Sensitivity Measures and Modeling Errors for YOULA Parameterization Based Regulators

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1 205 Thid Intenationa onfeence on Atificia Inteigence, Modeing and Simation Senitivit Meae and Modeing Eo fo YOULA aameteization Baed Regato ia Bánáz and Lázó Keviczk Intitte of ompte Science and Atomation MTABME onto Engineeing Reeach Gop Hngaian Academ of Science H Bdapet, Kende 37, HUNGARY cia.banaz@ztaki.mta.h and azo.keviczk@ztaki.mta.h Abtact Diffeent enitivit meae ae invetigated fo YOULApaameteized egato and the infence of a new obeve topoog i teated. The pape extend the obeve pincipe fo YOULA egato edcing the mode eo imia to the caica tate feedback/obeve topoogie. Kewod YOULA paameteization; YOULA egato; enitivit; obeve; tatefeedback; mode eo I. INTRODUTION The impe YOULA paameteization [5], [6] i not o wide known a the YOULAKUERA paameteization [4], [5]. The caica YOULA paameteization give a ve impe wa fo openoop tabe pocee when the egato can be anatica deigned b expicit foma. The YOULA paamete i, a a matte of fact, a tabe (b definition), ega tanfe fnction Q o hot Q whee i a tabiizing egato, and i the tanfe fnction of the tabe poce. It foow fom the definition of the YOULA paamete that the tcte of the eaizabe and tabiizing egato in the YOULApaameteized conto oop i fixed: Q Q o hot Q Q The YOULA paameteized conto oop i hown in Fig.. Q Q Fige. YOULApaameteized conto oop () (2) The YOULA paameteization can be extended fo twodegeeoffeedom conto tem and apping efeence mode fo the tacking and noie ejection popetie of the coedoop impe deign fomae can be deveoped fo the egato deign [], [2]. II. UNERTAINTIES OF ROESS MODELS AND LOSED LOO ARAMETERS The poce paamete ae neve known pecie and the poce i bject to change. The envionment can change, which can in tn change the paamete of the poce in a given egion. Negative feedback edce the enitivit of the tem to paamete change. Theefoe egato deign need to take poibe paamete change into accont. The eqied behavio of the conto oop mt be ffied not on fo the nomina paamete bt ao fo the poibe paamete change. The knowedge of a poce i neve exact, independent of the method whethe meaementbaed identification (ID) o phicachemica theoetica conideation b which it mode i detemined. The ncetaint of the pant can be expeed b the abote mode eo and the eative mode eo Δ ˆ (3) Δ ˆ ˆ ˆ whee ˆ i the avaiabe nomina mode ed fo egato deign and i the ea pant. Let now invetigate the behavio of the conto tem if the tanfe fnction of the poce change fom the (ea) vae to the nomina (mode) vae ˆ. The ovea tanfe fnction of the open oop i L. Fo ma change in the poce (4) ΔL L Δ Δ (5) Apping the eative change we obtain /5 $ IEEE DOI 0.09/AIMS

2 ΔL L L Δ Δ Δ ˆ Δ (6) ˆ e Ĉ ˆ ˆ ˆ The ovea tanfe fnction of the negative feedback coedoop i Fige 3. The nomina coed tem T (7) e Ĉ Fo ma change Fo eative change ΔT T Δ Δ (8) 2 ( ) ΔT T T Δ S Δ S (9) whee S i the enitivit fnction of the coedoop S ΔTT Δ (0) onide the foowing thee impe coed conto oop what can be ed in modebaed egato deign. The fit coedoop can be een in Fig. 2. Hee it i amed that the egato i compted fom the theoetica ea poce and i paced togethe with the ea poce in the coedoop. Obvio thi coedoop i not eaitic and epeent an idea cae on. e o o Fige 2. The theoetica coed tem The next veion can be een in Fig. 3, and i a appied in deign tak, name, when the egato Ĉ i detemined on the bai of the poce mode ˆ and the whoe coedoop i modebaed. Thi cae i a caed the nomina tem. Thi coedoop depend on on the deigne, on the knowedge of the poce and the ggeted egato. The cheme can be ed in imation, optimization and deign tak. o TABLE I. Fige 4. The ea coed tem appeaing in the pactice THE SENSITIVITY AND OMLEMENTARY SENSITIVITY FUNTIONS OF THE THREE SYSTEMS Stem idea nomina ea fnction T T Ĉˆ Ĉ ˆT T Ĉˆ Ĉ S S Ŝ S Ĉˆ Ĉ TABLE II. THE SENSITIVITY AND OMLEMENTARY SENSITIVITY FUNTIONS FOR THE YOULAARAMETERIZED ONTROL LOO Stem idea nomina ea fnction T Q ˆ ˆ S Q ˆ TABLE III. ˆ ˆ ˆ THE OTHER FORMS OF THE SENSITIVITY FUNTIONS FOR THE YOULAARAMETERIZED ONTROL LOO Stem idea nomina ea fnction T T Q ˆT ˆT ˆT S S Q Ŝ ˆ Ŝ ˆT The thid veion of the coed tem i what opeate in the eait. A modebaed egato i ed togethe with the ea poce in the coedoop a in Fig. 4. Ua meaement, veification and appication of identification method take pace in thee kind of coedoop. The enitivit and compementa enitivit fnction fo the above thee coed tem ae mmaized in 74

