State-space feedback 4 Ackermann s approach

Transcription

1 Stte-e feedbk 4 kerm roh J Roiter Slide by thoy Roiter

2 Itrodutio The reiou ideo howed how tte feedbk le ole reiely log the ytem u fully otrollble. x x Bu x x u Kx Both relied o otrol oil form. Thi ideo how how oe fid uitble tte feedbk without reortig to oil form withi the deig roedure. Slide by thoy Roiter

3 Remrk uh of thi reoure derie d exli the reult. If you oly wt to kow the reult go diretly towrd the ed. Slide by thoy Roiter

4 Pole olyomil The roh gie here relie o ler defiitio of two olyomil: the oe-loo ole olyomil. 4 o o the deired loed-loo ole olyomil. o Slide by thoy Roiter

5 Cyley-Hmilto Theorem mtrix tifie it ow hrteriti equtio. Thi imlie tht. Slide by thoy Roiter 5 I The roof i omitted but reltiely trightforwrd d ilble i mot text book.

6 Corollry The followig idetitie re kow to be true. Slide by thoy Roiter 6 I o o o I Fidig K uh tht thi lt idetify i tified imlie hieig trget ole.

7 kerm roh The itetio i to exloit the ltter of thee idetitie d full rk ture of the otrollbility mtrix. Firt defie: I The firt oet i to how tht i omoet of - d exloit thi i the lgebr. NOTE by defiitio: Slide by thoy Roiter I 7

8 kerm roh The firt oet i to exd - d uk the uderlyig truture. Firt oider eh term. Slide by thoy Roiter 8 Bf Bf Bf Deliberte ertio of term with ledig d ledig B.

9 kerm roh The ext te i to how tht oe exre eh term uig the otrollbility mtrix. Slide by thoy Roiter 9 K K f B B B B Bf ] [ ] [ K K f B B B B Bf The ZERO i key

10 kerm roh Next exd - d uk the uderlyig truture. Firt oider eh term. Slide by thoy Roiter Oly i o-zero i the bottom row! I ] [ I

11 kerm roh Show tht i omoet of - d exloit thi i the lgebr. Slide by thoy Roiter ] [ I ] [ ] [

12 kerm roh Exloit the truture of by extrtig oly the lt row of eh ide of the equtio d thu extrtig the feedbk gi K. Slide by thoy Roiter [ ] f K K K i by defiitio oe fid the required tte feedbk uig thi formule.

13 Summry of kerm roh Defie the deired loed-loo ole olyomil. o The required feedbk i gie from the formul where i the otrollbility mtrix. K Slide by thoy Roiter

14 4 EXPLES Slide by thoy Roiter

15 Exmle : Chooe K to et the loed-loo ole t - d -. Slide by thoy Roiter 5 Firt fid the required ole olyomil 4 ; ;.4 C B Fid

16 Exmle : Chooe K to et the loed-loo ole t - d -.. ; ;.4 B Defie the otrollbility mtrix [ B B].8 Ue kerm formule K..95 Slide by thoy Roiter

17 Slide by thoy Roiter 7

18 Exmle : Chooe K to et the loed-loo ole t d ;.4 B ; Firt fid the required ole olyomil C Fid [Ue TLB tediou] Slide by thoy Roiter.75.75I

19 Exmle : Chooe K to et the loed-loo ole t - d ; B..4. ;.46.4 Defie the otrollbility mtrix [ B.5 B B].6... Ue kerm formule K Slide by thoy Roiter

20 Summry Itrodued oet of ole lemet tte feedbk without otrol oil form.. Show tht umig full otrollbility oe ue kerm formule to defie the required feedbk to hiee y deired ole olyomil.. Not er/e exerie i geerl iole ubttil mtrix multilitio d ierio.. Numerilly oorly oditioed for wekly otrollble d lrge dimeio ytem. Slide by thoy Roiter u Kx K

21 thoy Roiter Dertmet of utomti Cotrol d Sytem Egieerig Uierity of Sheffield 6 Uierity of Sheffield Thi work i lieed uder the Cretie Commo ttributio. UK: Egld & Wle Liee. To iew oy of thi liee iit htt://retieommo.org/liee/by/./uk/ or ed letter to: Cretie Commo 7 Seod Street Suite S Frio Clifori 945 US. It hould be oted tht ome of the mteril otied withi thi reoure re ubjet to third rty right d y oyright otie mut remi with thee mteril i the eet of reue or reuroig. If there re third rty imge withi the reoure lee do ot remoe or lter y of the oyright otie or webite detil how below the imge. Plee lit detil of the third rty right otied withi thi work. If you ilude your ititutio logo o the oer lee ilude referee to the ft tht it i trde mrk d ll oyright i tht imge i reered.