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FLATNESS AND DEFECT OF NONLINEAR SYSTEMS:

INTRODUCTORY THEORY AND EXAMPLES ∗

Michel Fliess†Jean Lévine‡Philippe Martin§Pierre Rouchon¶

CAS internal report A-284, January 1994.

We introduce ﬂat systems, which are equivalent to linear ones via a special type of feedback

called endogenous. Their physical properties are subsumed by a linearizing output and they might be

regarded as providing another nonlinear extension of Kalman’s controllability. The distance to ﬂatness

is measured by a non-negative integer, the defect. We utilize differential algebra which suits well to

the fact that, in accordance with Willems’ standpoint, ﬂatness and defect are best deﬁned without

distinguishing between input, state, output and other variables. Many realistic classes of examples

are ﬂat. We treat two popular ones: the crane and the car with ntrailers, the motion planning of

which is obtained via elementary properties of planar curves. The three non-ﬂat examples, the simple,

double and variablelength pendulums, are borrowedfrom nonlinear physics. Ahigh frequency control

strategy is proposed such that the averaged systems become ﬂat.

∗This work was partially supported by the G.R. “Automatique” of the CNRS and by the D.R.E.D. of the “Minist`ere de

l’´

Education Nationale”.

1

1 Introduction

We present here ﬁve case-studies: the control of a crane, of the simple, double and variable length

pendulumsand themotion planningof thecar withn-trailers. They areall treatedwithin theframework

of dynamic feedback linearization which, contrary to the static one, has only been investigated by few

authors (Charlet et al. 1989, Charlet et al. 1991, Shadwick 1990). Our point of view will be probably

best explained by the following calculations where all vector ﬁelds and functions are real-analytic.

Consider

˙x=f(x,u)(x∈Rn,u∈Rm), (1)

where f(0,0)=0 and rank∂f

∂u(0,0)=m. The dynamic feedback linearizability of (1) means,

according to (Charlet et al. 1989), the existence of

1. a regular dynamic compensator

˙z=a(x,z,v)

u=b(x,z,v) (z∈Rq,v∈Rm)(2)

where a(0,0,0)=0, b(0,0,0)=0. The regularity assumption implies the invertibility1of

system (2) with input vand output u.

2. a diffeomorphism

ξ=(x,z)(ξ∈Rn+q)(3)

such that (1)and (2), whose (n+q)-dimensional dynamics is given by

˙x=f(x,b(x,z,v))

˙z=a(x,z,v),

becomes, according to (3), a constant linear controllable system ˙

ξ=Fξ+Gv.

Up to a static state feedback and a linear invertible change of coordinates, this linear system may

be written in Brunovsky canonical form (see, e.g., (Kailath 1980)),

y(ν1)

1=v1

.

.

.

y(νm)

m=vm

whereν1,...,νmarethecontrollability indicesand (y1,...,y(ν1−1)

1,...,ym,...,y(νm−1)

m)isanother ba-

sisof the vectorspacespanned bythe componentsofξ. Set Y=(y1,...,y(ν1−1)

1,...,ym,...,y(νm−1)

m);

1See (Li and Feng 1987) for a deﬁnition of this concept via the structure algorithm. See (Di Benedetto et al. 1989,

Delaleau and Fliess 1992) for a connection with the differential algebraic approach.

2

thus Y=Tξwhere Tis an invertible (n+q)×(n+q)matrix. Otherwise stated, Y=T(x,z).

The invertibility of yields x

z=−1(T−1Y). (4)

Thus from (2) u=b−1(T−1Y), v. From vi=y(νi)

i,i=1,...,m,uand xcan be expressed

as real-analytic functions of the components of y=(y1,...,ym)and of a ﬁnite number of their

derivatives: x=A(y,˙y,...,y(α))

u=B(y,˙y,...,y(β)). (5)

The dynamic feedback (2) is said to be endogenous if, and only if, the converse holds, i.e., if, and

only if, any component of ycan be expressed as a real-analytic function of x,uand a ﬁnite number of

its derivatives: y=C(x,u,˙u,...,u(γ )). (6)

Note that, according to (4), this amounts to expressing zas a function of (x,u,˙u,...,u(ρ) )for

some ρ. In other words, the dynamic extension does not contain exogenous variables, which are

independent of the original system variables and their derivatives. This justiﬁes the word endoge-

nous. Note that quasi-static feedbacks, introduced in the context of dynamic input-output decou-

pling (Delaleau and Fliess 1992), share the same property.

A dynamics (1) which is linearizable via such an endogenous feedback is said to be (differ-

entially) ﬂat;y, which might be regarded as a ﬁctitious output, is called a linearizing or ﬂat out-

put. The terminology ﬂat is due to the fact that yplays a somehow analogous role to the ﬂat co-

ordinates in the differential geometric approach to the Frobenius theorem (see, e.g., (Isidori 1989,

Nijmeijer and van der Schaft 1990)). A considerable amount of realistic models are indeed ﬂat. We

treatheretwocase-studies, namelythecrane(D’Andr´ea-Novel and L´evine 1990,Marttinen et al. 1990)

and the car with ntrailers (Murray and Sastry 1993, Rouchon et al. 1993a). Notice that the use of a

linearizing output was already known in the context of static state feedback (see (Claude 1986) and

(Isidori 1989, page 156)).

One major property of differential ﬂatness is that, due to formulas (5) and (6), the state and input

variables can be directly expressed, without integrating any differential equation, in terms of the ﬂat

outputand aﬁnite numberof itsderivatives. Thisgeneralidea canbe tracedback toworks byD. Hilbert

(Hilbert 1912) and E. Cartan (Cartan 1915) on under-determined systems of differential equations,

where the number of equations is strictly less than the number of unknowns. Let us emphasize on the

fact that this property may be extremely usefull when dealing with trajectories: from ytrajectories,

xand utrajectories are immediately deduced. We shall detail in the sequel various applications of

this property from motion planning to stabilization of reference trajectories. The originality of our

approach partly relies on the fact that the same formalism applies to study systems around equilibrium

points as well as around arbitrary trajectories.

As demonstrated by the crane, ﬂatness is best deﬁned by not distinguishing between input, state,

outputand other variables. Theequations moreovermight beimplicit. This standpoint, whichmatches

well with Willems’approach (Willems 1991), is here taken into account by utilizing differential

3

algebra which has already helped clarifying several questions in control theory (see, e.g., (Diop 1991,

Diop 1992, Fliess 1989, Fliess 1990a, Fliess and Glad 1993)).

Flatness might be seen as another nonlinear extension of Kalman’s controllability. Such an

assertion is surprising when having in mind the vast literature on this subject (see (Isidori 1989,

Nijmeijer and van der Schaft 1990)and thereferences therein). Remember, however,Willems’trajec-

tory characterization (Willems 1991) of linear controllability which can be interpreted as the freeness

of the module associated to a linear system (Fliess 1992). A linearizing output now is the nonlinear

analogue of a basis of this free module.

We know from (Charlet et al. 1989) that any single-input dynamics which is linearizable by a

dynamic feedback is also linearizable by a static one. This implies the existence of non-ﬂat systems

which verify the strong accessibility property (Sussmann and Jurdjevic 1972). We introduce a non-

negative integer, the defect, which measures the distance from ﬂatness.

These new concepts and mathematical tools are providing the common formalism and the under-

lying structure of ﬁve physically motivated case studies. The ﬁrst two ones, i.e., the control of a crane

and the motion planning of a car with n-trailers, which are quite concrete, resort from ﬂat systems.

The three others, i.e., the simple and double Kapitsa pendulums and the variable-length pendulum

exhibit a non zero defect.

The characterization of the linearizing output in the crane is obvious when utilizing a non-classic

representation, i.e., a mixture of differential and non-differential equations, where there are no dis-

tinction between the system variables. It permits a straightforward tracking of a reference trajectory

via an open-loop control. We do not only take advantage of the equivalence to a linear system but also

of the decentralized structure created by assuming that the engines are powerful with respect to the

masses of the trolley and the load.

The motion planning of the car with n-trailer is perhaps the most popular example of path planning

ofnonholonomicsystems(Laumond 1991,Murray and Sastry 1993, Monaco and Normand-Cyrot 1992,

Rouchon et al. 1993a, Tilbury et al. 1993, Martin and Rouchon 1993, Rouchon et al. 1993b). It is a

ﬂat system where the linearizing output is the middle of the axle of the last trailer. Once the linearizing

output is determined, the path planning problem becomes particularly easy: the reference trajectory

as well as the corresponding open-loop control can be expressed in terms of the linearizing output and

aﬁnite number of its derivatives. Let us stress that no differential equations need to be integrated to

obtain the open-loop control. The relative motions of the various components of the system are then

obtained thanks to elementary geometric properties of plane curves. The resulting calculations, which

are presented in the two-trailer case, are very fast and have been implemented on a standard personal

microcomputer under MATLAB.

