information set-based guidance algorithm against a decelerating maneuvering target

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:

[email protected])

B. Miller is with the Institute for Information Transmission Problems, 19 Bol'shoi Karetnyi per., 101447 Moscow,

Russia. (e-mail: [email protected])

July 25, 2002 DRAFT


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

July 25, 2002 DRAFT


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


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 =

July 25, 2002 DRAFT


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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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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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.

July 25, 2002 DRAFT


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)

July 25, 2002 DRAFT


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)

July 25, 2002 DRAFT


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)

July 25, 2002 DRAFT


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

July 25, 2002 DRAFT


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.

July 25, 2002 DRAFT


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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information set-based guidance algorithm against a decelerating maneuvering target

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