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Real-Time Stochastic Model and Control of EDM
Real-Time Stochastic Model and Control of EDM K. P. Rajurkar, W . M. Wang; University of Nebraska, Lincoln/USA - Submitted by R . P. Lindsay ( 1 ) Received on January 16,1990 T h i s paper proposes a new EDM s e r v o c o n t r o l system i n which a h i g h speed microcomputer, I B M 386. d i r e c t l y r e g u l a t e s t h e s e r v o f e e d speed, i n s t e a d of s e r v o r e f e r e n c e v o l t a g e , and t a k e s t h e p a r a m e t e r s of d i s c h a r g e time r a t i o s from an EDEl gap monitor a s t h e f e e d back s i g n a l . The c o n t r o l s t r a t e g y i s developed a c c o r d i n g t o t h e r e a l - t i m e achieved s t o c h a s t i c model t o minimize t h e v a r i a n c e s of t h e c o n t r o l l e d p r o c e s s and i n c r e a s e t h e machining speed w i t h the c o n t r o l i n t e r v a l time of 2 m i l l i s e c o n d s . Extensive experimental i n v e s t i g a t i o n shows t h a t w i t h t h i s c o n t r o l method the machining p r o d u c t i v i t y i s 15%h i g h e r t h a n the v a l u e s o b t a i n e d by t h e model r e f e r e n c e a d a p t i v e c o n t r o l l e r when machining small h o l e s and medium s i z e c a v i t i e s . This c o n t r o l system has been condensed i n t o a simple c o n t r o l c i r c u i t and can be implemented on any EDM u n i t .
K E Y WORDS:
EDM, Adaptive C o n t r o l , S t o c h a s t i c P r o c e s s , System I d e n t i f i c a t i o n .
INTRODUCTION The servo control unit in EDM machine is one of the most important Control devices for EDM process as it directly and in real-time regulates the dimensions of discharge gap. The dynamic features of EDM servo control unit strongly influences the stability of machining process. Conventional EDM servo control system takes only the average gap voltage Vs to be the feed back signal and to compare this signal with a so called servo reference voltage V,. The difference signal of this comparison drives the servo mechanism to adjust the gap size so that the average Vg is kept at the value specified by V,. It is known that the conventional servo system does not effectively respond either the parameters of different gap states or the dynamic and stochastic features of the on-going machining process. The machine operators only preset the V, according to his or her experiences. thus the machining process can not be optimum. In recent years, many EDM specialists have developed different kinds of monitoring apparatus to detect and measure the parameters of gap states, such as open. short, sparking. arcing and transient arcing etc. [2.9,10.14,19], and created many adaptive control strategies to on-line modify the servo reference voltage V, through a sub-feedback loop to search the optimum value of the parameters of every gap states for achieving maximum machining productivity and minimum arc damage [7,9,12,14l6.19.20]. The authors developed a model reference adaptive controller 1121 that improves the machining productivity and avoid the arc damage when machining deep blind hole with non-flushing. However those control methods only adjust the input of the servo control loop V,, without addressing the internal dynamic characteristics of this closed loop system and stochastic features of the machining process. EDM process is of very strong stochastic nature, particularly in cases of finishing or semi-finishing, machining with non-flushing condition. steel to steel machining and machining modern advanced materials (engineering ceramics and carbides). The process can not be very stable, and there are so many gap-open and gap-Short pulses and less sparking. The variations of average gap voltage are very frequent and large even with good flush condition. It is necessary to develop high stable EDM servo control system either to improve the current machining performance or to meet the future needs of machining advanced materials [4].
linearized [12.19.20], and therefore, the output of the process is defined as:
Y
2 f d / f it fe/l;
The output Y can correctly reflect the gap status. When gap is complete open, is 100% and when the gap is the Y equals to 2 as the gap-open ratio (I&,) optimal the Y is I because the spark ratio ( f e / f , ) is 100%. If the gap is highly polluted, there is no gap-open and normal spark in the gap. the Y equals to 0. Figure I shows the block diagram of the servo control unit and the EDM process. vsd is the driving signal of servo system mechanism. The relationship between and the process output Y can be described by a mathematical model as a difference equation that accounts for the dynamic and stochastic features of the process [1,3,6.8.10.12]. For correctly determining the structure of the mathematical model and designing an effective control strategy, it is necessary to understand the dynamic and stochastic characteristics of the controlled process. According to the theoretical analysis, the servo control and electrical discharge process is a high order system. However, in order to simplify the control algorithm for achieving better stability. this process was initially considered to be second order. Since the stable feed rate of the servo head is proportional to the servo driving signal V,d. and the changing velocity of the output of the process Y is almost
servo head
I I It
5
In this paper a new EDM servo control system is proposed. A high speed microcomputer directly regulates the servo feed rate instead of servo reference voltage. and takes the parameters of discharge time ratios from an EDM gap monitor as the feed back signal. This control system is designed for a hydraulic servo head and relative control circuit on the Eltee Pulsitron TRM-20 EDM machine. The on-line control strategy is made according to the real-time achieved stochastic model to minimize the variances of the controlled process and increase the machining speed with the control interval time of 2 millisecond. Extensive experimental investigation shows that with this control method the machining productivity is IS% higher than the values obtained by the model reference adaptive controller. reported in [12], when machining small holes and medium-sire cavities. The model reference controller ikelf has been found to be 4D-50% more productive than the existing control system of the machine 1121. The new control system has been condensed into a simple control circuit and can be implemented on any EDM unit. Description of the stochastic model of the EDM servo control is included in the second section. The principles of the self-tuning regulator for EDM servo adaptive control is described in third section, Fourth =tion includes the technology of development of the EDM servo adaptive control circuit. The experimental details and results are given in fifth section. Last section concludes this paper.
