Back Stepping

1

Ola HärkegårdBackstepping: From simple designs to take-off

Internal seminarJanuary 27, 2005

AUTOMATIC CONTROL

LINKÖPINGS UNIVERSITETCOMMUNICATION SYSTEMS

BacksteppingFrom simple designs to take-off

Ola HärkegårdControl & Communication

Linköpings universitet



AUTOMATIC CONTROL


Backstepping

� Constructive (=systematic) control design for nonlinear systems

( )( )

( )u,x,,x,x,xfx

x,x,xfx

x,xfx

n321nn

32122

2111

K&

M

&

&

=

==

(essentially same as forfeedback 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

2



AUTOMATIC CONTROL


Paper statistics

0

5

10

15

20

25

30

92 93 94 95 96 97 98 99 00 01 02 03

Conference papersJournal papers

� Conference papers: 194

� Not adaptive: 108

� Applied 2003: 16 of 25

IEEE Explore 1990-2003Backstepping in title



AUTOMATIC CONTROL


References

� Booksì Nonlinear and Adaptive Control Design, 1995 (Krstic,

Kanellakopolous, Kokotovic).

ì Constructive Nonlinear Control, 1997 (Sepulchre, Jankovic,Kokotovic).

ì Any recent textbook on nonlinear control.

� Papersì The joy of feedback: nonlinear and adaptive, 1991 Bode lecture

(Kokotovic).

ì Constructive nonlinear control: a historical perspective,Automatica, 2001 (Kokotovic, Arcak).

3



AUTOMATIC CONTROL


Lyapunov stability (geometric interpretation)

� Dynamics:

� Lyapunov function:

� For stability: 0fVV x ≤=&

( )xfx =&

( )xV

-4 -3 -2 -1 0 1 2 3 4-3

-2

-1

0

1

2

3

x1

x2

xVf

� Dynamics:

� is a CLF if

for some u

0guVfVV xx <+=&

( ) ( )uxgxfx +=&

( )xV



AUTOMATIC CONTROL


Example 1 (backstepping)

ux

xxx

2

2211

=+=

&

&

Step 1: 121d,2 xxx −−=

212

11 xV =

222

1212

12 x~xV +=

( ) ( )φ+++−= ux~x~xxV 22112&

Step 2: 1212d,222 xxxxxx~ ++=−=

0x~x 22

21 ≤−−=

( )φ+++−= uxx~x 1221

21 x~xu −φ−−=if

( )( )

+−++=

+−=

2112

211

x~x1x2ux~x~xx

&

& φ

0xxxV 21111 ≤−== &&

if d,22 xx =

4



AUTOMATIC CONTROL


Example 1 (feedback linearization)

ux

xxx

2

2211

=+=

&

&

uzz2z

zxxz

zxy

212

22211

11

+==+=

==

&

&

221121 zkzkzz2u −−−= gives stability

Which control lawshould I choose?

Which control lawshould I choose?�

� Same control law withSame control law with

2kk 21 ==



AUTOMATIC CONTROL


"Conclusion"

Backstepping with linearizing virtual control lawsand quadratic Lyapunov functions

Feedback linearization

Easier to tune

5



AUTOMATIC CONTROL


Example 1 (backstepping)

ux

xxx

2

2211

=+=

&

&

Step 1: 121d,2 xxx −−=

212

11 xV =

222

1212

12 x~xV +=

( ) ( )φ+++−= ux~x~xxV 22112&

Step 2: 1212d,222 xxxxxx~ ++=−=

0x~x 22

21 ≤−−=

( )φ+++−= uxx~x 1221

21 x~xu −φ−−=if

( )( )

+−++=

+−=

2112

211

x~x1x2ux~x~xx

&

& φ

0xxxV 21111 ≤−== &&

if d,22 xx =



AUTOMATIC CONTROL


Example 2 (adaptive backstepping)

Step 1: 121d,2 xxx −−=

212

11 xV =

0xxxV 21111 ≤−== &&

ux

xxx

2

2211

=+=

&

&22xθ+

Step 2: 1212d,222 xxxxxx~ ++=−=

( )( )

