Optimal Control. Lecture 6. Prof. Daniela Iacoviello
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1 Opmal Cnrl Lecure 6 Pr. Danela Iacvell
2 HESE SLIDES ARE NO SUFFICIEN FOR HE EXAM: YOU MUS SUDY ON HE OOS Par he sldes has been aken rm he Reerences ndcaed belw Pr. D.Iacvell - Opmal Cnrl 03//07 Pagna
3 Curse ulne Inrducn pmal cnrl Hamln-Jacb equan Pnryagn mnmum prncple he regular prblem he regular prblem n nne me nerval he pmal rackng prblem he pmal regular prblem wh null nal errr he pmal regular prblem wh lmed cnrl he mnmum me prblem r seady sae sysem Pr. D.Iacvell - Opmal Cnrl 03//07 Pagna 3
4 Pr. D.Iacvell - Opmal Cnrl 03//07 Pagna 4
5 he regular prblem he classcal descrpn a sysem s nrmally n erms s ranser uncn mar Ws. A classcal eedback arrangemen s: Plan wh ranser uncn Cnrller wh ranser uncn Pr. D.Iacvell - Opmal Cnrl 03//07 Pagna 5
6 he regular prblem he cnrller upu s assumed be an nsananeus uncn he plan sae : u H, Ineresng s he case lnear cnrl law: u H Pr. D.Iacvell - Opmal Cnrl 03//07 Pagna 6
7 he regular prblem I s necessary keep sme measure cnrl magnude bunded r even small durng he curse a cnrl acn. In denng ur pmal regular ask we seek ne ha has engneerng sense and yelds an pmal cnrller suable r mplemenan, preerably lnear Pr. D.Iacvell - Opmal Cnrl 03//07 Pagna 7
8 he regular prblem Nnquadrac measures d n lead, n general, a lnear eedback laws Pr. D.Iacvell - Opmal Cnrl 03//07 Pagna 8
9 Pr. D.Iacvell - Opmal Cnrl 03//07 Pagna 9
10 he regular prblem Le us cnsder he llwng lnear sysem: A u y C wh, ] ed, and ed [ A,, C are mar uncns me wh cnnuus enres; her dmensns are respecvely n n, nm, n p Pr. D.Iacvell - Opmal Cnrl 03//07 Pagna 0
11 he regular prblem A lnear cnrl law s baned we seek mnmze he quadrac perrmance nde: where Q and R have cnnuus enres, symmerc, nnnegave and psve dene respecvely; F s nnnegave dene mar smmerc, 0 F d u R u Q u J 03//07 Pr. D.Iacvell - Opmal Cnrl Pagna
12 he Rcca equan Rcca, Denn: he Rcca derenal equan asscaed a he regular prblem s: F Q A A R 03//07 Pr. D.Iacvell - Opmal Cnrl Pagna
13 he Rcca equan herem: he Rcca equan adms a unque slun, semdene psve, n he cnrl nerval. 03//07 Pr. D.Iacvell - Opmal Cnrl Pagna 3 F Q A A R
14 Pr: he man seps he pr are he llwng: he esence and unqueness herem r derenal equan may be appled nly lcally. herere here ess a subnerval, ] n whch he Rcca equan admds a unque slun Pr. D.Iacvell - Opmal Cnrl 03//07 Pagna 4
15 I mus be shwn ha desn es an elemen k j he mar such ha r any < resuls n: lm ' k j Fr he pr s useul ake n accun ha he ransn mar he mar A s lmed. Pr. D.Iacvell - Opmal Cnrl 03//07 Pagna 5
16 he Rcca equan Remark he slun he Rcca equan s symmerc Usually he Rcca equan culd be slved nly numercally Pr. D.Iacvell - Opmal Cnrl 03//07 Pagna 6
17 Slun he regular prblem herem: he pmal cnrl r he regular prblem s gven by he lnear eedback law: where: and R u R A u J, 03//07 Pr. D.Iacvell - Opmal Cnrl Pagna 7
18 Pr: We apply he resuls he Pnryagn prncple ur lnear sysem. Le us dene he Hamlnan: H, u,, Q u Ru A u he necessary cndns are: H H u Ru dg d Q 0 A Pr. D.Iacvell - Opmal Cnrl 03//07 Pagna 8
