Variations on a Flat Theme - lix.polytechnique.fr · Variations on a Flat Theme ... (Michel Fliess, Pierre Rouchon, Philippe Martin and myself) in 1991. About 400 citations of the

Variations on a Flat Theme

Jean LEVINE

CAS,Ecole des Mines de Paris

Dedicated to Michel Fliess for his 60th birthday

Jean LEVINE Variations on a Flat Theme

IntroductionContents

On Flatness-based DesignOn Necessary and Sufficient Conditions

Exogeneous Feedback LinearizationConclusion

Some Historical Considerations

Differential Flatness has been introduced by the gang of the four(Michel Fliess, Pierre Rouchon, Philippe Martin and myself) in 1991.

About 400 citations of the 1995 Int. J. Control paper (according toScholar Google).

Major contributions to the development of this concept have also beenmade by Richard Murray, Jean-Baptiste Pomet, Joachim Rudolph,Hebertt Sira-Ramirez and many others.

Happy birthday, Michel!Jean LEVINE Variations on a Flat Theme




A tribute to the Gang of the 4

Celebration of Flatness 10th anniversary in Mexico, November 2003.





Contents

1 Some Historical Considerations

2 On Flatness-based Design

3 On Necessary and Sufficient ConditionsThe Formalism of Manifolds of Jets of Infinite OrderPolynomial MatricesInterpretation in terms of Exact Sequences of ModulesFlatness Necessary and Sufficient ConditionsNecessary and Sufficient Conditions and Generalized MovingFrames

4 Exogeneous Feedback Linearization

5 Conclusions and Perspectives





Contents










Contents










Contents










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Contents










Recalls on Differentially Flat Systems

Definition

The nonlinear systemx = f (x,u), with x = (x1, . . . , xn): stateandu = (u1, . . . ,um): control, m≤ n.is (differentially) flatif and only if there existsy = (y1, . . . , ym) suchthat:

y and its successive derivativesy, y, . . . , are independent,

y = h(x,u, u, . . . ,u(r)) (generalized output),

Conversely,x andu can be expressed as:x = ϕ(y, y, . . . , y(α)), u = ψ(y, y, . . . , y(α+1))

with ϕ ≡ f (ϕ,ψ).The vectory is called aflat output .





Main advantages of Flatness

1 Direct open-loop trajectory computation, without integration noroptimization.

2 Local stabilization of any reference trajectoryusing theequivalence between the system trajectories and those of

y(α+1) = v.

“Flatness-Based Control” = Trajectory Planning+ Trajectory Tracking.

Alternative approach toPredictive Control(see e.g. Fliess and Marquez 2001, Delaleau and Hagenmeyer 2006,Devos and Levine 2006).





Consequence on motion planning

To every curvet 7→ y(t) enoughdifferentiable, there corresponds atrajectory

t 7→(

x(t)u(t)

)=(

ϕ(y(t), y(t), . . . , y(α)(t))ψ(y(t), y(t), . . . , y(α+1)(t))

)that identically satisfies the systemequations.

x = f (x ,x ,u)

y(α+1)+1) = v

Lie-BLie-Bäcklundklund

t → (x (t), u(t))))

t → (y(t), . . . , y(α+1)+1)(t))))

•

(ϕ,ψ)(ϕ,ψ)





Example: Linear Motor with Auxiliary Masses (I)Single Mass Case

Model:

Mx = F − k(x− z)− r(x− z)mz = k(x− z) + r(x− z)

Aims:

Rest-to-rest fast and high precisiondisplacements.

Measurements:

x andx measured,

z not measured.

mass

flexible beam

bumper

linear motorrail

In collaboration with Micro-Controle.





Flat output:

y =r2

mkx +

(1− r2

mk

)z− r

kz

x = y +rk

y +mk

y, z = y +rk

y

F = (M + m)(

y +rk

y(3) +Mm

(M + m)ky(4))





Videos

mass

flexible beam

bumper

linear motorrail

Mass=disturbance Input filtering

Flatness-based





Example: Linear Motor with Auxiliary Masses (II)The Case of Two Masses

Model:

Mx = F − k(x− z)− r(x− z)−k′(x− z′)− r ′(x− z′)

mz = k(x− z) + r(x− z)m′z′ = k′(x− z′) + r ′(x− z′)

linear motor

masses

bumpers

flexible beams

rail

In collaboration with Micro-Controle.





