Closed-Loop Compensation Method for Oscillations Caused by Control Valve Stiction

Transcription

1 Artle ub.a.org/iecr Cloe-Loo Coenaton Metho for Ollaton Caue by Control Valve Stton Janong Wang* College of Engneerng, Pekng Unverty, Bejng, Chna 0087 ABSRAC: h aer rooe a loe-loo oenaton etho to reove ollaton aue by ontrol valve tton. Wth the ontrol loo oeratng at the auto oe, the rooe etho a a hort-te retangular wave to the referene to ntroue two oveent for the ontrol valve to arrve at a ere oton. A yteat way to egn the araeter of the hort-te retangular wave eveloe. he rooe etho robut agant oelng error an eaureent noe. he effetvene of the rooe etho llutrate by laboratory an ulaton exale.. INRODUCION Inutral urvey 3 reorte that about 20 30% of ontrol loo n varou roe nutre erfor oorly ue to ontrol valve nonlnearte. One tyal abnoral henoenon the ollaton of gnal n a loe-ontrol loo aue by ontrol valve tton. Hene, t ratally ortant to etet the reene of tton an to quantfy the tton everty. One the tton ha been etete an quantfe, the ontrol valve an be heule for antenane or relaeent. However, t often erable to kee ung the tky ontrol valve, an to oenate the negatve effet of tton by teorarly reovng ollaton. In the lat a few year, the eteton an quantfaton of ontrol valve tton have been very atve reearh to. 4 However, there are relatvely fewer tue on the oenaton of ontrol valve tton. Gerry an Ruel 5 lte four uggeton for obatng tton onlne, e.g., reovng ntegral aton of the roortonal ntegral (PI) ontroller; ong o uually lea to large teay-tate ontrol error an work only for nonntegratng roee. 6 Ha gglun 7,8 reente a o-alle knoker etho to a hort ule to ontroller outut n orer to kee ontrol valve ovng. Srnvaan an Rengaway 9 ntegrate a tton etaton roeure wth the eleton of araeter for the knoker etho. Ivan an Lakhnarayanan 0 reve the knoker etho by relang the hort ule wth ontant renforeent ae to ontroller outut. Both the knoker etho 7,8 an the ontant renforeent etho 0 reue the varaton of roe outut at the ot of aggreve valve oveent that ay wear valve qukly. o avo the aggreve valve oveent, Srnvaan an Rengaway rooe a twoove etho an an otzaton-bae etho for tton oenaton. he two-ove etho a two oenaton oveent to the ontroller outut n orer to ake the ontrol valve eventually arrve at a ere teay-tate oton. Cuaro et al. 2 rove the two-ove etho by ntroung ore oveent to ontroller outut. Later, Cuaro et al. 3 rooe another oel-free oenaton aroah by ontorng the ontrol error after a knoker ntroue: f the knoker haen to ove the valve to a ere oton o that the ontrol error all for a ertan ero of te, then the ontroller able. Mohaa an Huang 4 elnate ollaton aue by ontrol valve tton or reue the agntue an frequeny of ollaton by tunng ontroller araeter. here are alo oe etho to oenate the negatve effet of valve frton for ontrol tak, 5,6 whh are not ae at teorarly reovng ollaton, an thu are not alable n th ontext. h aer nre by the two-ove etho an t rove veron. 2 However, thee two-ove etho have to wth to the oen-loo ontrol, whh ay not be allowe oete n rate owng to varou oeratonal requreent. By ontrat, a loe-loo oenaton etho rooe here, wthout wthng to the oen-loo ontrol. he rooe etho a a hort-te retangular wave to the referene to aheve the ae two valve oveent a n the two-ove etho. o aheve th, the araeter of the hort-te retangular wave are egne n a yteat anner: bae on the tton oel wth two araeter rooe by He et al., 7 gnal n fferent tage are analyze, analyt funton an onton to be atfe by the egn araeter are erve, an a roeure to obtan otal egn araeter eve. he rooe etho by egn ha a ertan level of robutne agant oelng error an eaureent noe; n aton, a eaban ntroue to gnore all ontrol error after oenaton. Aong the extng oenaton etho, the knoker etho, 7,8 the ontant renforeent etho, 0 the etho rooe by Cuaro et al. 3 an that by Mohaa an Huang 4 kee the ontrol loo oeratng at the auto oe; however, a tate earler, the knoker etho an the ontant renforeent etho uffer fro the rawbak of ntroung aggreve valve oveent. Hene, only the etho rooe by Cuaro et al. 3 an that by Mohaa an Huang 4 are oare here wth the rooe etho. he etho rooe by Cuaro et al. 3 bae on a rtal auton: f the varaton of the ontrol error after ung the knoker etho are ket all for a ertan te uraton, then t le that the valve oton arrve at the oton aoate wth a ere Reeve: January 27, 203 Reve: July 29, 203 Aete: Augut, 203 Publhe: Augut, Aeran Cheal Soety 3006 x.o.org/0.02/e400308z In. Eng. Che. Re. 203, 52,

