Probabilistic and Statistical Methods - Information Fusion in Signal and Image Processing - page 79

of great importance whether for the analysis of observability [TRE 96], or of the con-
vergence of the extended Kalman filter [SON 85]. As a result, the analysis of observ-
ability becomes much easier if we rewrite equations [6.11] and [6.13] as follows:

X k +1 = F X k + U

0 ≡ z k = H k X k

where:

F =Φ t k +1 ,t k = Id

α

t k

αId

t k +1 −

0 Id

[6.30]

X k X t k
H k = cos θ k , sin θ k , 0 , 0 −

≡

If we now assume that the observer is not maneuvering, U 0. We then have:

z 0 = H 0 X 0
z 1 = H 1 F

X 0

[6.31]

.

z k = H k F k

X 0

with:

F k = ccId

kαId

0 Id

so that the observability matrix O can be written:

⎛ ⎞ ⎛ ⎞

cH 0
H 1 F
.
H k F k

cos θ 0 − sin θ 0 0 0

⎝ ⎠ ⎝ ⎠

cos θ 1 − sin θ 1 α cos θ 1 − α sin θ 1

O = = [6.32]

.
cos θ k − sin θ k kα cos θ k − kα sin θ k

Factoring the observability matrix as follows is quite helpful:

⎛
⎝

⎞
⎠

−

r y (0) r x (0) 0 0

r y (1) − r x (1) αr y (1) − αr x (1)

O =Δ θ Δ r

.
r y ( k ) − r x ( k ) kαr y ( k ) − kαr x ( k )