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8/10/2019 Nonlinear_PSPI.pdf
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NonlinearControl
N. Marchand
References
Outline
Introduction
Linear/Nonlinear
The X4 example
Linearapproaches
Antiwindup
Linearization
Gain scheduling
Stability
NonlinearapproachesCLF
Sliding mode
Geometriccontrol
Recursivetechniques
X4 stabilization
Observers
Control of Nonlinear Systems
Nicolas Marchand
CNRS Associate researcher
Control Systems Department, gipsa-labGrenoble, France
Master PSPI 2009-2010
N. Marchand (gipsa-lab) Nonlinear Control Master PSPI 2009-2010 1 / 174
8/10/2019 Nonlinear_PSPI.pdf
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NonlinearControl
N. Marchand
References
Outline
Introduction
Linear/Nonlinear
The X4 example
Linearapproaches
Antiwindup
Linearization
Gain scheduling
Stability
NonlinearapproachesCLF
Sliding mode
Geometriccontrol
Recursivetechniques
X4 stabilization
Observers
References
Nonlinear systems Khalil - Prentice-Hall, 2002
Probably the best book to start with nonlinear control
Nonlinear systems S. Sastry - Springer Verlag, 1999
Good general book, a bit harder than Khalil’s
Mathematical Control Theory - E.D. Sontag - Springer, 1998
Mathematically oriented, Can be downloaded at:
http://www.math.rutgers.edu/ ~ sontag/FTP_DIR/sontag_mathematical_control_theory_springer98.pdf
Constructive nonlinear control - Sepulchre et al. - Springer, 1997 More
focused on passivity and recursive approaches
Nonlinear control systems - A. Isidori - Springer Verlag, 1995
A reference for geometric approach
Applied Nonlinear control - J.J. Slotine and W. Li - Prentice-Hall, 1991
An interesting reference in particular for sliding mode
“Research on Gain Scheduling” - W.J. Rugh and J.S. Shamma -
Automatica, 36 (10):1401–1425, 2000“Survey of gain scheduling analysis and design” - D.J. Leith and W.E.Leithead - Int. Journal of Control, 73:1001–1025, 1999Surveys on Gain Scheduling, The second reference can be downloaded at:
http://citeseer.ist.psu.edu/leith99survey.html
N. Marchand (gipsa-lab) Nonlinear Control Master PSPI 2009-2010 2 / 174
8/10/2019 Nonlinear_PSPI.pdf
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NonlinearControl
N. Marchand
References
Outline
Introduction
Linear/Nonlinear
The X4 example
Linearapproaches
Antiwindup
Linearization
Gain scheduling
Stability
Nonlinear
approachesCLF
Sliding mode
Geometriccontrol
Recursivetechniques
X4 stabilization
Observers
Outline
1 IntroductionLinear versus nonlinear
The X4 example2 Linear control methods for nonlinear systems
AntiwindupLinearization
Gain scheduling3 Stability
4 Nonlinear control methodsControl Lyapunov functions
Sliding mode controlState and output linearizationBackstepping and feedforwardingStabilization of the X4 at a position
5 Observers
N. Marchand (gipsa-lab) Nonlinear Control Master PSPI 2009-2010 3 / 174
8/10/2019 Nonlinear_PSPI.pdf
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NonlinearControl
N. Marchand
References
Outline
Introduction
Linear/Nonlinear
The X4 example
Linearapproaches
Antiwindup
Linearization
Gain scheduling
Stability
Nonlinear
approachesCLF
Sliding mode
Geometriccontrol
Recursivetechniques
X4 stabilization
Observers
Outline
1 IntroductionLinear versus nonlinear
The X4 example2 Linear control methods for nonlinear systems
AntiwindupLinearization
Gain scheduling3 Stability
4 Nonlinear control methodsControl Lyapunov functions
Sliding mode controlState and output linearizationBackstepping and feedforwardingStabilization of the X4 at a position
5 Observers
N. Marchand (gipsa-lab) Nonlinear Control Master PSPI 2009-2010 4 / 174
8/10/2019 Nonlinear_PSPI.pdf
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NonlinearControl
N. Marchand
References
Outline
Introduction
Linear/Nonlinear
The X4 example
Linearapproaches
Antiwindup
Linearization
Gain scheduling
Stability
Nonlinear
approachesCLF
Sliding mode
Geometriccontrol
Recursivetechniques
X4 stabilization
Observers
Introduction
Some preliminary vocable and definitions
What is control ?
A very short review of the linear case (properties,design of control laws, etc.)
Why nonlinear control ?
Formulation of a nonlinear control problem (model,representation, closed-loop stability, etc.)?
Some strange possible behaviors of nonlinear systems
Example : The X4 helicopter
N. Marchand (gipsa-lab) Nonlinear Control Master PSPI 2009-2010 5 / 174
8/10/2019 Nonlinear_PSPI.pdf
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NonlinearControl
N. Marchand
References
Outline
Introduction
Linear/Nonlinear
The X4 example
Linearapproaches
Antiwindup
Linearization
Gain scheduling
Stability
Nonlinear
approachesCLF
Sliding mode
Geometriccontrol
Recursivetechniques
X4 stabilization
Observers
Outline
1 IntroductionLinear versus nonlinear
The X4 example2 Linear control methods for nonlinear systems
AntiwindupLinearization
Gain scheduling3 Stability
4 Nonlinear control methodsControl Lyapunov functions
Sliding mode controlState and output linearizationBackstepping and feedforwardingStabilization of the X4 at a position
5 Observers
N. Marchand (gipsa-lab) Nonlinear Control Master PSPI 2009-2010 6 / 174
8/10/2019 Nonlinear_PSPI.pdf
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NonlinearControl
N. Marchand
References
Outline
Introduction
Linear/Nonlinear
The X4 example
Linearapproaches
Antiwindup
Linearization
Gain scheduling
Stability
Nonlinear
approachesCLF
Sliding mode
Geometriccontrol
Recursivetechniques
X4 stabilization
Observers
Some preliminary vocable
We consider a dynamical system:
ystem
u(t) y(t)x(t)
where:
y is the output: represents what is “visible” from outsidethe systemx is the state of the system: characterizes the state of thesystemu is the control input: makes the system move
N. Marchand (gipsa-lab) Nonlinear Control Master PSPI 2009-2010 7 / 174
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NonlinearControl
N. Marchand
References
Outline
Introduction
Linear/Nonlinear
The X4 example
Linearapproaches
Antiwindup
Linearization
Gain scheduling
Stability
Nonlinear
approachesCLF
Sliding mode
Geometriccontrol
Recursivetechniques
X4 stabilization
Observers
The 4 steps to control a system
y u y d
x
Observer
k (x )
Controller
1 ModelizationTo get a mathematical representation of the systemDifferent kind of model are useful. Often:
a simple model to build the control lawa sharp model to check the control law and the observer
2 Design the state reconstruction: in order to reconstructthe variables needed for control
3 Design the control and test it
4 Close the loop on the real system
N. Marchand (gipsa-lab) Nonlinear Control Master PSPI 2009-2010 8 / 174
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NonlinearControl
N. Marchand
References
Outline
Introduction
Linear/Nonlinear
The X4 example
Linearapproaches
Antiwindup
Linearization
Gain scheduling
Stability
Nonlinear
approachesCLF
Sliding mode
Geometriccontrol
Recursivetechniques
X4 stabilization
Observers
Linear dynamical systemsSome properties of linear system (1/2)
Definition: Systems such that if y 1 and y 2 are the
outputs corresponding to u 1 and u 2, then ∀λ ∈ R:y 1 + λy 2 is the output corresponding to u 1 + λu 2
Representation near the operating point:Transfer function:
y (s ) = h(s )u (s )
State space representation:x = Ax + Bu
y = Cx (+Du )
The model can be obtainedphysical modelization (eventually coupled withidentification)identification (black box approach)
both give h(s ) or (A, B , C , D ) and hence the model.
N. Marchand (gipsa-lab) Nonlinear Control Master PSPI 2009-2010 9 / 174
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NonlinearControl
N. Marchand
References
Outline
Introduction
Linear/Nonlinear
The X4 example
Linearapproaches
Antiwindup
Linearization
Gain scheduling
Stability
Nonlinear
approachesCLF
Sliding mode
Geometriccontrol
Recursivetechniques
X4 stabilization
Observers
Linear dynamical systemsSome properties of linear system (2/2)
x = Ax + Bu
y = Cx (+Du )
Nice properties:
Unique and constant equilibriumControllability (resp. observability) directly given byrank(B , AB , . . . , An−1B ) (resp. rank(C , CA, . . . , CAn−1))
Stability directly given by the poles of h(s )
or the eigenvaluesof A (asymptotically stable < 0)Local properties = global properties (like stability,stabilizability, etc.)The time behavior is independent of the initial conditionFrequency analysis is easyControl is easy: simply take u = Kx with K such that
(eig(A + BK )) < 0, the closed-loop system x = Ax + BKx willbe asymptotically stable· · ·
Mathematical tool: linear algebraThis is a caricature of the reality (of course problems due touncertainties, delays, noise, etc.)
N. Marchand (gipsa-lab) Nonlinear Control Master PSPI 2009-2010 10 / 174
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NonlinearControl
N. Marchand
References
Outline
Introduction
Linear/Nonlinear
The X4 example
Linearapproaches
Antiwindup
Linearization
Gain scheduling
Stability
Nonlinear
approachesCLF
Sliding mode
Geometriccontrol
Recursivetechniques
X4 stabilization
Observers
Why nonlinear control ?Why nonlinear control if linear control is so easy ?
All physical systems are nonlinear because of
Actuators saturationsViscosity (proportional to speed2)Sine or cosine functions in roboticsChemical kinetic in exp(temperature)
Friction or hysteresis phenomena
...
More and more the performance specification requiresnonlinear control (eg. automotive)
More and more controlled systems are deeply nonlinear
(eg. µ -nano systems where hysteresis phenomena, friction,discontinuous behavior)
Nonlinear control is sometimes necessary (oscillators,cyclic systems, . . . )
N. Marchand (gipsa-lab) Nonlinear Control Master PSPI 2009-2010 11 / 174
8/10/2019 Nonlinear_PSPI.pdf
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NonlinearControl
N. Marchand
References
Outline
Introduction
Linear/Nonlinear
The X4 example
Linearapproaches
Antiwindup
Linearization
Gain scheduling
Stability
Nonlinear
approachesCLF
Sliding mode
Geometriccontrol
Recursivetechniques
X4 stabilization
Observers
Nonlinear dynamical systemsHow to get a model ?
Representation:
State space representation:ODE Ordinary differential equation:
In this course, only:
x (t ) = f (x (t ), u (t ))
y (t ) = h(x (t ), u (t ))
PDE Partial differential equation (traffic, flow, etc.):
0 = g (x (t ), x (t ), ∂f (x ,... )
∂x u (t ))
0 = h(y (t ), x (t ), u (t ))
. . . Algebraic differential equations (implicit), hybrid (with
discrete or event based equations), etc.The model can be obtained
physical modelization and then nonlinear identification of the parameters (identifiability problems)
N. Marchand (gipsa-lab) Nonlinear Control Master PSPI 2009-2010 12 / 174
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NonlinearControl
N. Marchand
References
Outline
Introduction
Linear/Nonlinear
The X4 example
Linearapproaches
Antiwindup
Linearization
Gain scheduling
Stability
Nonlinear
approachesCLF
Sliding mode
Geometriccontrol
Recursivetechniques
X4 stabilization
Observers
Nonlinear dynamical systemsOpen-loop control versus closed-loop control ?
y u y d
x
States: x
Observer
k (x )
Controller
Open-loop controlfind u (t ) such that limt →∞ y (t ) − y d (t ) = 0
Widely used for path planning problems (robotics)
Closed-loop control
find u (x ) such that limt →∞ y (t ) − y d (t ) = 0
Better because closed loop control can stabilize systems and is robust w.r.t.perturbation.
In this course
Only closed-loop control problems are treated
N. Marchand (gipsa-lab) Nonlinear Control Master PSPI 2009-2010 13 / 174
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NonlinearControl
N. Marchand
References
Outline
Introduction
Linear/Nonlinear
The X4 example
Linearapproaches
Antiwindup
Linearization
Gain scheduling
Stability
Nonlinear
approachesCLF
Sliding mode
Geometriccontrol
Recursivetechniques
X4 stabilization
Observers
Nonlinear dynamical systemsAim of control ?
y u x d
x
States: x
Observer
k (x )
Controller
Tracking problemfind u (x ) such that limt →∞ x (t ) − x d (t ) = 0
Stabilization problem
find u (x ) such that limt →∞ x (t ) − x d (t ) = 0 for x d (t ) = constant
Null stabilization problemfind u (x ) such that limt →∞ x (t ) − x d (t ) = 0 for x d (t ) = 0
In any cases, a null stabilization problem of z (t ) = y (t ) − y d (t )
In this course
Only the stabilization problem will be treated.
N. Marchand (gipsa-lab) Nonlinear Control Master PSPI 2009-2010 14 / 174
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NonlinearControl
N. Marchand
References
Outline
Introduction
Linear/Nonlinear
The X4 example
Linearapproaches
Antiwindup
Linearization
Gain scheduling
Stability
Nonlinear
approachesCLF
Sliding mode
Geometriccontrol
Recursivetechniques
X4 stabilization
Observers
Some strange behaviors of nonlinear systemsThe undersea vehicle
Simplified model of the undersea vehicle (in one
direction):
v (t ) = −v |v | + u
Step answer:
No proportionalitybetween the inputand the output
0 1 2 3 4 5 6 7 8 9 100
0.5
1
1.5
u (s ) = 1
100s
u (s ) = 1s
u (s ) = 2s
N. Marchand (gipsa-lab) Nonlinear Control Master PSPI 2009-2010 15 / 174
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NonlinearControl
N. Marchand
References
Outline
Introduction
Linear/Nonlinear
The X4 example
Linearapproaches
Antiwindup
Linearization
Gain scheduling
Stability
Nonlinear
approachesCLF
Sliding mode
Geometriccontrol
Recursivetechniques
X4 stabilization
Observers
Some strange behaviors of nonlinear systemsThe Van der Pol oscillator (1/2)
Van der Pol oscillator
x 1 = x 2x 2 = −x 1 − εh(x 1)x 2
with h(x 1) = −1 + x 21
Oscillations: ε tunes the limit cycle
0 10 20 30 40 50−3
−2
−1
0
1
2
3
4
5
6
ε = 1, x (0) = (5, 5)
ε = 0.1, x (0) = (5, 5)
time
x
−4 −2 0 2 4 6−4
−3
−2
−1
0
1
2
3
4
5
6
ε = 1, x (0) = (5, 5)
ε = 0.1, x (0) = (5, 5)
x 1
x 2
N. Marchand (gipsa-lab) Nonlinear Control Master PSPI 2009-2010 16 / 174
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NonlinearControl
N. Marchand
References
Outline
Introduction
Linear/Nonlinear
The X4 example
Linearapproaches
Antiwindup
Linearization
Gain scheduling
Stability
Nonlinear
approachesCLF
Sliding mode
Geometriccontrol
Recursivetechniques
X4 stabilization
Observers
Some strange behaviors of nonlinear systemsThe Van der Pol oscillator (2/2)
Oscillations: stable limit cycles, fast dynamics
0 10 20 30 40 50−3
−2
−1
0
1
2
3
4
5
6
ε = 1, x (0) = (0, 1)
ε = 1, x (0) = (5, 5)
time
x
−3 −2 −1 0 1 2 3 4 5 6−3
−2
−1
0
1
2
3
4
5
6
ε = 1, x (0) = (0, 1)
ε = 1, x (0) = (5, 5)
x 1
x 2
0 0.1 0.2 0.3 0.4 0.5−3
−2
−1
0
1
2
3
4
5
6
ε = 1, x (0) = (0, 1)
ε = 1, x (0) = (5, 5)
time
x
N. Marchand (gipsa-lab) Nonlinear Control Master PSPI 2009-2010 17 / 174
S f
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NonlinearControl
N. Marchand
References
Outline
Introduction
Linear/Nonlinear
The X4 example
Linearapproaches
Antiwindup
Linearization
Gain scheduling
Stability
Nonlinear
approachesCLF
Sliding mode
Geometriccontrol
Recursivetechniques
X4 stabilization
Observers
Some strange behaviors of nonlinear systemsThe tunnel diode (1/2)
The tunnel diode model
C dV C
dt + i R = i L
E − Ri L = V C + Ldi L
dt
i R = h(v R )
It gives:
x 1(t ) = 1
C
(−h(x 1) + x 2)
x 2(t ) = 1
L(−x 1 − Rx 2 + u )
with x 1 = v C , x 2 = i L and u = E
N. Marchand (gipsa-lab) Nonlinear Control Master PSPI 2009-2010 18 / 174
S b h i f li
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NonlinearControl
N. Marchand
References
Outline
Introduction
Linear/Nonlinear
The X4 example
Linearapproaches
Antiwindup
Linearization
Gain scheduling
Stability
Nonlinear
approachesCLF
Sliding mode
Geometriccontrol
Recursivetechniques
X4 stabilization
Observers
Some strange behaviors of nonlinear systemsThe tunnel diode (2/2)
The tunnel diode behavior: bifurcation
with:i R = 17.76v R − 103.79v 2R + 229.62v 3R − 226.31v 4R + 83.72v 5R
0 2 4 6 8 100
0.2
0.4
0.6
0.8
1
1.2
1.4
x (0) = (0, 1.23)
x (0) = (0, 1.24)
time
x
0 0.2 0.4 0.6 0.8 1
0.4
0.5
0.6
0.7
0.8
0.9
1
1.1
1.2
1.3
x (0) = (0, 1.23)
x (0) = (0, 1.24)
x 1
x 2
N. Marchand (gipsa-lab) Nonlinear Control Master PSPI 2009-2010 19 / 174
S b h i f li
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NonlinearControl
N. Marchand
References
Outline
Introduction
Linear/Nonlinear
The X4 example
Linearapproaches
Antiwindup
Linearization
Gain scheduling
Stability
Nonlinear
approachesCLF
Sliding mode
Geometriccontrol
Recursivetechniques
X4 stabilization
Observers
Some strange behaviors of nonlinear systemsThe car parking
The car parking simplified model
x 1 = u 1
x 2 = u 2
x 3 = x 2u 1
System with a misleading simplicity
Theorem (Brockett (83))
There is no (u 1(x ), u 2(x )) continuous w.r.t. x such that x is
asymptotically stable
In practice:manoeuvresdiscontinuous controltime-varying control M
a r c h a n d a n d A l a m i r ( 2 0 0 3 )
N. Marchand (gipsa-lab) Nonlinear Control Master PSPI 2009-2010 20 / 174
N li d i l t
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NonlinearControl
N. Marchand
References
Outline
Introduction
Linear/Nonlinear
The X4 example
Linearapproaches
Antiwindup
Linearization
Gain scheduling
Stability
Nonlinear
approachesCLF
Sliding mode
Geometriccontrol
Recursivetechniques
X4 stabilization
Observers
Nonlinear dynamical systemsEverything is possible
x = f (x , u )
y = h(x , u )
Everything is possible:
Equilibrium can be unique, multiple, infinite or even not existControllability (resp. observability) are very hard to prove (it isoften even not checked)
Stability may be hard to proveLocal properties = global properties (like stability,stabilizability, etc.)The time behavior is depends upon the initial conditionFrequency analysis is almost impossibleNo systematic approach for building a control law: to each
problem corresponds it unique solution· · ·
Mathematical tool: Lyapunov and differential tools
N. Marchand (gipsa-lab) Nonlinear Control Master PSPI 2009-2010 21 / 174
Outline
8/10/2019 Nonlinear_PSPI.pdf
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NonlinearControl
N. Marchand
References
Outline
Introduction
Linear/Nonlinear
The X4 example
Linearapproaches
Antiwindup
Linearization
Gain scheduling
Stability
Nonlinear
approachesCLF
Sliding mode
Geometriccontrol
Recursivetechniques
X4 stabilization
Observers
Outline
1 IntroductionLinear versus nonlinear
The X4 example2 Linear control methods for nonlinear systems
AntiwindupLinearizationGain scheduling
3 Stability
4 Nonlinear control methodsControl Lyapunov functionsSliding mode controlState and output linearizationBackstepping and feedforwardingStabilization of the X4 at a position
5 Observers
N. Marchand (gipsa-lab) Nonlinear Control Master PSPI 2009-2010 22 / 174
The X4 helico te
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NonlinearControl
N. Marchand
References
Outline
Introduction
Linear/Nonlinear
The X4 example
Linearapproaches
Antiwindup
Linearization
Gain scheduling
Stability
Nonlinear
approachesCLF
Sliding mode
Geometriccontrol
Recursivetechniques
X4 stabilization
Observers
The X4 helicopterHow it works ?