2 Inutral & Engneerng Chetry Reearh referene value. However, there no guarantee for the auton to be alway hol. he reaon that the knoker etho ay aheve the all ontrol error by fat wthng aong everal valve oton; one the oenaton etho rooe by Cuaro et al. 3 able the PI ontroller, the valve ay tay a oton qute far away fro the ere one, reultng n a large ontrol error (ee exale 3 n eton 4 for an llutraton). heethorooebybymohaaanhuang 4 oe not oner where the ontrol valve tay after reovng the ollaton by tunng ontroller araeter. A a reult, the ontrol error oul be qute large even f t no longer ollatory after oenaton (ee exale 4 n eton 4). By ontrat, the rooe etho a at akng the ontrol valve arrve at a ere oton after oenaton, o that the ontrol error all after oenaton. Neverthele, the etho rooe by Cuaro et al. 3 an that by Mohaa an Huang 4 requre le nforaton than the rooe one an, thu, are eaer to leent. he ret of the aer organze a follow. Seton 2 erbe the oenaton roble. Seton 3 reent the loe-loo oenaton etho. he effetvene of the oenaton etho llutrate through laboratory an nueral exale n eton 4. Soe onlung reark are gven at eton PROBLEM DESCRIPION Coner a loe-ontrol loo wth a ontrol valve ete n Fgure, where r(t), e(t), (t), v(t), y (t), an w(t) are the Fgure. Dagra of a loe-ontrol loo. referene, ontrol error, ontroller outut, valve oton, eaure roe outut an roe noe/turbane, reetvely. he roe G() onfne to be a lnear tenvarant (LI) roe that erbe a θ 0 Ke G () = Π I ( τ + ) = 0, Suh a roe an be aroxate well by a frt-orer lu ea te (FOPD) oel 8 K θ G () = + e (2) A roortonal ntegral (PI) ontroller C() ue C () = K + he ontroller leente at an nutral trbute ontrol yte (DCS) latfor wth the alng ero.in Fgure, a real-value ontant to take are of the tat offet of y (t) o that the noe-free roe outut y(t) a evaton gnal. he ontrol objetve to let y (t) loe to a ere referene value r 0. he ontrol valve, enote by f( ) n Fgure, aue to have the tton roble. here are everal ata-rven tton () (3) Artle oel n the lterature, 4 aong whh the oel wth two araeter rooe by He et al. 7 aote here. he flowhart of the tton oel reente n Fgure 2. he araeter f an f n Fgure 2 tan for the tat an knet Fgure 2. Flowhart of the tton oel rooe by He et al. 7 Aate fro Fgure 2 of ref 7. Coyrght 2007 Aeran Cheal Soety. frton ban, reetvely. he varable r the reual fore atng on the valve whh ha not ateralze a valve oveent, an the varable u a urrent uulatve fore atng on the valve. Even though the referene r(t) take a ontant value r 0, the ontrol valve tton ay lea to ollaton n the loo, affete by oe nonzero ntal onton an/or rven by roe noe/ turbane w(t). h henoenon ha been oberve for any nutral ontrol loo (ee e.g., the nternatonal atabae [avalable onlne at htt:// bhuang/book2.ht] aoate wth the book 4 ), an an be theoretally hown va the erbng funton analy. For ntane, uh an analy ha been erfore for a artular ata-rven tton oel. 6 he oenaton roble to egn a oenaton etho to teorarly reove the ollaton aue by the ontrol valve tton, bae on the ollatory ata of y (t) an(t). 3. CLOSED-LOOP COMPENSAION MEHOD h eton rooe a loe-loo oenaton etho that olve the oenaton roble wth the ontrol loo oeratng n auto oe. o eae the reentaton, the roe noe/ turbane w(t) aue to be abent for the te beng. 3.. Man Iea. If the ontrol valve ha no tton, then the valve oton v(t) able to arrve at a teay-tate value, enote a v, to aheve zero ontrol error. he value of v r0 v = K (4) where r 0 the ere referene value, K the roe gan n eq 2, an the tat offet of y (t) (ee Fgure ). When the tton reent, however, v(t) annot reah v.intea,v(t) juaroun v n a ero anner, leang to ollaton n the loo. When the ollaton our, the haraterzaton of the tton oel eontrate n Fgure 3. When the valve ove, the relaton between (t) an v(t) v() t = () t ± f (5) Here the uaton (ubtraton) gn ue f (t) ereang (nreang). Owng to the tton, the ontrol valve an only ove along the nreang ath l a or the ereang ath l, e.g., the valve ay only tay at the oton arke by flle-rle n Fgure 3 when the ollaton our x.o.org/0.02/e400308z In. Eng. Che. Re. 203, 52,

3 Inutral & Engneerng Chetry Reearh Artle Fgure 3. Charaterzaton of the tton oel rooe by He et al. 7 an the ath ABC of the two valve oveent n the rooe oenaton etho. he an ea of the rooe loe-loo oenaton etho to a a hort-te retangular wave to the referene r(t) o that the ontrol valve tay at the ere oton v after two oveent, wth the ontrol loo oeratng n the auto oe. he two oveent are llutrate va Fgure 3: Suoe that the valve urrently tay at oton A. hen,thefrt oveent n the oenaton to ove the valve fro A to oton B, an the eon oveent fro B to the oton C that aoate wth the ere teay-tate value v.he varaton of (t) n eah oveent ha to be large enough to overoe the effet of tton n orer to ove the ontrol valve. he an ea alo llutrate by howng n Fgure 4 the gnal r(t), y (t), (t), an v(t) urng the oenaton. For larty, Fgure 5 reent an enlarge vewont of oe art of the gnal n Fgure 4, aroun a few ef te ntant efne a follow. Let t A be the te ntant the one ale before tartng the oenaton. he other te ntant are tb: = ta + tc: = tb + θ td: = tb + te: = td + tf: = te + θ (6) where the alng ero, θ the te elay n eq 2, an the uraton of the hort-te retangular wave. Wthout lo of generalty, t A an be hoen oewhere when (t) nreang. At the te ntant t B, the hort-te retangular wave wth altue r an uraton ae nto r(t) to ake the ontrol valve ju to a new oton v(t B ). he valve v(t) tay at the oton v(t B ) urng the tage t [t B, t D ] for the uraton. he referene r(t) goe bak to the noral value at the te ntant t E, an the valve v(t) ove to the ere teay-tate value v an tay at th value ever ne. herefore, the referene r(t) an be rereente a r() t = r0u() t + r( u( t tb) u( t tb )), t (7) where u(t) the unt te gnal an r 0 the ere referene value. heegnaraeterarethetentantt A, the altue r, an uraton. he egn requre a etale analy of the gnal y(t) (y (t)), (t), an v(t), whh gven n the next ubeton. Fgure 4. Sgnal for loe-loo oenaton etho: (a) eaure roe outut y (t) (ol) an referene r(t) (ah), (b) ontroller outut (t) (ol), an valve outut v(t) (ah). Fgure 5. Enlarge gnal for loe-loo oenaton etho: (a) eaure roe outut y (t) (ol) an referene r(t) (ah), (b) ontroller outut (t) (ol), an valve outut v(t) (ah) Sgnal at Dfferent Stage. he oenaton roe ont of everal tage enote by ABCDEF n Fgure 4. h ubeton analyze the gnal y(t)(y (t)), (t), an v(t) for eah tage. Stage AB an BC. Reall that t A efne a the te ntant the one ale before tartng the oenaton, whle (t) nreang. Let t be the latet te ntant before t A that y (t) hange the reton fro nreent to ereent. hen, the valve oton v(t) at the te ntant t, v(t θ) = (t θ) + f, aorng to the flowhart of the tton oel n Fgure 2. If the varaton of (t) n the te uraton fro t θ + to t A all uh that the nequalty (t A ) (t θ + ) f < f atfe, then the valve oton v(t) for t [t θ, t A ] oe not ove an tay at the value (t θ)+f. Hene, owng to the te elay θ, the roe outut y(t)urngthe te lot t [t A, t C ](t C efneneq6)uneroen-looontrol 3008 x.o.org/0.02/e400308z In. Eng. Che. Re. 203, 52,

