st Prss st Prss rt - - PowerPoint PPT Presentation
st Prss st Prss rt rt r trst st rs t tr
❘❡♠❛r❦ ■❢ s♦♠❡ ❞❡❧❛②❡❞ ♠♦♥✐t♦r❡❞ ✈❛r✐❛❜❧❡s ❛r❡ ❛♠♦♥❣ t❤❡ ❡①♣❧❛♥❛t♦r② ✈❛r✐❛❜❧❡s✱ t❤❡ s②st❡♠ ✐s ❞②♥❛♠✐❝✳ ❖t❤❡r✇✐s❡ ✐t ✐s st❛t✐❝✳
SYSTEM imput (control) disturbance noise
state ut vt et yt xt
❖✉t♣✉t✿ ▼♦♥✐t♦r❡❞ ✈❛r✐❛❜❧❡✳ ■♥♣✉t✿ ▼❛♥✐♣✉❧❛t❡❞ ✈❛r✐❛❜❧❡ ✲ ❝♦♥tr♦❧✳ ❉✐st✉r❜❛♥❝❡✿ ❈❛♥ ❜❡ ♠❡❛s✉r❡❞✱ ❝❛♥♥♦t ❜❡ ♠❛♥✐♣✉❧❛t❡❞✳ ❙t❛t❡✿ ❈❛♥♥♦t ❜❡ ♠❡❛s✉r❡❞✱ ✐s ❡st✐♠❛t❡❞ ❢r♦♠ ❞❛t❛✳ ◆♦✐s❡✿ ◆❡✐t❤❡r ❝❛♥ ❜❡ ♠❡❛s✉r❡❞ ♥♦r ♣r❡❞✐❝t❡❞✳
♣❛❣❡ ✽
❈♦♥❞✐t✐♦♥❛❧ ♣r♦❜❛❜✐❧✐t② ❞❡♥s✐t② ❢✉♥❝t✐♦♥ ✭♣❞❢✮ f
′
t, Θ
Θ = {θ, r}❀ θ = [b0, a1, b1, · · · , an, bn, k]′✱ θ ✲ r❡❣r❡ss✐♦♥ ❝♦❡✣❝✐❡♥ts✱ r ✲ ♥♦✐s❡ ✈❛r✐❛♥❝❡✳ ■t ✐s ❛ st♦❝❤❛st✐❝ ❞❡♣❡♥❞❡♥❝❡ ♦❢ yt ♦♥ ψt ✇✐t❤ r❡❧❛t✐♦♥s ❡①♣r❡ss❡❞ ❜② ♣r♦❜❛❜✐❧✐t② ❞❡♥s✐t② ❢✉♥❝t✐♦♥ ✭♣❞❢✮✳
♣❛❣❡ ✶✷
❚❤❡ ✈❛r✐❛❜❧❡s ❛r❡ ❝♦♥t✐♥✉♦✉s✱ ψ ❝❛♥ ❤❛✈❡ ❛❧s♦ s♦♠❡ ❞✐s❝r❡t❡ ♦♥❡s✳ ❚❤❡ ❛❜♦✈❡ ♣❞❢ ❡①♣r❡ss✐♦♥ ❝❛♥ ❜❡ ❣❡♥❡r❛t❡❞ ❜② t❤❡ st♦❝❤❛st✐❝ ❡q✉❛t✐♦♥ yt = b0ut + a1yt−1 + b1ut−1 + · · · + anyt−n + bnut−n + k + et = = ψ
′
tθ + et
✇❤❡r❡ et ✭♥♦✐s❡✮ ✐s ✐✳✐✳❞✳ ✭✐♥❞❡♣❡♥❞❡♥t✱ ✐❞❡♥t✐❝❛❧❧② ❞✐str✐❜✉t❡❞✮ r❛♥❞♦♠ ✈❛r✐❛❜❧❡ ✇✐t❤ ③❡r♦ ❡①♣❡❝t❛t✐♦♥ ❛♥❞ ✈❛r✐❛♥❝❡ r. E [yt|ψt, Θ] = b0ut + a1yt−1 + b1ut−1 + · · · + anyt−n + bnut−n + k✱ D [yt] = D [et] = r Pr♦❣r❛♠✿ ❚✶✶s✐♠❈♦♥t✳s❝❡ ✭♣❛❣❡ ✽✸✮
