Optimal Deployment Model for Near Space Bistatic Radar Network

Wang ShenShen 1,2,a, Che WanFang 2,b, Feng JinFu 1,c,

Wang FangNian3,d and Xie QiFeng3,e 1Engineering College, Air Force Engineering University, Xi’an 710038, China

2Key Lab of Complex Aviation System Simulation, Beijing 100076, China

3Guilin Air Force Academy, Guilin 541010, China

awangshenshen0616@163.com,

bchewf@163.com,

cjinfufeng@163.com,

dwfangnian@126.com,

exqifeng@163.com

Keywords: Bistatic radar network; Optimal deployment; Particle swarm

Abstract. The optimal deployment model and algorithm for near space bistatic radar network are

discussed in this paper. Firstly, the coverage rate is analyzed, and the optimal deployment model is

established. Then, the standard particle swarm algorithm is improved. Initial position and velocity of

every particle are created by the chaos algorithm, and the individual and global extremes are disturbed

with some probability in the optimization, which are advantageous in achieving a faster and

globalized search of particles. Finally, the improved particle swarm algorithm is used to solve the

optimal deployment issue. Simulation results demonstrate the validity of the proposed model and

algorithm.

Introduction

Near space has widely attracted attention because of its importance in development and application.

Currently, stealth plane mainly aims at detection of monostatic radar from some special area.

Construction of near space netted bistatic radar can sufficiently utilize the limitation of stealth plane.

The changing of radar cross section (RCS) of stealth plane from side and above area can reach 20~30

dB along with changing of angle of view [1]. If netted bistatic radars with different frequency in near

space are used to detect stealth plane and the information is combined by the data processing centre,

the detection probability can be effectively improved. Although some radar has been jammed or failed

to detect certain area, other radars can provide information and get more information than single

monostatic radar in common detection area.

Optimal deployment of near space radar network is important in the detection system, which is

directly related to the distribution of the detection ability in the space. The aim of optimal deployment

is to decide the position of every radar station to achieve the optimal performance. Presently, the

optimal deployment issue has been discussed in many references and some useful strategies have been

provided [2-6]. However, the research object in most references is monostatic radar network. Optimal

deployment of bistatic radar network is different from that of monostatic radar network. This paper

establishes the optimal deployment model of near space bistatic radar network, and an improved

particle swarm algorithm is proposed to solve the optimization issue. Some useful conclusions can be

drawn from the simulation results.

Optimal Deployment Model for Near Space Bistatic Radar Network

Suppose that near space bistatic radar network is composed of I transmitters 1 2

( , , , )I

T T T� and J

receivers 1 2

( , , , )J

R R R� , the responsible area includes surveillance area S and some highly concerned

area C, or called core area, azimuth angle of the target is min max

[ , ]θ θ θ∈ , elevation angle of the target is

min max[ , ]ϕ ϕ ϕ∈ . Optimal deployment for near space bistatic radar network can be described as when

performance parameters and numbers of transmitters and receivers are fixed, deploying the positions

of transmitters and receivers reasonably to achieve the higher coverage rate.

Applied Mechanics and Materials Vols. 157-158 (2012) pp 273-276Online available since 2012/Feb/27 at www.scientific.net© (2012) Trans Tech Publications, Switzerlanddoi:10.4028/www.scientific.net/AMM.157-158.273

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Coverage rate of radar network includes surveillance area coverage rate and core area coverage

rate. Surveillance area coverage rate is ratio between the surveillance area covered by radar network

and the surveillance area, and core area coverage rate is ratio between the core area covered by radar

network and the core area. As the target RCS changes with direction, whether the target can be

detected depends not only on the relative position of bistatic radar and the target, but also on the target

attitude. In order to make detection in core area has nothing to do with the target’s attitude, the

intersection of detection area corresponding to every attitude should be taken in core area coverage

computing. Surveillance area S and core area C can be divided into some sub-areas

{ ( ), 0,1,2, , }S t t I J= � and { ( ), 0,1,2, , }C t t I J= � , where ( )S t and ( )C t indicate the subarea with

overlap coefficient of t. Thus, we have [4]:

