IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS, VOL. XX, NO. Y, MONTH 2002 1
An Information Set-Based Guidance
Algorithm against a Decelerating
Maneuvering Target
Dmitry Emeliyanov, Evgeny Rubinovich , and Boris Miller
This material is based upon the work supported by the EOARD, Air Force O�ce of Scienti�c Research, Air
Force Research Laboratory, under Contract No. F61775-00-WE028, and extends results presented in the 40-th
IEEE Conference on Decision and Control, CDC01, Orlando, Florida.
D. Emeliyanov is with the Institute of Control Sciences, 117997 Moscow, Russia, and also with Deutsches
Elektronen-Synchrotron DESY, 22607 Hamburg, Germany (e-mail: [email protected])
E. Rubinovich is with the Institute of Control Sciences, 65 Profsoyuznaya str., 117997 Moscow, Russia. (e-mail:
B. Miller is with the Institute for Information Transmission Problems, 19 Bol'shoi Karetnyi per., 101447 Moscow,
Russia. (e-mail: [email protected])
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IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS, VOL. XX, NO. Y, MONTH 2002 2
Abstract
This paper describes a new guidance algorithm against maneuvering decelerating target based on the
concept of information sets (IS). As an example of guidance problem, a model of defense scenario against
reentering ballistic missile is considered. In this scenario, a maneuverable decelerating target is to be
destroyed by a hit-to-kill interceptor out�tted with an IR array seeker and lateral impulse thrusters. The
key element of the proposed approach is a description of the "interceptor-target" system state by means
of IS. The results of Monte Carlo tests of the developed algorithm are presented.
Keywords
Bearings-only guidance algorithm, Ballistic target, Optimal impulse control, Information set approach.
I. Introduction
The methodology of unknown but bounded errors is a well-known approach for the de-
scription of uncertainties in the control problems with incomplete data [1], [2]. The basic
assumption of this approach is that the errors on measurements of a controlled system
state vector are restricted by given boundaries, at the same time, the statistical properties
of the errors are assumed to be unknown. Then the totality of all possible states of the
controlled system can be completely described as a set in the system phase space. Such a
set (information set (IS)) is a bounding estimate of the system state and can be regarded
as a \generalized" system state. By de�nition, all points of an IS are to be geometrically
consistent with the system's dynamics, measurement model, and observation/control his-
tories. Considering the IS as a generalized system state, the original control problem can
be reduced to a problem of the IS evolution control with the aim to minimize a certain
performance index as a function of the information set.
This approach is especially suitable for problems of guidance with nonlinear (bear-
ings/range, bearings-only) measurements corrupted by additive non-Gaussian noise dis-
tributed within given boundaries. It is well known that a linearized Kalman-type estimator
incorporated in a guidance loop usually fails for such a measurement model, and overall
guidance performance deteriorates signi�cantly.
Using the IS-based approach, we investigate a three-dimensional problem of guidance
with incomplete information. An impulse-controlled interceptor is guided to a maneu-
verable decelerating target using passive angle-type measurements corrupted by additive
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IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS, VOL. XX, NO. Y, MONTH 2002 3
uniformly distributed noise. The aim of the guidance is a minimization of the relative dis-
tance to ensure target impact. Similar problems have been examined before using minimax
�ltration and the theory of di�erential games [3].
An important feature of the interceptor's dynamics is that the control impulses are
orthogonal to its longitudinal axis, so that the longitudinal component of the interceptor's
motion is out of control. Both, the interceptor and the target move under the in uence
of the aerodynamic drag forces with the drag coe�cients assumed to be known. For the
lateral target's motion with respect to the interceptor, two models of the target's dynamics
are considered:
1. pure deceleration caused by the drag force;
2. a sum of the deceleration and random disturbances which act in the direction orthogonal
to the deceleration vector and have magnitudes restricted by the given boundaries.
The second model describes the situation when a target decelerating in the atmosphere
performs jump-wise evasive maneuvers.