4 fixed rotors with controlled rotationspeed s i
s 1
s 2
s 3
s 4
N. Marchand (gipsa-lab) Nonlinear Control Master PSPI 2009-2010 23 / 174
The X4 helicopter
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NonlinearControl
N. Marchand
References
Outline
Introduction
Linear/Nonlinear
The X4 example
Linearapproaches
Antiwindup
Linearization
Gain scheduling
Stability
Nonlinear
approachesCLF
Sliding mode
Geometriccontrol
Recursivetechniques
X4 stabilization
Observers
The X4 helicopterHow it works ?
4 fixed rotors with controlled rotationspeed s i 4 generated forces F i
F 1
F 2
F 3
F 4
N. Marchand (gipsa-lab) Nonlinear Control Master PSPI 2009-2010 23 / 174
The X4 helicopter
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NonlinearControl
N. Marchand
References
Outline
Introduction
Linear/Nonlinear
The X4 example
Linearapproaches
Antiwindup
Linearization
Gain scheduling
Stability
Nonlinear
approachesCLF
Sliding mode
Geometriccontrol
Recursivetechniques
X4 stabilization
Observers
The X4 helicopterHow it works ?
4 fixed rotors with controlled rotationspeed s i 4 generated forces F i 4 counter-rotating torques Γ i
Γ 1
Γ 2
Γ 3
Γ 4
N. Marchand (gipsa-lab) Nonlinear Control Master PSPI 2009-2010 23 / 174
The X4 helicopter
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NonlinearControl
N. Marchand
References
Outline
Introduction
Linear/Nonlinear
The X4 example
Linearapproaches
Antiwindup
Linearization
Gain scheduling
Stability
Nonlinear
approachesCLF
Sliding mode
Geometriccontrol
Recursivetechniques
X4 stabilization
Observers
The X4 helicopterHow it works ?
4 fixed rotors with controlled rotationspeed s i 4 generated forces F i 4 counter-rotating torques Γ i Roll movement generated with adissymmetry between left and rightforces:
Γ r = l (F 4 − F 2)
F 2
F 4
l
N. Marchand (gipsa-lab) Nonlinear Control Master PSPI 2009-2010 23 / 174
The X4 helicopter
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NonlinearControl
N. Marchand
References
Outline
Introduction
Linear/Nonlinear
The X4 example
Linearapproaches
Antiwindup
Linearization
Gain scheduling
Stability
Nonlinear
approachesCLF
Sliding mode
Geometriccontrol
Recursivetechniques
X4 stabilization
Observers
The X4 helicopterHow it works ?
4 fixed rotors with controlled rotationspeed s i 4 generated forces F i 4 counter-rotating torques Γ i Roll movement generated with adissymmetry between left and rightforces:
Γ r = l (F 4 − F 2)Pitch movement generated with adissymmetry between front and rearforces:
Γ p = l (F 1 − F 3)
F 1
F 3
l
N. Marchand (gipsa-lab) Nonlinear Control Master PSPI 2009-2010 23 / 174
The X4 helicopter
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NonlinearControl
N. Marchand
References
Outline
Introduction
Linear/Nonlinear
The X4 example
Linearapproaches
Antiwindup
Linearization
Gain scheduling
Stability
Nonlinear
approachesCLF
Sliding mode
Geometriccontrol
Recursivetechniques
X4 stabilization
Observers
The X4 helicopterHow it works ?
4 fixed rotors with controlled rotationspeed s i 4 generated forces F i 4 counter-rotating torques Γ i Roll movement generated with adissymmetry between left and rightforces:
Γ r = l (F 4 − F 2)Pitch movement generated with adissymmetry between front and rearforces:
Γ p = l (F 1 − F 3)
Yaw movement generated with adissymmetry between front/rear andleft/right torques:
Γ y = Γ 1 + Γ 3 − Γ 2 − Γ 4
Γ 1
Γ 2
Γ 3
Γ 4
N. Marchand (gipsa-lab) Nonlinear Control Master PSPI 2009-2010 23 / 174
The X4 helicopter
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NonlinearControl
N. Marchand
References
Outline
Introduction
Linear/Nonlinear
The X4 example
Linearapproaches
Antiwindup
Linearization
Gain scheduling
Stability
Nonlinear
approachesCLF
Sliding mode
Geometriccontrol
Recursivetechniques
X4 stabilization
Observers
The X4 helicopterBuilding a model (1/3)
Electrical motor: A 2nd order system with friction and saturationusually approximated by a 1rst order system:
s i = − k 2mJ r R
s i − 1
J r τ load +
k m
J r R satU i
(U i ) i ∈ 1, 2, 3, 4 (1)
s i : rotation speedU i : voltage applied to the motor; real control variable
τ load: motor load: τ load = k gearbox κ |s i | s i with κ drag coefficient
Aerodynamical forces and torques: Very complex models exist
N. Marchand (gipsa-lab) Nonlinear Control Master PSPI 2009-2010 24 / 174
The X4 helicopter
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NonlinearControl
N. Marchand
References
Outline
Introduction
Linear/Nonlinear
The X4 example
Linearapproaches
Antiwindup
Linearization
Gain scheduling
Stability
Nonlinear
approachesCLF
Sliding mode
Geometriccontrol
Recursivetechniques
X4 stabilization
Observers
The X4 helicopterBuilding a model (1/3)
Electrical motor: A 2nd order system with friction and saturationusually approximated by a 1rst order system:
s i = − k 2mJ r R
s i − 1
J r τ load +
k m
J r R satU i
(U i ) i ∈ 1, 2, 3, 4 (1)
s i : rotation speedU i : voltage applied to the motor; real control variable
τ load: motor load: τ load = k gearbox κ |s i | s i with κ drag coefficient
Aerodynamical forces and torques: Very complex models existbut overcomplicated for control, better use the simplified model:
F i = bs 2i
Γ r = lb (s 24 − s
22 )
Γ p = lb (s 21 − s 23 )
Γ y = κ(s 21 + s 23 − s 22 − s 24 )
i ∈ 1, 2, 3, 4 (2)
b : thrust coefficient, κ: drag coefficient
N. Marchand (gipsa-lab) Nonlinear Control Master PSPI 2009-2010 24 / 174
The X4 helicopter
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NonlinearControl
N. Marchand
References
Outline
Introduction
Linear/Nonlinear
The X4 example
Linearapproaches
Antiwindup
Linearization
Gain scheduling
Stability
Nonlinear
approachesCLF
Sliding mode
Geometriccontrol
Recursivetechniques
X4 stabilization
Observers
The X4 helicopterBuilding a model (2/3)
Two framesa fixed frame E ( e 1, e 2, e 3)
a frame attached to the X4T ( t 1, t 2, t 3)
Frame change
a rotation matrix R from T to E
e 1
e 2
e 3 t 1
t 2
t 3
State variables:Cartesian coordinates (in E )
position p velocity v
Attitude coordinates:
angular velocity ω in the moving frame T either: Euler angles three successive rotations about t 3, t 1 and t 3 of angles angles φ, θ and ψ giving R or: Quaternion representation (q 0, q ) = (cosβ/2, u sinβ/2)
represent a rotation of angle β about u
N. Marchand (gipsa-lab) Nonlinear Control Master PSPI 2009-2010 25 / 174
The X4 helicopter
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NonlinearControl
N. Marchand
References
Outline
Introduction
Linear/Nonlinear
The X4 example
Linearapproaches
Antiwindup
Linearization
Gain scheduling
Stability
Nonlinear
approachesCLF
Sliding mode
Geometriccontrol
Recursivetechniques
X4 stabilization
Observers
The X4 helicopterBuilding a model (3/3)
Cartesian coordinates:
p = v m v = −mg e 3 + R (
i F i (s i ) t 3)
(3)
Attitude:
Euler angles formalism:
R = R ω×
J ω = − ω×J ω + Γ totwith ω× =
0 −ω3 ω2
ω3 0 −ω1
−ω2 ω1 0
(4)
ω× is the skew symmetric tensor associated to ω
Quaternion formalism:
q = 1
2 Ω( ω)q
= 12 Ξ(q ) ω
J ω = − ω×J ω + Γ tot
with
Ω( ω) =
0 − ωT
ω − ω×
Ξ(q ) =
− q
T
I 3×3q 0 + q × (5)
where Γ tot = −
i
I r ω× t 3s i
gyroscopic torque
+Γ pert +
Γ r (s 2, s 4)
Γ p (s 1, s 3)
Γ y (s 1, s 2, s 3, s 4)
N. Marchand (gipsa-lab) Nonlinear Control Master PSPI 2009-2010 26 / 174
The X4 helicopter
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NonlinearControl
N. Marchand
References
Outline
Introduction
Linear/Nonlinear
The X4 example
Linearapproaches
Antiwindup
Linearization
Gain scheduling
Stability
Nonlinear
approachesCLF
Sliding mode
Geometriccontrol
Recursivetechniques
X4 stabilization
Observers
pReview of the nonlinearities
s i = −
k 2mJ r R s i −
k gearbox κ
J r |s i | s i + k mJ r R satU i (U i )
p = v
m v = −mg e 3 + R
00
i F i (s i )
R = R ω×
J ω = − ω×J ω −
i I r ω×
00
i s i
+
Γ r (s 2, s 4)
Γ p (s 1, s 3)
Γ y (s 1, s 2, s 3, s 4)
In red: the nonlinearitiesIn blue: where the control variables act
N. Marchand (gipsa-lab) Nonlinear Control Master PSPI 2009-2010 27 / 174
The X4 helicopter
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NonlinearControl
N. Marchand
References
Outline
Introduction
Linear/Nonlinear
The X4 example
Linearapproaches
Antiwindup
Linearization
Gain scheduling
Stability
Nonlinear
approachesCLF
Sliding mode
Geometriccontrol
Recursivetechniques
X4 stabilization
Observers
pIdentification of the parameters
Electrical motor:
For small input steps, the system behaves very close to a linear
first order systemHence, use linear identification toolsU i is found on the data-sheet of the motor (damage avoidance)
Aerodynamical parameters: b and κ
b and κ measured with specific test beds,
N. Marchand (gipsa-lab) Nonlinear Control Master PSPI 2009-2010 28 / 174
The X4 helicopter
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NonlinearControl
N. Marchand
References
Outline
Introduction
Linear/Nonlinear
The X4 example
Linearapproaches
Antiwindup
Linearization
Gain scheduling
Stability
Nonlinear
approachesCLF
Sliding mode
Geometriccontrol
Recursivetechniques
X4 stabilization
Observers
pIdentification of the parameters
Electrical motor:
For small input steps, the system behaves very close to a linear
first order systemHence, use linear identification toolsU i is found on the data-sheet of the motor (damage avoidance)
Aerodynamical parameters: b and κ
b and κ measured with specific test beds, depends upon temperature,distance from ground, etc.
Mechanical parameters:
l length of an arm of the helicopter, easy to measurem total mass of the helicopter, easy to measureJ body inertia, hard to have preciselyI r rotor inertia, hard to have precisely
N. Marchand (gipsa-lab) Nonlinear Control Master PSPI 2009-2010 28 / 174
The X4 helicopter
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NonlinearControl
N. Marchand
References
Outline
Introduction
Linear/Nonlinear
The X4 example
Linearapproaches
Antiwindup
Linearization
Gain scheduling
Stability
Nonlinear
approachesCLF
Sliding mode
Geometriccontrol
Recursivetechniques
X4 stabilization
Observers
pValues of the parameters
Motor parameters:
parameter description value unit
k m motor constant 4.3 × 10−3 N.m/AJ r rotor inertia 3.4 × 10−5 J.g.m2
R motor resistance 0.67 Ω
k gearbox gearbox ratio 2.7 × 10−3 -U i maximal voltage 12 V
Aerodynamical parameters:
parameter description value
b thrust coefficient 3.8 × 10−6
κ drag coefficient 2.9 × 10−5
Body parameters:
parameter description value unit
J inertia matrix
14.6 × 10−3 0 00 7.8 × 10−3 00 0 7.8 × 10−3
kg.m2
m mass of the UAV 0.458 kgl radius of the UAV 22.5 cmg gravity 9.81 m/s2
N. Marchand (gipsa-lab) Nonlinear Control Master PSPI 2009-2010 29 / 174
The X4 helicopter
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NonlinearControl
N. Marchand
References
Outline
Introduction
Linear/Nonlinear
The X4 example
Linearapproaches
Antiwindup
Linearization
Gain scheduling
Stability
Nonlinearapproaches
CLF
Sliding mode
Geometriccontrol
Recursivetechniques
X4 stabilization
Observers
Open-loop behavior with roll initial speed
N. Marchand (gipsa-lab) Nonlinear Control Master PSPI 2009-2010 30 / 174
The X4 helicopter
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NonlinearControl
N. Marchand
References
Outline
Introduction
Linear/Nonlinear
The X4 example
Linearapproaches
Antiwindup
Linearization
Gain scheduling
Stability
Nonlinearapproaches
CLF
Sliding mode
Geometriccontrol
Recursivetechniques
X4 stabilization
Observers
Open-loop behavior with pitch initial speed
N. Marchand (gipsa-lab) Nonlinear Control Master PSPI 2009-2010 31 / 174
The X4 helicopter
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NonlinearControl
N. Marchand
References
Outline
Introduction
Linear/Nonlinear
The X4 example
Linearapproaches
Antiwindup
Linearization
Gain scheduling
Stability
Nonlinearapproaches
CLF
Sliding mode
Geometriccontrol
Recursivetechniques
X4 stabilization
Observers
Open-loop behavior with yaw initial speed
N. Marchand (gipsa-lab) Nonlinear Control Master PSPI 2009-2010 32 / 174
The X4 helicopter
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NonlinearControl
N. Marchand
References
Outline
Introduction
Linear/Nonlinear
The X4 example
Linearapproaches
Antiwindup
Linearization
Gain scheduling
Stability
Nonlinearapproaches
CLF
Sliding mode
Geometriccontrol
Recursivetechniques
X4 stabilization
Observers
Open-loop behavior with roll and yaw initial speed
N. Marchand (gipsa-lab) Nonlinear Control Master PSPI 2009-2010 33 / 174
The X4 helicopter
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NonlinearControl
N. Marchand
References
Outline
Introduction
Linear/Nonlinear
The X4 example
Linearapproaches
Antiwindup
Linearization
Gain scheduling
Stability
Nonlinearapproaches
CLF
Sliding mode
Geometriccontrol
Recursivetechniques
X4 stabilization
Observers
Open-loop behavior with pitch and yaw initial speed
N. Marchand (gipsa-lab) Nonlinear Control Master PSPI 2009-2010 34 / 174
Outline
8/10/2019 Nonlinear_PSPI.pdf
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NonlinearControl
N. Marchand
References
Outline
Introduction
Linear/Nonlinear
The X4 example
Linearapproaches
Antiwindup
Linearization
Gain scheduling
Stability
Nonlinearapproaches
CLF
Sliding mode
Geometriccontrol
Recursivetechniques
X4 stabilization
Observers
1 IntroductionLinear versus nonlinearThe X4 example
2 Linear control methods for nonlinear systemsAntiwindupLinearizationGain scheduling
3 Stability
4 Nonlinear control methodsControl Lyapunov functionsSliding mode control
State and output linearizationBackstepping and feedforwardingStabilization of the X4 at a position
5 Observers
N. Marchand (gipsa-lab) Nonlinear Control Master PSPI 2009-2010 35 / 174
Outline
8/10/2019 Nonlinear_PSPI.pdf
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NonlinearControl
N. Marchand
References
Outline
Introduction
Linear/Nonlinear
The X4 example
Linearapproaches
Antiwindup
Linearization
Gain scheduling
Stability
Nonlinearapproaches
CLF
Sliding mode
Geometriccontrol
Recursivetechniques
X4 stabilization
Observers
1 IntroductionLinear versus nonlinearThe X4 example
2 Linear control methods for nonlinear systemsAntiwindupLinearizationGain scheduling
3 Stability
4 Nonlinear control methodsControl Lyapunov functionsSliding mode control
State and output linearizationBackstepping and feedforwardingStabilization of the X4 at a position
5 Observers
N. Marchand (gipsa-lab) Nonlinear Control Master PSPI 2009-2010 36 / 174
Linear systems with saturated inputsI f h l f h X ’ ( / )
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NonlinearControl
N. Marchand
References
Outline
Introduction
Linear/Nonlinear
The X4 example
Linearapproaches
Antiwindup
Linearization
Gain scheduling
Stability
Nonlinearapproaches
CLF
Sliding mode
Geometriccontrol
Recursivetechniques
X4 stabilization
Observers
Impact of input saturations on the control of the X4’s rotors (1/4)
We go back to the X4 example and focus on the rotors:s i = −
k 2mJ r R
s i − 1
J r τ load +
k m
J r R satU i
(U i )
If one wants to act on the X4 with desired forces F d i , it
is necessary to be able to set the rotors speeds s i to s d i
with
s d i =
1
b F d
i
A usual way to control the electrical motor consist in
taking τ load as un unknown loadneglecting the voltage limitations U i
N. Marchand (gipsa-lab) Nonlinear Control Master PSPI 2009-2010 37 / 174
Linear systems with saturated inputsI f i i h l f h X4’ (2/4)
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NonlinearControl
N. Marchand
References
Outline
Introduction
Linear/Nonlinear
The X4 example
Linearapproaches
Antiwindup
Linearization
Gain scheduling
Stability
Nonlinearapproaches
CLF
Sliding mode
Geometriccontrol
Recursivetechniques
X4 stabilization
Observers
Impact of input saturations on the control of the X4’s rotors (2/4)
The so obtained system is linear
s i (s )
U i (s ) =
1k m
1 + J r R k 2m
s =
G
1 + τ s
Define a PI controller for it:
C (s ) = K p + K i
s
Taking K i = 1τCLG
and K p = τ K i , the closed loop systemis:
s i (s )
U i (s ) =
1
1 + τ CLs
N. Marchand (gipsa-lab) Nonlinear Control Master PSPI 2009-2010 38 / 174
Linear systems with saturated inputsI t f i t t ti th t l f th X4’ t (3/4)
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NonlinearControl
N. Marchand
References
Outline
Introduction
Linear/Nonlinear
The X4 example
Linearapproaches
Antiwindup
Linearization
Gain scheduling
Stability
Nonlinearapproaches
CLF
Sliding mode
Geometriccontrol
Recursivetechniques
X4 stabilization
Observers
Impact of input saturations on the control of the X4’s rotors (3/4)
Make a step that compensates the weight, that is such
that s
d
i = mg
4b so that i F
d
i = mg Taking τ CL = 50 ms, one gets with saturations
0 1 2 3 4 50
200
400
600
0 1 2 3 4 5−20
0
20
40
60
80
time
s i
U i
N. Marchand (gipsa-lab) Nonlinear Control Master PSPI 2009-2010 39 / 174
Linear systems with saturated inputsSat atio a ca se i stabilit
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NonlinearControl
N. Marchand
References
Outline
Introduction
Linear/Nonlinear
The X4 example
Linearapproaches
Antiwindup
Linearization
Gain scheduling
Stability
Nonlinearapproaches
CLF
Sliding mode
Geometriccontrol
Recursivetechniques
X4 stabilization
Observers
Saturation may cause instability
The result could be worse:
1
s-1
Transfer FcnSaturationPulse
Generator
7s+5
s
Control
For u ∈ [−1.2, 1.2 ], the closed-loop behavior is:
0 1 2 3 4 5−1
−0.5
0
0.5
1
1.5
2
2.5
3
3.5
4
without saturation
with saturation
Saturations may lead to instability especially in thepresence of integrators in the loop
N. Marchand (gipsa-lab) Nonlinear Control Master PSPI 2009-2010 40 / 174
Linear systems with saturated inputsKey idea of the anti windup scheme
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NonlinearControl
N. Marchand
References
Outline
Introduction
Linear/Nonlinear
The X4 example
Linearapproaches
Antiwindup
Linearization
Gain scheduling
Stability
Nonlinearapproaches
CLF
Sliding mode
Geometriccontrol
Recursivetechniques
X4 stabilization
Observers
Key idea of the anti-windup scheme
Consider a linear system with a PID controller:
Linear system
PID controller
Kd
Kp
KI 1/s
s
The instability comes from the integration of the errorKey idea: soften the integral effect when the control issaturated
N. Marchand (gipsa-lab) Nonlinear Control Master PSPI 2009-2010 41 / 174
Linear systems with saturated inputsPID controller with anti windup
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NonlinearControl
N. Marchand
References
Outline
Introduction
Linear/Nonlinear
The X4 example
Linearapproaches
Antiwindup
Linearization
Gain scheduling
Stability
Nonlinearapproaches
CLF
Sliding mode
Geometriccontrol
Recursivetechniques
X4 stabilization
Observers
PID controller with anti-windup
Structure of the PID controller with anti-windup:
Anti Windup
PID controller
Kd
Kp
KI 1/s
s
Linear system
Ks