19 akng n accun he equan Rcca and denng he csae he resul s prved. In ac:. R A A Q A R A Q A Q Pr. D.Iacvell - Opmal Cnrl 03//07 Pagna 9
20 J R R 0 F, u F Q R d Pr. D.Iacvell F - Opmal d Cnrl 03//07 Pagna 0 F F d R d A A d d
21 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//07 Pagna
22 Remarks he hyphess symmery he marces Q, R and F s n resrcve he slun he Rcca equan des n depend n he nal sae. herere s slun culd be und lne. he Rcca mar s a uncn curren me even he marces A,, Q and R are cnsan. Pr. D.Iacvell - Opmal Cnrl 03//07 Pagna
23 he regular prblem R r=0 + - u R A 03//07 Pr. D.Iacvell - Opmal Cnrl Pagna 3
24 Pr. D.Iacvell - Opmal Cnrl 03//07 Pagna 4
25 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 are bunded and wh elemens n C 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 class and bunded Pr. D.Iacvell - Opmal Cnrl 03//07 Pagna 5
26 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 mnmzng he cs nde J C,, u Q u R u d Pr. D.Iacvell - Opmal Cnrl 03//07 Pagna 6
27 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 R A R u, Q A A R 0 lm u J,, 03//07 Pr. D.Iacvell - Opmal Cnrl Pagna 7
28 Pr: Le us cnsder an arbrary ne nal nsan. Frm he prevus resul he mnmum value he cs nde s: J, u hs value depends n he value. Le us shw ha he value superrly lmed. J, u s In ac, he sysem s cmpleely cnrllable, whaever he nal sae s, here ess a cnrl u able ranser he rgn n a ne me, wh an evlun. Pr. D.Iacvell - Opmal Cnrl 03//07 Pagna 8
29 We can assume ha he cuple,u s dened r all >, assumng her values equal zer r >. herere, r any > : ' ' Q' u' Ru' d ' Q' u' Ru' J, u d I he sysem s epnenally sable, rm any nal sae wh null enry he crrespndng ree evlun he sae sases hs nequaly: Pr. D.Iacvell - Opmal Cnrl 03//07 Pagna 9
30 herere we have: y smple argumens can be shwn ha he uncn s mnncally nn decreasng n 0, e e c d e c d Q Qd J u J,0, u J, 03//07 Pr. D.Iacvell - Opmal Cnrl Pagna 30
31 ', u ' In ac, cnsder '', u '' w pmal sluns crrespndng and respecvely. Assume >. hen we have: J ' '' '' '' '' ' ', ' u Q u Ru d J u ' '', ' Pr. D.Iacvell - Opmal Cnrl 03//07 Pagna 3
32 herere he uncn : J, u has a lm as ends nny. eng arbrary, llws ha any mar has a lm as ends nny. As s arbrary, he slun he Rcca equan wh nal cndn =0 adms a lmed slun r any ne as ends nny. Esence, unqueness and he characerzan he slun can be jused as n he prevus herem, cnsderng he lm r he slun heren dened and recallng ha lm ' Pr. D.Iacvell - Opmal Cnrl 03//07 Pagna 3
33 Remark: I he mar Q has egenvalues greaer r equal a psve number α r each he pmal regular prblem s asympcally sable; n ac, r he pmal slun resuls n: herere:, d u R u Q d n R 0 lm 03//07 Pr. D.Iacvell - Opmal Cnrl Pagna 33
34 Pr. D.Iacvell - Opmal Cnrl 03//07 Pagna 34
35 he seady sae slun he deermnsc lnear pmal regular prblem herem: Le us cnsder he prevus prblem wh he addve hypheses : he marces A,, Q, R are cnsan he mar Q s psve dene hen here ess a unque pmal slun: u R r A R, r where: Pr. D.Iacvell - Opmal Cnrl 03//07 Pagna 35
