Games based Guidance in Anti Missile Defence
for High Order Participants
Ilan Rusnak*
*RAFAEL, POBox, 2250, Haifa, Israel.
Abstract—A three person game, the Target, the Missile and
the Defender (TMD) for high order players is formulated and
solved. The Missile that is attacking the Target has an objective
to minimize its miss distance to the Target, while the Target is
trying to maximize the miss distance from the Missile. The
Defender is trying to intercept the Missile prior to its arrival to
the Target. That is, the objective of the Defender is minimizing
its distance to the Missile while the objective of the Missile is to
maximize its distance from the Defender and at the same time
not to jeopardize its effort to intercept the Target.
An approach to the solution is presented. The approach to the
solution is based on the Multiple Objective Optimization and
Differential Games theories. A solution is derived for linear
system and quadratic criterion.
A closed loop saddle point solution in linear strategies is
derived for the three high order players. The solution is not
continuous. Sufficient conditions for existence of the solution are
presented.
An example demonstrates the result.
I. INTRODUCTION
The Target and the Missile is the classical zero-sum two
persons differential game [1,2]. The Target is the evader and
the Missile is the pursuer.
In this paper we consider the following three persons game:
the Target, the Missile and the Defender (TMD) game:
The Missile is pursuing the Target. Therefore, the Missile's
objective is to minimize its miss distance to the Target, while
the Target is trying to maximize the miss distance from the
Missile. The objective of the third player, the Defender, is to
intercept the Missile prior to its arrival in close vicinity of the
Target. That is, the objective of the Defender is minimizing its
distance to the Missile while the objective of the Missile is
maximizing its distance from the Defender and at the same
time not to jeopardize its effort to intercept the Target. During
this phase of the game the Missile is the evader and the
Defender is the pursuer. This is depicted on Figure 1.The
Defender has only one chance to intercept the Missile.
It is assumed that the success of the Defender is not
guaranteed. Therefore the game does not terminate at the
expected interception time of the Missile by the Defender, but
continues until the Missile is expected to intercept the Target.
The Defender ceases to exist after the Missile interception.
We assume a full information pattern.
"Three-body" problems are considered in [3,4,5,6]. Ref.
[4,6] consider solution under the assumption of collision
course for the vehicles.
An N-person differential game of fixed prescribed duration
of linear-quadratic type is considered in [7]. A solution and
978-1-4244-5794-6/10/$26.00 ©2010 IEEE
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conditions on the existence in the form of open loop and
closed loop Nash equilibrium in linear strategies are presented.
at tf2
Target
Missile
Defender
at tf1
Figure 1: Schematic description of the TMD scenario.
The objectives of the players of the TMD game are
formulated at different times. The author did not find a
formulation of N-person differential games where the
objectives are formulated at different time instants and on
different time intervals except of his previous publications
[14-19]. Moreover, he did not find differential games problem
where during the game the objective of the players change
their character from maximizer to minimizer.
In the paper the Target, the Missile, and the Defender game
is defined. In addition, an approach to the solution is presented
and a solution is derived for a linear system and a quadratic
criterion. The approach to the solution is based on the
Multiple Objective Optimization and Differential Games
theories. Part of the definition of the game and the solution
approach is repeated here for the completeness of the
presentation.
The main novelty of this paper is that a closed form
solution of the TMD game is derived here for arbitrary high
order transfer functions of the players. Optimal guidance laws
for a high order missile autopilot and a high order
representation of the target manoeuvre based on one sided
optimization are presented in [8-11]. Guidance laws for a two
person pursuer-evader game for linear arbitrary order
autopilots of the players are presented in [12].
The approach makes use of the impulse function in the
objective. This leads to a solution in the form of Riccati
equation that includes the impulse function in the indices. The
solution of the corresponding Riccati equation is not
continuous. Sufficient conditions for existence of the solution
are presented.
An example is presented.
812
II. STATEMENT OF THE TMD GAME PROBLEM
This section presents a formal definition of the game within
the domain of the linear systems and quadratic performance
indices. Let us introduce the following notations:
yM – the position of the Missile,
yT – the position of the Target,
yD – the position of the Defender.
There are two terminal objectives at different time instants
as follows from the previous section:
I)
at the expected interception instant of the Target by
the Missile, tf2,
(1)
2
[
GTM is the matrix weight on the target-missile state at tf2, GMD
is the matrix weight on the missile-defender state at tf1, and
RM, RT, RD are the weights on the energy expenditure of the
participants.
