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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

[email protected]

Control Systems Department, gipsa-labGrenoble, France

Master PSPI 2009-2010

N. Marchand (gipsa-lab) Nonlinear Control Master PSPI 2009-2010 1 / 174



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


http://www.math.rutgers.edu/~sontag/FTP_DIR/sontag_mathematical_control_theory_springer98.pdf








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




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







5 Observers




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




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







5 Observers




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




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




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.




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.)




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, . . . )




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)




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




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.




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




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




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


S f



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


S b h i f li



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


S b h i f li



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 li d i l t



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


Outline



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



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


The X4 helico te



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


The X4 helicopter



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


4 fixed rotors with controlled rotationspeed s i 4 generated forces F i

F 1

F 2

F 3

F 4


The X4 helicopter



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


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


The X4 helicopter



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


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


The X4 helicopter



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



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


The X4 helicopter



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



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


The X4 helicopter



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


The X4 helicopter



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


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


The X4 helicopter



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


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


The X4 helicopter



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


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)


The X4 helicopter



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


The X4 helicopter



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,


The X4 helicopter



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


The X4 helicopter



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


The X4 helicopter



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


The X4 helicopter



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


The X4 helicopter



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


The X4 helicopter



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


The X4 helicopter



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


Outline



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


Outline



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



3 Stability



5 Observers


Linear systems with saturated inputsI f h l f h X ’ ( / )



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


Linear systems with saturated inputsI f i i h l f h X4’ (2/4)



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


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


Linear systems with saturated inputsI t f i t t ti th t l f th X4’ t (3/4)



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


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


Linear systems with saturated inputsSat atio a ca se i stabilit



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


Linear systems with saturated inputsKey idea of the anti windup scheme



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


Linear systems with saturated inputsPID controller with anti windup



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


Linear systems with saturated inputsGeneral controller with anti-windup



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


Linear systems with saturated inputsGeneral controller with anti-windup



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 )|


Linear systems with saturated inputsA short Lyapunov explanation of the behavior



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)


Linear systems with saturated inputsBack to the unstable case



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


Linear systems with saturated inputsImpact of input saturations on the control of the X4’s rotors (4/4)



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





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


Linearization of nonlinear systemsImpact of the load on the control of the X4’s rotors



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 ?





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


Linearization of nonlinear systemsLinearization at the origin



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


Linearization of nonlinear systemsLinearization at a point



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


Linearization of nonlinear systemsSome properties of the linearization



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


Linearization of nonlinear systemsSome properties of the linearization



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)





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





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


1 IntroductionLinear versus nonlinearTh X4 l



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


3 Stability



5 Observers


Towards gain schedulingImpact of a change in the steady state speed of the X4’s rotors



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


Towards gain schedulingLocal linearization may be inadequate for large excursion



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 li

Towards gain schedulingLocal linearization may be inadequate for large excursion

Build a linear feedback law that places the poles at 1



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 li

Towards gain scheduling

For some systems, the radius of attraction of a



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


Nonlinear

Towards gain scheduling

Gain scheduling uses the idea of using a collection of



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


Nonlinear

Gain schedulingTake a nonlinear system



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


Nonlinear

Gain schedulingThe controller is then obtained by changing s in the apriori defined collection of points



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


Nonlinear

Gain schedulingDrawbacks of gain scheduling :

Con ergence onl if s aries slo l



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.


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



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


Nonlinear

Stability




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


3 Stability



5 Observers


Nonlinear

StabilityConsider the autonomous nonlinear system:



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


NonlinearC

StabilityGraphical interpretation



Control

N. Marchand

References

Outline

Introduction

Linear/Nonlinear

The X4 example

Linear


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 ?


NonlinearC l

StabilityUsing the linearization



Control

N. Marchand

References

Outline

Introduction

Linear/Nonlinear

The X4 example

Linear


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


NonlinearC t l

StabilityLyapunov theory: Lyapunov functions



Control

N. Marchand

References

Outline

Introduction

Linear/Nonlinear

The X4 example

Linear


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


NonlinearControl

StabilityLyapunov theory: Lyapunov theorem



Control

N. Marchand

References

Outline

Introduction

Linear/Nonlinear

The X4 example

Linear


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.


