Ph 10k The Physics of LIGO 29 April 1994

LECTURE 10. Control Systems for Test-Mass Position and Orientation

Lecture by Seiji Kawamura

Assigned Reading:

BB. S. Kawamura and M. E. Zucker, Applied Optics, in press. [This paper explains the influence of angular mirror orientation errors on the length of a Fabry-Perot resonator.]

Read either item CC. below or item DD. [Item DD. is highly reco=ended, since feedback loops are so important; but for some students it may entail a fair amount of work, and CC. might be preferred.]

CC. M. Stephens, P. Saulson, and J. Kovalik, "A double pendulum vibration isolation system for a laser interferometric gravitational wave antenna," Rev. Sci. Instrum., 62, 924- 932 (1991). [Here you are asked to focus on the control of the pendulum, rather than on the penedulum's role in vibration isolation.]

DD. Read, in your favorite control theory book [e.g., R. C. Dorf, Modern Control Sys­tems 5th editon (Addison-Wesley, 1989), cited as Dorfbelow] or elsewhere, about the following issues:

a. The relationship of Laplace transforms to Fourier transforms [e.g., Dorfpp. 264-266]. Control theory is often formulated in terms of Lapace transforms rather than Fourier transforms because Lapace transforms are more naturally suited to describing the transient response of a system to some input; the reason is that they entail only the behavior of the system between some initial time t = 0 and t = 00, by contrast with Fourier transforms which involve the behavior over all time. In this course we will probably not deal with any issues where the Laplace transform has an advantage; and we will most always discuss things in terms of Fourier transforms and thus in terms of the response of a system at some frequency w. However, in order to read control theory books on these issues, it is necessary to understand Laplace transforms and their relation to Fourier transforms. [Note that, although theoretical physicists normally use the form e-iwt for the time dependence of a Fourier component of frequency w, engineers, and control theorists normally use e+;wt (where i = j = A). In this course we shall use the engineers' conventions.]

b . The use of complex frequency-response plots to describe the ratio of the output amplitude Vout of a linear system such as a control loop, to its input amplitude "in, when the input and output have frequency w [e.g., read Dorf, pp. 266- 283] . In these plots, Vout/"in == G(w), which is a complex quantity, is plotted as a curve in the complex plane parametrized by w, for real w. Such a plot contains the same information as a Bode diagram, in which one gives two plots, one of IG{wli plotted upward and w horizontally; the other of the phase </I(w) of G plotted upward and w horizontally; for example:

I,.,., (G) I G I' I-~

010.------=====-6.

c. The Nyquist criterion for the stability of a control loop [e.g. , read DorE, pp. 309-333J. [The Nyquist criterion, in a nutshell, is this: Consider a simple feedback loop of form shown in (a) below. If the input and output ports are shut, the resulting closed loop shown in (b) can oscillate at certain complex eigenfrequencies without any stimulus. Those frequencies are easily deduced from the requirement that the amplitude y at the indicated point must satisfy y = G(w)H(w)y, and therefore y(1 + GH) = 0, and therefore the loop's frequencies of self oscillation are the zeroes of 1 + G(w)H(w).

Vi" .- \10....:1. = G V, ..... G-iG\ +-, HGI4 'J 1\ ,

'14 1 ~ I r

(a) ( b)

Since the time dependence of these oscillations is e+ j..,t , if there are any zeroes of 1 + G H in the lower-half complex frequency plane (any eigenfrequencies w with negative imaginary parts), then the amplitude of the closed loop's oscillations will grow in time; i.e., the closed loop will be unstable. The number of zeroes in the lower-half frequency plane can be inferred from the Cauchy theorem of complex variable theory: Construct the curve G(w)H(w) in the complex plane, with w running along the real axis from -00 to +00, and then swinging down around the lower half frequency plane and back to - 00; see drawing (a) below. The number of times that this curve, G(w)H(w) encircles clockwise the point GH = -1 (on the real axis) is the number of zeroes of 1 + GH minus the number of poles of 1 + GH; see drawing (b) below. For feedback loops there usually are no poles of 1 + GH [such a pole would give precisely zero output/input in the feedback loop of (a) aboveJ , so usually the number of clockwise trips around G H = -1 is the number of zeroes in the complex frequency plane. Thus, if there are no clockwise trips, the closed loop is stable; if there are some, it is unstable. This is the Nyquist criterion for stability.J

r ..... rv-l)