The problem being considered here is the optimization of J,
that is
(7)
max min max J
a Dc
a TC
(8)
]
II)
at the expected interception instant of the Missile by
the Defender, tf1,
[
]
min max y M (t f 1 ) − y D (t f 1 )
a Dc
a Mc
[
2
=
]
min max − y M (t f 1 ) − y D (t f 1 )
a Mc
a Dc
(2)
2
Further, each participant wishes, simultaneously, to
minimize its energy expenditure formulated as
III)
tf2
T
min ∫ a Mc RM a Mc dt
a Mc
tf2
IV) min a T R a dt
∫ Tc T Tc
aTc
V)
(4)
to
tf1
min ∫ a Dc RD a Dc dt
T
a Dc
(3)
to
(5)
y(t ) = Nx(t )
where
⎧ D, t ≤ t f 1 ,
(9)
D (t ) = ⎨
⎩ 0, t f 1 < t
Equation (9) reflects the assumption that the Defender
ceases to exist for t>tf1.
The objective (6) is rewritten by the use of the impulse
function as
(10)
⎡ x T (t f 2 )GTM x (t f 2 )
⎤
⎢
⎥
J = 12 ⎢ ∞ ⎧⎪− x T (t )G MDδ (t − t f 1 ) x (t )
⎫⎪ ⎥
⎢+ ∫ ⎨ T
⎬ dt ⎥
T
T
⎢⎣ to ⎪⎩+ a Mc RM a Mc − aTc RT aTc − a Dc RD a Dc ⎪⎭ ⎥⎦
to
where
aTc – the Target's control (evader)
aDc – the Defender's control (pursuer)
aMc – the Missile's control (pursuer-evader)
to – the moment the game starts
tf1 – the expected interception moment of the Missile by the
Defender
tf2 – the expected interception moment of the Target by the
Missile.
Notice that the energy expenditure of the game participants
is defined on different time intervals.
III. REFORMULATION OF THE TMD GAME PROBLEM
In order to solve the TMD game problem, following the
Multiple Objective Optimization, we scalarize the five
objectives that are defined in section II. As the criteria are not
convex, once a solution is derived the optimality of the
solution must be verified. This leads to conditions required for
the existence and the optimality of the solution.
Thus, the objective of the game is
(6)
a Tc
x (t ) = Ax(t ) + Ba Mc (t ) + CaTc (t ) + D(t )a Dc (t ), x o (t o ) = x o ,
min max y M (t f 2 ) − yT (t f 2 )
a Mc
a Mc
subject to the differential equation
where
GTM = N T gTM N , G MD = N T g MD N .
(11)
and gTM is the weight on the target-missile miss at tf2, and gMD
is the weight on the missile-defender miss at tf1.
The problem can be formulated in the discrete domain as in
[19] then all indices are finite and no generalized functions are
needed. The continuous solution presented in the following is
then derived by limiting the time interval by procedures
presented for example in [20], thus justifying the use of a
generalized functions.
IV. CANDIDATE SOLUTION OF THE TMD GAME
To arrive at a candidate solution we proceed by
constructing the Hamiltonian [2,13]
(12)
H=
⎫
⎧ x T (t f 2 )GTM x(t f 2 ) − x T (t f 1 )G MD x(t f 1 )
⎪⎪
⎪
⎪
tf2
t f1
J = 12 ⎨ t f 2
⎬
T
T
T
⎪+ ∫ a Mc RM a Mc dt − ∫ aTc RT aTc dt − ∫ a Dc RD a Dc dt ⎪
⎪⎭
⎪⎩ to
to
to
where x(t) is the augmented state of the game participants,
813
1
2
[− x
T
T
(t )G MD δ (t − t f 1 ) x (t ) + a TMc R M a Mc − aTc
RT aTc − a TDc RD a Dc
]
+ λ [Ax (t ) + Ba Mc ( t ) + CaTc (t ) + D(t )a Dc ( t )]
T
where λ(t) is the costate, and the terminal condition is
∂ 1 T
T
. (13)
λT (t f 2 ) =
2 x ( t f 2 )GTM x ( t f 2 ) = x ( t f 2 )GTM
∂x (t f )
Then we arrive at
H u = a TMc Ru + λT B = 0
T
H v = −aTc
Rv + λT C = 0
H w = −a Rw + λ D(t ) = 0
T
Dc
T
(14)
and we get
a Mc = − R B λ
−1
M
T
aTc = RT−1C T λ
.
(15)
a Dc = RD−1 D (t )T λ
Notice that the inclusion of the a Dc term under the integral
in (10) is justified by the last result (15) that shows that
aDc(t)=0 for tf1<t.
