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An Optimal End-Game Guidance Law with a Jerk Constraint
By Natan Grinfeld and Joseph Z. Ben-Asher
Many tactical and strategic missiles have been using
Proportional Navigation Guidance (PNG)
law for decades. Its simple implementation along with its high
accuracy has made PN the most
popular guidance law in many applications. Some specific
applications, however, may face
problems when trying to implement the PNG law. One such
application is the case of a physical
constraint on the jerk (first derivative of the acceleration)
for the intercept missile. This case may
occur mainly in exo-atmospheric, thrust vector maneuvering
missiles, in which the generation of
the required acceleration maneuver is less immediate then the
case of an endo-atmospheric
aerodynamic maneuvering missiles.
Though being brought into consideration indirectly through the
time constant of the flight
control, the jerk constraint has never been implemented directly
and explicitly when deriving an
optimal guidance law. The need for such a development of an
optimal, jerk constraint, guidance
law arises when introducing such constraint on the jerk on a
common PNG law. It was first
shown, as motivation for the study, that under a tight jerk
constraint the missile misses the target
using PN.
This research objective is to develop an end-game phase optimal
guidance law with jerk limit
considerations and investigate its performance in comparison to
PN and APN (Augmented
Proportional Navigation) laws and in the cases of a physical
constraint on the jerk.
By introducing a fourth state variable of mn (missile's
acceleration) and adding a "price" on its
derivative, thus obtaining the system of1 2 3 4
; ; ;t mx y x y x n x n , and solving the problem
of: 21
0, min2
f
o
t
f mt
y t J n dt
, using Cauchy-Schwartz inequality, an optimal, jerk
minimizing guidance law was developed.
The new guidance command is in the form of:
3
2
:
110; ;2
m
go
go go go f
N ZEMn
t
where
N ZEM y t y t y t t t
The new guidance law, named JLG (Jerk Limited Guidance), was
tested against both PN and
APN.
The results are interesting: JLG turns out to be more efficient
than PN under a jerk limit as
expected, and under certain jerk limitations, the missile hit
the target using JLG while missing
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using PN. Since it is not a fair game comparing JLG to PN
because JLG is using knowledge of
the targets maneuver while PN isnt, it was decided to compare
JLG to APN which also uses
that knowledge.
When comparing those methods against a constant target
acceleration, both methods hit the
target but with a very different behavior. While APN reacts to
the targets maneuver with aninstant acceleration peak in order to
cancel the targets advantage (while doing so, assuming
the target will continue with the same maneuver), JLG has a much
more moderate acceleration
reaction, building its acceleration slowly, to arrive at a
maximum level of acceleration only after
half of the end game time.
This encouraged the test of both methods against a bang-bang
maneuvering target. It turns out
that against a bang-bang maneuvering target, under a jerk limit,
JLG is much more efficient, at
some cases (depending on the acceleration switch time) hitting
the target while APN misses, thus
due to the moderate reaction which is much more suited to
changes in the maneuver than the
straight forward approach of APN.
Next to be studied in the research, is the guidance law
performance and sensitivity to changes
through an adjoint system representation.
