LI Bo,TIAN Linyu,CHEN Daqing,and LIANG Shiyang

1.School of Electronics and Information,Northwestern Polytechnical University,Xi’an 710129,China;2.School of Engineering,London South Bank University,London SE1 0AA,UK

Abstract: Real-time resource allocation is crucial for phased array radar to undertake multi-task with limited resources, such as the situation of multi-target tracking, in which targets need to be prioritized so that resources can be allocated accordingly and effectively. A three-way decision-based model is proposed for adaptive scheduling of phased radar dwell time. Using the model, the threat posed by a target is measured by an evaluation function,and therefore, a target is assigned to one of the three possible decision regions,i.e.,positive region,negative region,and boundary region. A different region has a various priority in terms of resource demand, and as such, a different radar resource allocation decision is applied to each region to satisfy different tracking accuracies of multi-target. In addition, the dwell time scheduling model can be further optimized by implementing a strategy for determining a proper threshold of three-way decision making to optimize the thresholds adaptively in real-time. The advantages and the performance of the proposed model have been verified by experimental simulations with comparison to the traditional twoway decision model and the three-way decision model without threshold optimization. The experiential results demonstrate that the performance of the proposed model has a certain advantage in detecting high threat targets.

Keywords: phased array radar resource scheduling, three-way decision,threat assessment.

1.Introduction

In recent years, the phased array radar technology has achieved a rapid development. Compared with the traditional mechanical radar, this type of radar can flexibly change the direction of the emitted wave to select a target to be illuminated, and adjust the parameters such as transmit power, dwell time and beam width. These features provide the possibility to manage and optimize radar resources, so that radar resources, such as dwell time, revisit time and beam width,can be managed reasonably to save radar power, improve accuracy of measurement and tracking,maximize the maximum number of tracking targets, save time and so on.Therefore,in a situation where only limited radar resources are available, such as multitarget tracking, it is indispensable to properly schedule radar resources in order to enable the phased array radar to track as many targets as possible with as fewer radar resources as possible.

There have been some studies on radar resource management or optimization. Common methods include resource scheduling methods based on covariance control[1–4],resource management algorithms based on tracking and filtering algorithms[5–7],and radar resource scheduling methods based on the target threat [8–10]. The resource scheduling method based on covariance control is computationally complex with slow operation speed and high system computational resource consumption. References[5]and[6] proposed an adaptive dwell time design method based on the interact multiple mode-probability data association (IMMPDA) tracking algorithm and the interacting multiple model particle filter (IMMPF) algorithm,respectively.In[7],a new method was proposed for calculating the target revisit time based on interacting multiple model (IMM) filtering algorithm.These three methods do not take into account factors such as target attributes and treat them equally.In contrast,radar resource scheduling methods based on the target threat do not have such problems.However, the target threat degree is mostly estimated and ranked according to ranking algorithms[11–13].In the case of a large number of targets,it is difficult to efficiently and accurately manage each target separately.Therefore, in this paper, it is proposed to classify targets according to the degree of threat to solve the above problems.

Among the classification methods based on target threats, the two-way classification method[14] is simple,but the classification results have low accuracy and poor results. The method based on Bayesian network[15–17]can directly output the probability of a target in a certain category.However,this method requires a large amount of training data or expert knowledge to obtain the conditional probability of target attributes.Therefore,the method cannot obtain the result of the target threat classification conveniently and accurately.In order to solve the problems of the existing methods and better classify the targets,this paper introduces the three-way decision theory[18–20].

Compared with the traditional two-way decision model with only two decision making options—a positive decision and a negative decision,the three-way decision model has a third option:a boundary decision.The three-way decision theory is frequently used to solve the problem of information uncertainty in various fields,and the problem of target uncertainty information exists in the classification process of the target threat assessment. Based on the three-way decision model,a target is assigned into one of the three regions, i.e., positive region, negative region, or boundary region.Compared with the Bayesian network algorithm, the three-way decision does not require a large amount of training data in principle.

One of the major issues of the three-way decision is to determine an appropriate set of thresholds. In the application literature[21–23], the thresholds of the three-way decision are mostly set to some fixed values obtained based on the degree of the classification loss set by experience or expert.This method is relatively subjective,and the accuracy of the results is not high. It cannot effectively adapt to changes in the environment.Therefore, this paper will use dynamic adaptive thresholds, and consider to include radar resource utilization in a cost function to calculate the optimal threshold at each time moment,and continuously update them.

Fig.1 shows the proposed radar time resource scheduling model.