Flatness:

x = y +(

rk

+r ′

k′

)y +

(mk

+m′

k′+

rr ′

kk′

)y

+(

mr′ + rm′

kk′

)y(3) +

mm′

kk′y(4)

z = y +(

rk

+r ′

k′

)y +

(m′

k′+

rr ′

kk′

)y +

rm′

kk′y(3)

z′ = y +(

rk

+r ′

k′

)y +

(mk

+rr ′

kk′

)y +

mr′

kk′y(3)

F = My + M

(rk

+r ′

k′

)y(3) +

(mk

M′ + Mm′

k′+ M

rr ′

kk′

)y(4)

+(

mr′

kk′M′ + M

rm′

kk′

)y(5) +

Mmm′

kk′y(6)

with M = (M + m+ m′), M = M + m andM′ = M + m′.





Videos

linear motor

masses

bumpers

flexible beams

rail

Masses=disturbance Input filtering

Flatness-based





Combined trajectory planning and output feedbackThe US Navy Crane Example (B. Kiss, J.L., P. Mullhaupt 2000)





Example of Infinite Dimensional Flat System: Riser ToWell-head Connection (R. Sabri et al., 2003)

Reduced scale set-up(in collaboration with IFP)

Motorsflexible

riser

synchronized

digital

cameras

actuatorsmotors

simulating

the wave

excitation

well-head

water tank





PDE system: flexible riser displacements in the water

r∂2u∂t2

= EA∂2u∂s2 + (EI − h0Te)

∂3w∂s3

∂2u∂s2 + Fτ

ra∂2w∂t2

= − (EI − h0Te)∂4w∂s4 + EA

(∂u∂s∂2w∂s2 + h0

(∂2u∂s2

∂3w∂s3 +

∂u∂s∂4w∂s4

))+h0rs

(s∂4w∂s4 +

∂3w∂s3

)+ (Te + rss)

∂2w∂s2 + rs

∂w∂s

+ Fν

Boundary conditionsw(L, t) = xA(t)

EI∂2w∂s2 (L) = 0

u(L, t) = zA(t)− z0

FτFν

s

(s-ds)p

φ

pipe

ST

u(s,t)

w(s,t)





Flatness-based control using a modal approximation of the PDEmodel (2 first horizontal modes and 0th vertical mode)





Jets of Infinite OrderPolynomial MatricesExact SequencesFlatness NSCFlatness and Generalized Moving Frames

Contents











References: Sufficient Conditions

Input affine systems:1 input: equivalence between static and dynamic feedbacklinearization(Charlet, Levine and Marino 89, 91);n− 1 inputs + [first-order controllability]: always flat(Charlet,Levine and Marino 89);2 inputs and 4 states:1-flatness(Pomet 97).

Driftless systems:2 inputs: rank conditions on the flag of codistributions(Martinand Rouchon 94) ;n−2 inputs: always flat if controllable(Martin and Rouchon 95).

Non affine in the input case:n− 1 inputs (Martin 93).






References: Necessary Conditions

Ruled manifold (Sluis 93, Rouchon 94).






References: General Formulations

Find output functions ψ1, . . . , ψm and multi-integersl = (l1, . . . , lm) and r = (r1, . . . , rm), y = ψ(x,u, u, . . . ,u(l))(1),such that the differential system

dxi ∧ dψ1 ∧ . . . ∧ dψ(r1)1 ∧ . . . ∧ dψm∧ . . . ∧ dψ(rm)

m = 0i = 1, . . . ,n

duj ∧ dψ1 ∧ . . . ∧ dψ(r1+1)1 ∧ . . . ∧ dψm∧ . . . ∧ dψ(rm+1)

m = 0j = 1, . . . ,m

dψ1 ∧ . . . ∧ dψ(r1)1 ∧ . . . ∧ dψm∧ . . . ∧ dψ(rm)

m 6= 0is integrable (Martin 91, Pereira da Silva 99).Find an integrable basis of the tangent module

non exact Brunovsky forms(Pomet, Moog and Aranda 92,Chetverikov 2000),using prolongations(Franch 99).

