differentially flat nonlinear control systems: overview of...

Contents

Differentially Flat Nonlinear Control Systems:Overview of the Theory and Applications,

and Differential Algebraic Aspects.

Jean LEVINE

CAS, Unite Mathematiques et Systemes, MINES-ParisTech, France.

DART-IV, Beijing, October 27–30, 2010

Jean LEVINE Flat Systems, Differential Algebraic Aspects

Contents

Contents

1 Introduction: Basic Notions of System Theory

2 Recalls on Differential FlatnessInformal PresentationDifferential Algebraic FormulationManifolds of Jets of Infinite OrderLie-Backlund Equivalence of Implicit SystemsInfinite Order Jets Geometric Formulation

3 Flatness Necessary and Sufficient ConditionsVariational PropertyPolynomial Matrices ApproachFlatness Necessary and Sufficient ConditionsExample


Introduction Recalls on Differential Flatness Flatness Necessary and Sufficient Conditions

Contents






Introduction: Basic Notions of System Theory

Consider a smooth n-dimensional manifold X and a family of vectorfields x 7→ f (x, u) ∈ TxX for all x ∈ X, indexed by u ∈ Rm, controlinput, 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 ).



After elimination of u (implicit function theorem), we get theequivalent 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

open 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 fieldgenerated by the variables x1, . . . , xn and the relation (2), assumedpolynomial.The system associated to (2) is the differential field D/K.Its differential transcendence degree is diff tr d◦D/K = n− m.



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 pointsxT ∈ X such that there exists u piecewise continuous on [0, T] and anintegral curve t 7→ Xt(x0; u) of the system, satisfying XT(x0; u) = xT .We say that the system is locally reachable if for every neighborhoodV of xT in X, RT(x0) ∩ V has non empty interior.

For linear systems, reachability is equivalent to controllability.



Motion Planning Problem:Given x0 ∈ X: initial state, xT ∈ X: final state, and T > 0: prescribedduration, find reference trajectories t 7→ x∗(t) and t 7→ u∗(t) satisfyingx∗(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 trajectoryt 7→ x∗(t) of (X, f ), find a feedback law x 7→ u(x) such that, notinge = 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 simplesolution may be obtained.



A simple example:

Fast rest-to-rest displacements of a pendulum without measuringthe 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).



Other simple examples:

Fast rest-to-rest displacements of a linear motor withperturbating oscillating masses

mass

flexible beam

bumper

linear motorrail

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).



More difficult:

Fast rest-to-rest displacements of a US Navy crane (reduced scalemodel – Kiss, Levine and Mullhaupt, 2000, Devos and Levine, 2006)



More and more difficult:

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



An infinite dimensional example:

Well-head / riser underwater connexion (reduced scale model – CAS /French Institute of Petroleum (IFP))

flexible

riser

synchronized

digital

cameras

actuatorsmotors

simulating

the wave

excitation

well-head

water tank


Introduction Recalls on Differential Flatness Flatness Necessary and Sufficient ConditionsInformal Presentation Differential Algebraic Formulation Manifolds of Jets of Infinite Order Lie-Backlund Equivalence Infinite Order Jets Geometric Formulation

Contents






Recalls on Differential Flatness (Fliess, Levine, 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.



In the mathematical literature, this concept may be traced back to twodifferent trends:

solution of underdetermined differential equations (Monge,Goursat, Zervos, etc.) and in particular Hilbert (“umkerhrbar,integrallos transformationen”, 1912) and Cartan (absoluteequivalence, 1914. See also Shadwick, 1990);

parameterization and uniformization in the sense of Hilbert’s22nd Problem (Poincare 1907).



In control, the idea is issued from:

the study of dynamic feedback linearization (Charlet, Levineand Marino 1989), after the characterization of static feedbacklinearization (Jakubczyk and Respondek 1980, Hunt, Su andMeyer 1983, Marino 1986) and the works on dynamicdecoupling (see the books of Isidori 1989, Nijmeijer and Vander Schaft 1990),

the works of Fliess since 1989, on the differential algebraicapproach of system theory.

Then followed by many contributions (Aranda-Bricaire, Moog andPomet 1995, Jakubczyk 1993, Pomet 1993, Sluis 1993, vanNieuwstadt, Rathinam and Murray 1994, Chetverikov 2001, etc.)



Books:

Sira-Ramirez and Agrawal, Differentially Flat Systems, MarcelDekker, 2004,

Levine, Analysis and Control of Nonlinear Systems: AFlatness-based Approach, Springer, 2009.