3 Tabe I. The comptation of each eement i ve diffeent and the mt not be mixed. Obvio, in the idea cae when ˆ the eement in the ame ow ae eqa. It i ea to check that Tabe I. fo the YOULApaameteized conto oop i changing to Tabe II. if i the mode baed YOULA egato. The can be Ĉ Ĉ ˆ ewitten in anothe fom, too (ee Tabe III.). Let invetigate how the ea tem appoximate the nomina one, which i awa the bai fo the deign. ompte the eative eo ˆ ŷ ε T ˆT T T ( ˆT ) Ŝ () Thi i an exceent popet, becae Ŝ attenate the eative mode eo at the ow feqenc domain. Ua the enitivit fnction i a high pa fite. III. INTRODUTION OF THE OBSERVERBASED YOULA REGULATOR It i we known that the mode baed YOULAegato coepond to the Intena Mode onto Stcte (IM), peented in Fig. 5. The eqivaent IM tcte baed YOULAegato pefom the feedback fom the mode eo ε Q. (a) (b) REGULATOR Ĉ ˆ YOULA ARAMETER ROESS ˆ INTERNAL MODEL Fige 5. The eqivaent IM tcte of a YOULAegato Simia to the caica StateFeedbackObeve (SFO) cheme it i poibe to contct an intena coedoop pefoming the feedback b fom ε (ee Fig. 6, [3]) to edce the mode eo ing the caica obeve pincipe. With taightfowad bock manipation the obeve baed IM topoog can be edced to the two coedoop tem hown in Fig. 7. ε Q i ε Fige 6. The obevebaed IM tcte The eationhip between the two eo in Fig. 5b and 6 ( ˆK ˆ ˆ ) ˆL ε Q Ĥ ε Q ; ˆL ˆ (2) i.e., the obeve pincipe vita edce the mode eo b Ĥ. Hee ˆL i the intena oop tanfe fnction. ˆ ˆ ˆ Fige 7. Eqivaent coedoop fo the obevebaed IM tcte The intodction of the obeve feedback change the YOULApaameteized egato to ˆ ( ) ˆ ˆ ( ˆ ) ˆ ˆ (3) The fom of Ĉ how that the egato vita conto a fictitio pant ˆ, what i demontated in Fig. 7. Hee the fictitio pant i ˆ ˆ ˆ Ĥˆ ˆL ˆ The fite attenating the eo i (4) 75

4 Ĥ ˆK H ˆ H whee H TABLE IV. THE OMLEMENTARY SENSITIVITY FUNTIONS WITH OBSERVERBASED YOULA REGULATOR Stem idea nomina ea fnction T T ˆ T T Q ˆ ˆ ˆĤ (5) The nomina compementa enitivit fnction in the obevebaed IM tcte i ˆT ˆ ˆĤ ˆ ˆT (6) o thi i eqa to the obeve fee cae. ompte the compementa enitivit fnction of the ea oop now T Ĉ ˆT ĈĤ ˆTĤ (7) and the eative eo T of T i, imia to () TABLE V. ˆT T T T ( ˆTĤ ) (8) THE OMLEMENTARY SENSITIVITY FUNTIONS WITH YOULA REGULATOR Stem idea nomina ea fnction T ( ) Q ˆ ˆ ompte the compementa enitivit fnction of the idea oop T Q (9) H what foow fom (6). The above et ae mmaized in Tabe IV. The mot impotant et of thi anai i that the obevebaed YOULA egato give the ame nomina and idea compementa enitivit fnction a the oigina YOULA egato. IV. REFERENE MODELBASED YOULA REGULATOR DESIGN The impet YOULA egato baed on efeence mode deign [], [2] i Ĉ R ˆ n ˆ ; Q Q whee the mode baed YOULA paamete Q ˆ (20) ; Q (2) wa appied, becae in pactica deign cae Q Q. Hee i the deied efeence mode fo the tacking. Apping thi egato, the Tabe I. wi be changed to Tabe V. acate now the eative deign eo x obtained with the diffeent compementa enitivit fnction. The obtained eationhip ae hown in Tabe VI., whee TABLE VI. x T x T x (22) THE RELATIVE DESIGN ERRORS WITH YOULA REGULATOR Stem idea nomina ea fnction x 0 0 Hee T ( ) S o (23) and S o i the enitivit fnction of the idea tem. Thi i an exceent popet, becae S o attenate the eative mode eo at the ow feqenc domain, ee (). The advantage of the efeence mode baed deign i that the ncetaint in the YOULA paamete i edced to ncetaint of the poce mode on. Theefoe the eative deign eo fo the idea and nomina tem ae zeo. It i inteeting to invetigate how thee tem fnction change ing an obevebaed YOULA egato, when ˆ ( ˆ ) ˆ ˆK ˆ Rn R ˆ n ( ) ˆK (24) ˆ Rn The obtained eationhip ae hown in Tabe VII. 76