♣❛❣❡ ✶✶✱ ✶✸
❆❧❧ ✈❛r✐❛❜❧❡s ❛r❡ ❞✐s❝r❡t❡ ✭✜♥✐t❡ ♥✉♠❜❡r ♦❢ ✈❛❧✉❡s✮ f (yt|ψt, Θ) = Θyt|ψt [ut, yt−1] yt = 1 yt = 2 ✶✱ ✶ Θ1|11 Θ2|11 ✶✱ ✷ Θ1|12 Θ2|12 ✷✱ ✶ Θ1|21 Θ2|21 ✷✱ ✷ Θ1|22 Θ2|22 2
i=1 Θi|jk = 1 ✲ ❝♦♥❞✐t✐♦♥❛❧ ♣r♦❜❛❜✐❧✐t✐❡s✳
❋♦r ❣✐✈❡♥ [ut, yt−1] t❤❡ ♦✉t♣✉t yt ✐s ❣❡♥❡r❛t❡❞ ✇✐t❤ t❤❡ ♣❞❢
Pr♦❣r❛♠✿ ❚✶✸s✐♠❉✐s❝✳s❝❡ ✭♣❛❣❡ ✽✻✮
♣❛❣❡ ✶✽
❚❤❡ ♦✉t♣✉t ✐s ❞✐s❝r❡t❡ ✭✵ ♦r ✶✮ ❛♥❞ ✐t ❞❡♣❡♥❞s ♦♥ ❝♦♥t✐♥✉♦✉s ✈❛r✐❛❜❧❡s✳ f (yt|ψt, Θ) = exp (ytzt) 1 + exp (zt) ✇❤❡r❡ zt = ψtΘ + et ❚❤❡ ♠♦❞❡❧ ❤❛s t❤❡ ❢♦❧❧♦✇✐♥❣ ❢♦r♠ ✲ tr❛♥s❢♦r♠❛t✐♦♥ ❢r♦♠ z t♦ p = f (yt = 1|zt)
1 P(yt = 1|zt) zt
♣❛❣❡ ✷✸
❉❡s❝r✐❜❡s t❤❡ st❛t❡ ✈❛r✐❛❜❧❡ xt ✕ st❛t❡ ♠♦❞❡❧ ✭st❛t❡ ♣r❡❞✐❝t✐♦♥✮ xt = Mxt−1 + Nut−1 + wt ✕ ♦✉t♣✉t ♠♦❞❡❧ ✭st❛t❡ ✜❧tr❛t✐♦♥✮ yt = Axt + But + vt M, N, A, B ❛r❡ ❦♥♦✇♥ ♠❛tr✐❝❡s✱ wt, vt ❛r❡ ♥♦✐s❡s ✇✐t❤ ③❡r♦ ❡①♣❡❝t❛t✐♦♥s ❛♥❞ ❦♥♦✇♥ ❝♦✈❛r✐❛♥❝❡s Rw, Rv
♣❛❣❡ ✺✵
❋♦r ✷nd ♦r❞❡r r❡❣r❡ss✐♦♥ ♠♦❞❡❧ yt = b0ut + a1yt−1 + b1ut−1 + a2yt−2 + b2ut−2 + k + et t❤❡ st❛t❡ ❢♦r♠ ✐s yt ut yt−1 ut−1 1 = a1 b1 a2 b2 k 1 1 1 yt−1 ut−1 yt−2 ut−2 1 + b0 1 ut + et Pr♦❣r❛♠✿ ❚✶✺s✐♠❙t❛t❡✳s❝❡ ✭♣❛❣❡ ✽✽✮
♣❛❣❡ ✶✹
◆♦t❛t✐♦♥✿ dt ❞❛t❛ ❛t t, d (t) = {d0, d1, · · · , dt} ❞❛t❛ ✉♣ t♦ t✱ d0 ♣r✐♦r✳ f (Θ|d (t − 1)) , f (Θ|d (t)) ❞❡s❝r✐♣t✐♦♥ ♦❢ ♣❛r❛♠❡t❡rs ✭♣r✐♦r✱ ♣♦st❡r✐♦r✮ ❇❛②❡s r✉❧❡ f (Θ|d (t))
