A study of advanced guidance laws for A study of advanced guidance laws for maneuvering target interceptionmaneuvering target interception

Student: Felix Vilensky Student: Felix Vilensky

Supervisor: Mark MoulinSupervisor: Mark Moulin

Control & Robotics LaboratoryControl & Robotics Laboratory

Introduction

This project deals with missile-target interception. This is a highly non-linear and non stable control problem.

We work with a simplified 2D model. We will discuss the following guidance laws:

PN (Proportional Navigation). Saturated PN. TDLQR(Time dependant LQR). OGL (Optimal guidance law).

Plant - Interception problem

2T Ta ca

.. . .

2.. .

2 sin( ) cos( )

cos( ) sin( )

T M

T M

r r a a

r r a a

0.0008c

PN controller

PlantPlant

PN ControllerPN Controller

rTarget Target

AccelerationAcceleration ModelModel

. .

Ma N r 3.85N

Ta

Ma

PN controller - references

Dhar,A.,and Ghose,D.(1993)Capture region for a realistic TPN guidance law.

Chakravarthy,A.,and Ghose,D.(1996)Capturability of realistic Generalized True Proportional Navigation.

Moulin,M.,Kreindler,E.,and ,Guelman,M(1996).Ballistic missile interception with bearings-only measurements.

PN controller-Command acceleration and relative distance vs. time

0 5 10 15 20 25 30 35 40 45 5030

40

50

60

70

80

90

100

110

120

130aM vs time

t[sec]

aM[m

/s^2

}

3.85, 0.1745 , 0.261 , 0.0051sec

1787 , 80000 , 0.96seci

i

r i i

radN rad rad

mV r m rad

0 5 10 15 20 25 30 35 40 45 500

1

2

3

4

5

6

7

8x 10

4

t[sec]

r[m

]

distance vs time

PN controller – Capturability limits

Initial parameters Capturability limits

<86977

<-1683

<= -0.82

>=0.04

>=0.174

(-0.0068,0.0232)

rV

r

PN controller -conclusions

The performance of the PN controller is quite good. It enables intercepting a target in a wide range of initial conditions.

Yet, the command acceleration is growing with time. And a real physical system cannot maintain an acceleration that is growing towards non physical values.

We seek to find a controller that will work under the constraint of limited (saturated) command

acceleration.

Saturated PN

The first and naive approach is to retain the PN controller and just to add at its output a lowpass filter and a saturation to get the command acceleration.

LP filterLP filter

We use a Butterworth LPF of order 30 with cutoff frequency of 8 rad/sec. This filter will ensure that the command acceleration won’t change too rapidly for the missile to follow.

Saturated PN controller - Command acceleration and relative distance vs. time

3.85, 0.1745 , 0.261 , 0.0051sec

2300 , 80000 , 0.96seci

i

r i i

radN rad rad

mV r m rad

0 5 10 15 20 25 30 35 400

10

20

30

40

50

60

t[sec]

aM[m

/s2 ]

aM vs time

0 5 10 15 20 25 30 35 400

1

2

3

4

5

6

7

8x 10

4

t[sec]

r[m

]

distance vs time

Linear controllers

Linear or partially linear controllers are easier to design than nonlinear ones.

Linear control can be more easily optimized than nonlinear one.

Recent papers used linear control design methods:

Hexner,G.,and Shima,T.(2007)Stochastic optimal control guidance law with bounded acceleration.

Hexner,G.,Shima,T.,and Weiss,H.(2008)LQG guidance law with bounded acceleration

command.

Calculate every fixed interval of time (T) a new infinite horizon LQR.

Use the following state variables:

Each time linearize the plant around:

i.e., around the relative speed and the distance at the time of calculation.

Using this LQR controller till the next calculation. LQR recalculation period:100ms.

Plant sampling period: about 50 ms.

Time dependant LQR

, ,r r

0 00, , 0r r r V

Time dependant LQR

This linear system we use in each calculation:

0

00

0 1 0 0

0 0 0 sin( ) ( )

2 cos( )0 0

M

r r

r r a

V

rr

The following J parameter is being minimized:2

0( ( ) )

0.001 0 0

0 0.01 0 , 1

0 0 1000000

TMJ x Qx a dt

Q

Time dependant LQR

LQR calculatorLQR calculator

PlantPlant

Target Target AccelerationAcceleration

ModuleModule

Ta

Ma

LQR controllerLQR controllerLPF andLPF and saturationsaturation

clockclock

State vector

Gain vector

controller

Time dependant LQR – Command acceleration behavior

0 5 10 15 20 25 30 35 400

10

20

30

40

50

60The command acceleration vs. time for saturated Time dependant LQR

t[sec]

aM[m

/s2 ]

8 8.5 9 9.5 10 10.5 11 11.5 12 12.5

18

20

22

24

26

28

30

32

34

36

38

t[sec]

aM[m

/s2 ]

A sketch of command acceleration vs. time for not saturated Time dependant LQR

3.85, 0.1745 , 0.261 , 0.0051sec

2300 , 80000 , 0.96seci

i

r i i

radN rad rad

mV r m rad

Pure LQR vs. TDLQR

The TDLQR is designed using methods and intuition of optimal linear control.

