Control systems

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Melvyn Flynn
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1 Last tim,.5 Cotrol sstms Cotrollabilit ad obsrvabilit (Chaptr ) Two approahs to stat fdbak dsig (Chaptr 8) Usig otrollabl aoial form B solvig matri quatios Toda, w otiu to work o fdbak dsig (Chaptr 8) Othr stat-fdbak problms Rgulatio ad trakig Robust trakig ad disturba rjtio Stabilizatio of uotrollabl sstms Full dimsioal stimator SISO as via obsrvabl aoial form MIMO as b solvig matri quatio Rdud-ordr stimator

2 Projt ad Fial Eam Du pm, Ma 8, 8 Fial am problms will b postd at lass wbsit at pm, Ma 7. Th writt part of th projt should b omplt with all th rsults larl prstd. poits out of will b giv o prstatio. ll th Matlab ad Simulik fils for th projt ad th fial am should b otaid i a CD for possibl vrifiatio. Rgulatio ad Trakig ( 8.) Grall, rgulatio is about brigig th output or stat to rtai dsird valu asmptotiall ad kp it thr. It a b trasformd ito a stabilizatio problm Trakig is a rlatioship btw th output ad ad rfr sigals. It dsribs th proprt how th output (t) follows a dsird rfr r(t). Th simplst trakig problm is to trak a stp sigal. mor ompliatd as is to trak a siusoidal sigal, a polomial sigal, or priodi sigals. W d mor advad tools to addrss th sod as.

3 Trakig a stp sigal Rall that w us u r k to stabiliz a sstm ad th rsultig losd-loop sstm is & ( bk) br, B hoosig k appropriatl, th trasfr futio from r to is g(s) (si bk) βs βs βs β b s ( k )s ( k )s ( k )s k Suppos that β. Th th DC gai from r to is g()β / ( k ). If g() is ad r(t) is a stp, th (t)-r(t). If g(), w d to itrodu a fdforward gai p, i, lt u pr-k, with p( k )/β. Th ŷ(s) βs βs βs β gf (s) p rˆ(s) s ( k )s ( k )s ( k )s k g f () Th output (t) a trak a stp sigal r(t). 5 Diagram & bu, u pr-k r u & p b Fdforward gai k g f ()pβ /( k ) ŷ(s) βs βs βs β gf (s) p rˆ(s) s ( k)s ( k)s ( k)s k Commts Th fdforward stratg should work wll wh th paramtrs ar aurat ad thr is o tral disturba. Howvr, if th paramtrs hav rrors, p ( k )/β,th th fial DC gai ma ot b atl o. lso, som disturba ma aus stad-stat trakig rrors. Robust trakig problm is formulatd to dal with ths issus.

4 Robust Trakig ad disturba rjtio Cosidr a op-loop sstm & bu bw, whr w is th disturba. ssum that (,b) is otrollabl. Suppos that thr ar urtaitis i, b ad δ, b bδb, δ. Robust trakig rquirs (t) to follow a stp r(t) i th prs of urtaitis ad disturbas. & bu bw, w r u & b How to driv u from r ad? 7 proposd ofiguratio & bu bw, w r & a a v u & /s k a b & a r u k k a a k & bk bk a a bw ( bk) bk a bw & bk bk a b r w, [ ] & a a a b bk bk a Lt L, bl. L bl[ k k a ] If (si-) - b has o zro at s, th ( L, b L ) is otrollabl Th th igvalus of L b L [k k a ] a b arbitraril assigd, 8 itral stabilit a b ahivd.

