Backstepping From simple designs to take-off Ola Hrkegrd Control - - PDF document

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Ola Härkegård Backstepping: From simple designs to take-off Internal seminar January 27, 2005

AUTOMATIC CONTROL LINKÖPINGS UNIVERSITET COMMUNICATION SYSTEMS

Backstepping

From simple designs to take-off

Ola Härkegård Control & Communication Linköpings universitet

Ola Härkegård Backstepping: From simple designs to take-off Internal seminar January 27, 2005

AUTOMATIC CONTROL LINKÖPINGS UNIVERSITET COMMUNICATION SYSTEMS

Backstepping

• Constructive (=systematic) control design for nonlinear systems

( ) ( ) ( )

u , x , , x , x , x f x x , x , x f x x , x f x

n 3 2 1 n n 3 2 1 2 2 2 1 1 1

• =

= =

(essentially same as for feedback linearization)

• Applies to systems of lower triangular form
• Can be used to avoid cancellation of "useful nonlinearities"

(unlike feedback linearization)

• Different flavours: adaptive, robust and observer backstepping

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Ola Härkegård Backstepping: From simple designs to take-off Internal seminar January 27, 2005

AUTOMATIC CONTROL LINKÖPINGS UNIVERSITET COMMUNICATION SYSTEMS

Paper statistics

5 10 15 20 25 30 92 93 94 95 96 97 98 99 00 01 02 03

Conference papers Journal papers

• Conference papers: 194
• Not adaptive: 108
• Applied 2003: 16 of 25

IEEE Explore 1990-2003 Backstepping in title

Ola Härkegård Backstepping: From simple designs to take-off Internal seminar January 27, 2005

AUTOMATIC CONTROL LINKÖPINGS UNIVERSITET COMMUNICATION SYSTEMS

References

Books

Nonlinear and Adaptive Control Design, 1995 (Krstic, Kanellakopolous, Kokotovic ). Constructive Nonlinear Control, 1997 (Sepulchre, Jankovic , Kokotovic ). Any recent textbook on nonlinear control.

Papers

The joy of feedback: nonlinear and adaptive, 1991 Bode lecture (Kokotovic). Constructive nonlinear control: a historical perspective, Automatica, 2001 (Kokotovic , Arcak).

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Ola Härkegård Backstepping: From simple designs to take-off Internal seminar January 27, 2005

AUTOMATIC CONTROL LINKÖPINGS UNIVERSITET COMMUNICATION SYSTEMS

Lyapunov stability (geometric interpretation)

• Dynamics:
• Lyapunov function:
• For stability:

f V V

x ≤

=

• ( )

x f x =

• ( )

x V

• 4
• 3
• 2
• 1

1 2 3 4

• 3
• 2
• 1

1 2 3 x1 x2

x

V f

• Dynamics:
• is a CLF if

for some u

gu V f V V

x x

< + =

• ( )

( )

u x g x f x + =

• ( )

x V

Ola Härkegård Backstepping: From simple designs to take-off Internal seminar January 27, 2005

AUTOMATIC CONTROL LINKÖPINGS UNIVERSITET COMMUNICATION SYSTEMS

Example 1 (backstepping)

u x x x x

2 2 2 1 1

= + =

• Step 1:

1 2 1 d , 2

x x x − − =

2 1 2 1 1

x V =

2 2 2 1 2 1 2 1 2

x ~ x V + =

( )

( )

φ + + + − = u x ~ x ~ x x V

2 2 1 1 2

• Step 2:

1 2 1 2 d , 2 2 2

x x x x x x ~ + + = − = x ~ x

2 2 2 1

≤ − − =

( )

φ + + + − = u x x ~ x

1 2 2 1 2 1

x ~ x u − φ − − =

if

( )(

)

     + − + + = + − =

2 1 1 2 2 1 1

x ~ x 1 x 2 u x ~ x ~ x x

• φ

x x x V

2 1 1 1 1

≤ − = =

• if

d , 2 2

x x =

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Ola Härkegård Backstepping: From simple designs to take-off Internal seminar January 27, 2005

AUTOMATIC CONTROL LINKÖPINGS UNIVERSITET COMMUNICATION SYSTEMS

Example 1 (feedback linearization)

u x x x x

2 2 2 1 1

= + =

• u

z z 2 z z x x z z x y

2 1 2 2 2 2 1 1 1 1

+ = = + = = =

• 2

2 1 1 2 1

z k z k z z 2 u − − − =

gives stability

Which control law should I choose? Which control law should I choose?