After formulation of a mathematicalmodel of the guidance problem, the information sets
for the considered dynamics are constructed. To facilitate operations with the introduced
IS so-called Grid-Polygon Approximation (GPA) of the IS is proposed. Using the GPA
technique, the recursive equations describing the evolution of the information set during
the guidance process are derived.
As a criterion for control of the IS evolution, the terminal miss averaged over the IS
is taken. The interceptor's control is chosen in the class of so-called "conditionally pro-
grammed controls", i.e. programmed controls which are updated after each measurement.
To update the control an auxiliary optimal impulse control problem is solved. Recently,
this technique has been successfully applied to similar guidance problems in [4], [5]. In the
auxiliary problem, a performance index has the meaning of expense of impulse control over
time-to-go, and a terminal condition provides zero average miss. The auxiliary problem is
transformated to the quadratic programming problem which is solved using Kuhn-Tucker
optimality conditions.
II. Problem statement
The guidance geometry is shown in Fig.1. It is assumed that the target T moves with
July 25, 2002 DRAFT
IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS, VOL. XX, NO. Y, MONTH 2002 4
Y
X
Z
M
T
O
FM
FT
VT
VM
u1
u2
2
3
3
3
3 1
X
ZY
Y
X
Z
FT
ψ
ϕ
VT
Fig. 1. The guidance geometry. 1 { interceptor, 2 { IR array seeker, 3 { lateral impulse thrusters
zero attack and sliding angles and the lift force can be neglected. Then the target's motion
is a deceleration caused by the drag force:
VT (t) = V0 � �T
Z t
t0
V 2T (s) ds; �T =
CxS�
2mT
: (1)
Here VT (t) { target's longitudinal velocity,mT { target's mass, � { atmospheric density, S
{ target's frontal area, Cx { drag coe�cient. The coe�cient �T is assumed to be known.
The relational motion of the interceptor M and the target T in the reference system
OXYZ �xed to the interceptor is described by the equations:
x(t) = x0 �Z t
t0
VM(s) ds �Z t
t0
kx VT (s) ds; (2)
y(t) = y0 �Z t
t0
Vy(s) ds�Z t
t0
ky VT (s) ds; (3)
z(t) = z0 �Z t
t0
Vz(s) ds�Z t
t0
kz VT (s) ds; (4)
Vy(t) =Xtk�t
u1(tk); (5)
Vz(t) =Xtk�t
u2(tk); (6)
VM (t) = W0 � �M
Z t
t0
V 2M (s) ds; (7)
where t0 { the initial instant of time, VM , Vy , Vz { interceptor's velocities, (u1; u2) =
(u1(t); u2(t)) { interceptor's impulse controls represented by a totality of impulses u1k =
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IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS, VOL. XX, NO. Y, MONTH 2002 5
u1(tk), u2k = u2(tk) which are applied at �xed instants tk = t0 + k�, k = 0; 1; : : :, and �
{ given time interval. The coe�cients kx, ky , and kz are equal to:
kx = cos' cos ; ky = sin'; kz = cos' sin :
The initial parameters x0, y0, z0, V0 and coe�cients ky, kz are assumed to be unknown
but restricted by the given inequalities.
During the guidance the interceptor performs measurements (�1k; �2k) described by the
equations
�1k =y(tk)
x(tk)+ �1k; �2k =
z(tk)
x(tk)+ �2k; k = 0; 1; : : : ;
where �1k, �2k { uniformly distributed noises:
j�1kj � c; j�2kj � c;
c { given constant. The observation process and control process are terminated at an
instant tf :
tf = infft : x(t) < "g: (8)
Here " { given small distance, " > 0. The condition (8) re ects the fact that, at low
distances, further observations are impossible due to the large angular size of the target
and overload of the seeker's IR receiver.
III. Construction of the information set
By de�nition, the information set (IS) consists of all points of the phase space whose
coordinates are consistent (in geometrical sense) with dynamics equations, measurement
model and restrictions imposed on the initial state vector.
For the problem under study, the IS I(t) is a 6-dimensional one: I(t) � fx; VT ; y; z; ky; kzg ;
assuming the coe�cient kx to be known.