4 Inutral & Engneerng Chetry Reearh at a ontant nut value (t θ)+f. On the ba of the roe oel eq 2, y(t) for[t A, t C ], enote a y AC (t), an be erbe a y () K t = θ + ( t ( ) f) y () t () AC AC (8) Here the uerrt () enote the frt-orer ervatve. Conerng the nonzero ntal value of y(t) att A,theLalae tranforaton of eq 8 θ + Y () y ( K t f t ) ( ( ) ) AC A = Y AC() o that Y AC K( ( t θ) + f ) yt ( A) () = K( ( t θ) + f ) = yt ( A) K( t ( θ) + f) + + (9) he te oan rereentaton of y AC (t) obtanefroeq9, y () t = K ( ( t θ) + f ) + [( y t ) AC A ( t t )/ K ( ( t θ) + f )]e A, t [ t, t ] A C (0) he ontroller outut (t) for [t A, t C ], enote a AC (t), obtane a follow. he ontroller C() n eq 3 aoate wth a fferental equaton K AC () () t = KAC e () () t + e AC() t () where e AC (t) tan for the ontrol error e(t) for [t A, t C ],.e., e () t = r() t y () t AC AC Conerng the nonzero ntal value at t A, the Lalae tranforaton of eq K MAC() t ( A) = KE AC() Ket ( A) + E AC() he te oan ounterart AC( t) = ( ta) Ke( ta) + KeAC( t) K t + eac() τ τ ta (2) he value of (t)att B an t C, to be ue later, are obtane fro eq 2. Conerng the varaton of r(t) n eq 7, eq 2 yel K tb ( tb) = ( ta) Ke( ta) + Ke( tb) + eac( τ)τ ta t ( A) K( r0 yt ( A) ) + K( r0 + r yt ( B) ) K + ( r 0 y ( t A) ) = C + K r where B (3) C : = t ( ) + K ( yt ( ) K yt ( )) B A A B + ( r 0 y( t A ) ) Note that the aroxaton n eq 3 owng to the nueral t ntegraton,.e., B ta e AC (τ) τ e(t A )= (r 0 y(t A ) ). he value (t C ) an be obtane analogouly to eq 3, K tc ( tc) = ( ta) Ke( ta) + Ke( tc) + eac( τ)τ ta t ( A) K( r0 yt ( A) ) K + K( r0 + r y( tc) ) + [ r 0 y ( t A) where tc + ( r + r y ( t) )] t= tb 0 AC Kθ = CC + Kr + r t (4) C K CC: = t ( A) + K( yt ( A) yt ( C)) + ( r0 y ( t) ) AC t= ta If the agntue of r uffent large, the altue hange of r(t) att B an ake the valve ove to a new oton. A r(t) nreae, fro eq 5, the new valve oton v( tb) = ( tb) f (5) Stage CD, DE, an EF. For ertan value of r an, the auulatve varaton of (t) fro t B to t D all o that the valve tay wth v(t B ) n eq 5,.e., v( t) = v( t ) = ( t ) f, for t [ t, t ] B B B D At the te ntant t D :=t B +, the hort-te retangular wave wth altue r an uraton aear. However, owng to the te elay θ, the roe outut y(t) uner oen-loo ontrol at a ontant nut value v(t B ) urng the te lot t [t C, t F ](t F efne n eq 6). hu, y(t) for [t C, t F ], enote a y CF (t), an be obtane analogouly to y AC (t) n eq 0, y ( t) = K ( ( t ) f ) + [ y( t ) CF B C ( t t )/ B C K ( ( t ) f )]e, t [ t, t ] C F (6) Analogouly to AC (t) neq2(t) for t [t C, t F ], enote a CF (t), CF( t) = ( tc) Ke( tc) + KeCF( t) K t + ecf() τ τ tc (7) where e () t = r() t y () t CF CF (8) For ertan value of r an, the agntue hange of r(t) at t E uffently large o that the valve ove to a new oton v( t ) = ( t ) f E E Artle 3.3. Degn of r an. On the ba of the analy of gnal n eton 3.2, th ubeton nvetgate the egn of the araeter r an Funton between r an to aheve a ere value (t E ). he valve exete to arrve at the ere teay-tate value v at the te ntant t E an tay at th 3009 x.o.org/0.02/e400308z In. Eng. Che. Re. 203, 52,

5 Inutral & Engneerng Chetry Reearh oton ever ne. Sne (t) ju own fro (t D )to (t E ), akng v(t) arrve at v requre ( t ) = v f E (9) Fro r(t) neq7, CF (t) n eq 7 an e CF (t) n eq 8, K te ( te) = ( td) Ke( td) + Ke( te) + ecf( τ)τ td ( td) K( r0 + r y( td) ) + K( r0 y( te) ) K + r ( 0 + r yt ( D) ) = t ( D) Kr + K( yt ( D) yt ( E)) K + ( + r 0 r y ( t D) ) (20) hu, t neeary to rereent y(t D ), y(t E ), an (t D )n eq 20 n ter of the egn araeter r an 0. Ung y CF (t) n eq 6 an the efnton of te ntant n eq 6, y(t D ) an y(t E ) are obtane a y( td) = K( ( tb) f ) + [ y( t ) C ( θ K ( ( t ) f )]e )/ (2) B y( te) = K( ( tb) f ) + [ y( t ) C θ K ( ( t ) f )]e ( + )/ B (22) Wth y CF (t) n eq 6, CF (t) n eq 7 an y(t D ) n eq 2, (t D ) an be rereente a K td ( td) = ( tc) Ke( tc) + Ke( td) + ecd( τ)τ tc = t ( C) + K( yt ( C) yt ( D)) K td + ( r0 + r y ( τ) )τ CF tc K = t ( C) + ( + θ r 0 r )( ) + K { y( t ) K ( ( t ) f ) C B ( θ)/ [ yt ( C) K( t ( B) f)]e } K td ( K( ( tb) f ) tc ( τ tc)/ + [ yt ( C) K( t ( B) f)]e )τ K = t ( C) + [ + r 0 r K ( ( t B) f )] ( θ) + K ( θ)/ [ yt ( ) K( t ( ) f)]( e ) C B (23) Subttutng y(t D ) n eq 2, y(t E ) n eq 22, an (t D )neq23 nto eq 20 yel K ( te) ( tc) + [ + θ r 0 r K ( ( t B) f )]( ) where ( θ)/ + K [ yt ( ) K( t ( ) f)]( e C B ) ( θ)/ Kr + K[ yt ( ) K( t ( ) f )](e C B ( θ+ K e )/ ) + [ + r 0 r K ( ( t B) f ) ] K ( θ)/ [( yt) K( t ( ) f)]e C B K = t ( C) + ( θ + )( r 0 y ( t C)) K + ( θ + + ) r F ( )[ y ( t C) K ( ( t B) f )] K θ F ( ): = [ + + θ + ( e / ( )/ )e ] (24) Ung v n eq 4, (t B ) n eq 3 an (t C ) n eq 4, 9, an 24 lea to an aroxate equalty r0 θ + + f C K r + θ + K K r K ( ) r C + F ( )[ yt ( ) K( C + Kr f)] C B (25) A take the value a an nteger ultle of, eq 25 rewrtten a r 0 r f C F( )[ y( t ) K ( C f )] C C B K K ( θ + )( r 0 y ( t C)) K + ( ) / F( ) KK (26) In wor, gven a ef value of, the value of r an be eterne fro the aroxate funton eq 26, n orer to ake v(t E ) arrve at v. Note that the ter C B an C C are rrelevant to r Conton o be Satfe. Bee eq 26, the araeter r an have to atfy ertan onton to ake the valve ove n the ae way a erbe n the above tage. Stage AB. Reall that t (efne at the begnnng of eton 3.2) the latet te ntant that y(t) hange the reton fro nreent to ereent, before ang the hort-te retangular wave, an (t θ) thevalueof(t) toake the latet oveent of the valve. Aorng to the flowhart of the tton oel n Fgure 2, the valve oveent at t B requre an f + ( t ) ( t θ) < f A (27) f + ( t ) ( t θ) + ( t ) ( t ) > f A B A (28) Equaton 27 onfne the altue of (t A ),.e., ( ta) < f + f + ( t θ) (29) Subttutng (t B )neq3ntoeq28yel f ( t θ) + C + K r f B Artle 300 x.o.org/0.02/e400308z In. Eng. Che. Re. 203, 52,

