Discrete Model Predic.ve Control

Transcription

1 Dcrete Model Predcve Control Lowell Brown

2 Outlne MPC: Bc Ide Stte Spce nd Augented Syte Model OpDzDon Wndow Recedng Horzon Control Exple

3 Reference Model PredcDve Control Syte Degn nd IpleentDon Ung MALAB Lupng Wng, Sprnger 29 Model Bed PredcDve Control-A PrcDcl Approch J A Roter, CRC Pre 24

4 MPC: Bc Ide

5 MPC: Bc Ide PredcDon rele on tte-pce odel OpDl trjecry et preter defne ncreentl oveent Δu ObjecDve yte error funcdon between et-pont gnl nd predcted output gnl

6 rckng Error Setpont r 8 6 e e 2 e 3 e 4 Output y te (ec)

7 Bc Ide: Increentl Control Error funcdon: eureent objecdve OpDl oludon obtned for et ncreentl oveent Δu(k ),Δu(k ),,Δu(k N c -) Only t ple Δu(k ) ued OpDzDon repet on next ple perod Bed on recedng horzon prncple Feedbck nturlly ncorported n yte degn

8 Bc Ide: Ebedded Integrr() Degn odel: ugented plnt odel wth ebedded ntegrr() Integrr() ke() gnl opdzed be Δu(k ) nted equence u(k ) Ebedded ntegrr() elnte() tedy tte error Stedy tte nfordon bout or tte vrble not requred Iportnt for plnt wth uldple nput nd uldple output (MIMO)

9 Stte Spce & Augented Syte Model

10 Stte Spce & Augented Syte Model Syte Model x (k )A x (k)b u(k), C x (k), Eq Model ue : ed D Dfference [ Syte Model ] x (k ) x (k) A (x (k) x (k )) B (u(k) u(k )) (k )A (k)b u(k) Eq y(k ) C (x (k ) x (k)) C (k ) y(k ) C A (k)c B u(k) Eq 2

11 Stte Spce & x(k) [ ] Augented Syte Model Stte Vrble x(k) [ Vecr (k) ] Cobne Eq nd Eq2 x(k) {}}{ [ ] A B {[ }} ]{ [ }} ]{{}}{ ]{ [{}} (k ) A o ]{{ [ } [ ] (k) B u(k) y(k ) C A C B C {[ }}{ o ] [ ] (k) Where: n [ {}}{ ] o wll be ued n th

12 Stte Spce & Aug Syte Model: Ebedded Integrr() Egenvlue Augented Syte? A x(k) ]{}}{ ]{{}} { [ [ ] [ ] A o ] { [ } [ (k) x(k) {[ }}{ (k ) y(k ) C A C {[ }}{ o ] [ ] [ ] [ ] (k) Where: ke Deternnt [ ] λi A o ρ(λ) det(λi A) det[ C A (λ ) (λ ) det(λi A ) At let one pure ntegrr {}}{ [ ] B {[ }} ]{ B u(k) C B n [ {}}{ ] o wll be ued n th

13 Opzon Wndow

14 Opzon Wndow Gol: Clculte predcted plnt output wth future gnl djutble vrble current De k opdzdon wndow N p ple long N_p predcdon horzon Future trjecry: Δu(k ),Δu(k ),,Δu(k N c -) where N c horzon N c dctte nuber preter ued cpture trjecry

15 Opzon Wndow Future tte vrble denoted : x(k k ),x(k 2 k ),, x(k k ),, x(k N p k ), where x(k k ) predcted tte vrble evluted t k gven current plnt nfo x(k ) ypclly elect nd predcdon horzon uch : N c N p

16 Opzon Wndow Future tte vrble propgted equendlly wth ugented tte pce odel x(k k )Ax(k )B u(k ) x(k 2 k )Ax(k k )B u(k ) A 2 x(k )AB u(k )B u(k ) x(k N p k )A N p x(k )A Np B u(k )A Np 2 B u(k ) A N p N c B u(k N c ) Predcted output vrble re lr n for e k : y(k k )CAx(k )CB u(k )

17 Opzon Wndow All predcted vrble re forulted n ter current tte vrble x(k ) nfordon nd future oveent Δu(k j) where j,,,n c - Defne Y vecr predcted output (N p ) Defne ΔU vecr predcted ncreentl tep (N c ) [ Y [ y(k k ) y(k 2 k ) y(k 3 k ) y(k N p k ) ] [ ] [ u(k ) u(k ) u(k 2) u(k N c ) ]

