第九讲 动态最优问题求解：（II....pdf

1Êù Ä •`¯K¦)µ £II¤•`›› •% E ŒÆ²LÆ •% (E ŒÆ²LÆ ) êþ²LÆ£1Êù¤ 1 / 15 ù̇SN 1. lÑžm •`››¯K 1.1 l.‚KF•{ •ŒŠ n 1.2 lÑžm•ŒŠ nµLㆲL¹Â 1.3 A^µlÑžm•`O•¯K 2. ëYžm •`››¯K 2.1 ëYžm •ŒŠ nµLㆲL¹Â 2.2 A^µè’?\{L 3. •ŒŠ •% n y²£ÀÖ¤ (E ŒÆ²LÆ ) êþ²LÆ£1Êù¤ 2 / 15 1. lÑžm •`››¯K 1.1 l.‚KF•{ •ŒŠ n lÑžm •`››µ±)·±Ï;O¯K•~ éu)·±Ï;O¯Kµ T X max β t ln ct t=1 s.t.kt − kt−1 = w + rkt−1 − ct k0 = kT = 0 .‚KF¼ê•µ L= T X β t ln ct − t=1 T X λt (kt − kt−1 − w − rkt−1 + ct ) t=1 Ù¥µ T X λt (kt − kt−1 ) = λ1 (k1 − k0 ) + λ2 (k2 − k1 ) + ... + λT (kT − kT −1 ) t=1 = T X kt (λt − λt+1 ) − λ1 k0 + λT +1 kT t=1 •% (E ŒÆ²LÆ ) êþ²LÆ£1Êù¤ 3 / 15 1. lÑžm Ç— î¼êÚ˜ •`››¯K 1.1 l.‚KF•{ •ŒŠ n ^‡ .‚KF•§Œ±U •eª§¿½ÂÇ— î¼ê£Hamiltonian¤µ   T X t  L= β ln ct + λt (w + rkt−1 − ct ) + kt (λt+1 − λt ) {z } | t=1 ≡H(kt ,ct ,λt ,t) + λ1 k0 − λT +1 kT Ä •`¯K ˜ 7‡^‡•µ ∂L ∂Ht =0⇒ =0 ∂ct ∂ct ∂Ht+1 ∂L =0⇒− = λt+1 − λt ∂kt ∂kt ∂L = 0 ⇒ kt − kt−1 = w + rkt−1 − ct ∂λt ±9Äu>.^‡ •% (E ŒÆ²LÆ ) (1) (2) (3) ¥ -©Ž^‡£Kuhn-Tucker condition§KTC¤µ λ1 k0 = λT +1 kT = 0 (4) êþ²LÆ£1Êù¤ 4 / 15 1. lÑžm •`››¯K 1.2 lÑžm•ŒŠ lÑžm•`››¯K •ŒŠ nµLㆲL¹Â n ˜„/§éulÑžm•`››¯K£discrete time optimal control problem¤ §Ù ¥ xt •G Cþ§ut •››Cþµ max U = T P f (xt , ut ) t=1 s.t. xt − xt−1 = g (xt−1 , ut ) x0 = xT = 0 ½ÂÇ— î¼ê H(x, u, λ, t) = f (xt , ut ) + λt g (xt−1 , ut )" Apä7•ŒŠ n£Pontryagin’s maximum principle¤•Ñ§{x∗t , u∗t }Tt=1 ´þã•`››¯KS:) ˜ 7‡^‡´§•3 λ∗t ¦ µ t | =0 1. é››Cþ ut £control variable¤¦ k ∂H ∂ut ∗ 2. éG Cþ xt £state variable¤¦ 3. é 4. ¥ •% ∂Ht k − ∂x |∗ = λ∗t − λ∗t−1 t−1 t Cþ λt £costate variable¤¦ k ∂H | = xt − xt−1 ∂λt ∗ -©Ž^‡£KTC¤^‡ λ1 x0 = λT +1 xT = 0 (E ŒÆ²LÆ ) êþ²LÆ£1Êù¤ 5 / 15 1. lÑžm ^‡ •`››¯K 1.2 lÑžm•ŒŠ nµLㆲL¹Â ²Lƹ Ǘ î¼ê´ ÏdŠ f ÚG CþCz‘5 dŠƒÚ§ ö uCzü ¦ ±Kfd‚ λt · (xt − xt−1 )" H(x, u, λ, t) = f (xt , ut ) + λt g (xt−1 , ut ) I •`ÀJA¦odŠ•Œ§džUC››Cþ I ∂Ht ∂xt−1 ‚ ´G CþCĘü ¤‘5 >SÂçλt − λt−1 K´G CþKfd Cz§Œ±n)•˜ï˜ü G Cþ ∂Ht ÎÒƒ‡"− ∂x |∗ = λt − λt−1 • t−1 I >S¤ §ˆ ¡•G ∂Ht | = g = xt − xt−1 ´½Â•§§~XýŽ ∂λt ∗ I c¡n‡^‡‰Ñ vkå:Úª: t >SÂÕ" ∂H | = 0" ∂ut ∗ •`´» Cþ •`žüöêþƒ ªÏÃ@|^‡" 啧" //G0 £þ,!eü½Ù¦Cz5K¤ § •›§„´káõ !ƒÓ/G ´»ÑÎÜ^‡"• XJ • ª(½,^´»§I‡\þªà^‡" •% (E ŒÆ²LÆ ) êþ²LÆ£1Êù¤ 6 / 15 1. lÑžm •`››¯K 1.3 A^µlÑžm•`O•¯K A^µlÑžm•`O•¯K éuÕÏ. •`O•¯Kµ max ∞ P β t ln ct t=1 α s.t. kt = zkt−1 − ct k0 = 1, lim β t+1 u0 (ct+1 ) kt = 0 t→∞ ÑÇ— α î¼ê Ht = β t ln ct + λt (zkt−1 − kt−1 − ct )§† |^•ŒŠ n ^‡¦Ñ•`)µ ∂Ht βt =0⇒ = λt ∂ct ct ∂Ht α−1 = λt − λt−1 ⇒ λt αzkt−1 = λt−1 ∂kt−1 ∂Ht α = g = kt − kt−1 ⇒ kt = zkt−1 − ct ∂λt − k0 = 1, lim β t+1 u0 (ct+1 ) kt = lim λt+1 kt = 0 t→∞ •% (E ŒÆ²LÆ ) t→∞ êþ²LÆ£1Êù¤ 7 / 15 2. ëYžm ëYžm •ŒŠ •`››¯K 2.1 ëYžm •ŒŠ nµLㆲL¹Â n éuëYžm•`››¯K£continuous time optimal control problem¤ §Ù¥ x(t) •G Cþ§u(t) •››Cþµ Z T f (x(t), u(t), t)dt max 0 dx s.t. ≡ ẋ = g(x(t), u(t), t) dt x(0) = x0 ½ÂÇ— ∗ î¼ê H (x, u, λ, t) = f (x, u, t) + λ (t) · g (x, u, t)§Šâ•ŒŠ ∗ {x (t), u (t)} ´þã•`››¯KS:) ˜ n§ ∗ 7‡^‡´•3 λ (t) ÷vµ |∗ = 0 1. é››Cþ u(t)£control variable¤¦ k ∂H(x,u,λ,t) ∂u 2. éG Cþ x(t)£state variable¤¦ k ∂H(x,u,λ,t) |∗ = −λ̇|∗ ∂x 3. é Cþ λ(t)£costate variable¤¦ k ∂H(x,u,λ,t) |∗ = ẋ|∗ ∂λ 4. ªà^‡ λ (T ) = 0 ½ H(T ) = 0 •% (E ŒÆ²LÆ ) êþ²LÆ£1Êù¤ 8 / 15 2. ëYžm ^‡ •`››¯K 2.1 ëYžm •ŒŠ nµLㆲL¹Â ²L¹Âµ±è’Ý] •`››¯K•~ b½è’¡ eã•`Ý]¯Kµ max Π (u) = RT 0 π (k, I, t) dt s.t. k̇ = f (k, I, t) , k (0) = k0 Ç— • H (k, I, λ, t) = π (k, I, t) + λ (t) f (k, I, t)"1˜‘ π •è’ Ï|d0§1 ‘¥ λ •G Cþ] Kfd‚§¦±] Czþ u/] / dŠ Cz0§ü‘ƒÚ•è’z˜Ï oÂÃ" I ∂H ∂u ∂f = 0 ⇒ ∂π = −λ ∂u µÝ]Y²ˆ ∂u I − ∂H = λ̇µ] ∂k I ∂H ∂λ •`ž§>S|d uÝ] Kfd‚ Cz u] = f = k̇ •ýŽ Ŭ¤ " >SoÂÃ" 啧" I éuR†ª(‚¯K§k λ(T ) = 0µè’¬¿©|^] §¦•ªž• ] d Š•"" I éuY²ª(‚¯K§k H(T ) = 0µL²è’ò¿©|^¤kI|Ŭ§¦• ªˆ •% k(T ) žoÂÕ"" (E ŒÆ²LÆ ) êþ²LÆ£1Êù¤ 9 / 15 2. ëYžm •`››¯K 2.2 A^µè’?\{L è’?\{L£&ŽÜA§SK10.3¤ ‰½½|oI¦ q (t) = a − bp (t) Ú è’ {Ž è’?\ þCzÇ ẋ§Œè’ÏLd‚üÑ •`››¯K•£b½ëê a, b, k > 0!2( β1 − 1) < kb < 2( β1 + 1)! pmin > c > 0¤ µ max R∞ e−βt [p (t) − c] [a − bp (t) − x (t)] dt t=0 s.t. ẋ (t) = k [p (t) − pmin ] Äk ÑÇ— G î¼ê H = e−βt (p − c) (a − bp − x) + λk (p − pmin )§Ù¥ x • Cþ§p •››Cþ"•ŒŠ n‰Ñ ˜ ^‡•µ −βt ∂H e =0⇒λ=− (a − 2bp + bc − x) ∂p k ∂H = −λ̇ ⇒ λ̇ = e−βt (p − c) ∂x üªéá2\þ½Â•§ ˜ ‡©•§|µ 1 ṗ = [−ẋ − β (a − 2bp + bc − x) + k (p − c)] 2b ẋ = k(p − pmin ) •% (E ŒÆ²LÆ ) êþ²LÆ£1Êù¤ 10 / 15 2. ëYžm •`››¯K 2.2 A^µè’?\{L - †=£Ä - ẋ = ṗ = 0§)Ñ µ    (a + bc) β + kc k ∗ ∗ − pmin + 2b p = pmin , x = k + 2bβ β w,§ .- Œè’ ½d p = pmin < (a+bc)β+kc ž§ k+2bβ è’¬?\ x∗ > 0"- ‡ ©•§|µ " ṗ # ẋ " = Q:´»•£Ù¥ λ = β β 2b k 0 √ 2 β− #" p x # " + k(pmin −c)−β(a+bc) 2b # −kpmin β +2βk/b ∈ (−1, 0)¤ µ 2 p (t) = (p0 − p∗ ) eλt + p∗ x (t) = (x0 − x∗ ) eλt + x∗ •% (E ŒÆ²LÆ ) êþ²LÆ£1Êù¤ 11 / 15 2. ëYžm •`››¯K 2.2 A^µè’?\{L ƒã ṗ = 0 ⇒ pss = − ẋ = 0 ⇒ •% (E ŒÆ²LÆ ) 1 ss k (pmin − c) − β (a + bc) x + 2b 2bβ ⇒ x > xss , ṗ > 0 pss = pmin ⇒ p > pss , ẋ > 0 êþ²LÆ£1Êù¤ 12 / 15 3. •ŒŠ n y²£ÀÖ¤ E.