases by 43%. In Figure 14, the adjusting time reduces from 360 ms to 275 ms, which is a 23% reduction; and the peak time structurebyhereafter) are shown in Figure 13b of every figures below. reduces 4.8%. the adjusting time reduces from 360 ms to 275 ms, which is a 23% reduction; and the peak time reduces by 4.8%. reduces by 4.8%. Improved Structure Traditional Structure Improved Structure Traditional Structure Improved Structure Traditional Structure 142ms 250ms 142ms 250ms 142ms 250ms
t [100ms/div] t [100ms/div] t [100ms/div] t [100ms/div] (a) (b) t [100ms/div] t [100ms/div] (a) (b) (a) error during the start-up of motors with load: (a) (b)Synchronization error Figure 13. Synchronization
210ms 210ms tp 210ms tp tp
Traditional Structure Traditional Structure Traditional Structure
n2 n2 n2
200ms 200ms p t200ms tp tp
n [100r/min/div] [100r/min/div] nn [100r/min/div]
n [100r/min/div] [100r/min/div] nn [100r/min/div]
Figure 13. Synchronization error during the start-up of motors with load: (a) Synchronization error in Figure 13. Synchronization error during the start-up of the motors with load: (a) Synchronization error in the traditional structure; Synchronization error improved structure. Figure 13. Synchronization error during the error start-up of motors with load: (a) Synchronization error the traditional structure; (b)(b) Synchronization in in the improved structure. in the traditional structure; (b) Synchronization error in the improved structure. in the traditional structure; (b) Synchronization error in the improved structure.
Improved Structure Improved Structure Improved Structure
n1 n2 n1 n1 n2 n1 n1 n2 n1 275ms 360ms 275ms 360ms ts ts 275ms 360ms ts ts tts[100ms/div] tst [100ms/div] t [100ms/div] t [100ms/div] ( a ) (b) t [100ms/div] t [100ms/div] (a) (b) (a)during the start-up of motors with load: (a) Output(bspeed ) Figure 14. Output speed in the traditional
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18N·m 18N·m 18N·m
T[6N·m/div] e1 [6N·m/div] Te1Te1 [6N·m/div]
Figure 14.(b) Output speed during start-up of motors with load: (a) Output speed in the traditional Figure speed during the start-up of structure; Output speed in thethe improved Figure 14. 14. Output Output speed during the start-upstructure. of motors motors with with load: load: (a) (a) Output Output speed speed in in the the traditional traditional structure; (b) Output speed in the improved structure. structure; structure; (b) (b) Output Output speed speed in in the the improved improved structure. structure.
140ms 140ms 140ms
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Improved Structure Improved Structure Improved Structure
III III III
t [100ms/div] t [100ms/div] t [100ms/div] t [100ms/div] ( a ) (b) t [100ms/div] t [100ms/div] (a) (b) (a) torque output of the first motor during the start-up (b) of motors with load: Figure 15. Electromagnetic
Figure 15. Electromagnetic torque output of the first structure; motor during the start-up of motors load: (a) Electromagnetic torque output in the traditional (b) Electromagnetic torque with output in Figure 15. Electromagnetic torque output of the first motor during the start-up of motors with load: Figure 15. Electromagnetic torque in output of the firststructure; motor during the start-up of motors load: (a) Electromagnetic torque output the traditional (b) Electromagnetic torque with output in the improved structure. (a) Electromagnetic Electromagnetictorque torqueoutput outputininthe the traditional structure; Electromagnetic torque output in (a) traditional structure; (b)(b) Electromagnetic torque output in the the improved structure. the improved structure. improved structure.
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III
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Te2 [6N·m/div]
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t [5ms/div] (b)
t [5ms/div] (a)
Figure 16. Electromagnetic torque output of the second motor during the start-up of motors with load:
Figure 16. Electromagnetic torque output of the second motor during the start-up of motors with load: (a) Electromagnetic torque output in the traditional structure; (b) Electromagnetic torque output in (a) Electromagnetic torque output in the traditional structure; (b) Electromagnetic torque output in the the improved structure. improved structure.
