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
Ola HärkegårdBackstepping: From simple designs to take-off
Internal seminarJanuary 27, 2005
AUTOMATIC CONTROL
LINKÖPINGS UNIVERSITETCOMMUNICATION SYSTEMS
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
Ola HärkegårdBackstepping: From simple designs to take-off
Internal seminarJanuary 27, 2005
AUTOMATIC CONTROL
LINKÖPINGS UNIVERSITETCOMMUNICATION SYSTEMS
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
Ola HärkegårdBackstepping: From simple designs to take-off
Internal seminarJanuary 27, 2005
AUTOMATIC CONTROL
LINKÖPINGS UNIVERSITETCOMMUNICATION SYSTEMS
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).
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Ola HärkegårdBackstepping: From simple designs to take-off
Internal seminarJanuary 27, 2005
AUTOMATIC CONTROL
LINKÖPINGS UNIVERSITETCOMMUNICATION SYSTEMS
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
Ola HärkegårdBackstepping: From simple designs to take-off
Internal seminarJanuary 27, 2005
AUTOMATIC CONTROL
LINKÖPINGS UNIVERSITETCOMMUNICATION SYSTEMS
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 =
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Ola HärkegårdBackstepping: From simple designs to take-off
Internal seminarJanuary 27, 2005
AUTOMATIC CONTROL
LINKÖPINGS UNIVERSITETCOMMUNICATION SYSTEMS
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 ==
Ola HärkegårdBackstepping: From simple designs to take-off
Internal seminarJanuary 27, 2005
AUTOMATIC CONTROL
LINKÖPINGS UNIVERSITETCOMMUNICATION SYSTEMS
"Conclusion"
Backstepping with linearizing virtual control lawsand quadratic Lyapunov functions
Feedback linearization
Easier to tune
5
Ola HärkegårdBackstepping: From simple designs to take-off
Internal seminarJanuary 27, 2005
AUTOMATIC CONTROL
LINKÖPINGS UNIVERSITETCOMMUNICATION SYSTEMS
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 =
Ola HärkegårdBackstepping: From simple designs to take-off
Internal seminarJanuary 27, 2005
AUTOMATIC CONTROL
LINKÖPINGS UNIVERSITETCOMMUNICATION SYSTEMS
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
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Ola HärkegårdBackstepping: From simple designs to take-off
Internal seminarJanuary 27, 2005
AUTOMATIC CONTROL
LINKÖPINGS UNIVERSITETCOMMUNICATION SYSTEMS
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:
Ola HärkegårdBackstepping: From simple designs to take-off
Internal seminarJanuary 27, 2005
AUTOMATIC CONTROL
LINKÖPINGS UNIVERSITETCOMMUNICATION SYSTEMS
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
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Ola HärkegårdBackstepping: From simple designs to take-off
Internal seminarJanuary 27, 2005
AUTOMATIC CONTROL
LINKÖPINGS UNIVERSITETCOMMUNICATION SYSTEMS
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)
Ola HärkegårdBackstepping: From simple designs to take-off
Internal seminarJanuary 27, 2005
AUTOMATIC CONTROL
LINKÖPINGS UNIVERSITETCOMMUNICATION 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 HärkegårdBackstepping: From simple designs to take-off
Internal seminarJanuary 27, 2005
AUTOMATIC CONTROL
LINKÖPINGS UNIVERSITETCOMMUNICATION SYSTEMS
Control problem
ω
V
Fu
Mu
� 6 states: V, ω
� 4 inputs: uF, uM
Ola HärkegårdBackstepping: From simple designs to take-off
Internal seminarJanuary 27, 2005
AUTOMATIC CONTROL
LINKÖPINGS UNIVERSITETCOMMUNICATION SYSTEMS
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
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Ola HärkegårdBackstepping: From simple designs to take-off
Internal seminarJanuary 27, 2005
AUTOMATIC CONTROL
LINKÖPINGS UNIVERSITETCOMMUNICATION SYSTEMS
Controlled variables
:V
� Angle of attack, α
� Sideslip angle, β
� Total velocity, |V|
β
αV
:V̂Tω � Velocity vector roll rate, pw
wp
Ola HärkegårdBackstepping: From simple designs to take-off
Internal seminarJanuary 27, 2005
AUTOMATIC CONTROL
LINKÖPINGS UNIVERSITETCOMMUNICATION SYSTEMS
Backstepping novelties
� No clear lower triangular form
� Vector states (not scalars)
� MIMO problem
( )M
F
uJJ
n̂ut,VfmVVm
+ω×ω−=ω++×ω−=
&
&
10
Ola HärkegårdBackstepping: From simple designs to take-off
Internal seminarJanuary 27, 2005
AUTOMATIC CONTROL
LINKÖPINGS UNIVERSITETCOMMUNICATION SYSTEMS
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:
Ola HärkegårdBackstepping: From simple designs to take-off
Internal seminarJanuary 27, 2005
AUTOMATIC CONTROL
LINKÖPINGS UNIVERSITETCOMMUNICATION 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 HärkegårdBackstepping: From simple designs to take-off
Internal seminarJanuary 27, 2005
AUTOMATIC CONTROL
LINKÖPINGS UNIVERSITETCOMMUNICATION SYSTEMS
Flight simulation
Backsteppingcontroller
Backsteppingcontroller
Matlab/Simulink FlightGear0
u,,
ref
Fref
=βγα
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