18 Opzon Wndow Output cn be wrfen where F CA CA 2 CA 3 CA N p [ ] Y Fx(k ), ; Eq 3 CB CAB CB CA 2 B CAB CB CA Np BCA Np 2 BCA Np 3 BCA N p N c B

19 Opzon Wndow Set pont gnl r(k ) ue et-pont gnl contnt n opdzdon {}}{ wndow [ ] Set-pont vecr wth n opt wndow R Cot FuncDon N p [{}}{] r(k ), J (R Y ) (R Y ) R, Eq 4

20 Opzon Wndow Cot FuncDon {}}{ [ ] J (R Y ) (R Y ) R, frt ter: n error between predcted output nd et-pont gnl Eq 4 econd ter reflect ze ΔU when J ll poble R dgonl trx for R r w I Nc N c (r w ) where r w tunng preter for cloed loop

21 Opzon Wndow Inert Output funcdon (Eq 3) n Cot funcdon J (Eq 4) nd J expreed : J (R Fx(k )) (R Fx(k )) 2 (R Fx(k )) ( R) Mnze J (Eq 5) wth repect ΔU uch : J 2 (R Fx(k )) 2( R),, OpDl gnl becoe: ( R) (R Fx(k )), Eq 6 Eq 5

22 Opzon Wndow ( R) Note { trx }} { clled Hen trx nd ued ext Recll R_ expreon et-pont nfordon: N p {}}{ R [ { ] }} r(k ){ R r(k ), he opdl oludon gnl lnked et-pont gnl r(k_) nd tte vrble x(k_) uch Eq 6 cn be rewrfen : ( R) ( R r(k ) Fx(k )) Eq 7

23 Recedng Horzon

24 Recedng Horzon OpDl preter vecr ΔU contn Δu(k ),Δu(k ),,Δu(k N c -) Only frt ple Δu(k ) pleented At next ple perod x(k ) ued clculte new equence ΔU Procedure repeted nd yeld recedng horzon lw

25 Illutron Recedng Horzon Illutrn recedng horzon Splng ntnt Horzon wndow

26 Relon LQR MPC lnk clcl lner qudrdc regulr (LQR) for uffcently long predcdon horzon Dfference between predcdve nd LQR predcdve olve opdzdon proble ung ovng De horzon wndow LQR olve e proble wthn fxed wndow

27 MPC Advntge over LQR Movng De horzon wndow Ablty perfor rel-de opdzdon wth hrd contrnt on plnt vrble

28 Cloed-Loop Control Syte

29 {}}{ Cloed-Loop Control Syte Recll ΔU olved ung Eq 7: ( R) ( R r(k ) Fx(k )) ( R) R Eq 7 where relte et-pont chnge nd ( R) F correpond tte feedbck ework wthn frework predcdve he recedng horzon yeld ncreentl u(k ) N c {}}{ [ ] ( R) ( R r(k ) Fx(k )) Eq 8

30 Cloed-Loop Control Syte {[ }}{] Rewrte Eq 8 u(k ) K y r(k ) K pc x(k ), where K_y frt eleent K_pc frt row ( R) R ( R) F ework Eq 9 K_pc tte feedbck gn vecr Inert Eq 9 n degn odel (Eq ) yeld x(k )Ax(k) BK pc x(k)bk y r(k) (A BK pc )x(k)bk y r(k)

31 Cloed-Loop Control Syte he cloed-loop Egenvlue cn be evluted fro cloed loop equdon: det[λi (A BK pc )] Note: Lt Colun F dendcl Rbr_ K_y dendcl lt eleent K_pc K_pc cn n be wrfen Recll tte vrble vecr y x(k )[ (k) ], K pc [K x K y ],

32 Cloed-Loop Control Syte (Exple Repone) x (k )x (k)bu(k) x (k), 8 b clrfn te-pce N p N c 4, pred [ (k ) ] [ ][ ][ ] (k) ][ ] b u(k) y(k ) b [ ] [ ] [ ] [ ] (k) F R