(a)

Suggested Supplementary Reading:

~~~~~~_ Re (G 14) w~o

T ... " ~ c.\~'rw:"'~ -trips ,h ~-c\ GI4::.-1

5. Read whichever of items 3. and 4. you did not do as "assigned reading" .

2

A Few Suggested ProbleID8

I. Use the Nyquist criterion for the stability of a feedback loop to show that, when the Bode diagram has the qualitative fonn shown on transparency 23 of Kawamura's lecture (where f = w/ 27r) , then the loop is stable if the phase of GH at the unity gain point is <I> > -180°, and unstable if <I> < -180°. [Hint: show that , becaUBe in the time domain the equations describing most any servo loop are real, when w is real then G( -w)H( -w) is the complex conjugate of G( +w)H( +w) . This permits you to construct the Nyquist curve in the frequency-response plot for both posit ive and negative w from Kawamura's positive-frequency Bode diagram.) For what shapes of Bode diagrams will this <I> > -180° stability criterion remain true?(Consider, for example, the issue of how many unity gain points there are).

2. Construct a complex frequency-re:Jponse curve and also a Bode diagram for the fol­lowing pass R - C circuit. From the Bode diagram infer that this circuit is a low-pass filter .

VI

3. In his lecture [transparencies numbered 15-17), Kawamura described the damping of the swing of a pendulum via a feedblack loop that produces a displacement ,;X = -"(dy/dt of the pendulum's support point , where "( is the damping constant and y is the horizontal position of the pendulum's mass. Of course, in order to implement this, one needs some fixed object with respect to which y is measured. In transparency 15 that object is the shadow sensor, but nothing is said about what that sensor is attached to. A practical approach is to attach the sensor to the pendulum's support point , as shown below. Then the feedback displacement is';x = -"(d(y- x)/dt , where x is the instantaneous horizontal position of the support point. Repeat Kawamura's analysis [transparencies 15-17)) for this feedback system.

~- :x:.

3

4. Suppose that one were to try to damp the (low-frequency, 1 Hz) swing of the pendulum in problem 3 not with a feedback displacement Ox = -,,(d(y - x) / dt, but instead with a feedback displacement that is -ay (for some constant a > 1) at low frequencies (near 1 Hz) but that shuts off at higher frequencies (above 10 Hz), where the gravity waves are to be measured. Suppose one implements this feedback displacement by simply passing a voltage, proportional to y , through a low-pass R- C filter of the sort discussed in problem 2. Show that the resulting damping system will be unstable.

5. Derive the relation 0/ = d1091 + d2092 on transparency 28 of Kawamura's lecture.

4

Lecture 10 Control Systems for Test-Mass Position and Orientation

by Seiji Kawamura, 29 April 1994

Kawamura lectured from the following transparencies. Kip has annotated them, based on Kawamura's lecture.

1