NonlinearControl

StabilityLyapunov theorem: a graphical interpretation



Control

N. Marchand

References

Outline

Introduction

Linear/Nonlinear

The X4 example

Linear


Linearization

Gain scheduling

Stability

Nonlinearapproaches

CLFSliding mode

Geometriccontrol

Recursivetechniques

X4 stabilization

Observers


NonlinearControl

StabilityStability and robustness



Control

N. Marchand

References

Outline

Introduction

Linear/Nonlinear

The X4 example

Linear


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


NonlinearControl

Outline




Control

N. Marchand

References

Outline

Introduction

Linear/Nonlinear

The X4 example

Linear


Linearization

Gain scheduling

Stability

Nonlinearapproaches

CLFSliding mode

Geometriccontrol

Recursivetechniques

X4 stabilization

Observers

The X4 example


3 Stability4 Nonlinear control methods

Control Lyapunov functionsSliding mode control


5 Observers


NonlinearControl

The X4 helicopterThe control problems



N. Marchand

References

Outline

Introduction

Linear/Nonlinear

The X4 example

Linear


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




N. Marchand

References

Outline

Introduction

Linear/Nonlinear

The X4 example

Linear


Linearization

Gain scheduling

Stability

Nonlinearapproaches

CLF

Sliding mode

Geometriccontrol

Recursivetechniques

X4 stabilization

Observers

The X4 example



Control Lyapunov functionsSliding mode control


5 Observers


NonlinearControl

Control Lyapunov functionsCharacterizing property (1/3)

Control Lyapunov functions (CLF) where introduced in the 80’s



N. Marchand

References

Outline

Introduction

Linear/Nonlinear

The X4 example

Linear


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


NonlinearControl


Control Lyapunov Functions: a tool for stabilization of



N. Marchand

References

Outline

Introduction

Linear/Nonlinear

The X4 example

Linear


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


V (x ) = 3

2x 21 + x 1x 2 + x 22


V (x (t ), u (t )) = ∇V (x ).f (x , u )

= −x 21 + x 1x 2 + x 22 + (x 1 + 2x 2)u


NonlinearControl


Control Lyapunov Functions: a tool for stabilization of



N. Marchand

References

Outline

Introduction

Linear/Nonlinear

The X4 example

Linear


Linearization

Gain scheduling

Stability

Nonlinearapproaches

CLF

Sliding mode

Geometriccontrol

Recursivetechniques

X4 stabilization

Observers

controlled systems


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


NonlinearControl

Control Lyapunov functionsFormal definitions

Definition: Lyapunov function



N. Marchand

References

Outline

Introduction

Linear/Nonlinear

The X4 example

Linear


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


NonlinearControl

Control Lyapunov functionsFormal definitions

Definition: Control Lyapunov Function



N. Marchand

References

Outline

Introduction

Linear/Nonlinear

The X4 example

Linear


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


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



N. Marchand

References

Outline

Introduction

Linear/Nonlinear

The X4 example

Linear


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


Theorem: Sontag’s universal formula (1989)

Consider a control affine system x = g0(x) + m ui gi (x) with f smooth and



N. Marchand

References

Outline

Introduction

Linear/Nonlinear

The X4 example

Linear


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


Definition: Small control property

A CLF V i id i fi h ll l if f



N. Marchand

References

Outline

Introduction

Linear/Nonlinear

The X4 example

Linear


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



N. Marchand

References

Outline

Introduction

Linear/Nonlinear

The X4 example

Linear


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



N. Marchand

References

Outline

Introduction

Linear/Nonlinear

The X4 example

Linear


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)



NonlinearControl

N M h d

Other structural toolsPassivity

Definition: Passivity



N. Marchand

References

Outline

Introduction

Linear/Nonlinear

The X4 example

Linear


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



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



N. Marchand

References

Outline

Introduction

Linear/Nonlinear

The X4 example

Linear


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



NonlinearControl

N Marchand

Outline




N. Marchand

References

Outline

Introduction

Linear/Nonlinear

The X4 example

Linear


Linearization

Gain scheduling

Stability

Nonlinearapproaches

CLF


Recursivetechniques

X4 stabilization

Observers

The X4 example



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



NonlinearControl

N Marchand

Sliding mode controlAn introducing example

Making V decrease as fast as possible is equivalent to minimizing V :

u = Argmin V



N. Marchand

References

Outline

Introduction

Linear/Nonlinear

The X4 example

Linear


Linearization

Gain scheduling

Stability

Nonlinearapproaches

CLF


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


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:



References

Outline

Introduction

Linear/Nonlinear

The X4 example

Linear


Linearization

Gain scheduling

Stability

Nonlinearapproaches

CLF


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


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:



References

Outline

Introduction

Linear/Nonlinear

The X4 example

Linear


Linearization

Gain scheduling

Stability

Nonlinearapproaches

CLF


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


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



References

Outline

Introduction

Linear/Nonlinear

The X4 example

Linear


Linearization

Gain scheduling

Stability

Nonlinearapproaches

CLF


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:



References

Outline

Introduction

Linear/Nonlinear

The X4 example

Linear


Linearization

Gain scheduling

Stability

Nonlinearapproaches

CLF


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)


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



References

Outline

Introduction

Linear/Nonlinear

The X4 example

Linear


Linearization

Gain scheduling

Stability

Nonlinearapproaches

CLF


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 −


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:



References

Outline

Introduction

Linear/Nonlinear

The X4 example

Linear


Linearization

Gain scheduling

Stability

Nonlinearapproaches

CLF


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


NonlinearControl

N. Marchand

Sliding mode controlSliding mode control for some affine systems: proof of convergence

Proof:




References

Outline

Introduction

Linear/Nonlinear

The X4 example

Linear


Linearization

Gain scheduling

Stability

Nonlinearapproaches

CLF


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


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



References

Outline

Introduction

Linear/Nonlinear

The X4 example

Linear


Linearization

Gain scheduling

Stability

Nonlinearapproaches

CLF


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


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.



References

Outline

Introduction

Linear/Nonlinear

The X4 example

Linear


Linearization

Gain scheduling

Stability

Nonlinearapproaches

CLF


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



References

Outline

Introduction

Linear/Nonlinear

The X4 example

Linearapproaches

Antiwindup

Linearization

Gain scheduling

Stability

Nonlinearapproaches

CLF


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


NonlinearControl

N. Marchand

Outline1 Introduction




References

Outline

Introduction

Linear/Nonlinear

The X4 example

Linearapproaches

Antiwindup

Linearization

Gain scheduling

Stability

Nonlinearapproaches

CLF


Recursivetechniques

X4 stabilization

Observers



Control Lyapunov functionsSliding mode controlState and output linearization


5 Observers


NonlinearControl

N. Marchand

State and output linearizationIntroducing examples

Consider nonlinear the system

x 1 = −2x 1 + ax 2 + sin(x 1)



References

Outline

Introduction

Linear/Nonlinear

The X4 example

Linearapproaches

Antiwindup

Linearization

Gain scheduling

Stability

Nonlinearapproaches

CLF


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))


NonlinearControl

N. Marchand


The system then becomes linear:

z =−2 1

z +0

v



References

Outline

Introduction

Linear/Nonlinear

The X4 example

Linearapproaches

Antiwindup

Linearization

Gain scheduling

Stability

Nonlinearapproaches

CLF


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)]


NonlinearControl

N. Marchand


Consider nonlinear the system

x 1 = sin(x 2) + (x 2 + 1)x 35



References

Outline

Introduction

Linear/Nonlinear

The X4 example

Linearapproaches

Antiwindup

Linearization

Gain scheduling

Stability

Nonlinearapproaches

CLF


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


NonlinearControl

N. Marchand


Potential problem if x 2 = −1

Here again, take the linear feedback v = y + 2y that



References

Outline

Introduction

Linear/Nonlinear

The X4 example

Linearapproaches

Antiwindup

Linearization

Gain scheduling

Stability

Nonlinearapproaches

CLF


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 )


NonlinearControl

N. Marchand


One has to check that the internal dynamics is stable. For this, weassume that y = y = 0 (which happens asymptotically):

→ →



References

Outline

Introduction

Linear/Nonlinear

The X4 example

Linearapproaches

Antiwindup

Linearization

Gain scheduling

Stability

Nonlinearapproaches

CLF


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


NonlinearControl

N. Marchand

State and output linearizationIntroduction

Consider again nonlinear systems affine in the control:

x = f (x ) + g (x )u



References

Outline

Introduction

Linear/Nonlinear

The X4 example

Linearapproaches

Antiwindup

Linearization

Gain scheduling

Stability

Nonlinearapproaches

CLF


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


NonlinearControl

N. Marchand

State and output linearizationThe SISO case

Take x = f (x ) + g (x )u

y = h(x )