33 Repon olunby 5 ) Fwe 83 ( R) 2 (R r(k x(k ))frt-order (7) Exple Suppoe yte decrbed , he optl t optl olun let r rble x(k followng equn: w 2 R F rble x(k v followng equn: ; )) v F 2 Exple yte I trx, ))72 (Suppoe R ) ( R frt-order r(k ) F x(k n: gven below, where 4 4 dentty 4 e 2 Suppoe frt-order yte decrbed by tte equf R ncreent ; gnf frt two ) he gntude n: ( R ) ( R r(k F x(k )) (7) ( R ) ( R r(k ) F x(k )) (7) Cloed-Loop Control Syte Exple 2 Suppoe frt-order yte decrbed by 6 x2 (k 4 )6 x bu(k) 3 (k) coponent no b longer 2 4 Fgure o 24lt4two re zero n: 2( I) (4 Splng RIntnt 6 Intnt F8x(k (22) ))2 Splng Intnt Splng Splng Intnt 2 by x (k ) x (k) uppoe frt-order yte decrbed tte equppoe frt-order yte decrbed by tte equ x (k), Splng Intnt Splng Intnt! " x (k ) x (k) bu(k) 2 een tte vrble Itdentcl output y dd not rech ecr R lt colun n trx F R Note vecr R dentcl lt colun n trx F vecr on Note te vrble wth no weght on (b) Stte vrble wth weght on u ble wth no weght on (b) Stte vrble wth weght u x (k), x (k ) x (k) bu(k) wth weght u te vrble wth nowhere weght on (b) Stte vrble on pproche x8 (k), (8), however, zero h becue lt colun F trx dentcl R nd b re clr Fnd ugented tt Gven: lt colun F trx dentcl R h becue lt colu x (k), At te, vecr x(k [ 2] In frt ce, xk (k choce, ) tte x (k) bu(k) th )where Aung tudy predcn N nd follow fro copron It ee f horzon 8 nd b wnt re clrhorzon Fnd N pwe x (k ) x (k) bu(k) Wth gntude frt two ncreent gnf 8 nd b re clr Fnd ugented tte-pce odel grror, tte vecr x(k [ 2] 4 In frt At te k8, outpu t )reduced wthout ny condern ce, between predcted Ytke nd 2 6horzon 2 nd 4 Copron R lte coponent for predcn future ; lne (2) y optl olun Key: lne () where 8 nd b re clr Fnd ugented tc Aung predcn N eng cuuly, n t longer for gnl cntly reduced, lo coponent re no zeropfgure b optl x (8) Splng predcn horzon N olun lt ndtwo horzon Nc 4,longer clcu; lne (2) y Copron Key: lne () p(k), Splng Intnt Intnt x (k), (8) he gntudey chnge Nely, rwf nd hen, optl (2) error between predcted Yk ;lne y not Copron optl olun Key: lne () predcn reduced wthout ny condern predcted nd R R, output Aung t te quntte Aung predcn horzon nd horzon becue lte coponent for predcn coponent for future YN,p nd how optl tte vrble It een output y dd rech te (e, vlue n decree ore lowly), ound through clculn ble wth no weght on (b) Stte vrble wth weght on u nd b clr ugented re Fnd odel x(k lte coponent for predcn outp th exple), r(k tte-pce nd tte vecr )future [ gntude ch chnge Nely, r hen, optl ) unng (No Input RetrcDon): R, F nd Aung t te k (k for e d b re clr Fnd ugented tte-pce odel R, F nd Aung t quntte w et-pont vlue, however, pproche zero energy dtrbuted over longer perod future te "!effect r(k decy zero o exne rrww on R, F [ nd Aung t wte k quntte edcn N nd horzon N 4, optl olun wth repect ce where r weght horzon,clcup c decy zero o exne effect weght on ple), ) nd tte vecr x(k ) 2] fnd found through clcult th exple), r(k ) nd tte vecr dcn horzon N nd horzon N 4, clcuan obervn follow fro copron tudy It ee f we wnt clculn reult p9, whle c 64 ) ( th R ntnng exple), F x(k ))r(k rw)72 tte vecr x(k ) [ N( he yhencreng c nd he optl olun r on l olun, we let r onent for predcn future output Y, nd nd copre reult w he olun l olun, let roptl longer nd rnd, olun wth repect we ce where rtke w w optl w decy zero o exne effect weght olun wth repect r ce ent for predcn future output Y, ove cuuly, n t for wgnl gntude eleent n reducng, but y re optl olun wth repect ce where w gven below, where I 4 4 dentty trx, R, F nd Aung t te k (k for! "ore pre reult ven 4 dentty trx, optl We wthout weghtng on ncreentl, lt two elenote tedy lne ; (2) y becue Copron Control Key: lne () nd copre reult t tte (e, vlue n decree lowly), rech olun R F nd Aung t te k (k for unng (Retrct Input): he optl olun olun, we let r nd copre reult w ) ( ( frt 8 eleent: 64 equn ) nd ( vecr F3)x(k [ r(k ( Solun ugented tte-pce tte ) 2] fnd )) ent 2) R nd u(k x(k, whle 72 frt two, eleent hve ) u(k he where optl %( dtrbuted longer perod future te tteenergy ) [ over (k )he nd vecr x(k 2], &fnd & % % & % & below, I 4 4 dentty trx, ( I) R F x(k )) (22) I) ( R F x(k )) (22) on ugented tte-pce equn r lrge gntude Fgure how chnge tte vrble n We wth repect by ncreng ce where rw nd rw ", (k ) rw b equ Solun he ugented 9, whle ntnng rtte-pce reult cn verfy th N Solun he ugented tte-pce equn he c w(k) nd, wth repect ce where r! " % & % & % & % & w" where we cn ee predcted output y h reched dered et-pont We note wthout wegh he reult thout weghtng on ncreentl, lt two % eleent & % & % & &ele%% u(k) & re % & % y(k ) b (k ) (k) b how gntude n reducng, but y reult decy zero o exne effect weght r on w bnd I) ( R u(k) F )ent % u(k ) (k (k) (k u(k &2)(22) ( x(k )) (k y(k ) b 2) nd u(k 3), whle frt two eleent hve gnfcnt for frt 8 eleent: ' ( y(k ")he b (k) unng Control hh ce(retrct where Nc Input): 4, we when N 9, optl olun olun, we let r & % note y(k ) c w!tte-pce ugented equn choce, frt two ncreent gnfgntude frt two ncreent gnf rr lrge Fg & % %gntude ' how ( (k)! " gntude Fgure chnge tte vrble ' ( ugented tte-pce equn % &dfferent % & 64 %re (k) b & 4 79 r n re lghtly fro prevou ce &trx, (9) educed, lo coponent re no longer zero Fgure b below, where I 4 dentty ' ( educed, lt two coponent no longer zero Fgure (k pr where we cn ee (k ) (k) b & % & % & % & et-pont ehe predcted output y h reched dered Bed onit F ndoutput trce tke followng for vrble u(k) optl tte It(2), (k) een output y dd dd not rech he een y not rech (k ) b y(k ) b n(2), wth trce tke followng for: epont n lterntve wy fnd nu cot two u(k) ncreent F nd nd vlue, however, pproche zero In copron ce where N 4, we note when N 9, ce, gntude frt gnf(2), Bed on trce tke followng for pont vlue F, however, pproche zero & % c c ( I) ( R F x(k )) (22) y(k ) b CA CB trce th qure ' fro ( (k) Bed on (2), F nd tke etng h n ntutve pproch, lo bervn follow copron tudy It ee f we wnt & lghtly frt four preter ncopron re dfferent fro prevou ce obervn follow tudy It f we wnt CB 2 CA no CAB!CAlt "ee d, lo two coponent re longer zero Fgure ( % (9) CA CB b CB ' fro (Exple: unng Preter)

