Optimal Control. Lecture. Prof. Daniela Iacoviello
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1 Opmal Cnrl Lecure Pr. Danela Iacvell
2 Gradng Prjec + ral eam Eample prjec: Read a paper n an pmal cnrl prblem 1 Sudy: backgrund mvans mdel pmal cnrl slun resuls 2 Smulans Yu mus gve me al leas en days bere he dae he eam: -A.dc dcumen -A pwer pn presenan - Malab smulan les Pr. D.Iacvell- Opmal Cnrl 03/12/2018 Pagna 2
3 Eample prjec: -Read a paper n an pmal cnrl prblem any eld wll d Yu can chse a pc prpsed n he papers I sen yu r rm he eamples prpsed n he bks suggesed.e. Evans r sme specc prblem yur neres. Pr. D.Iacvell- Opmal Cnrl 03/12/2018 Pagna 3
4 1 Sudy: backgrund mvans mdel pmal cnrl slun resuls Yu have ndcae clearly: -he backgrund he prblem yu are sudyng by a leraure research mvans why he prblem s mpran? -he mdel assumed ndcae varables parameers her meanng ec -he cs nde and he pmal cnrl nrduced he meanng wha hey represen deren pssble chces ec. 2 Smulans: Yu have mplemen he mdel and he cnrl by malab Descrbe he prgram prpse he resuls and dscuss hem. Pr. D.Iacvell- Opmal Cnrl 03/12/2018 Pagna 4
5 YOU MUS SUDY ON HE BOOKS HE SLIDES ARE NO SUFFICIEN Pr. D.Iacvell- Opmal Cnrl 03/12/2018 Pagna 5
6 Schedule Lecure 1: Inrducn pmal cnrl: mvans eamples. Denns Uncnsraned pman rs rder necessary cndns secnd rder necessary cndns Weersrass herem Cnsraned pman nrducn Lagrangan uncn Frs rder necessary cndns Secnd rder sucen cndns. Funcn spaces: Frs rder necessary cndn secnd rder necessary cndn necessary and sucen cndn Pr. D.Iacvell- Opmal Cnrl 03/12/2018 Pagna 6
7 Denns Cnsder a uncn And n D R R denes he Eucledan nrm D : A pn s a lcal mnmum ver n : R n D R I 0 such ha r all D sasyng Pr. D.Iacvell- Opmal Cnrl 03/12/2018 Pagna 7
8 Denns Cnsder a uncn And denes he Eucledan nrm A pn s a src lcal mnmum ver I n D R D 0 such ha : R r n : all R D sasyng D R n Pr. D.Iacvell- Opmal Cnrl 03/12/2018 Pagna 8
9 Denns Cnsder a uncn And n D R R denes he Eucledan nrm D : A pn s a glbal mnmum ver n : R n D R I 0 such ha r all D Pr. D.Iacvell- Opmal Cnrl 03/12/2018 Pagna 9
10 Uncnsraned pman - rs rder necessary cndns All pns sucenly near n are n R n D 1 Assume C and s lcal mnmum. Le an arbrary vecr. Beng n he uncnsraned case: + d D R clse enugh 0 n d R Le s cnsder: g : + d 0 s a mnmum g 0 Frs rder necessary cndn r pmaly Pr. D.Iacvell- Opmal Cnrl 03/12/2018 Pagna 10
11 Uncnsraned pman- secnd rder cndns 2 Assume C and s lcal mnmum. Le an arbrary vecr. Secnd rder aylr epansn g arund Snce 0 n d R g g0 + g'0 + g''0 + lm g' 0 0 g' ' Secnd rder necessary cndn r pmaly Pr. D.Iacvell- Opmal Cnrl 03/12/2018 Pagna 11
12 Uncnsraned pman- secnd rder cndns Le 2 C and s a src lcal mnmum Pr. D.Iacvell- Opmal Cnrl 03/12/2018 Pagna 12
13 Denns- Remarks d R n A vecr s a easble drecn a + d D r small enugh 0 I D n n he enre R whch s beng mnmed n hen D s he cnsran se ver Pr. D.Iacvell- Opmal Cnrl 03/12/2018 Pagna 13
14 Glbal mnmum Weersrass herem Le be a cnnuus uncn and D a cmpac se here es a glbal mnmum ver D Pr. D.Iacvell- Opmal Cnrl 03/12/2018 Pagna 14