Rudolph, Flatness Based Control of Distributed ParameterSystems, Shaker-Verlag, 2003.



Consequence on Motion Planning(parameterization by smooth curves)

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

t 7→(

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 systemequations.

x = f (x ,u)

y(s +1) = v

Lie-Bäcklund

t → (x (t), u(t))

t → (y(t), . . . , y(s +1)(t))

•

(ϕ,ψ)



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 7→ y(t) for t ∈ [ti, tf ] by interpolation,possibly satisfying further constraints.

3 Deduce the corresponding trajectory t 7→ (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 gety(ti) = . . . = y(r+1)(ti) = 0 and y(tf ) = . . . = y(r+1)(tf ) = 0.



Consequence on Trajectory Tracking (see e.g. Martin, 1992, Fliess,Levine, Martin and Rouchon, 1999)

There exists a linearizing endogeneous dynamic feedback , i.e. adynamic 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 7→ (yref (t), vref (t)) with vref (t) = y(s+1)

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

∑si=0 kiε

(i), the gainski, i = 0, . . . , s, being chosen such that all the roots of the polynomialλs+1 + ksλ

s + . . .+ k1λ+ k0 have negative real part.Thus ‖ε(t)‖ ≤ Ce−a(t−t0) and, by continuity, locally,

dist(x(t), xref (t))→ 0.



Example and Important Remark

Consider the celebrated “nonholonomic car” model:x = u cos θ,y = u sin θ,θ =

ul

tanϕ

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 = v1y2 = 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 theframeworks of Differential Algebra or of Differential Geometry ofJets of Infinite Order!!!



Differential Algebraic Formulation (polynomial case)

We consider as before the system associated to the n− m polynomialequations 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 thatD/K(y) is differentially algebraic.

In other words, y is a differential basis of D/K: for every elementξ ∈ D/K, there exists a polynomial Φ and a finite integer s such that ξsatisfies

Φ(ξ, y, . . . , y(s)) = 0.i.e. every system variable can be expressed in function of y and afinite number of its derivatives.

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



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

,ddt.

X is endowed with the infinite product topology, which implies thata continuous (resp. differentiable) function from X to R depends onlyon a finite number of derivatives of x, and is continuous (resp.differentiable) with respect to those variables.



Given the implicit representation (2) with F of class C∞

F(x, x) = 0

we consider its extension at any order k by

Lkτn

F(x, x, . . . , x(k+1)) =dkFdtk (x, x, . . . , x(k+1)) = 0

where Lτn denotes the Lie derivative along the Cartan field τn.

DefinitionThe system (2) is given by the triple (X, τn,F).



Definition of Lie-Backlund Equivalence

Consider two systems (X, τn,F) and (Y, τn′ ,G), with Y , Y × Rn′∞.

They are said Lie-Backlund 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. Lkτn′

G(y) = 0 for all k ≥ 0, thenx = Φ(y) satisfies Lk

τnF(x) = 0 for all k ≥ 0 and

conversely.

i.e. if x = Φ(y) with y = (y, y, . . .) satisfying Lkτn′

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 Lkτn

F(x) = 0for all k, then y = (y, y, . . .) satisfies Lk

τn′G(y) = 0 for all k.



Flatness Definition for Systems on Manifolds of Jets of Infinite Order

The system (X, τn,F) is flat iff it is Lie-Backlund equivalent to thetrivial 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 Lkτn

F(x) = 0for 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!


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

Contents






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∞ andinvertible mapping Φ : Rm

∞ → X such thatΦ∗dF = 0.



Polynomial Matrices Approach

dF =∂F∂x

dx +∂F∂x

dx =

(∂F∂x

+∂F∂x

ddt

)dx , P(F)dx

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

P(Φ0) ,∑j≥0

∂Φ0

∂y(j)

dj

dtj .

We thus haveΦ∗dF = P(F) P(Φ0)dy

Therefore, we have to find a polynomial matrix P(Φ0) solution toP(F) P(Φ0) = 0.

If we restrict to F meromorphic, P(Φ0) may be obtained via theSmith decomposition (or diagonal reduction) of P(F).



Notations:

K: field of meromorphic functions from X to RK[ d

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

Mp,q[ ddt ]: module of the p× q matrices over K[ d

dt ], with p and qarbitrary integers.