(I)
Servo feed
Workpiece
- Process OUtPUt
Figure I . Input and Output of the EDM Servo Control System
proportional to V,d. this controlled process can be considered as a second order integrator " foilowing:
Y(k+l)-(atI ) Y(k)-oY(k-I ) + h o ~ ~ ( k - m ) + h t ~ ~ k - ~ ~ - l ) t h ~ i e ( & t l )
(2)
k = 1.2.3... Above polynomial is an ARMA model. 1 given as: In order to let the adaptive control strategy be generated based upon the parameters of gap states, an EDM discharge parameters monitor was used to be the detecting unit in the control system. This monitoring system deteck the gap voltage, current and the high frequency signal generated by the to identify five basic EDM gap states and corresponding five analog voltages which are proportional to the time ratios of each gap states in the pulse duration These parameters are the time ratios of gap Open tdlfi, normal sparking f e / f i . transient arc f e , / f i , stable arc and short circuit For the having better performance, the process should be
Annals of the ClRP Vol. 3WlSk.I
Its transfer function with ;-transform is
'
(Z-'X'-o~~t)Y=(bo~hlZ~l'Z~mYSdtbc'~
Integer k is the instant time point. and m. another integer. is the restmme delay Steps O r the process. Coefficients a. bo and hi repreent the dynamics of the system, and their values are time variant with different machining conditions. Coefficient bc stands for the 0 f f - w of the whole procea, its value also changes with machining conditions, machining speed, the weight of the workpiece and the off-set of the servo driving circuit and servo valve. e(ktl) is the random disturbance with zero averane at time m i n t k+l in the controlled Drocess. Parameters of a. ho. h,, b, can be estimated on-line.
187
Equation (2) can be implemented to describe and analyze the transfer relation between Y and &d. However, it is not convenient to directly use this model to develop the control strategy as the value of &d(k) should be generated with the information of a future value Y(k+m) when m is larger than 0. At time point k. the value of Y ( k t m ) is not available, but it can be estimated by a filter. However, EDM is a typically stochastic process [7,9-201. and Y is a random variable [12]. It was found through a series of experiments that the future value of Y can not be precisely on-line estimated. Therefore, the equation (2) can be converted in the form as [I,3.5.61: Y ( k t l ) - ( l + u , ) Y(k-m)-u,Y(k-m- I )tboV,d(k-m)+b,V,d(k-m-I )+b,te(ktl)
(4)
The transfer function with z-transform can be written as: 12-(I + a , ) : - m t U ~ ~ ~ m ~ ' ] ~ - ( b ~ t ~ l ~ ~e l ) ~ ~ m ~ d + b c + ~
uncertainties of the parameter estimates are proportional to the variance of the identification algorithm [l,3.6]. The v(k) in equation (9) is the variance of the parameter estimator. This self-tuning regulator optimizes system feed forward gain based upon the state value Y . parameters bo and b, and as well as the variances of the estimates v(k). The error of the estimates is defined as:
a&)
&k)-Y(k)-Y(k-m- I )-b,(k)V,d(k-m- I )-b,(L.)
00)
The variance of the estimation can be achieved by a first order filter. v(k)-0.99 v(k-I)+O.Ol i'(k)
(11)
Extensive experiments show that the machining performance and stability of EDM process can be improved by this self-tuning regulator. (5)
m is an important factor and its value depends on the response delay time and
the cycle time of the control system. The system response delay time was found to be 8 millisecond in a special designed experiment. However it was also found that the control cycle time should be equal to or less than 2 millisecond to enable the process to be more stable. Therefore, the m should be taken as 4. This 4 step transfer delay actually increaser the order of the control system. On-line simulation experiments and theoretical analysis revealed that when the transfer delay is large enough and the order of the mathematical model is very high, the on-line estimated model might become non-minimum phase. Roots of the characteristic equation given by the left side of (5) can locate on the out-side of the unit circle of the r-plane. Thus the non-minimum phase feature may involve in the control strategy to cause the process unstable.
DEVELOPMENT O F EDM SERVO ADAPTIVE CONTROL In order to simplify the entire structure of the whole system, this control unit has been condensed into a simple analog circuit and can be implemented in any EDM unit. With this control circuit a fully automatic EDM equipment can be built. Since the digital computer has been replaced by the analog computing circuit, the hardware structure and the cost of this system is substantially reduced.
Experimental investigations revealed that the use of the model of equation ( 5 ) d a s not yield good machining stability and performance. Therefore, in order to obtain very steady mathematical model from process data, following polynomial was used to fit this controlled process as: Y(k+m+l) = ~ ( k ) t b o ~ d ( k ) t b c t e k +I)m +
(6)
This is a high order partially known integrator model with time-varying gain and off-set. Parameters bo and b, are uncertainty and can be on-line estimated by a system identification algorithm. F-TUNING REGULATOR FOR EDM SERVO CONTROL Figure 2 shows the block diagram of the self tuning regulator for EDM servo control. Out-side control signal Y , is generated by an optimization program [12,19,20]. An estimator algorithm was used to identify the uncertain parameters of the model, the control strategy was designed based upon the identified model. The minimum-variance control concept was used to design the controller.
Firure 3. Block Diagram of the EDM Servo Adaptive Control Circuit.
Figure 3 shows the block diagram of the control circuit. This control circuit should be considered as a continuous-time system. Equation (6) is converted as a continuous differential equation as:
Figure 2. Block Diagram of the Self-Tuning Regulator for EDM Servo Control.