TDLQR is linear only in each time slice between calculations.

There is a well known LQR guidance law, which is linear through all the engagement time. It is called OGL – Optimal Guidance Law.

While TDLQR is based on infinite horizon LQR, the OGL is a finite horizon LQR, which means that its control gain varies with time.

Optimal Guidance Law

The OGL is obtained using the following linearization of the plant:

Where:sin( )

is the target acceleration.

is the actual applied missile acceleration( ).

is the commanded acceleration(the output of the controller).

is the end of engagement time.

is the time

T

L M

c

F

y r

n

n a

n

t

T

constant of the guidance system.

Thangavelu,R.,and Pardeep,S.(2007)A differential evolution tuned Optimal Guidance Law.

Optimal Guidance Law

The OGL minimizes:

The optimal control is given by:

2

0, ( ) 0

Ft

c FJ n dt y t

The OGL output is then passed through LPF an saturation, as explained earlier to get the command acceleration.

2 22

2

3 2 2

[ 0.5 ( 1 )]

where

6 ( 1 )and

2 3 6 6 12 31

1

xc go T go L

go

go F

x

x x

L

c

Nn y yt n t n T e x

t

t t tx

T T

x e xN

x x x xe en

n sT

Performance Analysis

Miss distance Vs. intial relative speed in PN

0

10000

20000

30000

40000

50000

60000

-3500 -3000 -2500 -2000 -1500 -1000 -500 0

Vr[m/s]

Mis

s di

stan

ce[m

]

Miss distance vs. initial relative speed in saturated PN(-60,60)

0

10000

20000

30000

40000

50000

60000

-4500 -4000 -3500 -3000 -2500 -2000 -1500 -1000 -500 0

Vr[m/s]

Mis

s di

stan

ce[m

]

Miss distance vs. initial relative speed for PN (left) and saturated PN (right) controllers.

Performance Analysis

Miss distance vs. initial relative velocity for Time Dependant LQR, saturated PN and OGL controllers.

Miss distance vs. initial relative velocity in saturated PN,TDLQR and OGL(-60,60)

0

10000

20000

30000

40000

50000

60000

70000

-4500 -4000 -3500 -3000 -2500 -2000 -1500 -1000 -500 0

Vr[m/s]

Mis

s d

ista

nce

[m]

SatPN

TDLQR

OGL

Performance Analysis

Miss distance vs. maximal command acceleration for TDLQR and saturated PN

0

200

400

600

800

1000

1200

1400

1600

1800

2000

0 20 40 60 80 100 120 140 160 180

Maximal command acceleration[m/s^2]

Mis

s di

stan

ce[m

]

TDLQR

SatPN

Miss distance vs. maximal command acceleration for Time Dependant LQR and saturated PN (left) and for OGL (right).

Miss distance vs maximal command acceleration for OGL

18400

18450

18500

18550

18600

18650

18700

18750

18800

18850

0 20 40 60 80 100 120 140 160 180

Maximal command acceleration[m/s^2]

Mis

s di

stan

ce[m

]

Performance Analysis

Miss distance vs. initial distance for Time Dependant LQR, saturated PN and OGL controllers.

Miss distance vs. initial distance for OGL,TDLQR and saturated PN(-60,60)

0

10000

20000

30000

40000

50000

60000

70000

80000

90000

0 20000 40000 60000 80000 100000 120000 140000

Initial distance[m]

Mis

s d

ista

nce

[m]

OGL

SatPN

TDLQR

Performance Analysis

Miss distances vs. initial distances for Time Dependant LQR, saturated PN and OGL controllers (for relatively small initial distances).

Initial distance

Saturated PN (miss distance)

Time dependant LQR (miss distance)

OGL(miss distance)

10000 304.6893 301.162 250.6541

20000 231.6058 130.8969 18.3121

40000 26.603 182.5942 1811.2

Performance Analysis-Results

The effect of the lowpass filter and saturation block provides an optimal initial relative velocity for interception.

The capturability is improved when the maximal command acceleration is increased for TDLQR and saturated PN, and gets worse for OGL.

For large initial distances the miss distance grows monotonically with the initial distance.

For small initial distances an optimal initial distance results in a minimal miss distance.

Performance Analysis-Conclusions

TDLQR has a clear advantage in the performance evaluation over both saturated PN and OGL .

The miss distance for OGL is growing with the maximal command acceleration. OGL doesn’t update linearization and thus applies non optimal command acceleration.

Performance Analysis-Conclusions

The optimum of initial relative velocity is obtained, since for high velocities the command acceleration cannot be high enough to complete the maneuver needed to get the missile into collision course with the target.

The optimum of initial distance is obtained, since for small enough initial distances the missile covers too much distance (outruns the target) before the maneuver needed to get it into collision course with the target is completed.