Massachusetts Institute of Technology
Department of Mechanical Engineering
2.160 Identification, Estimation, and Learning
Spring 2006
Problem Set No. 3
Out: March 1, 2006 Due: March 8, 2006
For the first two problems below, hand calculation is recommended.
Problem 1
Consider a simple scalar random process governed by the following state
transition equation:
x
t
1
+
=
x
t
+
w
t
where
x
t
is state, and
w
t
is a zeromean process noise with
Q
>
0
∀
t
=
E
[
w w
]
t
s
=
s
.
0
≠
∀
s
t
The output of the process is observed with a single sensor having the following
properties:
+
y
t
=
x
t
v
t
R
>
0
=
∀
s
t
0
≠
∀
s
t
v
v
E
]
[
s
t
=
w
v
E
]
[
t
s
0
∀
∀
t
,
s
=
We want to build a Kalman filter and examine its characteristics with respect to the state
estimation error covariance and the sensor and process noise properties.
(a). Write out all the recursive equations of Kalman filter for this scalar process. Simplify
the equations as much as you can.
(b). Given
P
=
100 ,
Q
= 1, and
R
= 1, plot the values of a posteriori and a priori error
0
covariances,
P
,
P
10
,
P
,
P
21
,
P
,
P
32
,
P
,
"
against time. Also plot the Kalman gain
0
1
2
3
K
1
,
K
,
K
,
"
. Repeat it for different values of Q and R, and discuss how the process
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 Spring '06
 HarryAsada
 Mechanical Engineering, Heat, Signal Processing, Heat Transfer

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