∝ f (yt|ψt, Θ)
f (Θ|d (t − 1))
✕ ◆❛t✉r❛❧ ❝♦♥❞✐t✐♦♥s ♦❢ ❝♦♥tr♦❧ f (Θ|ut, d (t − 1)) = f (Θ|d (t − 1)) ✕ ❇❛t❝❤ ❡st✐♠❛t✐♦♥ f (Θ|d (t)) ∝ t
f (yτ|ψτΘ)
f (Θ|d (0)) ✕ ❙❡❧❢ r❡♣r♦❞✉❝✐♥❣ ♣r✐♦r f (Θ|d (t − 1)) → f (Θ|d (t)) ✲ t❤❡ s❛♠❡ ❢♦r♠✳
♣❛❣❡ ✷✻
❘❡s✉❧ts ♦❢ ❡st✐♠❛t✐♦♥
f (Θ|d (t)) ♣r♦❜❛❜✐❧✐t✐❡s ♦❢ ♣❛r❛♠❡t❡r ✈❛❧✉❡s
ˆ Θt = E [Θ|d (t)] = ∞
−∞
Θf (Θ|d (t)) dΘ
♣❛❣❡ ✷✾
❆♣♣❧✐❝❛t✐♦♥ ♦❢ ❇❛②❡s r✉❧❡ ✇✐t❤ r❡❣r❡ss✐♦♥ ♠♦❞❡❧ ❛♥❞ ♣r✐♦r✴♣♦st❡r✐♦r ✐♥ t❤❡ ❢♦r♠ ♦❢ ●❛✉ss✲✐♥✈❡rs❡✲ ❲✐s❤❛rt ❞✐str✐❜✉t✐♦♥ f (Θ|d (0)) ∝ r−0.5κ0 exp
θ
Vt = Vt−1 + Dt κt = κt−1 + 1 ✇❤❡r❡ Dt =
ψt
′
t
Pr♦❣r❛♠s✿ ❚✷✷❡st❈♦♥t❴❇✳s❝❡❀ ❚✷✷❡st❈♦♥t❴❇✷✳s❝❡❀ ❚✷✷❡st❈♦♥t❴❇✸✳s❝❡❀ ✭♣❛❣❡ ✾✸ ❛♥❞ ❢✉r✲ t❤❡r✮ ❚✷✷❡st❈♦♥t❴❇✹✳s❝❡ ✭❞❛t❛ ❢r♦♠ ❙tr❛❤♦✈ ❛r❡ ♦♥ ♦✉r ✇❡❜✮
♣❛❣❡ ✸✵✱ ✸✶
P♦✐♥t ❡st✐♠❛t❡s ♦❢ ♣❛r❛♠❡t❡rs ✕ ❞✐✈✐s✐♦♥ ♦❢ ✐♥❢♦r♠❛t✐♦♥ ♠❛tr✐① Vt =
V
′
yψ
Vyψ Vψ
|
ˆ θt = V −1
ψ Vyψ
✕ ❡st✐♠❛t❡ ♦❢ ♥♦✐s❡ ✈❛r✐❛♥❝❡ ˆ rt = Vy − V
′
yψV −1 ψ Vyψ
κt
♣❛❣❡ ✸✷
❇❛t❝❤ ❡st✐♠❛t✐♦♥ ❋♦r ✷nd ♦r❞❡r r❡❣r❡ss✐♦♥ ♠♦❞❡❧ yt = b0ut + a1yt−1 + b1ut−1 + a2yt−2 + b2ut−2 + k + et ❈♦♥str✉❝t Y = y1 y2 y3 · · · yN , X = u1 y0 u0 y−1 u−1 1 u2 y1 u1 y0 u0 1 u3 y2 u2 y1 u1 1 · · · · · · · · · · · · · · · · · · uN yN−1 uN−1 yN−2 uN−2 1 ❘❡❣r❡ss✐♦♥ ❝♦❡✣❝✐❡♥ts ❛r❡ ˆ θN = (X′X)−1 X′Y ✐♥ t❤❡ ♦r❞❡r ✐♥ ✇❤✐❝❤ t❤❡ r♦✇s ♦❢ X ❛r❡ ❝♦♥str✉❝t❡❞✳ Pr♦❣r❛♠✿ ❚✷✶❡st❈♦♥t❴▲❙✳s❝❡ ✭♣❛❣❡ ✾✵✮
