2.010 Fall 2000 Homework 3 Solution

Transcription

1 .00 Fall 000 Howork 3 Solution Probl # Ma drin by controlld-forc actuator a. Clod-loop tranfr function fro rfrnc oltag input to locity output. h firt tp i to forulat a odl. ranlating a: x Actuator: f act act Snor: nor f act pound-a (lb) D D 0 pound-forc/olt (lbf/olt) Controllr: act M M olt/inch/cond G ( rf nor ) A a tranfr function i ruird it i ait to work in th aplac doain (-doain). Cobining uation: D G ( M ) rf ( DG M ) DG h clod-loop tranfr function i: rf ( ) DG DG ( ) DG M rf DGM A anity chck i to aluat th tady-tat rpon to a unit tp input. rf ( ) DG ( ) DGM DG li ( t) li ( ) li t 0 0 DGM... which ak n and ha th right unit. M

2 hi tranfr function ha no zro and on pol, th lattr found by uating th dnoinator polynoial to zro. DG M λ b. Gain ruird to plac th clod-loop pol at,000 Hz Unit: λ D lbf olt G olt olt - c M olt inch Choic of unit i attr of prfrnc or conninc, a long a thy ar conitnt. Pound-forc, ft, cond and lug-a will work; lbf lug-ft/c. lbf olt olt λ 0 G olt olt inch λ 0 3. G rad λ π,000 c c - c lb inch foot lb lb 3. lug π,000 G 0 3. I prfr to u a pradht for calculation a it ak rcord-kping (and chang) air blow. howork 3 probl part b g 3. ft/c^ lb/lug inch/foot D 0 lbf/olt M olt/inch/c labda 000 Hz (cycl/c) *pi 6.3 rad/cycl G_.6 dinionl c. U th locity nor of howork, probl h potntiotr proid a oltag proportional to diplacnt. h ritor-capacitor ntwork r a an approxiat diffrntiator. It output oltag a proid a aur of locity. Fro th olution of howork 3, probl

3 V a ( ) ( ) C whr V 0 olt, 0 inch, /C 0,000 rad/c. h low-fruncy rpon of thi nor ay found by aluating th tady-tat oltag rpon to a unit tp locity. ( ) V a ( ) C V V li a ( t) li a ( ) li t 0 0 C C Ealuating, V C/ 0-4 olt/inch/c. At low frunci thi nor output i attnuatd 0,000 ti rlati to th idal nor abo. At high frunci th nor out put i furthr attnuatd. Subtitut into th odl uation in plac of th prfct locity nor. V D G rf C D G C C C DG DGV rf V D G DG V C rf C C h rid clod-loop tranfr function i: DG ( ) C DGV rf ( ) C rf A a chck, aluat th tady-tat rpon to a unit tp: 3

4 DG C li ( t) li ( ) li t 0 0 DGV C V C Priouly w obtaind th inr of th nor cofficint; thi ti w find th inr of th nor low-fruncy rpon, which ak n. h nor dynaic ha changd th clod-loop tranfr function in two iportant way: A zro ha bn introducd at th fruncy of th nor pol, z 0,000 rad/c h ingl pol ha bn rplacd by a pair of pol that ay b ral- or coplxalud, dpnding on paratr. o aluat, au th gnral cond ordr for and uat cofficint. C DGV rad c ζ C c ω rad n ω lbf D G olt olt V olt ζω n ω n lug inch lb [ olt] 3. n Ealuating with G incrad by 0,000: probl part c V_ 0 olt 0 inch /C 0000 rad/c.0e-04 olt-c/inch w_n^ 6.8E07 (rad/c)^ w_n^ 7.93E03 rad/c.6e03 zta 0.63 Hz inch foot lb Not that th daping ratio i l than unity. With an idal locity nor th clodloop rpon to a tp could not ocillat. Bcau of th nor dynaic, thi clodloop yt will xhibit dapd ocillation in rpon to a tp input. Although ngati locity fdback idally add th uialnt of daping, non-idal nor charactritic ay liit th aount of irtual daping that can b addd. Probl # Ball-crw dri with oltag-controlld otor. h olution to howork, probl dlopd th following odl of th ball-crw dri: 4