The control of the three non-ﬂat systems is based on high frequency control and approxima-

tions by averaged and ﬂat systems (for other approaches, see, e.g., (Baillieul 1993, Bentsman 1987,

Meerkov 1980)). Weexploithereanideaduetothe RussianphysicistKapitsa(Bogaevski and Povzner 1991,

Landau and Lifshitz 1982)for stabilizingthesethree systemsintheneighborhood ofquitearbitrary po-

sitionsand trajectories,and inparticular positionswhich arenot equilibriumpoints. Thisidea isclosely

related to a curiosity of classical mechanics that a double inverted pendulum (Stephenson 1908), and

even the Nlinked pendulums which are inverted and balanced on top of one another (Acheson 1993),

4

can be stabilized in the same way. Closed-loop stabilization around reference averaged trajecto-

ries becomes straightforward by utilizing the endogenous feedback equivalence to linear controllable

systems.

The paper is organized as follows. After some differentialalgebraic preliminaries, we deﬁneequiv-

alence by endogenous feedback, ﬂatness and defect. Their implications for uncontrolled dynamics

and linear systems are examined. We discuss the link between ﬂatness and controllability. In order

to verify that some systems are not linearizable by dynamic feedback, we demonstrate a necessary

condition of ﬂatness, which is of geometric nature. The last two sections are devoted respectively to

the ﬂat and non-ﬂat examples.

Firstdraftsofvariouspartsofthis articlehavebeenpresentedin(Fliess et al. 1991, Fliess et al. 1992b,

Fliess et al. 1992a, Fliess et al. 1993b, Fliess et al. 1993c).

2 The algebraic framework

We consider variables related by algebraic differential equations. This viewpoint, which possess

a nice formalisation via differential algebra, is strongly related to Willems’behavioral approach

(Willems 1991), where trajectories play a key role. We start with a brief review of differential ﬁelds

(see also (Fliess 1990a, Fliess and Glad 1993)) and we refer to the books of Ritt (Ritt 1950) and

Kolchin (Kolchin 1973) and Seidenberg’s paper (Seidenberg 1952) for details. Basics on the cus-

tomary (non-differential) ﬁeld theory may be found in (Fliess 1990a, Fliess and Glad 1993) as well

as in the textbook by Jacobson (Jacobson 1985) and Winter (Winter 1974) (see also (Fliess 1990a,

Fliess and Glad 1993)); they will not be repeated here.

2.1 Basics on differential ﬁelds

An (ordinary) differential ring R is a commutative ring equipped with a single derivation d

dt =•such

that

∀a∈R,˙a=da

dt ∈R

∀a,b∈R,d

dt(a+b)=˙a+˙

b

d

dt(ab)=˙ab +a˙

b.

Aconstant c ∈Ris an element such that ˙c=0. A ring of constants only contains constant elements.

An (ordinary) differential ﬁeld is an (ordinary) differential ring which is a ﬁeld.

Adifferential ﬁeld extension L/Kis given by two differential ﬁelds, Kand L, such that K⊆L

and such that the restriction to Kof the derivation of Lcoincides with the derivation of K.

An element ξ∈Lis said to be differentially K-algebraic if, and only if, it satisﬁes an algebraic

differential equation over K, i.e., if there exists a polynomial π∈K[x0,x1,...,xν], π= 0, such that

π(ξ, ˙

ξ,...,ξ(ν))=0. The extension L/Kis said to be differentially algebraic if, and only if, any

element of Lis differentially K-algebraic.

5

An element ξ∈Lis said to be differentially K -transcendental if, and only if, it is not differentially

K-algebraic. The extension L/Kis said to be differentially transcendental if, and only if, there exists

at least one element of Lthat is differentially K-transcendental.

A set {ξi|i∈I}of elements in Lis said to be differentially K -algebraically independent if,

and only if, the set of derivatives of any order, {ξ(ν)

i|i∈I,ν=0,1,2,...},isK-algebraically

independent. Suchan independentset whichis maximalwith respectto inclusionis calleda differential

transcendence basis of L/K. Two such bases have the same cardinality, i.e., the same number of

elements, which is called the differential transcendence degree of L/K: it is denoted by diff tr d0L/K.

Notice that L/Kis differentially algebraic if, and only if, diff trd0L/K=0.

Theorem 1 For a ﬁnitely generateddifferentialextension L/K, the next two properties are equivalent:

(i) L/K is differentially algebraic;

(ii) the (non-differential) transcendence degree of L/Kisﬁnite, i.e., trd0L/K<∞.

More details and some examples may be found in (Fliess and Glad 1993).

2.2 Systems 2

Let kbe a given differential ground ﬁeld. A system is a ﬁnitely generated differential extension D/k3.

Such a deﬁnition corresponds to a ﬁnite number of quantities which are related by a ﬁnite number of

algebraic differential equations over k4. We do not distinguish in this setting between input, state,

output and other types of variables. This ﬁeld-theoretic language therefore ﬁts Willems standpoint

(Willems 1991) on systems. The differential order of the system D/kis the differential transcendence

degree of the extension D/k.

Example Set k=R;D/kis the differential ﬁeld generated by the four unknowns x1,x2,x3,x4

related by the two algebraic differential equations:

˙x1+¨x3˙x4=0,˙x2+(x1+¨x3x4)x4=0.(7)

Clearly, diff trd0D/k=2: it is equal to the number of unknowns minus the number of equations.

Denote by k<u>the differential ﬁeld generated by kand by a ﬁnite set u=(u1,...,um)of

differential k-indeterminates: u1,...,umare differentially k-algebraically independent, i.e.,

2See also (Fliess 1990a, Fliess and Glad 1993).

3Two systems D/kand ˜

D/kare, of course, identiﬁed if, and only if, there exists a differential k-isomorphism between

them (a differential k-isomorphism commutes with d/dt and preserves every element of k).

4It is a standard fact in classic commutative algebra and algebraic geometry (c.f. (Hartshorne 1977)) that one needs

prime ideals for interpreting “concrete”equations in the language of ﬁeld theory. In our differential setting, we of course

need differential prime ideals (see (Kolchin 1973) and also (Fliess and Glad 1993) for an elementary exposition). The

veriﬁcation of the prime character of the differential ideals corresponding to all our examples is done in appendix A.

6

diff trd0k<u>/k=m.Adynamics with (independent) input u isaﬁnitely generated differentially

algebraic extension D/k<u>. Note that the number mof independent input channels is equal to

the differential order of the corresponding system D/k.Anoutput y =(y1,...,yp)is a ﬁnite set of

differential quantities in D.

According to theorem 1, there exists a ﬁnite transcendence basis x=(x1,...,xn)of

D/k<u>. Consequently, any component of ˙x=(˙x1,..., ˙xn)and of yis k<u>-algebraically

dependent on x, which plays the role of a (generalized) state. This yields:

A1(˙x1,x,u,˙u,...,u(α1))=0

.

.

.

An(˙xn,x,u,˙u,...,u(αn))=0

B1(y1,x,u,˙u,...,u(β1))=0

.

.

.

Bp(yp,x,u,˙u,...,u(βp))=0

(8)

where the Ai’s and Bj’s are polynomial over k. The integer nis the dimension of the dynamics

D/k<u>. We refer to (Fliess and Hasler 1990, Fliess et al. 1993a) for a discussion of such

generalized state-variable representations (8) and their relevance to practice.

Example (continued) Set u1=x3and u2=˙x4. The extension D/R<u>is differentially

algebraic and yields the representation

˙x1=−¨u1u2

˙x2=−(x1+¨u1x4)x4

˙x4=u2.

(9)

The dimension of the dynamics is 3 and (x1,x2,x4)is a generalized state. It would be 5 if we set

u1=¨x3and u2=˙x4, and the corresponding representation becomes causal in the classical sense.

Remark 1 Take the dynamics D/k<u>and a ﬁnitely generated algebraic extension D/D. The

two dynamics D/k<u>and D/k<u>, which are of course equivalent, have the same dimension

and can be given the same state variable representation (11). In the sequel, a system D/k<u>will

be deﬁned up to a ﬁnitely generated algebraic extension of D.

2.3 Modules and linear systems 5

Differential ﬁelds are to general for linear systems which are speciﬁed by linear differential equations.

They are thus replaced by the following appropriate modules.

5See also (Fliess 1990b).

7

Let kbe again a given differential ground ﬁeld. Denote by kd

dtthe ring of linear differential

operators of the type

ﬁniteaα

dα

dtα(aα∈k).