36 r s he cnsan mar, unque slun dene psve he algebrac Rcca equan: r R r r A A r Q 0 he mnmum value r he cs nde s: J, u r Pr. D.Iacvell - Opmal Cnrl 03//07 Pagna 36
37 Pr: Frm he prevus herem we have ha he value he cs nde r he cnsdered seady sae prblem s gven by, u J Due he sanary he prblem resuls ha s value s ndependen rm. Pr. D.Iacvell - Opmal Cnrl 03//07 Pagna 37
38 herere he unque slun he equan: s cnsan r. he esence and unqueness he pmal slun llw rm he prevus herem Q A A R 03//07 Pr. D.Iacvell - Opmal Cnrl Pagna 38
39 Remarks Cnsder he sysem: I can be dened an pmal regulan prblem rm he upu y, cnsderng he cs nde C y u A d u R u y Q y Fy y u y J, 03//07 Pr. D.Iacvell - Opmal Cnrl Pagna 39
40 he presen prblem s smlar he prevus ne, beng sased he hypheses ver he marces nvlved. Anyway he sae mus be accessble, ne wan realze he eedback acn: r=0 + - u A u C y R Pr. D.Iacvell - Opmal Cnrl 03//07 Pagna 40
41 Pr. D.Iacvell - Opmal Cnrl 03//07 Pagna 4
42 he pmal rackng prblem Prblem: le us cnsder he lnear sysem: A u, where A, are marces wh elemens uncns me wh enres C class. Cnsder he reerence varable Deermne he pmal cnrl 0 u C, and he sae C, sasyng he dnamcal cnsran and mnmzng he cs nde: r C, Pr. D.Iacvell - Opmal Cnrl 03//07 Pagna 4
43 where: Q s a symmerc sem-psve dene mar R s a symmerc psve dene mar he elemens Q and R are uncns C class d u R u r Q r u J, 03//07 Pr. D.Iacvell - Opmal Cnrl Pagna 43
44 he pmal rackng prblem he pmal rackng prblem can be asscaed he Rcca equan 0 Q A A R 03//07 Pr. D.Iacvell - Opmal Cnrl Pagna 44
45 herem: he pmal rackng prblem adms a unque pmal slun: g R R A g R u 03//07 Pr. D.Iacvell - Opmal Cnrl Pagna 45
46 where g s he slun he derenal equan he mnmum value he cs nde s: where s he slun he equan: 0 g r Q g A R g, v g u J v 0 v r Q r g R g v 03//07 Pr. D.Iacvell - Opmal Cnrl Pagna 46
47 Pr: I can be appled Pnryagn herem cnve case. Le us dene he Hamlnan: he necessary and sucen cndns are: u A Ru u r Q r u H,,, 0 0 Ru Qr A Q 03//07 Pr. D.Iacvell - Opmal Cnrl Pagna 47
48 Wh he llwng chce r he csae g he herem s prved. As ar as he cs nde s cncerned, s useul bserve ha: d d g g vd v Pr. D.Iacvell - Opmal Cnrl 03//07 Pagna 48
49 herem he pmal cnrl rackng prblem adms a unque pmal slun. Pr: he herem s prved by cnraddcn argumens. Pr. D.Iacvell - Opmal Cnrl 03//07 Pagna 49
50 g + - R u A u Pr. D.Iacvell - Opmal Cnrl 03//07 Pagna 50
51 Remarks Ne ha n he cs nde he quadrac errr n he nal nsan r s n presen. he realzan he pmal cnrl r he rackng prblem needs he slun he Rcca equan and he slun he derenal equan n g slve he rackng prblem he reerence varable r mus be knwn n advance n he nerval Pr. D.Iacvell - Opmal Cnrl 03//07 Pagna 5
52 he seady sae rackng prblem ver an nne nerval requres a seady sae slun r he derenal equan n he g uncn; r eample, r a cnsan reerence value r : g r R A Qr r Pr. D.Iacvell - Opmal Cnrl 03//07 Pagna 5
