Optimal Estimation of Dynamic Systems, 2nd Edition

Batch State Estimation 365

where Ψ(T −Δ,T) is the state transition matrix, which clearly must obey

d

dt

Ψ(T −Δ,T)=[F(T −Δ)+G(T −Δ)Q(T −Δ)G

T

(T −Δ)P

−1

f

(T −Δ)]

×Ψ(T −Δ,T), Ψ(t,t)=I

(5.179)

Note, the matrix Φ(t,T ) from §5.2.2 and the matrix Ψ(T −Δ,T) are related by

Ψ(T −Δ,T)=Φ(T −Δ,t)Φ(t, T) (5.180)

with Φ(0, T)=Ψ(0,T). Differentiating Equation (5.180) with respect to T and using

Equation (5.152) yields

d

dT

Ψ(T −Δ,T)=[F(T −Δ)+G(T −Δ)Q(T −Δ)

×G

T

(T −Δ)P

−1

f

(T −Δ)]Ψ(T −Δ,T )

−Ψ(T −Δ, T)[F(T)+G(T)Q(T )G

T

(T)P

−1

f

(T)]

(5.181)

Taking the derivative of Equation (5.178) with respect to T , and substituting the

forward-time state filter equation from Table 5.3 and Equation ...