15 Cnsraned pman Le D R C Equaly cnsrans n 1 p h 0 h : R R hc Inequaly cnsransg 0 q g : R R g C Regulary cndn: g h g a rank where g are wh a he dmensn q p + q a a acve cnsran 1 1 Lagrangan uncnl + h + g 0 0 I 0 0 he sanary pn s called nrmal and we can assume 0 1. Pr. D.Iacvell- Opmal Cnrl 03/12/2018 Pagna 15
16 Cnsraned pman Frm nw n 0 1 and herere he Lagrangan s L + h + g I here are nly equaly cnsrans he Lagrange mulplers are called I here are bh equaly and nequaly cnsrans we have Kuhn ucker mulplers Pr. D.Iacvell- Opmal Cnrl 03/12/2018 Pagna 16
17 Cnsraned pman Frs rder necessary cndns r cnsraned pmaly: Le D and h gc 1 he necessary cndns r mnmum are g be a cnsraned lcal I he uncns and g are cnve and he uncns h are lnear hese cndns are necessary and Pr. D.Iacvell- Opmal Cnrl 03/12/2018 Pagna 17 sucen!!! L 0 0 0
18 Cnsraned pman Secnd rder sucen cndns r cnsraned pmaly: Le and h gc 2 and assume he cndns D L 0 g 0 0 s a src cnsraned lcal mnmum d 2 L 2 d 0 d such ha dh d d p Pr. D.Iacvell- Opmal Cnrl 03/12/2018 Pagna 18
19 Funcn spaces Funcnal J : V Vecr space V R A V A s a lcal mnmum J ver A here ess an 0 such ha r all A sasyng J J Pr. D.Iacvell- Opmal Cnrl 03/12/2018 Pagna 19
20 Funcn spaces Cnsder uncn n V he rm he rs varan J a s he lnear uncn such ha and + V R J : V R J + J + J + Frs rder necessary cndn r pmaly: Fr all admssble perurban we mus have: J 0 Pr. D.Iacvell- Opmal Cnrl 03/12/2018 Pagna 20
21 Funcn spaces A quadrac rm J a we have: J 2 J and + : V J + J R s he secnd varan J + 2 secnd rder necessary cndn r pmaly: I A s a lcal mnmum J ver A V r all admssble perurban we mus have: 2 J 0 Pr. D.Iacvell- Opmal Cnrl 03/12/2018 Pagna 21
22 Funcn spaces he Weersrass herem s sll vald I J s a cnve uncnal and A V s a cnve se A lcal mnmum s aumacally a glbal ne and he rs rder cndn are necessary and sucen cndn r a mnmum Pr. D.Iacvell- Opmal Cnrl 03/12/2018 Pagna 22
23 Schedule Lecure 2: Calculus varans- Mvans he Lagrange prblem: Euler equan Weersrass Erdmann cndn ransversaly cndns Pr. D.Iacvell- Opmal Cnrl 03/12/2018 Pagna 23
24 he Lagrange prblem Prblem 1 Le us cnsder he lnear space C 1 R R R and dene he admssble se: D C 1 R R R : D Rv+ 1 D Rv+ 1 Inrduce he nrm: and cnsder he cs nde: wh L uncn C 2 class. Fnd he glbal mnmum pmum r J ver D : J sup + sup + + J L J D d Pr. D.Iacvell- Opmal Cnrl 03/12/2018 Pagna 24
25 SCHEME he herems herem Lagrange. I D s a lcal mnmum hen 1 L d L d 2 In any dscnnuy pn Weersrass-Erdmann cndn 0 Euler equan 3 ransversaly cndns deren cases dependng n he naure he bundary cndns Pr.Danela Iacvell- Opmal Cnrl 03/12/2018 Pagna 25
26 herem Lagrange. I D s a lcal mnmum hen L d d L 0 Euler equan In any dscnnuy pn he llwng cndns are vered: Weersrass- L L L L Erdmann L L cndn + + Pr. D.Iacvell- Opmal Cnrl 03/12/2018 Pagna 26
27 Mrever ransversaly cndns are sased: I are pen subse we have: I are clsed subses dened respecvely by such ha r D D L L L L D D 0 0 rg rg R R L L L L L L Pr. D.Iacvell- Opmal Cnrl 03/12/2018 Pagna 27
28 I he ses D and D are dened by he uncn ane wh respec such ha rg w w L w L w R Pr. D.Iacvell- Opmal Cnrl 03/12/2018 Pagna 28