Since the dynamics equations describing the motions along Y and Z axes as well as
the corresponding measurement equations are functionally independent of each other, the
dimension of the IS can be reduced. Namely, the 6-dim information set I(t) can be
replaced by two 4-dim IS Iy(t) and Iz(t) which are constructed separately in the subspaces
fx; VT ; y; kyg and fx; VT ; z; kzg.
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IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS, VOL. XX, NO. Y, MONTH 2002 6
In order to facilitate the numerical operations with IS, it is proposed to represent IS with
the help of so-called Grid-Polygon Approximation (GPA) illustrated in Fig.2. According
X−1
NV −1 VT
V = VT
x = x i
T j
ij
ky
Pij
P
1
23
4
56 T jT
x = x i
y
y��
V = V
N x
0
1
12
2
j
j+1
k
z
z
1
3
4
5
2
zi
i+1
Fig. 2. Grid-Polygon Approximation of IS.
to the GPA, IS Iy is represented by a set of its sections made at discrete values of x and
VT :
P yij(t) � fy; kyg ; i = 0; : : : ; Nx � 1; j = 0; : : : ; NV � 1;
P yij(t) = Iy(t)
\f(x; VT ; y; ky) : x = xi(t); VT = Vj(t)g ;
where (xi(t); Vj(t)) is a node of the 2-dim grid on the plane (x; VT ). All sections Pyij(t) are
2-dim polygons and each polygon corresponds to a certain node of the grid. The IS Iz(t)
is constructed in the same way.
The evolution of the GPA-described IS between measurements consists of motion of the
grid's nodes and evolution of the polygonal sections.
After getting a new measurements at the instant tl, the GPA polygons are updated
according to the equations
P yij(tl) = P y
ij(tl�)\Hyi (l); P z
ij(tl) = P zij(tl�)
\Hzi (l);
where Hyi (�), H
zi (�) are the sets of uncertainty related to the measurement �l
Hyi (l) = f(y; ky) :
y
xi2 [�1l � c; �1l + c]g;
Hzi (l) = f(z; kz) :
z
xi2 [�2l � c; �2l + c]g:
Here and later the sign "{" after the time argument means left-side limit, for example
P yij(tl�) = limP y
ij(t); t " tl:
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IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS, VOL. XX, NO. Y, MONTH 2002 7
The evolution of the information sets on the interval [tl; tl+1[ is given by the equations
Iy(tl+1�) = Ly(u1l)Iy(tl); Iz(tl+1�) = Lz(u2l)Iz(tl);
where Ly(�) and Lz(�) { one-step extrapolation operators, which can be found from (1)-(7)
by integrating over [tl; tl +�] with some controls u1l and u2l:
V 0T =
VT1 + �TVT�
; (9)
x0 = x�1
�Mln (1 + �MVM (tl)�)
�kx�T
ln (1 + �TVT�) (10)
y0 = y �ky�T
ln (1 + �TVT�)
� u1l�� Vy(tl)�; (11)
z0 = z �kz�T
ln (1 + �TVT�)
� u2l�� Vz(tl)�; (12)
k0y = ky; k0z = kz: (13)
Here, the values of Vy(tl) and Vz(tl) are calculated from Eqs.(5), (6), the sign " 0 " marks
variables related to the information sets Iy(tl+1�) and Iz(tl+1�), i.e.
�x0; V 0
T ; y0; k0y
�2 Iy(tl+1�) and (x; VT ; y; ky) 2 Iy(tl)
Similarly
(x0; V 0T ; z
0; k0z) 2 Iz(tl+1�) and (x; VT ; z; kz) 2 Iz(tl)
Let's de�ne an IS eI ly(tn�) extrapolated to the end of the time interval [tl; tn[:
eI ly(tn�) = Ly(u1n�1) : : :Ly(u1l)Iy(tl): (14)
The set eI lz(tn�) is de�ned in the same way. Let's denote the sets eI ly(tn�) and eI lz(tn�)after applying the terminal impulses u1n and u2n as eI ly(tn) and eI lz(tn), respectively.It is necessary to emphasize that the extrapolation of the sets eI ly(tn�) and eI lz(tn�) is
done regardless to the future measurements taking only the future controls fu1l; : : : u1n�1g
and fu2l; : : : u2n�1g into account.
