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Contract Nont-1866 (16) NR- 372-012
DIFFERENTIAL GAMES
ANDOPTIMAL PURSUIT-EVASION STRATEGIES
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By JA 0 1 0 oiYu-chi Ho, A.E. Bryson, Jr. &
DDCAIRA B
November 4, 1964
Technical Report No. 457
Cruft Laboratory
Division of Engineering and Applied Physics
Harvard University -Cambridge, Massachusetts
BEST
A n,-- . . ,A VA ILA LE 'y
Office of Naval Research
Contract Nonr-1866(16)
NR - 372 - 012
DIFFERENTIAL GAMES
AND OPTIMAL PURSUIT-EVASION STRATEGIES
by
*Yu-Chi Ho, A. E. Bryson, Jr., and S. Baron
November 4, 1964
The research reported in this document was made possible throughsupport extended to Cruft Laboratory, Harvard University, by theU. S. Army Research Office, the U. S. Air Force Office of ScientificResearch, and the U. S. Office of Naval Research under the JointServices Electronics Program by Contract Nonr-1866( 16). Repro-duction in whole or in part is permitted for any purpose of the UnitedStates Government.
NASA, Langley Field. On leave at Harvard University.
Technical Report No. 457
Cruft Laboratory
Division of Engineering and Applied Physics
Harvard University
Cambridge, Massachusetts
DIFFERENTIAL GAMES AND OPTIMAL PURSUIT-EVASION STRATEGIES
by
Yu-Chi Ho, A. E. Bryson, Jr., and S. Baron
Division of Engineering and Applied Physics
Harvard University, Cambridge, Massachusetts
ABSTRACT
In this report, we show that variational techniques can be applied to
solve games with differential constraints. Conditions for capture and for
optimality are derived for a class of optimal pursuit-evasion problems.
Results are used to demonstrate that the well known "proportional navigation
law" is actually an optimal intercept strategy.
-lili-
TR457 I. INTRODUCTION
The theory of differential games was initiated by Issacs in 1954 [1].
It was later studied in greater detail by Fleming and Berkovitz [2]. Recently,
because of the advances in the computational solution of variation problems,
renewed interest has developed in this subject. Stated simply, games are a
class of two-sided optimization problem. The simplest games are the type with
which we are most familiar. A function J of two discrete variables u and v
is given in tabular form. Each particular value of u or v is called a strategy.
The problem is the determination of the strategy for u and v such that J is
a saddle point (minimax). Because of the discrete nature of the available
strategies, it is generally not possible to realize a minimax with a simple
choice for u and v. Instead, a mixture of strategies must be employed to
realize a minimax on an average basis. The usual game theory devotes
considerable effort to the study of the properties of mixed strategies. On the
other hand, if u and v were to take on a continuous range of values, then a
saddle point could generally be realized by a pair of pure strategies for any
reasonably well behaved function J (u, v). The necessary and sufficient
conditions for a saddle point are simply,
JT 0 J -- 0()u v for all u , v
uu >0 v <0 (2)
This is in direct analogy to the one sided optimization problem of calculus.
A more involved version of the game problem can be stated as that of
It is the authors' understanding that Prof. Pontriagin lectured on the subjectduring the month of October 1964.
TR457 -2-
determining a saddle point for J(x, u, v) subject to the restriction that
i(x; u, v) = 0. In this case, u and v still have the usual meaning while
the variable x is an intermediate variable indirectly determined by the constraint
= 0. Of course, we can always eliminate x using the constraints and convert
the problem into a simple game by writing J = J (g- (u, v), u, v) where
x -1 (u, v) => { = 0. However, it is usually simpler both theoretically and
computationally to solve the problem by the introduction of a Lagrange
multiplier again in imitation of the one-sided calculus problem. From this
point, it is a straightforward conceptual generalization to consider the differential
game problem of determining a saddle point for
T
J = $(x(T), T) + L(x, u, v) dt (3)
0
subject to
x = f(x, u, v, t) ; x(t ) x0 (4)
where x plays the usual role of the state vector, and u, v the control vectors.
Extending our analogy with the one-sided optimizatj';n problem, we expect that
the techniques of variational calculus should be useful in this context. The
purpose of this report is to illustrate that this is indeed so by solving a class
of optimal pursuit-evasion problems and deriving conditions for optimality and
capture. As an interesting by-product, we shall show that the "proportional
navigation" law used in many missile guidance systems actually constitutes
a form of optimal pursuit strategy, under the usual simplifying approximations
to the equations of motion of the missile and the target.
TR457 -3-
II. A CLASS OF OPTIMAL PURSUIT-EVASION GAMES
We shall consider two dynamic systems
=F x +G u (5)
:e Fe xe + G v (6)
where the subscript p and e stand for pursuer and evader respectively.
