M. Rafeeyan. Keywords: MIMO, QFT, non-diagonal, control, uncertain

Transcription

1 IUST Ieraioal Joural of Eieeri Sciece, Vol. 9, No.5-, 008, Pae 37-4 QUANTITATIVE NON-IAGONAL REGULATOR ESIGN FOR UNCERTAIN MULTIVARIABLE SYSTEM WITH HAR TIME-OMAIN CONSTRAINTS owloae from ijiepr.iu.ac.ir a 5:43 IRST o Sauray Jauary 5h 09 M. Rafeeya Abrac: I hi paper a o-iaoal reulaor, bae o he QFT meho, i yheize for a ucerai MIMO pla whoe oupu a corol ial are ubjece o har ime-omai corai. Thi proceure iclue he ei of a o-iaoal pre-coroller bae o a ew imple approach, followe by he eueial ei of a iaoal QFT coroller. We pree a ew formulaio for he laer ae, which how he role of off-iaoal eleme i he ei proceure. A umerical example i ive o illurae he effecivee of he propoe meho. Keywor: MIMO, QFT, o-iaoal, corol, ucerai. Iroucio I he pa few year, everal corol echiue o ei o-iaoal coroller for ucerai MIMO yem have bee propoe by Yaiv[], Boje[], Y.H. Cha a J.C. Cha [3] a Garcia-Saz a Eaa[4]. Some of hee approache uch a [] have focue o he ei of off-iaoal eleme of coroller bae o he reucio of ieracio bewee he eleme. A how i [], improvi he iaoal omiace i o ecearily he be crierio for eii a o-iaoal coroller, iea reuci he bawih of he coroller i a more reaoable crierio. The work i [] ha cocerae oly o pla behavior a hih freuecie. I reuire ui he iiially evelope pla i a -ae eueial proceure. The, he uceraiy of he ex euivale SISO yem will o be iclue. Thu, aoher imple proceure o coribue all uceraiie i reuire. A ew approach i propoe i hi paper o are hi problem. Oe of he applie problem i ucerai MIMO yem which ha coiere before by ome reearcher uch a Frachek[5] i he ei of robu reulaor uer cerai har ime-omai corai o oupu a corol ial of he yem i repoe o he ep iurbace. Thi problem ha bee olve before i MIMO QFT framework bae o iaoal coroller. Sice o-iaoal eleme of he coroller ca improve he abiliy of he ei; hi Paper fir receive May. 0, 007 a i revie form Sep. 6, 009. M. Rafeeya, i wih he eparme of Mechaical Eieeri, Yaz Uiveriy, Yaz, Ira. rafeeya@yazui.ac.ir problem will be icue i more eail. Off-iaoal eleme of hi reulaor are yheize bae o he ew propoe meho a he iaoal eleme are yheize bae o Yaiv' approach [6]. I hi paper, we propoe a ew imple approach for eii o-iaoal coroller wihi he framework of he eueial MIMO QFT. Thi ivolve fir, he ei of a o-iaoal coroller, followe by he ei of a aar iaoal coroller o achieve abiliy a performace pecificaio. Alo, we pree he formulaio for he ei of oiaoal reulaor o mee imulaeouly har imeomai corai o boh he oupu a he corol ial a how he role of o-iaoal eleme i he ei proceure. A example i iclue o illurae he ew formulaio. The arraeme of he paper i a follow. I ecio II, he problem i ae. I ecio III, ei of oiaoal eleme of he reulaor i explaie. I ecio IV, he ew formulaio for he ei of iaoal eleme of he reulaor i evelope a ecio V illurae he meho by i applicaio o a example. Secio VI coclue he paper.. The Problem Saeme The problem aeme wihou lo of eeraliy i ive for a yem for impliciy. Coier he yem how i Fi., where P i a LTI pla beloi o a ep, i a ep iurbace vecor beloi o a ive e, a are coa vecor which irouce oupu a corol ial corai.