♣❛❣❡ ✸✸
❚❤❡ ♣❞❢ ♦❢ ♣❛r❛♠❡t❡r ❤❛s t❤❡ ❉✐r✐❝❤❧❡t ❢♦r♠ f (Θ|d (t)) ∝
Θ
νy|ψ;0 y|ψ
✇✐t❤ t❤❡ st❛t✐st✐❝s ✉♣❞❛t❡ νyt|ψt;t = νyt|ψt;t−1 + 1 ❚❤❡ ✉♣❞❛t❡ r✉♥s ❛s ❢♦❧❧♦✇s✿ ν ✐s ❛ ♠❛tr✐① ✇✐t❤ ❝♦❧✉♠♥s ❞❡♥♦t❡❞ ❜② ✈❛❧✉❡s ♦❢ yt ❛♥❞ r♦✇s ❝♦rr❡s♣♦♥❞✐♥❣ t♦ ❝♦♥✜❣✉r❛t✐♦♥s ♦❢ ✈❛❧✉❡s ♦❢ ψt ✭t❤❡ s❛♠❡ ❛s ✐♥ ♠♦❞❡❧✮✳ ■♥ t❤❡ ✉♣❞❛t❡ ✇❡ ✜♥❞ t❤❡ ❡♥tr② ❞❡♥♦t❡❞ ❜② yt ❛♥❞ t❤❡ r♦✇ ❝♦rr❡s♣♦♥❞✐♥❣ t♦ t❤❡ ❝♦♥✜❣✉r❛t✐♦♥ ♦❢ ψt ❛♥❞ ✇❡ ✐♥❝r❡❛s❡ ✐t ❜② ♦♥❡✳ P♦✐♥t ❡st✐♠❛t❡ ♦❢ t❤❡ ♣❛r❛♠❡t❡r ✐s ❣✐✈❡♥ ❜② ν ♥♦r♠❛❧✐③❡❞ s♦ t❤❛t t❤❡ s✉♠s ♦❢ r♦✇s ❛r❡ ❡q✉❛❧ t♦ ♦♥❡✳ Pr♦❣r❛♠✿ ❚✷✸❡st❉✐s❝✳s❝❡ ✭♣❛❣❡ ✶✵✶✮
♣❛❣❡ ✸✹✲✸✾
■t ✐s ♥♦t r❡❝✉rs✐✈❡ ✲ ✇❡ ♠✉st ❝♦♥str✉❝t ❧✐❦❡❧✐❤♦♦❞ ✭❢♦r ❛❧❧ ♠❡❛s✉r❡❞ ❞❛t❛✮ ❛♥❞ ♠❛①✐♠✐③❡ ✐t ♥✉♠❡r✐❝❛❧❧②✳ ▲✐❦❡❧✐❤♦♦❞ Lt =
t
exp {yτzτ} 1 + exp {zτ} ln Lt =
t
[yτzτ − ln (1 + exp {zt})] ˆ Θt = arg min
Θ ln Lt
♣❛❣❡ ✸✾
Pr❡❞✐❝t✐✈❡ ♣❞❢ ✭k✲st❡♣ ❛❤❡❛❞✮ f (yt+k|y (t − 1) , u (t + k)) → f (yt+k|y (t − 1)) P♦✐♥t ♣r❡❞✐❝t✐♦♥ ˆ yt = E [yt|y (t − 1)] =
t
ytf (yt|y (t − 1)) dyt
ut ❣✐✈❡♥ ❢♦r ❛❧❧ t ♥❡❡❞❡❞✳ ▼♦❞❡❧ f (yt|y (t − 1) , Θ) Pr❡❞✐❝t✐✈❡ ❞❡♥s✐t② f (yt|y (t − 1)) =
=
f (Θ|y (t − 1))
dΘ → →