5 Now w disuss trakig ad disturba rjtio si bk bk a Lt Δf ( s) dt(si L bl[k k a ]) dt s - Suppos (si - bk) b N(s)/D(s) Th Δ (s) sd(s) k N(s) Lt (s) r (s) w (s) ĝr (s)r(s) ĝ f r (s) k an(s) k an(s) ĝr (s) ĝ r () r(s) Δf (s) sd(s) k an(s) sn(s) ĝ w (s), ĝ w () If w(t) is a stp sigal, Δ (s) w (t) f (s)w(s) Colusios Th DC gai from r to is alwas. Trakig stp sigal asmptotiall v if paramtrs,b, hag. Th DC gai from w to is alwas. Stp disturba a b rjtd. w a 9 Eampl u s s (s) s s s go Dsig a robust trakig otrol stratg suh that traks a stp sigal r(t) asmptotiall. Stat spa ralizatio of g o (s), B, C, L [ ] b L b, Th basi rquirmt is stabilit. Th ovrg rat dpds o th igvalus of L b L k L (k L [k k a ]) W first hoos k L that assig th igvalus. 5

6 W us th sod approah of pol assigmt. Pik Pik k[ ]; F Tlap(L,-F,-bL*k) k L k*iv(t) k L [ ] k[. -..], ka85. (t) is plottd i rd urv. If w pik.. F 8 Th trakig prforma is improvd. S (t) plottd i blu tim(s) Th simulik modl w You a tr to hag th paramtrs, or b ad th trakig proprt is maitaid.

7 rspos i th prs of stp disturba.5 w(t) Problm st #. Dsig a robust trakig stratg for th sstm u s s go(s) s s s so that th output follows a stp sigal asmptotiall. Choos dsig paramtrs so that th losd-loop pols ar at -j, --j, -5 ad -. Simulat th sstm from ts to ts. (prit th simulik modl) ) with th giv g o (s) ) Kp all th dsig paramtrs but rpla g o (s) with s ( δ ) s g(s) for δ.5,, s s ( δ ) s Plot (t) for ah of th ass with iitial oditio for th stat. You ma plot all rsposs i th sam figur ad idtif thm with th valu of δ. What is th maimal δ to mak th losd-loop sstm ustabl? 7

8 Stabilizatio of uotrollabl sstms Rall If (,B) is otrollabl, th th igvalus of (BK) a b arbitraril assigd. What if (,B) is ot otrollabl? Ca th sstm b stabilizd? Th origial sstm & Bu Suppos that th sstm is trasformd (b z P ) ito th followig - z z& z Bu, PP, B PB, z z z& z B u, (, B ) otrollabl z & z Not ig() ig() ig( ) ig( ) 5 z& z & Not z B u, z (, B ) ig() ig() ig( ) ig( ) Cosidr a stat fdbak u r - K r - Kz r - z& BK z & BK otrollabl z B r, z [ ] z K K, K KP z Th ig( - BK) ig( - BK) ig( BK) ig( ) Colusio Si (, B ) is otrollabl, ig( BK) a b arbitraril assigd ig( ) aot b hagd b a stat fdbak. Th ar alld uotrollabl mods. For th sstm to b stabilizab l, th uotrollabl mods hav to b stabl, i, R(λ ( )) < for ah i. i 8

9 Toda s topis Othr stat-fdbak problms Rgulatio ad trakig Robust trakig ad disturba rjtio Stabilizatio of uotrollabl sstms Full dimsioal stimator SISO as via obsrvabl aoial form MIMO as b solvig matri quatio Rdud-ordr stimator 7 Stat Estimators Prviousl, w assumd that is availabl. Howvr, grall is ot availabl Q. What to do? Costrut a sstm to stimat ~ Stat stimator (or obsrvr) u LTI Sstm Stat Estimator ^ Wat (t) to b a "good" stimat of (t), i, (t) (t) should go to zro asmptotiall with fast ovrg rat. 8 9

10 Mai rsult If th sstm is obsrvabl, th b proprl dsigig th stat stimator, th pols of th damis assoiatd with (t) (t) a b arbitraril assigd Error gos to zro as quikl as possibl W will disuss svral tps of stimators Full-dimsioal stat stimator Sigl variabl as Multivariabl as Rdud-dimsioal stat stimator Fiall, otig stat stimator ad stat fdbak 9 Full-Dimsioal Stat Estimators How to stimat basd o u ad? & Bu; C ~ ssumig that D proposd ofiguratio u. B Itgrator C L & - B Itgrator C u Dupliat th sstm damis Mak orrtio o d /dt wh ad ar diffrt Pik th orrtio as a liar futio of ( )

