# Differential Flatness by Pure Prolongation: Necessary and Sufficient
Conditions

Jean Lévine Centre Automatique et Systèmes (CAS), MINES Paris - PSL, 75006
Paris, France, e-mail<EMAIL_ADDRESS>
(July 2023)

###### Abstract

In this article, we introduce the notion of _differential flatness by pure
prolongation_ : loosely speaking, a system admits this property if, and only
if, there exists a pure prolongation of finite order such that the prolonged
system is feedback linearizable. We obtain Lie-algebraic necessary and
sufficient conditions for a general nonlinear multi-input system to satisfy
this property. These conditions are comprised of the involutivity and relative
invariance of a pair of filtrations of distributions of vector fields. An
algorithm computing the minimal prolongation lengths of the input channels
that achieve the system linearization, yielding the associated flat outputs,
is deduced. Examples that show the efficiency and computational tractability
of the approach are then presented.

Keywords— differential flatness; Lie-Bäcklund isomorphism; Lie brackets;
distributions of vector fields; prolongation of vector fields; dynamic
feedback linearization.

##  1 Introduction

The notions of _static feedback linearization_ [17, 14] and _dynamic feedback
linearization_ of a nonlinear system, whose preliminary results were reported
in [6, 7, 32, 1] (see also [33, 34, 11, 2, 12]), were at the origin of a so
far uninterrupted thread of studies in nonlinear system theory. In particular,
during the last three decades, they gave birth to the concept of _differential
flatness_ , that plays a prominent part in motion planning and trajectory
tracking problems and their applications (see [24, 9, 10] and [21] for a
thorough presentation).

Sufficient or necessary conditions in special cases, as well as a general
differential flatness characterization, although without an upper bound on the
number of iterations of the corresponding algorithm, have been obtained (see
e.g. [21, 22] for a historical review). Nevertheless, the question of
obtaining _computationally tractable_ general necessary and sufficient
conditions for dynamic feedback linearization as well as for differential
flatness, remains open. In this paper, we restrict our study to the class of
_differentially flat systems by pure prolongation_ , i.e. , roughly speaking,
the class of $n$-dimensional systems with $m$ inputs

$\dot{x}=f(x,u_{1},\ldots,u_{m})$

for which there exists a multi-index
$\mathbf{j}\triangleq(j_{1},\ldots,j_{m})$ such that the prolonged system

$\displaystyle\dot{x}=f(x,u_{1},\ldots,u_{m}),$ (1.1) $\displaystyle
u_{i}^{(j_{i}+1)}=v_{i},\quad i=1,\ldots,m,$

of dimension $n+\sum_{i=1}^{m}j_{i}$, where we have denoted by
$u_{i}^{(j_{i})}$ the $j_{i}$-th order time derivative of $u_{i}$, is locally
static feedback linearizable.

The corresponding transformation is called _pure dynamic extension_ in [33,
34] in the context of finding upper bounds on the indices
$j_{1},\ldots,j_{m}$, if they exist. We prefer using here the word
_prolongation_ initially introduced in [7], in reference to É. Cartan [5] (see
also [32]).

This special class of differentially flat systems, initially introduced in the
framework of dynamic feedback linearization [6, 7, 32, 1], has also been
studied in [11, 12], under the name of _linearization by prolongation_ , again
in terms of upper bounds of prolongation orders, and in [2] where an
algorithm, yielding a sufficient condition, is obtained.

However, the necessary and sufficient conditions presented in this paper,
tackled via the here introduced concept of _equivalence by pure prolongation_
, are original to the author’s knowledge. Moreover, they are easily checkable
by an algorithm that only involves Lie bracket and linear algebra operations,
yielding the minimal pure prolongations if they exist.

Our main contributions are:

  * 1.

The introduction of the equivalence relation of systems by pure prolongation,
strictly coarser than the well-known equivalence by diffeomorphism and
feedback (see e.g. [17, 15, 16, 28, 20]) and strictly finer than the Lie-
Bäcklund equivalence (see e.g. [10, 21]). For this equivalence relation, the
set of flat systems by pure prolongation is identified with the equivalence
class of 0 (modulo the trivial vector field)(see definition 4.2);

  * 2.

Proposition 3.2, extending and specifying results implicitly present in
various forms in [7, 11, 2, 12], where we prove that, whatever the
prolongation $\mathbf{j}\triangleq(j_{1},\ldots,j_{m})$, the filtration of
prolonged distributions, built on the successive Lie brackets of the prolonged
drift with the prolonged control vector fields, is decomposable into the
direct sum of two filtrations of distributions. The first one, denoted by
$\\{\Delta_{k}^{(\mathbf{j})}\\}_{k\geq 0}$, is, at each point of the
prolonged manifold, generated by vector fields belonging to the original
tangent bundle, of dimension $n+m$ (in a suitably defined vertical bundle, see
(3.13)); The second one, denoted by $\\{\Gamma_{k}^{(\mathbf{j})}\\}_{k\geq
0}$, is generated by the sequence of prolonged control vector fields of
decreasing orders up to 1, i.e.
$\Gamma_{k}^{(\mathbf{j})}=\bigoplus_{i=1,\ldots,m}\\{\frac{\partial}{\partial
u_{i}^{(j_{i}-r)}}\mid r=0,\ldots,\min(j_{i}-1,k)\\}$ (see (3.12));

  * 3.

Theorem 4.1 giving the necessary and sufficient conditions: (i)
$\Delta_{k}^{(\mathbf{j})}$ must be involutive with locally constant dimension
and (ii) invariant by $\Gamma_{k}^{(\mathbf{j})}$ for all $k$, and (iii)
$\dim\Delta_{k}^{(\mathbf{j})}$ must be equal to $n+m$ for all $k$ large
enough (strong controllability);

  * 4.

Formula (4.11) and theorem 4.2 giving the prolongation lengths, if ithey
exist, or otherwise a criterion of non flatness by pure prolongation;

  * 5.

Algorithm 4.2 whose input comprises the system vector fields and whose output
is either the list of minimal prolongation lengths or the claim that the
system is not flat by pure prolongation, deduced in a finite number of steps.

The paper is organized as follows: In section 2, we present the necessary
recalls of basic results on differential flatness and feedback linearization.
Then we introduce and study the purely prolonged distributions and the
associated vertical ones in section 3. The equivalence by pure prolongation
and the definition of flatness by pure prolongation, followed by its necessary
and sufficient conditions and by the pure prolongation algorithm are then
presented in section 4. A series of five examples, four with two inputs
(sections 5.1, 5.2, 5.3 and 5.4) and one with three inputs (section 5.5) then
illustrate our results in section 5. Moreover, the pendulum example (section
5.4), is proven to be non flat by pure prolongation, though known to be
differentially flat [10, 21], thus proving that the set of flat systems by
pure prolongation is strictly contained in the set of differentially flat
ones. The paper ends with concluding remarks and an appendix establishing a
comparison formula between prolonged and non prolonged Lie brackets.

##  2 Recalls on Differential Flatness and Feedback Linearization

Consider a non-linear system over a smooth $n$-dimensional manifold $X$ given
by

$\dot{x}=f(x,u)$ (2.1)

where $x$ is the $n$-dimensional state vector, $u\in\mathbb{R}^{m}$ the input
or control vector, with $m\leq n$, and $f$ a $C^{\infty}$ vector field in the
tangent bundle $\mathrm{T}X$ of $X$ for each $u\in\mathbb{R}^{m}$ and whose
dependence on $u$ is of class $C^{\infty}$.

Before going further, we need the following notations, valid all along this
paper:

  * •

Boldface letters $\mathbf{j},\mathbf{k},\ldots,$ are systematically used to
denote multi-integers, i.e.

$\mathbf{j}\triangleq(j_{1},j_{2},\ldots,j_{m}),\quad\mathbf{k}\triangleq(k_{1},k_{2},\ldots,k_{m}),\quad\ldots$

  * •

We denote the minimum of two arbitrary integers $k$ and $l$ by $k\lor
l\triangleq\min\\{k,l\\}$, and their maximum by $k\wedge
l\triangleq\max(k,l)$.

Also, for every
$\mathbf{j}\triangleq\left(j_{1},\ldots,j_{m}\right)\in\mathbb{N}^{m}$ and
$\mathbf{k}\triangleq\left(k_{1},\ldots,k_{m}\right)\in\mathbb{N}^{m}$, we use
the _componentwise minimum_ notation

$\mathbf{j}\bigvee\mathbf{k}\triangleq\left(j_{1}\vee k_{1},\ldots,j_{m}\vee
k_{m}\right)=\left(\min(j_{1},k_{1}),\ldots,\min(j_{m},k_{m})\right).$ (2.2)

Accordingly, the _componentwise maximum_ is denoted by

$\mathbf{j}\bigwedge\mathbf{k}\triangleq\left(j_{1}\wedge
k_{1},\ldots,j_{m}\wedge
k_{m}\right)=\left(\max(j_{1},k_{1}),\ldots,\max(j_{m},k_{m})\right).$ (2.3)

If $k_{1}=\cdots=k_{m}=k \in\mathbb{N}$, we also use the notations
$\mathbf{j}\bigwedge k$ and $\mathbf{j}\bigvee k$, with $k$ in regular math
typeface.

  * •

Overlined symbols will denote a collection of successive time derivatives of a
time-dependent function as follows. Given a multi-integer
$\mathbf{k}=(k_{1},\ldots,k_{m})\in\mathbb{N}^{m}$ and a locally defined
$C^{\infty}$ function $t\mapsto\xi(t)\in\mathbb{R}^{m}$,

    * –

$\overline{\xi}^{(\mathbf{k})}$ denotes the vector
$\left(\xi_{1},\dot{\xi}_{1},\ldots,\xi_{1}^{(k_{1})},\ldots,\xi_{m},\dot{\xi}_{m},\ldots,\xi_{m}^{(k_{m})}\right)$
of dimension $m+|\mathbf{k}|$, with
$|\mathbf{k}|\triangleq\sum_{i=1}^{m}k_{i}$ and
$\xi_{i}^{(j)}\triangleq\frac{d^{j}\xi_{i}}{dt^{j}}$, $j=1,\ldots,k_{i}$,
$i=1,\ldots,m$;

    * –

$\overline{\xi}$ denotes the infinite sequence

$\overline{\xi}\triangleq(\xi,\dot{\xi},\ddot{\xi},\ldots)\triangleq\left(\xi_{i}^{(k)}\;;\;i=1,\dots,m\;;k\geq
0\right)\in\mathbb{R}^{m}_{\infty},$

where
$\mathbb{R}^{m}_{\infty}\triangleq\mathbb{R}^{m}\times\mathbb{R}^{m}\times\cdots$
is the product of an infinite number of copies of $\mathbb{R}^{m}$.

###  2.1 Recalls on Lie-Bäcklund Equivalence

For a detailed presentation of the topics of this section, the reader may
refer e.g. to [10, 29, 21, 22, 30].

######  Definition 2.1.

The infinite order jet space prolongation of system (2.1) is given by the pair
$(X\times\mathbb{R}^{m}_{\infty},C_{f})$, where
$X\times\mathbb{R}^{m}_{\infty}$ is the product of $X$ with an infinite number
of copies of $\mathbb{R}^{m}$, with coordinates $(x,\overline{u})$, endowed
with the Cartan field

$C_{f}=f(x,u)\frac{\partial}{\partial x}+\sum_{j\geq
0}\sum_{i=1}^{m}u_{i}^{(j+1)}\frac{\partial}{\partial u_{i}^{(j)}},$ (2.4)

defined on $\mathrm{T}X\times\mathrm{T}\mathbb{R}^{m}_{\infty}$, the tangent
bundle of $X\times\mathbb{R}^{m}_{\infty}$.

######  Definition 2.2 (Lie-Bäcklund equivalence).

Consider two systems:

$\dot{x}=g(x,u)\quad\mathrm{and}\quad\dot{y}=\gamma(y,v)$ (2.5)

and their respective prolongations $(X\times\mathbb{R}^{m}_{\infty},C_{g})$,
with coordinates $(x,\overline{u})$ and Cartan field

$C_{g}=g(x,u)\frac{\partial}{\partial x}+\sum_{j\geq
0}\sum_{i=1}^{m}u_{i}^{(j+1)}\frac{\partial}{\partial u_{i}^{(j)}},$ (2.6)

and $(Y\times\mathbb{R}^{\mu}_{\infty},C_{\gamma})$, with coordinates
$(y,\overline{v})$, and Cartan field

$C_{\gamma}=\gamma(y,v)\frac{\partial}{\partial y}+\sum_{j\geq
0}\sum_{i=1}^{\mu}v_{i}^{(j+1)}\frac{\partial}{\partial v_{i}^{(j)}}.$ (2.7)

We say that they are _Lie-Bäcklund equivalent_ at a pair of points
$(x_{0},\overline{u}_{0})$ and $(y_{0},\overline{v}_{0})$ if, and only if,
there exists neighborhoods ${\mathcal{N}}_{x_{0},\overline{u}_{0}}\subset
X\times\mathbb{R}^{m}_{\infty}$ and
${\mathcal{N}}_{y_{0},\overline{v}_{0}}\subset
Y\times\mathbb{R}^{\mu}_{\infty}$ and a $C^{\infty}$ isomorphism111Recall that
a continuous function and, _a fortiori_ , differentiable, resp. $C^{\infty}$,
depends, by definition of the source and target product topologies, on a
finite number of components of its variables, namely $\Phi$ (resp. $\Psi$)
depends an a finite number of components of $(y,\overline{v})$ (resp.
$(x,\overline{u})$) (see e.g. [19, 36, 21]).
$\Phi:{\mathcal{N}}_{y_{0},\overline{v}_{0}}\rightarrow{\mathcal{N}}_{x_{0},\overline{u}_{0}}$
satisfying $\Phi(y_{0},\overline{v}_{0})=(x_{0},\overline{u}_{0})$, with
$C^{\infty}$ inverse $\Psi$, such that the respective Cartan fields are $\Phi$
and $\Psi$ related, i.e. $\Phi_{\ast}C_{\gamma}=C_{g}$ in
${\mathcal{N}}_{x_{0},\overline{u}_{0}}$ and $\Psi_{\ast}C_{g}=C_{\gamma}$ in
${\mathcal{N}}_{y_{0},\overline{v}_{0}}$.

In other words, the two systems are _Lie-Bäcklund equivalent_ at the points
$(x_{0},\overline{u}_{0})$ and $(y_{0},\overline{v}_{0})$ if there exist
neighborhoods of these points where every integral curve of the first system
is mapped to an integral curve of the second one and conversely, with the same
time parameterization. Clearly, this relation is an equivalence relation.

We recall, without proof, a most important result from [24] (see also [9, 10,
21]) giving an interpretation of the Lie-Bäcklund equivalence in terms of
diffeomorphism in finite dimension and endogenous dynamic feedback, that will
be useful later on.

######  Theorem 2.1 (Martin [24]).

If the two systems (2.5) are Lie-Bäcklund equivalent at a given pair of
points, then (i) and (ii) must be satisfied:

  * (i)

$m=\mu$, i.e. they must have the same number of independent inputs;

  * (ii)

there exist

    * –

an endogenous dynamic feedback222A dynamic feedback is said _endogenous_ if,
and only if, the closed-loop system and the original one are Lie-Bäcklund
equivalent, i.e. if, and only if, the extended state $z$ can be locally
expressed as a smooth function of $x$, $u$ and a finite number of time
derivatives of $u$ (see [24, 9, 10, 21]).

$u=\alpha(x,z,w),\quad\dot{z}=\beta(x,z,w),$ (2.8)

where $z$ belongs to $Z$, a finite dimensional smooth manifold,

    * –

a multi-integer333Recall that we denote by
$v^{(\mathbf{r})}\triangleq\left(v_{1}^{(r_{1})},\ldots,v_{m}^{(r_{m})}\right)\triangleq\left(\frac{\mathrm{d}^{r_{1}}v_{1}}{\mathrm{d}t^{r_{1}}},\ldots,\frac{\mathrm{d}^{r_{m}}v_{m}}{\mathrm{d}t^{r_{m}}}\right)$.
$\mathbf{r}\triangleq\left(r_{1},\ldots,r_{m}\right)$,

    * –

and a local diffeomorphism $\chi:Y\times\mathbb{R}^{|\mathbf{r}|}\rightarrow
X\times Z$,

all defined in a neighborhood of the considered points, such that the extended
system

$\dot{y}=\gamma(y,v),\quad v^{(\mathbf{r})}=w$ (2.9)

and the closed-loop one

$\dot{x}=g(x,\alpha(x,z,w)),\quad\dot{z}=\beta(x,z,w)$ (2.10)

are $\chi$-related for all $w\in\mathbb{R}^{m}$, i.e.

$(x,z)=\chi(y,v,\dot{v},\ldots,v^{(\mathbf{r-1})}),\qquad(y,v,\dot{v},\ldots,v^{(\mathbf{r-1})})=\chi^{-1}(x,z)$
(2.11)

and

$\hat{g}=\chi_{\ast}\hat{\gamma},\qquad\hat{\gamma}=\chi^{-1}_{\ast}\hat{g}$
(2.12)

where we have denoted

$\displaystyle\hat{g}(x,z,w)\triangleq
g(x,\alpha(x,z,w))\frac{\partial}{\partial
x}+\beta(x,z,w)\frac{\partial}{\partial z}$
$\displaystyle\hat{\gamma}(y,v,\dot{v},\ldots,v^{(\mathbf{r-1})},w)\triangleq\gamma(y,v)\frac{\partial}{\partial
y}+\sum_{i=1}^{m}\sum_{j=0}^{r_{i}-1}v_{i}^{(j+1)}\frac{\partial}{\partial
v_{i}^{(j)}}+w_{i}\frac{\partial}{\partial v_{i}^{(r_{i})}}.$

###  2.2 Recalls on Differential Flatness

######  Definition 2.3.

We say that system (2.1) is _differentially flat_ (or, more shortly, _flat_)
at the pair of points $(x_{0},\overline{u}_{0})$ and
$\overline{y}_{0}\in\mathbb{R}^{m}_{\infty}$ if and only if, it is _Lie-
Bäcklund equivalent to the trivial system ( $\mathbb{R}^{m}_{\infty},\tau)$
where $\tau$ is the _trivial_ Cartan field_

$\tau\triangleq\sum_{j\geq
0}\sum_{i=1}^{m}y_{i}^{(j+1)}\frac{\partial}{\partial y_{i}^{(j)}}$ (2.13)

_at the considered points_.

Otherwise stated, the locally defined flat output $y=\Psi(x,\overline{u})$ is
such that
$(x,\overline{u})=\Phi(\overline{y})\triangleq(\Phi_{-1}(\overline{y}),\Phi_{0}(\overline{y}),\Phi_{1}(\overline{y}),\ldots)$,
with444The first component of $\Phi$, corresponding to the $x$ component, is
denoted by $\Phi_{-1}$ so that the component $\Phi_{i}$ effectively
corresponds to the $i$th time derivative of $u$.
$\frac{\mathrm{d}}{\mathrm{d}t}{\Phi}_{-1}(\overline{y})\equiv
f(\Phi_{-1}(\overline{y}),\Phi_{0}(\overline{y}))$ for all sufficiently
differentiable function $y:t\in\mathbb{R}\mapsto y(t)\in\mathbb{R}^{m}$.

This definition immediately implies that a system is flat if, and only if,
there exists a generalized output $y=\Psi(x,\overline{u})$ of dimension $m$,
depending at most on a finite number of derivatives of $u$, with independent
derivatives of all orders, such that $x$ and $\overline{u}$ can be expressed
in terms of $y$ and a finite number of its successive derivatives, i.e.
$(x,\overline{u})=\Phi(\overline{y})$, and such that the system equation
$\frac{\mathrm{d}}{\mathrm{d}t}{\Phi_{-1}}(\overline{y})=f\circ\Phi(\overline{y})$
is identically satisfied for all sufficiently differentiable
$y:\mathbb{R}\rightarrow\mathbb{R}^{m}$.

For a flat system, with the notations of theorem 2.1, the vector field
$\gamma$, or $\widehat{\gamma}$ indifferently, corresponds to the linear
system in Brunovský canonical form

$y_{i}^{(r_{i}+1)}=w_{i},\qquad i=1,\ldots,m,$ (2.14)

and $C_{\gamma}$, defined by (2.7), is the trivial Cartan field
$C_{\gamma}=\tau$, with $\tau$ given by (2.13).

######  Remark 2.1.

In view of Definition 2.2, since the trivial Cartan field (2.13) is the
infinite jet prolongation of $\dot{y}_{i}=u_{i}$, $i=1,\ldots,m$, system (2.1)
is flat if, and only if, it is Lie-Bäcklund equivalent to $m$ simple
integrators in parallel. We may also remark that, since the Cartan field is
equal to 0 (modulo $\tau)$, the class of flat systems may be identified with
the Lie-Bäcklund equivalence class of 0 (modulo $\tau$).

Theorem 2.1 reads:

######  Corollary 2.1.

If system (2.1) is flat at a given point, there exists an endogenous dynamic
feedback of the form (2.8), a multi-integer
$\mathbf{r}\triangleq\left(r_{1},\ldots,r_{m}\right)$ and a finite dimensional
local diffeomorphism $\chi$ such that systems (2.10), with $f$ in place of
$g$, and (2.14) are $\chi$-related for all $w\in\mathbb{R}^{m}$.

Consequently, a flat system can be transformed, by diffeomorphism and feedback
of a suitably extended space, in a linear controllable system, which motivates
the recalls of the next section.

###  2.3 Recalls on Feedback Linearization

Feedback linearizable systems [17, 15] (see also [16, 28, 23, 20]) constitute
a subclass of differentially flat systems. They correspond to the equivalence
class of linear controllable systems with respect to the following finer
equivalence relation, called _equivalence by diffeomorphism and feedback_.

######  Definition 2.4.

The two systems given by (2.5) are said equivalent by diffeomorphism and
feedback if, and only if, there exists

  * •

a local diffeomorphism $\varphi$ from a neighborhood of an equilibrium point
of $X$ (which may be chosen, without loss of generality, as the origin $0\in
X$) to a suitable neighborhood of $0\in Y$,

  * •

and a _static_ feedback $v=\alpha(x,u)$, $\alpha$ being invertible with
respect to $u$ for all $x$ in the above mentioned neighborhood of the origin,
i.e. $\mathsf{rank}\left(\frac{\partial\alpha}{\partial u}\right)(x,u)=m$ for
all $x$ and $u$ as above,

such that
$\varphi^{-1}_{\ast}(f(x,u))=\gamma\circ(\varphi\times\alpha)(x,u)=\gamma(y,v)$,

Indeed, this equivalence implies that $n=\dim X=\dim Y$ and that both $u$ and
$v$ are $m$-dimensional.

######  Definition 2.5 (Feedback Linearizability).

System (2.1) is said _static feedback linearizable_ or, shortly, _feedback
linearizable_ if the context allows, if, and only if, it is equivalent by
diffeomorphism and feedback to a linear system in Brunovský controllability
canonical form

$y_{i}^{(r_{i})}=v_{i},\quad i=1,\ldots,m,$ (2.15)

where the multi-integer $\mathbf{r}\triangleq(r_{1},\ldots,r_{m})$, whose
components $r_{i}$ are called the _controllability indices_ of system (2.15),
satisfies $|\mathbf{r}|\triangleq\sum_{i=1}^{m}r_{i}=n=\dim X$.

