User`s guide

9 Design Case Studies
9-50
Kalman Filtering
This final case study illustrates the use of the Control System Toolbox for
Kalman filter design and simulation. Both steady-state and time-varying
Kalman filters are considered.
Consider the discrete plant
with additive Gaussian noise on the input and data
A = [1.1269 –0.4940 0.1129
1.0000 0 0
0 1.0000 0];
B = [–0.3832
0.5919
0.5191];
C = [1 0 0];
Our goal is to design a Kalman filter that estimates the output given the
inputs and the noisy o utput measurements
where is some Gaussian white noise.
Discrete Kalman Filter
The equations of the steady-state Kalman filter for this problem are given as
follows.
Measurement update
Time update
xn 1
+[]
Ax n
[]
Bun
[]
wn
[]+()+=
yn[] Cx n[]=
wn
[]
un
[]
yn
[]
un
[]
y
v
n
[]
Cx n
[]
vn
[]+=
vn
[]
x
ˆ
nn[]x
ˆ
nn 1–[]My
v
n[] Cx
ˆ
nn 1–[]–()+=
x
ˆ
n1n+[]Ax
ˆ
nn[]Bu n[]+=