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IV. Auxiliary impulse control problem
The aim of the interceptor's control is to minimize the length of the terminal miss vector
� = (�y; �z). Approximately,
�2 = �2y + �2z = y2(T ) + z2(T );
where the terminal moment T is given by the condition x(T ) = 0.
Of course, the accurate values of the terminal miss and the moment T are unknown and
should be estimated from the current IS.
Let's introduce for each point A of the current IS a miss function �l(A) { a vector-
function with components �ly(A) and �lz(A). By de�nition, the function �l(A) gives the
terminal miss vector achieved on the controlled motion of the system which starts from
the point A at instant tl.
The idea is to average the function �l(A) over the volume of the extrapolated IS eI l(T )and take the obtained average �l = (�l
y;�lz) as an estimate of the real terminal miss.
Applying this idea to the IS Iy(t0) and Iz(t0) described by means of the GPA, let's
calculate �rst the expected time-to-go � 0ij for each node of the grid (and thus for each pair
of polygons P yij , P
zij). The integration of the dynamics equations gives
VT (t) =Vj
1 + �TVj(t� t0); VM (t) =
W0
1 + �MW0(t� t0):
The Eq.(2) gives for a node (xi; Vj) = (xi(t0); Vj(t0))
x(t) = xi �1
�Mln (1 + �MW0(t� t0))� (15)
�kx�T
ln (1 + �TVj(t� t0))
Then � 0ij is a root of the Eq.(15) when x(t) = 0, i.e.
xi =1
�Mln�1 + �MW0�
0ij
�+kx�T
ln�1 + �TVj�
0ij
�: (16)
Assuming that the values � 0ij are determined for all nodes having non-empty polygons
(for instance, by the numerical solution of the Eq.(16)) we can calculate the vector function
(�y(�); �z(�)) for arbitrary instant t = tl. Indeed, for an instant tl, expected time-to-go can
be recalculated as
� lij = � 0ij � (tl � t0): (17)
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If � lij < 0 for a certain node such a node is excluded from the further calculation. Let
(xi; Vj; y; ky) be a point A of the IS Iy(tl). Then the miss �ly(A) reads
�ly(A) = y � Vy(tl)�lij �
ky�T
ln�1 + �TVj�
lij
��
�X
tk2[tl;�l]
u1k�� lij � (tk � tl)
�;
where �l is a lower-bound estimate of the instant of the control process termination gov-
erned by condition (8). By de�nition,
�l = tl +�h� l"=�
i; (18)
where [a] { an integer part of a, � { a time interval between observations, � l" { a root of
the following equation
xlmin � " =ln�1 + �MVM(tl)� l"
��M
+kx�T
ln�1 + �TV
lmax�
l"
�;
where
xlmin = mini:P y
ij6=;;P z
ij6=;xi(tl); V l
max = maxj:P y
ij6=;;P z
ij6=;Vj(tl):
Then the expression for the averaged miss �ly (similarly for �l
z) takes the form
�ly =
1eQy
ZeIy(�l) �ly(x; VT ; y; ky) dx dVT dy dky; (19)
where eIy(�l) is the IS Iy(tl) extrapolated to the moment �l and eQy is a volume of the
extrapolated IS.
Now an auxiliary impulse control problem can be stated.
Problem 1: Let tl be a current instant and Iy(tl), Iz(tl) be information sets after mea-
surement �l. Let �l be the terminal instant estimated from the Eq.(18). It is required to
�nd the impulse control u(tk), tk 2 [tl; �l] in order to minimize the performance indices
�ly(u) =X
tk2[tl;�l]
u21k; �lz(u) =X
tk2[tl;�l]
u22k (20)
and satisfy the terminal conditions
�ly = 0; �l
z = 0: (21)
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V. Solution of the auxiliary control problem
Note that the auxiliary problem stated in the previous section can be decomposed into
two independent problems of control along Y and Z axes, respectively. Below we describe
the solution of the "Y"-problem only, the "Z"-problem is treated in the same way.
