Modification of Symmetric Optimum Method. 1 A synthesis of linear one-dimensional regulatory circuits

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1 XXX. ASR '5 Semnr, ntrument nd Cntrl, Otrv, Aprl 9, 5 9 Mdfctn f Symmetrc Optmum Methd MZERA, Rmn ng., Ktedr AŘ-5, VŠB-U Otrv, 7. ltpdu, Otrv rub, 78, rmn.mzer.f@vb.cz Abtrct: h cntrbutn del wth mdfctn f the ymmetrc ptmum methd. h methd degnted fr the ynthe f lner ne-dmennl regultry crcut whe tructure cn be dvded nt cntrller nd cntrlled ytem. t enble the degn f cntnuu cntrller nly. New equtn fr pecfc cmbntn f cntrller nd cntrlled ytem been derved, whch prvde n pprch fr degnng bth the cntnuu nd dcrete cntrller. Dervtn hve been crred ut bed n delt mdel prncple. Keywrd: ynthe, cntrller, delt mdel A ynthe f lner ne-dmennl regultry crcut Frt we hve t pecfy wht we undertnd wth the term regultry crcut nd ynthe f regultry crcut. n th ce we wll be delng wth the ynthe f lner ne-dmennl regultry crcut whe tructure cn be dvded nt cntrller nd cntrlled ytem [Blátě, ]. he ynthe prce del wth degnng dutble cntrller prmeter. Fgure Regultry crcut he ymmetrc ptmum methd he ymmetrc ptmum methd epeclly utble fr ce when the trnfer functn f n pen regultry crcut h thrd degree multnmnl plynml n the denmntr nd the number f ntegrtr q. Frt, we wll decrbe the dervtn prce f n equtn fr the clcultn f dutble cntrller prmeter by L-trnfrm. We wll hve cntrlled ytem whch wll be expreed by the trnfer functn: G S, ( ) nd we wll che cntrller fr th cntrlled ytem: G R. hen the trnfer functn f th pened regultry crcut

2 XXX. ASR '5 Semnr, ntrument nd Cntrl, Otrv, Aprl 9, 5 G, nd the trnfer functn f the cled regultry crcut G p p w. he ymmetrc ptmum methd bed n the generl equtn A. (5) t men, the determntn f the dutble cntrller prmeter wll be bed n the ytem f equtn A, m,...,,, (6) where m the number f chen dutble cntrller prmeter. n ur ce m nd we wll lve thee equtn A A, (7) where,, re ceffcent f chrctertc multnmnl. When we lve thee equtn we btn tw equtn fr clcultng the dutble cntrller prmeter, (8). (9) After ubttutn nt nd we btn the trnfer functn f the pen regultry crcut n tndrd frm 8 G nd the trnfer functn f the cled regultry crcut n tndrd frm fr ymmetrc ptmum methd 8 8 G w. he trnent chrctertc f cled regultry crcut hwn n fgure. Fgure - rnent chrctertc f cled regultry crcut degned by ymmetrc ptmum methd [ ] t t h wy,%, κ

3 XXX. ASR '5 Semnr, ntrument nd Cntrl, Otrv, Aprl 9, 5 Dervtn f equtn fr clcultn dutble prmeter f cntrller bed n δ mdel We wll hve the me trnfer functn f cntrlled ytem nd cntrller n prevu ce. We wll crry ut dcretztn f equtn. Dcretztn bed n th equtn γ Gw ( γ ) D L Gw. γ t hen we wll btn dcrete D - trnfer functn f the cntrlled ytem ( b ) γ γ ( γ b) b G S ( γ ), b e. Nw we need t btn D the trnfer functn f the cntrller. he reltn between the nput nd utput f the S cntrller ung bcwrd rectngle ummtn u( ) e( ) e( ). Fr trnfer we wll utlze th prperty f D - trnfrmtn γ D x( ) X ( γ ). (5) γ After trnfer we wll btn γ U ( γ ) E( γ ) γ. (6) We expre D trnfer functn f the S cntrller U ( γ ) γ G R γ. (7) E γ γ n rder t determne the chrctertc multnmnl nd t ceffcent we need t expre ether the trnfer functn f the cled regultry crcut r the trnfer functn f the pen regultry crcut. We wll ue ecnd wy [( b ) γ b]( γ γ ) γ ( γ b) M G ( γ ) GR ( γ ) GS ( γ ) N. (8) Nw we cn expre the chrctertc multnmnl whch generlly defned n n N( γ ) M N nγ n γ... γ. (9) n ur ce we btn N( γ ) γ [ b ( b b )] γ. ( b b b ) γ b We expre ceffcent,, nd frm th equtn b, ( b b b ), b ( b b ),. We cn ubttute ceffcent, nd nt the equtn fr clcultng A. We wll expre the generl equtn fr clcultng the gn f the cntrller frm th equtn. b b b b b b. (5)

