... . )26346 (

) ( 1385

- - ) SISO ( ) ( Bode ) 26346 ( 11

: : 1 2 . 0 t . : : . . . : :- : RC .- : ) ( .- : . : :- PFD-P&ID .- TAG Inventory Control .- Pairing .- .

. . :- ) ( - 2- . ) ( ) .( :- : / - : - : . ) 1 .( :- ) T ( / - / - ) V h ( . ) ( inT inF . inT inF ) ( . . ) T h ( . 1 - . ) 26346 ( 13 T V inT inF . ) ( ) ( V T . .. . ) T ( ) V h ( ) (! . ) (! . )inF ( ) F ( . inT inT ) .( ) ( . ) T ( )inF ( . inF ) ( ) ( . ) Feedforward ( . ) Feedback ( . . . 2 . . 2 ) ( - ) ( - ) ( - -

4 : ) PFD () P& ID ( ! . : :- ) (Controller- ) ( Stabilizer- ) ( Compensator- ) (Regulator - ) ( Tracker, Follower - ) ( Calculator, Estimator - ) ( Computer- ) ( Governor )26346 ( 2 1

. ) ( . ) ( . . ) ( . - t ( ) f t ) ( ( ) Fs s :) 1 (0( ) ( )stFs f t e dt+= :) 2 (1( ) { ( )} f t L Fs= 1( ) { ( )} f t L Fs= 1 : ( ) f t ( ) Fs ) t ( s . 2 : {} () [ ] . 3 : t 1 s s ( ) Fs . 4 : s p ). ( ( ) Fs ( ) f s . ) ( ( ) f t ) ( ) ( .

1 Dummy variable 2 1 - 1 ) ( = t f ) ( 1 ) ( t u t f = : ( ) 1 f t= . . 1 . ) ( ) ( { } {} { }s sedt e t u L L t f Lttstst1) 1 ( ) ( 1 ) (00= = = = = == + 1 : ) . 1 ) ( = t f ) ( 1 ) ( t u t f = :) 3 (0 0( ) : ( )1 0tf t utt= ,~ +0 ) 0 t ( . 0 = t ) 1 ) ( = t f ( ) ( " " . " " " " ) ( . ! 2 := = = 0 00) ( ) ( ) ( ) (~tt et f or e t u t f or e t fatat at : : { } { }a s a sed e e e t u L t f La ss a at+=+ = = = ==+ +1) ( ) ( ) (0) (0 2 : t : t . )26346 ( 2 3 3 : . . . ) 2 ( 0 = t ! 2 . ) ( ) ( 3 ) ramp( ) (0( ) ( ) ( ) ( )0 0t tf t t or f t ut t or f tt= = = ~, : : { } { }2 2001 1( ) ( ) ( ) (s sL f t L ut t e d es s s += = = = = + = , 3 . ) ( ) ( 4 : ) ( ) ( . ) ( . 4 4 ) ( ) w (sin ( ) 0( ) sin ( ) ( ) ( ) sin ( ) ( )0 0wt tf t wt or f t ut wt or f tt= = = ~, : ) :( { } { }02 2 2 20( ) ( ) sin ( ) (sin ( ))( sin ( ) cos ( )ssL f t L ut wt w e de ws w w ws w s w +=== = = = + = + +

, 4 . ) ( ) ( . (knowwhy) . ) (! (knowhow) . . ) .( ) ( : ) )26346 ( 2 5 ( . ) ( ! . . :1 2 3 .4 5 ) ( . ) (1 2 3 4 5 ( ( ) Shift) Fs6 ( ( ) Shift) f t7 ) ( ) (8 .1 . ) ( . ) ( ! (Diffeomorphism) . ) ( :1 2 1 21) { ( ) ( )} { ( )} { ( )}2) { ( )} { ( )}L f t f t L f t L f tL f t L f t + = += 6 . ) ( . ) ( . ) ( ) ( ) ( . ) ( .2 . ) ( . ) ( ) ( s . :( ){ } { ( )} ( ) (0)df tL L f t sFs fdt= = ` :0 000{ ( )}( )( ) ( ) ( ) (0) ( )dfst stL f t fe dt e dtdtste df Integration bypartst ste f t se f t dt f sFs + +++= == = = = + ` ` : s ) ( :23 2{ ( )} ( ) ( 0) ( 0){ ( )} ( ) (0) (0) (0)L f t s Fs sf t f tL f t sFs s f sf sf= = == `` ```` ` `` 5 : ( ) sFs 2( ) s Fs ... ) ( (0) f` (0) f`` n n s f` f`` ( ) f t ) ( ) Fs ( . . : )26346 ( 2 74 5 2 2(0) (0) (0) 0y y y yy y y+ + + == = =

( ) y t . : :3 22( ) (0) (0) (0)4[ ( ) (0) (0)]5[ ( ) (0)]2 ( ) 2/s Ys s y s y ysYs sy ysYs yYs s + + +=

6 : ) ( n . 7 : . ( ) ys s s ! ( ) y t 0 t = : ( ) ( ) y ( ) y t ) { ( )} Ly t ( :3 22( ) ( )( 4 5 2)y s y ss s s s=+ + + 8 : ( ) y t ! ( ) Ys 2 . ( ) Ys ( ) y t . . ! ) (rational) ( ) ( . ) ( . 8 . . ) ( : ) 1 - ( ( 1) s + . 3 2( 4 5 2) s s s + + + ( 1) s + :3 2 2( 4 5 2) ( 1)( 3 2) + + + = + + + s s s s s s 2 - :3 2 2( 4 5 2) ( 1) ( 2) s s s s s + + + = + + :3 2 2 22 21 2 ( 4 5 2) ( 1) ( 2) ( 1)A B C Ds s s ss s s ss s s= = + + ++ + + + + + + + :2 22 2( 1) ( 2) ( 1)( 2) ( 2) ( 1)1 2 ( 1) ( 1) ( 2)A B C D A s s Bs s s Css Dsss s s s ss s+ + + + + + + + ++ + + =+ + + + + :3 2 2 2=A( 4 5 2) ( 3 2) ( 2) ( 2 1) s s s Bss s Css Ds s s + + + + + + + + + + + :3 2 3 2( ) (4 3 2 ) (5 2 2 ) 2 0 0 0 2 A B Ds A B C Ds A B C Ds A s s s + + + + + + + + + + + + + + : ( ) 0 1(4 3 2 ) 0 0 (5 2 2 ) 0 22 2 1A B D AA B C D BA B C D CA D+ + = = + + + = = + + + = = = = 9 : ) ( :2 22( )1 2 ( 1) ( 2) ( 1)LH RHA B C DYss s s ss s s= = + + ++ + + + +_ _ ( ) RH ( ) LH s A 0 s = : 2 2002( 1) ( 2) 1 ( 1) 2ssB C DA ss s s s s== = + + + + + + + + )26346 ( 2 9 s s s ) , , D CB ( . A ( ) LH s . !221(0 1) (0 2)A A = =+ + , D C :22121( 1)22( 2)ssD DssC Css=== = += = + B : 222 ( 1)( 1) ( 1)( 2) 2A s DBs C sss s s+= + + + + ++ + 1 s = B . :2 2 2 22(2 2) 2(...) (...) ...( ) 0( 2) ( 2)sA B Ds s s s + = + + ++ + : ... ( ) RH . ( ) LH . ( ) LH 1 s = :

2 212(2 2)0( 2)ssB Bs s= += =+ :

3 2 22 1 2 1( )2 ( 4 5 2) ( 1)Yss s ss s s s= = + + + + + (!) : 111{ } ( ) Lsu t=, tte 211{ }( 1)( )tL tesu t =+, 2te 1 2( )1{ }( 2)tut L es =+, 10 (!) s ( ) Fs ( ) f t1 ( ) 1s ( ) u t, 1s + ( )te u t , !1( )nns ++ ( )n tt e u t , 2 2s + sin( ) ( ) t u t , 2 2ss + cos( ) ( ) t u t , ) ( :2( ) (1 2 ) ( )t tyt te e ut = , :2( ) (1 2 )t tyt te e = 10 : :) 1 ( t ) t ( .) 2 ( . ) ( !

3 : ( ) Fs ( ) f t ( ) f t .0lim ( ) lim ( )t sf t s Fs = 11 : !

)26346 ( 2 11 4 : ( ) Fs ( ) f t ( 0) f t = 0( )tf t= .0lim ( ) lim ( )t sf t s Fs = 12 : 1 ( ) u t,1s ( ) Fs ( ) f t . ( ) f t ( ) sFs ) 0 t s ( ) t 0 s ( .

5 : ) ( ) ( ) Fs ( !{ }( ) ( )te f t Fs= + / 13 : 1s ( ) ( ) f t u t = , 1( ) Fss =+) 1s ( ( ) ( )tf t e u t =,. 6 : s )d se ( (Dead Time) (Transportation lag) (Time Delay) ) ( .{ } ( ) ( )dsdL f t e Fs = 14 : d se ) ( ) ( . :(cos sin ) ,1jre r j j = + = 1 r = :1je = ) ( ) e ( j 1 ) ( 1 ! 12 7 ) ( . . . a ( ) duration b ) 5 :([ ] [ ] , ( ( )) ( ) ( ( )) ( ) ( ) pulsea b f t aut aut b aut ut b = = + = , , , , 5 . . :1( ) , 1 t pulse h dth+ = = 6 . . 1h h ) 1h( ) ( . 1 )26346 ( 2 13) 11 hh = .( ): 6 ([ ]01( ) lim( ) ( ) ( )ht ut u t hh = , , :{ } [ ]00 01( ) lim ( ) ( )1 1 0lim , ' lim 10hhs hsh hL t L ut ut hhe seLHopitalh s h = = = = = , , 8 . :{ }0( )( )tFsL f t dts= ( ) { ( )} Fs L f t = . ) 26346 ( 3 1 : . - - ) ( . - . . ) 1 ( . - . (Logic) (Heuristics) (Rule-Based) (Sup-star Algebra) . . . ) ( . ) ( . . ) ( ) Nu Re Pr ( . ) ( . ) ( . (Governing Equation) ) ( ) ( . - 2 ) ) , , ( x y z ) , , ( r z ( ) d Distribute ( (Lumped) . (bulk) . 1 . . space) (state (Marcov) . FIR) - Response Impulse (Finite FSR) - Response Step Finite ( . / ) ( . ) ( (s) ) ( (t) . ) ( . . ) 26346 ( 3 3 : ic) (Mechanist : ) ( ) () 1 (,ssT ) () 2 (,ssT ) (mT ... :1 (Lupmed) ) (mT ) (mT . ) (mT .2 ) ( ndt d ) / ( . . . .3 ) ( .4 .5 .6 ) ( . ) ( ) :(tt t t= + m m0 t p mT Tlim mC ) T hA(T (Lupmed) :) T hA(TTmCmmp =dtd 1 : r x y , , , . (Lupmed) . 2 : (t) . 3 : . T . - 4 A h, , C m,p ) ( T ) ( mT . 4 : . . . : 0 =dtd ss m ss ss ss , , m, ,T T 0 ) T hA(T = = :, , , mm m ss ssT T T T T T = = :) T T hA(TmCmmp =dtd : / =hAmC p ) pC m, ( ) A h, ( . 5 : ) " " ( ! : = + = T TT) T T (Tmmmmdtddtd ) t ( . ) ( = T Tm ! . : ) ( T ) ( T ) ( T sm ms s s= + 6 : ) 0 ( Tm= t mT . :( ) 1( )1 ( )mpT sG ss T s = + . ! . ) ( Tms ) 26346 ( 3 5 ) ( T s) ( ( )pG s ) (s Gp :) ( ) ( ) ( s T s G s Tp m = 7 : G Gain . 8 : G ! . 9 : (rational)pG (Transfer Function) Transform) ( . 10 : . . s 1 ) ( s ) ( . . 2 . 2 . .

- 6 1 1ks + . (First Order Lag) .2 . !3 . . (batch) !4 ) ( ( ) ( ) ( )pY s G s us = ( )pG s . ( ) us ( ) Y s .5 s . ) ( ) ( .6 ion) Superposit ( :) ( ) ( ) (2 2 1 1su asu as u + = :) ( ) ( ) ( ) ( ) (2 2 1 1sY asY as u s G s Yp+ = = Y1Y2 u12 u . ) ( 10c D 5c D .7 : ) ( :( ) 1 ( ) ( ) / ut ut us A s = =G 1) (+=sksGp ) ( :) 1 () (1) ( ) () (1) (+= += =+=s skAs usks Ys us YsksGp ) ( :/ 1{ } ( ) (1 )tY(s) Yt Ake L = = ) 3 ( ) 26346 ( 3 7 3 . . : ( . ( ) (t Y % 2 . 63 . ) 4 = t ( ) (t Y % 98 . 4 5 . ( ) (t Y = t ) ! 4 .( 4 . . - 8 ( ) ! 5 ( . 5 . . ( ) (t Y :) (lim,t YY t new ss = :

u uY YuYuYold ss new ssold ss new ss, ,, ,=== (D.C.Gain) :kAAkAeAkAYAYttnew ss new ss= == = ) 1 (lim00/, , = k Gp . ) ) 1 (/eAkt ( ! ) (t Y :kAs skAs s Y s s sFYst=+ = = == ) 1 () ( ) (lim0 k . ) 26346 ( 3 9 ) 1 = A ( ) ) (sGp( :) 0 (GuYp== 8 1 //( ) 1( ) ( )1 1/( ) { ( )} ( / )tk kus Ys Yss sYt L Ys k e = = = + += = 6 . 6 . . .dt t Y d t Y s Y ss skssks Ystep impulse step impulse/ ) ) ( ( ) ( ) () 1 ( 1) ( = =+ =+= !9 :2 2 2 21 /2 2 2 2 2 2 ( ) sin( )( ) ( )1 ( ) {( )} cos( ) sin( )1 1 1tA k Aut A t us Ysss sA k A k AkYt L Ys e t t = = = ++ += = ++ + + :q p q p rA r A q A p/ tan ,) sin( ) sin( ) cos(2 2 = + =+ = + :) ( tan) sin(11 ) (12 2/2 2 =++++=tAkek At Yt - 10 : ) : () sin(1) (2 2 ++= tAkt Yt : ( 2 211 + . ( . . ( ! ) 26346 ( 4 1 . . ) ( ) ( . . - . . ) ( ) .( - ) ( ) ( . - . : ( . ... ! : 1 . . 2 . . = 1 : . 2 : . : : . ::inoutt t tq tq tv v + :in outdv dhq q Adt dt = = h ) ( inq ) ( outq ) ( h . . ) ( :out vq c h = 3 : ) ( vc g . outq :v indhA c h qdt + = - . h . . :0 0220 0 0 21( ) ( ) ( ) ( ) ...2x x x xdf d ff x f x x x x xdx dx= == + + + :0 0 0 00 0 0 0 0 00 0 , 0 , 022 2 22, 0 , 0 0 , 0 2 2( , ) ( , ) ( ) ( )1 ( ) 2 ( )( ) ( ) ...2x y x yx y x y x yf ff xy f x y x x y yx yf f fx x x x y y y yx xy y = + + + + + + ) 26346 ( 4 3 x x y . h 12h :( )( )ssss h h ssd hh h h hdh== + ) :(,( )20 0v ssv ss inssv ss in ssc h h dhA c h qdt hc h q + + = + + = :, , = - ss in in inssh h h q q q = ) :(= ,2vinssc dhA c h q cdt h + :p/, 1/pAc k c = ) :(p ( )1pinkhss q =+ 4 : pk h ) L ( q ) 3 -1L T ( . . 5 : pk p( ) ssh . 2 . ) ( .: . : :in inout outout t t out tcq tc q tVc Vc+ - = :outin in out outdVccq c qdt = 4 2 . . : inc ) ( outc outincc . inc outc . inq outc :11 12 out in inc Gc Gq = + :11 12,out outin inc cG Gc q 2 1 ) ( 11G 12G out outc q ! ( ) MIMO : ) : 2 : 3 ( 11 1221 2231 32outinoutincG Gcq G GqG GT = ) : 3 : 3 ( 11 12 1321 22 2331 32 33out inout ininc cG G Gq G G G qG G GT T = ) 26346 ( 4 5 ) outq ( h outq . outq . . inc outc :, ,0outin outin outin ss out ssdccq c q Vq q q dtc q c q == = = p/ V q :p1=1outincs c + 6 : ! 7 : . V q . q V ! RC cv gv 3 . . ) ( :1gv Ri idtc= + cv ) : q c .(cqvc= dqidt= : 6 1( ) gccdqv R q tdt cdvRc vdt= += + :,,c c cssq q qssv v vv v v= = :pp( ) 1, 1 ( )cqv sRcs v s=+

8 : . RC ! ! . . 4 . . :1 1 1 122 2 2 ,1 1 1First Tank :Second Tank: inoutdhA c h qdtdhA c h q c hdt+ =+ = = ) 26346 ( 4 7inq 1 2( , ) h h . . 1h 2h . ) 1h ( . :1 1 1 1 12 2 2 2 1 1indhA c h R qdtdhA c h c hdt+ = + = :1 1 p,1 1 12 2 p,2 2 2 : R 1/, : R 1/ , c A Rc A R :11 1 1222 2 11 indhh R qdtR dhh hdt R+ =+ = 1 h !{ }1 11 122 2 121( ) ( ) ( )( ) ( ) ( )insH s H s R Q sRsH s H s H sRL+ =+ = " :1 1 111 1( )( ) . ( )1 1 ( )ininH s R RH s Q ss s Q s = =+ + 1( ) H s 2( ) H s ( )inQ s :2 21 2( ) 1.1 1 ( )inH s Rs s Q s =+ + 8 : 1( ) H s 2( ) H s 2 2 12 1( ) /1 ( )H s R Rs H s =+ 1( )( )inH sQ s :1 2 2 1 2 1 21 2 1 2 1/( ) ( ) ( )1 1 ( 1)( 1)in inH H H R R R Rs s s s Q H Q = =+ + + + 5 6 7 . (damp) ! 8 5 - . 6 - . 9 : : 7 - . ) 26346 ( 4 9 10 : 1R 1 . 11 : 2h 1h ) ( . . 8 - . 12 : . 7 0+ . ) 45D ( . . 13 : n Damp . 9 . ( ) convolution ( ) Stage Wised ) ( . 10 9 - . ) ( 10 . ) ( . :1 1 ,122 2 2 ,1First Tank :Second Tank: out inoutdhA q qdtdhA c h qdt+ =+ = ) 26346 ( 4 11 10 - . ,1 outq :) ( :,1 1 1 outq c h =) ( :,1 1 1 2 outq c h h = ) ( 2 2c h :2 2, 22,12ssssh h hh + :1, 2, 1, 2,1 2 1 21 2 1, 2, ,1 1, ,2 2,1 2( ) ( )( ) ( )ss ss ss ssss ss h h ss h h ssh h h hh h h h h h h hh h + + ) : (11 1 1 1 222 1 2 2 1 21, 2,2 2( )2 2inss ssss ssss ss ssc c dhA h h qdt h hc c dhA h h hdt h hh h h+ = + = :1 21 2 1, 2,1 1,2 2ss ssC CR R h h = = :11 2 1121 2 2 22 11inh h dhA qdt Rh h dhA hdt R R+ =+ = 12 inq ) ( 2 h .2 22 21 2 1 2 11( )( )( ) 1inH s RRQ ss sR =+ + + + 14 : :2 221 2 1 2( )( ) 1 ( )inH s Rs s Q s =+ + + + 2 1 1( / ) R R )1 2A R ( s . ) ( ) ( . :1 2 1 2 1 2, , A A A R R R = = = = = = :

