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Differentially Flat Nonlinear Control Systems: Overview of the Theory and Applications, and Differential Algebraic Aspects.

Jean L´ EVINE

CAS, Unit´ e Math´ ematiques et Syst` emes, MINES-ParisTech, France.

DART-IV, Beijing, October 27–30, 2010

Jean L´ EVINE Flat Systems, Differential Algebraic Aspects

Contents

Contents

1

Introduction: Basic Notions of System Theory

Jean L´ EVINE Flat Systems, Differential Algebraic Aspects

Contents

Contents

1

Introduction: Basic Notions of System Theory

2

Recalls on Differential Flatness Informal Presentation Differential Algebraic Formulation Manifolds of Jets of Infinite Order Lie-B¨ acklund Equivalence of Implicit Systems Infinite Order Jets Geometric Formulation

Jean L´ EVINE Flat Systems, Differential Algebraic Aspects

Contents

Contents

1

Introduction: Basic Notions of System Theory

2

Recalls on Differential Flatness Informal Presentation Differential Algebraic Formulation Manifolds of Jets of Infinite Order Lie-B¨ acklund Equivalence of Implicit Systems Infinite Order Jets Geometric Formulation

3

Flatness Necessary and Sufficient Conditions Variational Property Polynomial Matrices Approach Flatness Necessary and Sufficient Conditions Example

Jean L´ EVINE Flat Systems, Differential Algebraic Aspects

Introduction Recalls on Differential Flatness Flatness Necessary and Sufficient Conditions

Contents

1

Introduction: Basic Notions of System Theory

2

Recalls on Differential Flatness Informal Presentation Differential Algebraic Formulation Manifolds of Jets of Infinite Order Lie-B¨ acklund Equivalence of Implicit Systems Infinite Order Jets Geometric Formulation

3

Flatness Necessary and Sufficient Conditions Variational Property Polynomial Matrices Approach Flatness Necessary and Sufficient Conditions Example

Jean L´ EVINE Flat Systems, Differential Algebraic Aspects

Introduction Recalls on Differential Flatness Flatness Necessary and Sufficient Conditions

Introduction: Basic Notions of System Theory

Consider a smooth n-dimensional manifold X and a family of vector fields x → f(x, u) ∈ TxX for all x ∈ X, indexed by u ∈ Rm, control input, with m ≤ n, and rank

• ∂f

∂u

• = m in a suitable open dense set,

and the (ordinary) differential equation in X ˙ x = f(x, u) (1) with initial state x0 ∈ X at time t = 0. System (explicit representation) The system associated to (1) is the pair (X, f).

Jean L´ EVINE Flat Systems, Differential Algebraic Aspects

Introduction Recalls on Differential Flatness Flatness Necessary and Sufficient Conditions

After elimination of u (implicit function theorem), we get the equivalent set of n − m implicit equations F(x, ˙ x) = 0 (2) where F : TX → Rn−m, satisfies rank ∂F

∂˙ x

• = n − m in a suitable
• pen dense set, and with initial state x0 ∈ X at time t = 0.

System (implicit representation) The system associated to (2) is the pair (X, F). The two notions, explicit and implicit, are indeed equivalent! System (Differential algebraic definition for F polynomial) Let K = R be the ground field and D/K the differential field generated by the variables x1, . . . , xn and the relation (2), assumed polynomial. The system associated to (2) is the differential field D/K. Its differential transcendence degree is diff tr d◦D/K = n − m.

Jean L´ EVINE Flat Systems, Differential Algebraic Aspects

Introduction Recalls on Differential Flatness Flatness Necessary and Sufficient Conditions

Reachability (see e.g. Sussmann and Jurdjevic, J. Diff. Eq. 1972, Sussmann, SIAM J. Control, 1987) The reachable set at time T > 0, noted RT(x0), is the set of points xT ∈ X such that there exists u piecewise continuous on [0, T] and an integral curve t → Xt(x0; u) of the system, satisfying XT(x0; u) = xT. We say that the system is locally reachable if for every neighborhood V of xT in X, RT(x0) ∩ V has non empty interior. For linear systems, reachability is equivalent to controllability.

Jean L´ EVINE Flat Systems, Differential Algebraic Aspects

Introduction Recalls on Differential Flatness Flatness Necessary and Sufficient Conditions

Motion Planning Problem: Given x0 ∈ X: initial state, xT ∈ X: final state, and T > 0: prescribed duration, find reference trajectories t → x∗(t) and t → u∗(t) satisfying ˙ x∗(t) = f(x∗(t), u∗(t)) such that x∗(0) = x0, x∗(T) = xT. Reference Trajectory Tracking: Given a family of perturbations fp ∈ TX of f and a reference trajectory t → x∗(t) of (X, f), find a feedback law x → u(x) such that, noting e = x − x∗, the error equation ˙ e(t) = fp(e(t) + x∗(t), u(e(t) + x∗(t))) − f(x∗(t), u∗(t)) is asymptotically stable for all perturbations. Remark: These problems haven’t yet received a general answer. Our aim is to show that, for differentially flat systems, a simple solution may be obtained.

Jean L´ EVINE Flat Systems, Differential Algebraic Aspects

Introduction Recalls on Differential Flatness Flatness Necessary and Sufficient Conditions

A simple example:

Fast rest-to-rest displacements of a pendulum without measuring the pendulum position.

motor rotation axis pendulum bumper

vertical position indicator PID control on motor position input filtering flatness-based

Experiment realized thanks to Micro-Controle/Newport (France).