6 Inutral & Engneerng Chetry Reearh Artle o that r ha a lower boun f + f + ( t θ) C B r K (30) Stage BC. For the tage BC, the valve oe not ove o that the nreent of (t) nee to atfy the nequalty f + t ( ) t ( ) < f C B (3) Subttutng (t B ) n eq 3 an (t C ) n eq 4 nto the nequalty eq 3 gve Kθ f + C + K r + + r ( C K r) f C B whh oe an uer boun to r r f f C + C C B Kθ K (32) Stage CD. Slar to the tage BC, the valve oe not ove at the tage CD o that another nequalty ha to be atfe f + t ( ) t ( ) + t ( ) t ( ) f C B D C (33) On the ba of (t B ) n eq 3 an (t D ) n eq 23, the nequalty eq 33 beoe θ f + C C K r K [ r r C B 0 K( CB + Kr f )]( θ) + K ( θ)/ [ yt ( ) K( C + Kr f)]( e ) < f C B whh gve an uer boun of r beng eenent on ( ) ( ) ( θ)/ f f C + C [ r K ( C f )]( θ) K [ y( t ) K ( C f )]( e ) C B 0 B C B r K θ K 2 ( θ)/ + ( KK)( θ) K K ( e ) (34) In rate, (t) uually onfne to be le than an uer lt value, e.g., (t) 00. Sne (t D ) n eq 23 the larget value of (t) urng the oenaton, the next nequalty hol for an uer lt H of (t) K H t ( D) = t ( C) + [ + θ r 0 r K ( ( t B) f )]( ) ( θ)/ + K [ yt ( ) K( t ( ) f)]( e C B ) h lea to another uer boun of r, by ung (t B )n eq 3 an (t C ) n eq 4, r C K K H C 0 C C B C ( r yt ( )) [ yt ( ) K( C f)][ θ + ( )( e )] K ( θ)/ K + 2 [( θ) + ( )( e )] K K ( θ)/ (35) Stage DE. In the tage DE, (t) ereae wth a gnfant agntue o that the valve ove to a new oton. h requre an nequalty, f + t ( ) t ( ) + t ( ) t ( ) C B D C + t ( ) t ( ) < f E D whh beoe, bae on (t B ) n eq 3 an (t E ) n eq 24, θ f + C C + + θ + K r K ( )( r y ( t )) C B 0 C K + ( θ + + ) r F ( )[ y ( t C) K ( C + K r f )] f B hu, another lower boun of r obtane r f + f + C C + ( θ + )( r yt ( )) + F ( )[ yt ( ) K( C f)] K C B 0 C C B K F( ) KK ( + ) (36) Stage EF. In the tage EF, (t F ) aller than (t E ), an the valve oe not ove o that f + ( t ) ( t ) > f F E (37) Wth CF (t) n eq 7, K tf ( tf) ( te) = Ke( te) + Ke( tf) + ecf( τ) τ te K tf = K( y( te) y( tf)) + ( r0 y ( τ)) τ t CF Ung y CF (t) n eq 6 a well a t E an t F n eq 6, E (38) 30 x.o.org/0.02/e400308z In. Eng. Che. Re. 203, 52,

7 Inutral & Engneerng Chetry Reearh ( )/ + y( t) = K( t ( ) f) + [ yt ( ) K( t ( ) f)]e t F B C B (39) an the ntegraton n eq 38 beoe K tf ( r0 y ( τ)) τ CF te θ = K r K t f K [ 0 ( ( B) )] ( θ+ )/ ( + )/ [( yt) K( t ( f)](e e ) C B hu, wth y(t E ) n eq 22 an y(t F ) n eq 39, eq 38 rewrtten a ( tf) ( te) = K[ y( tc) K( ( tb) f )] ( θ+ + K θ )/ ( )/ (e e ) + K [ r0 K( ( tb) f )] ( θ+ )/ ( + )/ [ yt ( C) K( t ( B) f)](e e ) Kθ = ( + r 0 y ( t C)) F 2( ) [ yt ( ) K( t ( ) f)] where C B K θ+ + θ F( ) = ( + ( )/ ( )/ 2 )(e e ) K Ung (t B ) n eq 3, eq 37 yel an uer boun of r beng eenent on : Kθ r f f + ( + r y ( t )) F ( ) 0 C 2 [( ytc) K( CB f)]/[ F( KK ) ] 2 (40) 3.4. Seleton of t A. Bee the araeter r an, the te ntant t A another egn araeter to be elete. he eleton of t A eterne the value of (t A ) an y (t A ) o that the eleton affet eq 26, a well a the four uer boun of r n eq 32, 34, 35 an 40, an the two lower boun of r n eq 30 an 36. o aheve a goo oenaton erforane, t referre that y (t) equal to the ere teay-tate value r 0, rght after the aearane of the hort-te retangular wave,.e., y( t F ) = r 0 (4) Ung (t B ) n eq 3 an y(t F ) n eq 39, eq 4 beoe r r = K ( ( t ) f ) 0 0 B + [ yt ( ) K( t ( ) f)]e C B K ( C + K r f ) + [( yt) K( C + Kr f)]e = KKr( e ) + yt ( ) ( + )/ B ( + )/ C B ( + )/ C ( + )/ C B [ yt ( ) K( C f)]( e ) whh lea to another funton between r an r ( + )/ r0 yt ( C) + [ yt ( C) K( CB f)]( e ) ( + )/ KK( e ) Artle (42) Hene, t erable to elet t A an uh that r fro eq 26 an that fro eq 42 are loe to eah other Coenaton Ste. he loe-loo oenaton etho ont of the followng te. he flowhart of thee te gven n Fgure 6. Fgure 6. Flowhart for the onlne leentaton of the rooe oenaton etho. S. Etate the araeter K,, θ,, f, an f bae on the ollatory ata ale {y (t), (t)} N t=. S2. Loate the te ntant t n an ollaton ero n the at at whh y (t) hange the reton fro nreent to ereent, an fn the orreonng value (t θ) fro the urrent avalable ollatory ata of (t) an y (t). S3. For eah anate of t A = t + l for l =,2,, alulate r fro eq 26 an that fro eq 42 for a equene of wth = θ + k for k =0,,2,. he reultng two grou of et are enote a {t A,, r } (26) an {t A,, r } (42), reetvely. Fn the val et atfyng the uer boun of (t A ) n eq 29, the two lower boun of r n eq 30 an 36, an the four uer boun of r n eq 32, 34, 35, an eq 40. Aong the val et of {t A,, r } (26), elet the et whoe r ha the nal tane wth that fro {t A,, r } (42) a the otal hoe an alulate the ulate outut y(t A ) va eq 49 (aear later n eton 3.6). S4. Set t A a the te ntant that the two nequalte n eq 43 are atfe 302 x.o.org/0.02/e400308z In. Eng. Che. Re. 203, 52,

