[PPT] - Variations on a Flat Theme Jean L EVINE CAS, Ecole des Mines de PowerPoint Presentation

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Variations on a Flat Theme

Jean L´ EVINE

CAS, ´ Ecole des Mines de Paris

Dedicated to Michel Fliess for his 60th birthday

Jean L´ EVINE Variations on a Flat Theme

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Introduction Contents On Flatness-based Design On Necessary and Sufficient Conditions Exogeneous Feedback Linearization Conclusion

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

Happy birthday, Michel!

Jean L´ EVINE Variations on a Flat Theme

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Introduction Contents On Flatness-based Design On Necessary and Sufficient Conditions Exogeneous Feedback Linearization Conclusion

A tribute to the Gang of the 4

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

Jean L´ EVINE Variations on a Flat Theme

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Introduction Contents On Flatness-based Design On Necessary and Sufficient Conditions Exogeneous Feedback Linearization Conclusion

Recalls on Differentially Flat Systems

Definition The nonlinear system ˙ x = f(x, u), with x = (x1, . . . , xn): state and u = (u1, . . . , um): control, m ≤ n. is (differentially) flat if and only if there exists y = (y1, . . . , ym) such that: y and its successive derivatives ˙ y,¨ y, . . . , are independent, y = h(x, u, ˙ u, . . . , u(r)) (generalized output), Conversely, x and u can be expressed as: x = ϕ(y, ˙ y, . . . , y(α)), u = ψ(y, ˙ y, . . . , y(α+1)) with ˙ ϕ ≡ f(ϕ, ψ). The vector y is called a flat output.

Jean L´ EVINE Variations on a Flat Theme

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Introduction Contents On Flatness-based Design On Necessary and Sufficient Conditions Exogeneous Feedback Linearization Conclusion

Main advantages of Flatness

1

Direct open-loop trajectory computation, without integration nor

ptimization.

2

Local stabilization of any reference trajectory using the equivalence between the system trajectories and those of y(α+1) = v. “Flatness-Based Control” = Trajectory Planning + Trajectory Tracking. Alternative approach to Predictive Control (see e.g. Fliess and Marquez 2001, Delaleau and Hagenmeyer 2006, Devos and L´ evine 2006).

Jean L´ EVINE Variations on a Flat Theme

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Introduction Contents On Flatness-based Design On Necessary and Sufficient Conditions Exogeneous Feedback Linearization Conclusion

Consequence on motion planning To every curve t → y(t) enough differentiable, there corresponds a trajectory t → 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 system

equations.

x = f (x , x ,u) y(α+1)

+1) = v

Lie-B Lie-Bäcklund klund t → (x(t), u(t)) )) t → (y(t), . . . , y(α+1)

+1)(t))

))

(ϕ,ψ)

(ϕ,ψ) Jean L´ EVINE Variations on a Flat Theme

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Introduction Contents On Flatness-based Design On Necessary and Sufficient Conditions Exogeneous Feedback Linearization Conclusion

Example: Linear Motor with Auxiliary Masses (I)

Single Mass Case

Model: M¨ x = F − k(x − z) − r(˙ x − ˙ z) m¨ z = k(x − z) + r(˙ x − ˙ z) Aims: Rest-to-rest fast and high precision displacements. Measurements: x and ˙ x measured, z not measured.

mass flexible beam bumper linear motor rail

In collaboration with Micro-Contrˆ

le.

Jean L´ EVINE Variations on a Flat Theme

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Introduction Contents On Flatness-based Design On Necessary and Sufficient Conditions Exogeneous Feedback Linearization Conclusion

Flat output: y = r2 mkx +

1 − r2

mk

z − r

k ˙ z x = y + r k ˙ y + m k ¨ y, z = y + r k ˙ y F = (M + m)

¨

y + r ky(3) + Mm (M + m)ky(4)

Jean L´

EVINE Variations on a Flat Theme

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Introduction Contents On Flatness-based Design On Necessary and Sufficient Conditions Exogeneous Feedback Linearization Conclusion

Videos

mass flexible beam bumper linear motor rail Mass=disturbance Input filtering Flatness-based Jean L´ EVINE Variations on a Flat Theme

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Introduction Contents On Flatness-based Design On Necessary and Sufficient Conditions Exogeneous Feedback Linearization Conclusion

Example: Linear Motor with Auxiliary Masses (II)

The Case of Two Masses

Model: M¨ x = F − k(x − z) − r(˙ x − ˙ z) −k′(x − z′) − r′(˙ x − ˙ z′) m¨ z = 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-Contrˆ

le.

