a torque control method of three-inertia torsional system with backlash856

7/27/2019 A Torque Control Method of Three-Inertia Torsional System With Backlash856

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A Torque Control Method ofThree-Inertia Torsional Systemwith Backlash

Yu Nakayama, Kiyoshi Fujikawa, Hirokazu KobayashiResearch & Development Department, Toyo Electric.Mfg.Co.,Ltd.3-8Fukuura, Kanazawa-ku, Yokohama-shi, Kanagawa-ken, 236-0004, Japante1:+81-45-785-3702; fax:+81-45-790-3180;e-mail:[email protected]

Abstract: ?'his paper proposes a torque control method ofThree-Inertia Torsional System with gear (or coupling)backlash.The propo sed controller consists of a PID controller, anda proportional feed-forward compensation, and a propor-tional feedba ck com pensation of the gear torqu e estimatedby a disturbance observer. The param eters of he controllercan be designed by Coeficient Diagram Method (CDM).Analysis and experiment results show that the mechanicalvibrations of Three-Inertia Torsional System can besuppressed w ell, and a good torque transfer performanc ecan be got by using the propo sed controller.

1. IntroductionIn motor drive system, if link up motors and load by aflexible shaft, the motor drive system becom es a mechanicalresonance system called ,Two-Inertia Torsional system (2-1system). In the torque control of the 2-1 system, in order tosuppress the torsional vibration of the shaft, we proposed asimple control method called P-ID control in the paper [l].However, when a gear exist in the motor drive system with atorsional load, the motor drive system becomes a Three-Inertia Torsional System (3-1 system). If we still use a P -IDcontrol as so the torque control for the 2-1 system, thevibration caused by the gear backlash will can't be

suppressed. Therefore, in order to suppress the vibrationcaused by the gear b acklash , and ge,f a good transferperformance from the torque command T to the shaft torqueT,, in this paper, we propose a simple torque control method.The controller is designed by using a PID controller and afeed-forward proportional compensation and a feedbackproportional compensation of the gear torque estimated by adisturbance observer. The parameters of the controller canbe designed by Coefficient Diagram Method (CDM).Although the control design approach is easy, it is shownthat the vibration caused by the gear backlash can besuppre ssed well and a better transfer performa nce of torquecan be got. The effec tiveness of the propo sed method will beshown by simulation s and experim ents.2. Three-Inertia Torsional System and the Open-Loop

Frequency ResponseA typical structure of a motor drive system with a torsionalload is shown in Fig.1. The system consists of a drivingmotor and a load coupled to the driving motor through agear and a shaft. If we con sider the effects of the ge ar inertiaand backlash, the motor driver system can be modeled by aThree-Inertia Torsional System (3-1 system). Fig.2 shows ablock diagram represe ntation of the 3-1 system, where thebox containing a nonlinear element represents the model

used for the gear back lash. The 3-1 system paramete rs usedin this paper are listed on the Tablel.gear loadtorque, Shaft

Iloadspeedmotor motor ' gear I 'torque speed speed shafttorque

Fig.1 Three-Iner tia Torsio nal System

Tm

Fig.2 Blo ck diagra m of the 3-1 systemTablel 3-1System Parameters_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ - - - _ - _ - -

Symbol Description Valuemotor inertia 0.0641 Kgm 2J , gear inertia 0.0868 Kgm'load inertia 0.0523 Kgm'

K c shaf t stiffne ss 242 "/radD, shaft damping coefficient 0.1 Nmse dradgear stiffness 2000 Nm lradD , gear damp ing coefficien t 0.2 "sechad6 gear backlash 0.25 degree

Jr n

J L

K ,_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ - _ - - - - - - -

0-7803-5976-3/00/$10.002000 EEE.193

AMC2000-NAGOYA
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For the 3-1 system, when 6 =0, the open-loop transferfunction from the motor torque T, to the shaft torque T, isgiven by equ.(l), where, the damping coefficients D, and D,are omitted owing to the values are very small.

M ( s ) = J , K , K ,N ( s ) = J m J , J , s 4 + [ J , ( J , + J , ) K , +J , ( J , +J , ) K , ] s 2

+ ( J , + J , + J , ) K , K , = p 4 s 4 + p 2 s 2 + pO

[ ( p 2+ Jpi - 4p ,p , ) / 2 ~ , ] / ~ )re the natural resonantfrequencies respectively corresponding to the shaft-resonance and the gear- resonance.In Fig.3, the break lines show the plots of the open-loopfrequency response from the motor torque T, to the shafttorque T,. Because the values of the damping coefficients D,and D, are very small, so two high peaks arise in the gaincharacteristic plot at frequencies w and O 2, then atorsional vibration of the shaft with frequency O~ nd abacklash vibration of the gear with frequency o aresimultaneously easily caused.

