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Final Presentation Online-implementable robust optimal guidance law. Raghunathan T., Ph.D. student (On behalf of Late Dr. S Pradeep, Associate Professor, Aerospace Engineering Department). Two dimensional missile-target engagement model. - PowerPoint PPT Presentation
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15 March 2008 Final presentation - DRDO IISc Prgm on Adv. Res. in Math. Engg
1
Final Presentation
Online-implementable robust optimal guidance law
-Raghunathan T., Ph.D. student (On behalf of Late Dr. S Pradeep, Associate
Professor, Aerospace Engineering Department)
15 March 2008 Final presentation - DRDO IISc Prgm on Adv. Res. in Math. Engg
2
Two dimensional missile-target engagement model
15 March 2008 Final presentation - DRDO IISc Prgm on Adv. Res. in Math. Engg
3
Background and motivation:
Miss distances for the linear model
0 1 2 3 4 5 6 7 8 9 10-40
-20
0
20
40
60
80
100
tF, seconds
y(t
F),
feet
y(tF) for PN, APN & OGL; n
T = 3G, n
c max = 9G, single lag
PN
APN
OGL
15 March 2008 Final presentation - DRDO IISc Prgm on Adv. Res. in Math. Engg
4
Background and motivation
Optimal guidance law (OGL) Assumptions a) linear model of missile-target
engagement : b) unbounded control : infinite lateral acceleration c) tgo known accurately
d) constant target maneuver Yields an analytical/closed form
solution that is implementable online
UBXAX
15 March 2008 Final presentation - DRDO IISc Prgm on Adv. Res. in Math. Engg
5
Reality : how valid are the assumptions?
a) Missile-target engagement
kinematics is highly nonlinear b) Lateral acceleration is limited by saturation c) tgo cannot be known accurately
d) Constant target maneuver?
15 March 2008 Final presentation - DRDO IISc Prgm on Adv. Res. in Math. Engg
6
Result of applying OGL to the nonlinear
kinematic model Miss distances for the plant
0 2 4 6 8 10 12-40
-20
0
20
40
60
80
100
120
tF, seconds
Mis
s d
ista
nce
s, f
eet
Miss distances for PN, APN & OGL; nT
= 3G, nc max = 9G, single lag
PN
APN
OGL
15 March 2008 Final presentation - DRDO IISc Prgm on Adv. Res. in Math. Engg
7
Objective
An improved, robust guidance law i) that nullifies or at least mitigates the
effect of assumptions madeii) implementable in real-time
15 March 2008 Final presentation - DRDO IISc Prgm on Adv. Res. in Math. Engg
8
Solution Methodology
(i) Make use of the solution (i.e. OGL) that we know, as a starting point
(ii) Explore the solution space around this starting point for the best solution
15 March 2008 Final presentation - DRDO IISc Prgm on Adv. Res. in Math. Engg
9
The starting point: optimal guidance law (OGL)
Minimise
subject to
dtnJFt
c0
2
BuAXX
ty F
0)(
15 March 2008 Final presentation - DRDO IISc Prgm on Adv. Res. in Math. Engg
10
Linear model
c
L
T
L
T
n
Tn
n
y
y
Tn
n
y
y
/1
0
1
0
/1000
0000
0100
0010
15 March 2008 Final presentation - DRDO IISc Prgm on Adv. Res. in Math. Engg
11
The starting point: OGL (cont’d)
The solution/control input/lateral acceleration/OGL:
Cancellation of system
dynamics
])1(5.0[ 222
xeTntntyyt
Nn x
LgoTgogo
c
Ttx go
xx
x
eexxxx
xexN
223
2
3126632
)1(6
Tsn
n
c
L
1
1
15 March 2008 Final presentation - DRDO IISc Prgm on Adv. Res. in Math. Engg
12
Own problem formulation
Minimize subject to
free and free
Control input/guidance law :
)())(( FTMc tRtnJ
),,( tnXfX c
50;0 max, Nnn cc
,)( 00 XtX )( FtX Ft
)),(,,( TtNyynn Fcc
15 March 2008 Final presentation - DRDO IISc Prgm on Adv. Res. in Math. Engg
13
Nonlinear kinematic model
Tnn
VV
n
n
V
V
Vn
n
RR
V
V
R
R
cL
M
M
L
L
T
T
TT
L
M
M
M
M
T
T
/)(
cos
sin
sin
cos
/
2
1
2
1
2
1
2
1
15 March 2008 Final presentation - DRDO IISc Prgm on Adv. Res. in Math. Engg
14
Challenges
1) lack of optimal control methods to deal with inequality constraints
2) real-time implementation
15 March 2008 Final presentation - DRDO IISc Prgm on Adv. Res. in Math. Engg
15
Our approach:
The Differential Evolution Tuned
Optimal Guidance Law (DE-OGL):Control input/guidance law :
(Differential Evolution is one of the
evolutionary computation (EC) methods)
)),(,,( TtNyynn Fcc
15 March 2008 Final presentation - DRDO IISc Prgm on Adv. Res. in Math. Engg
16
Differential Evolution (DE) parameters used:
Crossover constant, CR = 0.9 Weighting factor, F = 0.8 Population size, NP = 12 Stopping criterion: max. no. of generations = 4 or solution < tolerance limit
15 March 2008 Final presentation - DRDO IISc Prgm on Adv. Res. in Math. Engg
17
Real-time implementation:The Optimal Control Problem
and evolutionary computation(EC)
In general, EC is computationally intensive!
