1࢔࢘ࢺࣛ࢖ࣥ2࢔࢘ࢺࣛ࢖ࣥ㟁ẼᏛ఍ ⏘ᴗᛂ⏝㒊㛛 ࣓࢝ࢺࣟࢽࢡࢫไᚚᢏ⾡ጤဨ఍⢭ᐦࢧ࣮࣎㸭ᐇୡ⏺ࣁࣉࢸ࢕ࢡࢫ㸭࣮ࣔࢩࣙࣥࢥࣥࢺ࣮ࣟࣝྡྂᒇᕤᴗ኱Ꮫ㸦ྡྂᒇᕷ㸧㸪2015 ᖺ 12 ᭶ 2 ᪥᭱㐺ไᚚࡢᇶ♏
᭱㐺ไᚚၥ㢟㸪೵␃᮲௳㠀⥺ᙧࣔࢹࣝணᐇ᫬㛫᭱㐺໬࡟ࡼࡿ㠀⥺ᙧᶵᲔࢩࢫࢸ࣒ࡢࣔࢹࣝண ไᚚไᚚࣔࢹࣝண ไᚚ࡜ࡣ㸽̭≉ᚩᐇ᫬㛫᭱㐺໬ၥ㢟ࡢᛶ㉁ࢆ฼⏝ Ѝ ᫬㛫ኚ໬ࡍࡿゎࡢ㏣㊧Ѝ ຠ⋡ⓗ࡞࢔ࣝࢦࣜࢬ࣒ி㒔኱Ꮫ኱Ꮫ㝔᝟ሗᏛ◊✲⛉ࢩࢫࢸ࣒⛉Ꮫᑓᨷᛂ⏝஦౛኱ሯ ᩄஅ㸰ࣜࣥࢡ࢔࣮࣒㸪࣍ࣂ࣮ࢡࣛࣇࢺ㸪⯪⯧࡯࠿

ohtsuka@i.kyoto-u.ac.jp⮬ືࢥ࣮ࢻ⏕ᡂᩘᘧฎ⌮࡟ࡼࡿ C ࣉࣟࢢ࣒ࣛࡢ⮬ື⏕ᡂ᭱㐺ไᚚࡢᇶ♏

3᭱㐺ไᚚࡢᇶ♏

࢔࢘ࢺࣛ࢖ࣥ᭱㐺ไᚚࡢᇶ♏
᭱㐺ไᚚၥ㢟㸪೵␃᮲௳

4᭱㐺ไᚚၥ㢟୚࠼ࡽࢀࡓືⓗࢩࢫࢸ࣒ࡢᩘᏛࣔࢹࣝ࡜ᣊ᮰᮲௳ࡢୗ࡛㸪ホ౯㛵ᩘࢆ ᭱ᑠ ࡟ࡍࡿไᚚධຊࢆồࡵࡼ㸬® 㠀⥺ᙧࣔࢹࣝண ไᚚࣔࢹࣝண ไᚚ࡜ࡣ㸽̭≉ᚩ® ᐇ᫬㛫᭱㐺໬ၥ㢟ࡢᛶ㉁ࢆ฼⏝ Ѝ ᫬㛫ኚ໬ࡍࡿゎࡢ㏣㊧Ѝ ຠ⋡ⓗ࡞࢔ࣝࢦࣜࢬ࣒® ᛂ⏝஦౛㸰ࣜࣥࢡ࢔࣮࣒㸪࣍ࣂ࣮ࢡࣛࣇࢺ㸪⯪⯧࡯࠿® ⮬ືࢥ࣮ࢻ⏕ᡂᩘᘧฎ⌮࡟ࡼࡿ C ࣉࣟࢢ࣒ࣛࡢ⮬ື⏕ᡂ౛ࣟࢣࢵࢺࢆᆅ⌫࠿ࡽᑠᝨᫍ࡬㣕ࡤࡍ࡜ࡁ㸪ᡤせ᫬㛫ࡲࡓࡣᾘ㈝⇞ᩱࢆ ᭱ᑠ ࡟ࡏࡼ㸬ࡓࡔࡋ㸪᥎ຊࡢ኱ࡁࡉ࡟ࡣไ㝈ࡀ࠶ࡿ㸬⮬ື㌴ࡀ࣮࢝ࣈࢆ᭤ࡀࡿ࡜ࡁࡢᡤせ᫬㛫ࢆ ᭱ᑠ ࡟ࡏࡼ㸬ࡓࡔࡋ㸪ࢫࢸ࢔ࣜࣥࢢゅ㸪㥑ື࣭ไືຊ㸪ࢱ࢖ࣖᦶ᧿ຊ࡟ࡣไ㝈ࡀ࠶ࡿ㸬࢚ࣥࢪࣥࡀጞືࡋ࡚┠ᶆᅇ㌿ᩘ࡟㐩ࡍࡿࡲ࡛ࡢ᫬㛫ࢆ ᭱ᑠ ࡟ࡏࡼ㸬ࡓࡔࡋ㸪ࢫࣟࢵࢺࣝ㛤ᗘ㸪⇞ᩱᄇᑕ㔞㸪᤼ฟ࢞ࢫᡂศ࡟ࡣไ㝈ࡀ࠶ࡿ㸬᭱㐺ไᚚࡢᇶ♏

5᭱㐺ไᚚၥ㢟᭱㐺ไᚚࡢᇶ♏

6೵␃᮲௳㸸࢜࢖࣮࣭ࣛࣛࢢࣛࣥࢪࣗ᪉⛬ᘧ୚࠼ࡽࢀࡓືⓗࢩࢫࢸ࣒ࡢᩘᏛࣔࢹࣝ࡜ᣊ᮰᮲௳ࡢୗ࡛㸪ホ౯㛵ᩘࢆ᭱ᑠ࡟ࡍࡿไᚚධຊࢆồࡵࡼ㸬ẋ = f (x, u),∂Hλ̇ = −(x, u, λ, μ),∂x∂H(x, u, λ, μ) = 0,∂u
C(x, u) = 0.