Fig.1 Radar time resource scheduling model

The key idea here is that based on the three-way decision model,a target is assigned to one of the three regions,and each region corresponds to a different tracking accuracy.Then,we determine dwell time based on the tracking accuracy. This model can optimize thresholds of the current time point according to the scheduling result of the radar time at the last time point to re-classify targets and reallocate the dwell time. The model adjusts the dwell time allocation in real-time based on situational changes of a target.

The remainder of this paper is organized as follows.Section 2 presents the three-way decision model and the evaluation function for assigning a target into a proper region.Section 3 discusses in detail how to make adaptive adjustment of the threshold and how the proposed scheduling method works.The analytical experiments and the relevant results obtained are given in Section 4,and finally,the concluding remarks are summarized in Section 5.

2.Target classification

In this paper, it is proposed to assign each target in a multiple-target tracking situation into an appropriate region of three possible regions before assigning radar resources.This section describes in detail how to determine an evaluation function of a target and how to assign a target into one of the three decision regions according to the value of the evaluation function of a target.

2.1 Three-way decision model

The three-way decision was proposed by Yao in 2010 on the basis of the decision-theoretic rough set [24], and it introduces a third option of decision-making, namely no commitment or delay,so potential loss caused by false rejection or false acceptance of decision-making could be avoided. In comparison, in the traditional two-way decision approach,only acceptance or rejection is considered.

In the three-way decision,there is a domain defined by a finite non-empty setU.LetAdonate a finite set of condition attributes.Based on the condition setA,the main task of the three-way decision is to divide the entity setUinto three disjoint regions,denoted asPOS,NEGandBND,respectively,indicating the positive region,the negative region and the boundary region.

In practical applications,it is indispensable to construct the evaluation function which reflects the extent to which an entity in the entity setUmeets the condition setAand the specified thresholds of the classification for the threeway decision model[25].

Definition 1Given a subsetX ? U, an evaluation functionμ(x) and a pair of thresholdsαandβwiththe positive region,the boundary region and the negative region are defined as follows:

It is assumed that there is a state spaceΩ={X,?X}.LetXdenote high threat and?Xdenote low threat. Ac-cording to Definition 1, we can use an evaluation function for classification of targets. In this paper, we use the threat level of a target as the evaluation function to classify each target into one of the three possible decision regions:high threat region,low threat region and boundary region.The threat degree as an evaluation function is affected by many measures relating to the target, such as speed, distance, altitude, heading angle, and interference ability of a target. Therefore, each target can be represented by a feature vectorxi= (gi1,gi2,...,gij,...,gim), wheregij(j= 1,2,...,m) represents thejth feature of a targetiaffecting a threat.We can find a target threatμ(xi)by the feature vectorxiof a target.The specific algorithm is given in the next section.Based on an evaluation functionμ(xi),if a pair of thresholds 0β <α1 is introduced,then options for decision-making are as follows:

(i) Ifμ(xi)α, choose to accept,xi ∈POS(X),belonging to high threat;

(ii) Ifμ(xi)β, choose to reject,xi ∈NEG(X),belonging to low threat;

(iii)Ifβ < μ(xi)< α, choose not to commit or delay the decision,xi ∈BND(X), belonging to the boundary region.

2.2 Evaluation function model

In order to deal with any uncertain information of a target,such as the uncertainty of target situation information,and the uncertainty in the environment and meteorology, the evaluation function (threat degree) is usually obtained by an evaluation method based on intuitionistic fuzzy reasoning[26–28].

Definition 2There exists a set of interval numbersThere are two types of interval numbers:the benefit type(the bigger the better)and the cost type(the smaller the better).The algorithm for converting the interval numbers of the benefit and the cost type interval into intuitionistic fuzzy numbers is as follows.

For the benefit type:

For the cost type:

The membership degree of the interval number converted into an intuitionistic fuzzy number is

and the non-membership degree is

Definition 3There exists a set of real numbers for benefit typesxij(i= 1,2,3,...,n;j= 1,2,3,...,m).The membership degree and the non-membership degree of the benefit type real number converted to the intuitionistic fuzzy number are

The main steps of the evaluation function algorithm(threat degree)are as follows.

Suppose that there arentargets,and each target hasmattributes.

Step 1The target information,namely the target threat factor,is detected by the radar to obtain the target information matrix:f= (xi)n×1= (gij)n×m, wherexirepresents the feature vector of theith target,andgijrepresents the value of thejth attribute of theith target. The speed,distance, altitude and heading angle of the target are represented by an interval numberand the interference is represented by a real numbergij=xij.

