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Chapter 7 / Missile Trajectory Optimisation
_ _
7.7 Gradients
The Euler-Lagrange equations are propagated backwards to obtain the
adjoint variables,
⎛
T
⎛ ∂ f ⎞ ⎞
k ∈ −
λ
⎜ ⎜ X ⎟ ⎟
⎝ ⎝ ∂ ⎠ k ⎠
X
[ N −1
( 1 ) n + 1 ] ⇒ λk
1 : = ⎜ I + ∆t
⋅ ⎜ ⎟ ⎟ ⋅ k
The elements of the reduced gradient,
g
≡
⎛ ∂ H ⎞
⎜
U
⎟
⎝ ∂ ⎠
T
: =
⎛
⎜ ∂ φ ∂ φ ⎛ ∂ f X ⎞
−
⎜ U X
⎜
X
⎟
∂ ∂
⎝
⎝ ∂ ⎠
7-15
−1
∂ f X ⋅
∂ U
⎛ ∂ H ⎞
⎛ ∂ f ⎞
k ∈
⎜
λ
U ⎟
⎜
k
U ⎟
⎝ ∂ ⎠
⎝ ∂ k ⎠
Equation 7.6-7
⎞
⎟
⎟
⎠
T
Equation 7.7-1
k
[ n ( 1 ) N −1
] ⇒ ⎜ ⎟ : = ∆t
⋅ ⎜ ⎟ ⋅ k
∂ H
∂ t
N
⎛ ∂ H ⎞
⎜
X ⎟
⎝ ∂ 0 ⎠
: =
∂ φ
∂ t
N
T
+
: =
1
N
⋅
T
⎛
⎜ ⎛ ∂ f 0 ⎞
I + ∆t
⋅
⎜ ⎜
X ⎟
0
⎝ ⎝ ∂ ⎠
∑ − N 1
k : = n
⎛
⎜
⎜
⎝
f
t
k
+ t
n
T
⎛ ∂ f
⋅ ⎜
⎝ ∂ t
⎞
⎟
⎟
⎠
k
k
⋅ λ
⎞
⎟
⎠
0
T
⎞
⎟
⎟
⎠
T
Equation 7.7-2
Equation 7.7-3
⋅ λ
k
Equation 7.7-4
Equation 7.7-3 is applicable when all the controls are available prior to
launch and the launcher angles are optimised. If the gradient at a particular
step results in a control exceeding the limit, the gradient for that control is
set to zero, the control to its limit, and optimisation proceeds over the
unbounded set of controls. The gradients with respect to optimiser states,
controls and terminal time are provided by GRADIENT. This invokes
OP_MS_DXDU for the variation in state derivatives with respect to the
controls, and OP_MS_DXDX for their variation with respect to the states.
The gradients are scaled in GRADIENT_SC, and the direction of steepest
descent in STEEPEST_DD.