16.322 Stochastic Estimation and Control, Fall 2004
Prof. Vander Velde
Page 1 of 3
Lecture 23
Last time:
(, ) () (, )
(,)
d
tAtt
dt
I
τ τ
ττ
Φ=Φ
Φ=
So the covariance matrix for the state at time t is
() () () ()
() ()
() ()
0 0
0
00 1 111 0 2 2 2 2
00
002 2 2 2
() () () () ()
() ()
, () , ()() () , () () ,
,()() ,
,()()() ,
T
T
tt
TT
TTT
T
T
t
T
TT
t
Xt xt xt xt xt
xtxt
Ettxt tBndxt t n B t d
tt xtxt tt
tt xt n B t d
t
τ τττ τ τ τ τ
ττ ττ
????
=? ?
????
=
? ?
=Φ +Φ Φ + Φ
??? ?
??? ?
=Φ Φ
+Φ Φ
+Φ
∫∫
∫
%%
%%%%
%%
%%
() ()
()
0
00
1110 0 1
12 1 11 2 2 2
, ( )( )() ,
,()()()()(,)
t
T
T
t
tt
TT T
tt
Bnxt ttd
dd t Bnn B t
τττ τ
ττ τ τττ τ τ
Φ
+Φ Φ
∫
∫∫
%%
%%
The two middle terms are zero:
- For
0
tτ > , ()n τ% and
0
()x t% are uncorrelated because ()n τ% is white (impulse
correlation function)
- For
0
tτ = , ()n τ% has a finite effect on
0
()x t% because ()n τ% is white. But the
integral of a finite quantity over one point is zero.
() () () ( )
() () ()
00
0
00 0 12 111212 2
00 0
() , ( ) , , ( ) ( ) ( ) (, )
,(), ,()()()(,)
tt
T
TT
tt
t
T
TT
t
Xt tt Xt tt d d t B N B t
tt Xt tt t B N B t d
τ ττττδτττ τ
ττττ ττ
=Φ Φ + Φ ? Φ
=Φ Φ + Φ Φ
∫∫
∫
This is an integral expression for the state covariance matrix. But we would
prefer to have a differential equation. So take the derivative with respect to time.
16.322 Stochastic Estimation and Control, Fall 2004
Prof. Vander Velde
Page 2 of 3
() ()
() ()
() ()
() ()
0
0
00 0
00 0
() () , ( ) ,
,(), ()
() , ( ) ( ) ( ) ,
,()()() , ()
() () ()
() () () () () () () ()
T
T
T
t
T
T
t
t
T
TT
t
T
TT
d
Xt At tt Xt tt
dt
tt Xt tt At
At t B N B t d
tBNB t Atd
BtNtBt
d
Xt AtXt XtAt BtNtBt
dt
τ τττ ττ
τ τττ τ τ
=Φ Φ
+Φ Φ
+Φ Φ
+Φ Φ
+
=+ +
∫
∫
This defines the first and second order statistics of the state.
Initial conditions
Often we wish to compute the time evolution of the statistics of a system which
starts from rest at time zero. If the input to this real system is being formed by a
shaping filter, then not all elements of X are zero at 0t = .
We want to model ()x t as a stationary process.
This situation is equivalent to:
where the white noise input has been applied for all past time. Thus at time zero:
- All elements of (0,0)X which are variances or covariances involving the
states of the system are zero.
- All elements of (0,0)X which are variances or covariances involving only
states of the shaping filter are at their steady state values for the shaping filter
alone driven by the white noise.
16.322 Stochastic Estimation and Control, Fall 2004
Prof. Vander Velde
Page 3 of 3
,
System states only System and filter states
System and filter states Filter states only
00
(0,0)
()0
T T
ss s
T T
s
vv
X
xx xv
vx vv
X
X
??
=
??
??
??
=??
??
??
=
??
∞
??
where
System states
Shaping filter states
s
x
x
v
??
==
??
????
With this initialization,
,
()
vv
X t will remain constant – which it should do if we
think of ()x t as a member of a stationary process.