MASSACHUSETTS INSTITUTE OF TECHNOLOGY
Depart,rnent of Electrical Engineering arlrl Cornpllt,er Science
6.302 Feetll);~ckSysterns
Spring Term 2007
Lat) 1.i
Iss~~ecl
: F e t ~ r ~ ~ 13,
a r y2007
Due : Fricltxy, F e t ~ r ~ ~ 23,
a r y2007
La11 grollp size is restri<:te(ito no rrlore t , l ~ a 2r ~people. Y ~ Iwill
I hand in inrlivicl~lalla11 \%-rit,e~~ps.
In your
~ v r i t , e ~please
~ p . in(iicate your la11 statior~m1rr11)er as rrell as t l ~ erlarrle of yollr la11 partner. Rerr~rrnl)erthat
l a l ~ swill not 1)e ac<:ept,eclfor credit rnithol~ta cornplete(i prelal), and everything rrlllst t)e t~lrneclin on tirr~e.
Background
The purpose of this la11 is t,o specify a. rnathern,ltical rnoclel for a p11ysi~:alsystem. Tlle physical syst,err~
The rnoclel (ievelope(i in t,l~isla11 will 1)e 11sec1t o rlesign a
explorer1 in t,l~isla11 is a sirr~pleservorr~ecl~ar~isrr~.
variety of feecll~ackcor~trolloops in s111)seclllentl a t ~ s(Labs 113, l C , and ID). Tlle electri<:alrrlorlel ernployecl
t,o c1escrit)e the rrlot,or is the following:
~vherethe 1)rxc:k voltage (or t)a<:kernf) of t l ~ espinning rrlotor is
e,,,, = IC,B
a1111t,lle o u t , p ~voltage
~t
of the t,acllorrlet,er is
i,,,= IC,<&,:,,,~
~vherethe o ~ ~ t t(>r(l~le
p ~ ~ of
t the ~ r ~ n t ,is
or
T = Il1,Im
t,lle position of t,lle o u t , p ~shaft
~ t is
Q,, = H / r r
a1111t,lle o u t , p ~voltage
~t
of the position serlsor is
Note that t,lle rrlotor sllaft arlgle is 0 , ancl t,lle o ~ ~ t shaft
p ~ ~angle
t
is Q,,. The fly\%-heel,with irlertia J I ,
is rrlollr~terldirectly 011 the rrlotcx shrxft. The o ~ ~ t p shaft
u t is geared to t,lle rrlotor shaft \%-it,ha gear rat,io n.
The potentiometer is connect,ecl to the o ~ ~ t shaft,
p ~ ~arlrl
t the voltage across t,lle potentiornet,er is Hp. This lat) <:onsistsof (iet,errr~iningvtx111esf ~ t,lle
r rr~o(ielprxrrxrrleters R,,, , L,,, , I'. IililCh.
J,,, , ,II, I!,, Ill,, arlrl
K, .
Cite as: James Roberge, course materials for 6.302 Feedback Systems, Spring 2007.
MIT OpenCourseWare (http://ocw.mit.edu/), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].
Equipment
Pick 1111 4 DNC <:onnectorsa t t l ~ erlesk. Note t,llat tllese are 1x connectors, llrllike t l ~ e10x scope prol~es;t)e
snre t,o check the s<:opei n p ~ set,ting.
~t
Setup
For this la11 me mill only l)e con<:erne(iwit11 t,lle Power Amp and Monitor se<:tions of t,lle cor~trol1)ox.
tVhen you arrive a t t l ~ elat) area, check t,o see t,llat the cor~trol1)ox is conne<:tr(ito t,lle 1 1 5 volt power s1111111y
a1111 that the rrlot,or cal~leis cor~r~r~ecterl
to the <:ontrolt)ox o ~ t , p ~Make
~ t . sure that yo^^ recorrl t l ~ emnnl~erof
t,lle rrlotor st,atiorl m11i~:hyou are using, as yo^^ will t)e usir~gt,l~isstatior~for all f o ~ l rof the s e r v c ~ ~ r ~ e d ~ a r ~ i s r r ~
l,ll~orat,ories.
Conne<:t a signal gerlerator o ~ ~ t to
p ~t l ~~ eti n p ~ ~
oft the Power Amp sect,ion. For t l ~ evolt,age-rlrive
rr~easl~rernents,
set the Power Amp gain select swit,ch to t,lle '.-2 Y/\';' set,tir~gso t,llat you car1 drive t l ~ e
rrlotor \%-it,ha volt,age. For the current-drive rneasl~rernents,l)e sure to change t l ~ esrrit<:hto t,lle "-0.4 A/\"'
setting.
