Chapter #5 EEE Control Systems

Transcription

1 Sprig EEE Chpter #5 EEE Cotrol Sytem Deig Bed o Root Locu Chpter /

2 Sprig EEE Deig Bed Root Locu Led Cotrol (equivlet to PD cotrol) Ued whe the tedy tte propertie of the ytem re ok but there i poor performce, i.e. we wt to move the pole of the ytem without gretly lterig the gi. Geerl form: Gc () c, b > b T Other form of Led cotroller: G () α T c c c, α <, o: T αt / T, / αt b Deig trtegy:. Expre the deig requiremet deired pole loctio.. Plot the ucompeted root locu d check if there i gi tht brig the ytem to tht loctio, i.e. if P cotroller i eough. 3. If ot, plce the zero t the rel prt of the deired pole loctio, i.e. ζω.. Fid the vlue of the pole by uig the gle criterio:, i.e. rg ( rel pole) rg( rel zero) π 5. Fid the vlue of c by uig the mgitude criterio, i.e. GH. 6. Check to ee if the ew competed ytem tified the deig pecifictio. 7. If ot repet by plcig the zero t differet loctio. Chpter /

3 Sprig EEE Exmple Led Cotrol Aume the ytem: G () ( ) Crete cloed loop (uity feedbck trtegy) uch the cloed loop ytem will hve ω rd d ζ. 5. Solutio / R E G c () U G() C If imple proportiol cotroller i ued the the CLTF i: ' G CL () ( ) ' E R () () E ' () R() E lim ' ' Ad the root locu i: um[] de[ ] rlocu(um,de) Chpter 3/

4 Sprig EEE.5 Root Locu Imgiry Axi Rel Axi Sice the required dmpig fctor i.5 thi implie gle of 6 o which will give ω ω d thi i chieved for : d Root Locu Imgiry Axi Sytem: y Gi: Pole: -.73i Dmpig:.5 Overhoot (%): 6.3 Frequecy (rd/ec): Rel Axi Chpter /

5 Sprig EEE But for thi gi d dmpig fctor the turl frequecy i ot the required, i.e. the ytem i lower. So we replce the proportiol cotroller with more geerl: The ew ope loop TF i: G b ( ) The loctio of the ope loop pole d zero d the deired cloed loop pole ( ± j 3) i: Chpter 5/

6 Sprig EEE Oe trtegy i to plce the ope loop zero uch it will be equl to the rel prt of the deired cloed loop, i.e. - (i thi exmple it jut hppe to coicide with the ope loop pole). Thi will leve u with the loctio of the pole. Let ume tht the cotroller pole i plced t x-b d crete gle ψ with the rel xi: ψ o 9 φ Chpter 6/

7 Sprig EEE I order for the cloed loop pole to be o the deired poit the ew chrcteritic equtio mut tify the gle coditio: φ ψ ψ 8 ψ 3 d hece b. So the OLTF i: G ( ) ( ) Now we c ue the mgitude criterio to fid the gi: ( ) j 3 ( GH ) j 3 j Chpter 7/

8 Sprig EEE 5 Root Locu Imgiry Axi Sytem: gol Gi: Pole: - 3.6i Dmpig:.5 Overhoot (%): 6.3 Frequecy (rd/ec): Rel Axi Lg Cotrol Ued whe the tedy tte propertie of the ytem re ot tifctory but the pole loctio i ok. Geerl form: Gc () c, > b b T Other form of Led cotroller: G () β T c c c, β >, o: Tβ βt / T, / βt b Deig trtegy:. Expre the deig requiremet gi (uully v ), pole loctio (dmpig fctor d turl frequecy). Chpter 8/

9 Sprig EEE. Check if it i poible to chieve thee with imple proportiol cotroller. 3. If the loctio i poible but t tht poit v i too mll (i.e. big error for rmp iput) the dd lg cotroller uch the cotroller pole will be cloe to the origi (for exmple t -.) d the zero to the left of the pole my time it i required to chieve the deired v.. Ue the mgitude criterio to fid the ecery cotroller gi. 5. Check to ee if you hve chieved the deig pecifictio. Exmple Lg Cotrol A plt trfer fuctio i G () tht will chieve the followig:.6 ( )( ). Crete cotrol tructure ζ.9, ω. 673 v 5 Solutio R E G c () U G() C Chpter 9/

10 Sprig EEE If imple proportiol cotroller i ued the the CLTF i: () ( )( ) ( )( ) ' '.6.6 G CL Hece the CE i: ( )( ) ' 3 ' 3 A thi i 3 rd order ytem the geeric CE c be writte : ( )( ) ( ) ( ) 3 ω ω ζω ζω ω ζω By equtig the correpodig coefficiet:.3.6, ' ' ω ζ ω ω ζω ζω Hece remrkbly the deired pole loctio i chieved eve without y extr proportiol cotroller. But: ( )( ).53.6 lim v Hece we eed to icree v time (more or le). Chpter /

11 Sprig EEE We ue Lg cotroller: We plce the pole t b.5 d hece the zero t. 5 ( we eed to icree the gi time).: G OL c c () c ( )( ).6 ( )( ) j The ew v i: v lim ( )( ) Ad the ew pole loctio i: um.79*[.5] decov(cov(cov([.5],[ ]),[ ]),[ ]) gclfeedbck(g,); dmp(gcl) Eigevlue Dmpig Freq. (rd/) -5.6e-.e 5.6e- -3.6e- 5.73e-i.7e- 6.5e- -3.6e e-i.7e- 6.5e- -.3e.e.3e Led - Lg Cotrol Thi i combitio of Led d lg cotroller (imilr to PID). Geerl form: / T Gc () c c / T / T / ct Chpter /

12 Sprig EEE / T c / T i the led cotroller d / T / ct i the lg cotroller. Exmple Led - Lg Cotrol A plt trfer fuctio i G ().5 will chieve the followig: ζ.5, ω 5 v 8 ( ). Crete cotrol tructure tht Solutio R E G c () U G() C Chpter /

13 Sprig EEE By uig the led cotroller deig tep: Deired pole loctio.33 ψ o 9 φ o φ.33 t φ φ o 65 o ( 8 65) ψ ψ t x.5 ( 35) x Hece c 8.68 c 3. 7 T By chooig ow T o tht the ecod pole will be cloe to the origi: Chpter 3/

14 Sprig EEE.5 v c c lim 8 c / T c / T / T / ct (.5) 8 Ad hece: G c (), G () OL (.5) >>um*cov([.5],[.]); decov(cov(cov([ 8.68], [.88]),[ ]),[.5]); >> gtf(um,de) Trfer fuctio: ^ ^ 9.9 ^3.6 ^.5 >> gclfeedbck(g,) Trfer fuctio: ^ ^ 9.9 ^3.6 ^. >> dmp(gcl) Eigevlue Dmpig Freq. (rd/) -.e-.e.e- -.3e.e.3e -.39e.6ei.97e-.8e -.39e-.6ei.97e-.8e.5. v lim (.5) Chpter /