WEE TRANSACTIONS ON INDUSTRIALJELECTRONICS AND CONTROL INSTRUMENTATION, VOL. iEci-22, NO. 3, AUGUST 1975

Control of the Laser Pattern GeneratorJAAN RAAMOT, MEMBER, IEEE

Abstract-The laser pattern generator uses a control algorithm de-.rived from information%theory. It is implemented by applying shapedinput waveforms and predicting the output. The performance indexof, this complementary control is several magnitudes better than thatachieved by the solution of the Riccati equation.

INTRODUCTIONAN optimal Control operates

systemtions to bring it from an unknown state to a known

state, and then maintains it there. The constraints on the'law are formulated to be in agreement with what thesystem can do. Thus, a bang-bang control law may bringthe system to a desired state in the least amount of time,or a control:may be implemented by optimizing a quad-ratic performance indx.The fact that such- control laws are optimal suggests

this,is the best that can be done. In these applications, theproblem is formulated to fit the solution. In other controlproblems, the application of an optimal control law maybe the right solution to the wrong problem.

This paper: describes the control of a laser patterngenerator [11. It has a performance index which is severalmagnitudes better than that obtainable by an optimalcontrol law. The optimal control law is not wrong, but thecontrol Rcan be redefined to exist outside the set of optimalcontrol solutions.

THE CONTROL PROBLEMThe laser pattern generator vaporizes a metallic thin

film from a substrate by directing a focused laser beamonto it, as is illustrated in Figs. 1 and 2. Patterns aremachined by vaporizing adjacent spots with 50 overlap[2], [3]. The beam movement is controlled by twogalvanometers with mirrors mounted on two orthogonalaxes so that there is no cross coupling aside from opticaleffects [4].,:The galvanometers are equivalent to small dc motors

Manuscript received February 3, 1975.The author is with the Western Electric Company, Inc., Princeton,

N.J. 08540.

Fig. 1. Functional block diagram of the laser pattern generator.

~ ~ YA

TH IN

PATTERN DETECTOR

Fig. 2. The optical components of the laser pattern generator.A pattern is machined on a thin-film plate by controlling the(xy) position of the focused YAG laser beam.

where the shaft angle is proportional to the current. Agood approximation to the beam position, xl, in one axis,as a function of the input current, u, is the matrix equation,

x - Ax +Bu ()where the state matrix is,

0 1 k 0.258 X 10-'(,.s)-

_k @2 15.15 X 105-(rad/s)2. (2)1- ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~~~~~. . .-f

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,UPATERIt is- a linear second order system of equations. Ne-

glected; is the, magnetic hysteresis in the galvanometeriron, and the characteristics of the drive amplifier for thegalvanometer. The latter was developed to drive the30 mH: galvanometer inductance up to 1.5 A with neg-ligible rise time or overshoot.The thin film, pattern is generated by machining fill lines

parallel to one axis [5], [6]. There are strict accuracy,requirements ( 1 part in 2 X 104) on all vaporized spots.:The beam position is sensed to a precision of + 1 part in8 X 104 [7].The control problem as stated so far appears to have a

ready made solution. The feedback gain, as determined bythe solution of the Riccati equation4will optimize a quad-ratic performance index. The desired results are obtainedby weighting the error penalty matrix. A sensitivityanalysis in this application is examined in [8].The solution of the optimal tracking problem indicates

that fill lines can be machined at a maximum rate of Ispot in 10 ms while maintaining the same degree ofaccuracy. The desired machining rate is 1 spot in 20 As.Otherwise the pattern generation time becomes too long;for practical applications.The accuracy-time relationship can be regarded as a

gain-bandwidth: product. Shown in Fig. 3 is the gain-bandwidth product of the galvanometer as compared tothe desired pattern machining rate [9]. A linear closedloop control can be executed to operate anywhere to theshaded region. The quadratic optimal control law extendsthe shaded region to better accuracies, but it cannot berealized at the desired operating point.

COMPLEMENTARY CONTROLAn alternate approach to the control problem is to

reexamine the control constraints. First, the optimal:control law brings the system of equations from anunknown state to a known'state. However, once the errorsignal goes to zero, there is nothing left to do in theoptimal control law. Second, in normal operation thepattern generator error signal is expected to be near zero.Third, if the error goes outside its limits, then machininghas to be terminated and restarted at the last good spot.The only ndesirable effect of that is a time penalty in sucha procedure. These control constraints are on the statevector and not on the conitrol. Therefore, the time-optimal control policy demonstrated by Luus [10] wasnot investigated.

