Optimal Estimation of Dynamic Systems, 2nd Edition

150 Optimal Estimation of Dynamic Systems

Substituting Equation (3.66) into Equation (3.65) gives

−[Φ

k

K

k

R

k

K

T

k

Φ

T

k

+ ϒ

k

Q

k

ϒ

T

k

]

×Φ

−T

k

[I −K

k

H

k

]

−T

P

−1

k

[I −K

k

H

k

]

−1

Φ

−1

k

< 0

(3.67)

Since Φ

−T

k

[I −K

k

H

k

]

−T

P

−1

k

[I −K

k

H

k

]

−1

Φ

−1

k

is positive definite, Equation (3.67) re-

duces down to

−[Φ

k

K

k

R

k

K

T

k

Φ

T

k

+ ϒ

k

Q

k

ϒ

T

k

] < 0 (3.68)

Clearly, if R

k

is positive definite and Q

k

is at least positive semi-definite, then the

Lyapunov condition is satisfied and the discrete-time Kalman filter is stable.

In the previous derivations of the discrete-time Kalman filter the covariance matrix

P

k

must remain positive definite. We now show that if P

k

is positive definite then P

k+1

is also positive definite. Assu