Final Presentation Online-implementable robust optimal guidance law

15 March 2008 Final presentation - DRDO IISc Prgm on Adv. Res. in Math. Engg

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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)


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Two dimensional missile-target engagement model


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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


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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


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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?


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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


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Objective

An improved, robust guidance law i) that nullifies or at least mitigates the

effect of assumptions madeii) implementable in real-time


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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


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The starting point: optimal guidance law (OGL)

Minimise

subject to

dtnJFt

c0

2

BuAXX

ty F

0)(


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Linear model

c

L

T

L

T

n

Tn

n

y

y

Tn

n

y

y

/1

0

1

0

/1000

0000

0100

0010


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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


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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


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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


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Challenges

1) lack of optimal control methods to deal with inequality constraints

2) real-time implementation


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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


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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


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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


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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 !


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Online Implementation

actuatorplant/guidance system

OGL

OGL model plant model

OGL

DE

+

+

DE- OGL

TF nTt ,,

)(),( tyty

)(),( tyty


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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


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Comparison of total acceleration

PN APN OGL DE-OGL

100 % 81.3 % 85.9 % 55.5 %dtnft

tc

0


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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


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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


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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?


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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


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Final Presentation Online-implementable robust optimal guidance law

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