If u = u , that is if u is not saturated, then the PIDcontroller with anti-windup is identical to the classicalPID controllerIf u is saturated (u = u ), K s tunes the reduction of theintegral effect of the PID
N. Marchand (gipsa-lab) Nonlinear Control Master PSPI 2009-2010 42 / 174
Linear systems with saturated inputsGeneral controller with anti-windup
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NonlinearControl
N. Marchand
References
Outline
Introduction
Linear/Nonlinear
The X4 example
Linearapproaches
Antiwindup
Linearization
Gain scheduling
Stability
Nonlinearapproaches
CLF
Sliding mode
Geometriccontrol
Recursivetechniques
X4 stabilization
Observers
General controller with anti-windup
Structure of the general dynamic controller:
++
++1
s
F
D
G C u e
with dynamics given by:
x c = Fx c + Ge u = Cx c + De
N. Marchand (gipsa-lab) Nonlinear Control Master PSPI 2009-2010 43 / 174
Linear systems with saturated inputsGeneral controller with anti-windup
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NonlinearControl
N. Marchand
References
Outline
Introduction
Linear/Nonlinear
The X4 example
Linearapproaches
Antiwindup
Linearization
Gain scheduling
Stability
Nonlinearapproaches
CLF
Sliding mode
Geometriccontrol
Recursivetechniques
X4 stabilization
Observers
General controller with anti windup
Anti-windup structure of the general dynamic controller:
++
++
+
1s
F − KC
D
G − KD C
K
u e
with dynamics given by:
x c = Fx c + Ge + K (u − Cx c − De )
= (F − KC )x c + (G − KD )e + Ku
Choice of the antiwindup parameters
Take K such that (F − KC ) is stable and |λmax(F − KC )| > |λmax(F )|
N. Marchand (gipsa-lab) Nonlinear Control Master PSPI 2009-2010 44 / 174
Linear systems with saturated inputsA short Lyapunov explanation of the behavior
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NonlinearControl
N. Marchand
References
Outline
Introduction
Linear/Nonlinear
The X4 example
Linearapproaches
Antiwindup
Linearization
Gain scheduling
Stability
Nonlinearapproaches
CLF
Sliding mode
Geometriccontrol
Recursivetechniques
X4 stabilization
Observers
A short Lyapunov explanation of the behavior
Consider a stable closed loop linear system: it isglobally asymptotically stable. A saturation on the inputmay:
transform the global stability into local stability. In thiscase, the aim of the anti-windup is to increase the radiusof attraction of the closed loop systemkeep the global stability property. In this case, the aim
of the anti-windup is to renders the saturated systemcloser to its unsaturated equivalent
However, there is no formal proof of stability of anti-windup strategies
Other approaches for saturated inputs
Optimal control
Nested saturation function (under development)
N. Marchand (gipsa-lab) Nonlinear Control Master PSPI 2009-2010 45 / 174
Linear systems with saturated inputsBack to the unstable case
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NonlinearControl
N. Marchand
References
Outline
Introduction
Linear/Nonlinear
The X4 example
Linearapproaches
Antiwindup
Linearization
Gain scheduling
Stability
Nonlinearapproaches
CLF
Sliding mode
Geometriccontrol
Recursivetechniques
X4 stabilization
Observers
ac to t e u stab e case
The unstable case:
1
s-1
Transfer FcnSaturationPulse
Generator
7s+5
s
Control
The closed-loop behavior is:
0 1 2 3 4 5−1
−0.5
0
0.5
1
1.5
2
2.5
3
3.5
4
without saturation
with saturation
with saturation and anti-windup
However, nothing is magic and divergent behavior may occurbecause of the level of the step input
N. Marchand (gipsa-lab) Nonlinear Control Master PSPI 2009-2010 46 / 174
Linear systems with saturated inputsImpact of input saturations on the control of the X4’s rotors (4/4)
8/10/2019 Nonlinear_PSPI.pdf
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NonlinearControl
N. Marchand
References
Outline
Introduction
Linear/Nonlinear
The X4 example
Linearapproaches
Antiwindup
Linearization
Gain scheduling
Stability
Nonlinearapproaches
CLF
Sliding mode
Geometriccontrol
Recursivetechniques
X4 stabilization
Observers
p p ( / )
Make a step that compensates the weight, that is such
that s d
i = mg
4b so that i
F d
i = mg
Taking τ CL = 50 ms, one gets with anti-windup
0 1 2 3 4 50
200
400
600
0 1 2 3 4 5−20
0
20
40
60
80
time
s i
U i
N. Marchand (gipsa-lab) Nonlinear Control Master PSPI 2009-2010 47 / 174
1 IntroductionLinear versus nonlinear
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NonlinearControl
N. Marchand
References
Outline
Introduction
Linear/Nonlinear
The X4 example
Linearapproaches
Antiwindup
Linearization
Gain scheduling
Stability
Nonlinearapproaches
CLFSliding mode
Geometriccontrol
Recursivetechniques
X4 stabilization
Observers
The X4 example
2
Linear control methods for nonlinear systemsAntiwindupLinearizationGain scheduling
3 Stability
4 Nonlinear control methodsControl Lyapunov functionsSliding mode controlState and output linearization
Backstepping and feedforwardingStabilization of the X4 at a position
5 Observers
N. Marchand (gipsa-lab) Nonlinear Control Master PSPI 2009-2010 48 / 174
Linearization of nonlinear systemsImpact of the load on the control of the X4’s rotors
8/10/2019 Nonlinear_PSPI.pdf
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NonlinearControl
N. Marchand
References
Outline
Introduction
Linear/Nonlinear
The X4 example
Linearapproaches
Antiwindup
Linearization
Gain scheduling
Stability
Nonlinearapproaches
CLFSliding mode
Geometriccontrol
Recursivetechniques
X4 stabilization
Observers
Make a step that compensates the weight, that is such that s d i =
mg 4b
so
that i F d i = mg
Taking τ CL = 50 ms, one gets with load
0 1 2 3 4 50
200
400
600
0 1 2 3 4 5−20
−10
0
10
20
time
s i
U i
The PI controller seems badly tuned: for t > 1.3 s , the control is not saturatedbut the convergence is very slow. What’s wrong ?
N. Marchand (gipsa-lab) Nonlinear Control Master PSPI 2009-2010 49 / 174
Linearization of nonlinear systemsImpact of the load on the control of the X4’s rotors
8/10/2019 Nonlinear_PSPI.pdf
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NonlinearControl
N. Marchand
References
Outline
Introduction
Linear/Nonlinear
The X4 example
Linearapproaches
Antiwindup
Linearization
Gain scheduling
Stability
Nonlinearapproaches
CLFSliding mode
Geometriccontrol
Recursivetechniques
X4 stabilization
Observers
Back to the rotor dynamical equation:
s i = − k 2mJ r R
s i − k gearbox κ
J r |s i | s i +
k m
J r R satU i
(U i )
Taking τ load as unknown implies that the PI control law
was tuned for:
s i = − k 2mJ r R
s i + k m
J r R satU i
(U i )
Implicitly, the second order terms were neglected
Unfortunately, these two systems behave similarly iff s i is
small, which is not the case for s d i =
mg 4b
≈544 rad s−1
N. Marchand (gipsa-lab) Nonlinear Control Master PSPI 2009-2010 50 / 174
Linearization of nonlinear systemsLinearization at the origin
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NonlinearControl
N. Marchand
References
Outline
Introduction
Linear/Nonlinear
The X4 example
Linearapproaches
Antiwindup
Linearization
Gain scheduling
Stability
Nonlinearapproaches
CLFSliding mode
Geometriccontrol
Recursivetechniques
X4 stabilization
Observers
Linearization at the origin
Take a nonlinear system x = f (x , u )
y = h(x , u )
with f (0, 0) = 0. Then, near the origin, it can be approximated by its
linear Taylor expansion at the first order:
x = ∂f
∂x
(x =0,u =0)
A
x + ∂f
∂u
(x =0,u =0)
B
u
y − h(0, 0) =
∂h
∂x
(x =0,u =0) C
x +
∂h
∂u
(x =0,u =0) D
u
N. Marchand (gipsa-lab) Nonlinear Control Master PSPI 2009-2010 51 / 174
Linearization of nonlinear systemsLinearization at a point
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NonlinearControl
N. Marchand
References
Outline
Introduction
Linear/Nonlinear
The X4 example
Linearapproaches
Antiwindup
Linearization
Gain scheduling
Stability
Nonlinearapproaches
CLFSliding mode
Geometriccontrol
Recursivetechniques
X4 stabilization
Observers
Linearization at a point
Take a nonlinear system x = f (x , u )
y = h(x , u )
then, near the equilibrium point (x 0, u 0), it can be approximated by itslinear Taylor expansion at the first order:
x ˙x
= ∂f
∂x
(x 0,u 0) A
(x − x 0) x
+ ∂f
∂u
(x 0,u 0) B
(u − u 0) u
y − h(x 0, u 0) = ∂h
∂x (x 0,u 0) C
(x − x 0) x
+ ∂h
∂u (x 0,u 0) D
(u − u 0) u
which is linear in the variables x = x − x 0 and u = u − u 0
N. Marchand (gipsa-lab) Nonlinear Control Master PSPI 2009-2010 52 / 174
Linearization of nonlinear systemsSome properties of the linearization
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NonlinearControl
N. Marchand
References
Outline
Introduction
Linear/Nonlinear
The X4 example
Linearapproaches
Antiwindup
Linearization
Gain scheduling
Stability
Nonlinearapproaches
CLFSliding mode
Geometriccontrol
Recursivetechniques
X4 stabilization
Observers
Controllability properties
Take a nonlinear system
x = f (x , u ) (6)
and its linearization at (x 0, u 0):
˙x = Ax + B u (7)
If (7) is controllable then (6) is locally controllableNothing can be concluded if (7) is uncontrollable
Example
The car is controllable but not its linearizationx 1 = u 1x 2 = u 2x 3 = x 2u 1
⇒
linearization at the origin
x 1
x 2x 3
=
1 0
0 10 0
u 1 u 2
N. Marchand (gipsa-lab) Nonlinear Control Master PSPI 2009-2010 53 / 174
Linearization of nonlinear systemsSome properties of the linearization
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NonlinearControl
N. Marchand
References
Outline
Introduction
Linear/Nonlinear
The X4 example
Linearapproaches
Antiwindup
Linearization
Gain scheduling
Stability
Nonlinearapproaches
CLFSliding mode
Geometriccontrol
Recursivetechniques
X4 stabilization
Observers
Stability properties
Take a nonlinear system
x = f (x , u ) (8)
and its linearization at (x 0, u 0):
˙x = Ax + B u (9)
Assume that one can build a feedback law u = k (x ) so that:
(9) is asymptotically stable ( < 0): then (8) is locallyasymptotically stable with u = u 0 + k (x )
(9) is unstable ( > 0): then (8) is locally unstable withu = u 0 + k (x )
Nothing can be concluded if (9) is simply stable ( ≤ 0)
N. Marchand (gipsa-lab) Nonlinear Control Master PSPI 2009-2010 54 / 174
Linearization of nonlinear systemsImpact of the load on the control of the X4’s rotors
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NonlinearControl
N. Marchand
References
Outline
Introduction
Linear/Nonlinear
The X4 example
Linearapproaches
Antiwindup
Linearization
Gain scheduling
Stability
Nonlinearapproaches
CLFSliding mode
Geometriccontrol
Recursivetechniques
X4 stabilization
Observers
Back to the rotor dynamical equation:
s i = −
k 2mJ r R s i −
k gearbox κ
J r |s i | s i +
k m
J r R satU i (U i )
Define U d i , the constant control that keeps the steady state speed:
U d i = k ms d
i + Rk gearbox κ
k m s d
i
s d
i
The linearization of the rotor dynamics near s d i gives:
˙s i = −
k 2mJ r R
+ 2k gearbox κ
J r
s d i
s i + k m
J r R satU i
(U i )
with:
U i = U i − U d i
s i = s i − s d i
satU i (U i ) =
U i if U i ∈ [−U i , +U i ]
U i if U i ≥ U i −U i if U i ≤ −U i
N. Marchand (gipsa-lab) Nonlinear Control Master PSPI 2009-2010 55 / 174
Linearization of nonlinear systemsImpact of the load on the control of the X4’s rotors
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NonlinearControl
N. Marchand
References
Outline
Introduction
Linear/Nonlinear
The X4 example
Linearapproaches
Antiwindup
Linearization
Gain scheduling
Stability
Nonlinearapproaches
CLFSliding mode
Geometriccontrol
Recursivetechniques
X4 stabilization
Observers
Make a step that compensates the weight, that is such that
s d i =
mg 4b
so that i F d i = mg
Taking τ CL = 50 ms and a PI controller tuned at s d i on the systemwith load, one has:
0 1 2 3 4 50
200
400
600
0 1 2 3 4 5−20
−10
0
10
20
time
s i
U i
N. Marchand (gipsa-lab) Nonlinear Control Master PSPI 2009-2010 56 / 174
1 IntroductionLinear versus nonlinearTh X4 l
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NonlinearControl
N. Marchand
References
Outline
Introduction
Linear/Nonlinear
The X4 example
Linearapproaches
Antiwindup
Linearization
Gain scheduling
Stability
Nonlinearapproaches
CLFSliding mode
Geometriccontrol
Recursivetechniques
X4 stabilization
Observers
The X4 example
2 Linear control methods for nonlinear systemsAntiwindupLinearizationGain scheduling
3 Stability
4 Nonlinear control methodsControl Lyapunov functionsSliding mode controlState and output linearization
Backstepping and feedforwardingStabilization of the X4 at a position
5 Observers
N. Marchand (gipsa-lab) Nonlinear Control Master PSPI 2009-2010 57 / 174
Towards gain schedulingImpact of a change in the steady state speed of the X4’s rotors
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NonlinearControl
N. Marchand
References
Outline
Introduction
Linear/Nonlinear
The X4 example
Linearapproaches
Antiwindup
Linearization
Gain scheduling
Stability
Nonlinearapproaches
CLFSliding mode
Geometriccontrol
Recursivetechniques
X4 stabilization
Observers
Take again τ CL = 50 ms and a PI controller tuned at s d i
Make speed steps of different level
0 2 4 6 8 10−200
0
200
400
600
0 2 4 6 8 10−20
−10
0
10
20
time
s i
U i
The controller is well tuned near s d i but not very good a large
range of use
N. Marchand (gipsa-lab) Nonlinear Control Master PSPI 2009-2010 58 / 174
Towards gain schedulingLocal linearization may be inadequate for large excursion
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NonlinearControl
N. Marchand
References
Outline
Introduction
Linear/Nonlinear
The X4 example
Linearapproaches
Antiwindup
Linearization
Gain scheduling
Stability
Nonlinearapproaches
CLFSliding mode
Geometriccontrol
Recursivetechniques
X4 stabilization
Observers
The result could be worse:
Take the inverted pendulum
θ
y
=
1
∆(θ)
m + M −ml cos θ
−ml cos θ I + ml 2
mgl sin θ
u + ml θ2 sin θ − k y
where
I is the inertia of the pendulum∆(θ) = (I + ml 2)(m + M ) − m2l 2 cos2 θ
Linearize it near the upper position (θ = 0) with m = M = I = g = 1,k = 0 and x = (θ, θ, y , y ):
x 1x 2x 3x 4
=
0 1 0 023 0 0 00 0 0 1
− 13 0 0 0
x 1x 2x 3x 4
+
0− 1
3023
u
N. Marchand (gipsa-lab) Nonlinear Control Master PSPI 2009-2010 59 / 174
N li
Towards gain schedulingLocal linearization may be inadequate for large excursion
Build a linear feedback law that places the poles at 1
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NonlinearControl
N. Marchand
References
Outline
Introduction
Linear/Nonlinear
The X4 example
Linearapproaches
Antiwindup
Linearization
Gain scheduling
Stability
Nonlinearapproaches
CLFSliding mode
Geometriccontrol
Recursivetechniques
X4 stabilization
Observers
Build a linear feedback law that places the poles at -1Starting from θ(0) = π/5, it converges, from θ(0) = 1.1 × π/5, it diverges
0 5 10 150
5
10
15
0 5 10 150
5
10
15
Inverted pendulum Linearization of the
inverted pendulum
Initial condition:
x (0) = (π/5, 0, 0, 0)
x (0) = (1.1π/5, 0, 0, 0)
x x
N. Marchand (gipsa-lab) Nonlinear Control Master PSPI 2009-2010 60 / 174
N li
Towards gain scheduling
For some systems, the radius of attraction of a
8/10/2019 Nonlinear_PSPI.pdf
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NonlinearControl
N. Marchand
References
Outline
Introduction
Linear/Nonlinear
The X4 example
Linearapproaches
Antiwindup
Linearization
Gain scheduling
Stability
Nonlinearapproaches
CLFSliding mode
Geometriccontrol
Recursivetechniques
X4 stabilization
Observers
stabilized linearization may be very small (less than onedegree on the double inverted pendulum for instance)
Linearization is not suitable for large range of use of the closed-loop system
For other systems, the controller must be very finelytuned in order to meet performance requirements (e.g.
automotive with more and more restrictive pollutionstandards)
Linearization is not suitable for high accuracy closedloop systems
Either for stability reasons or performance reasons, itmay be necessary to tune the controller at more thanone operating point
N. Marchand (gipsa-lab) Nonlinear Control Master PSPI 2009-2010 61 / 174
Nonlinear
Towards gain scheduling
Gain scheduling uses the idea of using a collection of
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NonlinearControl
N. Marchand
References
Outline
Introduction
Linear/Nonlinear
The X4 example
Linearapproaches
Antiwindup
Linearization
Gain scheduling
Stability
Nonlinearapproaches
CLFSliding mode
Geometriccontrol
Recursivetechniques
X4 stabilization
Observers
linearization of a nonlinear system at differentoperating points
N. Marchand (gipsa-lab) Nonlinear Control Master PSPI 2009-2010 62 / 174
Nonlinear
Gain schedulingTake a nonlinear system
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NonlinearControl
N. Marchand
References
Outline
Introduction
Linear/Nonlinear
The X4 example
Linearapproaches
Antiwindup
Linearization
Gain scheduling
Stability
Nonlinearapproaches
CLFSliding mode
Geometriccontrol
Recursivetechniques
X4 stabilization
Observers
x = f (x , u )
Define a family of equilibrium points
s = (x eq , u eq ) such that f (x eq , u eq ) = 0
At each point s , linearize the system assuming s is
constant:
˙x = A(s )x + B (s )u
with x = x − x eq and u = u − u eq
At each point s , define a feedback law:
u = u eq (s ) − K (s )(x − x eq (s ))
with (Eig(A(s ) − B (s )K (s ))) < 0
N. Marchand (gipsa-lab) Nonlinear Control Master PSPI 2009-2010 63 / 174
Nonlinear
Gain schedulingThe controller is then obtained by changing s in the apriori defined collection of points
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NonlinearControl
N. Marchand
References
Outline
Introduction
Linear/Nonlinear
The X4 example
Linearapproaches
Antiwindup
Linearization
Gain scheduling
Stability
Nonlinearapproaches
CLFSliding mode
Geometriccontrol
Recursivetechniques
X4 stabilization
Observers
priori defined collection of pointsThe change of s can be done discontinuously or continuously
by interpolation
N. Marchand (gipsa-lab) Nonlinear Control Master PSPI 2009-2010 64 / 174
Nonlinear
Gain schedulingDrawbacks of gain scheduling :
Con ergence onl if s aries slo l
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NonlinearControl
N. Marchand
References
Outline
Introduction
Linear/Nonlinear
The X4 example
Linearapproaches
Antiwindup
Linearization
Gain scheduling
Stability
Nonlinearapproaches
CLFSliding mode
Geometriccontrol
Recursivetechniques
X4 stabilization
Observers
Convergence only if s varies slowly The performance within a linearization area may be poor
The mapping may be prohibitive if the number of states islarge The stability issues are not clear
Recent evolution of gain scheduling:
Dynamic gain scheduling: improve the transition betweenthe different controllers, limits the importance of the slow
motion of s LPV: Linear parameter varying methods considers thenonlinear system as
x = A(ρ(x ))x + B (ρ(x ))u
and, under some conditions, uses LMI to build a feedback.Improves the performance within a linearization area.