From the above experiments, it should be noted that the starting process of the motor is divided into threebestages. In Figures 15a13, and 16a, the electromagnetic torque outputs theimproved two motors are As can seen from Figure the maximum synchronization error ofof the structure saturated within 7 ms in the first stage, and the synchronous compensator does not work at all. In the reduces from 157 r/min to 70 r/min, which is a reduction of 55% compared with the traditional secondMeanwhile, stage, the electromagnetic torque output of Motor 1 is still in saturation. As 43%. a result of the 14, structure. the dynamic response of synchronization error increases by In Figure compensation value, Motor 2 enters the linear region, and its working time is 173 ms. In the the adjusting time reduces from 360 ms to 275 ms, which is a 23% reduction; and the peak timethird reduces phase, both of the motors run in the linear working region under the action of compensation. by 4.8%. In Figures 15b and 16b, namely, in the improved structure, the working time of the first stage is From the above experiments, it should be noted that the starting process of the motor is divided shortened by 6 ms. Since the control period of the system is 0.4 ms, it enters the linear work area into three stages. In Figures 15a and 16a, the electromagnetic torque outputs of the two motors are 15 times in advance; in stage II, the electromagnetic torque output of Motor 1 stays in saturation for saturated the first stage, compensator does not work at all. In the 139 ms,within which7ams 19%inreduction, and isand whythe thesynchronous synchronization error significantly reduces.
199ms
∆n [50r/min/div]
∆n [50r/min/div]
second stage, the electromagnetic torque output of Motor 1 is still in saturation. As a result of the compensation value, Motor 2 enters linear region, itswith working time is 173ofms. 5.3. Performance Comparison of Twothe Structures in Steadyand State Sudden Changes LoadIn the third phase, both of the motors run in the linear working region under the action of compensation. In Figures 15b The given speed is 400 r/min, and the two motors both start without load. A sudden load of and 10 16b, namely, in theto improved the working the first stageofiseach shortened N·m is applied Motor 1 structure, in the stable operation.time Theofoutput speed motor by and6 ms. Since the control period of the system is 0.4 ms, it enters the linear work area 15 times in advance; synchronization error of the two structures are shown in Figures 17 and 18, respectively. in stage II, the electromagnetic torque output of Motor 1 stays in saturation for 139 ms, which a 19% reduction, and is why the synchronization error significantly reduces. Improved Structure Traditional Structure 100ms
95r/min
128r/min
5.3. Performance Comparison of Two Structures in Steady State with Sudden Changes of Load The given speed is 400 r/min, and the two motors both start without load. A sudden load of 10 N·m is applied to Motor 1 in the stable operation. The output speed of each motor and synchronization error of the two structures are shown in Figures 17 and 18, respectively. As can be seen from Figure 17, the maximum synchronous error between two motors reduces from 128 r/min to 95 r/min, which is a decrease of 25.7%, compared with the traditional structure. t [100ms/div] t [100ms/div] The convergence speed of synchronization error increased by 49%. (a) (b) In Figure 18, the adjusting time of the speed is shortened by 39%, the tracking error is reduced from 74 r/min to 70 r/min—almost unchanged—and the response time of the tracking error reduces from 69 ms to 60 ms.
