Thesis - Leigh Moody.pdf - Bad Request - Cranfield University

Chapter 5 / Missile State Observer
_ _
Propagating the state covariance matrix over the time interval (∆t),
−
k+
1
k+
1
k
4-11
−
k
k+
1 T
T
[ Φ ] + Γ ⋅ Q ⋅
C : = Φ ⋅ C ⋅
Γ
k
k
k
k
Equation 4.4-8
The symmetric, semi-definite, covariance matrix (Q) is computed between
measurement updates from,
t
∫ + k 1
t
k
Γ
⋅ Q
⋅ Γ
( τ ) ⋅ diag q ( τ )
k
: =
( ) ⋅ G ( τ )
T k+
1 T
[ ] ⋅ [ ] ⎟
⎞ dτ
⎜
⎛ k+
1
Φτ
⋅ G
Φ
⎝
k
Equation 4.4-9
If ∆t is small, the transition matrix collapses to the identity matrix and,
Q
k
: =
G
k
⋅ diag
T
k
T
( q ( k ) ) ⋅ G ⋅ ∆t
k
τ
⎠
Equation 4.4-10
Assuming that the sensor measurements ( Z ~ ) are non-linear functions of the
reference state vector at time (tk+1) plus zero mean Gaussian noise,
E
Z ~
: = h X
( ) + υ
T
( υ ) : = 0 ⇒ E ( υ ⋅ υ ) : = R
Equation 4.4-11
Equation 4.4-12
[R] is the symmetric, positive definite measurement covariance matrix. The
expected measurement is computed using the state approximation
propagated to the measurement reference time (tk+1),
Z ˆ
: =
h
⎜
⎛
⎝
−
Xk+1 ˆ
⎟
⎞
⎠
Equation 4.4-13
Expanding the measurement functions in a series about the estimated state,
h
( ) ⎛ − ⎞
−1
T
= X + H ⋅ ∆X
+ 2 ⋅ ∆X
⋅ G ⋅ ∆X
+ L
ˆ
X : h
k+
1
⎜
⎝
k+
1
⎟
⎠
Equation 4.4-14