פרוייקט בקרה – יירוט טילים
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Transcript פרוייקט בקרה – יירוט טילים
Optimal Missile
Guidance system
By Yaron Eshet & Alon Shtakan
Supervised by Dr. Mark Mulin
Equations of motion aT aM
rˆ :
r r 2 aT cos( ) aM ,r
ˆ :
r 2r aT sin( ) aM ,
y
ˆ
Target
aT
rˆ
aM
LOS
x
Missile
Interception - Overview
Parameters:
r, r, , , aT
Interception:
r 0
Necessary condition for interception for all initial conditions:
r 0 for t t0
The problem: non-linear and complex relation between the parameters
The solution:
a) Guidance law in ˆ (RTPN) to achieve
b) Guidance law in rˆ
0 const.
to complete the interception process
r 0
Test case
rˆ :
r r 2 aT cos( ) aM ,r
Simple maneuver
ˆ :
r 2r aT sin( ) aM ,
aT simulates realistic missile
aT xˆ (t ) 0
aT
Test case – Guidance law
RTPN: Realistic true proportional navigation
Guidance law perpendicular to line of sight (LOS(:
aM , r aT sin r
compensation of target missile acceleration
rˆ :
r r 2 aT cos aM ,r
ˆ :
r 2r aT sin aM ,
r ( 2)r
Test case – Guidance law
rˆ :
r r 2 aT cos aM ,r
ˆ :
r ( 2)r
distance decrease:
aT has a projection
in rˆ direction
( / 2 3 / 2)
y
rˆ
aT , xˆ
Target
aM ,ˆ
aM aM ,ˆ
aM ,r 0
LOS
x
Missile
Test Case: equations of motion
rˆ :
r r aT cos
ˆ :
r ( 2)r
Initial conditions
2
r0 8 10 4 m
r0 1787 m / sec
0 2.18 rad
0 0.0051 rad / sec
3.85
Interception in 37.57 sec
4
8
r(t)
x 10
dr(t)
-1600
-1800
6
-2000
4
-2200
2
0
-2400
0
10
20
30
40
-2600
0
10
-3
fe(t)
6
20
30
40
30
40
30
40
dfe(t)
x 10
4
2.25
2
0
2.2
0
10
20
30
40
-2
0
10
aT(t)
aM(t)
45
55
40
50
35
45
30
40
25
35
20
0
10
20
20
30
40
30
0
10
20
Interception time vs.
•The influence of depends
on the initial conditions
intercept time vs. lamda (ro=8e4)
38.1
t intercept
38
37.9
37.8
r0 8 104 m
37.7
37.6
37.5
0
20
40
60
80
100
120
lamda
difference in interception
time of order 0.1 sec
intercept time vs. lamda (for r0=8e5)
r0 8 105 m
230
t intercept
225
220
difference in
interception time of
order 10 sec
215
210
205
200
195
0
5
01
51
lamda
02
52
Interception time vs.
intercept time vs. lamda (ro=8e4)
•For values under a
certain bound, there is no
guarantee for interception
38.1
t intercept
38
37.9
37.8
37.7
37.6
37.5
0
20
40
60
80
100
120
lamda
Interception time
diverges for small
values of
intercept time vs. lamda (for r0=8e5)
230
t intercept
225
220
215
210
205
200
195
0
5
01
51
lamda
02
52
Interception time vs.
intercept time vs. lamda (ro=8e4)
•Saturation zone: minor
influence of . Critical
influence for initial
conditions and maneuver
38.1
t intercept
38
37.9
37.8
37.7
37.6
37.5
0
20
40
60
80
100
120
lamda
r0 8 104 m
interception time ~ 37 sec
intercept time vs. lamda (for r0=8e5)
230
t intercept
225
220
215
r0 8 105 m
210
205
200
195
0
5
01
51
lamda
02
52
interception time ~ 200 sec
Analytical analysis
Necessary condition for interception for all initial conditions
rˆ :
r r aT cos
ˆ :
r ( 2)r
2
(1)
(2)
d
(r 2 ) (2 3)r 2
dt
0
resulting condition
0
1.5
for r 0
Behavior of r 2 with respect to (comparison with theory)
)interception( =1.51
)miss( =0.9
extreme divergence
)interception( =3.85
)interception( =1.49
Edge of divergence
divergence occurs around r0 , as starts varying rapidly
rˆ :
r aT cos r
depends on maneuver
2
?
The solution: guidance law also in direction
0
ensures interception
for all initial conditions
rˆ
aM ,r aT cos K
rˆ :
r K r 2
r K
r 0.5 K t 2 r0 t r0
tintercept
r0 r0 2 2 K r0
K
0 0.016
d(fee0)=0.016
d(fee0)=0.0051
0 0.0051
Approx.
r00
K
?
2
Intercept time
Interception time vs. K (simulation vs. approx. calculation)
180
160
140
120
100
80
60
40
20
0
0
50
100
150
K
r K
r 0.5K t 2 r0 t r0
tintercept
r0 r0 2 2 K r0
K
1
K
( for r0 0)
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
r r 2 K [aT cos( ) aT ( delay ) cos( ( delay ) ( delay ) )] K (t )
rˆ :
K K (t )
K (t ) K original (t delay )
aT
r r 2 K (t ) K original
rˆ :
K (t )
K (t )
15
14
40
10
12
35
10
5
8
0
6
-5
30
25
4
5
10
15
20
25
30
35
40
45
Koriginal 5 delay 2 sec
5
10
15
20
25
30
35
40
limited sensitivity
45
50
5
tintercept :
10
15
20
25
30
35
40
212.08 203.73 sec
Gain Scheduling -
The case: adjusting for distance increase/decrease
r0 8 10 m
4
r 2
d
(r 2 ) (2 3)r 2
dt
r0 500 m / sec
0 2.15 rad
0 0.016 rad / sec
K
K 100
r002
not negligible
10
tintercept :
49.68 47.34 sec
Constraints on interception time: Optimal control
rˆ :
r r 2 K
ˆ :
r ( 2)r
tintercept
r0 r0 2 2 K r0
K
1
K
Optimal Control
4500
J [ y u ] dt
4000
2
tintercept ( K ) K 2
3500
cost function
2
3000
2500
2000
1500
optimal K
1000
500
0
0
5
10
15
20
K
rou = 0.5
rou = 1
rou = 10
25
Example: Limited angular acceleration
aM , aT sin r
aM ,
r0 8 104 m
r0 0 m / sec
0 2.15 rad
0 0.016 rad / sec
K 100
10
t intercept 41.66 sec
r (t )
No limit
r 50
r 40
t intercept 41.66 sec
t intercept 42.21 sec
No interception
r
r 2
Transition from failure to successful interception
(green plot – previous page)
r 2
r (t )
t intercept 62.64 sec
Conclusion:
Under realistic constraints, one gets an upper bound for K, which means
a lower bound for interception time
Ideal interception vs. interception under constraint
(blue vs. red plots)
No constraint
r 50
t intercept 41.66 sec
t intercept 42.21 sec
S 1.394 107
S 1.381107
Total Effort( M ) F dx m vM aM dt
vM (t ) aM (t )
Conclusion:
Interception with no constraint is faster indeed. However, it requires homing missile
with higher performance and greater control effort.
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