CN4227E Advanced Process Control

Tutorial Questions

,N1

'/. Consider a feedback control system with the following components:K

G, (s) = G, *t)Gr*t)

; G, (s) : G- (s) : I

where model parameters of Gr(s) are all positive.

(a) Design a PID controller using the IMC design procedure and

IMC fllter. ls this PID controller identical to that derived by

design method as shown in the Slide 13 of lecture note?

first-order

the direct

(b) Redo part (a) for the process G,(s)= .: "', where the time

delay e o is approximated by 1- 9s for both design methods.

(c) Redo part (a) for the process G, (s) =

.2. Consider the following second-order plus deadtime process,

G(s) =-4Je

(s+2[s+6)

(a) For a feedback control system consisting of G(s) and a proportional

controller K. (> 0), what is the smallest steady-state offset obtained for a

unit step change in the set-point?

(b) For a Smith Predictor embedded with a proportional controller K" (> 0),

what is the smallest steady-state offset obtained for a unit step change in

the set-point?

r{C)oo(!o.

,t 3'Aconventionalmulti-loopcontrolschemeconsistingoftwoPlcontrollersistobeusedtogelllgltheproductcompositionsgandlqofthedistillationcolumnshowninFigurel'Themanipulatedvariablesaretherefluxflowrate

Randthesteamflowratetothe,"uoit",!.Experimentaldataforanumberof

steady-state conditions are summarizeo il r'ote 1' Use this

information to do

the following:

(a)CalculatetheRGAanddeterminetherecommendedpairingbetween

controlled and manipulated variables'

(b) Does this pairing seem appropriate from dynamic

considerations?

Table 1

R

Run (lb/min)S

(lb/min) XaXo

1

2

3

4

5

125

150

175150

150

0.97

0.95

0.93

0.94

0.96

0.04

0.05

0.06

0.06

0.04

22

22

22

20

24

nLU

M

N

Figure 1

Al!. For the flow-pressure process shown in Figure 2, it is desired to controlboth pressure P1 and the flow rate F. The manipulated variables are the stempositions of the control valves, Mt and Mz. For simplicity, assume that theflow-head relations for the two valves are given by

F =20Mt(P, _r)F =30M2(p, - pr)

The nominal steady-state conditions are F = 100 gal/min, po = 20 psi, pr - 10psi, and Pz = 5 psi. What is the best pairing for decentralized controller designbaqed on the RGA analysis?

' t''Po}-o Fisure2 'ffi:fr|"..a-, ov I - L-'2"*.2"-f 6-: E= 1 l * !, 1-t { br L'rr ..t r,n r-n .@-{ c t -' )'a

unimg{. rt-!^ ;._r P \ un.ilri fi-s-t*'l-+ \ r!;-5|,ur, rErauuIr. \__\ 'J.alE il,rtelal t_--______-_ai-- I

where pn and e, denote controiler's output and error signars at t = !! ,respectively,y, y are positive integers, and {a,} and {r,}are constants.(a) What conditions must be satisfied for this digital controller to have a

steady-state gain and what is the expression for steady-state gain of this

(b) Suppose that this controller has integral action, what restrictions does thisplace on the {a, } and {a, } constants, if any?

It+ brf'+ b1t-:t.., rbr.r Z-" ) p(a-,) = 6,+AtL-'+.- + a*L-bgCt"l Etil

i'

c;

[3-r1 is norfartrf t

nqmrY^tov .

ffa d

cn0)ho(tt oeitne

,*

,{0.1=P(z-J ,zf P'-Kl 'L'v ?n'), k-t,t,3"' ln-}:j:::1"z te,"l-- [-t.-'1 . L lrr-oJ , z'" -(r-'1 , L'1,7-;? - J

PCz-')-- - br.-' :j:') - L, L''pS1) - ,. b" .-', l-!11 + 4ol'tz''tt4,t' E(z-') r,. r 4.z-uEct-')

7

?z

. ) ,( .du,1aty,lloritt {g{grmthO / pI ,ir\tt,ryil artwr ' \

\1dw'arn

W6. Consider a process given by:

HrGn(z-t) =az'

l+Bz-t *yr-'\Mat is the time delay of open-loop gf if f denotes the samplingtime?

\ffhat are the constraints imposed on o, p , and y arising from open-

loop stability condition?

(c) For a unit step change introduced in the process input at t = 0, what isthe output response up to f = 4T?