Td is the delay time of system transportation-lag. and f stands for the continuous time. The transfer function of equation (12) with Laplace-transform is: s Y(s)-b~-TdSVsd(s)+b,+e(S)
In order to assure the control cycle time to be less than 2 millisecond, a very high speed microcomputer, IBM 386, was used to set up this control system and a fast recursive least-square estimating algorithm [ I ] was implemented. Whole control software was written in C language. In this algorithm the estimated values of bo and b, are represented as b, and b,. The b, and b, at the time point k t l can be obtained by following q u a t i o n r
s is Laplace operator. Y(s). &d(s) and
(13)
4 s ) are the Laplace-transform of process
output Y ( f ) , input Vs,j(r)and noise df) respectively. bo and 6, are the uncertain parameters of the process. and can be estimated by filter circuits. Therefore. the method of designing the analog estimator is the key element for developing this adaptive controller. The estimated values of b, and bc are also represented as b, and b,. The differential equations of this filter are given as below:
And then the control signal can be achieved as:
Vsd(k)-
YAk)-YOc)-be(k) bdk)tO.Olv(k)
(9)
The transfer functions of above equations with Laplace-transform are: In equation (9) the control signal V,&k) is generated according to the minimumvariance and cautious control laws. In many machining cases the process noise is very intense to cause more uncertainty of parameter estimations and to increase the process variance. The cautious control strategy is derived with either the estimated parameters or their uncertainties. It can be derived that the
188
v(r) v(t)
(26)
;'(I)
-
In order to study the effect of the new EDM servo adaptive control system, experiments were designed to compare the on-line recorded data obtained by the self-tuning regulator with the data from a fixed-gain controller having same control feed-back loop, and as well as the model reference adaptive controller reported in [ 121. The fixed-gain controller means the control algorithm in Figure 2 is a fixed proportional unit and. the gain does not adaptively change with the machining condition.
(20)
Therefore the estimation of error signal &) given by equation (19) can simultaneously be approximated as:
.. wi h 9 Figure 4 shows the graph of autocovariances derived from the data obtained with the self-tuning regulator for the EDM servo adaptive control and with the fixedgain servo controller under good-flushing condition. These two experiments were conducted under similar conditions. It is evidently proved from Figure 4 that the variance of Y with this adaptive control strategy is 35% lower than that without adaptive control with same control loop. The autocovariance of Y with the adaptive control is closer to white noise, the machining stability has been obviously improved and the process was maintained in optimum condition.
The error of t h h approximation can be compensated by consequent circuits whose transfer functions have a numerator factor (Tlstl). The time-lag value of Qd can also be approximately obtained by a filter which is formed by first two terms of the series expansion of function e-TdSas: -T 5
-
The locations of closed-loop poles can be assigned by properly selecting parameters K, and K,. During machining, the control system adaptively adjusts the feed-forward gain to keep the closed-loop poles of entire process not be changed for achieving minimum process variance.
Above transfer functions have two technical difficulties for realizing in specific circuit. One of them is the measurement of differential value of Y due to the zero-off-set of analog devices, and the another is obtaining the time-lag value of Qd. First difficulty can be solved by using a passive first order high-pass filter to detect the approximate differential value Y' of Y as:
C
is the variance of estimation, and it can be obtained as:
Figure 5 shows the same parameters obtained under Door flushing situation. Since the discharge gap was easy to pollute by eroded debris, chips and the carbon blacks separated from the dielectric fluid, the process had intense disturbances, the covariance is much bigger than those with good flushing shown in Figure 4. Figure Sb shows the autocovariance without adaptive control. The data of Y can be found to have a big owillation in thh graph. the gap had more open and short circuit pulse, and l e u sparking. Once the self-tuning regulator algorithm was involved in the servo control. the data of Y had very good damping and thus (see Figure 5a) the stability was found to be improved.
Fd m (1-Td.V)
Whole transfer functions of the estimator are:
.. p tivitv with the Self-Tunin% B4pu Iat0r , The experiments for testing the improvement of machining productivity with the self-tuning regulator for EDM servo adaptive control were conducted by machining of small holes and medium-sire cavities. The results are compared with the fixed-gain servo controller and the model reference adaptive control system reported in 1121.
The controller is defined similar to equation ( 9 ) as:
E
.2 8 L.
k
i
-
4
6 4 2
0
-2 0
1
0
Z
O
k
~
4
0
- time point
5
0
~
7
0
~
9
o
l
~
0
1
O
~
m
4 0 5 0 6 k time point
-
0
7
0
e
J
3
~
1
m
a. with self-tuning regulator
a. with self-tuning regulator
IY
16 14
o i o m m a m m m m m i c n k
- time point
o
1
o
m
s
a ~ 6 0 7 k time point
-
0
e
J
3
9
o
l
b. with fixed-gain controller
b. with fixed-gain controller
Figure 4. Autocovariances of Data Y with Good-Flushing. (Test Condition: Y,-lOO%, on-time: 2 5 0 ~ .off-time: 250p, peak current: I5A. machining area: 93mm2. tool/workpiece: copper(+)/steel)
Figure 5. Autocovariances of Data Y without Flushing. (Test Condition: Y,-lOO%, on-time: 2 5 0 ~ off-time: . 250~s.peak current: I 5 A . machining area: 93mm2, machining depth > 2mm. twl/workpiece: copper(+)/steel)
m
189
Table I shows the comparison of the productivity achieved by the self-tuning regulator and the model reference control algorithm when machining a small hole. The table provides an average value of a series of data obtained during the experiments. Since this new EDM servo adaptive controller can provide more stable machining during small energy sparking, the productivity can be seen to have been improved approximately by 20%.
Table I.