1notation:u(l) = (u(l1)1 , . . . , u(lm)

m )Jean LEVINE Variations on a Flat Theme





System Representation

Consider the systemx = f (x,u)

with x ∈ X, dimX = n, u ∈ Rm andf smooth vector field onX

satisfyingrank(

∂f∂u

)= m.

It is locally equivalent to the underdetermined implicit system:

F(x, x) = 0

with x ∈ X, dimX = n, rank(

∂F∂x

)= n−m.

Remark

This implicit representation is invariant by endogeneous dynamicfeedback.






The Formalism of Manifolds of Jets of Infinite Order

We embedX in the manifoldX = X× Rn∞ of jets of infinite order

(Krasil’shchik, Lychagin, Vinogradov, 1986) with coordinates

xdef= (x, x, x, x(3), . . .)

endowed with thetrivial Cartan vector field

τX =n∑

i=1

∑j≥0

x(j+1)i

∂

∂x(j)i

.

We also noteLτXϕ =

∑ni=1

∑j≥0 x(j+1)

i∂ϕ

∂x(j)i

= dϕdt the Lie derivative

of ϕ alongτX andLkτXϕ its kth iterate. We have:

x(k)i = dkxi

dtk = LkτX

xi , i = 1, . . . ,n, k ≥ 0.

We say thatϕ : X → R is continuous(resp.differentiable) if ϕdepends only on a finite (but otherwise arbitrary) number of variablesand is continuous (resp. differentiable) with respect to these variables.






A regular implicit control system is a triple(X, τX,F) withX = X×Rn

∞, τX its trivial Cartan field, andF ∈ C∞(TX; Rn−m) withrank

(∂F∂x

)= n−m in an open subset of TX.

SetX0 = x ∈ X|LkτX

F(x) = 0, ∀k ≥ 0.

Definition (flatness)

The system(X, τX,F) is flat at (x0, y0) ∈ X0 × Rm∞ if and only if

there exist coordinatesy of Rm and a locally smooth mappingψ : X → Rm such thaty = ψ(x), with y0 = ψ(x0);there exists a locally smooth mappingϕ : Rm

∞ → X withx0 = ϕ(y0), such thatx = ϕ(y, y, . . .) satisfiesLk

τXF(x) = 0 for

every smooth trajectoryt 7→ y(t) and everyk ≥ 0.

Shortly, the system(X, τX,F) is flat iff it is Lie-Backlund equivalentat (x0, y0) to thetrivial implicit system(Rm

∞, τRm∞ ,0).






Denote

dF =∂F∂x

dx+∂F∂x

dx

and

P(F) =(∂F∂x

+∂F∂x

ddt

).

ThusdF = P(F)dx.

Theorem

The system(X, τX,F) is flat at(x0, y0) ∈ X0 × Rm∞ if and only if

there exists a local smooth invertible mappingΦ from Rm∞ to X0, with

smooth inverse, satisfyingΦ(y0) = x0, and such thatΦ∗dF = 0.

whereΦ∗dF is the (backward) image of the 1-formdF by Φ, namelyΦ∗dF = P(F)P(ϕ).






Polynomial matrices

Notations:

K: field of meromorphic functions from X to R.

K[ ddt]: (non commutative)principal ideal ring of polynomials

of ddt with coefficients inK.

Mp,q[ ddt]: module ofp× q matrices overK[ d

dt].

Up[ ddt]: group of p× p unimodular matrices (invertible with

inverse inMp,p[ ddt])






Right and Left Smith forms

Let M ∈ Mp,q[ ddt]. There existsU ∈ Uq[ d

dt] andV ∈ Up[ ddt] such that:

if p< q, V M U = (∆,0q−p) with ∆ = diagδ1, . . . , δp anddi

dividesdj for all 1≤ i ≤ j ≤ p;

if p> q, V M U =(

∆0p−q

)with ∆ = diagδ1, . . . , δq anddi

dividesdj for all 1≤ i ≤ j ≤ q.We say thatU ∈ R− Smith (M) andV ∈ L− Smith (M).

M is saidhyper-regular iff ∆ = I .

Remark

P(F) is hyper-regular iff the module associated to the variationalsystem of(X, τX,F) is free (thus controllable in the sense of Fliess(1990)).