• Same control law with

Same control law with

2 k k

2 1

= =

Ola Härkegård Backstepping: From simple designs to take-off Internal seminar January 27, 2005

AUTOMATIC CONTROL LINKÖPINGS UNIVERSITET COMMUNICATION SYSTEMS

"Conclusion"

Backstepping with linearizing virtual control laws and quadratic Lyapunov functions Feedback linearization Easier to tune

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Ola Härkegård Backstepping: From simple designs to take-off Internal seminar January 27, 2005

AUTOMATIC CONTROL LINKÖPINGS UNIVERSITET COMMUNICATION SYSTEMS

Example 1 (backstepping)

u x x x x

2 2 2 1 1

= + =

• Step 1:

1 2 1 d , 2

x x x − − =

2 1 2 1 1

x V =

2 2 2 1 2 1 2 1 2

x ~ x V + =

( )

( )

φ + + + − = u x ~ x ~ x x V

2 2 1 1 2

• Step 2:

1 2 1 2 d , 2 2 2

x x x x x x ~ + + = − = x ~ x

2 2 2 1

≤ − − =

( )

φ + + + − = u x x ~ x

1 2 2 1 2 1

x ~ x u − φ − − =

if

( )(

)

     + − + + = + − =

2 1 1 2 2 1 1

x ~ x 1 x 2 u x ~ x ~ x x

• φ

x x x V

2 1 1 1 1

≤ − = =

• if

d , 2 2

x x =

Ola Härkegård Backstepping: From simple designs to take-off Internal seminar January 27, 2005

AUTOMATIC CONTROL LINKÖPINGS UNIVERSITET COMMUNICATION SYSTEMS

Example 2 (adaptive backstepping)

Step 1:

1 2 1 d , 2

x x x − − =

2 1 2 1 1

x V = x x x V

2 1 1 1 1

≤ − = =

• u

x x x x

2 2 2 1 1

= + =

• 2

2

x θ +

Step 2:

1 2 1 2 d , 2 2 2

x x x x x x ~ + + = − =

( )(

)

     + − + + = + − =

2 1 1 2 2 1 1

x ~ x 1 x 2 u x ~ x ~ x x

• 2

2

x θ +

2 2 2 1 2 1 2 1 2

x ~ x V + =

( )

2 2 1

ˆ θ − θ +

( )

φ + + + − = u x x ~ x V

1 2 2 1 2

• 2

2

x θ +

( )

θ θ − θ −

• ˆ

ˆ

gives

2 1

x ~ x u − φ − − =

2 2

x ˆ θ −

2 2 2 1 2

x ~ x V − − =

• (

)

      θ − θ − θ +

• ˆ

x x ~ ˆ

2 2 2

≤

2 2 2x

x ~ ˆ − = θ

• if

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Ola Härkegård Backstepping: From simple designs to take-off Internal seminar January 27, 2005

AUTOMATIC CONTROL LINKÖPINGS UNIVERSITET COMMUNICATION SYSTEMS

Example 3 (useful nonlinearity)

u x x x x x

2 2 1 3 1 1

= + + − =

• 2
• 1

1 2

• 6
• 4
• 2

2 4 6

( )

2 2 1 1 2

x ~ x W V + =

( )(

) ( )

2 3 1 2 2 3 1 1 2

x ~ x u x ~ x ~ x x W V + − + + − ′ =

• ( )

( )

( )

2 3 1 1 2 3 1 1

x ~ x u x W x ~ x x W + − + ′ + ′ − =

x x3 + −

1 d , 2

x x − =

( )

1 1

x W V =

Step 1: With we can select

2

x ~ 3 u − =

( )

4 1 4 1 1

x x W =

to achieve

x ~ 2 x V

2 2 6 1 2

≤ − − =

• 1

2 d , 2 2 2

x x x x x ~ + = − =      + − = + − =

2 3 1 2 2 3 1 1

x ~ x u x ~ x ~ x x

• Step 2:

Ola Härkegård Backstepping: From simple designs to take-off Internal seminar January 27, 2005

AUTOMATIC CONTROL LINKÖPINGS UNIVERSITET COMMUNICATION SYSTEMS

Example 3 (control law properties)

2 1 3 1 1

x x x x + + − =

• u

x2 =

• 3

+− +−

1

x

2

x

2

x ~ − u

d , 2

x

• Cascaded control implementation
• No cancellations ⇒ gain margin (1/3, ∞)

( )