Torque command transfer performance

1wO

Fig.3 Frequency respo nse characteristics (3-1 system, 6=0)3. P-ID Control for Tw o-Inertia Torsional System [l]

In this chapter, a simple controller called P-ID controllerpresented in the paper [ l ]will be used for the torque controlof Two-Inertia Torsional system.From a fact that the gear stiffness constant Kg is far largerthan the shaft stiffness constant I&, if there is not backlash in

the gear (i.e. 6 =O), the 3-1 system can be approximated to a2-1 system as shown in Fig.4. In Fig.4, J,,(=J,+J,) is thesum of the motor inertia an d the gear inertia as an equivalentmotor inertia.For the torque control of 2-1 system, a P-ID controller asshown in Fig.4 can be easily designed with the CoefficientDiagram Method (CDM) presented by Prof. Manabe of

TOAKI UNI. JAPAN [3]. In the paper [ l ] , he P-IDcontroller can be designed as

2-1 systemP-ID controller

Fig.4 P-ID control of 2-1 system

I

where, wo= , / K , ( l / J , + l / J , ) is the natural resonantfrequency corresponding to the shaft-resonance of the 2-1system.

In Fig.5, the solid lines show the plots of the closed-loopfrequency response from the torque command T to the shafttorque T, with the P-ID controller. By the way, the breaklines show the plots of the open-loop frequency responsefrom the motor torque T, to the shaft torque T,.Torque command transfer performance

10 1W 1 00

FrequencyIraUsecl

Fig.5 Frequ ency respons e characteristics (2-1 system)

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Comparing the sol id h e s with the break lines, we can seecan be suppressed well, and a good torque transferperform ance of the 2-1 system can be got.However, when the P-ID controller is used for the 3-1system, the ch aracteristic polyn omial of the closed-loo psystem is given by equ.(3), where, it is supposed that 6 =Oand D,=O.

Fig.7.using the P-ID control, the torsional vibration of the shaft ...................................................................................................................................

A(s) = u s s s+ u 4 s 4+ u 3 s 3+u2s2 U , S + U ,U , = J L K c K , K iU , = ( J , + J , +J L ) K , K g +J , K c K g K , +D, JL K, Kiu 2 = D , ( J , + J , + J L ) K ,+J,K,K,K, + D , J L K , K ,u 3 = J , ( J , + J , ) K , + J L ( J , +J g ) K g D , J , K , K d' 4 = D c J m ( J , + J L )as = J , J g J L (3 )

Because the value of the shaft-damping coefficient D, svery small, the value of the coefficient a4 is nearly zero, sothat stabilizing the closed-lo op system become difficult.In Fig.3, the solid lines show the plots of the closed-loopfrequency response with a P-ID controller, owing to theclosed-loop system is hard to stabilize, the phase cha-racteristic plot become larger than 0 degree in the highfrequency range.Fig-6(a) and Fig.6(b) respectively show the simulated timerespon ses CorresPonding to the 2-1 system and the 3-1 ystem

with 6 =o, we can see the P-ID torque control can'tstabilize the 3-1 system.

i .i.................................................................................................Fig.7 Proposed torque control (3-1 system)

For the proposed controller can be designed easily, below,we take the filter time constant of the disturbance observer(Tf) as same as that of the differentiator (Td) in the PIDcontrolle r (i.e. Tf=Td=TO). Further more, w hen i t is supposedthat 6 =0, and D,=D,=O, an d J,,=J,, and Kff=O, he closed-loop transfer function from the torque command T' to theshaft torque T, is as follows( a ) ~~1............................... ~ ............. ............_... .................4 ........................... .> ......................z : 1 P-ID control for2-I system !....... ......... ............ ..........g 3 ._ _ :__,.. -. J , K , K , [ (K , + T,K ,) s* + ( K , + T o K i ) s + K i ] (4).................... j ..................... : .......................................... @*(s)= Us)........ .._....._..._ ....__....._.......... ........................... ..........OO 0 5 I 1.5 where, A(s) is the closed-loop characteristic polynomial,and give n byA(S) = u 6 s 6+ a i s s + a 4 s 4+ u 3 s 3+ u 2 s 2 + u , s + a oU , = J L K , K g K iU , = [ J , + J , + J , + ( J , +J , ) K , + J , ( K , + T o K i ) ] K c K ,' 2 = [ J L ( K d + T O K p ) + ( Jm + J g + J L )T O IK c K g

