Specifications

Kalman Filtering
10-51
In these equations:
• is the estimate of given past measurements up to
• is the updated estimate based on the last measurement
Given the current estimate , the time update predicts the state value at
the next sample (one-step-ahead predictor). The measurement update
then adjusts this prediction based on the new measurement . The
correction term is a function of the innovation, that is, the discrepancy.
between the measured and predicted values of . The innovation gain
is chosen to minimize the steady-state covariance of the estimation error
given the noise covariances
You can combine the time and measurement update equations into one
state-space model (the Kalman filter).
This filter generates an optimal estimate of . Note that the filter
state is .
Steady-State Design
You can design the steady-state Kalman filter described above with the
function
kalman. First specify the plant model with the process noise.
This is done by
% Note: set sample time to -1 to mark model as discrete
Plant = ss(A,[B B],C,0,-1,'inputname',{'u' 'w'},...
x
ˆ
nn 1–
[]
xn
[]
y
v
n 1–
[]
x
ˆ
nn
[]
y
v
n
[]
x
ˆ
nn
[]
n 1+
y
v
n 1+
[]
y
v
n 1+
[]
Cx
ˆ
n 1+ n
[]
– Cxn 1+
[]
x
ˆ
n 1+ n
[]
–
()
=
yn 1+
[]
M
Ewn
[]
wn
[]
T
()
Q ,= Evn
[]
vn
[]
T
()
R=
x
ˆ
n 1 n+
[]
AI MC–
()
x
ˆ
nn 1–
[]
BAM
un
[]
y
v
n
[]
+=
y
ˆ
nn
[]
CI MC–
()
x
ˆ
nn 1–
[]
CM y
v
n
[]
+=
y
ˆ
nn
[]
yn
[]
x
ˆ
nn 1–
[]
xn 1+
[]
Ax n
[]
Bu n
[]
Bw n
[]
++= (state equation)
yn
[]
Cx n
[]
= (measurement equation)