53 MALA lqr Lnear-quadrac LQ sae-eedback regular r saespace sysem Syna [,S,e] = LQRA,,Q,R,N Descrpn [,S,e] = lqrsys,q,r calculaes he pmal gan mar. Fr a cnnuus me sysem, he sae-eedback law u = mnmzes he quadrac cs uncn subjec he sysem dynamcs J, u Q u R u d Pr. D.Iacvell - Opmal Cnrl 03//07 Pagna 53
54 MALA In addn he sae-eedback gan, lqr reurns he slun S he asscaed Rcca equan SR S SA A S Q 0 aenn!!! and he clsed-lp egenvalues e = ega-*. s derved rm S usng R S Pr. D.Iacvell - Opmal Cnrl 03//07 Pagna 54
55 MALA A=[ 3; -4 ]; =[ ; 3 4] Q=[ 0; 0 3] R=[ 0;0 ] [ S ]=LQRA,,Q,R = S = Fr MALA: cnrl gan Fr MALA: slun he RICCAI equan e = Pr. D.Iacvell - Opmal Cnrl 03//07 Pagna 55
56 Pr. D.Iacvell - Opmal Cnrl 03//07 Pagna 56
57 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 mnmzng he llwng cs nde: 0,, u A C u, 0 C, 03//07 Pr. D.Iacvell - Opmal Cnrl Pagna 57
58 J, u Q u R u 0 d where: Q s symmerc semdene psve wh elemens C class R s symmerc dene psve wh elemens C class Pr. D.Iacvell - Opmal Cnrl 03//07 Pagna 58
59 herem: Le us nrduce he mar dmensn n n and ndcae wh A Q, s ransn mar parned n submarces dmensn n n. Assume ha he submar s n sngular. he pmal regular prblem wh null nal errr adms a unque pmal slun: R A,,,,, Pr. D.Iacvell - Opmal Cnrl 03//07 Pagna 59
60 u,,,, R,,,, Pr. We can apply herem 4 Pnryagng herem, cnve case. In parcular le us dene he Hamlnan: H, u,, Q u Ru A u he necessary and sucen cndns are: Ru Q A 0 u R Pr. D.Iacvell - Opmal Cnrl 03//07 Pagna 60
61 Recallng he denn Ω, we can cnsder he sysem: whse slun, akng n accun he nal cndn, s:,,,, 03//07 Pr. D.Iacvell - Opmal Cnrl Pagna 6
62 akng n accun he cndn n and he nn sngulary submar,, resuls n:,, and herere:,,,, Pr. D.Iacvell - Opmal Cnrl 03//07 Pagna 6
63 ha, subsued n he epressn cnrl yelds he pmal slun. u R Pr. D.Iacvell - Opmal Cnrl 03//07 Pagna 63
64 Remarks he pmal slun depends rm he nal sae Ne ha he mar are pssble:, s sngular w suans - n pmal slun ess r - here es nne nes dependng n he number slun he sysem:,, wh respec Pr. D.Iacvell - Opmal Cnrl 03//07 Pagna 64
65 Pr. D.Iacvell - Opmal Cnrl 03//07 Pagna 65
66 he pmal regular prblem wh lmed cnrl Prblem: Le us cnsder he pmal regular prblem n ne me nerval wh he addnal cnsran: u, j,,..., p,, j and he hyphess ha he weghng mar R s dagnal Pr. D.Iacvell - Opmal Cnrl 03//07 Pagna 66
67 herem: All he nrmal sluns can be und slvng he derenal sysem: assumng: F A Q R sa A R sa u 03//07 Pr. D.Iacvell - Opmal Cnrl Pagna 67
68 Pr: Le s apply Pnryagn herem cnve case; he necessary and sucen cndns becmes: u Q Ru A u R : j, j,,..., p F Pr. D.Iacvell - Opmal Cnrl 03//07 Pagna 68
69 y addng he quany R bh members he nequaly, we can wre as llws: u R R u R : j R R R, j,,..., akng n accun ha he mar R s dagnal and psve dene, he herem s prved. p Pr. D.Iacvell - Opmal Cnrl 03//07 Pagna 69
70 Remarks Wh he new cnsran he sysem sae-csae s nnlnear he esence he slun s n guardaneed I s pssble cnsder he same prblem wh he nal cndn he preceedng herem sll hlds whu he cndn n he csae. Pr. D.Iacvell - Opmal Cnrl 03//07 Pagna 70
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