Improvement of Productivity in Machining Small Hole with the Self-Tuning Regulator. (Testing Condition: Y,.-180%. pulse on-time: 100iis. pulse offtime: loop, peak current: 6A. machining area: 3mm*, tool/workpiece: graphite(+)lsteel)
-
Self-Tuning Regulators Some Connections, Proceedings OJ 7rh I F A C H'orld Congrers. Helsinki, pp. 1973-1980. Mahmoud, M. 5.. and Singh. M. G.. 1984. - Analvsis. Control and OotimizatiaQ. Springer-Verlag, Berlin, Heidelberg. New York. Tokyo. Pandit. S.M.. and Mueller, T.M.. 1987. Verification of On-line Computer Control of EDM by DDS. ASME Journal OJ Engineering Jar l n d u w y , Vol. 109. pp. 117-121. Pandit, S. M., Wu, S. M., 1983. Time AmlicationS. John Wiley and Sons, New York.
vsis with
Rajurkar. K. P,, and Pandit. S. M., 1988, Recent Progress in Adaptive Technology and Research of EDM. Proceedtngq OJ Control of EDM MonuJocturing 11rrernorrona1'88.ASME, Vol. I. pp. 2 19-226.
-
Control Mode
Self-Tuning Regulator
Productivity (mms/minute)
Model Reference Control
0.29
Improvement
0.23
26%
Rajurkar, K. P.. Pandit. S. M.. and Wittig, W. H., 1983, Pulse Current Signal as a Sensor for On-Line Computer control of EDM, Proceedings OJ l l r h NAMRC. pp. 379-385. Rajurkar. K. P.. Springer, 1. E., 1987. A Study of Electrical Discharges in EDM, Proceedings of 15th NAMRC. pp. 405-412.
Table 2 shows the improvement of the productivity in the machining of a medium-size cavity without flushing. As the process was maintained in optimum condition when non-flushing machining, and the process data Y has very good damping characteristics. more sparking less gap-open and gap-short were achieved. Therefore, the average improvement of the machining rate is more than 15%.
Rajurkar, K. P., Wang, W. M., 1989. A New Model Reference Adaptive Control of EDM. Annols OJ the C I R P . Vol. 3811, pp. 183-186. Schumacher, B. M., 1983, EDM Technology for Precision Workpieces with Excellent Surface Quality, Proc. OJ the 7th lnfernotionol Symposium on Elecfromachirring, Ed. Crookall, J. R., pp. 124- 135. Shaw, T.W.. Lee, L. C., and Crookall. J. R.. 1979, Automation of the EDM Process, Proceedings OJ 20fh M T D R , pp. 591 -598.
Table 2.
Machining Depth (mm)
Improvement of Productivity of Machining Medium-Size Cavity with the Self-Tuning Regulator. (Testing Condition: Y,-IOO%. pulse on-time: ZSOps, pulse offtime: 250~s. peak current ISA, machining a r e s 93mm'. tool/workpiece: copper(+)/steel) Self-Tuning Reaulator (mmJiminute)
1-3 3-4.6 4.6-5.6
Model Reference Control (mmslminute)
20.8 18 14.6
17.5 15.4 11.8
Fixed-Gain Control (mmJ/minute) 16.5
Improvement
Snoeys, R.. Dauw. D.. and Kruth, J. P., 1980, Improved Adaptive Control System for EDM. Annolr OJ lhe C I R P , Vol. 29/1, pp. 97-101. Snoeys, R., Staelens. F., and Dauw, D.. 1986, Adaptive Control Optimization as Basis for Intelligent EDM Die Sinking Machines, Advances in NonTroditiorrol Machining, ASME PED, Vol. 22. Ed. Rajurkar. K. P.. er al, pp. 63-78. Stackhouse. J.. 1986, Production EDM in the Jet Engine Industry. E D M Digest. SeptemberlOctober, pp. 8- 17.
15-19% 17% 25%
Stealens. F., Kruth, J. P., 1989, A Computer Integrated Machining Strategy for Planetary EDM, Annolr OJ the C I R P . Vol. 38/1. pp. 187-190. Wang, W. M., 1985. Investigations of Detecting Discharge Parameters and Adaptive Control for EDM. Ph.D. Thesis, Harbin Institute of Technology, Harbin. China.
SUMMARY AN D CONCLUSIOt i : This paper proposes a new EDM servo adaptive control system which directly regulates the servo feed rate in accordance with the stochastic control law. The control strategy is developed based upon a real-time achieved stochastic model to minimize the variances of the controlled process for improving the machining stability and increasing the productivity. The conclusions of development of this control system is listed as below. The self-tuning regulator for EDM servo adaptive control proposed in this paper can identify on-line the mathematical model which represents the relationship between the servo driving signal and the output value of EDM process formed by discharge parameters. With this control system the process stability was found to be significantly improved. The variance of the process output data is at least 10% lower than the that achieved by other control methods. As the new EDM servo adaptive controller improves the process stability to have more normal sparking and less gap-open and gap-short pulses than other control systems and as well as the model reference adaptive controller early developed by the authors, the machining productivity was found to be increased by 15% when machining small holes and medium-size cavities.
ACKNOWLEDGEMElrlT This work was partially supported by the National Science Foundation under grant number DDM-8906372.
REFERENCES [ I ] Astr6m. K. 1.. Wittenmark, 8.. 1989. -ti
ve Contrd, Addison-Wesley,
Reading Massachusetts. Snoeys, R., and Dekeyser. W., 1983, Advanced Pulse Discriminating System for EDM Process Analysis and Control, Annolr OJ fhe
[2] Dauw, D. F..