Jean L´ EVINE Flat Systems, Differential Algebraic Aspects

Introduction Recalls on Differential Flatness Flatness Necessary and Sufficient Conditions

Other simple examples:

Fast rest-to-rest displacements of a linear motor with perturbating oscillating masses

mass flexible beam bumper linear motor rail

Mass=perturbation input filtering flatness-based

linear motor masses bumpers flexible beams rail

Masses=perturbation input filtering flatness-based

Experiments realized thanks to Micro-Controle/Newport (France).

Jean L´ EVINE Flat Systems, Differential Algebraic Aspects

Introduction Recalls on Differential Flatness Flatness Necessary and Sufficient Conditions

More difficult:

Fast rest-to-rest displacements of a US Navy crane (reduced scale

model – Kiss, L´ evine and M¨ ullhaupt, 2000, Devos and L´ evine, 2006)

Jean L´ EVINE Flat Systems, Differential Algebraic Aspects

Introduction Recalls on Differential Flatness Flatness Necessary and Sufficient Conditions

More and more difficult:

2kPi: Juggling robot (Lenoir, Martin and Rouchon, CDC 98)

Jean L´ EVINE Flat Systems, Differential Algebraic Aspects

Introduction Recalls on Differential Flatness Flatness Necessary and Sufficient Conditions

An infinite dimensional example:

Well-head / riser underwater connexion (reduced scale model – CAS /

French Institute of Petroleum (IFP))

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

Jean L´ EVINE Flat Systems, Differential Algebraic Aspects

Introduction Recalls on Differential Flatness Flatness Necessary and Sufficient Conditions Informal Presentation Differential Algebraic Formulation Manifolds of

Contents

1

Introduction: Basic Notions of System Theory

2

Recalls on Differential Flatness Informal Presentation Differential Algebraic Formulation Manifolds of Jets of Infinite Order Lie-B¨ acklund Equivalence of Implicit Systems Infinite Order Jets Geometric Formulation

3

Flatness Necessary and Sufficient Conditions Variational Property Polynomial Matrices Approach Flatness Necessary and Sufficient Conditions Example

Jean L´ EVINE Flat Systems, Differential Algebraic Aspects

Introduction Recalls on Differential Flatness Flatness Necessary and Sufficient Conditions Informal Presentation Differential Algebraic Formulation Manifolds of

Recalls on Differential Flatness (Fliess, L´

evine, Martin and Rouchon, 1991)

Definition (informal) The nonlinear system ˙ x = f(x, u) is said (differentially) flat if and ony if there exists y = (y1, . . . , ym) such that: y and successive derivatives ˙ y,¨ y, . . . , are independent, y = h(x, u, ˙ u, . . . , u(r)) (generalized output), conversely, x and u are given by: x = ϕ(y, ˙ y, . . . , y(s)), u = ψ(y, ˙ y, . . . , y(s+1)) with ˙ ϕ ≡ f(ϕ, ψ). The vector y is called a flat output.

Jean L´ EVINE Flat Systems, Differential Algebraic Aspects

Introduction Recalls on Differential Flatness Flatness Necessary and Sufficient Conditions Informal Presentation Differential Algebraic Formulation Manifolds of

In the mathematical literature, this concept may be traced back to two different trends: solution of underdetermined differential equations (Monge, Goursat, Zervos, etc.) and in particular Hilbert (“umkerhrbar, integrallos transformationen”, 1912) and Cartan (absolute equivalence, 1914. See also Shadwick, 1990); parameterization and uniformization in the sense of Hilbert’s 22nd Problem (Poincar´ e 1907).

Jean L´ EVINE Flat Systems, Differential Algebraic Aspects

Introduction Recalls on Differential Flatness Flatness Necessary and Sufficient Conditions Informal Presentation Differential Algebraic Formulation Manifolds of

In control, the idea is issued from: the study of dynamic feedback linearization (Charlet, L´ evine and Marino 1989), after the characterization of static feedback linearization (Jakubczyk and Respondek 1980, Hunt, Su and Meyer 1983, Marino 1986) and the works on dynamic decoupling (see the books of Isidori 1989, Nijmeijer and Van der Schaft 1990), the works of Fliess since 1989, on the differential algebraic approach of system theory. Then followed by many contributions (Aranda-Bricaire, Moog and Pomet 1995, Jakubczyk 1993, Pomet 1993, Sluis 1993, van Nieuwstadt, Rathinam and Murray 1994, Chetverikov 2001, etc.)

Jean L´ EVINE Flat Systems, Differential Algebraic Aspects

Introduction Recalls on Differential Flatness Flatness Necessary and Sufficient Conditions Informal Presentation Differential Algebraic Formulation Manifolds of

Books: Sira-Ramirez and Agrawal, Differentially Flat Systems, Marcel Dekker, 2004, L´ evine, Analysis and Control of Nonlinear Systems: A Flatness-based Approach, Springer, 2009. Rudolph, Flatness Based Control of Distributed Parameter Systems, Shaker-Verlag, 2003.