Jean L´ EVINE Variations on a Flat Theme

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Introduction Contents On Flatness-based Design On Necessary and Sufficient Conditions Exogeneous Feedback Linearization Conclusion

Flatness: x = y + r k + r′ k′

˙

y + m k + m′ k′ + rr′ kk′

¨

y + mr′ + rm′ kk′

y(3) + mm′

kk′ y(4) z = y + r k + r′ k′

˙

y + m′ k′ + rr′ kk′

¨

y + rm′ kk′ y(3) z′ = y + r k + r′ k′

˙

y + m k + rr′ kk′

¨

y + mr′ kk′ y(3) F = ˆ M¨ y + ˆ M r k + r′ k′

y(3) +

m k ¯ M′ + ¯ Mm′ k′ + ˆ M rr′ kk′

y(4)

+ mr′ kk′ ¯ M′ + ¯ Mrm′ kk′

y(5) + Mmm′

kk′ y(6) with ˆ M = (M + m + m′), ¯ M = M + m and ¯ M′ = M + m′.

Jean L´ EVINE Variations on a Flat Theme

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Introduction Contents On Flatness-based Design On Necessary and Sufficient Conditions Exogeneous Feedback Linearization Conclusion

Videos

linear motor masses bumpers flexible beams rail Masses=disturbance Input filtering Flatness-based Jean L´ EVINE Variations on a Flat Theme

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Introduction Contents On Flatness-based Design On Necessary and Sufficient Conditions Exogeneous Feedback Linearization Conclusion

Combined trajectory planning and output feedback

The US Navy Crane Example (B. Kiss, J.L., P. M¨ ullhaupt 2000)

Jean L´ EVINE Variations on a Flat Theme

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Introduction Contents On Flatness-based Design On Necessary and Sufficient Conditions Exogeneous Feedback Linearization Conclusion

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

Reduced scale set-up (in collaboration with IFP) Motors

flexible riser synchronized digital cameras actuators motors simulating the wave excitation well-head water tank

Jean L´ EVINE Variations on a Flat Theme

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Introduction Contents On Flatness-based Design On Necessary and Sufficient Conditions Exogeneous Feedback Linearization Conclusion

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 conditions w(L, t) = xA(t) EI ∂2w ∂s2 (L) = 0 u(L, t) = zA(t) − z0

Fτ Fν s (s-ds) p φ

pipe

S T u(s,t) w(s,t)

Jean L´ EVINE Variations on a Flat Theme

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Introduction Contents On Flatness-based Design On Necessary and Sufficient Conditions Exogeneous Feedback Linearization Conclusion

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

Jean L´ EVINE Variations on a Flat Theme

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Introduction Contents On Flatness-based Design On Necessary and Sufficient Conditions Exogeneous Feedback Linearization Conclusion Jets of Infinite Order Polynomial Matrices Exact Sequences Flatness NSC Flatness and Generalized Moving Frames

References: Sufficient Conditions

Input affine systems:

1 input: equivalence between static and dynamic feedback linearization (Charlet, L´ evine and Marino 89, 91); n − 1 inputs + [first-order controllability]: always flat (Charlet, L´ evine and Marino 89); 2 inputs and 4 states: 1-flatness (Pomet 97).

