重复控制

International Journal of Machine Tools &Manufacture 47(2007)1807–1816

Repetitive control design and implementation for

linear motor machine tool

Shang-Liang Chen ?,Tsung-Hsien Hsieh

Institute of Manufacturing Engineering,National Cheng-Kung University,Taiwan,ROC Received 9August 2006;received in revised form 8April 2007;accepted 18April 2007

Available online 29April 2007

Abstract

The goal of this paper is to eliminate the period tracking error via the design of discrete-time domain repetitive controller.It increases the stabilizing range and enhances the robust performance by adopting the prototype repetitive controller design principle to compensate repetitive control.Furthermore,with the concept of command feedforward,it introduces the feedforward gains of speed and acceleration,which can forcefully enhance the tracking ability of the repetitive controller and improve on the errors of the system.Finally,it puts into practice the theory on a gantry type machinery platform with linear motors.The results prove that the theory can reduce period tracking error successfully.r 2007Elsevier Ltd.All rights reserved.

Keywords:Repetitive controller;Linear motor machine tool;Prototype repetitive controller;Command feedforward controller

1.Introduction

Most of the machines and equipments used in the manufacturing industry nowadays are working with rotat-ing motors,linear motors and manipulators to proceed on ?xed and repetitive movements,for examples,assembly,inspection,welding,paint-spraying,and drilling,etc.To achieve this repetitive movement,there has to be repetitive input of signals for the movement.The prosperous development in the electronic industry today insists on an ever-demanding requirement on the cleanliness of the working environment.However,the traditional rotating motors need to work with ball screws to achieve the purpose of positioning.The wearing away caused by the transmission of ball screws becomes a critical source of pollution in the working venue.Machine and equipment that are equipped with a linear motor with direct driver device have neither the above-mentioned pollution nor the problems of backlash and de?ection caused by the transmission of a ball screw.Furthermore,it has the

advantages of a higher acceleration and precise position-ing,added to which the ever-maturing controlling techni-que.All these have made linear machine tool more and more important in recent years.

The repetitive controller was classi?ed and described in [1].It can reduce the error after the ?rst Period [2].It can also be applied into lathe for improving the machining accuracy [3,4].It was also used in X –Y table for milling machine [5].Tsao and Tomizuka [6]implemented a repetitive controller for tool positioning a noncircular machining using a hydraulic linear actuating system.It was also applied in electrical injection molding machine [7]and micro-machine manufacture [8].There are some other repetitive controller applications are described in [9].

Two major goals for a machining process are quality and ef?ciency.Therefore,an excellent motion control device to meet the demands and can effectively demonstrate the ef?ciency of the machine tool is important.However,for the periodic reference input system,its tracking error is often a periodic error.The most representative solution is an excellent repetitive motion controller which is shown as Fig.1,where G p (z à1)is the closed-loop plant transfer function [10,11].Tomizuka and Chew adopt the accept of prototype repetitive controller [12,13]to verify that this method can

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E-mail address:slchen@https://www.360docs.net/doc/3413811477.html,.tw (S.-L.Chen).

effectively bring about asymptotic convergence of the errors,and has better robustness on a gantry type machine tool.Fig.2shows that a block diagram of the prototype repetitive controller,where G f (z à1)is a phase compensator.Q (z ,z à1)is a zero phase low-pass ?lter which is used to guarantee the stability of system.K r is the gain of repetitive controller.

Command feedforward controller was proposed by Masory [14].It added a velocity feedforward loop (VFFL)into the system and enhanced the precision of a CNC system.Lee and Chen [15]went further and added an acceleration feedforward command to improve the effects of delaying a servo system.Based on the concept of command feedforward,this research uses the feedforward gains of speed and acceleration,G ff (z à1),for enhancing the tracking ability of the repetitive controller and improving the positioning errors of the system.2.Theories

2.1.Repetitive controller

In Fig.1,a repetitive controller is added into the existing closed-loop system for reducing the effects of periodic input error.The system error can be shown as follow:E ez à1TR ez T?1àz àN

1àz tG ez T

(1)

Replace z à1?e àsT s ?e àj o T s into Eq.(1),we have E ej o TR ej o T?1àe àj o NT s

1àe s tG ez T

,(2)where NT s is error period,and T s is sample time.For reducing steady state error to 0,let E (j o )?0Then,1àe àj o NT s ?0,e àj o NT s ?1,o NT s ?2n p ,

o ?2p n NT s ?

2p nf s

.e3T

From Eq.(3),the steady-state error can be reduced to 0,if the frequency o is n times of reference frequency.Theoretically,E (s )will be 0,if the reference signal has the period of NT s [16].

