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Unformatted text preview: I. 79 Stark — Pupil Control System Introduction: The pupil control system is presented here as a classic case of biological modeling in which a
model of the system can be proposed, its parameters measured, and predictions made as to its
behavior in related experiments. In this case, the predictions turn out to be accurate! We'll use this system to illustrate the use of some concepts from control system stability analysis.
Schematic of the pupillary control system: Average
Light Intensity Size
\ Light Intensity
on Retina
Response
V. l of several
dilator C52; layers (if
muscle k— retina
lateral cells
geniculate
@D \/ Optic
nerve
superior
cervical
ganglion pretectal
Lsympathetic‘l/ area
system \
spinal cord
iris
sphincter Edinger
muscle Westphal
—contraction nucleus
parasympathetic
system
ciliary <_/
ganglion most control
is modulation
in E—W nucleus Reference: Stark, Neurological Control Systems. Although the work cited is from the 605,
more sophisticated versions are being pursued currently by researchers at McGill and at the University of Chicago. I. 80 The experimental technique is carred out under three conditions: (a) normal pupil: light is focused so as completely cover the pupil area, even when it is open
widest. In this state the pupil operates normally to control the total light reaching the retina. (b) Open Loop: light is focused so as to be completely within the pupil area, even when it is at
its narrowest. In this state the pupil does not control the light reaching the retina. (c) High Gain: light is focused in a small circle on the edge of the pupil. In this state small
changes in pupil diameter make very large changes in the light hitting the retina. Linearizing the System Like most sensory systems, the retina is decidedly non—linear, and in fact has a response which is
proportional to the logarithm of the input light intensity. This is the wellknown Weber—Fechner
Law, which states that equally perceived increments in light intensity correspond to equal
multiples of light intensity. The same is true with hearing and, to a lesser extent, touch and pain. Nonetheless, it is profitable to choose an average light intensity and to assume that, for modest
changes in light intensity, the system can be modeled as a linear system. The success of this
model demonstrates that, even when you know a system is non—linear, important principles can
be demonstrated using linear models. Formally, following Stark‘s development, the linear variables are developed as follows: ' choose I = average intensity = operating point ' choose A = area of light, focussed to a spot 0 AI = external change in light intensity
0 AA 2 change in pupil area Open Loop Gain Calculation Lin = external input = A  AI Low = output = I  AA '
size change in intensity area change
of spot intensity of light response of pupil The output is the change in total light falling on retina which the brain “thinks” it will achieve L . _
The transfer function G(s) is dimensionless: G(s) = fill = 1—913 = w i“ AAI AI/I Thus, the pupil area changes to achieve the fraction G(s) of the desired compensatory effect to
keep total light constant on the retina. However, it will suffice to understand the system if you qualitatively understand the output /
input functions and examine them for their open and closed loop gain properties. 1.81 Data are shown in which the Open Loop
frequency response follows thetransfer function:
016 6—018 5
G s = ————
ope“ ) (1 +0.1 s)3 Gain in dB It is stated that this transfer function agrees
with the step response data. 10.1 w W The Nyquist plot for this model follows. Note Frequency in Hz . ~ I .
the pos1tion of the pornt (1,0) (the cross hair
to the left in the graph. 0
UtSr— ‘ l v i ~17 ﬁ j
X”‘>K
100 m _ ' J
g .
e x
53200 ansr a!
D
.E '
3—300» g 0» i .1
E a i
a  ; 1  v in 05 4‘
400,”. . 7.2 l . ' 1
. _ . V '0‘ t ' I I l l I . . H
10" 10°. 10’ «015»
Frequency in HZ
'03 2 4‘ 08 as .65: .02 in 772' 04
RealAXis
Matlab plot of Gopen(s)
Closed Loop System
Closed Loop Gain: G (s)
FClosed (S) = 173%“? out High Gain System
Equivalent
KGo en(5) high gain due to spot .of /
light on edge of pupil Fclosed(s) = m
open I. 82 NYQUIST STABILITY CRITERION (Refer to the Notes on Stability of a Control System) The Nyquist criterion is one of the simplest means of determining a very complex phenomenon
is true. Full justification of the technique was given in the notes earlier. Here, your task is to
practice applying the simple criterion to determine when the pupil control system goes unstable. The system is unstable if either condition is met: Condition #1:
IKGQwM >1 when 31(KGQco)) =—180° Condition #2
31(KGQco)) <— 180° when KGQco) =1 =OdB Schematic Drawing of the Effect of High Gain on the Bode Plots of the Loop Gain Transfer
Function K cho). Before = Normal. After = High Gain. Before After l gain margin
gain less than 1
at—180' phase) 0.16 —180' l. 83 Normal Pupil: High Gain Pupil:
Bode Plot of Loop Gain Function Bode Plot of Loop Gain Function
.16
.13 .