Driftless systems:

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

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

Jean L´ EVINE Variations on a Flat Theme

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Introduction Contents On Flatness-based Design On Necessary and Sufficient Conditions Exogeneous Feedback Linearization Conclusion Jets of Infinite Order Polynomial Matrices Exact Sequences Flatness NSC Flatness and Generalized Moving Frames

References: Necessary Conditions

Ruled manifold (Sluis 93, Rouchon 94).

Jean L´ EVINE Variations on a Flat Theme

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Introduction Contents On Flatness-based Design On Necessary and Sufficient Conditions Exogeneous Feedback Linearization Conclusion Jets of Infinite Order Polynomial Matrices Exact Sequences Flatness NSC Flatness and Generalized Moving Frames

References: General Formulations

Find output functions ψ1, . . . , ψm and multi-integers l = (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

= 0 i = 1, . . . , n duj ∧ dψ1 ∧ . . . ∧ dψ(r1+1)

1

∧ . . . ∧ dψm ∧ . . . ∧ dψ(rm+1)

m

= 0 j = 1, . . . , m dψ1 ∧ . . . ∧ dψ(r1)

1

∧ . . . ∧ dψm ∧ . . . ∧ dψ(rm)

m

= 0 is 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

)

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Introduction Contents On Flatness-based Design On Necessary and Sufficient Conditions Exogeneous Feedback Linearization Conclusion Jets of Infinite Order Polynomial Matrices Exact Sequences Flatness NSC Flatness and Generalized Moving Frames

System Representation

Consider the system ˙ x = f(x, u) with x ∈ X, dim X = n, u ∈ Rm and f smooth vector field on X satisfying rank

∂f

∂u

= m.

It is locally equivalent to the underdetermined implicit system: F(x, ˙ x) = 0 with x ∈ X, dim X = n, rank ∂F

∂˙ x

= n − m.

Remark This implicit representation is invariant by endogeneous dynamic feedback.

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Introduction Contents On Flatness-based Design On Necessary and Sufficient Conditions Exogeneous Feedback Linearization Conclusion Jets of Infinite Order Polynomial Matrices Exact Sequences Flatness NSC Flatness and Generalized Moving Frames

The Formalism of Manifolds of Jets of Infinite Order

We embed X in the manifold X = X × Rn

∞ of jets of infinite order

(Krasil’shchik, Lychagin, Vinogradov, 1986) with coordinates x def = (x, ˙ x,¨ x, x(3), . . .) endowed with the trivial Cartan vector field τX =

n

i=1
j≥0

x(j+1)

i

∂ ∂x(j)

i

. We also note LτXϕ = n

i=1

j≥0 x(j+1)

i ∂ϕ ∂x(j)

i

= dϕ

dt the Lie derivative

f ϕ along τX and Lk

τXϕ its kth iterate. We have:

x(k)

i

= dkxi

dtk = Lk τXxi,

i = 1, . . . , n, k ≥ 0. We say that ϕ : X → R is continuous (resp. differentiable) if ϕ depends only on a finite (but otherwise arbitrary) number of variables and is continuous (resp. differentiable) with respect to these variables.

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Introduction Contents On Flatness-based Design On Necessary and Sufficient Conditions Exogeneous Feedback Linearization Conclusion Jets of Infinite Order Polynomial Matrices Exact Sequences Flatness NSC Flatness and Generalized Moving Frames

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

∞, τX its trivial Cartan field, and F ∈ C∞(TX; Rn−m) with

rank ∂F

∂˙ x

= n − m in an open subset of TX.

Set X0 = {x ∈ X|Lk

τXF(x) = 0, ∀k ≥ 0}.