Jean L´ EVINE Flat Systems, Differential Algebraic Aspects

Introduction Recalls on Differential Flatness Flatness Necessary and Sufficient Conditions Informal Presentation Differential Algebraic Formulation Manifolds of

Consequence on Motion Planning (parameterization by smooth curves) To every curve t → y(t) enough differentiable, there corresponds a trajectory t → x(t) u(t)

• =
• ϕ(y(t), ˙

y(t), . . . , y(s)(t)) ψ(y(t), ˙ y(t), . . . , y(s+1)(t))

• that identically satisfies the system

equations.

x = f (x ,u) y(s +1) = v Lie-Bäcklund t → (x(t), u(t)) t → (y(t), . . . , y(s +1)(t))

• (ϕ,ψ)

Jean L´ EVINE Flat Systems, Differential Algebraic Aspects

Introduction Recalls on Differential Flatness Flatness Necessary and Sufficient Conditions Informal Presentation Differential Algebraic Formulation Manifolds of

Practically:

1

Find the initial and final conditions of the flat output: given find (ti, x(ti), u(ti)) (y(ti), . . . , y(r+1)(ti)) (tf , x(tf ), u(tf )) (y(tf ), . . . , y(r+1)(tf ))

2

Build a smooth curve t → y(t) for t ∈ [ti, tf ] by interpolation, possibly satisfying further constraints.

3

Deduce the corresponding trajectory t → (x(t), u(t)). The important particular case of Rest-to-rest trajectories: If ˙ x(ti) = 0, ˙ u(ti) = 0 and ˙ x(tf ) = 0, ˙ u(tf ) = 0, we immediately get ˙ y(ti) = . . . = y(r+1)(ti) = 0 and ˙ y(tf ) = . . . = y(r+1)(tf ) = 0.

Jean L´ EVINE Flat Systems, Differential Algebraic Aspects

Introduction Recalls on Differential Flatness Flatness Necessary and Sufficient Conditions Informal Presentation Differential Algebraic Formulation Manifolds of

Consequence on Trajectory Tracking (see e.g. Martin, 1992, Fliess,

L´ evine, Martin and Rouchon, 1999)

There exists a linearizing endogeneous dynamic feedback , i.e. a dynamic feedback u = α(x, z, v), ˙ z = β(x, z, v) such that the closed-loop system is diffeomorphic to y(s+1) = v. Stabilization of the tracking error: Given a reference t → (yref (t), vref (t)) with vref (t) = y(s+1)

ref

(t), we assume that y, . . . , y(s) are measured or are suitably estimated. We set: ε = y − yref and ε(s+1) = v − vref = − s

i=0 kiε(i), the gains

ki, i = 0, . . . , s, being chosen such that all the roots of the polynomial λs+1 + ksλs + . . . + k1λ + k0 have negative real part. Thus ε(t) ≤ Ce−a(t−t0) and, by continuity, locally, dist(x(t), xref (t)) → 0.

Jean L´ EVINE Flat Systems, Differential Algebraic Aspects

Introduction Recalls on Differential Flatness Flatness Necessary and Sufficient Conditions Informal Presentation Differential Algebraic Formulation Manifolds of

Example and Important Remark

Consider the celebrated “nonholonomic car” model:      ˙ x = u cos θ, ˙ y = u sin θ, ˙ θ = u l tan ϕ Flat Output: (y1 = x, y2 = y)

x y θ ϕ l P Q O

This 3-dim system is equivalent to the 4-dim trivial one ¨ y1 = v1 ¨ y2 = v2 by the one-to-one smooth transformation with smooth inverse: x = y1, y = y2, θ = arctan ˙ y2 ˙ y1

• Apparent paradox: not a diffeomorphism ! Requires using the

frameworks of Differential Algebra or of Differential Geometry of Jets of Infinite Order!!!

Jean L´ EVINE Flat Systems, Differential Algebraic Aspects

Introduction Recalls on Differential Flatness Flatness Necessary and Sufficient Conditions Informal Presentation Differential Algebraic Formulation Manifolds of

Differential Algebraic Formulation (polynomial case)

We consider as before the system associated to the n − m polynomial equations F(x, ˙ x) = 0 and the associated differential field D/K. Differential Algebraic Definition of Flatness We say that the system is flat if there exists y = (y1, . . . , ym) such that D/K(y) is differentially algebraic. In other words, y is a differential basis of D/K: for every element ξ ∈ D/K, there exists a polynomial Φ and a finite integer s such that ξ satisfies Φ(ξ, y, . . . , y(s)) = 0. i.e. every system variable can be expressed in function of y and a finite number of its derivatives. Remark

• Limited to polynomial systems.
• No simple tools to characterize a flat output y.

Jean L´ EVINE Flat Systems, Differential Algebraic Aspects

Introduction Recalls on Differential Flatness Flatness Necessary and Sufficient Conditions Informal Presentation Differential Algebraic Formulation Manifolds of

Manifolds of Jets of Infinite Order (general C∞ case)

We introduce the manifold X X × Rn

∞ with global coordinates

x = (x, ˙ x,¨ x, . . . , x(k), . . .) endowed with the trivial Cartan field: τn =

• j≥0

n

• i=1

x(j+1)

i

∂ ∂x(j)

i

d dt. X is endowed with the infinite product topology, which implies that a continuous (resp. differentiable) function from X to R depends only

• n a finite number of derivatives of x, and is continuous (resp.

differentiable) with respect to those variables.