References

Outline

Introduction

Linear/Nonlinear

The X4 example

Linearapproaches

Antiwindup

Linearization

Gain scheduling

Stability

Nonlinearapproaches

CLF


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


NonlinearControl

N. Marchand

State and output linearizationThe SISO case

If Lg Lf h(x ) = 0, the control is

u = 1

(−L2f h(x ) + v )



References

Outline

Introduction

Linear/Nonlinear

The X4 example

Linearapproaches

Antiwindup

Linearization

Gain scheduling

Stability

Nonlinearapproaches

CLF


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


NonlinearControl

N. Marchand

State and output linearizationThe MIMO case

Takex = f (x ) + g 1(x )u 1 + g 2(x )u 2



References

Outline

Introduction

Linear/Nonlinear

The X4 example

Linearapproaches

Antiwindup

Linearization

Gain scheduling

Stability

Nonlinearapproaches

CLF


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


NonlinearControl

N. Marchand


One has:

y 1 = Lf h1(x ) + Lg 1 h1(x )

=0

u 1 + Lg 2 h1(x )

=0

u 2



References

Outline

Introduction

Linear/Nonlinear

The X4 example

Linearapproaches

Antiwindup

Linearization

Gain scheduling

Stability

Nonlinearapproaches

CLF


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


NonlinearControl

N. Marchand


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



References

Outline

Introduction

Linear/Nonlinear

The X4 example

Linearapproaches

Antiwindup

Linearization

Gain scheduling

Stability

Nonlinearapproaches

CLF


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


NonlinearControl

N. Marchand

State and output linearizationA block diagram representation



References

Outline

Introduction

Linear/Nonlinear

The X4 example

Linearapproaches

Antiwindup

Linearization

Gain scheduling

Stability

Nonlinearapproaches

CLF


Recursivetechniques

X4 stabilization

Observers

Linearization techniques can be interpreted as:

Nonlinear

system

Linear

control

Linearizing

feedbackLinear loop


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



References

Outline

Introduction

Linear/Nonlinear

The X4 example

Linearapproaches

Antiwindup

Linearization

Gain scheduling

Stability

Nonlinearapproaches

CLF


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


NonlinearControl

N. Marchand

R f



2 L l h d f l



References

Outline

Introduction

Linear/Nonlinear

The X4 example

Linearapproaches

Antiwindup

Linearization

Gain scheduling

Stability

Nonlinearapproaches

CLF


Recursivetechniques

X4 stabilization

Observers


3 Stability



5 Observers


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



References

Outline

Introduction

Linear/Nonlinear

The X4 example

Linearapproaches

Antiwindup

Linearization

Gain scheduling

Stability

Nonlinearapproaches

CLF


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)


NonlinearControl

N. Marchand

R f

BacksteppingMain results

Th h i l b li d



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


NonlinearControl

N. Marchand

References

ForwardingStrict-feedforward systems

Consider the class of strict-feedforward systems :



References

Outline

Introduction

Linear/Nonlinear

The X4 example

Linearapproaches


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


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)



References

Outline

Introduction

Linear/Nonlinear

The X4 example

Linearapproaches


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


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



References

Outline

Introduction

Linear/Nonlinear

The X4 example

Linearapproaches


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


NonlinearControl

N. Marchand

References

Backstepping/ForwardingAn interpretation of the names



References

Outline

Introduction

Linear/Nonlinear

The X4 example

Linearapproaches


Gain scheduling

Stability

Nonlinearapproaches

CLF

Sliding mode

Geometriccontrol

Recursivetechniques

X4 stabilization

Observers


NonlinearControl

N. Marchand

References



2 Linear control methods for nonlinear systems



References

Outline

Introduction

Linear/Nonlinear

The X4 example

Linearapproaches


Gain scheduling

Stability

Nonlinearapproaches

CLF

Sliding mode

Geometriccontrol

Recursivetechniques

X4 stabilization

Observers


3 Stability



5 Observers


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



Outline

Introduction

Linear/Nonlinear

The X4 example

Linearapproaches


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


NonlinearControl

N. Marchand

References

The X4 helicopterA two steps approach

s k 2m s k gearbox κ |s | s + k m sat (U )



Outline

Introduction

Linear/Nonlinear

The X4 example

Linearapproaches


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


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):



Outline

Introduction

Linear/Nonlinear

The X4 example

Linearapproaches


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


NonlinearControl

N. Marchand

References

The X4 helicopter: attitude stabilizationRecall of backstepping main result


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)



Outline

Introduction

Linear/Nonlinear

The X4 example

Linearapproaches


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)


NonlinearControl

N. Marchand

References


Define first the tracking error:

z1 = x1 − x1d = φ − φd



Outline

Introduction

Linear/Nonlinear

The X4 example

Linearapproaches


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 .


