Optimal Control. Lecture 5. Prof. Daniela Iacoviello
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1 Opimal Conrol ecre 5 Pro. Daniela Iacoviello
2 THESE SIDES ARE NOT SUFFICIENT FOR THE EXAM: YOU MUST STUDY ON THE BOOKS Par o he slides has been aken rom he Reerences indicaed below Pro. D.Iacoviello - Opimal Conrol 3//28 Pagina 2
3 Corse oline Inrodcion o opimal conrol Nonlinear opimizaion Dynamic programming Calcls o variaions Calcls o variaions and opimal conrol Q problem Minimm ime problem Pro.Daniela Iacoviello- Opimal Conrol Pro. D.Iacoviello - Opimal Conrol 3//28 Pagina 3
4 Pro. D.Iacoviello - Opimal Conrol 3//28 Pagina 4
5 Hamilon-acobi- Bellman eqaion 5
6 William Rowan Hamilon 4 Ags 85 2 Sepember 865 He was an Irish physicis asronomer and mahemaician who made imporan conribions o classical mechanics opics and algebra. His sdies o mechanical and opical sysems led him o discover new mahemaical conceps and echniqes. His greaes conribion is perhaps he reormlaion o Newonian mechanics now called Hamilonian mechanics. This work has proven cenral o he modern sdy o classical ield heories sch as elecromagneism and o he developmen o qanm mechanics. In mahemaics he is perhaps bes known as he invenor o qaernions.. 6
7 acobi Posdam 84- Berlin 85 He was sill only 2 years old ye he had reached he necessary sandard o ener Universiy in 87 Arond 825 acobi changed rom he ewish aih o become a Chrisian which now made niversiy eaching possible or him. By he academic year he was eaching a he Universiy o Berlin. acobi carried o imporan research in parial dierenial eqaions o he irs order and applied hem o he dierenial eqaions o dynamics. He also worked on deerminans and sdied he ncional deerminan now called he acobian. 7
8 Richard Bellman USA He was ineresed in dynamic programming wih applicaions in biology and medicine. 8
9 The Hamilon-acobi eqaion I is an approach in some sense alernaive o he Minimm Principle o Ponryagin combined wih he Eler agrange eqaion. Acally he Hamilon acobi eqaion has so ar rarely proved sel ecep or linear reglaor problems o which i seems pariclarly well sied. 9
10 The Hamilon-acobi eqaion The Hamilon-acobi H- eqaion is saisied by he opimal perormance inde nder siable diereniabiliy and coniniy assmpions. I a solion o he H- eqaion has cerain diereniabiliy properies hen his solion is he desired perormance inde.
11 The Hamilon-acobi eqaion - Sch a solion need no eis - No every opimal perormance inde saisies he Hamilon acobi eqaion The H eqaion represens only a sicien condiion on he opimal perormance inde
12 Principle o opimaliy Minimizing over [ ] is eqivalen o minimizing over [ ] and [ ] 2
13 The Hamilon-acobi eqaion Principle o opimaliy: OR: min min G d d min d Pro. D.Iacoviello - Opimal Conrol 3 3//28
14 The Hamilon-acobi eqaion Principle o opimaliy: min d The opimal cos or rajecories commencing a and inishing a is incrred by minimizing he sm o he cos in ransiing o and he opimal cos rom here onwards ha is: d and 4
15 Principle o opimaliy A conrol policy opimal over he inerval [ ] is opimal over all sbinervals [ ]
16 The Hamilon-acobi eqaion The Hamilon-acobi eqaion has so ar rarely proved sel ecep or linear reglaor problems 6
17 The Hamilon-acobi eqaion Assme he Hamilonian reglar: here eiss a niqe minimm wih respec o. The Hamilon-acobi eqaion: Pro. D.Iacoviello - Opimal Conrol 7 3//28 V V H V T T V V V V V T T T T
18 The Hamilon-acobi eqaion Problem: Consider he sysem: Find he opimal conrol given ha minimizes he cos: d G 8
19 Deine he Hamilonian: H T Assme: and G are coninosly diereniable in all argmens The Hamilonian reglar: here eiss a niqe minimm wih respec o 9
20 Derivaion o he Hamilon-acobi eqaion e s consider he sysem: ied ied or ree wih he cos inde: min min Noaion: min d G d G Iniial insan iniial sae 2
21 G d d G d d G d min min min Pro. D.Iacoviello - Opimal Conrol 2 3//28
22 min min ] [ d G d d Principle o opimaliy Idea: d min min ] [ d d d G d d d Pro. D.Iacoviello - Opimal Conrol 22 3//28
23 d small [ d] d d d d d d d d min [ d] d d d d min d d d 23
24 d d d min Divide or d and consider lim min Pro. D.Iacoviello - Opimal Conrol 24 3//28
25 is a solion o he Hamilon-acobi H- eqaion: wih bondary condiions: - coninos wih respec o is argmens -he irs derivaives o coninos wih respec o is argmens - - ' G Pro. D.Iacoviello - Opimal Conrol 25 T o 3//28
26 min is he opimal perormance inde or d G and he conrol: o Is he opimal conrol a ime or he class problems wih cos inde d G 26
27 The Hamilon-acobi eqaion Eample 2 rom Anderson Consider he sysem: Wih perormance inde: d wih 2 Applying he H- eqaion we have: 27
28 The minimizing conrol is: And we have: Wih bondary condiion: min In acal ac i is rarely possible o solve a Hamilon- acobi eqaion Pro. D.Iacoviello - Opimal Conrol 28 3//28
29 The Hamilon-acobi eqaion Main resls The Hamilon-acobi H- eqaion is saisied by he opimal perormance inde nder siable diereniabiliy and coniniy assmpions. I a solion o he H- eqaion has cerain diereniabiliy properies hen his solion is he desired perormance inde. Noe ha he H- eqaion represens only a sicien condiion on he opimal perormance inde 29
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