Uncertain Hybrid Multi-Sensor Alliance Dynamic Control Problem Using an Uncertain Ideal Point Approach Under the PEV Principle

Multi-sensor alliance has been the main means of gathering intelligence information in the complex battlefield environment. This paper investigates uncertain multi-sensor alliance dynamic control problem, which is a major concern in sensor management. Unlike earlier studies, the explored model explored here is based on uncertain circumstances instead of certain circumstances, and constructed which is established under the constraints of sensor tracking ability in order to improve the sensor resource utilization and target tracking accuracy. Specially, this uncertainty model is converted to a certainty model by using an uncertain ideal point approach under the PEV principle instead of the traditional solution approach. Also, the prediction and re-prediction mechanism is proposed to realize the dynamic control of the sensor alliance updating. The proposed methodology possesses high solution quality and a low computational cost; it also reduces the tracking errors of the target in uncertain battlefield environments when. Finally, simulation results are provided to demonstrate the feasibility and effectiveness of the proposed approaches.


I. INTRODUCTION
Cooperative multi-sensor detection has a wide range of applications in intelligence information gathering, surveillance, reconnaissance, and many other fields [1]- [4], and it also plays a significant role in target detecting and, target tracking, among other activities. An essential cooperative pattern of multi-sensors is to build a multi-sensor alliance, and the cooperative pattern is a dynamic process. Moreover, the multi-sensor alliance dynamic control problem becomes an uncertain nonlinear system control problem [5]- [9] when considering the complexity and uncertainty of the battlefield environment, including shifting targets.
With regard to the uncertain dynamic control problem, three steps should be taken: the first is to establish an uncertain multi-sensor alliance model-in other words, introduction of an objective function obeying particular rules. The second is to obtain the optimal scheme from the objective function by traversing through all potential solutions or using The associate editor coordinating the review of this manuscript and approving it for publication was Hasan S. Mir. optimization algorithms. The last step is alliance updating, using an alliance update mechanism after establishing the alliance. When modeling the objective functions, current existing approaches can be roughly classified into three types, namely, an approach using uncertain multi-objective programming, an approach establishing a multi-sensor alliance establishing, and an approach updating the multi-sensor alliance.
With regard to the first approach, the traditional solution approach [10]- [13] is to convert the original problem into a definite multi-objective programming problem, and then convert it into a definite single objective programming problem through the classical multi-objective programming method. In the second approach, most of the establishment of the multi-sensor alliance is based on serial structure [14]- [19]. The main problem of this structure is its low efficiency and the possibility of resource waste or excessive strain. Earlier work on multi-sensor alliance establishing problem has included studies of swarm intelligence algorithms [20]- [22], [22]- [26], represented by particle swarm optimization, and a series of other algorithms, such as a linear programming VOLUME 8, 2020 This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/ method [27] and an auction algorithm [28]. One study [29] proposed particle swarm optimization (PSO) as a method to solve the alliance formation problem. Similarly, another study [30] used discrete particle swarm optimization (DPSO) to realize alliance formation, which partially confirmed the effectiveness of swarm intelligence algorithm. In the third approach, in the process of investigating multi-sensor alliance updating mechanism, one group proposed a mechanism for measure and update, another group proposed a mechanism for prediction and update, which could be better compared with the previous one. After careful review of previous studies, it can be seen that three problems in the multi-sensor alliance dynamic control problem still need to be revolved. The first problem is that most of the cases using the traditional solution approach are applied for the none-related objective functions and the traditional approach ignores the uncertain nature of the uncertainty problems. The second problem is that swarm intelligence algorithms do not consider dynamic changes in the environment and the problem of consumption in the process of establishing the alliance. The third problem is that the existing alliance-updating mechanisms can lead to loss of the target because of rapid movement of the target in real-life applications; these mechanisms also do not take into account dynamic changes of the battlefield environment, which could increase the risk of losing targets in the updating process.
Based on the above analysis, the main contributions of this paper can be summarized as follows: 1) The uncertain multi-objective programming problem is transformed into the uncertain single objective programming problem by using an uncertain ideal point approach; and the uncertainty is the further realized under the P EV principle. 2) The fireworks algorithm based on the improved selection strategy is designed to acquire an effective solution to the process of establishing the alliance. 3) An updating mechanism of prediction and re-prediction is proposed to complete the updating task of the alliance and realize stable tracking of the target. The structure of the rest of the paper is organized as follows. In Section II, the uncertain hybrid multi-sensor alliance model is provided, including a description and mathematical model of the target motion model and of the objective functions and constraint conditions. Section III introduces solution methodology under the P EV principle to transform the uncertain model to a certain model. In Section IV, we present a fireworks algorithm, based on an improved selection strategy (ISSFA), to establish the alliance, and we propose a prediction and re-prediction mechanism to update the alliance. Finally, we conduct simulations a to verify the proposed model and algorithms, and present conclusions in Section V.