Definition (flatness) The system (X, τX, F) is flat at (x0, y0) ∈ X0 × Rm

∞ if and only if

there exist coordinates y of Rm and a locally smooth mapping ψ : X → Rm such that y = ψ(x), with y0 = ψ(x0); there exists a locally smooth mapping ϕ : Rm

∞ → X with

x0 = ϕ(y0), such that x = ϕ(y, ˙ y, . . .) satisfies Lk

τXF(x) = 0 for

every smooth trajectory t → y(t) and every k ≥ 0. Shortly, the system (X, τX, F) is flat iff it is Lie-B¨ acklund equivalent at (x0, y0) to the trivial implicit system (Rm

∞, τRm

∞, 0). Jean L´ EVINE Variations on a Flat Theme

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Introduction Contents On Flatness-based Design On Necessary and Sufficient Conditions Exogeneous Feedback Linearization Conclusion Jets of Infinite Order Polynomial Matrices Exact Sequences Flatness NSC Flatness and Generalized Moving Frames

Denote dF = ∂F ∂x dx + ∂F ∂˙ x d˙ x and P(F) = ∂F ∂x + ∂F ∂˙ x d dt

.

Thus dF = 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 mapping Φ from Rm

∞ to X0, with

smooth inverse, satisfying Φ(y0) = x0, and such that Φ∗dF = 0. where Φ∗dF is the (backward) image of the 1-form dF by Φ, namely Φ∗dF = P(F)P(ϕ).

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Introduction Contents On Flatness-based Design On Necessary and Sufficient Conditions Exogeneous Feedback Linearization Conclusion Jets of Infinite Order Polynomial Matrices Exact Sequences Flatness NSC Flatness and Generalized Moving Frames

Polynomial matrices

Notations: K: field of meromorphic functions from X to R. K[ d

dt]: (non commutative) principal ideal ring of polynomials

f d

dt with coefficients in K.

Mp,q[ d

dt]: module of p × q matrices over K[ d dt].

Up[ d

dt]: group of p × p unimodular matrices (invertible with

inverse in Mp,p[ d

dt])

Jean L´ EVINE Variations on a Flat Theme

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Introduction Contents On Flatness-based Design On Necessary and Sufficient Conditions Exogeneous Feedback Linearization Conclusion Jets of Infinite Order Polynomial Matrices Exact Sequences Flatness NSC Flatness and Generalized Moving Frames

Right and Left Smith forms

Let M ∈ Mp,q[ d

dt]. There exists U ∈ Uq[ d dt] and V ∈ Up[ d dt] such that:

if p < q, V M U = (∆, 0q−p) with ∆ = diag{δ1, . . . , δp} and di divides dj for all 1 ≤ i ≤ j ≤ p; if p > q, V M U =

∆

0p−q

with ∆ = diag{δ1, . . . , δq} and di

divides dj for all 1 ≤ i ≤ j ≤ q. We say that U ∈ R − Smith (M) and V ∈ L − Smith (M). M is said hyper-regular iff ∆ = I. Remark P(F) is hyper-regular iff the module associated to the variational system of (X, τX, F) is free (thus controllable in the sense of Fliess (1990)).

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Introduction Contents On Flatness-based Design On Necessary and Sufficient Conditions Exogeneous Feedback Linearization Conclusion Jets of Infinite Order Polynomial Matrices Exact Sequences Flatness NSC Flatness and Generalized Moving Frames

Theorem Assume that P(F) is hyper-regular.

1

Every solution of P(F)P(ϕ) = 0 is given by P(ϕ) = U 0n−m,m Im

W = ˆ

UW with U ∈ R − Smith (P(F)), ˆ U = U 0n−m,m Im

, and

W ∈ Um[ d

dt] arbitrary;

2

There exists Q ∈ L − Smith

ˆ

U

and Z ∈ Um[ d

dt] such that

QP(ϕ) =

Im

0n−m,m

Z

and the module generated by the K[ d

dt] combinations of the m

first lines of Qdx is free.

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Introduction Contents On Flatness-based Design On Necessary and Sufficient Conditions Exogeneous Feedback Linearization Conclusion Jets of Infinite Order Polynomial Matrices Exact Sequences Flatness NSC Flatness and Generalized Moving Frames

Interpretation in terms of Exact Sequences of Modules

We denote by Λ1

n the module generated by dx1, . . . , dxn,

by Λ1

n+q the module generated by dx1, . . . , dxn, dξ1, . . . , dξq

and by Λ1

m a suitable module of 1-forms of dimension m.