2 38 Quaiaive No-iaoal Reulaor ei for Ucerai Mulivariable ei he o-iaoal reulaor, G, uch ha for all P P; The yem i able; a, he pla oupu For all y T y a corol ial u u T y boue by u are owloae from ijiepr.iu.ac.ir a 5:43 IRST o Sauray Jauary 5h 09 y ( ), uk ( ) k k, () k k We aume ha / i a ive coa. 3. ei of No-iaoal Eleme of G Similar o Yaiv' meho [], we aume ha G G G, where G ia(, ), G Fi.. A MIMO feeback yem wih iurbace a he ipu o he pla A we kow from he baic of QFT, he effec of uceraiy i more effecive i low freuecie. The he miimizaio of imeio of he pla emplae i all freuecie i he moivaio of hi meho. By ui he orm of a marix A which i efie i [7] a: A max a () i, j ij We ca efie a fucio whoe value repree he emplae ize. Thi fucio i efie a k U ( ) c i u( ) (3) i where u( ) PG P0 G a P 0 i he omial pla. The weih c are ue for ui. The fucio i u () a illurae i Fi. meaure he maximum iace bewee all ucerai pla o he omial pla a ome freuecy. The fucio U um up a weih hem for all freuecie. Here, we elec ci / i The be elecio of hee coefficie ca be furher iveiae. Fi.. Templae of PG i ome freuecy. 4. ei of he iaoal Eleme of G I Fi., we ca wrie: y ( I PG) P (4) Sice we aume ha G G G, we have y ( I PG G ) PG G (5) Now, if we coier PG a a ew pla a G a a ew iurbace vecor, he we ca ue he aar eueial MIMO QFT proceure [6] a follow: B PG where / 0 0 / B 0 / / 0 y T, T, - T G ( / ) e( G ), T, (6) T =marix rafer fucio from oupu o iurbace. I eleme are: By ubiui he laer euaio i he above euaio for, we will e he followi euaio: y [ ( ) [ ( ) ] or y ( ). ( ) ( ) ( ) Ui raformaio lemma, y ( j), he we e he followi ieualiy:

3 M. Rafeeya 39 owloae from ijiepr.iu.ac.ir a 5:43 IRST o Sauray Jauary 5h 09..( ) ( ) ( ) (7) If we ue he followi riaular ieualiy: Fi. 3. Coour of coa uceraiy. a b a b a b he by mahemaical operaio;. (8) A imilar ieualiy i erive from y ( ) a follow:. (9) For ralai ime-omai corai o corol ial, we ubiue: u y y (0) By ubiui y, y from he cloe loop rafer fucio bewee oupu a iurbace i (0) yiel: u ( ) u ( ) ( ) u ( ) ( ) T ( ) ( ) () where T =rafer fucio from o u u ( ) ( ) ( ) ( ) u ( ) ( ) ( ) ( ) From raformaio lemma [8] a pecificaio (), a houl be eie o aify he followi ieualiie i all freuecie: f ( ) Where:. f. f ( ) f f( ) a. f ( ( ( ) b. ) ) a b () (3-4) From he ieualiie (8), (9) a (), he role of he o-iaoal par G become clear: i allow lower bou for he eie a, which reul i a bawih lower ha he oe achievable wih a iaoal coroller. We ca o ue he ieualiie () i compuaio of bou i QFT oolbox [9], becaue a appear i boh of hem. We have o olve () a fir a fi he limi o f ( ) a f ( ) for all uceraiie. Afer hi ep, we ca ue QFT oolbox a compue he bou o omial ope loop rafer fucio i Nichole char. I mu be oe ha he opimum o-iaoal coroller which ha eie i he fir ep ca o be opimum i he eco ep. 5. A Illuraive Example A. A Ucerai Pla Coier he feeback yem how i Fi., where he ucerai pla family i ive by: k k P ; 4,.0.8 k k (5) k k We aume ha he iurbace are he ame, i.e.. The performace i a follow y ) 0.5, y ( ) 0.5, u ( ), u ( ) ( Gai mari=6 b Thi example i iffere from [] oly i pecificaio.