C I R P . Vol. 32/l, pp. 541-49.
Adam ive Control: yheorv and ADD^^. Peter Peregrinus Ltd, Stevenage, UK. and New York.
131 Harris, C. J., and .Billings, S. A,, 1981, .
141 KOnig. W., Dauw, D. F., Levy, G., and Panten. U., 1988, EDM-Future Steps Towards the Machining of Ceramics, An~rolsOJ the C I R P , Vol. 3712.
IS] Ljung. L. and Landau, 1. D.. 1978, Model Reference Adaptive Systems and
190
1201 Wang. W. M., 1988, A New EDM Adaptive Control Plan Using Self-Tuning Control Algorithm, Proceedrngs OJ LfonuJocturntg 1nfernoftor1o1'88.ASME. Vol. I , pp. 227-233.
K E Y WORDS:
EDM, Adaptive C o n t r o l , S t o c h a s t i c P r o c e s s , System I d e n t i f i c a t i o n .
INTRODUCTION The servo control unit in EDM machine is one of the most important Control devices for EDM process as it directly and in real-time regulates the dimensions of discharge gap. The dynamic features of EDM servo control unit strongly influences the stability of machining process. Conventional EDM servo control system takes only the average gap voltage Vs to be the feed back signal and to compare this signal with a so called servo reference voltage V,. The difference signal of this comparison drives the servo mechanism to adjust the gap size so that the average Vg is kept at the value specified by V,. It is known that the conventional servo system does not effectively respond either the parameters of different gap states or the dynamic and stochastic features of the on-going machining process. The machine operators only preset the V, according to his or her experiences. thus the machining process can not be optimum. In recent years, many EDM specialists have developed different kinds of monitoring apparatus to detect and measure the parameters of gap states, such as open. short, sparking. arcing and transient arcing etc. [2.9,10.14,19], and created many adaptive control strategies to on-line modify the servo reference voltage V, through a sub-feedback loop to search the optimum value of the parameters of every gap states for achieving maximum machining productivity and minimum arc damage [7,9,12,14l6.19.20]. The authors developed a model reference adaptive controller 1121 that improves the machining productivity and avoid the arc damage when machining deep blind hole with non-flushing. However those control methods only adjust the input of the servo control loop V,, without addressing the internal dynamic characteristics of this closed loop system and stochastic features of the machining process. EDM process is of very strong stochastic nature, particularly in cases of finishing or semi-finishing, machining with non-flushing condition. steel to steel machining and machining modern advanced materials (engineering ceramics and carbides). The process can not be very stable, and there are so many gap-open and gap-Short pulses and less sparking. The variations of average gap voltage are very frequent and large even with good flush condition. It is necessary to develop high stable EDM servo control system either to improve the current machining performance or to meet the future needs of machining advanced materials [4].
linearized [12.19.20], and therefore, the output of the process is defined as:
Y
2 f d / f it fe/l;
The output Y can correctly reflect the gap status. When gap is complete open, is 100% and when the gap is the Y equals to 2 as the gap-open ratio (I&,) optimal the Y is I because the spark ratio ( f e / f , ) is 100%. If the gap is highly polluted, there is no gap-open and normal spark in the gap. the Y equals to 0. Figure I shows the block diagram of the servo control unit and the EDM process. vsd is the driving signal of servo system mechanism. The relationship between and the process output Y can be described by a mathematical model as a difference equation that accounts for the dynamic and stochastic features of the process [1,3,6.8.10.12]. For correctly determining the structure of the mathematical model and designing an effective control strategy, it is necessary to understand the dynamic and stochastic characteristics of the controlled process. According to the theoretical analysis, the servo control and electrical discharge process is a high order system. However, in order to simplify the control algorithm for achieving better stability. this process was initially considered to be second order. Since the stable feed rate of the servo head is proportional to the servo driving signal V,d. and the changing velocity of the output of the process Y is almost
servo head
I I It
5
In this paper a new EDM servo control system is proposed. A high speed microcomputer directly regulates the servo feed rate instead of servo reference voltage. and takes the parameters of discharge time ratios from an EDM gap monitor as the feed back signal. This control system is designed for a hydraulic servo head and relative control circuit on the Eltee Pulsitron TRM-20 EDM machine. The on-line control strategy is made according to the real-time achieved stochastic model to minimize the variances of the controlled process and increase the machining speed with the control interval time of 2 millisecond. Extensive experimental investigation shows that with this control method the machining productivity is IS% higher than the values obtained by the model reference adaptive controller. reported in [12], when machining small holes and medium-sire cavities. The model reference controller ikelf has been found to be 4D-50% more productive than the existing control system of the machine 1121. The new control system has been condensed into a simple control circuit and can be implemented on any EDM unit. Description of the stochastic model of the EDM servo control is included in the second section. The principles of the self-tuning regulator for EDM servo adaptive control is described in third section, Fourth =tion includes the technology of development of the EDM servo adaptive control circuit. The experimental details and results are given in fifth section. Last section concludes this paper.