Fig.6 Time responses4. Proposed Torque Control

To sta bilize the 3-1 system shown in Fig.2, and get a bettertorque transfer performance, in this chapter, we propose asimple control method. The proposed controller is composedof a PID con troller, and a feed-forward proportional com-pensatio n, and a feed back proportiona l compensation of thegear torque estimated by a disturbance observer as shown in

u 3 = J , ( J g + J , ) K , + J L ( J , + J , ) K g +J , J L K , K f b' 4 = [ J m ( J g + J , )Kc + L ( J m + g >Kg T'a6 = J,J,JLTo ( 5 )u s = J,J,J,

In the following, we will design the parameters of theproposed controller with using the Coefficient DiagramMethod (CDM) presented by Prof. Manabe of TOKAIUNIV. JAPAN 3]. The design approach is following:

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For equ.(5), the conditions of the Manabe polynomial [3]are

To = 0.004 = 4m ecK , = -0.6086K i =8.478K d =0.0088K , = -0.9881[*I = 0.0805

a2y i = , i =1 - 5 ;ai+lai-l

where, t is an equivalent time constant, and y i are thestability indexes.Finding the solutions of equ.(6), the gains I(p, Ki , &, Ktband the time constant Toof the proposed controller can befound as following

l ) 1y 1 y 2 y 3 y 4 y 5 -1 + ( J g + J , ) ( Y 4 Y 5 -K , =-{[*I[J L 2 3 4 4Y 1Y 2 Y 3 Y 4 Y 5 KCKJ: Y 4Y 5J 8J LKgI - J , - J , - J , )[*I

2 3 4 4i = Y IY 2 Y 3 Y 4 Y 5 J L K J J d 32 2 2 2

Y l Y 2 Y 3 Y 4 Y 5 - Y 1 Y2 Y 3 Y4 Y 5 + IY 1Y 2Y 3Y 4Y 5 J L K C K J O

( J 8 + J , ) ( Y 4 Y 5 - W OY 4Y 5J,JZK8

Kd =[*I[ 2 3 4 4

IK , = [ * I ( l - Y 4 Y s )

Y 4Y 5J g J LK g

(7)

For the system with the 6-th order, a standard form of thestability indexes y i is suggested in CDM, based onpractical experiences, as follows [3].

In Fig.8, the break lines show the plots of the*closed-loopfrequency response from the torque command T to the shafttorque T, with the proposed controller designed by equ.(9)and without feed-forward compensation (i.e. Kff=O).Where,the point lines show the plots of the open-loop frequencyresponse from the motor torque T, to the shaft torque T,.

T *-Using the stability indexes y given by equ.(8) and the 3-1system parameters on the Tablel, from equ.(7), theparameters of the proposed controller can be found asfollows

Fig.8 Frequency response characteristics (3-1system, 6 =0)Comparing the break lines with the point lines, we can seeusing the proposed controller, the mechanical vibration ofthe 3-1 system can be suppressed well. However, the gaincharacteristic plot is larger than 0 dB in frequency domain of10-50[rad/sec], and the phase characteristic plot is far later

than 0 degree in frequency domain of lO[rad/sec]-o,, sothe torque transfer perform ance isnt good.Therefore, to improve the torque transfer performance ofthe 3-1 system, we add a feed-forward compensation in theproposed controller as shown in Fig.7. In addition, for easyto explain the designing processes of the feed-forwardcompensator, we remake Fig.7 as Fig.9.

(9)

IFig.9 Proposed torque control (3-1system)

In Fig.9, Fl(s) is the transfer function of the PID controller,F2(s) is the transfer function of the feed-forward compen-sator. G m ( s ) is the transfer function with the 5 t h orderfrom the output of the PID controller Fm o the shaft torqueT, and given by

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With the feed-forward compensation, the closed-loop errortransfer function @e (s) from the torque command T to thetorque error AT(= T * - T c ) is

From equ.(ll), if we take the feed-forward compensationas F 2 ( s )= 1/6, s) , he closed-loop error transfer function@ e (s ) is @.,(s) = 0 , namely, a very good torque transferperformance can be got. However, from equ.(lO), the feed-forward compensator is a differentiator with the 5 t h order,so that the realization of the feed-forward compensator isntpossible. Therefore, from a practical point of view, if westress only on the improvement of the torque transferperformance in a lower frequency domain, the feed-forwardcompensator can be designed as a proportional compen-sation as

