positive definite (symmetric with positive eigenvalues) positive semi definite (symmetric with nonnegative eigenvalues)

Transcription

1 Chter Liner Qudrtic Regultor Problem inimize the cot function J given by J x' Qx u' Ru dt R > Q oitive definite ymmetric with oitive eigenvlue oitive emi definite ymmetric with nonnegtive eigenvlue ubject to x x y Cx Bu LQR SOLUTION: Find the oitive-definite olution P of the RE lgebric Rictti Eqution ' P P Q PBR B' P u Kx where K R B' P The oitive-definite olution of the RE reult in n ymtoticlly tble cloed-loo ytem if: the ytem i controllble R > Q C C q q where Cq, i obervble Thee condition re necery nd ufficient We cn define nother outut z where z C x controlled or regulted outut q Therefore LQR deign of double integrtor B C q ume Q R Q C ' C q q

2 , B i controllble C q, i obervble RE: olving 4 P ' P B R K The cloed loo ytem mtrix become BK Cloed loo root re: j dming rtio i.77 note P i ymmetric

3 The loo trnfer function i: K B K I B 65 he mrgin infinite gin mrgin R + - B + + C K mg L L j.77 j.77 j.44 z.77 c

4 .44 L jc.44 c c he Phe@ ω c = - c -8 tn.77 he m rg in -8 z z Phe mrgin = tn tn 6.4 USIN TLB TO ET EXCT RESULTS tlb num = qrt*[ qrt/ ] den = [ ] reult: mrginnum,den gm= ω=.554 Proertie of LQR deign From the RE we cn derive the reltion L j q j * i clr where L K B -loo gin I nd Q C ' C C B q q q q From * we ee L j Thi imlie tht the Nyquit lot of the loo trnfer function of n LQR deign lwy ty outide of unit circle centered t -,. 4

5 In SISO ce, LQR deign h > 6 he mrgin, infinite gin mrgin nd gin reduction tolernce of -6dB i.e. the gin cn be reduced by fctor of before intbility occur. Recll ole lcement doe not gurntee tbility mrgin. igh-frequency roll-off rte Cloed loo trnfer function T j K ji BK B lim T j KB R B' PB j j -db/dec roll off rte t high frequencie - not good for noie ureion Otiml Oberver Klmn Filter Stte etimtion lnt rereented x x Bu roce y Cx v meurement noie noie The otiml filter i given by xˆ xˆ Bu L y Cxˆ 5

6 where L C' R where i the oitive definite olution of ' Q C' R C Q o nd R o re noie covrince mtrice, which rereent the intenity of the roce nd enor noie inut. Require Q, R nd ytem to be obervble. If we combine the Klmn-Bucy Filter otiml etimtor with LQR deign, we hve LQ Liner Qudrtic uin. Let do LQ deign for double integrtor lnt. We lredy hve the LQR deign. For Klmn filter, ume nd R Solving Rictti eqution with b b c we find b b c b nd L C' R Trnfer function of comentor i given by Comrion of LQR nd LQ -LQR h gurnteed tbility mrgin -LQ h no gurnteed tbility mrgin 6

7 -high freq. roll off in LQ cn be > db/dec exhibited by LQR greter noie filtering in LQ -LQ i not robut uncertinty in lnt my cue ytem to go untble Loo Trnfer Recovery LTR LQR > 6 he mrgin infinite gin mrgin LQ no gurnteed mrgin The roertie of LQR cn be recovered ymtoticlly by uing Q o nd R tuning rmeter LQR loo gin, L K B K I B LQ Xˆ BK LC Xˆ LY ˆ I BK LC Y X loo gin, L LQ K I BK LC LC B L 7

8 If the following two condition hold then LQR loo roertie cn be recovered if Then it cn be hown i minimum he R = nd Q =q BB lim L LQ q L The vrible y tht i recovered my be different from the vrible z tht i to be controlled where y Cx nd z C x q Loo Shing Ste Determine the controlled vrible nd et Q=C C nd Q=C q C q et deired loo gin in LQR deign. Ue R tuning rmeter. Select clr q nd olve the filter Rictti eqution ' q BB' C' C L C' 4 Incree q until the reulting loo trnfer function i cloe to the LQR deign Do not mke q too high ince lrge gin in L re required the undeirble -db/dec high freq. roll-off of LQR will be recovered Exmle Double integrtor ytem with Q R= ve 65 he mrgin for LQR deign 8