Indeed, since the $n$-dimensional vector

$\overline{y}^{(\mathbf{r-1})}\triangleq(y_{1},\ldots,y_{1}^{(r_{1}-1)},\ldots,y_{m},\ldots,y_{m}^{(r_{m}-1)})$

and $x$ are diffeomorphic (again we have noted
$\mathbf{r-1}\triangleq(r_{1}-1,\ldots,r_{m}-1)$), it is immediate to verify
that $(y_{1},\ldots,y_{m})$ is a flat output and that every feedback
linearizable system is differentially flat.

These systems have been first characterized by [17, 14, 15] for control-affine
systems, i.e. systems given by $f(x,u)=f_{0}(x)+\sum_{i=1}^{m}u_{i}f_{i}(x)$.
More generally, it can be easily proven that systems of the form (2.1) are
feedback linearizable if, and only if, the first-order control-affine
prolongation

$\displaystyle\dot{x}=f(x,u),$ (2.16)
$\displaystyle\dot{u}_{i}=u_{i}^{(1)},\quad i=1,\ldots,m$

with state
$\overline{x}^{(\mathbf{0})}\triangleq(x,u)=(x_{1},\ldots,x_{n},u_{1},\ldots,u_{m})\in
X^{(\mathbf{0})}\triangleq X\times\mathbb{R}^{m}$ and control vector
$u^{(\mathbf{1})}\triangleq(u_{1}^{(1)},\ldots,u_{m}^{(1)})\in\mathbb{R}^{m}$,
is feedback linearizable (see e.g. [6, 31, 33, 20]).

Indeed, in the local coordinates555We introduce the superscript (0) from now
on to get ready to work with higher order prolongations (see section 3).
$\overline{x}^{(\mathbf{0})}$ of $X^{(\mathbf{0})}$, denoting the associated
vector fields by

$g_{0}^{(\mathbf{0})}(\overline{x}^{(\mathbf{0})})\triangleq\sum_{i=1}^{n}f_{i}(x,u)\frac{\partial}{\partial
x_{i}},\qquad
g_{i}^{(0)}(\overline{x}^{(\mathbf{0})})\triangleq\frac{\partial}{\partial
u_{i}},\quad i=1,\ldots,m,$ (2.17)

defined on the tangent bundle
$\mathrm{T}X^{(\mathbf{0})}=\mathrm{T}X\times\mathrm{T}\mathbb{R}^{m}$, system
(2.16) reads

$\dot{\overline{x}}^{(\mathbf{0})}=g_{0}^{(\mathbf{0})}(\overline{x}^{(\mathbf{0})})+\sum_{i=1}^{m}u_{i}^{(1)}g_{i}^{(0)}(\overline{x}^{(\mathbf{0})})$
(2.18)

with the usual abuse of notations identifying a vector field expressed in
local coordinates with its associated (Lie derivative) first order partial
differential operator.

Until now, for simplicity’s sake, a system (2.1) will always be considered in
the form (2.18), even if the vector-field $f$ is already given in control-
affine form. For the sake of coherence, we set
$u=(u_{1},\ldots,u_{m})\triangleq
u^{(\mathbf{0})}=(u_{1}^{(0)},\ldots,u_{m}^{(0)})$, so that
$\dot{u}^{(\mathbf{0})}=u^{(\mathbf{1})}$.

###  2.4 Recalls on Lie brackets and Distributions

The Lie bracket $[\eta,\gamma]$ of two arbitrary vector fields $\eta$ and
$\gamma$ of $\mathrm{T}X^{(\mathbf{0})}$ is given, in the
$\overline{x}^{(\mathbf{0})}$-coordinates, by
$[\eta,\gamma]\triangleq\sum_{i=1}^{n+m}\sum_{j=1}^{n+m}\left(\eta_{j}\frac{\partial\gamma_{i}}{\partial
x_{j}^{(0)}}-\gamma_{j}\frac{\partial\eta_{i}}{\partial
x_{j}^{(0)}}\right)\frac{\partial}{\partial x_{i}^{(0)}}$, with
$\overline{x}^{(\mathbf{0})}=(x,u^{(\mathbf{0})})\triangleq(x_{1}^{(0)},\ldots,x_{n+m}^{(0)})$.

For iterated Lie brackets, we use the notation
$\mathrm{ad}_{\eta}\gamma\triangleq[\eta,\gamma]$ and
$\mathrm{ad}_{\eta}^{k}\gamma\triangleq[\eta,\mathrm{ad}_{\eta}^{k-1}\gamma]$
for $k\geq 1$, with the convention that $\mathrm{ad}_{\eta}^{0}\gamma=\gamma$.
In addition, if $\Gamma$ is an arbitrary distribution of vector fields on
$\mathrm{T}X^{(\mathbf{0})}$, we note
$\mathrm{ad}_{\eta}^{k}\Gamma\triangleq\\{\mathrm{ad}_{\eta}^{k}\gamma:\gamma\in\Gamma\\}$.

The distribution $\Gamma$ is said _involutive_ if, and only if,
$[\eta,\gamma]\in\Gamma$ for every pair of vector fields
$\eta,\gamma\in\Gamma$, to which case we note $[\Gamma,\Gamma]\subset\Gamma$,
or $\overline{\Gamma}=\Gamma$, where $\overline{\Gamma}$ denotes the
involutive closure of $\Gamma$, i.e. the smallest involutive distribution
containing $\Gamma$.

If the distribution $\Gamma$ is locally generated by $p$ vector fields
$\gamma_{1},\ldots,\gamma_{p}$, with $p$ arbitrary, we write
$\Gamma\triangleq\\{\gamma_{1},\ldots,\gamma_{p}\\}$. We also denote by
$\Gamma(\xi)\triangleq\\{\gamma_{1}(\xi),\ldots,\gamma_{p}(\xi)\\}$ the vector
space generated by the vectors $\gamma_{1}(\xi),\ldots,\gamma_{p}(\xi)$ at a
point $\xi\in X^{(\mathbf{0})}$.

Consider the ($\mathbf{0}$th-order or non prolonged) filtration of
distributions built on the vector fields (2.17)666As before, the superscript
(0) is used to indicate that the distributions $G_{k}^{(\mathbf{0})}$ and the
related indices $\rho_{k}^{(\mathbf{0})}$ and $\kappa_{k}^{(\mathbf{0})}$ are
built on the non prolonged vector fields (2.17) and to distinguish them from
the prolonged distributions $G_{k}^{(\mathbf{j})}$ of arbitrary $\mathbf{j}$th
order, $\mathbf{j}\in\mathbb{N}^{m}$, and related indices,
$\rho_{k}^{(\mathbf{j})}$ and $\kappa_{k}^{(\mathbf{j})}$, later introduced in
sections 3 and 4.:

$G_{0}^{(\mathbf{0})}\triangleq\\{g_{1}^{(\mathbf{0})},\ldots,g_{m}^{(\mathbf{0})}\\},\qquad
G_{k+1}^{(\mathbf{0})}\triangleq
G_{k}^{(\mathbf{0})}+\mathrm{ad}_{g_{0}^{(\mathbf{0})}}G_{k}^{(\mathbf{0})},\quad\forall
k\geq 0,$ (2.19)

indeed satisfying $G_{0}^{(\mathbf{0})}\subset\cdots\subset
G_{k}^{(\mathbf{0})}\subset
G_{k+1}^{(\mathbf{0})}\subset\cdots\subset\mathrm{T}X^{(\mathbf{0})}$.

######  Theorem 2.2 ([17, 15]).

System (2.1), or equivalently system (2.16), is feedback linearizable in a
neighborhood of the origin of $X^{(\mathbf{0})}$ if, and only if, in this
neighborhood:

  * (i)

$G_{k}^{(\mathbf{0})}$ is involutive with constant dimension for all $k\geq
0$,

  * (ii)

there exists an integer $k^{(\mathbf{0})}_{\star}\leq n$ such that
$G_{k}^{(\mathbf{0})}=G_{k^{(\mathbf{0})}_{\star}}^{(\mathbf{0})}=\mathrm{T}X^{(\mathbf{0})}$
for all $k\geq k^{(\mathbf{0})}_{\star}$.

Note that, according to (2.17) and (2.19),
$G_{0}^{(\mathbf{0})}=\\{\frac{\partial}{\partial
u_{1}^{(0)}},\ldots,\frac{\partial}{\partial u_{m}^{(0)}}\\}$ is involutive
with constant dimension, equal to $m$, by construction.

Theorem 2.2 provides a construction of flat outputs via Frobenius theorem (see
e.g. [8]) and the list of the so-called _Brunovský’s controllability indices_
[4] as follows:

######  Definition 2.6.

Consider the sequence of integers

$\rho_{k}^{(\mathbf{0})}\triangleq\dim
G_{k}^{(\mathbf{0})}/G_{k-1}^{(\mathbf{0})}\quad\forall k\geq
1,\qquad\rho_{0}^{(\mathbf{0})}\triangleq\dim G_{0}^{(\mathbf{0})}=m.$

The Brunovský controllability indices $\kappa_{k}^{(\mathbf{0})}$’s are
defined by

$\kappa_{k}^{(\mathbf{0})}\triangleq\\#\\{l\mid\rho_{l}^{(\mathbf{0})}\geq
k\\},\quad k=1,\ldots,m,$

where $\\#A$ denotes the number of elements of an arbitrary set $A$.

It can be proven (see e.g. [17, 15, 16, 28, 20]) that, for a feedback
linearizable nonlinear system (2.1), or (2.16), we have:

  * •

the sequences $\\{\rho_{k}^{(\mathbf{0})}\\}_{k\geq 0}$ and
$\\{\kappa_{k}^{(\mathbf{0})}\\}_{k\geq 0}$ are non increasing,

  * •

$\rho_{k}^{(\mathbf{0})}\leq m$ for all $k,\quad\rho_{k}^{(\mathbf{0})}=0$,
for all $k\geq k^{(\mathbf{0})}_{\star}+1$,

  * •

$\kappa_{1}^{(\mathbf{0})}=k^{(\mathbf{0})}_{\star}+1,\quad\kappa_{m}^{(\mathbf{0})}\geq
1$,

  * •

$\displaystyle\sum_{k=0}^{k^{(\mathbf{0})}_{\star}}\rho_{k}^{(\mathbf{0})}=\sum_{k=1}^{m}\kappa_{k}^{(\mathbf{0})}=\dim
G_{k^{(\mathbf{0})}_{\star}}^{(\mathbf{0})}=n+m$.

The list $\kappa_{1}^{(\mathbf{0})},\ldots,\kappa_{m}^{(\mathbf{0})}$ is
uniquely defined up to input permutation, invariant by static state feedback
and state diffeomorphism, and is indeed equal to the list of controllability
indices of the associated linear system (2.15) with
$r_{i}=\kappa_{i}^{(\mathbf{0})}$, $i=1,\ldots,m$.

Moreover, for all $k$ and all $i=1,\ldots,m$, and possibly up to a suitable
input reordering, we have

$G_{k}^{(\mathbf{0})}=\bigoplus_{j=1}^{m}\\{\mathrm{ad}_{g_{0}^{(\mathbf{0})}}^{l}g_{j}^{(\mathbf{0})}\mid
l=0,\ldots,k\lor(\kappa_{j}^{(\mathbf{0})}-1)\\},\quad
G_{\kappa_{1}^{(\mathbf{0})}-1}^{(\mathbf{0})}=G_{k^{(\mathbf{0})}_{\star}}^{(\mathbf{0})}=\mathrm{T}X^{(\mathbf{0})}.$

Then, flat outputs $(y_{1},\ldots,y_{m})$ are locally non trivial solutions of
the system of PDE’s

$\displaystyle
L_{\mathrm{ad}_{g_{0}^{(\mathbf{0})}}^{k}g_{j}^{(\mathbf{0})}}\;y_{i}=0,\;k=0,\ldots,\kappa_{i}^{(\mathbf{0})}-2,\;j=1,\ldots,m,$
(2.20) $\displaystyle\mathrm{with\leavevmode\nobreak\
}L_{\mathrm{ad}_{g_{0}^{(\mathbf{0})}}^{\kappa_{i}^{(\mathbf{0})}-1}g_{i}^{(\mathbf{0})}}\;y_{i}\neq
0,$

for $i=1,\ldots,m$, where we have denoted by $L_{\eta}\varphi$ the Lie
derivative of a vector function $\varphi$ along the vector field $\eta$. These
solutions are such that the mapping

$\overline{x}^{(\mathbf{0})}\mapsto(y_{1},\ldots,y_{1}^{(\kappa_{1}^{(\mathbf{0})}-1)},\ldots,y_{m},\ldots,y_{m}^{(\kappa_{m}^{(\mathbf{0})}-1)})$

is a local diffeomorphism.

######  Remark 2.2.

Recall from [6] that, for single input systems, differential flatness and
feedback linearizability are equivalent.

##  3 System Pure Prolongation

###  3.1 Purely Prolonged Distributions

We now introduce higher order prolongations of the vector fields defined by
(2.17), called _pure prolongations_ [7] (see also [33, 34, 2, 12]).

Given a multi-integer
$\mathbf{j}\triangleq(j_{1},\ldots,j_{m})\in\mathbb{N}^{m}$, we note, as
before, $|\mathbf{j}|\triangleq\sum_{i=1}^{m}j_{i}$ and the prolonged state:

$\overline{x}^{(\mathbf{j})}\triangleq(x,\overline{u}^{(\mathbf{j})})\triangleq(x_{1},\ldots,x_{n},u_{1}^{(0)},\ldots,u_{1}^{(j_{1})},\ldots,u_{m}^{(0)},\ldots,u_{m}^{(j_{m})}),$

with $u_{i}^{(0)}\triangleq u_{i}$, $i=1,\ldots,m$.

Let $X^{(\mathbf{j})}\triangleq X\times\mathbb{R}^{m+|\mathbf{j}|}$ be the
associated $\mathbf{j}$-th order jet manifold of dimension $n+m+|\mathbf{j}|$
with coordinates $\overline{x}^{(\mathbf{\mathbf{j}})}$.

The pure prolongation of order $\mathbf{j}$ of system (2.16), or otherwise
said, of the vector fields (2.17), in the tangent bundle
$\mathrm{T}X^{(\mathbf{j})}=\mathrm{T}X\times\mathrm{T}\mathbb{R}^{m+|\mathbf{j}|}$,
is defined by

$\displaystyle g_{0}^{(\mathbf{j})}(\overline{x}^{(\mathbf{j})})$
$\displaystyle=\sum_{i=1}^{n}f_{i}(x,u)\frac{\partial}{\partial
x_{i}}+\sum_{i=1}^{m}\sum_{k=0}^{j_{i}-1}u_{i}^{(k+1)}\frac{\partial}{\partial
u_{i}^{(k)}}$ (3.1) $\displaystyle
g_{i}^{(\mathbf{j})}(\overline{x}^{(\mathbf{j})})$ $\displaystyle\triangleq
g_{i}^{(j_{i})}(\overline{x}^{(\mathbf{j})})\triangleq\frac{\partial}{\partial
u_{i}^{(j_{i})}},\quad i=1,\ldots,m,$

with the convention that
$\sum_{k=0}^{j_{i}-1}u_{i}^{(k+1)}\frac{\partial}{\partial
u_{i}^{(k)}}\triangleq 0$ if $j_{i}=0$.

They are naturally associated to the adjunction of $\mathbf{j}$ pure
integrators to $u=u^{(0)}$ in (2.16) (with the same usual abuse of notations
as in (2.18)):

$\dot{\overline{x}}^{(\mathbf{j})}=g_{0}^{(\mathbf{j})}(\overline{x}^{(\mathbf{j})})+\sum_{i=1}^{m}u_{i}^{(j_{i}+1)}g_{i}^{(j_{i})}(\overline{x}^{(\mathbf{j})})$
(3.2)

or

$\dot{x}=f(x,u),\qquad\dot{u}_{i}^{(k)}=u_{i}^{(k+1)},\quad
k=0,\ldots,j_{i},\leavevmode\nobreak\ \leavevmode\nobreak\ i=1,\ldots,m,$

$u^{(\mathbf{j+1})}=(u_{1}^{(j_{1}+1)},\ldots,u_{m}^{(j_{m}+1)})$ being the
new control vector of this _purely prolonged system_ , whose state is
$(x,\overline{u}^{(\mathbf{j})})=\overline{x}^{(\mathbf{j})}$.

######  Remark 3.1.

Note that the state of the $\mathbf{j}$-th prolonged system,
$\overline{x}^{(\mathbf{j})}$, coincides with the image of $\overline{x}$ by
the projection $p_{\mathbf{j}}:\overline{x}\in
X\times\mathbb{R}^{m}_{\infty}\mapsto
p_{\mathbf{j}}(\overline{x})=\overline{x}^{(\mathbf{j})}\in X^{(\mathbf{j})}$
for all $\mathbf{j}$. In addition, the family of projections
$p_{\mathbf{i},\mathbf{j}}:\overline{x}^{(\mathbf{i})}\in
X^{(\mathbf{i})}\mapsto\overline{x}^{(\mathbf{j})}=p_{\mathbf{i},\mathbf{j}}(\overline{x}^{(\mathbf{i})})\in
X^{(\mathbf{j})}$ for all $\mathbf{i},\mathbf{j}$ such that $i_{k}\geq j_{k}$
for all $k=1,\ldots,m$, that we note $\mathbf{i}\succeq\mathbf{j}$, indeed
satisfies $p_{\mathbf{i},\mathbf{j}}\circ
p_{\mathbf{j},\mathbf{k}}=p_{\mathbf{i},\mathbf{k}}$ for all
$\mathbf{i}\succeq\mathbf{j}\succeq\mathbf{k}$ and thus allows us to identify
the manifold $X\times\mathbb{R}^{m}_{\infty}$ with the _projective limit_ of
the family $(X^{(\mathbf{i})},p_{\mathbf{i},\mathbf{j}})$ for all $\mathbf{i}$
and all $\mathbf{j}$ such that $\mathbf{i}\succeq\mathbf{j}$, i.e.
$X\times\mathbb{R}^{m}_{\infty}\simeq\displaystyle\lim_{\leftarrow}X^{(\mathbf{i})}$
(see e.g. [3, Chap. I,§10]). A similar identification trivially holds for the
associated tangent bundles, i.e.
$\mathrm{T}X\times\mathrm{T}\mathbb{R}^{m}_{\infty}\simeq\displaystyle\lim_{\leftarrow}\mathrm{T}X^{(\mathbf{i})}$
relatively to the family $\mathrm{T}p_{\mathbf{i},\mathbf{j}}$ of tangent
projections, hence the identification of the Cartan field $C_{f}$, defined by
(2.4), with $\displaystyle\lim_{\leftarrow}g_{0}^{\mathbf{(j)}}$, the
projective limit of the vector fields $g_{0}^{\mathbf{(j)}}$. Nevertheless,
this property does not hold for the control vector fields
$g_{i}^{\mathbf{(j)}}$ since
$\mathrm{T}p_{\mathbf{j},\mathbf{k}}(g_{i}^{\mathbf{(j)}})$ is not equal to
$g_{i}^{\mathbf{(k)}}$, for $i=1,\ldots,m$ and $\mathbf{j}\succeq\mathbf{k}$.
Moreover, the Lie bracket of vector fields is not preserved by this family of
projections. This is one of the reasons why prolongations may enlarge the
range of the system transformations.

######  Remark 3.2.

Given an arbitrary point $\overline{x}_{0}\triangleq(x_{0},\overline{u}_{0})$
around which system (2.16) is defined, it is convenient to consider the shift

$\theta:(x,\overline{u})\in
X\times\mathbb{R}^{m}_{\infty}\mapsto\theta(x,\overline{u})=(x-x_{0},\overline{(u-u_{0})})\triangleq(z,\overline{v})\in
X\times\mathbb{R}^{m}_{\infty}$

such that $\overline{x}_{0}$ is mapped to the origin of
$\mathrm{T}X\times\mathrm{T}\mathbb{R}^{m}_{\infty}$, denoted by
$\overline{0}$, thus inducing the shift of vector fields:

$\theta_{\star}(g_{i}^{(\mathbf{j})})(z,\overline{v})\triangleq
g_{i}^{(\mathbf{j})}(z+x_{0},\overline{(v+u_{0})}),\quad i=0,\ldots m,$ (3.3)

now defined in a neighborhood of $\overline{0}$. For the sake of simplicity,
we will only consider such shifted vector fields in the sequel while keeping
the same notation $g_{i}^{(\mathbf{j})}$ as before, though abusive, but yet
unambiguous.

We now introduce the following filtration of $\mathbf{j}$-th order purely
prolonged distributions of $\mathrm{T}X^{(\mathbf{j})}$:

$G_{0}^{(\mathbf{j})}\triangleq\\{\frac{\partial}{\partial
u_{1}^{(j_{1})}},\ldots,\frac{\partial}{\partial u_{m}^{(j_{m})}}\\},\qquad
G_{k+1}^{(\mathbf{j})}\triangleq
G_{k}^{(\mathbf{j})}+\mathrm{ad}_{g_{0}^{(\mathbf{j})}}G_{k}^{(\mathbf{j})},\quad\forall
k\geq 0$ (3.4)

Indeed, for $\mathbf{j}=\mathbf{0}$, i.e. $j_{1}=\cdots=j_{m}=0$, this
filtration coincides with the $\mathbf{0}$th order one given by (2.19).
Similarly to the $\mathbf{0}$th order case, $G_{0}^{(\mathbf{j})}$ is
involutive with constant dimension, equal to $m$, by construction.

Moreover, since every $G_{k}^{(\mathbf{j})}\subset\mathrm{T}X^{(\mathbf{j})}$,
with $\dim\mathrm{T}X^{(\mathbf{j})}=n+m+\mid\mathbf{j}\mid$, we have

######  Proposition 3.1.

There exists a finite integer $k^{(\mathbf{j})}_{\star}$ such that
$G_{k}^{(\mathbf{j})}=G_{k^{(\mathbf{j})}_{\star}}^{(\mathbf{j})}$ for all
$k\geq k^{(\mathbf{j})}_{\star}$ and

$k^{(\mathbf{j})}_{\star}\leq n+\mid\mathbf{j}\mid.$ (3.5)

###### Proof.

Since

$n+m+\mid\mathbf{j}\mid\geq\dim
G_{k^{(\mathbf{j})}_{\star}}^{(\mathbf{j})}=\sum_{k=1}^{k^{(\mathbf{j})}_{\star}}\dim
G_{k}^{(\mathbf{j})}/G_{k-1}^{(\mathbf{j})}+\dim G_{0}^{(\mathbf{j})}\geq
k^{(\mathbf{j})}_{\star}+m,$

we get (3.5). ∎

######  Remark 3.3.

In full generality, $k^{(\mathbf{j})}_{\star}$ depends on the point where it
is evaluated. However, if $\dim G_{k}^{(\mathbf{j})}$ is constant in an open
dense subset of $X^{(\mathbf{j})}$ for all large enough $k$, so is
$k^{(\mathbf{j})}_{\star}$.

