Feedback Linearization Based Guidance

Feedback LinearizationFeedback LinearizationBased GuidanceBased Guidance

G. Weiss

Technion, Faculty of Electrical Engineering1

Project under the supervision ofDr. I. Rusnak

Control and Robotics LabJune, 2010

April 21, 2023

OutlineOutline

• Objectives

• Project milestones

• Planar scenario description

• Guidance problem formulation

• Feedback linearization

• Guidance strategies

• Performance example

• Summary

Technion, Faculty of Electrical Engineering2 April 21, 2023

ObjectivesObjectives

To examine the guidance problem of an acceleration-constrained homing missile when the initial missile heading is far from collision course using feedback linearization

The presented work is based on the paper S. Bezick, I. Rusnak, and W. S. Gray, “Guidance of a Homing Missile Via Nonlinear Geometric Control Methods” ,AIAA Journal of Guidance, Control, and Dynamics, vol. 18, no. 3, May-June 1995, pp. 441-448.



Project MilestonesProject Milestones

• Phase 1Phase 1– Study of the Feedback Linearization method– Validation of 2D simulation results

presented in the paper– Performance analysis

• Phase 2Phase 2– Derivation of 3D scenario guidance laws– Simulations– Performance analysis

Planar scenario descriptionPlanar scenario description


RTM

VT

aT

VM

aMΨM

ΨT

σReference direction

April 21, 2023

(xT , yT)

(xM , yM)

Target

Missile

Guidance problem Guidance problem formulationformulation

April 21, 2023Technion, Faculty of Electrical Engineering6

M M M

M M M

MM M MAX

M

T T T

T T T

TT

T

x V cos

y V sin

aa A

V

x V cos

y V sin

aV

M

TM

x f ( x ) g( x )a

y h( x ) R ( x )

Assumptions

• Ideal aerodynamics

• Constant velocities

• Ideal autopilot

• Full information

knowledge

1M

c

a sa s

,M TV V

Feedback LinearizationFeedback Linearization

Consider the single input single output system

find a state feedback control such that the input-output system from to is a linear mapping.

Using Lie Derivatives, such a satisfies


x f ( x ) g( x )uy h( x )

u x vx v y

u

11 r

frg f

u L h xL L h x

v

Guidance strategiesGuidance strategies

Large heading errorLarge heading error

closing velocity resulting from

Small heading errorSmall heading error


21M TM T T

M

c cd

cd

v

v

a R a sinsin

K V V

V

0LOSRV

M ca N V

FLGL

PN

Performance examplePerformance example


RTM

VT

VM

σ

ΨM

2

2

1000

1000

250

270

0

160

0

1000

TM

M

T

M

T

T

max

R m

V m / sec

V m / sec

a m / sec

A m / sec

Scenario ParametersTarget

Missile



-1000 -500 0 500 1000 1500 2000 2500-2500

-2000

-1500

-1000

-500

0

500

x axis [m]

y ax

is [

m]

Missile and Target Trajectories

PN

FLGLRGL

Target

April 21, 2023

0

0max

Mmax

M

c

c

c

A sign Va

A V

a N V

Ref Guidance Law

Large heading error

Small heading error



0 1 2 3 4 5 6 7

-1000

-800

-600

-400

-200

0

200

400

600

800

1000

Time [sec]

a m [m

/sec

2 ]

Missile acceleration vs. time

PN

FLGLRGL

tfinal = 6.5631 [sec]

tfinal = 6.5415 [sec]

April 21, 2023



0 1 2 3 4 5 6 7-70

-60

-50

-40

-30

-20

-10

0

10

Time [sec]

Sig

ma

dot

[deg

/sec

]LOS rate vs. Time

PN

FLGL

RGL

SummarySummary

• Feedback Linearization method has been studied and implemented

• Simulation results have been validated successfully

• 3D implementation of FLGL is on the way


Thank you all for listeningThank you all for listening


Backup SlidesBackup Slides

Outline Outline – Lie derivativesLie derivatives– End game geometryEnd game geometry– Guidance strategiesGuidance strategies– Feedback linearization guidance law Feedback linearization guidance law – Guidance loopGuidance loop– Minimum intercept time guidance Minimum intercept time guidance

lawlaw


Lie derivativesLie derivatives


Let be sufficiently smooth in , the Lie Derivative of with respect to is

denoted by f

dhL h x f x

dx

2

1

1

0

1

f

g f

f

f f f

kfk k

f f f

f

d L hL L h x g x

dx

d L hL h x L L h x f x

dx

d L hL h x L L h x f x , k

dx

L h x h x

Used notations:

f , h n h f

End game geometryEnd game geometry


Collision triangle

LOS LOSR T M R R

LOSR C TM

LOSR TM

V V V V V

V V R

V R

VT

VM

predicted intercept point

LOS

Guidance strategiesGuidance strategies


VT

LOS

predicted intercept point

VM leading the target

VM lagging the target

Case 2: VM lagging the Target

2

sin sin 0T

C CDV V

Case 1: VM leading the Target

1

sin sin 0T

C CDV V

FFeedbackeedback LLinearizationinearization GGuidanceuidance LLawaw


2

1 1

2 2

1

221

2

1

0

0

TM T T MMM

M

C CD T

C CD T

M M M ,pip

T TM ,pip

M

CD M T T T T

CD

C

v R a sin ,sina

N V ,

K V V , sin sinv

K V V , sin sin

V sinsin

V

V V V sin V cos

V V

M T TV cos

VCD1 = closing velocity resulted from 0LOS

RV

VCD2 = leads to the rotation of MV

Guidance loop descriptionGuidance loop description


Guidance Loop Block Diagram

- Known target dynamics

- Ideal missile autopilot

Minimum intercept time guidance Minimum intercept time guidance lawlaw

0

0m m ma V

April 21, 202321 Technion, Faculty of Electrical Engineering

0

0

th

maxM

max

th

M

c

c

c

if

A sign Va

A V

if

a N V

Feedback Linearization Based Guidance

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