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11 Robutne Robut tbility tble in the fce of lnt uncertintie Robut erformnce erformnce met even in the fce of lnt uncertintie Two imortnt roertie of feedbck enitivity reduction diturbnce rejection enerl feedbck ytem N D R Y Trcking error y - r e N D R E ctutor outut i.e. lnt inut i given by N D R U note: U E E U Define the following term J return difference S enitivity T comlementry enitivity note: T S Uing thee definition ytem outut: N R T D S Y trcking error: N D R S E

12 lnt inut: U S R D N From thee exreion we ee tht we need Diturbnce rejection: From Y exreion we ee we require S mll >> ince SD Trcking: S mll Noie ureion: From Y we hve TN require T mll 4 ctutor limit: From U exreion wnt S bounded Trcking nd Diturbnce rejection require mll S Noie ureion require mll T however S + T = however commnd inut nd diturbnce re low frequency where meurement noie i high frequency ignl kee S mll in low frequency rnge nd T mll in high frequency rnge lo S T mking T mll we reduce control energy Loo in Proertie Low Frequency id. Frequency igh Frequency Performnce R igh in Smooth Trnition for good mrgin Diturbnce Rejection D igh in Noie Sureion N Low in Uncertinty odeling Two ctegorie -- tructured uncertinty untructured uncertinty We will del with untructured uncertinty

13 ~ dditive uncertinty: ctul model ~ model uncertinty or error ultilictive uncertinty: ~ m inut uncertinty outut uncertinty Robut Stbility We y comentor robutly tbilize ytem if the cloed-loo ytem remin tble for the true ~ lnt. Robutne reult cn be derived uing the mll gin theorem. Smll in Theorem The cloed-loo ytem will remin tble if no ince

14 then cloed-loo tbility i gurnteed if There i no oibility of encirclement of -, oint by Nyquit lot. Two eqution tht the mll gin theorem cn hel u to nwer iven tht the uncertinty i tble nd bounded, will the cloed-loo ytem be tble for the given uncertinty? For given ytem, wht i the mllet uncertinty tht will detbilize the ytem? To nwer thee quetion we firt do ome block digrm mniultion With multilictive outut uncertinty where Determine, the trnfer function een by m 4

15 By mll gin theorem, cloed-loo ytem will be robutly tble if m i.e. m T comlementry enitivity T If the uncertinty i bounded by o tht m then the cloed-loo ytem will be tble if T or T Thi nwer the firt quetion Second quetion: find the ize of the mllet tble uncertinty tht will detbilize the ytem Becue the uncertinty mut be mller tht /T, it mut be mller tht the minimum of /T. We mut find the mximum of T. Define r ut j u = uremum let uer bound Then the mllet detbilizing uncertinty, we cll thi the multilictive tbility mrgin or S, i given by S r For dditive uncertinty cloed-loo will be robutly tble if or S 5

16 if uncertinty i tble nd bounded by then we gurntee cloed-loo tbility if S or S we cn define dditive tbility mrgin S by S u j S j Exmle 5, he mrgin: 8 gin mrgin:.8 9dB Find S nd S: S Find ek of T comlementry enitivity function ek =.5 S =.65 the ytem will be robutly tble gint unmodelled multilictive uncertintie with trnfer function mgnitude <.65 6

17 7 Problem.9. m ~ ~ m m b. c. ST: m m m

18 8 dditive uncertinty ~ ~ ST:

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20 Bic Bode gnitude Plot o o -db/dec o o o +db/dec o Q Q Q o -4dB/dec Q o o Q Q +4dB/dec o If Q < then root re rel. Fctor the exreion nd ue the reulting roduct of two firt order trnfer function to find mgnitude reone. o Q o

21 Exmle Q o o o o o Q Q Q o Q o

22 Exmle The tructure of ytem h been determined to be given by Q where nd Determine the condition under which robut tbility i ured. nwer By ST we require or From the bove digrm we cn ee tht we require nd Q o o Q Q or

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