Servo-modulator system used in a 2D shearing interferometer G. García-Torales *, J. L. Flores , J. G. Mateos Suárez , R. X. Muñoz . Dpto. of Electronic, Univ. of Guadalajara/CUCEI, Av. Revolución 1500, Guadalajara Jal., MX CP 44840;
ABSTRACT An improvement of a 2D shearing interferometer to measure small and large non-rotationally symmetrical wavefronts aberrations is described. This interferometric system encompasses large and differential wavefront displacements using a high accuracy rotation system incorporated in a Mach-Zehnder interferometer. The rotation of the prism arrangement is controlled with an electronic servomotor system by means of an auto-tuning using a Neural Network Algorithm, NNA. We describe the servo-mechanical system, the electronic interface and algorithms to control the performance of the rotation device with the aim of obtain accurate wave front position by a prism rotator system. Keywords: shearing interferometry, prism system, Neural Network Algorithm NNA, automatic control
1. INTRODUCTION There are many fields, scientific and technological, where the precision measurements of the wavefront are required. During the past several decades lateral shearing interferometers (LSIs) have been widely used for wavefront measurement. LSIs are self-referenced: they compare the wavefront under test with itself. They do not need an ideal reference. Usually, shearing is obtained by using a plane parallel plate to displace the wavefront in two orthogonal directions. Least squares numerical methods are usually employed to estimate the wavefront of these intensity patterns. Recently, we presented a 2D shearing interfometer based on a modified Mach−Zehnder configuration1,2.. We probe that in a particular condition the moving distance for a 2π phased shift was a fraction of millimeters. The introduction of a Risley prism makes the control on the position of the wavefront and thus, on the phase shift, easer to achieve by the cleaver rotation of the prisms system. Errors associated to the phase-shifter are determined mainly by the wedge angle error, which have been measured and well determined3. The interferometric Mach−Zehnder configuration is extensively used in optical testing. Figure 1 show our configuration 2D shearing configuration. Here, automatic calibration is possible and easier than other similar configurations that use just mirrors to shear the wavefront under test. Errors in the positioning of the mirrors are common. Each mirror has three axial degrees of freedom: two degrees of rotation (tip and tilt) and axial position. In our configuration, calibration (zero fringes at the detection plane), is reached only by the selection of each prism orientation independently. Moreover, many shearing interferometers are able to perform very small shear but not all of them are capable to generate large controlled displacements. Besides, the use of a laser source assures high contrast fringes even when the paths of the two beams are far from equal. However, the disadvantage of separated optical paths requires more care in the fabrication of duplicated components and generation of identical environmental conditions. In the laboratory environment, with the components mounted on an air−isolated optical table, these conditions are routinely achieved.
*
[email protected]; phone (052)33 394-259-20; fax (052)33 394-259-20 ext. 7744
Novel Optical Systems Design and Optimization IX, edited by José M. Sasian, Mary G. Turner, Proceedings of SPIE Vol. 6289, 628917, (2006) · 0277-786X/06/$15 · doi: 10.1117/12.681343
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Prism 1
ω2
ω1
M1
BS 2
Prism 2
Shearing System
A CCD camera Focusing lens
B
Prism 2
Compensation System Wavefront Under Test
Prism 1
A B
M2
BS1 Fig. 1 Experimental setup of the 2D shearing interferometer used in the wavefront test. Shearing and compensation systems are setting in a position of calibration.
The shearing system combines the angles of deviation of an aligned prism pair. Figure 2 depicts the deviation of an input ray collinear with the optical axis as it propagates through the shearing system. The direction of the ray emerging from the shearing system is given by director angles. The rotation angles of the first and second prism with respect to the optical axis. The relative angle between prisms is the difference of the rotation angles of the each prism. Additionally, the resulting deviation of the prism system yields a linear displacement in the detection plane. Therefore, we may represent the magnitude displacement, ρ0, generated by the shearing system, as a function of the deviation of the prism pair as:
⎛
⎛ ω1 − ω 2 ⎝ 2
ρ 0 = z i tan⎜⎜ 2(n − 1)ε cos⎜ ⎝
⎞⎞ ⎟ ⎟⎟ . ⎠⎠
(1)
Where, zi is the distance from the last surface of the shearing system to the detection plane, n the refraction index of the prisms, ε the apex angle of the wedge prisms, ω1 and ω2 are the rotation angles of each prism. Irradiance at the detection plane is proportional to the square of the amplitude of the electric field E0, and proportional to the deviation angle γ then, the reaibility of the systems depend on the exact performance of the shearing system.
I = 2 E02 + E02 exp(ik [x ′ sinγ + z i (cos γ − 1)]) + E02 exp(− ik [x ′ sinγ + z i (cos γ − 1)]) .