( ) ∫

∞

+ + +

2 6 1 2 2 1 2 1 6 1

dt u x x x

• Inverse optimal, minimizes

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Ola Härkegård Backstepping: From simple designs to take-off Internal seminar January 27, 2005

AUTOMATIC CONTROL LINKÖPINGS UNIVERSITET COMMUNICATION SYSTEMS

Design paths

System backstepping CLF, control law System backstepping CLF control law (inverse optimal) Sontag’s formula

( ) ( ) ( ) ( ) ( )

2 1 4 1 2 2 2 1 2 1 2 2 1 1 2 1 2 1

z cz cz z cz z z cz z z z 2 z cz z z z 2 u + + + + + + + + − − = Ex 1: System backstepping CLF control law (locally optimal, globally inv. opt.) H∞ (Ezal et al. 2000)

Ola Härkegård Backstepping: From simple designs to take-off Internal seminar January 27, 2005

AUTOMATIC CONTROL LINKÖPINGS UNIVERSITET COMMUNICATION SYSTEMS

Backstepping control of a rigid body

Plug-and-play flight controller Use vector description of dynamics for control design Plug-and-play flight controller Use vector description of dynamics for control design

(Extension of CDC 2002 paper by Glad & Härkegård.)

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Ola Härkegård Backstepping: From simple designs to take-off Internal seminar January 27, 2005

AUTOMATIC CONTROL LINKÖPINGS UNIVERSITET COMMUNICATION SYSTEMS

Control problem

ω V

F

u

M

u 6 states: V, ω 4 inputs: uF, uM

Ola Härkegård Backstepping: From simple designs to take-off Internal seminar January 27, 2005

AUTOMATIC CONTROL LINKÖPINGS UNIVERSITET COMMUNICATION SYSTEMS

Dynamics

ω V

F

u

M

u

( )

M F

u J J n ˆ u t , V f mV V m + ω × ω − = ω + + × ω − =

• Engine

Gravity Aerodynamics

• Stationary motion:

( )

V ˆ V g V V γ + = ω =

V ˆ γ

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Ola Härkegård Backstepping: From simple designs to take-off Internal seminar January 27, 2005

AUTOMATIC CONTROL LINKÖPINGS UNIVERSITET COMMUNICATION SYSTEMS

Controlled variables

: V

• Angle of attack, α
• Sideslip angle, β
• Total velocity, |V|

β α V

: V ˆ

T

ω

• Velocity vector roll rate, pw

w

p

Ola Härkegård Backstepping: From simple designs to take-off Internal seminar January 27, 2005

AUTOMATIC CONTROL LINKÖPINGS UNIVERSITET COMMUNICATION SYSTEMS

Backstepping novelties

• No clear lower triangular form
• Vector states (not scalars)
• MIMO problem

( )

M F

u J J n ˆ u t , V f mV V m + ω × ω − = ω + + × ω − =

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Ola Härkegård Backstepping: From simple designs to take-off Internal seminar January 27, 2005

AUTOMATIC CONTROL LINKÖPINGS UNIVERSITET COMMUNICATION SYSTEMS

Backstepping design ( )

n ˆ u t , V f mV V m

F

+ + × ω − =

• Step 1:

( ) ( )

T 2 m 1

V V V V W − − =

c d c F F F

u u u ω + ω = ω + =

cancel f(V,t)

( ) ( ) n

ˆ V V u V V m W

T F T 1

− + × ω =

• d

ω = ω

gives

W1 ≤

• if we select

( ) ( )

V ˆ V V K n ˆ V V k u

2 T 1 F

γ + × − = ω − − = ω ω + = ~ ~ cW W

T 2 1 1 2 d

~ ω − ω = ω W2 ≤

• if we select

ω − ω + ω × ω = ~ K J J u

3 d M

• M

u J J + ω × ω − = ω

• Step 2:

Ola Härkegård Backstepping: From simple designs to take-off Internal seminar January 27, 2005

AUTOMATIC CONTROL LINKÖPINGS UNIVERSITET COMMUNICATION SYSTEMS

Closed loop dynamics

Independent of nonlinear force f(V,t) (but not linear) Good decoupling Easy to tune locally linear dynamics of |V|, α, β and ω Singular at α = 90 deg Robustness?

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Ola Härkegård Backstepping: From simple designs to take-off Internal seminar January 27, 2005

AUTOMATIC CONTROL LINKÖPINGS UNIVERSITET COMMUNICATION SYSTEMS

Flight simulation

Backstepping controller Backstepping controller Matlab/Simulink FlightGear

u , ,

ref F ref

= β γ α