᭱ࡶᇶᮏⓗ࡞ၥ㢟タᐃ㸦⤊➃᫬้ T ᅛᐃ㸪⤊➃≧ែ x(T ) ⮬⏤㸧≧ែ᪉⛬ᘧ㸸 ẋ(t) = f (x(t), u(t))ึᮇ≧ែ㸸 x(0) = x0 (given)ホ౯㛵ᩘ㸸ᣊ᮰᮲௳㸸J = ϕ(x(T )) + T0㸦ᩘᏛࣔࢹࣝ㸧L(x(t), u(t))dt

C(x(t), u(t)) = 0x(t)㸸≧ែ࣋ࢡࢺࣝ㸪u(t)㸸ධຊ࣋ࢡࢺࣝ7࢔࢘ࢺࣛ࢖ࣥ® ᭱㐺ไᚚࡢᇶ♏
᭱㐺ไᚚၥ㢟㸪೵␃᮲௳㠀⥺ᙧࣔࢹࣝணไᚚࣔࢹࣝண ไᚚ࡜ࡣ㸽̭≉ᚩ® ᐇ᫬㛫᭱㐺໬ၥ㢟ࡢᛶ㉁ࢆ฼⏝ Ѝ ᫬㛫ኚ໬ࡍࡿゎࡢ㏣㊧Ѝ ຠ⋡ⓗ࡞࢔ࣝࢦࣜࢬ࣒® ᛂ⏝஦౛㸰ࣜࣥࢡ࢔࣮࣒㸪࣍ࣂ࣮ࢡࣛࣇࢺ㸪⯪⯧࡯࠿® ⮬ືࢥ࣮ࢻ⏕ᡂᩘᘧฎ⌮࡟ࡼࡿ C ࣉࣟࢢ࣒ࣛࡢ⮬ື⏕ᡂλ(T ) =∂ϕ∂x

T(x(T )),ึᮇ≧ែ࡜⤊➃ඹ≧ែࡀ୚࠼ࡽࢀ࡚࠸ࡿ㸰Ⅼቃ⏺್ၥ㢟Ѝ ᩘ್ゎἲ㸦཯᚟ἲ㸧ࡢィ⟬㔞ࡀከ࠸㸬[Animation]ホ౯㛵ᩘ J ࡣ㸪㛵ᩘ u ࡢ㛵ᩘ㸸ỗ㛵ᩘไᚚࣁ࣑ࣝࢺࢽ࢔ࣥ㸸 H(x, u, λ, μ) = L(x, u) + λT f (x, u) + μT C(x, u) ,ඹ≧ែ㸦㝶కኚᩘ㸧㸸 λ㸪ࣛࢢࣛࣥࢪࣗ஌ᩘ㸸 μ୙➼ᘧᣊ᮰ C ≤ 0 ࡢሙྜ㸸 C + v 2 = 0 㸦v: ࢲ࣑࣮ධຊ㸧㠀⥺ᙧࣔࢹࣝணx(0) = x0 ,T㠀⥺ᙧࣔࢹࣝண

ไᚚ

8ࣔࢹࣝண ไᚚ࡜ࡣ㸽᭷㝈᫬㛫ᮍ᮶ࡲ࡛ࡢືࡁࢆᐇ᫬㛫࡛ண ࣭᭱㐺໬ࡋ࡚㸪⌧ᅾࡢື࠿ࡋ
Model Predictive Control (MPC)᪉ࢆỴࡵࡿ㸬㠀⥺ᙧࣔࢹࣝணไᚚ9㠀⥺ᙧࣔࢹࣝணไᚚࣔࢹࣝண ไᚚ࡜ࡣ㸽10ไᚚධຊỴᐃࡢࡓࡵࡢ᭱㐺໬᭷㝈᫬㛫ᮍ᮶ࡲ࡛ࡢືࡁࢆᐇ᫬㛫࡛ண ࣭᭱㐺໬ࡋ࡚㸪⌧ᅾࡢື࠿ࡋ
Model Predictive Control (MPC)᪉ࢆỴࡵࡿ㸬≧ែ᪉⛬ᘧ㸸 ẋ(t) = f (x(t), u(t))
ホ౯㛵ᩘ㸸J = ϕ(x(t + T )) +ᣊ᮰᮲௳㸸C(x(t), u(t)) = 0

 t+TtL(x(τ ), u(τ ))dτ୙➼ᘧᣊ᮰ C ≤ 0 ࡢሙྜ㸸 C + v 2 = 0 㸦v: ࢲ࣑࣮ධຊ㸧ྛ᫬้࡟࠾࠸࡚㸪x(t) ࢆࠕึᮇ≧ែࠖ࡜ࡍࡿ᭱㐺ไᚚuopt (τ ; t, x(t)) (t ≤ τ ≤ t + T ) ࢆồࡵ㸪ࡑࡢࠕึᮇ್ࠖࡢࡳࢆᐇ㝿ࡢไᚚධຊ࡜ࡍࡿ㸬u(t) = uopt (t; t, x(t))

㠀⥺ᙧࣔࢹࣝணไᚚ

11ࣔࢹࣝண ไᚚࡢ≉ᚩ㸦㸯㸧u㠀⥺ᙧࣔࢹࣝண

· · · ≧ែࣇ࢕࣮ࢻࣂࢵࢡไᚚ

ไᚚ12ࣔࢹࣝண ไᚚࡢ≉ᚩ㸦㸰㸧ホ౯༊㛫ࡀ᭷㝈• ホ౯༊㛫ࢆ㞳ᩓ໬ࡋࡓᩘ್ィ⟬࡟㐺ࡍࡿ㸬• 㛢࣮ࣝࣉ⣔ࡢᏳᐃᛶࡣᚲࡎࡋࡶಖドࡉࢀ࡞࠸ࡀ㸪ホ౯㛵ᩘࡸᣊ᮰᮲௳࡟ࡼࡗ࡚ከᵝ࡞ၥ㢟ࢆᢅ࠼ࡿ㸬tt+Tホ౯༊㛫ࡀ⛣ື• ᫬୙ኚ࡞≧ែࣇ࢕࣮ࢻࣂࢵࢡไᚚࡀᚓࡽࢀࡿ㸬• ᭷㝈࡛ᅛᐃࡉࢀࡓホ౯༊㛫ࡔ࡜㸪᭷㝈᫬㛫࡛ไᚚࡀ⤊஢㸬ᐇ᫬㛫᭱㐺໬࢔ࣝࢦࣜࢬ࣒ࡀᚲせt⛣ືࡍࡿホ౯༊㛫 Ѝ Receding Horizon ไᚚ࡜ࡶ࿧ࡤࢀࡿ㸬recede · · · ᚋ㏥ࡍࡿ㸪㐲ࡊ࠿ࡿ• ࢜ࣥࣛ࢖࡛ࣥࡢ᭱㐺໬㸸໙㓄ἲࡢࡼ࠺࡞཯᚟ἲࡣ㠀ຠ⋡㸬• ࢜ࣇࣛ࢖࡛ࣥࡢ᭱㐺໬㸸ࢩࢫࢸ࣒ࡀ㧗ḟඖࡔ࡜グ᠈㔞ࡀ⭾኱㸬ᐇ᫬㛫᭱㐺໬13ᐇ᫬㛫᭱㐺໬