PH 1u3c: THE PHYSICS OF LTGO 29 APRIL 1994

LE C TURE 10

CONTROL SYSTEMS

FOR TEST-MASS

POSITION AND ORIENTATION

Seiji Kawamura

APR. 29, 1994

You will learn ...

1. What is test mass position / orientation control?

2. How to damp a test mass without adding extra noise?

3. How to predict test mass orientation noise in a Fabry-Perot cavity?

1. W~+ is t~.s-!: j-y,;;>~ p,.d,'~,..../one."'bCl-blo" 0<l " .. :6. 0 I ~

Without control .....

MIRRORS

BEAMSPLITTER

MIRRORS -- ----- --.

(( )) (( PHOTODETECTOR

It doesn't work !

TEST MASS POSITION / ORIENTATION CONTROL SYSTEM

))

2

MIRROIZ BEAn

1 Vc;W;~y AXIS

CENTER 0 r= CURVATURE

RESONATE D /

r? ~ f>EAH -r: > • ------/) CAVITY "-

A'f..13 '-= CAN"T BE RESONATED! ,

ORlENTA-rION

3

,POS(TION CONTROL)

~--~ ~--?

y

MOTIDN SHALL LAI<G:t:.

v

LO:k EA~Y D\F~Ic..ULT

4

1Hz: STOP /

\

FREQUENCY DEPENDENcE

OF- CONTROL

LONGITUDI '~'-.'--4-+-­POSITION ~ 1)..1;"

~"YAW

TRANSVERSE POSITION

POSITION AND ORIENTATION

5 -e.nsor- a-'" J .Ac.-bIAe>"t:or

(See he..,t.

tra ... ~cd)

DAMPING 1--------1

CONTROL

TEST MASS DAMPING CONTROL SYSTEM 1

7

COIL

C8J LED V

VANE! PO V

C8J Attached to Stand

,

" :,~,

' c ,

SENSOR AND ACTUATOR

PZT

,----_--, EDGE SENSOR!

OPTICAL LEVER

SENSOR

poe

ODe

TEST MASS DAMPING CONTROL SYSTEM 2

2. ,.,.,~~ .. {5

d l:;\. iI ~ \...e ~

-b ~ "" ~ s..s

'+.r c .. ,,;I.)'

1o "k;~

BEAM SPOT

Pitch (A+B) - (C+O)

Yaw (A+C) - (8+0)

QUADRANT PHOTOOIOOE

/0

L-0W PASS 'FILlER

KN05 I<NOB

TES, NoN ITO R

2}~ jec.t ·h.t s;~..-Q \

OR \f::. N TAT \ ON DAMP INGr

I I

Ga/n.

20~J~1 [dB]

z. . f-.i ovJ .. h Dd ....... p d rest H;;;l.ss t-v,i:1..o",t,

A--J4~ Ex'Lr;;,. ~':Se.

--- i".. 2 0 d G ;~ ~ +&c..{,<>o- 10

{" d E, I'.!. d. .fa ,,-6- 2.

b~(Wi+¢)

/2

a~wt

~TSINE

S

b~wt

CIi.l CH.2 ~------, 0 p\.)o D

00 - -

ANAL Yi-fR

[3

( '2. 9-

Wo :: I )

~I 11---'

¢ ¢ = 0

0

( w < oJo) tll--_w::;:,=,,-o ___ -;:>w

-I~oo ~ > (.00) -1'toO • - - • L-__ _

> __ .fr1" JC..-------.

•• • z

~ + ~o ~ + Wo d-I

(G- -= rwo ) QlAG\[;+;- Foctol-

•

feedktck tt>~

PaW\r; 1I\ca-

[,(filE 1<ESPONS'£~~ FREOUENCY Rf~ftJNS§ , I) C}."7 O};; ( Unoler -Ja£4l\ p I'd- )

q.', ,,-Q. - - - ~

/1------"

G-/{( o/C'les ~Y Ye "-Wo , ,

II) ~ =- 0, ~ (Cr-;{) u,d - daVnp:kj.)

~i

I

I t()\ I f

16

1

+ ') ~- > 10~,

H

7

I \ I \

/ , I ,

,/ '-

~ /

'c \o.sU )0-'1" ::1"';"''' IJ a ~ Il«""~ \I~l.Aoe c:....l"....c¥"t.;

i t ~ e.p"-' ~ ..... . w he.r ~ • """ c:-l-~oso.s ~ '~p~"1: ~J. Ot.-by>v:t- j e.~. i'-: , ~Pv."t U -X d ..... .l 0",,'1. p~"'- iJ, Y ) b-(..",.,

1----T--fl\--~--7 cJOH . .! loop ~".v. is

Y _ G

/ )C - 1+G» ,/

./ ~ tv-;{ ~~l- dal"\p;"'~

.. \ UnJe\-- kVY\p;~ /7

I

r

1

(

o.s C CI- ~~ ~ca. \ Gh""p;~)

~Wh~ 4" e<-<L "'" .. Ie. l oo p is 0Fa{:.~

A.s d<l'"'1';~-!:l IJ ;"'crV;;>JeJ (Q ,-!, <:b.r"'UeJ) I '/r.--,s J<?tr .... -.s. Not ,""uc-\.... I ·~ ";lc.-h ' ~<4 ~ 'J o;"':j i; ~ t'] I,,,?,<l~ J (Q '" /'2..) ~ oV"<!.r..ldA-.p~ r ';5 \~ .