34 Repo whch [ for Begnner, ], refore denote lt ele2 y dentcl B C rete-te q ntegrr 2Kodule n MPC new equence gnl procedure repeted q dcrete-te n: e, vrble x(k followng equn: h ) v n: q 4 Notng tte vrble vecr x(k, ) [ (k) ]! " u(k) te gve recedng horzon Aby cloed-loop yte obtned lw u(k) ubttutng (29) y(, we ],F where K K pc x )K y x corre (R r(k [K K K ( R ) ( ) cn wrte R )y Exple x(k )) (7) F R ; efnn 5 h exple wll exne cloed-loop feedbc x (k ) x (k) bu(k) 3 x (k ) x (k) bu(k) x (k relted ) u(k) q eedbck gnvecr x (k), nd Ky correpond yte equn; chngng ndex k k, ledng c (k) ce generted fro Exple 2 nd egenvlue cloed le 4 We llutrte th procedure by contnung Exple 2, where we block x n cloed-loop dbrelted q obtn x Suppoe hen, frt-order yte C decrbed by tte equ K x (k), (8) (k), A y K wth weght r nd r order wth decrpn denote edctveyte yte ntte-pce Fgure 3 where q x w w! ) x(k) BKy r x(k! " " Ax(k) BKpc A u(k) odel he8 dgr tte feedbck tructure for ( q re 8 clr ( R ) Fugented ) clr 8 Fnd K q where nd b re ugented tte-p )operr pc where nd bhow Fnd tte-pce (A BK )x(k) BK pc y r(k) r n whch x odel predcn (DMPC) wth ntegrl cn (k ) x (k) bu(k) q x (k ) 8x (k) u(k) Aung predcn horzon N nd horzon Aung predcn horzon N nd horzon N 4, clcu R dentcl lt n, we hve Solun Whenyte colun weght rpw Hence, egenvlue cloed-loop re clculted by evlutng c p ctrx F N K Gven: cr denote dcrete-te ntegrr 6 qcrete-te predctve 2output Y A K xcloed-loop (8) trx A cloed-loop BK, future where, fro Exple 4 egenvlue 6 2 yte x lte coponent for predcn output Y, nd lte coponent for predcn future (k), pc hu, egenvlue cn be evluted thro Fg 3 Block dgr dcrete-t lt colun F trx dentcl R Splng Intnt Splng Intnt h wll cloed-loop feedbck gn u(k) n exple wll ce he ntl exne We ) t for R conder rt q te 2 coputn w * F ) tr*equn:, F nd Aung te k (k quntte chrctertc R, nd Aung quntte k (k odelclculte Egenvlue 8 u(k) 8 nd b re clr Fnd ugented tte-pce q ro Exple 2 nd egenvlue cloed-loop yte K y, tte vecr x(k )(b) [ tte frt ce, nd u(9) on re x() [ 2] u r(k And ; )B vecr x(k!2] "In[ Augented Syte: ble wth no weght on Stte vrble wth weght on th exple), r(k ) tte ) 2], fnd 7 th exple), nd vecr x(k ) [ 2] nd r w predcn horzon N nd horzon N 4, clcu8 7 on c ( R ) k)λ2 649 p, pproxtely K ( q hey re λ 649 nd λ2 64 r (A r BK )]) ywthout Fg 3 Block where dgr det[λi predctve yte dcrete-te pc nd, optl olun wth repect ce r reduced ny condern redcted Y nd R K w w nd optl olun wth repect ce where x ponent for predcn future Y, nd outpurgn plne w K coplex x on ple kcopre optl w prevouly co te, reult we hve en weght rw At, nd copre R RetrcDon):, F nd x nd Aung t w te k (khen, for reult pecl unng (No Input f chnge Nely, r optl nd A, K Becue tructure trce C we 7 7 u(9) hve When λweght rw, hve u() 72 Aung, n gnl hey re λ 649 nd 649, pproxtely 2 e), r(k ) nd tte vecr, fnd [, ], refore K de x(k ) [2] dentcl R, whch ; lne (2) y! y " Copron optl olun Key: lne () orgn wth coplex plne "y()!nd 2, he 72 nt clculn u() " u(9) u() nd x () we! wth Solun he ugented tte-pce equn r, un repect ce where r w w he ugented tte-pce equn 8 ent pc K dgr Notng x(ky Block ( tte R ) vrble (predctve F )vecr ) K pc 3 Block dcrete-te predctve When weght rw dgr, we hve ulted K (Solun R )plnt ( tte Fg )Fg y next 3 dcrete-te yt te vrble % & % & % & % & Block dgr dcrete-te predctve yte e3 reult & % b & K % y, we & [Kx K % nd wth!defnn! & %"Kpc cn wrte " (k ) (k) cloed-loop u(k) (k) reb(clculted Ky yte R ) b( Hence, egenvlue ( ) 2453 (k ) 64 R ) pond feedbck gn vecr relted u(k) 72 ) R F x(k )) ( 7 ( (k), nd 649 8x ) λy(k " 7"b 649 λ 649, re pproxtely on 2y(k 7! () () u() 88 (26) egenvlue he ugented tte-pce equn ) br, 2453! x"nd! hey nd λ 649 BK pc cloed-loop trx A, pprox where, fr hey re λ 649 nd λ 649,pproxt % K 2 2 o exne decy zero weght on &effect relted ( R ) ( ) feedbck hen, we obtn cl y gn coplex plne w 8 ( R ) ( F ) K pc & % % & % &2 %orgnorgn & ' &(%coplex plne (k) coplex plne ) * ) (9) * Fgure 3 wh ' ( weght r, we hve w (k) 68 (k ) nforn (k) gr predctve yte n bfor unng (Retrct Control Input): nvlue cloed-loop yte re clculted by evlutng, new plnt () 88 2 nd he optl olun olun, we let r u(k) w 8! " When weght r, we hve thout weghtng on ncreentl, lt two elewhen weght r, we hve w w!pc, where, " Exple fee ) A BK bfro bckwrd hft operr how tte A Khe ;dgr B y(k cloed-loop trx ( R ) (! " pc hen we obtn 88, whch for x() 8! "! " & % below, where I 4 4 dentty trx,! " )KR ) ( nd, nteg hvewth *'6939 ) 3) dcrete odel 2453 Bed ) u(k whle ( followng frt predcn 2453 Kpc ( two for: R )eleent ( F ) (DMPC) 6939 (2), F)( F( nd tke R ) ) (k) trce 2453 * y on Wth th gn vecr, egenvlue t 8 Bed on (2), F nd trce tke followng for: (9) A ; B denote dcrete-te ntegrr odule! "! " Wth re cloed-loop! q "egenvlue gntude Fgure chnge j542, tte vrble th gn vecr, yte re 8 lue cloed-loop yte λ 853 ± ndctng dyn how ( K R ) ( ) 2453 CB,2 y ( ) ( R CA F x()) F KyR ndctng ( )) R ) ( ) (22) 2453yte cloed-loop 853 ± j542, dync ( I) ( x(k CA CB n f! "! " Exple 5 h exple wll exne cloed-loop uch lower repone thn one 2 he dync cloed-loop yte hve output y h reched dered et-pont 2), F nd trce tke followng for: predcted CA CAB CB 2repone uch thn one n ce when pc (! R ) ( F3 ) lower fro " rw CA CAB CB ce generted Exple 2 nd egenvlue 2 d optl u() u() u() 296 h new ne n ce when r CA CA B Suppoe CAB CB Suppoe w n vecr, CA egenvlue cloed-loop yte re λ65 Exple 6 contnuou-t,2 CB Exple 6 contnuou-te yte decrbed "CAr B CAB CB! " wthca weght r! nd Cloed-Loop Control Syte (Exple: unng Preter)