(I)
Servo feed
Workpiece
- Process OUtPUt
Figure I . Input and Output of the EDM Servo Control System
proportional to V,d. this controlled process can be considered as a second order integrator " foilowing:
Y(k+l)-(atI ) Y(k)-oY(k-I ) + h o ~ ~ ( k - m ) + h t ~ ~ k - ~ ~ - l ) t h ~ i e ( & t l )
(2)
k = 1.2.3... Above polynomial is an ARMA model. 1 given as: In order to let the adaptive control strategy be generated based upon the parameters of gap states, an EDM discharge parameters monitor was used to be the detecting unit in the control system. This monitoring system deteck the gap voltage, current and the high frequency signal generated by the to identify five basic EDM gap states and corresponding five analog voltages which are proportional to the time ratios of each gap states in the pulse duration These parameters are the time ratios of gap Open tdlfi, normal sparking f e / f i . transient arc f e , / f i , stable arc and short circuit For the having better performance, the process should be
Annals of the ClRP Vol. 3WlSk.I
Its transfer function with ;-transform is
'
(Z-'X'-o~~t)Y=(bo~hlZ~l'Z~mYSdtbc'~
Integer k is the instant time point. and m. another integer. is the restmme delay Steps O r the process. Coefficients a. bo and hi repreent the dynamics of the system, and their values are time variant with different machining conditions. Coefficient bc stands for the 0 f f - w of the whole procea, its value also changes with machining conditions, machining speed, the weight of the workpiece and the off-set of the servo driving circuit and servo valve. e(ktl) is the random disturbance with zero averane at time m i n t k+l in the controlled Drocess. Parameters of a. ho. h,, b, can be estimated on-line.
187
Equation (2) can be implemented to describe and analyze the transfer relation between Y and &d. However, it is not convenient to directly use this model to develop the control strategy as the value of &d(k) should be generated with the information of a future value Y(k+m) when m is larger than 0. At time point k. the value of Y ( k t m ) is not available, but it can be estimated by a filter. However, EDM is a typically stochastic process [7,9-201. and Y is a random variable [12]. It was found through a series of experiments that the future value of Y can not be precisely on-line estimated. Therefore, the equation (2) can be converted in the form as [I,3.5.61: Y ( k t l ) - ( l + u , ) Y(k-m)-u,Y(k-m- I )tboV,d(k-m)+b,V,d(k-m-I )+b,te(ktl)
(4)
The transfer function with z-transform can be written as: 12-(I + a , ) : - m t U ~ ~ ~ m ~ ' ] ~ - ( b ~ t ~ l ~ ~e l ) ~ ~ m ~ d + b c + ~
uncertainties of the parameter estimates are proportional to the variance of the identification algorithm [l,3.6]. The v(k) in equation (9) is the variance of the parameter estimator. This self-tuning regulator optimizes system feed forward gain based upon the state value Y . parameters bo and b, and as well as the variances of the estimates v(k). The error of the estimates is defined as:
a&)
&k)-Y(k)-Y(k-m- I )-b,(k)V,d(k-m- I )-b,(L.)
00)
The variance of the estimation can be achieved by a first order filter. v(k)-0.99 v(k-I)+O.Ol i'(k)
(11)
Extensive experiments show that the machining performance and stability of EDM process can be improved by this self-tuning regulator. (5)
m is an important factor and its value depends on the response delay time and
the cycle time of the control system. The system response delay time was found to be 8 millisecond in a special designed experiment. However it was also found that the control cycle time should be equal to or less than 2 millisecond to enable the process to be more stable. Therefore, the m should be taken as 4. This 4 step transfer delay actually increaser the order of the control system. On-line simulation experiments and theoretical analysis revealed that when the transfer delay is large enough and the order of the mathematical model is very high, the on-line estimated model might become non-minimum phase. Roots of the characteristic equation given by the left side of (5) can locate on the out-side of the unit circle of the r-plane. Thus the non-minimum phase feature may involve in the control strategy to cause the process unstable.
DEVELOPMENT O F EDM SERVO ADAPTIVE CONTROL In order to simplify the entire structure of the whole system, this control unit has been condensed into a simple analog circuit and can be implemented in any EDM unit. With this control circuit a fully automatic EDM equipment can be built. Since the digital computer has been replaced by the analog computing circuit, the hardware structure and the cost of this system is substantially reduced.
Experimental investigations revealed that the use of the model of equation ( 5 ) d a s not yield good machining stability and performance. Therefore, in order to obtain very steady mathematical model from process data, following polynomial was used to fit this controlled process as: Y(k+m+l) = ~ ( k ) t b o ~ d ( k ) t b c t e k +I)m +
(6)
This is a high order partially known integrator model with time-varying gain and off-set. Parameters bo and b, are uncertainty and can be on-line estimated by a system identification algorithm. F-TUNING REGULATOR FOR EDM SERVO CONTROL Figure 2 shows the block diagram of the self tuning regulator for EDM servo control. Out-side control signal Y , is generated by an optimization program [12,19,20]. An estimator algorithm was used to identify the uncertain parameters of the model, the control strategy was designed based upon the identified model. The minimum-variance control concept was used to design the controller.
Firure 3. Block Diagram of the EDM Servo Adaptive Control Circuit.
Figure 3 shows the block diagram of the control circuit. This control circuit should be considered as a continuous-time system. Equation (6) is converted as a continuous differential equation as:
Figure 2. Block Diagram of the Self-Tuning Regulator for EDM Servo Control.
