Optimal Estimation of Dynamic Systems, 2nd Edition

Batch State Estimation 385

d

dT

ˆ

x(T −Δ|T)=[F(T −Δ)+G(T −Δ)Q(T −Δ)

×G

T

(T −Δ)P

−1

f

(T −Δ)]

ˆ

x(T −Δ|T)

−G(T −Δ) Q(T −Δ)G

T

(T −Δ)P

−1

f

(T −Δ)

ˆ

x

f

(T −Δ)

+ Ψ(T −Δ, T)P

f

(T)H

T

(T)R

−1

(T)[

˜

y(T) −H(T)

ˆ

x

f

(T)]

d

dT

P(T −Δ|T)=[F(T −Δ)+G(T −Δ)Q(T −Δ)

×G

T

(T −Δ)P

−1

f

(T −Δ)]P(T −Δ|T)+P(T −Δ|T)

×[F(T −Δ)+G(T −Δ)Q(T −Δ)G

T

(T −Δ)P

−1

f

(T −Δ)]

T

−Ψ(T −Δ, T)P

f

(T)H

T

(T)R

−1

(T)P

f

(T)Ψ(T −Δ,T)

−G(T −Δ) Q(T −Δ)G

T

(T −Δ)

• Innovations Process (Discrete-Time)

λ

k

≡−

N

∑

i=k

Z

T

fi,k

H

T

i

E

−1

fi

e

fi

Z

fi,k

=

&

Z

fi−1

Z

fi−2

···Z

fk

for i > k

I for i = k

Z

fk

≡ Φ

k

[I −K

fk

H

k

]

E

fk

≡ H

k

P

fk

H

T

k

+ R

k

e

fk

≡

˜

y

k

−H

k

ˆ

x

fk

λ

k

= Z

T

fk

λ

k+1

−H

T

k

E

−1

fk

e

fk

, λ

N

= 0

• Innovations Process (Continuous-Time)

λ(t) ≡−



T

t

Ψ

T

(

τ

, t)H

T

(

τ

)R

−1

(

τ

)e

f

(

τ

) d

τ

d

d

τ

Ψ(

τ

, t)=[F(

τ

) −K