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❙②st❡♠

Pr♦❝❡ss ✲ ❙②st❡♠ ✲ ▼♦❞❡❧ • Pr♦❝❡ss ✲ ♣❛rt ♦❢ r❡❛❧✐t② ✇❡ ❛r❡ ✐♥t❡r❡st❡❞ ✐♥✳ • ❙②st❡♠ ✲ ✈❛r✐❛❜❧❡s ✇✐t❤ t❤❡✐r r❡❧❛t✐♦♥s✳ • ▼♦❞❡❧ ✲ ♠❛t❤❡♠❛t✐❝❛❧ r❡❧❛t✐♦♥ ♦❢ t❤❡ ♠♦♥✐t♦r❡❞ ✈❛r✐❛❜❧❡ ❛♥❞ ♦t❤❡r ❡①♣❧❛♥❛t♦r② ✈❛r✐❛❜❧❡s✳ ❘❡♠❛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✐❝ ✳

❱❛r✐❛❜❧❡s ✐♥ t❤❡ s②st❡♠ noise e t imput (control) u t SYSTEM output y t disturbance v t state x t ❖✉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❡❞✳ ♣❛❣❡ ✽

▼♦❞❡❧

❇❛②❡s✐❛♥ ✈✐❡✇ ♦♥ ♠♦❞❡❧ ❈♦♥❞✐t✐♦♥❛❧ ♣r♦❜❛❜✐❧✐t② ❞❡♥s✐t② ❢✉♥❝t✐♦♥ ✭♣❞❢✮ � � ′ f y t | ψ t , Θ ψ t = [ u t , y t − 1 , u t − 1 , · · · , y t − n , u t − n , 1] ′ ✲ r❡❣r❡ss✐♦♥ ✈❡❝t♦r❀ Θ = { θ, r } ❀ θ = [ b 0 , a 1 , b 1 , · · · , a n , b n , k ] ′ ✱ θ ✲ r❡❣r❡ss✐♦♥ ❝♦❡✣❝✐❡♥ts✱ r ✲ ♥♦✐s❡ ✈❛r✐❛♥❝❡✳ ■t ✐s ❛ st♦❝❤❛st✐❝ ❞❡♣❡♥❞❡♥❝❡ ♦❢ y t ♦♥ ψ t ✇✐t❤ r❡❧❛t✐♦♥s ❡①♣r❡ss❡❞ ❜② ♣r♦❜❛❜✐❧✐t② ❞❡♥s✐t② ❢✉♥❝t✐♦♥ ✭♣❞❢✮✳ ♣❛❣❡ ✶✷

❘❡❣r❡ss✐♦♥ ♠♦❞❡❧ ❚❤❡ ✈❛r✐❛❜❧❡s ❛r❡ ❝♦♥t✐♥✉♦✉s✱ ψ ❝❛♥ ❤❛✈❡ ❛❧s♦ s♦♠❡ ❞✐s❝r❡t❡ ♦♥❡s ✳ ❚❤❡ ❛❜♦✈❡ ♣❞❢ ❡①♣r❡ss✐♦♥ ❝❛♥ ❜❡ ❣❡♥❡r❛t❡❞ ❜② t❤❡ st♦❝❤❛st✐❝ ❡q✉❛t✐♦♥ y t = b 0 u t + a 1 y t − 1 + b 1 u t − 1 + · · · + a n y t − n + b n u t − n + k + e t = ′ = ψ t θ + e t ✇❤❡r❡ e t ✭♥♦✐s❡✮ ✐s ✐✳✐✳❞✳ ✭✐♥❞❡♣❡♥❞❡♥t✱ ✐❞❡♥t✐❝❛❧❧② ❞✐str✐❜✉t❡❞✮ r❛♥❞♦♠ ✈❛r✐❛❜❧❡ ✇✐t❤ ③❡r♦ ❡①♣❡❝t❛t✐♦♥ ❛♥❞ ✈❛r✐❛♥❝❡ r. E [ y t | ψ t , Θ] = b 0 u t + a 1 y t − 1 + b 1 u t − 1 + · · · + a n y t − n + b n u t − n + k ✱ D [ y t ] = D [ e t ] = r Pr♦❣r❛♠✿ ❚✶✶s✐♠❈♦♥t✳s❝❡ ✭♣❛❣❡ ✽✸✮ ♣❛❣❡ ✶✶✱ ✶✸