0

( )I J

t

S S t×

== ∪ , ( ) ( )S i S j φ=∩ ,

0

( )I J

t

C C t×

== ∪ , ( ) ( )C i C j φ=∩ , , , [0, ]i j i j I J≠ ∈ × . (1)

If being covered by T radars is regarded as effective coverage, the surveillance area coverage rate

and core area coverage rate of radar network are defined as:

[ ]( )

( )S

Volume S t T S

Volume Sη

≥=

∩,

[ ]( )

( )C

Volume C t T C

Volume Cη

≥=

∩. (2)

where ( )Volume i indicates the corresponding volume of the area; ( )S t T≥ and ( )C t T≥ indicate the

area with overlap coefficient of T. I transmitters and J receivers can form a total of I J× bistatic radar.

Suppose the k-th bistatic radar is composed of the ki -th transmitter and the

kj -th receiver, ( )S t T≥

and ( )C t T≥ are given by:

1 1

( )

TI J

t

C T

kk t

S t T S×

= =≥ = ∪ ∩ ,

max max

min min

( , )k kk i j

S Sθ ϕ

θ θ ϕ ϕθ ϕ

= == ∪ ∪ ,

1 1

( )

TI J

t

C T

kk t

C t T C×

= =≥ = ∪ ∩ ,

max max

min min

( , )k kk i j

C Sθ ϕ

θ θ ϕ ϕθ ϕ

= == ∩ ∩ . (3)

where T

I JC × is combination number of selecting T out of I J× ; 1 2, , , Tk k k� are the selected radar

number of the k-th combination; ( , )k ki jS θ ϕ is the detection area of the k-th bistatic radar related to

target with attitude ( , )θ ϕ . It is decided by the detection performance of bistatic radar and the RCS of

target, which can be obtained by the radar equation.

Since coverage rates of bistatic radar network have been calculated, optimal deployment model

can be established by minimizing the weighted sum of penalty functions:

1

,min ( , , , ) [ (1 ) (1 )(1 ) ]i j

c c c

S CD E R

L D E c w w wη η∈

= − + − − . (4)

where w and c are the weighted coefficient and weighted index; iD is position of the i-th transmitter

and jE is position of the j-th receiver; R is the range of position.

Improved Particle Swarm Optimization Algorithm

Improved particle swarm optimization algorithm is used to solve the optimal deployment issue. In

standard particle swarm optimization algorithm (PSO) [7], the speed and position of particle are given

by:

1 1 2() ( ) () ( )t t best t best tw r rand r rand+ = + − + −v v p x g xi i i i , 1t t t+ = +x x v . (5)

where w is inertia weight; 1r and 2r are acceleration constants; ()rand is a random number uniformly

distributed between 0 and 1; bestp is the best position of particle at time t; bestg is the global best

position at time t; 1 2( , , , )t t t tnx x x= �x and 1 2( , , , )t t t tnv v v= �v are position and speed of particle at time t.

274 Mechatronics and Applied Mechanics

Standard PSO has the following deficiencies: Random particle initialization can ensure uniform

distribution of the initial solution, but cannot guarantee the quality of the individual; when the global

extreme bestg or individual extreme

bestp is in the local extreme, the particle cannot re-search in the

searching space, and other particles will quickly move closer to the local optimal solution, leading to

early convergence phenomenon.

The basic idea of improved PSO algorithm contains the following aspects:

(1) Use chaos algorithm to initialize particle positions and velocities:

Step 1: Use ()rand function to generate a n-dimensional vector 1 11 12 1( , , , )

ns s s= �s , where

1is is a

random number uniformly distributed between 0 and 1, n is the number of variables in the objective

function. According to the chaos iterative equation, N chaos vectors 1 2, , ,

N�s s s can be obtained.

Step 2：Map the n components of ( 1,2, , )ii N= �s to the corresponding variable interval to

generate the initial positions of particles:

max min

min min

max min

( ), 1,2, ,i i

x xx s s i N

s s

−= + − =

−x � . (6)

where ix is position of particle;

maxs and

mins are upper and lower bounds of chaotic sequence;

maxx

and minx are upper and lower bounds of the particles.

Step 3: Calculate the fitness value of particles ( 1,2, , )ii N= �x , and select m particles which have

the best performance as the initial particles. Generate the initial particle velocity randomly.

