Optimal Estimation of Dynamic Systems, 2nd Edition

538 Optimal Estimation of Dynamic Systems

Since S

k+1

is assumed to have an inverse, then Equation (8.92) can be rewritten as

S

k

= Φ

T

k



S

−1

k+1

+ Γ

k

R

−1

k

Γ

T

k



−1

Φ

k

+ Q

k

(8.93)

Using the matrix inversion lemma in Equation (1.70) with A = S

−1

k+1

, B = Γ

k

, C =

R

−1

k

,andD = Γ

T

k

gives

S

k

= Φ

T

k

S

k+1

Φ

k

−Φ

T

k

S

k+1

Γ

k



Γ

T

k

S

k+1

Γ

k

+ R

k



−1

Γ

T

k

S

k+1

Φ

k

+ Q

k

(8.94)

with terminal boundary condition

S

N

= S

f

(8.95)

Equation (8.94) represents the discrete-time matrix Riccati equation, which is prop-

agated backward in time. The discrete-time LQR gain for the time-varying linear

feedback form is more complicated than the continuous-time case. We first substi-

tute Equation (8.86) into Equation (8.84c) ...