8 Inutral & Engneerng Chetry Reearh y ( t ) > y ( t ) an y ( t) < y ( t ) A A (43) an nreae the value of the referene r(t) fro r 0 to r 0 + r at the te ntant t B := t A +. S5. Dereae the value of r(t) bak to the orgnal value r 0 at the te ntant t E := t B + + an ntroue a eaban wth the level ε to the ontroller at t E to gnore ontrol ontrol error havng all abolute value. hat, the nut to the ontroller C() beoe the outut e ε (t) of the eaban et (), et () ε eε() t = 0, et ( ) < ε (44) he te S S3 eterne the egn araeter {t A,, r } bae on the at ollatory ata ale of y (t) an (t). Note that the te ntant t an be elete a the eak value of y (t) n any ollaton ero n the at, o that the value of (t A ) an y (t A ) are avalable for the egn. One t A eterne, the ulate outut y(t A ) obtane, an the te ntant that the two nequalte n eq 43 are atfe et to be the atual te ntant t A for the onlne leentaton te S4 an S5. hu, the fnal egn araeter for the rooe oenaton etho are {y(t A ), r, } Ileentaton Iue. h ubeton ue the leentaton ue for the rooe oenaton etho, naely, the etaton of requre araeter an the eternaton of the eaban level ε. Frt, the loe-loo oenaton etho requre the nforaton of the roe araeter K,, an θ n eq 2, the ontant araeter (ee Fgure ), the ontroller araeter K an n eq 3, an the araeter f an f of the tton oel n Fgure 2. Wth the exeton of the ontroller araeter K an, the other araeter ay not be avalable a ror. here ext a few aroahe n the lterature for entfaton of the requre roe an tton oel. Srnvaan et al. 9 ale the ea of earable leat-quare etho to tton oel araetrze by one araeter for ontrol valve. he ae ea wa generalze by Jelal, 20 Chouhury et al., 2 Karra an Kar, 22 an Farenzena an rerweler 23 to entfy Haerten yte ontng of a two-araeter tton oel onnete wth a lnear roe oel. In th ontext, the araeter K,, an θ n eq 2, the ontant araeter an the araeter f an f of the tton oel are etate bae on the eaureent of (t) an y (t) va a two-te aroah. he frt te take the ea of the earable leat-quare etho, analogou to the two-tage etho rooe by Jelal, 20 to etate the araeter f an f of the tton oel, an a rete-te autoregreon wth extra nut (ARX) oel for the lnear roe,.e., na nb j = j= y () t = ay ( t ) + bv( t j n ) + + et () k (45) It a well-known fat that the etaton of ARX oel only requre the tanar lnear leat-quare etho, an a hgh-orer ARX oel aable of aroxatng any LI yna yte arbtrarly well 24 (age 336). A hgh-orer ARX oel an yel very aurate oel etaton even f the eaureent noe heavly olore. 25,26 Wth the etate fŝ an f, the etate of v(t) an be obtane fro the He tton oel ete n Fgure 2 rven by (t),.e., v( t; f, f ) = f( ( t); f, f ) Relang v(t) byv (t; fŝ, f), eq 45 beoe y () t = φ (; t f, f ) γ + e() t where y ( t ) y ( t n) a φ(; t f, f ) = vt ( n; f, f ), k vt ( n n; f, f b k ) a an a γ = b b n b Gven the ata ale {y (t), (t)} N t=, the araeter vetor γ an be etate va the leat-quare etho a γ = Φ Φ ( f, f ) ( ( f, f ) ( f, f )) ( Φ ( f, f ) Y ) (46) where Y y () φ (; f, f ) =, Φ( f, f ) = y ( N) φ ( N; f, f ) he oel truture araeter (n a, n b, n k ) an the otal etate of f an f are obtane a the one ahevng the nu value of a lo funton aoate wth the ulaton error,.e., o o o ( n, n, n, f, f ) a b k o o N = arg n ( y ( t) φ ( t; f, f ) γ( f, f )) n, n, n, f, f N a b k t= 2 (47) he otzaton n eq 47 an be olve, e.g., by the global otzaton va a gr earh over fve enonal ae of n a, n b, n k, fŝ, an f. Wth fŝo an f o n eq 47, the otal etate of γ obtane a y(fŝo, f o ) fro eq 46. he eon te to etate the araeter K,, an θ of the ontnuou-te oel G() n eq 2 an the ontant araeter bae on the entfe rete-te ARX oel enote a G(q). If the te reone of G(q) reveal whether a FOPD oel an erbe the roe well, then the etate of G() an be obtane by alyng the reton-error etho (PEM) 27 to the te reone of G(q) (ee, e.g., the Matlab funton e ). Note that the elay araeter θ nee to be etate a an nteger ultle of the alng ero. he ontant araeter etate bae on the araeter a î an n y(fŝo, f o )a = na + o a = Artle 303 x.o.org/0.02/e400308z In. Eng. Che. Re. 203, 52,