The chaos initialization method is advantageous for choosing the best initial particles, which is

helpful for searching.

(2) Disturb the individual and global extreme with probability Pe and the new individual extreme

bestfp and global extreme

bestfg after disturbing are taken as the new optimization direction. Suppose

1 2( , , , )

ny y y= �y is a point to be disturbed, defined ( )F y as the degree of constraint violation

function, 2

1( ) (max{0, ( )})

m

i iiF w g y

==∑y , and disturbance of y is as follows:

Step 1: If ( ) 0F =y , the disturbed point is 1 2

( , , , )n

fy fy fy= �fy , where ( , )i i i i

fy y y N y δ= + ,

( 1,2, , )i n= � ; ( , )i

N y δ is normal distribution with mean iy and variance δ . If the value of δ is small,

it can be ensured that the disturbed point is not far from the feasible solution.

Step 2: If ( ) 0F >y , it means a constraint violation, and the disturbance operation should minimize

( )F y . The finally disturbed result can be obtained by doing one-dimensional search along the

negative gradient direction of ( )F y .

The negative gradient direction is 1 2

( , , , )n

d d d= �d , where [ ( ) ( )]i id F y v F yε ε= + − , ( 1,2, , )i n= � ,

iv is a n-dimensional unit vector (the i-th component is 1 and the remaining components are 0), 0ε >

and is sufficiently small. This ensures that the point after the disturbance is not far from the feasible

region and the quality of disturbance is improved. Disturbance probability is defined as:

1exp tan

max 2e

tP

r

π − = − ×

. (7)

where max r is the total iteration number. The disturbance probability is high at the beginning, which

is advantageous for extending the search scope. As the search progresses, the disturbance probability

decreases. The disturbance probability is close to zero in the end, which ensures that the algorithm

converges to the global optimum. So evolution equations of improved PSO become:

1 1 2rand() ( ) rand() ( )

t t best t best tw r r+ = + − + −v v fp x fg xi i i i ,

1t t t+ = +x x v . (8)

Applied Mechanics and Materials Vols. 157-158 275

Simulation Results

Suppose the bistatic radar network is compose of one transmitter and four receivers; RCS of the target

is 2 m2; the maximum distance products of bistatic radar 1 (the transmitter and receiver 1), bistatic

radar 2 (the transmitter and receiver 2), bistatic radar 3 (the transmitter and receiver 3) and bistatic

radar 4 (the transmitter and receiver 4) are 80000 km2, 65000 km

2, 90000 km

2 and 70000 km

2,

respectively; the surveillance area is an area of 1000 km × 600 km (x-coordinate 0~1000 km, y-coordinate 0~600 km); the core area is an area of 200 km× 230 km (x-coordinate 470~670 km, y-coordinate 220~450 km).

The changes of objective function obtained by the standard PSO and the improved PSO are shown

in Fig. 1. The optimal deployment of bistatic radar network and the coverage are shown in Fig. 2.

Fig. 1 Changes of objective function Fig. 2 Optimal deployment scheme

It can be seen from the simulation that the proposed algorithm provides better performances than

the standard PSO algorithm, because the proposed algorithm selects better initial particles and

disturbs the individual and global extreme. In addition, the proposed model can effectively solve the

optimal deployment problem for bistatic radar network. The weight may be adjusted to achieve

different forms of optimal deployment.

Summary

In this paper, optimal deployment model of bistatic radar network is established by the coverage rates.

The improved particle swarm algorithm is used to solve the optimal deployment issue. Simulation

results show that the proposed model and algorithm can effectively solve the optimization issue for

bistatic radar network, and the proposed algorithm provides better performance than the standard

PSO. The model can be further extended, e.g. considering the detection probability factor, location

accuracy factor, frequency factor, polarization factor, viability factor, et al. The proposed algorithm

can be applied if the index system is extended.

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276 Mechatronics and Applied Mechanics

Mechatronics and Applied Mechanics 10.4028/www.scientific.net/AMM.157-158 Optimal Deployment Model for near Space Bistatic Radar Network 10.4028/www.scientific.net/AMM.157-158.273