Frequency Domain Design of a Complex Controller under Constraints on Robustness and Sensitivity to Measurement Noise

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1 40 ELECTRONICS VOL 5 NO JUNE 0 Frqucy Domai Dig of a Comlx Cotrollr udr Cotrait o Robut ad Sitivity to Maurmt Noi Tomilav B Šara Miloš B Trifuović ad Vida ovdarica Abtract Nw gral rul hav b dvlod for digig comlx cotrollr udr cotrait o robut ad itivity to maurmt oi Th dig i bad o a comromi btw robut ad rformac Thi olutio ma oibl obtaiig ractically ralizabl comlx cotrollr It i how that th rood mthod rult ito coidrably bttr rformac ad robut idic comard to tho obtaid by th otimal PID cotrollr Idx Trm Comlx cotrollr PID cotrollr Robut Frqucy domai T I INTRODUCTION HE ar rt gral rul for digig a comlx cotrollr C() itdd for th roc ivolvig tim dlay ad havig a arbitrary ordr ad multil ataticim Th roblm of cotrol of comlx roc (multil itabiliti multil ataticim domiat tim dlay) caot b olvd adquatly by alyig PID cotrollr which i th mai rao for dvloig th mthod for dig of comlx cotrollr It i wll ow that about 94% of fdbac i idutry ar ralizd by PI/PID cotrollr [] whil i trochmical idutry thi rctag i 97% [3] Owig to a high igificac of PI/PID vry fficit ad iml rocdur for tuig aramtr of idutrial cotrollr hav b dvlod [456] a wll a otimizatio rocdur [7-] for digig PI/PID cotrollr o th IAE (Itgral of Abolut Error) i miimizd udr cotrait o robut which atifi th critrio dfid i [3] I additio to th mtiod mthod thr ar mthod for dig of PID cotrollr which ar bad o th IMC (Itral Modl Cotrol) cotrollr [4-6] Th IMC mthod of cotrollr dig cotai o adutabl aramtr λ which for a arrow cla of roc ha dirct ifluc o th tim cotat of th clod loo ytm Ro of a roc rgulatd by alyig a IMC mthod to a Haviid-ty diturbac i ddt o th domiat dyamic of th roc Eg if a roc i domiatd by ocillatory dyamic ro to ay diturbac will b ocillatory For th uro of accomlihig adquat idic of robut ad rformac for a widr cla of tabl ad utabl roc w mthod of digig comlx cotrollr bad o th modifid IMC tructur [7-9] hav b dvlod Howvr th dig rul for comlx cotrollr by alyig th mthod hav ot b digd for th gral form of trafr fuctio of th roc () but oly for crtai cla of roc () [7-9] Comlx cotrollr C() dfid i thi wor i digd for th gral form of trafr fuctio of th roc giv i a ratioal form icludig dlay ()H()x(-τ)/Q() udr cotrait o robut ad itivity to maurmt oi Th adutabl aramtr of th comlx cotrollr C() ar th tim cotat λ ad rlativ damig factor ζ of th domiat ol of th roc i th clod loo with th comlx cotrollr C() [7] By adutig aramtr ζ o ca accomlih a comromi btw th robut ad rformac idic which i ot oibl for comlx cotrollr digd by IMC [4-6] or modifid IMC [7-89] Th rood comlx cotrollr C() i comard with th PID cotrollr [7] through a ri of imulatio of a wid cla of idutrial roc It i how that th alicatio of th rood mthod rult i coidrably bttr idic of robut ad rformac comard to tho obtaid with th mthod dcribd i [7] II DESIN OF THE COMPLEX CONTROLLER FOR A PROCESS HAVIN TRANSFER FUNCTION P (S) Th cotrol tructur ivolvig comlx cotrollr C() i rtd i Fig d + + y u + ff + y + T B Šara i with th Faculty of Elctrical Egirig Uivrity of Blgrad Srbia (-mail: tomi@tfr) M B Trifuović i with th Faculty of Elctrical Egirig Uivrity of Blgrad Srbia (-mail: milohtrifuovic@gmailcom) V ovdarica i with th Faculty of Elctrical Egirig Uivrity of Eatr Saravo Boia ad Hrzgovia (-mail: vidagov@yahoocom) Fig Th cotrol tructur C

