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

Chapter 7 / Missile Trajectory Optimisation
_ _
7.5 Dynamic Constraints
The cost function is minimised subject to the dynamic equality constraints,
7-13
( f ) ⋅ ∆t
: = 0
X X +
k
k − k + 1
X
k
Equation 7.5-1
The high incidence cost is already accounted for in the control effort. An
active set strategy is used in conjunction with control hard limiting for the
gradients. The remaining constraints are of the form,
h ( X ) ≤ c
Equation 7.5-2
This can be converted to an equality constraint using a slack variable,
s
>
0
⇒
h
2 ( X ) + s − c : = 0
Equation 7.5-3
Of the many constraints that may be applied two stand out in importance for
air-defence:
7.5.1 Up-link communications
To maintain up-link communication the LOS to the missile must remain
within the missile receiver beam width. Assuming that the missile is
travelling away from the launcher and the limit for communication purposes
is (ξC),
1
−1
A 147 MV 2
( cos ( nˆ ( P ) • ϕ ) ) ≤ C
h : =
ξ
m
A
Equation 7.5-4
The cost is squared to heavily penalise excursions beyond the threshold.
7.5.2 Avoidance regions
If the region to be avoided is centred at point (j) and extends spherically a
distance RJ,
−
XA 2 YA 2 ZA 2
2 A
A A
( P ) − ( P ) − ( P ) ≤ − R ; P : = P − P
m,
j
m,
j
m,
j
h
2
: =
J
m,
j
o,
j
o,
m
Equation 7.5-5