‚KF¼ê Äk 1 Ñ.‚KF¼ê: Z T Z T L= f (x(t), u(t), t)dt + λ(t) [g(x(t), u(t), t) − ẋ] dt 0 0 RT R |^©ÚÈ©úª 0 λ(t)ẋdt = λ(T )x(T ) − λ(0)x(0) − 0T λ̇x(t)dt kµ Z Th i H(x(t), u(t), λ(t), t) + λ̇x(t) dt + λ(0)x(0) − λ(T )x(T ) L= 0 duÈ©ØUé˜]m Cþ¦ §Ïd·‚3 x∗ (t), u∗ (t) NCÚ\?¿ !žm ¢Š¼ ꣡•/6Ä-‚0¤ p(t), q(t)§Ú?¿¢ê ∆T, ∆xT "- ε ´˜‡?¿ x(t) = x∗ (t) + εp(t) u(t) = u∗ (t) + εq(t) T = T + ε∆T x(T ) = x∗ (T ) + ε∆xT ¢ê§·‚Pµ .‚KF¼êŒ±- •µ Z T (ε) h i L= H (x∗ + εp (t) , u∗ + εq (t) , λ(t), t) + λ̇ · (x∗ + εp (t)) dt 0 − λ (T + ε∆T ) (x∗ (T ) + ε∆xT ) + λ (0) x0 1 ö¥˜Í§1999µ5Ä •`zÄ:6§ •% (E ŒÆ²LÆ ) [÷ȧûÖ<Ö, êþ²LÆ£1Êù¤ 13 / 15 3. •ŒŠ n y²£ÀÖ¤ ˜ ^‡ L= Z T (ε) h i H (x∗ + εp (t) , u∗ + εq (t) , λ(t), t) + λ̇ · (x∗ + εp (t)) dt 0 − λ (T + ε∆T ) (x∗ (T ) + ε∆xT ) + λ (0) x0 Z T  ∂H ∂H · p (t) + · q (t) + λ̇ · p (t) dt ∂x ∂u 0     ∂T ∂x (T ) ∂T + H(T ) + λ̇x(T ) − λ̇ (T ) x(T ) + λ (T ) =0 ∂ε ∂ε ∂ε   Z T  ∂H ∂H ⇒ + λ̇ · p (t) + · q (t) dt + H(T )∆T − λ (T ) ∆xT = 0 ∂x ∂u 0 ∂L =0⇒ ∂ε du p(t), q(t) ´?¿-‚§Ïdk ∂H + λ̇ = ∂H = 0§=•ŒŠ ∂x ∂u ^‡", •% 3 E.‚KF•§ž®²^ (E ŒÆ²LÆ ) ∂H ∂λ = g = ẋ êþ²LÆ£1Êù¤ n ü‡˜ ½Â•§" 14 / 15 3. •ŒŠ n y²£ÀÖ¤ ªà^‡ H(T )∆T = λ (T ) · ∆xT I ªàžm ½!G CþØ(½ R†ª(‚¯K ∆T = 0, ∆xT 6= 0 ‡¦ λ (T ) = 0§AO/µ I éuN\G Cþe•‡¦ x(T ) > xmin / äR†ª(‚0¯K§ƒ A^‡• λ (T ) (x(T ) − xmin ) = 0" I éuÕÏ.¯K / äR†ª(‚0¯K§kî 5^‡µ lim λ (t) (x(t) − xmin ) = 0 t→∞ I ªàžmØ ½!G Cþ(½ Y²ª(‚¯K ∆T 6= 0, ∆xT = 0 ‡¦ éukžmþ•‡¦ T > Tmax H(T ) = 0"d / äY²ª(‚0¯K§ª à^‡• H(T )(T − Tmax ) = 0" I ªàžmÚG CþÑØ(½! k ª å x(T ) = φ (T ) ª(-¡¯K‡¦ 0 H(T ) − λ(T )φ (T ) = 0" •% (E ŒÆ²LÆ ) êþ²LÆ£1Êù¤ 15 / 15