The Two Point Boundary Value Problem is
(16)
[
reference direction; σTM - the Target-to-Missile line-of-sight
angle with respect to the reference direction; σMD - the
Missile-to-Defender line-of-sight angle with respect to the
reference direction; VTM - the closing velocity between the
Target and the Missile; and VMD - the closing velocity
between the Missile and the Defender.
Close to collision course
y - direction
yT - Target
]
x (t ) = Ax (t ) − BRM−1 B T − CRT−1C T − D(t ) RD−1 D(t )T λ (t )
λ (t ) = G δ (t − t ) x (t ) − AT λ (t )
f1
MD
σTM
x (t o ) = x o , λ (t t f 2 ) = GTM x (t t f 2 ),
yM - Missile
As the uniqueness of the feedback Nash (saddle point)
equilibrium for linear systems and quadratic indices has been
verified only when the solution belongs to the set of linear
strategies [7], we seek a solution within this set. That is, we
assume the existence of a matrix, P(t), such that
λ (t ) = P (t ) x (t ).
The formal solution is given by the time varying Riccati
equation with indices that include the impulse function.
(17)
− P = PA + AT P − G MD δ ( t − t f 1 )
[
]
− P BR M−1 B T − CRT−1C T − D ( t ) R D−1 D ( t )T P ,
P ( t f 2 ) = GTM .
Therefore, the optimal linear strategies in closed loop are
a ∗M = − RM−1 B T P (t ) x (t )
aT∗ = RT−1C T P (t ) x (t )
∗
D
−1
D
.
(18)
reference direction
Figure 2: Example of a TMD scenario close to the collision course.
We have the state space representation of the equations of
motion
⎡ξ TM ⎤ ⎡0 1 0 0⎤ ⎡ξ TM ⎤ ⎡ 0 ⎤
⎡0⎤
⎡0⎤
⎢ξ ⎥ ⎢0 0 0 0⎥ ⎢ξ ⎥ ⎢− 1⎥
⎢1 ⎥
⎢ ⎥
d ⎢ TM ⎥ ⎢
⎥ ⎢ TM ⎥ + ⎢ ⎥ a + ⎢ ⎥ a + ⎢ 0 ⎥ a
=
M
T
⎢0⎥ D
⎢0⎥
dt ⎢ξ MD ⎥ ⎢0 0 0 1⎥ ⎢ξ MD ⎥ ⎢ 0 ⎥
⎢ ⎥ ⎢
⎢ ⎥
⎥⎢ ⎥ ⎢ ⎥
⎢ ⎥
⎣0⎦
⎣ξ MD ⎦ ⎣0 0 0 0⎦ ⎣ξ MD ⎦ ⎣ b4 ⎦
⎣d 4 ⎦
⎧1, t ≤ t f 1
⎧− 1, t ≤ t f 1
(21)
b4 = ⎨
; d4 = ⎨
t
t
0
,
<
f1
⎩
⎩ 0, t f 1 < t
a = R D(t ) P ( t ) x (t )
GTM
V. EXAMPLE OF A TMD GAME FOR HIGH ORDER PLAYERS
This example is the differential games formulation of the
problem introduced in [4, 5, 6]. The example is solved
explicitly. This will give the guidance laws for the target, the
missile and the defender. We denote
RT = rT , RM = rM , RD = rD
(19)
Figure 2 depicts a planar TMD scenario close to the
collision course. The equations of motion perpendicular to a
reference direction are
ξ TM = yT − y M ; ξ1 = aT − a M
ξ MD = y M − y D ; ξ2 = a M − a D
.
(20)
where yT, yM and yD represent the deviations of the Target,
the Missile and the Defender, respectively, from the reference
direction. aT, aM, aD are the accelerations of the Target, the
Missile and the Defender, respectively, perpendicular to the
VMD(tf1-t)
VTM(tf2-t)
T
Sufficient condition for existence of a solution can be found
in [14].
σMD
yD - Defender
⎡ g TM
⎢ 0
=⎢
⎢ 0
⎢
⎣ 0
0 0 0⎤
⎡0
⎥
⎢0
0 0 0⎥
; G MD = ⎢
⎢0
0 0 0⎥
⎥
⎢
0 0 0⎦
⎣0
0
0
0
0
0
g MD
0
0
0⎤
0⎥⎥
0⎥
⎥
0⎦
The autopilots of the players, from the commanded
acceleration to the players’ acceleration, are
a β ( s)
(22)
H ( s) =
, β = T , M , D. .
β
a βc ( s )
The order of the players’ autopilots is nM, nT, and nD,
respectively. We further assume that the state space
realization of each H β (s) is such that the players’
accelerations, a β (s) , are the first state, as in [8].