Theorem

Assume thatP(F) is hyper-regular.1 Every solution ofP(F)P(ϕ) = 0 is given by

P(ϕ) = U

(0n−m,m

Im

)W = UW

with U ∈ R− Smith (P(F)), U = U

(0n−m,m

Im

), and

W ∈ Um[ ddt] arbitrary;

2 There existsQ ∈ L− Smith(

U)

andZ ∈ Um[ ddt] such that

QP(ϕ) =(

Im

0n−m,m

)Z

and the module generated by theK[ ddt] combinations of them

first lines ofQdx is free.






Interpretation in terms of Exact Sequences of Modules

We denote byΛ1n the module generated bydx1, . . . ,dxn,

by Λ1n+q the module generated bydx1, . . . ,dxn,dξ1, . . . ,dξq

and byΛ1m a suitable module of 1-forms of dimensionm.

The previous result corresponds to the construction of the exactsequences:

0 −→ Λ1m

U−→ ker(P(F)) ∩ Λ1n

P(F)−→ Λ1n−m

and

0 −→ Λ1m

U−→ ker(P(F)) ∩ Λ1n

Q−→ Λ1m −→ 0

whereQ =(

Im

0n−m,m

)Q.






Flatness Necessary and Sufficient Conditions

Denote byωi = Qidx the ith line ofQdx, i = 1, . . . ,m, andω = (ω1, . . . , ωm)T.DefineΩ : K[ d

dt] ideal generated by the 1-formsω1, . . . , ωm.We say thatΩ is strongly closedif there existsM ∈ Um[ d

dt] such that

d(Mω) = 0.






Theorem (JL, NOLCOS 2004)

Assume thatP(F) is hyper-regular. The three assertions areequivalent:

(i) The system(X, τX,F) is flat;

(ii) The idealΩ = span ω1, . . . , ωm is strongly closed;

(iii) There exists anm×m matrixµ whose entries arepolynomials of d

dt with coefficients inΛ1n, and a matrix

M ∈ Um[ ddt] such that the“generalized moving frame

structure equations” hold true:

dω = µ ∧ ω, dµ = µ ∧ µ, dM = −Mµ.

with ω = (ω1, . . . , ωm)T.






Sketch of proof of (ii)⇐⇒ (iii)

(ii) =⇒ (iii)If there existsM such thatd(Mω) = 0, we havedM ∧ ω + Mdω = 0,i.e. dω = −M−1dM ∧ ω. Settingµ = −−M−1dM gives

dω = µ ∧ ω and dM = −Mµ.Then, taking the exterior derivative of the latter expression,Mdµ−Mµ ∧ µ = M (dµ− µ ∧ µ) = −d2M = 0, ordµ = µ ∧ µ.

(iii) =⇒ (ii)Since there existM ∈ Um[ d

dt] andµ such thatdM = −Mµ, we haveµ = −M−1dM and thusdω = −M−1dM ∧ ω, orMdω + dM ∧ ω = d(Mω) = 0 which proves the strong closedness.






Necessary and Sufficient Conditions and GeneralizedMoving Frames

Assume that The system(X, τX,F) is flat.Define the manifoldΘ ⊂ Ω× Um[ d

dt], Θ 6= ∅:

Θ =

(ω,Mω)∣∣ω =

ω1...ωm

,Ω = span ω1, . . . ωm ,d(Mωω) = 0

Group of transformations:

T = f : Θ 7→ Θ | f (ω,Mω) = (ω′,Mω′) ω ω'

Mω Mω'

f

Θ

f ∈ T ⇐⇒ ∃T ∈ Um[ ddt] such thatω′ = Tω, Mω′ = MωT−1.

Thus:T identified withUm[ ddt] (unimodular group).






Recalls on Cartan’s Method of Moving Frames

Consider the Euclidean spaceRN, the set of orthogonal frames(p,e1, . . . ,eN) and the groupSO(N,R). The group right action sendsa given frame(O, δ1, . . . , δN) to a frame(p,e1, . . . ,eN) with

−→Op = aδ, e = Aδ, δ = (δ1, . . . , δN)T, e = (e1, . . . ,eN)T.