f (yt|y (t − 1) , θi) f (θi|y (t − 1)) · · · ❛✈❡r❛❣❡ ✭❡①♣❡❝t❛t✐♦♥✮ ♦❢ ❛❧❧ ♣♦ss✐❜❧❡ ♠♦❞❡❧s ✇❡✐❣❤t❡❞ ❜② t❤❡✐r ♣r♦❜❛❜✐❧✐t✐❡s✳
♣❛❣❡ ✹✵✲✹✷
ut ❣✐✈❡♥ ❢♦r ❛❧❧ t ♥❡❡❞❡❞✳ ▼♦❞❡❧ f (yt|y (t − 1) , Θ) Pr❡❞✐❝t✐✈❡ ❞❡♥s✐t② f (yt+1|y (t − 1)) =
t
= f (yt+1|y (t) , Θ) f (yt|y (t − 1) , Θ) f (Θ|y (t − 1)) dytdΘ = (∗) (♠♦❞❡❧ (yt+1)) (♠♦❞❡❧ (yt)) (♣♦st❡r✐♦r (t − 1)) dytdΘ
♣♦✐♥t ❡st✐♠❛t❡s ♦❢ ♣❛r❛♠❡t❡rs · · · f (Θ|y (t − 1)) . = δ
Θt−1
f (yt+1|y (t) , Θ) f (yt|y (t − 1) , Θ) δ
Θt−1
= . =
Θt−1
Θt−1
♣♦✐♥t ❡st✐♠❛t❡s ♦❢ ♦✉t♣✉ts · · · f
Θt−1 . = δ (yt, ˆ yt) (∗∗) =
Θt−1
yt) dyt = f
yt, y (t − 1) , ˆ Θt−1
ˆ yt+1 = E [yt+1|y (t − 1)] =
· · · ❡①♣❡❝t❛t✐♦♥ ❝♦♥❞✐t✐♦♥❡❞ ❜② y (t − 1) .
♣❛❣❡ ✹✸
P♦✐♥t ♣r❡❞✐❝t✐♦♥ ✲ r❡♣❡t✐t✐✈❡ s✉❜st✐t✉t✐♦♥ ♦❢ ♠♦❞❡❧✳ ❊①❛♠♣❧❡ ❢♦r ♠♦❞❡❧ yt = ayt−1 + but + et Pr❡❞✐❝t✐♦♥ ˆ yt = ayt−1 + but ˆ yt+1 = aˆ yt + but+1 ˆ yt+2 = aˆ yt+1 + but+2 etc. Pr♦❣r❛♠s✿ ❚✸✶♣r❡❈♦♥t✳s❝❡❀ ❚✸✷♣r❡❈♦♥t❴❆❞❛♣t✳s❝❡❀ ❚✸✷♣r❡❈♦♥t❴❆❞❛♣t✷✳s❝❡❀ ✭♣❛❣❡ ✶✵✹✮ ❚✸✷♣r❡❈♦♥t❴❆❞❛♣t✸✳s❝❡ ✭✇✐t❤ t❤❡ ❞❛t❛ ♦♥ ✇❡❜✮
♣❛❣❡ ✹✸
❋✉❧❧ ♣r❡❞✐❝t✐♦♥ ❢♦r ♥♦r♠❛❧ ♠♦❞❡❧ yt = ayt−1 + but + et yt+1 = ayt + but+1 + et+1 = = a (ayt−1 + but + et) + but+1 + et+1 = = a2yt−1 + abut + but+1 + aet + et+1 yt+2 = ayt+1 + but+2 + et+2 = = a3yt−1 + a2but + abut+1 + but+2 + a2et + aet+1 + et+2 ❛♥❞ ♣r❡❞✐❝t✐✈❡ ♣❞❢ ✐s Nyt+2 (ˆ µ, ˆ r) ✇❤❡r❡ ˆ µ = E [yt+2|y (t − 1)] = a3yt−1 + a2but + abut+1 + but+2 ˆ r = D [yt+2|y (t − 1)] = D [a2et + aet+1 + et+2] = (a4 + a2 + 1) r
♣❛❣❡ ✹✺
Pr❡❞✐❝t✐✈❡ ♣❞❢ ✐s ❛ r♦✇ ♦❢ t❤❡ ♠♦❞❡❧ ♠❛tr✐①✳ P♦✐♥t ♣r❡❞✐❝t✐♦♥ ✐s ❣❡♥❡r❛t❡❞ ❢r♦♠ t❤❡ ♣r❡❞✐❝t✐✈❡ ♣❞❢✳ ❊①❛♠♣❧❡✿ ▼♦❞❡❧ f (yt|ut, yt−1) ; yt ∈ {1, 2, 3} , ut ∈ {1, 2} ut, yt−1 yt = 1 yt = 2 yt = 3 ✶✱ ✶ ✵✳✷ ✵✳✺ ✵✳✸ ✶✱ ✷ ✵✳✶ ✵✳✸ ✵✳✻ ✶✱ ✸ ✵✳✼ ✵✳✷ ✵✳✶ ✷✱ ✶ ✵✳✸ ✵✳✸ ✵✳✹ ✷✱ ✷ ✵✳✺ ✵✳✷ ✵✳✸ ✷✱ ✸ ✵✳✻ ✵✳✶ ✵✳✸ ❋♦r ♠❡❛s✉r❡❞ ut = 1 ❛♥❞ yt−1 = 3 t❤❡ ♣r❡❞✐❝t✐✈❡ ♣❞❢ ✐s f (yt|ut = 1, yt−1 = 3) → yt ✶ ✷ ✸ f (yt) ✵✳✼ ✵✳✷ ✵✳✶
♣❛❣❡ ✹✻
■t ✐s ❣❡♥❡r❛t❡❞ ❛s ❛ ✈❛❧✉❡ ❢r♦♠ ❝❛t❡❣♦r✐❝❛❧ ❞✐str✐❜✉t✐♦♥ ✇✐t❤ t❤❡ ♣r❡❞✐❝t✐✈❡ ♣❞❢✳ ❚❤❡ ❣❡♥❡r❛t✐♦♥ ✐♥ ❙❝✐❧❛❜ ❝❛♥ ❜❡ ❞♦♥❡ ✐♥ t❤❡ ❢♦❧❧♦✇✐♥❣ ✇❛②✿ ✕ ♠♦❞❡❧ ♠❛tr✐① Θ = 0.2 0.5 0.3 0.1 0.3 0.6 · · · 0.6 0.1 0.3 ✕ ✜♥❞ r♦✇ r ❝♦rr❡s♣♦♥❞✐♥❣ t♦ ut, yt−1 ✭ut/yt−1 ❤❛✈❡ nu/ny ✈❛❧✉❡s✮ r = ny ∗ (ut − 1) + yt−1 ✕ ❣❡♥❡r❛t❡ ❢r♦♠ t❤✐s r♦✇ yt = (s✉♠ (r❛♥❞✭✶✱✶✱✬✉✬) > ❝✉♠s✉♠ (Θ (r, :))) + 1 Pr♦❣r❛♠s✿ ❚✸✸♣r❡❈❛t❴❖✛✳s❝❡❀ ❚✸✹♣r❡❈❛t❴❖✛❊st✳s❝❡❀ ❚✸✺♣r❡❈❛t❴❖♥❊st✳s❝❡ ✭♣❛❣❡ ✶✶✹ ❛♥❞ ❢✉rt❤❡r✮
♣❛❣❡ ✹✽
❙t❛t❡✲s♣❛❝❡ ♠♦❞❡❧ ✕ st❛t❡ ♠♦❞❡❧ ✭st❛t❡ ♣r❡❞✐❝t✐♦♥✮ xt = Mxt−1 + Nut−1 + wt ✕ ♦✉t♣✉t ♠♦❞❡❧ ✭st❛t❡ ✜❧tr❛t✐♦♥✮ yt = Axt + But + vt M, N, A, B ❛r❡ ❦♥♦✇♥ ♠❛tr✐❝❡s✱ wt, vt ❛r❡ ♥♦✐s❡s ✇✐t❤ ③❡r♦ ❡①♣❡❝t❛t✐♦♥s ❛♥❞ ❦♥♦✇♥ ❝♦✈❛r✐❛♥❝❡s Rw, Rv