Then the solution is:
I)
The time period t f 1 < t < t f 2
In this time interval there are only two players. This has been
solved in [12]. The solution is
(23)
1
a Mc (t ) = −
rM
814
⎡ 1 a ( s) ⎤
L−1 ⎢ 2 M
⎥
⎣ s a Mc ( s) ⎦ t f 2 − t
1
+
gTM
tf2
∫
t
⎡
⎢1
⎢r
M
⎣⎢
2
⎡ ⎡
⎤
⎡
⎤
⎢ L−1 ⎢ 1 a M ( s ) ⎤⎥
⎥ − 1 ⎢ L−1 ⎡⎢ 1 aT ( s ) ⎤⎥
⎥
⎢ ⎣ s 2 a Mc ( s ) ⎦
⎥ rT ⎢ ⎣ s 2 aTc ( s) ⎦
⎥
t f 2 −t ⎦
t f 2 −t ⎦
⎣
⎣
2
⎤
⎥ dτ
⎥
⎦⎥
ZTM (t )
aTc (t ) =
1
rT
⎡ 1 a ( s) ⎤
L−1 ⎢ 2 T
⎥
⎣ s aTc ( s ) ⎦ t f 2 − t
1
+
gTM
tf2
∫
t
⎡
⎢1
⎢r
⎢⎣ M
′ =
gTM
2
⎡ ⎡
⎤
⎡ ⎡
⎤
⎤
⎤
⎢ L−1 ⎢ 1 a M ( s ) ⎥
⎥ − 1 ⎢ L−1 ⎢ 1 aT ( s ) ⎥
⎥
⎢ ⎣ s 2 a Mc ( s ) ⎦
⎥ rT ⎢ ⎣ s 2 aTc ( s) ⎦
⎥
−
−
t
t
t
t
f
2
f
2
⎣
⎦
⎣
⎦
2
⎤
⎥ dτ
⎥
⎥⎦
ZTM (t )
⎡ 1 a (s) ⎤
L ⎢ 2 T ⎥
⎣ s aT (0) ⎦ t f 2 − t
−1
⎡ 1 a ( s) ⎤
L ⎢ 2 T
⎥
⎢⎣ s ρ T (0) ⎥⎦
t f 2 −t
−1
⎡ 1 a ( s) ⎤
−L ⎢ 2 M ⎥
⎣ s a M (0) ⎦ t f 2 − t
−1
⎡ξ TM ( t )⎤
⎢ξ ( t )⎥
⎢ TM ⎥
⎤
⎡
⎤
⎢ aT ( t ) ⎥
−1 1 a M ( s )
⎥
−L ⎢ 2
⎥
⎥ ⎢⎢ ρ ( t ) ⎥⎥
⎢⎣ s ρ T (0) ⎥⎦
⎥⎢ T ⎥
t f 2 −t ⎦
a M (t )
⎥
⎢
⎣⎢ ρ M ( t ) ⎦⎥
where the notations of the vectors are adopted from [8] and
L-1 is the inverse Laplace Transform.
II) The time period t < t f 1 (prior to defender-missile
intercept)
In this time interval there are three players. An approach to
derive this solution is outlined in [22]. The solution is
t f1
t f1
⎡⎛
⎤
⎛ S2 S2 ⎞ ⎞
S S
⎢⎜ g 1 + ∫ ⎜⎜ w1 − u1 ⎟⎟dτ ⎟ Su 2 + Su1 ∫ u1 u 2 dτ ⎥ ZTM
MD
⎜
⎟
rM
rM ⎠ ⎠
⎢⎣⎝
⎥⎦
t ⎝ rD
t
a Mc ( t ) =
1
rM ⎛
⎜ 1
′
⎜ gTM
⎝
t f1
⎡ tf1 S S
⎛
⎛ Su22 Sv22 ⎞ ⎞⎟ ⎤
1
⎜
⎟dτ Su1 ⎥ Z MD
+ ⎢ S u 2 ∫ u1 u 2 dτ + ⎜ gTM
−
′ + ∫⎜
⎜
rM
rT ⎟⎠ ⎟⎠ ⎥
⎢⎣
t
t ⎝ rM
⎝
⎦
2
tf1
tf1
t
⎛ S w21 Su21 ⎞ ⎞⎟ ⎡ f 1 Su1S u 2 ⎤
⎛ Su22 Sv22 ⎞ ⎞⎟⎛⎜ 1
⎟⎟dτ + ⎢ ∫
⎟⎟dτ g + ∫ ⎜⎜
−
d
τ
+ ∫ ⎜⎜
−
⎥
rM ⎠ ⎟⎠ ⎣⎢ t rM
rT ⎠ ⎟⎠⎜⎝ MD t ⎝ rD
t ⎝ rM
⎦⎥
(24)
aTc (t ) = −
1
rT ⎛
⎜ 1
′
⎜ gTM
⎝
tf1
tf1