22 2( )2 1 ( 1)( 1) ( )inH s R Rs s s s Q s = =+ + + +: 22 2( )3 1 (0.382 1)(2.618 1) ( )inH s R Rs s s s Q s = =+ + + +: : . 15 : :( 0) ( 0)interacting non interactingG s G s= = = 16 : 11 . 11 - . ) 26346 ( 5 1 . ) ( . 1 . 1 . . . ) ( . M :( )idvF M Madt= = :

( ) ( ) ( ) ( )External Force Spring Force Damper ForceFt Ky t Cv t Ma t = y :2222,( )dy dv d yv adt dt dtMd y Cdy FtyK Kdt K dt = = = + + = ) y ( y . ) :(2( ), 2 , ( )M C FtutK K K = = = :2222 ( )d y dyy utdt dt + + = :2 2( ) 12 1 ( )yss s us =+ + 2 1 : .1 21 2( )( ) ( )( ) ,( )( ) ( )mns z s z s zGs m ns p s p s p+ + += + + +

: 11 011 0( ) ,m mn ns b s bGs m ns a s a+ + += + + +

: 1 21 2( 1)( 1) ( 1)( ) ,( 1)( 1) ( 1)mns s sGs K m ns s s + + += + + +

: s 3s . . :2 21 1 1( ) ( )2 1us yss s s s = = + + :1 2 1 221 221 1 1( )1,A B Cyss s p s p s s p s pp p = = + + = 1p 2p :1 - 1 1p 2p . ) ( . S . overdamped .2 -1 = ) ( . (!) . critically damped . .3 - 1 . . underdamped . . 1 ) : 2 C K ( . . ) ( . ) 26346 ( 5 3 ! : 1 : / 2 22( ) 1 cosh 1 sinh 11tt ty t e = + 1 = :/( ) 1 (1 )tty t e= + 1 : 2 /2 121( ) 1 sin 11te ty t tg = + 2 : 1 :2 2( )2 1 ( )ys gains s us =+ + ) ( ) ( . 3 :211tg . . 4 : / . ) / te ( . ) 1/ ( . / . :222 2 / // 2 /C K C C M KM K M M C K = = = = = CM M . ) ( ) ( . . . . ) ( ! 2 . . 4 2 . . ) overshoot ( : . AB100AB :2exp1overshoot = 5 : 21 21 1 . 6 : 1 0 . 100% . 1 . 16% ) 0.504 = ( 25% )0.404 = ( . 7 : ) ! / M K ( ) 26346 ( 5 5 ) undershoot ( : . 100DB DB= ) decayratio ( : ) ( . 100CA CA= 22exp1 = 8 : : 50% . ) rise time ( : .21211risetgt = ) peak time ( : .:21peakt= ) settling time ( : . 95 ) 5% ( 98 ) 2% ( .3 / 54 / 2ssfortfor 9 : )riset ( )sst settlingt ( . : : 21wt t w ) ( f ) ( :2 21 1( 2 )2w w f f = = = ) ( :21 21Tf= = 10 : ! 6 ) .( . :201 1 122natural natural naturalw f T == = = = 11 : 0.2 .21naturalff =

) 1 ( :{ }12 22/21( ) 1 ( ) ( ) ( )2 11 1( ) sin1tus ys Gs us Ls syt e t = = = + + =

:1 1 12 2 21 12 1 22( ) ( ) ( ) ( ) 1/ ( ),( ) ( ) ( ) ( ) 1 ( )( ) ( )( ) ( ) ( )( ) ( )y s Gs u s u s s step inputy s Gs u s u s impulse inputy s step response d yy s sy s y ty s impulse response dt= = = = = = == . 12 : ) ( :2 221 1sin 0 , 0,1, 2,1kt t k t k = = = =

0 k = :100 0 0tdyk tdt== = = 0 t = ) ( ! 1 k = :211peakk t t= = 2 k = :2221k t= = ) 26346 ( 5 7 . 3 k = .2331k t= =

1 = 1 = :/( )ttyt e =

1 . 1 :2/21 1( ) sinh1tyt e t = 13 : . 14 : . 15 : . 1 :( ) { }( )12 2 2 2 2 22 2/1 21( ) sin ( ) ( ) ( ) ( )2 11 1( ) sin sin sintAw Awut A wt us ys Gs us Ls w s w s syt B wt e C t C t = = = = + + + + = + + +

1C 2C t ) / te ( . ( ) ( ) sin yt B wt = + ) t ( 12 222 2 2 2 22,11 4A wB tgww w = = + 16 : ) BA( . BA 1 (!) 8 . 17 : 180

90

. 18 : 1 = 1 ) ( . ) 26346 ( 6 . . .

(pure gain) ) ( . . ) ( . ) t ( y ) ( ) ( :( ) ( ) yt K ut = :( ) ( ) ( ) ys K us Gs K = = K . RTD . . 1 : . ) ( ) ( . (pure capacitiy) ) ( ) ( . :( )KGss= (lead/lag) :1( )1sGs Ks+=+ . (strictlyproper) (rational) G(S) . ) ( . :101( )1 1A sGs K K As s + = = + + + :0 1, 1 1 A A = = = = :( ) ( ) ( ) ( ) 1 ( )1Kys Gs us K uss = = + + :1 - ( ) ys .2 - ( ) ys ) 1 ( . (lead-to-lag-ratio) . ) 1 ( lead-lag . V . FA A FB B ) ( . B A . CB,out A ) FA( :( ) ( ),, ,1 1B tankB Bfeed A B B tankdCV FC F F Cdt = + : B 1 . . ) 26346 ( 6 ) ( ) ( :( ), ,,1A B B tank B BfeedB outA BF F C FCCF F + + =+ A B (FA>>FB) : AVF = :( ) 1A B A B AF F F F F + + ) ( :( ),, ,, , ,1B tank BBfeed B tankABB out B tank BfeedAdC FC Cdt FFC C CF = = + :, ,, , ,, ,B BBfeed BfeedA AssB out B out B outssB ank B tankssF Fu C CF FC y C Cx C C= = = :( )( )1 ( ) ( )( ) ( ) ( )d xtut xtdtyt xt ut = = + :1( ) ( )11( ) ( ) ( ) ( )1xs ussys xs us uss =+ = + = + + ) ( ) :(( ) 1( ) 1ys sus s+=+ 2) 2 ( . :) ( 1 21 2( )( ) ( )( )( )( ) ( )mns z s z s zGs Ks p s p s p =

1111( )m mmn nns b s bGss a s a+ +=+ +

1 21 2( 1)( 1) ( 1)( )( 1)( 1) ( 1)mns s sGs Ks s s + + +=+ + +

) ( n m n-m . - . ) 1 1 ( ) ( ) ( ) 1 2 ( . (rational) . ) 1 2 ( :11 2( 1) ( )( )( ) ( 1)( 1)K s ysGsus s s += =+ + :0 1 1 21 2 1 2( 1) 1( )( 1)( 1) 1 1A K s A Ays Ks s s s s s += = + + + + + + :1 1 1 2 2 10 1 21 2 2 1( ) ( )1 , , A A A = = = y :1 21 1 2 11 2 2 1( ) 1t tyt K e e = ) 2 ( 1( 0) = ) 1 ( :1 21 21 2 2 1( ) 1t tyt K e e = 1 ! 1 2 1 2 < 1 : :1 2 > K ! ( ) yt ! :1 2 = 1 1 = ) ( ) 1 2 ( ) 1 ( . 1 2 . ) 26346 ( 6 :1 20 < < ) 1 ( . 1 1 . 11 = 24 = 10 K = 1 2 . 2 . ) 1 2 ( . (Inverse-reponse systems) ) ( ) ( . . ) ( ( ) Gs :1 2( ) ( ) ( ) Gs G s G s = . ) 2( ) G s ( ) 1( ) G s ( . 1( ) G s 2( ) G s . 3 :1 21 2( )1 1K KGss s = + + ) ( 1K 2K 1K 2K . 3 . . 1 2 . ) ( . :1 2 1 2( ) (0) (0) (0) y G G G K K = = = 1 2K K . :1 2 1 21 2 00ttd y d y K K d ydt dt dt == = = 11k 12k ) 4 ( . ) 121kk > ( 2 1 < :1 2 1 12 11 2 2 2( 1) 1K K KK 4 . . ) 26346 ( 6 . . ) 2 ( 1y ) ( 2y . (pure lag system) . (transportdelay) (measurementlag) ) GC ( . Lv L v ) 5 ( . . 5 . . ) 5 ( s s . d ) 6 .( dse . 8 (time delay) ) 7 8 ( . 6 . . 7 . . 8 . . ) 26346 ( 6 Pade - dse . Pade : / 2/ 2122212ddddss dsddss eeses+= =++ . ) 26346 ( 7 . ) ( ) ( . ) ( . . - . . : . . . - ) 1 ( . ) ( spoutT . 1 - ) ( . 1 - . ) ( ) ( . ) ( - . ) SISO ( . . . . - ) PFD ( ) P&ID ( . . PFD ) 1 ( . . 1 outT h ) V ( outq inq inT aT . satT ). satP ( . outT . - outT . . outT ) ( outT outT )spoutT ( . ( outT ! ( outT )aT ( )inT ( ( ) ( ) ( . outT )inT ( ( ( ( . outT inT ( ( - )outT ( ) inT ( ( . - - . ) spoutT ( ) ( ) ( . . . ) .( . GC (on-line) 250000 . (inferential) ) ( ) 26346 ( 7 . ) ( . GC RTD ) ( . ) ( )outT ( , outmT . (ManipulatedVariable) . inq (pairing) ( )out inT q ) ( . ( ). out satT q outT . satq . - ) ( PFD P&ID . ) ( . ( )out inT q inq outT . PFD ) P&ID ( . PFD ) P&ID ( . (P&ID) ) PFD ( ) P&ID ( ! ... . 1 . 2 - SISO . ) SISO ( ) 2 ( - ) ( . - ( ). out satT q outT . . outT inq inT aT ) ( spoutT . ) ( ) ( . ) 3 ( - RTD . ) . satq ( - ( )out heaterT Q . TIC Temperature Indicator & Control . P&ID . 3 - SISO . ) 4 ( - . ( )out inT q . ) # ( . ) 5 ( - (pairing) ( )out inT T . inT . )inq ( . ! 10 % 10 % . ) 26346 ( 7 . 4 - SISO . 5 - SISO . ) 6 ( - - . ) ( ) ( inT . . )spoutT ( ) SISO ( ) ( . aT inq inT . . (MISO) ) (! . . 6 - MISO . - outT outq . LIC Level Indicator & Control . outq ) ( . - ( ). out satT q ( )inh q - ) - ( ! ) 26346 ( 7 7 - SISO MISO . P&ID PFD . . 8 ) ( SISO . ) ( . :1 - y my . ) ( . outT .2 - u u . . . satq ) ( heaterQ ) (inq ) ( inT ) ( .3 - id n . . . ) ( inT inq aT . ) SISO ( 8 - . 4 - - ) .( ) ( (deterministic) . ) ( .5 - - . spy ) my ( spy my ) spy .( :1 - - V3 ! ) ( .2 - - . ) ( .3 - ) ( - (KV) .4 - - u u ) y my ( )spy ( ) ( . : . (setpoint ) 26346 ( 7 tracking) (servoMechanism) (command following) u u y my spy . (startup) (shutdown) (batch) . (loadrejection) (Disturbance Compensation) . u u )my y ( ) spy ( . 9 ) ( . 10 ) ( . 9 - .