29 he Lagrange prblem Prblem 2 Cnsder Prblem 1 wh and ed v+ 1 D I D are clsed ses n dened by he C 1 R uncns 0 dmensn v + 1 wh and rg 0 dmensn v + 1 ane uncns and rg Pr.Danela Iacvell- Opmal Cnrl 03/12/2018 Pagna 29
30 I he ses wh σ cmpnens C 1 class such ha D D rg w are dened by he uncn w ane wh respec he uncn L mus be cnve wh respec Fnd he glbal mnmum pmum r J ver D: J D J Pr.Danela Iacvell- Opmal Cnrl 03/12/2018 Pagna 30
31 herem 2. s he pmum and nly In any dscnnuy pn he llwng cndns are vered: Mrever ransversaly cndns are sased.. D L d d L 0 Euler equan L L L L L L + + Weersrass- Erdmann cndn Pagna 31 Pr.Danela Iacvell- Opmal Cnrl 03/12/2018
32 he Lagrange prblem Prblem 1_bs Le us cnsder he admssble se: cnsder he cs nde: wh L uncn C 2 class. D C 1 [ ]: Fnd he glbal mnmum pmum r J ver D J L d Pr. D.Iacvell- Opmal Cnrl 03/12/2018 Pagna 32 Pr.Danela Iacvell- Opmal Cnrl
33 he Lagrange prblem herem Legendre Necessary cndn r D be lcal mnmum s ha 2 L 2 0 [ ] Pr. D.Iacvell- Opmal Cnrl 03/12/2018 Pagna 33 Pr.Danela Iacvell- Opmal Cnrl
34 he Lagrange prblem Prblem 3 Le us cnsder he lnear space and dene he admssble se dmensn Cnsder he cs nde: R R R C k d h g R D R D R R R C D v v 0 : g v d L J Pr. D.Iacvell- Opmal Cnrl 03/12/2018 Pagna 34
35 Dene he augmened lagrangan: h g L Pr. D.Iacvell- Opmal Cnrl 03/12/2018 Pagna 35
36 herem 3 Lagrange. Le be such ha I s a lcal mnmum r J ver D hen here es n smulaneusly null n such ha: wh RANSVERSALIY CONDIIONS D g rank 0 d d R C R 0 0 ] [ ] [ 03/12/2018 Pagna 36 Pr.Danela Iacvell- Opmal Cnrl + + k k k k where are cuspd pns r k
37 he Lagrange prblem Prblem 4 Le us cnsder he lnear space and dene he admssble se C 1 R R R D 1 C D D g 0 h d k [ ] g dmensn v ed g and h ane uncns n [ ] L C 2 uncn cnve wh respec [ Cnsder he cs nde: Pr.Danela Iacvell- Opmal Cnrl J L 03/12/2018 Pagna 37 d ]
38 Dene he augmened lagrangan: L 0 + g + h Pr. D.Iacvell- Opmal Cnrl 03/12/2018 Pagna 38
39 herem 4 Lagrange. Le D such ha D g rank s an pmal nrmal slun and nly d d 0 n he nsans are pen we have: and/r r whch 0 D and/r D Pr.Danela Iacvell- Opmal Cnrl 03/12/2018 Pagna 39
40 Schedule Lecure 3: Calculus varans and pmal cnrl: he Hamlnan uncn- Necessary cndns- Necessary and sucen cndns Pr. D.Iacvell- Opmal Cnrl 03/12/2018 Pagna 40
41 Prblem 1 Le us cnsder he dynamcal ssem descrbed by: u uncn C 2 class. u cnrl vecr n R p sae vecr n R n knwn Pr. D.Iacvell- Opmal Cnrl 03/12/2018 Pagna 41
42 0 q u 0 Vecral uncn C 1 class dmensn n +1 Vecral uncn C 2 class dmensn Assume he nrm: u sup + sup + sup u d + sup u + Dene he cs nde wh L uncn C 2 class u J u L d Pr.Danela Iacvell- Opmal Cnrl 03/12/2018 Pagna 42
43 AIM: Fnd ess he nsan he cnrl he sae u C 0 R C 1 R ha sasy he prevus equans and mnme he cs nde Pr.Danela Iacvell- Opmal Cnrl 03/12/2018 Pagna 43
44 DEFINE he scalar uncn he Hamlnan uncn 0 0 u u L u H + 03/12/2018 Pagna 44 Pr.Danela Iacvell- Opmal Cnrl
45 herem 1 Le u rk be an admssble slun such ha q rk u acve dmensn a IF u s a lcal mnmum here es C [ ] C [ ] n smulaneusly null n [ ] such ha: Pr.Danela Iacvell- Opmal Cnrl 03/12/2018 Pagna 45
46 q H 03/12/2018 Pagna 46 0 u q u H j u q j j j R H he dscnnuy may ccurr nly n he pns where has a dscnnuy and and u + H H Pr.Danela Iacvell- Opmal Cnrl