II. UNCERTAIN HYBRID MULTI-SENSOR ALLIANCE MODEL A. TARGET MOTION MODEL
The uncertain hybrid multi-sensor alliance assumes that each sensor has the ability of independent decision-making. The model can also change dynamically according to changes in the environment to take the greatest advantage of the alliance and to accomplish tasks efficiently. In the process of building an alliance with multi-sensors, there are two main modes. One is that a sensor can join multiple sensor alliances or that the task can be performed by multiple sensor alliances, [ Figure 1(a) below]. The other is that a sensor alliance can handle multiple tasks at the same time [ Figure 1 Suppose that the state of the target is X k = [x (k) , y (k) ,ẋ (k) ,ẏ (k)] T , the state transition matrix of the target is X k+1 = X k + W k , and the observation equation of the sensor is Z k = HX k + V k , where , is the state transition matrix of the target and the distribution matrix of the process noise; W k , V k is the system and the observation noise, the W k covariance matrix is Q k , and the V k covariance matrix is R k . The initial sensor alliance formed when the sensor tracks the target at a certain time k is s initial = {s 1 , s 2 , . . . , s M } In the process of target tracking, extended Kalman filter (EKF) is used to estimate the target state. The iterative formula of Kalman filter is: (1) The uncertain objective function f j (t, ξ ) and uncertain constraint conditions g i (t, ξ ) are the functions under consideration when forming the alliance, where the objective function is the binary function of the time and the uncertain variable.
Because the process of forming an uncertain hybrid multi-sensor alliance will be affected by many non-determinants, it can be analyzed based on the expert reliability of the actual frequency of the event. The performance indicator of sensor performance in a complex dynamic environment is based on the relative energy-consumption value of the sensor, the detection range of the sensor, and the threat level of the sensor, where the three kinds of uncertainty are defined as the uncertain variable in the uncertain space ( , L, M), and then the expert reliability is given by the expert.

C. OBJECTIVE FUNCTIONS AND CONSTRAINT CONDITIONS
The objective function of an uncertain hybrid multi-sensor alliance model is as follows:

1) THE SHORTEST TIME REQUIRED TO ESTABLISH AN ALLIANCE
The minimum time it takes to form a sensor alliance to perform a task, indicating that each task is implemented by the recent sensor alliance, greatly improving the allocation of sensor resources and completion of the task. Based on expert reliability analysis, the sensor's relative energy value and the threat level detected by the sensor will have a large impact on the time required for forming an alliance, and the following target functions can be established as follows, where t ij represents the time every formed alliance used.