2 ELECTRONICS VOL 5 NO JUNE 0 4 I gral ( ) ( ) τ / ( ) H Q whr Q() ad H() ar olyomial of th ordr dg Q dg H m ad H(0)h 0 0 I ordr to facilitat th roc of drivig with o lo i grality it i aumd that dgh() 0 i H()h 0 Th comlmtary itivity fuctio of th cotrolld roc () of Fig i giv by rlatio T ( ) L ( ) /( + L ( )) with th fdbac fuctio of th form L C Lt th dird comlmtary itivity fuctio T() b giv by τ N T P N + P + λ () with N ad adutabl aramtr λ > 0 R which ar dtrmid o th bai of th dird rformac of th clod loo ytm From rlatio () th cotrollr C() of th roc havig trafr fuctio () rultig i maximum urio of diturbac d or i T Q N C () T h F 0 τ whr F P N I gral aramtr ar dtrmid o that th ol of roc () ar caclld by th zro of fuctio F() Lt th ol of roc () ar: v 0 v+ v+ v+ r 0 i zro of olyomial Q() of th ordr v ad r Lt th rmaiig zro of olyomial Q() v+ r+ b iml th aramtr ar dtrmid accordig to th followig rul Rul If th zro of olyomial Q() ar v 0 aramtr v ar dtrmid from coditio d F d 0 0 v (3) Rul If th zro of olyomial Q() ar v+ v+ v+ r 0 aramtr v+ r ar dtrmid from coditio d F d v+ 0 r (4) Rul 3 If th zro of olyomial Q() ar iml aramtr v+ r+ ar dtrmid from coditio v+ r+ F 0 for v r (5) v + r + Polyomial P() i uually ta i th form P ( λ (6) For th uro of achivig bttr comromi rformac/robut i thi wor a w form of olyomial P() i rood ( ) ( + ζ O() (7) P λ ζλ Rul 4 If i Rul or Rul 3 om of th zro of olyomial Q() ha a oitiv ral art (utabl roc) i cotrollr () caclig of th zro i th domiator ad zro i th umrator ha to b carrid out (limiatio of diol) Rmar If dgh() > 0 th rlatio giv by () - (7) rmai th am ad olyomial N() i () bcom N ( H / h h 0 0 N (0) 0 O th bai of (7) fr aramtr of th comlx cotrollr () ar th tim cotat λ>0 ad rlativ damig factor ζ>0 of th clod loo ytm li i [7] Th damig factor which i itroducd i th dig of comlx cotrollr lay a igificat rol i accomlihig a comromi btw th rformac ad robut idic It i how latr that through th damig factor o ca xrt ifluc uo itivity to maurmt oi at high frquci M M C(i ω) lim (8) ω + C (i ω ) (i ω ) I ordr to tri a comromi btw dird rformac IAE ad M max /(+L(iω)) tim cotat λ ω hould atify coditio max /( + C(i ω) (i ω)) M (9) ω λ For giv ζ ad M (9) tim cotat λ i dtrmid by olvig two oliar algbraic quatio li i [7] + C( iω) ( iω) / M 0 (0) ( + C(i ω) (i ω) ) 0 () ω Iitially aramtr ζ hould b ta a ζ ad aramtr λ clo to th timatd traort dlay By dtrmiig tim cotat λ for diffrt valu of aramtr ζ o accomlih a comromi btw th valu IAE M ad M A comario of th qualiti of cotrol for diffrt valu of aramtr ζ i aalyzd i dtail i th xt ctio III COMPARATIVE ANALYSIS AND SIMULATIONS A comario of th rood mthod for dig of cotrollr C() () for diffrt valu of aramtr ζ i giv i Tabl for ixt rrtativ tyical dyamic charactritic: (0 (5 ( 4

3 4 ELECTRONICS VOL 5 NO JUNE (07 (0 8 5 ( ( (38 (84 (59 4 ( (0 (7 (3 ( ( ) 6( 05 0 (3 3 ( ( ) 4 (5 )( ( ( ) Th cotrollr for all roc xct th utabl o ( 3 () 4 () i 5 ()) ar of th form () with th corrodig aramtr λ ad ζ from Tabl For utabl roc o th bai of rul 4 i ordr to limiat utabl diol of th cotrollr tim dlay -τ i olyomial F() i aroximatd by Pad aroximatio of th ordr N/N whr N i cho o that th robut ad rformac idic ar rrvd For roc 4 () ad 5 () it i ufficit to ta N τ 6τ + τ () + 6τ + τ i N3 for roc 3 () τ 0 60τ + τ τ τ + τ + τ (3) Eg o th bai of () (7) gral form of th cotrollr () for utabl roc 3 () i giv by (4 )( C 4(( λ + ζλ ( ) For λ335 ad ζ obtaid o th bai of rlatio (9)-() ad rlacig traort dlay - by aroximatio (3) for τ o obtai th cotrollr 3( + 3)( ) ( 05) ( ) C ( ) ( 0499) ( ) ad uo caclig