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

Appendix H / Steady State Tracking Filters
_ _
X ˆ
X ˆ&
X ˆ &
⎛
2
2
⎜ ω ⋅
( ) ( 2 ⋅ s + 2 ⋅ ωE
⋅ s + ωE
)
s : =
⎜
2
2
( s + ω ) ⋅ ( s + ω ⋅ s + ω )
⎝
E ⋅
E
⎛
2 3
⎜ s ⋅
( ) ( 2 ⋅ ω ⋅ s + ωE
)
s : =
⎜
2
2
( s + ω ) ⋅ ( s + ω ⋅ s + ω )
( s )
: =
⎝
⎛
⎜
⎜
⎝
E
21-7
s
E
E
E
E
⎞
⎟
⎟
⎠
E ⋅
2
2
( s + ω ) ⋅ ( s + ω ⋅ s + ω )
E
2
⋅ ω
3
E
E
E
X ~
Equation 21.3-18
⎞
⎟
⎟
⎠
X ~
Equation 21.3-19
⎞
⎟
⎟
⎠
⋅
X ~
Equation 21.3-20
When expressing filter gains in terms of the Butterworth frequency (ωE),
the filter poles are identical and constrained to lie on the unit circle with a
damping of 0.5. Its relationship with the KF and equivalent α−β−γ gains,
⎛
⎜
1 − exp
⎝
K
: =
⎛
⎜
⎝
β
∆t
γ
∆t
α , , 2
( α , β , γ )
⎞
⎟
⎠
Equation 21.3-21
β
4 ⋅ α
( ) ( ) ⎟ − 2 ⋅ ω ⋅ ∆t
, 2 ⋅ 2 − α − 4 ⋅ 1 − α ,
E
Equation 21.3-22
The relationship between the tracking index and butterworth frequency
(ωE),
E
: =
21.4 First Order Correlated Noise Filter
ω
3
σS
σ ⋅ ∆t
M
: =
2
⎞
⎠
Equation 21.3-23
Arcasory [A.5] extended their work to include 1 st order Gauss-Markov
dynamics driven by piecewise continuous jerk. Using the same PI as in
§21.2,