47 Calculus varan and pmal cnrl Prblem 3 Le us cnsder he dynamcal sysem descrbed by: u uncn C 2 class u cnrl vecr n R p knwn sae vecr n R n Pr.Danela Iacvell- Opmal Cnrl 03/12/2018 Pagna 47
48 0 h u d K Vecral uncn C 1 class dmensn n +1 Vecral uncn C 2 class dmensn Assume he nrm: u sup + sup + sup u d + sup u + Dene he cs nde wh L uncn C 2 class u J u L d Pr.Danela Iacvell- Opmal Cnrl 03/12/2018 Pagna 48
49 AIM: Fnd he nsan he cnrl he sae ha sasy he prevus equans and mnme he cs nde u C 0 R C 1 R Pr.Danela Iacvell- Opmal Cnrl 03/12/2018 Pagna 49
50 DEFINE he scalar uncn he Hamlnan uncn u h u u L u H /12/2018 Pagna 50 Pr.Danela Iacvell- Opmal Cnrl
51 herem 3 Le IF u u be an admssble slun such ha s a lcal mnmum rk 1 0 ] here es R C [ smulaneusly null n [ ] such ha: R n H H 0 u H R Pr.Danela Iacvell- Opmal Cnrl 03/12/2018 Pagna 51
52 he dscnnuy may ccurr nly n he pns where has a dscnnuy u k and H H k k + Pr.Danela Iacvell- Opmal Cnrl 03/12/2018 Pagna 52
53 Prblem 4: Cnsder he lnear sysem A + B u Assume: q u D beng 0 ed D Dene he cs nde a ed pn n r R Funcns C 2 class Vecral uncn C 2 class dmensn CONVEX J u L u d + G Funcns C 3 class- CONVEX Funcns C 2 class CONVEX Pr.Danela Iacvell- Opmal Cnrl 03/12/2018 Pagna 53
54 herem 4 Le u u s a nrmal pmal slun be an admssble slun such ha qacve rk a u AND IF D R n dg d 0 j j H H u q j 0 q + u u q 0 Pr.Danela Iacvell- Opmal Cnrl 03/12/2018 Pagna 54
55 Schedule Lecure 4: Pnryagn mnmum prncple- Necessary cndns- he cnve case- Eample Pr. D.Iacvell- Opmal Cnrl 03/12/2018 Pagna 55
56 he Pnryagn prncple Prblem 1: Cnsder he dynamcal sysem: wh: Assume ed he nal cnrl nsan and he nal and nal values : Dene he perrmance nde : wh u n U R C R U u R n p n Pagna 56 03/12/2018 Pr.Danela Iacvell- Opmal Cnrl G d u L u J C G n U R C L L n
57 Deermne: he value he cnrl he sae u C 0 R C 1 R ha sasy: he dynamcal sysem he cnsran n he cnrl he nal and nal cndns and mnme he cs nde Pr.Danela Iacvell- Opmal Cnrl 03/12/2018 Pagna 57
58 Hamlnan uncn H u L u + u 0 0 Pr.Danela Iacvell- Opmal Cnrl 03/12/2018 Pagna 58
59 he Pnryagn prncple herem 1 necessary cndn: Assume he admssble slun u here es a cnsan 0 0 and a n-dmensnal vecr C n smulaneusly null such ha : 1 s a mnmum H H H u U 0 H 0 0 Pr.Danela Iacvell- Opmal Cnrl 03/12/2018 Pagna 59
60 he Pnryagn prncple Prblem 2 herem 2 Prblem 3 herem 3 Pr. D.Iacvell- Opmal Cnrl 03/12/2018 Pagna 60
61 he Pnryagn prncple Prblem 4: Cnsder he dynamcal sysem: wh: R n u U R and nal nsan and sae ed p Pr.Danela Iacvell- Opmal Cnrl C 0 n R U R u Fr he nal values assume: where s a uncn dmensn Assume he cnsran Dene he perrmance nde : wh L L 0 L C 0 n +1 C 1 class. h u d k h n R U R h h C J 0 wh n R U R n u L u d 03/12/2018 Pagna 61
62 Deermne he value he cnrl u C 0 R and he sae C 1 R ha sasy he dynamcal sysem he cnsran n he cnrl he nal and nal cndns and mnme he cs nde. Pr.Danela Iacvell- Opmal Cnrl 03/12/2018 Pagna 62
63 Hamlnan uncn H u L u + u + h u 0 0 Pr.Danela Iacvell- Opmal Cnrl 03/12/2018 Pagna 63