10 - .

) 26346 ( 8

. :1.Instrument Engineers' Handbook, by B. G. Liptak, ehilto, Radnor, PA, 1970 2.Measurement Fundamentals, by Moore, ISA Publication, Res. Triangle park, NC, 1989 . ) ( . :1 ) :( .2 ) :( ) ( .3 ) :( .4 ) :( (workstation) . (CentralizedControl) ) ( (Distributed Control System DCS) .5 ) :( DCS (IT) DCS ) ( . (Bus) (Data Bus) (Address Bus) (Field Bus) (Process Bus) .

(online) . ) ( . ) ( pH . . . . 20 200 . . ) ( . ) ( (Vortex Shedding) . . ) ( . .

(Thermowell) . ) ( . 0-1300 F . (Filled Bulb) . . . (Resistant Thermometer Diode RTD) . . . ) ( !

1 ( (Bourden Tube)2 ( (Bellows) 3 ( . ) ( .

:1 .2 ) (3

) 26346 ( 8 (Transmitters) ) ( . . (Transducer) . . . :: 0-1, 0-10, 10, 5 : 4-20 mA : 3-15 psig 1 . 100 1000 4 20 . (range) ) ( 100-1000 . (span) 900 (zero) 100 .

1 . ) ( . 16900 mAkPa :( ) .20 4 161000 100 900transmAKkPa= = ) (

2 ) 1 ( ) ( 50 250 4 20 . 50 250F 200F 50F . :( ) .20 4 161000 100 900transmAKkPa= =

2 ) .( :( )1transmmKG ss =+ 3 P . . 100 2000kghr. P ) ( 16100mAin water . .

) 26346 ( 8

3 . P ) ( P :( )24 162000FPV = +

pv mA F kghr . :( )2max32ssFPV FF= FSS Fmax ). 2000kghr .( . ) plug ( ) stem ( ) seat ( ) 4 5 .( . . ) ( ) action ( ) characteristics ( ) size .( ) (

4 .

5 .

) ( . . . . ) 26346 ( 8 ) ( . 4 5 . ) ( ) Air-to-close,AC ( . ) ( ) Air-to-open,AO ( . ) ( AO ) ( AC .

) ( . ) ( ) ( . ) ( :( )vvPF Cf x= :F : ) gpm (vC : x : ) (( ) f x : . ) ( . : ) (vP : ) psi ( . ) ( :Masonielan Handbook for Control Valve Sizing, Dresser Industries, 6th Ed., 1977. ) ( . .

) ( : : ) :( 100 gpm : HP : 40 psi : 150 psi ) :( 1

size . case :Case1 ): ( (head) ) (.Case2 : ) ( ) Rangeability ( . ) 50 % ( 20 = Case 1: .150 20 40 210Reservoir Valve HExchangerpsi + + =

: ) 50 % ( 80 = Case 2: .150 80 40 270Reservoir Valve HExchangerpsi + + =

: : ( )vvPF Cf x=,144.72vC =(%50 )20100 0.501vopenC =

Case 1:,222.36vC =(%50 )80100 0.501vopenC =

Case 2: case 2 case 1 .

( ) 1.0 f x = : Fmax Fmax . . F :)))2240100Hmax max maxHmaxH designdesignPF FPP F = =

) 26346 ( 8 : : .total v H v total HHExchanger ValveP P P P P P = + = 2max,max1: 40 20 60 , 44.72 60 40100total v vFCase P C P = + = = = ( )( )2max,144.72 1.0 60 40 115100max maxFF F gpm = = :

2 : 40 80 120 , 22.36total vCase P C = + = =( )( )2max,222.36 1.0 120 40 141100max maxFF F gpm = = :

( ) 0.1 f x = : ) . pop (....2,11: 44.72 0.1 60 40 33.3100minmin minFCase F F gpm = = 2,22 : 22.36 0.1 120 40 24.22100minmin minFCase F F gpm = = : max min ) ( turn-downratio ) ( rangeability .

: Case 1 : : ) .(Case 2 : : . : 50 % ) ( . : ) :( . ) % 20 :( . :1 (sizing) FmaxFmin.2 ) ( ) ( ) ( : FmaxFmin : Cvalve )PP .( : . 50gpm . ) ( . . 10 psi ) ( . . 2 . . :210 50coilP F = :21050valve total coil totalFP P P P = = FmaxFmin :

22150150 (1.0) 1052525 (1.0) 105v totalv totalC PC P = =

vC totalP :21.3 , 139.2 2 139.2 2 141.2v total pump totalC P P P psi = = = + = + = ( )designf x :50 21.3 ( ) 139.2 10 ( ) 0.206design designf x f x = = : Fmin :2maxmax ..2minmin min ..(1.0) ( )( ) ( )v total h desdesv total h desdesFF C P PFFF C f P PF = =

totalP .( )h desP ) ( . . desF minf . totalP :{ }( )2 2max min2.min maxmin2.( )1desF FFtotalf Fh desFPP= ! ) 26346 ( 8 Rangeability : :min maxminf FRF= R 1 .

: 50 % 40 % totalP 2 / 139 202psi 40 35 totalP 355 30 R 1 totalP :maxminmin50 30.1 150 0.3FR fF= = = turn-down ratio 1 / 0 . 1 / 0 . splitrange full open . .

- (plug & seat) . :1: ( ) ,: ( )xLinear TrimValves f x xEqual Percentage TrimValves f x == ) Inherent Characteristic ( 6 .

6 . ) ( 50 50 ) 50 = ( 14 . (Installed Characteristic) 7 ) : HP vP (

7 Equal P. Trim ) ( Linear Trim ) .(

trim . trim . . . : . . ). Q F = T :pQ C T = ( Equal p. trim . . )vHigh P ( HLow P :..HvalvedesHvalvedesPHigh whenflowis highPPLow whenflowis lowP - Inherent char. Installed Char. . 0HP = ) ( . ) 26346 ( 8 Valve Positioner : (Sticking) I/P ) ( . ) ( . : positioner . positioner . positioner split-range . : : ) ( . 20 40 ). 4 10 ( ) ( Gain . : . : ) .( : ) :( ) ( ) ( Transmitter . ) ( . actuator ) ( . : . . : (discrete) . ) ( - 7 . . ) ( / )bourden ( T ) ( . ) (

7 . ) ( - 8 . ) stepper motor ( ) ( . .

8 . ) 26346 ( 8 9 . ) I/P ( .

9 ) ( . ) ( ) ( ) ( ) .(

(proportional) : ) ( ) ( ) ( . :( ) ( )C sp mut bias K y y = ( ) ut :Bias : ) ( . ) ( cK : . ) ( . cK . :{ }, ,( )( ) ( ) ( ) ( )( )( ) ( ) , ( ) ( ) , ( ) ( )C sp m C C Csp sp sp ss m m m ssusut K y y K e t L G s Keswhere ut ut bias y t y t y y t y t y= = = = = = = cK . (proportional band) cK :100%CPBK= PB )cK ( ) ( . ) ( cK . cK ) PB ( ActionMode ) / (Reverse/Direct) ( cK ) PB ( :( )( ) ( ) ( )( )C C Cusut G es or G s Kes= = = CK . CK . CK ) ( ) AC AO ( ) ( . . )offset ( . On/Off (bang-bang controller) : :0CPB or K = (limit cycle) . (ProportionalIntegralControllers) : ) 80 ( ) ( ) ( . . ) 26346 ( 8 ) ( . ) ( PI :0 0 01( ) ( ) ( ) ( ) ( ) ( ) ( )1 1 1 ( )( ) ( ) ( ) 1 ( ) ( ) ,(1/ ) ( ) 1 1( )( )t t tCC c C II IC CC c C I II I IC IC C C I CIKut Ke t e d K e t e d Ke t K e dK K esus Kes es K es Kes K Ks s sK s usG s K K K Kes s s s = + = + = + = + = + = + = += = + = + = cK I . :1s : . PI . . (ProportionalIntegral-Derivative Controllers) . . ) ( . ) ( PID :0 0( ) ( )( ) ( ) ( ) ( ) ( )1( ) ( ) ( ) ( ) , ,( ) 1 1( )( )t tCC C D C I DIC CC C D I D C DI ICC C C D C I DIK de t de tut Ke t e d K Ke t K e d Kdt dtK Kus Kes es K ses K K KsK usG s K K s K K K ses s s = + + = + + = + + = == = + + = + + cK I D . ) ( ) ( . - ) ( . : : T ) :(inT ) ( :. satq 10 12 P PI PID 10 t = . ) (

10 .