C .. ; + t ca ( da.""p: ~d- \ s e >\0vf-

-1t) \-- -t La, 1-0W\. \" +erte'vO~e+e'v- /

x

'( N

+ -

H

'y

N ~ ~V\50)'- N ois e )

t . e.3' shot h O'S" i,., Se.-Hrer

p\" .~ • .l.. ~d~) Or po ; ~ii~ " .\se eo+ se.r..so .... ' ."s "'e~

\ 0 ,,'f '::J iP' o~U'

(GH I

~

! GH \ <:

10""" - - - - -

10-20 m/JH;

at (00 f./"t:

G

~ ) y

....::::. , / ,J

x / 1\-

H~ HI

/" / ~-ot 1"-I I't

Steep ~ Low - .pqs.) - ri lte ~

L-------------------------~21

~o.: '" f 1-----

- - - - - - - -

[ P~ase

~---~ ----""'\

/1-----

f dl--------~

-90D ~ - - - - - - ---

t ~"'d-l: I .... ~~"'1'-bdbl e ph~se. d<-f:; {"'nsi:OI\Je\

"22

+

d b---_--I---;7

-W> -- - - -~- U",s-1Clbie ( I ,

T~ ¢< -(?S'oo cJ-- -tUq. c::==f> UV\slo. b (€-

~M"'7 poss: i.le

{. 0 d "'.." ~

1 ... :,-\, .... r L / b D

rev '" ·I <""'~~-t. b/ -\:-h .,-,

h-~t-ho.l.

-tor-- stab·le ~$+-€VVL

24-

'(

N'

Nt OUTPUT

y

w~ "" .... s-t re~ce C

1" \l S"'fp ,-es s. /O"' -p;O<,s ~J ~,.~ ov.:bpVl1 ",~ l.se

2~

R 8)' ck ; ""«] .s 0""",, s .... " ( \ -b-t-., ~ ) l:k.e p'" -bIc 'i::, ,~ -1:=-"-" re.s ' ~;",,,,,, 'R} w'" ~ ........ I::~ t:-4o~~-+ h O '5-e ~' Ow ~ {'~ w oo -r,::':'t I;"'-kr~

~7 S~Q\ \ r~v.''': 'e. .w-...~~

(~oocl ( )

l-k w -bv -P~k'b l'''.s"i. N ~ CJ l'""I '~ -l: ~-1,;~ '" ")~ 15-£ , 'VI Cl r-;;) \",.../ 'P~ vi. C -;;)"')~

How tes + l'Yli.ts;s- (J'r:evtfcd ,'Ovt

Vto; se Cl.ff.ec-+~ Co v; ~ feVf;:lh?

se __ - - ---.~

- -- -1 - I i-

f. L tc

\

hlo w ~ 0 e.s, +}<.< c:1, V\:> bl i", d-e- o{'

t €s-to ""-;l.s.s Or-',~*= C>{ '~"'" d +~t­~v~ ~y le"';j-t-t- Q ?

,<.

27

CC

l--l~ uV e 2Ss."' ......... a,....c!. "'·,rr o ,.. .. P .. "t -t I.-. 0 t-I-- e..- CA-I r v~.I; VI ~ C' '" I,") <! 1'<0

LASE.. c.v..rveJ.!Cv.,.-veJ .

~ 8',-~~~

)

Iolea \ CaS"e '. Cd-v·,")' C>)f ;S "j.~S -\:-\"'''.'''b\'' c _b<>..rs.~ ~w~ ""'d$> .. ..s:

CM

e ::::: ~e + ~ ~

00 100 H l "\.....-""

~ prodM.u.. ... d, ~~ d '-~ -

d, set + d2.. S02..

[ L: "ea y r:c-H-ec+ of- o.-r",3ct { " "'" No I')e \

2~

Ai r\eNe Laser Cawera..

a~ A2.

~

Measure. S£ /S 9 to\-- d i+te l-eV\ t

beC{M S'po+ PO); +I' OVL

2S0H~

S,~~ W~~

-tv ...... ~ke

de-O~II""e

I'

J;

1

/

/

~' / /

/ /

/ /

/

1

-1

,

"

+ /

/ /

/

+/ /

/

,

+ , /

/

/ ,

/" .... ~

4

30

sv

0 ... , ;....,~<1-1.,.~.., ..{:: Iv,"! ... :;,1". ~

6 e '" /-'- r ad> pr 0 .l.. '-lit. ".)."d)-f:.~o""J ....... 0.2",,_ o{ \:.e-Q.~ I~ o.,,'bl~~f le~"e,-

e V

X I I

I ""/00

®

Jan. 94 10-1•

Best Displacement Spectrum

10-13 Predicted Orientation Noise

-~ ~N

::r: 10-14 --E SVY&SEY ~ - 10-'s c:

Q)

E Q) 10-1• U co a.

.en 10-'7

0 , " I " " . I

1 0-18 " v •

\ . I ~ ",", II

" \\1,.,,, II ", SEY r-~ ... ' I, '

10-'9 ... v ..... y II

0 50 100 150 200

Frequency (Hz)

I I

se

q,l

'$ ~ "- dLF

(+) ~ (1 l ~-t) dj I .. 'V -w ~ 'to h,~"'-~<t,"""':1 w'OOI<:oo<"";r .. '" t low '~~IA"""'''J "",,,b· .. ~ cr\- lo~ 5p<f\: ~;. .... ~ ...,.or

J (IDH~) S'e UOOth)

33

Al

A2