Hence,the repetitive controller is suitable for applying into a system with repetitive input reference signal.

In the design of a repetitive controller system,the biggest challenge is the system stability.The minimum gain principle is used for stability analysis in this research [17].The minimum gain principle adopts Nyquist criterion to ensure the stability of the system.From Fig.1,the equation of tracking error can be shown as

E ez à1T?R ez à1TàD ez à1T??1àG PC ez à1T??

e1àz àN T

1àz 1àG PC ez T?

.e4T

From Eq.(4)and the minimum gain principle,the stability can be classi?ed into three conditions for a plant with a stable closed loop system.They are,

(1)The original closed loop G PC (z à1)is stable,so

1àG PC (z à1)will be stable.

(2)If R (z à1)and D (z à1)are both harmonic periodic

signals,then [R (z à1)àD (z à1)][1àG PC (z à1)](1àz àN )will have the ability to eliminate disturbance and make the error convergence.However,if D (z à1)is not periodic nor the harmonic frequency of R (z à1),the system will not be able to eliminate D (z à1)effectively and the system stability might be decayed by high-frequency disturbance.(3)The last item of the errors,1 1àz àN ?1àG PC ez à1T àá,

also needs to be stable in order to verify the stability of the entire system.With Nyquist criterion,the stable

Fig.1.Repetitive controller [10,11].

Linear Fig.2.Developed linear motor machine tool for the experiments.

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condition can be shown as z àN ?1àG PC ez à1T

o 1;o 2?0;p (5)and

G PC ez à1T?P eo T Te j y eo T T;P eo T TX 0,

(6)1àP eo T Te j yo T o 1,

(7)

where P (o T )and y are the magnitude and phase of G PC (z à1),respectively.

Then,the stability condition will be 0p P eo T To 2cos y eo T T.

(8)

Therefore,the stable range of y in Eq.(6)is between 7901.The limit for magnitude is P (o T )o 2cos y (o T ).From Eqs.(8)and (5),two results about the stability in a repetitive controlling system can be obtained:

(1)Repetitive control system has poor stability,and the

poles are interfered with unstable disturbance.

(2)The stable range of magnitude and phase for the plant

is limited.

2.2.Prototype repetitive controller

Because repetitive controller has problems of stable range,it is necessary to explain the structure of prototype repetitive controller,which is proposed by Tomizuka for improving the stable range and robustness [12,13].There-fore the transfer function of prototype repetitive controller,G R (z à1),in Fig.3,can be shown as G R ez à1

K r Q ez ;z à1TG f ez à1Tz àN

1àQ ez ;z à1Tz àN

,(9)

where

Q ez ;z à1

e1tz à1T2t z t

2.(10)The minimum gain principle adopts Nyquist criterion to ensure the stability of the system.To make the analysis simple,rewrite Fig.5to get the equation of tracking error

E ez à1T?R ez à1TàD ez à1T??1àG P ez à1T??

e1àQ ez ;z à1Tz àN T

1àQ ez ;z à1Tz àN 1àK r G f ez à1TG P ez à1T?

.e11TIn Eq.(11),it is interesting to know that the Q (z ,z à1)is a zero-phase low-pass ?lter,so its low-frequency gain is smaller or equal to 1,thus it can enhance the robustness of the system to restrains high-frequency disturbance.There-fore,R ez à1TàD ez à1T??1àG P ez à1T??1àQ ez ;z à1Tz àN ??will

be stable.Furthermore,let G P (z à1)?P (o T )e

j y (o T )

,where P (o T )and y are the magnitude and phase of G P (z à1),G f (z à1)?F (o T )e j f (o T )where F (o T )and f are the magnitude and phase of G f (z à1),and Q (z ,z à1)?Q (o T )e j0?Q (o T ),where Q (o T )is the magnitude of Q (z ,z à1).Then,the stability condition will be

0p P eo T To cos ey tf Tt???????????????????????????????????????????????

1àQ 2eo T Tsin 2ey tf T

q K r F eo T T.(12)

From Eq.(12),P (o T )can be stable and the stable range is varied by adjusting K r ,F (o T ),and f (o T ).Also,(y +f )will be stable in any degrees.That is because co-s 2(y +f )?1àsin 2(y +f ).We have ???????????????????????????????????????????????

1àQ 2eo T Tsin 2ey tf Tq X j cos ey tf Tj .(13)After adding the zero-phase low-pass ?lter,and proving

Q (z ,z à1)to enhance the strength of the system and increase its stability,the design of a prototype repetitive controller is then completed.

https://www.360docs.net/doc/3413811477.html,mand feedforward controller

It is known that a basic repetitive controller cannot compensate the ?rst period tracking error.Therefore,a feedforward controller G ff (s )is usually used to improve the ?rst period tracking performance.Fig.3is the control structure of command feedforward with a repetitive controller G R .The closed loop transfer function can be obtained as T es T?