5 π π π ( I I c ) x B x K i () a a a n K n x whr a of nut I otor rotor ont of inrtia I c crw ont of inrtia a crw pitch B otor icou daping contant K otor torqu contant K n poition nor contant x nut poition i otor currnt n nor oltag Whn th otor input i a controlld oltag i whr K π K x a otor oltag otor back EMF contant; in conitnt unit K K total lctrical ritanc of otor Subtitut: π a π a π a K ( I I ) x B x K x c π a π π K π K a a a h controllr uation i: ( I I c ) x B x otor G ( ) rf n hr ar ral way to find th alu of gain, G, for which th clod-loop yt i tabl; on good thod i to aluat th gain argin. o do o, w firt find th opnloop tranfr function. For conninc, introduc o notation. 5

6 π a ( I I ) crw rotation pr unit nut diplacnt (lik a gar ratio) c uialnt a of nut, crw and rotor; not >> K b B uialnt tranlational daping du to lctrical ritanc and otor rotary daping K D actuator forc pr olt Expr th abo uation in th aplac doain and ubtitut to find th phyical yt tranfr function, P(): x( ) D P( ) ( ) b In thi probl th aurnt yt tranfr function, M(), i a contant. n ( ) M ( ) K n x( ) h controllr (or copnator) tranfr function, C(), i alo a contant. C( ) rror ( ) G ( ) whr rror rf n h opn-loop tranfr function, OF(), i th product of th thr tr. DGK n OF ) C( ) P( ) M ( ) b ( DGK b n b b o find th gain argin, au G, aluat th raining contant. Uing SI unit: 6

7 probl a 0. inch 5.08E-03 tr.4e03 rad/tr.63 kg I_c 9.08E-06 kg-^ I_.98E-05 kg-^ _ 4.67E0 kg not _ >> K_ 3.55E-0 N-/ap.5 oh olt/ap B_ 3.98E-05 N--c/rad b_ 8.6E0 N-c/ D 7.4 N/olt K_n olt/inch olt/tr (hough MAAB can aily aluat th paratr I oftn find th layout proidd by a pradht air to follow and rbr.) MAAB' function MAGIN idntifi croor frunci and gain and pha argin on a Bod plot..00 Fall 000 howork 3 probl Find th gain argin of a ball-crw dri drin by a oltagcontrolld actuator paratr D 7.4; N/olt K_n 39.36; olt/tr b_ 8.60; N-c/ _ 4.670; kg la b_/_; pol oltftf((d*k_n/b_),)*tf(,[ 0])*tf(la,[ la]) Chck howork olution argin(oltf) Coput croor frunci, gain & pha argin 7

8 Bod Diagra 0 G Inf, P87.36 dg. (at rad/c) Pha (dg); Magnitud (db) Fruncy (rad/c) Figur. Opn-loop Bod plot of oltag-drin ball-crw actuator. h gain argin i infinit bcau th pha croor fruncy (at which th pha rach 80 dgr) i infinit. At infinit fruncy, th opn loop tranfr function agnitud approach zro, hnc th clod loop yt would rain tabl n with infinit gain. h clod-loop yt will b tabl for all poiti alu of gain, G. Effct of otor inductanc hat concluion i only a rliabl a th odl on which it i bad. A indicatd in th anufacturr' pcification, th otor lctrical dynaic includ inductanc, albit all ( < 00 icrohnry). o undrtand it' ffct, ri th phyical yt odl to includ an inductanc of 0-4 Hnry. A diffrntial uation for otor currnt i found by writing a oltag balanc uation. di π i K x dt a U thi with uation () abo. π π π a a a ( I I ) x B x K i c 8

9 9 Expr in th aplac doain (uing th notation abo) and ol for currnt. Subtitut to find th phyical yt tranfr function, P() ( ) x K i x K i ( ) x K K x B ( )( ) K x K x B K x K B B 3 D x b B 3 b B D x P ) ( ) ( ) ( o chck, not that b D P ) ( li 0... which ak n; a tady oltag cau a tady currnt which i oppod by th total ffcti daping to yild a tady locity. U MAAB to find th gain argin:.00 Fall 000 howork 3 probl Find th gain argin of a ball-crw dri drin by a oltagcontrolld actuator hi rion includ otor inductanc paratr D 7.4; N/olt K_n 39.36; olt/tr b_ 8.60; N-c/ _ 4.670; kg B_ ; N--c/rad.403; rad/tr.5; oh ; Hnri la_ /;