N. Marchand (gipsa-lab) Nonlinear Control Master PSPI 2009-2010 65 / 174
Nonlinear
Gain schedulingHandling changes in the steady state speed of the X4’s rotors
Take again τCL = 50 ms and a PI controller tuned at si
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Control
N. Marchand
References
Outline
Introduction
Linear/Nonlinear
The X4 example
LinearapproachesAntiwindup
Linearization
Gain scheduling
Stability
Nonlinearapproaches
CLFSliding mode
Geometriccontrol
Recursivetechniques
X4 stabilization
Observers
Take again τ CL = 50 ms and a PI controller tuned at s i Make speed steps of different level
0 2 4 6 8 10−200
0
200
400
600
0 2 4 6 8 10−20
−10
0
10
20
time
s i
U i
The rotors are now well controlled
N. Marchand (gipsa-lab) Nonlinear Control Master PSPI 2009-2010 66 / 174
Nonlinear
Stability
1 IntroductionLinear versus nonlinear
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Control
N. Marchand
References
Outline
Introduction
Linear/Nonlinear
The X4 example
LinearapproachesAntiwindup
Linearization
Gain scheduling
Stability
Nonlinearapproaches
CLFSliding mode
Geometriccontrol
Recursivetechniques
X4 stabilization
Observers
Linear versus nonlinearThe X4 example
2 Linear control methods for nonlinear systemsAntiwindupLinearizationGain scheduling
3 Stability
4 Nonlinear control methodsControl Lyapunov functionsSliding mode control
State and output linearizationBackstepping and feedforwardingStabilization of the X4 at a position
5 Observers
N. Marchand (gipsa-lab) Nonlinear Control Master PSPI 2009-2010 67 / 174
Nonlinear
StabilityConsider the autonomous nonlinear system:
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Control
N. Marchand
References
Outline
Introduction
Linear/Nonlinear
The X4 example
Linear
approachesAntiwindup
Linearization
Gain scheduling
Stability
Nonlinearapproaches
CLFSliding mode
Geometriccontrol
Recursivetechniques
X4 stabilization
Observers
x = f (x , u (x )) = g (x ) (10)
StabilitySystem (10) is said to be stable at the origin iff:∀R > 0, ∃r (R ) > 0 such that ∀x 0 ∈ B(r (R )), x (t ; x 0), solutionof (10) with x 0 as initial condition, remains in B(R ) for allt >
0.
Attractivity
The origin is said to be attractive iff:limt →∞ x (t ; x 0) = 0.
Asymptotic stability
System (10) is said to be asymptotically stable at the originiff it is stable and attractive
N. Marchand (gipsa-lab) Nonlinear Control Master PSPI 2009-2010 68 / 174
NonlinearC
StabilityGraphical interpretation
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Control
N. Marchand
References
Outline
Introduction
Linear/Nonlinear
The X4 example
Linear
approachesAntiwindup
Linearization
Gain scheduling
Stability
Nonlinearapproaches
CLFSliding mode
Geometriccontrol
Recursivetechniques
X4 stabilization
Observers
For linear systems: Attractivity → Stability
For nonlinear systems: Attractivity Stability
Stability and attractivity : properties hard to check ?
N. Marchand (gipsa-lab) Nonlinear Control Master PSPI 2009-2010 69 / 174
NonlinearC l
StabilityUsing the linearization
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Control
N. Marchand
References
Outline
Introduction
Linear/Nonlinear
The X4 example
Linear
approachesAntiwindup
Linearization
Gain scheduling
Stability
Nonlinearapproaches
CLFSliding mode
Geometriccontrol
Recursivetechniques
X4 stabilization
Observers
Asymptotic stability and local linearizatonConsider x = g (x ) and its linearization at the origin
x = ∂g ∂x
x =0x . Then:
Linearization with eig< 0 Nonlinear system is locally
asymptotically stableLinearization with eig> 0 Nonlinear system is locallyunstable
Linearization with eig= 0: nothing can be concluded on
the nonlinear system (may be stable or unstable)
Only local conclusions
N. Marchand (gipsa-lab) Nonlinear Control Master PSPI 2009-2010 70 / 174
NonlinearC t l
StabilityLyapunov theory: Lyapunov functions
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Control
N. Marchand
References
Outline
Introduction
Linear/Nonlinear
The X4 example
Linear
approachesAntiwindup
Linearization
Gain scheduling
Stability
Nonlinearapproaches
CLFSliding mode
Geometriccontrol
Recursivetechniques
X4 stabilization
Observers
Aleksandr Mikhailovich LyapunovMarkov’s school friend, Chebyshev’s student Master Thesis : On the stability of ellipsoidal forms of equilibrium of a rotating liquid in 1884
Phd Thesis : The general problem of the stability of motionin 1892
6 June 1857 - 3 Nov 1918
Definition: Lyapunov function
Let V : Rn → R be a continuous function such that:
1 (definite) V (x ) = 0
⇔ x = 0
2 (positive) ∀
x , V (x ) ≥
0
3 (radially unbounded) limx →∞ V (x ) = +∞Lyapunov functions are energies
N. Marchand (gipsa-lab) Nonlinear Control Master PSPI 2009-2010 71 / 174
NonlinearControl
StabilityLyapunov theory: Lyapunov theorem
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Control
N. Marchand
References
Outline
Introduction
Linear/Nonlinear
The X4 example
Linear
approachesAntiwindup
Linearization
Gain scheduling
Stability
Nonlinearapproaches
CLFSliding mode
Geometriccontrol
Recursivetechniques
X4 stabilization
Observers
Aleksandr Mikhailovich LyapunovMarkov’s school friend, Chebyshev’s student
Master Thesis : On the stability of ellipsoidal forms of equilibrium of a rotating liquid in 1884Phd Thesis : The general problem of the stability of motion in 1892
6 June 1857 - 3 Nov 1918
Theorem: (First) Lyapunov theorem
If ∃ a Lyapunov function V : Rn → R+ s.t.
(strictly decreasing) V (x (t )) is strictly decreasing for allx (0) = 0
then the origin is asymptotically stable.
If ∃ a Lyapunov function V : Rn → R+ s.t.
(decreasing) V (x (t )) is decreasing
then the origin is stable.
N. Marchand (gipsa-lab) Nonlinear Control Master PSPI 2009-2010 72 / 174
NonlinearControl
StabilityLyapunov theorem: a graphical interpretation
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Control
N. Marchand
References
Outline
Introduction
Linear/Nonlinear
The X4 example
Linear
approachesAntiwindup
Linearization
Gain scheduling
Stability
Nonlinearapproaches
CLFSliding mode
Geometriccontrol
Recursivetechniques
X4 stabilization
Observers
N. Marchand (gipsa-lab) Nonlinear Control Master PSPI 2009-2010 73 / 174
NonlinearControl
StabilityStability and robustness
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Control
N. Marchand
References
Outline
Introduction
Linear/Nonlinear
The X4 example
Linear
approachesAntiwindup
Linearization
Gain scheduling
Stability
Nonlinearapproaches
CLFSliding mode
Geometriccontrol
Recursivetechniques
X4 stabilization
Observers
Stability holds robustness:
x = g (x ) stable ⇒ V =
∂V
∂x g (x ) < 0
⇒ ∂V
∂x (g (x ) + ε(x )) < 0
⇒ x = g (x ) + ε(x ) stable
N. Marchand (gipsa-lab) Nonlinear Control Master PSPI 2009-2010 74 / 174
NonlinearControl
Outline
1 IntroductionLinear versus nonlinear
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Control
N. Marchand
References
Outline
Introduction
Linear/Nonlinear
The X4 example
Linear
approachesAntiwindup
Linearization
Gain scheduling
Stability
Nonlinearapproaches
CLFSliding mode
Geometriccontrol
Recursivetechniques
X4 stabilization
Observers
The X4 example
2 Linear control methods for nonlinear systemsAntiwindupLinearizationGain scheduling
3 Stability4 Nonlinear control methods
Control Lyapunov functionsSliding mode control
State and output linearizationBackstepping and feedforwardingStabilization of the X4 at a position
5 Observers
N. Marchand (gipsa-lab) Nonlinear Control Master PSPI 2009-2010 75 / 174
NonlinearControl
The X4 helicopterThe control problems
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N. Marchand
References
Outline
Introduction
Linear/Nonlinear
The X4 example
Linear
approachesAntiwindup
Linearization
Gain scheduling
Stability
Nonlinearapproaches
CLFSliding mode
Geometriccontrol
Recursivetechniques
X4 stabilization
Observers
s i = − k 2mJ r R
s i − k gearbox κ
J r |s i | s i +
k mJ r R
satU i (U i )
p = v
m v = −mg e 3 + R 00i F i (s i )
R = R ω×
J ω = − ω×J ω −
i I r ω×
00
i s i
+
Γ r (s 2, s 4)
Γ p (s 1, s 3)
Γ y (s 1, s 2, s 3, s 4)
Position control problem
Attitude control problem
N Marchand (gipsa-lab) Nonlinear Control Master PSPI 2009-2010 76 / 174
NonlinearControl
Outline
1 IntroductionLinear versus nonlinear
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N. Marchand
References
Outline
Introduction
Linear/Nonlinear
The X4 example
Linear
approachesAntiwindup
Linearization
Gain scheduling
Stability
Nonlinearapproaches
CLF
Sliding mode
Geometriccontrol
Recursivetechniques
X4 stabilization
Observers
The X4 example
2 Linear control methods for nonlinear systemsAntiwindupLinearizationGain scheduling
3 Stability4 Nonlinear control methods
Control Lyapunov functionsSliding mode control
State and output linearizationBackstepping and feedforwardingStabilization of the X4 at a position
5 Observers
N Marchand (gipsa-lab) Nonlinear Control Master PSPI 2009-2010 77 / 174
NonlinearControl
Control Lyapunov functionsCharacterizing property (1/3)
Control Lyapunov functions (CLF) where introduced in the 80’s
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N. Marchand
References
Outline
Introduction
Linear/Nonlinear
The X4 example
Linear
approachesAntiwindup
Linearization
Gain scheduling
Stability
Nonlinearapproaches
CLF
Sliding mode
Geometriccontrol
Recursivetechniques
X4 stabilization
Observers
CLF are for controllability and control what Lyapunov functions
are for stabilityLyapunov: a tool for stability analysis of autonomoussystems
Take
x 1 = x 2
x 2 = −x 1 − x 2
Take the Lyapunov function
V (x ) = 3
2x 21 + x 1x 2 + x 22
Along the trajectories of the system:
V (x (t )) = ∇V (x ).f (x ) = − x 2
hence, V 0 and the system is asymptotically stable
N Marchand (gipsa-lab) Nonlinear Control Master PSPI 2009-2010 78 / 174
NonlinearControl
Control Lyapunov functionsCharacterizing property (2/3)
Control Lyapunov Functions: a tool for stabilization of
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N. Marchand
References
Outline
Introduction
Linear/Nonlinear
The X4 example
Linear
approachesAntiwindup
Linearization
Gain scheduling
Stability
Nonlinearapproaches
CLF
Sliding mode
Geometriccontrol
Recursivetechniques
X4 stabilization
Observers
controlled systems
Take now the controlled system
x 1 = x 2
x 2 = −x 1 + u
Take the Lyapunov function
V (x ) = 3
2x 21 + x 1x 2 + x 22
Along the trajectories of the system:
V (x (t ), u (t )) = ∇V (x ).f (x , u )
= −x 21 + x 1x 2 + x 22 + (x 1 + 2x 2)u
N Marchand (gipsa-lab) Nonlinear Control Master PSPI 2009-2010 79 / 174
NonlinearControl
Control Lyapunov functionsCharacterizing property (3/3)
Control Lyapunov Functions: a tool for stabilization of
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N. Marchand
References
Outline
Introduction
Linear/Nonlinear
The X4 example
Linear
approachesAntiwindup
Linearization
Gain scheduling
Stability
Nonlinearapproaches
CLF
Sliding mode
Geometriccontrol
Recursivetechniques
X4 stabilization
Observers
controlled systems
Along the trajectories of the system:
V (x (t ), u (t )) = ∇V (x ).f (x , u )
= −x 21 + x 1x 2 + x 22 + (x 1 + 2x 2)u
henceif x 1 + 2x 2 = 0, there exists u such that V otherwise (if x 1 + 2x 2 = 0)
V (x (t ), u (t )) = −4x 22 − 2x 22 + x 22 = −5x 22
which is negative unless x 2 = 0 = − 1
2
x 1: V
In conclusion: characterizing property of CLF
For any x = 0, there exists u such that V (x (t ), u (t )) < 0
N Marchand (gipsa-lab) Nonlinear Control Master PSPI 2009-2010 80 / 174
NonlinearControl
Control Lyapunov functionsFormal definitions
Definition: Lyapunov function
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N. Marchand
References
Outline
Introduction
Linear/Nonlinear
The X4 example
Linear
approachesAntiwindup
Linearization
Gain scheduling
Stability
Nonlinearapproaches
CLF
Sliding mode
Geometriccontrol
Recursivetechniques
X4 stabilization
Observers
A continuous function V : Rn
→ R≥0 is called a Lyapunov
function if 1 [positive definiteness] V (x ) ≥ 0 for all x and V (0) = 0 if and
only if x = 02 [radially unbounded] limx →∞ V (x ) = +
∞or:
1 [positive definiteness] V (x ) ≥ 0 for all x and V (0) = 0 if andonly if x = 0
2 [proper] for each a ≥ 0, the set x |V (x ) ≤ a is compact
or, alternatively:
(∃α, α ∈ K∞) α (x ) ≤ V (x ) ≤ α (x ) ∀x ∈ Rn
N Marchand (gipsa-lab) Nonlinear Control Master PSPI 2009-2010 81 / 174
NonlinearControl
Control Lyapunov functionsFormal definitions
Definition: Control Lyapunov Function
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N. Marchand
References
Outline
Introduction
Linear/Nonlinear
The X4 example
Linear
approachesAntiwindup
Linearization
Gain scheduling
Stability
Nonlinearapproaches
CLF
Sliding mode
Geometriccontrol
Recursivetechniques
X4 stabilization
Observers
Definition: Control Lyapunov Function
A differentiable control Lyapunov function denotes adifferentiable Lyapunov function being infinitesimally
decreasing , meaning that there exists a positive continuousdefinite function W : Rn
→R≥0 and such that:
supx ∈Rn minu ∈Rm ∇V (x )f (x , u ) + W (x ) ≤ 0
Roughly speaking a CLF is a Lyapunov function that onecan force to decrease
Extensions to nonsmooth CLF exist and are necessary forthe class of system that can not be stabilized by means of smooth static feedback
N Marchand (gipsa-lab) Nonlinear Control Master PSPI 2009-2010 82 / 174
NonlinearControl
Control Lyapunov functionsControl affine systems
Definition: Directional Lie derivative
For any f and g with values in appropriate sets, the Lie derivative of g along f is
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N. Marchand
References
Outline
Introduction
Linear/Nonlinear
The X4 example
Linear
approachesAntiwindup
Linearization
Gain scheduling
Stability
Nonlinearapproaches
CLF
Sliding mode
Geometriccontrol
Recursivetechniques
X4 stabilization
Observers
For any f and g with values in appropriate sets, the Lie derivative of g along f isdefined by:
Lf g (x ) := ∇g (x ) · f (x )
Iteratively, one defines:Lk
f g (x ) := Lf Lk −1f g (x )
Theorem
Consider a control affine system x = g 0(x ) + m
i =1 u i g i (x ) with f smooth andf (0) = 0. Assume that the set
S =
x |Lf V (x ) = 0 and Lk f Lg i V (x ) = 0 for all k ∈ N, i ∈ 1, . . . , m
= 0
then, the feedback
k (x ) := − (∇V (x ) · G (x ))T =
Lg 1 V (x ) · · · Lg m V (x )
T
globally asymptotically stabilizes the system at the origin.