⎛
⎛ S2
S2 ⎞ ⎞
S S
Sv 2 ⎜ g11 + ∫ ⎜⎜ w1 − u1 ⎟⎟dτ ⎟ ZTM + S v 2 ∫ u1 u 2 dτ Z MD
⎜
⎟
r
r
rT
M ⎠
t ⎝ D
t
⎝
⎠
2
t f1
t
tf1
f
1
⎤
⎛ S2
⎛ S2
S 2 ⎞ ⎞⎛ 1
S2 ⎞ ⎞ ⎡ S S
+ ∫ ⎜⎜ u 2 − v 2 ⎟⎟dτ ⎟⎜ g MD
+ ∫ ⎜⎜ w1 − u1 ⎟⎟dτ ⎟ + ⎢ ∫ u1 u 2 dτ ⎥
r
rT ⎠ ⎟⎠⎜⎝
r
rM ⎠ ⎟⎠ ⎣⎢ t rM
t ⎝ M
t ⎝ D
⎦⎥
tf1
S w1 ∫
a Dc (t ) = −
tf1
⎛
⎛ S u22 Sv22 ⎞ ⎞⎟
S u1 S u 2
1
⎜
⎟dτ Z MD
−
dτ ZTM + S w1 ⎜ gTM
′ + ∫⎜
⎜
rM
rT ⎟⎠ ⎟⎠
t ⎝ rM
⎝
t
1
2
t f1
tf1
rD ⎛
2
2
2
2
⎞⎛
⎞ ⎡t f 1
⎤
⎛
⎜ 1 + ⎜ Su 2 − Sv 2 ⎞⎟dτ ⎟⎜ 1 + ⎛⎜ S w1 − S u1 ⎞⎟dτ ⎟ + ⎢ S u1 S u 2 dτ ⎥
′
∫t ⎜⎝ rM rT ⎟⎠ ⎟⎜ g MD ∫t ⎜⎝ rD rM ⎟⎠ ⎟ ⎢ ∫t rM ⎥
⎜ gTM
⎝
⎠⎝
⎠ ⎣
⎦
Z TM (t ) =
⎡
⎢1 t − t
f2
⎢
⎢⎣
⎡ 1 a ( s) ⎤
L−1 ⎢ 2 T ⎥
⎣ s aT (0) ⎦ t f 2 − t
⎡ 1 a (s) ⎤
L−1 ⎢ 2 T
⎥
⎣⎢ s ρ T (0) ⎦⎥
t f 2 −t
⎡ 1 a ( s) ⎤
− L−1 ⎢ 2 M ⎥
⎣ s a M (0) ⎦ t f 2 − t
⎡ξTM ( t )⎤
⎢ξ ( t )⎥
TM
⎥
⎤ ⎢⎢
⎡
⎤
aT ( t ) ⎥
−1 1 a M ( s )
⎥
−L ⎢ 2
⎥
⎢
⎥
⎥
ρ
(
t
)
⎣⎢ s ρ T (0) ⎦⎥ t f 2 − t ⎥⎦ ⎢ T ⎥
⎢ a M (t ) ⎥
⎢
⎥
⎣⎢ ρ M (t ) ⎦⎥
⎡ 1 a ( s) ⎤
− L−1 ⎢ 2 D ⎥
⎣ s a D (0) ⎦ t f 1 −t
⎡ξ MD (t )⎤
⎢ξ (t )⎥
⎢ MD ⎥
⎤ ⎢ a (t ) ⎥
⎡
⎤
M
−1 1 a D ( s )
⎥
−L ⎢ 2
⎥
⎢
⎥
⎢⎣ s ρ D (0) ⎥⎦ t −t ⎥⎥ ⎢ ρ M (t ) ⎥
f1
⎦⎢
a D (t ) ⎥
⎥
⎢
⎢⎣ ρ D (t ) ⎥⎦
⎡ 1 a ( s) ⎤
L−1 ⎢ 2 M ⎥
⎣ s a M (0) ⎦ t f 1−t
⎡ 1 a ( s) ⎤
L−1 ⎢ 2 M
⎥
⎢⎣ s ρ M (0) ⎥⎦ t − t
f1
⎤
⎥ dτ
⎥
⎦⎥
One can see that the features that are important from the
guidance law point of view are the ramp responses of the
autopilots and of the initial conditions, exactly as in the one
sided optimal guidance laws. Sufficient condition for
optimality is that the denominators in (23,24) are positive for
the whole intercept periods. There are limiting cases solution
for the parameters of the game ( rT , rM , rD , g MD , gTM and
the poles and zeros of the Target, Missile and Defender
transfer functions). Some of these cases are considered here.