4 40 Quaiaive No-iaoal Reulaor ei for Ucerai Mulivariable owloae from ijiepr.iu.ac.ir a 5:43 IRST o Sauray Jauary 5h 09 B. ei of G The o-iaoal eleme of marix rafer fucio G were choe a: k, 3 k 3 a ci /. Fi. 3 how i U () a a fucio of he off-iaoal ai of eleme (x-axi) a i offiaoal ai of eleme (y-axi). A oluio i: G 3 3 Fi. 3 how ha k, k ca be elece i he rae [ 0..4].We ca o ome ajume for improvi ome pecificaio uch a, abiliy, bawih, a o o. We elec hem here for beer loop hapi i he ex ep (loop hapi for iaoal eleme). Thi elecio epe o he problem. C. ei of G Fi. 4 a 5 how he QFT bou a he ope loop freuecy repoe for he iaoal ei. Afer he loop hapi proce, he iaoal coroller eleme are eie a: 4.65 ( ) ( / / ).83 ( ) ( / 7.65 )( /9.3 ) There are hree roup bou i Fi. 4 a 5, i.e. robu abiliy, robu performace. Upper bou are how oe curve a lower bou are how full curve. The ierecio reio of hee bou i he allowable reio for loop hapi of he coroller. If here i o ierecio reio for ome freuecie, he loop hapi i o poible a here i o oluio. We ca elec aoher e of oiaoal eleme a icue i ecio B, or if i i poible, ime-omai corai o he oupu a he corol ial mu be chae.. Reul a icuio Fi. 6 o 9 how he imulaio reul of he yem wih he eie coroller. A i i ee, all eire pecificaio are aifie. Boh ipu iurbace are he ui ep fucio. The reul how ha he reulaor are omewha coervaive. Templae of (P) a ( PG ) a 0. ra/ are how i Fi. 0 a ypically. We ca ee emplae of oher eleme of P a PG. Thee fiure how ha he ize of he emplae i Fi. i lower ha Fi. 0. Thi ifferece i o coierable here bu i may e more value i oher example. The maller ize of he pla emplae, he lower bawih of he ope loop i achive. Fi. 4. Upper a lower bou a L 0( j) i Nichol char. Fi. 5. Upper a lower bou a L 0 ( j) i Nichol char. Fi. 6. Time-omai imulaio of oupu ial y o ui ep iurbace.

5 M. Rafeeya 4 owloae from ijiepr.iu.ac.ir a 5:43 IRST o Sauray Jauary 5h 09 Fi.7. Time-omai imulaio of oupu ial y o ui ep iurbace. Fi.8. Time-omai imulaio of corol ial u ue o ui ep iurbace. Fi. 9. Time-omai imulaio of corol ial u ue o ui ep iurbace. Fi. 0. Templae of (P) a 0. ra/. Fi.. Templae of ( PG ) a 0. ra/. 6. Cocluio A ew a imple meho wa irouce o ei off-iaoal eleme of o-iaoal coroller for ucerai MIMO LTI yem by aumi ha he coroller of he yem i he marix prouc of he wo marice: oe i iaoal a oher i o-iaoal. Thi paper offere a ew imple approach o ei o-iaoal par of he coroller bae of he miimizaio of he emplae ize epecially i mall freuecie. The, he ei of iaoal par of he coroller wa oe. The elaorm efiiio of he marice wa ue o uaify he emplae ize. The reul how he uccefully a impliciy of he meho. Alo a ew formulaio wa evelope o he ei of o-iaoal robu reulaor uer cerai har ime-omai corai o oupu a corol ial of he yem i repoe o he ep iurbace. The role of off-iaoal eleme of he coroller wa eablihe clearly i he formulaio. Thi heoreical ehaceme ha o bee

6 4 Quaiaive No-iaoal Reulaor ei for Ucerai Mulivariable irouce before. Simulaio of he reuli coroller how he aifacio of he pecificaio bu hey were omewha coervaive. Referece [] Yaiv, O., "MIMO QFT Ui No-iaoal Coroler," Ieraioal Joural of Corol, Vol. 6, No., 995, pp owloae from ijiepr.iu.ac.ir a 5:43 IRST o Sauray Jauary 5h 09 [] Boje, E., No-iaoal Coroller i MIMO Quaiaive Feeback ei," Ieraioal Joural of Robu a Noliear Corol", Vol., 00, pp [3] Cha, Y.H., Cha, J.C., "Quaiaive ei for Mulivariable Syem wih Uceraiy," Ieraioal Joural of Syem Sciece, Vol. 3, No. 3, 00, pp [4] Garcia-Saz, Eaa, I., "Quaiaive No-iaoal Coroller ei for Mulivariable Syem wih Uceraiy," Ieraioal Joural of Robu a Noliear Corol, Vol., 00, pp [5] Frachek, M., Jayauriya, S., "Coroller ei for Performace Guaraee i Ucerai Reulai Syem," Ieraioal Joural of Corol, Vol. 6, 995, pp [6] Yaiv, O., Quaiaive Feeback ei of Liear a Noliear Corol Syem, Kluwer Acaemic Publiher, 999. [7] Golub, G.H., Va Loa, C.F., Marix Compuaio, The Joh Hopki Uiveriy Pre, 996. [8] Sobhai, M., Jayauriya, S., "Coroller ei for Maximizi he Size of a Sep iurbace i No- Miimum Phae Ucerai Syem, "Ieraioal Joural of Corol, Vol. 59, No., 994, pp [9] Boreai, C., Chai, Y., Yaiv, O., Malab TM Quaiaive Theory Toolbox, Mahwork Ic.,998.