The previous result corresponds to the construction of the exact sequences: 0 − → Λ1

m ˆ U

− → ker(P(F)) ∩ Λ1

n P(F)

− → Λ1

n−m

and 0 − → Λ1

m ˆ U

− → ker(P(F)) ∩ Λ1

n ˆ Q

− → Λ1

m −

→ 0 where ˆ Q =

Im

0n−m,m

Q.

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Introduction Contents On Flatness-based Design On Necessary and Sufficient Conditions Exogeneous Feedback Linearization Conclusion Jets of Infinite Order Polynomial Matrices Exact Sequences Flatness NSC Flatness and Generalized Moving Frames

Flatness Necessary and Sufficient Conditions

Denote by ωi = Qidx the ith line of Qdx, i = 1, . . . , m, and ω = (ω1, . . . , ωm)T. Define Ω : K[ d

dt] ideal generated by the 1-forms ω1, . . . , ωm.

We say that Ω is strongly closed if there exists M ∈ Um[ d

dt] such that

d(Mω) = 0.

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Introduction Contents On Flatness-based Design On Necessary and Sufficient Conditions Exogeneous Feedback Linearization Conclusion Jets of Infinite Order Polynomial Matrices Exact Sequences Flatness NSC Flatness and Generalized Moving Frames

Theorem (JL, NOLCOS 2004) Assume that P(F) is hyper-regular. The three assertions are equivalent: (i) The system (X, τX, F) is flat; (ii) The ideal Ω = span {ω1, . . . , ωm} is strongly closed; (iii) There exists an m × m matrix µ whose entries are polynomials of d

dt with coefficients in Λ1 n, and a matrix

M ∈ Um[ d

dt] such that the “generalized moving frame

structure equations” hold true: dω = µ ∧ ω, dµ = µ ∧ µ, dM = −Mµ. with ω = (ω1, . . . , ωm)T.

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Introduction Contents On Flatness-based Design On Necessary and Sufficient Conditions Exogeneous Feedback Linearization Conclusion Jets of Infinite Order Polynomial Matrices Exact Sequences Flatness NSC Flatness and Generalized Moving Frames

Sketch of proof of (ii) ⇐ ⇒ (iii)

(ii) = ⇒ (iii) If there exists M such that d(Mω) = 0, we have dM ∧ ω + Mdω = 0, i.e. dω = −M−1dM ∧ ω. Setting µ = − − 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, or dµ = µ ∧ µ. (iii) = ⇒ (ii) Since there exist M ∈ Um[ d

dt] and µ such that dM = −Mµ, we have

µ = −M−1dM and thus dω = −M−1dM ∧ ω, or Mdω + dM ∧ ω = d(Mω) = 0 which proves the strong closedness.

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Introduction Contents On Flatness-based Design On Necessary and Sufficient Conditions Exogeneous Feedback Linearization Conclusion Jets of Infinite Order Polynomial Matrices Exact Sequences Flatness NSC Flatness and Generalized Moving Frames

Necessary and Sufficient Conditions and Generalized Moving Frames

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

dt], Θ = ∅:

Θ =      (ω, Mω)

ω =

   ω1 . . . ωm    , Ω = span {ω1, . . . ωm} , d(Mωω) = 0      Group of transformations: T = {f : Θ → Θ | f(ω, Mω) = (ω′, Mω′)}

ω ω' Mω Mω' f Θ

f ∈ T ⇐ ⇒ ∃T ∈ Um[ d

dt] such that ω′ = Tω,

Mω′ = MωT−1. Thus: identified with [ d ] (unimodular group).

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Introduction Contents On Flatness-based Design On Necessary and Sufficient Conditions Exogeneous Feedback Linearization Conclusion Jets of Infinite Order Polynomial Matrices Exact Sequences Flatness NSC Flatness and Generalized Moving Frames

Recalls on Cartan’s Method of Moving Frames

Consider the Euclidean space RN, the set of orthogonal frames (p, e1, . . . , eN) and the group SO(N, R). The group right action sends a 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-forms ω and µ are the relative components of the moving frame. It can be seen that ω and µ are characterized by the structure equations dω = µ ∧ ω, dµ = µ ∧ µ.