Which leads to the second set of challenges :
System dynamics slow enough A ‘good enough’ (suboptimal) solution Massively parallel implementation
15 March 2008 Final presentation - DRDO IISc Prgm on Adv. Res. in Math. Engg
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EC for the Missile Guidance Problem
Fast dynamics Acceptable: almost the best
solution Limited onboard computation
power Must be available in real-time !
15 March 2008 Final presentation - DRDO IISc Prgm on Adv. Res. in Math. Engg
19
Online Implementation
actuatorplant/guidance system
OGL
OGL model plant model
OGL
DE
+
+
DE- OGL
TF nTt ,,
)(),( tyty
)(),( tyty
15 March 2008 Final presentation - DRDO IISc Prgm on Adv. Res. in Math. Engg
20
Acceleration signatures
0 2 4 6 8 10 12-4
-3
-2
-1
0
1
2
3
time t (s)
n c/nT
nc requirement of all laws
PN
APN
OGL
DE-OGL
15 March 2008 Final presentation - DRDO IISc Prgm on Adv. Res. in Math. Engg
21
Comparison of total acceleration
PN APN OGL DE-OGL
100 % 81.3 % 85.9 % 55.5 %dtnft
tc
0
15 March 2008 Final presentation - DRDO IISc Prgm on Adv. Res. in Math. Engg
22
Miss distances for all guidance laws
0 2 4 6 8 10 12-50
0
50
100
150
t F
, seconds
Mis
s d
ista
nce
s, f
eet
Miss distances for PN, APN, OGL & DE-OGL; nT = 3G, n
c = 9G, single lag
PN
APN
OGL
DE-OGL
15 March 2008 Final presentation - DRDO IISc Prgm on Adv. Res. in Math. Engg
23
N’ for OGL and DE-OGL
0 2 4 6 8 10 123.6
3.8
4
4.2
4.4
4.6
4.8
5
time t (s)
N'
N' for OGL and DE-OGL
DE- OGL
OGL
For PN and APN, N' = 4
15 March 2008 Final presentation - DRDO IISc Prgm on Adv. Res. in Math. Engg
24
Convergence of the solution
0 1 2 30
10
20
30
40
50
60
70
generation/iteration
co
st
(mis
s d
ista
nc
e, R
TM
(tF))
, fe
et
best
population average
29 Oct 2007 25
Future work
For more practical maneuvers of target
More complex model? Applicability to a larger range of
initial conditions?
15 March 2008 Final presentation - DRDO IISc Prgm on Adv. Res. in Math. Engg
26
Publications Papers: (a) Raghunathan T. and S. Pradeep, “A Differential Evolution Tuned
Optimal Guidance Law,” in The 15th Mediterranean Conference on Control and Automation - MED’07 held at Athens, Greece during June 27-29, 2007.
(b) Raghunathan T. and S. Pradeep, “An online Implementable Differential Evolution Tuned Optimal Guidance Law,” in Genetic and Evolutionary Computation Conference - GECCO 2007, held at London, United Kingdom, during July 7-11, 2007.
Technical Report: Raghunathan T. and S. Pradeep, “Online-implementable Robust
Optimal Guidance Law,” Technical Report No. TR-PME-2007-12 dated 20 December 2007, under DRDO-IISc Programme on Advanced Research in Mathematical Engineering.
The financial support provided for the above by DRDO-IISc Program on Advanced Research in Mathematical Engineering is gratefully acknowledged