Jean L´ EVINE Flat Systems, Differential Algebraic Aspects

Introduction Recalls on Differential Flatness Flatness Necessary and Sufficient Conditions Informal Presentation Differential Algebraic Formulation Manifolds of

Given the implicit representation (2) with F of class C∞ F(x, ˙ x) = 0 we consider its extension at any order k by Lk

τnF(x, ˙

x, . . . , x(k+1)) = dkF dtk (x, ˙ x, . . . , x(k+1)) = 0 where Lτn denotes the Lie derivative along the Cartan field τn. Definition The system (2) is given by the triple (X, τn, F).

Jean L´ EVINE Flat Systems, Differential Algebraic Aspects

Introduction Recalls on Differential Flatness Flatness Necessary and Sufficient Conditions Informal Presentation Differential Algebraic Formulation Manifolds of

Definition of Lie-B¨ acklund Equivalence Consider two systems (X, τn, F) and (Y, τn′, G), with Y Y × Rn′

∞.

They are said Lie-B¨ acklund equivalent iff there exists a locally C∞ mapping Φ : Y → X, with locally C∞ inverse Ψ s.t. (i) Φ∗τn′ = τn and Ψ∗τn = τn′; (ii) for every y s.t. Lk

τn′G(y) = 0 for all k ≥ 0, then

x = Φ(y) satisfies Lk

τnF(x) = 0 for all k ≥ 0 and

conversely. i.e. if x = Φ(y) with y = (y, ˙ y, . . .) satisfying Lk

τn′G(y) = 0 for all k,

then x = (x, ˙ x, . . .) satisfies Lk

τnF(x) = 0 for all k.

Conversely, if y = Ψ(x) with x = (x, ˙ x, . . .) satisfying Lk

τnF(x) = 0

for all k, then y = (y, ˙ y, . . .) satisfies Lk

τn′G(y) = 0 for all k.

Jean L´ EVINE Flat Systems, Differential Algebraic Aspects

Introduction Recalls on Differential Flatness Flatness Necessary and Sufficient Conditions Informal Presentation Differential Algebraic Formulation Manifolds of

Flatness Definition for Systems on Manifolds of Jets of Infinite Order The system (X, τn, F) is flat iff it is Lie-B¨ acklund equivalent to the trivial system (Rm

∞, τm, 0).

In other words, if x = Φ(y) with y = (y, ˙ y, . . .), then x = (x, ˙ x, . . .) satisfies Lk

τnF(x) = 0 for all k.

Conversely, if y = Ψ(x) with x = (x, ˙ x, . . .) satisfying Lk

τnF(x) = 0

for all k, then y = (y, ˙ y, . . .) is a global coordinate system of Rm

∞,

satisfying Lk

τmy = y(k) for all k.

We thus recover the previous definition!

Jean L´ EVINE Flat Systems, Differential Algebraic Aspects

Introduction Recalls on Differential Flatness Flatness Necessary and Sufficient Conditions Variational Property Polynomial Matrices Approach Flatness NSC Example

Contents

1

Introduction: Basic Notions of System Theory

2

Recalls on Differential Flatness Informal Presentation Differential Algebraic Formulation Manifolds of Jets of Infinite Order Lie-B¨ acklund Equivalence of Implicit Systems Infinite Order Jets Geometric Formulation

3

Flatness Necessary and Sufficient Conditions Variational Property Polynomial Matrices Approach Flatness Necessary and Sufficient Conditions Example

Jean L´ EVINE Flat Systems, Differential Algebraic Aspects

Introduction Recalls on Differential Flatness Flatness Necessary and Sufficient Conditions Variational Property Polynomial Matrices Approach Flatness NSC Example

Flatness is equivalent to the existence of Φ : Rm

∞ → X smooth and

invertible such that F(Φ(y), ˙ Φ(y)) = 0 ∀y ∈ Rm

∞

Hence Variational Property The system (X, τn, F) is flat iff there exists a locally C∞ and invertible mapping Φ : Rm

∞ → X such that

Φ∗dF = 0.

Jean L´ EVINE Flat Systems, Differential Algebraic Aspects

Introduction Recalls on Differential Flatness Flatness Necessary and Sufficient Conditions Variational Property Polynomial Matrices Approach Flatness NSC Example

Polynomial Matrices Approach

dF = ∂F ∂x dx + ∂F ∂˙ x d˙ x = ∂F ∂x + ∂F ∂˙ x d dt

• dx P(F)dx

Recall that x = Φ(y). Thus x = Φ0(y) and dx = P(Φ0)dy, where we have noted P(Φ0)

• j≥0

∂Φ0 ∂y(j) dj dtj . We thus have Φ∗dF = P(F) P(Φ0)dy Therefore, we have to find a polynomial matrix P(Φ0) solution to P(F) P(Φ0) = 0. If we restrict to F meromorphic, P(Φ0) may be obtained via the Smith decomposition (or diagonal reduction) of P(F).

Jean L´ EVINE Flat Systems, Differential Algebraic Aspects

Introduction Recalls on Differential Flatness Flatness Necessary and Sufficient Conditions Variational Property Polynomial Matrices Approach Flatness NSC Example

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

dt]: principal ideal ring of K-polynomials of d dt = Lτn.

Mp,q[ d

dt]: module of the p × q matrices over K[ d dt], with p and q

arbitrary integers. Up[ d

dt]: group of unimodular matrices of Mp,p[ d dt].

Smith Decomposition Let M ∈ Mp,q[ d

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

VMU = (∆p, 0p,q−p) if p ≤ q and VMU =

• ∆q

0p−q,q

• if p ≥ q.