2) THE LEAST CONSUMPTION REQUIRED TO ESTABLISH AN ALLIANCE
The consumption of hybrid multi-sensor alliance to complete the task mainly includes the consumption of sensor itself and the consumption of each sensor. Assume that every sensor's consumption is c j (j = 1, . . . , M ), and that every sensor alliance is marked as A M ×N , where a ij = 1 represents the sensors tracks the target and a ij = 0 represents the sensors don't tracks the target. The sensor's detection distance and the threat of the sensor have a large impact on the formation of a dynamic alliance, which can be established as follows: 3

) THE HIGHEST TRACKING ACCURACY REQUIRED TO ESTABLISH AN ALLIANCE
Assuming that {p 1 , p 2 , . . . , p M } is the tracking accuracy factor corresponding to each sensor, the tracking accuracy factor is defined as follows: where P i (i = 1, 2, . . . , n) is the error matrix of least squares estimation of the target state based on the i th sensor, and tr (·) is the trace of the matrix. Based on expert reliability analysis, it can be concluded that the degree of threat level and the relative energy consumption value of the sensor have a large t influence on formation of the alliance, with the objective function defined as follows: The constraint conditions are as follows: (1) Whether the sensors join the hybrid multi-sensor alliance to realize the target tracking process, where a ij = 1 represents the situation where sensors track the target and a ij = 0 represent the situation where the sensors do not track the target.
(2) Each sensor alliance can undertake a maximum number of tasks, where N max represents the maximum alliance number, M max represents the maximum task number.
(3) The number of targets that each sensor can track is limited and cannot exceed the maximum allowable range σ M . VOLUME 8, 2020 The mathematical expression of the constraints is shown in Eq. (10).

III. SOLUTION METHODOLOGY UNDER P EV PRINCIPLE
The uncertain multi-objective programming model can be expressed as the equation below.
where x ∈ R is the decision variable, suppose each objective function has the same uncertain variable, ξ i1 , ξ i2 , . . . , ξ im is an uncertain vector, in which each component is an independent uncertain variable, and all uncertainties are defined in uncertain space.
Due to the fact that the uncertainty factors remain constant in the process of building the uncertainty hybrid multi-sensor alliance, the uncertain vectors are the same, that is, the objective functions are related, where the uncertain variables could have only two possible effects on the objective functions. One effect is a monotonic increase at the same time, and the other is monotonic decrease at the same time; for example, the detection range of the sensor change will lead to the consumption and time of sensors establishing; the change in threat level of the sensors will affect the tracking accuracy and the time of formation of the sensor alliance.
In practical engineering applications, the specific eigenvalues of different objective functions have different meanings; thus, it is necessary to determine the specific significance of the specific eigenvalues of different objective functions through several order-relation criteria. Because the uncertain objective function itself is also an uncertain variable, this paper uses the symbols ≺ and ≺= to define the relationship between the uncertain variables. The order relation principle usually includes an expectation principle and an expectation variance principle, which are defined as follows: Definition 1 (Expectation Principle, P E Principle): Suppose ξ and η are two uncertain variables, if and only if Definition 2 (Expectation-Variance Principle, P EV Principle): Suppose ξ and η are two uncertain variables, if and Because the P EV principle has a wide range of practical engineering applications, this study has incorporated the P EV principle in designing a solution to the uncertain hybrid multisensor alliance problem.
However, uncertain programming problems often have inherent uncertainty and the objective functions are not completely independent of each other. Therefore, we have used the uncertain ideal point approach to solve this kind of problem.
The uncertain ideal point approach is used to transform the uncertain multi-objective programming problem to a uncertain single objective programming problem by optimizing the distance from each objective function point to the optimal ideal point. Applying the approach of uncertain ideal point, we obtain: In which uncertain constraint conditions g i (x, ξ ) can be transformed into certain constraint conditions by uncertain measurement and expert reliability, and f 0 i is the lower boundary when the objective function f j (x, ξ ) does not consider other objective functions.
Under the P EV principle, the definition of the effective solution to the uncertain multi-objective programming problem can be expressed as theorem 1.
Theorem 1: The optimal solution x * to uncertain single-objective programming problem under P EV principle is the P EV effective solution to the uncertain multi-objective programming problem.
We can obtain that When j = j 0 ,we can obtain that Then, we find that where, is an integer greater than one. That is to say, U (x, ξ ) ≺ U (x * , ξ ), which is contradicts with previous hypothesis, The Proof Is Completed: From the proof above, we can obtain the solution process of the uncertain hybrid multi-sensor alliance in Figure 2.
Through the approach of uncertain ideal point, the multi-objective programming problem can be transformed into the single objective programming problem. The specific form is as follows: (17) where, T 0 , C 0 and P 0 are the lower boundary of T (t, ξ ), C (t, ξ ) and P (t, ξ ) on the sequence of the effective solution sets.
The uncertain single objective programming problem can be further transformed into the corresponding equivalent deterministic model under P EV principle.
Taking ξ 1 as an example, after inviting experts to evaluate the impact of the relative energy consumption of sensors on the formation of multi-sensor alliances, they think that with 100% reliability, the impact of the relative energy consumption of sensors on formation of alliance will be less than 18. At the same time, experts believe that with 0% reliability, the relative energy consumption of the sensor will have less than a 1.5 impact on formation of the alliance. It is assumed that the degree of influence of relative energy consumption on the sensor is linear, with expert reliability t within the numerical range, and the specific uncertainty distribution is as follows: The same can be obtained: In order to describe it clearly, we denote: (17) can be transformed into a problem of determining single objective planning under the P EV principle: According to the three objective functions, T (t, ξ ) , C (t, ξ ) , P (t, ξ ) can increase strictly monotonically with respect to ξ 2 and ξ 3 , and decrease with respect to ξ 3 . Thus, we can plug the lower bound of ξ 2 and ξ 3 , and the upper bound of ξ 1 into three objective functions.