N Marchand (gipsa-lab) Nonlinear Control Master PSPI 2009 2010 83 / 174
NonlinearControl
Control Lyapunov functionsControl affine systems
Theorem: Sontag’s universal formula (1989)
Consider a control affine system x = g0(x) + m ui gi (x) with f smooth and
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N. Marchand
References
Outline
Introduction
Linear/Nonlinear
The X4 example
Linear
approachesAntiwindup
Linearization
Gain scheduling
Stability
Nonlinearapproaches
CLF
Sliding mode
Geometriccontrol
Recursivetechniques
X4 stabilization
Observers
Consider a control affine system x = g 0(x ) +
i =1 u i g i (x ) with f smooth and
f (0) = 0. Assume that there exists a differentiable CLF, then the feedback
k i (x ) = −b i (x )ϕ(a(x ), β(x )) (= 0 for x = 0)
a(x ) := ∇V (x )f (x ) and b i (x ) := ∇V (x )g i (x ) for i = 1, . . . , m
B (x ) = b 1(x ) · · · b 2(x ) and β(x ) = B (x )2
q : R→ R is any real analytic function with q (0) = 0 and bq (b ) > 0 for anyb > 0ϕ is the real analytic function:
ϕ(a, b ) =
a+
√ a2+bq (b )
b if b = 0
0 if b = 0
is such that k (0) = 0, k smooth on Rn\0 and
supx ∈Rn
∇V (x )f (x , k (x )) + W (x ) ≤ 0
N Marchand (gipsa-lab) Nonlinear Control Master PSPI 2009 2010 84 / 174
NonlinearControl
Control Lyapunov functionsControl affine systems
Definition: Small control property
A CLF V i id i fi h ll l if f
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References
Outline
Introduction
Linear/Nonlinear
The X4 example
Linear
approachesAntiwindup
Linearization
Gain scheduling
Stability
Nonlinearapproaches
CLF
Sliding mode
Geometriccontrol
Recursivetechniques
X4 stabilization
Observers
A CLF V is said to satisfies the small control property if for any ε,
there is some δ so that:
supx ∈B(δ)
minu ∈B(ε)
∇V (x )f (x , u ) + W (x ) ≤ 0
The small control property implicitly means that if ε is small (hence the
control), the system can still be controlled as long as it is sufficiently close to
the origin
Theorem: Sontag’s universal formula (1989)
Consider an control-affine system x = g 0(x ) + i = 1mu i g i (x )
with f smooth and f (0) = 0. Assume that there exists adifferentiable CLF, then the previous feedback k with q (b ) = b issmooth on Rn\0 and continuous at the origin
N Marchand (gipsa lab) Nonlinear Control Master PSPI 2009 2010 85 / 174
NonlinearControl
Control Lyapunov functionsRobustness issues
Model Real systemx = f (x u) x = f (x u)+ε
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N. Marchand
References
Outline
Introduction
Linear/Nonlinear
The X4 example
Linear
approachesAntiwindup
Linearization
Gain scheduling
Stability
Nonlinearapproaches
CLF
Sliding mode
Geometriccontrol
Recursivetechniques
X4 stabilization
Observers
x = f (x , u )
u = k (x ) x = f (x , u )+ε
u = k (x +δ)
Definition: Robustness w.r.t. model errors
A feedback law k : Rn → Rm is said to be robust with respect to model errors if
limt →∞ x (t ) ∈ B(r (ε)) with limε→0 r (ε) = 0 (x (t ) denotes the solution of x = f (x , k (x )) + ε)
Definition: Robustness w.r.t. measurement errors
A feedback law k : Rn → Rm is said to be robust with respect to measurement
errors if limt →∞ x (t ) ∈ B(r (δ)) with limδ→0 r (δ) = 0 (x (t ) denotes the solutionof x = f (x , k (x + δ)))
Theorem: Smooth CLF = robust stability (1999)
The existence of a feedback law robust to measurement errors and model errorsis equivalent to the existence of a smooth CLF
N Marchand (gipsa-lab
) Nonlinear Control Master PSPI 2009 2010 86 / 174
NonlinearControl
N M h d
Control Lyapunov functionsConclusion
A CLF is th ti l t l t i l t l
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N. Marchand
References
Outline
Introduction
Linear/Nonlinear
The X4 example
Linear
approachesAntiwindup
Linearization
Gain scheduling
Stability
Nonlinearapproaches
CLF
Sliding mode
Geometriccontrol
Recursivetechniques
X4 stabilization
Observers
A CLF is a theoretical tool, not a magical tool
A CLF is not a constructive tool
Finding a stabilizing control law and finding a CLF are atbest the same problems. In all other cases, finding astabilizing control law is easier than finding a CLF
Even if one can find a CLF for a system, the universalformula often gives a feedback with poor performances
However, this is an important field of research in thecontrol system theory community that proved importanttheoretical results, for instance the equivalence betweenasymptotic controllability and stabilizability of nonlinearsystems (1997)
N Marchand (gipsa-lab
) Nonlinear Control Master PSPI 2009 2010 87 / 174
NonlinearControl
N M h d
Other structural toolsPassivity
Definition: Passivity
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N. Marchand
References
Outline
Introduction
Linear/Nonlinear
The X4 example
Linear
approachesAntiwindup
Linearization
Gain scheduling
Stability
Nonlinearapproaches
CLF
Sliding mode
Geometriccontrol
Recursivetechniques
X4 stabilization
Observers
A system x = f (x , u )y = h(x , u )
is said to be passive if there exists some function S (x ) ≥ 0 withS (0) = 0 such that
S (x (T )) − S (x (0)) ≤ T
0u T (τ )y (τ )d τ
Roughly speaking, S (x (0)) (called the storage function)denotes the largest amount of energy which can be extractedfrom the system given the initial condition x (0).Passivity eases the design of control law and is a powerful toolfor interconnected systemsImportant literature
N Marchand (gipsa-lab
) Nonlinear Control Master PSPI 2009 2010 88 / 174
NonlinearControl
N Marchand
Other structural toolsInput-to-State Stability (ISS)
Class K A continuous function γ : R≥0 → R≥0 is said in K if it is strictly increasingand γ(0) = 0
Cl KL A ti f ti β R × R → R i id i KL if β( ) ∈ K f
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N. Marchand
References
Outline
Introduction
Linear/Nonlinear
The X4 example
Linear
approachesAntiwindup
Linearization
Gain scheduling
Stability
Nonlinearapproaches
CLF
Sliding modeGeometriccontrol
Recursivetechniques
X4 stabilization
Observers
Class KL A continuous function β : R≥0 × R≥0 → R≥0 is said in KL if β(·, s ) ∈ K foreach fixed s and β(r , ·) is decreasing for each fixed r and lims →∞ β(r , s ) = 0
Definition: Input-to-State Stability
A systemx = f (x , u )
is said to be input-to-state stable if there exists functions β ∈ KL
and γ ∈ Ksuch that for each bounded u (·) and each x (0), the solution x (t ) exists and is
bounded by:x (t ) ≤ β(x (0) , t ) + γ( sup
0≤τ≤t u (τ ))
Roughly speaking, ISS characterizes a relation between the norm of thestate and the energy injected in the system: the system can not diverge infinite time with a bounded controlISS eases the design of control lawImportant literature
N Marchand (gipsa-lab
) Nonlinear Control Master PSPI 2009 2010 89 / 174
NonlinearControl
N Marchand
Outline
1 IntroductionLinear versus nonlinearThe X4 example
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N. Marchand
References
Outline
Introduction
Linear/Nonlinear
The X4 example
Linear
approachesAntiwindup
Linearization
Gain scheduling
Stability
Nonlinearapproaches
CLF
Sliding modeGeometriccontrol
Recursivetechniques
X4 stabilization
Observers
The X4 example
2 Linear control methods for nonlinear systemsAntiwindupLinearizationGain scheduling
3 Stability4 Nonlinear control methods
Control Lyapunov functionsSliding mode controlState and output linearizationBackstepping and feedforwardingStabilization of the X4 at a position
5 Observers
N M h d (gipsa-lab
) N li C t l M st PSPI 2009 2010 90 / 174
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NonlinearControl
N Marchand
Sliding mode controlAn introducing example
Making V decrease as fast as possible is equivalent to minimizing V :
u = Argmin V
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N. Marchand
References
Outline
Introduction
Linear/Nonlinear
The X4 example
Linear
approachesAntiwindup
Linearization
Gain scheduling
Stability
Nonlinearapproaches
CLF
Sliding modeGeometriccontrol
Recursivetechniques
X4 stabilization
Observers
−x 21 + x 1x 2 + x 22 can not be changed
(x 1 + 2x 2)u minimal for u =−sign(x 1+2x 2)
u = Argminu ∈[−1,1]
V
= Argmin
u ∈[−1,1]
− x 21 + x 1x 2 + x 22 + (x 1 + 2x 2)u
= − sign(x 1 + 2x 2)
Simulating the closed loop system with initial condition x (0) = (2, −3), it gives:
0 2 4 6 8 10−4
−2
0
2
0 2 4 6 8 10−1.5
−1
−0.5
0
0.5
1
1.5
x 1x 2
u
Why does it work ?
N M h d ( i-lab
) N li C t l M t PSPI 2009 2010 92 / 174
NonlinearControl
N. Marchand
Sliding mode controlAn introducing example
Define σ(x ) = x 1 + 2x 2. The set S (x ) = x |σ(x ) = 0 ∩ |x 1| ≤ 0.6, |x 2| ≤ 0.3 is attractive:
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References
Outline
Introduction
Linear/Nonlinear
The X4 example
Linear
approachesAntiwindup
Linearization
Gain scheduling
Stability
Nonlinearapproaches
CLF
Sliding modeGeometriccontrol
Recursivetechniques
X4 stabilization
Observers
x 1 x 2
V
S =
x | x 1 +
2 x 2 =
0
N M h d ( i-lab
) N li C t l M t PSPI 2009 2010 93 / 174
NonlinearControl
N. Marchand
Sliding mode controlAn introducing example
Define σ(x ) = x 1 + 2x 2. The set S (x ) = x |σ(x ) = 0 ∩ |x 1| ≤ 0.6, |x 2| ≤ 0.3 is attractive:
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References
Outline
Introduction
Linear/Nonlinear
The X4 example
Linear
approachesAntiwindup
Linearization
Gain scheduling
Stability
Nonlinearapproaches
CLF
Sliding modeGeometriccontrol
Recursivetechniques
X4 stabilization
Observers
x 1 x 2
S S < 0S S < 0
S S > 0
S S > 0
S =
x | x 1 +
2 x 2 =
0
N M h d (gipsa-lab
) N li C l M PSPI 2009 2010 93 / 174
NonlinearControl
N. Marchand
Sliding mode controlAn introducing example
The set S (x ) = x |σ(x ) = 0 ∩ |x 1| ≤ 0.6, |x 2| ≤ 0.3 isattractiveO h i j i d h l i h h i
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References
Outline
Introduction
Linear/Nonlinear
The X4 example
Linear
approachesAntiwindup
Linearization
Gain scheduling
Stability
Nonlinearapproaches
CLF
Sliding modeGeometriccontrol
Recursivetechniques
X4 stabilization
Observers
Once the set is joined, the control is such that x remains onS (x ), that is:
S (x ) = 0 = x 1 + 2x 2 = x 2 − x 1 + u
Hence, on S (x ), u = − sign(x 1 + 2x 2) has the same influence
on the system as the control u eq =
x 1 −
x 2On S (x ), the system behaves like:
x 1 = x 2
x 2 = −x 1 + u = −x 2
x 2 and hence also x 1 clearly exponentially go to zero withoutleaving S (x ) = x |x 1 + 2x 2 = 0 ∩ |x 1| ≤ 0.6, |x 2| ≤ 0.3
N M h d (gipsa-lab
) N li C l M PSPI 2009 2010 94 / 174
NonlinearControl
N. Marchand
Sliding mode controlPrinciple of sliding mode control
Principle:
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References
Outline
Introduction
Linear/Nonlinear
The X4 example
Linear
approachesAntiwindup
Linearization
Gain scheduling
Stability
Nonlinearapproaches
CLF
Sliding modeGeometriccontrol
Recursivetechniques
X4 stabilization
Observers
Principle:
Define a sliding surface S (x ) = x |σ(x ) = 0
A stabilizing sliding mode control is a control law
discontinuous in on S (x )
that insures the attractivity of S (x )
such that, on the surface S (x ), the states “slides” tothe origin
Main characteristic of sliding mode control:
Robust control law
Discontinuities may damage actuators (filtered versionsexist)
N. Marchand (gipsa-lab) Nonlinear Control Master PSPI 2009-2010 95 / 174
NonlinearControl
N. Marchand
Sliding mode controlNotion of sliding surface and equivalent control
With a sliding mode control, the system takes the form:
˙ f +(x ) if σ(x ) > 0
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References
Outline
Introduction
Linear/Nonlinear
The X4 example
Linear
approachesAntiwindup
Linearization
Gain scheduling
Stability
Nonlinearapproaches
CLF
Sliding modeGeometriccontrol
Recursivetechniques
X4 stabilization
Observers
x = ( ) ( )
f −
(x ) if σ(x ) < 0
σ(x ) = 0
σ(x )> 0
σ(x ) < 0
f + (x )
f − (x )f −n (x )
f +n (x )
What happens for σ(x ) = 0 ?The solution on σ(x ) = 0 is the solution of x = α f +(x ) + (1 − α )f −(x )
where α satisfies α f +n (x ) + (1 − α )f −n (x ) = 0 for the normal projections of f + and f −
N. Marchand (gipsa-lab) Nonlinear Control Master PSPI 2009-2010 96 / 174
NonlinearControl
N. Marchand
Sliding mode controlA sliding mode control for a class of affine systems
Theorem: Sliding mode control for a class of nonlinear systems
Take the nonlinear system:
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References
Outline
Introduction
Linear/Nonlinear
The X4 example
Linear
approachesAntiwindup
Linearization
Gain scheduling
Stability
Nonlinearapproaches
CLF
Sliding modeGeometriccontrol
Recursivetechniques
X4 stabilization
Observers
Take the nonlinear system:
x 1x 2...
x n
=
f 1(x ) + g 1(x )u
x 1...
x n−1
= f (x ) + g (x )u
Choose the control law
u = −p T f (x )
p T g (x ) −
µ
p T g (x ) sign(σ(x ))
with µ > 0, σ(x ) = p T
x and p T
=
p 1 · · · p n
is a stablepolynomial (all roots have strictly negative real parts)Then the origin of the closed loop system is asymptoticallystable
N. Marchand (gipsa-lab) Nonlinear Control Master PSPI 2009-2010 97 / 174
NonlinearControl
N. Marchand
Sliding mode controlSliding mode control for some affine systems: proof of convergence
Proof:
Take the Lyapunov function
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References
Outline
Introduction
Linear/Nonlinear
The X4 example
Linear
approachesAntiwindup
Linearization
Gain scheduling
Stability
Nonlinearapproaches
CLF
Sliding modeGeometriccontrol
Recursivetechniques
X4 stabilization
Observers
V (x ) = 12
x T pp T x
Note that V (x ) = σ2(x )
2Along the trajectories of the system, one has:
V (x ) = σT (x )σ(x ) = x T p
p T f (x ) + p T g (x )u
with the chosen control, it gives:
V (x ) = −µσ(x ) sign(σ(x )) = −µ |σ(x )|
x tends to S = x |σ(x ) = 0µ tunes how fast the system converges to S = x |σ(x ) = 0
N. Marchand (gipsa-lab) Nonlinear Control Master PSPI 2009-2010 98 / 174
NonlinearControl
N. Marchand
Sliding mode controlSliding mode control for some affine systems: proof of convergence
On S , the system is such that
σ(x ) = 0 = p 1x 1 + · · · + p n−1x n−1 + p nx n
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References
Outline
Introduction
Linear/Nonlinear
The X4 example
Linear
approachesAntiwindup
Linearization
Gain scheduling
Stability
Nonlinearapproaches
CLF
Sliding modeGeometriccontrol
Recursivetechniques
X4 stabilization
Observers
But:
x n−1 = x n
x n−2 = x n−1 = x (2)n
...
x 1 = x 2 = · · · = x (n−1)n
That can be written with z i = x n−i +1 in the form:
z =
0 1 0 · · · 0
0 0 1 . . .
......
.. .
.. . 0
0 · · · · · · 0 1− p 2
p 1− p 3
p 1· · · · · · − p n
p 1
z
Hence, x asymptotically goes to the origin
N. Marchand (gipsa-lab) Nonlinear Control Master PSPI 2009-2010 99 / 174
NonlinearControl
N. Marchand
Sliding mode controlSliding mode control for some affine systems: time to switch
Consider an initial point x 0 such that σ(x 0) > 0.
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References
Outline
Introduction
Linear/Nonlinear
The X4 example
Linear
approachesAntiwindup
Linearization
Gain scheduling
Stability
Nonlinearapproaches
CLF
Sliding modeGeometriccontrol
Recursivetechniques
X4 stabilization
Observers
Sinceσ(x )σ(x ) = −µσ(x ) sign(σ(x ))
it follows that as long as σ(x ) > 0
σ(x ) = −µ
Hence, the instant of the first switch is
t s = σ(x 0)
µ < ∞
Moreover, t s → 0 as µ →∞N. Marchand (gipsa-lab) Nonlinear Control Master PSPI 2009-2010 100 / 174
NonlinearControl
N. Marchand
Sliding mode controlRobustness issues
Assume that one knows only a model
x = f (x ) + g (x)u
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References
Outline
Introduction
Linear/Nonlinear
The X4 example
Linearapproaches
Antiwindup
Linearization
Gain scheduling
Stability
Nonlinearapproaches
CLF
Sliding modeGeometriccontrol
Recursivetechniques
X4 stabilization
Observers
x = f (x ) + g (x )u
of the true system
x = f (x ) + g (x )u
The time derivative of the Lyapunov function is:
V (x ) = σ(x )
p T (f g − f g T )p
p T g − µ
p T g
p T g sign(σ(x ))
If sign(p T g ) = sign(p T g ) and µ > 0 sufficiently large, V < 0The closed-loop system is robust against model errors
N. Marchand (gipsa-lab) Nonlinear Control Master PSPI 2009-2010 101 / 174
NonlinearControl
N. Marchand
Outline1 Introduction
Linear versus nonlinearThe X4 example
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References
Outline
Introduction
Linear/Nonlinear
The X4 example
Linearapproaches
Antiwindup
Linearization
Gain scheduling
Stability
Nonlinearapproaches
CLF
Sliding modeGeometriccontrol
Recursivetechniques
X4 stabilization
Observers
2 Linear control methods for nonlinear systemsAntiwindupLinearizationGain scheduling
3 Stability4 Nonlinear control methods
Control Lyapunov functionsSliding mode controlState and output linearization
Backstepping and feedforwardingStabilization of the X4 at a position
5 Observers
N. Marchand (gipsa-lab) Nonlinear Control Master PSPI 2009-2010 102 / 174
NonlinearControl
N. Marchand
State and output linearizationIntroducing examples
Consider nonlinear the system
x 1 = −2x 1 + ax 2 + sin(x 1)
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References
Outline
Introduction
Linear/Nonlinear
The X4 example
Linearapproaches
Antiwindup
Linearization
Gain scheduling
Stability
Nonlinearapproaches
CLF
Sliding modeGeometriccontrol
Recursivetechniques
X4 stabilization
Observers
x 2 = −x 2 cos(x 1) + u cos(2x 1)
Take the new set of state variables
z 1 = x 1z 2 = ax 1 + sin(x 1)
x 1 = z 1
x 2 = z 2−sin(z 1)
a
The state equations become:
z 1 = −2z 1 + z 2
z 2 = −2z 1 cos(z 1) + cos(z 1) sin(z 1) + au cos(2z 1)
The nonlinearities can then be canceled taking the new control v :
u = 1
a cos(2z 1) (v − cos(z 1) sin(z 1) + 2z 1 cos(z 1))
N. Marchand (gipsa-lab) Nonlinear Control Master PSPI 2009-2010 103 / 174
NonlinearControl
N. Marchand
State and output linearizationIntroducing examples
The system then becomes linear:
z =−2 1
z +0
v
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References
Outline
Introduction
Linear/Nonlinear
The X4 example
Linearapproaches
Antiwindup
Linearization
Gain scheduling
Stability
Nonlinearapproaches
CLF
Sliding modeGeometriccontrol
Recursivetechniques
X4 stabilization
Observers
z = 0 0
z + 1 v
Taking v = −2z 2 places the poles of the closed-loop in −2, −2Hence, limt →∞ z = 0Looking back to the transformation:
x 1 = z 1
x 2 = z 2 − sin(z 1)
a
one also have limt →∞ x = 0In the original coordinates, the control writes:
u = 1
a cos(2x 1)[−2ax 2−2sin(x 2)−cos(x 1) sin(x 1)+2x 1 cos(x 1)]
N. Marchand (gipsa-lab) Nonlinear Control Master PSPI 2009-2010 104 / 174
NonlinearControl
N. Marchand
State and output linearizationIntroducing examples
Consider nonlinear the system
x 1 = sin(x 2) + (x 2 + 1)x 35
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References
Outline
Introduction
Linear/Nonlinear
The X4 example
Linearapproaches
Antiwindup
Linearization
Gain scheduling
Stability
Nonlinearapproaches
CLF
Sliding modeGeometriccontrol
Recursivetechniques
X4 stabilization
Observers
x 2 = x 51 + x 3
x 3 = x 21 + u
Take x 1 as “output”: y = x 1When not imposed, the choice of the appropriate output is often delicate
Compute the first time derivative of y :
y = x 1 = sin(x 2) + (x 2 + 1)x 3
Compute the second time derivative of y :
y = (x 2 + 1)u + (x 51 + x 3)(x 3 + cos(x 2)) + (x 2 + 1)x 21 m(x )
Taking u = v −m(x )
x 2+1 gives the linear system: y = v
N. Marchand (gipsa-lab) Nonlinear Control Master PSPI 2009-2010 105 / 174
NonlinearControl
N. Marchand
State and output linearizationIntroducing examples
Potential problem if x 2 = −1
Here again, take the linear feedback v = y + 2y that
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References
Outline
Introduction
Linear/Nonlinear
The X4 example
Linearapproaches
Antiwindup
Linearization
Gain scheduling
Stability
Nonlinearapproaches
CLF
Sliding modeGeometriccontrol
Recursivetechniques
X4 stabilization
Observers
places the poles of the closed loop system in −1, −1Hence, y and y asymptotically converges to the origin
The state of the system is of dimension 3. Only twovariables have been brought to the origin, what about the
third one ?Write the system with the new coordinates(z 1, z 2, z 3) = (y , y , x 3):
Linearized sub-system: z 1 = z 2
z 2 = v Internal dynamics:
z 3 = z 21 + u (z )
N. Marchand (gipsa-lab) Nonlinear Control Master PSPI 2009-2010 106 / 174
NonlinearControl
N. Marchand
State and output linearizationIntroducing examples
One has to check that the internal dynamics is stable. For this, weassume that y = y = 0 (which happens asymptotically):
→ →
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References
Outline
Introduction
Linear/Nonlinear
The X4 example
Linearapproaches
Antiwindup
Linearization
Gain scheduling
Stability
Nonlinearapproaches
CLF
Sliding modeGeometriccontrol
Recursivetechniques
X4 stabilization
Observers
z 3 =
→0 z 21 +
→0 v −((
→
0 z 51 +z 3)(z 3 + cos x 2) +
→
0 (x 2 + 1)z 21 )
x 2 + 1
= −z 3(z 3 + cos x 2)
x 2 + 1
If x 2 and z 3 are small enough, z 3 ≈ −z 3(z 3 + 1) ≈ −z 3: the internaldynamics is locally asymptotically stable
The approach can be applied if and only if the internal dynamics isstable
If the internal dynamics is unstable, the system is called non-minimumphase
Change the inputUse additional inputs to stabilize the internal dynamicsOther approach in the literature like approximate linearization
N. Marchand (gipsa-lab) Nonlinear Control Master PSPI 2009-2010 107 / 174
NonlinearControl
N. Marchand
State and output linearizationIntroduction
Consider again nonlinear systems affine in the control:
x = f (x ) + g (x )u
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References
Outline
Introduction
Linear/Nonlinear
The X4 example
Linearapproaches
Antiwindup
Linearization
Gain scheduling
Stability
Nonlinearapproaches
CLF
Sliding modeGeometriccontrol
Recursivetechniques
X4 stabilization
Observers
y = h(x )
The linearization approach tries to find
a feedback control law
u = α (x ) + β(x )v
and a change of variablez = T (x )
that transforms the nonlinear systems into:
Input-State linearization
z = Az + Bv
No systematic approach
Input-Output linearization
y (r ) = v
Systematic approach
N. Marchand (gipsa-lab) Nonlinear Control Master PSPI 2009-2010 108 / 174
NonlinearControl
N. Marchand
State and output linearizationThe SISO case
Take x = f (x ) + g (x )u
y = h(x )
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References
Outline
Introduction
Linear/Nonlinear
The X4 example
Linearapproaches
Antiwindup
Linearization
Gain scheduling
Stability
Nonlinearapproaches
CLF
Sliding modeGeometriccontrol
Recursivetechniques
X4 stabilization
Observers
The time derivative of y is then given by:
y = ∂h
∂x (f (x ) + g (x )u ) = Lf h(x ) + Lg h(x )u
If Lg h(x ) = 0, the control is
u = 1
Lg h(x )(−Lf h(x ) + v )
yielding y = v
Otherwise (that is Lg h(x ) ≡ 0), differentiate once more:
y = ∂Lf h
∂x (f (x ) + g (x )u ) = L2
f h(x ) + Lg Lf h(x )u
N. Marchand (gipsa-lab) Nonlinear Control Master PSPI 2009-2010 109 / 174
NonlinearControl
N. Marchand
State and output linearizationThe SISO case
If Lg Lf h(x ) = 0, the control is
u = 1
(−L2f h(x ) + v )
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References
Outline
Introduction
Linear/Nonlinear
The X4 example
Linearapproaches
Antiwindup
Linearization
Gain scheduling
Stability
Nonlinearapproaches
CLF
Sliding modeGeometriccontrol
Recursivetechniques
X4 stabilization
Observers
Lg Lf h(x )
yielding to y = v
. . .