࢔࢘ࢺࣛ࢖ࣥ

14ᩘ್ゎἲࡢࡓࡵࡢ㞳ᩓ㏆ఝ
᫬้ t ࠿ࡽ T ࡔࡅᮍ᮶ࡲ࡛ࡢไᚚ㸪≧ែ᪉⛬ᘧ㸪ホ౯㛵ᩘࢆ㞳ᩓ㏆ఝ㸬® ᭱㐺ไᚚࡢᇶ♏

∗T∗T∗T∗T∗TTᮍ▱㔞 U (t) = [u∗T0 (t), μ0 (t), u1 (t), μ1 (t), . . . , uN −1 (t), μN −1 (t)] 㸬ᐇ㝿ࡢධຊ u(t) = u∗0 (t)㸪ᣊ᮰᮲௳ C = 0 ࡢࣛࢢࣛࣥࢪࣗ஌ᩘ μ∗i 㸬᭱㐺ไᚚၥ㢟㸪೵␃᮲௳® 㠀⥺ᙧࣔࢹࣝண ไᚚi 㞳ᩓ໬ࡢ้ࡳ Δτ = ไᚚ࿘ᮇ Δt㸬 Ѝ ᰂ㌾࡞タᐃࣔࢹࣝண ไᚚ࡜ࡣ㸽̭≉ᚩᐇ᫬㛫᭱㐺໬ၥ㢟ࡢᛶ㉁ࢆ฼⏝ Ѝ ᫬㛫ኚ໬ࡍࡿゎࡢ㏣㊧Ѝ ຠ⋡ⓗ࡞࢔ࣝࢦࣜࢬ࣒® ᛂ⏝஦౛㸰ࣜࣥࢡ࢔࣮࣒㸪࣍ࣂ࣮ࢡࣛࣇࢺ㸪⯪⯧࡯࠿® ⮬ືࢥ࣮ࢻ⏕ᡂᩘᘧฎ⌮࡟ࡼࡿ C ࣉࣟࢢ࣒ࣛࡢ⮬ື⏕ᡂᐇ᫬㛫᭱㐺໬

15

㞳ᩓ㏆ఝࡉࢀࡓၥ㢟ホ౯㛵ᩘ㸸 J(U (t), (t), t) = ϕ(x∗N (t)) +೵␃᮲௳㸸x∗0 (t) = x(t)x∗i+1 (t) = x∗i (t) + f (x∗i (t), u∗i (t))ΔτN −1i=0L(x∗i (t), u∗i (t))Δτ∂ϕ T(x∗N (t))∂x T(x∗i (t), u∗i (t), λ∗i+1 (t), μ∗i (t))Δτλ∗i (t) = λ∗i+1 (t) + ∂H
∂xλ∗N (t) =

∂H(x∗i (t), u∗i (t), λ∗i+1 (t), μ∗i (t)) = 0∂u
C(x∗i (t), u∗i (t)) = 0

ࣁ࣑ࣝࢺࢽ࢔ࣥ㸸 H(x, λ, u, μ) = L + λT f + μT C≧ែ x∗i (t), ඹ≧ែ λ∗i (t) ࡣ U (t) ࡜ x∗0 (t) = x(t) ࡢ㛵ᩘ㸬ᐇ᫬㛫᭱㐺໬16ධຊ⣔ิࡢ㠀⥺ᙧ௦ᩘ᪉⛬ᘧ㠀⥺ᙧ௦ᩘ᪉⛬ᘧ㸸⎡∂H(x∗0 (t), u∗0 (t), λ∗1 (t), μ∗0 (t))∂u⎤⎢⎥⎥⎢∗∗⎥⎢
C(x0 (t), u0 (t))
⎥⎢⎥
⎢..
⎥
⎢
⎥=0F (U (t), x(t), t) = ⎢.
⎥⎢⎥⎢⎢ ∂H (x∗ (t), u∗ (t), λ∗ (t), μ∗ (t)) ⎥⎥⎢ ∂u N −1
N −1NN −1⎦⎣

C(x∗N −1 (t), u∗N −1 (t))
∗T
∗T
∗T
∗T
∗T
T
U (t) = [u∗T0 (t), μ0 (t), u1 (t), μ1 (t), . . . , uN −1 (t), μN −1 (t)]ࢽ࣮ࣗࢺࣥἲ㸸F (U, x, t) ࡀ 0 ࡟࡞ࡿࡼ࠺ U ࡢಟṇࢆ཯᚟㸬ィ⟬㔞ࡀከࡃ㸪ィ⟬᫬㛫ࡀ཯᚟ᅇᩘ࡟౫Ꮡࡍࡿ㸬཯᚟ἲࡣᐇ᫬㛫࡛ࡢࣇ࢕࣮ࢻࣂࢵࢡไᚚ࡟㐺ࡉ࡞࠸㸬ᐇ᫬㛫᭱㐺໬

17ᐇ᫬㛫᭱㐺໬18ᐇ᫬㛫᭱㐺໬࡬ࡢ࢔ࣉ࣮ࣟࢳࣔࢹࣝண㐃⥆ኚᙧἲ࡜ࡣ㸽᫬้ t ࡟౫Ꮡࡍࡿᮍ▱㔞 U (t) ࡢ᪉⛬ᘧ㸸 F (U (t), x(t), t) = 0ไᚚࡢ≉ᚩ㸸

dF (U (t), x(t), t) = 0, F (U (0), x(0), 0) = 0dt
d⇔
F (U (t), x(t), t) = −ζF (U (t), x(t), t), F (U (0), x(0), 0) = 0dtၥ㢟ࡀ᫬้ࡸ≧ែ࡜࡜ࡶ࡟⁥ࡽ࠿࡟ኚ໬ࡋ࡚࠸ࡃ㸬⇔᫬㛫౫Ꮡࡢ㠀⥺ᙧ᪉⛬ᘧ F (U (t), x(t), t) = 0ࡑࡇ࡛㸸ζ > 0 ࡟ࡼࡗ࡚ㄗᕪ㸦F = 0㸧ࡀῶ⾶㸦F = 0 ࡢᏳᐃ໬㸧ၥ㢟ࡢኚ໬࡟ᛂࡌࡓ᭱㐺ゎࡢኚ໬ࢆṇ☜࡟㏣㊧࡛ࡁࢀࡤ㸪཯᚟ゎἲࡣ୙せ࡛ࡣ㸽⇔Ў㐃⥆ኚᙧἲ㸦࣍ࣔࢺࣆ࣮ἲ㸧−1 

∂F−ζF − ∂Fẋ − ∂F,U̇ = ∂U∂x∂tF (U (0), x(0), 0) = 0

UU(t)཯᚟ィ⟬࡞ࡋ࡟ U̇ (t) ࢆᩘ್✚ศࡋ࡚
U (t) ࡀồࡵࡽࢀࡿ㸬F(U,x(t),t)=0t㐃⥆ኚᙧἲࢆᐇ᫬㛫࡛౑࠺ࡢࡀ࣏࢖ࣥࢺ㸬ᐇ᫬㛫᭱㐺໬19ᐇ᫬㛫᭱㐺໬