Td is the delay time of system transportation-lag. and f stands for the continuous time. The transfer function of equation (12) with Laplace-transform is: s Y(s)-b~-TdSVsd(s)+b,+e(S)
In order to assure the control cycle time to be less than 2 millisecond, a very high speed microcomputer, IBM 386, was used to set up this control system and a fast recursive least-square estimating algorithm [ I ] was implemented. Whole control software was written in C language. In this algorithm the estimated values of bo and b, are represented as b, and b,. The b, and b, at the time point k t l can be obtained by following q u a t i o n r
s is Laplace operator. Y(s). &d(s) and
(13)
4 s ) are the Laplace-transform of process
output Y ( f ) , input Vs,j(r)and noise df) respectively. bo and 6, are the uncertain parameters of the process. and can be estimated by filter circuits. Therefore. the method of designing the analog estimator is the key element for developing this adaptive controller. The estimated values of b, and bc are also represented as b, and b,. The differential equations of this filter are given as below:
And then the control signal can be achieved as:
Vsd(k)-
YAk)-YOc)-be(k) bdk)tO.Olv(k)
(9)
The transfer functions of above equations with Laplace-transform are: In equation (9) the control signal V,&k) is generated according to the minimumvariance and cautious control laws. In many machining cases the process noise is very intense to cause more uncertainty of parameter estimations and to increase the process variance. The cautious control strategy is derived with either the estimated parameters or their uncertainties. It can be derived that the
188
v(r) v(t)
(26)
;'(I)
-
In order to study the effect of the new EDM servo adaptive control system, experiments were designed to compare the on-line recorded data obtained by the self-tuning regulator with the data from a fixed-gain controller having same control feed-back loop, and as well as the model reference adaptive controller reported in [ 121. The fixed-gain controller means the control algorithm in Figure 2 is a fixed proportional unit and. the gain does not adaptively change with the machining condition.
(20)
Therefore the estimation of error signal &) given by equation (19) can simultaneously be approximated as:
.. wi h 9 Figure 4 shows the graph of autocovariances derived from the data obtained with the self-tuning regulator for the EDM servo adaptive control and with the fixedgain servo controller under good-flushing condition. These two experiments were conducted under similar conditions. It is evidently proved from Figure 4 that the variance of Y with this adaptive control strategy is 35% lower than that without adaptive control with same control loop. The autocovariance of Y with the adaptive control is closer to white noise, the machining stability has been obviously improved and the process was maintained in optimum condition.
The error of t h h approximation can be compensated by consequent circuits whose transfer functions have a numerator factor (Tlstl). The time-lag value of Qd can also be approximately obtained by a filter which is formed by first two terms of the series expansion of function e-TdSas: -T 5
-
The locations of closed-loop poles can be assigned by properly selecting parameters K, and K,. During machining, the control system adaptively adjusts the feed-forward gain to keep the closed-loop poles of entire process not be changed for achieving minimum process variance.
Above transfer functions have two technical difficulties for realizing in specific circuit. One of them is the measurement of differential value of Y due to the zero-off-set of analog devices, and the another is obtaining the time-lag value of Qd. First difficulty can be solved by using a passive first order high-pass filter to detect the approximate differential value Y' of Y as:
C
is the variance of estimation, and it can be obtained as:
Figure 5 shows the same parameters obtained under Door flushing situation. Since the discharge gap was easy to pollute by eroded debris, chips and the carbon blacks separated from the dielectric fluid, the process had intense disturbances, the covariance is much bigger than those with good flushing shown in Figure 4. Figure Sb shows the autocovariance without adaptive control. The data of Y can be found to have a big owillation in thh graph. the gap had more open and short circuit pulse, and l e u sparking. Once the self-tuning regulator algorithm was involved in the servo control. the data of Y had very good damping and thus (see Figure 5a) the stability was found to be improved.
Fd m (1-Td.V)
Whole transfer functions of the estimator are:
.. p tivitv with the Self-Tunin% B4pu Iat0r , The experiments for testing the improvement of machining productivity with the self-tuning regulator for EDM servo adaptive control were conducted by machining of small holes and medium-sire cavities. The results are compared with the fixed-gain servo controller and the model reference adaptive control system reported in 1121.
The controller is defined similar to equation ( 9 ) as:
E
.2 8 L.
k
i
-
4
6 4 2
0
-2 0
1
0
Z
O
k
~
4
0
- time point
5
0
~
7
0
~
9
o
l
~
0
1
O
~
m
4 0 5 0 6 k time point
-
0
7
0
e
J
3
~
1
m
a. with self-tuning regulator
a. with self-tuning regulator
IY
16 14
o i o m m a m m m m m i c n k
- time point
o
1
o
m
s
a ~ 6 0 7 k time point
-
0
e
J
3
9
o
l
b. with fixed-gain controller
b. with fixed-gain controller
Figure 4. Autocovariances of Data Y with Good-Flushing. (Test Condition: Y,-lOO%, on-time: 2 5 0 ~ .off-time: 250p, peak current: I5A. machining area: 93mm2. tool/workpiece: copper(+)/steel)
Figure 5. Autocovariances of Data Y without Flushing. (Test Condition: Y,-lOO%, on-time: 2 5 0 ~ off-time: . 250~s.peak current: I 5 A . machining area: 93mm2, machining depth > 2mm. twl/workpiece: copper(+)/steel)
m
189
Table I shows the comparison of the productivity achieved by the self-tuning regulator and the model reference control algorithm when machining a small hole. The table provides an average value of a series of data obtained during the experiments. Since this new EDM servo adaptive controller can provide more stable machining during small energy sparking, the productivity can be seen to have been improved approximately by 20%.
Table I.
Improvement of Productivity in Machining Small Hole with the Self-Tuning Regulator. (Testing Condition: Y,.-180%. pulse on-time: 100iis. pulse offtime: loop, peak current: 6A. machining area: 3mm*, tool/workpiece: graphite(+)lsteel)
-
Self-Tuning Regulators Some Connections, Proceedings OJ 7rh I F A C H'orld Congrers. Helsinki, pp. 1973-1980. Mahmoud, M. 5.. and Singh. M. G.. 1984. - Analvsis. Control and OotimizatiaQ. Springer-Verlag, Berlin, Heidelberg. New York. Tokyo. Pandit. S.M.. and Mueller, T.M.. 1987. Verification of On-line Computer Control of EDM by DDS. ASME Journal OJ Engineering Jar l n d u w y , Vol. 109. pp. 117-121. Pandit, S. M., Wu, S. M., 1983. Time AmlicationS. John Wiley and Sons, New York.
vsis with
Rajurkar. K. P,, and Pandit. S. M., 1988, Recent Progress in Adaptive Technology and Research of EDM. Proceedtngq OJ Control of EDM MonuJocturing 11rrernorrona1'88.ASME, Vol. I. pp. 2 19-226.
