Feedback Control Systems - Lecture Notes | EL ENG 128, Study notes of Electrical and Electronics Engineering

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I EEC.S J2f5 LECTURE NOTES f I

GOALS:

. C-CTvv1fQ.VL T!CJ T~ vUl~ SS

· in J--,o dvu CL .ftv.. m a. J, ; X -tKf O'YIUl'h ~

eAt

. haw fa crrwlpvlL eAt?

7-f

Ve-j r!: A set oj va.via.bles iYl a moNL

of a.?~:; J'crU Stfshwv oxe, 'fh~

Sfo.+e-s oj Ct /1IIAdu-L If:

(i) joY (). YUj "hYvt '" t ('j ) 'lit.~ sfa t U; (). i to

aLO'Vlj /til i'!h- Yh~ I'\JrAVe-j01 m S oj (). {;(

input Vo.vrabltS Jov i~to ddUlYVlM1.e..

'Htt sflA.te-s o.-G tl<J v,iru- -6>-60 ~

(ll) '-I--tt( Sf-aft at "-h'YV.L tan d %e

Va Iu VJ of Yh e., iYlf vt (At Vt'm f., -t

defljJrYlA~ f.- UJ1A.9~Uj ..f'k{ va Lw.- oj

~y VM lo.h It ir1 ~ modLL. J2em~v k S

1.. STafe.-s ave.. not UVI.A-?!At-)

how€.,VUl .fh.£j arf.- u~u(}.., Sd.L(':tul

-/0 have- 7' >rt'eo1 WlLOYU'/-'

_-. ¥leW 0 L- T F Step Response 1.41------,---------=Fro~m:~U(1~) ---r---~

1.. 1 .... 0.. 5 10 15 Time (sec.) IS 20

1- Ai (J ~ C e- t's L 00 It (1. t IA! ho.. t we IVt- ) iJ S t done- ) bvt iVlJ "-litL 5, f- () --tL sr aJ.1- : _ K(~) G(~) ~ f 5 - JL }lL 1 I \ e i s 52 -J1.z. d1,V10t~ inpJt to (JLTF a.s

U. No vJ w~ kY10 ()J "--fhat: , .. Xl ::. ?Gz.. e = JL 2. e + 1,(, t Xl. =- 51...1· xI T Ul o..L so x,:: Et x·- e U = (s -sft ) Q C ~ SLL== SGl-J2CC

". _ ass lJ rvu. VlJ.

'> ~ = 7Av - fl u ( ltvo i rUV-t a1 em a&'!-t CYY'/) 0'Yl- -u)u) , "" Yl uri fo ole I /v1.L ~ n l.vtJ State VOJll<k'b iL t ~ V( J v.. r4'n amiUJ ; A:: 3

: = V(, - vL

:. ~;j ~ -f2 -u.,

l-b

C/osecL Loof systel'1'1J ctm1fufUJ( en follollJs:.:

I< U I" x=- A X + 15 D II rJ" y•ex • 'I = e "". u == 12 - Y := R-CX .'. close-Ii I oop s~ sfe.wc, ; . X ::: AX + 15 cR.-ex) " ... X=[A-EC]X-t-BQ ..c y = C X

C La <j ui L 0 @? shp rlj) p IYYl.JlL : sys- cL = 5S { A -1St}l!: J c) D ) j stq::> CSjS-CL).; Step Response From: U(1) 1.4r-------,-------~----.--------------.

1.. Q) 0.8· " .~ T"" c.. >: E ~ « 0.6 ·-txa..c~ihi-sa.W1i OJ)

0.4 - btr··.

• ( VL 0 Sv rfl I <.<.., ) •

2. .•............ o~--~:-------L----.l...----~ o 5 10 15 20 Time (sec.)

bu i '"

7- '1he- unsfa.bu.- role, is sfillv ~ut / anal rYIOj bt exCA'tu{,:.

- nm - uyO /1'1 / Y-t cx1 C rYVl 00.-l/1 0'rV) - dA"$ \f..u( ban CJJJ. -rlavtf uVLwda."Y7. ~. " dt. [--rHUS]. S+a+~ Spa.Ct rvwr!J-{S yovid..e us W/~ (A, til) ivtiolAJ Info Ytte infUlno d:r-awu;c.s of. tl 51 ShWL ('-!has t ~cx.mA.C-S iN tuck mrMj bt ?u.d.rMVL ffYl/Vt- ~ 1YlpJT /0 l/yt UT 'kaVLsfUL fuEer::Jmp ~) He( r coull havt l)(~d}':CTh.ot '-!tLf.; uJ/lsfo.-btL wwdL by CO'Wlft'tt J : ~qeV\v(,lWJ.n (A):= {1)-1/3] ~j~VlVo1M.w (A- -gC):: [-i'" .nj ) /} 1

*/-* o~~ of. ( A ivf~oJv S+~tt VtDp~ obse.xVe,d.-.·· W~/ \ '6\tL (^) olAl WhAj Soan ... ~iJ rJl t-4. A.f<: 'f: S +0<it. S po- cJ.- yrl)) ~LD /-ua. v ~. SaM~ Gu:AvQ.Yl+~ti) oVUl ~1kV1.S feY fYlC'ftO'YL VV;oc1.L Ls : i. IV (lVI. &} _Qf_ -rLJ V\ ~'kem, (tf {.~ft.t.Xl ta.'tt 01'V is rQ,s'-\V1c+~il -to ~ iVtpv+-oul..rvt ~I'VtlC~ (JY\lj) (}..Vl ~ '-f'1;t \J S h IcLe. ~ :