Infinitesimal motions satisfy

dp = ωe, de= µe

where the 1-formsω andµ are therelative components of themoving frame. It can be seen thatω andµ are characterized by thestructure equations

dω = µ ∧ ω, dµ = µ ∧ µ.






Back to Generalized Moving Frames for Flat Systems

The 1-formω and the polynomial valued 1-formµ may be seen as the(generalized) relative componentsof the generalized frames(ω,Mω) of Θ and

dω = µ ∧ ω, dµ = µ ∧ µ

are their(generalized) structure equations, with µ = −M−1ω dMω.

Remark

For non flat systems, we have:

dω = µ ∧ ω +$, d$ = µ ∧$, dµ = µ ∧ µ, dM = −Mµ

which suggests a (formal) characterization of flat submanifolds ofX

(see e.g. Chern, Chen and Lam).





Contents










Exogeneous Feedback Linearization

Consider the systemx = f (x,u) with rank(

∂f∂u

)= m in a suitable

neighborhood, and a1st order regular exogeneous dynamicfeedback: u = α(x, ξ, v), ξ = β(x, ξ, v) with ξ ∈ Rq, v ∈ Rm and

rank

(∂α∂v

∂β∂v

)= m.

The systemx = f (x,u) is saidfeedback linearizable by 1st orderregular exogeneous dynamic feedbackif and only if there existq ∈ Nfinite,α andβ satisfying the above rank condition, such that theclosed-loop systemx = f (x, α(x, ξ, v)), ξ = β(x, ξ, v) is flat.

Theorem

The systemx = f (x,u) is feedback linearizable by 1st orderregular exogeneous dynamic feedback if and only if it is flat.





Sketch of Proof

The closed-loop system admits the implicit representation

F(X, X) =(

F(x, x)G(x, ξ, x, ξ)

)= 0

with F(x, x) = 0 implicit representation ofx = f (x,u),dimF = n−m, dimG = q andX = (x, ξ)T, and its variational systemis

P(F)dX =(

Px(F) 0Px(G) Pξ(G)

)(dxdξ

)= 0

with Px(.) = ∂∂x + ∂

∂xddt andPξ(.) = ∂

∂ξ + ∂∂ξ

ddt.

As before, we denote byΛ1n the module generated bydx1, . . . ,dxn,

by Λ1n+q the module generated bydx1, . . . ,dxn,dξ1, . . . ,dξq

and byΛ1m a suitable module of 1-forms of dimensionm.





ConsiderU ∈ R− Smith(P(F)

), U = U

(0n−m,m

Im

),

Q ∈ L− Smith(

U)

andQ =(

Im

0n−m,m

)Q.

We have the followingexact sequence:

0−→ Λ1m

U−→ ker(P(F)) ∩ Λ1n+q

Q−→ Λ1m −→ 0

and there existU0 ∈ R− Smith (Px(F)), Q0 ∈ L− Smith(

U0

),

with U0 = U0

(0n−m,m

Im

), Q0 =

(Im

0n−m,m

)Q0, andT ∈ Um[ d

dt]

such that:

0 −→ Λ1m

U−→ ker(P(F)) ∩ Λ1n+q

Q−→ Λ1m −→ 0

Im ↓ πn ↓ T ↓

0 −→ Λ1m

U0−→ ker(Px(F)) ∩ Λ1n

Q0−→ Λ1m −→ 0.





Thus, since the closed-loop system is flat, there existsM ∈ Um[ ddt]

such thatd(MQdX) = 0, and sinceQ = T−1Q0πn, we have

d(M′Q0dx) = 0

with M′ = MT−1 ∈ Um[ ddt].

Therefore, the idealΩ0 generated by the lines ofQ0dx is stronglyclosed if the coefficients of the matrixM′ depend only onx (not onξ).This results from the fact thatdM′ ∧ Q0dx = −M′d(Q0dx) (bycontradiction). Q.E.D.





Contents










Conclusions and Perspectives

Numerous industrial applications:

asynchronous motors (Schneider-Electric)

mechatronics (Bosch)

automotive equipements (Valeo, Bosch, PSA,Siemens Automotive (IFAC Congress Applications Paper Prize2002 to Horn, Bamberger, Michau, Pindl))

underwater applications (IFP)

high-precision positionning (Micro-Controle)

magnetic bearings (Alcatel, Axomat GmbH)

chemical reactors (Total-Fina-Elf)

biotechnological processes (Ifremer)

. . .Jean LEVINE Variations on a Flat Theme




ReferencesI

R. Baron, J. Levine, and M. Mastail.Modelling and control of a fish extrusion process.In Proc. 1st IMACS/IFAC Conf. on Mathematical Modelling andSimulation in Agriculture and Bio-industries, Bruxelles, May1995.