Y es TR es T

G ff es TG P es TtC R es TG P es T

1tC R es TG P eS T.e14T

Let T (s )?1,and Y (s )?R (s ),therefore G ff es T?

1G P

?1tK fv s tK fa s 2tK ff s 3t...

e15T

From Eq.(15),G ff (s )is only related to G P .So,when the parameters or the controller structure is changed,the

Fig.3.Prototype repetitive controller [12,13].

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design of the feedforward controller will not be affected.The coef?cients in Eq.(15)are used to be the coef?cients of the position feedforward controller.Therefore,combining the feedforward controller and a repetitive controller will improve the performance of a repetitive control system.However,most of the coef?cients of s 3or higher orders are too small,their compensation can be ignored.Fig.4shows the last controller structure in this research,which K fv and K fa represent the speed and acceleration feedforward gains of input command,respectively.3.Experimental results

Fig.5shows the kinematics diagram of a linear motor machine tool for the experimental setup.The type of the linear motor is MITSUBISHI LM-HP1F-15M and its driver is MR-J2S-60A-S040U502[18].The Commands which are calculated and send by a PC-based motion control card [19].3.1.System identi?cation

The open-loop measurement results and LSF estimation principle are used in system identi?cation.The data of chirp voltage input signals and velocity (mm/s)output signals are collected.Finally,the System Identi?cation Toolbox in MATLAB is used to work out the linear plant transfer function through the curve ?tting method.Hence,the Y -axis closed loop transfer function,in Fig.5,from voltage to position can be shown as G P_Y es T?

2;596;000

S t330:2S t27;260S t2;596;000

(16)

Through the zero-order hold and a sampling rate of 5ms,the Y -axis discrete-time domain plant transfer function will become

G P_Y ez à1

z à1e0:03632t0:09798z à1t0:01599z à2T

1à1:781z t1:123z à0:1919z ?0:03632z à1e1t2:5232z à1Te1t0:1745z à1T1à1:781z t1:123z à0:1919z .

e17T

With the same concept,the Z -axis closed loop transfer function,in Fig.5,can be shown as G P_Z es T?

14;620S t905;100

S t168S t18;359:5S t905;100

(18)

G P_Z ez à1

z à1e0:1506t0:01561z à1à0:09256z à2T

1à2:091z à1t1:596z à2à0:4317z à3?0:1506z à1e1t0:8375z à1Te1à0:7339z à1T1à2:091z à1t1:596z à2à0:4317z à3

.e19T

3.2.Selection of system parameters

3.2.1.Phase compensator (G f )

From the discrete Y-and Z -axes transfer functions,G f_y (z à1)and G f_z (z à1)can be obtained by adopting the PTC or ZPETC method to compensate the repetitive controller.However,because Y -axis contains an unstable zero,the ZPETC model must be adopted in G f_y (z à1),the PTC model can use in Z -axis directly to design G f_z (z à1),which are shown in Eqs.(20)and (21)G f_y ez à1T?

z 2e5:59à7:74z à1t2:332z à2t1:415z à3à0:4251z à4T

1t0:1745z ,

e20T

G f_z ez à1

z e6:64à13:884z à1t10:6z à2à2:867z à3T

1à0:1037z à0:6146z .(21)

Fig. 4.Structure of command feedforward controller with repetitive

controller.

Fig.5.Structure of the repetitive controller with command feedforward.

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3.2.2.Zero-phase low-pass ?lter (Q(z,z à1))

From the presenting equation of a zero-phase low-pass ?lter in Eq.(10),both Y-and Z -axes choose t ?1to proceed the designing.Thus,the zero-phase low-pass ?lter can be shown as

Q ez ;z à1

T?z t2tz à1

.(22)

The zero phase low-pass ?lter can increase the stability and robustness of the system or to lower down the in?uence from disturbance.

https://www.360docs.net/doc/3413811477.html,mand feedforward controller (G ff (z à1))

The parameters of command feedforward controller on Y-and Z -axes can be obtained by replacing Eqs.(16)and (18)into Eq.(15),and the results are shown in Table 1.3.3.Experimental results

A linear machine tool shown in Fig.5is used to verify the proposed theory.The experiments are divided into two parts:single axis motion control and Y +Z motion control.Furthermore,the structure of the controller is divided into two parts.The ?rst part is the loop of prototype repetitive controller.In the second part,the ?nal control structure is a repetitive controller added with a command feedforward controller.