As follows from Eqs.(9)-(13), the extrapolation of the information set Iy(tl) according
to (14) reads
eVT =VT
1 + �TVT � l;
exi = xi �ln�1 + �MV
lM�
l�
�M�kx�T
ln�1 + �TVT �
l�;
ey = y � V ly �
l �ky�T
ln�1 + �TVT �
l��
�X
tk2[tl;�l]
u1k�� l � (tk � tl);
�eky = ky;
where (x; VT ; y; ky) 2 Iy(tl), (ex; eVT ; ey; eky) 2 eIy(�l), V lM = VM (tl), V l
y = Vy(tl) and � l =
� l(x; VT ) is calculated from Eqs.(16), (17) after substitution xi = x and Vj = VT .
The obtained extrapolation formulas de�ne a change of variables in the integral (19)
with the jacobian
Dl(x; VT ) =�1 + �TVT �
l(x; VT )��2
:
Then the �rst of the conditions (21) takes the form
ZIy(tl)
ey(x; VT ; y; ky)Dl(x; VT ) dx dVT dy dky = 0: (22)
Since the IS Iy(tl) is represented by the GPA, the integral (22) can be calculated in
two steps: �rst, Green's formula is applied to calculate integrals over polygons and then
the other two integrations (over dx and dVT ) are done on the grid fxi; Vjg using Euler's
formula.
First, let's calculate two auxiliary double integrals:
J1m =Z Z�m
(� � �) d� d�; J2m =Z Z�m
C d� d�; (23)
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where �m is a two-dimensional polygon de�ned by a set of its vertices f�p; �pg, p = 1; : : : ;m
and C is a constant. According to Green's formula,
J1m =Z Z�m
(� � �) d� d� =Z�m
��(d� + d�);
where �m is a boundary of �m. The last integral is a sum of m integrals J1m(p)
J1m =mXp=1
J1m(p) =mXp=1
Z p
��(d� + d�); (24)
where J1m(p) is an integral over a segment p with ends f�p; �pg and f�p+1; �p+1g, p =
1; : : : ;m. It is assumed that f�m+1; �m+1g = f�1; �1g. A segment p admits a parametric
description with � 2 [0; 1]
� = �(�p+1 � �p) + �p; � = �(�p+1 � �p) + �p: (25)
Then, it follows from (25) that
J1m(p) =Z 1
0(���p + �p)(���p + �p)(��p +��p)d� =
= (��p +��p)� (26)
�
"�p�p +
�p��p + �p��p
2+
��p��p
3
#;
where ��p = �p+1 � �p and ��p = �p+1 � �p. Similarly, Green's formula gives for the
integral J2m
J2m =Z Z�m
C d� d� =C
2
Z�m
(�d� � �d�) =mXp=1
J2m(p); (27)
where
J2m(p) =C
2
Z p
(�d� � �d�) =C
2(�p��p � �p��p) : (28)
To facilitate the further calculations let's introduce the following notations.
alij(k) =�� lij � (tk � tl)
� �1 + �TVj�
lij
��2;
blij =1
�Tln�1 + �TVj�
lij
�;
f lij =Z ZPyi;j(tl)
�y � kyb
lij
� 1�1 + �TVj� lij
�2 dy dky;hlij =
Z ZPyi;j(tl)
Vy(tl)�lij
1�1 + �TVj� lij
�2 dy dky ;
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where P yi;j(tl) is a polygon with vertices fyp; kypg, p = 1; : : : ;m; in the plane (y; ky) corre-
sponding to the GPA node with number (i; j) (Fig.2). Under �xed l; i; j and substitution
� = y and � = kyblij, the polygon P y
i;j(tl) transforms to the polygon �m with vertices
(�p; �p), where �p = yp and �p = kypblij, p = 1; : : : ;m;. In these notations,
f lij =1
blij�1 + �TVj� lij
�2 J1m; hlij =Vy(tl)� lij�
1 + �TVj� lij�2 J2m;
where J1m is de�ned by the Eq.(23) and calculated by formulas (24), (26), J2m is given by
the Eqs.(27), (28) with constant C = 1. Let's denote
glij(k) =Z ZPyi;j (tl)
alij(k) dy dky = J2m;
where J2m is given by Eqs.(27), (28) with constant C = alij(k) and substitution �p = yp,
�p = kyp.