In the alternate approach, the control system isdefined as an information channel with the galvanometercurrent, u x3, being the input and the laser beamposition, xl, the output. Control is achieved when

information. uncertainty =. 0. (3)The control equation (3) is realized when to the output

is assigned the binary state, v, which is zero for falseposition and one for true position. In addition the outputis taken to have the probabilities between p 1 for zero:error signal (true position) and p 0 for nonzero error(false position). Then the control equation becomes,

CLOSED LOOP CONTROL

z

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0

4

4c

10

TIMESECONDS

PATTERN,-0-l GENERATION

I

II

I1,,

102 lo3 1-4 15 O6(FREQUENCY)

Fig. 3. The galvanometer gain bandwidth encloses the shadedregion in this accuracy-time graph. The laser pattern generatormachines at an accuracy of 1 part in 104 per 20 p.s

1(v) - -Y3p(v) log p(v)

--plogp-(1 p)log(1-p)l ( 4)It follows from these definitions that for p = 0, the

closed loop control is active, and for p = lit is in the deadband and is inactive. The control equation (3) also impliesthe existence of a complementary control where p = 0 is_an abort state and p = 1 is the active state. The com-plementary control can be realized by predicting the stateof the system of equations. If the prediction is true, thenpi 1 and the information uncertainty is zero [11].The complementary control characteristic is illustrated

in Fig. 4. Given any state of the system, its future state canbe predicted with good accuracy over short intervals oftime, as is indicated by the shaded region. The demarca-tion between the unknown state and the predicted state isdetermined by channel noise, which includes all non-linearities.The desired operation point is well within the shaded

region. The complementary control exists outside all gain-bandwidth limitations. As a matter of fact, a faster ma-chining rate would ifnprove its performance. The com-plementary control is realized by shaping the galvanometercurrent waveforms to produce an oscillation free. response.This makes it easy to predict the beam position.

IMPLEMENTATION OF THECOMPLEMENTARY

CONTROL

One shaped waveform is illustrated in Fig. 5-. It is a twostep used to change the system between two states at rest[12]. On applying the first step, the galvanometer over-shoots. At the peak of the overshoot, the input is,broughtto its steady-state value at that overshoot, which forces

POSITIONVELOCITY

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RAAM r: %LASER: PATERN GENERATOR253

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7.

5.

2.

0.0.0 2.5 5.0

time7.5 10.0

Fig. 6. The galvanometer position in response to a ramp segment.The position osillates about the steady-state values.

~~-4 -5 -6TIME 10 10-3 to0 10 10(FREQUENY)-

Fig. 4. The complementary control accuracy-time characteristic.At zero time, given an accurate state of the system, i.e., positionand velocity, then its future state can be predicted over a shortinterval of time.

0

u

0-

5.0time

Fig. 5. The galvanometer position in response to a steady-stateinput change fom 2.0 to 3.0, and the response to a two-step from0, to 1.85.

system tocome

to

rest. The time optimal solutionnot implemented because of the difficulty in realizing thebang-bang waveformls with sufficient accuracy. Also, atbest it shortens the time by 50% [13].

In the two-step positioning, each step may be approxi-mated by a short ramp. Simulations have shown that ifthe ramp duration is less than 3% of the period of oscil-lation,: then the system response of a ramp two step isindistinguishable from an ideal two step to within theaccuracy constraints. For example, 10 volts appliedacross a 30 mH galvanometer for 100 As will produce a.current change of 3.3 mA. An overshoot of 90% implies thesecond step is 3.0 mA, and the maximum two-step rangeis: 6.3 mA.The galvanometer ramp response is shown in Fig. 6. The

initial output. oscillates about the steady-state rampreponse. The input:waveform can be shaped similarly tothe tuwo step to produce an oscillation free ramp. Simula-

0al

0.0 2.5 5.0 7.5: 10.0timb

Fig. 7. The galvanometer response to a shaped ramp waveform.

tions and practice-showed this to be the case. The result isshown in Fig. 7.The shaped ramp response was simulated to investigate

the sensitivity of the waveform parameters. The results aredisplayed in Fig. 8. The upper accuracy bound is 10 psfor a 20 p.s machining rate. This translates -to a 2.53%variation in the parameters. The latter figure is the sum ofall variations, but nevertheless is easily realizable.The rqverse slype can ajAything ketwbae a vertical

and equal negative slope. A longer reverse slope cannotproduce an oscillation free response. Further ivestiga..tions showed that the expected nonlinearities do notproduce oscillations.

THE CONTROL ALGORITHMGood closed-loop control systems do not fall apart when

the error signal goes-to zero. Likewise the comiplementarycontrol does not have to abort at all instances when theprediction is not true. Some dynamic corrections can beinitiated which reestablish the correct prediction.The control algorithm is a two-step to the start of a fill

line acceleration point. If the position is false, then two-:step corrections are initiated until the prediction: is true.Subsequently, a shaped ramp is: applied, and after theinitial reverse slope the actual position is ompared withthe-predicted one. A lead error implies the position is pastthe intended one and an abort is initiated. However, a4lag.