We note V ∈ L − Smith (M) and U ∈ R − Smith (M). Hyper-regularity A matrix M ∈ Mp,q[ d

dt] is said hyper-regular iff its

Smith-decomposition leads to ∆ = Ip if p ≤ q and to ∆ = Iq if p ≥ q.

Jean L´ EVINE Flat Systems, Differential Algebraic Aspects

Introduction Recalls on Differential Flatness Flatness Necessary and Sufficient Conditions Variational Property Polynomial Matrices Approach Flatness NSC Example

Variational System Module Consider an arbitrary ξ = (ξ1, . . . , ξn) = 0. [ξ]: the K[ d

dt]-module generated by the components of ξ;

[P(F)ξ]: the K[ d

dt]-submodule generated by the components of P(F)ξ;

The quotient [ξ]/[P(F)ξ] is called the variational system module. Following Fliess (1990), we say that the variational system is F-controllable iff the associated variational system module is free.

Jean L´ EVINE Flat Systems, Differential Algebraic Aspects

Introduction Recalls on Differential Flatness Flatness Necessary and Sufficient Conditions Variational Property Polynomial Matrices Approach Flatness NSC Example

Variational System Module Consider an arbitrary ξ = (ξ1, . . . , ξn) = 0. [ξ]: the K[ d

dt]-module generated by the components of ξ;

[P(F)ξ]: the K[ d

dt]-submodule generated by the components of P(F)ξ;

The quotient [ξ]/[P(F)ξ] is called the variational system module. Following Fliess (1990), we say that the variational system is F-controllable iff the associated variational system module is free. Proposition If System (X, τn, F) is flat, then:

• Its variational system is F-controllable (⇐

⇒ is flat)

• P(F) is hyper-regular, i.e.

∃V ∈ L − Smith (P(F)) and ∃U ∈ R − Smith (P(F)) such that VP(F)U = (In−m, 0n−m,m) .

Jean L´ EVINE Flat Systems, Differential Algebraic Aspects

Introduction Recalls on Differential Flatness Flatness Necessary and Sufficient Conditions Variational Property Polynomial Matrices Approach Flatness NSC Example

From now on, we assume that P(F) is hyper-regular. A flat output of the variational system can be explicitly constructed: Proposition Let U ∈ R − Smith (P(F)), U = U 0n−m,m Im

• ,

Q ∈ L − Smith

• U
• , R ∈ R − Smith
• U
• ,

Q = R (Im, 0m,n−m) Q. The vector 1-form ω = (ω1, . . . , ωm) given by dx = Uω =

• rd(

U)

• α=0
• U(α)ω(α),

ω = Qdx =

• rd(

Q)

• α
• Qαdx(α)

is a flat output of the variational system.

Jean L´ EVINE Flat Systems, Differential Algebraic Aspects

Introduction Recalls on Differential Flatness Flatness Necessary and Sufficient Conditions Variational Property Polynomial Matrices Approach Flatness NSC Example

From now on, we assume that P(F) is hyper-regular. A flat output of the variational system can be explicitly constructed: Proposition Let U ∈ R − Smith (P(F)), U = U 0n−m,m Im

• ,

Q ∈ L − Smith

• U
• , R ∈ R − Smith
• U
• ,

Q = R (Im, 0m,n−m) Q. The vector 1-form ω = (ω1, . . . , ωm) given by dx = Uω =

• rd(

U)

• α=0
• U(α)ω(α),

ω = Qdx =

• rd(

Q)

• α
• Qαdx(α)

is a flat output of the variational system. Moreover, we have dωi =

• rd(Γ)
• α,β=0

m

• j,k=1

Γj,k

i,α,β ω(α) j

∧ ω(β)

k

.

Jean L´ EVINE Flat Systems, Differential Algebraic Aspects

Introduction Recalls on Differential Flatness Flatness Necessary and Sufficient Conditions Variational Property Polynomial Matrices Approach Flatness NSC Example

Flatness Necessary and Sufficient Conditions

Theorem ((L´ evine (2004), L´ evine (2006))) Given a flat output ω of the variational system, the system (X, τn, F) is flat iff there exists an m × m polynomial matrix µ whose entries are

d dt-polynomials with coefficients in Λ1(X), and a matrix M ∈ Um[ d dt]

satisfying the generalized Cartan’s moving frame structure equations: dω = µω, d (µ) = µµ, d (M) = −Mµ (3) with d extension of the exterior derivative d to polynomial matrices with entries in Λ(X).

Jean L´ EVINE Flat Systems, Differential Algebraic Aspects

Introduction Recalls on Differential Flatness Flatness Necessary and Sufficient Conditions Variational Property Polynomial Matrices Approach Flatness NSC Example

Flatness Necessary and Sufficient Conditions

Theorem ((L´ evine (2004), L´ evine (2006))) Given a flat output ω of the variational system, the system (X, τn, F) is flat iff there exists an m × m polynomial matrix µ whose entries are

d dt-polynomials with coefficients in Λ1(X), and a matrix M ∈ Um[ d dt]

satisfying the generalized Cartan’s moving frame structure equations: dω = µω, d (µ) = µµ, d (M) = −Mµ (3) with d extension of the exterior derivative d to polynomial matrices with entries in Λ(X). Moreover, a flat output y of system (X, τn, F) is obtained by integration of dy = Mω.