th diol 05 th fial trafr fuctio of cotrollr from Tabl for roc 3 () i C 3( + 3)( )( ) ( + 986)( ) By uig th quivalt rocdur all othr cotrollr of utabl roc hav b dtrmid with aroximatio () alid for roc 4 () ad 5 () I ordr to rduc th ordr of th cotrollr of tabl roc obtaid by alyig rul to 3 it i dirabl to aly th dcribd rductio by cacllatio of diol For tabl roc havig domiat dlay thi rductio of TABLE I THE RESULTS OBTAINED BY THE PROPOSED METHOD FOR DIFFERENT VALUES OF PARAMETER Ζ OF CONTROLLER C(S) MSCONST Proc λ ζ M IAE M M () () () () () () () () () () () () () () () () () () () () () () () () () () () () () () () () () () () () () () () () () () () () () () () () a comlx cotrollr may lad to dgradatio of th robut ad rformac idic thu thi rductio i ot rcommdabl Such a xaml i roc 4 () From Tabl it i clar that for all roc () 6 wh alyig C() for th am M th adutabl aramtr ζ allow accomlihig a comromi btw IAE M ad M Thi aramtr i of y igificac ic by it u o ca dcra or icra valu of M ad imrov th robut ad rformac idic (Fig ad 3)

4 ELECTRONICS VOL 5 NO JUNE ( 05 (3 Fig Ro to a Haviid-ty diturbac of roc () i clod loo with cotrollr C() for M Th rood mthod for dig of th comlx cotrollr C() () will b comard to th PID cotrollr [7] which a ha b how i [7] accomlihd th robut ad rformac idic th am a th otimal PID [] Th comario of th mthod aumig th am valu of M ad M i rtd i tabl for th roc () 6 05 ( ) 4 (5 )( (05 TABLE II THE COMPARISON OF THE METHOD PROPOSED FOR COMPLEX CONTROLLER C(S) WITH THE OPTIMALLY TUNED PID CONTROLLER [7] Proccotrollr λ ζ M IAE M M ()-C ()-PIDtu ()-C ()-PIDtu ()-C ()-PIDtu ()-C ()-PIDtu ()-C ()-PIDtu ()-C ()-PIDtu ()-C ()-PIDtu ()-C ()-PIDtu ()-C ()-PIDtu ()-C ()-PIDtu ()-C ()-PIDtu ()-C ()-PIDtu ()-C ()-PIDtu ()-C ()-PIDtu ()-C ()-PIDtu ()-C ()-PIDtu Fig 3 Ro to a Haviid-ty diturbac of roc () i clod loo with cotrollr C() for M 6 I Tabl aramtr ζ for all comlx cotrollr ha b dtrmid o that ractically th am valu of M a i th ca of th PID cotrollr i obtaid It ca alo b from Tabl that th comlx cotrollr ur a coidrably bttr quality of cotrol comard to that of th PID cotrollr Th followig figur how th ro to a Haviid ty of diturbac of th comlx cotrollr C() ad PID cotrollr [7] From Fig 4-7 it i obviou that alicatio of th comlx cotrollr rult i a igificatly lowr IAE with ractically th am robut a th o obtaid with th PID cotrollr For utabl roc th comlx cotrollr giv coidrably highr idic of robut ad rformac comard to tho of th PID cotrollr It hould b mtiod that for th roc of highr ordr of itability ad ataticim comlx cotrollr ca b uccfully digd a dmotratd with roc 5 () ad 6 () 5 ( 4 3 Fig 4 Ro to a Haviid-ty diturbac of roc 4() i clod loo with cotrollr from Tabl for M M 87 ( Fig 5 Ro to a Haviid-ty diturbac of roc 9() i clod loo with cotrollr from Tabl for M M 38

5 44 ELECTRONICS VOL 5 NO JUNE C() M M6 PID M M6 ( Tim[c] Fig 6 Ro to a Haviid-ty diturbac of roc () i clod loo with cotrollr from Tabl for M M Fig 7 Ro to a Haviid-ty diturbac of roc 3() i clod loo with cotrollr from Tabl for M 4 IV THE CONCLUSION Dig of comlx cotrollr i aimd at icraig th robut ad rformac idic comard to tho obtaiabl with covtioal cotrollr For digig comlx cotrollr a adquat owldg of trafr fuctio of th roc i rquird Th ar rt gral rul for digig comlx cotrollr which hav b ttd o a wid cla of roc By alyig uitabl aroximatio of comlx cotrollr adquat covtioal cotrollr ar obtaid for crtai cla of roc Th comarativ aalyi ad imulatio gav th xctd rult REFERENCES [] S Yamamoto ad I Hahimoto "Prt