Proportional
Navigation (PN) is a guidance law for homing missiles that has been used
successfully for the last 50 years. According to this law, the maneuver of the
missile is proportional to the line of sight rate. It is based on the fact that
if the target and the missile have constant velocities then on a collision
course the line of sight rate is zero .
We revisit the basic block diagrams in PN, and study the effect of initial
conditions on miss-distance. Using the important case of multiple targets, we
compare two PN block diagrams. The first, the ideal PN, applies the true line
of sight rate, which is calculated by an analytic derivative of the line of
sight angle. The block diagram resulting from this guidance loop is minimal.
The second applies an approximated derivative of a LOS angle. The block diagram
resulting from this guidance loop is unobservable with respect to target
displacement due to zero-pole cancellation at the origin. The consequence is
that for the target displacement initial condition the two diagrams do not
result in the same miss-distance. A missile-target engagement with a step in
target displacement is a basic initial condition scenario in the neighborhood
of a collision course. In particular it serves as a model for the multiple
targets problem. In multiple targets scenario a missile is guided by PN on the
power centroid of two targets flying in close formation. At the moment where
the missile seeker can separate the targets, it appears to the missile that the
target has instantaneously shifted from the power centroid to the location of
the resolved target. Since the separation occurs at a late moment of the
engagement, this shift may induce a large miss-distance. In this research we
show that target displacement and heading error are interchangeable. By
choosing a certain reference line, one may interpret a multiple target scenario
as a target displacement. By choosing a different reference line one may
interpret it as a heading error. Nevertheless, since miss distance is invariant
with respect to the choice of reference line, it follows that heading error and
Target displacement must result in the same miss distance. Moreover, we suggest
a few solutions that can resolve the above difficulty; One of the solutions is
to reset the derivative network with the correct initial conditions. By doing
so target displacement and heading error yield the same miss distance again.