❙t❛t❡ ❡✈♦❧✉t✐♦♥✿ ♣r❡❞✐❝t✐♦♥ → ✜❧tr❛t✐♦♥ f (xt−1|d (t − 1)) →
f (xt|d (t − 1)) →
f (xt|d (t)) Pr❡❞✐❝t✐♦♥ f (xt|d (t − 1)) =
t−1
f (xt|xt−1, ut−1) f (xt−1|d (t − 1)) dxt−1 ❋✐❧tr❛t✐♦♥ f xt
|d (t) ∝ f (yt|xt, ut)
f xt
|d (t − 1)
♣❛❣❡ ✺✵
❑❛❧♠❛♥ ✜❧t❡r ❋♦r ♥♦r♠❛❧ ♠♦❞❡❧ ❛♥❞ ✐♥✐t✐❛❧ ❝♦♥❞✐t✐♦♥s ✇❡ ❣❡t ❑❛❧♠❛♥ ✜❧t❡r ❬①t✱❘①✱②♣❪❂❑❛❧♠❛♥✭①t✱②t✱✉t✱▼✱◆✱❋✱❆✱❇✱●✱❘✇✱❘✈✱❘①✮ ①t ✲ st❛t❡ ❡st✐♠❛t❡ ✭❡①♣❡❝t❛t✐♦♥✮ ❘① ✲ st❛t❡ ❝♦✈❛r✐❛♥❝❡ ♠❛tr✐① ②♣ ✲ ♦✉t♣✉t ♣r❡❞✐❝t✐♦♥ ②t✱ ✉t ✲ ♦✉t♣✉t✱ ✐♥♣✉t ▼✱ ◆✱ ❋✱ ❆✱ ❇✱ ● ✲ st❛t❡ ♠♦❞❡❧ ♣❛r❛♠❡t❡rs ✭❋✱● ✲ ❝♦♥st❛♥ts✮ ❘✇✱ ❘✈ ✲ ♠♦❞❡❧ ♥♦✐s❡ ❝♦✈❛r✐❛♥❝❡s Pr♦❣r❛♠✿ ❚✹✻st❛t❊st❴❑❋✳s❝❡❀ ❚✹✼st❛t❊st❴◆♦✐s❡✳s❝❡ ✭♣❛❣❡ ✶✷✷ ❛♥❞ ❢✉rt❤❡r✮
♣❛❣❡ ✺✷
◆♦♥❧✐♥❡❛r st❛t❡ ❡st✐♠❛t✐♦♥ ▼♦❞❡❧ xt = g (xt−1, ut) + wt yt = h (xt, ut) + vt ▼♦❞❡❧ ❧✐♥❡❛r✐③❛t✐♦♥ ✭❚❛②❧♦r ❡①♣❛♥s✐♦♥✮ g (x, ut) . = g (ˆ xt−1, ut) + g′ (ˆ xt−1, ut) (x − ˆ xt−1) h (x, ut) . = h (ˆ xt, ut) + h′ (ˆ xt, ut) (x − ˆ xt) ✇❤❡r❡ ˆ x ✐s t❤❡ ❧❛st ♣♦✐♥t ❡st✐♠❛t❡✳
❘❡s✉❧t xt = ¯ Mxt−1 + F + wt yt = ¯ Axt + G + vt ✇❤❡r❡ ¯ M = g′ (ˆ xt−1, ut) , F = g (ˆ xt−1, ut) − g′ (ˆ xt−1, ut) ˆ xt−1, ¯ A = h′ (ˆ xt, ut) , G = h (ˆ xt, ut) − h′ (ˆ xt, ut) ˆ xt.