IV. DYNAMIC CONTROL PROCESS OF HYBRID MULTI-SENSOR ALLIANCE MODEL A. ALLIANCE ESTABLISHING PROCESS
When the original problem is transformed into a deterministic hybrid multi-sensor alliance formation problem, the fireworks algorithm based on an improved selection strategy (ISSFA) is designed to solve the problem.
The fireworks algorithm [33]- [35] is a mathematical model based on the abstraction of the phenomenon of fireworks explosion in nature. In this algorithm, the fireworks lack of information exchange mechanism between fireworks and insufficient use of the guidance information of optimal fireworks, but the improved algorithm ensures that high quality sparks are selected for the next generation of fireworks and the diversity of fireworks. The fireworks chosen under this strategy are the most marginal sparks, and can continuously explore out to the edge.
Suppose X (t) = [x 1 , x 2 , . . . , x i , . . . , x N ] is the initial fireworks set of the first iteration, where N is the number of the fireworks; x i ∈ R D is the information of the first firework in the solution space, and its fitness is f (x i ) . Each firework particle x i is exploded to produce a spark particle set Y i (t) = [Y i,1 , Y i,2 , . . . , Y s,i ] with the number of s i . y i,j ∈ R D has the same dimension as x i , the formation process of spark particle y i,j can be expressed by Eq. (20): where B is a 1 × D -dimensional random matrix, the value of matrix elements is 0 or 1, and A i is the explosion amplitude of firework particle x i , which can be expressed as: whereÂ is a constant to restrict the biggest explosion amplitude of firework particle, ε is a minimal constant to avoid zero. The sparks number s i produced by each firework x i is decided by Eq. (22) as follows, where m is a constant to restrict the total number of the sparks in order to avoid too much or too few sparks from the explosion.
where s i is the number of sparks that the first fireworks would eventually produce, round[] is the rounding bracket function, a and b are given constants. In order to further improve the diversity of spark population, the Gauss mutation process is introduced in the process of solving the fireworks algorithm. p(0 < p ≤ N ) sparks are randomly selected from the X (t) set, and the Gauss mutation operation is carried out according to Eq.(24) to generate the Gauss mutation spark particle set.
where x h is the fireworks selected randomly from the X (t) set; B is a 1 × D -dimensional random matrix, the value of matrix elements is 0 or 1; g is a random number obey g ∼ N (0, 1) . In order to prevent the newly generated spark particles from exceeding the search range, the fireworks algorithm uses the mapping rule of modular operation to pull the sparks beyond the feasible range. When the sparks particle y i,j exceeds the feasible range, Eq.(25) is used to calculate. (25) where y d i,j is the position of spark particle y i,j on the ddimension, y d min and y d max are the upper and lower search boundaries of the d -dimension, and %is the modular operation.
At the end of each iteration, the fireworks algorithm selects N particles from the set W (t) = {X (t)∪Y (t)∪Z (T )} as the initial fireworks of the next iteration. The best individual in the particle swarm is retained and the other N − 1 fireworks are selected by roulette gambling. The probability of individual ω i ∈ W (t) being selected is based on distance. The calculated equations are as follows.
where L(ω i ) is the sum of the distance between individual ω i and other individuals, using a Euclidean distance measure; and P(ω i ) is the selected probability of individual ω i . Assuming that f i (i = 1, 2, . . . , n) denotes the fitness value of the x i spark, and makes it normalization.
where f i is the transferred meaning fitness value; as in a minimize optimization problem, the larger the transferred meaning fitness value, the better the sparks. The definition of the distances among the sparks rely on the transferred meaning fitness value as follows.
Defining the sparks distance normalization and calculating the product of the transferred meaning fitness value and  the sparks distance: We can call the sparks as the exploratory sparks when they follow the Eq. (33).
According to Eq.(32-33), the improved selected strategy can select the next generation's N fireworks. In these fireworks, the best fitness value of previous generation is selected, and the area where more sparks exist and has the lowest fitness value could be selected, avoiding selecting the same similar spark; due to the existence of the exploratory spark, the sparks with better global search ability will also be selected.
The algorithm pseudo code is shown in Table 1 below. According to the pseudo code of the ISSFA above, it is necessary to analyze the complexity of the algorithm.
According to the pseudo code of the ISSFA above, it is necessary to analyze the complexity of the algorithm.   Assuming that the number of the sparks is m, and the complexity of step13, 15, 17 is O (m), step 14 should estimate the fitness value first, the complexity is O (m log m), next is to calculate the distance among the sparks, after calculating, the complexity now is O m 2 , so the final complexity of step 14 is O m 2 . Because the sequence has already been arranged, the time to select the peak spark only costs a few constant times. To sum up, the complexity of ISSFA is O m 2 .