NonlinearControl

N. Marchand

References


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



Outline

Introduction

Linear/Nonlinear

The X4 example

Linearapproaches


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


NonlinearControl

N. Marchand

ReferencesO li


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



Outline

Introduction

Linear/Nonlinear

The X4 example

Linearapproaches


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:


...

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


NonlinearControl

N. Marchand

ReferencesO tli


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



Outline

Introduction

Linear/Nonlinear

The X4 example

Linearapproaches


Gain scheduling

Stability

Nonlinearapproaches

CLF

Sliding mode

Geometriccontrol

Recursivetechniques

X4 stabilization

Observers


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


NonlinearControl

N. Marchand

ReferencesOutline


Trying to write

x 2 − x 1d = −α 1z 1

ideal case

+? (x 2 − x 1d + α 1z 1)

difference between realand ideal cases



Outline

Introduction

Linear/Nonlinear

The X4 example

Linearapproaches


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


NonlinearControl

N. Marchand

ReferencesOutline


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



Outline

Introduction

Linear/Nonlinear

The X4 example

Linearapproaches


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


NonlinearControl

N. Marchand

ReferencesOutline


Simulations: apply successive steps of 45 in roll, pitch andyaw

60



Outline

Introduction

Linear/Nonlinear

The X4 example

Linearapproaches


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


NonlinearControl

N. Marchand

References

Outline


Adjusting the α ’s: with α i = 0.2

60 1



Outline

Introduction

Linear/Nonlinear

The X4 example

Linearapproaches


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


NonlinearControl

N. Marchand

References

Outline


Adjusting the α ’s: with α i = 2

40

60 5



Introduction

Linear/Nonlinear

The X4 example

Linearapproaches


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


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


Seems to be a good choice




NonlinearControl

N. Marchand

References

Outline




Introduction

Linear/Nonlinear

The X4 example

Linearapproaches


Gain scheduling

Stability

Nonlinearapproaches

CLF

Sliding mode

Geometriccontrol

Recursivetechniques

X4 stabilization

ObserversSuccessive steps of 45 in roll, pitch and yaw with α i = 2


NonlinearControl

N. Marchand

References

Outline


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:



Introduction

Linear/Nonlinear

The X4 example

Linearapproaches


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


NonlinearControl

N. Marchand

References

Outline




Introduction

Linear/Nonlinear

The X4 example

Linearapproaches


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


NonlinearControl

N. Marchand

References

Outline


Adjusting the α ’s: with α 1,...,4 = 0.4 and α 5,6 = 8

40

60 1



Introduction

Linear/Nonlinear

The X4 example

Linearapproaches


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


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


The controls are still too large


NonlinearControl

N. Marchand

References

Outline


Adjusting the α ’s: with α 1,...,4 = 0.2 and α 5,6 = 8

40

60 1



Introduction

Linear/Nonlinear

The X4 example

Linearapproaches


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


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


Oscillations are now present


NonlinearControl

N. Marchand

References

Outline


Taking α 1,...,4 = 0.2 and α 5,6 = 8 and applying it on thesystem with saturations, it gives:

60 0.5



Introduction

Linear/Nonlinear

The X4 example

Linearapproaches


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


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


Seems to work


NonlinearControl

N. Marchand

References

Outline




Introduction

Linear/Nonlinear

The X4 example

Linearapproaches


Gain scheduling

Stability

Nonlinearapproaches

CLF

Sliding mode

Geometriccontrol

Recursivetechniques

X4 stabilization

ObserversThe control law with the saturated system


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:



Introduction

Linear/Nonlinear

The X4 example

Linearapproaches


Gain scheduling

Stability

Nonlinearapproaches

CLF

Sliding mode

Geometriccontrol

Recursivetechniques

X4 stabilization

Observers


NonlinearControl

N. Marchand

References

Outline


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:



Introduction

Linear/Nonlinear

The X4 example

Linearapproaches


Gain scheduling

Stability

Nonlinearapproaches

CLF

Sliding mode

Geometriccontrol

Recursivetechniques

X4 stabilization

Observers


NonlinearControl

N. Marchand

References

Outline


This problem is referred in the literature as the PVTOL

aircraft stabilization problem (Planar Vertical Take Off and Landing aircraft)



Introduction

Linear/Nonlinear

The X4 example

Linearapproaches


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.