When the feed-orward compensator designed by equ.(l2)is added in the proposed controller, the plots of the closed-loop frequency response from the torque comm and T to theshaft torque T, are shown by the solid lines in Fig.8.Comparing the solid lines with the break lines, we can seethe torque transfer performance cab be improved well byusing the feed-forward compensation.In addition, Fig.lO(a) and Fig.lO(b) respectively show thesimulated time responses corresponding to 6 =O[degree] and6 =0.25[degree], we can see .the proposed torque control

can supp ress the mech anical vibration of the 3-1 system evenif there is a smaller backlash in the gear. Where, the input inthe simulation s istorque com mand : T is a step signal 6 at t=0.2[sec]

Fig.10 Time responses5.Experimental Results

In this chapter, we give some experim ental results.Fig.11 shown the experimental result of the open-loopfrequency response, where the 3-1 experimental systemparameters are also in the Table1 (i.e. 6 =0.25[degree]). Inthe same way as the sim ulation result given by the point linein Fig.8, at the natural resonant frequency 0 corres-ponding to the shaft vibration, a high peak arises in the gaincharacteristic plot, however, at the natural resonant fre-quency O 2 corresponding to the gear backlash vibration,only a low peak arises in the gain characteristic plot.Fig.12 show n the experimental result of the closed-loopfrequency response by using the proposed torque controller,corresponding to 6 =0.25[degree], where the controller isimplemented with the digital signal processor.It is similar to the simulation result of Fig.8, in Fig.12, wecan see even if there is a smaller backlash in the gear, usingthe proposed torque control, the mechanical vibration of the3-1 system can be suppressed well, and a better torquetransfer performance can be got.Fig.l3(a) and Fig.l3(b) shown the experimental results ofthe time response, respectively corresponding to the open-loop and th e closed-loop with backlash 6 =0.25[degree], allresults are similar to the simulation results shown inFig.lO(b).

6. ConclusionIn this paper, from a practical point of view, we proposed asimple torque control method of 3-Inertia Torsional Systemwith backlash. The torque controller is designed using a PIDcontroller and a feed-forward proportional compensation,and a feedback proportio nal compensation of the gear torque

estimated by a disturbance observer. The parameters of theproposed controller can be designed by Coefficient DiagramMethod (CDM).The analysis results on the closed-loop frequency responseshow that although the structure of the proposed controller issimple, the mechanical vibrations can be suppressed well,and the better torque transfer performance can be got.The effectiveness of the proposed control method is shownby simulations and experiments.

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20

HA GdB

2 0 d W-60

200

PHASEde9

-7nn-- 0.25 X f r FUNC r c t LO G lOOHz& 12.50Hz V, -117.6deg

20

M A GdaEU/EU20dB/

-60

200

PHASEde9

--3nn

Fig.11 Experimental result of the open-loop(6=0.25[degree])

- 0.25 X f r FUNC rct LOG lOOHz& 2. OOHZ v_ l -13.3deg

Fig.12 Experimental result using the proposed controller(6=0.25[degree])

ReferencesY.Wu, K.Fujikawa and H.Kobayashi, A Torque ControlMethod of Two-Mass Resonant System with PID-PController IEEE 4th AMC Workshop, pp.240-245,1998.Y.Wu, K.Fujikawa and H.Kobayashi, A ControlMethod of Speed Control Drive System with BacklashIEEE 3rd AM C Workshop, pp.631-636,1996.[3] Y.Wu, K.Fujikawa and HXobayashi, VibrationSuppression Control for Motor Drive System withTorsional Shaft and Backlash SICE, vo1.34, No.11,

[4] S.Manabe, Controller Design of Two-Mass ResonantSystem by Coefficient Diagram Method T.IEE Japan,pp.1639-1644, 1998.

vol.l18-D7No.1, pp.58-66, 1998.

_.

T * J ... . . - - . . . . . . . - . .

. . . .~ . .

Fig.l3( a) Experimental result of the open-loop( 6 =0.25[degree])

Fig.l3(b) Experimental result using the proposed controller( 6 =0.25[degree])

[5] S.Manabe, Coefficient Diagram Method 14th IFACSynp. on Automatic Control in Aerospace, pp.199-210,1998.[6] M.Odai, Y.Hori, Speed Control of 2-Inertia Systemwith Gear Backlash using Gear Torque CompensatorIEEE 4th AMCWorkshop, pp.234-239, 1998.[7] M.Odai, Y.Hori, Speed Control of 2-Inertia Systemwith Gear Backlash based on Gear Torque Compen-

sationT.IEE Japan, vol.l20-D , No.1, pp.5-10,2000.[8] M.Odai, Y.Hori, Controller Design Robust to NonlinearElements based on Fractional Order Control SystemT.IEE Japan, vol.l20-D, No.1, pp.l l-18 ,20 00.

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a torque control method of three-inertia torsional system with backlash856

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