B. ALLIANCE UPDATING PROCESS
After the establishment of the uncertain hybrid multi-sensor alliance, each sensor sometimes can not respond to the operational requirements in time because of the large coverage of the early-warning detection network composed of multisensors,. At the same time, estimation of the state of the target is changing constantly in a dynamic environment. When the actual tracking accuracy of the time target can not reach the predicted accuracy, it is easy to lose the target in the process of alliance updating. The prediction is made by filtering VOLUME 8, 2020   algorithm, and the moving state of the target at any time will appear in the prediction range, which is unable to track the target. Therefore, we propose a mechanism, called prediction and re-prediction, to reduce the probability of losing the target and better to achieve updating of the alliance, that is, at moment k, because the state estimation of sensor and target is dynamic. When target tracking cannot be achieved within the prediction range, the prediction fails, thus, the dichotomy approach should be used to reduce sampling time, and predictions should be repeated until tracking accuracy is sufficient to meet the conditions. The updating algorithm of the hybrid multi-sensor alliance is shown in Table 2.

V. SIMULATION
A. PARAMETERS SETTING Figure 3 (below) shows the deployment information of the three targets and eight sensors, and Figure 4 shows the performance indicators of each sensor and target. Table 3 shows the remaining parameters of the deployed sensors.
The initial parameters of the EKF algorithm are set as follows: Covariance matrix of observation noise Parameters settings of the deployed sensors are shown in Table 3. Parameters settings of the improved selection strategy of fireworks algorithm (ISSFA) are shown in Table 4.

