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Chapter 6 / Missile Guidance
_ _
offset with respect to an arbitrary frame. The feed forward acceleration
comprises weave tuned α−β−γ filter derivatives F2(s) and F3(s).
6.6.2 Guidance Loop Stability
In Figure 6-4 the stability filter F(s), and its companion noise filter N(s),
compensate for the destabilising effect of the double integrator I(s). The
autopilot A(s), with a damping ratio of 0.5 and a bandwidth (ωA) of 6 Hz, is
assumed to be insensitive to dynamic pressure, speed and incidence
variations. For steady-state analysis the delay D(s) was modelled using a
2 nd order Tustin approximation equivalent to an update rate of 100 Hz. The
missile dynamics M(s) are defined by the incidence lag (TI).
The design criteria for the stability filter was a bandwidth (ωG) of 2 Hz. The
filter was designed using pole placement ignoring the missile dynamics,
using pure double integration, and a 1 st order Padé delay approximation,
s 2
⋅ G
( s ) : = F(
s ) ⋅ N ( s ) ⋅ A ( s ) ⋅ D ( s )
6-21
Equation 6.6-4
The 5th order stability filter was reduced to 3rd order by gain and phase
approximation and split into a phase advance element F(s), and a 1 st order
lag N(s) applied to both the feed back and feed forward demands.
⎛ 13.
6 + 5.
2605 ⋅ s + 0.
08770 ⋅ s
⎜
⎝ 1 + 0.
0427 ⋅ s + 0.
00144 ⋅ s
⎛ 63.
4
⎜
⎝ 63.
4 + s
( ) ( ) ⎟ s ⋅ N s : = 13.
6 ⋅ ⎜
⎟ ⋅ ⎜
F 2
2
⎞
⎟
⎠
⎞
⎠
Equation 6.6-5
The stability loop used for pole placement provides static margins of 8 dB
and 45° shown in Figure 6-5, margins similar to those using the full stability
loop at the steady-state condition:
• missile velocity ( V_OM ) := 550 m/s
• missile acceleration ( A_OM ) := -115 m/s 2
• impact range ( P_OM ) := 7800 m
• missile incidence lag ( TI ) := 0.25 s
Compare the pole placement results with the double phase advance filter
proposed by Fox [F.1] ,
F
( s )
: =
⎛ ω
23 ⋅
⎜
⎝ ω
A
A
+ 5.
9276 ⋅ s ⎞
1.
8745 s ⎟
+ ⋅ ⎠
2
Equation 6.6-6