E!LI!tl---Q-Ilr$rg *@l-'7 . Consider P control of the following process (with zero-order hold):

G,,oGr(z):-iffi

and sampling time is f = 0.25,

(a)

(b)

ELi

r a"Qt, QPR - t+ CrchuQ? t

Arc

(a) What are the values of K" (> 0) that result in a stable closed-loop system?

(b) \Men K":05 is designed, calculate process output response up tot = 2T for a unit step change introduced in the set-point at t = 0.

8. Consider a digital control system consisting of the following first-order process

and a proportional controller K, =3 .

G(s) = -1-s+1Suppose that a zero-order hold is used, what are the values of sampling time

resulting in a stable closed-loop system?

+0,)ho

o.

$rctcr WN'

9. A process plus holding device is represented by

G,,nGr(z)=a z+ulo (, - b,)(z - b.)

where ao=0.3, ar=0.9. b, =0.8, bz=0.6. This process is placed in a digital

negative feedback loop with negligible sensor and final control element

dynamics. lt is desired to develop a control algorithm to give closed-loopresponse-as shown in Figure 3 corresponding to a Unit step changeintroduced in the set-point at t = 0.

(a) For what values of x, and -r, is the closed-loop system stable?

(b) Derive the transfer function for the corresponding controller.

@'. the controller derived in (b) physically realizable or not?

(d) Consider the controller derived in (b) alone, what is the steady state value

of controller's output when a unit step-change introduced in controller's

input at t = 0? Assume that the initial value of controller output is zero.

output

I

Figure 3

tnAJbo(6o.

oi(eslSIN1HES tS :

10. Consider the following second-order process plus zero-order hold:

17+ z t)z-l

cttl=#-*u

ryvl'tr- (#), = drZ-t + arz-' + arz't + arz-o

\ w)a-- L-N-\

6'u% H oG o(z) = l- z-t +0.22-2To apply direct synthesis method to design a controller, the desirable servo

response (y(z)lr(z))6is given by:

(a) Wlrat condition(s) must be satisfied to ensure zero steadv-state offset in

the servo response?

(b) ln addition to the requirement given in (a), what are the restrictions (if any)

placed on the coefficients dl, d2, and as resulting from no overshoot in

servo response requirement?

-

(c) Derive the transfer function of th rresponding to (y(z)l(z))aspecified above. \Mat is the condition to ensure that this controller has an

integral action? ls this condition identical to that obtained in (a)?

(d) ln addition to achieve zero steady-state offset for serve responsecorresponding to a unit step change introduced in the set-point at t = 0,process output response reaches 0.8 and 1.2att=2T (T is the sampling

time) and t = 3T, respectively, what is the output response at t = T?

\oOJho(!o.

I - (?r'ld

I a'l

0 to g-os@ zoro (Kut)

n(s\= t)K'o [Cov]smnrl

9, t:) 'tr,st1)(trsttl

Qte1. Gr(s) q-(s)

srEp I 6r (s) = |

slg2 0 (rMc contoucr)-t?-q',

Q-tr)=*=e

= trtrl't (trtG)stl--------k

-e Q,4uivq1s11 F B eortrollcr ;

Qc le) = "h= G"f

I

Gnf q *t,,*_\qr rt- laj c.JLo.flni, F,

, E;f

. \t r sFtdl(Hlir}ri.\j. y k.t ) rnorc 4gr{ss\ve

ohl+au|, l; O /

., pov$vnnen tc,"lo-sls otfwt

I-ftrt\gn.

kc is n* {lxtd .{\xcdt t, 1T' 7 kiom'oflo${onS,

Stabitit, on$rrneA

rfhDr If 0 1 Xc{ @Iri@ I Vt4O

... 0( Kc ( 00

Kg gtt tf + tot )A3'+Bs+C . ('+= B(r+**-t,

I

Ic8 tlI

\stl

[q) i5, trbg:+ C:.tr.)stl

_ # _ i=fu*,---T-T-

Is{l-} =fiN

' ni-P L '*,*rai-ilF",]L rcJ tal- ta, j(iitt,)

(rs)l+ G. gc'20

. u tw(t tr*+() r_I t ft4(tl*rt1 * ---7- ' nt -- o

I t fr ;!Its+l __O

s. * . o. dv a[ r>o).--/-JCtoEed tooP ?o\a

7 xc= B\ ." *1.\ t', ft/e

)

e.L'

limr De.\A1 + Rr\ P Ztvo .

k.Qrts) = t ,r g-t" =

t g-os' l- gS

G. (s)= l-9sE+ (s-0) = I l4ain=r\ -i oKiq-6)= * -#;;o.,

n- f rtr- Gl- T- trlr 9' 4 (tr +1.) t +t

ts Is*\

4

kc ( Kqrnrx t" "t) r

lbl

zero tRuF) rn(s): o

t-es = 0a_ I

Atts 'Oh^?rvin5 '.