9 Inutral & Engneerng Chetry Reearh he qualty of the etate araeter an be quanttatvely eaure by the ftne between the eaure outut y (t) an the ulate one y(t), = y () t y () t 2 ftne 00 y () t E{ y ()} t 2 (48) where, 2 the Eulean nor an y(t) obtane a y () t = θ Ke f t f f + ( ( );, ) (49) Seon, t unavoable that the etate araeter are aoate wth oelng error, an the roe noe/ turbane w(t) reent. A a reult, the elete et of {r, } fro eq 26 an 42 woul be evate fro the atual one, o are the lower boun of r n eq 30 an 36 an the uer boun of r n eq 32, 34, 35 an 40. Even o, there ay tll be oe ae fro the elete value r to the atual lower an uer boun. hu, the two oveent of v(t) oul be reahe a egne, ubjet to oelng error an/or roe noe. Exerental an nueral llutraton are rove later n exale an 2 n eton 4. Wth the oelng error an roe noe, the an onern that v(t) after oenaton ay tay at a oton not exatly equal to v, leang to nonzero ontrol error between r(t) an y (t). In th ae, the ntegral aton of the PI ontroller woul rve the valve to ove agan, o that the loe-ontrol loo goe bak to ollaton. h roble an be olve by ung the eaban n eq 44 to freeze the outut of the PI ontroller C() when the abolute value of the ontrol error all. he eaban uually avalable n ot nutral trbute ontrol yte (DCS) an ha been oonly ue n rate, e.g., the fourth uggeton ae by Gerry an Ruel. 5 he eaban level ε n eq 44 hoen to be roortonal to the ale tanar evaton of the etate roe noe w(t) :=y (t) y(t),.e., ε = 3 N N N wt () wt () N t= t= 2 (50) 4. EXAMPLES h eton rove laboratory an ulaton exale to llutrate the rooe loe-loo oenaton etho an to oare wth two extng oenaton etho. Exale. A laboratory exerent arre out at Pekng Unverty, wth the exerent onfguraton heatally ete n Fgure 7. In the exerent, the lnear roe a water tank yte, whoe ro-etonal area about he oenng oton of the outlet valve fxe. he water level of the tank yte ontrolle by ajutng the nlet flow va a ontrol valve rven by a roortonal ntegral (PI) ontroller, C ( ) = (5) he rete-te ounterart of the PI ontroller C() leente wth the alng ero 0.5 n a DCS latfor of Seen PCS7. hu, the loe-ontrol loo ete n Fgure forulate, where y (t), (t), an r(t) tan for the water level of the tank yte, the ontroller outut, an the referene, reetvely. he tton nreae by tghtenng the valve te akng rew. A a reult, y (t) an (t) hown n Fgure 8 are ollatory ue to the ontrol valve tton, even f the referene r(t) ket at a ontant value r 0 = 35 an no external turbane other than eaureent noe reent a w(t) n Fgure. Note that the valve oton v(t) n Fgure 8 avalable n the exerent, but t ue only for valaton, not for etatng the araeter of roe an tton oel. [he ata an entfe oel for the exerent are avalable for aae tue: htt:// n/robot/teaher/wangjanong/reearh.ht.] 7200 Frt, bae on the ollatory ata ale {y (t), (t)} t=600 (ne the retangular box n Fgure 8 an rereente n Fgure 2), the two-te aroah n eton 3.6 exlote to yel an ARX oel G(q) wth the truture araeter n o â = 5, nbo = 5, an nko =. Fro the te reone of G(q) hown n Fgure 9, the roe oel entfe, G () = e 2.5 Artle Fgure 7. Dagra of the feebak ontrol loo for a water tank yte. Fgure 8. Sgnal n exale : (a) y (t) (ol) an r(t) (ah), (b) (t) (ol), an v(t) (ah). (52) A hown n Fgure 9, the te reone of G() ft well wth that of G(q), exet for a few ont at the begnnng tage of 304 x.o.org/0.02/e400308z In. Eng. Che. Re. 203, 52,

10 Inutral & Engneerng Chetry Reearh Artle Fgure 9. Ste reone of the entfe oel G(q) (ol) an G() (ah) n exale : (a) the entre te reone an (b) the enlarge art for the begnnng tage of the te reone. Fgure. Meaure valve oton v(t) (ol) an t etate v(t) (ah) n exale (ol). the te reone. he ontant araeter (ee Fgure ) an the araeter f an f for the tton oel are etate = 92.34, f = 8.4, f = (53) Here the valve oton v(t) avalable o that the etate araeter fŝ an f n eq 53 an be valate fro a valve gnature obtane by lottng (t) v v(t) n Fgure 0, where Fgure 2. Ollatory ata ale for oel entfaton n exale : (a) y (t) (ol) an y(t) (ah), (b) (t) (ol). Fgure 0. Valve gnature (t) v v(t) n exale. the horzontal an vertal lne have the length loe to (fŝ + f) an (fŝ f), reetvely (ee alo Fgure 3). Fgure alo oare the eaure valve oton v(t) an t etate v(t) fro the entfe He tton oel ung fŝ an f n eq 53. he He tton oel a loe aroxaton of the atual valve harater, ete oe nor reany between v(t) an v(t) n Fgure. In rate, the eaureent of v(t) uually unavalable, an the valaton of the entfe oel an rely on the oaron between the eaure outut y (t) an the ulate outut y(t)obtanevaeq49.ahownnfgure2a, y(t) an well ature the an yna varaton of y (t), wth the ftne n eq 48 equal to %. Hene, the etate araeter of the roe an tton oel are atfatory. Seon, the egn araeter for the rooe oenaton etho are eterne. he lat eak value of the ata egent y (t) hown n Fgure 2 elete a the ale y (t ),.e., t = Ung the etate of θ equal to 2.5 n eq 52, (t θ) = obtane fro the avalable ale of (t). he otal hoe of t A lea to (t A )= an y (t A ) = , an the egn araeter for onlne leentaton of the rooe oenaton etho, y ( t ) = 34, = 6.5, r = A (54) Fgure 3 reent the otal hoe of {r, }, enote by the ybol *, a well a the aoate two funton n eq 26 an 42, two lower boun of r n eq 30 an 36, an two uer boun of r n eq 34 an 35. Note that the other two uer boun of r n eq 32 an eq 40 are far away fro the otal hoe an thu are not hown n Fgure 3. he eaban level ε alulate fro eq 50, ε = x.o.org/0.02/e400308z In. Eng. Che. Re. 203, 52,