3.3.1.Y-axis experimental results

For the Y -axis experiments,a 2Hz sin wave input is used to test the performance of https://www.360docs.net/doc/3413811477.html,ing the repetitive controller shown in Fig.1only,the response signal of Y -axis can be obtained and shown in Fig.6.It is found that the plant of Y -axis is not located in the stable range.Therefore,the system needs to be compensated.

Fig.7(a)is the response error of Y -axis with the prototype repetitive controller shown in Fig.3with 2Hz sin wave input,and the amplitude at 730mm.It shows that the proposed method can effectively eliminate the tracking error caused by periodic input.Fig.7(b)is the response error of Y -axis with a command feedforward added (from Table 1).Fig.7(b)shows the command feedforward controller can signi?cantly reduce the ?rst period tracking error to make the system converge faster,and the tracking error will ?nally converge to 715.8m m in amplitude.

The effects of the amplitudes and frequency of input signal are very interesting for a repetitive controller.Therefore,the response errors with different input ampli-tudes and input frequency will be investigated in the following section.Fig.8(a)shows the tracking error result without any controllers used and the input frequency is 2Hz.Fig.8(b)shows the tracking error result after adding a compensated repetitive controller and the command feedforward controller.From Fig.8(b),the tracking errors caused by different amplitudes can be compensated with the add-in command feedforward controller.The tracking errors will be signi?cantly changed with the varied input signal https://www.360docs.net/doc/3413811477.html,pare Fig.8(a)and (c),it is found that the adding of the prototype repetitive controllers into the Y -axis system has no signi?cant effect on the improvement of the ?rst period tracking errors.However,the tracking errors in the ?rst period can be greatly reduced after adding the command feedforward controller.

3.3.2.Z-axis experimental results

For the Z -axis experiments,a 2Hz sin wave input is also used to test the performance of controllers.

Fig.9(a)shows the tracking error result of Z -axis without any controllers used and the input frequency is 2Hz.Fig.9(b)shows the tracking error result of Z -axis after adding a compensated repetitive controller and the command feedforward controller.From Fig.9(b),the tracking errors caused by different amplitudes can be compensated with the add-in command feedforward controller.The tracking errors will be signi?cantly changed with the varied input signal https://www.360docs.net/doc/3413811477.html,pare Fig.9(a)

Table 1

Parameters of command feedforward Axis K fv K fa Y 0.01050.000127Z

0.0041

0.000119

20E r r o r (m m )

151050-5-10-15-20

Time(sec)

Fig.6.Experiment result of Y -axis with repetitive controller.

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20E r r o r (m m )

E r r o r (m m )

151050-5-10-15-20

Time(sec)

1015

Time(sec)

1015

5-5

4-43-32-21-10Fig.7.Response error of Y -axis:(a)with prototype repetitive controller;(b)with prototype repetitive and command feedforward controllers.

18161412

1086420E r r o r (m m )

181614121086420E r r o r (m m )

E r r o r (m m )

51515202025253035

1010Input Amplitude of Y-Axis(mm)

1520253035

10Input Amplitude of Y-Axis(mm)

1520253035

Input Amplitude of Y-Axis(mm)

2Hz

Without Controller

2Hz RC+Gf+Q

2Hz

RC+Gf+Q+Gff

2Hz

RC+Gf+Q+Gff

Fig.8.Response tracking errors of Y -axis on different amplitudes:(a)without controllers;(b)with controllers and (c)?rst period error.

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and (c),it is found that the adding of the prototype repetitive controllers into the Z -axis system has no signi?cant effect on the improvement of the ?rst period tracking errors.However,the tracking errors in the ?rst period can be greatly reduced after adding the command feedforward controller.

In Fig.9(b),the adding of command feedforward controller into the Z -axis experiments did not signi?cant improve the tracking errors compared with the results of adding prototype repetitive controller.The possible reason is that a air pressure cylinder is used in the Z -axis to compensate the Z -axis weight from the gravity,which may make the transfer function and the coef?cient of the command feedforward in the system worse than those of the Y -axis.Therefore,the tracking error after compensated is affected.

Fig.10(a)is the response error of Z -axis with the prototype repetitive controller shown in Fig.3with 2Hz sin wave input,and the amplitude at 730mm.It shows that the proposed method can effectively eliminate the tracking error caused by periodic input.Fig.10(c)is the

response error of Z -axis with a command feedforward added (from Table 1).Fig.10(c)shows the command feedforward controller can signi?cantly reduce the ?rst period tracking error to make the system converge faster,and the tracking error will ?nally converge to 730m m in amplitude.