compensation value, Motor 2 enters the linear region, and its working time is 173 ms. In the third phase, both of the motors run in the linear working region under the action of compensation. In Figures 15b and 16b, namely, in the improved structure, the working time of the first stage is shortened by 6 ms. Since the control period of the system is 0.4 ms, it enters the linear work area 15 times in advance; in stage II, the electromagnetic torque output of Motor 1 stays in saturation for Energies 2018, 11, 282 14 of 16 139 ms, which a 19% reduction, and is why the synchronization error significantly reduces. 5.3. Performance Comparison of Two Structures in Steady State with Sudden Changes Loadimproves the To summarize, the improved structure proposed in this paper not ofonly synchronization performance and dynamic response twowithout motorsload. in theA starting The given speed is 400 r/min, and the two motorsbetween both start sudden process, load of it also the synchronization of motors The whenoutput the steady state interrupted by a 10 N·mimproves is applied to Motor 1 in performance the stable operation. speed of iseach motor and sudden change of load. synchronization error of the two structures are shown in Figures 17 and 18, respectively. Improved Structure 100ms
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Figure 17. Synchronization error of motors under sudden changes of a load during steady Figure 17. Synchronization error of motors under sudden changes of a load during steady operation: operation: (a) Synchronization error in the traditional structure; (b) Synchronization error in the (a) Synchronization error in the traditional structure; (b) Synchronization error in the improved improved structure. structure.
Improved Structure
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n1
82ms ts
n2 n1
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n2
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n [25r/min/div]
Traditional Structure 69ms
n2 n1
t [15ms/div]
(a)
t [100ms/div]
t [15ms/div]
(b)
Figure 18. Speed output of motors under sudden changes of a load during steady operation: (a) Speed
Figure 18. Speed output of motors under sudden changes of a load during steady operation: (a) Speed output in the traditional structure; (b) Speed output in the improved structure. output in the traditional structure; (b) Speed output in the improved structure.
As can be seen from Figure 17, the maximum synchronous error between two motors reduces
6. Conclusions from 128 r/min to 95 r/min, which is a decrease of 25.7%, compared with the traditional structure. The convergence of synchronization increased by control 49%. Aiming at thespeed shortcomings of the error cross-coupling structure, which has a large In Figure 18, the adjusting time of the speed is shortened by 39%, the tracking error is reduced synchronization error and a long starting time in the multi-motor system during the start-up of from 74 r/min to 70 r/min—almost unchanged—and the response time of the tracking error reduces motors with load, this paper presents a new coupling control structure with a fuzzy self-adjusting from 69 ms to 60 ms. filter andToansummarize, advanced synchronization compensator. the improved structure proposed in this paper not only improves the synchronization performance and dynamic response motorsmodes in the of starting process, it 1. The structure adopts the mode selector to switch between betweentwo working the fuzzy controller. also improves the synchronization performance of motors when the steady state is interrupted by a It automatically adjusts the softened coefficient according to different given speeds and maximum sudden change of load.
load torques, and then adjusts the softened speed so that each motor follows the trajectory of the
6. Conclusions
Aiming at the shortcomings of the cross-coupling control structure, which has a large synchronization error and a long starting time in the multi-motor system during the start-up of motors with load, this paper presents a new coupling control structure with a fuzzy self-adjusting filter and an advanced synchronization compensator.
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softened speed in the starting process. These adjustments effectively reduce the synchronization error during the start-up of motors with load. At the same time, the advanced synchronization compensator is put forward to improve the system’s dynamic response. The synchronization performance of the fuzzy self-adjusting cross-coupling control structure under the steady state with a sudden change of load is analyzed. The experimental results show that the structure proposed in this paper can improve the system synchronization performance and dynamic response both in the starting process, and in the steady state with a sudden change of load.
Acknowledgments: This work is supported by The National Natural Science Foundation of China (No. 51777135), The Key Project in the Science & Technology Pillar Program of Tianjin (No. 16YFZCSF00580). Author Contributions: Wei Chen, Jiaojiao Liang and Tingna Shi put forward the idea and designed the simulation and the experiments; Jiaojiao Liang performed the simulation and the experiments; all authors analyzed the results; all authors contributed to the writing of the paper; Jiaojiao Liang and Tingna Shi reviewed and edited the manuscript. Conflicts of Interest: The authors declare no conflict of interest.
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