11 Inutral & Engneerng Chetry Reearh Fgure 3. Degn for exale. he funton n eq 26 (rle ah lne) an eq 42 (quare ah lne), the lower boun of r n eq 30 (ol lne) an eq 36 (ah lne at the botto), an the uer boun of r n eq 34 (ah lne at the rght-uer orner) an eq 35 (otte lne). Next, the rooe oenaton etho leente n an onlne anner, a ete n the flowhart n Fgure 6. hat, when the two nequalte n eq 43 are atfe, then the hort-te retangular wave ae to the referene r(t) at the next ale. Fgure 8 how the gnal before an after the oenaton. Fgure 4 reent an enlarge von of gnal Fgure 4. Enlarge von of gnal at the tage of ang the hortte retangular wave to r(t) n the laboratory exale : (a) y (t) (ol) an r(t) (ah), (b) (t) (ol) an v(t) (ah). at the tage of ang the hort-te retangular wave to r(t). A exete, the valve oton tay at a ere oton o that the water level y (t) loe to the ere referene value r 0 = 35, an the ollaton aear after oenaton. Fnally, f the referene r(t) exerene a varaton or a loa turbane (a w(t) n Fgure ) beng atve to the roe outut y (t) reent, then the loe-loo loo after oenaton lke that n Fgure 8 ay go bak to ollaton. In th ae, the rooe oenaton etho an be ale agan to reove the ollaton. A an llutraton, another nlet flow ontrolle by a frequeny onverter njete to the water tank yte to nue a te loa turbane w(t). Fgure 5 reent the te hange of the oan to the frequeny onverter an the varaton of the orreonng nlet flow. A a reult, the loa turbane lea to the reaearane of ollaton, a hown n Fgure 5. In orer to reove the reaeare ollaton, the egn araeter for the rooe oenaton etho an be eterne analogouly to thoe n eq 54. By aung that the roe oel n eq 52 an the araeter f an f n eq 53 are ket the ae, the ontant araeter nee to be uate by oarng the eaure roe outut an the ulate one y(t) n eq 49, bae on 0584 the ollatory ata ale {y (t), (t)} t=799 (ne the retangular box n 5), naely, = he otal egn araeter are eterne a y ( t ) = , = 8.5, r = A Artle (55) A exete, ung thee egn araeter, the onlne leentaton of the rooe oenaton etho reove the ollaton a hown n Fgure 5. Here the ae eaban level ε = 3.2 ue. In exale, even though both the oelng error an eaureent noe are reent a reveale n Fgure 2, the rooe oenaton etho tll aheve atfatory reult. hat, the orgnal ollaton aear an the roe outut loe to the ere referene value after oenaton. A ulaton exale rove next, n orer to further exlan the ratonale tate n eton 3.6 for the rooe etho havng a ertan level of robutne agant oelng error an eaureent noe. Exale 2. A ulaton exerent erfore for the loe-ontrol loo ete n Fgure, where the PI ontroller C(), the roe oel G(), the ontant araeter, an the araeter f, f for the tton oel are the ae a thoe n eq 5, 52 an 53 n exale, reetvely. In other wor, the etate araeter of the roe an tton oel are regare a the atual one n th ulaton exale. he eaureent noe w(t) abent. he ret onfguraton the ae a exale. A oelng error n ter of the te elay θ onere. hat, a bae value of the te elay θ = 5, ntea of the atual one θ = 2.5 n eq 52, ue for the egn of the rooe oenaton etho. A a reult, the egne araeter beoe y ( t ) = 34, = 5, r = A (56) Fgure 6 the ounterart of Fgure 3. Note that the axe of Fgure 6 are et to be the ae a thoe of Fgure 3 for oaron uroe, an the nu value of tart fro the te elay θ = 5 n Fgure 6. he otal hoe of {r, } n eq 56, enote by the ybol * n Fgure 6, fferent fro that n eq 54 (the ybol * n Fgure 3). However, there qute a large ae fro the otal hoe of r n Fgure 6 to the atual lower an uer boun n Fgure 3. Hene, the two oveent of v(t) an be reahe a egne, whh valate by the oenaton reult hown n Fgure 7. Due to the oelng error n θ, the valve oton n Fgure 7 after oenaton tay at v(t) = , beng fferent fro the ere oton v =(r 0 )/K = hu, the roe outut after oenaton ha a all evaton fro the ere referene value r 0. he eaban wth ε =3.2 (the ae a exale ) tolerate uh a all evaton an freeze the outut of the PI ontroller, a hown n Fgure 7. he next two ulaton exale oare the rooe loe-loo oenaton etho wth the etho rooe 306 x.o.org/0.02/e400308z In. Eng. Che. Re. 203, 52,

12 Inutral & Engneerng Chetry Reearh Artle Fgure 5. Sgnal n the laboratory exale wth a loa turbane: (a) y (t) (ol) an r(t) (ah), (b) (t) (ol) an v(t) (ah), () the oan to the frequeny onverter (ol) an the varaton of the orreonng nlet flow (ah). Fgure 6. Degn for exale 2 wth a oelng error n the te elay: eq 26 (rle ah lne) an eq 42 (quare ah lne), the lower boun of r n eq 30 (ol lne) an 36 (ah lne at the botto), an the uer boun of r n eq 34 (ah lne at the rghtuer orner) an eq 35 (otte lne). Fgure 7. Sgnal n exale 2: (a) y (t) (ol) an r(t) (ah), (b) (t) (ol) an v(t) (ah). by Cuaro et al. 3 an that by Mohaa an Huang, 4 reetvely. Analogouly to exale 2, ulaton exerent n exale 3 an 4 are erfore for the loe-ontrol loo ete n Fgure by takng the etate araeter of the roe an tton oel a the atual one. o eae the oaron, the eaureent noe w(t) abent. he ret of the onfguraton, nlung the PI ontroller, the ae a exale. Exale 3. he oenaton etho rooe by Cuaro et al. 3 bae on the knoker etho, 7 whh a a oenaton gnal k (t) totheontrolleroutut(t), naely, agn( t ( ) t ( )), t t + hk + τ k() t = 0, t > t + hk + τ By followng the guelne gven n ref 9, the three araeter are elete a a = 0.5( f + f ) = τ = 2 =, h = 5 = 2.5 k he oenaton etho 3 able the PI ontroller f the varaton of a fltere ontrol error e f (t) all,.e., e f (t)/t < δ for a all value δ = 0.2, urng the te ero of 4h k.athenoe w(t) abent, the ontrol error e(t) uenteaofe f (t). For larty, the ehan of ablng the PI ontroller et to be atve after the knoker ha been enable for a whle. he oenaton reult gven n Fgure 8, wth oe enlarge art n Fgure 9. It reveale fro Fgure 9 that the knoker etho 307 x.o.org/0.02/e400308z In. Eng. Che. Re. 203, 52,