5 TABLE VII. THE OMLEMENTARY SENSITIVITY FUNTIONS WITH OBSERVERBASED YOULA REGULATOR Stem idea nomina ea fnction TABLE VIII. T Q ˆ ( ) Ĥ ˆ THE RELATIVE DESIGN ERRORS WITH OBSERVERBASED YOULA REGULATOR Stem idea nomina ea fnction x 0 0 Ĥ acate now the eative deign eo x obtained fo obevebaed YOULA egato, which ae mmaized in Tabe VIII. V. SENSITIVITY REDUTION BY DIFFERENT OBSERVER REGULATORS Invetigate the enitivit edction fo thee impe egato fo. Fit eect an integating (I) egato, when A The fite attenating the eo i Ĥ( jω) ˆ A ˆ jω Fo a popotiona integating (I) egato A T (25) 0 ; ω 0 (26) ; ω (27) Fo a popotiona () egato the fite attenating the eo i Ĥ( ˆ 0 jω) ˆK ˆ ˆ ω A (29) ; ω 0 ; ω (30) Thi mean that zeo enitivit at the ow feqenc domain ( ω 0 ) can be eached b chooing age obeve egato gain within the tabiit domain. Fo a phae ead/ag egato the fite i A T 2 T A ˆ ( 0 ) Ĥ( jω) ˆK ˆ T 2 A T ˆ ; ω 0 ; ω (3) (32) The above egato tpe mean that no caica egato can datica edce the mode eo in the impotant medim feqenc domain. Fo ch ppoe pecia egato oophaping methodoog mt be appied. VI. SIMULATION EXAMLES onide a impe fit ode poce and it mode a A T 0 Â.5 ; ˆ ˆT 20 (33) the fite become Ĥ( jω) ˆK ˆ T A jω jω 0 ; ω 0 ; ω ˆ (28) Seect the deign goa to peed p the opeation five time, i.e. eect a efeence mode T n 2 (34) The above imit vae mean that I tpe obeve egato can povide zeo enitivit at the ow feqenc domain ( ω 0 ), o the can toeate age eo in the poce gain. The YOULA egato baed on efeence mode deign [], [2] i 77

6 .5 Ampitde 0.5 Ĉ Q ˆ ˆ ˆT T n whee the mode baed YOULA paamete i appied. T ˆ Â Fige ( ˆT ) 20 T n ˆ.5 20 A T Itpe obevebaed YOULA egato Obeve baed Yoa egato ˆ ŷ ε (35) (36) The obevebaed YOULA egato i hown in Fig. 8. Thi cheme can be fthe impified a Fig. 7 how. It i inteeting to how the tep epone of the diffeent eement in thi cheme. Fig. 9 how thee fnction fo the te poce, the mode, the efeence mode and the obevebaed coed conto oop T. The eached eo: T i ao hown in the fige. VII. ONLUSIONS The YOULA paamete baed egato deign i an exceent too fo cae when the openoop poce i tabe. Thi appoach give expicit anatica foma fo the deign pocede. Unfotnate the diffeent enitivit meae fo ch egato ae miing fom the conto efeence. Thi pape tie to eiminate thi gap giving a detaied anai fo the eative enitivit meae of thee egato. The pape ao incde the extenion of the obeve pincipe fo YOULA egato edcing the mode eo imia to the caica tate feedback/obeve topoogie. The infence of the diffeent obeve egato fo the fite attenating the eo i ao hown. Fina a impe imation et i hown whee the mode eo i 00 % in the time contant and 50 % in the gain of the ea poce. The imation cea how that ve good et can be obtained combining the YOULA egato and the obeve pincipe. AKNOWLEDGMENT Thi wok wa ppoted in pat b the MTABME onto Engineeing Reeach Gop of the HAS, at the Bdapet Univeit of Technoog and Economic. 0 T Fige 9. The mot impotant tep epone if the efeence igna i a nit tep Fit eect a Itpe obeve egato A T (37) REFERENES [] Keviczk, L. (995). ombined identification and conto: anothe wa, (Invited pena pape.) 5th IFA Smp. on Adaptive onto and Signa oceing, AAS'95, Bdapet, H, 330. [2] Keviczk, L. and. Bánáz (200). Geneic twodegee of feedom conto tem fo inea and noninea pocee, J. Stem Science, Vo. 26, 4, pp [3] Keviczk, L. and. Bánáz (20). Mode eo in obeve baed tate feedback and Yoapaametized egato, 9th Mediteanean onf on onto and Atomation MED20, of, GR, pp [4] Kčea, V. (975). Stabiit of dicete inea feedback tem, 6th IFA onge, Boton, MA, USA. [5] Maciejowki, J.M. (989). Mtivaiabe Feedback Deign, Addion Wee. [6] Yoa, D.., Bongiono, J.J. and. N. L (974). Singeoop feedback tabiization of inea mtivaiabe dnamica pant, Atomatica, Vo. 0, 2, pp