Triggering
For DC i n p ~ rvar~eforrr~s
~t
(frorn the signal generat,or). j7ou sl~ol~lrl
t,rigger off of the <:hannel yo^^ are recorrling.
t , s . the '.norrna17' t,rigger rnocle for tirne-varying
Use t l ~ e" a ~ l t olevel" trigger mode for t,lle DC o ~ ~ t p l ~ and
For sqllare wave arlrl sine wrxe i r ~ p ~ lfr0111
t s the w a v e f ~ r ~
g re~~ ~ e r a tit~ ~rlay
r , l ~ e11sef111tn ~ O I I I I ~ ( : ~
o~itpl~t,s.
t,lle '.TTL OUT" sigrlal on the waveform gerlerator to cllarlrlel 3 or 4 on t l ~ escope, and trigger off of that
<:hannel. Again. use the .'norrr~al'; trigger mode. If your rlisplaji is l~lankor interrnitt,ent, you may rleerl
aclj~lstthe t,rigger level.
Cite as: James Roberge, course materials for 6.302 Feedback Systems, Spring 2007.
MIT OpenCourseWare (http://ocw.mit.edu/), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].
Measurements: Part I
Part I of the lat) is ~ O I I Pat 011e nf the st,arlclr%r(i
r r ~ ~la11
t ~stat,inr~s
r
thr%tl~avea flywheel. These statior~smill
11e the s~~l)je(:t
lal~s.Make sure illat yo^^ re<:or(it l ~ emlrnt~erof t,lle rrlot,or statior~\%-hichyou
nf all slll~seq~lent
1at)oratories.
are using, as you sho111(iuse the sarrle stat,ior~for all fo~lrof the servorr~ecl~ar~isrr~
1. Drive the rrlotor wit11 a. cor~st,ar~t
(DC) voltage. Make sure t l ~ ePower Amp gain select swit<:l~
is in the
.'-2 V / Y set,ting. '4 DC volt,age can t)e ot)taine(i frorr~the o ~ l t p l ~oft , t l ~ esigrlal
ger~eratort)y rr~akir~g
sure that rlorle nf t,lle wave shape l ~ ~ ~ t t are
o n sp~lshe(i
in. To vary t,l~isDC voltage. tsvi(irl1e t,lle "DC
Offset?' knol). Make sure illat t l ~ e0 dB t)11tto11is 1111s1ledin. The Hp sl~ol~lrl
lonk like this:
Kote t,llat t l ~ e8, rvar~eforrr~
is rliscor~tir~l~o~~s
- t l ~ e
rrlor~itorvoltage "wraps arolln(i'; when the o u t , p ~ ~ t
shaft passes throllgh an angle of i~radians (the rr~eas~~recl
volt,age Or, is c o r ~ t i r ~ l ~
foro ~o11tp11t
~ s shaft
angles in t,lle range -T < H,, < +T). hleas~lret,lle size of t l ~ e(iiscontim~ity. 2. Drive the rrlotor wit11 a <:onstant (DC) volt,age. Measllre and recorrl t l ~ errlorlitor vahles I;,,,, I,, arlrl
H,,,,. .4lso rrleasure the n~ltpl~t,
shaft speerl 1)~'recorcling At, t l ~ tirr~e
e t)et,meen wraps, in the Hrl w a v e f ~ r ~ r ~ .
Repeat illis r r ~ e a s ~ ~ r e r rfor
~ e rfive
~ t rlifferer~tvolt,ages in t l ~ erarlge 0 Y I;,,
10 V. To what cloes t l ~ e
tirrle t)et,meen the wraps corrrspon(i'!
<
<
Kote illat all of the rrlorlitor o~ltpl~t,s
are voltages. '4 reading of 0.5 \' on the I,,, o ~ ~ t <:orrrspon(is
p ~ ~ t to
a rrlotor <:~lrrent
of 0.S A, since t,lle scale ft%ct,orfor t l ~ eI,,, rrlor~it,oris 1 \'/A. Recall illat the rrlor~itorvoltage d m is relater1 to the rrlotor shaft speerl d t)y t l ~ escale factor I',,,,,,.
t,lle value of I<,,,,,,; rlote that it has ur~itsof \'-sec/racl.
part of t,l~islat) j7ou will rleterrr~ir~e
As
Cite as: James Roberge, course materials for 6.302 Feedback Systems, Spring 2007.
MIT OpenCourseWare (http://ocw.mit.edu/), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].
3. Drive t,lle rrlotor wit,h a sqllare mrwe of current (rvith no DC offset). Rerr~ernt)erto s ~ v i t c the
l ~ Power
Amp gain set,tir~gto t,lle .'-0.4 d/V" set,tir~g.The rrlot,or currer~tand ar~gularvelocity sho111~1
look
sornet,hing like t l ~ efolio\%-ing pict~lre:
1Ieas11rearlrl recorrl the va111es nf Im ruld $ill,,.
De sure t h r ~ t~ ~ 0 are
1 1 riot sat,llrat,irlg t,lle amplifier when
(I,,,,
sl~ol~lrl
l)e a clean square mrxve). Drive t,lle systerrl fast erln~lgl~
thr~t
rr~akir~g
tllese rr~easl~rerr~er~ts
t l ~ ed m mr~veforrnis made 1111of relat,ively straigl~tline segments. Repeat t,l~isfor five different val~les
in t l ~ erarlge 0 .i i
1.: '4.
< <
Cite as: James Roberge, course materials for 6.302 Feedback Systems, Spring 2007.
MIT OpenCourseWare (http://ocw.mit.edu/), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].
Measurements: Part I1
Part I1 requires t,llat yo^^ use a rrlotor s e t ~ l pm l ~ i c ldries
~ riot irl<:l~~(ie
the fly\%-heel.Go over to a s e t ~ qin~t l ~ e
6.302 area illat is lal~eled"NO FLYtYHEEL7' arlrl take the following rr~eas~~rerr~er~t,s
at that stat,ion.
1. Drive t,lle rrlotor \%-it,ha relatively slow square wave of voltage (with no DC offset,). Make sure t l ~ e
Power Amp gain select srrit<:his in the "-2 V/V" setting. The rrlotor voltage and <:~lrrrnt
maveforrns
sho111cl look sorr~etl~ir~g
like the following art,ist's <:oncrption:
look at t h ~ tramlhon
s
on a faster hme scale
I
I
90%)
-
rise tirrle
1Ieas11reand recorrl t,lle \rahles of 4i and Ao. .ilso rrleasllre t,lle 10%,-90%)riset,irrle of t l ~ rrlotor
e
<:~lrrent,
rrar~eforrr~.
Not,e t,llat it is easier to rrleasure this risetirr~et)y srvit<:hingt,o a. rr111c11fast,er tirr~e-t~ase
setting
is quite noisy. you will prol~al~ly
on your oscilloscope. Sir~cet,l~isrrar~eforrr~
\%-antto select t,lle i~i~f'nigf:
rlisplay rrlorle 1)~;
pressi11g t l ~ eDISPLAY l)l~tt,onon the scope. Make sure illat yo^^ are rr~easl~ring
tl~e
in the figure, and not t l ~ efall tirr~e.
rise tirne, as sllo~vr~
Repeat this rr~easl~rerr~er~t
for five (lifferent 1
1:~x111esin t,lle range 0 \'
< I r ; < 10 \'.
2. Drive t l ~ ;notor
e
with a square wave of volt,age and rrleasllre t l ~ e10%-90% riset,irrle of the rrlnt,or velncity
rnonitor, H,,,,. Repeat illis for five rlifferent volt,ages.
3. Drive t,lle rrlotor \%-it,ha sqllare wave of 1:urrent (rvith no DC offset). Rerr~ernt)erto s ~ v i t c tl ~
l ~ ePower
Amp gain sett,ir~gto t l ~ e'.-0.4 d/V" set,ting. Repeat t l ~ err~easl~rerr~er~t
frorr~Part 1-3.
Cite as: James Roberge, course materials for 6.302 Feedback Systems, Spring 2007.
MIT OpenCourseWare (http://ocw.mit.edu/), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].
Data Reduction
!Ye will use a lir~ear-least-scl~~arecl
error approximation techni(l~~e
to rer111ce the rrleasllrerrler~t(iatrx
nl~tairl
t,lle rlesire(i rrlorlel pararneter val~les.In a perfect worlrl (like rntxyl~eon an exarn) illis t,e<:hniqllrmol~lrln'tt)e
in t,lle lat) rro111~1sl~fficientlyand ilniqllely (leterrnine t l ~ epararneter
necessary; a single set of rr~ras~~rerr~er~t,s
val~les.Horrever. r r ~ e a s ~ ~ r e r rerrors
~ e r ~ tand modeling (1efi~:iencieslirrlit the accuracy of pararr~eter<:alclllation
i r ~t,lle real ~vnrlcl.Tn irrlprnve our >L<:<:~I~>LC~.