♣❛❣❡ ✺✹
❈r✐t❡r✐♦♥✿ E N
t=1 (Jt) |d (0)
Jt = y2
t + ωu2 t ♦r y2 t + λ (ut − ut−1)2
❈r✐t❡r✐♦♥ ❝❛♥ ❜❡ ♠✐♥✐♠✐③❡❞ s❡q✉❡♥t✐❛❧❧② ❢r♦♠ t❤❡ ❡♥❞✳ ❚❤❡ r❡❝✉rs✐♦♥ ✭❇❡❧❧♠❛♥ ❡q✉❛t✐♦♥s✮ ❛r❡ ϕN+1 = 0 ❢♦r t = N, N − 1, · · · , 1 ϕt = E
t+1 + Jt|ut, d (t − 1)
ϕ∗
t = min ut ϕt
♠✐♥✐♠✐③❛t✐♦♥ u∗
t = arg min ϕt ❝♦♥tr♦❧
❡♥❞
♣❛❣❡ ✻✵
■t ✐s ♣❡r❢♦r♠❡❞ ❢♦r st❛t❡ ❢♦r♠ ♦❢ t❤❡ ♠♦❞❡❧✳ RN+1 = 0 ❢♦r t = N, N − 1, · · · , 1 U = Rt+1 + Ω A = N ′UN B = N ′UM C = M ′UM St = A−1B Rt = C − S
′
tASt
❡♥❞ ❍❡r❡✱ t❤❡ ✈❡❝t♦rs St ❛r❡ ❝♦♠♣✉t❡❞ ❛♥❞ t❤❡♥ t❤❡② ❛r❡ ✉s❡ ❢♦r ❝♦♥tr♦❧ ❛♣♣❧✐❝❛t✐♦♥ ✭✐♥ t✐♠❡ ❞✐r❡❝t✐♦♥✮ ❢♦r t = 1, 2, · · · , N, ut = ut = −Stxt−1; yt = ❣❡♥❡r(ut)❀ ❡♥❞ Pr♦❣r❛♠✿ ❚✺✸❝tr❧❳✳s❝❡❀ ❚✺✹❝tr❧❳❊st✳s❝❡ ✭♣❛❣❡ ✶✷✽ ❛♥❞ ❢✉rt❤❡r✮
♣❛❣❡ ✻✷
❘❡♠❛r❦s ✶✳ ■❢ ✐♥ ❝r✐t❡r✐♦♥ (yt − st)2 ✐s ✉s❡❞ t❤❡ ♦✉t♣✉t ❢♦❧❧♦✇s t❤❡ s❡t♣♦✐♥t st ✷✳ ■❢ Jt = y2
t + λ (ut − ut−1)2 ✐s ✉s❡❞✱ st❡❛❞②✲st❛t❡ ❞❡✈✐❛t✐♦♥ ✐s ❛✈♦✐❞❡❞✳
✸✳ ■❢ t❤❡ ♠♦❞❡❧ ♣❛r❛♠❡t❡rs ❛r❡ ♥♦t ❦♥♦✇♥✱ ✇❡ ♠✉st ✉s❡ s✉❜✲♦♣t✐♠❛❧ ❝♦♥tr♦❧ ✇✐t❤ r❡❝❡❞✐♥❣ ❤♦r✐③♦♥✿ ✭❛✮ ❢♦r ❡①✐st✐♥❣ ♣❛r❛♠❡t❡r ❡st✐♠❛t❡❞ ❞❡s✐❣♥ t❤❡ ❝♦♥tr♦❧ ❛♥❞ ✉s❡ ♦♥❧② t❤❡ ✜rst st❡♣✱ ✭❜✮ ❛♣♣❧② t❤❡ ❝♦♠♣✉t❡❞ ❝♦♥tr♦❧❀ ✭❝✮ ♠❡❛s✉r❡ ♥❡✇ ♦✉t♣✉t❀ ✭❞✮ ✇✐t❤ ♥❡✇ ❞❛t❛ r❡❝♦♠♣✉t❡ ♣❛r❛♠❡t❡r ❡st✐♠❛t❡s ✭❡✮ ❣♦ t♦ ✭❛✮✳
♣❛❣❡ ✻✸
■t ✐s ♣❡r❢♦r♠❡❞ ❡①❛❝t❧② ✐♥ t❤❡ s❛♠❡ ✇❛② ❛s ❝♦♥t✐♥✉♦✉s ✇✐t❤ t❤❡ ❞✐s❝r❡t❡ ♠♦❞❡❧✳ ❍♦✇❡✈❡r✱ t❤❡ ♦♣❡r❛✲ t✐♦♥s ✇✐t❤ t❛❜❧❡s ❛r❡ s♦♠❡✇❤❛t ✉♥✉s✉❛❧✳ ❨♦✉ ❝❛♥ ❧♦♦❦ ❛t t❤❡♠ ✐♥t♦ t❤❡ t❡①t✳ Pr♦❣r❛♠✿ ❚✺✷❝tr❧❉✐s❝✳s❝❡ ✭♣❛❣❡ ✶✸✺✮
♣❛❣❡ ✻✺✲✻✼