Optimal Missile Guidance system

By Yaron Eshet & Alon ShtakanSupervised by Dr. Mark Mulin

Equations of motion MT aa

2,

,

ˆ : cos( )

ˆ : 2 sin( )

T M r

T M

r r r a a

r r a a

LOS

Target

Missile

r

Ta

Ma

y

x

Interception - Overview

Parameters:

Interception:

, , , , Tr r a

0r

0r

0 .const

r

Necessary condition for interception for all initial conditions:

The problem: non-linear and complex relation between the parameters

The solution:

a) Guidance law in (RTPN) to achieve

b) Guidance law in to complete the interception process

00r for t t

Test case

2,

,

ˆ : cos( )

ˆ : 2 sin( )

T M r

T M

r r r a a

r r a a

Simple maneuver

Ta

ˆ ( ) 0Ta x t

simulates realistic missile Ta

Test case – Guidance law

2,

,

ˆ : cos

ˆ : 2 sin ( 2)

T M r

T M

r r r a a

r r a a r r

RTPN: Realistic true proportional navigation

, sinM Ta r a r

compensation of target missile acceleration

Guidance law perpendicular to line of sight (LOS(:

Test case – Guidance law

2,ˆ : cos

ˆ : ( 2)

T M rr r r a a

r r

( / 2 3 / 2) Target

Missile

r ˆ,T xa

ˆ,Ma

y

x

Ta

r

,

,

ˆ

0

M M

M r

a a

a

LOS

distance decrease: has a projection in direction

Test Case: equations of motion

rr

arrr T

)2(:ˆ

cos:ˆ 2

40

0

0

0

8 10

1787 / sec

2.18

0.0051 / sec

3.85

r m

r m

rad

rad

Initial conditions

0 10 20 30 400

2

4

6

8x 10

4 r(t)

0 10 20 30 40-2600

-2400

-2200

-2000

-1800

-1600dr(t)

0 10 20 30 40

2.2

2.25

fe(t)

0 10 20 30 40-2

0

2

4

6x 10

-3 dfe(t)

0 10 20 30 4020

25

30

35

40

45aT(t)

0 10 20 30 4030

35

40

45

50

55aM(t)

Interception in 37.57 sec

Interception time vs. intercept time vs. lamda (ro=8e4)

37.5

37.6

37.7

37.8

37.9

38

38.1

0 20 40 60 80 100 120

lamda

t in

terc

ept

intercept time vs. lamda (for r0=8e5)

195

200

205

210

215

220

225

230

0 5 10 15 20 25

lamda

t in

terc

ept

•The influence of depends on the initial conditions

difference in interception time of order 0.1 sec

difference in interception time of

order 10 sec

40 8 10r m

50 8 10r m

Interception time vs. intercept time vs. lamda (ro=8e4)

37.5

37.6

37.7

37.8

37.9

38

38.1

0 20 40 60 80 100 120

lamda

t in

terc

ept

intercept time vs. lamda (for r0=8e5)

195

200

205

210

215

220

225

230

0 5 10 15 20 25

lamda

t in

terc

ept

•For values under a certain bound, there is no guarantee for interception

Interception time diverges for small

values of

Interception time vs. intercept time vs. lamda (ro=8e4)

37.5

37.6

37.7

37.8

37.9

38

38.1

0 20 40 60 80 100 120

lamda

t in

terc

ept

intercept time vs. lamda (for r0=8e5)

195

200

205

210

215

220

225

230

0 5 10 15 20 25

lamda

t in

terc

ept

•Saturation zone: minor influence of . Critical influence for initial conditions and maneuver

interception time ~ 37 sec

interception time ~ 200 sec

40 8 10r m

50 8 10r m

Analytical analysis

2ˆ : cos

ˆ : ( 2)

0Tr r r a

r r

2 2( ) (2 3) 0dr r

dt

Necessary condition for interception for all initial conditions

1.5

0for r

resulting condition)1(

)2(

=1.49 (interception) =0.9 (miss)

=3.85 (interception) =1.51 (interception)

Behavior of with respect to (comparison with theory)

extreme divergenceEdge of divergence

divergence occurs around r0 , as starts varying rapidly

2r

2ˆ : cos 0Tr r a r

?

The solution: guidance law also in direction r

, cosM r Ta a K

2ˆ :r r K r r K

ensures interception for all initial conditionsdepends on maneuver

20 0

20 0 0

0.5

2intercept

r K t r t r

r r K rt

K

Interception time vs. K (simulation vs. approx. calculation)

020406080

100120140160180

0 50 100 150

K

Inte

rce

pt

tim

e

d(fee0)=0.016

d(fee0)=0.0051

Approx.

20 0

20 0 0

0

0.5

2 1( 0)intercept

r K t r t r

r r K rt for r

K K

r K

0 0.0051 0 0.016

20 0K r

?

Summery: sufficient conditions for interception

K>0

1.5>

These conditions ensure interception for all initial conditions and for any target missile maneuver.

Gain Scheduling - K The case: delay in data acquisition about the target missile maneuver

5 10 15 20 25 30 35 40 45 504

6

8

10

12

14

2( ) ( ) ( )

2

[ cos( ) cos( )] (ˆ :

ˆ( )

)

( ) : ( )

( )

T T delay delay delay

original original

r r r K K

K t K t delay r r r

a a

K

K

K t

t

K t

5 10 15 20 25 30 35 40 45

25

30

35

40

5 10 15 20 25 30 35 40

-5

0

5

10

15

( )K t( )K tTa

5 2 secoriginalK delay limited sensitivity sec 203.73 212.08interceptt :

Gain Scheduling -

The case: adjusting for distance increase/decrease

sec 47.34 49.68

2 2( ) (2 3)dr r

dt

0

0

0

4

0

500 /

8 10

2.15

100

1

sec

0.016 / se

0

c

r m

r

r m

ad

K

rad

20 0K r

2r

interceptt :

not negligible

Constraints on interception time: Optimal control

2ˆ :

ˆ : ( 2)

r r r K

r r

20 0 02 1

intercept

r r K rt

K K

Optimal Control

0

500

1000

1500

2000

2500

3000

3500

4000

4500

0 5 10 15 20 25

K

cost

fu

nct

ion

rou = 0.5 rou = 1 rou = 10

optimal K

2 2

2

[ ]

( )intercept

J y u dt

t K K

Example: Limited angular acceleration

, sinM Ta a r

40

0

0

0

8 10

0 / sec

2.15

0.016 / sec

100

10

r m

r m

rad

rad

K

interceptt 41.66 sec

,Ma

r 2r

( )r tNo limit

50

40

r

r

intercept

interceptt

t 41.66 sec

42.21 sec

No interception

( )r t 2r

Transition from failure to successful interception(green plot – previous page)

Conclusion:

Under realistic constraints, one gets an upper bound for K, which meansa lower bound for interception time

interceptt 62.64 sec

Ideal interception vs. interception under constraint (blue vs. red plots)

interceptt 41.66 sec interceptt 42.21 sec

50r ( )M M MTotal Effort F dx m v a dt

No constraint

71.394 10S 71.381 10S

Conclusion:

Interception with no constraint is faster indeed. However, it requires homing missile with higher performance and greater control effort.

( ) ( )M Mv t a t

Project summery

• Analysis of the equations of motion of the system• Introduction of guidance laws and study of their

function in ensuring interception• Applying “Gain Scheduling” methods for improved

performance • Analysis of the system behavior under realistic

constraints and restrictions