tatu ad futur d: th viw from Jaa idutry" I Aru ad Ray Ed Chmical Proc Cotrol-CPCIV Proc 4 th Itr Cof o Chmical Proc Cotrol TX 99 [] L Dbourough R Millr "Icraig cutomr valu of idutrial cotrol rformac moitorig Hoywll xric" i: Sixth Itratioal Cofrc o Chmical Proc Cotrol AIChE Symoium Sri Numbr 36 Vol [3] KJ Ǻtröm T Hägglud Rviitig th Ziglr-Nichol t ro mthod for PID cotrol Joural of Proc Cotrol vol Stmbr 004 [4] M R Matauš S Kvaščv "A uifid t ro rocdur for autotuig of PI cotrollr ad Smith rdictor for tabl roc" Joural of Proc Cotrol Vol [5] S Sogtad "Siml aalytic rul for modl rductio ad PID cotrollr tuig" Joural of Proc Cotrol Vol [6] M Shamuzzoha M L IMC-PID cotrollr dig for imrovd diturbac rctio of tim-dlayd roc Idutrial & Egirig Chmitry Rarch vol 46 o [7] MR Matauš TB Šara PID cotrollr frqucy-domai tuig for tabl itgratig ad utabl roc icludig dad-tim J Proc Cotrol vol Iu [8] TB Šara MR Matauš Claificatio of dyamic roc ad PID cotrollr tuig i a aramtr la J Proc Cotrol vol Iu [9] H Paagooulo KJ Ǻtröm ad T Hägglud Dig of PID cotrollr bad o cotraid otimizatio IEE Procdig- Cotrol Thory ad Alicatio vol Jauary 00 [0] A Wallé KJ Ǻtröm ad T Hägglud Loo-haig dig of PID cotrollr with cotat T i/t d ratio Aia Joural of Cotrol vol Dcmbr 00 [] C Hwag ad C-Y Hiao Solutio of o-covx otimizatio ariig i PI/PID cotrol dig Automatica vol Novmbr 00 [] B Kritiao ad B Larto Evaluatio ad iml tuig of PID cotrollr with high-frqucy robut Joural of Proc Cotrol vol6 9-0 Fbruary 006 [3] B Kritiao ad B Larto Robut tuig of PI ad PID cotrollr: uig drivativ actio dit or oi IEEE Cotrol Sytm Magazi Fbruary 006 [4] AJ Iao ad SF rab Drivativ filtr i a itgral art of PID dig IEE Procdig-Cotrol Thory ad Alicatio vol Jauary 00 [5] A Karimi M Kuz ad R Logcham Robut cotrollr dig by liar rogrammig with alicatio to a doubl-axi oitioig ytm Cotrol Egirig Practic vol Fbruary 007 [6] T B Šara M R Matauš "A iml ffctiv mthod to obtai a wll-tud PID cotrollr" Procdig of th 5t Cofrc o ETRAN AU3 007 (i Srbia) [7] TB Šara MR Matauš Otimal tuig of a PI/PID cotrollr for roc dfid by a ratioal trafr fuctio INFOTEH Vol 6 Par A- 6-9 Jahoria March 007 (i Srbia) [8] TB Šara ad MR Matauš Otimizatio of PID cotrollr bad o maximizatio of th roortioal gai udr cotrait o robut ad itivity to maurmt oi IEEE Tra Automatic Cotrol vol 54 o Ja 009 [9] TB Šara ad MR Matauš Rviitig th Ziglr-Nichol roc dyamic charactrizatio J Proc Cotrol Vol 0 Iu [0] T B Šara M R Matauš "A four-aramtr otimizatio of a PID cotrollr" Procdig of 5 Cof ETRAN Vol Palić Ju 008 (i Srbia) [] TB Šara MR Matauš Otimal tuig of a PID cotrollr i frqucy domai INFOTEH Par A Boia ad Hrzgovia Jahoria March 009 (i Srbia) [] TB Šara MB Trifuović Otimal tuig of a PID cotrollr havig a diffrtial comator coctd i ri i frqucy domai Procdig of INDEL 58-6 Baa Lua 4-6 Novmbr 00 (i Srbia) [3] F Shiy How good ar our cotrollr i abolut rformac ad robut? Maurmt ad Cotrol vol 3 4- May 990 [4] Chi I L IMC-PID cotrollr dig-a xtio IFAC Procdig Sri vol 6 ( 47-5) 988 [5] Fruhauf P S Chi I L & Laurit M D Simlifid IMC-PID tuig rul ISA Traactio 33() [6] L Y L M Par S & Broilow C PID cotrollr tuig for dird clod-loo ro for SI/SO ytm AIChE Joural 44() [7] Liu T Zhag W u D Aalytical dig of two-dgr-of-frdom cotrol chm for o-loo utabl roc with tim dlay J Proc Cotrol 005; 5:559-7 [8] M Shamuzzoha M L Aalytical dig of PID cotrollr cacadd with a lad-lag filtr for tim-dlay roc Kora J Chm Eg vol [9] M Shamuzzoha M L Ehacd diturbac rctio for oloo utabl roc with tim dlay ISA Traactio vol