20

Continuation/GMRES ࢔ࣝࢦࣜࢬ࣒ྛ᫬้࡛≧ែ x(t) ࢆᐇ᫬㛫᭱㐺໬ࡢせⅬ㸦㸯㸧ゎࡃ࡭ࡁၥ㢟㸸ᐃ㸬U̇ ࡟㛵ࡍࡿ㐃❧ 1 ḟ᪉⛬ᘧ㸸

∂Fẋ − ∂FU̇ = −ζF − ∂F∂U∂x∂tࢆゎࡁ㸪

U̇ (t) ࢆᩘ್✚ศࡋ࡚ U (t) ࢆ᭦᪂㸸 U (t + Δt) = U (t) + U̇ (t)Δtྛ᫬้࡛ࡣ㸪㐃❧ 1 ḟ᪉⛬ᘧࢆ 1 ᅇゎࡃࡢࡳ㸪཯᚟࡞ࡋ᭱㐺໬ၥ㢟 Ѝ ධຊࡢ⣔ิ࡟㛵ࡍࡿ㠀⥺ᙧ᪉⛬ᘧ㸦㸰㸧㐃⥆ኚᙧἲ㸸ゎࡢ㏣㊧ Ѝ ゎࡢ᫬㛫ᚤศ࡟㛵ࡍࡿ㐃❧ 1 ḟ᪉⛬ᘧЎ㐃❧ 1 ḟ᪉⛬ᘧࡢຠ⋡ⓗ࡞ゎἲ࡛࠶ࡿ GMRES ࢆ㐺⏝i ホ౯༊㛫ࡢ㛗ࡉ T ࡣ 0 ࠿ࡽ୍ᐃ್࡬⁥ࡽ࠿࡟ቑࡸࡍ㸦U (0) ࡀ⡆༢࡟ồࡵࡽࢀࡿ㸧㸬(኱ሯ ’02; O. ’04)

㸦㸱㸧GMRES㸸኱つᶍ㐃❧ 1 ḟ᪉⛬ᘧࢆຠ⋡Ⰻࡃゎࡃᛂ⏝஦౛

21ᛂ⏝஦౛

22ᛂ⏝㸦㸯㸧
㸸2 ࣜࣥࢡ࣭࢔࣮࣒ࡢ᣺ࡾୖࡆ࢔࢘ࢺࣛ࢖ࣥ® ᭱㐺ไᚚࡢᇶ♏

u1᭱㐺ไᚚၥ㢟㸪೵␃᮲௳≧ែ࣋ࢡࢺࣝ㸸s1

J1® 㠀⥺ᙧࣔࢹࣝண ไᚚmq1

l1x = [q1 q2 q̇1 q̇2 ]T1ࣔࢹࣝண ไᚚ࡜ࡣ㸽̭≉ᚩu2® ᐇ᫬㛫᭱㐺໬

s2

l2
mJ2q

ไᚚࢺࣝࢡࡢ࣋ࢡࢺࣝ㸸2u = [u1 u2 ]T2ၥ㢟ࡢᛶ㉁ࢆ฼⏝ Ѝ ᫬㛫ኚ໬ࡍࡿゎࡢ㏣㊧Ѝ ຠ⋡ⓗ࡞࢔ࣝࢦࣜࢬ࣒ホ౯㛵ᩘ㸸ᛂ⏝஦౛J =㸰ࣜࣥࢡ࢔࣮࣒㸪࣍ࣂ࣮ࢡࣛࣇࢺ㸪⯪⯧࡯࠿® ⮬ືࢥ࣮ࢻ⏕ᡂ1(x(t + T ) − xf )T Sf (x(t + T ) − xf )
2

1 t+T
((x − xf )T Q(x − xf ) + uT Ru)dτ+2 tᩘᘧฎ⌮࡟ࡼࡿ C ࣉࣟࢢ࣒ࣛࡢ⮬ື⏕ᡂ(኱ሯ ’02; O. ’04)ᛂ⏝஦౛23ᛂ⏝஦౛

24ࢩ࣑࣮ࣗࣞࢩࣙࣥ⤖ᯝࡢ౛







X>1P@

[>UDGV@

X

[>UDGV@





Truncated Newton__)__μ
7LPH>V@



7LPH>V@ᣊ᮰ࡉࢀࡓ⫪㛵⠇ࢺࣝࢡ u1 ࡢࡳ࡟ࡼࡿ 2 ࣜࣥࢡ࣭࢔࣮࣒ࡢ᣺ࡾୖࡆ㸸|u1 | ≤ u1max[Animation 1: |u1 | ≤ 5 ] [Animation 2: |u1 | ≤ 2.5 ]Average Computational Time per Update [ms]

100


[>UDG@[>UDG@ィ⟬᫬㛫ࡢẚ㍑
9080
7060
L−BFGS

50403020C/GMRES
1000204060
Number of Grids on the Horizon, N80100ホ౯༊㛫ࡢศ๭ᩘ N ࡟ᑐࡍࡿไᚚධຊ᭦᪂ࡢᖹᆒィ⟬᫬㛫 [ms]L-BFGS, Truncated Newton · · · ỗ⏝ࡢ཯᚟ゎἲ

CPU: PentiumII 300 MHz, ホ౯༊㛫 Tf = 0.2 [s], GMRES ࡢ཯᚟ k = 2ᛂ⏝஦౛25ᛂ⏝஦౛26ᛂ⏝㸦㸰㸧㸸࣍ࣂ࣮ࢡࣛࣇࢺࡢ఩⨨ไᚚ࣍ࣂ࣮ࢡࣛࣇࢺࡢࣔࢹࣝண ไᚚ≧ែ࣋ࢡࢺࣝ㸸x = [x y θ ẋ ẏ θ̇]T ∈ R6᥎ຊࡢ࣋ࢡࢺࣝ㸸u = [u1 u2 ]T ∈ {umin , 0, umax }2㠀࣍ࣟࣀ࣑ࢵࢡ 㸩 㞳ᩓ್ධຊホ౯㛵ᩘ㸸1(x(t + T ) − xf )T Sf (x(t + T ) − xf )2
1  t+T+((x − xf )T Q(x − xf ) + uT Ru)dτ
2 t

J =࣍ࣂ࣮ࢡࣛࣇࢺᶍᆺ㸸඲㛗 356 [mm]㸪඲ᖜ 212 [mm], ඲㧗 142 [mm], ㉁㔞 0.894 [kg](Seguchi, O. ’03)