10 oltftf(k_n*la_*d/_,)*tf(,[ 0])*tf(,[ ((B_*^/_)la_) la_*b_/_]) argin(oltf) Coput croor frunci, gain & pha argin Bod Diagra 00 G db (at rad/c), P87.36 dg. (at rad/c) Pha (dg); Magnitud (db) Fruncy (rad/c) Figur. Opn-loop Bod plot of oltag-drin actuator including inductanc. h ffct of th inductanc i to introduc non-idal dynaic in th actuator rpon. hi add pha lag uch that th clod-loop yt would bco untabl at 668 rad/c (06 Hz) if th fdback loop wr clod with a gain of 89 db (G 30,000). hi again illutrat that apparntly ngligibl apct of yt dynaic can b a liiting factor in controllr dign. Which odl i bttr? h anwr dpnd on what "bttr" an. h or dtaild odl can dcrib phnona that th iplr odl can not, but it i or coplx to analyz and ruir or phyical yt paratr. Probl #3 Mchanical traniion dynaic riitd. a. Modl with output rotor angular locity. h olution to howork, probl dlopd th following odl of th chanical traniion yt. 0

11 d x Ax Bu dt y Cx Du whr, in SI unit, ω.4 x ω l A 0.4 θ B u { τ (t)} h output in that ca wa th load angular locity, y { (t)}, hnc C { 0 0} and D 0. o obtain a odl with th rotor angular locity a output, iply chang th out ctor a follow: y { ω (t)} hnc { 0 0} C and D 0. b. ranfr function, pol and zro. h ait way i to u MAAB. I wrot th following cript:.00 Fall 000 howork 3 probl 3 Mchanical traniion of howork, probl, thi ti with output rotor locity A [ ; ; - 0 ]; B [ 0000 ; 0 ; 0 ]; C [ 0 0 ]; y (A,B,C,0); tfy tf(y) [p,z] pzap(y) pzap(y)... which rturnd th following rult: ranfr function: 0000 ^ ^3 4.8 ^ ω l p i i z i

12 i Not that th dnoinator polynoial and hnc th yt pol ar idntical to th rult found in howork, probl. Howr, bcau w conidr a diffrnt output ariabl, th nurator polynoial and hnc th yt zro ar diffrnt. A polzro plot i hown in figur 3. Pol zro ap Iag Axi al Axi Figur 3. Pol and zro of tranfr function fro otor torqu to load angular locity. Not th unual axi cal. c. Bod plot... ar producd by MAAB coand bod(y) and hown in figur 4.

13 Bod Diagra Pha (dg); Magnitud (db) Fruncy (rad/c) Figur 4. Bod plot of tranfr function fro otor torqu to load angular locity. d. ang of gain for clod-loop tability... i bt found a bfor by coputing th gain argin. In thi ca th controllr tranfr function C() and th aurnt tranfr function, M(), ar both unity, o th opn-loop tranfr function i idntical to th phyical yt tranfr function, P(). Figur 5 how th rult uing MAAB coand argin(y). 3

14 Bod Diagra 75 G Inf, P dg. (at 0000 rad/c) Pha (dg); Magnitud (db) Fruncy (rad/c) Figur 5: Gain and pha argin for tranfr function fro otor torqu to load angular locity. In thi ca th gain argin i infinit bcau th pha lag nr rach or xcd 80 dgr at any finit fruncy. A fruncy approach infinity th pha lag approach 80 dgr but at th a ti th opn loop tranfr function agnitud approach zro. h clod-loop yt will b tabl for all poiti alu of gain, G. Not that thi i quit diffrnt fro th rult found in howork, probl, whn w xaind th tranfr function fro otor torqu to load angular locity. On phyical xplanation i that, according to thi odl, th (idal) aurnt of th rotor angular locity and th (idal) application of otor torqu ar co-locatd -- aurnt and actuation occur at th a point in th yt. A a rult, an idal proportional ngati locity fdback controllr i uialnt to adding "irtual daping" at that point. Gin th idal auption, intraction btwn that "irtual dapr" and th phyical yt dynaic cannot add nrgy to th yt and hnc cannot induc intability. 4