64 herem 4 necessary cndn: Cnsder an admssble slun IF s a lcal mnmum u such ha rank here es a cnsan 0 0 R n smulaneusly null such ha : 1 C H 0 H u 0 H U Mrever here ess a vecr such ha: R H H + d k H k R Pr.Danela Iacvell- Opmal Cnrl 03/12/2018 Pagna 64
65 he dscnnues may ccur nly n he nsans n whch u has a dscnnuy and n hese nsans he Hamlnan s cnnuus Pr.Danela Iacvell- Opmal Cnrl 03/12/2018 Pagna 65
66 Remark I he se U cncdes wh R p cndn reduces : he mnmum H u 0 Pr.Danela Iacvell- Opmal Cnrl 03/12/2018 Pagna 66
67 he Pnryagn prncple - cnve case Prblem 5: Cnsder he dynamcal lnear sysem: A + B u wh A and B uncn C 1 class; assume ed he nal and nal nsans and he nal sae and Assume ed r u U R p R n where U s a cnve se. Pr.Danela Iacvell- Opmal Cnrl 03/12/2018 Pagna 67
68 Dene he perrmance nde : wh L L cnve uncn wh respec u n per gn L L C 0 J u L u d + G n R U n R n U G s a scalar uncn C 2 respec. class and cnve wh Pr.Danela Iacvell- Opmal Cnrl 03/12/2018 Pagna 68
69 Deermne: he cnrl and he sae 0 u C 1 C ha sasy he dynamcal sysem he cnsran n he cnrl he nal and nal cndns and mnme he cs nde. Pr.Danela Iacvell- Opmal Cnrl 03/12/2018 Pagna 69
70 herem 5 necessary and sucen cndn: Cnsder an admssble slun I s a mnmum nrmal.e.λ 0 1 such ha and nly here ess an n-dmensnal vecr such ha : H u u rank 1 C H H u U Mrever R n dg d Pr.Danela Iacvell- Opmal Cnrl 03/12/2018 Pagna 70
71 Schedule Lecure : he Hamln- Jacb equan- Dervan he H-J equan- Prncple pmaly- Eample Pr. D.Iacvell- Opmal Cnrl 03/12/2018 Pagna 71
72 Prncple pmaly A cnrl plcy pmal ver he nerval [ ] s pmal ver all subnervals [ 1 ] Rme PARIS MILAN I he shres pah rm Rme Pars passes hrugh Mlan HEN he OBAINED pah beween Mlan Pars s he shres ne beween Mlan Pars Pr. D.Iacvell - Opmal Cnrl 03/12/
73 Schedule Lecure 8: he pmal regular prblem: he pmal rackng prblem he pmal regular prblem wh null nal errr he pmal regular prblem wh lmed cnrl Eamples Pr. D.Iacvell- Opmal Cnrl 03/12/2018 Pagna 73
74 he regular prblem 1 K B R r0 + - u K B R B A 1 03/12/2018 Pr. D.Iacvell - Opmal Cnrl Pagna 74
75 he regular prblem Le us cnsder he llwng lnear sysem: A + B u y C wh ] ed and ed [ A B C are mar uncns me wh cnnuus enres; her dmensns are respecvely n n nm n p Pr. D.Iacvell - Opmal Cnrl 03/12/2018 Pagna 75
76 he regular prblem A lnear cnrl law s baned we seek mnme he quadrac perrmance nde: 1 J u 2 0 Q + u R u 1 + F 2 where Q and R have cnnuus enres symmerc nnnegave and psve dene respecvely; F s nnnegave dene mar smmerc d Pr. D.Iacvell - Opmal Cnrl 03/12/2018 Pagna 76
77 he Rcca equan herem: he Rcca equan adms a unque slun symmerc semdene psve n he cnrl nerval. 03/12/2018 Pr. D.Iacvell - Opmal Cnrl Pagna 77 F K Q K A A K K B R B K K 1
78 Slun he regular prblem herem: he pmal cnrl r he regular prblem s gven by he lnear eedback law: where: and 1 K B R u K B R B A 1 K u J /12/2018 Pr. D.Iacvell - Opmal Cnrl Pagna 78
79 he regular prblem herem: he regular prblem adms a unque slun. Pr: he unqueness prpery s shwn by cnraddcn argumens. Pr. D.Iacvell - Opmal Cnrl 03/12/2018 Pagna 79