Up[ ddt ]: group of unimodular matrices of Mp,p[ d

dt ].

Smith Decomposition

Let M ∈Mp,q[ ddt ]. There exist V ∈ Up[ d

dt ] and U ∈ Uq[ ddt ] 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[ ddt ] is said hyper-regular iff its

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



Variational System ModuleConsider an arbitrary ξ = (ξ1, . . . , ξn) 6= 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 isF-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) .



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 =

ord(U)∑α=0

U(α)ω(α), ω = Qdx =

ord(Q)∑α

Qαdx(α)

is a flat output of the variational system.

Moreover, we have

dωi =

ord(Γ)∑α,β=0

m∑j,k=1

Γj,ki,α,β ω

(α)j ∧ ω(β)

k .



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 =

ord(U)∑α=0

U(α)ω(α), ω = Qdx =

ord(Q)∑α

Qαdx(α)

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

dωi =

ord(Γ)∑α,β=0

m∑j,k=1

Γj,ki,α,β ω

(α)j ∧ ω(β)

k .



Flatness Necessary and Sufficient Conditions

Theorem ((Levine (2004), Levine (2006)))

Given a flat output ω of the variational system, the system (X, τn,F) isflat iff there exists an m× m polynomial matrix µ whose entries areddt -polynomials with coefficients in Λ1(X), and a matrix M ∈ Um[ d

dt ]satisfying the generalized Cartan’s moving frame structureequations:

dω = µω, d (µ) = µµ, d (M) = −Mµ (3)

with d extension of the exterior derivative d to polynomial matriceswith entries in Λ(X).

Moreover, a flat output y of system (X, τn,F) is obtained byintegration of dy = Mω.

Remark The differential system (3) is algebraically closed byconstruction.



Flatness Necessary and Sufficient Conditions

Theorem ((Levine (2004), Levine (2006)))

Given a flat output ω of the variational system, the system (X, τn,F) isflat iff there exists an m× m polynomial matrix µ whose entries areddt -polynomials with coefficients in Λ1(X), and a matrix M ∈ Um[ d

dt ]satisfying the generalized Cartan’s moving frame structureequations:

dω = µω, d (µ) = µµ, d (M) = −Mµ (3)

with d extension of the exterior derivative d to polynomial matriceswith entries in Λ(X).Moreover, a flat output y of system (X, τn,F) is obtained byintegration of dy = Mω.

Remark The differential system (3) is algebraically closed byconstruction.



Remark The general solution µ =(µk

i

)i,k=1,...,m of dω = µω is given

by

µki =

m∑j=1

ord(µ)∑α,β=0

(Γj,k

i,α,β + ν j,ki,α,β

)ω

(α)j ∧ dβ

dtβ,

with

ν j,k

i,α,β = νk,ji,β,α ∀j = 1, . . . ,m, ∀α, β = 0, . . . ,ord(µ), α 6= β,

ν j,ki,α,α = νk,j

i,α,α ∀j = 1, . . . ,m, j 6= k, ∀α = 0, . . . ,ord(µ),

νk,ki,α,α arbitrary, ∀α = 0, . . . ,ord(µ).

for all j, k = 1, . . . ,m, with ord(µ) ≥ ord(Γ), the ν j,ki,α,β’s being

arbitrary meromorphic functions.



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 ν j,k

i,α,β’s by solving thePDE’s obtained from d (µ) = µµ;if the differential system d (µ) = µµ is not compatible, replaceord(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) byord(M) + 1.

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



Comments and Open Questions

This procedure requires using symbolic computation. A generalprogram in Maple has been realized by Antritter and Levine(ISSAC 2008). It only allows to solve simple examples!

Does there exist a bound on the differential orders of µ and Mand, 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 nondifferential equations?Note that for non-flat examples, the conditions, though µ and Mexist, are violated by the fact that no unimodular solution Mexists.