Definition: Relative degree
There exists an integer r ≤ n such that Lg Li f h(x ) ≡ 0 for alli ∈ 1, . . . , r − 2 and Lg L
r −1f h(x ) = 0 that is called the relative degree of
the system
The control law
u = 1Lg L
r −1f h(x )
(−Lr f h(x ) + v )
yields the linear system y (r ) = v
N. Marchand (gipsa-lab) Nonlinear Control Master PSPI 2009-2010 110 / 174
NonlinearControl
N. Marchand
State and output linearizationThe MIMO case
Takex = f (x ) + g 1(x )u 1 + g 2(x )u 2
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References
Outline
Introduction
Linear/Nonlinear
The X4 example
Linearapproaches
Antiwindup
Linearization
Gain scheduling
Stability
Nonlinearapproaches
CLF
Sliding modeGeometriccontrol
Recursivetechniques
X4 stabilization
Observers
y 1 = h1(x )y 2 = h2(x )
The relative degree r 1, r 2 is then given by:
Lg 1 Li −1f h1(x ) = Lg 2 Li −1
f h1(x ) = 0
∀i < r 1
Lg 1 Li −1f h2(x ) = Lg 2 Li −1
f h2(x ) = 0 ∀i < r 2
Assume that the Input-Output decoupling condition issatisfied, that is:
rank
Lg 1 Lr 1−1f h1(x ) Lg 2 Lr 1−1f h1(x )Lg 1 Lr 2−1
f h2(x ) Lg 2 Lr 2−1f h2(x )
= 2
N. Marchand (gipsa-lab) Nonlinear Control Master PSPI 2009-2010 111 / 174
NonlinearControl
N. Marchand
State and output linearizationThe MIMO case
One has:
y 1 = Lf h1(x ) + Lg 1 h1(x )
=0
u 1 + Lg 2 h1(x )
=0
u 2
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References
Outline
Introduction
Linear/Nonlinear
The X4 example
Linearapproaches
Antiwindup
Linearization
Gain scheduling
Stability
Nonlinearapproaches
CLF
Sliding modeGeometriccontrol
Recursivetechniques
X4 stabilization
Observers
y 1 = L2f h1(x ) + Lg 1 Lf h1(x ) =0
u 1 + Lg 2 Lf h1(x ) =0
u 2
...
y (r 1)1 = Lr 1
f h1(x ) + Lg 1 Lr 1−1f h1(x )
either =0
u 1 + Lg 2 Lr 1−1f h1(x )
or =0
u 2
Make the same for y 2:
y 2 = Lf h2(x ) + Lg 1 h2(x ) =0
u 1 + Lg 2 h2(x ) =0
u 2
y 2 = L2f h2(x ) + Lg 1 Lf h2(x )
=0
u 1 + Lg 2 Lf h2(x )
=0
u 2
...
y (r 2)2 = Lr 2
f h2(x ) + Lg 1 Lr 2−1f h2(x )
either =0
u 1 + Lg 2 Lr 2−1f h2(x ) or =0
u 2
N. Marchand (gipsa-lab) Nonlinear Control Master PSPI 2009-2010 112 / 174
NonlinearControl
N. Marchand
State and output linearizationThe MIMO case
Hence, one has:
y (r 1)1 Lr 1
f h1(x ) Lg 1 Lr 1−1f h1(x ) Lg 2 Lr 1−1
f h1(x ) u 1
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References
Outline
Introduction
Linear/Nonlinear
The X4 example
Linearapproaches
Antiwindup
Linearization
Gain scheduling
Stability
Nonlinearapproaches
CLF
Sliding modeGeometriccontrol
Recursivetechniques
X4 stabilization
Observers
y (r 2)
2
=
Lr 2f h2(x )
b (x )
+
Lg 1 Lr 2−1f h2(x ) Lg 2 Lr 2−1f h2(x )
A(x )
u 2
u
Thanks to the decoupling condition, one can take:
u = −A(x )−1
b (x ) + A(x )−1
v
which gives two chains of integrators:
y (r 1)1 = v 1 y
(r 2)2 = v 2
This approach can be extended to any output and input sizes
N. Marchand (gipsa-lab) Nonlinear Control Master PSPI 2009-2010 113 / 174
NonlinearControl
N. Marchand
State and output linearizationA block diagram representation
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References
Outline
Introduction
Linear/Nonlinear
The X4 example
Linearapproaches
Antiwindup
Linearization
Gain scheduling
Stability
Nonlinearapproaches
CLF
Sliding modeGeometriccontrol
Recursivetechniques
X4 stabilization
Observers
Linearization techniques can be interpreted as:
Nonlinear
system
Linear
control
Linearizing
feedbackLinear loop
N. Marchand (gipsa-lab) Nonlinear Control Master PSPI 2009-2010 114 / 174
NonlinearControl
N. Marchand
State and output linearizationThe general case
The internal dynamics or zero-dynamics is then of dimensionn − r ; its stability has to be checkedFor linear systems r =number of poles-number of zeros
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References
Outline
Introduction
Linear/Nonlinear
The X4 example
Linearapproaches
Antiwindup
Linearization
Gain scheduling
Stability
Nonlinearapproaches
CLF
Sliding modeGeometriccontrol
Recursivetechniques
X4 stabilization
Observers
This approach took a rapid development in the 80’s Maybe the only “general” approach for nonlinear system with
predictive control Can be extended to MIMO systems with a decoupling
condition Many extensions in particular with the notion of flatness Power tool for path generation Non robust approach since it is based on coordinate changes
that can be stiff The control may take too large values “Kills” nonlinearities even if they are good
N. Marchand (gipsa-lab) Nonlinear Control Master PSPI 2009-2010 115 / 174
NonlinearControl
N. Marchand
R f
Outline1 Introduction
Linear versus nonlinearThe X4 example
2 L l h d f l
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References
Outline
Introduction
Linear/Nonlinear
The X4 example
Linearapproaches
Antiwindup
Linearization
Gain scheduling
Stability
Nonlinearapproaches
CLF
Sliding modeGeometriccontrol
Recursivetechniques
X4 stabilization
Observers
2 Linear control methods for nonlinear systemsAntiwindupLinearizationGain scheduling
3 Stability
4 Nonlinear control methodsControl Lyapunov functionsSliding mode controlState and output linearization
Backstepping and feedforwardingStabilization of the X4 at a position
5 Observers
N. Marchand (gipsa-lab) Nonlinear Control Master PSPI 2009-2010 116 / 174
NonlinearControl
N. Marchand
R f
BacksteppingMain results
Theorem: backstepping design
Assume that the system ˙ χ = f ( χ, v ) with f (0, 0) = 0 can asymptotically be stabilized withv = k ( χ). Let V denote a C1 Lyapunov function (definite, positive and radiallyunbounded) such that ∂V
∂χf ( χ, k ( χ)) < 0 for all χ
= 0. Then, the system
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References
Outline
Introduction
Linear/Nonlinear
The X4 example
Linearapproaches
Antiwindup
Linearization
Gain scheduling
Stability
Nonlinearapproaches
CLF
Sliding modeGeometriccontrol
Recursivetechniques
X4 stabilization
Observers
˙ χ = f ( χ, ξ)
ξ = h( χ, ξ) + u
with h(0, 0) = 0 is also asymptotically stabilizable with the control:
u ( χ, ξ) = −h( χ, ξ) +
∂k
∂χ f ( χ, ξ) − ξ + k ( χ) −∂V
∂χ G ( χ, ξ − k ( χ))T
with
G ( χ, ξ) =
10
∂f
∂ξ( χ, k ( χ) + λξ)d λ
The Lyapunov function
W ( χ, ξ) = V ( χ) + 1
2 ξ − k ( χ)2
is then strictly decreasing for any ( χ, ξ) = (0, 0)
N. Marchand (gipsa-lab) Nonlinear Control Master PSPI 2009-2010 117 / 174
NonlinearControl
N. Marchand
R f
BacksteppingMain results
Th h i l b li d
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References
Outline
Introduction
Linear/Nonlinear
The X4 example
Linearapproaches
Antiwindup
Linearization
Gain scheduling
Stability
Nonlinearapproaches
CLF
Sliding mode
Geometriccontrol
Recursivetechniques
X4 stabilization
Observers
The theorem can recursively be applied to
˙ χ = f ( χ, ξ1)
ξ1 = a1( χ, ξ1) + b 1( χ, ξ1)ξ2
ξ2 = a2( χ, ξ1, ξ2) + b 2( χ, ξ1, ξ2)ξ3
...
ξn−1 = an−1( χ, ξ1, ξ2, . . . , ξn−2) + b n−1( χ, ξ1, ξ2, . . . , ξn−2)ξn−1
ξn = an( χ, ξ1, ξ2, . . . , ξn−1) + b n( χ, ξ1, ξ2, . . . , ξn−1)u
N. Marchand (gipsa-lab) Nonlinear Control Master PSPI 2009-2010 118 / 174
NonlinearControl
N. Marchand
References
ForwardingStrict-feedforward systems
Consider the class of strict-feedforward systems :
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References
Outline
Introduction
Linear/Nonlinear
The X4 example
Linearapproaches
AntiwindupLinearization
Gain scheduling
Stability
Nonlinearapproaches
CLF
Sliding mode
Geometriccontrol
Recursivetechniques
X4 stabilization
Observers
x 1 = x 2 + f 1(x 2, x 3, . . . , x n, u )
x 2 = x 3 + f 2(x 3, . . . , x n, u )
...
x n−1 = x n + f n−1(x n, u )
x n = u
Strict-feedforward systems are in general no feedbacklinearizable
Backstepping is not applicable
N. Marchand (gipsa-lab) Nonlinear Control Master PSPI 2009-2010 119 / 174
NonlinearControl
N. Marchand
References
ForwardingForwarding procedure
At step 0: Begin with stabilizing the system x n = u n.Take e.g. u n = −x n and the corresponding Lyapunov function V n = 1
2 x 2nAt step 1: Augment the control law
un−1(xn−1 xn) un(xn) + vn−1(xn−1 xn)
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References
Outline
Introduction
Linear/Nonlinear
The X4 example
Linearapproaches
AntiwindupLinearization
Gain scheduling
Stability
Nonlinearapproaches
CLF
Sliding mode
Geometriccontrol
Recursivetechniques
X4 stabilization
Observers
u n−1(x n−1, x n) = u n(x n) + v n−1(x n−1, x n)
such that u n−1 stabilizes the cascade
x n−1 = x n + f n−1(x n, u n−1)
x n = u n−1
At step k: Augment the control law
u n−k (x n−k , . . . , x n) = u n−k +1(x n−k +1, . . . , x n) + v n−k (x n−k , . . . , x n)
such that u n−k stabilizes the cascade
ξn−k = x n−k +1 + f n−k (x n−k +1, . . . , x n, u n−k )
.
..x n−1 = x n + f n−1(x n, u n−k )
x n = u n−k
N. Marchand (gipsa-lab) Nonlinear Control Master PSPI 2009-2010 120 / 174
NonlinearControl
N. Marchand
References
ForwardingForwarding procedure
The control law u n−k always exists for allk ∈ 1, . . . , n − 1
The L ap no f nction at step k can be gi en b
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References
Outline
Introduction
Linear/Nonlinear
The X4 example
Linearapproaches
AntiwindupLinearization
Gain scheduling
Stability
Nonlinearapproaches
CLF
Sliding mode
Geometriccontrol
Recursivetechniques
X4 stabilization
Observers
The Lyapunov function at step k can be given by:
V k = V k +1 + 1
2x 2k +
∞
0x k (τ )f k (x k +1(τ ), . . . , x n(τ ))d τ
To avoid computations of the integrals, low-gain controlcan be used. It gave rise to an important litterature onbounded feedback control (Teel (91), Sussmann et al.(94))
Various extensions for more general system exist but are
still based on the same recursive constructionForwarding can be interpreted with passivity, ISS oroptimal control
N. Marchand (gipsa-lab) Nonlinear Control Master PSPI 2009-2010 121 / 174
NonlinearControl
N. Marchand
References
Backstepping/ForwardingAn interpretation of the names
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References
Outline
Introduction
Linear/Nonlinear
The X4 example
Linearapproaches
AntiwindupLinearization
Gain scheduling
Stability
Nonlinearapproaches
CLF
Sliding mode
Geometriccontrol
Recursivetechniques
X4 stabilization
Observers
N. Marchand (gipsa-lab) Nonlinear Control Master PSPI 2009-2010 122 / 174
NonlinearControl
N. Marchand
References
Outline1 Introduction
Linear versus nonlinearThe X4 example
2 Linear control methods for nonlinear systems
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References
Outline
Introduction
Linear/Nonlinear
The X4 example
Linearapproaches
AntiwindupLinearization
Gain scheduling
Stability
Nonlinearapproaches
CLF
Sliding mode
Geometriccontrol
Recursivetechniques
X4 stabilization
Observers
2 Linear control methods for nonlinear systemsAntiwindupLinearizationGain scheduling
3 Stability
4 Nonlinear control methodsControl Lyapunov functionsSliding mode controlState and output linearization
Backstepping and feedforwardingStabilization of the X4 at a position
5 Observers
N. Marchand (gipsa-lab) Nonlinear Control Master PSPI 2009-2010 123 / 174
NonlinearControl
N. Marchand
References
The X4 helicopterX4 position stabilization
The aim is now to apply nonlinear control techniques to the stabilizationproblem of an X4 helicopter at a pointInput-Ouput Linearization techniques can not be applied since the system
is non minimum phaseTh b k t i h i i i d f
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Outline
Introduction
Linear/Nonlinear
The X4 example
Linearapproaches
AntiwindupLinearization
Gain scheduling
Stability
Nonlinearapproaches
CLF
Sliding mode
Geometriccontrol
Recursivetechniques
X4 stabilization
Observers
is non minimum phaseThe backstepping approach is inspired fromS. Bouabdallah and R. Siegwart, ‘‘Backstepping and sliding-mode
techniques applied to an indoor micro quadrotor’’, at the EPFL
(Lausanne, Switzerland) presented at the International Conference on
Robotics and Automation 2005 (ICRA’05, Barcelonna, Spain)
The PVTOL control strategy comes fromA. Hably, F. Kendoul, N. Marchand and P. Castillo, Positive Systems,
chapter: "Further results on global stabilization of the PVTOL
aircraft", pp 303-310, Springer Verlag , 2006
The saturated control law is taken in:N. Marchand and A. Hably, "Nonlinear stabilization of multiple
integrators with bounded controls", Automatica, vol. 41, no. 12, pp2147-2152, 2005.
Revealing example of what often happens in nonlinear: a solution comesfrom the combination of different methods
N. Marchand (gipsa-lab) Nonlinear Control Master PSPI 2009-2010 124 / 174
NonlinearControl
N. Marchand
References
The X4 helicopterA two steps approach
s k 2m s k gearbox κ |s | s + k m sat (U )
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Outline
Introduction
Linear/Nonlinear
The X4 example
Linearapproaches
AntiwindupLinearization
Gain scheduling
Stability
Nonlinearapproaches
CLF
Sliding mode
Geometriccontrol
Recursivetechniques
X4 stabilization
Observers
s i = − m
J r R s i −
g
J r |s i | s i + m
J r R satU i
(U i )
p = v
m v = −mg e 3 + R
00
i
F i (s i )
R = R ω×
J ω = − ω×J ω −
i I r ω×
0
0
i s i
+
Γ r (s 2, s 4)
Γ p (s 1, s 3)
Γ y (s 1, s 2, s 3, s 4)
Position control problem
Attitude control problem
N. Marchand (gipsa-lab) Nonlinear Control Master PSPI 2009-2010 125 / 174
NonlinearControl
N. Marchand
References
The X4 helicopter: attitude stabilizationA backstepping approach
The aim of this first step is to be able to bring (φ,θ,ψ) toany desired configuration (φd , θd , ψd )
Attitude equations (with an appropriate choice of Euler
angles):
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Outline
Introduction
Linear/Nonlinear
The X4 example
Linearapproaches
AntiwindupLinearization
Gain scheduling
Stability
Nonlinearapproaches
CLF
Sliding mode
Geometriccontrol
Recursivetechniques
X4 stabilization
Observers
angles):
x 1 = x 2x 2 = a1x 4x 6 + b 1Γ r
x 3 = x 4x 4 = a2x 2x 6 + b 2Γ p
x 5 = x 6x 6 = a3x 2x 4 + b 3Γ y
with (x 1, . . . , x 6) = (φ, φ,θ, θ, ψ, ˙ ψ), a1 = J 2−J 3J 1
, a2 = J 3−J 1J 2
,
a3 = J 1−J 2J 3
, b 1 = 1J 1
, b 2 = 1J 2
and b 3 = 1J 3
The system would be trivial to control if (x 2, x 4, x 6) was thecontrol instead of (Γ r , Γ p , Γ y )
This is precisely the philosophy of backstepping
N. Marchand (gipsa-lab) Nonlinear Control Master PSPI 2009-2010 126 / 174
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N. Marchand
References
The X4 helicopter: attitude stabilizationRecall of backstepping main result
Theorem: backstepping design
Assume that the system ˙ χ = f ( χ, v ) with f (0, 0) = 0 can asymptotically be stabilized withv = k ( χ). Let V denote a C1 Lyapunov function (definite, positive and radiallyunbounded) such that ∂V
∂χf ( χ, k ( χ)) < 0 for all χ
= 0. Then, the system
˙ f ( ξ)
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Outline
Introduction
Linear/Nonlinear
The X4 example
Linearapproaches
AntiwindupLinearization
Gain scheduling
Stability
Nonlinearapproaches
CLF
Sliding mode
Geometriccontrol
Recursivetechniques
X4 stabilization
Observers
˙ χ = f ( χ, ξ)
ξ = h( χ, ξ) + u
with h(0, 0) = 0 is also asymptotically stabilizable with the control:
u ( χ, ξ) = −h( χ, ξ) +
∂k
∂χ f ( χ, ξ) − ξ + k ( χ) −∂V
∂χ G ( χ, ξ − k ( χ))T
with
G ( χ, ξ) =
10
∂f
∂ξ( χ, k ( χ) + λξ)d λ
The Lyapunov function
W ( χ, ξ) = V ( χ) + 1
2 ξ − k ( χ)2
is then strictly decreasing for any ( χ, ξ) = (0, 0)
N. Marchand (gipsa-lab) Nonlinear Control Master PSPI 2009-2010 127 / 174
NonlinearControl
N. Marchand
References
The X4 helicopter: attitude stabilizationA backstepping approach
Define first the tracking error:
z1 = x1 − x1d = φ − φd
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Outline
Introduction
Linear/Nonlinear
The X4 example
Linearapproaches
AntiwindupLinearization
Gain scheduling
Stability
Nonlinearapproaches
CLF
Sliding mode
Geometriccontrol
Recursivetechniques
X4 stabilization
Observers
z x x d φ φz 2 = x 3 − x 3d
= θ − θd
z 3 = x 5 − x 5d = ψ − ψd
It gives the subsystem
z 1 = x 2 − x 1d
z 2 = x 4 − x 3d
z 3 = x 6 − x 5d
where, at this step, (x 2, x 4, x 6) is the control vectorTake first V 1 = 1
2 z 21 .