As was mentioned above, the integrations over x and VT in (22) are performed on the
grid with the help of Euler's formula. Applying this formula and using the introduced
notations we obtain from (22)
�ly = F l �
Xtk2[tl;�l]
u1kGlk = 0; (29)
where
Glk =
Xi;j
�glij(k) + gli+1j(k) + glij+1(k) + gli+1j+1(k)
�;
F l =Xi;j
�F lij + F l
i+1j + F lij+1 + F l
i+1j+1
�;
where F lij = f lij � hlij.
Thus, we get a quadratic programming problem with the payo� function given by the
"Y"-part of Eq.(20) and the linear condition (29). This problem is solved in a standard
way by minimizing a Lagrangian
L(u; �) =X
tk2[tl;�l ]
u21k + �
0@F l �X
tk2[tl;�l]
u1kGlk
1A ;
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where � � 0 is a Lagrange multiplier. It follows from Kuhn-Tucker theorem that the
optimality condition for the control is
@L(u; �)
@u1k= 0; k : tk 2 [tl; �
l]: (30)
Then the optimal control sequence fu�1(tk)g is obtained from Eqs.(30) and (29)
u�1(tk) =GlkF
l
Sl; tk 2 [tl; �
l]; Sl =X
tk2[tl;�l]
�Glk
�2:
In particular, the optimal impulse to be applied immediately after the current measurement
is
u�1l =GllF
l
Sl:
VI. The guidance to a maneuvering target
So far the only one model of the target's motion is employed { the non-maneuvering
target whose dynamics is a pure deceleration due to the drag force. In this section,
we consider the case of a maneuvering decelerating target and propose how to treat the
target's maneuvers in the framework of the information-set approach. The basic idea
is to arti�cially increase size of the IS during the exrapolation between two neighboring
measurements in order to account for possible target's maneuver.
Let's suppose that the target can slightly change attack and sliding angles trying to
avoid collision with the interceptor. This can be described by introducing time-dependent
coe�cients ky and kz:
ky(t) = sin'+Z t
t0
ay(s)ds; (31)
kz(t) = cos' sin +Z t
t0
az(s)ds: (32)
In this model, ay(t), az(t) account for possible maneuvers. These functions are assumed
to be random and restricted by inequalities:
jay(t)j � Ay; jaz(t)j � Az: (33)
According to the proposed approach, the information sets are constructed as before
except for the extrapolation between measurements. The new extrapolation procedure is
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described by
Iy(tl+1�) = MyLy(u1l)Iy(tl); (34)
Iz(tl+1�) = MzLz(u2l)Iz(tl); (35)
where Ly(�) and Ly(�) are operators given by (9)-(13). The operators My, Mz perform
the linear transformation of the GPA's polygons in order to take target's maneuvers into
account. As follows from the model of maneuvers (31){(33), each point (y; ky) of a polygon
is transformed into an interval [(y�; k�y ); (y+; k+y )] due to the possible maneuver:
y� = y �AyL(VT ); k�y = ky �Ay�;
y+ = y +AyL(VT ); k+y = ky +Ay�;
where function L(�) is obtained by integrating equation (3):
L(VT ) =�
�T�
1
�2TVTln (1 + �TVT�) :
Geometrically, such transformation is simply stretching of the GPA's polygons along a
straight line with a slope L(VT )=� as it is shown in Fig.3
k y
L(V )T
y
∆
Fig. 3. Transformation of a polygon.
