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
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
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)
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 9 5 6 8b. (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 6 8 7 6 6. (9) R ( ) Nw we need t mplfy equtn (8) nd (9). We cn rewrte equtn (9) nt th frm R R 89 8 5 6Q 6Q 6 6. After nly f the frt three member f equtn we cn expre th equtn n fllwng frm 5 5 6 6. 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
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.
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 8 778 57 5. VÍEČEK, A., VÍEČKOVÁ, M., Sbrní vědecých prcí Vyé šly báňé echncé unverzty Otrv. Otrv,, 6., SBN 8-8-65-. VÍEČKOVÁ, M., VÍEČEK A., SMUNÝ, L., FARANA, R., WÁGNEROVÁ R., Metdy yntézy ytémů řízení zlžené n delt mdelech. echncá zpráv grntvéh pretu //86,.