N. Marchand (gipsa-lab) Nonlinear Control Master PSPI 2009-2010 128 / 174
NonlinearControl
N. Marchand
References
The X4 helicopter: attitude stabilizationA backstepping approach
Along the trajectories of the system, the Lyapunovfunction gives:
V 1 = z 1(x 2 − x 1d )
Hence taking as fictive control x2 = x1 α1z1 with
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Outline
Introduction
Linear/Nonlinear
The X4 example
Linearapproaches
AntiwindupLinearization
Gain scheduling
Stability
Nonlinearapproaches
CLF
Sliding mode
Geometriccontrol
Recursivetechniques
X4 stabilization
Observers
Hence taking as fictive control x 2 = x 1d − α 1z 1 with
α 1 > 0 insures the decrease of V 1:
V 1 = −α 1z 21
x 1 will asymptotically converge to x 1d ... unfortunately x 2is not the control and we have to build thanks tobackstepping approach “ the Γ r that will force x 2 to makex 1 converge to x 1d
”
For this, define the tracking error for x 2:
z 2 = x 2 − (x 1d − α 1z 1)
input of subsystem in z 1we would like to put
N. Marchand (gipsa-lab) Nonlinear Control Master PSPI 2009-2010 129 / 174
NonlinearControl
N. Marchand
ReferencesO li
The X4 helicopter: attitude stabilizationA backstepping approach
One hasz 2 = a1x 4x 6 + b 1Γ r
x 2
−x 1d + α 1x 2 − α 1x 1d
α1z 1
Recall we have to build “ the Γ r that will force x 2 to make x 1 converge
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Outline
Introduction
Linear/Nonlinear
The X4 example
Linearapproaches
AntiwindupLinearization
Gain scheduling
Stability
Nonlinearapproaches
CLF
Sliding mode
Geometriccontrol
Recursivetechniques
X4 stabilization
Observers
r 2 1 gto x 1d
”. For this, we use the formula of the backstepping theorem:
Theorem: backstepping design
...
u ( χ, ξ) = −h( χ, ξ) + ∂k
∂χf ( χ, ξ) − ξ + k ( χ) −
∂V
∂χ G ( χ, ξ − k ( χ))
T
...
with in our case:
χ = z 1ξ = z 2h = a1x 4x 6 − x 1d
+ α 1x 2 − α 1x 1d
k = x 1d − α 1z 1
f = x 2 − x 1d
V = 12 z 21
N. Marchand (gipsa-lab) Nonlinear Control Master PSPI 2009-2010 130 / 174
NonlinearControl
N. Marchand
ReferencesO tli
The X4 helicopter: attitude stabilizationA backstepping approach
Or more simply, we construct it:
W (z 1, z 2) = V (z 1) + 1
2 x 2 − (x 1d
− α 1z 1)2=
1
2(z 21 + z 22 )
Al th t j t i s f th s st
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Outline
Introduction
Linear/Nonlinear
The X4 example
Linearapproaches
AntiwindupLinearization
Gain scheduling
Stability
Nonlinearapproaches
CLF
Sliding mode
Geometriccontrol
Recursivetechniques
X4 stabilization
Observers
Along the trajectories of the system:
W = z 1(x 2 − x 1d ) + z 2
a1x 4x 6 +
control
b 1Γ r −x 1d
+ α 1x 2 − α 1x 1d
Taking v as new control variable:
v = a1x 4x 6 + b 1Γ r − x 1d + α 1x 2 − α 1x 1d
b 1Γ r = v − a1x 4x 6 + x 1d − α 1x 2 + α 1x 1d
gives:W = z 1(x 2 − x 1d
) + z 2v
N. Marchand (gipsa-lab) Nonlinear Control Master PSPI 2009-2010 131 / 174
NonlinearControl
N. Marchand
ReferencesOutline
The X4 helicopter: attitude stabilizationA backstepping approach
Trying to write
x 2 − x 1d = −α 1z 1
ideal case
+? (x 2 − x 1d + α 1z 1)
difference between realand ideal cases
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Outline
Introduction
Linear/Nonlinear
The X4 example
Linearapproaches
AntiwindupLinearization
Gain scheduling
Stability
Nonlinearapproaches
CLF
Sliding mode
Geometriccontrol
Recursivetechniques
X4 stabilization
Observers
= −α 1z 1 + x 2 − x 1d + α 1z 1
z 2
Hence, it gives:
W = −α 1z 21 + z 1z 2 +z 2v
can be compensated with v
Taking:v = −z 1 − α 2z 2
where α 2 > 0 in order to insure
W = −α 1z 21 − α 2z 22 < 0
Repeating this for θ and ψ gives the wanted control law
N. Marchand (gipsa-lab) Nonlinear Control Master PSPI 2009-2010 132 / 174
NonlinearControl
N. Marchand
ReferencesOutline
The X4 helicopter: attitude stabilizationA backstepping approach
Attitude stabilization
The control law given by
b 1Γ r = −z 1 − α 2z 2 − a1x 4x 6 + x 1d − α 1x 2 + α 1x 1d
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Outline
Introduction
Linear/Nonlinear
The X4 example
Linearapproaches
AntiwindupLinearization
Gain scheduling
Stability
Nonlinearapproaches
CLF
Sliding mode
Geometriccontrol
Recursivetechniques
X4 stabilization
Observers
b 2Γ p = −z 3 − α 4z 4 − a2x 2x 6 + x 3d − α 3x 4 + α 3x 3d
b 3Γ y = −z 5 − α 6z 6 − a3x 2x 4 + x 5d − α 5x 6 + α 5x 5d
with z 1 = x 1 − x 1d , z 2 = x 2 − x 1d
+ α 1z 1, z 3 = x 3 − x 3d ,
z 4 = x 4 − x 3d + α 3z 3, z 5 = x 5 − x 5d and z 6 = x 6 − x 5d + α 5z 5asymptotically stabilizes (x 1, x 3, x 5) to their desired position(x 1d
, x 3d , x 5d
)
This kind of approach appeared in the literature in the last
90’sNow better approach based on forwarding are appearingtaking into account practical saturation of the control torques
N. Marchand (gipsa-lab) Nonlinear Control Master PSPI 2009-2010 133 / 174
NonlinearControl
N. Marchand
ReferencesOutline
The X4 helicopter: attitude stabilizationA backstepping approach
Simulations: apply successive steps of 45 in roll, pitch andyaw
60
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Outline
Introduction
Linear/Nonlinear
The X4 example
Linearapproaches
AntiwindupLinearization
Gain scheduling
Stability
Nonlinearapproaches
CLF
Sliding mode
Geometriccontrol
Recursivetechniques
X4 stabilization
Observers
0 10 20 30 40 50 60 70−20
0
20
40
60
0 10 20 30 40 50 60 70−20
0
20
40
60
0 10 20 30 40 50 60 70−20
0
20
40
60
N. Marchand (gipsa-lab) Nonlinear Control Master PSPI 2009-2010 134 / 174
NonlinearControl
N. Marchand
References
Outline
The X4 helicopter: attitude stabilizationA backstepping approach
Adjusting the α ’s: with α i = 0.2
60 1
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Outline
Introduction
Linear/Nonlinear
The X4 example
Linearapproaches
AntiwindupLinearization
Gain scheduling
Stability
Nonlinearapproaches
CLF
Sliding mode
Geometriccontrol
Recursivetechniques
X4 stabilization
Observers
0 10 20 30 40 50 60 70−20
0
20
40
0 10 20 30 40 50 60 70−20
0
20
40
60
0 10 20 30 40 50 60 70−20
0
20
40
60
Roll, pitch and yaw answers
0 10 20 30 40 50 60 70−1
0
0 10 20 30 40 50 60 70−0.5
0
0.5
0 10 20 30 40 50 60 70−2
−1
0
1
Γ r , Γ p and Γ y controls
Too many oscillations
N. Marchand (gipsa-lab) Nonlinear Control Master PSPI 2009-2010 135 / 174
NonlinearControl
N. Marchand
References
Outline
The X4 helicopter: attitude stabilizationA backstepping approach
Adjusting the α ’s: with α i = 2
40
60 5
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Introduction
Linear/Nonlinear
The X4 example
Linearapproaches
AntiwindupLinearization
Gain scheduling
Stability
Nonlinearapproaches
CLF
Sliding mode
Geometriccontrol
Recursivetechniques
X4 stabilization
Observers
0 10 20 30 40 50 60 70−20
0
20
40
0 10 20 30 40 50 60 70−20
0
20
40
60
0 10 20 30 40 50 60 70−20
0
20
40
60
Roll, pitch and yaw answers
0 10 20 30 40 50 60 70−5
0
0 10 20 30 40 50 60 70−2
0
2
0 10 20 30 40 50 60 70−2
0
2
Γ r , Γ p and Γ y controls
Seems to be a good choice
N. Marchand (gipsa-lab) Nonlinear Control Master PSPI 2009-2010 136 / 174
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NonlinearControl
N. Marchand
References
Outline
The X4 helicopter: attitude stabilizationA backstepping approach
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Introduction
Linear/Nonlinear
The X4 example
Linearapproaches
AntiwindupLinearization
Gain scheduling
Stability
Nonlinearapproaches
CLF
Sliding mode
Geometriccontrol
Recursivetechniques
X4 stabilization
ObserversSuccessive steps of 45 in roll, pitch and yaw with α i = 2
N. Marchand (gipsa-lab) Nonlinear Control Master PSPI 2009-2010 138 / 174
NonlinearControl
N. Marchand
References
Outline
The X4 helicopter: attitude stabilizationA backstepping approach
The saturation of the possible control torques Γ r , Γ p , Γ y
may be problematic
On the X4 system, the maximum speed of the rotors is
given by the solution of:
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Introduction
Linear/Nonlinear
The X4 example
Linearapproaches
AntiwindupLinearization
Gain scheduling
Stability
Nonlinearapproaches
CLF
Sliding mode
Geometriccontrol
Recursivetechniques
X4 stabilization
Observers
k 2ms max + k gearbox κRs max2 − k mU = 0
which gives: s max = 604 rad.s−1
The maximum roll and pitch torques are when one rotor isat rest, the other one at is maximum speed (we assumethat the rotation is in only one direction):
Γ maxr ,p = lbs 2max = 0.31 N.m
The maximum yaw torque is obtained when two oppositerotors turn at the s max the two others are at rest:
Γ maxy = 2κs 2max = 21.2 N.m
N. Marchand (gipsa-lab) Nonlinear Control Master PSPI 2009-2010 139 / 174
NonlinearControl
N. Marchand
References
Outline
The X4 helicopter: attitude stabilizationA backstepping approach
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Introduction
Linear/Nonlinear
The X4 example
Linearapproaches
AntiwindupLinearization
Gain scheduling
Stability
Nonlinearapproaches
CLF
Sliding mode
Geometriccontrol
Recursivetechniques
X4 stabilization
Observers
Adapt the parameters to the saturation:
Take larger α 5,6 than the α 1,...,4Take small enough α i to fulfill the saturation constraint
N. Marchand (gipsa-lab) Nonlinear Control Master PSPI 2009-2010 140 / 174
NonlinearControl
N. Marchand
References
Outline
The X4 helicopter: attitude stabilizationA backstepping approach
Adjusting the α ’s: with α 1,...,4 = 0.4 and α 5,6 = 8
40
60 1
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Introduction
Linear/Nonlinear
The X4 example
Linearapproaches
AntiwindupLinearization
Gain scheduling
Stability
Nonlinearapproaches
CLF
Sliding mode
Geometriccontrol
Recursivetechniques
X4 stabilization
Observers
0 10 20 30 40 50 60 70−20
0
20
0 10 20 30 40 50 60 70
−20
0
20
40
60
0 10 20 30 40 50 60 70−20
0
20
40
60
Roll, pitch and yaw answers
0 10 20 30 40 50 60 70−1
0
0 10 20 30 40 50 60 70
−0.5
0
0.5
0 10 20 30 40 50 60 70−50
0
50
Γ r , Γ p and Γ y controls
The controls are still too large
N. Marchand (gipsa-lab) Nonlinear Control Master PSPI 2009-2010 141 / 174
NonlinearControl
N. Marchand
References
Outline
The X4 helicopter: attitude stabilizationA backstepping approach
Adjusting the α ’s: with α 1,...,4 = 0.2 and α 5,6 = 8
40
60 1
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Introduction
Linear/Nonlinear
The X4 example
Linearapproaches
AntiwindupLinearization
Gain scheduling
Stability
Nonlinearapproaches
CLF
Sliding mode
Geometriccontrol
Recursivetechniques
X4 stabilization
Observers
0 10 20 30 40 50 60 70−20
0
20
0 10 20 30 40 50 60 70
−20
0
20
40
60
0 10 20 30 40 50 60 70−20
0
20
40
60
Roll, pitch and yaw answers
0 10 20 30 40 50 60 70−1
0
0 10 20 30 40 50 60 70
−0.5
0
0.5
0 10 20 30 40 50 60 70−50
0
50
Γ r , Γ p and Γ y controls
Oscillations are now present
N. Marchand (gipsa-lab) Nonlinear Control Master PSPI 2009-2010 142 / 174
NonlinearControl
N. Marchand
References
Outline
The X4 helicopter: attitude stabilizationA backstepping approach
Taking α 1,...,4 = 0.2 and α 5,6 = 8 and applying it on thesystem with saturations, it gives:
60 0.5
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Introduction
Linear/Nonlinear
The X4 example
Linearapproaches
AntiwindupLinearization
Gain scheduling
Stability
Nonlinearapproaches
CLF
Sliding mode
Geometriccontrol
Recursivetechniques
X4 stabilization
Observers
0 10 20 30 40 50 60 70−20
0
20
40
60
0 10 20 30 40 50 60 70−20
0
20
40
60
0 10 20 30 40 50 60 70−20
0
20
40
60
Roll, pitch and yaw answers
0 10 20 30 40 50 60 70−0.5
0
0.5
0 10 20 30 40 50 60 70−0.5
0
0.5
0 10 20 30 40 50 60 70−50
0
50
Γ r , Γ p and Γ y controls
Seems to work
N. Marchand (gipsa-lab) Nonlinear Control Master PSPI 2009-2010 143 / 174
NonlinearControl
N. Marchand
References
Outline
The X4 helicopter: attitude stabilizationA backstepping approach
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Introduction
Linear/Nonlinear
The X4 example
Linearapproaches
AntiwindupLinearization
Gain scheduling
Stability
Nonlinearapproaches
CLF
Sliding mode
Geometriccontrol
Recursivetechniques
X4 stabilization
ObserversThe control law with the saturated system
N. Marchand (gipsa-lab) Nonlinear Control Master PSPI 2009-2010 144 / 174
NonlinearControl
N. Marchand
References
Outline
The X4 helicopter: position controlSimplification of the problem
First use the attitude control to adjust the X4 in the rightdirection:
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Introduction
Linear/Nonlinear
The X4 example
Linearapproaches
AntiwindupLinearization
Gain scheduling
Stability
Nonlinearapproaches
CLF
Sliding mode
Geometriccontrol
Recursivetechniques
X4 stabilization
Observers
N. Marchand (gipsa-lab) Nonlinear Control Master PSPI 2009-2010 145 / 174
NonlinearControl
N. Marchand
References
Outline
The X4 helicopter: position controlSimplification of the problem
There exists a relation between (α,β,ψ) and (φ,θ,ψ) (arotation about the yaw axis)Use the attitude control to drive and keep α and ψ to the
originTake now the system in the plane:
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Introduction
Linear/Nonlinear
The X4 example
Linearapproaches
AntiwindupLinearization
Gain scheduling
Stability
Nonlinearapproaches
CLF
Sliding mode
Geometriccontrol
Recursivetechniques
X4 stabilization
Observers
N. Marchand (gipsa-lab) Nonlinear Control Master PSPI 2009-2010 146 / 174
NonlinearControl
N. Marchand
References
Outline
The X4 helicopter: position controlSimplification of the problem
This problem is referred in the literature as the PVTOL
aircraft stabilization problem (Planar Vertical Take Off and Landing aircraft)
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Introduction
Linear/Nonlinear
The X4 example
Linearapproaches
AntiwindupLinearization
Gain scheduling
Stability
Nonlinearapproaches
CLF
Sliding mode
Geometriccontrol
Recursivetechniques
X4 stabilization
Observers
The “generic” equations are:
x = − sin(θ)u
y = cos(θ)u − 1θ = v
u v
θ
x
y
where ’-1’ represent the normalized gravity, v represents
the equivalent control torque.
N. Marchand (gipsa-lab) Nonlinear Control Master PSPI 2009-2010 147 / 174
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N. Marchand
References
Outline
I d i
The X4 helicopter: position controlA saturation based control
Define z (z 1, z 2, z 3, z 4, z 5, z 6)T (x , x , y , y , θ, θ)T
The system becomes:
Translational part:
z 1 = z 2z 2 = −u sin(z 5)
Rotational part:z 5 = z 6
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Introduction
Linear/Nonlinear
The X4 example
Linearapproaches
Antiwindup
Linearization
Gain scheduling
Stability
Nonlinearapproaches
CLF
Sliding mode
Geometriccontrol
Recursivetechniques
X4 stabilization
Observers
z 3 = z 4z 4 = u cos(z 5) − 1
z 6 = v
The idea is to use v to drive z 5 to z d 5 such that:
z 5d arctan(
−r 1
r 2 + 1)
with ε < 12 (tuning parameter) and σ(·) = max(min(·, 1), −1):
r 1 = −εσ(z 2) − ε2
σ(εz 1 + z 2)r 2 = −εσ(z 4) − ε2σ(εz 3 + z 4)
N. Marchand (gipsa-lab) Nonlinear Control Master PSPI 2009-2010 148 / 174
NonlinearControl
N. Marchand
References
Outline
I t d ti
The X4 helicopter: position controlA saturation based control
When z 5 reaches z 5d and applying the thrust control input u
u = r 21 + (r 2 + 1)2
the translational subsystem takes the form of two independent
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Introduction
Linear/Nonlinear
The X4 example
Linearapproaches
Antiwindup
Linearization
Gain scheduling
Stability
Nonlinearapproaches
CLF
Sliding mode
Geometriccontrol
Recursivetechniques
X4 stabilization
Observers
second order chain of integrators
z 1 = z 2z 2 = r 1
z 3 = z 4z 4 = r 2
If we could prove that
r 1 = −εσ(z 2) − ε2σ(εz 1 + z 2)
r 2 = −εσ(z 4) − ε2σ(εz 3 + z 4)
insures (z 1, z 2, z 3, z 4) → 0Then, it will follow that once (z 1, z 2, z 3, z 4) = 0, z 5d
= 0 andz 5d
= z 6d = 0 hence also (z 5, z 6) → 0
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References
Outline
I t od ctio
The X4 helicopter: position controlA saturation based control
Remain two problems:Prove that
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Introduction
Linear/Nonlinear
The X4 example
Linearapproaches
Antiwindup
Linearization
Gain scheduling
Stability
Nonlinearapproaches
CLF
Sliding mode
Geometriccontrol
Recursivetechniques
X4 stabilization
Observers
Prove that
r 1 = −εσ(z 2) − ε2σ(εz 1 + z 2)
r 2 = −εσ(z 4) − ε2σ(εz 3 + z 4)
brings (z 1, z 2, z 3, z 4) to zero
Build a control so that (z 5, z 6) tends to (z 5d , z 5d
= z 6d )
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NonlinearControl
N. Marchand
References
Outline
Introduction
The X4 helicopter: position controlA saturated control law for linear systems
An integrator chain is defined by:
x =
0 1 0 . . . 0...