%—20 g . ..
.24 . E‘s“: "2%2 04 06 oje 1 12 14 16 “llz 73424— 076 0'8 1T 1‘2 T? 1is Frequency In Hz Frequency In HZ Phase in Degrees
Phase in Degrees '25%,2 0‘4 ‘35. oie i 1:2 1‘4 1'6 £5912 34 0:5 0.? i 1i2 1:4 1.6
Frequency In Hz Frequency In Hz
bodelaln(10,[.001 .03 .3 1],.18,.2,1.5); bodelaln(2,[.001 .03 .3 1],.18,.2,1.5);
0166—0185 206—0185
(l+0.ls)3 (l+0.1s)3 We have created Matlab plots to reinforce the ideas sketched above. Note that the choice of
limits on the frequency axis was manipulated to enhance the region around the frequency where
the phase shift is —l80 degrees. The Matlab function, bodelaln (the 8 character truncation of
bode plot with delay on a linear axis) was created for use with ECE 375. Its syntax is:
bodela(numerator, denominator, time delay, start frequency, end frequency). The frequencies are optional. The "critical" frequency, where the phase shift is —l80 degrees, is approximately 1.1 Hz, and
does not change with gain. In the normal pupil the loop gain (which, in this case, equals the
open loop gain) is never as great as 0 db (dB scale) or 1 (linear scale). However, in the high gain
example (with the gain K set to 2.0), the gain is approximately 0.5 dB (dB scale) or 1.06 (linear
scale). Thus, this plot predicts that the system will be unstable when the open loop gain constant
is 2.0 instead of its normal 0.16. The Nyquist criterion can be employed to determine if the system is unstable. The short version
of this criterion, which is fully justified in the notes earlier, is that, when the gain is plotted on a
polar plot, if the point (—10) (the negative one point along the real axis in the complex plane), is
encircled, then the system is unstable. I. 84 Normal Pupil: High Gain Pupil:
Nyquist Plot of Loop Gain Function Nyquist Plot of Loop Gain Function
O15 . v . i; . 2 ‘r v
5’ ‘ ............... 01 _ f/ i‘. _' 1 5  '
f 3
0,05» i 1F c
4. lmag Axus
o
o
u.
m
5",
Imag AXIS
o
o m .01, 015» 4 ~1S_ 'Oiz i 70‘s 43's “girl1m 0‘2 6 0:2 04 '35 i ~o‘s Bneawlsols i is 2
nyq2(.16,[.001 .03 .3 1],.18) nyq2(2.0,[.001 .03 .3 1],.18) We have repeated the Nyquist plot, using another function created for ECE 375:
nyq2(numerator, denominator, delay time). Note that the point (—1,0) is not circled in the low
gain, normal case, whereas this point lies inside the curve for the high gain case. This is a
convenient way to describe when the system is unstable. It is entirely equivalent to the analysis
with the Bode plots above. I. 85 What happens when the system goes unstable? In this case, the system breaks into oscillations at the “critical” frequency. It is beyond the scope
of this course to try to analyze why the frequency of oscillation should be very close to that at
which the loop gain has a 180 degree phase shift. There are three predictions of the control systems model of the pupillary system:
(a) the system should go unstable when the gain is increased (b) the critical gain can be estimated quantitatively
(c) the critical frequency, at which unstable oscillations take place, is predictable The data show that, for a number of patients , the pupillary system did show oscillations and that
those oscillations took place near the critical frequency. No data were presented showing
whether or not the gain was near the estimated critical gain. Further experiments, in which the gain was increased such that:
(a) there were two frequencies where the phase shift was 180 degrees when the gain was
greater than 0 dB.
and, equivalently,
(b) the point (—1,0) was encircled twice.
With the predictable result that the pupil showed a superposition of two oscillations, one at each
of the two critical frequencies. A final experiment was done where topical application was made of amphetamine (which dilated
the pupil) and then eserine (which constricted the pupil again) so that a drugged pupil with
approximately the same diameter and DC gain was achieved. However, the pupil showed
considerably greater phase lag, with the result that the 180 degree crossover was achieved at a
much lower critical frequency. The drugged pupils showed a change in oscillatory frequency
down to the new, lower frequency. ...
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 Spring '08
 PRINCIPE
 Signal Processing

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