Jean L´ EVINE Flat Systems, Differential Algebraic Aspects

Introduction Recalls on Differential Flatness Flatness Necessary and Sufficient Conditions Variational Property Polynomial Matrices Approach Flatness NSC Example

Flatness Necessary and Sufficient Conditions

Theorem ((L´ evine (2004), L´ evine (2006))) Given a flat output ω of the variational system, the system (X, τn, F) is flat iff there exists an m × m polynomial matrix µ whose entries are

d dt-polynomials with coefficients in Λ1(X), and a matrix M ∈ Um[ d dt]

satisfying the generalized Cartan’s moving frame structure equations: dω = µω, d (µ) = µµ, d (M) = −Mµ (3) with d extension of the exterior derivative d to polynomial matrices with entries in Λ(X). Moreover, a flat output y of system (X, τn, F) is obtained by integration of dy = Mω. Remark The differential system (3) is algebraically closed by construction.

Jean L´ EVINE Flat Systems, Differential Algebraic Aspects

Introduction Recalls on Differential Flatness Flatness Necessary and Sufficient Conditions Variational Property Polynomial Matrices Approach Flatness NSC Example

Remark The general solution µ =

• µk

i

• i,k=1,...,m of dω = µω is given

by µk

i = m

• j=1
• rd(µ)
• α,β=0
• Γj,k

i,α,β + νj,k i,α,β

• ω(α)

j

∧ dβ dtβ , with      νj,k

i,α,β = νk,j i,β,α

∀j = 1, . . . , m, ∀α, β = 0, . . . , ord(µ), α = β, νj,k

i,α,α = νk,j i,α,α

∀j = 1, . . . , m, j = k, ∀α = 0, . . . , ord(µ), νk,k

i,α,α arbitrary, ∀α = 0, . . . , ord(µ).

for all j, k = 1, . . . , m, with ord(µ) ≥ ord(Γ), the νj,k

i,α,β’s being

arbitrary meromorphic functions.

Jean L´ EVINE Flat Systems, Differential Algebraic Aspects

Introduction Recalls on Differential Flatness Flatness Necessary and Sufficient Conditions Variational Property Polynomial Matrices Approach Flatness NSC Example

A sequential procedure for flat output computation:

1

find a flat output ω of the variational system and determine µ according to the previous formula;

2

choose ord(µ) = ord(Γ) and compute the νj,k

i,α,β’s by solving the

PDE’s obtained from d (µ) = µµ; if the differential system d (µ) = µµ is not compatible, replace

• rd(µ) by ord(µ) + 1;

3

choose ord(M) ≥ 0 and compute M solution of d (M) = −Mµ; if M so obtained is not unimodular, replace ord(M) by

• rd(M) + 1.

If the system is flat, this procedure ends in a finite number of steps.

Jean L´ EVINE Flat Systems, Differential Algebraic Aspects

Introduction Recalls on Differential Flatness Flatness Necessary and Sufficient Conditions Variational Property Polynomial Matrices Approach Flatness NSC Example

Comments and Open Questions

This procedure requires using symbolic computation. A general program in Maple has been realized by Antritter and L´ evine (ISSAC 2008). It only allows to solve simple examples! Does there exist a bound on the differential orders of µ and M and, if yes, how are they related to n and m? The NSC exterior differential system is closed by construction. Are there examples of incompatibilities due to the non differential equations? Note that for non-flat examples, the conditions, though µ and M exist, are violated by the fact that no unimodular solution M exists.

Jean L´ EVINE Flat Systems, Differential Algebraic Aspects

Introduction Recalls on Differential Flatness Flatness Necessary and Sufficient Conditions Variational Property Polynomial Matrices Approach Flatness NSC Example

A Simple Example (See Chetverikov 2001, Schlacher and Sch¨

• berl 2007)

Explicit system: ˙ x1 = u1 ˙ x2 = u2 ˙ x3 = sin u1 u2

• Jean L´

EVINE Flat Systems, Differential Algebraic Aspects

Introduction Recalls on Differential Flatness Flatness Necessary and Sufficient Conditions Variational Property Polynomial Matrices Approach Flatness NSC Example

A Simple Example (See Chetverikov 2001, Schlacher and Sch¨

• berl 2007)

Explicit system: ˙ x1 = u1 ˙ x2 = u2 ˙ x3 = sin u1 u2

• Implicit system:

F(x1, x2, x3, ˙ x1, ˙ x2, ˙ x3) ˙ x3 − sin ˙ x1 ˙ x2

• = 0

Jean L´ EVINE Flat Systems, Differential Algebraic Aspects

Introduction Recalls on Differential Flatness Flatness Necessary and Sufficient Conditions Variational Property Polynomial Matrices Approach Flatness NSC Example

A Simple Example (See Chetverikov 2001, Schlacher and Sch¨

• berl 2007)

Explicit system: ˙ x1 = u1 ˙ x2 = u2 ˙ x3 = sin u1 u2

• Implicit system:

F(x1, x2, x3, ˙ x1, ˙ x2, ˙ x3) ˙ x3 − sin ˙ x1 ˙ x2

• = 0

Variational system: P(F) =

• −
• ˙

x−1

2

cos ˙ x1 ˙ x2 d dt

• ˙

x1˙ x−2

2

cos ˙ x1 ˙ x2 d dt d dt

• Jean L´

EVINE Flat Systems, Differential Algebraic Aspects

Introduction Recalls on Differential Flatness Flatness Necessary and Sufficient Conditions Variational Property Polynomial Matrices Approach Flatness NSC Example