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Back to Generalized Moving Frames for Flat Systems

The 1-form ω and the polynomial valued 1-form µ may be seen as the (generalized) relative components of the generalized frames (ω, Mω) of Θ and dω = µ ∧ ω, dµ = µ ∧ µ are their (generalized) structure equations, with µ = −M−1

ω dMω.

Remark For non flat systems, we have: dω = µ ∧ ω + ̟, d̟ = µ ∧ ̟, dµ = µ ∧ µ, dM = −Mµ which suggests a (formal) characterization of flat submanifolds of X (see e.g. Chern, Chen and Lam).

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Exogeneous Feedback Linearization

Consider the system ˙ x = f(x, u) with rank

∂f

∂u

= m in a suitable

neighborhood, and a 1st order regular exogeneous dynamic feedback: u = α(x, ξ, v), ˙ ξ = β(x, ξ, v) with ξ ∈ Rq, v ∈ Rm and rank ∂α

∂v ∂β ∂v

= m.

The system ˙ x = f(x, u) is said feedback linearizable by 1st order regular exogeneous dynamic feedback if and only if there exist q ∈ N finite, α and β satisfying the above rank condition, such that the closed-loop system ˙ x = f(x, α(x, ξ, v)), ˙ ξ = β(x, ξ, v) is flat. Theorem The system ˙ x = f(x, u) is feedback linearizable by 1st order regular exogeneous dynamic feedback if and only if it is flat.

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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 of ˙ x = f(x, u), dim F = n − m, dim G = q and X = (x, ξ)T, and its variational system is P(F)dX = Px(F) Px(G) Pξ(G) dx dξ

= 0

with Px(.) = ∂

∂x + ∂ ∂˙ x d dt and Pξ(.) = ∂ ∂ξ + ∂ ∂ ˙ ξ d dt.

As before, we denote by Λ1

n the module generated by dx1, . . . , dxn,

by Λ1

n+q the module generated by dx1, . . . , dxn, dξ1, . . . , dξq

and by Λ1

m a suitable module of 1-forms of dimension m.

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Consider U ∈ R − Smith

P(F)
, ˆ

U = U 0n−m,m Im

,

Q ∈ L − Smith ˆ U

and ˆ

Q =

Im

0n−m,m

Q.

We have the following exact sequence: 0 − → Λ1

m ˆ U

− → ker(P(F)) ∩ Λ1

n+q ˆ Q

− → Λ1

m −

→ 0 and there exist U0 ∈ R − Smith (Px(F)), Q0 ∈ L − Smith

ˆ

U0

,

with ˆ U0 = U0 0n−m,m Im

, ˆ

Q0 =

Im

0n−m,m

Q0, and T ∈ Um[ d

dt]

such that: − → Λ1

m ˆ U

− → ker(P(F)) ∩ Λ1

n+q ˆ Q

− → Λ1

m

− → Im ↓ πn ↓ T ↓ − → Λ1

m ˆ U0

− → ker(Px(F)) ∩ Λ1

n ˆ Q0

− → Λ1

m

− → 0.

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Thus, since the closed-loop system is flat, there exists M ∈ Um[ d

dt]

such that d(M ˆ QdX) = 0, and since ˆ Q = T−1 ˆ Q0πn, we have d(M′ ˆ Q0dx) = 0 with M′ = MT−1 ∈ Um[ d

dt].

Therefore, the ideal Ω0 generated by the lines of ˆ Q0dx is strongly closed if the coefficients of the matrix M′ depend only on x (not on ξ). This results from the fact that dM′ ∧ ˆ Q0dx = −M′d(ˆ Q0dx) (by contradiction). Q.E.D.