. . trt[.rXr( n-o-

I

thr 0[uiv6lcnt fB

G"cc)--fu

:

_ (trrtz'lK[l+ 9)

- trc-)

cAh YAv$.

Che,& st0lrit\ '. 9r

tontol\tf i-al- t.- .61a

- f,rlr8'+ CIrtIrl Stl IK m' r-?iTs)TFTT

L.-l-J

i( O =O , No tme dt\ol ,I Nc Art ?Arf [4\ 4ns$/cv

r.t[lJTg =

L,i - cic.,t-ffiir];,-r?L l-orrtto rs Aif

flolrr (a)

[,* 1rS+ Iosl

tlrs) s

Assqmc K ?0 lis thcrr. ryfer / towcr boutdar$ &r kc "

fixd , t, V t, y' 1. (a$ ye ^hl 1" e'iNt r^\rc'

nhtn ttr o , ({ask aC$on) 1K. 4 r,*'ttt's\

6Lr\v'|r' I oa *, affi 6-r^ly;. ot- k0 I

rNhrn \+ ro ( crow ^cti!l\) \1" StlHi*)u stabitit',

r, ro JIc tra)

0 tn -- 4o' r ltroo -..v{htYr

1I ." h=l,lr3,C.., , ^ K0'os Cl,itt\(T'et0

I- r'ffi1' ----K- YGGjJ =o

\l:

" duq.to trme delo( /, +7 ' t*t**

usinq 0\r00t silbstttut\oh.' r0 po(eoaro juicir6inag) 6 c1s tgrn) -0 ..0\r0ct.e\rbs\tu\0h"n$;il[]tl,t,*-*f'+';:,:;r.,.0 . +=4[rto)'jro*coDj'g J 'u*'(E-] '12rr711"'

'4f ]roc

()tt o) juc * cos iolo) - je,,ntohr) = o t 0J \ tr,* 0)ro- s\n(rrol.Orl-y* " #o\ d;;s\n deggyr: ena ordort\mr do\aJ(lt0)uc - s\n l,0r,rr) =0 I -U _.ls

t o,tgluol\"0 I ,. r- rrrt, -T,tr.(ro^) I ttn- trs+l

= ' *t -\ rhtr' n?1'|)'3 " = ,]*r, ru +f I = r,(rsrr !rrr(,)E+r r Nhrrr doj r-&s-n--t;- gl I -------*-'ffir-e.)Kc=:- # ..,kc(K,,*o\*.Xi r.f,b./. Y\ "re;er re.ktrNq A of(\nuoqs $0. L ^:t ",:U \ ^r.l " gJ

=l , El f,t'l 0==,' ,' ' i *,t \iu"y :: , { kr,r*, , Nwnn--l

cs"attes+)--3r .e) vf )

rct- og)[Gstr'lterq)

fJo deod time. ; -o PHP eevo.t(o)

ea rs1 , ###,r **' T']ii , o

Q' G, q-G. = t- TsG.=g- e+ ' ct,g_Ott s+D-

O o-a.f_ (rrstr ) Cbgtt .t I-

--,FTi-FB Corttro\ler i

o4l9c'p* =-q_r]GT. (trstl ) (T,stl)

K

_ (trt Iz) rrtrrtrt L 1+

Ierl 1- t-tcIS +l

Iurr)r

l1T.Tr1[trrr,)G'*ffitsJ

qain= I oKl

oj .urp!.r! t- Ts,drP = I.rT j

-ltTsfs t tl '+ $0|'n -' -1

wrll ngf AchQiv( ter{e rt ,c6trc\ .

A, . tq * o/rfl, - (rr st) (rlsfl) .**' t- o/5e K[r-rs) ' rlE' I tts+T

= [Irs$)(f,.! il \ * Trr: 9t r LTrt tr\ g ilK(tts+lt'ts -\') -K Irc t T ) s

[utt') [,. -*.,1t*

p.,rrl(Tc+1) x L' (r'vrhrn \lF

^"., ;.1t,

0 < Kc ( :'t!- tr rto ' tonstavrfkt

/ StaUiVl, Su,av1ltet)tt.