Let us inductively define the $n$-dimensional vector functions
$\gamma_{k,i}^{(\mathbf{j})}$, for $k\geq 1$, $i=1,\ldots,m$, and arbitrary
$\mathbf{j}=(j_{1},\ldots,j_{m})$ as follows:

$\displaystyle\gamma_{k+1,i}^{(\mathbf{j})}$ $\displaystyle\triangleq
L_{g_{0}^{(\mathbf{j})}}\gamma_{k,i}^{(\mathbf{j})}-\gamma_{k,i}^{(\mathbf{j})}\frac{\partial
f}{\partial
x}=L_{g_{0}^{(\mathbf{0})}}\gamma_{k,i}^{(\mathbf{j})}+\sum_{p=1}^{m}\sum_{l=0}^{j_{p}-1}u_{p}^{(l+1)}\frac{\partial\gamma_{k,i}^{(\mathbf{j})}}{\partial
u_{p}^{(l)}}-\gamma_{k,i}^{(\mathbf{j})}\frac{\partial f}{\partial x}$ (3.6)

with

$\gamma_{1,i}^{(\mathbf{j})}=(-1)^{(j_{i}+1)}\frac{\partial f}{\partial
u_{i}^{(0)}}.$ (3.7)

For an arbitrary $\mathbf{j}$ and given $i=1,\ldots,m$, thanks to (3.7), it is
readily seen that $\gamma_{1,i}^{(\mathbf{j})}$ depends at most of
$\overline{x}^{(\mathbf{0})}$ and thus, if $k\leq j_{i}-1$, thanks to (3.6),
$\gamma_{k+1,i}^{(\mathbf{j})}$ depends at most of
$\overline{x}^{(\mathbf{j}\bigvee k)}$.

###  3.2 Vertical Distributions of Purely Prolonged Ones

Before stating the next Lemma, we need to recall the definition of _vertical
bundle_. Given an arbitrary $\mathbf{r}\in\mathbb{N}^{m}$ and the fiber bundle
$\pi_{\mathbf{r}}:
X^{(\mathbf{r})}\rightarrow\mathbb{R}^{m+\mid\mathbf{r}\mid}$, with
$\pi_{\mathbf{r}}(\overline{x}^{(\mathbf{r})})=\overline{u}^{(\mathbf{r})}$,
its vertical space at $\overline{x}^{(\mathbf{r})}$, denoted by
$\mathrm{V}_{\overline{x}^{(\mathbf{r})}}X^{(\mathbf{r})}$, is the tangent
space $\mathrm{T}_{x}X$. Its vertical bundle, denoted by
$\mathrm{V}X^{(\mathbf{r})}$, is the vector bundle made of the vertical spaces
at each $\overline{x}^{(\mathbf{r})}$, i.e. the set of linear combinations
$\sum_{i=1}^{n}\alpha_{i}(\overline{x}^{(\mathbf{r})})\frac{\partial}{\partial
x_{i}}$ whose coefficients $\alpha_{i}$ are smooth functions that depend at
most on $\overline{x}^{(\mathbf{r})}$ and where $(x_{1},\ldots,x_{n})$ are
local coordinates of $X$.

The same definition indeed holds for the vertical bundle
$\mathrm{V}(X\times\mathbb{R}^{m}_{\infty})$ associated to the fiber bundle
$\pi:X\times\mathbb{R}^{m}_{\infty}\rightarrow\mathbb{R}^{m}_{\infty}$, i.e.
the set of linear combinations of $\frac{\partial}{\partial
x_{1}},\ldots,\frac{\partial}{\partial x_{n}}$ whose coefficients are smooth
functions of $\overline{x}$.

We now establish some comparison formulae between Lie brackets of the vector
fields of the purely prolonged system and those of the original (non
prolonged) one. More complete formulae may be found in Lemma A.1 of the
Appendix A.

######  Lemma 3.1.

For all $\mathbf{j}=\left(j_{1},\ldots,j_{m}\right)\in\mathbb{N}^{m}$
satisfying $0\leq j_{1}\leq\ldots\leq j_{m}$, for all $k\leq j_{i}$ and
$i=1,\ldots,m$, we have:

$\mathrm{ad}_{g_{0}^{(\mathbf{j})}}^{k}\frac{\partial}{\partial
u_{i}^{(j_{i})}}=(-1)^{k}\frac{\partial}{\partial u_{i}^{(j_{i}-k)}}$ (3.8)

and for all $k\geq 1$:

$\mathrm{ad}_{g_{0}^{(\mathbf{j})}}^{j_{i}+k}\frac{\partial}{\partial
u_{i}^{(j_{i})}}=(-1)^{j_{i}}\mathrm{ad}_{g_{0}^{(\mathbf{j})}}^{k}\frac{\partial}{\partial
u_{i}^{(0)}}=\gamma_{k,i}^{(\mathbf{j})}\frac{\partial}{\partial
x}\in\mathrm{V}X^{(\mathbf{j}\bigvee(k-1))},$ (3.9)

Moreover, we have

$\left[\frac{\partial}{\partial
u_{p}^{(j_{p}-k)}},\mathrm{ad}_{g_{0}^{(\mathbf{j})}}^{l-j_{q}}\frac{\partial}{\partial
u_{q}^{(0)}}\right]=0,\quad\forall k<j_{p},\;\;\forall l\geq
j_{q}\leavevmode\nobreak\ \mathrm{s.t.\leavevmode\nobreak\
}k+l<j_{p}+j_{q}+1.$ (3.10)

###### Proof.

It is immediately seen that

$\mathrm{ad}_{g_{0}^{(\mathbf{j})}}\frac{\partial}{\partial
u_{i}^{(j_{i})}}=\left[f\frac{\partial}{\partial x}+\sum_{k=1}^{m}\sum_{l\geq
0}u_{k}^{(l+1)}\frac{\partial}{\partial u_{k}^{(l)}},\frac{\partial}{\partial
u_{i}^{(j_{i})}}\right]=-\frac{\partial}{\partial u_{i}^{(j_{i-1})}}.$

Iterating this computation up to $k=j_{i}$ yields (3.8):

$\mathrm{ad}^{j_{i}}_{g_{0}^{(\mathbf{j})}}\frac{\partial}{\partial
u_{i}^{(j_{i})}}=(-1)^{j_{i}}\frac{\partial}{\partial u_{i}^{(0)}}.$

Then, for $k=j_{i}+1$, using the fact that $[\frac{\partial}{\partial
u_{k}^{(l)}},\frac{\partial}{\partial u_{i}^{(0)}}]=0$ for all $i$, $k$ and
$l\geq 0$, we have:

$\displaystyle\mathrm{ad}_{g_{0}^{(\mathbf{j})}}^{j_{i}+1}\frac{\partial}{\partial
u_{i}^{(j_{i})}}$ $\displaystyle=(-1)^{j_{i}}\left[f\frac{\partial}{\partial
x}+\sum_{k=1}^{m}\sum_{l=0}^{j_{k}-1}u_{k}^{(l+1)}\frac{\partial}{\partial
u_{k}^{(l)}},\frac{\partial}{\partial u_{i}^{(0)}}\right]$ (3.11)
$\displaystyle=(-1)^{j_{i}}\mathrm{ad}_{g_{0}^{(\mathbf{0})}}\frac{\partial}{\partial
u_{i}^{(0)}}=(-1)^{(j_{i}+1)}\frac{\partial f}{\partial
u_{i}^{(0)}}(\overline{x}^{(\mathbf{0})})\frac{\partial}{\partial
x}=\gamma_{1,i}^{(\mathbf{j})}\frac{\partial}{\partial x},$

which proves that
$\mathrm{ad}_{g_{0}^{(\mathbf{j})}}^{j_{i}+1}\frac{\partial}{\partial
u_{i}^{(j_{i})}}\in\mathrm{V}X^{(\mathbf{0})}$ and that (3.9) holds at the
order $k=1$.

Assuming that (3.9) holds up to $k=\nu$, with $\gamma_{\nu,i}^{(\mathbf{j})}$
depending at most on $\overline{x}^{(\mathbf{j}\lor(\nu-1))}$, we have

$\mathrm{ad}_{g_{0}^{(\mathbf{j})}}^{j_{i}+\nu+1}\frac{\partial}{\partial
u_{i}^{(j_{i})}}=(-1)^{j_{i}}\left[g_{0}^{(\mathbf{j})},\mathrm{ad}_{g_{0}^{(\mathbf{j})}}^{\nu}\frac{\partial}{\partial
u_{i}^{(0)}}\right]=\left(L_{g_{0}^{(\mathbf{j})}}\gamma_{\nu,i}^{(\mathbf{j})}-\gamma_{\nu,i}^{(\mathbf{j})}\frac{\partial
f}{\partial x}\right)\frac{\partial}{\partial
x}=\gamma_{\nu+1,i}^{(\mathbf{j})}\frac{\partial}{\partial x},$

and, according to (3.6)-(3.7), for all $j_{p}+1\leq\nu<j_{p+1}$,
$p=1,\ldots,m-1$, since we differentiate $\gamma_{\nu,i}^{(\mathbf{j})}$ with
respect to
$\sum_{k=1}^{m}\sum_{l=0}^{j_{k}-1}u_{k}^{(l+1)}\frac{\partial}{\partial
u_{k}^{(l)}}$, it is immediate to verify that
$\gamma_{\nu+1,i}^{(\mathbf{j})}$ depends at most on

$\overline{x}^{(\mathbf{j}\bigvee\nu)}=\left(x,u_{1}^{(0)},\ldots,u_{1}^{(j_{1})},\ldots,u_{p}^{(0)},\ldots,u_{p}^{(j_{p})},u_{p+1}^{(0)},\ldots,u_{p+1}^{(\nu)},\ldots,u_{m}^{(0)},\ldots,u_{m}^{(\nu)}\right)$

and on $\overline{x}^{(\mathbf{j})}=\overline{x}^{(\mathbf{j}\bigvee\nu)}$ if
$\nu\geq j_{m}$, hence (3.9).

Concerning (3.10), since
$\mathrm{ad}_{g_{0}^{(\mathbf{j})}}^{l-j_{q}}\frac{\partial}{\partial
u_{q}^{(0)}}=\pm\gamma_{l-j_{q},q}^{(\mathbf{j})}\frac{\partial}{\partial x}$
depends at most on $\overline{x}^{(\mathbf{j}\bigvee(l-j_{q}-1))}$, the
derivative of $\gamma_{l-j_{q},q}^{(\mathbf{j})}$ with respect to
$\frac{\partial}{\partial u_{p}^{(j_{p}-k)}}$ is indeed null if
$j_{p}-k>l-j_{q}-1$, hence the result. ∎

###  3.3 Decomposition of Purely Prolonged Distributions

Let us assume, without loss of generality, that the control components have
been reordered in such a way that $j_{1}\leq j_{2}\leq\cdots\leq j_{m}$.
Moreover, we may suppose that $j_{1}=0$, as shown to be sufficient in the next
section.

We now introduce two new filtrations of $\mathrm{T}X^{(\mathbf{j})}$, noted
$\Gamma_{k}^{(\mathbf{j})}$ and $\Delta_{k}^{(\mathbf{j})}$, for $k\geq 0$, as
follows

$\Gamma_{k}^{(\mathbf{j})}\triangleq\bigoplus_{p=2}^{m}\left\\{\frac{\partial}{\partial
u_{p}^{(j_{p}-l)}}\mid l=0,\ldots,k\vee(j_{p}-1)\right\\},$ (3.12)
$\Delta_{k}^{(\mathbf{j})}\triangleq\sum_{p=1}^{m}\left\\{\mathrm{ad}_{g_{0}^{(\mathbf{j})}}^{l-j_{p}}\frac{\partial}{\partial
u_{p}^{(0)}}\mid l=j_{p},\ldots,k\right\\}.$ (3.13)

with the convention that
$\mathrm{ad}_{g_{0}^{(\mathbf{j})}}^{k-j_{p}}\frac{\partial}{\partial
u_{p}^{(0)}}=0$ if $k<j_{p}$, $p=1,\ldots,m$.

We indeed have
$\Gamma_{k}^{(\mathbf{j})}=\Gamma_{j_{m}-1}^{(\mathbf{j})}=\bigoplus_{p=2}^{m}\\{\frac{\partial}{\partial
u_{p}^{(j_{p})}},\ldots,\frac{\partial}{\partial u_{p}^{(1)}}\\}$ for all
$k\geq j_{m}-1$. Thus, $\dim\Gamma_{k}^{(\mathbf{j})}\leq|\mathbf{j}|$ for all
$k$.

The definitions (3.12)–(3.13) and Lemma 3.1 readily yield, for all $k$, $l$
and $\mathbf{j}$,

$\displaystyle\overline{\Gamma_{k}^{(\mathbf{j})}}=\Gamma_{k}^{(\mathbf{j})},\qquad[\Gamma_{k}^{(\mathbf{j})},\Gamma_{l}^{(\mathbf{j})}]\subset\Gamma_{k\wedge
l}^{(\mathbf{j})},$ (3.14)
$\displaystyle\Delta_{l}^{(\mathbf{j})}\subset\mathrm{V}(X\times\mathbb{R}^{m}_{\infty}),\quad[\Gamma_{k}^{(\mathbf{j})},\Delta_{l}^{(\mathbf{j})}]\subset\mathrm{V}(X\times\mathbb{R}^{m}_{\infty}),\quad[\Gamma_{k}^{(\mathbf{j})},\Delta_{l}^{(\mathbf{j})}]\cap\Gamma_{k}^{(\mathbf{j})}=\\{0\\}.$

######  Remark 3.4.

Note that, unlike the filtration $\\{G_{k}^{(\mathbf{j})}\\}_{0\leq k\leq
k^{(\mathbf{j})}_{\star}}$ that is increasing for $k\leq
k_{\star}^{(\mathbf{j})}$, the mapping $k\mapsto\dim\Delta_{k}^{(\mathbf{j})}$
is only non-decreasing in general.

######  Remark 3.5.

In our definition of $\Delta_{k}^{(\mathbf{j})}$, we consider Lie brackets of
the form $\mathrm{ad}_{g_{0}^{(\mathbf{j})}}^{k}\frac{\partial}{\partial
u_{p}^{(0)}}$, as opposed to [2] where Lie brackets of the form
$\mathrm{ad}_{g_{0}^{(\mathbf{0})}}^{k}\frac{\partial}{\partial u_{p}^{(0)}}$
are used.

######  Proposition 3.2.

For all $\mathbf{j}$ such that $0=j_{1}\leq\ldots\leq j_{m-1}\leq j_{m}$, with
$j_{m}$ finite, if $\mathbf{j}=(j_{1},\ldots,j_{i},j_{i+1},\ldots,j_{m})$ and
$\mathbf{j^{\prime}}=(j_{1},\ldots,j_{i},j^{\prime}_{i+1},\ldots,j^{\prime}_{m})$
for some $(j^{\prime}_{i+1},\ldots,j^{\prime}_{m})$, we have
$\Delta_{k}^{(\mathbf{j})}=\Delta_{k}^{(\mathbf{j^{\prime}})}$ for all
$k=0,\ldots,(j_{i+1}\vee j^{\prime}_{i+1})-1$.

Moreover, for all $k\geq 0$,

$\dim\Gamma_{k}^{(\mathbf{j})}=\left\\{\begin{array}[]{ll}\displaystyle\sum_{p=1}^{i}j_{p}+(k+1)(m-i)&\displaystyle\mathrm{if\leavevmode\nobreak\
}j_{i}\leq k<j_{i+1},\;i=1,\ldots,m-1\\\
\displaystyle\mid\mathbf{j}\mid&\displaystyle\mathrm{if\leavevmode\nobreak\
}k\geq j_{m}\end{array}\right.$ (3.15)
$\dim\Delta_{k}^{(\mathbf{j})}\leq\left\\{\begin{array}[]{ll}\displaystyle((k+1)i-\sum_{p=1}^{i}j_{p})\vee(n+i)&\displaystyle\mathrm{if\leavevmode\nobreak\
}j_{i}\leq k<j_{i+1},\;i=1,\ldots,m-1\\\
\displaystyle\left((k+1)m-\mid\mathbf{j}\mid\right)\vee(n+m)&\displaystyle\mathrm{if\leavevmode\nobreak\
}k\geq j_{m}\end{array}\right.$ (3.16)

and we have

$G_{k}^{(\mathbf{j})}=\Gamma_{k}^{(\mathbf{j})}\oplus\Delta_{k}^{(\mathbf{j})},\quad\forall
k\geq 0.$ (3.17)

Furthermore, the finite integer $k_{\star}^{(\mathbf{j})}$, satisfying (3.5),
is such that
$\Delta_{k}^{(\mathbf{j})}=\Delta_{k_{\star}^{(\mathbf{j})}}^{(\mathbf{j})}$
and
$\Gamma_{k}^{(\mathbf{j})}=\Gamma_{k_{\star}^{(\mathbf{j})}}^{(\mathbf{j})}$
for all $k\geq k_{\star}^{(\mathbf{j})}$.

If, in addition, $\dim\Delta_{k_{\star}^{(\mathbf{j})}}^{(\mathbf{j})}=m+n$,
then

$n+\mid\mathbf{j}\mid\geq k_{\star}^{(\mathbf{j})}\geq
j_{m}\wedge\frac{n+\mid\mathbf{j}\mid}{m}.$ (3.18)

###### Proof.

By definition, the generators of $\Gamma_{k}^{(\mathbf{j})}$ are independent
for all $k$ and thus their number is equal to $\dim\Gamma_{k}^{(\mathbf{j})}$,
hence (3.15). The dimension of $\Delta_{k}^{(\mathbf{j})}$, in turn, is lesser
than, or equal to, the number of its generators, in number
$(k+1)+\ldots+(k-j_{i}+1)=(k+1)i-\sum_{p=1}^{i}j_{p}$, if $j_{i}\leq
k<j_{i+1}$ (respectively $(k+1)+\ldots+(k-j_{m}+1)=(k+1)m-\mid\mathbf{j}\mid$,
if $k\geq j_{m}$), and, since, according to (3.9) of Lemma 3.1,
$\Delta_{k}^{(\mathbf{j})}$ is contained in $\\{\frac{\partial}{\partial
u_{1}^{(0)}},\ldots,\frac{\partial}{\partial
u_{i}^{(0)}},\frac{\partial}{\partial x_{1}},\ldots,\frac{\partial}{\partial
x_{n}}\\}$ if $j_{i}\leq k<j_{i+1}$, $i=1,\ldots,m-1$ (respectively in
$\\{\frac{\partial}{\partial u_{1}^{(0)}},\ldots,\frac{\partial}{\partial
u_{m}^{(0)}},\frac{\partial}{\partial x_{1}},\ldots,\frac{\partial}{\partial
x_{n}}\\}$ if $k\geq j_{m}$), its dimension is bounded above by $i+n$ (resp.
$m+n$), hence (3.16)

The proof of (3.17) is by induction. For $k=0$, by (3.12)-(3.13), we indeed
have $G_{0}^{(\mathbf{j})}=\left\\{\frac{\partial}{\partial
u_{1}^{(0)}},\frac{\partial}{\partial
u_{2}^{(j_{2})}},\ldots,\frac{\partial}{\partial
u_{m}^{(j_{m})}}\right\\}=\left\\{\frac{\partial}{\partial
u_{1}^{(0)}}\right\\}\oplus\left\\{\frac{\partial}{\partial
u_{2}^{(j_{2})}},\ldots,\frac{\partial}{\partial
u_{m}^{(j_{m})}}\right\\}=\Delta_{0}^{(\mathbf{j})}\oplus\Gamma_{0}^{(\mathbf{j})}$.
Thus, (3.17) is valid at the order 0.

Assume now that (3.17) holds true up to the order $\nu>0$ with
$j_{r}\leq\nu<j_{r+1}$ for some $r\in\\{1,\ldots,m\\}$, assuming that
$j_{r}<j_{r+1}$. If $j_{r}=j_{r+1}$, the reader may immediately go to the case
$\nu+1=j_{r+1}$ below.

At the order $\nu+1$, two cases are possible: either $j_{r}\leq\nu+1<j_{r+1}$
or $\nu+1=j_{r+1}$. In the first case, using Lemma 3.1, we get:

$\displaystyle G_{\nu+1}^{(\mathbf{j})}$
$\displaystyle=G_{\nu}^{(\mathbf{j})}+ad_{g_{0}^{(\mathbf{j})}}G_{\nu}^{(\mathbf{j})}=\Gamma_{\nu}^{(\mathbf{j})}\oplus\Delta_{\nu}^{(\mathbf{j})}+ad_{g_{0}^{(\mathbf{j})}}\Gamma_{\nu}^{(\mathbf{j})}+ad_{g_{0}^{(\mathbf{j})}}\Delta_{\nu}^{(\mathbf{j})}$
$\displaystyle=\Gamma_{\nu}^{(\mathbf{j})}\oplus\Delta_{\nu}^{(\mathbf{j})}+\left\\{ad_{g_{0}^{(\mathbf{j})}}^{\nu+1}\frac{\partial}{\partial
u_{1}^{(0)}},\ldots,ad_{g_{0}^{(\mathbf{j})}}^{\nu+1-j_{r}}\frac{\partial}{\partial
u_{r}^{(0)}},\frac{\partial}{\partial
u_{r+1}^{(j_{r+1}-\nu-1)}},\ldots,\frac{\partial}{\partial
u_{m}^{(j_{m}-\nu-1)}}\right\\}$
$\displaystyle=\Gamma_{\nu+1}^{(\mathbf{j})}\oplus\Delta_{\nu+1}^{(\mathbf{j})}.$

In the second case, namely if $\nu+1=j_{r+1}$,

$\displaystyle G_{\nu+1}^{(\mathbf{j})}$
$\displaystyle=\Gamma_{\nu}^{(\mathbf{j})}\oplus\Delta_{\nu}^{(\mathbf{j})}$
$\displaystyle\hskip
28.45274pt+\left\\{ad_{g_{0}^{(\mathbf{j})}}^{\nu+1}\frac{\partial}{\partial
u_{1}^{(0)}},\ldots,ad_{g_{0}^{(\mathbf{j})}}^{\nu+1-j_{r}}\frac{\partial}{\partial
u_{r}^{(0)}},\frac{\partial}{\partial u_{r+1}^{(0)}},\frac{\partial}{\partial
u_{r+2}^{(j_{r+2}-\nu-1)}},\ldots,\frac{\partial}{\partial
u_{m}^{(j_{m}-\nu-1)}}\right\\}$
$\displaystyle=\Gamma_{\nu+1}^{(\mathbf{j})}\oplus\Delta_{\nu+1}^{(\mathbf{j})}.$

The case $j_{m}\leq\nu+1$ follows the same lines:

$G_{\nu+1}^{(\mathbf{j})}=\Gamma_{\nu}^{(\mathbf{j})}\oplus\Delta_{\nu}^{(\mathbf{j})}+\left\\{ad_{g_{0}^{(\mathbf{j})}}^{\nu+1}\frac{\partial}{\partial
u_{1}^{(0)}},\ldots,ad_{g_{0}^{(\mathbf{j})}}^{\nu+1-j_{m}}\frac{\partial}{\partial
u_{m}^{(0)}}\right\\}=\Gamma_{\nu+1}^{(\mathbf{j})}\oplus\Delta_{\nu+1}^{(\mathbf{j})}$

hence (3.17) is proven and the property of the number of iterations
$k_{\star}^{(\mathbf{j})}$ to simultaneously saturate the dimensions of
$\Gamma_{k}^{(\mathbf{j})}$ and $\Delta_{k}^{(\mathbf{j})}$ immediately
follows.