A Simple Example (See Chetverikov 2001, Schlacher and Sch¨

• berl 2007)

Explicit system: ˙ x1 = u1 ˙ x2 = u2 ˙ x3 = sin u1 u2

• Implicit system:

F(x1, x2, x3, ˙ x1, ˙ x2, ˙ x3) ˙ x3 − sin ˙ x1 ˙ x2

• = 0

Variational system: P(F) =

• −
• ˙

x−1

2

cos ˙ x1 ˙ x2 d dt

• ˙

x1˙ x−2

2

cos ˙ x1 ˙ x2 d dt d dt

• r, using the system equation:

P(F) =

• −
• ˙

x−1

2

• 1 − ˙

x2

3

d dt

• ˙

x−1

2

arcsin (˙ x3)

• 1 − ˙

x2

3

d dt d dt

• Jean L´

EVINE Flat Systems, Differential Algebraic Aspects

Introduction Recalls on Differential Flatness Flatness Necessary and Sufficient Conditions Variational Property Polynomial Matrices Approach Flatness NSC Example

Computation of U, Q and ω

The right Smith decomposition of P(F) yields:

• U =

     

• ¨

x−1

3

arcsin2 (˙ x3)

• 1 − ˙

x2

3

d dt − ˙ x1 ¨ x3 d dt 1 +

• ¨

x−1

3

arcsin (˙ x3)

• 1 − ˙

x2

3

d dt − ˙ x2 ¨ x3 d dt 1       ,   dx1 dx2 dx3   = U ω1 ω2

• ,

Jean L´ EVINE Flat Systems, Differential Algebraic Aspects

Introduction Recalls on Differential Flatness Flatness Necessary and Sufficient Conditions Variational Property Polynomial Matrices Approach Flatness NSC Example

Computation of U, Q and ω

The right Smith decomposition of P(F) yields:

• U =

     

• ¨

x−1

3

arcsin2 (˙ x3)

• 1 − ˙

x2

3

d dt − ˙ x1 ¨ x3 d dt 1 +

• ¨

x−1

3

arcsin (˙ x3)

• 1 − ˙

x2

3

d dt − ˙ x2 ¨ x3 d dt 1       ,   dx1 dx2 dx3   = U ω1 ω2

• ,

Then, the left Smith decomposition of U gives:

• Q =
• − (arcsin (˙

x3))−1 1 1

• ,

ω1 ω2

• =
• −

1 arcsin(˙ x3)dx1 + dx2

dx3

• ,

Jean L´ EVINE Flat Systems, Differential Algebraic Aspects

Introduction Recalls on Differential Flatness Flatness Necessary and Sufficient Conditions Variational Property Polynomial Matrices Approach Flatness NSC Example

Taking exterior derivatives and combining with the expressions of dx previously obtained: dω1 dω2

• =
• −¨

x−1

3

˙ ω1 ∧ ˙ ω3

• =

− 1

¨ x3 ˙

ω1 ∧ d

dt

ω1 ω2

• Jean L´

EVINE Flat Systems, Differential Algebraic Aspects

Introduction Recalls on Differential Flatness Flatness Necessary and Sufficient Conditions Variational Property Polynomial Matrices Approach Flatness NSC Example

Taking exterior derivatives and combining with the expressions of dx previously obtained: dω1 dω2

• =
• −¨

x−1

3

˙ ω1 ∧ ˙ ω3

• =

− 1

¨ x3 ˙

ω1 ∧ d

dt

ω1 ω2

• which suggests the choice ord(µ) = 1 with

µ = µ1

1

µ2

1

• ,
• µ1

1

= ν2,1

1,1,0 ˙

ω2∧ µ2

1

=

• −ν2,1

1,1,0ω1 − 1 ¨ x3 ˙

ω1 + ν2,2

1,1,1 ˙

ω2

• ∧ d

dt

according to the general formula.

Jean L´ EVINE Flat Systems, Differential Algebraic Aspects

Introduction Recalls on Differential Flatness Flatness Necessary and Sufficient Conditions Variational Property Polynomial Matrices Approach Flatness NSC Example

Taking exterior derivatives and combining with the expressions of dx previously obtained: dω1 dω2

• =
• −¨

x−1

3

˙ ω1 ∧ ˙ ω3

• =

− 1

¨ x3 ˙

ω1 ∧ d

dt

ω1 ω2

• which suggests the choice ord(µ) = 1 with

µ = µ1

1

µ2

1

• ,
• µ1

1

= ν2,1

1,1,0 ˙

ω2∧ µ2

1

=

• −ν2,1

1,1,0ω1 − 1 ¨ x3 ˙

ω1 + ν2,2

1,1,1 ˙

ω2

• ∧ d

dt

according to the general formula. µ must satisfy: d(µ) = d(µ1

1)

d(µ2

1)

• = µµ =

µ1

1µ2 1

• Jean L´

EVINE Flat Systems, Differential Algebraic Aspects

Introduction Recalls on Differential Flatness Flatness Necessary and Sufficient Conditions Variational Property Polynomial Matrices Approach Flatness NSC Example

We get dν2,1

1,1,0 ∧ ˙

ω2 = 0

• dν2,2

1,1,1 − x(3)