NonlinearControl

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



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)


NonlinearControl

N. Marchand

References

Outline

I t d ti


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



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


NonlinearControl

N. Marchand

References

Outline

I t od ctio


Remain two problems:Prove that



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 )


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)



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


NonlinearControl

N. Marchand

References

Outline

Introduction


First compute

n

i =1

(λ + εi ) = p 0 + p 1λ +

· · ·+ p n−1λn−1 + λn

Apply the coordinate change



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


NonlinearControl

N. Marchand

References

Outline

Introduction


Theorem: An efficient saturated controlLet ε(n) denote the only root of εn − 2ε + 1 = 0 in ]0, 1[.



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.


NonlinearControl

N. Marchand

References

Outline

Introduction


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



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


NonlinearControl

N. Marchand

References

Outline

Introduction


Remain two problems:

Prove that

r 1 = −εσ(z 2) − ε2σ(εz 1 + z 2)

r2 = −εσ(z4) − ε2σ(εz3 + z4)



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


NonlinearControl

N. Marchand

References

Outline

Introduction

The X4 helicopterSecond step: position control



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)


NonlinearControl

N. Marchand

References

Outline

Introduction

The X4 helicopterSecond step: position control



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)


NonlinearControl

N. Marchand

References

Outline

Introduction

Outline


2 Linear control methods for nonlinear systemsAntiwindupLinearization



Linear/Nonlinear

The X4 example

Linearapproaches

Antiwindup

Linearization

Gain scheduling

Stability

Nonlinearapproaches

CLF

Sliding mode

Geometriccontrol

Recursivetechniques

X4 stabilization

Observers

Gain scheduling

3 Stability



5 Observers


NonlinearControl

N. Marchand

References

Outline

Introduction

Nonlinear observers

y u y d

x

Observer

k (x )

Controller



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


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



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 )


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)



Linear/Nonlinear

The X4 example

Linearapproaches

Antiwindup


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




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)



Linear/Nonlinear

The X4 example

Linearapproaches

Antiwindup


Stability

Nonlinearapproaches

CLF

Sliding mode


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


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



Linear/Nonlinear

The X4 example

Linearapproaches

Antiwindup


Stability

Nonlinearapproaches

CLF

Sliding mode


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




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



/

The X4 example

Linearapproaches

Antiwindup


Stability

Nonlinearapproaches

CLF

Sliding mode


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)


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:



/

The X4 example

Linearapproaches

Antiwindup


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 ))


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))



The X4 example

Linearapproaches

Antiwindup


Stability

Nonlinearapproaches

CLF

Sliding mode

Geometric


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


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



The X4 example

Linearapproaches

Antiwindup


Stability

Nonlinearapproaches

CLF

Sliding mode

Geometric


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


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:



The X4 example

Linearapproaches

Antiwindup


Stability

Nonlinearapproaches

CLF

Sliding mode

Geometric


X4 stabilization

Observers

x = Arg minx (·)

t 1

t 0

h(x (τ )) − y (τ )) d τ

Sliding mode observers




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



The X4 example

Linearapproaches

Antiwindup


Stability

Nonlinearapproaches

CLF

Sliding mode

Geometric


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


NonlinearControl

N. Marchand

References

Outline

Introduction

Linear/Nonlinear

The X4 example


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



The X4 example

Linearapproaches

Antiwindup


Stability

Nonlinearapproaches

CLF

Sliding mode

Geometric


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


NonlinearControl

N. Marchand

References

Outline

Introduction

Linear/Nonlinear

The X4 example


Block diagram of the observer



The X4 example

Linearapproaches

Antiwindup


Stability

Nonlinearapproaches

CLF

Sliding mode

Geometric


X4 stabilization

Observers


nonlinear_pspi.pdf

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