COMPLEMENTARY CHARACTERISTIC

.50 .

00-

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.00

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IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS AND CONTROL INSTRUMENTATION, AUGUST 1975

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Fig. 9. Short line segments machined with a 25 pm spot. The topleft frame illustrates the sequence in which the segments weremachined. Below it, the lines are machined with no corrections.In subsequent frames, the correction tolerance is 480 am, b40jAm, -420 rm, and 4-10 pm.

. .

TI ME

T * RETURN PULSE WIDTHA * RETURN PULSE AMPLITUDEt * RETURN PULSE START TIMEW * GALVANOMETER RESONANT FREQUENCY

Fig. 8. The sensitivity of the conditioned ramp response to thegalvanometer and input parameters.

error is a typical occurrence caused by hysteresis. Then theposition difference is monitored until it is within theacceptable tolerance and machining can commence.During machining, both lead and lag errors can be

corrected dynamically, but an error in the orthogonalaxis forces an abort. The results are illustrated in Fig. 9.The frames show the galvanometer positioning sequenceand the results. The frame below the positioning sequenceshows machining with no corrections, and in subsequentframes the error tolerance is i80 jAm, t40 Mm, 20 Am,and i 10 jAm. Actual pattern generation occurs at a toler-ance of +5 Am.The system simulation showed that the complementary

control has to execute the predictions in 2.5 As incrementsto 17 bit accuracy. This forced most of the control al-gorithm to be implemented in hardware. It consists ofmore than 500 TTL integrated circuits.

RESULTSFig. 10 shows four machined squares, each 0.5 cm X 0.5

cm and separated by 5.78 cm. The pattern contains 2000fill lines and 106 machined spots. It was machined in 76seconds. Thus, the average machining rate is 76 Ms perspot even though the actual rate was 20 Ms per spot. Thedifference is due to initial positioning and aborts.The galvanometers used in this example have a period of

oscillation of 8 ms. The corresponding optimal controlmachining rate is approximately 20 ms per spot, whichimplies a machining time for the same pattern of 2 X 104

..

Fig. 10. Four machined squares spaced 5.78 em. The squares are0.5 cm X 0.5 cm and contain a total of 106 spots.

seconds. Let the quadratic performance index beT

PI = (e'Qe + u'Ru) dt (5)

where e is the error and u is the control. A comparison ofthese variables in Figs. and 7 with the corresponding onesin [8] shows that both amplitudes are approximately thesame. Assign the same weighting matrices Q and R to both,the optimal and complementary control systems. Then, themain difference is the time in which the task is completed.It is directly proportional to the performance index, andtherefore the complementary control operation is better bya factor of more than 102.

CONCLUSIONThere is no value to knowing what the performance

index is, because it is only a means to an end, i.e., optimalcontrol law. Still the results appear to be somewhat startl-ing. They suggest that a time optimal solution, augmentedby the control equation (3) at p = 1, has a far better per-formance than the quadratic solution. The results do showthat a sub-time optimal solution is adequate to improvingthe performance.

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tEFERENCES'

[11 Raamot-XandV. Zaleckas, Laser pattern generation using;xN-Y beam deflection," Applied Optics, vol. 13, pp. 1179-1183,

ay 19 .M. I. Cohen, A. Unger, and J. F. Milkosky, "Laser machin::-hingof nfilms and integrated circuits," Bell System TechnicalJournal, vl1. 47, pp. 385-405, Mar. 1968.

[31 S. Carshan (d.) Lasers in Indstry. New York: VanNostrand-Reinhold Inc. 1973.

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Ipp. 57-61.10 A. G. Gross, J. Ramot and Mrs. S. B. Watkins, "Computer

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[81 . B. O'Briein, "Sensitivity analysis of the optimal regulatorcontrol of a galvanometer beam deflectionsystem," MastersThesis, Lehigh University, 1973.[91 J. D. Zook, "Light beam deflector performance: A comparativanalysis," Applied Optics, vol. 13, 875-887, 1974[10] R. Luus, "Time-optimal control inear systems," The Cans.dian Journal of Chemical Engineering, vol. 52, pp. 98-10Feb. 1974.[111 C. E. Shannon and W. Weaver, The Mathematical Theory ofCommunication. Urbana, Ill.: University of Illnois Press,1949.

[121 P. J. Brosens, "Fast retrace optical scanning,": Electro-OptiealSystems Design, pp. 21-24, Apr. t971.[131 B. N. Szabados, N. K. Sinha, and C. D. diCenzo "Practicalswitching characteristics for minimum-time position controlusing a permanent magnet motor," IEEE Trans. Ind. Eectron.Contr. nstrum., vol.-IECI-19, 69-73, Aug. 1973.^

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