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Automatica 39 (2003) 607 – 618
www.elsevier.com/locate/automatica
Missile guidance laws based on pursuit–evasion game formulations
Vladimir Turetsky
1
, Josef Shinar
∗
Faculty of Aerospace Engineering, Technion, Israel Institute of Technology, Haifa 32000, Israel
Received 15 November 2001; received in revised form 23 June 2002; accepted 10 November 2002
Abstract
The increasing maneuverability potential of tactical ballistic missiles motivates the development of improved interceptor guidance
laws. In this paper two guidance laws are compared, both based on a pursuit–evasion dierential game formulation. The ÿrst game is a
linear-quadratic one. The second one is formulated with bounded controls and the miss distance as the performance index. The comparison
shows that for the same values of physical parameters the capture zone (guaranteeing zero miss distance) of the guidance law using
bounded controls, as well as the control eort associated with it, are larger. However, for short duration end games and for large values
of interceptor maneuverability advantage the capture zones nearly coincide.
? 2003 Elsevier Science Ltd. All rights reserved.
Keywords: Guidance systems; Control laws; Dierential games
1. Introduction
Missile guidance problems in general and interception of
airborn targets in particular, have been an important part of
aerospace control activities. They have received an increased
attention in the last decade in relationship with anti-ballistic
missile defense. The Gulf War introduced a new type of
target, namely the tactical ballistic missile (TBM), able to
carry non-conventional warheads. Successful interception
of a TBM, much less vulnerable than an aircraft, requires a
very small miss distance or even a direct hit (to hit a “bul-
let” with a “bullet”), creating a new challenge for the guided
missile community. Although a good homing accuracy was
recently demonstrated against non-maneuvering targets, in
the future highly maneuverable tactical ballistic missiles
are anticipated. Recent simulation studies of anti-ballistic
missile defense scenarios indicated that currently used
guidance laws are unable to guarantee an adequate hom-
ing accuracy for a kinetic “hit-to-kill” in the interception
of the expected highly maneuvering target. The need for
This paper was presented at the 15th IFAC Symposium on Automatic
Control in Aerospace, Bologna/Forli, Italy, September 2–7. This paper
was recommended for publication in revised form by Associate Editor
Kenko Uchida under the direction of Editor Tamer Basar.
∗
Corresponding author.
E-mail address: aer4301@aerodyne.technion.ac.il (J. Shinar).
1
Partly supported by the Center of Absorption in Science, Ministry of
Immigrant Absorption, State of Israel.
developing improved guidance laws for such a task has been
recently acknowledged.
Interceptor missiles of the early generations were guided
by Proportional Navigation. Later, “modern” guidance laws
were developed based on a linearized kinematical model
and a linear quadratic optimal control formulation (Zarchan,
1990; Cottrel, 1971). Analysis showed that Proportional
Navigation by itself is the optimal guidance law for a missile
of ideal dynamics against a non-maneuvering target using a
linear quadratic control formulation (Kreindler, 1973).
There is a basic deÿciency in formulating the interception
of a maneuverable target as an optimal control problem.
Target maneuvers are independently controlled. Thus, they
cannot be predicted. Since future target maneuver time his-
tory, or strategy, cannot be known, the optimal control for-
mulation is not appropriate. The mathematical framework
for analyzing conicts controlled by two independent agents
is in the realm of dynamic games. Thus, the scenario of in-
tercepting a maneuverable target has to be formulated as a
zero-sum pursuit–evasion game. The roles of the players are
clearly deÿned, the interceptor is the pursuer and the target
is the evader. The natural cost function of such zero-sum
game is the miss distance (the distance of the closest ap-
proach, or in other words the smallest norm of the separa-
tion vector), to be minimized by the pursuer and maximized
by the evader. The game solution provides simultaneously
the missiles guidance law (the optimal pursuer strategy),
the “worst” target maneuver (the optimal evader strategy)
0005-1098/03/$ - see front matter ? 2003 Elsevier Science Ltd. All rights reserved.
doi:10.1016/S0005-1098(02)00273-X