16.322 Stochastic Estimation and Control, Fall 2004
Prof. Vander Velde
Lecture 23
Last time:
d
Φ (t ,τ ) = A(t )Φ (t ,τ )
dt
Φ (τ ,τ ) = I
So the covariance matrix for the state at time t is
X (t ) = ⎡⎣ x (t ) − x (t ) ⎤⎦ ⎡⎣ x (t ) − x (t ) ⎤⎦
T
= x% (t ) x% (t )T
t
t
⎡
⎤⎡
⎤
T
T
= E ⎢Φ ( t , t0 ) x% (t0 ) + ∫ Φ ( t ,τ 1 ) B(τ 1 ) n% (τ 1 )dτ 1 ⎥ ⎢ x% (t )T Φ ( t , t0 ) + ∫ n% (τ 2 )T B(τ 2 )T Φ ( t ,τ 2 ) dτ 2 ⎥
⎢⎣
⎥⎦ ⎢⎣
⎥⎦
t0
t0
= Φ ( t , t0 ) x% (t ) x% (t )T Φ ( t , t0 )
T
t
+ ∫ Φ ( t , t0 ) x% (t0 ) n% (τ 2 )T B(τ 2 )T Φ ( t ,τ 2 ) dτ 2
T
t0
t
+ ∫ Φ ( t ,τ 1 ) B(τ 1 ) n% (τ 1 ) x% (t0 )T Φ ( t , t0 ) dτ 1
T
t0
t
t
t0
t0
+ ∫ dτ 1 ∫ dτ 2Φ ( t ,τ 1 ) B(τ 1 ) n% (τ 1 ) n% (τ 2 )T B(τ 2 )T Φ (t ,τ 2 )T
The two middle terms are zero:
- For τ > t0 , n% (τ ) and x% (t0 ) are uncorrelated because n% (τ ) is white (impulse
correlation function)
- For τ = t0 , n% (τ ) has a finite effect on x% (t0 ) because n% (τ ) is white. But the
integral of a finite quantity over one point is zero.
t
t
t0
t0
X (t ) = Φ ( t , t0 ) X (t0 )Φ ( t , t0 ) + ∫ dτ 1 ∫ dτ 2Φ ( t ,τ 1 ) B(τ 1 ) N (τ 1 )δ (τ 2 − τ 1 ) B(τ 2 )T Φ (t ,τ 2 )T
T
t
= Φ ( t , t0 ) X (t0 )Φ ( t , t0 ) + ∫ Φ ( t ,τ ) B (τ ) N (τ ) B(τ )T Φ (t ,τ )T dτ
T
t0
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.
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16.322 Stochastic Estimation and Control, Fall 2004
Prof. Vander Velde
d
T
X (t ) = A(t )Φ ( t , t0 ) X (t0 )Φ ( t , t0 )
dt
+Φ ( t , t0 ) X (t0 )Φ ( t , t0 ) A(t )T
T
t
+ ∫ A(t )Φ ( t ,τ ) B(τ ) N (τ ) B(τ )T Φ ( t ,τ ) dτ
T
t0
t
+ ∫ Φ ( t ,τ ) B(τ ) N (τ ) B(τ )T Φ ( t ,τ ) A(t )T dτ
T
t0
+ B ( t ) N ( t ) B ( t )T
d
X (t ) = A(t ) X (t ) + X (t ) A(t )T + B(t ) N (t ) B(t )T
dt
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 t = 0 .
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 X (0,0) which are variances or covariances involving the
states of the system are zero.
- All elements of X (0,0) 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.
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16.322 Stochastic Estimation and Control, Fall 2004
Prof. Vander Velde
System and filter states⎤
⎡ System states only
X =⎢
Filter states only ⎥⎦
⎣System and filter states
⎡ x x T xs v T ⎤
=⎢ s s
⎥
⎢⎣ vxsT
vv T ⎥⎦
0 ⎤
⎡0
X (0, 0) = ⎢
⎥
⎣ 0 X v ,v ( ∞ ) ⎦
where
⎡ x ⎤ ⎡ System states ⎤
x = ⎢ s⎥ = ⎢
⎥
⎣ v ⎦ ⎣Shaping filter states⎦
With this initialization, X v ,v (t ) will remain constant – which it should do if we
think of x (t ) as a member of a stationary process.
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