The pursuing system desires to intercept or rendezvous with the evading
system at time T while the latter attempts to do the opposite. Both systems
have limited energy sources. Hence, a reasonable criterion would be
Tx()IA+I SHuI - ]dtll
SL11x(T)-x (T)11- + -LI [1ulIR 2 ] dt (7)p e
where A> 0 and R and Re > 0. The minus sign in front of the llvll 2 term
comes about due to the fact that we shall attempt to maximize J w. r. t. to the
variable v . Following the usual variational procedure, we introduce at this
point the multiplier functions X (t) and X (t) which are used to adjoin (5)p e
and (6) to (7). Now let us take a particular pair of strategies u(t) and
v(t) with the associated trajectories x (t) and x (t) and considerp e
variations 6u(t) and bv(t). The change in J to second order is
6J = [(X (T)-x (T).TAT_ % T(T)]6x (T)+[(X (T)-x (T))TAT- T(T)]6xe(T)p e p p e A e e
T+.- 115xp(T) (6xe(T) 112 + [ +H x 6X +HI'I 6Xe
0
T ] 0 6ul
+ H 6u + H 6v]dt + -1 5 [6 u 6 v]T dt (8a)0 0 - R 6v
e
TR457 -4-
where H is the Hamiltonian
H = 1 112- Ilv~il ) + X T (F x+G u)+%T (Fx + G V) (8b)Z(u R p p p e e e e
In order that (8) vanish to first order, we immediately derive the necessary
conditions
T X T F XpT(T) (x (T)- xe(T))T A T (9)p PP p p e
T- XT F X T (T) (xe(T) - Xp(T))TAT (10)
U 0 => u p p - GpTXp-1 T
Hv =0 => v =+ Re Ge Te (12)
Equations (5, 6, 9- 12 ) represent a linear two-point boundary value problem
which can be solved in terms of the fundamental matrix solution of the "p1 and
"e" system, P p(t, T ) and 4e(t, r). We have,
XpM) =1p (T, t) A (x p(T)-x e (T))
(13)
ke (t) = DeT (T, t) A(xe (T)-xp (T))
Substituting (13) into (5-6) and using (11-12), we obtain
x p(T) = (T, t)xp (t) - M pA (x p(T) - xe(T))p p p) p (14)
Xe( ) :@e(, t Xet). Me (p(W- e ()
TR457 -5-
where
T
M (Tt) j (T, 7) G R G T T(TIr)dT (15)p p p Pt
TM e(Tt) A= S e (T I T ) GeRe " G e T e T (T,.T)d (16)
t
Finally, (13) and (14) combine to yield the optimal pursuit and evasive
strategies
u(t) = -R-IG T PT(T,t)A[I+(M -M )A]- [o (T,t)x (t)- e(Tt)Xe (t)](17)
p p p p e P p
v(t) = -R-le Ge T ' (Tt)A[I+(Mp-M)A] -l[ (T, t)xp(t). e(Tt)xe(t)] (18)
Since 0 (T, t) x (t) has the interpretation of the predicted terminal state of a
dynamic system, the optimal pursuit-evasive strategies are simply linear
combinations of the predicted terminal miss - a very reasonable result.
Examination of the second-order terms in (8) shows that the analogous
Legendre-Clebsch condition for the saddle point is satisfied.
Huu Rp vv e= - R <0 (19)
Furthermore, the accessory minimax problem is
"determine a saddle point for T6J= 16xp(T)-6Xe(T) 112+ ' [11 b ll 2 1 -II vll- dt
0 P e(p)
subject to 6x = F 5x + G 6u; 6 x (0)= 0
6c =F 6x +G 6v ; 6 x (0) 0e e e e e
TR457 -6-
By exactly the same argument leading to the derivation of the conjugate point
conditions for the one sided variational problem [3], we find that
6J> 0 V 6u/0, 6v fixed,and 6J<0 V 6v /0, 6ufixed, if and only if
the matrix solution X (t) is nonsingular during the interval (0, T) where
X (t) obeys
-I TX F 0 -G CRG T 0 X X(T) = I
0 F 0 G R-IG Te e e ee T (20)
0 0 -F 0p T A -A')
A 0 0 0 -F e A A-
But from (20) ,
X (t) -p(t)T) 0e(tT K p-Me (21)
0 () L e A I-Me jThus, the nonsingularity of X (t) (i. e., nonexistence of conjugate point) is
equivalent to the condition
de +Mp- I = det[I+(M ^M )A]/0 (22)A Me Ae p e
which in view of (17) and (18) also guarantees the boundedness of the
strategies.
