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hw 4 solutions - LlL'I—EH—EUUH llfll4'fi P‘llzL'H E...

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Unformatted text preview: LlL'I—EH—EUUH llfll4'fi P‘llzL'H E: HtHLI tNLillelelNLi {Zr-Yd HES-4 DUEM 5' f 'Zu-D K IBOWI Homework Set #4: MAE 4720/7720 — Modern Control Fall 2008 Due October 28, 2008 Problem 1: Given the quadratic scalar function F to be minimized: Minimize F(u) = 2n]2 + 6n: + 0.3a.2 subject to the two linear equality constraints 1121(Il)=ul —4-u2 =0 h2(u)=3ul flu: +2143 —8=0 Determine the optimal solution vector u* and show your steps. Problem 2: A fixed-time optimal control problem is shown below: Determine the optimal control u*(t), O 5 t 5 T, which minimizes Minimize J = 502 + as: + ax: )dr 0 subject to the dynamical equations 2:. =3c2 i2 : 5:1:I + 221. where the initial state is x(0) = [ 2 43.5 ]T and enddtime T = 4 (fixed). :1) Define the Hamiltonian, H(x, u) b) Present all first-order necessary conditions for an optimal solution (i.e., derive the _,_..-""" ZPBVP). 6‘ o) Is the Hamiltonian constant? Explain. 1-"! |-'. [£11 LIL' | —E":J—Eldldti 11:1: 4? MtL'H E: HtHLI IzNLillelelNLi {Zr-Yd HES-4 DUEM |-‘ . [£13 . ...________.. _.__i_! *9 HWLWWI m..__._I_U£M ............. _—%F I MW. # 4' i x""- :62. ’1. M. F(Zi)~2 ;._ (£1. 032.; ‘P 55’ “’1 - Lt, -t- bra + — H3 !‘ 5-“.be +19: 7:037): “1.4“: = [:j‘ a; firm Layman“ j at“ amfimw per-farm“; (ado M: F =- F + > T5 '-'-‘ Eur-L + Eu: + 03%;— l '4- )‘l (L4! fill-LIL) + }2(3q,—¢r1+2.q3-9> ‘ N ~ - 1c; - 3i? 7* uefim‘; 92¢“:th l. Malamth If; __ O f"? I N =35. :1 0 xv )\ ' 3F .. 4-H, + 1, 4* 3.}; 1 Q “I (a? = (2:41 #4.)” ”A; = «:3 $13? but; (fi‘nflr) [ as UMkAawns . #- QH—w = “I HF 4L2. = ('3 (‘4')th “3;; :‘BA, A' 9 Al) 2351., 3a,*—H_,_ +2“; ”8—20 ”A .312 EU 3 HEETS 22-1 41 LIL' | {dB—EUUH 11:1: 4? MtL'H E: HtHLI tNLillelelNLi {Zr-Yd HES-4 DUEM |-‘ . __—.___."d...__.,.‘_.__M.‘.._..M.._H.., ML. ._..... _ __ __.._— 1 xx. x . ‘ SOtU‘Q Adm,- fiysmfit ; 4- c: a 1 3 M, o E D 12. a -4. *3 Liz a g a a am, a z ”a = 0 E9 L / ‘5‘ 5’ a ‘3‘ >‘r ‘3 II E; I 3 *l 1 o o A: 3 gm - . .M I: C it: FE ($2; -=~ ... 056,02. 3) 1% L = fl :2 = 0H0?— L 3.2.2.3? | 0.6sz > #03587 F“ . .2»? 9.5609 ” M : mil-#02, 3-1189 :fi'TB (ta-EDA gq‘glpf‘LfELT mfldq'fi‘aHJ Me. 2.“? paw—Mauls! a. I 4" 0 <5 am- (a. 3” fix 0 CD 0. x - If Mr". UEI—E‘j—EUUH 11548 P'II:L'H ES: HtHLI IzNLillelelNLi . b'l’j titi4 DUEM E—'. 1:14 ___E—.#_* mmmmmmmmmmmmmmmmm __E_ _ _ _. . .J - _.._______ fl FREE- 2. 2%me “*(t ) 2 (-7 E {- E T I th‘cji flum'mE'afl. fl“ ‘7' L ‘ . MM 3- :. £5 [“7" 4. 33;; + 9X2 )fi/‘E o E E :2.ij +u “>2, = x,__ Egg; 7:; 2 5x, + 2H, E Imam. saga) = 2- T = 4 mm, M WA) , I ”I" ’ ; E”: . Q; a.) Ham/4mm .. H = L + A Tat m L — —‘ Z .1. “=- 2 L f“ — ’E‘ E LL) "' 3. L4 + 3k; + K}... ) j “'“ x1 rm; .3 E H: 3E:[M1*+3xf‘+9x;> + MK; +)\1(5x,+2aE¢) E9) ".1“ 51‘ amffir fleet-335mg, GamnEffi-‘ms ( LPEUP )E X! E E ii.— = —._a.—tl-. = '— 9 x1 - >‘I E i Carl E E M 3351‘) = :121/ E E A 3x {WT E Jow+ qéf/YZTH 7-. o E E E E So A; (T) = O E E f...‘ E EC 5 E ALCT) O “ E E E E 22441 BUSHEETS 22442 1UDSHEETE 22144 ZUGSHEETS @1313 UEI—EH—EUUH lUl4H _d—A—I—_d\_d_d‘dv—dd .. “mm.“ MtEH & HtHU tNUthtHlNU , .,1. .. .. . . _..._ _..._ . ___..____ __ ___\_H.A_I_Jlm P208 2. (egg-pf) firm/fl iii: M+2J1=CS ; 3a. 1 # LN" - "2}; affi‘mue mmfiyz/ ASImMe. zPEVP: 75: == 5’ (DEE. is- ”:f‘ ET: 4. XI(D)= 2 5% h“ 7‘2. xf (T) - fireep ramps); a = 5x, um: w) = 79% mam? X, = “329-51; Aim—J = o ALdfl)=? X3 3 —3X, —A] A; CT"): C) Rifle. u: Ceca Salue. fl} 2-PEUP a‘nmL £56 099:? are their”. C) If; H. Cons‘F-zfih? JH xi _ an "d"? “ 37- giucfi Lea-MC/fim-an H data: m EKPGW¥9 dapw M +7‘Ma 1:, {142“ fi * O I , .313 " arm? 14 = 0 27% H (I COWSHIzZL um...wm..m.umwuuw v-w H! 5T5 HH4 DUHU F.Ub —!'7fi1 Mb I-‘. 5'35 554 bUHU MtL'H #5: HtHLI IzNLi 1 NI:le 1 NH 1111143 LIL' | —E‘:J-EUUH Eur—W nmn HEAR. mkmmim F: mgrmm mkmmzw mm Eu wxmm m meéwwmnw (WW TEITFIL F' . IZIE ...
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