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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 where 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 n trailers, the motion planning of which is obtained via elementary properties of plane curves. The three non-flat examples, the simple, double and variable length pendulums, are borrowed from non-linear physics. A high frequency control strategy is proposed such that the averaged systems become flat.
Michel FliessJean LévinePhilippe 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
Education Nationale”.
1 Introduction
We present here ve 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 elds and functions are real-analytic.
˙x=f(x,u)(xRn,uRm), (1)
where f(0,0)=0 and rankf
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
u=b(x,z,v) (zRq,vRm)(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
such that (1)and (2), whose (n+q)-dimensional dynamics is given by
becomes, according to (3), a constant linear controllable system ˙
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)),
whereν1,...,νmarethecontrollability indicesand (y1,...,y11)
m)isanother ba-
sisof the vectorspacespanned bythe componentsofξ. Set Y=(y1,...,y11)
1See (Li and Feng 1987) for a denition 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.
thus Y=Tξwhere Tis an invertible (n+q)×(n+q)matrix. Otherwise stated, Y=T(x,z).
The invertibility of yields x
z=1(T1Y). (4)
Thus from (2) u=b1(T1Y), v. From vi=yi)
i,i=1,...,m,uand xcan be expressed
as real-analytic functions of the components of y=(y1,...,ym)and of a nite 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 nite 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 justies 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 ctitious 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 at 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 at. We
treatheretwocase-studies, namelythecrane(DAndr´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 atness 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 at
outputand anite 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, atness is best dened by not distinguishing between input, state,
outputand other variables. Theequations moreovermight beimplicit. This standpoint, whichmatches
well with Willemsapproach (Willems 1991), is here taken into account by utilizing differential
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 Kalmans 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,Willemstrajec-
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-at systems
which verify the strong accessibility property (Sussmann and Jurdjevic 1972). We introduce a non-
negative integer, the defect, which measures the distance from atness.
These new concepts and mathematical tools are providing the common formalism and the under-
lying structure of ve physically motivated case studies. The rst 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 at 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
at 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
anite 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-at systems is based on high frequency control and approxima-
tions by averaged and at 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),
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
The paper is organized as follows. After some differentialalgebraic preliminaries, we deneequiv-
alence by endogenous feedback, atness and defect. Their implications for uncontrolled dynamics
and linear systems are examined. We discuss the link between atness and controllability. In order
to verify that some systems are not linearizable by dynamic feedback, we demonstrate a necessary
condition of atness, which is of geometric nature. The last two sections are devoted respectively to
the at and non-at 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 Willemsbehavioral approach
(Willems 1991), where trajectories play a key role. We start with a brief review of differential elds
(see also (Fliess 1990a, Fliess and Glad 1993)) and we refer to the books of Ritt (Ritt 1950) and
Kolchin (Kolchin 1973) and Seidenbergs paper (Seidenberg 1952) for details. Basics on the cus-
tomary (non-differential) eld 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
dt R
dt(ab)ab +a˙
Aconstant c Ris an element such that ˙c=0. A ring of constants only contains constant elements.
An (ordinary) differential eld is an (ordinary) differential ring which is a eld.
Adifferential eld extension L/Kis given by two differential elds, Kand L, such that KL
and such that the restriction to Kof the derivation of Lcoincides with the derivation of K.
An element ξLis said to be differentially K-algebraic if, and only if, it satises an algebraic
differential equation over K, i.e., if there exists a polynomial πK[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.
An element ξLis 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|iI}of elements in Lis said to be differentially K -algebraically independent if,
and only if, the set of derivatives of any order, {ξ(ν)
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 nitely generateddifferentialextension L/K, the next two properties are equivalent:
(i) L/K is differentially algebraic;
(ii) the (non-differential) transcendence degree of L/Kisnite, 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 eld. A system is a nitely generated differential extension D/k3.
Such a denition corresponds to a nite number of quantities which are related by a nite number of
algebraic differential equations over k4. We do not distinguish in this setting between input, state,
output and other types of variables. This eld-theoretic language therefore ts 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 eld generated by the four unknowns x1,x2,x3,x4
related by the two algebraic differential equations:
Clearly, diff trd0D/k=2: it is equal to the number of unknowns minus the number of equations.
Denote by k<u>the differential eld generated by kand by a nite 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, identied 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 concreteequations in the language of eld 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
verication of the prime character of the differential ideals corresponding to all our examples is done in appendix A.
diff trd0k<u>/k=m.Adynamics with (independent) input u isanitely 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 nite set of
differential quantities in D.
According to theorem 1, there exists a nite 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:
where the Ais and Bjs 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 u2x4. The extension D/R<u>is differentially
algebraic and yields the representation
The dimension of the dynamics is 3 and (x1,x2,x4)is a generalized state. It would be 5 if we set
u1x3and u2x4, and the corresponding representation becomes causal in the classical sense.
Remark 1 Take the dynamics D/k<u>and a nitely 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 dened up to a nitely generated algebraic extension of D.
2.3 Modules and linear systems 5
Differential elds are to general for linear systems which are specied by linear differential equations.
They are thus replaced by the following appropriate modules.
5See also (Fliess 1990b).
Let kbe again a given differential ground eld. Denote by kd
dtthe ring of linear differential
operators of the type
This ring is commutative if, and only if, kisaeld of constants. Nevertheless, in the general
non-commutative case, kd
dtstill is a principal ideal ring and the most important properties of left
dt-modules mimic those of modules over commutative principal ideal rings (see (Cohn 1985)).
Let Mbe a left kd
dt- module. An element mMis said to be torsion if, and only if, there exists
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 nitely 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 nite.
Anitely generated module Mis free if, and only if, its torsion submodule Tis trivial, i.e., T={0}6.
Any nitely generated module Mcan be written M=Twhere 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 denition, a nitelygenerated leftkd
dt-module. Wearethus dealingwith
anite number of variables which are related by a nite number of linear homogeneous differential
equations and our setting appears to be strongly related to Willemsapproach (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
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 denition of free modules, but a characterization which holds for nitely generated modules over
principal ideal rings, where any torsion-free module is free (see (Cohn 1985)).
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:
nite Dν
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 Willemstrajectory characterization (Willems 1991).
When the system is uncontrollable, thetorsion submodule corresponds to theKalman uncontrollability
Remark 2 The relationshipwiththe generaldifferentialeldsetting 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 eld D, which is a differential eld,
corresponds to the nonlinear eld 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).
Toanitely generated differential extension L/K, associate a mapping dL/K:LL/K, called
ahler) differential 7and where L/Kisanitely generated left Ld
dt-module, such that
dt =d
dt dL/Ka
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=1,...,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 aL,dL/Kashould be intuitively understood, likein analysis anddifferentialgeometry, as a smallvariation
of a.
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
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, atness 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
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 eld generated by Dand ˜
D:K/Dand K/˜
Dare algebraic
extensions. Therefore,
diff trd0D/k=diff trd0K/k=diff trd0˜
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
and ˜
˜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 denition of equivalence can also be read as follows: two systems D/kand ˜
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.
between (11) and (12):
˜u,..., ˜ui))=0i=1,...,m
˜u,..., ˜uα))=0α=1,...,n
˜σα(˜xα,x,u,˙u,...,u(˜µα))=0α=1,..., ˜n
where the ϕis, σαs, ˜ϕis and ˜σαs 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, arealmostidentical. Taketwo equivalentsystems D1/k andD2/k and denoteby
Dthesmallest algebraicextensionofD1and D2. Itis straightforwardto checkthatthethreeleft Dd
modulesD/k,DD1D1/kand DD2D2/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|iI}of L/Ksuch
that L=K>. A system D/kis called purely differentially transcendental if, and only if, the
extension D/kis so.
A system D/kis called (differentially) at 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 at output of the system D/k.
Example (continued) Let us prove that y=(y1,y2)with
is a linearizing output for (7). Set σ=x1x3x4. Differentiating y1=x2+σ2/2y(3)
2, we have, using
(7), σ2=−
. Thus x2=y1σ2
is an algebraic function of (y1,˙y1,y(3)
2). Since
σand x1=σ−¨y2x4,x4and x1are algebraic functions of (y1,˙y1,¨y1,¨y2,y(3)
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)
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/kisat 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 eld-theoretic language (Fliess 1990a), a nitely 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 denition, the defect of its associated differential eld 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 at if, and only if, it is
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 at. When T= {0}, the differential eld 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:
where the νis 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:
where M is a d ×d matrix over R.
3.4 A necessary condition for atness
Consider the system D/kwhere D=k<w>is generated by a nite set w=(w1,...,w
q). The
wis are related by a nite set, (w, ˙w,...,w
(ν) )=0, of algebraic differential equations. Dene the
algebraic variety Scorresponding to (ξ0,...,ξν)=0inthe+1)q-dimensional afne space with
q), j=0,1,...,ν.
Theorem 3 If the system D/kisat, the afne algebraic variety S contains at each regular point a
straight line parallel to the ξν-axes.
Proof The components of w,˙w,...,w 1)are algebraically dependent on the components of a
linearizing output y=(y1,..., ym)and a nite number of their derivatives. Let µbe the highest order
of these derivatives. The components of w ) depend linearly on the components of y+1), which play
the role of independent parameters for the coordinates ξν
Theabove conditionis not sufcient. Considerthe systemD/Rgenerated by(x1,x2,x3)satisfying
˙x1=(˙x2)2+(˙x3)3. This system does not satisfy the necessary condition: it is not at. The same
system Dcan be dened via the quantities (x1,x2,x3,x4)related by ˙x1=(x4)2+(˙x3)3and x4x2.
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
denition 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 denition is invariant under our equivalence via endogenous feedback.
Proposition 4 Aat system is controllable
Proof It sufces 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
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 nite 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 innitesimalgeneralization of linear controllability, atness should
be viewed as a more globaland, 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 at systems for which the control design is straightforward.
4 Examples and control of at systems
The verication of the prime character of the differential ideals corresponding to all our examples
is done in appendix A. This means that the equations dening all our examples can be rigorously
interpreted in the language of differential eld theory.
4.1 The 2-D crane
Figure 1: The two dimensional crane.
Consider the crane displayed on gure 1 which is a classical object of control study (see, e.g.,
(DAndr´ea-Novel and L´evine 1990), (Marttinen et al. 1990)). The dynamics can be divided into two
parts. The rst 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 xed 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:
(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:
that is
Thus, system (14) is at with (x,z)as linearizing output.
Remark 5 Assume that the modeling equations (14) are completed with the following traversing and
hoisting dynamics:
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)are constant physical parameters. Then (14,16)
is also at with the same linearizing output (x,z). This explains without any additional computation
whythe systemconsideredin(DAndr´
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 dened, for t[t1,t2], by
D(t)=α(t)¨α(t)γ (t)
R(t)=γ2(t)+¨α(t)γ (t)
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 tt2. 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):
Dcosθgsin 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 nal conguration close to the steady-state dened by D2and R2.
In the simulations displayed here below, we have veried that the addition of reasonable fast and
stable regulator dynamics modies only slightly the nal 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/g0.3 s) (see (Fliess et al. 1991)).
010 20
x (m)
z (m)
load trajectory
05 10 15
time (s)
trolley position
05 10 15
time (s)
rope length
05 10 15
time (s)
vertical deviation angle
Figure 2: Simulation of the control dened by (17) without (solid lines) and with (dot lines) ideal
low-level controllers for Dand R.
For the simulations presented in gure 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 nal
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
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
This system is clearly at with the cartesian coordinates of the load, (x1,x2,z),asat output.
Remark 7 In (DAndr´
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 at as a consequence of the following
considerations. According to an old result due to Huygens (see, e.g. (Whittaker 1937, p. 131132)),
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)):
(ux)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, Huygensoscillation 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). Theatness 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 i1 are considered in
(Martin and Rouchon 1993, Rouchon et al. 1993b).
Figure 3: The kinematic car with ntrailers.
of the motion planning problem without obstacles. Notice that most of nonholonomic mobile robots
are at (DAndr´ea-Novel et al. 1992a, Campion et al. 1992).
The hitch of trailer iis attached to the center of the rear axle of trailer i1. 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
simplied 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 gure 3):
˙y0=u1sin θ0
j=1cosj1θj)sini1θi)for i=1,...,n
where (x0,y0
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 gure 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).
4.2.2 Cartesian coordinates of Pnas at output
Denote by (xi,yi)the cartesian coordinates of Pi,i=0,1,...,n:
A direct computation shows that tanθi=˙yi
˙xi. Since, for i=0,...,n1, xi=xi+1+di+1cosθi+1
and yi=yi+1+di+1sinθi+1, the variables θn,xn1,yn1,θn1,...,θ1,x0,y0and θ0are functions of xn
and ynand their derivatives up to the order n+1. But u1x0/cos θ0, tanφ=d0˙
θ0/u1and u2=˙
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 at 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 θiyi/˙xi, can be avoided by
the following developments.
4.2.3 Motion planing using atness
In (Rouchon et al. 1993a, Rouchon et al. 1993b), the following result was sketched.
Proposition 5 Consider (18)and twodifferentstate-space congurations: ˜p=(˜x0,˜y0,˜
φ, ˜
θ0,..., ˜
and p=(x0,y0,φ, θ0,...,θn). Assume that the angles ˜
θi,i =1,...,n, ˜
i=1,...,n, and φbelong to ]π/2/2[. 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 anglesθi1θi,
i=1,...,n, and φ(i =1,...,n) always remain in ]π/2/2[ and such that (u1(t), u2(t)) =0
for t =0,T.
The conditions θi1θi]π/2/2[ (i=1,...,n) and φ]π/2/2[ are meant for avoiding
some undesirable geometric congurations: trailer ishould not be in front of trailer i1.
The detailed proof is given in the appendix and relies basically on the fact that the system is at. It
is constructive and gives explicitly (u1(t), u2(t)). The involved computations are greatly simplied 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 gure 4, the point Pi1
belongs to the tangent to Ciat Piand at the xed distance difrom 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
where νiis the unitary vector orthogonal to τiand κiis the curvature of Ci. Since d
dsiPi1gives the
tangent direction to Ci1,wehave tani1θi)=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 gures 5 and 6 have been written in MATLAB. They
can be obtained upon request from the fourth author via electronic mail (
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 nal angles 2
0) =0. We consider the two smooth curves
CAB and CCB of the gure 5, dened by their natural parameterizations [0,LAB]sPAB(s)and
[0,LCB]sPCB(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 nal system conguration in AandCimpose κAB(0)=d
and κCB(0)=d
ds2κCB(0)=0. We impose additionally that AB and CB are tangent at B
Figure 5: parking the car with two trailers from Ato Bvia C.
It is straightforward to nd curves satisfying such conditions. For the simulation of gure 6, we take
polynomial curves of degree 9.
Proposition 5 implies that, if P2follows CAB and CCB as displayed on gure 5, then the initial and
nalstates willbe asdesired. Takeasmooth function[0,T]ts(t)[0,LAB]suchthat s(0)=0,
s(T)=LAB and ˙s(0)s(T)=0. This leads to smooth control trajectories [0,T]tu1(t)0
and [0,T]tu2(t)steering the system from Aat time t=0toBat time t=T. Similarly,
[T,2T]ts(t)[0,LCB] such that s(T)=LCB,s(2T)=0 and ˙s(T)s(2T)=0 leads to
control trajectories [T,2T]tu1(t)0 and [T,2T]tu2(t)steering the system from B
to C. This gives the motions displayed on gure 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
gure 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
Figure 6: the successive motions of the car with two trailers.
We have
and ds1=1+d2
2ds2. Similarly,
and ds0=1+d2
1ds1. Thus u1is given explicitly by
dt =1+d2
where[0,T]tx(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) =d0κ0,weget
dt =d0
Here, we are not actuallyconcerned with obstacles. The fact that the internalcongurationdepends
only on the curvature results from the general following property: a plane curve is entirely dened (up
to rotation and translation) by its curvature. For the n-trailer case, the angles θnθn1,...,θ1θ0and
φdescribing the relative conguration of the system are only functions of κnand its rst n-derivatives
with respect to sn.
Consequently, limitations due to obstacles can be expressed up to a translation (dened by Pn)
and a rotation (dened by the tangent direction dP
dsn) via κnand its rst n-derivatives with respect to
sn. Such considerations can be of some help in nding 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 at: the at 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 gure 4.
5 High-frequency control of non-at systems
We address here a method for controlling non-at systems via their approximations by averaged
and at 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 eld 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
stabilizedwhen 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
Figure 7: The Kapitsa pendulum: the suspension point oscillates rapidly on a vertical axis.
The notation are summarized on gure 7. We assume that the vertical velocity ˙z=uof the suspension
point is the control. The equations of motion are:
l2cosαsin αu
where pis proportional to the generalized impulsion; gand lare physical constants. This sys-
tem is not at 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 l/g. It is then natural to consider the following
averaged control system:
2l2cosαsin αu1
It admits two control variables, u1and u2, whereas the original system (19) admits only one, u.
Moreover (20) is at with (α, z)as linearizing output.
The endogenous dynamic feedback
cosαsin αv2
transforms (20) into ¨
where the parameters τ1
2>0 and αsp ]π/2/2[/{0}. Then, the closed-loop averaged system
(20,21,23) admits an hyperbolic equilibrium point characterized by (zsp
sp)that is asymptotically
Consider now (19) and the high-frequency control u=u1+u2sin(t/ε) with 0 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)=sp,0,zsp)(Guckenheimer and Holmes 1983, theorem 4.1.1, page
168): such control maintains (z) near (zsp
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 gure 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 ms2. The design control parameters are
ε=0.025/2πs and τ1=τ2=0.10 s. For the two upper graphics of gure 8, no error is introduced:
control is computed with l=0.10 m and g=9.81 ms2. For the two lower graphics of gure 8,
parameter errors are introduced: control is computed with with l=0.11 m and g=9.00 ms2.
01 2
time (s)
no parameter error
01 2
time (s)
v (m)
no parameter error
01 2
time (s)
parameter error
01 2
time (s)
v (m)
parameter error
Figure 8: Robustness test of the high-frequency control for the inverted pendulum.
5.2 The variable-length pendulum
Figure 9: pendulum with variable-length.
Let us consider the variable-length pendulum of (Bressan and Rampazzo 1993). The notations are
summarized on gure 9. We assume as in (Bressan and Rampazzo 1993) that the velocity ˙u=vis
the control. The equations of motion are:
where mass and gravity are normalized to 1.
This system is not at 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
where v1and v2are auxiliary controls, 0 1 .We consider the averaged control system:
This system is obviously linearizable via static feedback with (q,u)as linearizing output.
The static feedback
transforms (25) into ˙
with τ1
2>0, usp ]π/2/2[, 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),01;
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 gure 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.
= 0.02
= 0.02
= 0.04
= 0.04
Figure 10: high-frequency control for the variable-length pendulum.
5.3 The inverted double pendulum
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 gure 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˙α2cos1α2)+n1˙xcos α1n1˙zsin α1
p2=I˙α1cos1α2)+I2˙α2+n2˙xcos α2n2˙zsin α2
˙p1=n1gsinα1n1˙α1˙xsinα1n1˙α1˙zcos α1
˙p2=n2gsinα2n2˙α2˙xsinα2n2˙α2˙zcos α2
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:
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,isnotat.
Proof The proof is just an application of the necessary atness condition of theorem 3. Since ux
and vz, (29) is at if, and only if, the reduced system,
p1=I1˙α1+I˙α2cos1α2)+n1˙xcos α1n1˙zsin α1
p2=I˙α1cos1α2)+I2˙α2+n2˙xcos α2n2˙zsin α2
˙p1=n1gsinα1n1˙α1˙xsinα1n1˙α1˙zcos α1
˙p2=n2gsinα2n2˙α2˙xsinα2n2˙α2˙zcos α2
is at. Denote symbolically by F, ˙
ξ) =0 the equations (30) where ξ=1
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 λR,F, ζ +λa)=0. The second order conditions, d2
dλ2λ=0F, ζ +λa)=0, lead
to aα1(axsinα1+azcos α1)=0,aα2(axsinα2+azcos α2)=0.
Two rst order conditions, d
dλλ=0F, ζ +λa)=0, are
axcosα1+azsin α1=I1
axcosα2+azsin α2=I
Simple computations show that, if I
n1= I2
n2and I1
n1= I
n2(these conditions are always satised for
homogeneous identical beams), then (aα1,aα2,ax,az)=0. The two remaining rst order conditions
imply that (ap1,ap2)=0. Thus a=0 and the inverted double pendulum is not at.
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-at system (29) by a at one, we set u=u1+u2cos(t/ε) and v=
v1+v2cos(t/ε) where 0 min I1
n2gand u1,u2,v1,v2are new control variables. This
leads to a at averaged system with 1
2,x,z)as the linearizing output. The endogenous dynamic
feedback that linearized the averaged system provides then (u1,u2,v
2). For the simulations of
gure 12, the angles α1and α2follow approximately prescribed trajectories whereas, simultaneously,
the suspension point (x,z)is maintained approximately constant.
time (s)
vertical deviation of beam 1
time (s)
vertical deviation of beam 2
Figure 12: Simulation of the inverted double pendulum via high-frequency control.
6 Conclusion
Our ve examples, as well as other ones in preparation in various domains of engineering, indicate that
atness and defect ought to be considered as physical and/or geometric properties. This explains why
at 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 atness and defect bring a new theoretical and practical
insight in control. We briey list some important open problems:
Ritts work (Ritt 1950) shows that differential algebra provides powerful algorithmic means (see
(Diop 1991, Diop 1992) for a survey and connections with control). Can atness 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
at (see, e.g., (Coron and DAndr´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 rst step in this
direction and the references therein).
the two averaged systems associated to high-frequency control are at. Can this result be
generalized to a large class of devices?
differentialalgebrais nottheonly possiblelanguage forinvestigatingatnessand 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.
Acheson, D. 1993. A pendulum theorem. Proc. R. Soc. Lond. A 443, 239245.
Baillieul,J. 1993.Stable averagemotionsof mechanical systemssubject toperiodic forcing.preprint.
Bentsman, J. 1987. Vibrational control of a class of nonlinear multiplicative vibrations. IEEE Trans.
Automat. Control 32, 711716.
Bogaevski, V. and A. Povzner 1991. Algebraic Methods in Nonlinear Perturbation Theory. Springer,
New York.
Bressan, A. and F. Rampazzo 1993. On differential systems with quadratic impulses and their
applications to Lagrangian mechanics. SIAM J. Control Optimization 31, 12051220.
Bushnell, L., D. Tilbury and S. Sastry 1993. Steering chained form nonholonomic systems using
sinusoids: the retruck example.. In Proc. ECC93, Groningen. pp. 14321437.
Campion, G., B. DAndr´ea-Novel, G. Bastin and C. Samson 1992. Modeling and feedback control
of wheeled mobile robots. In Lecture Note of the Summer School on Theory of Robots, Grenoble.
Cartan, E. 1915. Sur lint´egration de certains syst`emes ind´etermin´es d’´equations diff´erentielles. J.
ur reine und angew. Math. 145,8691. also in Oeuvres Compl`etes, part II, vol 2, pp.11641174,
CNRS, Paris, 1984.
Charlet, B., J. L´evine and R. Marino 1989. On dynamic feedback linearization. Systems Control
Letters 13, 143151.
Charlet, B., J. L´evine and R. Marino 1991. Sufcient conditions for dynamic state feedback lin-
earization. SIAM J. Control Optimization 29,3857.
Claude, D. 1986. Everything you always wanted to know about linearization. In M. Fliess and
M. Hazewinkel (Eds.). Algebraic and Geometric Methods in Nonlinear Control Theory. Reidel,
Dordrecht. pp. 181226.
Cohn, P. 1985. Free Rings and their Relations. second edn. Academic Press, London.
Coron, J. 1992. Global stabilization for controllable systems without drift. Math. Control Signals
Systems 5, 295312.
Coron, J. 1994. Linearized control systems and applications to smooth stabilization. SIAM J. Control
Coron, J. and B. DAndr´ea-Novel 1992. Smooth stabilizing time-varying control laws for a class of
nonlinearsystems: applicationto mobilerobots.In Proc. IFAC-SymposiumNOLCOS92,Bordeaux.
pp. 658672.
DAndr´ea-Novel, B. and J. L´evine 1990. Modelling and nonlinear control of an overhead crane. In
M. Kashoek, J. van Schuppen and A. Rand (Eds.). Robust Control of Linear and Nonlinear Systems,
MTNS89. Vol. II. Birkh¨auser, Boston. pp. 523529.
DAndr´ea-Novel, B., G. Bastin and G. Campion 1992a. Dynamic feedback linearization of nonholo-
nomic wheeled mobile robots. In IEEE Conference on Robotics and Automation, Nice. pp. 2527
DAndr´ea-Novel, B., Ph. Martin and R. S´epulchre 1992b. Full dynamic feedback linearization of a
class of mechanical systems. In H. Kimura and S. Kodama (Eds.). Recent Advances in Mathematical
Theory of Systems, Control, Network andSignal Processing II(MTNS-91, Kobe, Japan), Mita Press,
Tokyo. pp. 327333.
Delaleau, E. and M. Fliess 1992. Algorithme de structure, ltrations et d´ecouplage. C.R. Acad. Sci.
Paris I315, 101106.
Di Benedetto, M., J.W. Grizzle and C.H. Moog 1989. Rank invariants of nonlinear systems. SIAM
J. Control Optimization 27, 658672.
Diop, S. 1991. Elimination in control theory. Math. Control Signals Systems 4,1732.
Diop, S. 1992. Differential-algebraic decision methods and some applications to system theory.
Theoret. Comput. Sci. 98, 137161.
Dubrovin, B., A.T. Fomenko and S.P. Novikov 1984. Modern Geometry Methods and Applications
- Part I. Springer, New York.
Fliess, M. 1989. Automatique et corps diff´erentiels. Forum Math. 1, 227238.
Fliess, M. 1990a. Generalized controller canonical forms for linear and nonlinear dynamics. IEEE
Trans. Automat. Control 35, 9941001.
Fliess, M. 1990b. Some basic structural properties of generalized linear systems. Systems Control
Letters 15, 391396.
Fliess, M. 1992. A remark on Willems trajectory characterization of linear controllability. Systems
Control Letters 19,4345.
Fliess, M. and M. Hasler 1990. Questioning the classical state space description via circuit examples.
In M. Kashoek, J. van Schuppen and A. Ran (Eds.). Realization and Modelling in System Theory,
MTNS89. Vol. I. Birkh¨auser, Boston. pp. 112.
Fliess,M. and S.T.Glad1993.Analgebraic approachtolinearand nonlinear control.InH.Trentelman
andJ.Willems(Eds.).Essays onControl: PerspectivesintheTheoryand itsApplications.Birkh¨auser,
Boston. pp. 223267.
Fliess, M., J. L´evine and P. Rouchon 1991. A simplied approach of crane control via a generalized
state-space model. In Proc. 30th IEEE Control Decision Conf., Brighton. pp. 736741.
Fliess, M., J. L´evine and P. Rouchon 1993a. A generalized state variable representation for a sim-
plied crane description. Int. J. Control 58, 277283.
Fliess, M., J. L´evine, Ph. Martin and P. Rouchon 1992a. On differentially at nonlinear systems. In
Proc. IFAC-Symposium NOLCOS92, Bordeaux. pp. 408412.
Fliess, M., J. L´evine, Ph. Martin and P. Rouchon 1992b. Sur les syst`emes non lin´eaires
diff´erentiellement plats. C.R. Acad. Sci. Paris I315, 619624.
Fliess, M., J. L´evine, Ph. Martin and P. Rouchon 1993b.D´efaut dun syst`eme non lin´eaire et com-
mande haute fr´equence. C.R. Acad. Sci. Paris I-316, 513518.
Fliess, M., J. L´evine, Ph. Martin and P.Rouchon 1993c.Differential atnessand defect: anoverview.
In Workshop on Geometry in Nonlinear Control, Banach Center Publications, Warsaw.
Fliess, M., J. L´evine, Ph. Martin and P. Rouchon 1993d. Lin´earisation par bouclage dynamique et
transformations de Lie-B¨acklund. C.R. Acad. Sci. Paris I-317, 981986.
Fliess, M., J. L´evine, Ph. Martin and P. Rouchon 1993e. Towards a new differential geometric setting
in nonlinear control. In Proc. Internat. Geometric. Coll., Moscow.
Guckenheimer, J. and P. Holmes 1983. Nonlinear Oscillations, Dynamical Systems and Bifurcations
of Vector Fields. Springer, New York.
Hartshorne, R. 1977. Algebraic Geometry. Springer, New York.
Hilbert, D. 1912. ¨
Uber den Begriff der Klasse von Differentialgleichungen. Math. Ann. 73,95108.
also in Gesammelte Abhandlungen, vol. III, pp. 8193, Chelsea, New York, 1965.
Isidori, A. 1989. Nonlinear Control Systems. 2nd edn. Springer, New York.
Jacobson, N. 1985. Basic Algebra, I and II. 2nd edn. Freeman, New York.
Jakubczyk, B. 1992. Remarks on equivalence and linearization of nonlinear systems. In Proc. IFAC-
Symposium NOLCOS92, Bordeaux. pp. 393397.
Johnson, J. 1969. K¨ahler differentials and differential algebra. Ann. Math. 89,9298.
Kailath, T. 1980. Linear Systems. Prentice-Hall, Englewood Cliffs, NJ.
Kolchin, E. 1973. Differential Algebra and Algebraic Groups. Academic Press, New York.
Landau, L. and E. Lifshitz 1982. Mechanics. 4th edn. Mir, Moscow.
Laumond, J. 1991. Controllability of a multibody mobile robot. In IEEE International Conf. on
advanced robotics, 91 ICAR. pp. 10331038.
Laumond, J., P.E. Jacobs, M. Taix and R.M. Murray 1993. A motion planner for nonholonomic
mobile robots. preprint.
Li, C.-W. and Y.-K. Feng 1987. Functionalreproducibility of generalmultivariableanalytic nonlinear
systems. Int. J. Control 45, 255268.
Martin, P. 1992. Contribution `al’´etude des syst`emes diff`erentiellement plats. PhD thesis. ´
Ecole des
Mines de Paris.
Martin, P. 1993. An intrinsic condition for regular decoupling. Systems Control Letters 20, 383391.
Martin, P. and P. Rouchon 1993. Systems without drift and atness. In Proc. MTNS 93, Regensburg,
Marttinen, A., J. Virkkunen and R.T. Salminen 1990. Control study with a pilot crane. IEEE Trans.
Edu. 33, 298305.
Meerkov, S. 1980. Principle of vibrational control: theory and applications. IEEE Trans. Automat.
Control 25, 755762.
Monaco, S. and D. Normand-Cyrot 1992. An introduction to motion planning under multirate digital
control. In Proc. 31th IEEE Control Decision Conf.,Tucson. pp. 17801785.
Moog, C., J. Perraud, P. Bentz and Q.T. Vo 1989. Prime differential ideals in nonlinear rational
control systems. In Proc. NOLCOS89, Capri, Italy. pp. 178182.
Murray, R. and S.S. Sastry 1993. Nonholonomic motion planning: Steering using sinusoids. IEEE
Trans. Automat. Control 38, 700716.
Nijmeijer, H. and A.J. van der Schaft 1990. Nonlinear Dynamical Control Systems. Springer, New
Pomet, J. 1993. A differential geometric setting for dynamic equivalence and dynamic linearization.
In Workshop on Geometry in Nonlinear Control, Banach Center Publications, Warsaw.
Ritt, J. 1950. Differential Algebra. Amer. Math. Soc., New York.
Rouchon, P. 1994. Necessary condition and genericity of dynamic feedback linearization. J. Math.
Systems Estim. Control. In press.
Rouchon, P., M. Fliess, J. L´evine and Ph. Martin 1993a. Flatness and motion planning: the car with
n-trailers.. In Proc. ECC93, Groningen. pp. 15181522.
Rouchon, P., M. Fliess, J. L´evine and Ph. Martin 1993b. Flatness, motion planning and trailer
systems. In Proc. 32nd IEEE Conf. Decision and Control, San Antonio. pp. 27002705.
Sagdeev, R., D.A. Usikov and G.M. Zaslavsky 1988. Nonlinear Physics. Harwood, Chur.
Seidenberg, A. 1952. Some basic theorems in differential algebra (characteristic p, arbitrary). Trans.
Amer. Math. Soc. 73, 174190.
Shadwick, W. 1990. Absolute equivalence and dynamic feedback linearization. Systems Control
Letters 15,3539.
Sontag, E. 1988. Finite dimensional open loop control generator for nonlinear control systems.
Internat. J. Control 47, 537556.
Sontag, E. 1992. Universal nonsingular controls. Systems Controls Letters 19, 221224.
Stephenson, A. 1908. On induced stability. Phil. Mag. 15, 233236.
Sussmann, H. and V. Jurdjevic 1972. Controllability of nonlinear systems. J. Differential Equations
Sussmann,H. andW.Liu 1991.Limitsof highlyoscillatorycontrols andthe approximationofgeneral
paths by admissible trajectories. In Proc. 30th IEEE Control Decision Conf., Brighton. pp. 437442.
Tcho´n, K. 1994. Towards robustness and genericity of dynamic feedback linearization. J. Math.
Systems Estim. Control.
Tikhonov, A., A. Vasileva and A. Sveshnikov 1980. Differential Equations. Springer, New York.
Tilbury, D. and A. Chelouah 1993. Steering a three-input nonholonomic using multirate controls. In
Proc. ECC93, Groningen. pp. 14281431.
Tilbury, D., O. Sørdalen, L. Bushnell and S. Sastry 1993. A multi-steering trailer system: conver-
sion into chained form using dynamic feedback. Technical Report UCB/ERL M93/55. Electronics
Research Laboratory, University of California at Berkeley.
Whittaker, E. 1937. A Treatise on the Analytical Dynamics of Particules and Rigid Bodies (4th
edition). Cambridge University Press, Cambridge.
Willems, J. 1991. Paradigms and puzzles in the theory of dynamical systems. IEEE Trans. Automat.
Control 36, 259294.
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)(n0) and u=(u1,...,um)(m0), the differential ideal corresponding to
where the ais and bis 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 ve case-studies.
Kapitsa pendulum (19) Let us replace αby σ=tan(α/2). Then, using
2˙α, cos α=1σ2
the equations (19) become explicit and rational
Theassociateddifferentialidealis thusprimeandleadsto anitelygenerateddifferentialeld 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=tan1/2)and σ2=tan2/2)prove that
the associated differential ideal is prime.
Car with n-trailers (18) Similar computations with σ=tan(ϕ/2)and σi=tani/2)prove that
the associated differential ideal is prime.
Crane (17) Analogous calculations on the generalized state variable equation R¨
Dcosθgsin θgiven in (Fliess et al. 1991, Fliess et al. 1993a) lead to a prime differential ideal.
Another more direct way for obtaining the differential eld corresponding to the crane is the
following. Take (17) and consider the differential eld 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 denes the system.
B Dynamic feedbacks versus endogenous feedbacks
Adynamic feedback between two systems D/kand ˜
D/kconsists in a nitely differential extension
E/ksuch that DEand ˜
DE. Assume moreover that the extension E/˜
Dis differentially algebraic.
According to theorem 1, the (non-differential) transcendence degree of E/˜
Dis nite, say ν. Choose a
transcendence basis z=(z1,...,zν)of E/˜
D. It yields like (8):
B, z)=0
where ξis any element of Eand the Aαs and Bare polynomials over ˜
The above formulas are the counterpart in the eld theoretic language of the usual ones for den-
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.,
C Proof of proposition 5
Figure 13: Frénet frame , 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 gure 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,
=1 where denotes the Euclidian norm. For smooth curves with a natural parameterization
sP(s), its signed curvature κis dened 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 tP(t)</