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Introduction Contents On Flatness-based Design On Necessary and Sufficient Conditions Exogeneous Feedback Linearization Conclusion

Conclusions and Perspectives

Numerous industrial applications: asynchronous motors (Schneider-Electric) mechatronics (Bosch) automotive equipements (Valeo, Bosch, PSA, Siemens Automotive (IFAC Congress Applications Paper Prize 2002 to Horn, Bamberger, Michau, Pindl)) underwater applications (IFP) high-precision positionning (Micro-Contrˆ

le)

magnetic bearings (Alcatel, Axomat GmbH) chemical reactors (Total-Fina-Elf) biotechnological processes (Ifremer) . . .

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References I

R. Baron, J. L´

evine, and M. Mastail. Modelling and control of a fish extrusion process. In Proc. 1st IMACS/IFAC Conf. on Mathematical Modelling and Simulation in Agriculture and Bio-industries, Bruxelles, May 1995.

B. Charlet, J. L´

evine, and R. Marino. On dynamic feedback linearization. Systems & Control Letters, 13:143–151, 1989.

B. Charlet, J. L´

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

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SLIDE 49

Introduction Contents On Flatness-based Design On Necessary and Sufficient Conditions Exogeneous Feedback Linearization Conclusion

References II

I.K. Chatjigeorgiou, S.A. Mavrakos. An investigation of the non-linear transverse vibrations of parametrically excited vertical marine risers and cables under tension. Proceedings of the 21st International Conference on Offshore Mechanics and Artic Engineering. Oslo, Norway, 2002. A.A.R. Fehn, R. Rothfuß, and M. Zeitz. Flatness-based torque ripple free control of switched reluctance servo machines. In R.M. Parkin, A. Al-Habaibeh, and M.R. Jackson, editors, ICOM 2003, International Conference on Mechatronics, pages 209–214, 2003.

Jean L´ EVINE Variations on a Flat Theme

SLIDE 50

Introduction Contents On Flatness-based Design On Necessary and Sufficient Conditions Exogeneous Feedback Linearization Conclusion

References III

M. Fliess, J. L´

evine, Ph. Martin, P. Rouchon. On differentially flat nonlinear systems. Comptes Rendus des s´ eances de l’Acad´ emie des Sciences I-315: 619-624, 1992.

M. Fliess, J. L´

evine, Ph. Martin, P. Rouchon. Flatness and defect of nonlinear systems: Introductory theory and applications.

Internat. J. Control 61: 1327-1361, 1995.
M. Fliess, J. L´

evine, Ph. Martin, P. Rouchon. A Lie B¨ acklund approach to equivalence and flatness of nonlinear systems. IEEE Trans. Automat. Control 44 (5): 922-937, 1999.

Jean L´ EVINE Variations on a Flat Theme

SLIDE 51

Introduction Contents On Flatness-based Design On Necessary and Sufficient Conditions Exogeneous Feedback Linearization Conclusion

References IV

M. Fliess, R. Marquez.

Continuous-time linear predictive control and flatness: a module-theoretic setting with examples.

Internat. J. Control, 73: 606–623, 2001.
V. Hagenmeyer and E. Delaleau.

Robustness analysis of exact feedforward linearization based on differential flatness. Automatica, 39:1941–1946, 2003.

J. Horn, J. Bamberger, P. Michau, and S. Pindl.

Flatness-based clutch control for automated manual transmissions. Control Engineering Practice, 11(12):1353–1359, 2003.

Jean L´ EVINE Variations on a Flat Theme

SLIDE 52

Introduction Contents On Flatness-based Design On Necessary and Sufficient Conditions Exogeneous Feedback Linearization Conclusion

References V

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. L´

evine, 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. L´

evine, and Ph. Mullhaupt. Modeling and motion planning for a class of weight handling equipment.

J. Systems Science, 26, 4: 79–92, 2000.

Jean L´ EVINE Variations on a Flat Theme

SLIDE 53

Introduction Contents On Flatness-based Design On Necessary and Sufficient Conditions Exogeneous Feedback Linearization Conclusion

References VI

B. Kiss, J. L´

evine, 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. L´

evine, and Ph. Mullhaupt. Global stability without motion planning may be worse than local tracking. In Proc. ECC’01, Porto, 2001.