80 Slun he deermnsc lnear pmal regular prblem n Prblem: le us cnsder he regular prblem ver he nne me nerval. Assume: he marces A and B are bunded and wh elemens n C 1 class he dynamcal sysem s cmpleely cnrllable r epnenally sable he marces Q and R are symmerc semdene and psve dene respecvely wh elemens n C 1 class and bunded Pr. D.Iacvell - Opmal Cnrl 03/12/2018 Pagna 80
81 Slun he deermnsc lnear pmal regular prblem n Fnd: he cnrl u and he sae 0 C sasyng he dynamcal sysem he nal cndn and mnmng he cs nde J 1 C 1 u Q + u R u 2 d Pr. D.Iacvell - Opmal Cnrl 03/12/2018 Pagna 81
82 Slun he deermnsc lnear pmal regular prblem n herem: he prblem adms a unque pmal slun ha can be epressed as llws: where s he slun he Rcca equan wh nal cndn: and K B R B A K B R u 1 1 K 1 Q K A A K K B R B K K 0 lm K K u J /12/2018 Pr. D.Iacvell - Opmal Cnrl Pagna 82
83 he seady sae slun he deermnsc lnear pmal regular prblem herem: Le us cnsder he prevus prblem wh he addve hypheses : he marces A B Q R are cnsan he mar Q s psve dene hen here ess a unque pmal slun: u R 1 B K r 1 A BR B K r where: Pr. D.Iacvell - Opmal Cnrl 03/12/2018 Pagna 83
84 K r s he cnsan mar unque slun dene psve he algebrac Rcca equan: K r BR 1 B K r K r A A K r Q 0 he mnmum value r he cs nde s: J 1 2 u K r Pr. D.Iacvell - Opmal Cnrl 03/12/2018 Pagna 84
85 Remarks Cnsder he sysem: I can be dened an pmal regulan prblem rm he upu y cnsderng he cs nde C y u B A d u R u y Q y Fy y u y J /12/2018 Pr. D.Iacvell - Opmal Cnrl Pagna 85
86 he presen prblem s smlar he prevus ne beng sased he hypheses ver he marces nvlved. Anyway he sae mus be accessble ne wan reale he eedback acn: r0 + u A + C y Bu - R 1 B K Pr. D.Iacvell - Opmal Cnrl 03/12/2018 Pagna 86
87 he pmal rackng prblem Prblem: le us cnsder he lnear sysem: A + B u where A B are marces wh elemens uncns me wh enres C 1 class. Cnsder he reerence varable Deermne he pmal cnrl 0 u C 1 and he sae C sasyng he dnamcal cnsran and mnmng he cs nde: 1 r C Pr. D.Iacvell - Opmal Cnrl 03/12/2018 Pagna 87
88 where: Q s a symmerc sem-psve dene mar R s a symmerc psve dene mar he elemens Q and R are uncns C 1 class + d u R u r Q r u J /12/2018 Pr. D.Iacvell - Opmal Cnrl Pagna 88
89 he pmal rackng prblem he pmal rackng prblem can be asscaed he Rcca equan 0 1 K Q K A A K K B R B K K 03/12/2018 Pr. D.Iacvell - Opmal Cnrl Pagna 89
90 herem: he pmal rackng prblem adms a unque pmal slun: g B R B K B R B A K g B R u /12/2018 Pr. D.Iacvell - Opmal Cnrl Pagna 90
91 where g s he slun he derenal equan he mnmum value he cs nde s: where s he slun he equan: 0 1 g r Q g A B R B K g 2 1 v g K u J + v v r Q r g B R B g v 03/12/2018 Pr. D.Iacvell - Opmal Cnrl Pagna 91
92 g + - R 1 B u A + Bu K Pr. D.Iacvell - Opmal Cnrl 03/12/2018 Pagna 92
93 he pmal regular prblem wh null nal errr Prblem: Le us cnsder he lnear sysem: Deermne he cnrl and he sae n rder sasy he dynamcal sysem he nal and nal cndns and mnmng he llwng cs nde: 0 + u B A C u 0 C 1 03/12/2018 Pr. D.Iacvell - Opmal Cnrl Pagna 93
94 J 1 2 u Q u R u + 0 d where: Q s symmerc semdene psve wh elemens C 1 class R s symmerc dene psve wh elemens C 1 class Pr. D.Iacvell - Opmal Cnrl 03/12/2018 Pagna 94
