Flight Control Design Using Backstepping Ola Hrkegrd, T orkel Glad - - PDF document

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Flight Control Design Using Backstepping

Ola Härkegård, T orkel Glad Linköping University

Background

Avoid tedious gain-scheduling. Previous work: feedback linearization. Can you design a single c

• ntroller that

will give stability and performance throughout the entire flight regime?

• T ry backstepping!

T ry backstepping!

2

Controller overview

Control alloc.

r

δ S tate feedback u x Pre- filter

Control objectives

α

ref

α = α = β

β

ref s s

p p =

p

s

3

Angle of attack dynamics

V L α M δ

J ) , q , ( M q q mV mg ) ( L δ α = + − α − = α

• Aerodynamic efforts
• 40
• 20

20 40

• 0.5

0.5 α (deg) δ (deg) Cm

• 40
• 20

20 40

• 1

1 2 α (deg) δ (deg) CL

4

Ideas for control design

Linearize the airc raft dynamic s for a set

• f flight cases (gain-scheduling).

Cancel the nonlinear system behaviour using feedback (feedback linearization). Refrain from canc elling “ harmless” nonlinearities (backstepping).

2-stage control design

Dynamic s: 1.Design . 2.Allocate control surfac es.

J ) , q , ( M q q mV mg ) ( L δ α = + − α − = α

• ( )

u q f = + α =

( )

q , k u α =

5

Backstepping

First c

• nsider

( )

q f + α = α

• Virtual c
• ntrol law:

( ) ( )

ref 1 ref des

k f q α − α − α − = ( )

α f α

ref

α

Backstepping, contd.

Create Lyapunov function

( ) ( )

2 des ref

q q F V − + α − α =

Demand

W dt dV < − =

and solve for u.

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Resulting control law

Feedback linearization Backstepping

( ) ( ) ( ) ( ) ( )

α + α ′ + − α − α − = f q f k k u

2 ref 1

( ) ( ) ( )

ref 2 ref 1

f q k k u α + − α − α − =

Step responses

2 4 6 8 20 40 60 α [deg] 2 4 6 8

• 200

200 q [deg/s] 2 4 6 8

• 1000

1000 u [deg/s 2] time [s] 2 4 6 8 10 20 30 α [deg] 2 4 6 8

• 100

100 q [deg/s] 2 4 6 8

• 500

500 u [deg/s 2] time [s]

deg 20

ref =

α deg 60

ref =

α

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Step responses

deg 20

ref =

α deg 60

ref =

α

2 4 6 8 10 20 30 α (deg) 2 4 6 8

• 500

500 u (deg/s

2)

time (s) 2 4 6 8 20 40 60 80 2 4 6 8

• 1000

1000 time (s) α (deg) u (deg/s

2)

Control law properties

Global stability. Does not involve dL/ dα. Inverse optimal ⇒ gain margin.

( ) ( ) ( ) ( )

ref 2 ref 1

f q k k J , q , M α + − α − α − = δ α

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Full controller

=f1(α, β, p

s)+q

=u1 =f2(β, α)+rs =u2 =u3

q

• s

r

• α
• β
• s

p

• (

)

s ref 1 1

p , , q , , k u β α α = ⇒   

( )

s 2 2

r , k u β = ⇒   

}

( )

s ref s 3

p p K u − = ⇒

( ) ( )

ω × ω − δ =

−

J M SJ u

1

Simulations

5 10 10 20 α (deg) 5 10

• 100

100 200 ps (deg/s) 5 10

• 5

5 β (deg) 5 10

• 40
• 20

20 time (s) δ (deg)

Aileron Elevator Rudder