User`s guide

Kalman Filtering
9-51
In these equations:
• is the estimate of given past measuremen ts 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- st ep- ahe ad predict or). 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 me asured 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 t he time and measurement update equations into one
state-space model (the Kalman filter).
This filt er generates an optim al 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.
x
ˆ
nn 1–[]
xn
[]
y
v
n 1
–[]
x
ˆ
nn[]
y
v
n
[]
x
ˆ
nn[]
n1
+
y
v
n1
+[]
y
v
n1+[]Cx
ˆ
n 1+ n[]– Cxn 1+[]x
ˆ
n1+n[]–()=
yn 1
+[]
M
Ewn[]wn[]
T
()Q,=Evn[]vn[]
T
()R=
x
ˆ
n1n+[]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)