11 Will th proposd ofiguratio work? Th origial sstm & Bu; C Th dupliatd sstm (stimator) & ( ), C Bu L Th orrtio trm L(- ) plas th sstial rol. Ca w hoos L appropriatl to mak approah? W ow aalz th sstm First, osidr th stimator damis & Bu L( ), C Bu L[ C C ] ( LC) LC Bu Nt, osidr th damis of & & & ( Bu) ( LC) LC Bu ( LC)( ) ( LC) & ( LC) [ ] If th igvalus of ( - LC) hav gativ ral parts, a rror will ovrg to,. Ca th igvalus of -LC b arbitraril assigd?

12 Thorm. If th sstm (,C) is obsrvabl, th all th igvalus of ( - LC) a b arbitraril plad, providd that ompl igvalus appar i pairs Proof Th rsult follows from dualit Th igvalus of (-LC) ad ( -C L ) ar th sam. B Thorm.5 (, C) is obsrvabl iff (, C ) is otrollabl (pag 55) Now, si (, C) is obsrvabl, (, C ) is otrollabl Eigvalus of C L a b arbitraril plad Not ll th stat fdbak dsig produr a b usd to dsig stat stimator, ad som of thm will b highlightd t Full Dimsioal Stat Estimator, SISO Cas Thorm. If a SISO sstm is obsrvabl, th it a b trasformd, b a quivalt trasformatio, to a obsrvabl aoial form Obsrvabl Caoial Form β β & β β u; [ ] du (si ) βs βs β b d s s s d

13 Th trasformatio matri is formd as follows q ' q 'q ' '' ' i i q i 'q i i' (') (') L i- ' q 'q ' (') (') L - ' For ampl, with Q [(') ' (') ' '' ' ] With P Q, th (PP -, Pb, P - ) is i th obsrvabl aoial form. 5 Eampl & u; [ ] Is th sstm obsrvabl? C o G C C G ~ Obsrvabl λ ( ) ( λ ) λ Δ( λ) λ λ λ λ λ λ 9λ ~, -9,

14 7 ; ' q ' 'q q 7 9 ' 'q q ; 7 Q 9 Pb b [ ]; ; 7 Q' P 8 Now, how to slt L? Cosidr th quivalt sstm i Obsrvabl Caoial Form [ ] u; β β β β & ( ) L & [ ] L l l l l

15 L ( l ) ( l ) ( l ) ( l ) Th haratristi polomial Δ d (s) ( l ) s ( l ) s ( l ) s Suppos that th dsird stimator pols ar { λ } i i ad th dsird haratristi futio is Δ (s) s ˆ s ˆ s ˆ d Th it is lar that l ˆ ; l ˆ ; l ˆ ; l ˆ ˆ 9 ˆ ˆ L ˆ Fiall, Lt L P - L, th LC P P P LCP P ( LC)P -LC also has th dsird igvalus. L is uiqul dtrmid b th dsird igvalus of th stimator. 5

16 Produr for dsigig th stimator gai Stp. Choos th dsird igvalu st {λ i, i,, } of -L ad obtai th offiits of d s s s ) λ (s ) λ )(s λ (s (s) Δ L L Stp. Comput th haratristi polomial of s s s (s) Δ L ad th trasformatio matri,.g., for [ ] Q' P, ' '' ' (') ' (') Q [ ], CP, PP Th - - bk - l l l l Stp i i i Choos l Th - L Stp L. P L Comput -,,.,} i, {λ th dsird igvalus L)P has ( P - L th i -