+−++=

+−=

2112

211

x~x1x2ux~x~xx

&

&

22xθ+

222

1212

12 x~xV += ( )22

1 θ̂−θ+

( )φ+++−= uxx~xV 12212

& 22xθ+ ( )θθ−θ− &̂ˆ

gives21 x~xu −φ−−= 22xθ̂−

22

212 x~xV −−=& ( )

θ−θ−θ+ &̂xx~ˆ 2

22 0≤

222xx~ˆ −=θ&if

6



AUTOMATIC CONTROL


Example 3 (useful nonlinearity)

ux

xxxx

2

21311

=++−=

&

&

-2 -1 0 1 2-6

-4

-2

0

2

4

6

( ) 221

12 x~xWV +=

( )( ) ( )23122

3112 x~xux~x~xxWV +−++−′=&

( ) ( )( )23112

311 x~xuxWx~xxW +−+′+′−=

xx3 +−

1d,2 xx −=

( )11 xWV =

Step 1:With we can select

2x~3u −=

( ) 414

11 xxW =

to achieve 0x~2xV 22

612 ≤−−=&

12d,222 xxxxx~ +=−=

+−=

+−=

2312

2311

x~xux~x~xx

&

&

Step 2:



AUTOMATIC CONTROL


Example 3 (control law properties)

21311 xxxx ++−=&ux2 =&0 3+

−+

− 1x2x2x~− ud,2x

� Cascaded control implementation

� No cancellations ⇒ gain margin (1/3, ∞)

( )( )∫∞

+++0

2612

21216

1 dtuxxx

� Inverse optimal, minimizes

7



AUTOMATIC CONTROL


Design paths

Systembackstepping

CLF, control law

Systembackstepping

CLFcontrol law

(inverse optimal)Sontag’s formula

( ) ( ) ( )( ) ( )21

412

22121221121

21 zcz

czzczzzczzzz2zczzzz2u

++++++++

−−=Ex 1:

Systembackstepping

CLFcontrol law

(locally optimal,globally inv. opt.)

H∞

(Ezal et al. 2000)



AUTOMATIC CONTROL


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.)

8



AUTOMATIC CONTROL


Control problem

ω

V

Fu

Mu

� 6 states: V, ω

� 4 inputs: uF, uM



AUTOMATIC CONTROL


Dynamics

ω

V

Fu

Mu

( )M

F

uJJ

n̂ut,VfmVVm

+ω×ω−=ω++×ω−=

&

&

Engine

GravityAerodynamics

� Stationary motion:

( ) 00

0

V̂Vg

VV

γ+=ω

=

0V̂γ

9



AUTOMATIC CONTROL


Controlled variables

:V

� Angle of attack, α

� Sideslip angle, β

� Total velocity, |V|

β

αV

:V̂Tω � Velocity vector roll rate, pw

wp



AUTOMATIC CONTROL


Backstepping novelties

� No clear lower triangular form

� Vector states (not scalars)

� MIMO problem

( )M

F

uJJ

n̂ut,VfmVVm

+ω×ω−=ω++×ω−=

&

&

10



AUTOMATIC CONTROL


Backstepping design

( ) n̂ut,VfmVVm F++×ω−=&Step 1:

( ) ( )0T

02m

1 VVVVW −−=

cd

cFFF uuu

ω+ω=ω

+=cancel f(V,t)

( ) ( ) n̂VVuVVmW T0F0

T1 −+×ω=&

dω=ω gives

0W1 ≤& if we select ( )( ) V̂VVK

n̂VVku

02

T01F

γ+×−=ω

−−=

ωω+= ~~cWW T21

12

d~ ω−ω=ω

0W2 ≤& if we select

ω−ω+ω×ω= ~KJJu 3dM &

MuJJ +ω×ω−=ω&Step 2:



AUTOMATIC CONTROL


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?

11



AUTOMATIC CONTROL


Flight simulation

Backsteppingcontroller

Backsteppingcontroller

Matlab/Simulink FlightGear0

u,,

ref

Fref

=βγα

Back Stepping

Documents

Transcript of Back Stepping