This ring is commutative if, and only if, kisaﬁeld of constants. Nevertheless, in the general

non-commutative case, kd

dtstill is a principal ideal ring and the most important properties of left

kd

dt-modules mimic those of modules over commutative principal ideal rings (see (Cohn 1985)).

Let Mbe a left kd

dt- module. An element m∈Mis said to be torsion if, and only if, there exists

π∈kd

dt,π= 0, such that π·m=0. The set of all torsion elements of Mis a submodule T, which

is called the torsion submodule of M. The module Mis said to be torsion if, and only if, M=T. The

following result can regarded as the linear counterpart of theorem 1.

Proposition 1 For a ﬁnitely generated left k d

dt-module M, the next two properties are equivalent:

(i) M is torsion;

(ii) the dimension of M as a k-vector space is ﬁnite.

Aﬁnitely generated module Mis free if, and only if, its torsion submodule Tis trivial, i.e., T={0}6.

Any ﬁnitely generated module Mcan be written M=T⊕where Tis the torsion submodule of M

and is a free module. The rank of M, denoted by rk M, is the cardinality of any basis of . Thus,

Mis torsion if, and only if, rk M=0.

Alinearsystem is, by deﬁnition, a ﬁnitelygenerated leftkd

dt-module. Wearethus dealingwith

aﬁnite number of variables which are related by a ﬁnite number of linear homogeneous differential

equations and our setting appears to be strongly related to Willems’approach (Willems 1991). The

differential order of is the rank of .

Alinear dynamics with input u =(u1,...,um)is a linear system which contains usuch

that the quotient module /[u] is torsion, where [u] denotes the left kd

dt-module spanned by the

components of u. The input is assumed to be independent, i.e., the module [u] is free. This implies

that the differential order of is equal to m. A classical Kalman state variable representation is always

possible:

d

dt

x1

.

.

.

xn

=A

x1

.

.

.

xn

+B

u1

.

.

.

um

(10)

where

•the dimension nof the state x=(x1,...,xn), which is called the dimension of the dynamics, is

equal to the dimension of the torsion module /[u]asak-vector space.

6This is not the usual deﬁnition of free modules, but a characterization which holds for ﬁnitely generated modules over

principal ideal rings, where any torsion-free module is free (see (Cohn 1985)).

8

•the matrices Aand B, of appropriate sizes, have their entries in k.

An output y =(y1,...,yp)is a set of elements in . It leads to the following output map:

y1

.

.

.

yp

=C

x1

.

.

.

xn

+

ﬁnite Dν

dν

dtν

u1

.

.

.

um

.

The controllability of (10) can be expressed in a module-theoretical language which is independent

of any denomination of variables. Controllability is equivalent to the freeness of the module . This

just is an algebraic counterpart (Fliess 1992) of Willems’trajectory characterization (Willems 1991).

When the system is uncontrollable, thetorsion submodule corresponds to theKalman uncontrollability

subspace.

Remark 2 The relationshipwiththe generaldifferentialﬁeldsetting isobtained byproducinga formal

multiplication. The symmetrictensor product(Jacobson 1985)of alinearsystem , where isviewed

as a k-vector space, is an integral differential ring. Its quotient ﬁeld D, which is a differential ﬁeld,

corresponds to the nonlinear ﬁeld theoretic description of linear systems.

2.4 Differentials and tangent linear systems

Differential calculus, which plays such a role in analysis and in differential geometry, admits a nice

analoguein commutativealgebra(Kolchin 1973,Winter 1974),which hasbeenextendedto differential

algebra by Johnson (Johnson 1969).

Toaﬁnitely generated differential extension L/K, associate a mapping dL/K:L→L/K, called

(K¨

ahler) differential 7and where L/Kisaﬁnitely generated left Ld

dt-module, such that

∀a∈Ld

L/Kda

dt =d

dt dL/Ka

∀a,b∈Ld

L/K(a+b)=dL/Ka+dL/Kb

dL/K(ab)=bdL/Ka+adL/Kb

∀c∈Kd

L/Kc=0.

Elements of Kbehave like constants with respect to dL/K. Properties of the extension L/Kcan be

translated into the linear module-theoretic framework of L/K:

•A set ξ=(ξ1,...,ξ

m)is a differential transcendence basis of L/Kif, and only if, dL/Kξ=

(dL/Kξ1,...,dL/Kξm)is a maximal set of Ld

dt-linearly independent elements in L/K. Thus,

diff trd0L/K=rk L/K.

7For any a∈L,dL/Kashould be intuitively understood, likein analysis anddifferentialgeometry, as a “small”variation

of a.

9

•The extension L/Kis differentially algebraic if, and only if, the module L/Kis torsion. A set

x=(x1,...,xn)is a transcendence basis of L/Kif, and only if, dL/Kx=(dL/Kx1,...,dL/Kxn)

is a basis of L/Kas L-vector space.

•The extension L/Kis algebraic if, and only if, L/Kis trivial, i.e., L/K={0}.

The tangent (or variational) linear system associated to the system D/kis the left Dd

dt-module

D/k. To a dynamics D/k<u>is associated the tangent (or variational) dynamics D/kwith the

tangent (or variational) input dL/Ku=(dL/Ku1,...,dL/Kum). The tangent (or variational) output

associated to y=(y1,...,yp)is dL/Ky=(dL/Ky1,...,dL/Kyp).

3 Equivalence, ﬂatness and defect

3.1 Equivalence of systems and endogenous feedback

Two systems D/kand ˜

D/kare said to be equivalent or equivalent by endogenous feedback if, and

only if, any element of D(resp. ˜

D) is algebraic over ˜

D(resp. D)8. Two dynamics, D/k<u>and

˜

D/k<˜u>, are said to be equivalent if, and only if, the corresponding systems, D/kand ˜

D/k, are

so.

Proposition 2 Two equivalent systems (resp. dynamics) possess the same differential order, i.e., the

same number of independent input channels.

Proof Denote by Kthe differential ﬁeld generated by Dand ˜

D:K/Dand K/˜

Dare algebraic

extensions. Therefore,

diff trd0D/k=diff trd0K/k=diff trd0˜

D/k.

Consider two equivalent dynamics, D/k<u>and ˜

D/k<˜u>. Let n(resp. ˜n) be the dimension

of D/k<u>(resp. ˜

D/k<˜u>). In general, n= ˜n. Write

Ai(˙xi,x,u,˙u,...,u(αi))=0,i=1,...,n(11)

and ˜

Ai(˙

˜xi,˜x,˜u,˙

˜u,..., ˜u(˜αi))=0,i=1,..., ˜n(12)

the generalized state variable representations of D/k<u>and ˜

D/k<˜u>, respectively. The

algebraicity of any element of D(resp. ˜

D) over ˜

D(resp. D) yields the following relationships

8According to footnote 3, this deﬁnition of equivalence can also be read as follows: two systems D/kand ˜

D/kare

equivalent if, and only if, there exist two differential extensions D/Dand D/Dwhich are algebraic (in the usual sense),

and a differential k-automorphism between D/kand D/k.

10

between (11) and (12):

ϕi(ui,˜x,˜u,˙

˜u,..., ˜u(νi))=0i=1,...,m

σα(xα,˜x,˜u,˙

˜u,..., ˜u(µα))=0α=1,...,n

˜ϕi(˜ui,x,u,˙u,...,u(˜νi))=0i=1,...,m

˜σα(˜xα,x,u,˙u,...,u(˜µα))=0α=1,..., ˜n

(13)

where the ϕi’s, σα’s, ˜ϕi’s and ˜σα’s are polynomials over k.

The two dynamic feedbacks corresponding to (13) are called endogenous as they do not necessitate

the introduction of any variable that is transcendental over Dand ˜

D(see also (Martin 1992)). If we

know ˜x(resp. x), we can calculate u(resp. ˜u) from ˜u(resp. u) without integrating any differential

equation. The relationship with general dynamic feedbacks is given in appendix B.

Remark 3 The tangent linear systems (see subsection 2.4) of two equivalent systems are strongly

related and, infact, are“almostidentical”. Taketwo equivalentsystems D1/k andD2/k and denoteby

Dthesmallest algebraicextensionofD1and D2. Itis straightforwardto checkthatthethreeleft Dd

dt-

modulesD/k,D⊗D1D1/kand D⊗D2D2/kareisomorphic (see(Hartshorne 1977,Jacobson 1985)).

3.2 Flatness and defect

Like in the non-differential case, a differential extension L/Kis said to be purely differentially tran-

scendental if, and only if, there exists a differential transcendence basis ξ={ξi|i∈I}of L/Ksuch

that L=K<ξ>. A system D/kis called purely differentially transcendental if, and only if, the

extension D/kis so.

