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16.322 Stochastic Estimation and Control, Fall 2004
Prof. Vander Velde
Page 1 of 7
Lecture 24
Last time:
()
() ()
() () ()
TT
d
Xt
AtXt XtAt
BtNtBt
dt
=+
+
If the system reaches a statistical steady state
, the covariance matrix would be
constant.
The system
would have to be invariant
,
0
d
X t
AX
XA
BNB
dt
+
=
and the steady state covariance matrix can be solved for from this algebraic
equation.
Note that:
X t
is symmetric.
If
[ ]
,
A B
is controllable and
0
() 0
>
,
>
for all
t
.
If, in addition,
[ ]
,
AC
is observable, then
Cov
( )
0
yt
⎡⎤
>
⎣⎦
for all
.
t
Kalman Filter
Our text treats two forms of the Kalman filter:
•
Discrete time filter
Based on the model
o
System:
1(
)
kk
x
kx
k
w
φ
+=
+
o
Measurements:
k
zH
x
kv
•
Continuous time filter
o
System:
x
Fx Gw
&
o
Measurements:
xv
The most common practical situation is neither of these.
In the aerospace area,
and many other application areas as well, we are interested in dynamic systems
that evolve continuously in time
, so we must at least start with a continuous
system model.
The system may be driven by a known command input and may
be subject to random disturbances.
If the given disturbances cannot be well
approximated as white, then a shaping filter
must be added to shape the given
disturbance spectrum from white noise.
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View Full Document16.322 Stochastic Estimation and Control, Fall 2004
Prof. Vander Velde
Page 2 of 7
The augmented system is modeled as
()
x
At x Gtu Btn
=++
&
where
nt
is an unbiased white noise with correlation function
() ( )
T
Entn
Nt t
τ
δτ
⎡⎤
=−
⎣⎦
We allow the system matrices to be time varying.
The most common reason
for
dealing with time varying linear
dynamics is because we have linearized
nonlinear dynamics
about some trajectory.
But although the system dynamics are continuous in time, we do not process
measurements continuously.
The Kalman filter involves significant
computations, and those computations are universally executed in a digital
computer
.
Since a digital computer works on a cyclic basis, the measurements
are processed at discrete points in time
.
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 Fall '04
 WallaceVanderVelde

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