(2)
In practice, accuracy in the determination of the wave−front displacement is limited by the error in the deflection angle γ that depends on three essential parameters: (a) the apex angle ε , which depends on the fabrication process, (b) the index of refraction n that depends on the quality of the glass, and (c) the relative angle between prisms ϖ which depends on the accuracy of the rotary holders. All surfaces of the wedges must be of high quality, e. g. figure error less than λ/20, and apex angles with tolerances on the order of arc seconds. Nowadays, transparent materials with high degree of homogeneity are commercially available, e.g., ultra pure synthetic fused silica.
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y Prism 2
x
Prism 1
Shearing System
z
Fig. 2 Shearing system: The relative angle between prisms determines the deviation angle and also the position on the plane of the ray.
Therefore, the deciding parameter over the net deviation is the relative angle between prisms ϖ = ω1-ω2. Hence, a precise opto−mechanical design for mutual orientation and rotation of the wedges is necessary. The precision and accuracy of the mechanical system determine the degree of control upon the wedge rotation. Rotary holders perform rotations in the order of minutes or even seconds of arc, are expensive.
2. PID CONTROLLER New technologies have improved the accuracy on control positioning. Servomotors are extensively used to perform highly precise positioning systems. The control of the servomotors is commonly achieved by some of the several PID methods. PID stands for Proportional, Integral, and Derivative. Controllers are designed to eliminate the need for continuous operator attention 4. Controllers are used to automatically adjust some variable to hold the measurement (or process variable) at the set-point. The set-point is where you would like the measurement to be. Error is defined as the difference between set-point and measurement. The variable being adjusted is called the manipulated variable which usually is equal to the output of the controller. The output of PID controllers will change in response to a change in measurement or set-point 5. Manufacturers of PID controllers use different names to identify the three modes. Depending on the manufacturer, integral or reset action is set in either time/repeat or repeat/time. One is just the reciprocal of the other. Note that manufacturers are not consistent and often use reset in units of time/repeat or integral in units of repeats/time. Derivative and rate are the same. Tuning methods are divided in open loop and feedback loop methods, where the Ziegler and Nichol method (feedback loop) is widely used. Theoretical PID control structures become very useful since Minorsky publications in 1922. In spite of the plenty of work published about PID control systems are still used practically in most of the industry controlling more than the 95 % of the feedback control process. Besides, one of the problems that the engineer in instrumentation confronts is to accomplish the tuning of controllers PID. There are logic programmable controls on the market that permit the use of mathematical tools using digital controls for servo-drivers. A PID tuning controller with a linear adaptive element is compounded by a servo-driver of type multi-axle programmed on its own programming language, it makes use of learning algorithms with NNA, it learns to make the tuning simpler and automatic in its function, no only of constants. Nets with a RNA’s process discreet signals, in order to use adaptive filters, the analog signal converts by analogical to digital to digital values, and processed by a microprocessor.
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c(s)
R(s)
Fig. 3. PID system controller placed in the feedback loop.
Tuning a PID controller involves adjustment of three parameters: the proportionality constant KI, the integral time constant TI and the anticipative time constant Td. First to all, the dynamic parameters of the process must be identified in order to obtain a reliable and robust performance of those parameters of the system for. The variables of in loop control are the reference value wish r(t), the output c(t) from a transmitter that generates the wish value, the feedback signal y(t) the output of the controller that modifies the final control element, ands the error xW(t) defined as the difference between the real value and the expected value. Considering these variables in the complex plane domain, we define GC(s) as the transfer function of the controller, GP(s) as the transfer function of the process. Thus from the feedback loop system shown in Fig. 1, the behavior of the system can be described as
C (s ) =
GC (s )G P (s ) G P (s ) R (s ) + S (s ) 1 + GC (s )G P (s ) 1 + GC (s )G P (s )
(3)
Regarding these two conditions according with the superposition theorem, a first condition a wish value when S(s) = 0, then
C (s ) =
G P (s ) R (s ) 1 + GC (s )G P (s )
(4)
A second condition, when the reference R(S) = 0,
C (s ) =
GC (s )G P (s ) S (s ) 1 + GC (s )G P (s )
(5)
Another important consideration is perturbation sensitivity of the system. The common process implies to establish a high signal to noise figure. Particularly in industrial environments, controllers have to be able to respond quickly and reliable to any transitory signal in order to reach the optimal output value. When the system has a large time constant implies that some perturbations has not been eliminated efficiently. Eqs. 4 and 5, the numerators are different, then it must be necessary a perfect adjustment in the controller Gc(S) in order to reach the dynamical condition of the system, thus it is necessary implement tuning controllers. According with the Figure 3, the standard basic control structure of the transfer function can be expressed in the time domain by
g C (t ) = K P xw (t ) + K D