A system D/kis called (differentially) ﬂat if, and only if, it is equivalent to a purely differentially

transcendental system L/k. A differential transcendence basis y=(y1,...,ym)of L/ksuch that

L=k<y>is called a linearizing or ﬂat output of the system D/k.

Example (continued) Let us prove that y=(y1,y2)with

y1=x2+(x1+¨x3x4)2

2x(3)

3

,y2=x3.

is a linearizing output for (7). Set σ=x1+¨x3x4. Differentiating y1=x2+σ2/2y(3)

2, we have, using

(7), σ2=−

2˙y1(y(3)

2)2

y(4)

2

. Thus x2=y1−σ2

2y(3)

2

is an algebraic function of (y1,˙y1,y(3)

2,y(4)

2). Since

x4=−˙x2

σand x1=σ−¨y2x4,x4and x1are algebraic functions of (y1,˙y1,¨y1,¨y2,y(3)

2,y(4)

2,y(5)

2).

Remark there exist many other linearizing outputs such as ˜y=(˜y1,˜y2)=(2y1y(3)

2,y2), the inverse

transformation being y=(˜y1/2˜y(3)

2,˜y2).

11

Take an arbitrary system D/kof differential order m. Among all the possible choices of sets

z=(z1,...,zm)of mdifferential k-indeterminates which are algebraic over D, take one such that

trd0D<z>/k<z>is minimum, say δ. This integer δis called the defect of the system D/k. The

next result is obvious.

Proposition 3 A system D/kisﬂat if, and only if, its defect is zero.

Example The defect of the system generated by x1and x2satisfying ˙x1=x1+(˙x2)3is one. Its

general solution cannot be expressed without the integration of, at least, one differential equation.

3.3 Basic examples

3.3.1 Uncontrolled dynamical systems

An uncontrolled dynamical system is, in our ﬁeld-theoretic language (Fliess 1990a), a ﬁnitely gen-

erated differentially algebraic extension D/k: diff trd0D/k=0 implies the non-existence of any

differential k-indeterminate algebraic over D. Thus, the defect of D/kis equal to tr d0D/k, i.e., to

the dimension of the dynamical system D/k, which corresponds to the state variable representation

Ai(˙xi,x)=0, where x=(x1,...,xn)is a transcendence basis of D/k. Flatness means that D/kis

algebraic in the (non-differential) sense: the dynamics D/kis then said to be trivial.

3.3.2 Linear systems

The defect of is, by deﬁnition, the defect of its associated differential ﬁeld extension D/k(see

remark 2).

Theorem 2 The defect of a linear system is equal to the dimension of its torsion submodule, i.e.,

to the dimension of its Kalman uncontrollable subspace. A linear system is ﬂat if, and only if, it is

controllable.

Proof Take the decomposition =T⊕, of section 2.3, where Tis the torsion submodule and

a free module. A basis b=(b1,...,bm)of plays the role of a linearizing output when is

free: the system then is ﬂat. When T= {0}, the differential ﬁeld extension T/kgenerated by Tis

differentially algebraic and its (non-differential) transcendence degree is equal to the dimension of T

as k-vector space. The conclusion follows at once.

Remark 4 The above arguments can be made more concrete by considering a linear dynamics over

R. If it is controllable, we may write it, up to a static feedback, in its Brunovsky canonical form:

y(νi)=ui,(i=1,...,m)

12

where the νi’s are the controllability indices and y =(y1,...,ym)is a linearizing output. In the

uncontrollable case, the defect d is the dimension of the uncontrollable subspace:

d

dt

ξ1

.

.

.

ξd

=M

ξ1

.

.

.

ξd

where M is a d ×d matrix over R.

3.4 A necessary condition for ﬂatness

Consider the system D/kwhere D=k<w>is generated by a ﬁnite set w=(w1,...,w

q). The

wi’s are related by a ﬁnite set, (w, ˙w,...,w

(ν) )=0, of algebraic differential equations. Deﬁne the

algebraic variety Scorresponding to (ξ0,...,ξν)=0inthe(ν +1)q-dimensional afﬁne space with

coordinates

ξj=(ξ j

1,...,ξj

q), j=0,1,...,ν.

Theorem 3 If the system D/kisﬂat, the afﬁne algebraic variety S contains at each regular point a

straight line parallel to the ξν-axes.

Proof The components of w,˙w,...,w(ν −1)are algebraically dependent on the components of a

linearizing output y=(y1,..., ym)and a ﬁnite number of their derivatives. Let µbe the highest order

of these derivatives. The components of w(ν ) depend linearly on the components of y(µ+1), which play

the role of independent parameters for the coordinates ξν

1,...,ξν

q.

Theabove conditionis not sufﬁcient. Considerthe systemD/Rgenerated by(x1,x2,x3)satisfying

˙x1=(˙x2)2+(˙x3)3. This system does not satisfy the necessary condition: it is not ﬂat. The same

system Dcan be deﬁned via the quantities (x1,x2,x3,x4)related by ˙x1=(x4)2+(˙x3)3and x4=˙x2.

Those new equations now satisfy our necessary criterion.

3.5 Flatness and controllability

Sussmann and Jurdjevic (Sussmann and Jurdjevic 1972) have introduced in the differential geometric

settingthe conceptof strongaccessibilityfor dynamicsof theform ˙x=f(x,u). Sontag(Sontag 1988)

showedthat strong accessibility implies the existence ofcontrols suchthat thelinearized systemaround

atrajectory passingthrough a pointaof thestate-spaceis controllable. Coron (Coron 1994)andSontag

(Sontag 1992) demonstrated that, for any a, those controls are generic.

The above considerations with those of section 2.3 and 2.4 lead in our context to the following

deﬁnition of controllability, which is independent of any distinction between variables: a system D/k

issaid to becontrollable(or stronglyaccessible) if, and only if, its tangent linearsystem is controllable,

i.e., if, and only if, the module D/kis free.

Remark 3 shows that this deﬁnition is invariant under our equivalence via endogenous feedback.

Proposition 4 Aﬂat system is controllable

13

Proof It sufﬁces to prove it for a purely differentially transcendental extensions k<y>/k, where

y=(y1,...,ym). The module k<y>/ k, which is spanned by dk<y>/ky1,...,dk<y>/ kym, is necessarily

free.

The converse is false as demonstrated by numerous examples of strongly accessible single-input

dynamics ˙x=f(x,u)which are not linearizable by static feedback and therefore neither by dynamic

ones (Charlet et al. 1989).

Flatness which is equivalent to the possibility of expressing any element of the system as a func-

tion of the linearizing output and a ﬁnite number of its derivatives, may be viewed as the nonlinear

extension of linear controllability, if the latter is characterized by free modules. Whereas the strong

accessibility property only is an “inﬁnitesimal”generalization of linear controllability, ﬂatness should

be viewed as a more “global”and, perhaps, as a more tractable one. This will be enhanced in section

5 where controllable systems of nonzero defect are treated using high-frequency control that enables

to approximate them by ﬂat systems for which the control design is straightforward.

4 Examples and control of ﬂat systems

The veriﬁcation of the prime character of the differential ideals corresponding to all our examples

is done in appendix A. This means that the equations deﬁning all our examples can be rigorously

interpreted in the language of differential ﬁeld theory.

4.1 The 2-D crane

DR

x

z

X

Z

θ

m

g

O

Figure 1: The two dimensional crane.

14

Consider the crane displayed on ﬁgure 1 which is a classical object of control study (see, e.g.,

(D’Andr´ea-Novel and L´evine 1990), (Marttinen et al. 1990)). The dynamics can be divided into two

parts. The ﬁrst part corresponds to the motor drives and industrial controllers for trolley travels and

rolling up and down the rope. The second part is relative to the trolley load, the behavior of which is

very similar to the pendulum one. We concentrate here on the pendulum dynamics by assuming that

•the traversing and hoisting are control variables,

•the trolley load remains in a ﬁxed vertical plane OXZ,

•the rope dynamics are negligible.

A dynamic model of the load can be derived by Lagrangian formalism. It can also be obtained,

in a very simple way, by writing down all the differential (Newton law) and algebraic (geometric

constraints) equations describing the pendulum behavior:

m¨x=−Tsinθ

m¨z=−Tcosθ+mg

x=Rsinθ+D

z=Rcosθ

(14)

where

•(x,z)(the coordinates of the load m), T(the tension of the rope) and θ(the angle between the

rope and the vertical axis OZ) are the unknown variables;

•D(the trolley position) and R(the rope length) are the input variables.