B. SIMULATION OF ALLIACNE ESTABLISHING
In order to verify the effectiveness of the proposed algorithm, we used the basic fireworks algorithm [34] (FWA) and discrete dynamic particle swarm optimization algorithm [35] (DDPSO) for comparative analysis and discussion. Applying the parameter settings of the improved selection strategy of fireworks algorithm, the solutions are shown in Figure 5.
As shown in Figure 5, the three algorithms have the same downward trend, and finally converge to 3.09596 min, 1.20661 and 0.54308 respectively. But the convergence rate of these three algorithms is different for each. The ISSFA algorithm converges to the optimal value quickly, but the FWA algorithm converges slowly, the optimization effect is poor, and it is easy to fall into the local optimal solution. Like the ISSFA algorithm, the DDPSO algorithm can find the global optimal solution, but its convergence speed is not as fast as that of the ISSFA algorithm. The results show that the ability of ISSFA to prevent falling into a local optimum is better than that of DDPSO and FWA, and thus more conducive to solving complex optimization problems.
After the solution of the three parameters is obtained, the three parameter values can be used to calculate the formation scheme of multi-sensor alliance under the P EV principle. Figure 6 shows the sensor response sequence number within the multi-sensor alliance using the traditional approach. Figure 7 shows the sensor response number in a multi-sensor alliance using the uncertain ideal point approach.
It can be seen from Figures 6 and 7 that both the traditional solution approach and the uncertain ideal point approach can generate hybrid multi-sensor alliances, and the sensors play some role in the entire process of detection. In comparison, the hybrid multi-sensor alliance scheme obtained by the uncertain ideal point approach is better than that obtained by the traditional solution approach. For each alliance scheme, there are almost no idle eight sensors, and the sensor utilization rate in the scheme is higher, which effectively avoids the waste of sensor resources; therefore, the established alliance will greatly improve the tracking accuracy of the target, and that the detection effect will be higher in the subsequent detection process. For the same uncertain problem, it can also be seen that different ideal point approaches yield different results, mostly because each approach has a different processing order for the uncertain problem; under the P EV principle, however, all of these are effective multi-sensor alliance schemes.

C. SIMULATION OF ALLIANCE UPDATING
In order to verify the effectiveness of the ''prediction and re-prediction'' mechanism used by the sensor alliance in the updating process, the mechanism proposed in this paper is compared with the two updating mechanisms-the ''measurement and re-update (M&R-P)'' and ''prediction is update (P&U)'' mechanisms. The sampling number is set to 200. The simulation results are shown in Figure 8. The position root mean square error (RMSE) of target tracking is shown in Table 5. The running time of different update mechanisms is shown in Figure 9. Figures 8-9 and Table 5 demonstrate that the proposed updating mechanism has smaller error and better convergence than the other two handover mechanisms. The hybrid alliance established by the state estimation of the target by the sensor in the dynamic environment of the target can provide accurate  measurement values, thus greatly reducing the tracking error and running time.

VI. CONCLUSION
In this paper, the uncertain hybrid multi-sensor alliance control problem is solved by using an uncertain ideal point approach under the P EV principle. First, after the definition of the relationship between uncertain variables is given, the ideal point method of uncertainty is applied to the model, and the uncertain multi-objective programming problem is transformed into the definite single objective programming problem. Second, the fireworks algorithm based on the improved selection strategy is proposed to obtain the effective solution of the model. The solution to the algorithm is obtained by comparing FWA and DDPSO, proving that the algorithm can avoid selecting particles with similar performance and poor quality for iteration. Next, the scheme of multi-sensor hybrid alliance is obtained by calculation. Third, compared with the traditional approach and the uncertain ideal point approach, the uncertain ideal point approach can solve the problem related to objective function, so that the final alliance scheme is superior to the traditional approach overall. Finally, by comparing and analyzing the tracking errors of three targets and the running time of the algorithm, the validity of the prediction and re-prediction mechanism is verified.