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 n trailers, 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 borrowedfromnonlinear physics. A high 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
pendulums andthemotionplanningof the car with n-trailers. Theyare all treated within the framework
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 ∈ R
n
, u ∈ R
m
), (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 ∈ R
q
,v∈ R
m
)
(2)
where a(0, 0, 0) = 0, b(0, 0, 0) = 0. The regularity assumption implies the invertibility
1
of
system (2) with input v and output u.
2. a diffeomorphism
ξ = (x, z)(ξ∈ R
n+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
(ν
1
)
1
= v
1
.
.
.
y
(ν
m
)
m
= v
m
where ν
1
, ..., ν
m
are the controllability indices and (y
1
,...,y
(ν
1
−1)
1
,...,y
m
,...,y
(ν
m
−1)
m
) is another ba-
sisofthe vectorspacespannedby the componentsof ξ . SetY = (y
1
,...,y
(ν
1
−1)
1
,...,y
m
,...,y
(ν
m
−1)
m
);
1
See (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 = T ξ where T is an invertible (n + q) × (n + q) matrix. Otherwise stated, Y = T(x, z).
The invertibility of yields
x
z
=
−1
(T
−1
Y). (4)
Thus from (2) u = b
−1
(T
−1
Y), v
. From v
i
= y
(ν
i
)
i
, i = 1,...,m, u and x can be expressed
as real-analytic functions of the components of y = (y
1
,...,y
m
) 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 y can be expressed as a real-analytic function of x, u and a finite number of
its derivatives:
y = C(x, u, ˙u,...,u
(γ )
). (6)
Note that, according to (4), this amounts to expressing z as 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 y plays 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 n trailers (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
output and a finite number of itsderivatives. This general idea can be traced back to worksby D. 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 y trajectories,
x and u trajectories 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,
output and other variables. The equations moreovermight be implicit. This standpoint, which matches
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 the references 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
a finite 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)). WeexploithereanideaduetotheRussianphysicistKapitsa(Bogaevski and Povzner 1991,
Landau and Lifshitz 1982)for stabilizing these three systemsin the neighborhood of quitearbitrary po-
sitionsand trajectories, and in particularpositions which arenot equilibrium points. This ideais closely
related to a curiosity of classical mechanics that a double inverted pendulum (Stephenson 1908), and
even the N linked 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 organizedas follows. After some differentialalgebraic preliminaries, we define equiv-
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.
Firstdraftsofvariouspartsofthisarticlehavebeenpresentedin(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.
A constant c ∈ R is 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.
A differential field extension L/K is given by two differential fields, K and L, such that K ⊆ L
and such that the restriction to K of the derivation of L coincides with the derivation of K.
An element ξ ∈ L is 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[x
0
, x
1
,...,x
ν
], π = 0, such that
π(ξ,
˙
ξ,...,ξ
(ν)
) = 0. The extension L/K is said to be differentially algebraic if, and only if, any
element of L is differentially K-algebraic.
5