&re '.txver>~ge(Y
severtxl ~r~etxs~lre~rler~t,s
t,n yield the desired 1r1nrIe1
pararnet,ers.
II;,,:,,,calculation: Ideally, ~I<~,,,~,
= 4,. 11-e can ot)t,ainseveral rr~ras~~rerr~er~t,s
of 0, fro111t,lle tirrle 1)etsreen
in the Hli ~vrxefnrrr~
((:nrrespor~(iir~g
shrxft rnt,atin11Bo eq~lalt o 2 ~ ) Ue
. sure
rrraparo~~nrls
to a11 ollt,p~~t
to <:onvert 0, to B by taking the gear ratio ir~t,oac<:o~mt.!Ye >also l~avet,lle correspon(iing t,a<:horr~rter
rrlor~itorvoltage rrleasllrerrler~tsof B,,, . T1111s. in vector forrn, me can write:
QI~l,"Ci, =
Det,errr~irlet l ~ el~est-fitvahle of
om
Ill,,,i,11~7
rrlir~irrlizatior~
of
t,lle sq~lareclerror. a la
R,,, calculation: Or1c:e agair~in an ideal morlcl, R,Ai = 4'1:.
Frorn Part 11-1 we have several rrleasllrerrlerlts
of AI. alor~g\%-it,ht l ~ ecorrespon(iing i r ; r~txl~les;
calclllate t l ~ et)est,-fit vtx111e fnr R,,, usir~gthe lerxst
squares est,irrlatior~t,e<:hniqllras \%-asilse(i for IililCh.
L,, calculation: Tlle t,irrle cor~st,ar~t
of an exponential response is eq~lalto t l ~ e10%-90% risetirrle rliviclerl
t)y a ftxctor of 2.2. Average t l ~ efive riset,irr~err~ras~~rerr~er~t,s
arlrl divide 1)~.2.2 to o l ~ t a i rt,lle
~ average
tirrle constant, m11i~:hcorrespon(is to the L,,,/R,,, tirrle corlstar~tin t l ~ errlotor rnoclel. The \-able of L,
is calcl~laterlby rmlltiplying t,lle average t,irr~ecor~st,ar~t
by the l~est-fitR,,,, vahle.
I<; calculation: BII, sho111cl eq~lal(P'h - I,,,,R,,,). 11-e llave ol)tairle(i several rrleasllrerrler~tsof B for t l ~ e
I i l , , , h calc~~lat,ior~
at)ove. 11-e also have the correspon(iing P'h arlrl I, rneasllrernents, and t,lle t)est,-fit
va111e of R,,,,. Once again, use t l ~ eLLSE to (iet,errr~irlrt l ~ el~est-fitvahle for I<,.
Gear Ratio ( n ) calculation: Co~mtinggear feet11 isn't t,errit)ly ir~terest,ir~g
or e(il~<:ational.
so Ire llave clone
illis for yell. The gear or1 the rrlotor shaft has 32 teeth. arlrl t l ~ eollt,p~~t
sllaft gear has 216 teet,h. Tlllls
t l ~ evtx111eof I I is 6.7:.
Il;, calculation: The rliscontim~ityat an o ~ ~ t shaft
p ~ a angle of fT radians <:orrespor~clsto a j ~ n r ~ofp 2~
r.ar 1'.
larls. T111ls the vall~eof
is l r ; / 2 ~ .
.JJ calculation: The flywheel ir~ertia.JJ is calci~lat,e~l
frorr~l~asicphysics (yo11 rnay need to find yollr 8.01
t)ook). The flyrrl~eelis a cylinrler rrlacle of al~nnimlrn,m11i~:hhas a (iensit,y of 2.7 g/crn? T11e tllickr~ess
of t,lle flyrrl~eelis 6.35 rnrn. and its (iiarneter is 63.5 rnrn.
II; and J , calculation: Cor~sirlerfirst the data set taker^ in Part 11-3 rvitllo~~t
t,lle extra fly\%-heelinert,ia.