ᛂ⏝஦౛

27

ᛂ⏝஦౛

28

ࢩ࣑࣮ࣗࣞࢩࣙࣥ࡜ᐇ㦂ࡢ⤖ᯝᛂ⏝㸦㸱㸧㸸⮬ື᧯⯪ࢩࢫࢸ࣒

0.2

00

−0.2

−0.2

−0.4

−0.4010

Time [s]2030

00.20.2

10Time [s]20300−0.2010

Time [s]2030−1010Time [s]2030

10Time [s]20yi(x,y)X

Target PointShipv

WPi+1Lyyi+130ධຊ୙➼ᘧᣊ᮰㸩ᑠࡉ࠸ධຊ㔜ࡳЍ ㏆ఝⓗ࡟ࣂࣥࣂࣥไᚚࢆィ⟬ Ѝ 㞳ᩓ್ධຊࢆỴᐃ

CPU: Athlon 900 MHz, ホ౯༊㛫 Tf = 1 [s],ࢧࣥࣉࣜࣥࢢ࿘ᮇ 1/120 [s] 㸦఩⨨ィ ⏝ CCD ࣓࢝ࣛࡢไ⣙㸧
[Movie 1: θ(0) = −π ]Guide Pointxi+1N◚⥺㸸ࢩ࣑࣮ࣗࣞࢩࣙࣥ
ᐇ⥺㸸ᐇ㦂

2

0−0.2
0xi

u

Y

0−0.5

u [N]

0.4

u1 [N]

0.4

WPi

0.5

θ [rad]

0.2

y [m]

x [m]

0.4

Ψx1

0.4

[Movie 2: θ(0) = π/2 ][Movie 3: disturbance ]≧ែ࣋ࢡࢺࣝ㸸x = [x, u, y, v, ψ, r, X, Y, N ]Tไᚚධຊ㸸u = [Xr , Yr , Nr ]T 㸦࢔ࢡࢳ࢚࣮ࣗࢱᣦ௧್㸧ホ౯㛵ᩘ㸸J = 12 (x(t + T ) − xref )T Sf (x(t + T ) − xref )+ 21 tt+T ((x − xref )T Q(x − xref ) + uT Ru)dτ(Ἑ㔝ࡽ ’02, ’08)

ᛂ⏝஦౛

29

ᛂ⏝஦౛ᶍᆺ⯪ヨ㦂⤖ᯝ
-2-2

0

0

2

2y [m]

-4

y [m]

-4

4

6

8

8-20

2

4x [m]

6ࢩ࣑࣮ࣗࣞࢩࣙࣥ81012

ᾏὒᕤ஦సᴗ⯪࡛ᐇ⏝໬

4610-430

10-4-202

4x [m]

68

1012

ᐇ㦂

ᶍᆺ⯪㸸⯪㛗 3.5 [m], ⯪ᖜ 0.59 [m], ㉁㔞 200 [kg]㸹̭ไᚚ࿘ᮇ㸸0.1 [sec], ホ౯༊㛫㸸30 [sec]㟁ຊࢣ࣮ࣈࣝᕸタ⯪ (90 × 30m㸪ႚỈ 3m) (Ἑ㔝ࡽ ’05, ’08)

ᛂ⏝஦౛

31

ᛂ⏝஦౛

ᛂ⏝㸦㸲㸧㸸㟁ຊ㟂⤥ࡢᐇ᫬㛫౯᱁ไ࡟ࡼࡿไᚚ⊂❧⣔⤫㐠⏝ᶵ㛵 (Independent System Operator; ISO) ࠿ࡽࡢ౯᱁ಙྕ࡛㟁ຊࡢ㟂せ࣭౪⤥ࢆไᚚ ⇒ ㈇Ⲵ࿘Ἴᩘࢆไᚚ

[More]

32n ᆅᇦ㟁ຊ⣔⤫ࣔࢹࣝ㟂せᐙ࣭౪⤥ᐙ• ⮬㌟ࡢ฼┈ࢆ᭱኱໬ࡍࡿࡼ࠺࡟ᾘ㈝㔞 d㸪౪⤥㔞 s ࢆỴᐃ• 㟂せᐙࡢຠ⏝㛵ᩘ v(d)㸪౪⤥ᐙࡢ㈝⏝㛵ᩘ c(s)• HEMS (Home Energy Management System) ࡢ㧗㏿࡞ᛂ⟅ࢆ᝿ᐃⓎ㟁ᶵ• ௚ᆅᇦ࡜᥋⥆ࡉࢀࡓ㠀⥺ᙧືᦂ᪉⛬ᘧ㸦ᆅᇦࡢᖹᆒࣔࢹࣝ㸧⊂❧⣔⤫㐠⏝ᶵ㛵㸦ISO㸧• ྛᆅᇦࡢ㟁ຊ౯᱁ λ ࢆỴᐃ• ㈇Ⲵ࿘Ἴᩘኚືࢆᩚᐃ㸪㟂⤥ࢆ୍⮴㟂せᐙ࣭౪⤥ᐙࡢ⤒῭⾜ື࡜Ⓨ㟁ᶵ࣭㏦㟁ࡢ≀⌮⌧㇟ࡀΰᅾ(டᮏࡽ ’14)

• 㟂せᐙ࣭౪⤥ᐙࡢ⥲฼┈ࢆ᭱኱໬ᛂ⏝஦౛33ᛂ⏝஦౛34ISO ࡢホ౯㛵ᩘ10 ᆅᇦࡢࢩ࣑࣮ࣗࣞࢩࣙࣥไᚚ┠ⓗⓎ㟁ᶵࡢ┦ᕪゅ࣭┦ᕪゅ㏿ᗘࢆ 0 ࡟ࡍࡿ㸬㟂⤥ᕪࢆ 0 ࡟ࡍࡿ㸬♫఍ࡢ⥲฼┈ࢆ᭱኱໬ࡍࡿ㸬┦ᕪゅ θi(i = 1, · · · , 10)ホ౯㛵ᩘJ = x (t + T )Q1 x(t + T ) +TⓎ㟁ᶵ