d xw (t ) + K I ∫ xw (t )dt dt
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(6)
Then, Eq. 6 can be transform in the Laplace domain as:
⎛ K s2 + KPs + KI K ⎞ ⎛ GC (s ) = ⎜ K P + K D s + I ⎟ Xw(s ) = ⎜⎜ D s ⎠ s ⎝ ⎝
⎞ ⎟⎟ Xw(s ) ⎠
(7)
There are several techniques to achieve the adjustment the controllers, that means o select the appropriate value of the constants: proportional gain KP, the integration time KI, and the action time KD, in order to match with the transmitter, the process and the final control element. These adjustments, force to the controller to generate a stable recuperation curve, typically a dumping constant rate about of 0.25 between two successive valleys of the wave over dumping. The dynamics of the process is often identifying analytically and experimentally. The analytical method must solve the mathematical expression that describes the system as a function of time. This method is very complex in most of the real systems applications. In most cases mathematical functions are included o digital controllers and the distributed control due to their calculus power. The experimental method, the static and dynamical features are obtained from direct measurements, where the Ziegler and Nichols is commonly used in two ways limit gain and the reaction curve. Ziegler and Nichols technique is one of the first tuning methods applied to the feedback loop systems [2]. The adjustment begins only with the proportional gain constant, increasing slowly the gain until to get a sustained oscillation due to an step function with an amplitude that defines the reference value. We define Ku as the last gain and Tu as the last period at the sustained oscillation point. In this method the gain of the controller must be proportional to one half of the last gain Ku. The values of KP should be in the 6.0 KU ≤ KP ≤2KU. It must be known the structure of the PID system (serial or parallel) in order to apply the ZieglerNichols. In the limit gain method calculates the three adjustments PID constants getting the values of the constant from a fast test. The band of proportional gain must be stretched with the integral and derivative adjustments until the system begin to oscillate continuously. This band is called PBU= Ku, then Tu is obtained (minutes). The adjustments of the system produce an amplitude rate of 0.25[3]. The tuning equations of the PID controller are shown in the table 1. The process of tuning is described as follows. KD and KI are set equal to zero. Set the gain KP in a small value. Increase the KP value until the system oscillates sustained. The obtain Ku and Tu. Set the parameters of the PID controller according with the table 1. Table 1. Parameter of the controller: using the limit gain method.
Action of control
KP
KD
Proportional
0.5KU
Proportional-Integral
0.45 KBU
2. 1 TU
Proportional-Integral-Derivative
0.60 KBUB
2 TU
KI
8 TU
R= Slope at the inflection point
to
ti
Fig. 4. Response to the step function applying the reaction curve method.
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In this method the feedback loop must be open just before the final control element. The final element is directly operated with the controller in the manual fashion, and makes a fast and small step at the input of the system. The output can be displayed and finding out the inflection point of the curve, it should be traced a line tangent to this point in order to measure the slope at the inflection point R and the time delay L, (t1-t0), as is shown in the Figure 4. Set the parameters of the PID controller according with the table 2, here ∆p is the percent of variation in the final element of control.
Table 2. Parameter of the controller: using reaction curve method.
Action of control [%]
KP
KD
Proportional
100 RL ∆p
Proportional-Integral
110 RL ∆p
L 0.3
Proportional-Integral-Derivative
83RL ∆p
L 0.5
KI
0 .5 L
3. NEURAL NETWORK ADALINE The first meaning for ADALINE was ADAptive LInear NEuron but its meaning change to o Adaptive LInear Element and only has one neural element; it has been used to solve linearly separable systems 6. This one element adds the product of the input vectors and its weights applying an output function in order to obtain one value response. Using a NNA, it is possible to establish a proceeding to modify the weight function and get a correct value to a given input. Signal to be proceeded by a NNA must be digitalized with sample times and the variables are the set point the reference signal w(t), a feedback signal x(t) (the output of system) then is possible to asses the error e(t), that correspond to the input GC(s). The PID controller generates an output gC(k), where Kp is the proportional gain, Ti is the integration time, and Td is the derivative time, where the discrete representation of the output of the PID controller in the time domain is given by n ⎛ e − en −1 ⎞ ⎟ g C (k ) = ⎜⎜ K p e(k ) + K I ∑ e j Ts + K D n ⎟ Ts = j 1 ⎝ ⎠
(8)
The tuning process for discrete signals using the Ziegler-Nichols method is similar to those before explained and as a result only preserve the error and proportional constant KP as is shown next.
g C (k ) = (K p xW (k ) = K P [x(k ) − w(k )])
(9)
At the output of the ADALINE controller we obtain the signal that is represented by vector u(k) that becomes in the sequential input u(k-1), generating the knowledge matrix W(Kp). This matrix information is useful to calculate the new Adaline control outputs. Then, the constant for the conventional system are obtained applying the limit gain method
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regarding the transfer function of he servo-driver. The learning rule of Adaline needs aprori information about the input parameters. Once the NNA have learned, the error is reduced by an α factor, at the same time, the weights of the system change according with the input values. The weights are renewed every time until the system converges.