95 herem: Le us nrduce he mar dmensn 2n 2n and ndcae wh A Q s ransn mar parned n submarces dmensn n n. 12 Assume ha he submar s n sngular. he pmal regular prblem wh null nal errr adms a unque pmal slun: B R 1 A B Pr. D.Iacvell - Opmal Cnrl 03/12/2018 Pagna 95
96 B R u /12/2018 Pr. D.Iacvell - Opmal Cnrl Pagna 96
97 Schedule Lecure 9: he mnmum me prblem- he mnmal me pmal prblem characeran he pmal slununqueness resul- Esence- Mnmum me prblem n case seady sae sysem Swchng pns Pr. D.Iacvell- Opmal Cnrl 03/12/2018 Pagna 97
98 he lnear me-pmal cnrl Le us cnsder he prblem pmal cnrl a lnear sysem wh ed nal and nal sae cnsrans n he cnrl varables and wh cs nde equal he lengh he me nerval Pr. D.Iacvell- Opmal Cnrl 03/12/2018 Pagna 98
99 SINGULAR SOLUIONS Denn: Le u and 0 be an pmal slun he abve prblem he crrespndng mulplers. he slun s sngular here ess a subnerval ' '' '' ' n whch he Hamlnan H 0 s ndependen rm a leas ne cmpnen n ' '' Pr. D.Iacvell - Opmal Cnrl 03/12/2018 Pagna 99
100 SINGULAR SOLUIONS herem: Assume he Hamlnan he rm H Le u be an eremum and 0 he crrespndng mulplers such ha N cmpnen n any subnerval s dependen n any A necessary cndn r u be a sngular eremum s ha here ess a subnerval ] such ha: 0 u 0 H1 0 + H2 0 N u H 0 [ ' ''] [ ] '' ' [ ' ''] [ Pr. D.Iacvell - Opmal Cnrl 03/12/2018 Pagna 100
101 he lnear mnmum me pmal cnrl Le us cnsder he prblem pmal cnrl a lnear sysem wh ed nal and nal sae cnsrans n he cnrl varables and wh cs nde equal he lengh he me nerval Pr. D.Iacvell - Opmal Cnrl 03/12/2018 Pagna 101
102 Prblem: Cnsder he lnear dynamcal sysem: Wh A + B u n R u R p and he cnsran u j 1 j p R Marces A and B have enres wh elemens class C n-2 and C n-1 respecvely a leas C 1 class. he nal nsan s ed and als: 0 Pr. D.Iacvell - Opmal Cnrl 03/12/2018 Pagna 102
103 he am s deermne he nal nsan R he cnrl u C R and he sae C 1 R ha mnme he cs nde: J d Pr. D.Iacvell - Opmal Cnrl 03/12/2018 Pagna 103
104 herem: Necessary cndns r be an pmal slun are ha here es a cnsan and an n-dmensnal uncn 1 n smulaneusly null and such ha: u 0 C 0 A B Bu R p : j 1 j p Pssble dscnnues n he pns n whch u can appear nly n has a dscnnuy. Pr. D.Iacvell - Opmal Cnrl 03/12/2018 Pagna 104
105 Mrever he asscaed Hamlnan s a cnnuus uncn wh respec and resuls: H Pr: he Hamlnan asscaed he prblem s: H Applyng he mnmum prncple he herem s prved. 0 u + A+ Bu 0 0 Pr. D.Iacvell - Opmal Cnrl 03/12/2018 Pagna 105
106 SRONG CONROLLABILIY Le us ndcae wh he j-h clumn he mar B. he srng cnrllably s guaraneed by he cndn: where : b j n j j j j p j b b b G de de 2 1 n k b A b b b b k j k j k j j j Generc clumn mar B 03/12/2018 Pr. D.Iacvell - Opmal Cnrl Pagna 106
107 Remark: Srng cnrllably crrespnds he cnrllably n any nsan n any me nerval and by any cmpnen he cnrl vecr Pr. D.Iacvell - Opmal Cnrl 03/12/2018 Pagna 107
108 Remark: In he NON-seady sae case he abve cndn s sucen r srng cnrllably. In he seady case s necessary and sucen and may be wren n he usual way: de n1 b Ab A b 0 j p j j j Pr. D.Iacvell - Opmal Cnrl 03/12/2018 Pagna 108
109 Characeran he pmal slun herem: Le us cnsder he mnmal me pmal prblem. I he srng cnrllably cndn s sased and an pmal slun ess s nn sngular. Mrever every cmpnen he pmal cnrl s pecewse cnsan assumng nly he ereme values and he number dscnnuy nsans s lmed. 1 Pr. D.Iacvell - Opmal Cnrl 03/12/2018 Pagna 109