 t+T x (τ )Q2 x(τ ) +T



t

Ⓨ㟁ᶵ

Φi = −r1 λi si − ci (si ) + vi (di ) − λi di౪⤥ᐙࡢ฼┈

vi (di ) = 2 di ,㟂せᐙࡢ฼┈+ r2ni=1Φi (λi ) dτs i −d i㟁ຊ౯᱁ λi

2(i = 1, · · · , 10)㟂⤥ࡢᕪไᚚධຊ᭦᪂ࡢィ⟬᫬㛫㸸 0.33 ms ⇒ ᐇ᫬㛫ィ⟬ࡀྍ⬟ci (si ) = s2i /2ᛂ⏝஦౛

35⮬ືࢥ࣮ࢻ⏕ᡂ௚ࡢᛂ⏝36࢔࢘ࢺࣛ࢖ࣥ

ࢸࢨ࣮⾨ᫍ㸪ࣟ࣎ࢵࢺ㸪࣊ࣜࢥࣉࢱ࣮ࡢ⦅㝲㣕⾜࡞࡝㣕⾜ᶵࡢ࢜ࣥࣛ࢖ࣥ㣕⾜⤒㊰⏕ᡂ㸦㡿ᇦᅇ㑊ㄏᑟ๎ࡢ㣕⾜ᐇド㸧® ᭱㐺ไᚚࡢᇶ♏
᭱㐺ไᚚၥ㢟㸪೵␃᮲௳® 㠀⥺ᙧࣔࢹࣝண ไᚚࣔࢹࣝண ไᚚ࡜ࡣ㸽̭≉ᚩࢸࢨ࣮⾨ᫍ
(Okazaki, O. ’06)ᐇ㦂⏝⯟✵ᶵMuPAL-α(Ọሯࡽ ’05, ’06, ’09)㗰ᯈࡢ෭㛫ᅽᘏ࡟࠾ࡅࡿᯈཌ࣭ᙇຊไᚚ (ᑿᓮࡽ ’10)
⢏Ꮚຍ㏿ჾ㸦LHC㸧㉸ఏᑟ☢▼ࡢᗘไᚚ (Noga et al. ’10, ’11)೫ᚤศ᪉⛬ᘧ࡛グ㏙ࡉࢀࡿࢩࢫࢸ࣒ࡢไᚚ (Hashimoto et al. ’10, ’12)㠀⥺ᙧ Moving Horizon ᥎ᐃ㸪Receding Horizon ᚤศࢤ࣮࣒࡞࡝® ᐇ᫬㛫᭱㐺໬ၥ㢟ࡢᛶ㉁ࢆ฼⏝ Ѝ ᫬㛫ኚ໬ࡍࡿゎࡢ㏣㊧Ѝ ຠ⋡ⓗ࡞࢔ࣝࢦࣜࢬ࣒® ᛂ⏝஦౛㸰ࣜࣥࢡ࢔࣮࣒㸪࣍ࣂ࣮ࢡࣛࣇࢺ㸪⯪⯧࡯࠿⮬ືࢥ࣮ࢻ⏕ᡂᩘᘧฎ⌮࡟ࡼࡿ C ࣉࣟࢢ࣒ࣛࡢ⮬ື⏕ᡂ⮬ືࢥ࣮ࢻ⏕ᡂ37⮬ືࢥ࣮ࢻ⏕ᡂᩘᘧฎ⌮࡟ࡼࡿ⮬ືࢥ࣮ࢻ⏕ᡂ38࣮࣡ࢡࢩ࣮ࢺ࡬ࡢධຊ

Maple ࡢ᭩ᘧ࡛୚࠼ࡓ≧ែ᪉⛬ᘧࡸホ౯㛵ᩘ࠿ࡽ㸪C ゝㄒࡢࢩ࣑࣮ࣗࣞࢩ࣭ࣙࣥࣉࣟࢢ࣒ࣛࢆ⮬ື⏕ᡂ㸦Mathematica ∧ࡶ࠶ࡾ㸧ၥ㢟タᐃࡸࢩ࣑࣮ࣗࣞࢩࣙࣥ᮲௳ࢆࢸ࢟ࢫࢺ࣎ࢵࢡࢫ࡬ධຊධຊ㸦≧ែ᪉⛬ᘧ࡜ホ౯㛵ᩘࡢ㒊ศ㸧⮬ືࢥ࣮ࢻ⏕ᡂ39⮬ື⏕ᡂࡉࢀࡓ C ࢯ࣮ࢫࣇ࢓࢖ࣝ⮬ືࢥ࣮ࢻ⏕ᡂ40⮬ືࢥ࣮ࢻ⏕ᡂࢩࢫࢸ࣒ࡢ≉㛗࢔ࣝࢦࣜࢬ࣒ࢆヲࡋࡃ▱ࡽ࡞ࡃ࡚ࡶ౑⏝࡛ࡁࡿ㸬Ѝ ᑓ㛛ᐙ࡛࡞ࡃ࡚ࡶ฼⏝ࡀᐜ᫆࢔ࣝࢦࣜࢬ࣒୰ࡢ↹㞧࡞೫ᚤศィ⟬➼ࢆ⮬ື໬࡛ࡁࡿ㸬Ѝ ࣉࣟࢢ࣒ࣛ㛤Ⓨປຊࡢ኱ᖜ࡞పῶ
⧞ࡾ㏉ࡋ⌧ࢀࡿィ⟬ᘧࢆ⮬ືⓗ࡟୰㛫ኚᩘ࡛⨨ࡁ᥮࠼ࡿ㸬Ѝ ຠ⋡ⓗ࡞ࢥ࣮ࢻࢆ⏕ᡂᩘᘧฎ⌮࡟ࡼࡿ≧ែ᪉⛬ᘧᑟฟࡢ⤖ᯝࢆ฼⏝࡛ࡁࡿ㸬Ѝ 」㞧࡞ࢩࢫࢸ࣒ࡢᢅ࠸ࡶᐜ᫆ฟຊࣇ࢓࢖ࣝ㸦∂H/∂u ࡟㛵ࡍࡿ㒊ศ㸧ࢥࣥࣃ࢖ࣝ࡜ᐇ⾜㸪ࢩ࣑࣮ࣗࣞࢩࣙࣥ⤖ᯝࡢࢢࣛࣇᥥ⏬ࡶࡍ࡭࡚࣮࣡ࢡࢩ࣮ࢺ࠿ࡽᐇ⾜ྍ⬟ࣁ࣮ࢻ࢙࢘࢔ᐇ⿦⏝ࣉࣟࢢ࣒ࣛ࡬ᐜ᫆࡟ᨵ㐀࡛ࡁࡿ㸬ࢯ࣮ࢫࢥ࣮ࢻࢆ↓ᩱ㓄ᕸ㸬ࡲ࡜ࡵ41ཧ⪃ᩥ⊩ࡲ࡜ࡵཧ⪃ᩥ⊩ࣔࢹࣝண ไᚚ㸸ྛ᫬้࡛᭷㝈᫬㛫ᮍ᮶ࡲ࡛ࡢ᭱㐺ไᚚၥ㢟ࢆゎࡃࣇ࢕࣮ࢻࣂࢵࢡไᚚ᫬㛫࡜࡜ࡶ࡟ኚ໬ࡍࡿゎࡢ㏣㊧㸦㐃⥆ኚᙧἲ㸧࡟ࡼࡗ࡚㸪㠀⥺ᙧࣔࢹࣝண ไᚚࡢࡓࡵࡢ㧗㏿࡞ᐇ᫬㛫᭱㐺໬ࡀྍ⬟᥎ᐃࡸᚤศࢤ࣮࣒ࢆྵࡵᐇ᫬㛫᭱㐺໬ࡢᛂ⏝⠊ᅖࡣᗈ࠸㸬ᩘᘧฎ⌮ࢆ⏝࠸ࡓ⮬ືࢥ࣮ࢻ⏕ᡂ࡟ࡼࡗ࡚㸪」㞧࡞ၥ㢟࡛ࡶᐜ᫆࡟ࢩ࣑࣮ࣗࣞࢩࣙࣥࡀ⾜࠼ࡿ㸬