- No Algvorirnatron

- \M0 o\cSO3n ) o?ch \ool st4vl.{,

.9\m'\qr r"o La")

C'E' I t g,Gc,o

\ r XC\-Ts) [(,Stt][r'str) . IGrstr') ft,3111 " k ^ (\tT)s

tl-Tel r (r,+r)s --0

}|s t I --0

-tS=-1-

0t;tnd ovdtv r\us deod trrnc r /. fi}l,lli1? muet be stable !

*--rffi-r'f^-4sQ (s) ' Gftt?S--6,)

a) Fbls.s offset = t-:r** - 5'ct-ro)

=l-{,s=01 t!!---1--J

c#**wt': r- ft+;. (s,o)

' t4zKc

t-@; ='*t

^eor4'iii6.*

wYlat \s +he cmallesf offset Z- NhQn kc io targei

1to-ar) -+

tg:t) i

(r. zo)

ctossi I FB -

Yt6-)t' S? '

G. Qe-otr* Q" QLo'

?er fertrmdcl

stab\trtrt e4S

c.E r \* c*fotur Kc = o (t6'rr)J AW\ !\r{ct sqbsntution Ls.jr,c)

Cjto +"1 (jro t6) + cos (4o) - js\r, t0o) = o- 0j

b) Smtn Prcdictor.8.8 offser = r- { cs=o) ' !

lL = *. #rrl,*o)l ?

t+ Kgxpirytq

s+ohtr+U :

c.E + t+ ffir1 -- o(t+6)Cst)) + Kc --o

S)+ 89 r 12 r Kc=Otvr. wQ L *u.J

Rouir'r Avral oY v"tWc)q

closed tool statilty 6uaravttedNo con3tra\fits on '\lrs Kc

P0vfect Mdcl

to-)2,+.r= #

-y Q srnalltsf offset is t^lhgn' +his eqrl a?Proa$es DAnsr 0,y

03 5'k'ur Xlr . kt,l-** J= L r,,

Krr.ffi* Carsrt slUt a i 4

Cts tlert i3 ait\g

. AXDKtt - A8

\$Yul/

ll ut_ o 4b- o 41

llo - l?5

_ o 43-o,t1t19 - \1,

0 46 - 0.4$Q\ -)0

g t16, -o4E-il-r1

0.0s - 0'0\tbo - ttb

Kt,Rr Kre SKzrP* krrS

L\atet on Plr. { I

(baseA n Rur.

Xo --

XB,

= - 0 olaq

0

- - o. o,.l60

a s!)

n r { eB)

- I xB(zz AS

o-02

t/7

CLegeJ or ?vrrF4 f S 6)

Le r { o a)

\(rr :outPui ''ott

L9 \ L ''t)

- + 1'* r, tr)

=:.o.lt f*r,te)4

BY\!to\ AnQ\Utris, I ,

0 ort -0.0Lc9-20

i tru l'l rt 'lI t-I" \,, J

Ir\r' \- ffi; t-&ffir,t-

l-!' ' L

='l Jt-- Li ; ] go$ dra3ona\ c1!,,ri') AC{"u')

tH) u,t-.q\gutg.r t

k'ro-A9- o'|?.

160 - \3to-4{ - o 43t50 - l+s

_ s ot/z\[c\osg.l-1og 5*'At1 l)

. set 9t _o.ot | 7s,.-, C S45.t

At^r)

f_

t*&,)' =

r ))q.\\ aB, xg

Pvr ...1F.O0 I^rCPo-Pr)F'3oUa tPr-P")

Sls , F.lo0 3at/mth?o , )0 ?giPr ' l0 ?sip, , bgei

r ts'o\ ' 166 = ooM, ( 90 - l0)too -_ c oo M,V1, =5

F(S.0) . l0o , 3orvl: (tO- 9), . loot4, = --r5o

--L7

Me+hd3,-ia\oMr)

F -- 6oo tvtryl Po - P'[3oM> + :0M,;

K5,

*" '*, :

r-i-i :o(?o-Pt)'T 'ut

:

QrrL rrrio X,'')A.'/z >o'

?6irnvra I - U,J^ U"Ir' \'{2

)t/i

6ooM, hl** -[oou,vr,ffiu*r,yl ,,4oo:

5 too

600 Mr #:,*, - | ,*M,M, c***,,,1 ^6o0rqr C:cNr+ a0Mr) (Po - Pr) - taooo nr,u,(pu- P')

C)oM,t3or.lr)t

con'ia I

. Art = Krr^Krrl-#_' ku Kzz

_

-r l)rsftrolS)(rro)

-- 50

JDMTPI + JOM: P'

aOMil 10M)..- l5c,tlr. )ol\l,-\l

6Dfo Mz r zqrM, Po - +a!ui?o - 6qr.l, PL

-- r/.' nlt-'/z)

,t,Ll"','lt]"1/xl,trtto*ni^fu)

?ain19',

4ooM:P: + 6o0M'P: - 6grr, 9o - 4oo,Xi?r

tloM: " lor'a'17

, (94 - Po')Goot'r, 1.[l[-ru,)-

-5

? gt)

-[r.)

[ ]Ot,tt + ?o Mr'] 2

= 6coM,@- - .2o(3our rz'aurf T

39 72. (lour.t rc Mr'\ - (lor.a, h + 30}rr?

[3ovr,1 ?o rr^,)]

"3b

0b.e.-@---r ;gi

il'- arQ.y.rt A,0n-r t "' + Am?n-rn - b,pn-r

Ho Qp[a-t) :fiz-3

l* pzr{ tr;,

Ca) T de,oot?,s .rarngtirg tin'e ,depl &ctor = 3 , A ctml {trne detaY = .3T csar'rgr6g toe)

aT = p'fir',?rr* .' [i*l.ix],l"r1 ] r"*;3^*"-(N+t)

, a -r.t-t

N'+Pb58s tmg , &T

o =o clto r':_99J,8ee an$ r{!gongs '.gou do not

(b) No coMrion irngosed on (-Apl\tl .'Jut'8 st4bifitt cir\sta rr 6srrdg64)

Cef-) denornlnotor : t+ pz-r + t t-' --O

p)

?(,z) = a:t 4r L+ AtItn{. etab,ititu er\trrla apglred -t" dr qr.d At 6tQ. t

P (,t'l) -- t'r ar -r o: > oP [t,-r) = t - 4, to5 > O 6Sr cven n. ,o,])laol I Az , 6, > I

dzl + pt + I'o

ls artFl resFn.ge up t, t,4T Z

b< -'t+ (4r-t)z{ + 1a,-a,)z-2

-2.-AtZ-

asP &,r 4't '

J SRf o\W '."pr Q 1=9 / (Dhot

, us{, Lorg fivtsion: :HD(.)=#fi:h-" T;

b z-e _ b U,-r)z-4 + . .. 4- sro? 0 4 Ono 4ns',i!i11,..:).-q4zt'la,a - - b{-3 + bta,-tU-+ + btnr-ar)a-5{.,,

o - bt^-r\t-4- -b(a,-lla-4

$(o)= 0,

Utr) -- o$(rr) = o

$(3r) = bg t4t), -b Cffr -t)

-- b( t- o,)r

.l .

zeYD- order hotd '

QuoGe,.l = fffifiS4rnpllq6*ln T, o.15

Ca) KcrO 1.

C.E' l*k.GhoQett) =f,, . -t + t':Zt1 Z-t

r - o.tl88z-t

1.:112. - zIt 7- oi,[,P( t-rtt:'7\Y.

Nc --o

. r.r18B - t

_ 0fif9 -r.'tztlhc,1188.lttt'ttctl- lc )

Kc '-o

Q - r. >r1z 1-') 1cc

Kc + 't'L7o EZ

-r.22r2 kc -t 0.1?99z

_ t. ,rt ! Kr r O. 1{9,

-2kc +I

E:

| - o'1188z

a--kq

tu) 4''11= o uftr s.f Uww..t

tJtt")'

{or urirt &t44 a -

-t t t.r2 l: z-l-f=oT[-ET:r x o'5

tt {ffi- ro'r r-4-r

-l -l '

-_l (kc(0181 ..

gl Lon$ divi sroa;

o.E-o66p2r,J - o'ttEZ z-t + 0.t82 z-z r...1o .rb82 r-2 / - o.S t O6lo61,t

o r -o oE{62:;oJEE,

- 0.5 t 0-6106L-l

tt -b .r) + t 0.61tlo-0.1r88)7-t

0.5 + {* 0.6682)ir+ 0.\68rr-'

$to) = -;

1(r) - - o'llsz

I

t-t',

o.ost st'\ 0.qtr6q z-L +...