Moreover, if $\dim\Delta_{k_{\star}^{(\mathbf{j})}}^{(\mathbf{j})}=m+n$, we
must have $j_{m}\leq k_{\star}^{(\mathbf{j})}$ since otherwise, using
definition (3.13) for $k_{\star}^{(\mathbf{j})}<j_{m}$,
$\Delta_{k_{\star}^{(\mathbf{j})}}^{(\mathbf{j})}$ would not contain
$\\{\frac{\partial}{\partial u_{m}^{(0)}}\\}$ and its dimension would not
exceed $m-1+n$. Consequently, applying once more (3.16) with
$\dim\Delta_{k_{\star}^{(\mathbf{j})}}^{(\mathbf{j})}=m+n$, we get
$(k_{\star}^{(\mathbf{j})}+1)m-\mid\mathbf{j}\mid\geq m+n$, which, combined
with (3.5), immediately yields (3.18). ∎

######  Remark 3.6.

The inequality (3.18) reads
$k_{\star}^{(\mathbf{j})}-\frac{\mid\mathbf{j}\mid}{m}\geq\frac{n}{m}$ and may
thus be interpreted as an estimate of the gap between
$k_{\star}^{(\mathbf{j})}$ and the average value
$\frac{\mid\mathbf{j}\mid}{m}$ of the prolongation lengths
$j_{1},j_{2},\ldots,j_{m}$, provided that the prolonged system satisfies the
strong accessibility rank condition
$\dim\Delta_{k_{\star}^{(\mathbf{j})}}^{(\mathbf{j})}=m+n$.

##  4 Flatness by Pure Prolongation

###  4.1 Equivalence by Pure Prolongation

Consider the two systems (2.5) with $\dim x=n$, $\dim y=n^{\prime}$ and $\dim
u=\dim v=m$.

Given arbitrary $\mathbf{j}$ and $\mathbf{l}$, we recall that the associated
prolonged vector fields are

$g^{(\mathbf{j})}\triangleq g\frac{\partial}{\partial
x}+\sum_{i=1}^{m}\sum_{p=0}^{j_{i}}u_{i}^{(p+1)}\frac{\partial}{\partial
u_{i}^{(p)}},\qquad\gamma^{(\mathbf{l})}\triangleq\gamma\frac{\partial}{\partial
y}+\sum_{i=1}^{m}\sum_{p=0}^{l_{i}}v_{i}^{(p+1)}\frac{\partial}{\partial
v_{i}^{(p)}}.$

The prolonged states are, respectively, $\overline{x}^{(\mathbf{j})}$ and
$\overline{y}^{(\mathbf{l})}$, and the control inputs $u^{(\mathbf{j+1})}$ and
$v^{(\mathbf{l+1})}$.

######  Definition 4.1.

The systems (2.5) are equivalent by pure prolongation (in short
$P^{2}$-equivalent) at a point $\overline{x}_{0}\in
X\times\mathbb{R}^{m}_{\infty}$ if, and only if, there exist finite
$\mathbf{j}$ and $\mathbf{l}$ such that the prolonged systems of order
$\mathbf{j}$ and $\mathbf{l}$ respectively are equivalent by diffeomorphism
and feedback, i.e. if, and only if, there exists a local diffeomorphism
$\varphi$ and a feedback $W$:

$\overline{y}^{(\mathbf{l})}=\varphi(\overline{x}^{(\mathbf{j})})\qquad
v^{(\mathbf{l+1})}=W(\overline{x}^{(\mathbf{j})},u^{(\mathbf{j+1})})$

with $W$ invertible with respect to $u^{(\mathbf{j+1})}$ for all
$\overline{x}^{(\mathbf{j})}$ in a suitable neighborhood of
$\overline{x}_{0}$, such that

$\gamma^{(\mathbf{l})}\circ(\varphi\times W)=\varphi_{\star}g^{(\mathbf{j})}.$

This equivalence indeed implies that
$n+\mid\mathbf{j}\mid=n^{\prime}+\mid\mathbf{l}\mid$.

######  Remark 4.1.

The equivalence relation by _diffeomorphism and feedback_ is easily seen to be
strictly finer than the _$P^{2}$ -equivalence_ (take
$\mathbf{j}=\mathbf{l}=\mathbf{0}$), which in turn is strictly finer than the
_Lie-Bäcklund equivalence_ (see e.g. Example 5.4).

######  Definition 4.2.

We say that system (2.16) is _flat by pure prolongation_ (in short
$P^{2}$-flat) at a point $\overline{x}_{0}\in X\times\mathbb{R}^{m}_{\infty}$
if, and only if, it is $P^{2}$-equivalent to $\dot{y}_{i}=v_{i}$,
$i=1,\ldots,m$ and $y$ is called $P^{2}$-flat output.

It is therefore immediate to remark that a system is $P^{2}$-flat if, and only
if, there exists a pure prolongation of finite order $\mathbf{j}$ such that
the prolonged system is feedback linearizable at $\overline{x}_{0}$, thus
recovering the definition of linearization by prolongation already introduced
in [11, 12, 2].

Moreover, a $P^{2}$-flat output being obviously a flat output and pure
prolongations being particular cases of Lie-Bäcklund isomorphisms, the class
of $P^{2}$-flat systems is indeed contained in the class of Lie-Bäcklund
equivalence to $0$ (modulo the trivial field $\tau$), i.e. constitutes a
subclass of differentially flat systems.

The next Lemma extends a well-know result (see e.g. [6, 33]) to our context
(see also [12]).

######  Lemma 4.1.

We consider system (3.2), denoted by $\Sigma^{(\mathbf{j})}$, with
$\mathbf{j}=(j_{1},\ldots,j_{m})$, assuming, without loss of generality, that
$0\leq j_{1}\leq\ldots\leq j_{m}$, possibly up to input renumbering. We denote
by $\mathbf{j^{\prime}}=\mathbf{j-j_{1}}=(0,j_{2}-j_{1}\ldots,j_{m}-j_{1})$,
and by $\Sigma^{(\mathbf{j^{\prime}})}$ the corresponding system. Then
$\Sigma^{(\mathbf{j})}$ is $P^{2}$-flat at a given point
$(x_{0},\overline{u}_{0})$ if, and only if, $\Sigma^{(\mathbf{j^{\prime}})}$
is also $P^{2}$-flat at this point. Moreover, every $P^{2}$-flat output of
$\Sigma^{(\mathbf{j})}$ at $(x_{0},\overline{u}_{0})$ is a $P^{2}$-flat output
of $\Sigma^{(\mathbf{j^{\prime}})}$ at the same point, and conversely.

###### Proof.

Since $\Sigma^{(\mathbf{j})}$ is $P^{2}$-flat, there exists $\mathbf{j_{0}}$
and $\mathbf{l_{0}}$ such that $\Sigma^{(\mathbf{j+j_{0}})}$ is feedback
equivalent to the linear system $y^{(\mathbf{l_{0}+1})}=w$. Therefore, since,
by assumption,
$\Sigma^{(\mathbf{j+j_{0}})}=\Sigma^{(\mathbf{j^{\prime}+j_{1}+j_{0}})}$,
$\Sigma^{(\mathbf{j^{\prime}})}$ is also $P^{2}$-flat. The converse is trivial
as well as the fact that $\Sigma^{(\mathbf{j+j_{0}})}$ and
$\Sigma^{(\mathbf{j^{\prime}+j_{1}+j_{0}})}$ have the same $P^{2}$-flat
output. ∎

######  Remark 4.2.

In the single input case ($m=1$), this lemma shows that a $P^{2}$-flat system
is equivalent, without prolongation, to a linear system. We thus trivially
recover the fact that $P^{2}$-flatness and feedback linearizability are
equivalent in this case (see [6] and remark 2.2)

###  4.2 Necessary and Sufficient Conditions

The following Proposition is a straightforward adaptation of Theorem 2.2 for
an arbitrary order $\mathbf{j}$. Note that, at this stage, nothing is said
about a possible choice of $\mathbf{j}$, a question that will be dealt with in
subsection 4.3, theorem 4.2.

######  Proposition 4.1.

The prolonged system of order $\mathbf{j}$ is feedback linearizable at
$\overline{0}$ if, and only if, $G_{k}^{(\mathbf{j})}$ is involutive with
locally constant dimension for all $k$ and such that
$G_{k^{(\mathbf{j})}_{\ast}}^{(\mathbf{j})}=\mathrm{T}X^{(\mathbf{j})}$.

Again, flat outputs can be computed via Frobenius theorem, once established
the list of _Brunovský’s controllability indices of order $\mathbf{j}$_, as
follows:

######  Definition 4.3.

Consider the sequence of integers

$\rho_{k}^{(\mathbf{j})}\triangleq\dim
G_{k}^{(\mathbf{j})}/G_{k-1}^{(\mathbf{j})}\quad\forall k\geq
1,\qquad\rho_{0}^{(\mathbf{j})}\triangleq\dim G_{0}^{(\mathbf{j})}=m.$

The Brunovský controllability indices of order $\mathbf{j}$ are defined by

$\kappa_{k}^{(\mathbf{j})}\triangleq\\#\\{l\mid\rho_{l}^{(\mathbf{j})}\geq
k\\},\quad k=1,\ldots,m,$

As in the $0$th order case, if the prolonged system of order $\mathbf{j}$ is
feedback linearizable at $\overline{0}$, we have:

  * •

$\rho_{k}^{(\mathbf{j})}$’s and $\kappa_{k}^{(\mathbf{j})}$’s are non
increasing sequences,

  * •

$\rho_{k}^{(\mathbf{j})}\leq m$ for all $k$ and $\rho_{k}^{(\mathbf{j})}=0$
for all $k\geq k^{(\mathbf{j})}_{\ast}+1$,

  * •

$\kappa_{1}^{(\mathbf{j})}=k^{(\mathbf{j})}_{\ast}+1,\quad\kappa_{m}^{(\mathbf{j})}\geq
1$,

  * •

$\displaystyle\sum_{k=0}^{k^{(\mathbf{j})}_{\ast}}\rho_{k}^{(\mathbf{j})}=\sum_{k=1}^{m}\kappa_{k}^{(\mathbf{j})}=\dim
G_{k^{(\mathbf{j})}_{\ast}}^{(\mathbf{j})}=n+m+\mid\mathbf{j}\mid$.

The list $\kappa_{1}^{(\mathbf{j})},\ldots,\kappa_{m}^{(\mathbf{j})}$ is
uniquely defined up to input permutation, invariant by prolonged state
feedback and prolonged state diffeomorphism, and is thus equal to the list of
controllability indices of the associated linear system (2.15) with
$r_{i}=\kappa_{i}^{(\mathbf{j})}$, $i=1,\ldots,m$.

Moreover, for all $k$ and all $i=1,\ldots,m$, and possibly up to a suitable
input reordering, we have

$G_{k}^{(\mathbf{j})}=\bigoplus_{p=1}^{m}\left\\{\mathrm{ad}_{g_{0}^{(\mathbf{j})}}^{l}\frac{\partial}{\partial
u_{p}^{(j_{p})}}\mid l=0,\ldots
k\lor(\kappa_{p}^{(\mathbf{j})}-1)\right\\},\quad
G_{\kappa_{1}^{(\mathbf{j})}-1}^{(\mathbf{j})}=G_{k^{(\mathbf{j})}_{\ast}}^{(\mathbf{j})}=\mathrm{T}X^{(\mathbf{j})}$

and flat outputs $(y_{1},\ldots,y_{m})$ are locally non trivial solutions of
the system of PDE’s

$\left<G_{k}^{(\mathbf{j})},dy_{i}\right>=0,\;k=0,\ldots,\kappa_{i}^{(\mathbf{j})}-2,\quad\mathrm{with\leavevmode\nobreak\
}\quad\left<G_{\kappa_{i}^{(\mathbf{j})}-1}^{(\mathbf{j})},dy_{i}\right>\neq
0,\quad i=1,\ldots,m.$ (4.1)

Finally, the mapping

$\overline{x}^{(\mathbf{j})}\mapsto(y_{1},\ldots,y_{1}^{(\kappa_{1}^{(\mathbf{j})}-1)},\ldots,y_{m},\ldots,y_{m}^{(\kappa_{m}^{(\mathbf{j})}-1)})$

is a local diffeomorphism.

In virtue of Lemma 4.1, it suffices to restrict our analysis to prolongations
of order $\mathbf{j}=(j_{1},\ldots,j_{m})$ such that $0=j_{1}\leq\ldots,\leq
j_{m}$.

We are now ready to state our main result.

######  Theorem 4.1.

A necessary and sufficient condition for $P^{2}$-flatness at $\overline{0}$ is
that there exists $\mathbf{j}=(j_{1},\ldots,j_{m})\in\mathbb{N}^{m}$,
$0=j_{1}\leq\ldots,\leq j_{m}$, such that

  * (i)

$\overline{\Delta_{k}^{(\mathbf{j})}}=\Delta_{k}^{(\mathbf{j})}$ with
$\dim\Delta_{k}^{(\mathbf{j})}$ locally constant for all $k\geq 0$,

  * (ii)

$[\Gamma_{k}^{(\mathbf{j})},\Delta_{k}^{(\mathbf{j})}]\subset\Delta_{k}^{(\mathbf{j})}$
for all $k\geq 0$,

  * (iii)

$k^{(\mathbf{j})}_{\ast}$ is such that
$\Delta_{k}^{(\mathbf{j})}=\mathrm{T}X\times\mathrm{T}\mathbb{R}^{m}$ and
$\Gamma_{k}^{(\mathbf{j})}=\mathrm{T}\mathbb{R}^{\mid\mathbf{j}\mid}$ for all
$k\geq k^{(\mathbf{j})}_{\ast}$.

###### Proof.

By (3.17) of Proposition 3.2, we have
$G_{k}^{(\mathbf{j})}=\Gamma_{k}^{(\mathbf{j})}\oplus\Delta_{k}^{(\mathbf{j})}$
for all $k\geq 0$. Then,
$G_{k}^{(\mathbf{j})}=\overline{G_{k}^{(\mathbf{j})}}$ implies that
$[\Gamma_{k}^{(\mathbf{j})}\oplus\Delta_{k}^{(\mathbf{j})},\Gamma_{k}^{(\mathbf{j})}\oplus\Delta_{k}^{(\mathbf{j})}]\subset\Gamma_{k}^{(\mathbf{j})}\oplus\Delta_{k}^{(\mathbf{j})}$.
Since $\Gamma_{k}^{(\mathbf{j})}=\overline{\Gamma_{k}^{(\mathbf{j})}}$ for all
$k$, and since
$[\Gamma_{k}^{(\mathbf{j})},\Delta_{k}^{(\mathbf{j})}]\cap\Gamma_{k}^{(\mathbf{j})}=\\{0\\}$
by Lemma 3.1 and (3.14), we deduce that
$[\Gamma_{k}^{(\mathbf{j})},\Delta_{k}^{(\mathbf{j})}]+[\Delta_{k}^{(\mathbf{j})},\Delta_{k}^{(\mathbf{j})}]\subset\Delta_{k}^{(\mathbf{j})}$,
hence
$[\Gamma_{k}^{(\mathbf{j})},\Delta_{k}^{(\mathbf{j})}]\subset\Delta_{k}^{(\mathbf{j})}$
and
$[\Delta_{k}^{(\mathbf{j})},\Delta_{k}^{(\mathbf{j})}]\subset\Delta_{k}^{(\mathbf{j})}$,
i.e. $\Delta_{k}^{(\mathbf{j})}=\overline{\Delta_{k}^{(\mathbf{j})}}$, for all
$k\geq 0$.

Conversely,
$[\Gamma_{k}^{(\mathbf{j})},\Delta_{k}^{(\mathbf{j})}]\subset\Delta_{k}^{(\mathbf{j})}$
and $\Delta_{k}^{(\mathbf{j})}=\overline{\Delta_{k}^{(\mathbf{j})}}$ for all
$k\geq 0$ trivially implies that
$G_{k}^{(\mathbf{j})}=\overline{G_{k}^{(\mathbf{j})}}$ for all $k\geq 0$.

Moreover, since $\dim\Gamma_{k}^{(\mathbf{j})}$ is constant by construction,
the fact that $G_{k}^{(\mathbf{j})}$ has locally constant dimension is
equivalent to the fact that $\Delta_{k}^{(\mathbf{j})}$ has locally constant
dimension too for all $k\geq 0$, hence (i).

Finally, (iii) is an immediate consequence of the condition that
$G_{k}^{(\mathbf{j})}=\mathrm{T}X\times\mathrm{T}\mathbb{R}^{m+\mid\mathbf{j}\mid}$
for all $k\geq k^{(\mathbf{j})}_{\ast}$, and the theorem is proven. ∎

######  Remark 4.3.

Expressed in words, the above necessary and sufficient conditions read (i)
involutivity with locally constant dimension of the
$\Delta_{k}^{(\mathbf{j})}$’s, (ii) invariance of $\Delta_{k}^{(\mathbf{j})}$
by $\Gamma_{k}^{(\mathbf{j})}$ and (iii) strong controllability rank condition
(see e.g. [35, 13]).

###  4.3 The Pure Prolongation Algorithm

From now on, for every sequence
$\mathbf{l}=\left(l_{1},\ldots,l_{m}\right)\in\mathbb{N}^{m}$, we
systematically re-order the indices $\\{1,\ldots,m\\}$ by a suitable
permutation $\alpha:\\{1,\ldots,m\\}\rightarrow\\{1,\ldots,m\\}$ such that
$0=l_{\alpha(1)}\leq\ldots\leq l_{\alpha(m)}$. Moreover, for simplicity’s
sake, the permutation $\alpha$ will be omitted. We will thus abusively replace
$l_{\alpha(i)}$ by $l_{i}$, i.e. $0=l_{1}\leq\ldots\leq l_{m}$.

In the following algorithm we assume that the computations are done in a
suitable open dense neighborhood of $\overline{0}$ where all the distributions
involved have constant dimension.

####  4.3.1 Initialization

Consider the filtration $\\{G_{k}^{(0)}\mid k\geq 0\\}$ defined by (2.19). If
every $G_{k}^{(0)}$ satisfies the conditions of theorem 2.2, the system is
feedback linearizable and no prolongation is needed. In particular, for $m=1$,
a case where flatness and feedback linearizability are equivalent, the results
of this section are pointless.

Otherwise, we have the following alternative:

  * •

either there must exist $k_{0}\leq n$ such that $G_{k_{0}}^{(0)}$ is not
involutive while every $G_{k}^{(0)}$ is involutive for $k<k_{0}$ (note that
$k_{0}\geq 1$ since $G_{0}^{(\mathbf{0})}$ is always involutive),

  * •

or the $G_{k}^{(0)}$’s are all involutive but with $\max_{k\geq 0}\dim
G_{k}^{(0)}=\dim G_{k_{\star}^{(\mathbf{0})}}^{(0)}<n+m$.

##### First Case.

There exists $k_{0}\in\\{1,\ldots,n\\}$, first index for which
$G_{k_{0}}^{(0)}$ is not involutive.

Let $H_{1}^{(\mathbf{0})}\neq\\{0\\}$ be any involutive distribution included
in $G_{1}^{(\mathbf{0})}$, of the form

$H_{1}^{(\mathbf{0})}\triangleq\\{\frac{\partial}{\partial
u_{1}^{(0)}},\ldots,\mathrm{ad}_{g_{0}^{(0)}}\frac{\partial}{\partial
u_{1}^{(0)}},\ldots,\frac{\partial}{\partial
u_{p_{0}}^{(0)}},\ldots,\mathrm{ad}_{g_{0}^{(0)}}\frac{\partial}{\partial
u_{p_{0}}^{(0)}}\\}$ (4.2)

for some $p_{0}\leq m$.

By (A.3), $\mathrm{ad}_{g_{0}^{(0)}}\frac{\partial}{\partial
u_{p}^{(0)}}=\mathrm{ad}_{g_{0}^{(\mathbf{l})}}\frac{\partial}{\partial
u_{p}^{(0)}}$ for all $\mathbf{l}$ and, thanks to (3.10), we have
$[\frac{\partial}{\partial
u_{q}^{(l_{q}-k)}},\mathrm{ad}_{g_{0}^{(0)}}\frac{\partial}{\partial
u_{p}^{(0)}}]=0$ for all $l_{q}-k>1$ with $q>p_{0}$ and all
$p=1,\ldots,p_{0}$. Therefore we can choose

$\Delta_{1}^{(\mathbf{l})}\triangleq\\{\frac{\partial}{\partial
u_{1}^{(0)}},\ldots,\mathrm{ad}_{g_{0}^{(\mathbf{l})}}\frac{\partial}{\partial
u_{1}^{(0)}},\ldots,\frac{\partial}{\partial
u_{p_{0}}^{(0)}},\ldots,\mathrm{ad}_{g_{0}^{(\mathbf{l})}}\frac{\partial}{\partial
u_{p_{0}}^{(0)}}\\}=H_{1}^{(\mathbf{0})}$ (4.3)

and

$\Gamma_{1}^{(\mathbf{l})}\triangleq\\{\frac{\partial}{\partial
u_{p_{0}+1}^{(l_{p_{0}+1})}},\frac{\partial}{\partial
u_{p_{0}+1}^{(l_{p_{0}+1}-1)}},\ldots,\frac{\partial}{\partial
u_{m}^{(l_{m})}},\frac{\partial}{\partial u_{m}^{(l_{m}-1)}}\\}$ (4.4)

for any

$\mathbf{l}=\left(\underbrace{0,\ldots,0}_{p_{0}},l_{p_{0}+1},\ldots,l_{m}\right),\quad
2=l_{p_{0}+1}\leq\ldots\leq l_{m}$ (4.5)

since they naturally satisfy
$\overline{\Delta_{1}^{(\mathbf{l})}}=\overline{H_{1}^{(\mathbf{0})}}=H_{1}^{(\mathbf{0})}=\Delta_{1}^{(\mathbf{l})}$
and
$[\Gamma_{1}^{(\mathbf{l})},\Delta_{1}^{(\mathbf{l})}]\subset\Delta_{1}^{(\mathbf{l})}$
for all $\mathbf{l}$ given by (4.5), i.e. with $l_{p_{0}+1}=2$.

However if, for $l_{p_{0}+1}=1$,

$\Delta_{1}^{(\mathbf{l})}\triangleq\\{\frac{\partial}{\partial
u_{1}^{(0)}},\ldots,\mathrm{ad}_{g_{0}^{(\mathbf{l})}}\frac{\partial}{\partial
u_{1}^{(0)}},\ldots,\frac{\partial}{\partial
u_{p_{0}}^{(0)}},\ldots,\mathrm{ad}_{g_{0}^{(\mathbf{l})}}\frac{\partial}{\partial
u_{p_{0}}^{(0)}},\frac{\partial}{\partial u_{p_{0}+1}^{(0)}}\\}$ (4.6)

and

$\Gamma_{1}^{(\mathbf{l})}\triangleq\\{\frac{\partial}{\partial
u_{p_{0}+1}^{(1)}},\frac{\partial}{\partial
u_{p_{0}+2}^{(l_{2})}},\frac{\partial}{\partial
u_{p_{0}+2}^{(l_{2}-1)}},\ldots,\frac{\partial}{\partial
u_{m}^{(l_{m})}},\frac{\partial}{\partial u_{m}^{(l_{m}-1)}}\\}$ (4.7)

also satisfy $\overline{\Delta_{1}^{(\mathbf{l})}}=\Delta_{1}^{(\mathbf{l})}$
and
$[\Gamma_{1}^{(\mathbf{l})},\Delta_{1}^{(\mathbf{l})}]\subset\Delta_{1}^{(\mathbf{l})}$,
the previous choice (4.3)-(4.4)-(4.5) maybe replaced by (4.6)-(4.7) with
$1=l_{p_{0}+1}\leq\ldots\leq l_{m}$.

These choices, namely (4.3)-(4.4)-(4.5), or (4.6)-(4.7) with $l_{p_{0}+1}=1$,
for any $H_{1}^{(\mathbf{0})}$ given by (4.2), constitutes the initialization
of the algorithm.

Note that there are at most
$\sum_{p=1}^{m-1}{\mathrm{C}}_{m}^{p}=\sum_{p=1}^{m-1}\frac{m!}{p!(m-p)!}$
possibilities of such initialization.