http://www.symlab.sys.i.kyoto-u.ac.jp/˜ohtsuka/index j.htm

ཧ⪃ᩥ⊩

42

43

Guidance, Control, and Dynamics, Vol. 21, No. 1, pp. 174–176 (1998)[13] T. Ohtsuka, “Quasi-Newton-Type Continuation Method for Nonlinear Receding HorizonControl,” J. of Guidance, Control, and Dynamics, Vol. 25, No. 4, pp. 685–692 (2002)[14] T. Ohtsuka, “A Continuation/GMRES Method for Fast Computation of Nonlinear
Receding Horizon Control”, Automatica, Vol. 40, No. 4, pp. 563–574 (2004)[15] Y. Shimizu, T. Ohtsuka, and M. Diehl, “A Real-Time Algorithm for Nonlinear Receding
Horizon Control Using Multiple Shooting and Continuation/Krylov Method,” Int. J. ofRobust and Nonlinear Control, Vol. 19, No. 8, pp. 919–936 (2009)[16] ㉥ᒣ៞ኴ, ኱ሯᩄஅ, “࢜ࣇࣛ࢖࡛ࣥࡢ≉␗್ศゎ࡟ᇶ࡙ࡃᣊ᮰᮲௳௜ࡁ㠀⥺ᙧ Receding Horizonไᚚࡢᐇ᫬㛫࢔ࣝࢦࣜࢬ࣒,” ࢩࢫࢸ࣒ไᚚ᝟ሗᏛ఍ㄽᩥㄅ, Vol. 25, No. 5, pp. 126–133 (2012)ᛂ⏝㸸࣍ࣂ࣮ࢡࣛࣇࢺ㸪⯪⯧[17] H. Seguchi, and T. Ohtsuka, “Nonlinear Receding Horizon Control of an Underactuated
Hovercraft,” Int. J. of Robust and Nonlinear Control, Vol. 13, Nos. 3–4, pp. 381–398 (2003)[18] ὾ᯇṇ඾, ຍ㈡㇂༤᫛, Ἑ㔝⾜ఙ, “㠀⥺ᙧ Receding Horizon ไᚚࡢ⮬ື᧯⯪ࢩࢫࢸ࣒࡬ࡢ㐺⏝,”ィ ⮬ືไᚚᏛ఍ㄽᩥ㞟, Vol. 44, No. 8, pp. 685–691 (2008)ᛂ⏝㸸⮬ື㌴[19] T. Kawabe, H. Nishira, and T. Ohtsuka, “An Optimal Path Generator Using a Receding
Horizon Control Scheme for Intelligent Automobiles,” Proc. of the 2004 IEEE Int. Conf.
on Control Applications, pp. 1597–1602 (2004)[20] す⨶ග, 㧗ᮌⰋ㈗, ฟཱྀḠ㧗, “㞀ᐖ≀ᅇ㑊ࢩࢫࢸ࣒ࡢᅇ㑊⤒㊰ᐇ᫬㛫᭱㐺໬₇⟬ࡢタィ,” ➨ 11 ᅇィ ⮬ືไᚚᏛ఍ไᚚ㒊㛛኱఍ CD-ROM, ㅮ₇␒ྕ 163-3-2 (2011)[21] M. Nanao, and T. Ohtsuka, “Vehicls Dynamics Control for Collision AvoidanceConsidering Physical Limitations,” Proc. of SICE Annual Conf. 2011, pp. 688–693 (2011)[22] M. A. S. Kamal, M. Mukai, J. Murata, and T. Kawabe, “Model Predictive Control ofVehicles on Urban Roads for Improved Fuel Economy,” IEEE Trans. on Control Systems᭱㐺ไᚚ඲⯡[1] A. E. Bryson, Jr. and Y.-C. Ho, Applied Optimal Control, Hemisphere (1975)[2] J. T. Betts, Practical Methods for Optimal Control Using Nonlinear Programming, 2ndEd., SIAM (2010)[3] ኱ሯᩄஅ, 㠀⥺ᙧ᭱㐺ไᚚධ㛛, ࢥࣟࢼ♫ (2011)ࣔࢹࣝண ไᚚ඲⯡[4] J. M. Maciejowski ⴭ, ㊊❧ಟ୍, ⟶㔝ᨻ᫂ ヂ, ࣔࢹࣝண ไᚚ, ᮾி㟁ᶵ኱Ꮫฟ∧ᒁ (2005)[5] L. Magni, D. M. Raimondo, and F. Allgöwer (Eds.), Nonlinear Model Predictive Control:Towards New Challenging Applications, Springer (2009)[6] J. B. Rawlings, and D. Q. Mayne, Model Predictive Control: Theory and Design, Nob Hill
Publishing (2009)[7] ኱ሯᩄஅ ⦅ⴭ, ᐇ᫬㛫᭱㐺໬࡟ࡼࡿไᚚࡢᐇᛂ⏝, ࢥࣟࢼ♫ (2015)㐃⥆ኚᙧἲ࡜ GMRES[8] S. L. Richter and R. A. DeCarlo, “Continuation Methods: Theory and Applications,”
IEEE Trans. on Automatic Control, Vol. AC-28, No. 6, pp. 660–665 (1983)[9] C. T. Kelley, Iterative Methods for Linear and Nonlinear Equations, SIAM (1995)[10] H. A. van der Vorst, Iterative Krylov Methods for Large Linear Systems, Cambridge
University Press (2003)ᐇ᫬㛫࢔ࣝࢦࣜࢬ࣒
[11] T. Ohtsuka, and H. A. Fujii, “Real-Time Optimization Algorithm for Nonlinear
Receding-Horizon Control”, Automatica, Vol. 33, No. 6, pp. 1147–1154 (1997)[12] T. Ohtsuka, “Time-Variant Receding-Horizon Control of Nonlinear Systems”, J. ofཧ⪃ᩥ⊩