11 .

12 . ) ( . satq . . ) 26346 ( 9 1 1 . ) Nested ( . 1 . . ) s ( mG mG : 1mG= . valve plant valveK valveK CK valveK . comparator ) ( combiner ) summer!!( . . - (Forward path) +.- (Feed Forward) ++ . my . )( ) 0my t = ( . 2 . . : y ) ( ) d u (

( )p d p d py G Gd u y GGd Gu = + = + .... ...dpv Gd uy Gv= += 0p d p dyy GGd GGd= + = : d y0p pyy Gu Gu= + = : u y 2 2 . . : y ) ( ) d spy ( ( ) (1 )p d p C sp p C p d p C spy GGd GG y y y GG GGd GGy = + + = + dCspv Gd uu Gee y y= += = 1 1p d p Cspp C p CGG GGy d yGG GG= ++ + 3 4 . 3 . ) ( . ) 26346 ( 9 3 4 . ) ( . :( ),( ),( ) ( )( ), !!!( ) ( ) ( ) 1iforwardpathfrom X to Y iiall blocks in the loop iGminus forpositive feedbackYspositive for minus feedback Xs G + =+ : ) ( 5 : 1 2( ) ( ) ( ), , ?( ) ( ) ( )spys ys ysy s d s d s= 5 . . 4 :12( )( ) 1( )1 ( )( )1 ( )C v d psp C v d p m Td pC v d p m Tm T pC v d p m TGG GGysy s GG GGGGGGysGG GGGG d sGGGysGG GGGG d s=+=+=+ : ) Nested branched ( . ) ( . (VisualizingsystemDynamics) . . (Visualization) . (IMC) ) ( ... . Visual Editing Computer-Aided Software Design CASE . (Drag & Drop) . Simulink )MATLAB ( . . :1 ) ( . .2 ) ( .3 (Task) . ) 26346 ( 9 54 .5 . . 6 : 1 : ) Summing Points ( . 12 : . 13 ) : ( 1 1Y XX YY X= = + 6 . . . (Nested loop) ! 6 : : : 6 1 13 13 13 : (signal Flow Graphs) . Mason . . :Mason, S.J. Feedback Theory: Some Properties of Signal Flow Graph, Proc. IRE, 44, (1965), pp. 920-926. ) 26346 ( 9 7 . - ) ( : pG 1pppKGs =+ ( 0)CK :1 1( )( ) 1 11 11 1( )( ) , ,( ) 1 1 1p C pCC p p C p C pp psp C p p C pCp C pclC p pcl cl clclsp C p C pK K KKKG s K K K KysKy s KG s K KK ss K KK Kys KG s Ky s K K K K s + += = = = + + ++ ++ += = = =+ + + y spy spy :111 1C pC p C pK KK K K K =+ += ) (pK CK ) ( . p 1C pK K + 1 1C pK K + CK . : ( )( )( )2/( )( ) 1 /1C I p C p I psp C I pp C p I pK K s G K Ks KKysy s K K s Gs K K s KK + += =+ ++ + + ! ) 26346 ( 10 1

. ) Stabilizer ( . . - ) Performance-Oriented ( - ) Stability-Oriented .( . . ) ( 16 % ) ( . . ) ( ) ( ) ( . . .

. ) Sustainability ( . . ) bounded ( ) Unbounded ( (!) . . 2 ) ( :( ) ( 1) ( )0 1 1 0,n n mn n may a y a y a b u b m n+ + + + = + + y ) ( u ) ( . . ) ( ( )i i j t pt e + ) 0,1, 2, , p n =.( i i ite p ( 0) p = ( 1) p . ( )1pi is j + p ( )i is j + . ) ( ) i ( ) ( .

) - ( :1 21 2( )( )( )C C BsGsA s s s s s= = + +

( ) Bs s m ( ) A s s n :010 1 1( ) ,( )mmn nn nBs b s bA s as a s a s a= + += + + + +

) ( :iiCs s )1s=( . iiCs s is te is . is is . BIBO : (Left Half plane-LHP) RHP ) ( . BIBO ! ) 26346 ( 10 3 ) ( : : : . ) Decarte's Rule of sign :( . : . ) ( . . RHP . )Routh'scriteria :( . ) ( ) ( . Hurwitz : ) ( . Jury : ) . ( Routh ) ( (RHP) . : :10 1 10n nn nas a s a s a+ + + + = : : 0 2 41 3a a aa a

: .1 2 3b b b 4 :0 2 0 41 3 1 51 2 0 3 1 4 0 51 21 1 1 1,a a a aa a a a a a aa a a aab or b ora a a a = = : . : ) ( . :1 ) ( LHP .2 RHP . ) Tips & Tricks (- ) ( .- .- . .- .- . .- ) (- .- . ) ( Pade . : . .20 14 3 2( )3 5 4 2b s bGss s s s+=+ + + + : :4 3 23 5 4 2 0 s s s s + + + + = : ) 26346 ( 10 5 . is :4321 5 23 4 01 5 1 23 4 3 03 3sss 0s :43211 5 23 4 011 63 411 6011ssss :432101 5 23 4 011 626 026 626sssss . : CK ) ( . :( 1)(0.5 1): 1 0( 1)(0.5 1)( / 3 1)1( 1)(0.5 1)( / 3 1)CCCspKK y s scharacteristic equationKy s s ss s s+ += + =+ + +++ + + : 63 2( 1)(0.5 1)(( / 3) 1) 0 6 11 6(1 ) 0C cs s s K s s s K + + + + = + + + + = :3 26(1 ) 6 11 0cK K s s s K= + + + + = :32101 116660: 1 , 10 666006C CsKs kStability Criteria K KKsKs K

) 1CK ( 0CK 10 . 1 - . : 3 22 2 0 s s s + ++ = . :32101 12 20 022ssss= ) ( . : ) ( ) ( . : ) ( jw w :jw = : 3 223( ) 2( ) ( ) 2 0 02 2 0 100jw jw jw jw www w+ + + + + = = + Hurwitz . ) 26346 ( 10 72 - : : 33 2 0 s s + = .! 3 . :3 23 2 ( 1) ( 2) 0 s s s s + = + = :32101 30 223 02ssss ! : 1s 0 1 . ) RHP ( . 1 s = + . 3 - ) ( . . ) ( (Auxiliary polynomial) . ) ( ) ( s . . :5 4 3 22 24 48 25 50 0 s s s s s + + + =! ) -50 25 ( . :5431 24 252 48 5500 0sss 4s 4 2 3( )( ) 2 48 50 0 8 96dpsps s s s sds= + = = + 8 ( ) dpsds) ( 3s :5432101 24 252 48 508 96 .24 50112.7 050ss Auxiliary Polynomials Derivative Coeff of Auxiliary Polynomialsss . :5 4 3 22 24 48 25 50 ( 1)( 1)( 5 )( 5 )( 2) 0 s s s s s s s s j s j s + + + = + + + = : :4 2 2 2( ) 2 48 50 0 1, 251, 5ps s s s ss s j= + = = = = =

. . 11 p = 110 p = :) A ( ( )( 1)( 10)AGss s=+ + ) ( te 10te . ) ( te0.37 10te10-54.5 . 8150 . ) ( . (!) . 5 . damp (DominantPole) . . ) PID ( (Tuning) ) ( . 3 PI ) ( . . KC . KC ) 26346 ( 10 9 ) ( . Load ) .( . ) ( ) ( (!). ) ( ) ( . . . (steady state) . . - : N ) ( ) SN .( . N=0 N=1 1 . 1/S PI ) ( . ) N=2 ( ... - : ( ) ( ) ( ) ( ) 111 ( ) ( ) ( ) ( ) 1 ( ) ( )sp sp spy Gs es ysHsy GsHs y s y s GsHs= = = + +0( )( ) ( )1 ( ) ( )spsst s tsy se lime t limses limGsHs = = =+ 10 PK (staticPositionerrorconstant) : : 1spys=11 (0) (0)sseG H=+ PK 1(0) (0)1P ssPK G H eK= =+ . : 000( 1) ( 1) 1( 1) ( 1) 1mP sssnKb s b sK lim K eas as K+ +== = =+ + +

0 type000( 1) ( 1)0( 1) ( 1)mP ssNsnKb s b sK lim eS as as + +== = =+ +

1 type ) 1 N ( ) ( . VK (staticVelocityerrorconstant) : :20 01 11 ( ) ( ) (1 ( ) ( ))sss sse lim limGsHs s GsHs s = =+ + VK 0( ) ( )VsK lim sGsHs=000( 1) ( 1) 10( 1) ( 1)mV sssn VsKb s b sK lim eas as K+ +== = = = + +

0 type000( 1) ( 1) 1 1( 1) ( 1)mV sssn VsKb s b sK lim K esas as K K+ +== = = =+ +

1 type000( 1) ( 1) 10( 1) ( 1)mV ssNsV nsKb s b sK lim eK s as as + +== = = =+ +

2 type ) 2 N ( . . ) 26346 ( 10 11 AK (staticAccelerationerrorconstant) : )2( )2spty t = ( :3 20 01 11 ( ) ( ) (1 ( ) ( ))sss sse lim limGsHs s s GsHs = =+ + sse . : 2( ) / 2spAcceleration Inputy t t = ( )spRamp Inputy t t = ( ) 1spStep Inputy t = 11 K + 0 type1K0 1 type1K0 0 2 type ) 26346 ( 11 1 : (Root locus) . ) ( ) ( . ) ( ) ( . . .1 (rational) s . )dse ( .2 .3 s ) ( .4 ) ( .5 (Spirule) ) MATLAB- Control Toolbox ( . ) (... ) (... . 1 : : ) 0CK ~ ( ) ( . CK CK . u spy 1 . 2 : ( )1111( ) 1 11 1CCC CCsp CCKKK K ysKy s KsK s++= = =+ ++ ++ + ) ( :(1 ) 0 ( ) (1 )C Cs K Closed loop Poles K + + = = + CK . ) ( . 1 - : CK : ) 11CK +( y spy .