35 Nuercl Intblty Ebedded ntegrr() reult n t let one egenvlue on unt crcle A predcdon horzon N p becoe lrge nuercl ntblty occur Stblty cnnot be gurnteed wth ll predcdon N p nd horzon N c preter Stblty cn be checked N p nd N c becoe tunng preter odfcdon cn overcoe ue nd tblty cn be gurnteed (Ch 4)

36 Exple ] x (k ) Syte [ ] x (k) [ ] x (k) t do N c 4 n f N p 2 [ 5 ] u(k) Effect r_w on output repone? Effect r_w on nput? Look t r_w v r_w 2

37 Exple R_w Repone efle t roughly 5 tep Control nput between 6 nd -4

38 Exple R_w 2 Repone efle t roughly 8 tep Control nput between 3 nd -8

39 Iportnt MPC opc Cover n Future OpDzDon nd QudrDc Progrng MPC nd LQR Input Contrnt Output contrnt Lguerre B FuncDon nd MPC MIMO MPC Stblty nd MPC

40 Contrnt Convex OpDzDon

41 Contrnt nd Generl Convex Opzon Mnze cot funcdon Subject J 2 x Ex x F 2 Mx γ, Convex opdzdon technque requred: Pre-Dul Method Hldreth QudrDc Progrng Procedure

42 Forulte Contrnt Contrnt nert n cot funcdon Becoe nequlde: In trx for: n x [ I I ] n x [ ] n x Note th ft generl contrnt for 2 Mx γ,

43 QueDon? hnk You!