##### Second Case.

######  Lemma 4.2.

Assume that the $G_{k}^{(0)}$’s are all involutive with $\max_{k\geq 0}\dim
G_{k}^{(\mathbf{0})}<n+m$. Then
$G_{k}^{(\mathbf{j})}=\overline{G_{k}^{(\mathbf{j})}}=G_{k}^{(\mathbf{0})}$
for all $k$ and $\mathbf{j}$ and $\max_{k\geq 0}\dim G_{k}^{(\mathbf{j})}<m+n$
for all $\mathbf{j}$.

###### Proof.

According to (A.3) of lemma A.1, every
$\mathrm{ad}_{g_{0}^{(\mathbf{j})}}^{k}\frac{\partial}{\partial
u_{p}^{(j_{p})}}$ is equal to
$\pm\mathrm{ad}_{g_{0}^{(\mathbf{0})}}^{k-j_{p}}\frac{\partial}{\partial
u_{p}^{(0)}}$ \+ terms in
$\overline{G_{k-1}^{(\mathbf{0})}}=G_{k-1}^{(\mathbf{0})}$, thanks to the
involutivity of the $G_{k}^{(0)}$’s. Therefore, $G_{k}^{(\mathbf{j})}\subset
G_{k}^{(\mathbf{0})}$, the converse inclusion being immediate using (A.3).
Thus $\dim G_{k}^{(\mathbf{0})}=\dim G_{k}^{(\mathbf{j})}$ for all $k$ and
$\mathbf{j}$, Q.E.D. ∎

We immediately conclude that, in this case, the strong controllability rank
condition does not hold, which contradicts (iii) of theorem 2.2, hence the
non-flatness by pure prolongation. Therefore, this case must be discarded.

####  4.3.2 Recursion

For all $k\geq 1$ and all $\mathbf{l}$ given by:

$\mathbf{l}=\left(\underbrace{0,\ldots,0}_{p_{0}},l_{p_{0}+1},\ldots,l_{m}\right),\quad
1\leq l_{p_{0}+1}\leq\ldots\leq l_{m},$ (4.8)

and denoting by $C\\!\\!-\\!\min_{\mathbf{l}\in L}(l_{p_{0}+1},\ldots,l_{m})$
(resp. $C\\!\\!-\\!\max_{\mathbf{l}\in L}(l_{p_{0}+1},\ldots,l_{m})$) the
componentwise minimum (resp. maximum) of a collection of $(m-p_{0})$-tuples
$(l_{p_{0}+1},\ldots,l_{m})$ in a set $L$, with $1\leq
l_{p_{0}+1}\leq\ldots\leq l_{m}$, let us introduce the following numbers

$\displaystyle\sigma_{\Delta}(k)$
$\displaystyle\triangleq\left(\sigma_{p_{0}+1,\Delta}(k),\ldots,\sigma_{m,\Delta}(k)\right)$
(4.9) $\displaystyle\triangleq C\\!\\!-\\!\min\\{1\leq
l_{p_{0}+1},\leq\ldots\leq
l_{m}\mid\overline{\Delta_{k}^{(\mathbf{l})}}=\Delta_{k}^{(\mathbf{l})}\\},$
$\displaystyle\sigma_{\Gamma,\Delta}(k)$
$\displaystyle\triangleq\left(\sigma_{p_{0}+1,\Gamma,\Delta}(k),\ldots,\sigma_{m,\Gamma,\Delta}(k)\right)$
(4.10) $\displaystyle\triangleq C\\!\\!-\\!\min\\{1\leq
l_{p_{0}+1}\leq\ldots\leq
l_{m}\mid[\Gamma_{k}^{(\mathbf{l})},\Delta_{k}^{(\mathbf{l})}]\subset\Delta_{k}^{(\mathbf{l})}\\}$

and

$\displaystyle\left(j_{p_{0}+1},\ldots,j_{m}\right)$ (4.11)
$\displaystyle\qquad\triangleq C\\!\\!-\\!\max_{k\geq
1}\left(\sigma_{p_{0}+1,\Delta}(k)\wedge\sigma_{p_{0}+1,\Gamma,\Delta}(k)\leq\ldots,\leq\sigma_{m,\Delta}(k)\wedge\sigma_{m,\Gamma,\Delta}(k)\right).$

######  Remark 4.4.

Thanks to (4.5)-(4.3)-(4.4), we get

$\sigma_{p_{0}+1,\Delta}(1)\wedge\sigma_{p_{0}+1,\Gamma,\Delta}(1)\leq 2$

and if $\overline{\Delta_{1}^{(\mathbf{l})}}=\Delta_{1}^{(\mathbf{l})}$ and
$[\Gamma_{1}^{(\mathbf{l})},\Delta_{1}^{(\mathbf{l})}]\subset\Delta_{1}^{(\mathbf{l})}$,
with (4.6)-(4.7) and $l_{p_{0}+1},\ldots,l_{m}\geq 1$, we get

$\sigma_{p_{0}+1,\Delta}(1)\wedge\sigma_{p_{0}+1,\Gamma,\Delta}(1)=1.$

######  Remark 4.5.

In view of the definition (3.13) of $\Delta_{k}^{(\mathbf{l})}$, if $l_{q}\leq
k<l_{q+1}$ for some $q\in\\{1,\ldots,m\\}$ and $k\geq 1$,
$\Delta_{k}^{(\mathbf{l})}$ does not depend on $l_{q+1},\ldots,l_{m}$. Thus
$\sigma_{\Delta}(k)=\left(\sigma_{1,\Delta}(k),\ldots,\sigma_{q,\Delta}(k),\underbrace{\sigma_{q,\Delta}(k),\ldots,\sigma_{q,\Delta}(k)}_{m-q}\right)$.

######  Proposition 4.2.

We have

$[\Gamma_{k}^{(\mathbf{l})},\Delta_{k}^{(\mathbf{l})}]=\\{0\\},\quad\mathrm{and}\quad\sigma_{p_{0}+1,\Gamma,\Delta}(k)\leq
2k\quad\forall l_{p_{0}+1}>2k-1,\;\forall k\geq 0.$ (4.12)

Moreover, $\Delta_{k}^{(\mathbf{l})}$ is independent of $l_{p_{0}+1}$ if
$l_{p_{0}+1}>k$. Therefore, if $\Delta_{k}^{(\mathbf{l})}$ is involutive for
all $k$, we get

$\sigma_{p_{0}+1,\Delta}(k)\wedge\sigma_{p_{0}+1,\Gamma,\Delta}(k)\leq 2k$
(4.13)

and $j_{p_{0}+1}\leq\ldots\leq j_{m}<+\infty$. The minimal prolongation orders
are thus obtained by minimizing (4.11) over all initializations built on
(4.2).

###### Proof.

(4.13) is an immediate consequence of (3.10) and of the definitions of
$\Delta_{k}^{(\mathbf{l})}$ and $\Gamma_{k}^{(\mathbf{l})}$. Moreover, since,
according to remark 3.4, $k\mapsto\dim\Delta_{k}^{(\mathbf{l})}$ is non
decreasing and bounded by $n+m$, then there exists a finite $K^{\star}$ such
that $\Delta_{K^{\star}}^{(\mathbf{l})}=\Delta_{k}^{(\mathbf{l})}$ for all
$k\geq K^{\star}$ and thus the maximum with respect to $k$ in (4.11) is
achieved for $k=K^{\star}$. The C-minimal prolongation orders are thus
obtained by varying the initializations built on (4.2). ∎

We thus have proven:

######  Theorem 4.2.

We have the following alternative:

  1. 1.

If, for every choice of initialization built on (4.2), there exists a $k$ for
which $\max_{p\in\\{p_{0}+1,\ldots,m\\}}\sigma_{p,\Delta}(k)=+\infty$ or if
$\max_{k}\dim\Delta_{k}^{(\mathbf{j})}<n+m$ for all $\mathbf{j}$, then the
system is not flat by pure prolongation.

  2. 2.

Otherwise, $\mathbf{j}$ is finite and given by (4.11) and the minimal
prolongation orders are obtained by C-minimizing (4.11) over all
initializations.

###### Proof.

Straightforward from what precedes. ∎

We immediately deduce the following algorithm:

Algorithm 1. flatness by pure prolongation

Input:

the vector fields $g_{0}^{(\mathbf{0})},g_{1}^{(0)}=\frac{\partial}{\partial
u_{1}^{(0)}},\ldots,g_{m}^{(0)}=\frac{\partial}{\partial u_{m}^{(0)}}$ (see
(2.17))

output:

the minimal lengths $0=j_{1}\leq\ldots\leq j_{m}$ and
$k_{\star}^{(\mathbf{j})}$ or fail if the system is not flat by pure
prolongation.

Initialization.

Construct $G_{k}^{(0)}$ for all $k\leq n$. If $\exists k_{0}\leq n$ such that
$G_{k_{0}}^{(0)}$ is not involutive, choose an involutive subdistribution
$H_{1}^{(\mathbf{0})}\subset G_{k_{0}}^{(0)}$, given by (4.2), with $p_{0}\leq
m-1$ and compute $\Delta_{1}^{(\mathbf{l})}$ and $\Gamma_{1}^{(\mathbf{l})}$
by (4.3)-(4.4)-(4.5), or (4.6)-(4.7) with $l_{p_{0}+1}=1$.

Step $k\geq 1$.

Compute $\Delta_{k}^{(\mathbf{l})}$, $\Gamma_{k}^{(\mathbf{l})}$,
$\sigma_{\Delta}(k)$ and $\sigma_{\Gamma,\Delta}(k)$ by (4.9) and (4.10).
Continue up to the first $k_{1}$ such that
$\sigma_{\Gamma,\Delta}(k_{1})\bigwedge\sigma_{\Delta}(k_{1})$ is C-maximal.
Then $\mathbf{j}$ is given by (4.11).
If $\Delta_{k}^{(\mathbf{l})}$ is non involutive for some $k\leq k_{1}$,
change the initialization and restart and if $\Delta_{k}^{(\mathbf{l})}$ is
non involutive for all initializations, then fail.

Step $k_{\star}^{(\mathbf{j})}$.

Determine $k_{\star}^{(\mathbf{j})}$. If
$\dim\Delta_{k_{\star}^{(\mathbf{j})}}^{(\mathbf{j})}=n+m$, stop. Otherwise,
fail.

##  5 Examples

All the examples of this paper concern two input systems, i.e. $m=2$, except
example 5.5, with $3$ inputs, resuming the example 2 of [18].

In the two input examples, the prolongation index $\mathbf{j}=(j_{1},j_{2})$
is supposed to be such that $0=j_{1}\leq j_{2}$, up to a suitable input
permutation. However, for the sake of readability, we will keep the original
input numbering unchanged and thus consider that $\mathbf{j}=(0,j_{2})$ or
$(j_{1},0)$ depending on the context. At the exception of this modification,
we strictly apply algorithm 4.2 in all the examples.

The first example gives a detailed presentation of the application of this
algorithm, in particular the role played by $\sigma_{\Gamma,\Delta}(k)$ and
$\sigma_{\Delta}(k)$. The second one shows the importance of the sole number
$\sigma_{1,\Delta}(k)$ to determine the prolongation length, and the third
one, borrowed from [7], and carried on again in [2, Section 3.1], is reported
here to compare our approach with the one of [2]. Then, the pendulum example
is presented to show that non flat systems by pure prolongation can be
detected in a finite number of steps. To conclude this section, the last
example, with 3 inputs, compares the constuction proposed in [18, Example 2]
with our algorithm leading to a minimal prolongation, strictly smaller than
the one of [18].

###  5.1 Chained System [25]

$\displaystyle x_{1}^{(3)}=u_{1}$ (5.1) $\displaystyle\ddot{x}_{2}=u_{2}$
$\displaystyle\dot{x}_{3}=u_{1}u_{2}$

This system has been proven to be flat in [25, section 3.1.1] with the
following flat output

$y_{1}=x_{3}-\ddot{x}_{1}u_{2}+\dot{x}_{1}\dot{u}_{2}-x_{1}\ddot{u}_{2},\quad
y_{2}=x_{2}.$ (5.2)

####  5.1.1 Associated Non Prolonged Distributions

Let us start this section by showing that system (5.1) is not static feedback
linearizable. We denote the state coordinates by
$(x_{1,1},x_{1,2},x_{1,3},x_{2,1},x_{2,2},x_{3},u_{1}^{(0)},u_{2}^{(0)})$
($n=6$ and $m=2$), with $x_{1,j}\triangleq x_{1}^{(j)}$, $j=1,2,3$,
$x_{2,j}\triangleq x_{2}^{(j)}$, $j=1,2$ and $u_{i}^{(0)}=u_{i}$, $i=1,2$.
The two input variables are $u_{i}^{(1)}\triangleq\dot{u}_{i}$, $i=1,2$.

The system vector fields are

$\displaystyle g_{0}^{(0)}$ $\displaystyle\triangleq
x_{1,2}\frac{\partial}{\partial x_{1,1}}+x_{1,3}\frac{\partial}{\partial
x_{1,2}}+u_{1}^{(0)}\frac{\partial}{\partial
x_{1,3}}+x_{2,2}\frac{\partial}{\partial
x_{2,1}}+u_{2}^{(0)}\frac{\partial}{\partial
x_{2,2}}+u_{1}^{(0)}u_{2}^{(0)}\frac{\partial}{\partial x_{3}}$ (5.3)
$\displaystyle g_{1}^{(0)}$ $\displaystyle\triangleq\frac{\partial}{\partial
u_{1}^{(0)}},\qquad g_{2}^{(0)}\triangleq\frac{\partial}{\partial
u_{2}^{(0)}}$

One can verify that

$\displaystyle\mathrm{ad}_{g_{0}^{(0)}}g_{1}^{(0)}=-\frac{\partial}{\partial
x_{1,3}}-u_{2}^{(0)}\frac{\partial}{\partial
x_{3}},\quad\mathrm{ad}_{g_{0}^{(0)}}^{2}g_{1}^{(0)}=\frac{\partial}{\partial
x_{1,2}},\quad\mathrm{ad}_{g_{0}^{(0)}}^{3}g_{1}^{(0)}=-\frac{\partial}{\partial
x_{1,1}},\quad\mathrm{ad}_{g_{0}^{(0)}}^{4}g_{1}^{(0)}=0$

and

$\displaystyle\mathrm{ad}_{g_{0}^{(0)}}g_{2}^{(0)}=-\frac{\partial}{\partial
x_{2,2}}-u_{1}^{(0)}\frac{\partial}{\partial
x_{3}},\quad\mathrm{ad}_{g_{0}^{(0)}}^{2}g_{2}^{(0)}=\frac{\partial}{\partial
x_{2,1}},\quad\mathrm{ad}_{g_{0}^{(0)}}^{3}g_{2}^{(0)}=0.$

Therefore

$G_{0}^{(0)}=\left\\{\frac{\partial}{\partial
u_{1}^{(0)}},\frac{\partial}{\partial
u_{2}^{(0)}}\right\\}=\overline{G_{0}^{(0)}},$
$G_{1}^{(0)}=G_{0}^{(0)}+\left\\{-\frac{\partial}{\partial
x_{1,3}}-u_{2}^{(0)}\frac{\partial}{\partial x_{3}},-\frac{\partial}{\partial
x_{2,2}}-u_{1}^{(0)}\frac{\partial}{\partial
x_{3}}\right\\}\neq\overline{G_{1}^{(0)}},$

since, e.g. ,
$[g_{2}^{(0)},\mathrm{ad}_{g_{0}^{(0)}}g_{1}^{(0)}]=-\frac{\partial}{\partial
x_{3}}\not\in G_{1}^{(0)}$, and $\dim G_{1}^{(0)}=4$ whereas
$\dim\overline{G_{1}^{(0)}}=5$,

$G_{2}^{(0)}=G_{1}^{(0)}+\left\\{\frac{\partial}{\partial
x_{1,2}},\frac{\partial}{\partial
x_{2,1}}\right\\}\neq\overline{G_{2}^{(0)}},\quad\dim\overline{G_{2}^{(0)}}=7,$
$G_{3}^{(0)}=G_{2}^{(0)}+\left\\{\frac{\partial}{\partial
x_{1,1}}\right\\}\neq\overline{G_{3}^{(0)}}$

and $G_{k}^{(0)}=G_{3}^{(0)}$ for all $k\geq 3$. Moreover,
$\dim{G_{3}^{(0)}}=7<\dim{\overline{G_{3}^{(0)}}}=n+m=8$. We conclude that the
system is not feedback linearizable.

Since $G_{1}^{(0)}$ is not involutive, there are two possible initializations:

$H_{1,1}^{(\mathbf{0})}=\\{g_{1}^{(0)},\mathrm{ad}_{g_{0}^{(0)}}g_{1}^{(0)}\\}=\\{\frac{\partial}{\partial
u_{1}^{(0)}},-\frac{\partial}{\partial
x_{1,3}}-u_{2}^{(0)}\frac{\partial}{\partial
x_{3}}\\}=\overline{H_{1,1}^{(\mathbf{0})}}$ (5.4)

or

$H_{1,2}^{(\mathbf{0})}=\\{g_{2}^{(0)},\mathrm{ad}_{g_{0}^{(0)}}g_{2}^{(0)}\\}=\\{\frac{\partial}{\partial
u_{2}^{(0)}},-\frac{\partial}{\partial
x_{2,2}}-u_{1}^{(0)}\frac{\partial}{\partial
x_{3}}\\}=\overline{H_{1,2}^{(\mathbf{0})}}.$ (5.5)

We use the second possibility, which amounts to prolonging the first input.

####  5.1.2 Flatness by Pure Prolongation of the First Input

Let us now apply theorem 4.1 and algorithm 4.2 with $j_{2}=0$, i.e.
$g_{2}^{(j_{2})}\triangleq g_{2}^{(0)}=\frac{\partial}{\partial u_{2}^{(0)}}$
to determine if this system is flat by pure prolongation and compute
$j_{1}\geq 1$. Recall that we have set
$g_{0}^{(l_{1},0)}=g_{0}^{(0)}+\sum_{p=0}^{l_{1}-1}u_{1}^{(p+1)}\frac{\partial}{\partial
u_{1}^{(p)}}$.

##### $\bullet\leavevmode\nobreak\ \mathbf{k=0.}$

We have $\Gamma_{0}^{(l_{1},0)}=\left\\{\frac{\partial}{\partial
u_{1}^{(l_{1})}}\right\\}$,
$\Delta_{0}^{(l_{1},0)}=\left\\{\frac{\partial}{\partial
u_{2}^{(0)}}\right\\}=\overline{\Delta_{0}^{(l_{1},0)}}$ and
$\left[\Gamma_{0}^{(l_{1},0)},\Delta_{0}^{(l_{1},0)}\right]\subset\Delta_{0}^{(l_{1},0)}$
for all $l_{1}\geq 1$.

##### $\bullet\leavevmode\nobreak\ \mathbf{k=1.}$

If $l_{1}\geq 2$,

$\Gamma_{1}^{(l_{1},0)}=\left\\{\frac{\partial}{\partial
u_{1}^{(l_{1})}},\frac{\partial}{\partial u_{1}^{(l_{1}-1)}}\right\\}$

and

$\Delta_{1}^{(l_{1},0)}=\left\\{\frac{\partial}{\partial
u_{2}^{(0)}},\mathrm{ad}_{g_{0}^{(l_{1},0)}}\frac{\partial}{\partial
u_{2}^{(0)}}\right\\}=\left\\{\frac{\partial}{\partial
u_{2}^{(0)}},\frac{\partial}{\partial
x_{2,2}}+u_{1}^{(0)}\frac{\partial}{\partial
x_{3}}\right\\}=\overline{\Delta_{1}^{(l_{1},0)}}.$

Moreover,
$\left[\Gamma_{1}^{(l_{1},0)},\Delta_{1}^{(l_{1},0)}\right]=\\{0\\}\subset\Delta_{1}^{(l_{1},0)}$.

Now, for $l_{1}=1$, we have
$\Gamma_{1}^{(1,0)}=\left\\{\frac{\partial}{\partial u_{1}^{(1)}}\right\\}$
and $\Delta_{1}^{(1,0)}=\left\\{\frac{\partial}{\partial
u_{1}^{(0)}},\frac{\partial}{\partial
u_{2}^{(0)}},\mathrm{ad}_{g_{0}^{(1,0)}}\frac{\partial}{\partial
u_{2}^{(0)}}\right\\}=\left\\{\frac{\partial}{\partial
u_{1}^{(0)}},\frac{\partial}{\partial u_{2}^{(0)}},\frac{\partial}{\partial
x_{2,2}}+u_{1}^{(0)}\frac{\partial}{\partial x_{3}}\right\\}$ which is not
involutive.

Thus $\sigma_{1,\Delta}(1)=2$ and $\sigma_{1,\Gamma,\Delta}(1)=0$ which
implies that

$j_{1}=\max_{k\geq
0}\sigma_{1,\Delta}(k)\wedge\sigma_{1,\Gamma,\Delta}(k)\geq\sigma_{1,\Delta}(1)\wedge\sigma_{1,\Gamma,\Delta}(1)=2.$

##### $\bullet\leavevmode\nobreak\ \mathbf{k=2.}$

If $l_{1}\geq 3$, we have

$\Gamma_{2}^{(l_{1},0)}=\left\\{\frac{\partial}{\partial
u_{1}^{(l_{1})}},\frac{\partial}{\partial
u_{1}^{(l_{1}-1)}},\frac{\partial}{\partial u_{1}^{(l_{1}-2)}}\right\\}$

and

$\displaystyle\Delta_{2}^{(l_{1},0)}$
$\displaystyle=\left\\{\frac{\partial}{\partial
u_{2}^{(0)}},\mathrm{ad}_{g_{0}^{(l_{1},0)}}\frac{\partial}{\partial
u_{2}^{(0)}},\mathrm{ad}_{g_{0}^{(l_{1},0)}}^{2}\frac{\partial}{\partial
u_{2}^{(0)}}\right\\}$ $\displaystyle=\left\\{\frac{\partial}{\partial
u_{2}^{(0)}},\frac{\partial}{\partial
x_{2,2}}+u_{1}^{(0)}\frac{\partial}{\partial x_{3}},\frac{\partial}{\partial
x_{2,1}}-u_{1}^{(1)}\frac{\partial}{\partial
x_{3}}\right\\}=\overline{\Delta_{2}^{(l_{1},0)}}.$

Moreover, it is readily verified that
$\left[\Gamma_{2}^{(l_{1},0)},\Delta_{2}^{(l_{1},0)}\right]\subset\Delta_{2}^{(l_{1},0)}$
only if $l_{1}\geq 4$, condition (ii) of theorem 4.1 being violated if
$l_{1}=3$ and we have $\sigma_{1,\Delta}(2)=0$ and
$\sigma_{1,\Gamma,\Delta}(2)=4$ which implies that $j_{1}=\max_{k\geq
0}\sigma_{1,\Delta}(k)\wedge\sigma_{1,\Gamma,\Delta}(k)\geq\max_{r=1,2}\sigma_{1,\Delta}(r)\wedge\sigma_{1,\Gamma,\Delta}(r)\geq
4$.

