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Chapter 7 / Missile Trajectory Optimisation
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7.8 Search Direction
Rapid initial convergence from the PN trajectory and in response to
changing boundary conditions is required. Slow convergence to an optimal
trajectory thereafter is less import compared with the ability to respond
rapidly to target manoeuvres. The selection of the search direction is a
crucial factor in the stability and efficiency of optimisation techniques for
minimising the Hamiltonian function. A review of the development of
optimisation algorithms is provided by Bazaraa [B.4] , including an extensive
bibliography.
Techniques for selecting the search direction using derivative information
are better suited to the current application, although direct searches are more
robust in an off-line capacity. Gradient-based techniques update the system
states and controls by taking a step of length (α) in the direction (d) defined
by the inverse Hessian [B] -1 and gradient vector (g),
X
k + 1
X k 1 : = X k − d
+
: =
7-16
X
k
− α ⋅
−1
[ B ] ⋅ g
Equation 7.8-1
Equation 7.8-2
Techniques requiring Hessians such as the class of Variable Metric Methods
converge more rapidly than gradient based alternatives. However, they
involve numerical differencing of non-linear aerodynamic functions, and
possible ill-conditioning if penalty or barrier functions are used. Early
optimisation algorithms were prone to premature termination, divergence
and cycling. Wolfe introduced three rules associated with the length
travelled along the search direction to prevent these effects, conditions that
ensure that the magnitude of the gradient (g) reduces for twice differentiable
convex functions.
WOLFE CONDITION 1
T
−1
[ B ] ⋅ g > - ε ⋅ g ⋅ d
g • d = - g ⋅
1
Equation 7.8-3
This condition ensures that the direction chosen (d) is close to the initial
gradient (g) by selecting a small value, typically (ε1 := 0.001).
WOLFE CONDITION 2
g • d ≤ ε ⋅ g • d
k + 1
k
2
k
k
Equation 7.8-4