) 2 ( : ) ( ) ( : : :22 ( 3) ( 1)( 2)2 3( 1)( 2)( 3) 61( 1)( 2( 3)CCCsp CKK s y s sKy s s s Ks s s+ + += =+ + + +++ + + :3 2( 1)( 2)( 3) 6 0 6 11 6 6 0C Cs s s K s s s K + + + + = + + + + = ) ( CK ) ( CK : ) 26346 ( 11 3 CK -1 -2 -3 0 -1.15 -1.75 -3.1 0.038 -1.28+0.75j -1.28-0.75j -3.45 0.263 -0.45+2.5j -0.45-2.5j -5.1 4.42 0.0+3.32j 0.0-3.32j -6.0 10.0 0.35+4j 0.35-4j -6.72 16.67 CK . ) ( . ) ( . ! : CK .! . 4 Evans ) ( : ) ( :1 1C p pspC p m C p mGG Gy y dGGG GGG= ++ + 1 G + ) C p mG GGG= ( . y ) ( d spy . 1 0 G + = ) G ( . . : ) G ( :( )( )NsG gain rational function KDs= = N(s) D(s) :1 21 2.( ) ( )( ) ( )( ) ( )( ) ( ) ,mnK ConstNs s z s z s zDs s p s p s p m n== =

iz ) ( ip ) ( . : 1 0 G + =( ) ( )1 1 0 1( ) ( )Ns NsG K KDs Ds+ = + = = ) 26346 ( 11 5 s 1 0 G + = ) ( ( )( )NsKDs 1 - . :2 21(cos( ) sin( )( )( )( )jjr s u vs re r js u jv Cartezianvs s s s re Polars tgu = = + = = + = + = == = XX 1 :1 (1) cos( ) sin( ) 1 0je j j = = + = + s ( )1( )NsKDs= ( )1 ( )( )( )(2 1) ( )( )NsK Magnitude RuleDsNsK k Angle RuleDs= = + X :( ) ( ) ( ) { }( ) ( ) ( ) { }( ) ( )1 211 211 21 21 1( )1( )( )( )(2 1) , 0,1, 2,...miminniimnm ni ii is zs z s z s zNsK K KDss p s p s ps pNsK s z s z s zDss p s p s ps z s p k k = = = = = = = = + + + + + + = = + =

X X XXX XXX X : K s . ) : ( ) ( . . 1 : . 2 : ) 0 K = ( ) K ( . n m ) ( . 6 ) q ( q . q ) q ( . 3 : (Real) . 4 : :1 1( ) ( )(2 1)( )n mj jj jp zpoles zerosn m relative orderkangle with real axisrelative order= == =+ = = < 5 : ) 2 ( (Im) :1 11 1m ni ii is z s p = == s ip iz . 6 : q q :1 11(2 1) ( ) ( ) , 0,1, 2,..., 1m ni ji jk p z p p k qq = = = + + = X X v :1 11(2 1) ( ) ( ) , 0,1, 2,..., 1m ni ji jk z z z p k vv = = = + + = X X 7 : - :( ) ( ) 0 Ds K Ns + =! : ) n-m ( 2 ) ( K . : ) ( ) SISO ( . . : ) ( ) ( Evans . . . . ) 26346 ( 11 7 : ) I ( ) ( . K 0.5 ) ( . 0.5 = 16 . .) ( :.( 1)( 2)( _)( 1)( 2)1( 1)( 2)clKK ss sG sKss s Kss s+ += =+ + +++ + 1 : ) ( . 3 : 1 2 . ) ( : :

.( 1) ( 2)( 1)( 2)olKG s s sss s = = + + + + X X X X X ) (. .1 ( ) 0 ( ) 1ol olG s G s + = = . ) ( . 2 s= + :( 2) 0 , ( 1) ( 2 1) 0 , ( 2) ( 2 2) 0 s s s = + = + = + + = + = + + = X X X X X X 2 s= + . 0 1 . )milestones ( ! 8 4 : 3 ) n m = ( :( ) ( )( 0 1 2) 013(2 1)3 5( ) , ,3 3 3poles zerosrelative orderkangle with real axisrelative order = = = += = = < ) ( : s ) s ( :. . 31 ( ) 0 lim(1 ( )) lim(1 ) 0 lim(1 ) 0( 1)( 2)ol ols s sK KG s G sss s s + = + = + = + =+ + .

(2 1)3 (2 1) , 0,1, 2,...3ks k k s angel of asymptotes + = + = = = X X : ) ( : : s 3 2( 1)( 2) 0 3 2 0 ss s K s s s K + + + = + + + = : 3( ) ( 1) 0 1 s = + = = 3 2 33 2 ( 1) ( 0) 0 ( 1) 0 s s s s s + + + + + = + = 5 : (break-away point) : 1 - :2 2 2 21,21 1 1 ( 1)( 2) ( 2) ( 1)0 01 2 ( 1)( 2)( 3 2) ( 2) ( ) 0 3 6 2 03 31.577,3 9 6333 30.4233s s ss sss s s ss ss s s s s s s sunreasonables+ + + + + += + + = + + + ++ + + + + + = + + = = = = += ) ( : ) ( K .) ) ( ( :( )1 ( ) 0 1 0 ( ) ( ) ( ) 0( )NsGs K f s Ds KNsDs+ = + = = + = ( ) f s :( )0df sds= : 1s s = . 1 2( ) ( )( )...( )2rnf s s s s s s swhere r = ) 26346 ( 11 9{ }111 2 1 2 1( ) ( )( ) [( )...( )] ( ) ( )...( ) ( ) 0r rn ns sdf s d df srs s s s s s s s s s s s let s sds ds ds== + = =) ... (( ) ( ) ( )( ) ( ) 0 ( ) ( ) ( ) 0( ) ( )df s Ds DsDs KN s K f s Ds Nsds N s N s = + = = = = s . : ) ( :) (2( ) ( )( ) ( ) ( ) 0 0( ) ( )0( ). . : ( ) ( ) 0( )Ds DsK f s Ds Ns DN D NN s N s dKDs dK D N DN dscharaceq Ds KNs KNs ds N = = = = = + = = = :3 2 3 2 20.4233 2 0 ( 3 2) (3 6 2)1.577 :dKs s s K K s s s s s sunacceptable ds + + + = = + + = + + = 0.385 K = 0 K~ . : 0.385 K = + : : Routh3 23 2 0 s s s K + + + =32101 23603ss KKss K16 : 0 0 if K row s Auxiliarypolynomial = 2 2 2( ) 3 0 0 3 3 6 0 2 ps s s K s K s s j = + + + = + = + = = : s jw= :3 2 2 2( ) 3( ) 2( ) 0 ( 3 ) (2 3 ) 0 02 6 0 0jw jw jw K K w j w w jw for K or w for K+ + + = + + = = = = . K ) ( 16 : :2 21,21 1cos( ) sin( ) ( ) 1.732 1.047( ) 60 p j r r j tg rad = + = = = =

10 K ) ( 60

) ( . : ) 60

( :s j = + ) K (.( ) ( 1) ( 2) s s s + + = X X X111( ) ( )1 ( 1)( 1) ( ( 1))2 ( 2)( 2) ( ( 2))s tg xs js j s tg ys js tg z = = + = + += + + = + = + = + + + = + =XXX 1 1 1: ( ) ( ( 1)) ( ( 2)) ( ) ( ) 0x y zAngel Criteria tg tg tg tgx y z tg + + + + = + + = =_ _ _ ( ) ( ) ( ) ( ) ( ) ( )( ) 01 ( ) ( ) ( ) ( ) ( ) ( )tgx tgy tgz tgx tg ytgztgx y ztgx tg y tgx tgz tg ytgz+ + + + = = 32 22 2 21 2 ( 1)( 2)0 3 6 21( 1) ( 2) ( 1)( 2) + + + + + += + + = + + + + :1( ) 120 ( ) 1.732 1.732 tg = = =

:1,21 17323 3000s j= K :1 1 2 1.037( 1)( 2)KK s s sss s = = + + =+ +: ) 26346 ( 12 1 CG ) .( 1 - .

:0.03 0.03,(2 1)( 1) ( 0.5)( 1) 2C CK K KG Ks s s s= = =+ + + + 2 . 2 - . 2 :1 ) . (2 .3 K K ) ( K . ) ( . 3 . 3 - . :0.03 0.03 ( 1 ) 11 ,(2 1)( 1) ( 0.5)( 1) 2C C IIK K KsG Ks s s ss s += + = = + + + + : PI ) ( . : 1 2 0,5 1 PI 1 ) ( 0.5 2Inew zero = = ) -2 ( 4 . ) 26346 ( 12 3 4 - - ) 0.5I = .( : ) 5 ( . : 1 - K 0,5 1 . ... 5 - ) 0.5I = .( 42 ) ( I 1 0,5 : ) 6 ( 6 - - ) 4 3I = .( : ! ! !3 ) ( ) ( 0,5 : ) 7 ( 7 - - ) 4I = .( ) 26346 ( 12 5 : ) . ( : ) 4 3I = 4I = ( . 8 . 8 - ) I 1 K = .( ) PD PID ( . CK . 10CK= 50CK= . : Pade :2( )2 221212dd ddddssssdsee ese ++= =+ : Pade dse ) ( .1 ) 26346 ( 13 ) ( : : 15 . . . ) ( . . : ) ( . ) Processing Signal ( )Design Filter ( . . .