Step 2For normalization, different types of data values need to be transformed into intuitionistic fuzzy numbers[29], and an intuitionistic fuzzy decision matrixF= (sij)n×mis obtained from (4), (5), and (6), wheresij= (μij,νij).μijrepresents the membership degree of thejth attribute of theith target, andνijrepresents the non-membership of thejth attribute of theith target.

Step 3Calculate the weights of target attributeω=[ω1,ω2,...,ωj,...,ωm]using the entropy method:

whereEjis the intuitionistic fuzzy entropy.

Step 4Calculate the weighted intuitionistic fuzzy matrixR= ([aij,bij])n×m, where [aij,bij] =ωj ·[μij,υij]=

Step 5Calculate the positive and negative ideal solutions of the weighted intuitionistic fuzzy matrixR. The positive ideal solution is the best solution of each attribute.The negative ideal solution is the worst solution of each attribute.

Positive ideal solution:

Negative ideal solution:

Step 6Calculate the target threat degree.Calculate the Hamming distancesandof each target to the positive ideal and the negative ideal according to the distance formula:

3.Phased array radar resource scheduling method based on three-way decision

3.1 Thresholds adaptation

The three-way decision can produce a pair of thresholds according to a cost function.The cost function in the threeway decision is a function relating to the classification cost.However,in practical applications,the cost function must consider not only the classification cost but also the cost of implementation of the decision rules[30].

This paper considers the cost of resource allocation of phased array radar after three-way classification,and therefore,the cost function of resource allocation costfis added to the cost function of the three-way classification costs:costsis related to the thresholdsα,βand the factorγbetween the thresholds;costfis expressed as the ratio of the sum of radar dwell timeTeffor each target to the working cycle timeTof the phased array radar.Hence,the cost function can be expressed as

The process of deriving the threshold is a typical problem of solving an optimization problem of a cost function.Simulated annealing is one of the popular methods to solve this kind of problems.The basic principle of the approach is to start from an initial solutioniand an initial valuetof the control parameter(temperature),and repeat the following steps for the current solution: (i) generate a new solution; (ii) calculate the difference in the fitness function; and(iii) accept or discard the solution, while gradually attenuating the valuet.The solution at the end of the algorithm results in an optimal approximate solution.The specific steps of solving the threshold are as follows:

Step 1Determine the initial temperatureT(sufficiently large), the lower limit temperatureTmin(sufficiently small), and the number of iterationsLfor eachTvalue. The fitness function of this paper is cost(α,β,γ)according to(7).

Step 2Randomly generate the initial solutionx0=(α0,β0,γ0), as the current best solutionxopt=x0, and calculate the fitness function value cost(xopt).

Step 3Do Steps 4–6 forl=1,2,...,L.

Step 4Make random changes to the current best solution to generate a new solutionxk, and then calculate the fitness function value of the new solution cost(xk)and the increment of the fitness function value Δcost =cost(xk)?cost(xopt).

Step 5When Δ<0, accept the new solution as the current best advantagexopt;otherwise accept the new solution as the current best advantage with a certain probability

Step 6If the termination condition is satisfied(l>Lor several consecutive solutions have not been accepted),the current solution is output as the optimal solution to obtain the thresholdsαandβ,and the program is terminated.

Step 7Tis gradually reduced by the rules ofTi+1=rTi(r=0.93)andT >Tmin,then go to Step 3.

Here we set the constraint of the solution to

3.2 Phased array radar resource scheduling rules

This section discusses different allocation rules for phased array radar resources allocation according to the three regions into which a target has been assigned.If a target has been detected[31,32],then certain radar resources should be allocated to the detected target to make it conform to a certain tracking accuracy.The greater the threat,the greater the tracking accuracy to be met;and in the meantime,certain resources should be allocated to other undetected targets so that they can be detected.

A radar detects, locates, and identifies the target based on the received echo energy. Assuming that the radar shares an antenna for transmitting and receiving targets,the echo power of a target with a distanceRtfrom the radar is

wherePtis the transmit power,Gtis the antenna gain of the radar,λis the wavelength of the laser,σis radar crosssection,andLris the radar loss factor.

For the phased array radar, the target echo power, calculated by a single phased array cell, is multiplied by the power partition coefficientTeof the purpose unit,accordingly the echo power of the target unit is

The power partition coefficient

whereCis the detection period coefficient.

