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AOE 5244
Optimization Techniques
HW Set 7
unit acceleration
unit speed
unit force
normal load-factor
e.g.
h x
v γ
h t v γ
x t v γ
v t τ D v, n γ
γ t n γ /v.
g
V
W
τ D v, n
D v, n E v n/v ,
E
n
n t n
h γ , v .
E , τ .
Consider the problem (class of problems) of optimal vertical plane motion
for an aircraft. The usual model includes state variables: - altitude, -
range, - speed, and - flight-path angle. Our dynamic model is
˙( ) = sin( )
˙( ) = cos( )
˙( ) = ( ) sin( )
˙ ( ) = ( cos( ))
In this form the variables have been non-dimensionalized based on
is the standard sea-level value
is the speed for minimum-drag in level flight at standard
sea-level
is the (constant) weight.
The symbol is the ratio of thrust to weight and the function ( ) is the
ratio of drag to weight. For a simple parabolic drag polar this is given by
( ) = (2 ) ( + ( ) )
where is an aerodynamic performance parameter - the maximum lift-to-drag
ratio.
The control for the problem is the . For simplicity we
consider only a simple bound ( ) . Consider the (not completely
defined) problem of maximizing the altitude beginning from = = 0 = 1 2
in a fixed time with = 10 and = 0 7. Discuss the effects of different final
boundary conditions ( final speed free or specified, final path-angle free or
specified, ??).
Your complete analysis should include a direct transcription solution and a
solution based on the Minimum Principle. Present (graphical) comparisons of
the solutions and include a discussion about implementation issues.
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