-

o r-to4iJ;-T-_ f o.qt7.->r.^

tt 8u.0p k c

- - 0.5 + 0.668)Z-t _0.16,1"

lC:r) > o'182

t_

*-h-,1 u*F---,f 'J'-l-t

,'t -',unt ,o,+)-'.-1,,

9. lst ordQr Kc=B Zort,

ets) = s;0 ({)Y (t)

= q GxeL+l - zI#, \t-'1z(.ihil) -.(ffi)

= (r-z-,1 Z (,6ii)cr-u-') z(t - t") .=r

__ z_rCt_e_r)I - q-rz-t

( stideee)

C.E' vt+ -e Cl - e- t) 1-li--?-a-r =$Lrb Q:3 ,

,r* Qj-).-! _ nt - g-r3-l 'v

r - 0-rz-r + Bz-' _3e-te-t ce

a-t (49-r _B) = |

r:4e-1-o y

tor staYi14, ,

f, / -a

o5< L-I ( I

c _rn o.F.

&4. ?i4r-Qv,o qr[z) -- 4o ffi;,)

4o' 0 '3e,- 0.4b,= 0.8[, = 0.6

^') ilnt gte? 6 i at {:oidoa\ rrsfonse :

Cra(t'') -- 0r xr La + \>yta l7'3'lz-1 * ""

k1 (z-'J ra I Xr z-' t x) ?-' t z-3 r " )r(r-,)=F

'[ ,-'(ty+xr1'') + z'?(ltt-r+e-'t"')] f r- e-')

: (z-Lz-'Xx,+xrt'') + Lz-t - z-+ ) (t+ t-'+t'2+"')

= xt?-r+ (x2-x1) LL - \'z'3 + Z-3

xr?-" + (Xr-I''){. - lr J--;r-. Cz - l) Cxrz +Xr) r -7:

Ho Qp Ct) =

al 0(z) e

tffitr = dr t-' r (,tr-z + o(e r-3 + xr &-4

(r?-' *o(:t-2+ Vgt- 3 + ilt 7-a (, _*-1,)

Lon4 division:=)

(,fl* [Ur+Ur\Z-'+ (t(rtdr +Xs)?i + (r/, to{,1vs )1-4r- z-rJdrt-'+o(rTzto(tTs+ 91r Z-4

- (,1,- (,t-,

- Qrt(rlt-2 - (Y1+(r)23

-

, [nr* Ue to( r)7-3 + d r 1.t

- [U'r+o(ztD(rJT3

[]1,1(r+o( s)Z-l- ()o{r*VztVg)t-1 t

0E)a : Xr t-t * (o{,to(r)t-,.t Cs{,tv.to{3)a-? +

f?O{r{* r .ti{s) t- 3

(cv, +o{a + o(s) CZ-+.r ?-s + Z-6t .., )

Sbeadg stat4 olt?"t,

v,

8l\

o\ F'lo,r,t , *,_*fi-' ,"EEf -F* l" H, (e- f, ) rr, --F=IoH.{fl_nJ * Fr

'r[*- f = {o,o , F. = rg, p,= {a. F. - I,l[_rr= o.g I 4-+

H"l d{ freuiu Jo,^ H S,*"1. fL,f

^ lFr f-v I ,'rrre 'r

) -[ ;J=IX; ;"JIT/

ll"n di J Le! ?tt.rF= ,*Hr{B-tq). }* rq.{R-tJ

(}ata, r ?r F{r} f, r ?o r*,p,+ }o r,t, p.

': P -- 1" 4Prt 1a Hrh ii ," H; r" ft tit

Lt#t tr.g t. o, .,q$ q. {,J J+frF.- f,o* tr*, u, . hj-l' f4 r

0u\qut in?ut5r. t 1.Ar--M'1>'P, il"M'

0,,, Hl", = fi,1,,

J

&-31 -"{ - }},t, -

=

k,* = Jl --lHr

r. -t

-

It6d&x,ri.rr*/

i{q "

Itr =

,,friHr

TiPl*

tra

1

,f

tr90

Jt'

,13

t( 0k\'

"' ]"=

_lt_.- [

II

, f in[-#

Kx*

4 r-{, - {dIr, {r

lLJ- =Tlho'F

=-'-I

1 + {. F/tl, , P'/,'r.}JIffiqr*Jd P^'$,'f

\Itt6t\ qr

Htt * t" h"*l +, $ollo*,r '

