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Chapter 7 / Missile Trajectory Optimisation
_ _
m
m
m
m
⋅ P&
⋅ fΘ
o,
m
: =
MV
⋅ P&
⋅ T
Ψ
o,
m
7.2.2 Dynamic Constraint - State Jacobians
A
F
B
m
7-6
⋅
369
ϕ
MV
B
+ m
m
⋅ g
ZA
m
⋅ T
MV
A
( 9 )
Equation 7.2-10
XB MV
YB MV
( 9 ) ⋅ f : = ∆F
⋅ T ( 7 ) + F ⋅ T ( 9 )
m
B
L
B
Equation 7.2-11
The TPBVP solution requires the variation in the dynamic constraints with
the optimiser states. For on-line applications analytical derivatives are more
efficient than performing functional evaluations and differencing. The
automatic computation of partial differentials using Table methods has been
studied by Wengert [W.1] and Bellman [B.3] . For the missile position vector,
∂ P
∂ X
MV
MV
T
( − P&
⋅ ( ) ( ) ) ⎟
⎞
o,
m TA
4 , P&
o,
m TA
1 , 0
⎠
⎜
⎛ 147 MV
369 MV
03x3
M ϕ M − P&
o,
m ⋅ ϕ M
⋅
⎝
A
A
The missile speed variation with respect to the optimiser states,
∂ f
A
m
∂ X
: =
Equation 7.2-12
T
ZA
ZA MV
( 0 M ∂ f ∂ P M ∂ f ∂ P&
M − g ⋅ T ( 9 ) M 0 )
2
V
m
V
V
o,
m
: =
B
ZA
∂ f 1 ⎛ F ⎞
V 147 MV g MV
: ⎜
∂ m
∂ m
= ⋅ ⎟ ⋅ ϕ + ⋅ T
ZA
ZA
ZA A
P m
B
∂
⎜
m
m P ⎟
⎝ ∂ m ⎠
∂ Pm
V
o,
m
T
B
T 147 MV
( ∂ F ∂ P ) ⋅
∂ f ∂ P&
: = m
ϕ
−1 m ⋅ &
m o,
m
m
A
Equation 7.2-13
( 7 )
Equation 7.2-14
B
Equation 7.2-15
The missile pitch incidence angle variation with respect to optimiser states,
⎛ T ⎜
∂ fΘ
∂ fΘ
∂ fΘ
∂ fΘ ∂ X : = 0
⎜ 2 M M M
ZA
MV
⎝ ∂ P ∂ P&
m
o,
m ∂ ΘA
M
⎞
0 ⎟
⎠
Equation 7.2-16