13 Inutral & Engneerng Chetry Reearh Artle Fgure 8. Sgnal n exale 3: (a) y (t) (ol) an r(t) (ah), (b) (t) (ol) an v(t) (ah). Fgure 20. Sgnal n exale 4: (a) y (t) (ol) an r(t) (ah), (b) (t) (ol) an v(t) (ah). Fgure 9. Enlarge von of gnal for exale 3: (a) y (t) (ol) an r(t) (ah), (b) (t) (ol) an v(t) (ah). an reue the altue of ollaton gnfantly an ake the ontrol error to be all, by fat wthng aong everal ontrol oton. When the ehan of ablng the PI ontroller et to be atve, the valve tay a oton that however not guarantee to be loe to the ere valve oton. In Fgure 9, the valve tay at a oton qute far away fro the ere one, reultng n a large ontrol error. By ontrat, a hown n Fgure 8, the rooe oenaton etho oe not reult n a arfe on the fat varaton of valve oton, an the ontrol error uh aller after oenaton. Exale 4. For the FOPD oel n eq 52, the oenaton etho rooe by Mohaa an Huang 4 ay that reung the ntegral aton an reove the reene of ollaton. hu, the araeter of C()neq5etto = 2000 after the aearane of ollaton, wth the araeter K = 0.25 unhange. he valve oton n Fgure 20 after oenaton qute fferent fro the ere oton. A a reult, the ontrol error large even f the ollaton aear after oenaton. By ontrat, the rooe oenaton etho ove the valve loe to the ere oton, o that the ontrol error uh aller, a hown n Fgure CONCLUSION h aer rooe a loe-loo oenaton etho to reove the ollaton aue by ontrol valve tton. he rooe etho ae a hort-te retangular wave to the referene to enable the ontrol valve to arrve at the ere oton wth two oveent. he araeter of the hortte retangular wave were egne n a yteat anner. he te of the rooe loe-loo oenaton etho were gven at the flowhart n Fgure 6. Soe laboratory an ulaton exale llutrate the effetvene of the rooe etho. he rooe etho wa reente for the FOPD roe oel n eq 2 an the PI ontroller n eq 3. A known fro the egn roeure, t feable to generalze the rooe etho for ore general roe oel an ontroller. Bee the tton oel 7 aote here, there are other ata-rven tton oel avalable n lterature. he egn rnle of the rooe etho an be followe to egn new oenaton etho for thee tton oel. A a future work, a theoretal analy nee to be erfore to tuy the effet of the unertante of the etate araeter on the oenaton etho. he ffulty for uh a theoretal analy le at the olex relatonh between the egn araeter an etate araeter n eq 26 an 42, the lower boun of r n eq 30 an 36, an the uer boun of r n eq 32, 34, 35 an 40. Another future work to ntegrate the rooe oenaton etho wth an onlne ollaton etho uh a the one n our reent tuy 28 to etet the reaearane of ollaton an to ntate the oenaton te n an autoat anner. hat, one the ollaton etete onlne, the requre araeter of the roe an tton oel an be re-etate f neeary, an the rooe oenaton etho wth the uate egn araeter an be leente agan to reove the ollaton. AUHOR INFORMAION Correonng Author *o who orreonene houl be aree E-al: janong@ku.eu.n. Phone: +86(0) Fax: +86(0) x.o.org/0.02/e400308z In. Eng. Che. Re. 203, 52,

14 Inutral & Engneerng Chetry Reearh Note he author elare no oetng fnanal nteret. ACKNOWLEDGMENS he author thank the Natonal Natural Sene Founaton of Chna for fnanal grant No an No REFERENCES () Balkowk, W. Drea veru realty: a vew fro both e of the ga. Pul Pa. Can. 993, 94, (2) Deborough, L.; Norh, P.; Mller, R. Control yte relablty: roe out of ontrol. In. Cout. 200, 8, (3) Paulon, M.; Cox, J. A ratal aroah for large-ale ontroller erforane aeent, agno, an roveentl. J. Proe Control 2003, 3, (4) Jelal, M.; Huang, B. Deteton an Dagno of Stton n Control Loo: State of the Art an Avane Metho; Srnger Verlag, 200. (5) Gerry, J.; Ruel, M. How to eaure an obat valve tton onlne. 20 ISA ehnal Conferene, Houton, X, Seteber, 200. (6) Chouhury, M.; Shah, S.; hornhll, N. Moelng valve tton. Control Eng. Prat. 2005, 3, (7) Ha gglun,. A frton oenaton for neuat ontrol valve. J. Proe Control 2002, 2, (8) Ha gglun,. Autoat on-lne etaton of baklah n ontrol loo. J. Proe Control 2007, 7, (9) Srnvaan, R.; Rengaway, R. Stton oenaton n roe ontrol loo: A fraework for ntegratng tton eaure an oenaton. In. Eng. Che. Re. 2005, 44, (0) Ivan, L.; Lakhnarayanan, S. A new unfe aroah to valve tton quantfaton an oenaton. In. Eng. Che. Re. 2009, 48, () Srnvaan, R.; Rengaway, R. Aroahe for effent tton oenaton n roe ontrol valve. Cout. Che. Eng. 2008, 32, (2) Cuaro, M.; Munaro, C.; Munareto, S. Irove tton oenaton n neuat ontrol valve. Cout. Che. Eng. 202, 38, (3) Cuaro, M.; Munaro, C.; Munareto, S. Novel oel-free aroah for tton oenaton n ontrol valve. In. Eng. Che. Re. 202, 5, (4) Mohaa, M.; Huang, B. Coenaton of ontrol valve tton through ontroller tunng. J. Proe Control 202, 22, (5) Artrong-Heĺouvry, B.; Duont, P.; e Wt, C. A urvey of oel, analy tool an oenaton etho for the ontrol of ehne wth frton. J. Proe Control 994, 30, (6) Kayhan, A.; Doyle, F. Frton oenaton for a roe ontrol valve. Control Eng. Prat. 2000, 8, (7) He, Q.; Wang, J.; Pottann, M.; Qn, S. A urve fttng etho for etetng valve tton n ollatng ontrol loo. In. Eng. Che. Re. 2007, 46, (8) Seborg, D.; Egar,.; Mellha, D. Proe Dyna an Control, 2n e.; John Wley & Son: Hoboken, NJ, (9) Srnvaan, R.; Rengaway, R.; Narahan, S.; Mller, R. Control loo erforane aeent. 2. Haerten oel aroah for tton agno. In. Eng. Che. Re. 2004, 44, (20) Jelal, M. Etaton of valve tton n ontrol loo ung earable leat quare an global earh algorth. J. Proe Control 2008, 8, (2) Chouhury, M.; Jan, M.; Shah, S. Stton efnton, oelng, eteton an quantfaton. J. Proe Control 2008, 8, (22) Karra, S.; Kar, M. Corehenve ethoology for eteton an agno of ollatory ontrol loo. Control Eng. Prat. 2009, 7, (23) Farenzena, M.; rerweler, J. Valve tton etaton ung global otzaton. Control Eng. Prat. 202, 20, Artle (24) Ljung, L. Syte Ientfaton: heory for the Uer, 2n e.; Prente Hal: Englewoo Clff, N.J., 999. (25) Zhu, Y. Multvarable Syte Ientfaton for Proe Control; Elever Sene: Oxfor, 200. (26) jarntro, F.; Ljung, L. Varane roerte of a two-te ARX etaton roeure. Eur. J. Control 2003, 9, (27) Ljung, L.; Wll, A. Iue n alng an etatng ontnuou-te oel wth tohat turbane. Autoata 200, 46, (28) Wang, J.; Huang, B.; Lu, S. Irove DC-bae etho for onlne eteton of ollaton n unvarate te ere. Control Eng. Prat. 203, 2, x.o.org/0.02/e400308z In. Eng. Che. Re. 203, 52,