Note that the restrictions (33) are symmetrical with respect to zero. Due to this, the
miss averaged over the IS extrapolated according to (34), (35) is equal to the averaged miss
calculated for a non-maneuvering target. Therefore the same statement of the auxiliary
impulse control problem can be used for both maneuvering and non-maneuvering targets.
It means that the previously found optimal solution is valid for a maneuvering target too.
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VII. Monte Carlo (MC) tests of the guidance algorithm
The proposed algorithm was implemented in C program module and tested using Monte
Carlo method on PC Pentium III 500 MHz. The tests have been performed on a model
with parameters from Tabl.I:
TABLE I
Parameters used on the tests
x0, m VM , m/s VT , m/s �M �T
71000 3000 4750 1:5 � 10�5 2:0 � 10�5
The grid of the employed GPA consists of 51�51=2601 nodes:
xi = x� + i�x; Vj = V � + j�V ; i; j = 0; : : : ; 50;
where the start node (x�; V �) and the grid steps are chosen according to Tabl.II. The
TABLE II
Parameters of the GPA
x�, m V �, m/s �x, m �V
65000 4000 200 200
initial GPA polygons are rectangles de�ned by the restrictions:
y 2 [y�0 + �y�y ��y; y�0 + �y�y +�y] ;
z 2 [z�0 + �z�z ��z; z�0 + �z�z +�z] ;
ky 2 [sin('��'); sin('+�')] ;
kz 2 [cos('+ �') sin('� �'); cos('� �') sin('+ �')] ;
where y�0, z�0 are the initial positions such that they provide zero terminal miss on free
(i.e. with zero controls) motion of the "target-interceptor" system, �y, �z { random values
uniformly distributed over [�1; 1] interval. The other used parameters are presented in
Tabl.III.
For the Monte Carlo tests, the target's maneuver have been simulated as follows. At
some instants t , t', the angles , ' which determine a position of the target's velocity
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TABLE III
Parameters of the initial GPA polygons
�y, m �z, m �y, m �z, m
100 100 3000 3000
', o , o �', o � , o
{20 20 5.0 5.0
vector start changing with constant rates _ , _' (Fig.4). The maneuver durations are T ,
tψ
(t)ψ
ϕ(t)tϕtψ
tϕt t
ψ ϕ
+ ψT
+ ϕT
Fig. 4. Evolution of and ' during target's maneuver.
T' for angles , ' respectively. The parameters governed the target maneuver simulation
and chosen restrictions (33) are summarized in Tabl.IV.
TABLE IV
Parameters of the target maneuver simulation
t , s T , s _ , o=s Ay,s�1
5.0 0.2 0.2 3:5 � 10�4
t', s T', s _', o=s Az,s�1
5.0 0.2 0.2 3:5 � 10�4
The MC tests have been conducted for three di�erent values of restrictions imposed
on measurement errors (Tabl.V). For all the three tests, the time interval � between
measurements is the same { 0.1 s, the number of the Monter Carlo runs for each tests is
also the same { 1000.
The performance of the guidance algorithm has been evaluated according to the criteria:
1. the accuracy of guidance { r.m.s and bias of the terminal miss distributions My, Mz;
2. the impulse control consumption { r.m.s and means of the distributions of values Ry
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TABLE V
Measurement errors used on the tests
Test 1 Test 2 Test 3
cy, rad 0.001 0.002 0.003
cz, rad 0.001 0.002 0.003
and Rz:
Ry =Xtk<tf
ju1kj; Rz =Xtk<tf
ju2kj;
where tf is the instant of guidance process termination;
The computational properties of the guidance algorithm have been also investigated:
1. stability and rate of GPA convergence to the true values of x0 and VT ;
2. computational speed of the algorithm.
The rate and stability of the GPA convergence have been checked by investigating so-
called average GPA pro�les for di�erent instants of time. The GPA pro�le is de�ned as a
two-dimensional distribution:
Pij(t) =1
Nruns
NrunsXk=0
Ek(i; j; t);
where Nruns is the number of MC runs taken for averaging, Ek(i; j; t) equals 1 if, for k�th
MC run, both (Y and Z) polygons of the GPA node (i; j) are non-empty at the moment
t, and equals 0 otherwise.