6.r Fl , :f'{L - {ou f"i u. L-- f"---=r3"?+lf,+fuHe - (LU,*r.+J 'bu

3' It. - f' -fr - ; x zo =[?4H,-httJi

- P. -f,, -9o '--"- + ?o FLlrS{1r lr hl*

Ft++h"l ?,

F= =o *r, t r=, -O

F-. tu ,v', {@ r-ttt)

ttJ

F"r't 6t tt) fi*r=

t+)r, ' ,o {a-r,}-:- H, ['S, ]o= lo. - 1,p F{q *, ,* r

lt

l*ln" tIL-[1)=zoc] {tfi"^ 1, tu 1*r'

L+)*= ?' ro'$*,)o. F,rE*. + frr I te] 6[tr'ot'

+. nn Ii! )* = o.o - z.Fr, (#,)*

.',H,)*= offi,)*,-'T ryr,,=ffi=**7-.[*=tt 11

4e 1 iAr) --r 4-3 3ss

.1WlrAn= T'7y=T

'(

\,r'r-l9l\\ D u' / ur f'rrdr )t'l' 1t 0ur J lrfl,trd

f-

Crl nfro I.Q5

6i- 7lt a*fu fi'wfiw -fn $/''ld! c'nil"ttl- nD;/fi,,t,

Zt-doruau.

t'X'ntlilt" g tz I=

e tt)

-l ^ :tvlC+o'f a, i\ " ' ! a*' '

@;L*-. -rfininfi{tt't

Lh)

Det)

fLr- con'rpltr ha* ;w^L **Yr":k ikr.!,ah,,,r5Cv1lJ,hot4 ltte Jvrl'r tost

$(z*t)=O ,

ftt*=l)+ o4l-.tt

7tt )*Lo'bf^ii&tu t tP n*'.-ry#,##,#' .6',ial l.o"( .p){

* z=gtT9- 4orn0iv1

3d,n--9 ts'o)

Z - domaM

lain , QCt= t)

tt A)+ AF':t t "' am {u { (,-z'')' Q Cr -')

rlnu,tt7 c^i;1t*

wrlil"fi"a Bn*hi

s(trt9

orisra

tlt*

o(

,r,t

24i.T-InL bi

l" I Y{i47r'0

A, yw. W ia dN *tI f- 6+^tb ?w@r'Yl*C^sl{i.,tr"fu [ bi] nlu,sl

?'torrl I T,tf ,h,b,ltrrrttna,

rCrL') Z^n, !_(,!: P -e(t) B Ct1

Ao+ 7'1*. - .* 0n

*.1, tb>t " +Ll

a;*0

*'u'*It-6,n t -'t dornain)qCz-') . '-'(t-

e-'lt);r7ir | -1=;;

$I{f; )r- e-fr2-') 5it-') - z-' Lt- g-'/') *ct-')

t',:,TdT- -rlc ) Xn -lgn - e'lt 9n-' =

(t'e

\tadnltttm/

eo oJQD- 5J* =0}+rt

6$*- 55r-z -- tl lt- r

65(a-') - sz-?$til = r' !)

-t -9Cz-') z'\utz-,) 6- 57--a

4 tz-') F,:L-= (u IL-')

-'1 = 9-6

llrcdehy{octw'3 , Acttul rtur de ta5 = sr (sSSlta0af

- $)i'

tll Y*b.

a1?

l0

aYv-/ =)

frfrre - UrI

N=Z

,77

lc) \h o,"..-1ur' re{y onk n i,,w gq6r) uf.+r-- Lz'l / '$o),rr.)...grr) UG)-_

,o ., _ ,. L 1o(z-rr o((p-r)e-. ll

at/-l+ 6ziL I - z-l

"?Fi " B.r !^a &v;srp,t--61p"1n-a-

J ( - v'r''v-Y I

-l(P-t)t-r' '' , , .. .r ., .r- \6 b)= o , Y (T) = o-J.fl1r)=0, 7Gt1=Lfl(a,7= brt-a t)