• vVI sto.b Lt ~().. Yv'Ll(..s eiYh V1 V1. (l,t c Of ().. bIe. o{ btiYl~ Ik-jftde,cl by 'fW.. lvtpV+ oy. J II l1V1.c,~-Wolla.bLL ~(}..vY\Ac.s II
• vy"\st().. bIt ryYUC$ YJ,ot Vi sib It oj ~ ou'lfut lJ Y. 0 '0 s Vl v(}...'b LL JrCAyy\J.~ II rl II ~ 2.. sto-tt- sP.a.cJ.., m.o<hl~ a..rt yYlOre.., GWlvmie. + o'L yY1. V+, _\npvt (^1) m \J I ti - 00 j-rvt S4 s+e..1'VlA. UJ\fV.o) J ) { D. Ir1 {-i a.l stillt (~) P (YYl ~ e., l q iVtVl bj S to..tt, Spo..cL- rnO ~fl~ but ~ot hilt '-\¥6:' VL SfljlfuV..c'-1 (Yl"\), I ~ 0 teVL ~J J . ) 'j

7- SO&U noN X it) -to ~ (t)

x = AX+"B tt STATE S"PAC,E MOl:>ELS: ~= ~XTbU X iO) =Xo

COI1 ',) ; it r '-I1tr.L S colAx' dt ffuu. t1. y.., 01 t? lJ {Ll/1 tTYL:

")0= a..z+bu a}h)c)rlell(.

~ OU+fvt. ~.IL J == C, x, .,.. ti u ; ini~ 0.1 ~fa.+~ ,y ( ) ?(. : = x.l.t)

. -lIf 11.., ~ 0 ::.?G o Lte. rJ tel ()(..t'~ "Ih ().. t ..fi-tL soIv'H ~ x (-6) fo 'I1t~ S fo.. tL J.iI U ()..,Y-t (YVV vU i ~ ?t (0) :. Xo is: I i; X- (:c) '= eat X + J e a. (i--r? b·u (-r) a..'r o i~·lfiQ.T s~tl. Q -. t re.spGv'\se... 't1,fU Y'POYl How -fo CMcJ: Vh.ai '--ftN.,s 1£ ~

COf(t,ct solv '11m 7

1. CVtuk '1iuLf it S{L'!-isf'e.~ 'p. E. 1.. CMWt %.Clt i f SCl'h':f,i~ ;V1.i '/1 oL eoYldA'H~'

1. if '-tttevtv ~ (t) ~ tte a,t x'o I a.t a.lo) '#1 J6lt)::=- ~e M + 1-. e b~lt;) - 0 _i

• f_ (a.e tJ.lt-,..) b U l't~cL"'" o t = (l (e a.i ?Go + J e, a..(-/:-T ) bu t:1'" ) d,..) + hUlt) c = a. x Lt)+h 7A.,t-t) .' • x. l t) s fA.'H sf UJ YIv.- ]). F. ...

7- Vtf fj, '--rtv. YY1.a>fv ix eKf fYY\J..Y1..'h'o.1 e, At is riel~ to be: e itt = I + A t + N· t 2. + _.. II .. Z

I wh Uti- ,{ € 12 n.x.1'f.., ) r i5 "!-ttL. n x. Y/.,.. / rkvr.Y.;. & m.,,~ f..y « )(. I cr (it. eAt if att n.J(.~ m~ fyix) .. _7l.° f Ul^ iu-D i ~ UIA.Ir,',/- EJ<f-fYN-rlf.a..{:

1. eO = I 2.. eA l-t-t S) =- e A.t e As

3. e, (A-t E)t = e Ai- ~ ~t iff A:E =:"EA.
4. (eAt)-1 - e _-At
5. i_ eAt = Ae At :0 eAt. A 6. L ~ 1: l (t) b ~ ~Y'L, nXfL n-vaJr/)L , /lttVl '#u., So ItJ'/-1'% to

itt) = A ~ ct)

z Co) = I i~ :z (f.) = e At

{-/ Le,t 's ~ow iN~ 6. is ¥V~: 1.. chulG ~t l Lt) =- ~ At S a../-·i l' f./) 'th ~ iVL " ¥--t oL con tit 'ft cnu : It. lit) = e Ai J = e- 0 = I It=o V

2. cltect 'HuJ.. t l ti) '" ~ At s(J..~·sflDJ ~ d.iffWvt~d UJlJ()J/~G7v: · de At I~. l::- CJI = A. eAt (r-(}pen~ 5) '" A. Vi). V How -to do..1w.1atu ~Ai?

l-I~

Kllt~ is e At =- /-1 (ST -10 -I?

-:g~C~ : ~e)V'~ ~wrv Wto.-t '-h 6ol()Y1.~ to .

Z Lt) == A:z Lt)] - ) At

:z (.0 ') = I I S .=z L -t :: e-

So iala.. GItf LfA1.L I./ra + 1% oj i == A ~ :

S l L~) - -c La ') : A ~ (S) :. 8:C (~) - A ~ L &) =: ~ (0) =- r .... (sT-A) ~ (S) :: I

. "^ .. £^ (s)^ =^ l^ sT^ -^ A)-l 1 il::CLi)) II t[eAt^ ) = [SI-A)-I ....~~-__

t Xli) = eAt}(o + Je Mi -"")1?ut-,..-) eLy o fANDJ 'iiu- OlJ~vt YLt) ("IS ~Yltp11 ex l t)+ "DlALt~

Crl"ECk: : blJl cVteeb'YLq 1. XLt) so..~sf'e.s J:>E

J J 2. Xlt) so..iiSj'U) T-C.

-t

1. Xlt)= Ae At Xo + A feAlt--r-)-g -u.L1-)oh o. .. + B1,t.(t;) :. XLt)== AXlt)+"1?lA.-Lt) V- a

2. X(0 ') ::; ef.O )( 0 t i-rAl = Xo °v ...