The TMD game degenerates to the classical evader-pursuer
game [1] in the following cases:
I) Two person game between the Target (evader) and the
Missile (pursuer) g MD → 0 , and gTM → ∞ , (no Defender
has been launched);
II) Two person game between the Missile (evader) and the
Defender (pursuer) g MD → ∞ , and gTM → 0 , (intercept of
the Missile by the Defender, nobody cares about the Target).
The case of instantaneous response of the players has been
considered in [14,18], is used for normalization and is
presented here for the completeness of the presentation.
We have:
A) Two person game between the Target (evader) and the
Missile (pursuer)
g MD → 0 , and gTM → ∞ , (no Defender has been launched)
a Tc (t ) =
[
]
[
]
3
ξ TM + (t f 2 − t )ξTM , t < t f 2
⎞
⎛ rT
2
⎜⎜
− 1⎟⎟(t f 2 − t )
⎠
⎝ rM
3
ξ TM + (t f 2 − t )ξTM , t < t f 2
⎛ rM ⎞
⎜⎜1 −
⎟⎟(t f 2 − t )2
rT ⎠
⎝
(25)
a Dc (t ) = 0, t < t f 2
This is the classical two person game between the Target
and the Missile [21].
Z MD ( t ) =
⎡
⎢1 t − t
f1
⎢
⎢⎣
2
⎡ 1 a ( s) ⎤
⎡ 1 a ( s) ⎤
S v 2 = L−1 ⎢ 2 M ⎥
; S w1 = γ L−1 ⎢ 2 D
⎥
⎣ s a Dc ( s) ⎦ t f 1 − t
⎣ s a Tc ( s) ⎦ t f 2 −t
a Mc (t ) =
where the respective Zero Effort Miss terms, ZTM and ZMD are
2
⎡ ⎡
⎤
⎡ ⎡
⎤
⎤
⎤
⎥
⎢ L−1 ⎢ 1 a M ( s) ⎥
⎥ − 1 ⎢ L−1 ⎢ 1 aT ( s) ⎥
2
2
⎥
⎢ ⎣ s a Mc ( s) ⎦
⎥ rT ⎢ ⎣ s aTc ( s) ⎦
t
t
t
t
−
−
f
2
f
2
⎣
⎦
⎣
⎦
⎡ 1 a ( s) ⎤
⎡ 1 a ( s) ⎤
S u 2 = L−1 ⎢ 2 M
; S u1 = L−1 ⎢ 2 M
⎥
⎥
a
s
(
)
s
Mc
⎣ s a Mc ( s ) ⎦ t f 1 −t
⎦ t f 2 −t
⎣
where ZTM(t) is the miss without effort in target-missile
encounter [21], called the zero effort miss, given by
⎡
Z TM (t ) = ⎢1 t f 2 − t
⎢
⎢⎣
1
⎡
1
1
+ ∫ ⎢⎢
gTM t f 1 rM
⎣⎢
tf2
where ZMD(t) is the miss without effort in missile-defender
encounter [21], called the zero effort miss.
