1. 1st Annual Israel Multinational BMD Conference & Exhibition
ALLOCATION OF RADAR
RESOURCES TO SEARCH
AND TRACKING MISSIONS
Presented By:
Prof. J. Ben-Asher
WALES, Ltd., Israel

2. ALLOCATION OF RADAR TRACKING RESOURCES
OUTLINE
Introduction
Problem Formulation
Single Target allocation
Multiple Targets Allocation
Conclusions

3. ALLOCATION OF RADAR TRACKING RESOURCES
INTRODUCTION
The problem of assigning pulses of an array radar which is engaged in tracking multiple targets is discussed
In a multiple target scenario time and duration of tracking each target determine the information state for this target
Classically, the objective of the sensor allocation process has been to maximize the information state of all relevant targets

4. ALLOCATION OF RADAR TRACKING RESOURCES
PROBLEM MODELING
Let
N’ be the final discrete time of the tracking period (the actual final time is dt•N’ where dt is the KF time increment)
Xj be the covariance of the j-th target at a certain predicted time N≥N’ (representing the uncertainty in the position and velocity)
n be the total number of targets
uj be a unit vector in the j-th direction
Then we can approximate (by assuming an EKF)
Gj represents the improvement in the uncertainty which would result from the assignment of the i-th pulse to the j-th target

5. PROBLEM FORMULATION (Cont.)
ALLOCATION OF RADAR TRACKING RESOURCES
PROBLEM FORMULATION (Cont.)
The optimization objective function can be written as
Where:
wj is the relative importance of the j-th target
f0 is a function of the covariance matrix at the future discrete time N≥N’
We minimize J subject to

6. SINGLE TARGET ALLOCATION
ALLOCATION OF RADAR TRACKING RESOURCES
SINGLE TARGET ALLOCATION
CASE SCENARIO
The study is performed in the ballistic plane (2 D)
A representative target is detected at a certain range
The investigated period is assumed to be 20 time units, out of which only 50% are allocated to tracking the target ; the rest are for the search mission
The allocation time interval is taken as 1 time unit
Tracking the target is given the highest priority hence the allocation slots can take place everywhere over the tracking period
We want to minimize the position error (i.e. uncertainty) at a certain time which may be as soon as the tracking period is over, or as late as required

7. SINGLE TARGET ALLOCATION (Cont.)
ALLOCATION OF RADAR TRACKING RESOURCES
SINGLE TARGET ALLOCATION (Cont.)
RESULTS
Position Error [m]
Allocation (p=1)
Allocation (p=0)
The time-of-interest is at 20 time units. Most resources are near the end; the rest is at the beginning

8. SINGLE TARGET ALLOCATION (Cont.)
ALLOCATION OF RADAR TRACKING RESOURCES
SINGLE TARGET ALLOCATION (Cont.)
RESULTS
Allocation (p=0.3)
Position Error [m]
Allocation (p=0.7)
The time-of-interest is at 50 time units. More resources are near the beginning. Partial allocation (p<1) is obtained

9. ALLOCATION OF RADAR TRACKING RESOURCES
SINGLE TARGET ALLOCATION (Cont.)
RESULTS
Position Error [m]
Asymptotic Allocation
The time-of-interest is at 100 time units. Resources are equally shared at the beginning and at the end . This result does not change with further increasing the time of interest

10. SINGLE TARGET ALLOCATION (Cont.)
ALLOCATION OF RADAR TRACKING RESOURCES
SINGLE TARGET ALLOCATION (Cont.)
RESULTS
Velocity Error [m/s]
Same as Asymptotic Allocation
The asymptotic allocation for large time-of-interest is obtained as the results of minimizing the velocity error at 20 time units

11. SINGLE TARGET ALLOCATION SUMMARY
ALLOCATION OF RADAR TRACKING RESOURCES
SINGLE TARGET ALLOCATION SUMMARY
There is always an allocation at the beginning and at the end
There is an asymptotic allocation for large enough prediction period
The explanation is from the following observation in
1-D:
Clearly the second term dominants the solution for sufficiently large N
This reduces the computation time by several orders of magnitude (the prediction time slows down the optimization in a non-linear way)

12. MULTIPLE TARGETS ALLOCATION
ALLOCATION OF RADAR TRACKING RESOURCES
MULTIPLE TARGETS ALLOCATION
CASE SCENARIO
Two representative targets are detected at the two different ranges: short range and long range
The tracking period is assumed to be 20 time units, out of which only 50% are allocated to tracking both targets; the rest are for the search mission
The allocation time interval is taken as 1 time unit
Tracking both targets has the same priority
We want to minimize the squared sum of the position errors at a certain prediction time
At a later step we want to study the effect of the detection range separation

13. MULTIPLE TARGETS ALLOCATION (Cont.)
ALLOCATION OF RADAR TRACKING RESOURCES
MULTIPLE TARGETS ALLOCATION (Cont.)
RESULTS
Position Error
Long range
Short range
Prediction Time
20 time units
The allocation resembles the single target allocation (mostly bang-bang) but prefers the distant target (less accurate) and provides it with the “sweet spots”

14. MULTIPLE TARGETS ALLOCATION (Cont.)
ALLOCATION OF RADAR TRACKING RESOURCES
MULTIPLE TARGETS ALLOCATION (Cont.)
RESULTS
Position Error
Long range
Short range
Dead Zone
Prediction Time
50 time units
The allocation resembles the single target allocation. Joint dead zone is maintained

15. MULTIPLE TARGETS ALLOCATION (Cont.)
ALLOCATION OF RADAR TRACKING RESOURCES
MULTIPLE TARGETS ALLOCATION (Cont.)
RESULTS
Position Error
Prediction Time time units
Long range
Short range
The allocation resembles the single target allocation. Asymptotic allocation is observed

16. MULTIPLE TARGETS ALLOCATION (Cont.)
ALLOCATION OF RADAR TRACKING RESOURCES
MULTIPLE TARGETS ALLOCATION (Cont.)
RESULTS
Velocity Error
Long range
Short range
Asymptotic behavior is maintained. One can minimize the squared sum of the velocity error at the end of the tracking period to get the asymptotic allocation

17. MULTIPLE TARGETS ALLOCATION (Cont.)
ALLOCATION OF RADAR TRACKING RESOURCES
MULTIPLE TARGETS ALLOCATION (Cont.)
RESULTS
Velocity Error
Very Long range
Short range
Reverse Priority
Increasing the range between targets. Further priority is given to the distant target

18. MULTIPLE TARGETS ALLOCATION (Cont.)
ALLOCATION OF RADAR TRACKING RESOURCES
MULTIPLE TARGETS ALLOCATION (Cont.)
RESULTS
Velocity Error
Same range
Closely spaced targets. Equal priority is given to both targets. The solution equalizes the predicted error

19. MULTIPLE TARGETS ALLOCATION (Cont.)
ALLOCATION OF RADAR TRACKING RESOURCES
MULTIPLE TARGETS ALLOCATION (Cont.)
SUMMARY
The double period mostly bang-bang allocation is maintained along with the single targets’ original joint dead-zone
The asymptotic allocation for large enough prediction period is also maintained
Minimizing the velocity error is a viable approach for large enough prediction times
More distant targets are given higher priority; Closely space targets optimal error values are equalized
Based on the above observations, approximate solutions might be constructed from the single target allocation