B. Kiss, J. L´

evine, and B. Lantos. On motion planning for robotic manipulation with permanent rolling contacts.

Int. J. of Robotics Research, Vol. 21, 5-6, special issue

“Non-holonomy on purpose”, 443–461, May 2002.

Jean L´ EVINE Variations on a Flat Theme

SLIDE 54

Introduction Contents On Flatness-based Design On Necessary and Sufficient Conditions Exogeneous Feedback Linearization Conclusion

References VII

J. L´

evine, 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. L´

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

Jean L´ EVINE Variations on a Flat Theme

SLIDE 55

Introduction Contents On Flatness-based Design On Necessary and Sufficient Conditions Exogeneous Feedback Linearization Conclusion

References VIII

J. L´

evine, D.V. Nguyen. Flat output characterization for linear systems using polynomial matrices. Systems & Control Letters 48: 69-75, 2003.

J. L´

evine. On the synchronization of a pair of independent windshield wipers. IEEE Trans. on Control Systems Technology, Vol. 12, 5: 787-795, 2004.

J. L´

evine. On necessary and sufficient conditions for differential flatness.

Proc. of IFAC NOLCOS 2004 Conference, Stuttgart, 2004.

Jean L´ EVINE Variations on a Flat Theme

SLIDE 56

Introduction Contents On Flatness-based Design On Necessary and Sufficient Conditions Exogeneous Feedback Linearization Conclusion

References IX

J. L´

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

J. L´

evine and B. R´ emond. Flatness based control of an automatic clutch. In Proc. MTNS 2000, Perpignan, 2000.

J. von L¨
wis, J. Rudolph, J. Thiele, and F. Urban.

Flatness-based trajectory tracking control of a rotating shaft. In 7th International Symposium on Magnetic Bearings, Z¨ urich, pages 299–304, 2000.

Jean L´ EVINE Variations on a Flat Theme

SLIDE 57

Introduction Contents On Flatness-based Design On Necessary and Sufficient Conditions Exogeneous Feedback Linearization Conclusion

References X

Ph. Martin.

Contribution ` a l’ ´ Etude des Syst` emes Diff` erentiellement 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-tank systems. IEEE Trans. Automatic Control, 47(4):594–609, 2002.

Jean L´ EVINE Variations on a Flat Theme

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Introduction Contents On Flatness-based Design On Necessary and Sufficient Conditions Exogeneous Feedback Linearization Conclusion

References XI

N. Petit, P. Rouchon, J.-M. Boueilh, F. Gu´

erin 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 and dynamic linearization. In B. Jakubczyk, W. Respondek, and T. Rze˙ zuchowski, editors, Geometry in Nonlinear Control and Differential Inclusions, pages 319–339. Banach Center Publications, Warsaw, 1993.

Jean L´ EVINE Variations on a Flat Theme

SLIDE 59

Introduction Contents On Flatness-based Design On Necessary and Sufficient Conditions Exogeneous Feedback Linearization Conclusion

References XII

J.-B. Pomet. On dynamic feedback linearization of four-dimensional affine control 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 by

ne 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.

Jean L´ EVINE Variations on a Flat Theme

SLIDE 60

Introduction Contents On Flatness-based Design On Necessary and Sufficient Conditions Exogeneous Feedback Linearization Conclusion

References XIII

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 technological domains. Shaker Verlag, Aachen, 2003.

R. Sabri, J. L´

evine, 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.

Jean L´ EVINE Variations on a Flat Theme

SLIDE 61

Introduction Contents On Flatness-based Design On Necessary and Sufficient Conditions Exogeneous Feedback Linearization Conclusion

References XIV

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.

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Flat output computation: linear motor with a single auxiliary mass

Model (recall) M¨ x = F − k(x − z) − r(˙ x − ˙ z) m¨ z = k(x − z) + r(˙ x − ˙ z) Setting s = d

dt:

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

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THANK YOU!

Jean L´ EVINE Variations on a Flat Theme