Optimal Estimation of Dynamic Systems, 2nd Edition

210 Optimal Estimation of Dynamic Systems

3.12

Consider the following first-order discrete-time system:

x

k+1

=

φ

x

k

+ w

k

where w

k

is a zero-mean Gaussian noise process with variance q.Derivea

closed-form expression for the variance of x

k

, where p

k

≡E



x

2

k



. What is the

steady-state variance? Also, discuss the properties of the steady-state value

in terms of the stability of the system (i.e., in terms of

φ

).

3.13 Consider the following discrete-time model:

x

k+1

= x

k

˜y

k

= x

k

+ v

k

where v

k

is a zero-mean Gaussian noise process with variance r. Note that

this system has no process noise, so Q = 0. Using the discrete-time Kalman

filter equations in Table 3.1 derive a closed-form recursive solution for the

gain K in terms of r, P

0

(the initial error variance), and k (the