The effective interference power spectral density is

wherePjis the interference power transmitted by the target aircraft,Gjis the antenna gain of the jammer,and Δfcis the interference spectral density. Sincethe interference signal enters from the main lobe of the radar antenna.

Therefore,the signal-to-noise ratio(SNR)is

whereNrrepresents the sum of the total noise power such as the radar internal noise power and the background noise power.

The target detecting probability is

This paper is mainly concerned with determining the radar dwell time of each target.The major steps are as follows:

Step 1Determine the detection period coefficient factorCand the azimuth tracking accuracyδθ:

wherea1,a2,a3,b1,b2,b3are constants.Candδθare determined according to the results of the three-way classifications:the greater the threat,the smallerCandδθ.Therefore,we havea1

Step 2TakeTef0=Tefmin.Tefminas the minimum value of the retention timeTef.

Step 3Calculate the expected SNR and detect probabilityPd.

Teis calculated by(10),and the SNR and the probability of discovery are obtained by (9), (11), (12) and (13),successively.IfPd

wherecis a constant, to recalculate the SNR andPd, untilis met,and then proceed to the next step.If the maximum value of the dwell time still does not meet the limit value of the detecting probability,proceed directly to the next step.

Step 4Select a proper dwell time.

The azimuth tracking accuracy is determined according to the classification result of the three-way decision on the target threat degree. Then calculate the azimuth standard deviationσΔθaccording toand compare it with the set azimuth tracking accuracyδθ. IfσΔθδθ, save the time in an array and go to the next step, or increase the dwell according to (7), and return to Step 3.If the maximum time of the dwell time is still not met, take the minimum dwell time and go directly to the next step.

Step 5Calculate the total dwell time of all targets.

Step 2 to Step 4 are repeated to obtain the dwell times corresponding to the respective targets and add them.

Step 6Calculate the threshold according to Section 3.1.

Step 7Reclassify the target based on the updated threshold and return to Step 1.

4.Simulation and analysis

This section provides experimental simulations to demonstrate the adaptive scheduling process of phased array radar dwell time in a multi-target environment based on the proposed model in Section 1. The process includes radar detection classification and dwell time allocation of multitarget. This experiment testifies the effectiveness of the adaptive model based on the three-way decision on phased array radar dwell time scheduling when phased array radar detects and tracks multiple targets.

In the experiment,a ground-radar model is established and 10 different maneuvering targets in the air are considered, each of which is conducting self-defense jamming to avoid the detection by the phased array radar. Earthfixed coordinate system with ground-radar as the origin is established.The target status is shown in Table 1.

Table 1 Status of an air target

It is known from Step 1 in Section 2.2 that the distance,speed,and angle of the target are expressed by interval values.This involves measurement errors of a radar.Interval values are obtained by superimposing error signals on the basis of real values,and these errors satisfy white Gaussian noise.

The standard deviation of the radar distance error is

wherecis the speed of light andτis the pulse width.

If the pulse Doppler identification is used to measure the speed,the standard deviation of the speed error is

where Δfis the resolution of the Doppler frequency.

The standard deviation of the angle error is

whereθ0.5is the half-power width of the antenna beam.

A random numberlobeyingN(0,σ)is generated.The interval number of target measurement information is expressed as[x?|l|,x+|l|],whereσis the standard deviation of each error,andxis the real value of targets.

In order to demonstrate the effect of the three-way decision and the selection of the threshold,the performance of the phased array radar has been compared in the following three modes of radar resource scheduling:

(i)Using the three-way decision threshold model in Section 3.1 to realize threshold adaptive change,the targets are assigned into three regions under the real-time threshold:

i) The positive region(high threat)with a required azimuth tracking accuracy of 0.1;

ii)The boundary region with a tracking accuracy of 0.2;

iii)The negative region(low threat)with a tracking accuracy of 0.3;

(ii)The three-way decision classifies the target at a fixed threshold (0.6, 0.4), which requires the azimuth tracking accuracy to be the same as 1;

(iii)Under the traditional two-way decision,the targets are simply assigned into high threat and low threat according to the threshold of 0.5, and the standard deviations of the tracking angle error are required to be 0.1 and 0.3,respectively.

During the 120 s simulation, we assume that the radar transmit peak power is 16 MHz,the radar center frequency is 3 GHz,the antenna gain is 35 dB,and the loss factorsLrandLjare 4 and 3.The dwell time meets0.2,and the increment of adjusting is 0.01 s.

Fig.2 shows the adaptive change of the three-way decision thresholds in Mode(i).