##### $\bullet\leavevmode\nobreak\ \mathbf{k=3.}$

Again, if $l_{1}\geq 4$, we have:

$\Gamma_{3}^{(l_{1},0)}=\left\\{\frac{\partial}{\partial
u_{1}^{(l_{1})}},\frac{\partial}{\partial
u_{1}^{(l_{1}-1)}},\frac{\partial}{\partial
u_{1}^{(l_{1}-2))}},\frac{\partial}{\partial u_{1}^{(l_{1}-3)}}\right\\}$

and

$\displaystyle\Delta_{3}^{(l_{1},0)}$
$\displaystyle=\left\\{\frac{\partial}{\partial
u_{2}^{(0)}},\mathrm{ad}_{g_{0}^{(l_{1},0)}}\frac{\partial}{\partial
u_{2}^{(0)}},\mathrm{ad}_{g_{0}^{(l_{1},0)}}^{2}\frac{\partial}{\partial
u_{2}^{(0)}},\mathrm{ad}_{g_{0}^{(l_{1},0)}}^{3}\frac{\partial}{\partial
u_{2}^{(0)}}\right\\}$ $\displaystyle=\left\\{\frac{\partial}{\partial
u_{2}^{(0)}},\frac{\partial}{\partial
x_{2,2}}+u_{1}^{(0)}\frac{\partial}{\partial x_{3}},\frac{\partial}{\partial
x_{2,1}}-u_{1}^{(1)}\frac{\partial}{\partial
x_{3}},u_{1}^{(2)}\frac{\partial}{\partial x_{3}}\right\\}$
$\displaystyle=\left\\{\frac{\partial}{\partial
u_{2}^{(0)}},\frac{\partial}{\partial x_{2,2}},\frac{\partial}{\partial
x_{2,1}},\frac{\partial}{\partial
x_{3}}\right\\}=\overline{\Delta_{3}^{(l_{1},0)}}$

provided that $u_{1}^{(2)}\neq 0$. We also indeed have
$\left[\Gamma_{3}^{(l_{1},0)},\Delta_{3}^{(l_{1},0)}\right]\subset\Delta_{3}^{(l_{1},0)}$
for $l_{1}\geq 4$ hence $\sigma_{1,\Gamma,\Delta}(3)=4$ and
$j_{1}\geq\max_{r=1,2,3}\sigma_{1,\Delta}(r)\wedge\sigma_{1,\Gamma,\Delta}(r)\geq
4$.

##### $\bullet\leavevmode\nobreak\ \mathbf{k=4}$

If $l_{1}\geq 5$, we have
$\Gamma_{4}^{(l_{1},0)}=\left\\{\frac{\partial}{\partial
u_{1}^{(l_{1})}},\ldots,\frac{\partial}{\partial u_{1}^{(l_{1}-4)}}\right\\}$
and $\Delta_{4}^{(l_{1},0)}=\Delta_{3}^{(l_{1},0)}$ since
$\mathrm{ad}_{g_{0}^{(l_{1},0)}}^{4}\frac{\partial}{\partial
u_{2}^{(0)}}=-u_{1}^{(3)}\frac{\partial}{\partial
x_{3}}\in\Delta_{3}^{(l_{1},0)}$.

If now $l_{1}=4$, $\Gamma_{4}^{(4,0)}=\left\\{\frac{\partial}{\partial
u_{1}^{(4)}},\ldots,\frac{\partial}{\partial u_{1}^{(1)}}\right\\}$ and

$\displaystyle\Delta_{4}^{(4,0)}$
$\displaystyle=\left\\{\frac{\partial}{\partial
u_{1}^{(0)}},\frac{\partial}{\partial
u_{2}^{(0)}},\mathrm{ad}_{g_{0}^{(4,0)}}\frac{\partial}{\partial
u_{2}^{(0)}},\ldots,\mathrm{ad}_{g_{0}^{(4,0)}}^{4}\frac{\partial}{\partial
u_{2}^{(0)}}\right\\}$ $\displaystyle=\left\\{\frac{\partial}{\partial
u_{1}^{(0)}},\frac{\partial}{\partial u_{2}^{(0)}},\frac{\partial}{\partial
x_{2,2}},\frac{\partial}{\partial x_{2,1}},\frac{\partial}{\partial
x_{3}}\right\\}=\overline{\Delta_{4}^{(4,0)}}.$

We thus immediately get
$\left[\Gamma_{4}^{(l_{1},0)},\Delta_{4}^{(l_{1},0)}\right]\subset\Delta_{4}^{(l_{1},0)}$
for all $l_{1}\geq 4$.

##### $\bullet\leavevmode\nobreak\ \mathbf{k\geq 5}$

Finally, the reader may easily check that
$\Gamma_{k}^{(4,0)}=\Gamma_{4}^{(4,0)}$ for all $k\geq 5$ and

$\displaystyle\Delta_{5}^{(4,0)}$
$\displaystyle=\Delta_{4}^{(4,0)}+\left\\{\mathrm{ad}_{g_{0}^{(4,0)}}\frac{\partial}{\partial
u_{1}^{(0)}}\right\\}=\left\\{\frac{\partial}{\partial
u_{1}^{(0)}},\frac{\partial}{\partial x_{1,3}},\frac{\partial}{\partial
u_{2}^{(0)}},\frac{\partial}{\partial x_{2,2}},\frac{\partial}{\partial
x_{2,1}},\frac{\partial}{\partial x_{3}}\right\\}$
$\displaystyle=\overline{\Delta_{5}^{(4,0)}}$
$\displaystyle\Delta_{6}^{(4,0)}$
$\displaystyle=\Delta_{5}^{(4,0)}+\left\\{\mathrm{ad}_{g_{0}^{(4,0)}}^{2}\frac{\partial}{\partial
u_{1}^{(0)}}\right\\}=\left\\{\frac{\partial}{\partial
u_{1}^{(0)}},\frac{\partial}{\partial x_{1,3}},\frac{\partial}{\partial
x_{1,2}},\frac{\partial}{\partial u_{2}^{(0)}},\frac{\partial}{\partial
x_{2,2}},\frac{\partial}{\partial x_{2,1}},\frac{\partial}{\partial
x_{3}}\right\\}$ $\displaystyle=\overline{\Delta_{6}^{(4,0)}}$
$\displaystyle\Delta_{7}^{(4,0)}$
$\displaystyle=\Delta_{6}^{(4,0)}+\left\\{\mathrm{ad}_{g_{0}^{(4,0)}}^{3}\frac{\partial}{\partial
u_{1}^{(0)}}\right\\}=\mathrm{T}\mathbb{R}^{8}.$

We also indeed have
$\left[\Gamma_{k}^{(4,0))},\Delta_{k}^{(4,0))}\right]\subset\Delta_{k}^{(4,0)}$
for all $k\geq 5$.

Hence, for all $k\geq 0$, the minimal $\mathbf{j}$ is equal to $(4,0)$ and we
conclude that system (5.1), with the first input channel prolonged up to
$j_{1}=4$, i.e. controlled by $u_{1}^{(5)}$, is feedback linearizable.

Note that the bounds (3.5) and (3.18) are indeed satisfied. They read
$n+j_{1}=10\geq k_{\star}^{(4,0)}=7\geq\frac{n+j_{1}}{2}\wedge j_{1}=5$, but
they are not tight.

Accordingly, evaluating the bound on the number of integrators needed to
linearize the system, proposed by [34] for $m=2$, we find $2n-3=9$, and the
one proposed by [12] gives $2n-\frac{1}{6}(8+24-14)=2n-3=9$ as well.

####  5.1.3 Flat output computation

The prolonged system is now expressed in the state coordinates

$\overline{x}^{(4,0)}\triangleq(x_{1,1},x_{1,2},x_{1,3},x_{2,1},x_{2,2},x_{3},u_{1}^{(0)},u_{1}^{(1)},u_{1}^{(2)},u_{1}^{(3)},u_{1}^{(4)},u_{2}^{(0)})$

still with $x_{1,j}\triangleq x_{1}^{(j-1)}$, $j=1,2,3$, $x_{2,j}\triangleq
x_{2}^{(j-1)}$, $j=1,2$ and $u_{i}^{(0)}=u_{i}$, $i=1,2$. We indeed still have
$n=6$ but the prolonged state dimension is now equal to $n+m+|j|=12$ with the
two input variables $u_{1}^{(5)}$ and $u_{2}^{(1)}$.

The prolonged system vector fields are

$g_{0}^{(4,0)}\triangleq
g_{0}^{(0)}+\sum_{j=0}^{3}u_{1}^{(j+1)}\frac{\partial}{\partial
u_{1}^{(j)}},\qquad g_{1}^{(4)}\triangleq\frac{\partial}{\partial
u_{1}^{(4)}},\qquad g_{2}^{(0)}\triangleq\frac{\partial}{\partial
u_{2}^{(0)}}$

and the corresponding distributions are, according to (3.17),

$\displaystyle G_{0}^{(4,0)}$ $\displaystyle=\left\\{\frac{\partial}{\partial
u_{1}^{(4)}},\frac{\partial}{\partial
u_{2}^{(0)}}\right\\}=\overline{G_{0}^{(4,0)}},$ $\displaystyle G_{1}^{(4,0)}$
$\displaystyle=G_{0}^{(4,0)}+\left\\{\frac{\partial}{\partial
u_{1}^{(3)}},-\frac{\partial}{\partial
x_{2,2}}-u_{1}^{(0)}\frac{\partial}{\partial
x_{3}}\right\\}=\overline{G_{1}^{(4,0)}},$ $\displaystyle G_{2}^{(4,0)}$
$\displaystyle=G_{1}^{(4,0)}+\left\\{\frac{\partial}{\partial
u_{1}^{(2)}},\frac{\partial}{\partial
x_{2,1}}-u_{1}^{(1)}\frac{\partial}{\partial
x_{3}}\right\\}=\overline{G_{2}^{(4,0)}},$ $\displaystyle G_{3}^{(4,0)}$
$\displaystyle=G_{2}^{(4,0)}+\left\\{\frac{\partial}{\partial
u_{1}^{(1)}},-u_{1}^{(2)}\frac{\partial}{\partial
x_{3}}\right\\}=\overline{G_{3}^{(4,0)}},$ $\displaystyle G_{4}^{(4,0)}$
$\displaystyle=G_{3}^{(4,0)}+\left\\{\frac{\partial}{\partial
u_{1}^{(0)}}\right\\}=\overline{G_{4}^{(4,0)}},$ $\displaystyle G_{5}^{(4,0)}$
$\displaystyle=G_{4}^{(4,0)}+\left\\{\frac{\partial}{\partial
x_{1,3}}\right\\}=\overline{G_{5}^{(4,0)}},$ $\displaystyle G_{6}^{(4,0)}$
$\displaystyle=G_{5}^{(4,0)}+\left\\{\frac{\partial}{\partial
x_{1,2}}\right\\}=\overline{G_{6}^{(4,0)}},$ $\displaystyle G_{7}^{(4,0)}$
$\displaystyle=G_{6}^{(4,0)}+\left\\{\frac{\partial}{\partial
x_{1,1}}\right\\}=\overline{G_{7}^{(4,0)}}=\mathrm{T}\mathbb{R}^{12}.$

This confirms that the system (5.1) is flat by pure prolongation in any
neighborhood excluding $u_{1}^{(2)}=0$, with $k^{(4,0)}_{\ast}=7$. The reader
may easily verify that $\rho_{k}^{(4,0)}=2$ for $k=0,\ldots,3$ and
$\rho_{k}^{(4,0)}=1$ for $k=4,\ldots,7$, which yields $\kappa_{1}^{(4,0)}=8$
and $\kappa_{2}^{(4,0)}=4$.

The corresponding flat outputs are thus obtained by solving the set of
P.D.E.’s

$\displaystyle\left<G_{k}^{(4,0)},dy_{1}\right>=0,\quad
k=0,\ldots,6,\qquad\left<G_{7}^{(4,0)},dy_{1}\right>\neq 0$ (5.6)
$\displaystyle\left<G_{k}^{(4,0)},dy_{2}\right>=0,\quad
k=0,\ldots,2,\qquad\left<G_{3}^{(4,0)},dy_{2}\right>\neq 0$

whose solution is

$y_{1}=x_{1},\quad
y_{2}=x_{3}-x_{2,2}u_{1}^{(0)}+x_{2,1}u_{1}^{(1)}=x_{3}-\dot{x}_{2}u_{1}+x_{2}\dot{u}_{1}.$
(5.7)

######  Remark 5.1.

Using the first intialization (5.4), which amounts to prolonging the second
input, we leave to the reader the verification that one obtains a prolongation
of order 6, thus non minimal, with flat outputs given by (5.2).

######  Remark 5.2.

In [25, section 3.1.1], the authors consider a dual notion of minimality,
called $r$-flatness, where $r$ is the minimal number over all possible flat
outputs of the maximal number of derivatives of the inputs that appear in the
flat outputs, i.e.

$r=\min_{Y:\leavevmode\nobreak\ \mathrm{flat\leavevmode\nobreak\
output}}\leavevmode\nobreak\
\max_{i=1,\ldots,m}\\{s_{i}\mid\mathbf{s}=(s_{1},\ldots,s_{m}),\leavevmode\nobreak\
y=Y(x,\overline{u}^{(\mathbf{s})})\\}.$

They conjectured that $r$ should be equal to 1 in the present case (with their
notations, $\alpha_{1}=3$, $\alpha_{2}=2$ and
$\min(\alpha_{1},\alpha_{2})-1=1$). As the reader may easily verify, it is
indeed minimal since the minimal $\mathbf{j}$ is $(4,0)$ and moreover since,
by the equations of the first line of (5.6), $y_{1}$ neither can depend on
$\overline{u}_{1}^{(4)}$ nor on $u_{2}^{(0)}$, and, by the second line,
$y_{2}$ cannot depend on $u_{1}^{(4)},u_{1}^{(3)},u_{1}^{(2)}$ but explicitly
depends on $u_{1}^{(1)}$ by the definition of $G_{2}^{(4,0)}$.

###  5.2 4-dimensional Driftless Bilinear System [25, 26, 27]

$\displaystyle\dot{x}_{1}$ $\displaystyle=u_{1}$ (5.8)
$\displaystyle\dot{x}_{2}$ $\displaystyle=x_{3}u_{1}$
$\displaystyle\dot{x}_{3}$ $\displaystyle=x_{4}u_{1}$
$\displaystyle\dot{x}_{4}$ $\displaystyle=u_{2}$

It is immediate to verify that this system is flat with flat output

$y_{1}=x_{1},\quad y_{2}=x_{2}$ (5.9)

(see [26, 27] and [25, theorems 4 and 5]) but not static feedback
linearizable.

According to our formalism, we consider the state
$(x_{1},x_{2},x_{3},x_{4},u_{1}^{(0)},u_{2}^{(0)})$ of dimension 6, with $n=4$
and $m=2$, and the new inputs $(u_{1}^{(1)},u_{2}^{(1)})$. The associated
vector fields are

$g_{0}^{(0)}=u_{1}^{(0)}\left(\frac{\partial}{\partial
x_{1}}+x_{3}\frac{\partial}{\partial x_{2}}+x_{4}\frac{\partial}{\partial
x_{3}}\right)+u_{2}^{(0)}\frac{\partial}{\partial x_{4}},\quad
g_{1}^{(0)}=\frac{\partial}{\partial u_{1}^{(0)}},\quad
g_{2}^{(0)}=\frac{\partial}{\partial u_{2}^{(0)}}.$

The reader may easily verify that $G_{1}^{(\mathbf{0})}$ is not involutive and
that there are two possible initializations:

$H_{1,1}^{(\mathbf{0})}=\\{g_{1}^{(0)},\mathrm{ad}_{g_{0}^{(\mathbf{0})}}g_{1}^{(0)}\\}=\\{\frac{\partial}{\partial
u_{1}^{(0)}},\frac{\partial}{\partial x_{1}}+x_{3}\frac{\partial}{\partial
x_{2}}+x_{4}\frac{\partial}{\partial
x_{3}}\\}=\overline{H_{1,1}^{(\mathbf{0})}}$

or

$H_{1,2}^{(\mathbf{0})}=\\{g_{2}^{(0)},\mathrm{ad}_{g_{0}^{(\mathbf{0})}}g_{2}^{(0)}\\}=\\{\frac{\partial}{\partial
u_{2}^{(0)}},\frac{\partial}{\partial
x_{4}}\\}=\overline{H_{1,2}^{(\mathbf{0})}}.$

We use the second one, which amounts to prolonging the first input
$u_{1}^{(1)}$.

####  5.2.1 the distributions $\Gamma_{k}^{(\mathbf{l})}$ and
$\Delta_{k}^{(\mathbf{l})}$

We set $\mathbf{l}=(l_{1},0)$, $l_{1}\geq 1$. We thus have

$g_{0}^{(l_{1},0)}=g_{0}^{(0)}+\sum_{p=0}^{l_{1}-1}u_{1}^{(p+1)}\frac{\partial}{\partial
u_{1}^{(p)}},\quad g_{1}^{(l_{1})}=\frac{\partial}{\partial
u_{1}^{(l_{1})}},\quad\quad g_{2}^{(0)}=\frac{\partial}{\partial u_{2}^{(0)}}$

and

$\Gamma_{0}^{(l_{1},0)}=\left\\{\frac{\partial}{\partial
u_{1}^{(l_{1})}}\right\\},\quad\Delta_{0}^{(l_{1},0)}=\left\\{\frac{\partial}{\partial
u_{2}^{(0)}}\right\\}$

for all $l_{1}\geq 1$.

##### $\bullet\leavevmode\nobreak\ \mathbf{k=1.}$

If $l_{1}\geq 2$, we have

$\Gamma_{1}^{(l_{1},0)}=\left\\{\frac{\partial}{\partial
u_{1}^{(l_{1})}},\frac{\partial}{\partial
u_{1}^{(l_{1}-1)}}\right\\},\quad\Delta_{1}^{(l_{1},0)}=\Delta_{0}^{(l_{1},0)}+\\{\mathrm{ad}_{g_{0}^{(l_{1},0)}}\frac{\partial}{\partial
u_{2}^{(0)}}\\}=\left\\{\frac{\partial}{\partial
u_{2}^{(0)}},\frac{\partial}{\partial x_{4}}\right\\}$

and if $l_{1}=1$,

$\Gamma_{1}^{(1,0)}=\left\\{\frac{\partial}{\partial
u_{1}^{(1)}}\right\\},\quad\Delta_{1}^{(1,0)}=\Delta_{0}^{(1,0)}+\\{\frac{\partial}{\partial
u_{1}^{(0)}},\mathrm{ad}_{g_{0}^{(1,0)}}\frac{\partial}{\partial
u_{2}^{(0)}}\\}=\left\\{\frac{\partial}{\partial
u_{1}^{(0)}},\frac{\partial}{\partial u_{2}^{(0)}},\frac{\partial}{\partial
x_{4}}\right\\}.$

We thus have $\Delta_{1}^{(l_{1},0)}=\overline{\Delta_{1}^{(l_{1},0)}}$ and
$\left[\Gamma_{1}^{(l_{1},0)},\Delta_{1}^{(l_{1},0)}\right]=\\{0\\}\subset\Delta_{1}^{(l_{1},0)}$
for all $l_{1}\geq 1$, hence
$\sigma_{1,\Gamma,\Delta}(1)=\sigma_{1,\Delta}(1)=0$.

##### $\bullet\leavevmode\nobreak\ \mathbf{k=2.}$

If $l_{1}\geq 3$, we have

$\begin{array}[]{c}\Gamma_{2}^{(l_{1},0)}=\left\\{\frac{\partial}{\partial
u_{1}^{(l_{1})}},\frac{\partial}{\partial
u_{1}^{(l_{1}-1)}},\frac{\partial}{\partial
u_{1}^{(l_{1}-2)}}\right\\},\quad\Delta_{2}^{(l_{1},0)}=\left\\{\frac{\partial}{\partial
u_{2}^{(0)}},\frac{\partial}{\partial x_{4}},\frac{\partial}{\partial
x_{3}}\right\\},\vspace{10pt}\\\
\left[\Gamma_{2}^{(l_{1},0)},\Delta_{2}^{(l_{1},0)}\right]=\\{0\\}\end{array}$

and if $l_{1}=2$,

$\begin{array}[]{c}\Gamma_{2}^{(2,0)}=\left\\{\frac{\partial}{\partial
u_{1}^{(2)}},\frac{\partial}{\partial
u_{1}^{(1)}}\right\\},\quad\Delta_{2}^{(2,0)}=\left\\{\frac{\partial}{\partial
u_{1}^{(0)}},\frac{\partial}{\partial u_{2}^{(0)}},\frac{\partial}{\partial
x_{4}},\frac{\partial}{\partial x_{3}}\right\\},\vspace{10pt}\\\
\left[\Gamma_{2}^{(2,0)},\Delta_{2}^{(2,0)}\right]=\\{0\\}.\end{array}$

Finally, if $l_{1}=1$,

$\displaystyle\Gamma_{2}^{(1,0)}=\left\\{\frac{\partial}{\partial
u_{1}^{(1)}}\right\\},\quad\Delta_{2}^{(1,0)}$
$\displaystyle=\left\\{\frac{\partial}{\partial
u_{1}^{(0)}},-\frac{\partial}{\partial x_{1}}-x_{3}\frac{\partial}{\partial
x_{2}}-x_{4}\frac{\partial}{\partial x_{3}},\frac{\partial}{\partial
u_{2}^{(0)}},\frac{\partial}{\partial x_{4}},\frac{\partial}{\partial
x_{3}}\right\\}$ $\displaystyle\neq\overline{\Delta_{2}^{(1,0)}}.$

We thus have $\Delta_{2}^{(l_{1},0)}=\overline{\Delta_{2}^{(l_{1},0)}}$ and
$\left[\Gamma_{2}^{(l_{1},0)},\Delta_{2}^{(l_{1},0)}\right]\subset\Delta_{2}^{(l_{1},0)}$
for all $l_{1}\geq 2$ but, for $l_{1}=1$, $\Delta_{2}^{(1,0)}$ is not
involutive. Therefore, $\sigma_{1,\Delta}(2)=2$ and our search may be
restricted to $l_{1}\geq 2$.

##### $\bullet\leavevmode\nobreak\ \mathbf{k=3.}$

If $l_{1}\geq 4$, we have

$\Gamma_{3}^{(l_{1},0)}=\left\\{\frac{\partial}{\partial
u_{1}^{(l_{1})}},\frac{\partial}{\partial
u_{1}^{(l_{1}-1)}},\frac{\partial}{\partial
u_{1}^{(l_{1}-2)}},\frac{\partial}{\partial
u_{1}^{(l_{1}-3)}}\right\\},\quad\Delta_{3}^{(l_{1},0)}=\left\\{\frac{\partial}{\partial
u_{2}^{(0)}},\frac{\partial}{\partial x_{4}},\frac{\partial}{\partial
x_{3}},\frac{\partial}{\partial x_{2}}\right\\}.$

If $l_{1}=3$,

$\Gamma_{3}^{(3,0)}=\left\\{\frac{\partial}{\partial
u_{1}^{(3)}},\frac{\partial}{\partial u_{1}^{(2)}},\frac{\partial}{\partial
u_{1}^{(1)}}\right\\},\quad\Delta_{3}^{(3,0)}=\left\\{\frac{\partial}{\partial
u_{1}^{(0)}},\frac{\partial}{\partial u_{2}^{(0)}},\frac{\partial}{\partial
x_{4}},\frac{\partial}{\partial x_{3}},\frac{\partial}{\partial
x_{2}}\right\\}.$

If $l_{1}=2$,

$\Gamma_{3}^{(2,0)}=\left\\{\frac{\partial}{\partial
u_{1}^{(2)}},\frac{\partial}{\partial
u_{1}^{(1)}}\right\\},\quad\Delta_{3}^{(2,0)}=\left\\{\frac{\partial}{\partial
u_{1}^{(0)}},\frac{\partial}{\partial u_{2}^{(0)}},\frac{\partial}{\partial
x_{4}},\frac{\partial}{\partial x_{3}},\frac{\partial}{\partial
x_{2}},\frac{\partial}{\partial x_{1}}\right\\}=\mathrm{T}\mathbb{R}^{6}$

Therefore, $\Delta_{k}^{(2,0)}=\overline{\Delta_{k}^{(2,0)}}$ and
$\left[\Gamma_{k}^{(2,0)},\Delta_{k}^{(2,0)}\right]\subset\Delta_{k}^{(2,0)}$
for all $k\geq 0$ and $\Delta_{3}^{(2,0)}=\mathrm{T}\mathbb{R}^{6}$.