) Substitution Rule ( ) Amplitude Ratio-AR ( . ) : Angle Phase (. AR ) ( .2 AR ) s ( G . jw s : ( ) AR Gjw =) (( ) Gjw = () (

( ) Gjw : ( ) Gs ) ( ) ( s ( ) Gs ) ( ) ( . s jw w . ) ( w ) ( . ( ) Gjw . Bode ( ) Gjw ) w ( . .-40-35-30-25-20-15-10Magnitude (dB)10-1100101102-90-450Phase (deg)Bode DiagramFrequency(rad/sec) 1 - . 3-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1-1-0.8-0.6-0.4-0.200.20.40.60.81Nyquist DiagramReal AxisImaginary Axis 2 - . ( ) Gjw . . ) 26346 ( 14 1 ) 26346 ( 14 Bode : 1 1( )1Gss =+ . ) ( . ) ( ( ) Gjw :2 2 21 1 ( ) 1 ( ) 1 ( )( )( ) 1 (1 ( ))(1 ( )) 1 ( ) 1 ( ) 1 ( )Real part Imaginarypartjw wj wGjw jjw jw jw w w w = = = = ++ + + + + 1( )( ) 1Gjwjw =+ :( )2 22 2221 121 ( ) 1( )1 ( ) 1 ( )1 ( )( )1 ( )( ) ( )11 ( )wGjww wwwwGjw tg tg ww = + = + ++ + = = + ( :( ( ))2 21 1 ( )( ) ( )( ) 1 1 ( ) 1 ( )G jw jReal part ImaginarypartwGjw j Gjwejw w w += = + =+ + +( w :( ) 1 , ( ) 0 Gjw Gjw (: w 1( ) , ( ) ( / 2) 90 Gjw Gjw orw D(: w ( ) Gjw ( ) Gjw ( w log semi log log . log :) 1 ( w ) 2 ( damp ) ( . 1 ) AR ( log w . :20 log db AR = 2 1 - . :( )( )2 221( ) log log 1 ( ) 0.5log1 ( )1 ( )Gjw AR AR w ww = = = + = ++ :) (( )20 log 0.5log1 (0) 0 for w AR = + =) (( ) ( ) ( )20 log log 0.5log1 (0) logp pfor w AR k k = + = ) -1 ( :( ) ( ) ( ) ( )2log 0.5log( ) log log log for w AR w w w = = = :( )10 log 1 w w w = = = w ) corner frequency ( . :( )( ) ( )21log log 1 1 1/ 2log2 0.7072AR AR = + = = ) 26346 ( 14 3 2 - ) w ( AR . : ( ) Gjw ( )iG jw ) AR ( :{ }1 11 221 1 11 2 1 222 2 2j j zj z zj j zz r e z ez z z z ez r e z e+ = = == =(( (( : AR Modulus . : ) ( . : ) ( 1( )( 1)( 5)Gss s=+ + ) :( ( ) Gs :12 31 2 3( ) 1 (1/ 5)( )( 1)( 5) ( 1)(0.2 1) ( ) ( )1 1( ) (1/ 5) , ( ) , ( )( 1) (0.2 1)G sGss s s s G s G sG s G s G ss s = = =+ + + += = =+ + 4 ( ) z Gjw = 1z 2z 3z :11 1 2 2 3 32 3( ) , ( ) , ( ) , ( )zz Gjw z G jw z G jw z G jwzz = = = = = ( ) z Gjw = 1z 2z 3z :1 2 31 2 3log( ) ( ) ( ) ( )AR logAR logAR logARGjw G jw G jw G jw= = ( ( ( ( ) 1( ) 1/ 5 G jw = ( ) ( :11(1/ 5) 5( ) 0logAR log logG jw= = =( ) .( :2 2221 1211 1log log( ) log( ) log( ) log(1) log( 1 ) 0.5log(1 )1 111( ) ( ) (1) ( 1) 0 ( 1) ( ) ( )1 1AR w wjw jwwwG jw jw jw tg tg wjw = = = = + = ++ + += = + = + = = +( ( ( ( ( 2321 1311 1log log( ) log( ) log( ) 0.5log(1 (0.2 ) )(0.2 ) 1 (0.2 ) 11 (0.2 )1 0.2( ) ( ) (1) ((0.2 ) 1) 0 ((0.2 ) 1) ( ) (0.2 )(0.2 ) 1 1AR wwj wjwwG jw wj wj tg tg wwj = = = = ++ + += = + = + = = +( ( ( ( ( :2 21 1log log(1/ 5) 0.5log(1 ) 0.5log(1 (0.2 ) )( ) 0 ( ) (0.2 )AR w wGjw tg w tg w = + += ( : 3 . : : scale ) ) . ( ( : AR log ) AR ( ) ain g ( log scale ain g /AR ) .( : AR log AR log . : /1 = . : . : 1 - . ) 26346 ( 14 5-80-60-40-200Magnitude (dB)10-210-1100101102-180-135-90-450Phase (deg)Bode DiagramFrequency(rad/sec)ARAR1AR2AR3132 3 - . ) ( ) 1 ( . :2 2 22 2121( )(1 ( ) ) (2 ) 1( )2 12( ) ( )1 ( )Gjw ARw wGss swGjw tgw = = + = + += =( scale . AR ) ( . 4 . 1 w = :121( ) 0 : 1 , ( 2 ) 01( ) : 0 ,( )( ) 1: ( ) / 2w AR tg w orw AR or ARww tg = = = : 1 w = 2 / ! : AR . : 2 . 6-80-60-40-20020Magnitude (dB)10-210-1100101102-180-135-90-450Phase (deg)Bode DiagramFrequency(rad/sec)=0.2=2=2=0.2 4 - ) .( ) ( :( ) 1: 1/cw w = = : AR ) AR ( ) ( :21 2rw= : 0,707 < ! ) lag pure ( ) 5 .(( ) 1( )( )dsdGjw ARGs eGjw w = == = = ( ) 26346 ( 14 7-1-0.500.51Magnitude (dB)100101-720-540-360-1800Phase (deg)Bode DiagramFrequency(rad/sec) 5 - ) 1d = .( ) ( ) 6 ( .( )( )( ) 0ccGjw AR KGs KGjw = == = =( -1-0.500.51Magnitude (dB)100101-1-0.500.51Phase (deg)Bode DiagramFrequency(rad/sec) 6 - ) 1cK = .(

8 PI PI - ) 7 .(211( ) 1( )11( ) 11( )cIcIIGjw AR KwGs KsGjw tgw= = + = + = = ( : 2 / .0102030405060Magnitude (dB)10-210-1100101102-90-450Phase (deg)Bode DiagramFrequency(rad/sec) 7 - - ) 1 , 1c IK = = .( PD ) 1 + s /( 1 ! ) 8 ( ( )( )21( ) 1 ( )( ) 1( )c Dc dDGjw AR K wGs K sGjw tg w = = += + = =( ) 26346 ( 14 9010203040Magnitude (dB)10-210-110010110204590Phase (deg)Bode DiagramFrequency(rad/sec) 8 - - ) 1 , 1c DK = = .( PID PID - ) 9 .(211( ) 1 ( )1( ) 11( )c DIc dIDIGjw AR K wwGs K ssGjw tg ww = = + = + + = = ( 0102030405060Magnitude (dB)10-310-210-1100101102103104-90-4504590Phase (deg)Bode DiagramFrequency(rad/sec) 9 - - - ) 1 , 10 , 0.01c I DK = = = .( 10

: ) ( . : 10 , 0.5c DK = = : 1 0OLG + = OLG ) ( ) ( . 1 10.10.1 0.12 2 23 3 2 20.5 1 , (0.5 1)10(0.5 1)( ) , ( )( 1) (0.1 1)1 1, ( )( 1) (0.1 1) ( 1) (0.1 1)sOL jw jwAR jw jws eG s AR e es sARjw jw jw jw = + = ++ = = =+ += =+ + + +((( 1AR PD 120.5cw = = . 2AR ) ( . 3AR ) ( :1 21 11 , 101.0 0.1c cw w = = = = 10 iAR i .1 2 31 2 3( ) ( ) ( ) ( ) dBAR dBAR dBAR dBAR = + + = + + ) 26346 ( 14 1110-1100101-540-360-1800180Phase (deg)Bode DiagramFrequency(rad/sec)-100-50050Magnitude (dB)AR2AR1ARAR3231 10 - PD )10 , 0.5c DK = = .( : . . cK ) ( ) ( . 12 cK . ) sin( )spy A wt = ( ) my y ( . ) cross-over frequency ( cow . cK ) y ( ) ( cK ) ( ! ) y ( . ) ( )spy ( . ) ( 180 ) .( ) ( sin( ) y B wt = + BARA= sin( ) sin( ) y B w t B w t = = . spy . BARA= ) snow ball effect ( ! : ) ( AR . - :0.120.1 10.5 11 0.250.5 11 0.5( ) ( ) 0.1 ( )0.5 1 1CCsOL m Csp jwK KARjw y KewGy swe w tgjw = = + += = + = = +( ( ) ( )OLAR G jw = ( ) ( )OLG jw = ( ( w AR w CK . ( ) w = . ( )cow ww == :10.1 (0.5 ) w tg w = = :1( ) 0.1 (0.5 ) 0 f w w tg w = + = ) 26346 ( 14 13 :1 ( )1 ( )(0.5 )0.1 (0.5 ) 00.1prvnxt cocotg ww tg w w+ = = 17.0 ( . / .)cow rad sec . CK ) ( . CK ) ultimateCK ( . CK 1 AR = ultimateCK . 11 . 11 - . ) 26346 ( 15 1 - ! . . . . . . . ) Ziegler & Nichols ( ) Taylor Instrument Co. ( PI PID ) lag-dominant ( . ZN ) deadtime-dominant ( . ) Margin Gain & Phase ( . : : ) PI PD lead-lag 20 ( ) ( tuning .