B. Charlet, J. Levine, and R. Marino.On dynamic feedback linearization.Systems & Control Letters, 13:143–151, 1989.

B. Charlet, J. Levine, and R. Marino.Sufficient conditions for dynamic state feedback linearization.SIAM J. Control and Optimization, 29(1):38–57, 1991.





ReferencesII

I.K. Chatjigeorgiou, S.A. Mavrakos.An investigation of the non-linear transverse vibrations ofparametrically excited vertical marine risers and cables undertension.Proceedings of the 21st International Conference on OffshoreMechanics and Artic Engineering. Oslo, Norway, 2002.

A.A.R. Fehn, R. Rothfuß, and M. Zeitz.Flatness-based torque ripple free control of switched reluctanceservo machines.In R.M. Parkin, A. Al-Habaibeh, and M.R. Jackson, editors,ICOM 2003, International Conference on Mechatronics, pages209–214, 2003.





ReferencesIII

M. Fliess, J. Levine, Ph. Martin, P. Rouchon.On differentially flat nonlinear systems.Comptes Rendus des seances de l’Academie des Sciences I-315:619-624, 1992.

M. Fliess, J. Levine, Ph. Martin, P. Rouchon.Flatness and defect of nonlinear systems: Introductory theory andapplications.Internat. J. Control 61: 1327-1361, 1995.

M. Fliess, J. Levine, Ph. Martin, P. Rouchon.A Lie Backlund approach to equivalence and flatness of nonlinearsystems.IEEE Trans. Automat. Control 44 (5): 922-937, 1999.





ReferencesIV

M. Fliess, R. Marquez.Continuous-time linear predictive control and flatness: amodule-theoretic setting with examples.Internat. J. Control, 73: 606–623, 2001.

V. Hagenmeyer and E. Delaleau.Robustness analysis of exact feedforward linearization based ondifferential flatness.Automatica, 39:1941–1946, 2003.

J. Horn, J. Bamberger, P. Michau, and S. Pindl.Flatness-based clutch control for automated manualtransmissions.Control Engineering Practice, 11(12):1353–1359, 2003.





ReferencesV

F. Jadot, Ph. Martin, and P. Rouchon.Industrial sensorless control of induction motors.In A. Isidori, F. Lamnabhi-Lagarrigue, and W. Respondek,editors,Nonlinear Control in the Year 2000, volume 1: 535–544.Springer, 2000.

B. Kiss, J. Levine, and Ph. Mullhaupt.Modelling, flatness and simulation of a class of cranes.Periodica Polytechnica, Ser. El. Eng. 43(3):215–225, 1999.

B. Kiss, J. Levine, and Ph. Mullhaupt.Modeling and motion planning for a class of weight handlingequipment.J. Systems Science, 26, 4: 79–92, 2000.





ReferencesVI

B. Kiss, J. Levine, and Ph. Mullhaupt.A simple output feedback PD controller for nonlinear cranes.In Proc. of the 39th IEEE CDC 2000, Sydney, 2000.

B. Kiss, J. Levine, and Ph. Mullhaupt.Global stability without motion planning may be worse than localtracking.In Proc. ECC’01, Porto, 2001.

B. Kiss, J. Levine, and B. Lantos.On motion planning for robotic manipulation with permanentrolling contacts.Int. J. of Robotics Research, Vol. 21, 5-6, special issue“Non-holonomy on purpose”, 443–461, May 2002.





ReferencesVII

J. Levine, J. Lottin, J.C. Ponsart.A nonlinear approach to the control of magnetic bearings.IEEE Trans. on Control Systems Technology, 4, 5: 524–544,1996.

J. Levine.Are there new industrial perspectives in the control of mechanicalsystems?.in “Advances in Control (ECC99)”, P.M. Frank ed., 197–226,Springer, 1999.