We conclude that the conditions of theorem 4.1 hold true whenever $j_{1}\geq
2$, which proves that system (5.8) with the pure prolongation of order
$\mathbf{j}=(2,0)$ is feedback linearizable.

The reader may easily check that, using the first initialization
$H_{1,1}^{(\mathbf{0})}$, since $\Delta_{2}^{(0,l_{2})}$ is not involutive for
all $l_{2}\geq 3$, no pure prolongation of the second input channel,
$u_{2}^{(1)}$, can lead to the linearizability conditions. Therefore the
minimal prolongation is $\mathbf{j}=(2,0)$.

####  5.2.2 Flat output computation

The prolonged state is now
$\overline{x}^{(2,0)}\triangleq(x_{1},x_{2},x_{3},x_{4},u_{1}^{(0)},u_{1}^{(1)},u_{1}^{(2)},u_{2}^{(0)})$
of dimension 8, and the new inputs are $(u_{1}^{(3)},u_{2}^{(1)})$.

$G_{0}^{(2,0)}=\left\\{\frac{\partial}{\partial
u_{1}^{(2)}},\frac{\partial}{\partial
u_{2}^{(0)}}\right\\}=\overline{G_{0}^{(2,0)}},$
$G_{1}^{(2,0)}=\left\\{\frac{\partial}{\partial
u_{1}^{(2)}},\frac{\partial}{\partial u_{1}^{(1)}},\frac{\partial}{\partial
u_{2}^{(0)}},\frac{\partial}{\partial
x_{4}}\right\\}=\overline{G_{1}^{(2,0)}},$
$G_{2}^{(2,0)}=\left\\{\frac{\partial}{\partial
u_{1}^{(2)}},\frac{\partial}{\partial u_{1}^{(1)}},\frac{\partial}{\partial
u_{1}^{(0)}},\frac{\partial}{\partial u_{2}^{(0)}},\frac{\partial}{\partial
x_{4}},\frac{\partial}{\partial x_{3}}\right\\}=\overline{G_{2}^{(2,0)}},$
$\displaystyle G_{3}^{(2,0)}$ $\displaystyle=\left\\{\frac{\partial}{\partial
u_{1}^{(2)}},\frac{\partial}{\partial u_{1}^{(1)}},\frac{\partial}{\partial
u_{1}^{(0)}},\frac{\partial}{\partial x_{1}}+x_{3}\frac{\partial}{\partial
x_{2}}+x_{4}\frac{\partial}{\partial x_{3}},\frac{\partial}{\partial
u_{2}^{(0)}},\frac{\partial}{\partial x_{4}},\frac{\partial}{\partial
x_{3}},\frac{\partial}{\partial x_{2}}\right\\}$
$\displaystyle=\overline{G_{3}^{(2,0)}}=\mathrm{T}\mathbb{R}^{8}$

hence the feedback linearizability of the purely prolonged system with
$\rho_{0}^{(2,0)}=\rho_{1}^{(2,0)}=\rho_{2}^{(2,0)}=\rho_{3}^{(2,0)}=2$ and
$\kappa_{1}^{(2,0)}=\kappa_{2}^{(2,0)}=4$.

Flat outputs $(y_{1},y_{2})$ are locally non trivial solutions of the system
(5.10), i.e. :

$\left<G_{k}^{(2,0)},dy_{i}\right>=0,\;k=0,1,2,\quad\mathrm{with\leavevmode\nobreak\
}\quad\left<G_{3}^{(2,0)},dy_{i}\right>\neq 0,\quad i=1,2.$ (5.10)

It is immediate to verify that

$y_{1}=x_{1},\qquad y_{2}=x_{2}$

is a solution of (5.10) and that the mapping

$\overline{x}^{(2,0)}\mapsto\left(y_{1},\ldots,y_{1}^{(3)},y_{2},\ldots,y_{2}^{(3)}\right)$

is a local diffeomorphism.

###  5.3 An Example from [7]

In our formalism, considering the inputs
$(u_{1},u_{2})\triangleq(u_{1}^{(0)},u_{2}^{(0)})$ as part of the state, with
$n=4$ and $m=2$, this example from [7, Example 2] reads:

$\displaystyle\dot{x}_{1}=x_{2}+x_{3}u_{2}^{(0)}$ (5.11)
$\displaystyle\dot{x}_{2}=x_{3}+x_{1}u_{2}^{(0)}$
$\displaystyle\dot{x}_{3}=u_{1}^{(0)}+x_{2}u_{2}^{(0)}$
$\displaystyle\dot{x}_{4}=u_{2}^{(0)}$
$\displaystyle\dot{u}_{1}^{(0)}=u_{1}^{(1)}$
$\displaystyle\dot{u}_{2}^{(0)}=u_{2}^{(1)}.$

It is shown in [7] that this sytem does not satisfy the sufficient, but
otherwise not necessary, condition for dynamic linearization of Theorem 4.2 of
this paper. Nevertheless, it satisfies the algorithm of [2, section 3.1], that
constitutes a sufficient condition for flatness by pure prolongation, without
proof of minimality of the obtained prolongation. We show here that it is
linearizable by pure prolongation by application of our algorithm, thus
providing the minimal prolongation.

The non prolonged vctor fields are

$\displaystyle
g_{0}^{(\mathbf{0})}=(x_{2}+x_{3}u_{2}^{(0)})\frac{\partial}{\partial
x_{1}}+(x_{3}+x_{1}u_{2}^{(0)})\frac{\partial}{\partial
x_{2}}+(u_{1}^{(0)}+x_{2}u_{2}^{(0)})\frac{\partial}{\partial
x_{3}}+u_{2}^{(0)}\frac{\partial}{\partial x_{4}}$ $\displaystyle
g_{1}^{(\mathbf{0})}=\frac{\partial}{\partial u_{1}^{(0)}},\qquad
g_{2}^{(0)}=\frac{\partial}{\partial u_{2}^{(0)}}.$

and it is easily seen that

$G_{1}^{(\mathbf{0})}=\\{\frac{\partial}{\partial
u_{1}^{(0)}},\frac{\partial}{\partial u_{2}^{(0)}},\frac{\partial}{\partial
x_{3}},-x_{3}\frac{\partial}{\partial x_{1}}-x_{1}\frac{\partial}{\partial
x_{2}}-x_{2}\frac{\partial}{\partial x_{3}}-\frac{\partial}{\partial
x_{4}}\\}\neq\overline{G_{1}^{(\mathbf{0})}}$

and that the subdistributions

$H_{1,1}^{0}=\\{g_{1}^{(\mathbf{0})},\mathrm{ad}_{g_{0}^{(\mathbf{0})}}g_{1}^{(\mathbf{0})}\\}=\\{\frac{\partial}{\partial
u_{1}^{(0)}},\frac{\partial}{\partial x_{3}}\\}=\overline{H_{1,1}^{0}}$

or

$H_{1,2}^{0}=\\{g_{2}^{(\mathbf{0})},\mathrm{ad}_{g_{0}^{(\mathbf{0})}}g_{2}^{(\mathbf{0})}\\}=\\{\frac{\partial}{\partial
u_{2}^{(0)}},-x_{3}\frac{\partial}{\partial
x_{1}}-x_{1}\frac{\partial}{\partial x_{2}}-x_{2}\frac{\partial}{\partial
x_{3}}-\frac{\partial}{\partial x_{4}}\\}=\overline{H_{1,2}^{0}}$

can be taken as possible initializations.

Choosing $H_{1,1}^{0}$ amounts to prolonging the second input at an arbitrary
order $l_{2}\geq 1$ and set $\mathbf{l}=(0,l_{2})$. For $l_{2}\geq 1$, we
denote, as before,

$\displaystyle
g_{0}^{(0,l_{2})}=(x_{2}+x_{3}u_{2}^{(0)})\frac{\partial}{\partial
x_{1}}+(x_{3}+x_{1}u_{2}^{(0)})\frac{\partial}{\partial
x_{2}}+(u_{1}^{(0)}+x_{2}u_{2}^{(0)})\frac{\partial}{\partial x_{3}}$
$\displaystyle\hskip 170.71652pt+u_{2}^{(0)}\frac{\partial}{\partial
x_{4}}+\sum_{p=0}^{l_{2}-1}u_{2}^{(p+1)}\frac{\partial}{\partial u_{2}^{(p)}}$
$\displaystyle g_{1}^{(\mathbf{0})}=\frac{\partial}{\partial
u_{1}^{(0)}},\qquad g_{2}^{(l_{2})}=\frac{\partial}{\partial u_{2}^{(l_{2})}}$

and we indeed have

$\Gamma_{0}^{(0,l_{2})}=\left\\{\frac{\partial}{\partial
u_{2}^{(l_{2})}}\right\\},\quad\Delta_{0}^{(0,l_{2})}=\left\\{\frac{\partial}{\partial
u_{1}^{(0)}}\right\\}=\overline{\Delta_{0}^{(0,l_{2})}},\quad\left[\Gamma_{0}^{(0,l_{2})},\Delta_{0}^{(0,l_{2})}\right]\subset\Delta_{0}^{(0,l_{2})},\quad\forall
l_{2}\geq 1.$

##### $\bullet\leavevmode\nobreak\ \mathbf{k=1.}$

For all $l_{2}\geq 2$,
$\displaystyle\mathrm{ad}_{g_{0}^{(0,l_{2})}}\frac{\partial}{\partial
u_{1}^{(0)}}=-\frac{\partial}{\partial x_{3}}$ and

$\begin{array}[]{c}\displaystyle\Gamma_{1}^{(0,l_{2})}=\left\\{\frac{\partial}{\partial
u_{2}^{(l_{2})}},\frac{\partial}{\partial
u_{2}^{(l_{2}-1)}}\right\\},\qquad\Delta_{1}^{(0,l_{2})}=\left\\{\frac{\partial}{\partial
u_{1}^{((0))}},-\frac{\partial}{\partial
x_{3}}\right\\}=\overline{\Delta_{1}^{(0,l_{2})}},\vspace{1em}\\\
\displaystyle\left[\Gamma_{1}^{(0,l_{2})},\Delta_{1}^{(0,l_{2})}\right]\subset\Delta_{1}^{(0,l_{2})}.\end{array}$

For $l_{2}=1$:

$\begin{array}[]{c}\displaystyle\Gamma_{1}^{(0,1)}=\left\\{\frac{\partial}{\partial
u_{2}^{(1)}}\right\\},\quad\Delta_{1}^{(0,1)}=\left\\{\frac{\partial}{\partial
u_{2}^{(0)}},\frac{\partial}{\partial u_{1}^{(0)}},-\frac{\partial}{\partial
x_{3}}\right\\}=\overline{\Delta_{1}^{(0,1)}},\vspace{1em}\\\
\displaystyle\left[\Gamma_{1}^{(0,1)},\Delta_{1}^{(0,1)}\right]\subset\Delta_{1}^{(0,1)}.\end{array}$

##### $\bullet\leavevmode\nobreak\ \mathbf{k=2.}$

For all $l_{2}\geq 3$,
$\displaystyle\mathrm{ad}_{g_{0}^{(0,l_{2})}}^{2}\frac{\partial}{\partial
u_{1}^{(0)}}=u_{2}^{(0)}\frac{\partial}{\partial
x_{1}}+\frac{\partial}{\partial x_{2}}$ and

$\begin{array}[]{c}\displaystyle\Gamma_{2}^{(0,l_{2})}=\left\\{\frac{\partial}{\partial
u_{2}^{(l_{2})}},\frac{\partial}{\partial
u_{2}^{(l_{2}-1)}},\frac{\partial}{\partial
u_{2}^{(l_{2}-2)}}\right\\},\vspace{1em}\\\
\Delta_{2}^{(0,l_{2})}=\left\\{\frac{\partial}{\partial
u_{1}^{(0)}},-\frac{\partial}{\partial
x_{3}},u_{2}^{(0)}\frac{\partial}{\partial x_{1}}+\frac{\partial}{\partial
x_{2}}\right\\}=\overline{\Delta_{2}^{(0,l_{2})}},\vspace{1em}\\\
\displaystyle\left[\Gamma_{2}^{(0,l_{2})},\Delta_{2}^{(\mathbf{0},l_{2})}\right]\subset\Delta_{2}^{(0,l_{2})}.\end{array}$

But for $l_{2}=2$:

$\Gamma_{2}^{(0,2)}=\left\\{\frac{\partial}{\partial
u_{2}^{(2)}},\frac{\partial}{\partial
u_{2}^{(1)}}\right\\},\quad\Delta_{2}^{(0,2)}=\left\\{\frac{\partial}{\partial
u_{2}^{(0)}},\frac{\partial}{\partial u_{1}^{(0)}},-\frac{\partial}{\partial
x_{3}},u_{2}^{(0)}\frac{\partial}{\partial x_{1}}+\frac{\partial}{\partial
x_{2}}\right\\}\neq\overline{\Delta_{2}^{(0,2)}}$

therefore, $\sigma_{2,\Delta}(2)=2$ and we must exclude $j_{2}=2$.

##### $\bullet\leavevmode\nobreak\ \mathbf{k=3.}$

For all $l_{2}\geq 4$,
$\displaystyle\mathrm{ad}_{g_{0}^{(0,l_{2})}}^{3}\frac{\partial}{\partial
u_{1}^{(0)}}=(u_{2}^{(1)}-1)\frac{\partial}{\partial
x_{1}}-\left(u_{2}^{(0)}\right)^{2}\frac{\partial}{\partial
x_{2}}-u_{2}^{(0)}\frac{\partial}{\partial x_{3}}$ and, if we exclude the
points where $u_{2}^{(0)}=0$ and $u_{2}^{(1)}=1$,

$\Gamma_{3}^{(0,l_{2})}=\left\\{\frac{\partial}{\partial
u_{2}^{(l_{2})}},\frac{\partial}{\partial
u_{2}^{(l_{2}-1)}},\frac{\partial}{\partial
u_{2}^{(l_{2}-2)}},\frac{\partial}{\partial u_{2}^{(l_{2}-3)}}\right\\},$
$\displaystyle\Delta_{3}^{(\mathbf{0},l_{2})}$
$\displaystyle=\left\\{\frac{\partial}{\partial
u_{1}^{(0)}},-\frac{\partial}{\partial
x_{3}},u_{2}^{(0)}\frac{\partial}{\partial x_{1}}+\frac{\partial}{\partial
x_{2}},(u_{2}^{(1)}-1)\frac{\partial}{\partial
x_{1}}+\left(u_{2}^{(0)}\right)^{2}\frac{\partial}{\partial
x_{2}}-u_{2}^{(0)}\frac{\partial}{\partial x_{3}}\right\\}$
$\displaystyle=\left\\{\frac{\partial}{\partial
u_{1}^{((0))}},\frac{\partial}{\partial x_{3}},\frac{\partial}{\partial
x_{2}},\frac{\partial}{\partial
x_{1}}\right\\}=\overline{\Delta_{3}^{(0,l_{2})}},$
$\left[\Gamma_{3}^{(0,l_{2})},\Delta_{3}^{(0,l_{2})}\right]\subset\Delta_{3}^{(0,l_{2})}.$

The reader may then easily check that the same holds for $l_{2}=3$:

$\Gamma_{3}^{(0,3)}=\left\\{\frac{\partial}{\partial
u_{2}^{(3)}},\frac{\partial}{\partial u_{2}^{(2)}},\frac{\partial}{\partial
u_{2}^{(1)}}\right\\},$ $\Delta_{3}^{(0,3)}=\left\\{\frac{\partial}{\partial
u_{2}^{(0)}},\frac{\partial}{\partial u_{1}^{(0)}},\frac{\partial}{\partial
x_{3}},\frac{\partial}{\partial x_{2}},\frac{\partial}{\partial
x_{1}}\right\\}=\overline{\Delta_{3}^{(0,3)}}.$

##### $\bullet\leavevmode\nobreak\ \mathbf{For\leavevmode\nobreak\
all\leavevmode\nobreak\ k\geq 4.}$

Since $\mathrm{ad}_{g_{0}^{(0,l_{2})}}^{k}\frac{\partial}{\partial
u_{1}^{(0)}}$ and
$\mathrm{ad}_{g_{0}^{(0,l_{2})}}^{k-3}\frac{\partial}{\partial u_{2}^{(0)}}$
are linear combinations of $\frac{\partial}{\partial x_{1}}$,
$\frac{\partial}{\partial x_{2}}$ and $\frac{\partial}{\partial x_{3}}$ only,
we have, for all $l_{2}\geq 4$:

$\Delta_{k}^{(0,l_{2})}=\overline{\Delta_{k}^{(0,l_{2})}},\quad\left[\Gamma_{k}^{(0,l_{2})},\Delta_{k}^{(0,l_{2})}\right]\subset\Delta_{k}^{(0,l_{2})},$

and, for $l_{2}=3$, using the fact that
$\mathrm{ad}_{g_{0}^{(0,3)}}\frac{\partial}{\partial
u_{2}^{(0)}}=-x_{3}\frac{\partial}{\partial
x_{1}}-x_{1}\frac{\partial}{\partial x_{2}}-x_{2}\frac{\partial}{\partial
x_{3}}-\frac{\partial}{\partial x_{4}}$, we have

$\Gamma_{4}^{(0,3)}=\left\\{\frac{\partial}{\partial
u_{2}^{(3)}},\frac{\partial}{\partial u_{2}^{(2)}},\frac{\partial}{\partial
u_{2}^{(1)}}\right\\},$ $\displaystyle\Delta_{4}^{(0,3)}$
$\displaystyle=\left\\{\frac{\partial}{\partial
u_{2}^{(0)}},-x_{3}\frac{\partial}{\partial
x_{1}}-x_{1}\frac{\partial}{\partial x_{2}}-x_{2}\frac{\partial}{\partial
x_{3}}-\frac{\partial}{\partial x_{4}},\frac{\partial}{\partial
u_{1}^{(0)}},\frac{\partial}{\partial x_{3}},\frac{\partial}{\partial
x_{2}},\frac{\partial}{\partial x_{1}}\right\\}$
$\displaystyle=\left\\{\frac{\partial}{\partial
u_{2}^{(0)}},\frac{\partial}{\partial u_{1}^{(0)}},\frac{\partial}{\partial
x_{4}},\frac{\partial}{\partial x_{3}},\frac{\partial}{\partial
x_{2}},\frac{\partial}{\partial
x_{1}}\right\\}=\overline{\Delta_{4}^{(0,3)}}=\mathrm{T}\mathbb{R}^{6}$

and

$\Delta_{k}^{(0,3)}=\mathrm{T}\mathbb{R}^{6}\quad\forall k\geq 4,$

hence $j_{2}=3$ and $k_{\ast}^{(0,3)}=4$.

We conclude that the conditions of theorem 4.1 are satisfied for all $k$
provided that $j_{2}=3$.

On the contrary, initializing the algorithm by $H_{1,2}^{0}$, which amounts to
prolonging the first input $u_{1}^{(1)}$, we may easily check that, for all
$l_{1}\geq 3$,

$\Delta_{2}^{(l_{1},0)}=\left\\{\frac{\partial}{\partial
u_{2}^{(0)}},-x_{3}\frac{\partial}{\partial
x_{1}}-x_{1}\frac{\partial}{\partial x_{2}}-x_{2}\frac{\partial}{\partial
x_{3}}-\frac{\partial}{\partial
x_{4}},(x_{1}-u_{1}^{(0)})\frac{\partial}{\partial
x_{1}}-x_{3}\frac{\partial}{\partial x_{3}}\right\\}$

is not involutive, thus contradicting condition (i) of Theorem 4.1, which
proves that the minimal prolongation of the second input is equal to 3.

Let us finally give the construction of the flat output and prolonged state
diffeomorphism. The prolonged state is
$(x_{1},x_{2},x_{3},x_{4},u_{1}^{(0)},u_{2}^{(0)},u_{2}^{(1)},u_{2}^{(2)},u_{2}^{(3)})$
of dimension $9=n+m+\mid\mathbf{j}\mid$.

We get

$\displaystyle G_{0}^{(0,3)}=\left\\{\frac{\partial}{\partial
u_{1}^{(0)}},\frac{\partial}{\partial
u_{2}^{(3)}}\right\\}=\overline{G_{0}^{(0,3)}},\quad
G_{1}^{(0,3)}=\left\\{\frac{\partial}{\partial x_{3}},\frac{\partial}{\partial
u_{2}^{(2)}}\right\\}\oplus G_{0}^{(0,3)}=\overline{G_{1}^{(0,3)}},$
$\displaystyle G_{2}^{(0,3)}=\left\\{u_{2}^{(0)}\frac{\partial}{\partial
x_{1}}+\frac{\partial}{\partial x_{2}},\frac{\partial}{\partial
u_{2}^{(1)}}\right\\}\oplus G_{1}^{(0,3)}=\overline{G_{2}^{(0,3)}},$
$\displaystyle G_{3}^{(0,3)}=\left\\{(u_{2}^{(1)}-1)\frac{\partial}{\partial
x_{1}}-u_{2}^{(0)}\frac{\partial}{\partial x_{3}},\frac{\partial}{\partial
u_{2}^{(0)}}\right\\}\oplus G_{2}^{(0,3)}=\overline{G_{3}^{(0,3)}},$
$\displaystyle G_{4}^{(0,3)}\left\\{-x_{3}\frac{\partial}{\partial
x_{1}}-x_{1}\frac{\partial}{\partial x_{2}}-x_{2}\frac{\partial}{\partial
x_{3}}-\frac{\partial}{\partial x_{4}}\right\\}\oplus
G_{4}^{(0,3)}=\mathrm{T}\mathbb{R}^{9}$

The Brunovský’s controllability indices are $\kappa_{1}^{(0,3)}=5$ and
$\kappa_{2}^{(0,3)}=4$ and the system of PDE’s that the flat outputs must
satisfy is :

$\displaystyle\left<G_{k}^{(0,3)},dy_{1}\right>=0,\quad
k=0,\ldots,3,\qquad\left<G_{4}^{(0,3)},dy_{1}\right>\neq 0$
$\displaystyle\left<G_{k}^{(0,3)},dy_{2}\right>=0,\quad
k=0,\ldots,2,\qquad\left<G_{3}^{(0,3)},dy_{2}\right>\neq 0.$

Its solution is given by $y_{1}=x_{4}$, $y_{2}=x_{1}-u_{2}^{(0)}x_{2}$.

###  5.4 The Pendulum Example [10, section II. C]

This model of pendulum in the vertical plane has been studied in [10, section
II. C], [21, section 6.2.3],[22, section 5.3] where it is shown to be flat. We
prove here that it is not flat by pure prolongation.