From (14), it is clear that sinθ,T,Dand Rare algebraic functions of (x,z)and their derivatives:

sinθ=x−D

R,T=mR(g−¨z)

z,(¨z−g)(x−D)=¨xz,(x−D)2+z2=R2

that is

D=x−¨xz

¨z−g

R2=z2+¨xz

¨z−g2

.

(15)

Thus, system (14) is ﬂat with (x,z)as linearizing output.

Remark 5 Assume that the modeling equations (14) are completed with the following traversing and

hoisting dynamics:

M¨

D=F−λ˙

D+Tsinθ

J

ρ2¨

R=C−µ

ρ˙

R−Tρ(16)

15

where the new variables Fand Care, respectively the external force applied to the trolley and the

hoisting torque. The other quantities (M,J,ρ,λ,µ)are constant physical parameters. Then (14,16)

is also ﬂat with the same linearizing output (x,z). This explains without any additional computation

whythe systemconsideredin(D’Andr´

ea-Novel and L´

evine 1990)is linearizableviadynamicfeedback.

Let us now address the following question which is one of the basic control problems for a crane:

how can one carry a load mfrom the steady-state R=R1>0 and D=D1at time t1, to the

steady-state R=R2>0 and D=D2at time t2>t1?

It is clear that any motion of the load induces oscillations that must be canceled at the end of the

load transport. We propose here a very simple answer to this question when the crane can be described

by (14). This answer just consists in using (15).

Consider a smooth curve [t1,t2]t→(α(t), γ (t)) ∈R×]0,+∞[ such that

•for i=1,2, (α(ti), γ (ti)) =(Di,Ri), and dr

dtr(α, γ )(ti)=0 with r=1,2,3,4.

•for all t∈[t1,t2], ¨γ(t)<g.

Then the solution of (14) starting at time t1from the steady-state D1and R1, and with the control

trajectory deﬁned, for t∈[t1,t2], by

D(t)=α(t)−¨α(t)γ (t)

¨α(t)−g

R(t)=γ2(t)+¨α(t)γ (t)

¨γ(t)−g2(17)

and, for t>t2,by(D(t), R(t)) =(D2,R2), leads to a load trajectory t→(x(t), z(t)) such that

(x(t), z(t)) =(α(t), γ (t)) for t∈[t1,t2] and (x(t), z(t)) =(D2,R2)for t≥t2. Notice that, since for

all t∈[t1,t2], ¨z(t)<g, the rope tension T=mR(g−¨z)

zremains always positive and the description

of the system by (14) remains reasonable.

This results from the following facts. The generalized state variable description of the system is

the following (Fliess et al. 1991, Fliess et al. 1993a):

R¨

θ=−2˙

R˙

θ−¨

Dcosθ−gsin θ.

Since αand γare smooth, Dand Rare at least twice continuously differentiable. Thus, the classical

existenceand uniquenesstheorem ensures thatthe aboveordinary differentialequationadmits a unique

smooth solution that is nothing but θ(t)=arctan(α(t)−D(t))/γ (t)).

The approximation of the crane dynamics by (14) implies that the motor drives and industrial low-

levelcontrollers(trolley travels and rollingup anddownthe rope)produce fastand stable dynamics(see

remark 5). Thus, if these dynamics are stable and fast enough, classical results of singular perturbation

theory of ordinary differential equation (see, e.g., (Tikhonov et al. 1980)), imply that the control (17)

leads to a ﬁnal conﬁguration close to the steady-state deﬁned by D2and R2.

16

In the simulations displayed here below, we have veriﬁed that the addition of reasonable fast and

stable regulator dynamics modiﬁes only slightly the ﬁnal position (R2,D2). Classical proportional-

integral controller for Dand Rare added to (14). The typical regulator time constants are equal to

one tenth of the period of small oscillations ( 1

10 2πR/g≈0.3 s) (see (Fliess et al. 1991)).

-10

-9

-8

-7

-6

-5

010 20

x (m)

z (m)

load trajectory

0

5

10

15

20

05 10 15

time (s)

(m)

trolley position

5

6

7

8

9

10

05 10 15

time (s)

(m)

rope length

-0.4

-0.2

0

0.2

0.4

05 10 15

time (s)

(rd)

vertical deviation angle

Figure 2: Simulation of the control deﬁned by (17) without (solid lines) and with (dot lines) ideal

low-level controllers for Dand R.

For the simulations presented in ﬁgure 2, the transport of the load mmay be considered as a rather

fast one: the horizontal motion of Dis of 10 m in 3.5 s; the vertical motion of Risupto5min3.5s.

Compared with the low-level regulator time constants (0.1 and 0.3 s), such motions are not negligible.

This explains the transient mismatch between the ideal and non-ideal cases. Nevertheless, the ﬁnal

control performances are not seriously altered: the residual oscillations of the load after 7 s admit less

than 3 cm of horizontal amplitude. Such small residual oscillations can be canceled via a simple PID

17

regulator with the vertical deviation θas input and the set-point of Das output.

The simulations illustrate the importance of the linearizing output (x,z). When the regulations

of Rand Dare suitably designed, it is possible to use the control given in (17) for fast transports of

the load mfrom one point to another. The simplicity and the independence of (17) with respect to the

system parameters (except g) constitute its main practical interests.

Remark 6 Similar calculations can be performed when a second horizontal direction O X2, orthogo-

nal to O X1=O X, is considered. Denoting then by (x1,x2,z)the cartesian coordinates of the load, R

the rope length and (D1,D2)the trolley horizontal position, the system is described by

(¨z−g)(x1−D1)=¨x1z

(¨z−g)(x2−D2)=¨x2z

(x1−D1)2+(x2−D2)2+z2=R2.

This system is clearly ﬂat with the cartesian coordinates of the load, (x1,x2,z),asﬂat output.

Remark 7 In (D’Andr´

ea-Novel et al. 1992b), the control of a body of mass m around a rotation

axle of constant direction is investigated. This system is ﬂat as a consequence of the following

considerations. According to an old result due to Huygens (see, e.g. (Whittaker 1937, p. 131–132)),

the equations describing the motion are equivalent to those of a pendulum of the same mass m and of

length l =J

md where d = 0is the vertical distance between the mass center G and the axle ,Jis

the inertial moment around . Denoting by u and v, respectively, the vertical and horizontal positions

of , the equations of motion are the following (compare to (15)):

¨u

u−x=¨v−g

v−z

(u−x)2+(v −z)2=l2

where (x,z)are the horizontal and vertical coordinates of the Huygens oscillation center. Clearly

(x,z)is a linearizing output.

Remark 8 The examples corresponding to the crane, Huygens’oscillation center (see remark 7) and

the car with n-trailers here below, illustrate the fact that linearizing outputs admit most often a clear

physical interpretation.

4.2 The car with n-trailers

4.2.1 Modeling equations

Steeringacarwithntrailers isnowtheobjectofactiveresearches(Laumond 1991,Murray and Sastry 1993,

Monaco and Normand-Cyrot 1992,Rouchon et al. 1993a, Tilbury et al. 1993). Theﬂatness of abasic

model9of this system combined with the use of Frénet formula lead to a complete and simple solution

9More realistic models where trailer iis not directly hitched to the center of the axle of trailer i−1 are considered in

(Martin and Rouchon 1993, Rouchon et al. 1993b).

18

xn

yn

Pndn

θn

Pn−1

θn−1

P1

d1

θ1

P0

d0Q

φ

θ0

Figure 3: The kinematic car with ntrailers.

of the motion planning problem without obstacles. Notice that most of nonholonomic mobile robots

are ﬂat (D’Andr´ea-Novel et al. 1992a, Campion et al. 1992).

The hitch of trailer iis attached to the center of the rear axle of trailer i−1. The wheels are aligned

with the body of the trailer. The two control inputs are the driving velocity (of the rear wheels of the

car) and the steering velocity (of the front wheels of the car). The constraints are based on allowing

the wheels to roll and spin without slipping. For the steering front wheels of the car, the derivation is

simpliﬁed by assuming them as a single wheel at the midpoint of the axle. The resulting dynamics

are described by the following equations (the notations are those of (Murray and Sastry 1993) and

summarized on ﬁgure 3):

˙x0=u1cosθ0

˙y0=u1sin θ0

˙

φ=u2

˙

θ0=u1

d0tanφ

˙

θi=u1

dii−1

j=1cos(θj−1−θj)sin(θi−1−θi)for i=1,...,n

(18)

where (x0,y0,φ,θ

0,...,θ

n)∈R2×]−π/2,+π/2[×(S1)n+1is the state, (u1,u2)is the control and

d0,d1,...,dnare positive parameters (lengths). As displayed on ﬁgure 3, we denote by Pi, the medium

point of the wheel axle of trailer i, for i=1,...,n. The medium point of the rear (resp. front) wheel

axle of the car is denoted by P0(resp. Q).