❉✐s❝r❡t❡ ♠♦❞❡❧ ❆❧❧ ✈❛r✐❛❜❧❡s ❛r❡ ❞✐s❝r❡t❡ ✭✜♥✐t❡ ♥✉♠❜❡r ♦❢ ✈❛❧✉❡s✮ f ( y t | ψ t , Θ) = Θ y t | ψ t [ u t , y t − 1 ] y t = 1 y t = 2 ✶✱ ✶ Θ 1 | 11 Θ 2 | 11 ✶✱ ✷ Θ 1 | 12 Θ 2 | 12 ✷✱ ✶ Θ 1 | 21 Θ 2 | 21 ✷✱ ✷ Θ 1 | 22 Θ 2 | 22 � 2 i =1 Θ i | jk = 1 ✲ ❝♦♥❞✐t✐♦♥❛❧ ♣r♦❜❛❜✐❧✐t✐❡s✳ � � ❋♦r ❣✐✈❡♥ [ u t , y t − 1 ] t❤❡ ♦✉t♣✉t y t ✐s ❣❡♥❡r❛t❡❞ ✇✐t❤ t❤❡ ♣❞❢ Θ 1 | u t ,y t − 1 Θ 2 | u t ,y t − 1 . Pr♦❣r❛♠✿ ❚✶✸s✐♠❉✐s❝✳s❝❡ ✭♣❛❣❡ ✽✻✮ ♣❛❣❡ ✶✽

▼♦❞❡❧ ♦❢ ❧♦❣✐st✐❝ r❡❣r❡ss✐♦♥ ❚❤❡ ♦✉t♣✉t ✐s ❞✐s❝r❡t❡ ✭✵ ♦r ✶✮ ❛♥❞ ✐t ❞❡♣❡♥❞s ♦♥ ❝♦♥t✐♥✉♦✉s ✈❛r✐❛❜❧❡s✳ exp ( y t z t ) f ( y t | ψ t , Θ) = 1 + exp ( z t ) ✇❤❡r❡ z t = ψ t Θ + e t ❚❤❡ ♠♦❞❡❧ ❤❛s t❤❡ ❢♦❧❧♦✇✐♥❣ ❢♦r♠ ✲ tr❛♥s❢♦r♠❛t✐♦♥ ❢r♦♠ z t♦ p = f ( y t = 1 | z t ) P ( y t = 1 | z t ) 1 z t ♣❛❣❡ ✷✸

❙t❛t❡✲s♣❛❝❡ ♠♦❞❡❧ ❉❡s❝r✐❜❡s t❤❡ st❛t❡ ✈❛r✐❛❜❧❡ x t ✕ st❛t❡ ♠♦❞❡❧ ✭st❛t❡ ♣r❡❞✐❝t✐♦♥✮ x t = Mx t − 1 + Nu t − 1 + w t ✕ ♦✉t♣✉t ♠♦❞❡❧ ✭st❛t❡ ✜❧tr❛t✐♦♥✮ y t = Ax t + Bu t + v t M, N, A, B ❛r❡ ❦♥♦✇♥ ♠❛tr✐❝❡s✱ w t , v t ❛r❡ ♥♦✐s❡s ✇✐t❤ ③❡r♦ ❡①♣❡❝t❛t✐♦♥s ❛♥❞ ❦♥♦✇♥ ❝♦✈❛r✐❛♥❝❡s R w , R v ♣❛❣❡ ✺✵

❙t❛t❡ ❢♦r♠ ♦❢ r❡❣r❡ss✐♦♥ ♠♦❞❡❧ ❋♦r ✷ nd ♦r❞❡r r❡❣r❡ss✐♦♥ ♠♦❞❡❧ y t = b 0 u t + a 1 y t − 1 + b 1 u t − 1 + a 2 y t − 2 + b 2 u t − 2 + k + e t t❤❡ st❛t❡ ❢♦r♠ ✐s           y t a 1 b 1 a 2 b 2 k y t − 1 b 0 e t           u t 0 0 0 0 0 u t − 1 1 0                     = + u t + y t − 1 1 0 0 0 0 y t − 2 0 0                               u t − 1 0 1 0 0 0 u t − 2 0 0           1 0 0 0 0 1 1 0 0 Pr♦❣r❛♠✿ ❚✶✺s✐♠❙t❛t❡✳s❝❡ ✭♣❛❣❡ ✽✽✮ ♣❛❣❡ ✶✹