. . . 1 . . .
...
... . . . . . . 0
.. . . 1
x +
0......0
u (11)
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Introduction
Linear/Nonlinear
The X4 example
Linearapproaches
Antiwindup
Linearization
Gain scheduling
Stability
Nonlinearapproaches
CLF
Sliding mode
Geometriccontrol
Recursivetechniques
X4 stabilization
Observers
. . 10 . . . . . . . . . 0
=:A
01
=:B
Problem: stabilizing (11) with
−u ≤ u ≤ u
and u is a positive constant
N. Marchand (gipsa-lab) Nonlinear Control Master PSPI 2009-2010 151 / 174
NonlinearControl
N. Marchand
References
Outline
Introduction
The X4 helicopter: position controlA saturated control law for linear systems
First compute
n
i =1
(λ + εi ) = p 0 + p 1λ +
· · ·+ p n−1λn−1 + λn
Apply the coordinate change
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Introduction
Linear/Nonlinear
The X4 example
Linearapproaches
Antiwindup
Linearization
Gain scheduling
Stability
Nonlinearapproaches
CLF
Sliding mode
Geometriccontrol
Recursivetechniques
X4 stabilization
Observers
Apply the coordinate change
y := ni =1 εi
u Tx with
T n = B
T n−1 = (A + p n−1I )B
T n−2 = (A2 + p n−1A + p n−2I )B ...
T 1 = (An−1 + p n−1An−2 + · · · + p 1I )B
The system becomes normalized in
y =
0 εn−1 εn−2 . . . ε
0 0 εn−2 . . . ε...
. . . . . .
...0 . . . . . . 0 ε
0 . . . . . . 0 0
y +
1......
...1
v
with −n
i =1 εi ≤ v ≤ ni =1 εi
N. Marchand (gipsa-lab) Nonlinear Control Master PSPI 2009-2010 152 / 174
NonlinearControl
N. Marchand
References
Outline
Introduction
The X4 helicopter: position controlA saturated control law for linear systems
Theorem: An efficient saturated controlLet ε(n) denote the only root of εn − 2ε + 1 = 0 in ]0, 1[.
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Introduction
Linear/Nonlinear
The X4 example
Linearapproaches
Antiwindup
Linearization
Gain scheduling
Stability
Nonlinearapproaches
CLF
Sliding mode
Geometriccontrol
Recursivetechniques
X4 stabilization
Observers
( ) y + ] , [
Then for all ε with 0 < ε < ε(n),
u = − ¯
u ni =1 εi
ni =1
εn−i +1 sat1(y i )
globally asymptotically stabilizes the integrator chain.
N. Marchand (gipsa-lab) Nonlinear Control Master PSPI 2009-2010 153 / 174
NonlinearControl
N. Marchand
References
Outline
Introduction
The X4 helicopter: position controlA saturated control law for linear systems
Sketch of the proof:
Assume that |y n| > 1 and take V n := 12 y 2n
Then
V n = −y n ε sat1(y n) =ε
−y n
ε2 sat1(y n−1) + · · · + εn sat1(y 1)
| |<ε2+ +εn
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t oduct o
Linear/Nonlinear
The X4 example
Linearapproaches
Antiwindup
Linearization
Gain scheduling
Stability
Nonlinearapproaches
CLF
Sliding mode
Geometriccontrol
Recursivetechniques
X4 stabilization
Observers
=ε |·|<ε2+···+εn
So V is decreasing if
ε > ε2
+ · · · + εn ⇔ 1 − 2ε + ε
n
> 0 ⇔ ε < ε(n)
y n joins [−1, 1 ] in finite time and remains thereDuring that time, y n−1 to y 1 can not blow upRepeating this scheme for y n−1 to y 1 gives y ∈ B(1) aftersome time
In B(1), the system is linear and stable ⇒ GASThe construction of this feedback law is based on feedforward
N. Marchand (gipsa-lab) Nonlinear Control Master PSPI 2009-2010 154 / 174
NonlinearControl
N. Marchand
References
Outline
Introduction
The X4 helicopter: position controlA saturation based control
Remain two problems:
Prove that
r 1 = −εσ(z 2) − ε2σ(εz 1 + z 2)
r2 = −εσ(z4) − ε2σ(εz3 + z4)
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Linear/Nonlinear
The X4 example
Linearapproaches
Antiwindup
Linearization
Gain scheduling
Stability
Nonlinearapproaches
CLF
Sliding mode
Geometriccontrol
Recursivetechniques
X4 stabilization
Observers
r 2 = εσ(z 4) ε σ(εz 3 + z 4)
brings (z 1, z 2, z 3, z 4) to zero
direct application of the saturated control lawBuild a control so that (z 5, z 6) tends to (z 5d
, z 5d = z 6d
)
take:
v = σβ(z 5d ) − εσ(z 6 − z 5d
) − ε2σ(ε(z 5 − z 5d ) + (z 6 − z 5d
))
that work applying the saturated control law on thevariables z 5 − z 5d
and z 6 − z 5d
N. Marchand (gipsa-lab) Nonlinear Control Master PSPI 2009-2010 155 / 174
NonlinearControl
N. Marchand
References
Outline
Introduction
The X4 helicopterSecond step: position control
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Linear/Nonlinear
The X4 example
Linearapproaches
Antiwindup
Linearization
Gain scheduling
Stability
Nonlinearapproaches
CLF
Sliding mode
Geometriccontrol
Recursivetechniques
X4 stabilization
Observers
(φ,θ,ψ,x ,y ,z ) = (pi 2 ,
pi 2 ,
pi 4 ,5,5,5)
(φ, θ, ψ, x , y , z ) = (0,0,0,1,2,0)
N. Marchand (gipsa-lab) Nonlinear Control Master PSPI 2009-2010 156 / 174
NonlinearControl
N. Marchand
References
Outline
Introduction
The X4 helicopterSecond step: position control
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Linear/Nonlinear
The X4 example
Linearapproaches
Antiwindup
Linearization
Gain scheduling
Stability
Nonlinearapproaches
CLF
Sliding mode
Geometriccontrol
Recursivetechniques
X4 stabilization
Observers
Zoom on the 10 first seconds with a snapshot every second
Initial conditions:
(φ,θ,ψ,x ,y ,z ) = ( pi 2 ,
pi 2 ,
pi 4 ,5,5,5)
(φ, θ, ψ, x , y , z ) = (0,0,0,1,2,0)
N. Marchand (gipsa-lab) Nonlinear Control Master PSPI 2009-2010 157 / 174
NonlinearControl
N. Marchand
References
Outline
Introduction
Outline
1 IntroductionLinear versus nonlinearThe X4 example
2 Linear control methods for nonlinear systemsAntiwindupLinearization
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Linear/Nonlinear
The X4 example
Linearapproaches
Antiwindup
Linearization
Gain scheduling
Stability
Nonlinearapproaches
CLF
Sliding mode
Geometriccontrol
Recursivetechniques
X4 stabilization
Observers
Gain scheduling
3 Stability
4 Nonlinear control methodsControl Lyapunov functionsSliding mode controlState and output linearization
Backstepping and feedforwardingStabilization of the X4 at a position
5 Observers
N. Marchand (gipsa-lab) Nonlinear Control Master PSPI 2009-2010 158 / 174
NonlinearControl
N. Marchand
References
Outline
Introduction
Nonlinear observers
y u y d
x
Observer
k (x )
Controller
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Linear/Nonlinear
The X4 example
Linearapproaches
Antiwindup
Linearization
Gain scheduling
Stability
Nonlinearapproaches
CLF
Sliding mode
Geometriccontrol
Recursivetechniques
X4 stabilization
Observers
Observer
1 ModelizationTo get a mathematical representation of the systemDifferent kind of model are useful. Often:
a simple model to build the control lawa sharp model to check the control law and the observer
2 Design the state reconstruction: in order to reconstructthe variables needed for control
3 Design the control and test it
Close the loop
N. Marchand (gipsa-lab) Nonlinear Control Master PSPI 2009-2010 159 / 174
NonlinearControl
N. Marchand
References
Outline
Introduction
Linear systems: Simple observerx (t ) = Ax (t ) + Bu (t )
y (t ) = Cx (t )
Observability given by rank(C , CA, . . . , CAn−1)
O fi
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Linear/Nonlinear
The X4 example
Linearapproaches
Antiwindup
LinearizationGain scheduling
Stability
Nonlinearapproaches
CLF
Sliding mode
Geometriccontrol
Recursivetechniques
X4 stabilization
Observers
Once the observability has been checked, define theobserver:
˙x (t ) = Ax + Bu (t ) + L(y (t ) − y (t ))y (t ) = C x (t )
with L such that (eig(A + LC )) < 0.Then x tends to x asymptotically with a speed
related to eig(A + LC )
N. Marchand (gipsa-lab) Nonlinear Control Master PSPI 2009-2010 160 / 174
NonlinearControl
N. Marchand
References
Outline
Introduction
Li /N li
Linear systems (cntd.):
convergence of the observer
Define the error e (t ) := x (t ) − x (t ). Then:
e(t) = Ae(t) + L(y (t) − y (t)) = (A + KC )e(t)
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Linear/Nonlinear
The X4 example
Linearapproaches
Antiwindup
LinearizationGain scheduling
Stability
Nonlinearapproaches
CLF
Sliding mode
GeometriccontrolRecursivetechniques
X4 stabilization
Observers
e (t ) = Ae (t ) + L(y (t ) − y (t )) = (A + KC )e (t )
Hence, if (eig(A + LC )) < 0, limt →∞ x (t ) = x (t )
Separation principle: the controller and the observer canbe designed separately, if each are stable, their associationwill be stable
N. Marchand (gipsa-lab) Nonlinear Control Master PSPI 2009-2010 161 / 174
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NonlinearControl
N. Marchand
References
Outline
Introduction
Linear/Nonlinear
Time-varying linear systems:x (t ) = A(t )x (t ) + B (t )u (t )
y (t ) = C (t )x (t )
Kalman filter:˙x = A(t )x (t ) + B (t )u (t ) + L(t )(y (t ) − y (t ))
y (t) C (t)x(t)
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Linear/Nonlinear
The X4 example
Linearapproaches
Antiwindup
LinearizationGain scheduling
Stability
Nonlinearapproaches
CLF
Sliding mode
GeometriccontrolRecursivetechniques
X4 stabilization
Observers
y (t ) = C (t )x (t )
with: P = AP + PAT − PC T W −1CP + V + δP
P (0) = P 0 = P T 0 > 0
W = W T > 0
L = −PC T W −1
δ > 2
A
or V = V T > 0
δ enables to tunes the speed of convergence of theobserver
N. Marchand (gipsa-lab) Nonlinear Control Master PSPI 2009-2010 163 / 174
NonlinearControl
N. Marchand
References
Outline
Introduction
Linear/Nonlinear
The observer can also be computed as L = −S −1
C T
W −1
where
−S = AT S + SA − C T W −1C + SVS + δS
S (0) = S 0 = S T 0 > 0
The observer is optimal in the sense that it minimizes
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Linear/Nonlinear
The X4 example
Linearapproaches
Antiwindup
LinearizationGain scheduling
Stability
Nonlinearapproaches
CLF
Sliding mode
GeometriccontrolRecursivetechniques
X4 stabilization
Observers
t
0[C (τ )z (τ ) − y (τ )]T W −1 [C (τ )z (τ ) − y (τ )] +
[B (τ )u (τ )]T V −1 [B (τ )u (τ )] d τ +
(x 0 − x 0)T P −10 (x 0 − x 0)
The observer is also optimal for
x is affected by a white noise w x of variance V
y is affected by a white noise w y of variance W
w x and w y are uncorrelated
N. Marchand (gipsa-lab) Nonlinear Control Master PSPI 2009-2010 164 / 174
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NonlinearControl
N. Marchand
References
Outline
Introduction
Linear/Nonlinear
Extended Kalman filter (EKF):˙x = f (x (t ), u (t )) − L(t )(y (t ) − y (t ))
y (t ) = h(x (t ))
with P = AP + PAT − PC T W −1CP + V + δP
P (0) = P 0 = P T 0 > 0
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/
The X4 example
Linearapproaches
Antiwindup
LinearizationGain scheduling
Stability
Nonlinearapproaches
CLF
Sliding mode
GeometriccontrolRecursivetechniques
X4 stabilization
Observers
( ) 0 0
W = W T > 0
L = PC T W −1
δ > 2 A or V = V T > 0
A = ∂f
∂x (x (t ), u (t )) C =
∂h
∂x (x (t ))
No guarantee of convergence (except for specificstructure conditions)
N. Marchand (gipsa-lab) Nonlinear Control Master PSPI 2009-2010 166 / 174
NonlinearControl
N. Marchand
References
Outline
Introduction
Linear/Nonlinear
Luenberger-Like observers
For a system of the form:x (t ) = Ax (t ) + φ(Cx (t ), u )
y (t ) = Cx (t )
with (A, C ) observable, if A − KC is stable, an observer is:
˙x (t ) = Ax (t ) + φ(y (t ), u (t )) − K (C x (t ) − y (t ))
For a system of the form:
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/
The X4 example
Linearapproaches
Antiwindup
LinearizationGain scheduling
Stability
Nonlinearapproaches
CLF
Sliding mode
Geometric
controlRecursivetechniques
X4 stabilization
Observers
x (t ) = A0x (t ) + φ(x (t ), u (t ))
y (t ) = C 0x (t )
with (A0, C 0) are in canonical form, if A0 − K 0C 0 is stable, λ
sufficiently large, φ global lipschitz and
∂φi
∂x j
(x , u ) = 0 for j ≥ i + 1
an observer is:˙x (t ) = A0x (t ) + φ(x (t ), u (t )) − diag(λ, λ2, . . . , λn)K 0(C 0x (t ) − y (t ))
N. Marchand (gipsa-lab) Nonlinear Control Master PSPI 2009-2010 167 / 174
NonlinearControl
N. Marchand
References
Outline
Introduction
Linear/Nonlinear
Kalman-Like observersFor a system of the form:
x (t ) = A(u (t ))x (t ) + B (u (t ))
y (t ) = Cx (t )
an observer is:
˙x(t) = A(u(t))x(t) + B (u(t)) − K (t)(C x (t) − y (t))
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The X4 example
Linearapproaches
Antiwindup
LinearizationGain scheduling
Stability
Nonlinearapproaches
CLF
Sliding mode
Geometric
controlRecursivetechniques
X4 stabilization
Observers
x (t ) = A(u (t ))x (t ) + B (u (t )) − K (t )(C x (t ) − y (t ))
with
P = A(u (t ))P + PAT (u (t )) − PC T W −1CP + V + δP
P (0) = P 0 = P T 0 > 0, W = W T > 0
δ > 2 A(u (t )) or V = V T > 0
L = PC T W −1
N. Marchand (gipsa-lab) Nonlinear Control Master PSPI 2009-2010 168 / 174
NonlinearControl
N. Marchand
References
Outline
Introduction
Linear/Nonlinear
Kalman-Like observers (cntd.)
For a system of the form:x (t ) = A0(u (t ), y (t ))x (t ) + φ(x (t ), u (t ))
y (t ) = C 0x (t )
with:
A0(u , y ) =
0 a12(u , y ) 0
. . .
an−1n(u , y )
0 0
bounded
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The X4 example
Linearapproaches
Antiwindup
LinearizationGain scheduling
Stability
Nonlinearapproaches
CLF
Sliding mode
Geometric
controlRecursivetechniques
X4 stabilization
Observers
0 0
C 0 =
1 0 · · · 0
∂φi
∂x j
(x , u ) = 0 for j
≥ i + 1
an observer is:
˙x = A(u , y )x + φ(x , u ) − diag(λ, λ2, . . . , λn)K 0(t )(C 0x − y )
with P = λ
A(u (t ))P + PAT (u (t )) − PC T W −1CP + δP
P (0) = P 0 = P
T 0 > 0, W = W
T
> 0δ > 2 A(u (t )) and λ large enough
L = P C T W −1
N. Marchand (gipsa-lab) Nonlinear Control Master PSPI 2009-2010 169 / 174
NonlinearControl
N. Marchand
References
Outline
Introduction
Linear/Nonlinear
Not mentionned:
Optimal observers (robust, very efficient and easy to
tune but costly)Based on something like:
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The X4 example
Linearapproaches
Antiwindup
LinearizationGain scheduling
Stability
Nonlinearapproaches
CLF
Sliding mode
Geometric
controlRecursivetechniques
X4 stabilization
Observers
x = Arg minx (·)
t 1
t 0
h(x (τ )) − y (τ )) d τ
Sliding mode observers
N. Marchand (gipsa-lab) Nonlinear Control Master PSPI 2009-2010 170 / 174
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NonlinearControl
N. Marchand
References
Outline
Introduction
Linear/Nonlinear
The X4 example
ObserversApplication to the attitude estimation
The gyrometers give:
b gyr = ω + ηgyr 1 noise
+ νgyr 1 bias
where hmag are the coordinates of the magnetic field in
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The X4 example
Linearapproaches
Antiwindup
LinearizationGain scheduling
Stability
Nonlinearapproaches
CLF
Sliding mode
Geometric
controlRecursivetechniques
X4 stabilization
Observers
g
the fixed frame.The bias drift is the main error and it deteriorates the
accuracy of the rate gyros on the low frequency band.We take:
b gyr = ω + ηgyr 1 + νgyr 1
˙ νgyr 1 = − 1τ νgyr 1 + ηgyr 2
noise
N. Marchand (gipsa-lab) Nonlinear Control Master PSPI 2009-2010 172 / 174
NonlinearControl
N. Marchand
References
Outline
Introduction
Linear/Nonlinear
The X4 example
ObserversApplication to the attitude estimation
Nonlinear observer for attitude estimation
An observer of the attitude can be: ˙q = 1
2 Ω( b gyr − ^ νgyr 1 + K 1ε)q ˙ν = −T 1ν − K2ε
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The X4 example
Linearapproaches
Antiwindup
LinearizationGain scheduling
Stability
Nonlinearapproaches
CLF
Sliding mode
Geometric
controlRecursivetechniques
X4 stabilization
Observers
ν T ν K 2ε
where T is a diagonal matrix of time constant, K i are positive
definite matrices, ε is given by:
ε = q e sign(q e 0 )
where q e = (q e 0 , q e ) is the quaternion error between the
estimate q and a direct projection obtained with b mag and b acc
N. Marchand (gipsa-lab) Nonlinear Control Master PSPI 2009-2010 173 / 174
NonlinearControl
N. Marchand
References
Outline
Introduction
Linear/Nonlinear
The X4 example
ObserversApplication to the attitude estimation
Block diagram of the observer
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The X4 example
Linearapproaches
Antiwindup
LinearizationGain scheduling
Stability
Nonlinearapproaches
CLF
Sliding mode
Geometric
controlRecursivetechniques
X4 stabilization
Observers
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