Download - ETM 607 Application of Monte Carlo Simulation: Scheduling Radar Warning Receivers (RWRs)

August 27, 2012 ETM 607 Slide 1

ETM 607Application of Monte Carlo

Simulation:Scheduling Radar Warning

Receivers (RWRs)

Scott R. Schultz Mercer University


Problem Statement

Develop an RWR scheduler that minimizes the time to detect multiple threats across multiple frequency bands.


RWR Scheduling Definitions

Pulse Width (PW)

Revisit Time (RT)

IlluminationTime (IT)

Pulse RepetitionInterval (PRI)

Beam Width (BW)

Definitions:

Revisit Time (RT) – time to rotate 360 degrees (rotating radar)

Illumination Time (IT) – function of RT and BW

Pulse Width (PW) – length of time while target is energized

Pulse Repetition Interval (PRI) – time between pulses

Time


Example RWR Schedule

RWR Schedule – a series of dwells on different frequency bands: sequence and length


RWR Scheduling Problem

Objective – detect all threats as fast as possible (protect the pilot)

How to sequence dwells?How to determine dwell length?How to evaluate / score schedules?

Meta-Heuristics

Simulation


Need for Simulation

Given that the offset for each threat pulse train is unknown.

Determine: MTDAT - expected time to detect all threats, MaxDAT - maximum time to detect all threats

Threat 1Band 2

Band 1

Band 2

Band 3

Band 1

Band 2

Band 3

Band 1

Band 2

Band 3

Threat 1Band 2

300 360 390 420 450 480 510 540 ......

Time (milliseconds)

RWR Schedule

Threat

Band 1

Band 2

Band 3

Band 1

Band 2

Band 3

Band 1

Band 2

Band 3

Threat 1Band 2

Threat 1Band 2

300 360 390 420 450 480 510 540 ......

Time (milliseconds)

RWR Schedule

Threat

Note different offsets

Threat detected in

cycle 1

Threat detected in

cycle 2


Simulation Algorithm

n = 1

i = 1

Generate offset for threat i ~ U(0,RTi)

Determine time when RWR schedule

coincides with threat i

i = i + 1

i < I

Objective: Evaluate / Score a single RWR schedule.

N – number of iterations

I – number of threatsn = n + 1

Update MTDAT, MaxDAT

n < N

Done

Yes

Yes

NoNo


Simulation iterations - N

When does the MTDAT running average begin to converge?

Running Average - 3 Threats

760

780

800

820

840

860

0 10000 20000 30000 40000 50000 60000

Number of Iterations

MTD

AT

MTDAT running average: 3 threats


4400

4500

4600

4700

4800

4900

5000

5100

0 10000 20000 30000 40000 50000 60000


MTD

AT


16000

16400

16800

17200

17600

18000

0 10000 20000 30000 40000 50000 60000

Number of IterationsM

TDA

T




Simulation Run-Time

How long does simulation run to evaluate a single schedule?

CPU Run Time

0

2

4

6

8

10

12

14

0 10000 20000 30000 40000 50000 60000


Sec

on

ds 3 Threats

5 Threats

10 Threats


Empirical Density Functions

Can we take advantage of the distribution function of MTDAT to avoid costly simulation?

Histogram - 3 Threats

0100200

300400500600

700800

020

040

060

080

010

0012

0014

00M

ore

Time to Detect All Threats

Nu

mb

er o

f O

ccu

ren

ces


0

100

200

300

400

500

600

700

Time to Deect All Threats

Nu

mb

er o

f O

ccu

ren

ces


0100200300400500600700800

050

00

1000

0

1500

0

2000

0

2500

0

3000

0

3500

0

4000

0M

ore

Time to Detect All ThreatsN

um

ber

of

Ocu

rren

ces


POI Theory – 2 pulse trains

Enter: Kelly, Noone, and Perkins (1996) – The probability of intercept can be divided into four regions, associated with

the pulse count, n, of the shorter periodic pulse train.

where P1 = (1 + 2 - 2d + 1)/T2, and assumes T1 < T2.

* Note, Kelly Noone and Perkins did not add the 1, we believe trailing edge triggered.

nnn

nnnnn

nnPnnnnnPn

nnnnnPn

nnnP

nPPOI

c

cc

cccc

ccc

c

',1

''))'((1

)'()'(

',)(

,

)( 11

1

1

nnn

nnnnn

nnPnnnn

nnP

np

c

cc

cc

c

c

',0

''))'((1

',

,

)( 1

1


MTD - Mean Time to Detect

Our contribution:

Knowing that,

MTD = E[n] = ,

where t(n) is the intercept time for pulse n.

What is t(n) for all n?

cnnn

n

ntnp'

1

)()(



Observation:

When threat starts in positions -1,0,1, or 2, intercept occurs on pulse 1 of RWR.

When threat starts in position 3, 4 or 5 intercept occurs on pulse 4, 7 and 10 respectively.

Intercept time occurs at T1(n-1) + d + i, where n is the RWR pulse count and i is 0 if threat starts before start of cycle, else i is amount of time elapsed between start of RWR pulse and start of threat.

d = 2 T2 = 20 T1 = 7

Time of Intercep

t Start Time of Threat

Threat RWR

2 -1

2 0 0 20 40 602 1

1

3 24

24 3

7

45 4 1066 5

9 6 9 7 9 8

2

10 9 5

31 10

8

52 11 1173 12

16 13 16 14 16 15

3

17 16 6

38 17

9

59 18 12- 19

- 20



Expected times t(n) per cycle n:

where is an indeterminate error bounded by:

and, MTD =

where E is the total error bounded by:

,12

)1()(21

11

1

1

d

idnTnt

d

i

nn v

,)1()( 1'

dnTntnnnn vc

.1

1

1

d

i

i

Entnpcnnn

n

'

1

)()(

.)1(1

11

1

d

ic

iPnE



Is there error, E, a concern?RWR Threat Coincidence MTD

T1 tau1 T2 tau2 d RPT Enumeration (1) Simulation (2) %Error (3)E[n]

t(n)p(n) %Error (4)

41 7 307 25 1 400.4 209.95 214 1.93% 209.95 0.00%

41 7 307 25 2 428.0 225.77 227 0.54% 225.77 0.00%

41 7 307 25 6 591.0 289.28 290 0.25% 289.28 0.00%

41 7 307 25 7 653.3 348.36 349 0.18% 348.36 0.00%

41 7 321 10 1 818.2 592.12 597 0.82% 590.47 0.28%

41 7 321 10 2 935.1 655.98 665 1.37% 654.67 0.20%

41 7 321 10 3 1090.9 728.04 731 0.41% 727.07 0.13%

41 7 277 40 1 240.8 115.57 116 0.37% 115.19 0.33%

41 7 277 40 2 251.8 118.22 115 2.72% 117.95 0.23%

(1) Enumeration - by hand, correct value

(2) Simulated for 10,000 iterations

(3) %Error = ABS(Enumeration - Simulation)/Enumeration

(4) %Error = (Enumeration - t(n)p(n))/Enumeration

Note: calculated E[n] is better and faster than Simulated value.


Summary and Limitations

Summary:• An innovative closed form approach for determining the mean time for coincidence of periodic pulse trains has been developed using POI theory and insight on the coincidence of periodic pulse trains.

• The approach is computationally faster and more accurate than a previous presented Monte Carlo simulation approach.

Limitations:• This method is limited to threats which exhibit strictly periodic pulse train behavior (e.g. rotating beacons).• Still need method to determine MaxDAT

Future:• An enumerative approach is being evaluated for non-periodic pulse trains.

Download - ETM 607 Application of Monte Carlo Simulation: Scheduling Radar Warning Receivers (RWRs)

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