VIII. Results and discussion
The results on the observed terminal misses and impulse control expenses for all three
tests are summarized in Tabl.VI. The distributions of the terminal misses My, Mz and
impulse control expenses Ry, Rz for Test 1 (measurement errors cy = cz = 0:001 rad
� 3:440) are shown in Fig.5. In Fig.5, the �lled histograms show distributions of the
impulse consumptions for the optimal guidance with perfect information, i.e. when target's
motion parameters are taken from Monte Carlo truth and then the interceptor follows the
trajectory that provides zero terminal miss and the minumum expense of the impulses.
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TABLE VI
Terminal miss and expense of control impulses
Value Test 1 Test 2 Test 3
My, cm 0:5� 9:8 0:6 � 19:2 0:7� 25:9
Mz, cm �0:4� 9:8 0:2 � 19:5 �0:5 � 25:2
Ry, m/s 43:8 � 8:4 58:2 � 11:8 66:5� 12:2
Rz, m/s 38:8 � 7:7 52:5 � 11:4 61:9� 13:0
Fig. 5. Test 1: distributions of My, Mz and Ry, Rz.
The results on the computing time are given in Tabl.VII. The correct convergence of
the GPA is illustrated by Fig.6, where the average GPA pro�les for Test 1 and di�erent
time instants are shown.
The obtained results show that
� The algorithm provides precise and unbiased guidance, the distributions of terminal
misses are symmetrical without pronounced tails for both, OY and OZ, control directions.
� The consumption of control impulses during guidance depends on measurement accuracy
and tends to the optimal solution with perfect information when the measurement precision
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TABLE VII
Total and partial CPU time consumption
Time, s Test 1 Test 2 Test 3
Mean total time 10:81 13:35 16:27
Polygon intersection 2:88 3:54 4:31
Auxiliary problem solution 4:42 5:46 6:56
Stretching and extrapolation 3:48 4:32 5:28
Fig. 6. Test 1: average GPA pro�les.
is increased.
� The terminal average GPA pro�le has a pronounced single peak corresponding to the
true values of x0, VT . It means that the algorithm is able to identify these unknown
parameters correctly and the GPA converges to them.
� The mean computing time of the algorithm depends on the measurement accuracy and
decreases when the accuracy is increased. It can be explained by faster convergence of
GPA in case of more accurate measurements.
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IX. Conclusion
The three-dimensional problem of guidance against a decelerating target with bearings-
only measurements has been considered. To solve the problem an approach based on the
information-set concept has been employed. The state of the "interceptor-target" system
has been described by an information set and the original control problem has been reduced
to an auxiliary impulse control problem to be solved after each measurement. The guidance
algorithm against both, maneuvering and non-maneuvering, targets has been developed.
The performance and computational properties of the algorithm have been evaluated using
Monte Carlo simulation. The results of the Monte Carlo tests have been presented and
discussed.
References
[1] M. Milanese, J. Norton, H. Piet-Lahanier, and E.Walter, Bounding approaches to system identi�cation, Plenum
Press, New York, 1996.
[2] A. Kurzhanski and I. Valyi, Ellipsoidal calculus for estimation and control, Birkhauser, Boston, 1997.
[3] S. Kumkov and V.Patsko, \Information sets in a model problem of homing," J. Optimis. Theory and Appl.,
vol. 108, no. 3, pp. 499{526, Mar. 2001.
[4] D. Emeliyanov, \Optimal impulse control of an information set in guidance with incomplete information,"
Automation and Remote Control, vol. 59, no. 3, pp. 28{35, Jan. 1998.
[5] D. Emeliyanov, E. Rubinovich, and B. Miller, \Advanced guidance law design based on the information-set
concept," in Proceedings of the 40th IEEE Conference on Decision and Control, Orlando, Florida, USA, Dec.
2001, IEEE, pp. 652{657.
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