Kote t,llat t l ~ irlertias
e
of t l ~ rrlot,or
e
arrnat~lre.t l ~ ge ears, t,lle potentiometer. et,c.. are all 11nnpeclt,oget,ller
ir~t,oa sirlgle rffect,ive ir~ert,irxm l ~ i c Ire
l ~ call t,l~e'rrlot,or inert,ia', I;,, . Once agair~cor~sirlerir~g
the irleal
scenario, ICii = I;,, $4. 11-e 11txve several rrleasllrerrlerlts of
(\%-llicl~
car1 l)e corlverterl t~ $4 va111es
tllrollgh division 1)~.t,lle rrlorlel pararr~eterIClilCh).> ~ l o rwit11
~ g the c~rrespnr~dir~g
j vahles. Cal<:~~late
tl~e
t)est,-fit vtx111ef ~ t,lle
r ratio 111,/1,,,,
11sir1g t,lle LLSE t,e<:hniqllr.
Sir,,,,
Cite as: James Roberge, course materials for 6.302 Feedback Systems, Spring 2007.
MIT OpenCourseWare (http://ocw.mit.edu/), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].
Repeat t,lle al~oveprocecl~~re
for the rlat,a set taker1 in Part 1-3 with t l ~ eflywheel irlertia Ji on t l ~ e
rrlotor shaft. Kote that rlorv y o l ~are solvir~gfor t l ~ erat,io Ill,/(.J,,, 71). !Ye rlow have an expression for
Ii,/J, and and expression for IG/(,J,,,, I S ) . Since t,lle vallle for ISis known. we ht~vetwo eqllations
in trro ~~nknomns.
Solve tllese e q l l a t i c ~ ~for
~ sIll,txrld J,,, .
+
T,
+
calculation: .Iverage t,lle five risetirr~er r ~ e a s ~ ~ r e r r ~for
e r ~Q,,,
t s fro111 Part 11-2 and divide l1y 2.2 to (iet,errrlir~et,lle rrlotor tirr~ecor~st,ar~t
T,. Explain why illis calclllatior~gives y o l ~the rrlot,or tirrle constant.
Questions:
1. In t,lle prelal], you folln(i an expression for t,lle r r l o t ~ rtittle ~:or~st,ar~t.
HOW\%-elldoes the rr~easllredva111e
agree wit11 a. c:alc:lllate(i vt~111e1)asecl 11pon your model parameters? (Rernernl~erthat t,lle flywheel mas
r.f:rrioof:ti for t,llese r r ~ e a s ~ ~ r e r r ~ e ~ ~ t s . )
3. Horr rloes t,lle inert,ia. of t,lle flyrrl~eelcornpare to that of t l ~ ebig gear? 11-11y?
Cite as: James Roberge, course materials for 6.302 Feedback Systems, Spring 2007.
MIT OpenCourseWare (http://ocw.mit.edu/), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].
Just to give you an idea.. .
Here al'e
S O J ~ I Porder
of r ~ ~ r % g r ~i i~t p~p~~d~e o x i r ~ ~for
r %the
t i r npr%rr%rIletel.s
yo11 %%-ill
be cor~(:el.r~ed
with:
R,,,
L,,,
I
I,",,:,,
.J1
I,,,,
IC,
IC,
-
100
lo-"
lo-"
lo-"
lo-"
lo-"
lo-"
lo0
Tllese are list,ecl in MKS urlits (x%-tlichyou ~leterrnine(iin t,he prelal]. . . )
Write Up
T h e write 111)for the la11 st~ol~lrl
l ~ short,
e
sirnple, and informal. A11 Ire req~lireis your rrxm rnea.;l~rerlrlat,a.
or concl~~sions
m11i~:h
your re(ill(:eil prxrrxrneter vahles (in a rleat t a l ~ l eon t,he first page), and arly l~riefir~sigl~ts
you feel are relevant. Inrlivi~l~lal
rrhi<:h rrlotor
rrrite 1111sxe req~lire(ifrorr~each st~l(ient.Please <lo<:~nnent
.
yinlr bat) I I ~ L I ~ I was,
I P ~ and wklo did rrl~at,.
setllp TOII I I S P ~who
Be sure to make a copy of your motor parameters from this lab to use in calculations for
the subsequent motor labs. Yc111'll ttlar~ky o l ~ r s ~llrf x t ~ ~ .
Cite as: James Roberge, course materials for 6.302 Feedback Systems, Spring 2007.
MIT OpenCourseWare (http://ocw.mit.edu/), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].