19

4.2.2 Cartesian coordinates of Pnas ﬂat output

Denote by (xi,yi)the cartesian coordinates of Pi,i=0,1,...,n:

xi=x0−

i

j=1djcosθj

yi=y0−

i

j=1djsinθj.

A direct computation shows that tanθi=˙yi

˙xi. Since, for i=0,...,n−1, xi=xi+1+di+1cosθi+1

and yi=yi+1+di+1sinθi+1, the variables θn,xn−1,yn−1,θn−1,...,θ1,x0,y0and θ0are functions of xn

and ynand their derivatives up to the order n+1. But u1=˙x0/cos θ0, tanφ=d0˙

θ0/u1and u2=˙

φ.

Thus, the entire state and the control are functions of xnand ynand their derivatives up to order n+3.

This proves that the car with ntrailers described by (18) is a ﬂat system: the linearizing output

corresponds to the cartesian coordinates of the point Pn, the medium point of the wheel axle of the

last trailer.

Flatness implies that for generic values of the state, the strong accessibility rank associated to the

controlsystem (18) ismaximum andequal to itsstate-space dimension: thesystem is thus controllable.

The singularity which might occur when dividing by ˙xi=0 in tan θi=˙yi/˙xi, can be avoided by

the following developments.

4.2.3 Motion planing using ﬂatness

In (Rouchon et al. 1993a, Rouchon et al. 1993b), the following result was sketched.

Proposition 5 Consider (18)and twodifferentstate-space conﬁgurations: ˜p=(˜x0,˜y0,˜

φ, ˜

θ0,..., ˜

θn)

and p=(x0,y0,φ, θ0,...,θn). Assume that the angles ˜

θi−1−˜

θi,i =1,...,n, ˜

φ,θi−1−θi,

i=1,...,n, and φbelong to ]−π/2,π/2[. Then, there exists a smooth open-loop control [0,T]

t→(u1(t), u2(t)) steeringthe systemfrom ˜p at time 0to p at time T >0,such thatthe anglesθi−1−θi,

i=1,...,n, and φ(i =1,...,n) always remain in ]−π/2,π/2[ and such that (u1(t), u2(t)) =0

for t =0,T.

The conditions θi−1−θi∈]−π/2,π/2[ (i=1,...,n) and φ∈]−π/2,π/2[ are meant for avoiding

some undesirable geometric conﬁgurations: trailer ishould not be in front of trailer i−1.

The detailed proof is given in the appendix and relies basically on the fact that the system is ﬂat. It

is constructive and gives explicitly (u1(t), u2(t)). The involved computations are greatly simpliﬁed by

a simple geometric interpretation of the rolling without slipping conditions and the use of the Frénet

formula. Here, we just recall this geometric construction and give the explicit formula for parking a

car with two trailers. The Frénet formula are recalled in the appendix.

Denote by Cithe curve followed by Pi,i=0,...,n. As displayed on ﬁgure 4, the point Pi−1

belongs to the tangent to Ciat Piand at the ﬁxed distance difrom Pi:

Pi−1=Pi+diτi

20

Ci−1

Ci

Pi

θi

Pi−1

θi−1

θi−1

Pi−1

θi

Pi

Figure 4: The geometric interpretation of the rolling without slipping conditions.

with τithe unitary tangent vector to Ci. Differentiating this relation with respect to si, the arc length

of Ci, leads to d

dsiPi−1=τi+diκiνi

where νiis the unitary vector orthogonal to τiand κiis the curvature of Ci. Since d

dsiPi−1gives the

tangent direction to Ci−1,wehave tan(θi−1−θi)=diκi.

4.2.4 Parking simulations of the 2-trailer system

We now restrict to the particular case n=2. We show how the previous analysis can be employed to

solve the parking problem. The simulations of ﬁgures 5 and 6 have been written in MATLAB. They

can be obtained upon request from the fourth author via electronic mail (rouchon@cas.ensmp.fr).

The car and its trailers are initially in Awith angles θ2=θ1=θ0=π/6, φ=0. The objective

is to steer the system to Cwith ﬁnal angles (θ2,θ

1,θ

0,φ) =0. We consider the two smooth curves

CAB and CCB of the ﬁgure 5, deﬁned by their natural parameterizations [0,LAB]s→PAB(s)and

[0,LCB]s→PCB(s),respectively(PAB(0)=A,PCB(0)=C,LAB isthelengthof CAB and LCB the

length of CCB). Their curvatures are denoted by κAB(s)and κCB(s). These curves shall be followed by

P2. The initial and ﬁnal system conﬁguration in AandCimpose κAB(0)=d

dsκAB(0)=d2

ds2κAB(0)=0

and κCB(0)=d

dsκCB(0)=d2

ds2κCB(0)=0. We impose additionally that AB and CB are tangent at B

21

CAB

CCB

A

B

C

begin

end

o

o

o

Figure 5: parking the car with two trailers from Ato Bvia C.

and

κAB(LAB)=d

dsκAB(LAB)=d2

ds2κAB(LAB)=κCB(LCB)=d

dsκCB(LCB)=d2

ds2κCB(LCB)=0.

It is straightforward to ﬁnd curves satisfying such conditions. For the simulation of ﬁgure 6, we take

polynomial curves of degree 9.

Proposition 5 implies that, if P2follows CAB and CCB as displayed on ﬁgure 5, then the initial and

ﬁnalstates willbe asdesired. Takeasmooth function[0,T]t→s(t)∈[0,LAB]suchthat s(0)=0,

s(T)=LAB and ˙s(0)=˙s(T)=0. This leads to smooth control trajectories [0,T]t→u1(t)≥0

and [0,T]t→u2(t)steering the system from Aat time t=0toBat time t=T. Similarly,

[T,2T]t→s(t)∈[0,LCB] such that s(T)=LCB,s(2T)=0 and ˙s(T)=˙s(2T)=0 leads to

control trajectories [T,2T]t→u1(t)≤0 and [T,2T]t→u2(t)steering the system from B

to C. This gives the motions displayed on ﬁgure 6 with forwards motions from Ato B, backwards

motions from Bto Cand a stop in B.

Let us detail the calculation of the control trajectories for the motion from Ato B. Similar

calculations can be done for the motion from Bto C. The curve CAB corresponds to the curve Ciof

ﬁgure 4 with i=2. Assume that CAB is given via the regular parameterization, y=f(x)((x,y)are

the cartesian coordinates and fis a polynomial of degree 9). Denote by sithe arc length of curve Ci,

i=0,1,2. Then ds2=1+(df/dx)2dx and the curvature of C2is given by

κ2=d2f/dx2

(1+(df/dx)2)3/2.

22

A

B

C

begin

end

o

o

o

ooooooooooooooooooooooo

o

o

o

o

o

o

o

o

o

o

o

o

o

o

o

o

o

oo

Figure 6: the successive motions of the car with two trailers.

We have

κ1=1

1+d2

2κ2

2κ2+d2

1+d2

2κ2

2

dκ2

ds2

and ds1=1+d2

2κ2

2ds2. Similarly,

κ0=1

1+d2

1κ2

1κ1+d1

1+d2

1κ2

1

dκ1

ds1

and ds0=1+d2

1κ2

1ds1. Thus u1is given explicitly by

u1=ds0

dt =1+d2

1κ2

11+d2

2κ2

21+(df/dx)2˙x(t)

where[0,T]t→x(t)isanyincreasingsmoothtime function. (x(0), f(x(0))) (resp. (x(T), f(x(T))))

are the coordinates of A(resp. B) and ˙x(0)=˙x(T)=0. Since tan(φ) =d0κ0,weget

u2=dφ

dt =d0

1+d2

0κ2

0

dκ0

ds0u1.

Here, we are not actuallyconcerned with obstacles. The fact that the internalconﬁgurationdepends

only on the curvature results from the general following property: a plane curve is entirely deﬁned (up

to rotation and translation) by its curvature. For the n-trailer case, the angles θn−θn−1,...,θ1−θ0and

φdescribing the relative conﬁguration of the system are only functions of κnand its ﬁrst n-derivatives

with respect to sn.

23

Consequently, limitations due to obstacles can be expressed up to a translation (deﬁned by Pn)

and a rotation (deﬁned by the tangent direction dP

n

dsn) via κnand its ﬁrst n-derivatives with respect to

sn. Such considerations can be of some help in ﬁnding a curve avoiding collisions. More details on

obstacle avoidance can be found in (Laumond et al. 1993) where a car without trailer is considered.