~~~~~---r--------~---

MeC{,5lAi-~ :s 1 Ov-olAV'tA 2->0 He QVle\

d L.F <; 'I Y'n u.[ + C{ \Il€ Ot.D Y

r-. ~

'-.-J

CI) .E A

~ V\)

lh,~ C-vI,-v-t 'u

I;" ~+"rO_"~ y.,O f.se .5'P .. c.4-.-"' .... . ( 1,01-1-0"" ~ S~)

T)...1C: . .k etA rve 1.s

"'" "'" S'< r ~ ~ p. w£or 5P"c+r~_ f' (Hz) o~ be~ spo-t ~h ... ~-4,;>-t,o~

o (b.,., s c...t<. b" "~o 10- 11 ~~--~---r--.---,---r--'---r--~~ 10-2

10-12 ~'" Vif 10-3

'.f' . ~e- 0 J...'" "" "'.J-.. ~

10-13 ~ ~'" .J" 10-4 >L u<>-<"

t.J . -< ~

10- 14 10-5 Vl

~

10-15 d2 10-6 ~ '-"

1 0-16 10-7

1 0 -17 L-----L __ --'-__ --'-__ -'--__ -'----...J __ --1. __ ---'-__ -'-~

200 250 300

f (Hz)

\

~..,.----J{'v

~seJ ra~1t\. flobe

Heus(,o-e Jlls e- 10i-- d~~eYe~ t

bectw\ .>po+ fOS ; -h OVL S

36

~

~ f ~

'--"

1.5

cD V"v 1

""-~ C/1V

-1.5 -1 -0.5 a 0.5 1 1.5

J2 (l-itw\ ) 't,

6Q.'a""- sr~t p"5'";~''' ( Dc.. ~ ... oS ~t -\.,...;,,,,, , ~";' \ loc""O)'iI''''' )

d"Z rtrl5 /\..... 0.2 MWL 1 0 re M..c-<e 4::-\-, ;,; hod .. /

~ wE' n-.",st. r e. ti.kC.~

d 2. ........ .s d5 w~t\ ~~

You have learned ...

1. Test mass position / orientation control is necessary !

2. To damp a test mass without adding extra noise is possible !

3. To predict test mass orientation noise in a Fabry-Perot cavity is fun!

38