3

¨ x3

3 ˙

ω1 + 1

¨ x2

3 ¨

ω1

• ∧ ˙

ω2 −

• dν2,1

1,1,0 −

• ν2,1

1,1,0

2 ˙ ω2

• ∧ ω1 = 0

Jean L´ EVINE Flat Systems, Differential Algebraic Aspects

Introduction Recalls on Differential Flatness Flatness Necessary and Sufficient Conditions Variational Property Polynomial Matrices Approach Flatness NSC Example

We get dν2,1

1,1,0 ∧ ˙

ω2 = 0

• dν2,2

1,1,1 − x(3)

3

¨ x3

3 ˙

ω1 + 1

¨ x2

3 ¨

ω1

• ∧ ˙

ω2 −

• dν2,1

1,1,0 −

• ν2,1

1,1,0

2 ˙ ω2

• ∧ ω1 = 0

Lengthy computations lead to the solution: ν2,1

1,1,0

= − 1 arcsin (˙ x3)

• 1 − ˙

x2

3

ν2,2

1,1,1

= − 1 arcsin (˙ x3)   ˙ x2˙ x3 1 − ˙ x2

3

+ ˙ x2 ¨ x3

• 1 − ˙

x2

3

− η(˙ x3)   with η arbitrary function of ˙ x3 only.

Jean L´ EVINE Flat Systems, Differential Algebraic Aspects

Introduction Recalls on Differential Flatness Flatness Necessary and Sufficient Conditions Variational Property Polynomial Matrices Approach Flatness NSC Example

Then we look for a unimodular matrix M: M = M1

1

M2

1 d dt

1

• (ord(M) = 1), satisfying

d(M) = dM1

1∧

dM2

1 ∧ d dt

• = −Mµ =
• −M1

1ν2,1 1,1,0 ˙

ω2∧ M1

1

• ν2,1

1,1,0ω1 + 1 ¨ x3 ˙

ω1 − ν2,2

1,1,1 ˙

ω2

• ∧
• Jean L´

EVINE Flat Systems, Differential Algebraic Aspects

Introduction Recalls on Differential Flatness Flatness Necessary and Sufficient Conditions Variational Property Polynomial Matrices Approach Flatness NSC Example

Then we look for a unimodular matrix M: M = M1

1

M2

1 d dt

1

• (ord(M) = 1), satisfying

d(M) = dM1

1∧

dM2

1 ∧ d dt

• = −Mµ =
• −M1

1ν2,1 1,1,0 ˙

ω2∧ M1

1

• ν2,1

1,1,0ω1 + 1 ¨ x3 ˙

ω1 − ν2,2

1,1,1 ˙

ω2

• ∧
• which yields (again after lengthy computations!)

M1

1 = − arcsin (˙

x3) , M2

1 = −

  x2

• 1 − ˙

x2

3

− κ (˙ x3)   with κ arbitrary function of ˙ x3 only.

Jean L´ EVINE Flat Systems, Differential Algebraic Aspects

Introduction Recalls on Differential Flatness Flatness Necessary and Sufficient Conditions Variational Property Polynomial Matrices Approach Flatness NSC Example

Finally, we integrate dy = Mω, i.e. dy1 = − arcsin (˙ x3) ω1 −   x2

• 1 − ˙

x2

3

− κ (˙ x3)   ˙ ω2, dy2 = ω2

Jean L´ EVINE Flat Systems, Differential Algebraic Aspects

Introduction Recalls on Differential Flatness Flatness Necessary and Sufficient Conditions Variational Property Polynomial Matrices Approach Flatness NSC Example

Finally, we integrate dy = Mω, i.e. dy1 = − arcsin (˙ x3) ω1 −   x2

• 1 − ˙

x2

3

− κ (˙ x3)   ˙ ω2, dy2 = ω2 and obtain the flat output: y1 = x2 − x1 1 arcsin(˙ x3) + σ(x3, ˙ x3), y2 = x3 with σ arbitrary.

Jean L´ EVINE Flat Systems, Differential Algebraic Aspects

Introduction Recalls on Differential Flatness Flatness Necessary and Sufficient Conditions Variational Property Polynomial Matrices Approach Flatness NSC Example

For more readings:

Analysis and Control

• f Nonlinear

Systems

฀1 23

A Flatness-based Approach Jean Lévine

MATICALENGINEERINGMATHEMATICALENGINEERINGMATHEMATI

1

Analysis and Control

• f Nonlinear Systems

This is the first book on a hot topic in the field of control of nonlinear

• systems. It ranges from mathematical system theory to practical industrial

control application and addresses two fundamental questions in Systems and Control: how to plan the motion of a system and track the correspon- ding trajectory in presence of perturbations. It emphasizes on structural aspects and in particular on a class of systems called differentially flat. Part 1 discusses the mathematical theory and part 2 outlines applications

• f this new method in the fields of electric drives (DC motors and linear

synchronous motors), magnetic bearings, automative equipments, and automatic flight control systems. The author offers web-based videos illustrating some dynamical aspects and case studies in simulation (Scilab and Matlab).

Lévine 56320 WMXDesign GmbH Heidelberg – Bender 23.2.09

Dieser pdf-file gibt nur annähernd das endgültige Druckergebnis wieder !

› springer.com

ISBN 978-3-642-00838-2

Jean L´ EVINE Flat Systems, Differential Algebraic Aspects

Introduction Recalls on Differential Flatness Flatness Necessary and Sufficient Conditions Variational Property Polynomial Matrices Approach Flatness NSC Example

Thank you for your attention!

Jean L´ EVINE Flat Systems, Differential Algebraic Aspects