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Technology, Vol. 21, No. 3, pp. 831–841 (2013)ᛂ⏝㸸ࣟ࣎ࢵࢺ
[23] J. Lee, and M. Yamakita, “Nonlinear Model Predictive Control for Constrained
Mechanical Systems with State Jump,” Proc. of the 2006 IEEE Int. Conf. on ControlApplications, pp. 585–590 (2006)[24] T. Murao, H. Kawai, and M. Fujita, “Visual Motion Observer-based Stabilizing Receding
Horizon Control via Image Space Navigation Function,” Proc. of the 19th IEEE Int. Conf.
on Control Applications, pp. 1648–1653 (2010)ᛂ⏝㸸⯟✵Ᏹᐂ
[25] M. Okazaki, and T. Ohtsuka, “Switching Control for Guaranteeing the Safety of a Tethered
Satellite,” J. of Guidance, Control, and Dynamics, Vol. 29, No. 4, pp. 822-830 (2006)[26] Ọሯ‶, ᏷ᡞ⣖ᙪ, ቑ஭࿴ஓ, ෠⏣༤ྐ, “⪏ᨾ㞀㣕⾜ไᚚࢩࢫࢸ࣒ — ⥭ᛴ㡿ᇦᅇ㑊ไᚚ,” ᪥ᮏ⯟✵ᏱᐂᏛ఍ㄅ, Vol. 57, No. 669, pp. 285-287 (2009)[27] M. Saﬀarian, and F. Fahimi, “Non-Iterative Nonlinear Model Predictive Approach Applied
to the Control of Helicopters’ Group Formation,” Robotics and Autonomous Systems, Vol.
57, pp. 749–757 (2009)ᛂ⏝㸸㕲㗰㸪ࣉࣟࢭࢫ[28] ᑿᓮ᪸ᖹ, ኱ሯᩄஅ, ⸨ᮏ೺἞, ໭ᮧ❶, ୰ᒣ୓ᕼᚿ “ᯈཌྍኚ࡞෭㛫ᅽᘏᶵ࡟࠾ࡅࡿᯈཌ࡜ᙇຊࡢ
㠀⥺ᙧ Receding Horizon ไᚚ,” 㕲࡜㗰, Vol. 96, No. 7, pp. 459–467 (2010)[29] M. Takekawa, J. Aoki, M. Nakaya, T. Ohtani, and T. Ohtsuka, “An Application ofNonlinear Model Predictive Control Using C/GMRES Method to a pH Neutralization
Process,” Proc. of SICE Annual Conf. 2010, pp. 1494–1496 (2010)[30] R. Noga, T. Ohtsuka, C. de Prada, E. Blanco, and J. Casas, “Simulation Study on
Application of Nonlinear Model Predictive Control to the Superﬂuid Helium CryogenicCircuit,” Reprints of the 18th IFAC World Congress, pp. 3647–3652 (2011)

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[31] T. Hashimoto, Y. Yoshioka, and T. Ohtsuka, “Receding Horizon Control for Hot Strip Mill
Cooling Systems,” IEEE/ASME Trans. on Mechatronics, Vol. 18, No. 3, pp. 998–1005(2013)[32] ኱ゅὯᖹ, ኱ሯᩄஅ, ᖹ⏣ග⏨, ሷ㇂ᨻ඾, “⢏Ꮚࣔࢹࣝண ไᚚ࡟ࡼࡿ㗰ᯈ ᗘࡢࡤࡽࡘࡁไᚚ,”㕲࡜㗰, Vol. 99, No. 4, pp. 275–282 (2013)௚ࡢၥ㢟タᐃ
[33] ኱ሯᩄஅ, ▼㇂㞞ᏹ, “㠀⥺ᙧᅄ㍯㌴୧ࣔࢹࣝࡢ Receding-Horizon ᚤศࢤ࣮࣒,” ᪥ᮏᶵᲔᏛ఍ㄽᩥ
㞟㸦C ⦅㸧, Vol. 66, No. 652, pp. 3962-3969 (2000)[34] Y. Azuma, and T. Ohtsuka, “Receding Horizon Nash Game Approach for Distributed
Nonlinear Control,” Proc. of SICE Annual Conf. 2011, pp. 380–384 (2011)[35] T. Ohtsuka, and H. A. Fujii, “Nonlinear Receding-Horizon State Estimation by Real-Time
Optimization Technique,” J. of Guidance, Control, and Dynamics, Vol. 19, No. 4,pp. 863–870 (1996)[36] T. Ohtsuka, “Nonlinear Receding-Horizon State Estimation with Unknown Disturbances,”ィ ⮬ືไᚚᏛ఍ㄽᩥ㞟, Vol. 35, No. 10, pp. 1253–1260 (1999)[37] T. Ohtsuka, and K. Ohata, “Hardware Experiment of Nonlinear Receding Horizon
Adaptive Control,” Proc. of the 38th IEEE Conf. on Decision and Control, pp. 1232-1233(1999)[38] Y. Soneda, and T. Ohtsuka, “Nonlinear Moving Horizon State Estimation withContinuation/Generalized Minimum Residual Method,” J. of Guidance, Control, andDynamics, Vol. 28, No. 5, pp. 878–884 (2005)[39] K. Ohsumi, and T. Ohtsuka, “Particle Model Predictive Control for Probability DensityFunctions,” Reprints of the 18th IFAC World Congress, pp. 7993–7998 (2011)[40] Ḉ஭ඃኴ, ኱ሯᩄஅ, “እ஘᥎ᐃ࡟ࡼࡿ㐃⥆᫬㛫ࣔࢹࣝண ไᚚࡢ࢜ࣇࢭࢵࢺ⿵ൾ,” ࢩࢫࢸ࣒ไᚚ᝟ሗᏛ఍ㄽᩥㄅ, Vol. 25, No. 7, pp. 172–180 (2012)

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[41] N. Fujii, and T. Ohtsuka, “Nonlinear Adaptive Model Predictive Control via Immersion
and Invariance Stabilizability,” ࢩࢫࢸ࣒ไᚚ᝟ሗᏛ఍ㄽᩥㄅ, Vol. 25, No. 10, pp. 281–288(2012)[42] T. Hashimoto, Y. Yoshioka, and T. Ohtsuka, “Receding Horizon Control with NumericalSolution for Nonlinear Parabolic Partial Diﬀerential Equations,” IEEE Trans. on
Automatic Control, Vol. 58, No. 3, pp. 725–730 (2013)[43] J. Marutani, and T. Ohtsuka, “A Real-Time Algorithm for Nonlinear Inﬁnite Horizon
Optimal Control by Time Axis Transformation Method,” Int. J. of Robust and Nonlinear
Control, Vol. 23, No. 17, 2013, pp. 1955–1971 (2013)[44] F. Tahir, and T. Ohtsuka, “Tuning of Performance Index in Nonlinear Model Predictive
Control by the Inverse Linear Quadratic Regulator Design Method,” SICE J. of Control,
Measurement, and System Integration, Vol. 6, No. 6, pp. 387–395 (2013)⮬ືࢥ࣮ࢻ⏕ᡂ[45] T. Ohtsuka, and A. Kodama, “Automatic Code Generation System for Nonlinear Receding
Horizon Control,” ィ ⮬ືไᚚᏛ఍ㄽᩥ㞟, Vol. 38, No. 7, pp. 617–623 (2002)[46] T. Ohtsuka, AutoGenU, http://www.symlab.sys.i.kyoto-u.ac.jp/˜ohtsuka/code/index j.htm, (2000)[47] ᯇỌዉ⨾, ▼ሯ┿୍, ኱ሯᩄஅ, “ᩘᘧฎ⌮ࢶ࣮ࣝ࡟ࡼࡿ㠀⥺ᙧࣔࢹࣝண ไᚚࡢタィ࣭ࢩ࣑࣮ࣗࣞ
ࢩࣙࣥ⎔ቃࡢᵓ⠏,” ➨ 56 ᅇ⮬ືไᚚ㐃ྜㅮ₇఍, No. 908 (2013)[48] http://www.maplesoft.com/applications/view.aspx?SID=153555