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Chapter 7 / Missile Trajectory Optimisation
_ _
7.6 TPBVP Formulation
Minimising the cost function by adjusting the control sequence is replaced
by the simpler problem of minimising a scalar Hamiltonian that depends on
boundary conditions only. Thus the constrained TPBVP is reduced to the
unconstrained minimisation of a Hamiltonian function. Pontryagin theory is
used to reformulate the open-loop solution into a classical control law by
relating the controls to state observer parameters, in this case using Equation
7.3-4. The Hamiltonian is formed by combining the cost function, dynamic
constraints using Lagrangian multipliers (λ), and inequality constraints
using co-state variables (µ),
H
: =
φ + λ
T
⋅ f
7-14
X
+ µ
T
⋅
( h + s ⊗ s − c )
Equation 7.6-1
The Kuhn-Tucker 1 st order conditions to be satisfied on an optimum
trajectory,
∂
∂ X
∂
∂ U
H T
T
: =
: =
∂ φ
+
∂ X
λ
∂ f X ⋅
∂ X
H T
T
∂
∂ s
∂ φ
+
∂ U
∂ H
∂ λ
λ
: =
∂ f X ⋅
∂ U
f
X
H T
: =
0
⇒
µ
+
+
: =
µ
µ
0
∂ h
⋅
∂ X
∂ h
⋅
∂ U
: =
: =
( c − h ) : = 0
0
Equation 7.6-2
0
Equation 7.6-3
Equation 7.6-4
Equation 7.6-5
The optimiser states are propagated using Euler integration over the
remaining controls [n(1)N-1], control limits applied, and the Hamiltonian
Jacobians determined for each control. For proof of principle the discrete
equations that follow involve only the dynamic equality constraints.
Modifying the discrete equations in Vorley [V.4] to deal with the remaining
controls, the transversality conditions at impact,
λ N − 1
( ) T
∂ φ X
: = ∂
N
Equation 7.6-6