The efficiency of risk control and the accuracy of risk prediction are verified in this section. Furthermore, the performance of PMA and the comparison with three other methods are simulated and analyzed as follows.
5.2.1. Simulation of PMA
(1) Determine the number of steps H
Calculation accuracy and time complexity are influenced by the number of steps
H in (19). According to analysis, the larger the number of steps
H, the more the cumulative joint risk, the better the multi-step risk prediction, but the time complexity increases. To this end, it is important to select the right number of steps
H for balancing time complexity and calculation accuracy. The change of the cumulative joint risk with number of steps
H is shown in
Figure 4.
In
Figure 4, it is obvious that the cumulative joint risk decreases with the increase of the number of steps
H. It can be found that the cumulative joint risk happens with a small change from step
H = 5 to
H = 8, meaning fewer optimizations in the sensor scheduling scheme. Considering time complexity increases with a bigger number of steps
H, this paper takes the number of steps
H = 5.
- (2)
Solution of joint risk
The joint risk assessment changes with a target move. The joint risk comparison of the real and prediction value is shown in
Figure 5.
In
Figure 5, the prediction risk and the real risk describe the change of joint risk. In
Figure 5a,b, symbol “A” expresses the same turning point under two different physical indicators (time, distance). Moreover, before “A”, there is a greater distance between sensors and the target is longer, as expected. The loss risk of the target is the main factor of the joint risk. However, with the distance between sensor and target gradually decreasing, the loss risk probability of the target decreases, resulting in a downward trend of the joint risk. Similarly, after “A”, the loss risk probability of the target decreases too, while the threat risk of the target is the main factor of the joint risk. Additionally, the threat risk probability of the target keeps increasing, leading to an increasing trend of the joint risk. It can be seen that the curves of risk prediction and real risk are roughly the same, it can reasonably explain the risk assessment problems; moreover, it can verify the accuracy of risk prediction.
- (3)
Scheme of sensor scheduling
Through the convex optimization, the minimization problem of joint risk is solved; however, the solution is a discrete 0–1 scheme. To cope with the problem, the D-L method is necessary to obtain an optimal scheme of sensor scheduling. As an example, the sensor scheduling matrix is discretized by threshold by using the D-L method as:
It can be seen in
Figure 6, the elements in the matrix that are greater than or equal to the threshold are set to 1, otherwise 0. Therefore, a continuity matrix is transformed into a discrete 0–1 matrix. According to the given constraint, in which a sensor can only track one target and a target can only be tracked by one sensor, the sensor scheduling matrix is also legitimized. Since the elements of the first and second row in the first column are both 1, and the corresponding optimization value
, the first-row element in the first column is set to 0, so as to complete the legalization.
After D-L processing, the sensor scheduling scheme can be obtained by solving the optimization problem. As expected, some sensors can be scheduled to track the target scheme as shown in
Figure 7.
In this figure, taking the period from 40 s to 50 s as an example, during the moment of 40–41 s, 42–43 s, 44–45 s, 46–47 s and 48–50 s, , , , and were assigned to detect and track target, respectively. Indeed, the above scheme of sensor scheduling can be obtained in real time by solving the minimum multi-step prediction joint risk problem, leading to reduced joint risk of the detection system.
5.2.2. Comparison of Different Methods
Comparing the proposed method PMA with the other three methods CMA, RMA and MMA, we analyze their joint risk assessment, which can be seen in
Figure 8 and
Figure 9. Here,
Figure 8 is the joint risk comparison of four methods, and
Figure 9 is the comparison of cumulative joint risk, cumulative threat risk and cumulative loss risk.
In
Figure 8, the cumulative risk of PMA is lower than other methods (RMA, CMA and MMA) during the total simulation time. Taking the moment 60 s as an example, compared with methods RMA, CMA and MMA, the joint risk of method PMA reduces by 30.43%, 11.24% and 7.18%, respectively. Moreover, it can be seen from
Figure 9, compared with three methods, the cumulative joint risk of method PMA is reduced by 10.05%, 12.96% and 3.14%, the cumulative threat risk reduced by 6.63%, 9.21% and 1.95%, the cumulative loss risk reduced by 47.81%, 56.57% and 20.34%, respectively. The reason behind this observation is that the method RMA randomly assigns sensors. The risk prediction model is not considered to control the risk, leading to the highest risk. Although the CMA method can guarantee better sensor measurement information, it has high threat risk. Moreover, the risk prediction model is also not considered, the resulting risk is controlled with delay. The MMA method considers risk as a one-step prediction model to better predict and control risk, but it is not considered to estimate risk, leading to the best risk control effect hardly being obtained. According to the above analysis, compared with methods RMA, CMA and MMA, the PMA method can better balance the target threat risk and target loss risk, controlling joint risk and improving system safety.
To better illustrate the advantage of the PMA method,
Figure 10 shows the cumulative risk comparison of four methods on the joint risk, loss risk and threat risk.
As expected, the PMA method has the smallest cumulative joint risk, cumulative loss risk and cumulative threat risk in
Figure 10. For example, at moment 60 s, compared with methods RMA, CMA and MMA, the cumulative loss risk of PMA reduces by 68.29%, 43.72% and 11.59%, respectively. Moreover, the cumulative threat risk of PMA respectively reduces by 11.23%, 6.72% and 1.91%, and the cumulative joint risk of PMA respectively reduces by 17.29%, 11.32% and 3.73%. These results show that the PMA method is effective in risk assessment, controlling the joint risk and providing a better scheme of sensor scheduling.
- (2)
Tracking error comparison
Obviously, the tracking error of the sensor is one important indicator to evaluate the scheme of sensor scheduling, i.e., the smaller the tracking error, the higher the tracking accuracy of the sensor. We regard the maximum tracking error (MTE) and average tracking error (ATE) as indicators to evaluate four methods, which can be seen in
Table 5.
It can be seen from this table, compared with the other three methods, the proposed PMA method is not minimum at MTE. However, compared with RMA, CMA and MMA, the average tracking error of PMA reduces by 51.9%, 7.4% and 22.2%, respectively. Moreover, compared with the RMA method, the PMA method decreased by 34.1% with respect to ATE. These results illustrate that the PMA method does not reduce target tracking accuracy, while emphasizing risk control.
The complexity of the proposed PMA method is mainly determined by the number of sensors and targets. The optimization problem (21) contains
N sensors and
M targets, and the complexity of multi-sensor scheduling by convex optimization theory is
. Since the CMA and MMA methods use exhaustive methods to solve sensor scheduling, their complexity both are
. Therefore, the complexity of the PMA method is significantly lower than that of CMA and MMA. Furthermore, the RMA method randomly schedules sensor resources, in which the optimization problem of multi-sensor scheduling is not involved, leading to complexity, is the lowest. However, it can be analyzed from (1) and (2) mentioned above that compared with RMA, PMA has a great improvement in risk control and target tracking accuracy. For comparison, the complexities of the proposed PMA method and the other three methods are shown in
Table 6.