Application of Kronecker Summation Method in Computation of Robustly Stabilizing PI Controllers for Interval Plants

Transcription

1 Applicatio of Kroecer Summatio Method i Computatio of Robutly Stabilizig Cotroller for terval lat RADEK MATUŠŮ, ROMAN ROKO Departmet of Automatio ad Cotrol Egieerig Faculty of Applied formatic Toma Bata Uiverity i Zlí Nad Stráěmi 4511, 765 Zlí CZECH REUBLC {rmatuu; proop}@fai.utb.cz KATARÍNA MATEJČKOVÁ, MONKA BAKOŠOVÁ Departmet of formatio Egieerig ad roce Cotrol Faculty of Chemical ad Food Techology Slova Uiverity of Techology i Bratilava Radliého 9, Bratilava SLOVAKA {ataria.matejicova; moia.baoova}@tuba. Abtract: - The cotributio deal with deig of cotiuou-time robutly tabilizig cotroller for iterval ytem uig the combiatio of Kroecer ummatio method, ixtee plat theorem ad a algebraic approach to cotroller tuig. The effectivee ad practical applicability of the propoed method i demotrated i cotrol of a 3rd order oliear electroic plat. Key-Word: - Robut Stabilizatio, Cotroller, terval Sytem, Kroecer Summatio Method 1 troductio Depite the developmet of may advaced cotrol techologie, the egieer from practice till clearly prefer the applicatio of cotroller with imple or D tructure. Thi id of cotroller i very popular becaue of their eay implemetatio ad ufficiet performace at the ame time, eve uder coditio of ucertaity, ad thu the ivetigatio of a effective tuig method remai very topical. A poible approach to robut cotrol deig for ytem with iterval ucertaity [1], [2] coit of computatio of all robutly tabilizig cotroller ad coequetly the electio of the fial oe o the bai of uer demad. The calculatio of robutly tabilizig cotroller ca be doe uig the tability boudary locu a publihed i [3], [4] or alteratively with the aitace of Kroecer ummatio method [5]. The approach from [3], [4] ha bee aalyzed i [6], [7], while thi paper tudie alterative method [5] ad verifie it o the ame laboratory apparatu a i [6], [7]. Furthermore, a techique for cotroller choice itelf ca be adopted from algebraic approach [8], [9], [1]. Thi method i baed maily o geeral olutio of Diophatie equatio i the rig of proper ad Hurwitz table ratioal fuctio (R S ). A advatage i that the cotroller ca be further tued through the oly poitive calar tuig parameter m. The cotributio i focued o computatio of cotiuou-time robutly tabilizig cotroller for iterval plat uig Kroecer ummatio method, ixtee plat theorem ad everal algebraic tool. Origiality of the propoed approach lie i combiatio of Kroecer ummatio method for obtaiig the tability boudary ad the choice of the fial cotroller via a algebraic methodology. However, the wor deal ot oly with theoretical bacgroud but alo with the practical applicatio i laboratory coditio. A oliear electroic plat, coidered a the 3rd order iterval ytem, ha bee cotrolled i variou operatioal poit with the aitace of the deiged algorithm which have bee realized uig the Simatic automatio ytem by Sieme Compay. 2 Computatio of Stabilizig Cotroller Uig Kroecer Summatio Coider the traditioal cloed-loop cotrol ytem a depicted i fig. 1. SSN: SBN:

2 w(t) e(t) u(t) y(t) C () G () - Fig. 1: Feedbac cotrol loop The cotrolled plat i decribed by: B() G () = (1) A () ad cotroller i uppoed to be i a form: + C () = + = (2) The iitial ta i to determie the parameter of cotroller which guaratee tability of the feedbac ytem. A approach to computatio of tabilizig cotroller which i baed o iteretig feature of Kroecer ummatio ha bee publihed i [5]. Firt, remid that Kroecer ummatio of two quare matrice Y (of ize -by-) ad Z (l-by-l) i geerally defied a Y Z = Y + Z (3) where, l are idetity matrice of ize -by- ad l-byl, repectively, ad where deote the Kroecer product [11], e.g. cociely: Y l = l y11l y1 l y1 l y l The mometou property of the obtaied quare matrix Y Z (l-by-l) i that it ha l eigevalue which are pair-wie combiatoric ummatio of the eigevalue of Y ad l eigevalue of Z. t mea the Kroecer ummatio operatio iduce the eigevalue additio feature to the matrice. Oe ca exploit thi attribute to obtai the equatio for which all pair (, ) leadig to purely imagiary root comply. The characteritic equatio of the cloed-loop ytem from fig. 1 i: CL = A() + B() ( + ) = (5) = f (, ) + + f (, ) + f (, ) = Defie: 1 (4) x 1 = x2 x 2 = x3 f (, ) f (, ) f (, ) x = x x x f (, ) f (, ) f (, ) (6) ad traform (5) ito matrix differetial equatio: X = MX (7) where M i -by- matrix: M = 1 f(, ) f1(, ) f2(, ) f 1(, ) f(, ) f(, ) f(, ) f(, ) ad X = [ x x x ] T, X [ x x x ] T (8) 1, 2,, = 1, 2,,. The equatio (5) ad (7) are lied via: ( ) = f (, )det M = (9) CL Obviouly, the ame complex variable i both the root of (5) ad the eigevalue of M. Owig to the fact that M i a cotat matrix, the complex cojugate of mut alo atify (9). * ( M) det = (1) O that accout, a it ha bee preeted i [5], if = jω i the root of (5) it mut be the eigevalue of M. * Moreover, = jω i alo the root of (5) ad the eigevalue of M. A the um of two eigevalue = jω * ad = jω equal to zero, the Kroecer ummatio of two matrice mut be igular whe uch correpodece of, ad ω occur. Thu: ( M M) det = (11) defie the tability boudary i (, ) every couple of (, ) plae, becaue atifyig (11) mea that the ame couple ierted ito (5) will lead to the pair of cojugate purely imagiary root or zero root. Thoe are the oly poitio where the ytem tability ca hift. Geerally, the tability boudary plit the (, ) plae ito the table ad utable regio. The determiatio of the tabilizig area (or area) ca be doe via a tet poit, leadig to a polyomial to verify, withi each regio. SSN: SBN:

3 3 Robut Stabilizatio of terval lat The previou ectio ha outlied calculatio of regio of tabilizig compeator parameter oly for a ytem with fixed coefficiet. Neverthele, the wor [3], [4], [5] have embellihed a arbitrary tabilizatio techique alo for iterval plat imply by uig it combiatio with the ixtee plat theorem [1], [12], [13]. Accordig to thi rule, a firt order cotroller robutly tabilize a iterval plat: m + i bi, bi Bb (, ) i= (,, ) ; Aa (, ) + i ai, a i i= Gba = = m< (12) + + where bi, bi, ai, ai repreet repectively lower ad upper boud for parameter of umerator ad deomiator if ad oly if it tabilize it 16 Kharitoov plat, which are defied a: Bi () Gi, j() = (13) A () where i, j { 1,2,3,4} ; ad B 1 () to B () ad A () to 4 1 A () 4 are the Kharitoov polyomial for the umerator ad deomiator of the iterval ytem (12). Remid that the cotructio of Kharitoov polyomial e.g. for the umerator iterval polyomial: m i= j + i B(, b) = bi ; b i (14) i baed o ue of the lower ad upper boud of iterval parameter i compliace with the priciple [14]: B() = b + b + b + b + B = b + b + b + b + B = b + b + b + b + B = b + b + b + b () () () (15) A ca be ee, the tabilizatio of a iterval plat directly follow from the imultaeou tabilizatio of all 16 fixed Kharitoov plat. Thu the fial area of tability for origial iterval plat i give by iterectio of all 16 related partial area obtaied idividually uig the Kroecer ummatio method from the previou ectio. 4 Algebraic Deig of Cotroller So far, the methodologie from ectio 2 ad 3 allow calculatig all robutly tabilizig combiatio of proportioal ad itegral gai i cotroller. Noethele, the fial electio of a cotroller i aother problem. A effective olutio i repreeted by algebraic approach to cotrol deig [8], [9], [1], which i baed o geeral olutio of Diophatie equatio i R S, Youla-Kučera parameterizatio ad coditio of diviibility i the pecific rig. A merit of the techique i that the cotroller ca be tued by the oly poitive calar parameter m. Due to the limited pace the paper ca ot provide full detail o thi method [7], [1]. t exploit oly oe pecific reult, i.e. the coefficiet of feedbac cotroller (2) ca be computed accordig to: 2 2 m a = ; m = (16) b b where the parameter a ad b of the firt order omial cotrolled plat: b G () = + a (17) are uppoed to be ow ad where the tuig parameter m ca be choe o the bai of everal approache uch a trivial trial-ad-error, uer owledge ad experiece, or uig recommedatio [15]: m= a (18) Appropriate coefficiet deped o the ize of firt overhoot of the output (cotrolled) variable. Some of it value ca be foud i table 1. Table 1: Relatio betwee ad overhoot Overhoot [%] K Real Cotrol Experimet The preeted theoretical tool have bee teted i laboratory coditio durig robut cotrol of a oliear electroic model while the cotrol loop ha bee realized uig Simatic S7-3 automatio ytem. The utilized plat, cotructed at Slova Uiverity of Techology i Bratilava, ha icluded a 3rd order ytem with a variable time cotat, adjutable from 5 to 2, ad a model of oliear valve. The real viual appearace of thi model i how i fig. 2 ad the bloc SSN: SBN:

4 cheme of the proce i i fig. 3, where igal are deoted a follow: V cotrol igal for valve opeig ( 1V) F igal repreetig the valve opeig ( 1V) output of the proce ( 1V) U diturbace ( 1V) A ca be verified, the tabilizig cotroller for the plat (2) are i the ier pace Fig. 4: Stability boudary for the plat (2) V Fig. 2: Electroic laboratory model U VALVE T 1 T 2 T 3 F Fig. 3: Bloc cheme of laboratory model Geerally, oe mut repeat a aalogical procedure for all 16 Kharitoov plat (13). Noethele, i uch pecific cae, oly 8 plat are eough to tet. t i tha to the fact that the umerator of (19) repreet jut zero order polyomial with two extreme value ad thu it i ot eceary to deal with all 4 Kharitoov polyomial for thi umerator. The regio of tability regio for all 8 Kharitoov plat are plotted i fig. 5. The plat ha bee idetified a the third order iterval ytem which ha led to the approximate mathematical model [6], [7], [16]: [.35, 5.5] G (,, b a) = , , , (19) [ ] [ ] [ ] The firt of it 16 Kharitoov plat (13) ca be imply cotructed:.35 G1,1() = (2) The cloed-loop characteritic equatio (5) i: ( ) = (21) From here, the matrix (8) follow: 1 1 M = (22) The tability boudary i give by (11). The poitio of, which fulfil (11) are how i fig. 4. uch pair ( ) Fig. 5: Stability area for 8 Kharitoov plat The iterectio of all thee tability area i zoomed ad depicted i fig. 6. t determie the fial regio of robutly tabilizig cotroller for the origial iterval model (19). SSN: SBN:

5 C Regio of Stability.2 C Fig. 6: Stability regio for the iterval ytem (19) Quite aturally, the followig tep brig the quetio of how to fid the practically coveiet cotroller from the obtaied robut tability regio. Amog poible method, the algebraic approach from the art 4 ha bee utilized for thi purpoe. However, thi algebraic ythei require the model of cotrolled ytem i the form of firt order trafer fuctio i order to obtai the fial cotroller of appropriate (firt) order ( type). So the implet approximatio of (19) ha bee applied. t reult i: GA [.35, 5.5] [ ] + (, b, a) = 19, 25 1 (23) Computig the average value of iterval parameter the lead to the omial plat for cotrol deig: GN () = (24) Firt, the aumptio of % firt overhoot i output variable for the cae of omial ytem, applicatio of appropriate parameter from table 1, ad furthermore equatio (18) ad (16) give the trafer fuctio of the cotroller: % m=.4545 C1( ) (25) The aalogically, 1% firt overhoot requiremet reult i: % m.7363 C2( ) (26) The fig. 7 depict the poitio of the cotroller (25) ad (26) i the tability area from fig. 6. A ca be ee, they lie o the curve hypothetically coectig the other potetial cotroller tued by variou parameter m > Fig. 7: oitio of cotroller (25) ad (26) i tability regio Fially, three cotrol experimet have bee executed uder differet worig poit uig the choe cotroller ad LC Simatic S7-3. Referece, Cotrolled Variable(%) Referece Sigal Cotroller 1 Cotroller Time(mi) Fig. 8: Real cotrol reult (for 15% referece poit) Referece, Cotrolled Variable(%) Referece Sigal Cotroller 1 Cotroller Time(mi) Fig. 9: Real cotrol reult (for 6% referece poit) SSN: SBN:

6 Referece, Cotrolled Variable(%) Referece Sigal Cotroller 1 Cotroller Time(mi) Fig. 1: Real cotrol reult (for 9% referece poit) The omially precribed overhoot have ot bee meaured i real coditio. Actually it wa expected, becaue the cotrolled plat ha had highly oliear behaviour ad thee recommedatio trictly hold true oly for the omial liear ytem. Fig. 8-1 idicate that the le aggreive cotroller C 1 provide very good reult maily i the mea et poit, but it ha comparatively log ettlig time i higher operatioal area. O the other had, the cotroller C 2 i much fater here, however it i more ocillatig i the lower level. Altogether, both compeator have bee able to cotrol the oliear proce robutly table ad with acceptable performace. The defiitive electio of the cotroller would deped o the mai operatioal area. 6 Cocluio The paper ha dealt with a approach to computatio of robutly tabilizig cotroller. The propoed method ha bee baed o combiatio of calculatig the tability boudary via Kroecer ummatio, it exteio for iterval ytem uig 16 plat theorem, ad the choice of the fial regulator through the igleparameter tuig algebraic approach. The developed ythei repreet eay but effective way of deigig the cotroller for iterval ytem. O the other had, coicidet omial performace ad robut tability ca ot be aured i advace. They have to be verified durig the deig proce which ca be coidered a a method demerit. However, the applicability ha bee how o laboratory experimet i which a oliear 3rd order electroic model ha bee uccefully cotrolled i variou operatioal poit. Acowledgemet: The author would lie to gratefully acowledge the upport from the Miitry of Educatio, Youth ad Sport of the Czech Republic uder Reearch la No. MSM ad the Scietific Grat Agecy of the Slova Republic uder Grat No. 1/537/1 ad 1/71/9. Referece: [1] B. R. Barmih, New Tool for Robute of Liear Sytem, Macmilla, New Yor, USA, [2] S.. Bhattacharyya, H. Chapellat, L. H. Keel, Robut cotrol: The parametric approach, retice Hall, Eglewood Cliff, NJ, USA, [3] N. Ta,. Kaya, Computatio of tabilizig cotroller for iterval ytem, : roceedig of the 11th Mediterraea Coferece o Cotrol ad Automatio, Rhode, Greece, 23. [4] N. Ta,. Kaya, C. Yeroglu, D.. Atherto, Computatio of tabilizig ad D cotroller uig the tability boudary locu, Eergy Coverio ad Maagemet, Vol. 47, No , 26, pp [5] J. Fag, D. Zheg, Z. Re, Computatio of tabilizig ad D cotroller by uig Kroecer ummatio method, Eergy Coverio ad Maagemet, Vol. 5, No. 7, 29, pp [6] R. Matušů, K. Vaeová, R. roop, M. Baošová, Robut proportioal-itegral cotrol of a laboratory model uig programmable logic cotroller SMATC S7-3, : roceedig of the 6th FAC Sympoium o Robut Cotrol Deig, Haifa, rael, 29. [7] R. Matušů, K. Vaeová, R. roop, M. Baošová, Deig of robut cotroller ad their applicatio to a oliear electroic ytem, Joural of Electrical Egieerig, Vol. 61, No. 1, 21, pp [8] M. Vidyaagar, Cotrol ytem ythei: a factorizatio approach, MT re, Cambridge, M.A., USA, [9] V. Kučera, Diophatie equatio i cotrol - A urvey, Automatica, Vol. 29, No. 6, 1993, pp [1] R. roop, J.. Corriou, Deig ad aalyi of imple robut cotroller, teratioal Joural of Cotrol, Vol. 66, No. 6, 1997, pp [11] D. S. Bertei, Matrix Mathematic: Theory, Fact, ad Formula with Applicatio to Liear Sytem Theory, riceto Uiverity re, NJ, USA, 25. SSN: SBN:

7 [12] B. R. Barmih, C. V. Hollot, F. J. Krau, R. Tempo, Extreme poit reult for robut tabilizatio of iterval plat with firt order compeator, EEE Traactio o Automatic Cotrol, Vol. 37, No. 6, 1992, pp [13] M. T. Ho, A. Datta, S.. Bhattacharyya, Deig of, ad D cotroller for iterval plat, : roceedig of America Cotrol Coferece, hiladelphia, USA, [14] V. L. Kharitoov, Aymptotic tability of a equilibrium poitio of a family of ytem of liear differetial equatio, Differetial'ye Uraveiya, Vol. 14, 1978, pp [15] R. Matušů, R. roop, Sigle-parameter tuig of cotroller: from theory to practice. : roceedig of the 17th FAC World Cogre, Seoul, Korea, 28. [16] K. Vaeová, M. Baošová, J. Závacá, Robut cotrol of a laboratory proce uig cotrol ytem Simatic, : roceedig of 16th teratioal Coferece o roce Cotrol, Štrbé leo, Slovaia, 27. SSN: SBN: