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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