Fig.2 Change of three-way decision thresholds

Then, a comprehensive comparison can be made with regard to the working efficiency of the radar from various aspects including the cumulative detection probability of the target and the utilization of the radar resources under three-way decision and traditional two-way decision. We also explore the effect of threshold on three-way decision,the necessity of resource management allocation and the feasibility of the allocation method.

For the situation of target detection, the time taken for the cumulative detection probability of the target to be 1 is used as the judgment standard,due to the large simulation time.Fig.3 shows the result.

Fig. 3 Time taken for the cumulative detection probability to be 1 in three modes

The total dwell time reflects the scheduling situation of radar time resources. The results in the three modes are shown in Fig.4.

Fig.4 Utilization of radar time resources in three modes

As shown in Fig. 3, in addition to Target 4, the time taken for the cumulative detection probability of the target to be 1 is longer and the target detection and tracking has a poor effect under the two-way decision for the other targets. Fig. 4 shows that the radar dwell time of the two decisions is much less than that of the three-way decision.To summarize,the results indicate that the radar dwell time scheduling method under the two-way decision has a problem in that it cannot allocate enough resources to the target,and the radar time resource cannot be effectively utilized,which would be harmful to the tracking and interception of the target under the two-way decision.

In modes(i) and (ii), the results of the target threat assessment under the three-way decision are shown in Fig.5 and Fig.6,where the threat degree of 0 means that the target is lost at that moment.

Fig.5 Target threat in Mode(i)

Fig.6 Target threat in Mode(ii)

We compare Mode(i)and Mode(ii)to explore the role of thresholds in the three-way decision.As can be seen in Fig.4,the change of the radar dwell time under the threeway decision with adaptive thresholds change is stable.However, the radar dwell time curve under the three-way decision with fixed thresholds is a bit turbulent within 0–40 s, and then the radar dwell time curve tends to be smooth.This is because the classification results for targets are different in the two modes: In Mode (i), the adaptive optimization of the threshold can find the optimal threshold in real time to classify targets and allocate the dwell time reasonably,so that the radar can track the target stably.In Mode(ii),as shown in Fig.6,the targets are divided into three regions according to the threshold(0.6,0.4),and the detection and tracking of the targets is unstable within 0–40 s.After 40 s in the simulation,all targets are in the boundary region due to the change of threat degrees, and the target tracking is more stable,so the dwell time curve is relatively smooth.The above analyses show that the selection of the thresholds will affect the classification of targets and the radar dwell time allocation,and this consequently will affect the target tracking.

In addition, comparing the dwell time curves under Mode(i)and Mode(ii)in Fig.4,it is apparent that in 0–40 s, the dwell time used in Mode(ii)is significantly less than the time used in Mode(i).The key to seizing the initiative in modern warfare is the right to information,so it is crucial to get effective information quickly in the early stages.As shown in Fig.3,in Mode(i)and Mode(ii),there is no significant difference in terms of the time taken for the cumulative detection probability of the target to be 1 except for Target 6 that needs more time in Mode(ii).Fig.5 shows that Target 6 has the greatest threat within 0–20 s.It can be seen that Mode(i)can make full use of time resources to enable the radar to quickly detect targets in the early stage, especially the targets with high threats, and facilitate the rapid acquisition of the initiative on the battlefield, so the adaptive optimization of thresholds has a certain advantage in detecting and tracking high threat targets.

In summary, the algorithm proposed in this paper can make full use of radar time resources to make phased array radar detect and track targets more stably and efficiently,especially for high-threat targets.Therefore,the simulation results in all the three modes have collectively illustrated the effectiveness of the proposed phased array radar dwell time scheduling model.

5.Conclusions

This study aims to address the problem of dwell time scheduling of phased array radar in target tracking, and a phased array radar dwell time scheduling model based on three-way decision has been proposed.Compared with the traditional two-way decision, the proposed model can potentially avoid waste of resources and/or insufficient of radar time resource scheduling,so that the radar can track targets better and prevent the target from being lost. In order to further optimize the dwell time of phased array radar, an adaptive optimization model of three-way decision thresholds is also established,which implements realtime scheduling of phased array radar dwell time.In addition, the adaptive optimization of thresholds has a certain advantage in detecting high threat targets.The simulation results show that the model can improve the accuracy of phased array radar dwell time scheduling in multi-target tracking effectively.

Note that, compared with the method in Mode (ii), the proposed algorithm has no obvious advantage for the detection and tracking of the target with less threat. Further improvements, such as the improvement of the threshold solving algorithm, will be made to improve the performance in the future.