❊st✐♠❛t✐♦♥

❇❛②❡s✐❛♥ ❡st✐♠❛t✐♦♥ ◆♦t❛t✐♦♥✿ d t ❞❛t❛ ❛t t, d ( t ) = { d 0 , d 1 , · · · , d t } ❞❛t❛ ✉♣ t♦ t ✱ d 0 ♣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 ( y t | ψ t , Θ) f (Θ | d ( t − 1)) � �� � � �� � � �� � ♣♦st❡r✐♦r ♠♦❞❡❧ ♣r✐♦r ✕ ◆❛t✉r❛❧ ❝♦♥❞✐t✐♦♥s ♦❢ ❝♦♥tr♦❧ f (Θ | u t , d ( t − 1)) = f (Θ | d ( t − 1)) ✕ ❇❛t❝❤ ❡st✐♠❛t✐♦♥ � t � � f (Θ | d ( t )) ∝ f ( y τ | ψ τ Θ) f (Θ | d (0)) τ =1 � �� � ▲✐❦❡❧✐❤♦♦❞ L t (Θ) ✕ ❙❡❧❢ r❡♣r♦❞✉❝✐♥❣ ♣r✐♦r f (Θ | d ( t − 1)) → f (Θ | d ( t )) ✲ t❤❡ s❛♠❡ ❢♦r♠✳ ♣❛❣❡ ✷✻

❘❡s✉❧ts ♦❢ ❡st✐♠❛t✐♦♥ • P♦st❡r✐♦r ♣❞❢ f (Θ | d ( t )) ♣r♦❜❛❜✐❧✐t✐❡s ♦❢ ♣❛r❛♠❡t❡r ✈❛❧✉❡s • P♦✐♥t ❡st✐♠❛t❡ ♦❢ ♣❛r❛♠❡t❡r ✭❡①♣❡❝t❛t✐♦♥✮ � ∞ ˆ Θ t = E [Θ | d ( t )] = Θ f (Θ | d ( t )) d Θ −∞ ♣❛❣❡ ✷✾

❊st✐♠❛t✐♦♥ ♦❢ r❡❣r❡ss✐♦♥ ♠♦❞❡❧ ❆♣♣❧✐❝❛t✐♦♥ ♦❢ ❇❛②❡s r✉❧❡ ✇✐t❤ r❡❣r❡ss✐♦♥ ♠♦❞❡❧ ❛♥❞ ♣r✐♦r✴♣♦st❡r✐♦r ✐♥ t❤❡ ❢♦r♠ ♦❢ ●❛✉ss✲✐♥✈❡rs❡✲ ❲✐s❤❛rt ❞✐str✐❜✉t✐♦♥ � � �� − 1 f (Θ | d (0)) ∝ r − 0 . 5 κ 0 exp [ − 1 , θ ′ ] V 0 θ ❙t❛t✐st✐❝s ✉♣❞❛t❡ V t = V t − 1 + D t κ t = κ t − 1 + 1 � � y t � � ′ ✇❤❡r❡ D t = y t , ψ ✐s ❞❛t❛ ♠❛tr✐①✱ V t ✐s ✐♥❢♦r♠❛t✐♦♥ ♠❛tr✐① ❛♥❞ κ t ✐s ❝♦✉♥t❡r✳ t ψ 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 � � � � ′ V y V • −− yψ V t = · · · V yψ V ψ | � ✕ ❡st✐♠❛t❡s ♦❢ r❡❣r❡ss✐♦♥ ❝♦❡✣❝✐❡♥ts ˆ θ t = V − 1 ψ V yψ ✕ ❡st✐♠❛t❡ ♦❢ ♥♦✐s❡ ✈❛r✐❛♥❝❡ ′ yψ V − 1 r t = V y − V ψ V yψ ˆ κ t ♣❛❣❡ ✸✷