It is also interesting to consider the limiting case where A is
positive definite and infinite (e. g., infinite diagonal matrix). This is the
situation where u attempts to capture v with minimal energy. Condition
(22) becomes
TR457 -7-
det [I+ (Mp - M e)A]/O<:>det(A-I + (Mp -M e )) /0 (23)
<=> det (M -M ) /0p e
In terms of the usual definition of controllability, (23) simply says that the
pursuer must be "more controllable" than the evader - an eminently reasonable
conclusion. Since both M and M > 0 if the individual systems arep e
controllable, then a sufficient condition for capture is simply M - M > 0.p e
Furthermore, we have the following
Proposition: Let Rp and Re in (7) be scalar, and the optimal pursuit and~p e
T Tevasion energy for A= o be T jjujj 2 dr= cp jjvjjdt= ce) respectively,
0 C
then a necessary and sufficient condition for capture of an evader with energy
resources ce by a pursuer with energy resourses c is M -M > 0. Theep p e
proof of this proposition is a direct consequence of the utility interpretation
of Lagrange multipliers [4].
III. GUIDANCE LAW FOR TARGET INTERCEPTION
A special case of the class of problems treated in Section II can be
formulated as follows: the equations of motion (kinematic) in space for an
interceptor and a target are
v f + ap p p
r = VP P (24)v =f + a
e e e
r ve e
TR457 -8-
where
v = velocity of a body in three dimensions
r = position vector of a body in three dimensions
= external force per unit mass exerted on the body
a= control acceleration of a body
We assume that the altitutude difference between the pursuer and the evader
is small enough such that f = f . Hence, if we are only interested in thep e
difference Y- (t) - "- (t) , the effect of the external forces can be ignored.p e
Now consider the criterion
J = -l r(T) -- re(T))("r (T) - "r(T))
Zp e p e
T (25)
+ ) - c (a e'a)]dt2 [c ( e ep e
0
where c and c e represent the energy capacity of the pursuer and thepe
evader, respectively. Applying the result of the last section, it can be
directly verified that (17) and (18) become in this case
-e + () Va= p p e p e (26)
-- + -c ) (T - t) /3a (p e
- C (T-t)[% (t)-1" (t) + (e(t)-"Ve(t))(T-t)]a e 3 (27)
- + (c p-c e ) (T.-t) /3
TR457 -9-
We note immediately that if
(i) if c > ce (i. e., pursuer has more energy than the evader)p e
then the feedback control gain is always of one sign.
(ii) if c < ce (i. e., pursuer has less energy than the evader)p e
then the feedback gain will change sign at
1 + (c -c ) (T- t) 3/3 = 0 (28)a Cp e
for T sufficiently large.
But (28) is simply the conjugate point condition (22) specialized for this
problem. Hence, for case (ii), (26) is no longer optimal for large T. This
fact is, of course, obvious to startwith, particularly in the case when a = oo.
In that case, interception is not possible when c < ce (cf. M < M ). Assumingp e p e
(i) and letting a = co , the control strategy for the pursuer simplifies to
p 2 p - p (
(1- e) (T-t)Cp
Let the pursuer and the target be on a nominal collision course with range
R and closing velocity V = R/(T-t). Let x -xe represent the lateralc p e
deviation from the collision course as shown in Fig. 1
TR457 -10-
x ip _x-e Line of sight angle
R-Xe I (T Nominal line of sight
H(-"R = V(T-t) -)Pursuer Evader
Fig. 1 Geometry of Proportional Navigation.
Then the lateral control acceleration to be applied by the pursuer according
to (29) is
3a (lateral) V V c (30)C C
(1- -e)
Cp
which is simply proportional navigation with the effective navigation constant
K = 3 From experience it has been found that the "best" valuee c
Cp
for K ranges between 3 to 5 [5]. In view of (30), we see that the value of 3e
corresponds to the case when the target is not maneuverable[6] (ce = 0) ) whileC
the value of 5 corresponds to e 2c 5
p
TR457 -11-
REFERENCES
1. R. Issacs, "Differential Games I, II, III, IV, " RAND Corporation ResearchMemorandum RM-1391, 1399, 1411, 1468, 1954-56.
2. L. D. Berkovitz and W. H. Fleming, "On Differential Games with IntegralPayoff, " Annal. of Math. Study No. 39, pp. 413-435, PrincetonUniversity, 1957.
3. J. V. Breakwell and Y. C. Ho, " On the Conjugate Point Condition for theControl Problem," Int. Journal of Engineering Science (to appear) 1964.Also, Cruft Laboratory Technical Report No. 441, Harvard University,March 1964.
4. R. Bellman, Adaptive Control-A Guided Tour, Princeton University Press,Chapter VI, pp. 103-104, 1961.
5. A. Puckett and S. Ramo, Guided Missile Engineering, pp. 176-180,McGraw-Hill Book Company, New York, 1959.
6. A. E. Bryson, "Optimal Guidance Laws for Injection, Interception,Rendezvous, and Soft Landing, " AIAA Journal (to appear).
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