Page 729 - The Mechatronics Handbook

0066_Frame_C23 Page 37 Wednesday, January 9, 2002 1:55 PM

which, unfortunately, is not realizable as it stands because w k is not available to measurement. Therefore,
the noise sequence {w k } has to be substituted by some function of the observed variable {y k }. A linear
predictor chosen according to Eq. (23.76) is

1
(
–
c 1 –
y ˆ k+1 k = Gz ) --------------------y (23.82)
---------------y k =
a 1
Cz ( – 1 ) 1 + c 1 z – 1 k
The Kalman Filter
Consider the linear state-space model
x k+1 = Φx k + v k , x k ∈ n
(23.83)
y k = Cx k + w k , y k ∈ m

where {v k } and {w k } are assumed to be independent zero-mean white-noise processes with covariances
, respectively. It is assumed that {y k }, but not {x k }, is available to measurement and that it is
Σ v and Σ w
desirable to predict {x k } from measurements of {y k }.
Introduce the state predictor,

x ˆ k+1 k = Φx ˆ kk−1 – K k y ˆ k – y k ), x ˆ kk−1 ∈ n
(
(23.84)
y ˆ k = Cx ˆ kk−1, y k ∈ m
Φ
The predictor of Eq. (23.84) has the same dynamics matrix as the state-space model of Eq. (101.83)
and, in addition, there is a correction term K k (y ˆ k – y k ) with a factor K k to be chosen. The prediction
error is

x ˜ k+1 k = x ˆ k+1 k – x k+1 (23.85)

The prediction-error dynamics is

x ˜ k+1 = ( Φ K k C)x ˜ k + v k – K k w k (23.86)
–
The mean prediction error is governed by the recursive equation

{
x ˜ k+1} = ( Φ K k C) x ˜ k (23.87)
{}
–
and the mean square error of the prediction error is governed by
T
T
{
x ˜ k+1x ˜ k+1} = [ ( Φ K k C)x ˜ k + v k – K k w k ] Φ K k C)x ˜ k + v k – K k w k ] }
{
[
(
–
–
T
= ( Φ – K k C) x ˜ k x ˜ k } Φ –( K k C) + Σ v + K k Σ w K k (23.88)
T
{
If we denote
P k = x ˜ k x ˜ k }, Q k = Σ w + CP k C T (23.89)
T
{
then Eq. (23.88) is simplified to
P k+1 = ΦP k Φ – K k CP k ΦΦ P k C K k + Σ v + K k Q k K k T (23.90)
T
T
T
T
–
©2002 CRC Press LLC