17 7 Eampl (Cotiud) Fid L s.t. th stimator pols ar all at -, -, - s disussd arlir, th sstm is obsrvabl, ad [ ] u; & ( ) 9 λ λ λ λ Δ ~, -9, Th dsird stimator haratristi futio ( ) ( ) 9 9s s s s s (s) Δ d 9; 9,, L From th prvious ampl, ; 7 P T Q 7 P L P L Vrifiatio [ ] L

18 λ ( L) λ λ 9 8 λ ( λ )( λ ) 78 7( λ ) ( λ ) ( λ λ λ ) 78 ( 7λ 7) ( λ 9) λ λ 9λ 9 ~ s dsird 5 Toda s topis Othr stat-fdbak problms Rgulatio ad trakig Robust trakig ad disturba rjtio Stabilizatio of uotrollabl sstms Full dimsioal stimator SISO as via obsrvabl aoial form MIMO as b solvig matri quatio Rdud-ordr stimator 8

19 Gral MIMO Stat Estimator Dual to MIMO stat fdbak ~ Mthods disussd arlir has a outrpart hr W will disuss th mthod basd o matri quatio, assumig that th sstm is obsrvabl u LTI Sstm full-dimsioal stat stimator with D & Stat stimator ( ), C Bu L Lt -, th rror damis & ( LC) 7 Similar to th stat fdbak dsig, th objtiv is to pik a obsrvr gai L suh that -LC is quivalt to a rtai F whih has th dsird igvalus, i, LC T - FT for a osigular T T FT TLC, Lt L TL T FT LC, Th produr Giv,C ad F. Pik L, solv T FT L C for T. If T is osigular, lt LT - L. Th LC has th dsird igvalus. 8 9

20 9 bout th solutio to T FT L C, w hav Th matri quatio has a uiqu solutio iff ad F hav o ommo igvalus; Th solutio T is osigular ol if (,C) is obsrvabl ad (F,L ) is otrollabl; I as that C has o row, T is osigular if ad ol if (,C) is obsrvabl ad (F,L ) is otrollabl. Th abov rsult a b drivd from th rsults for stat fdbak dsig from dualit takig traspos of th quatio, w obtai T T F C L as ompard with T T F B K for th stat fdbak dsig Eampl u; & Fid T s.t. th stimator pols ar at -5, -5, - Chk obsrvabilit C C C G o ~ Obsrvabl

21 Slt F Slt L 5 F 5 L {F, L } otrollabl Solv T-FTL C with matlab Tlap(-F,,-L*C) T , L T L Vrif ig( LC) { 5, 5, } If w pik L - -7 L B Bu C B Bu C L Tak to b th origial, BI, CI, D B(),C(),Dzros();

22 Eplaatio u & Bu C Ca b quivaltl ralizd with u B v & v C v Bu, C C Th purpos of doig this is to gt ad u B v & v C L(- ) v BuL(- ), C C.5 Th stimatio rror ( ), () tim(s).5.5 Th output

23 Problm st #. For th sstm & u; - Dsig a obsrvr to stimat th stat. Th pols of th obsrvr ar -j, --j ad -7. Simulat th sstm from t to t5 with ()[ ], ()[ ] ad u. Plot th stat (t) i o figur ad (t)- (t) i aothr figur. lso prit th simulik modl. 5 Toda s topis Othr stat-fdbak problms Rgulatio ad trakig Robust trakig ad disturba rjtio Stabilizatio of uotrollabl sstms Full dimsioal stimator SISO as via obsrvabl aoial form MIMO as b solvig matri quatio Nt Tim Rdud-ordr stimator Cotio of stat-fdbak with stat stimatio LQR optimal otrol Problm st # s slids, 5

Control systems (Lecture #11)

Control systems (Lecture #11) .5 Cotro sstms (Lctur #) Last tim, Cotroabiit ad obsrvabiit (Chaptr ) Two approachs to stat fdback dsig (Chaptr 8) Usig cotroab caoica form B sovig matri quatios Toda, w cotiu to work o fdback dsig (Chaptr