Domain Freq Domain Time ) 1 ( :1. .. . 180G M Gain MarginMP M Phase Margin P= == =

. :) : ( . . 1.7 , . . 30 G M P M

) :(. . 2 , . . 45 G M P M

2 1 . 1 : . . G M : 210.12 . .0.121 0.25(17)CCCKAR K G MK= = =+ ) ( : . . 1.7 G M = :1. . 1.7 4.90.12CCG M KK= = = 0.10 0.15 ) %50 : ( : AR cow 0.15( )0.5 1sOL CKeG ss= + 24.9( ) 11.6 ( ) 0.83 11 (0.5 11.6)OL OLcoG jw w G jw = = = =+ ) CK . . G M ( % 50 ) ( ) setpoint ( . 2 :. . P M : :( )12, 0.1 0.51 0.25CKAR w tg ww= = + ) 26346 ( 15 3 . . P M 30

:( )1(180 30 ) 150 150 0.1 0.5 12.5180w tg w w= = = =

21 6.331 0.25(12.5)CCKAR K == =+ % 50 ) ( :( ) ( )0.15112.5( ) ( ) 0.5 12.5 0.15 12.5 1880.5 1sOL OL CwKeG s G s tgs== = + = +

detune . : . . G M . . P M point starting .

- . . . ) manual ( ) auto ( . . . . . ) PI ( :(PI)(PI) Integral of Time-weighted Absolute Error - ITAE dt e tIntegral of Squared Error ISE dt e2 Integral of Time-weighted Squared Error - ITSEdt te2 Integral of Absolute Error IAE dt e 4 - . ) ( ) ( ) validation ( . ) finetune ( . ) ( ) sub-optimal ( .

1 - ) .(DIPControllerProcess 5 . 0 K 250PIDead-time-dominant 0 . 4mK / 106PILag-dominant 45 . 0 8 . 1mK / 77PID (non-interacting) 55 . 0 5 . 1mK / 106PID (interacting) 0 . 4. int/ 106 PINon-self-regulating 48 . 0 9 . 1. int/ 78 PID (non-interacting) 58 . 0 6 . 1. int/ 108 PID (interacting) 5 . 0 K 20PIDistributed lags 09 . 0 3 . 0 K 10PID (non-interacting) 1 . 0 25 . 0 K 15PID (interacting) 1 IAE . ) K ( )m ( . . int ) non-self-regulating ( . . ) 26346 ( 15 5 . m K . int ) ( 2 . TimeResponsemodeldata 2 . 2 3 .

2 - PI ) .(Proportional-Integral (PI) Controller:)11 ( ) (sK s GIC c+ =ITAEIAE 0.5860.758 1a-0.916-0.861 1b1.031.02 2a-0.165-0.323 2b) / (2 211mmIbmCb aKaK +==

C 63 . 0u K C = u . intm 6 3 - PID ) .(Proportional-Integral (PI) Controller:)11 ( ) ( ssK s GDIC c+ + =ITAEIAE 0.9651.086 1a-0.855-0.869 1b0.7960.740 2a-0.147-0.130 2b0.3080.348 3a0.92920.914 3b3132 21) / (bmm DmmIbmCab aKaK=+==

) ( - . . ) 2 ( . 4 .

) PID ( : . ) PD ( ) PID ( . PD - PID : bdtdeK dt e K e KCdtdedt e ePudtdedt e e K uD I CDIss DIC+ + + + + + + + + =

)1(100)1( ) 26346 ( 15 7 D . PID dedt . PID ) ( u . : dydtdedt . . PID . ) ( . . . : , . , . ,1(1 ) , ,1 1DCeff C I eff I D DeffII DK K = + = + =+ PID ) . ,eff CK . ,eff I . ,eff D ( PID ) CK I D ( . PID . ) ( . dydtdedt . PID PI . 4 - PID - ) ( . DI P Controller --- 5 . 3m / 150 PI 63 . 0 1 . 2m / 75 PID (non-interacting) 7 . 0 8 . 1m / 113 PID (interacting) 8 ZN ) ( . )n ( uK uP 5 . 5 - PID - ) ( . DI P Controller ------ uP 0 . 2 P --- n 81 . 0uP 7 . 1 PI n 11 . 0n 48 . 0uP 3 . 1 PID (non-interacting) n 14 . 0n 39 . 0uP 8 . 1 PID (interacting) ) 26346 ( 16 1 ) 26346 ( 16 - ) ( ( ) Gjw w . ) - - .( . 1 ) ( ) 0, w + ( . ) ( AR ) ( . 2 ) ( ) , w + ( . . ( )OLG jw : ) ( ) ([ ] [ ] ( ) Re ( ) Im ( ) Gjw Gjw j Gjw = +( )( ) ( )G jwGjw Gjwe=

1 ) .( 2 ) .( ( ) Gs : 1( )1Gjwwj =+ :10 :00:2ARwARw 2 ) 3 ( ) 02 ( ) 0 1 AR .(

( ) Gs ) ( :1 00 : , :0AR ARw w ) 4 ( ) 0 ( 0.7 . 3 . 4 . ( ) Gs ) ( :010 : , :302ARARw w ) 5 ( )302 ( .

( ) Gs ) 6 ( :1 ,dAR w = = ) 26346 ( 16 3 5 . 6 .

) capacitive pure ( ) 7 ( :01 1 1( ) ) ) , : , :2 2AR ARGs Gjw j w ws jw w = = = 7 . 8 ) ( . 4 P ) 8 ( ! 9 - . 10 - . PI ) ( PD ) ( PID ) ( - - - - 9 10 11 ) ( . 11 - . ) 26346 ( 16 5 . : ( )OLG jw w ( ) 1, 0 ( )CLG s . . : ( ) Fs z ) ( p ) ( s z p ( ) Fs ) ( ) Fs ( . : . . ( ) y f x = x y x y . ) ( : ( ) Fs :1 21( )( )( )s z s zFss p = 12 . 12 ( ) Fs s . ) ( ( ) Fs :1 2 1( ) Fs z z p = + s ) ( . 2 s 2z . s 6 2 2 ( ) Fs 2 . 1 ) s 1z ( ( ) Fs 2 ) ( . ( ) Fs 2 . : ( ) Fs ) ( ) 13 14 ( : 13 C 3 0 3 N z p = = = . 14 C 0 2 2 N z p = = = ) C .( ( ) Fs ( ) 1 ( )OLFs G s = + . RPH . C ( ) Fs ) ( s . ) 26346 ( 16 7 1 ( )OLG s + ( ) Fs . ( ) Fs ) ( RPH ( ) 1 ( )OLFs G s = + : N z p = :N : z : ( ) Fs p : ( ) Fs : z N p = + : ( ) Gs ( ) 1 ( )OLFs G s = + ( ) Fs ( ) Fs ( ) F s : ( ) ( ) 1 ( )OLF s Fs G s = = . . . ( ) 0, 0 . .. . . . ( ) 1, 0 . . . . 1 ( ) Gs + ( ) Gs ) ( ) Gs 1 1 ( ) Gs + ( ) Gs .( : ( ) Fs RPH RPH RPH . : . s ( ) F s ( ) Gs . ( ) Gs ) ( ( ) 1, 0 ) N ( ) ( ) 1, 0 0 N = 2 N = .( RPH ) :OLC pG GG = ( P 0 p = . N P :z N p = + ) ( : ) ( ( ) Gjw ( ) G jw . s s . ( ) Gjw ( ) G jw jw 8 jw . . : . . ) .( 1 : ) ( :3( )( 1)OL CKG ss=+ : ) RPH ( 0 p = . s ) 15 (. 15 . C+RC C C+ s ) + (RC C - : C+: C+ s s jw = w . ( ) Gs :3( )( 1)OL CKG sjw=+ ( )OLG jw ( )OLG jw ) ! 16 ( ) 26346 ( 16 9 RC : ) ( s Rejs= R 2 / 2 / s ( )OLG s Rej: 3( ) 0 0( 1)OL j CjRKG Re jRe= = ++ ) ! 16 .( C: C+ s - ) 16 .( 16 . 16 . 10 16 . 17 . CK ( ) 1, 0 . ( ) 1, 0 CK ( ) 1, 0 0 N = . p . CK ( ) 1, 0 2 N = 2 z N p = + = . 17 ) w .( CK .CK CK ( ) 1, 0 ] [ :2 33 2 3 2 2 3 2 2 2 3 2(1 3 ) ( 3 )( )( 1) (1 3 ) (3 ) (1 3 ) (3 ) (1 3 ) (3 )OL C C C CK K K w K w wG jw jjw w j w w w w w w w w = = = ++ + + + ) 26346 ( 16 11 1 0 j + :330( 3 )0 ( 0) ( 3 )3CCw intersection withpostive sesctionK w wK w ww intersection with negative sesction= = =

w 3 :2 23 ( )3u u uuw T or P time unitw = = = w CK ) uK ( :22 2 3 23(1 3 )1 1 88 (1 3 ) (3 )C CC uwK w KK Kw w w== = = = + 8CK= ) 8CK( ) 2 N = . ( 2 : ) ( : 1 2( )( 1)( 1)OL CKG ss s s =+ + ) ( PI PID . C+C 2 4( 1) ( 2) s s + +

j 42 j . s r

) 0 r

( s RPH 1 G + ) 18 .( s : C+RC CC

: ( )OLG jw ) ( ) 0, w + :( ( 0 & ) r Rs jwas w goesfrom rto R =

: C+ :( ): 0j js Re GRe = = : RC( )OLG jw : ( ) Rs jwas w goesfrom to r= : C :2 2js r e where r goes zero and variesfrom to = : C

) : 1 10 0( ) lim lim( 1)( 1)OL j j C Cj j jr rK KG r e er r e r e r e = = + +

( 12 18 s . 18 . CK ( ) 1, 0 . uK :1 21 2 1 21,u uK w += = C+ . . ( ) 1, 0 . convex concave . !!! C+ 0 w = w = w = . . . 3 :) ( :1 2 3 4( 1)( 1)( 1)( 1)( 1)CK ss s s s ++ + + + CK ( ) 1, 0 ) ( ) .( ) 26346 ( 16 13 19 3 . ( ) 1, 0 : ( ) 1, 0 : ( ) 1, 0 : ( ) 1, 0 :