.zSlrll\+" Ul^alk 6+aa',rb, firtst

r r F E, + te-' -6 +,,[tt+FlE + I=O

bi3I t (a,-r) z-l h,-d)iL

- Arz'3 )\-/

dAlJ i

y-toof Sdabilti"J, il,* ,> wo unA;liow tpo*, o*

o1j\n !'.rV1- *bAry $r*,yt;

fl

J-

-t { 1.22 !1 g. -1l+ 4. Lt.0' | - o. t't BBI.-Iv

8 @) 7ta C.E, b 6[!'tt*l an/ + -tlr l""t"-. Kc = o

Z - o,17 gg

s

= 0J)? t - /,2>lz k6

a 77 9 E-/, z> lz kc

3T . r*i,i,n!".Z-(r{{) =ra-U-'

., N- f

I r -l + t,zz tz il-, ',,, l_oil.lgg/

Ia-.

r -u-l

Ca4rrplalr4+t ri s)-

^D @ t=oevun$v0=oCNo +th,l,

(t-K, ) *

f=l- Kc

7l* e*auit;tX s"*rahlt ; .

-l < o.7J 8g -t,z>lrKcl.-Kc

o. ?gl= ?'t'' ) zzl> = /,4f 7

-l-ltl.ezlza. .yo,s@

-1 t xrt'' *lI--

o,{ ' o. /d8 z z-t

-o,5 + o,blobZ-t

0 .9 - o. LLtz r't+ o. lB g 2 z4

/r,W W ol;,ntin,.

(

.-t-

"-l

o,)82?'d (rr )J

\ b ) Frr s".t-Pot'tr d^a/nt" I

Jf.-')' l+ Gh"a?.k

-6..5t o,bloL ?'l

rre dslaYfaoor'3 , A(tual ri,xe dtlal ' 3T

?Liiililv + r

, ?ro3s sttmc Gdrtar;

DT Aqr{1.brud"fr'O?,A

fto1u^eLl

'&tf-.,

f,G)='-l ) I@c-orilg?/

l

@

4 + Ih{ }ran{or fi,,r*li,^ 6tr t}+ T** fl* W-"rfu^ hbtL i,

=" {,( t _ .-r ). ( f,** l:!,^turr,te )t - er z-'

I sr'ae r

11" C,t, .E ff* Vl

I+ xrct-i') it = oI ^a' 3-l

, ..|: Z= 4.e -3 (,;Kc,3)

Fr sktb',,,U

o,f1i1

&q

-d'6 Lq Tlu rd-aa!. onirtyelava& t; 7:,*^ h1 :

f,re'l = - o r tt ll* xr ,-'+ ii* i4*,, r \

2\ ,"., )I

-

)- ?-l-t -3-'-l -zXzZ')t 4-(t+z*Z+'

*lls3

:7det= O)o)O + 6lasol-toup s+vrbl4

By +t"r d,,;v.d |ynlbt:s \ra*ho[,

'l -l.. )).r,- ? )= V'(nY*

rb>

Cci\= +.66f6rt*7

Ya

-Tl- v4r= (2-r,8 )(l-0,d1 l+ (z-tt(x,2tx")

a,3 (2t a?) ZL t - lz-tt(rtzt/,2)

= (Z'$;8) (*a,l)o,j (*+ ".q )

a

E- t) .( ut, C )t 2112 t I + l*z)

Jlv^r{r,* , lk closd- l,,of tranEt* 6*hn f"-5.?f- p ol nI ch"anXt t4 i

W = ('';'* *'i'r biTri4nI^ ( rir;

I-

d6- cC;

udl

4re. c-*rclh ,,;,, pwa $l cqx bL re-ury;{l,a * ,

CCf,1 -' (r-oy{)tt-,,/i)) . (J- i5Q,+n,l)+f,Lo j(t+u,72-,)

En-Q- -,fu b6{'1" ru*nevo!fuv a",,t, )"s*o*a*Lq F)ffi.H1, 0n i, sc*f'4ln ')eonElartr " rfrms, rha a.,^tvor,te^- ; pLyf i c*t/tr+ al.i;onr .

At t)rq tuNwtlry hw illral, a^d-r,ra, it a (// f &tfh.l't v[o cc nfi"ltav ,u*?.rrt W * p Ltva y,c.,r-ato .h, vt

C,tuf*lV ilordp r,4 :. l" U C CZ1 -* rc

Z-> I

0\aofa

b 7hl out74 ratyorw b h* al i

t o

- r{)

U

5. lni,X 11,! above etwafrr,, p^

dl= -0.2,, dr= I , 4J= o,e

1*f*, $ne a*g,t w,l s\^-,1?(r)= xr= -o,L

a

a'