ReferencesVIII

J. Levine, D.V. Nguyen.Flat output characterization for linear systems using polynomialmatrices.Systems& Control Letters 48: 69-75, 2003.

J. Levine.On the synchronization of a pair of independent windshieldwipers.IEEE Trans. on Control Systems Technology, Vol. 12, 5:787-795, 2004.

J. Levine.On necessary and sufficient conditions for differential flatness.Proc. of IFAC NOLCOS 2004 Conference, Stuttgart, 2004.





ReferencesIX

J. Levine, L. Praly, and E. Sedda.On the control of an electromagnetic actuator of valvepositionning on a camless engine.In Proc. AVEC 2004 Conference, Arnhem, 2004.

J. Levine and B. Remond.Flatness based control of an automatic clutch.In Proc. MTNS 2000, Perpignan, 2000.

J. von Lowis, J. Rudolph, J. Thiele, and F. Urban.Flatness-based trajectory tracking control of a rotating shaft.In 7th International Symposium on Magnetic Bearings, Zurich,pages 299–304, 2000.





ReferencesX

Ph. Martin.Contributiona l’ Etude des Systemes Differentiellement Plats.PhD thesis,Ecole des Mines de Paris, 1992.

Ph. Martin, R.M. Murray, P. Rouchon.Flat systems,Plenary Lectures and Minicourses, Proc. ECC 97, Brussels, G.Bastin and M. Gevers eds., 211–264, 1997.

N. Petit and P. Rouchon.Dynamics and solutions to some control problems for water-tanksystems.IEEE Trans. Automatic Control, 47(4):594–609, 2002.





ReferencesXI

N. Petit, P. Rouchon, J.-M. Boueilh, F. Guerin and P. Pinvidic.Control of an industrial polymerization reactor using flatness.J. of Process Control, 12: 659–665, 2002.

J.-B. Pomet.A differential geometric setting for dynamic equivalence anddynamic linearization.In B. Jakubczyk, W. Respondek, and T. Rzezuchowski, editors,Geometry in Nonlinear Control and Differential Inclusions, pages319–339. Banach Center Publications, Warsaw, 1993.





ReferencesXII

J.-B. Pomet.On dynamic feedback linearization of four-dimensional affinecontrol systems with two inputs.ESAIM-COCV, 1997.http://www.emath.fr/Maths/Cocv/Articles/articleEng.html.

M. Rathinam and R.M. Murray.Configuration flatness of Lagrangian systems underactuated byone control.SIAM J. Control and Optimization, 36(1):164–179, 1998.

R. Rothfuss, J. Rudolph, and M. Zeitz.Flatness based control of a nonlinear chemical reactor model.Automatica, 32:1433–1439, 1996.





ReferencesXIII

J. Rudolph.Flatness based control of distributed parameter systems.Shaker Verlag, Aachen, 2003.

J. Rudolph, J. Winkler, and F. Woittenek.Flatness based control of distributed parameter systems:examples and computer exercises from various technologicaldomains.Shaker Verlag, Aachen, 2003.

R. Sabri, J. Levine, F. Biolley, Y. Creff, C. Le Cunff and C. Putot.Connection of the riser to the well-head using active control.Proc. of the Deep Offshore Technology Conf., Marseille, Nov.2003.





ReferencesXIV

H. Sira-Ramirez and S. Agrawal.Differentially Flat Systems.Marcel Dekker, New York, 2004.

H.L. Trentelman.On flat systems behaviors and observable image representations.Systems & Control Letters, 19:43–45, 1992.





Flat output computation: linear motor with a singleauxiliary mass

Model (recall)

Mx = F − k(x− z)− r(x− z)mz = k(x− z) + r(x− z)

Settings = ddt:

(Ms2 + rs + k)x = (rs + k)z+ F(ms2 + rs + k)z = (rs + k)x

Definition polynomials:

x = Px(s)y, z = Pz(s)y, F = Q(s)y(ms2 + rs + k)Pz(s) = (rs + k)Px(s)

thusPx(s) = (ms2 + rs + k)P0, Pz(s) = (rs + k)P0.





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Variations on a Flat Theme - lix.polytechnique.fr · Variations on a Flat Theme ... (Michel Fliess, Pierre Rouchon, Philippe Martin and myself) in 1991. About 400 citations of the