110 A cnrl uncn ha assumes nly he lm values s called bang-bang cnrl and he nsans dscnnuy are called cmmuan nsans Pr. D.Iacvell - Opmal Cnrl 03/12/2018 Pagna 110
111 herem UNIQUENESS: I he hyphess srng cnrllably s sased an pmal slun ess s unque. Pr: he herem s prved by cnradcn argumens. Pr. D.Iacvell - Opmal Cnrl 03/12/2018 Pagna 111
112 Remark hs resul hlds als he case n whch he cnrl s n he space measurable uncn dened n he space real number R. Pr. D.Iacvell - Opmal Cnrl 03/12/2018 Pagna 112
113 Esence he pmal slun herem: he cndn srng cnrllably s sased and an admssble slun ess hen here ess a unque pmal slun nnsngular and he cnrl s bang-bang. Pr: he esence herem s prved n he bass he llwng resuls. Pr. D.Iacvell - Opmal Cnrl 03/12/2018 Pagna 113
114 he llwng resul hlds: herem: Le assume ha he cnrl uncns belng he space measurable uncns. I an admssble slun ess hen he pmal slun ess. Pr. D.Iacvell - Opmal Cnrl 03/12/2018 Pagna 114
115 he resuls abu he characeran he cnrl may be eended he case n whch he cnrl belng he space measurable uncns: herere resuls n: I he srng cnrllably cndn s sased and an pmal cnrl n he space measurable uncn ess hen hs slun s unque nn sngular and he cnrl s bang bang ype Remark he esence he pmal slun s guaraneed nly r he cuples r whch he admssble slun ess Pr. D.Iacvell - Opmal Cnrl 03/12/2018 Pagna 115
116 Esence he pmal slun herem: he cndn srng cnrllably s sased and an admssble slun ess wh he cnrl n he space measurable uncns hen here ess a unque pmal slun nnsngular and he cnrl s bang-bang. Pr. D.Iacvell - Opmal Cnrl 03/12/2018 Pagna 116
117 Mnmum me prblem r seady sae sysem Le us cnsder he mnmum me prblem n case seady sae sysem. In hs case s pssble deduce resuls abu he number cmmuan pns Pr. D.Iacvell - Opmal Cnrl 03/12/2018 Pagna 117
118 Prblem: Cnsder he lnear dynamcal sysem: A + Bu Wh n R u R p and he cnsran u j 1 j p R he nal nsan s ed and als: 0 Pr. D.Iacvell - Opmal Cnrl 03/12/2018 Pagna 118
119 he am s deermne he nal nsan R he cnrl u C R and he sae C 1 R ha mnme he cs nde: J d Pr. D.Iacvell - Opmal Cnrl 03/12/2018 Pagna 119
120 herem: Le us cnsder he abve me mnmum prblem and assume ha he cnrl uncn belng he space measurable uncn ; he sysem s cnrllable here ess a neghbr Ω he rgn such ha r any here ess an pmal slun. Pr. D.Iacvell - Opmal Cnrl 03/12/2018 Pagna 120
121 herem: Le us cnsder he abve me mnmum prblem n he seady sae case and assume ha he cnrl uncn belngs he space measurable uncn MR. I he sysem s cnrllable and he egenvalues mar A have negave real par here ess an pmal slun whaever he nal sae s. Pr. D.Iacvell - Opmal Cnrl 03/12/2018 Pagna 121
122 herem: I he cnrllably cndn s sased and all he egenvalues he mar A are real nn psve hen he number cmmuan nsans r any cmpnens cnrl s less r equal han n-1 whaever he nal sae s. Pr. D.Iacvell - Opmal Cnrl 03/12/2018 Pagna 122
123 Schedule Lecure he mnmum me prblemeamples: duble negrar; harmnc scllar Pr. D.Iacvell- Opmal Cnrl 03/12/2018 Pagna 123
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