-
Control Mode
Self-Tuning Regulator
Productivity (mms/minute)
Model Reference Control
0.29
Improvement
0.23
26%
Rajurkar, K. P.. Pandit. S. M.. and Wittig, W. H., 1983, Pulse Current Signal as a Sensor for On-Line Computer control of EDM, Proceedings OJ l l r h NAMRC. pp. 379-385. Rajurkar. K. P.. Springer, 1. E., 1987. A Study of Electrical Discharges in EDM, Proceedings of 15th NAMRC. pp. 405-412.
Table 2 shows the improvement of the productivity in the machining of a medium-size cavity without flushing. As the process was maintained in optimum condition when non-flushing machining, and the process data Y has very good damping characteristics. more sparking less gap-open and gap-short were achieved. Therefore, the average improvement of the machining rate is more than 15%.
Rajurkar, K. P., Wang, W. M., 1989. A New Model Reference Adaptive Control of EDM. Annols OJ the C I R P . Vol. 3811, pp. 183-186. Schumacher, B. M., 1983, EDM Technology for Precision Workpieces with Excellent Surface Quality, Proc. OJ the 7th lnfernotionol Symposium on Elecfromachirring, Ed. Crookall, J. R., pp. 124- 135. Shaw, T.W.. Lee, L. C., and Crookall. J. R.. 1979, Automation of the EDM Process, Proceedings OJ 20fh M T D R , pp. 591 -598.
Table 2.
Machining Depth (mm)
Improvement of Productivity of Machining Medium-Size Cavity with the Self-Tuning Regulator. (Testing Condition: Y,-IOO%. pulse on-time: ZSOps, pulse offtime: 250~s. peak current ISA, machining a r e s 93mm'. tool/workpiece: copper(+)/steel) Self-Tuning Reaulator (mmJiminute)
1-3 3-4.6 4.6-5.6
Model Reference Control (mmslminute)
20.8 18 14.6
17.5 15.4 11.8
Fixed-Gain Control (mmJ/minute) 16.5
Improvement
Snoeys, R.. Dauw. D.. and Kruth, J. P., 1980, Improved Adaptive Control System for EDM. Annolr OJ lhe C I R P , Vol. 29/1, pp. 97-101. Snoeys, R., Staelens. F., and Dauw, D.. 1986, Adaptive Control Optimization as Basis for Intelligent EDM Die Sinking Machines, Advances in NonTroditiorrol Machining, ASME PED, Vol. 22. Ed. Rajurkar. K. P.. er al, pp. 63-78. Stackhouse. J.. 1986, Production EDM in the Jet Engine Industry. E D M Digest. SeptemberlOctober, pp. 8- 17.
15-19% 17% 25%
Stealens. F., Kruth, J. P., 1989, A Computer Integrated Machining Strategy for Planetary EDM, Annolr OJ the C I R P . Vol. 38/1. pp. 187-190. Wang, W. M., 1985. Investigations of Detecting Discharge Parameters and Adaptive Control for EDM. Ph.D. Thesis, Harbin Institute of Technology, Harbin. China.
SUMMARY AN D CONCLUSIOt i : This paper proposes a new EDM servo adaptive control system which directly regulates the servo feed rate in accordance with the stochastic control law. The control strategy is developed based upon a real-time achieved stochastic model to minimize the variances of the controlled process for improving the machining stability and increasing the productivity. The conclusions of development of this control system is listed as below. The self-tuning regulator for EDM servo adaptive control proposed in this paper can identify on-line the mathematical model which represents the relationship between the servo driving signal and the output value of EDM process formed by discharge parameters. With this control system the process stability was found to be significantly improved. The variance of the process output data is at least 10% lower than the that achieved by other control methods. As the new EDM servo adaptive controller improves the process stability to have more normal sparking and less gap-open and gap-short pulses than other control systems and as well as the model reference adaptive controller early developed by the authors, the machining productivity was found to be increased by 15% when machining small holes and medium-size cavities.
ACKNOWLEDGEMElrlT This work was partially supported by the National Science Foundation under grant number DDM-8906372.
REFERENCES [ I ] Astr6m. K. 1.. Wittenmark, 8.. 1989. -ti
ve Contrd, Addison-Wesley,
Reading Massachusetts. Snoeys, R., and Dekeyser. W., 1983, Advanced Pulse Discriminating System for EDM Process Analysis and Control, Annolr OJ fhe
[2] Dauw, D. F..
C I R P . Vol. 32/l, pp. 541-49.
Adam ive Control: yheorv and ADD^^. Peter Peregrinus Ltd, Stevenage, UK. and New York.
131 Harris, C. J., and .Billings, S. A,, 1981, .
141 KOnig. W., Dauw, D. F., Levy, G., and Panten. U., 1988, EDM-Future Steps Towards the Machining of Ceramics, An~rolsOJ the C I R P , Vol. 3712.
IS] Ljung. L. and Landau, 1. D.. 1978, Model Reference Adaptive Systems and
190
1201 Wang. W. M., 1988, A New EDM Adaptive Control Plan Using Self-Tuning Control Algorithm, Proceedrngs OJ LfonuJocturntg 1nfernoftor1o1'88.ASME. Vol. I , pp. 227-233.