B) Two person game between the Missile (evader) and the
Defender (pursuer)
g MD → ∞ , and gTM → 0 , (intercept of the Missile by the
Defender, nobody cares about the Target)
t < t f1
⎧0,
(26)
a Tc ( t ) = ⎨
⎩0, t f 1 < t < t f 2
815
]
where τi are the time constant of the players’ autopilots,
respectively. The full state guidance law is,
⎧
⎧⎪
⎫⎪
⎛tf2 −t ⎞
⎛tf2 −t ⎞
⎪ N M ⎨VTMσTM + ψ ⎜⎜
⎟⎟a M (t )⎬⎪
⎟⎟aT (t ) −ψ ⎜⎜ τ
τ
⎪
⎝ M ⎠
⎝ T ⎠
⎪
⎩
⎭
t < tf1
⎪
⎧⎪
⎫⎪
⎪
⎛ t f1 − t ⎞
⎛ t f1 − t ⎞
aMc (t ) = −⎨
⎟⎟a M (t ) −ψ ⎜⎜
⎟⎟a D (t )⎬,
+ ΓM ⎨VMDσ MD + ψ ⎜⎜
⎪⎩
⎪⎭
⎪
⎝ τM ⎠
⎝ τD ⎠
⎪
⎧⎪
⎫⎪
⎛tf2 −t ⎞
⎛tf2 − t ⎞
⎪
⎟⎟aT (t ) −ψ ⎜⎜
⎟⎟a M (t )⎬,
N M ⎨VTMσTM + ψ ⎜⎜
tf1 < t < t f 2
⎪⎩
⎪⎩
⎪⎭
⎝ τT ⎠
⎝ τM ⎠
⎧ ⎧⎪
⎫⎪
⎛tf2 −t ⎞
⎛t −t ⎞
⎟⎟aT (t ) −ψ ⎜⎜ f 2
⎟⎟a M (t ) ⎬
⎪ N T ⎨VTM σ TM + ψ ⎜⎜
⎪⎭
⎝ τT ⎠
⎝ τM ⎠
⎪ ⎪⎩
t < t f1
⎪
⎧⎪
⎫⎪
⎪
⎛ t f1 − t ⎞
⎛ t f1 − t ⎞
aTc (t ) = ⎨
⎟⎟a M (t ) −ψ ⎜⎜
⎟⎟a D ( t )⎬,
+ ΓT ⎨VMDσ MD + ψ ⎜⎜
⎪⎩
⎪⎭
⎪
⎝ τM ⎠
⎝ τD ⎠
⎪
⎧⎪
⎫⎪
⎛tf2 −t ⎞
⎛ tf2 −t ⎞
⎪
N T ⎨VTM σ TM + ψ ⎜⎜
t f1 < t < t f 2
⎟⎟aT (t ) −ψ ⎜⎜ τ
⎟⎟a M (t )⎬⎪,
⎪⎩
⎪⎩
⎝ τT ⎠
⎝ M ⎠
⎭
⎧ ⎧⎪
⎫⎪
⎛tf2 −t ⎞
⎛t −t ⎞
⎟⎟aT (t ) −ψ ⎜⎜ f 2 ⎟⎟aM (t )⎬
⎪ΓD ⎨VTMσTM +ψ ⎜⎜
τ
τ
⎪
⎪⎭
⎪ ⎩
⎝ T ⎠
⎝ M ⎠
t < t f1
⎪
aDc (t ) = ⎨
⎧⎪
⎫⎪
⎛t −t ⎞
⎛t −t ⎞
+ ND ⎨VMDσ MD +ψ ⎜⎜ f 1 ⎟⎟aM (t ) −ψ ⎜⎜ f 1 ⎟⎟aD (t )⎬,
⎪
⎪⎩
⎪⎭
⎪
⎝ τM ⎠
⎝ τD ⎠
⎪
0
tf1 < t < t f 2
⎩
where
Figure 3 presents the Guidance gains.
(29)
Target gain to σ′ TM
T
T
NT
f1
2
20
10
0
0
1
2
3
4
0
-2
5
0
Missile gain to σ′ TM
NM
20
10
0
0
1
2
3
4
5
0
ΓD
3
4
5
1
2
3
4
5
Defender gain to σ′ MD
10
30
5
0
2
15
10
5
0
Defender gain to σ′ TM
2
1
Missile gain to σ′ MD
30
T
2
ND
ΓD = (t f 1 − t ) LD
f1
D
We have the Target-to-Missile line-of-sight angle and lineof-sight rate
ξ TM + (t f 2 − t )ξTM
ξ TM
(31)
σ TM =
,σ TM =
( t f 2 − t )VTM
(t f 2 − t ) 2VTM
and the Missile-to-Defender line-of-sight angle and line-ofsight rate
Target gain to σ′ MD
30
where Kβ and Lβ, β=T,M,D, are the gains from the solution
of the DRME (Differential Riccati Matrix Equation) (16).
We define the Effective Navigation gains, Nβ, β=T,M,D,
and the Effective Navigation Feed-Forward gains, Γβ,
β=T,M,D, as
2
2
NM = t f2 − t KM
ΓM = (t f 1 − t ) LM
2
2
(30)
N = t −t K
Γ = (t − t ) L
f2
(33)
⎛tf −t ⎞
1
⎡1 τ ⎤
⎟⎟ =
L−1 ⎢ 2
,
2
⎥
⎣ s sτ + 1 ⎦ t − t
⎝ τ ⎠ (t f − t )
f
ψ ⎜⎜
a Dc (t ) = LD Z TM + K D Z MD
T
(32)
(t f 1 − t ) 2V MD
σ TM - the Target-to-Missile line-of-sight angle with
respect to reference direction.
σ TM - the inertial Target-to-Missile line-of-sight rate
σ MD - the Missile-to-Defender line-of-sight angle with
respect to reference direction.