The multi-steering trailer systems considered in (Bushnell et al. 1993), (Tilbury et al. 1993),

(Tilbury and Chelouah 1993) are also ﬂat: the ﬂat output is then obtained by adding to the Cartesian

coordinates of the last trailer, the angles of the trailers that are directly steered. This generalization is

quite natural in view of the geometric construction of ﬁgure 4.

5 High-frequency control of non-ﬂat systems

We address here a method for controlling non-ﬂat systems via their approximations by averaged

and ﬂat ones. More precisely, we develop on three examples an idea due to the Russian physicist

Kapitsa (Bogaevski and Povzner 1991, Landau and Lifshitz 1982,Sagdeev et al. 1988). He considers

the motion of a particle in a highly oscillating ﬁeld and proposes a method for deriving the equations

of the averaged motion and potential. He shows that the inverted position of a single pendulum is

“stabilized”when the suspension point oscillates rapidly. Notice that some related calculations may

be found in (Baillieul 1993). For the use of high-frequency control in different contexts see also

(Bentsman 1987, Meerkov 1980, Sussmann and Liu 1991).

(Acheson 1993, Stephenson 1908)

5.1 The Kapitsa pendulum

z

αlg

m

Figure 7: The Kapitsa pendulum: the suspension point oscillates rapidly on a vertical axis.

24

The notation are summarized on ﬁgure 7. We assume that the vertical velocity ˙z=uof the suspension

point is the control. The equations of motion are:

˙α=p+u

lsinα

˙p=g

l−u2

l2cosαsin α−u

lpcosα

˙z=u

(19)

where pis proportional to the generalized impulsion; gand lare physical constants. This sys-

tem is not ﬂat since it admits only one control variable and is not linearizable via static feedback

(Charlet et al. 1989). However it is strongly accessible.

We state u=u1+u2cos(t/ε)

where u1and u2are auxiliary control and 0 <ε√l/g. It is then natural to consider the following

averaged control system:

˙

α=p+u1

lsinα

˙

p=g

l−(u1)2

l2cosα−(u2)2

2l2cosαsin α−u1

lpcosα

˙

z=u1.

(20)

It admits two control variables, u1and u2, whereas the original system (19) admits only one, u.

Moreover (20) is ﬂat with (α, z)as linearizing output.

The endogenous dynamic feedback

˙

ξ=v1

u1=ξ

u2=2l

cosα(g+v1)−2l2

cosαsin αv2

(21)

transforms (20) into ¨

z=v1

¨

α=v2.(22)

Set

v1=−

1

τ1+1

τ2ξ−1

τ1τ2(z−zsp)

v2=−

1

τ1+1

τ2p+ξ

lsinα−1

τ1τ2(α −αsp)

(23)

where the parameters τ1,τ

2>0 and αsp ∈]−π/2,π/2[/{0}. Then, the closed-loop averaged system

(20,21,23) admits an hyperbolic equilibrium point characterized by (zsp,α

sp)that is asymptotically

stable.

25

Consider now (19) and the high-frequency control u=u1+u2sin(t/ε) with 0 <ε√l/gand

(u1,u2)givenby(21,23)where α,pandzare replacedbyα,pand z. Then,the correspondingaveraged

system is nothing but (22) with v1and v2given by (23). Since the averaged system admits a hyperbolic

asymptotically stable equilibrium, the perturbed system admits an hyperbolic asymptotically stable

limit cycle around (α, p,z)=(αsp,0,zsp)(Guckenheimer and Holmes 1983, theorem 4.1.1, page

168): such control maintains (z,α) near (zsp,α

sp). Moreover this control method is robust in the

following sense: the existence and the stability of the limit cycle is not destroyed by small static errors

in the parameters land gand in the measurements of α,p,zand u.

As illustrated by the simulations of ﬁgure 8, the generalization to trajectory tracking for αand z

is straightforward. These simulations give also a rough estimate of the errors that can be tolerated.

The system parameter values are l=0.10 m and g=9.81 ms−2. The design control parameters are

ε=0.025/2πs and τ1=τ2=0.10 s. For the two upper graphics of ﬁgure 8, no error is introduced:

control is computed with l=0.10 m and g=9.81 ms−2. For the two lower graphics of ﬁgure 8,

parameter errors are introduced: control is computed with with l=0.11 m and g=9.00 ms−2.

26

0.4

0.6

0.8

1

1.2

01 2

time (s)

(rd)

no parameter error

0

0.5

1

01 2

time (s)

v (m)

no parameter error

0.4

0.6

0.8

1

1.2

01 2

time (s)

(rd)

parameter error

0

0.5

1

01 2

time (s)

v (m)

parameter error

z

αα

z

Figure 8: Robustness test of the high-frequency control for the inverted pendulum.

27

5.2 The variable-length pendulum

gravity

u

q

O

Figure 9: pendulum with variable-length.

Let us consider the variable-length pendulum of (Bressan and Rampazzo 1993). The notations are

summarized on ﬁgure 9. We assume as in (Bressan and Rampazzo 1993) that the velocity ˙u=vis

the control. The equations of motion are:

˙q=p

˙p=−cosu+qv2

˙u=v

(24)

where mass and gravity are normalized to 1.

This system is not ﬂat since it admits only one control variable and is not linearizable via static

feedback (Charlet et al. 1989). It is, however, strongly accessible.

As for the Kapitsa pendulum, we set

v=v1+v2cos(t/ε)

where v1and v2are auxiliary controls, 0 <ε1 .We consider the averaged control system:

˙

q=p

˙

p=−cosu+q(v1)2+q(v2)2/2

˙

u=v1.

(25)

This system is obviously linearizable via static feedback with (q,u)as linearizing output.

The static feedback

v1=w1

v2=2w2+cosu

q−(w1)2(26)

28

transforms (25) into ˙

u=w1

¨

q=w2.(27)

Set

w1=−

u−usp

τ1

w2=−

1

τ1+1

τ2p−1

τ1τ2(q−qsp)

(28)

with τ1,τ

2>0, usp ∈]−π/2,π/2[, qsp >0 . The closed-loop averaged system (25,26,28) admits an

hyperbolic equilibrium point (usp,qsp), which is asymptotically stable.

Similarly to the Kapitsa pendulum, the control law is as follows: v=v1+v2sin(t/ε),0<ε1;

(v1,v

2)is given by (26,28) where q,pand uare replaced by q,pand u. This control strategy leads to

a small and attractive limit cycle. As illustrated by the simulations of ﬁgure 10, the size of these limit

cycle is an increasing function of εand tends to 0 as εtends to 0+. The design control parameters are

τ1=0.5, τ2=0.4.

29

1

1.2

1.4

1.6

1.8

2

0246

time

q

= 0.02

-0.5

0

0.5

1

1.5

0246

time

u

= 0.02

0.5

1

1.5

2

0246

time

q

= 0.04

-0.5

0

0.5

1

1.5

0246

time

u

= 0.04

ε

εε

ε

Figure 10: high-frequency control for the variable-length pendulum.

30

5.3 The inverted double pendulum

α1

α2

g

u

v

x

z

O

beam 1

beam 2

Figure 11: The inverted double pendulum: the horizontal velocity u and vertical velocity vof the

suspension point are the two control variables.

The double inverted pendulum of ﬁgure 11 moves in a vertical plane. Assume that u(resp. v) the

horizontal (resp. vertical) velocity of the suspension point (x,z)is a control variable. The equations

of motion are (implicit form):

p1=I1˙α1+I˙α2cos(α1−α2)+n1˙xcos α1−n1˙zsin α1

p2=I˙α1cos(α1−α2)+I2˙α2+n2˙xcos α2−n2˙zsin α2

˙p1=n1gsinα1−n1˙α1˙xsinα1−n1˙α1˙zcos α1

˙p2=n2gsinα2−n2˙α2˙xsinα2−n2˙α2˙zcos α2

˙x=u

˙z=v

(29)

where p1and p2are the generalized impulsions associated to the generalized coordinates α1and α2,

respectively. The quantities g,I,I1,I2,n1and n2are constant physical parameters:

I1=m1

3+m2(l1)2,I2=m2

3(l2)2,I=m2

2l1l2,n1=m1

2+m2l1,n2=m2

2l2,

where m1and m2(resp. l1and l2) are the masses (resp. lengths) of beams 1 and 2 which are assumed

to be homogeneous.

Proposition 6 System (29) with the two control variables u and v,isnotﬂat.

31

Proof The proof is just an application of the necessary ﬂatness condition of theorem 3. Since u=˙x

and v=˙z, (29) is ﬂat if, and only if, the reduced system,

p1=I1˙α1+I˙α2cos(α1−α2)+n1˙xcos α1−n1˙zsin α1

p2=I˙α1cos(α1−α2)+I2˙α2+n2˙xcos α2−n2˙zsin α2

˙p1=n1gsinα1−n1˙α1˙xsinα1−n1˙α1˙zcos α1

˙p2=n2gsinα2−n2˙α2˙xsinα2−n2˙α2˙zcos α2

(30)

is ﬂat. Denote symbolically by F(ξ, ˙

ξ) =0 the equations (30) where ξ=(α1,α

2,x,z,p1,p2).

Consider (ξ , ζ ) such that F(ξ , ζ ) =0. We are looking for a vector a=(aα1,aα2,ax,az,ap1,ap2)such

that, for all λ∈R,F(ξ, ζ +λa)=0. The second order conditions, d2

dλ2λ=0F(ξ, ζ +λa)=0, lead

to aα1(axsinα1+azcos α1)=0,aα2(axsinα2+azcos α2)=0.

Two ﬁrst order conditions, d

dλλ=0F(ξ, ζ +λa)=0, are

−axcosα1+azsin α1=I1

n1aα1+I

n1cos(α1−α2)aα2

−axcosα2+azsin α2=I

n2cos(α1−α2)aα1+I2

n2aα2

Simple computations show that, if I

n1= I2

n2and I1

n1= I

n2(these conditions are always satisﬁed for

homogeneous identical beams), then (aα1,aα2,ax,az)=0. The two remaining ﬁrst order conditions

imply that (ap1,ap2)=0. Thus a=0 and the inverted double pendulum is not ﬂat.

The same control method as the one explained in details for the Kapitsa pendulum (19) can be

also used for the double pendulum. The only difference relies on the calculations that are here more

tedious. We just sketch some simulations (Fliess et al. 1993b).

To approximate the non-ﬂat system (29) by a ﬂat one, we set u=u1+u2cos(t/ε) and v=

v1+v2cos(t/ε) where 0 <εmin I1

n1g,I2

n2gand u1,u2,v1,v2are new control variables. This

leads to a ﬂat averaged system with (α1,α

2,x,z)as the linearizing output. The endogenous dynamic

feedback that linearized the averaged system provides then (u1,u2,v

1,v

2). For the simulations of

ﬁgure 12, the angles α1and α2follow approximately prescribed trajectories whereas, simultaneously,

the suspension point (x,z)is maintained approximately constant.

32

-0.5

-0.45

-0.4

-0.35

-0.3

-0.25

-0.2

-0.15

-0.1

0510

time (s)

(rd)

vertical deviation of beam 1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0510

time (s)

(rd)

vertical deviation of beam 2

α1

α2

Figure 12: Simulation of the inverted double pendulum via high-frequency control.

33

6 Conclusion

Our ﬁve examples, as well as other ones in preparation in various domains of engineering, indicate that

ﬂatness and defect ought to be considered as physical and/or geometric properties. This explains why

ﬂat systems are so often encountered in spite of the non-genericity of dynamic feedback linearizability

in some customary mathematical topologies (Tcho´n 1994, Rouchon 1994).

We hope to have convinced the reader that ﬂatness and defect bring a new theoretical and practical

insight in control. We brieﬂy list some important open problems:

•Ritt’s work (Ritt 1950) shows that differential algebra provides powerful algorithmic means (see

(Diop 1991, Diop 1992) for a survey and connections with control). Can ﬂatness and defect be

determined by this kind of procedures?

•great progress have recently been made in nonlinear time-varying feedback stabilization (see,

e.g., (Coron 1992, Coron 1994)). Most of the examples which were considered happen to be

ﬂat (see, e.g., (Coron and D’Andr´ea-Novel 1992)). The utilization of this property is related to

the understanding of the notion of singularity (see, e.g., (Martin 1993) for a ﬁrst step in this

direction and the references therein).

•the two averaged systems associated to high-frequency control are ﬂat. Can this result be

generalized to a large class of devices?

•differentialalgebrais nottheonly possiblelanguage forinvestigatingﬂatnessand defect. The ex-

tensionofthedifferentialalgebraicformalismto smoothandanalyticfunctions(Jakubczyk 1992)

and the differential geometric approach (Martin 1992, Fliess et al. 1993d, Fliess et al. 1993e,

Pomet 1993) should also be examined in this context.

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Willems, J. 1991. Paradigms and puzzles in the theory of dynamical systems. IEEE Trans. Automat.

Control 36, 259–294.

Winter, D. 1974. The Structure of Fields. Springer, New York.

A Prime differential ideals

We know from (Diop 1992, lemma 5.2, page 158) (see also (Moog et al. 1989)) that, for x=

(x1,...,xn)(n≥0) and u=(u1,...,um)(m≥0), the differential ideal corresponding to

˙xi=ai(x,u,˙u,...,u(αi))

bi(x,u,˙u,...,u(βi)),i=1,...,n,

where the ai’s and bi’s are polynomials over k, is prime. It is then immediate that the differential ideal

corresponding to the tutorial example (7) is prime: set x=(x1,x2)and u=(x3,x4). Let us now list

our ﬁve case-studies.

39

Kapitsa pendulum (19) Let us replace αby σ=tan(α/2). Then, using

˙σ=1+σ2

2˙α, cos α=1−σ2

1+σ2,sinα=2σ

1+σ2,

the equations (19) become explicit and rational

˙σ=1+σ2

2p+2σu

l(1+σ2)

˙p=g

l−u2(1−σ2)

l2(1+σ2)2σ

1+σ2−pu(1−σ2)

l(1+σ2)

˙z=u.

Theassociateddifferentialidealis thusprimeandleadsto aﬁnitelygenerateddifferentialﬁeld extension

over R.

Variable-length pendulum (24) Similar computations with σ=tan(u/2)prove that the associated

differential ideal is prime.

Double pendulum (29) Similar computations with σ1=tan(α1/2)and σ2=tan(α2/2)prove that

the associated differential ideal is prime.

Car with n-trailers (18) Similar computations with σ=tan(ϕ/2)and σi=tan(θi/2)prove that

the associated differential ideal is prime.

Crane (17) Analogous calculations on the generalized state variable equation R¨

θ=−2˙

R˙

θ−

¨

Dcosθ−gsin θgiven in (Fliess et al. 1991, Fliess et al. 1993a) lead to a prime differential ideal.

Another more direct way for obtaining the differential ﬁeld corresponding to the crane is the

following. Take (17) and consider the differential ﬁeld R<x,z>generated by the two differential

indeterminates xand z. The variable Dbelongs to R<x,z>and the variable Rbelongs to an

obvious algebraic extension Dof R<x,z>, which deﬁnes the system.

B Dynamic feedbacks versus endogenous feedbacks

Adynamic feedback between two systems D/kand ˜

D/kconsists in a ﬁnitely differential extension

E/ksuch that D⊂Eand ˜

D⊂E. Assume moreover that the extension E/˜

Dis differentially algebraic.

According to theorem 1, the (non-differential) transcendence degree of E/˜

Dis ﬁnite, say ν. Choose a

transcendence basis z=(z1,...,zν)of E/˜

D. It yields like (8):

Aα(˙zα,z)=0α=1,...,ν

B(ξ, z)=0

40

where ξis any element of Eand the Aα’s and Bare polynomials over ˜

D.

The above formulas are the counterpart in the ﬁeld theoretic language of the usual ones for deﬁn-

ing general dynamic feedbacks (see, e.g., (Isidori 1989, Nijmeijer and van der Schaft 1990)). The

dynamic feedback is said to be regular if, and only if, E/Dand E/˜

Dare both differentially algebraic.

The following generalization of proposition 2 is immediate: the systems D/kand ˜

D/kpossess the

same differential order, i.e., the same number of independent input channels.

The situation of endogenous feedbacks is recovered when E/Dand E/˜

Dare both algebraic, i.e.,

ν=0.

C Proof of proposition 5

1/κ

P(s)τ

ν

Figure 13: Frénet frame (τ, ν) and curvature κof a smooth planar curve.

The Frénet formula Let us recall some terminology and relations relative to planar smooth curves

that are displayed on ﬁgure 13 (see, e.g., (Dubrovin et al. 1984)). A curve parameterization Rs→

P(s)∈R2is called regular if, and only if, for all s,dP

ds = 0. A curve is called smooth if, and only

if, it admits a regular parameterization. A parameterization is called natural if, and only if, for all s,

dP

ds

=1 where denotes the Euclidian norm. For smooth curves with a natural parameterization

s→P(s), its signed curvature κis deﬁned by dτ

ds =κν, where τ=dP

ds is the unitary tangent vector

and νis the oriented normal vector ((τ, ν ) is a direct orthonormal frame of the oriented Euclidian plane

R2). Notice that dν

ds =−κτ. Every smooth curve admits a natural parameterization: every regular

parameterization t→P(t)</