A Simple Example (See Chetverikov 2001, Schlacher and Schoberl 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−12 cos

(x1

x2

))ddt

(x1x−2

2 cos(

x1

x2

))ddt

ddt

]or, using the system equation:

P(F) =

[−(

x−12

√1− x2

3

)ddt

(x−1

2 arcsin (x3)√

1− x23

)ddt

ddt

]




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−12 cos

(x1

x2

))ddt

(x1x−2

2 cos(

x1

x2

))ddt

ddt


P(F) =

[−(

x−12

√1− x2

3

)ddt

(x−1

2 arcsin (x3)√

1− x23

)ddt

ddt

]




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−12 cos

(x1

x2

))ddt

(x1x−2

2 cos(

x1

x2

))ddt

ddt

]

or, using the system equation:

P(F) =

[−(

x−12

√1− x2

3

)ddt

(x−1

2 arcsin (x3)√

1− x23

)ddt

ddt

]




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−12 cos

(x1

x2

))ddt

(x1x−2

2 cos(

x1

x2

))ddt

ddt


P(F) =

[−(

x−12

√1− x2

3

)ddt

(x−1

2 arcsin (x3)√

1− x23

)ddt

ddt

]Jean LEVINE Flat Systems, Differential Algebraic Aspects


Computation of U, Q and ω

The right Smith decomposition of P(F) yields:

U =

(

x−13 arcsin2 (x3)

√1− x2

3

)ddt

− x1

x3

ddt

1 +

(x−1

3 arcsin (x3)√

1− x23

)ddt− x2

x3

ddt

0 1

,dx1

dx2dx3

= U[ω1ω2

],

Then, the left Smith decomposition of U gives:

Q =

[− (arcsin (x3))−1 1 0

0 0 1

],

[ω1ω2

]=

[− 1

arcsin(x3)dx1 + dx2

dx3

],



Taking exterior derivatives and combining with the expressions of dxpreviously obtained:[

dω1dω2

]=

[−x−1

3 ω1 ∧ ω30

]=

[0 − 1

x3ω1 ∧ d

dt0 0

] [ω1ω2

]

which suggests the choice ord(µ) = 1 with

µ =

[µ1

1 µ21

0 0

],

{µ1

1 = ν2,11,1,0ω2∧

µ21 =

(−ν2,1

1,1,0ω1 − 1x3ω1 + ν2,2

1,1,1ω2

)∧ d

dt

according to the general formula.

µ must satisfy:

d(µ) =

[d(µ1

1) d(µ21)

0 0

]= µµ =

[0 µ1

1µ21

0 0

]



Taking exterior derivatives and combining with the expressions of dxpreviously obtained:[

dω1dω2

]=

[−x−1

3 ω1 ∧ ω30

]=

[0 − 1

x3ω1 ∧ d

dt0 0

] [ω1ω2

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

µ =

[µ1

1 µ21

0 0

],

{µ1

1 = ν2,11,1,0ω2∧

µ21 =

(−ν2,1

1,1,0ω1 − 1x3ω1 + ν2,2

1,1,1ω2

)∧ d

dt

according to the general formula.

µ must satisfy:

d(µ) =

[d(µ1

1) d(µ21)

0 0

]= µµ =

[0 µ1

1µ21

0 0

]



We get

dν2,11,1,0 ∧ ω2 = 0[

dν2,21,1,1 −

x(3)3x3

3ω1 + 1

x23ω1

]∧ ω2 −

[dν2,1

1,1,0 −(ν2,1

1,1,0

)2ω2

]∧ ω1 = 0

Lengthy computations lead to the solution:

ν2,11,1,0 = − 1

arcsin (x3)√

1− x23

ν2,21,1,1 = − 1

arcsin (x3)

x2x3

1− x23

+x2

x3

√1− x2

3

− η(x3)

with η arbitrary function of x3 only.



Then we look for a unimodular matrix M: M =

[M1

1 M21

ddt

0 1

](ord(M) = 1), satisfying

d(M) =

[dM1

1∧ dM21 ∧

ddt

0 0

]= −Mµ =

[−M1

1ν2,11,1,0ω2∧ M1

1

(ν2,1

1,1,0ω1 + 1x3ω1 − ν2,2

1,1,1ω2

)∧

0 0

]

which yields (again after lengthy computations!)

M11 = − arcsin (x3) , M2

1 = −

x2√1− x2

3

− κ (x3)

with κ arbitrary function of x3 only.



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 − x11

arcsin(x3)+ σ(x3, x3), y2 = x3

with σ arbitrary.



For more readings:

Analysis and Control of Nonlinear Systems

1 23

A Flatness-based Approach

Jean Lévine

maticalengineeringmathematicalengineeringmathemati1

Analysis and Control of Nonlinear System

s

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 of 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 WMX

Design G

mbH

Heidelberg – Bender 23.2.09

D

ieser pdf-file gibt nur annähernd das endgültige Druckergebnis w

ieder !

› springer.com

ISBN 978-3-642-00838-2



Thank you for your attention!


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