σ MD - the inertial Missile-to-Defender line-of-sight rate.
Substitution gives the final result
aTc (t ) = K T Z TM + LT Z MD
)
)
− t) K
ξ MD + ( t f 1 − t )ξMD
ΓT
]
II) rT > rM > 0, gTM → ∞ from (25), respectively.
It is difficult to arrive at a simple expression for sufficient
conditions for existence of the optimal solution even for
thissimple example. From the above we asserted that for the
case of instantaneous responses the conditions
rT > rM > rD > 0, , gTM → ∞, and g MD → ∞. (27)
are sufficient conditions. The assertion (27) has been checked
numerically to be true for gMD= 108, gTM= 108, rM = 1, rM >
rD > 0, and 20 > rT > 1 in [18]. Moreover, the verification
process revealed, as expected, that these conditions are not
necessary, and that the domain of parameters for an optimal
solution is larger than (27).
In the following we assume first order players’ response
that is the players’ autopilots are
a β ( s)
a β ( s)
τβ
1
(28)
=
;
=
,β = T, M, D
a βc ( s ) sτ i + 1 a β (0) sτ β + 1
(
(
= (t
( t f 1 − t )V MD
,σ MD =
where
3
⎧
ξ MD + (t f 1 − t )ξMD ,
t < t f1
⎪⎛
⎞
rD
⎪
2
⎟⎟(t f 1 − t )
a Dc ( t ) = ⎨ ⎜⎜1 −
⎪ ⎝ rM ⎠
0,
t f1 < t < t f 2
⎪⎩
These cases are used for the normalization of the gains in
the following.
In the preceding limiting solutions we have the following
sufficient conditions:
I) rM > rD > 0, g MD → ∞ from (26); and
a Mc (t ) = K M Z TM + LM Z MD
ξ MD
ΓM
[
σ MD =
ND
[
3
⎧
ξ MD + (t f 1 − t )ξMD ,
t < t f1
⎪⎛ r
⎞
⎪ M
2
a Mc ( t ) = ⎨ ⎜⎜
− 1⎟⎟(t f 1 − t )
⎠
⎪ ⎝ rD
0,
t f1 < t < t f 2
⎪⎩
0
1
2
3
4
time-to-go [sec]
5
20
10
0
0
1
2
3
4
time-to-go [sec]
5
Figure 3: The Effective Navigation gains, Nβ, β=T,M,D,
and the Effective Navigation Feed-forward gains, Γβ,
β=T,M,D, versus time-to-go, for tf1=3sec, tf2=5sec, rM = 1, rT
= 5/3, rD = 3/5, gMD= 1012, gTM= 1012, τT = 0.2sec, τM = 0.1sec,
and τD =0.05sec.
816
One can see that for 0≤tgo≤tf2-tf1 the effective navigation
gains/ratios of the Target and the Missile are the ones for a
two person game [12].
The following sections present an example of the
trajectories and the accelerations of the players. The players
apply the optimal guidance-laws/strategies (24), as presented
in Figure 3. Figure 4 presents the Target, Missile and
Defender trajectories, i.e. the distance perpendicular to the
reference line versus the range for the initial conditions
yT (t o ) = 0 m, y T (t o ) = 50 m / sec, y M (t o ) = 0 m
(34)
y M (t o ) = 100 m / sec, yD (t o ) = 0 m, y D (t o ) = 48 m / sec
In this case the Defender is launched when the Missile is
close to the collision course with the Target. The achieved
miss are very small. They are not zero as gMD and gTM are
finite. Figure 5 presents the players’ accelerations.
VMissile/VTarget=1, VDefender/VMissile=2
120
Target
Missile
Defender
players perpendicular separation [m]
100
80
60
40
20
0
0
500
1000
1500
2000
2500 3000
range [m]
3500
4000
4500
5000
Figure 4: The trajectories of the Target, Missile and Defender.
0
-10
acceleration [m/sec 2]
-20
-30
-40
-50
Target
Missile
Defender
-60
-70
0
0.5
1
1.5
2
2.5
3
time [sec]
3.5
4
4.5
5
Figure 5: The acceleration of the players.
VI. CONCLUSION
A three person game, the Target, the Missile and the
Defender (TMD) has been defined, an approach to the
solution has been presented and a solution has been derived
for high order autopilots of the participants. This is a closed
loop saddle point equilibrium solution in linear strategies for
the three players. The solution is not continuous. Sufficient
conditions for the existence of the solution are presented. It
follows that the Target and the Defender must be coordinated
in order to achieve the optimal cost.
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