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Chapter 7 / Missile Trajectory Optimisation
_ _
7.2 Dynamic Model
The position of the target at the impact time (tF) is propagated from its
current position provided by the missile state observer,
( t )
⎛ A
A
2
P ⎞
t F 03x3
M t F I ⎛
3 P ⎞ ⎛ t 0.
5 t F I ⎞
⎜ ⎟ ⎛ ⋅ ⎞
⋅ ⋅ 3
⎜
⎟ ⎜ ⎟ ⎜
⎟
⎜ L ⎟ : = ⎜ L M L ⎟ ⋅ ⎜ L ⎟ + ⎜ L ⎟ ⋅ P&
&
A
A
⎜
P&
t ( t F ) ⎟ ⎜ 03x3
M 0 ⎟ ⎜
3x3
P&
⎟ ⎜
⎟
⎝
⎠ t ⎝
t F ⋅ I
⎝ ⎠
⎝ ⎠
3 ⎠
7-4
A
t
Equation 7.2-1
For real-time optimisation the size of the optimiser state vector must be as
small as possible. Here missile position, speed and direction, and the
launcher orientation with respect to the Alignment frame are used. In
literature mass-flow is often included however, when using solid propellant
boosters mass can be treated as a quasi-static variable providing the ambient
pressure is accounted for in the thrust equation.
( ) T
XA YA ZA
MV MV
P , P , P , P , Θ ,
X : = &
Ψ
m
m
m
o, m
A
A
Equation 7.2-2
These states are propagated from the initial conditions provided by the
missile state observer according to the difference equations (fX),
k+
1
k
k
( X , U , t k ) ⋅ ∆t
k ≡ X + f ⋅ t k
X : = X + X&
∆
k
k
k
X
Equation 7.2-3
Euler integration is the obvious choice, balancing speed and the number of
control steps. Runge-Kutta algorithms, although more accurate, impose a
greater computation load and require a single-step start up algorithm
following a discontinuity in the derivatives. The 1 st order Newton-Cotes
numerical integration formula (trapezoidal rule), is an alternative to Euler
integration with 2 nd order convergence and none of the problems associated
with higher-order algorithms.
7.2.1 Dynamic Model Propagation Equations
The following parameters, and their derivatives, with respect to the
optimiser states are defined in §6 and §20. The time, optimiser states and
control dependencies are listed,
• Missile mass mm ( t )
• Missile thrust force
ZA
F T ( Pm
)
• Missile drag force F
ZA ( P , P&
MV MV
, Θ , Ψ )
D
m
o,
m
B
B