4 XXX. ASR '5 Semnr, ntrument nd Cntrl, Otrv, Aprl 9, 5 We ubttute th equtn nt, nd nd ceffcent, nd ubttute nt equtn fr clcultn A. We wll expre generl equtn fr clcultn frm th equtn 8 8 b O 9 b b b O O 7b b 8b 7b 8 6b b. (6) Frm equtn (5) nd (6) t evdent tht the dervtn f thee equtn qute cmplcted nd dng tht mnully te t much tme. he ymblc mthemtc tlbx n MALAB w ued fr th dervtn. n rder t mplfy thee equtn we ued the pprxmtn b e. (7) ( ) We btned the equtn fr clcultng n th frm fter pprxmtn (7) nt (5) 8 (8) ( 8 5 ) nd equtn fr clcultn n th frm fter pprxmtn (7) nt (6) [ Q 6 R R 8 Q 5 Q ] 6 Q Q (9) R ( ) Nw we need t mplfy equtn (8) nd (9). We cn rewrte equtn (9) nt th frm R R Q 6Q 6 6. After nly f the frt three member f equtn we cn expre th equtn n fllwng frm We wll btn the mplfed equtn fr clcultn fter lt mdfctn. Nw we mplfy (8). Frt we neglect 5 nd then we ubttute nd we wll btn mplfed equtn fr clcultn ( 8. ) f we cnder the lmt ce then we btn ytem f equtn me we btned n dervtn when we ued L - trnfrmtn

5 Exmple XXX. ASR '5 Semnr, ntrument nd Cntrl, Otrv, Aprl 9, 5,. (5) We wll hve the mthemtc mdel f cntrlled ytem n th frm G S. (6) ( ) Frt we wll che cntrller. he mplng perd wll be n th ce. Cmprn f clculted dutble prmeter f cntrller tted n ble. ble. Clculted ptml dutble prmeter f cntrller equtn (9) equtn equtn (8),5 equtn,5 Nw we che S cntrller nd the mplng perd wll be,. Cmprn f clculted dutble prmeter f cntrller tted n ble. ble. Clculted ptml dutble prmeter f S cntrller equtn (9),895 equtn,85 equtn (8),75 equtn,95 Subequently we perfrmed numercl multn n the MALAB envrnment. We ee the trnent chrctertc n Fgure. We cn ee preumptve verht,% n ce when we ued cntrller nd mplng perd w. n the ecnd ce when we the ued S cntrller the verht bgger. h lgcl becue f the mplng perd.

6 XXX. ASR '5 Semnr, ntrument nd Cntrl, Otrv, Aprl 9, 5,8 y t, t[ ] 5 Fgure rnent chrctertc regultry crcut degned by ymmetrc ptmum methd 5 Cnclun n th cntrbutn we delt wth the mdfctn f the ymmetrc ptmum methd. We explned wht we undertnd wth the term regultry crcut nd ynthe f regultry crcut. Subequently we explned when the ymmetrc ptmum methd utble. hen we hwed the dervtn f the equtn fr clcultng the dutble cntrller prmeter ung L trnfrmtn nd ubequently by the men f delt mdel. he next gl t derve the equtn fr clcultng the dutble cntrller prmeter fr ther cmbntn f cntrlled ytem nd cntrller. 6 Reference BALÁĚ, J. Autmtcé řízení. BEN techncá ltertur, RAHA,, SBN 8 7. HALÁSEK,. Syntéz lneárních ytémů řízení n záldě delt mdelů. Otrv: FS VŠB - UO. Dertční práce, veducí D: M. Vítečvá,. MNDEKOVÁ, D., užtí delt mdelů př yntéze lneárních regulčních bvdů. Otrv: FS VŠB-UO. Dplmvá práce, veducí D: M. Vítečvá, 995 VÍEČKOVÁ, M., Seřízení regulátrů metdu nverze dynmy. Otrv: Srpt FS VŠB UO,. VÍEČKOVÁ, M., Mtemtcé metdy v řízení. L- Z-trnfmce. Otrv, 999, SBN VÍEČEK, A., VÍEČKOVÁ, M., Sbrní vědecých prcí Vyé šly báňé echncé unverzty Otrv. Otrv,, 6., SBN VÍEČKOVÁ, M., VÍEČEK A., SMUNÝ, L., FARANA, R., WÁGNEROVÁ R., Metdy yntézy ytémů řízení zlžené n delt mdelech. echncá zpráv grntvéh pretu //86,.