Though naturally control-affine, it is presented here in its prolonged form
(2.16):

$\begin{array}[]{cclcccl}\displaystyle\dot{x}_{1}&=&\displaystyle
x_{2}&\nobreak\leavevmode\hfil&\displaystyle\dot{x}_{2}&=&\displaystyle
u_{1}^{(0)}\\\ \displaystyle\dot{y}_{1}&=&\displaystyle
y_{2}&\nobreak\leavevmode\hfil&\displaystyle\dot{y}_{2}&=&\displaystyle
u_{2}^{(0)}\\\
\displaystyle\dot{\theta}_{1}&=&\displaystyle\theta_{2}&\nobreak\leavevmode\hfil&\displaystyle\dot{\theta}_{2}&=&\displaystyle-\frac{u_{1}^{(0)}}{\varepsilon}\cos\theta_{1}+\frac{u_{2}^{(0)}+1}{\varepsilon}\sin\theta_{1}\\\
\displaystyle\dot{u}_{1}^{(0)}&=&\displaystyle
u_{1}^{(1)}&\nobreak\leavevmode\hfil&\displaystyle\dot{u}_{2}^{(0)}&=&\displaystyle
u_{2}^{(1)}.\end{array}$ (5.12)

The state is
$(x_{1},x_{2},y_{1},y_{2},\theta_{1},\theta_{2},u_{1}^{(0)},u_{2}^{(0)})$, of
dimension $n+m=6+2=8$. The associated non prolonged vector fields are

$\displaystyle g_{0}^{(0)}$ $\displaystyle=x_{2}\frac{\partial}{\partial
x_{1}}+y_{2}\frac{\partial}{\partial
y_{1}}+\theta_{2}\frac{\partial}{\partial\theta_{1}}+\frac{1}{\varepsilon}\sin\theta_{1}\frac{\partial}{\partial\theta_{2}}$
(5.13) $\displaystyle+u_{1}^{(0)}\left(\frac{\partial}{\partial
x_{2}}-\frac{1}{\varepsilon}\cos\theta_{1}\frac{\partial}{\partial\theta_{2}}\right)+u_{2}^{(0)}\left(\frac{\partial}{\partial
y_{2}}+\frac{1}{\varepsilon}\sin\theta_{1}\frac{\partial}{\partial\theta_{2}}\right)$
$\displaystyle g_{1}^{(0)}$ $\displaystyle=\frac{\partial}{\partial
u_{1}^{(0)}}\qquad g_{2}^{(0)}=\frac{\partial}{\partial u_{2}^{(0)}}$

We have

$G_{0}^{(\mathbf{0})}=\\{g_{1}^{(0)},g_{2}^{(0)}\\}=\\{\frac{\partial}{\partial
u_{1}^{(0)}},\frac{\partial}{\partial u_{2}^{(0)}}\\}$

and

$\displaystyle G_{1}^{(\mathbf{0})}$
$\displaystyle=\\{g_{1}^{(0)},g_{2}^{(0)},\mathrm{ad}_{g_{0}^{(\mathbf{0})}}g_{1}^{0},\mathrm{ad}_{g_{0}^{(\mathbf{0})}}g_{2}^{0}\\}$
$\displaystyle=\\{\frac{\partial}{\partial
u_{1}^{(0)}},\frac{\partial}{\partial u_{2}^{(0)}},-\frac{\partial}{\partial
x_{2}}+\frac{1}{\varepsilon}\cos\theta_{1}\frac{\partial}{\partial\theta_{2}},-\frac{\partial}{\partial
y_{2}}-\frac{1}{\varepsilon}\sin\theta_{1}\frac{\partial}{\partial\theta_{2}}\\}=\overline{G_{1}^{(\mathbf{0})}}$

but

$\displaystyle G_{2}^{(\mathbf{0})}$
$\displaystyle=G_{1}^{(\mathbf{0})}+\\{\frac{\partial}{\partial
x_{1}}-\frac{1}{\varepsilon}\cos\theta_{1}\frac{\partial}{\partial\theta_{1}}-\frac{\theta_{2}}{\varepsilon}\sin\theta_{1}\frac{\partial}{\partial\theta_{2}},\frac{\partial}{\partial
y_{1}}+\frac{1}{\varepsilon}\sin\theta_{1}\frac{\partial}{\partial\theta_{1}}-\frac{\theta_{2}}{\varepsilon}\cos\theta_{1}\frac{\partial}{\partial\theta_{2}}\\}$
$\displaystyle\neq\overline{G_{2}^{(\mathbf{0})}}.$

As before, we may initialize the pure prolongation algorithm by choosing

$H_{1,1}^{0}=\\{g_{1}^{(0)},\mathrm{ad}_{g_{0}^{(\mathbf{0})}}g_{1}^{0}\\}=\\{\frac{\partial}{\partial
u_{1}^{(0)}},-\frac{\partial}{\partial
x_{2}}+\frac{1}{\varepsilon}\cos\theta_{1}\frac{\partial}{\partial\theta_{2}}\\}=\overline{H_{1,1}^{0}}$

which amounts to prolonging $u_{2}^{(1)}$, or

$H_{1,2}^{0}=\\{g_{2}^{(0)},\mathrm{ad}_{g_{0}^{(\mathbf{0})}}g_{2}^{0}\\}=\\{\frac{\partial}{\partial
u_{2}^{(0)}},-\frac{\partial}{\partial
y_{2}}-\frac{1}{\varepsilon}\sin\theta_{1}\frac{\partial}{\partial\theta_{2}}\\}=\overline{H_{1,2}^{0}}$

which amounts to prolonging $u_{1}^{(1)}$.

It is noticeable that the inputs $u_{1}^{(1)}$ and $u_{2}^{(1)}$ play a
symmetric role . Thus, one may choose indifferently one of them as the non
prolonged input. Let us choose $u_{1}^{(1)}$ as non prolonged input, with the
initialization $H_{1,1}^{0}$. Thus, the vector fields associated to a
prolongation of length $l_{2}$ on the second input read:

$\displaystyle g_{0}^{(0,l_{2})}$ $\displaystyle=x_{2}\frac{\partial}{\partial
x_{1}}+y_{2}\frac{\partial}{\partial
y_{1}}+\theta_{2}\frac{\partial}{\partial\theta_{1}}+\frac{1}{\varepsilon}\sin\theta_{1}\frac{\partial}{\partial\theta_{2}}+u_{1}^{(0)}\left(\frac{\partial}{\partial
x_{2}}-\frac{1}{\varepsilon}\cos\theta_{1}\frac{\partial}{\partial\theta_{2}}\right)$
(5.14) $\displaystyle+u_{2}^{(0)}\left(\frac{\partial}{\partial
y_{2}}+\frac{1}{\varepsilon}\sin\theta_{1}\frac{\partial}{\partial\theta_{2}}\right)+\sum_{p=0}^{l_{2}-1}u_{2}^{(p+1)}\frac{\partial}{\partial
u_{2}^{(p)}}$ $\displaystyle g_{1}^{(0)}$
$\displaystyle=\frac{\partial}{\partial u_{1}^{(0)}}\qquad
g_{2}^{(l_{2})}=\frac{\partial}{\partial u_{2}^{(l_{2})}}.$

##### $\bullet\leavevmode\nobreak\ \mathbf{k=1.}$

For all $l_{2}\geq 2$,

$\Gamma_{1}^{(0,l_{2})}=\\{\frac{\partial}{\partial
u_{2}^{(l_{2})}},\frac{\partial}{\partial
u_{2}^{(l_{2}-1)}}\\},\quad\Delta_{1}^{(0,l_{2})}=\\{\frac{\partial}{\partial
u_{1}^{(0)}},-\frac{\partial}{\partial
x_{2}}+\frac{1}{\varepsilon}\cos\theta_{1}\frac{\partial}{\partial\theta_{2}}\\},$

and if $l_{2}=1$,

$\Gamma_{1}^{(0,1)}=\\{\frac{\partial}{\partial
u_{2}^{(1)}}\\},\quad\Delta_{1}^{(0,l_{2})}=\\{\frac{\partial}{\partial
u_{1}^{(0)}},-\frac{\partial}{\partial
x_{2}}+\frac{1}{\varepsilon}\cos\theta_{1}\frac{\partial}{\partial\theta_{2}},\frac{\partial}{\partial
u_{2}^{(0)}}\\}.$

Thus, we have $\overline{\Delta_{1}^{(0,l_{2})}}=\Delta_{1}^{(0,l_{2})}$ and
$[\Gamma_{1}^{(0,l_{2})},\Delta_{1}^{(0,l_{2})}]\subset\Delta_{1}^{(0,l_{2})}$
for all $l_{2}\geq 1$.

##### $\bullet\leavevmode\nobreak\ \mathbf{k=2.}$

For all $l_{2}\geq 3$,

$\displaystyle\Gamma_{2}^{(0,l_{2})}$
$\displaystyle=\\{\frac{\partial}{\partial
u_{2}^{(l_{2})}},\frac{\partial}{\partial
u_{2}^{(l_{2}-1)}},\frac{\partial}{\partial u_{2}^{(l_{2}-2)}}\\},$
$\displaystyle\Delta_{2}^{(0,l_{2})}$
$\displaystyle=\\{\frac{\partial}{\partial
u_{1}^{(0)}},-\frac{\partial}{\partial
x_{2}}+\frac{1}{\varepsilon}\cos\theta_{1}\frac{\partial}{\partial\theta_{2}},\frac{\partial}{\partial
x_{1}}-\frac{1}{\varepsilon}\cos\theta_{1}\frac{\partial}{\partial\theta_{1}}-\frac{1}{\varepsilon}\theta_{2}\sin\theta_{1}\frac{\partial}{\partial\theta_{2}}\\}.$

If $l_{2}=2$,

$\displaystyle\Gamma_{2}^{(0,2)}$ $\displaystyle=\\{\frac{\partial}{\partial
u_{2}^{(2)}},\frac{\partial}{\partial u_{2}^{(1)}}\\},$
$\displaystyle\Delta_{2}^{(0,2)}$ $\displaystyle=\\{\frac{\partial}{\partial
u_{1}^{(0)}},-\frac{\partial}{\partial
x_{2}}+\frac{1}{\varepsilon}\cos\theta_{1}\frac{\partial}{\partial\theta_{2}},\frac{\partial}{\partial
x_{1}}-\frac{1}{\varepsilon}\cos\theta_{1}\frac{\partial}{\partial\theta_{1}}-\frac{1}{\varepsilon}\theta_{2}\sin\theta_{1}\frac{\partial}{\partial\theta_{2}},\frac{\partial}{\partial
u_{2}^{(0)}}\\}.$

If $l_{2}=1$,

$\displaystyle\Gamma_{2}^{(0,1)}$ $\displaystyle=\\{\frac{\partial}{\partial
u_{2}^{(1)}}\\},$ $\displaystyle\Delta_{2}^{(0,1)}$
$\displaystyle=\\{\frac{\partial}{\partial
u_{1}^{(0)}},-\frac{\partial}{\partial
x_{2}}+\frac{1}{\varepsilon}\cos\theta_{1}\frac{\partial}{\partial\theta_{2}},\frac{\partial}{\partial
x_{1}}-\frac{1}{\varepsilon}\cos\theta_{1}\frac{\partial}{\partial\theta_{1}}-\frac{1}{\varepsilon}\theta_{2}\sin\theta_{1}\frac{\partial}{\partial\theta_{2}},$
$\displaystyle\hskip 113.81102pt\frac{\partial}{\partial
u_{2}^{(0)}},-\frac{\partial}{\partial
y_{2}}-\frac{1}{\varepsilon}\sin\theta_{1}\frac{\partial}{\partial\theta_{2}}\\}.$

Again, we have
$[\Gamma_{2}^{(0,l_{2})},\Delta_{2}^{(0,l_{2})}]\subset\Delta_{2}^{(0,l_{2})}$
for all $l_{2}\geq 1$, but
$\overline{\Delta_{2}^{(0,l_{2})}}\neq\Delta_{2}^{(0,l_{2})}$ for all
$l_{2}\geq 1$ since, e.g.

$[-\frac{\partial}{\partial
x_{2}}+\frac{1}{\varepsilon}\cos\theta_{1}\frac{\partial}{\partial\theta_{2}},\frac{\partial}{\partial
x_{1}}-\frac{1}{\varepsilon}\cos\theta_{1}\frac{\partial}{\partial\theta_{1}}-\frac{1}{\varepsilon}\theta_{2}\sin\theta_{1}\frac{\partial}{\partial\theta_{2}}]=-\frac{1}{\varepsilon^{2}}\sin
2\theta_{1}\frac{\partial}{\partial\theta_{2}}\not\in\Delta_{2}^{(0,l_{2})}$

for all $l_{2}\geq 1$. Changing the non prolonged input $u_{1}^{(0)}$ in
$u_{2}^{(0)}$, as previously announced, a similar calculation, left to the
reader, shows that $\Delta_{2}^{(l_{1},0)}$ is not involutive for all
$l_{1}\geq 1$. Thus, according to the first item of theorem 4.2, system (5.12)
is not flat by pure prolongation, though differentially flat, as shown in [10,
section C].

###  5.5 An example with 3 inputs [18]

$\begin{array}[]{ccl}\dot{x}_{1}&=&u_{1}^{(0)}\\\
\dot{x}_{2}&=&x_{3}u_{1}^{(0)}\\\
\dot{x}_{3}&=&x_{4}u_{1}^{(0)}+x_{1}u_{3}^{(0)}\\\
\dot{x}_{4}&=&u_{2}^{(0)}.\end{array}$ (5.15)

The state is $(x_{1},\ldots,x_{4},u_{1}^{(0)},u_{2}^{(0)},u_{3}^{(0)})$ (of
dimension $n+m=4+3=7$) and the control inputs are
$(u_{1}^{(1)},u_{2}^{(1)},u_{3}^{(1)})$. The reader may easily see that
$G_{1}^{(\mathbf{0})}$ is not involutive and thus that this system is not
static feedback linearizable.

This example may be found in [18] in the context of control affine systems
with $n$ states and $n-1$ inputs with the property that there exists $i=1,2,3$
such that

$\dim\\{f_{1},f_{2},f_{3},\mathrm{ad}{g}f_{i}\\}=4,$ (5.16)

with

$f_{1}=\frac{\partial}{\partial x_{1}}+x_{3}\frac{\partial}{\partial
x_{2}}+x_{4}\frac{\partial}{\partial x_{3}},\quad
f_{2}=\frac{\partial}{\partial x_{4}},\quad
f_{3}=x_{1}\frac{\partial}{\partial x_{3}},\quad
g=u_{1}^{(0)}f_{1}+u_{2}^{(0)}f_{2}+u_{3}^{(0)}f_{3},$

a property satisfied for $i=3$ since

$\mathrm{ad}{g}f_{3}=u_{1}^{(0)}[f_{1},f_{3}]=u_{1}^{(0)}\left(\frac{\partial}{\partial
x_{3}}-x_{1}\frac{\partial}{\partial
x_{2}}\right)\not\in\\{f_{1},f_{2},f_{3}\\}.$

Therefore, in [18, Theorem 3], flat outputs are exhibited as 3 independent
first integrals of $f_{3}$, yielding the prolongation
$\dot{u}_{1}^{(1)}=u_{1}^{(2)}$, $\dot{u}_{2}^{(1)}=u_{2}^{(2)}$, with
$\mathbf{j}=(1,1,0)$ and thus $\mid\mathbf{j}\mid=2$.

We show here that this prolongation is not minimal and we compute the minimal
one.

We initialize the algorithm by remarking that $\\{f_{2},f_{3}\\}$ is the
unique maximal involutive subdistribution of $\\{f_{1},f_{2},f_{3}\\}$, thus
indicating that $u_{2}^{(1)}$ and $u_{3}^{(1)}$ are suitable candidates of non
prolonged inputs. We thus, consider the following vector fields, corresponding
to prolongations of $u_{1}^{(1)}$, for all $l\geq 1$:

$g_{0}^{(l,0,0)}\triangleq u_{1}^{(0)}\left(\frac{\partial}{\partial
x_{1}}+x_{3}\frac{\partial}{\partial x_{2}}+x_{4}\frac{\partial}{\partial
x_{3}}\right)+u_{2}^{(0)}\frac{\partial}{\partial
x_{4}}+u_{3}^{(0)}x_{1}\frac{\partial}{\partial
x_{3}}+\sum_{p=0}^{l-1}u_{1}^{(p+1)}\frac{\partial}{\partial u_{1}^{(p)}},$
$g_{1}^{(l)}=\frac{\partial}{\partial u_{1}^{(l)}},\qquad
g_{2}^{(0)}=\frac{\partial}{\partial u_{2}^{(0)}},\qquad
g_{3}^{(0)}=\frac{\partial}{\partial u_{3}^{(0)}}.$

The reader may easily check that
$\Delta_{k}^{(l,0,0)}=\overline{\Delta_{k}^{(l,0,0)}}$ and
$[\Gamma_{k}^{(l,0,0)},\Delta_{k}^{(l,0,0)}]\subset\Delta_{k}^{(l,0,0)}$ for
all $l\geq 1$ and $k=0,1,2$, and that $\dim\Delta_{2}^{(1,0,0)}=n+m=7$.
Therefore, the prolongation $\mathbf{j}=(1,0,0)$, with $\mid\mathbf{j}\mid=1$,
is minimal and the minimally purely prolonged system is equivalent by
diffeomorphism and feedback to

$y_{1}^{(3)}=u_{1}^{(2)},\qquad\ddot{y}_{2}=u_{2}^{(1)}\qquad
y_{3}^{(3)}=W(y_{1},\ldots,y_{1}^{(3)},y_{2},\ldots,\ddot{y}_{2},y_{3},\ldots,\ddot{y}_{3},u_{3}^{(1)})$

with flat outputs

$y_{1}=x_{1},\quad y_{2}=x_{4},\quad y_{3}=x_{2}$

and with

$\begin{array}[]{l}\displaystyle
W(y_{1},\ldots,y_{1}^{(3)},y_{2},\ldots,\ddot{y}_{2},y_{3},\ldots,\ddot{y}_{3},u_{3}^{(1)})\triangleq
y_{2}\dot{y}_{1}\left(\ddot{y}_{1}-\frac{\left(\dot{y}_{1}\right)^{2}}{y_{1}}\right)+\dot{y}_{2}\left(\dot{y}_{1}\right)^{2}\\\
\displaystyle\hskip
56.9055pt+y_{3}\frac{y_{1}^{(3)}}{\dot{y}_{1}}-\dot{y}_{3}\ddot{y}_{1}\left(\frac{1}{y_{1}}+2\frac{\ddot{y}_{1}}{(\dot{y}_{1})^{2}}\right)+\ddot{y}_{3}\left(\frac{\dot{y}_{1}}{y_{1}}+2\frac{\ddot{y}_{1}}{\dot{y}_{1}}\right)+y_{1}\dot{y}_{1}u_{3}^{(1)}.\end{array}$

##  6 Concluding Remarks

We have established necessary and sufficient conditions for a system to be
flat by pure prolongation, i.e. belonging to the equivalence class of $0$ with
respect to the _equivalence by pure prolongation relation_. These conditions
extend preliminary results of [6, 7, 33, 34, 2, 12] thanks to a thorough study
of purely prolonged vector fields. We then deduce a computationally tractable
algorithm giving the minimal prolongation in a finite number of steps using
only Lie brackets and linear algebra.

Possible extensions of this work towards general flatness necessary and
sufficient conditions are under study.

Acknowledgements— The author wishes to express his warm thanks to Ph. Martin
and Y. Kaminski for many fruitful discussions.

## References

  * [1] E. Aranda-Bricaire, C.H. Moog, and J.-B. Pomet, _A linear algebraic framework for dynamic feedback linearization_ , IEEE Trans. Automat. Contr. 40 (1995), no. 1, 127–132.
  * [2] S. Battilotti and C. Califano, _A constructive condition for dynamic feedback linearization_ , Systems & Control Letters 52 (2004), 329–338.
  * [3] N. Bourbaki, _Algèbre_ , Éléments de Mathématiques, Chapitres 1-3, Hermann, Paris, 1970.
  * [4] P. Brunovský, _A classification of linear controllable systems_ , Kybernetika 6 (1970), 173–178.
  * [5] É. Cartan, _Sur l’équivalence absolue de certains systèmes d’équations différentielles et sur certaines familles de courbes_ , Bull. Soc. Math. France 42 (1914), 12–48, in Oeuvres Complètes, part II, vol 2, pages 1133–1168, CNRS, Paris, 1984.
  * [6] B. Charlet, J. Lévine, and R. Marino, _On dynamic feedback linearization_ , Systems & Control Letters 13 (1989), 143–151.
  * [7] , _Sufficient conditions for dynamic state feedback linearization_ , SIAM J. on Control and Optimization 29 (1991), no. 1, 38–57.
  * [8] C. Chevalley, _Theory of Lie Groups_ , Princeton University Press, 1946\.
  * [9] M. Fliess, J. Lévine, Ph. Martin, and P. Rouchon, _Flatness and defect of nonlinear systems: introductory theory and examples_ , Int. J. Control 61 (1995), no. 6, 1327–1361.
  * [10] , _A Lie-Bäcklund approach to equivalence and flatness of nonlinear systems_ , IEEE Trans. Automat. Contr. 44 (1999), no. 5, 922–937.
  * [11] J. Franch, _Flatness, tangent systems and flat outputs_ , Ph.D. thesis, Universitat Politecnica de Catalunya, 1999.
  * [12] J. Franch and E. Fossas, _Linearization by prolongations: New bound on the number of integrators_ , European Journal of Control 11 (2005), 171–179.
  * [13] R. Hermann and A.J. Krener, _Nonlinear controllability and observability_ , IEEE Trans. Automat. Contr. 22 (1977), 728–740.
  * [14] L.R. Hunt, R. Su, and G. Meyer, _Design for multi-input nonlinear systems_ , Differential Geometric Control Theory (R.W. Brockett, R.S. Millman, and H.J. Sussmann, eds.), Birkhäuser, Boston, 1983, pp. 268–298.
  * [15] , _Global transformations of nonlinear systems_ , IEEE Trans. Automat. Contr. 28 (1983), 24–31.
  * [16] A. Isidori, _Nonlinear Control Systems. An Introduction_ , 1st ed., Springer, Berlin, New York, 1985.
  * [17] B. Jakubczyk and W. Respondek, _On linearization of control systems_ , Bull. Acad. Pol. Sci. Ser. Sci. Math. 28 (1980), no. 9–10, 517–522.
  * [18] Y. J. Kaminski, J. Lévine, and F. Ollivier, _On singularities of flat affine systems with $n$ states and $n-1$ controls_, International Journal of Robust and Nonlinear Control 30 (2020), no. 9, 3547–3565.
  * [19] I. S. Krasil’shchik, V. V. Lychagin, and A. M. Vinogradov, _Geometry of Jet Spaces and Nonlinear Partial Differential Equations_ , Gordon and Breach, New York, 1986.
  * [20] J. Lévine, _Static and dynamic state feedback linearization_ , Nonlinear Systems (A. Fossard and D. Normand-Cyrot, eds.), vol. 3, Chapman & Hall, 1997, pp. 93–126.
  * [21] , _Analysis and Control of Nonlinear Systems: A Flatness-based Approach_ , Mathematical Engineering, Springer, 2009.
  * [22] , _On necessary and sufficient conditions for differential flatness_ , Applicable Algebra in Engineering, Communication and Computing 22 (2011), 47–90.
  * [23] R. Marino, _On the largest feedback linearizable subsystem_ , Systems & Control Letters 6 (1986), 345–351.
  * [24] Ph. Martin, _Contribution à l’étude des systèmes diffèrentiellement plats_ , Ph.D. thesis, École des Mines de Paris, 1992\.