3.3.3.Y+Z-axes motion experimental results

Table 1shows the parameters of command feedforward for the Y +Z -axes motion experiments.Table 2shows the comparison of the max.steady-state response error of single Y -axis and single Z -axis experiments.In Table 2,the single Y -axis and single Z -axis experiments means that the experimental parameters Y -axis are ?xed and separately input with the input parameters of Z -axis varied.Therefore,a ?xed input command is send into Y -axis and variant input commands are sent into Z -axis of the gantry machine tool at the same time.In the situation of no controller used,some disturbances between Y -axis and Z -axis motion are created.Under the situation,it is very easy to make the poles of repetitive controller

876543210E r r o r (m m )

E r r o r (m m )

876543210E r r o r (m m )

51515202025253030

35351010Input Amplitude of Z-Axis(mm)

1520253035

10Input Amplitude of Z-Axis(mm)

1520253035

Input Amplitude of Z-Axis(mm)

402Hz

Without Controller

2Hz RC+Gf+Q

2Hz

RC+Gf+Q+Gff

2Hz

RC+Gf+Q+Gff

Fig.9.Response tracking errors of Z -axis on different amplitudes:(a)without controllers;(b)with controllers and (c)?rst period error.

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8-8

6-64-42-20E r r o r (m m )

E r r o r (m m )

012345678910

Time(sec)

0.03-0.03-0.04

0.02-0.020.01-0.010E r r o r (m m )

0.030.04-0.03-0.04

0.02-0.020.01-0.01011.5

12.5

1313.5

14.5

4-4

3-32-21-1005

1015

13.213.413.6

13.81414.214.414.6

Fig.10.Response error of Z -axis:(a)with prototype repetitive controller;(b)amplitude of the response error of (a);(c)with prototype repetitive and command feedforward controllers and (d)amplitude of the response error of (c).

Table 2

Max.steady-state response error of single axis Y and Z Axis

Input signal Single axis expt.Freq.(Hz)

Amp.(mm)Closed loop (mm)With controllers (m m)Y 2730715.8715Z

273077.25740571076.0974710

74.98

772

Table 3

Max.steady-state response error of Y +Z -axes motion Axis

Input signal Y +Z motion expt.Freq.(Hz)

Amp.(mm)Closed loop (mm)With controllers (m m)Y 2730716.00718.8Z 273077.26743.5Y 2730715.96732.0Z 571076.10748.0Y 2730715.94716.2Z

75.02

775.0

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fallen beyond the unit circle and become unstable.However,through the using of both prototype repetitive controller and command feedforward controller,the steady state tracking errors can be ?ltered and decreased down to 735m m.

Table 3shows the max.steady-state response error of Y +Z -axes motion experiments.The disturbances pro-duced by Y +Z -axes movement are related to the both input signals,and most of them are harmonic frequencies of the input signals which will affect the performance of the

E r r o r (m m )

45-4-5

3-32-21-1005

1015

Time(sec)

E r r o r (m m )

0.03-0.03-0.04

0.02-0.020.01-0.010Time(sec)

11.5

12.513

13.5

4-43-32-21-10E r r o r (m m )

0.030.040.05-0.03-0.04

0.02-0.020.01-0.01012.2

12.4

12.6

12.81313.2

13.4

Fig.11.Response errors of Y +Z -axes with prototype repetitive and command feedforward controllers:(a)with input signal Y -axis:2Hz,30mm;Z -axis:2Hz,30mm;(b)amplitude of the response error of (a);(c)input signal Y -axis:2Hz,30mm;Z -axis:5Hz,10mm;(d)amplitude of the response error of (c).

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control system.From Table2,the tracking errors of separated input signals(Y?2Hz,30mm,Z?5Hz, 10mm)are(Y?715m m;Z?747m m).From Table3, the tracking errors of simultaneous input signals(Y?2Hz,30mm,Z?5Hz,10mm)are(Y?732m m; Z?748m m).It is found that the steady state tracking error of Y-axis is slightly increased in the Y+Z simulta-neous motion situation.(Fig.11)

4.Conclusion

A control strategy of repetitive controller is proposed in this research.This proposed method takes advantage on prototype repetitive controller and a command feedfor-ward controller to compensate various problems of repetitive control.At the end,linear motors are used in a gantry type machine tool as the experimental setup to verify the performance of the proposed control theory.The results show that the prototype repetitive controller combining a command feedforward controller can effec-tively reduce the errors in the system with periodic input control signals.The robustness of the control system is then better.

Acknowledgments

Parts of the research results were supported by the NSC of R.O.C(NSC93-2212-E-006-106)and ITRI of R.O.C. These?nancial supports are gratefully acknowledged. References

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