∆e(k ) = −α
e(k ) p T (k ) p(k ) p(k )
2
= −αe(k )
(10)
The best selection of α, helps to manage the training process stability and velocity of convergence. Stability condition in achieve if 0.1 > α < 2. The selection of α does not depends on the input magnitude, every weight actualized is collinear respect the inputs parameters and its magnitude is inversely proportional to
p(k ) . 2
4. RESULTS In order to implement the rotary holder control, it was regarded a security factor ranging from 40% to 100% in some cases, e.g. load effects. We used two servomotors with resolution 131,072 pulses per revolution without considering the reducer screw, by a factor of 100. Figure 5 shows the display of the design program and the main parameters used.
Seflilig Data [Hall scrw, Hr7
Coupling yI+Ext Red Gear In]
Calculate C Set Mtr
Pr's. ctrl. mode
—
MR-U? S-NtCP
Amplifier:
Amplifier (MR-J2S-CP series is 7KW or smaller cap.)
-J
_—. Motor. M4o, —-
F-IC-MFS 3000 rmin
No Reduction Gear Option
No Erake Option
Optn
it—
Iin if 0 rm Ac:cjDec md in All S':t. o1
Poe Cr' Mode Oper. Pattern Aeceler
I
10,000 (mrrusec2)
LMU Seuiuq Mess of table 5 of load
Thrustlod Guide tigMenincj force Coupling inertia Inertia 01 the others
Leod ot ball screw Diameter 'jt ball :rcreR: Length ot bell screw
Dr e efficiency Coefficierfl of friction
Vt
0•500
0100
Sizing Result
kg N
Fc F'S
1,000 U !5C10
N
JC
O•000
J!::!
Pb
0,000 10,000
kg-cm2 ku-cm? mm
Load Inertia:
D6
2CI!DIJD
mm
Lfl
500000
mm
eta mu
0,900 0,100
PeakTorque : RMS Torque: Pegen. Pwr :
AmpliIer : MR-J2S-l UNBIOR Regeneration needless
• Guide tighteninu force
F0
0,500 IN
_______________________
Motor: HC-MFSI 3 [I DOW]
I
0,629 [kg-cm2] 0,055 [N-mi
20QTimes
0021 [N-mi 0,000 [V\
17,5%
66% 0,0%
Tt,e :ni:inçj sc'ft-,vare calc:ulated the system wth theoretical equations and can only be used as a guide ton suable soltdion. ,:heck the ne:rL,Rs against your own requiremeots ensUring that Satety Tiaroin is Ok from selected svsteni in reserve
'Show Graph
Show Calculations
Fig. 5. Servomotor parameters used to design the rotary holder controller.
Proc. of SPIE Vol. 6289 628917-7
po t iw.
—S
,i
)
Ve,oc,l
Vol oc
H__
V,Focit
2 0
5
S
p
it'
P!ili.,
The matrix of weight is formed applying the conditions before exposed tuning with the ADALINE. Figure 6 shows the response to the step function applying the auto tuner ADALINE. Note that the velocity does not change, but the position is corrected smoothly.
Fig. 6. Response to the step function applying the AUTO TUNER ADALINE
5. CONCLUSIONS Tuning process was implemented in a rotary holder servo-controller. Tuning was achieved using the neural network ADALINE. Algorithms were programmed in Motion Arquitect Program (c) in order to facilitate the introduction of constants. PID tuning can be used in SISO process automatically. The rotary holder can be programmed until find the proper position of the prism system in order of perform scanning applications.
REFERENCES 1. García-Torales G., M. Strojnik, and G. Paez, “Simulations and experimental results with a vectorial shearing interferometer,” Opt. Eng. 40 (5), pp. 767-773 (2001). 2. Garcia-Torales G., Paez G, Strojnik M, Villa J., Flores J.L., Alvarez A., “Experimental intensity patterns obtained from a 2D shearing interferometer with adaptable sensitivity, ” Opt. Commun. 257 (1): 16-26, (2006). 3. Xu RW, Liu HZ, Luan Z, et al., “A phase-shifting vectorial-shearing interferometer with wedge plate phase-shifter, “ J. Opt. A: Pure Appl. Opt 7 (11): 617-623, (2005). 4. K.J. Åström and T. Hägglund, PID Controllers: Theory, Design, and Tuning.Research Triangle Park, NC: Instrum. Soc. Amer. (1995). 5. K. Ogata, “Modern Control Engineering,” Pearson-Prentice Hall, 4 Edición (2004). 6. James, A. Freeman-David M. Skapura, Artifcial Neural Networks, Addison Wesley, (1991).
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