# Differential Flatness by Pure Prolongation: Necessary and Sufficient Conditions

Jean Levine

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Jean Levine. Differential Flatness by Pure Prolongation: Necessary and Sufficient Conditions. 2023. [hal-04065868](https://hal.science/hal-04065868)

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# Differential Flatness by Pure Prolongation:
## Necessary and Sufficient Conditions

Jean Lévine*

February 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 notion of *dynamic feedback linearization* of a nonlinear system, whose preliminary results where reported in [5, 6] (see also [27, 28, 1, 10]), was at the origin of a long chain of studies on nonlinear system theory. In particular, during the last three decades, it gave birth to the concept of *differential flatness*, that plays a prominent role in motion planning and trajectory tracking problems and their applications (see [19, 8, 9] and [17] for a thorough presentation).

Sufficient or necessary conditions in special cases, as well as general differential flatness characterizations, though unable to detect if a system is non flat in a finite number of steps, have been obtained (see e.g. [17, 18] for a historical review). Nevertheless, the question of obtaining computationally tractable 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*

<footnote>*Centre Automatique et Systèmes (CAS), MINES Paris - PSL, 75006 Paris, France, e-mail: jean.levine@minesparis.psl.eu</footnote>
DIFFERENTIAL FLATNESS BY PURE PROLONGATION

systems by pure prolongation, i.e. , roughly speaking, the class of n -dimensional systems with m inputs

$$
\dot{x} = f(x, u_1, \dots, u_m)
$$

for which there exists a multi-index **j** ≜ (j₁, . . . , jₘ) such that the prolonged system, denoting by u<sub>i</sub><sup>(j<sub>i</sub>)</sup> the j<sub>i</sub>-th order time derivative of u<sub>i</sub>,

$$
\dot{x} = f(x, u_1, \dots, u_m),
$$

$$
u_i^{(j_i+1)} = v_i, \quad i = 1, \dots, m,
$$

of dimension n + ∑ m i =1 j$_{i}$ is locally diffeomorphic to the trivial system

$$
y_i^{(\alpha_i+1)} = w_i, \quad i = 1, \dots, m,
$$

with

$$
y = \psi(x, u_1, \dots, u_1^{(j_1)}, \dots, u_m, \dots, u_m^{(j_m)}),
$$

$$
w_i = W_i(x, u_1, \dots, u_1^{(j_1)}, v_1, \dots, u_m, \dots, u_m^{(j_m)}, v_m), \quad i = 1, \dots, m,
$$

and ∑ m i =1 α$_{i}$ = n + ∑ m i =1 j$_{i}$ . This diffeomorphism indeed locally implies that

$$
x = \varphi(y_1, \dots, y_1^{(\alpha_1)}, \dots, y_m, \dots, y_m^{(\alpha_m)}),
$$

$$
u = U(y_1, \dots, y_1^{(\alpha_1)}, w_1, \dots, y_m, \dots, y_m^{(\alpha_m)}, w_m).
$$

Note that such transformations are called *pure dynamic extensions* in [27, 28],
in particular in the context of finding bounds on the indices *j*₁, ..., *j*ₘ, if they exist.
We use here the word *prolongation* initially introduced in [6], in reference to É.
Cartan [4] (see also [1, 10]).

We stress that even for this special class of differentially flat systems, no necessary
and sufficient conditions were known until now to the author’s knowledge.

Our main contributions are:

1. proposition 2, extending results implicitly present in [1, 10], where we prove that, whatever the prolongation **j** ≜ (j₁, ..., jₘ), the filtration made of 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 {Δ<sub>k</sub><sup>(j)</sup>}<sub>k</sub>, is included in the original tangent bundle (of dimension n + m) and the other one, denoted by {Γ<sub>k</sub><sup>(j)</sup>}<sub>k</sub>, is only made of the decreasing orders of the prolonged control vector fields ⊕<sub>i=1,...,m</sub> {g<sub>i</sub><sup>(jᵢ-r)</sup> | r = 0,...,max(jᵢ-1,k)},

2. theorem 3 giving the necessary and sufficient conditions: Δ<sub>k</sub><sup>(j)</sup> must be involutive with locally constant dimension and invariant by Γ<sub>k</sub><sup>(j)</sup> for all k, and dim Δ<sub>k</sub><sup>(j)</sup> must be equal to n + m for all k large enough (strong controllability),
3. **formula (4.8) and theorem 4** giving the *i*-th prolongation length, $j_i$, knowing the $i-1$ first minimal prolongation lengths $j_1 = 0 \le \dots \le j_{i-1}$, or the criterion of non flatness by pure prolongation,

4. **algorithm 1** whose input is 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, finishing in a finite number of steps.

The paper is organized as follows: In section 2, we present short recalls of basic results on differential flatness and feedback linearization. Then we introduce and study the pure prolongation distributions and the associated vertical ones in section 3. The necessary and sufficient conditions for flatness by pure prolongation are then presented in section 4, followed by the pure prolongation algorithm. A series of four examples then illustrate our results in section 5, one of them, the pendulum example, being proven to be non flat by pure prolongation, though known to be differentially flat [9, 17]. The paper ends with concluding remarks.

## 2 Recalls on Flatness and Feedback Linearization

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

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

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

In the sequel, we systematically use boldface letters $j, k, \dots$, to denote multi-integers $(j_1, j_2, \dots, j_m)$, $(k_1, k_2, \dots, k_m)$, $\dots$ and overlined symbols to denote a collection of successive time derivatives of a time-dependent function as follows: given a multi-integer $k = (k_1, \dots, k_m) \in \mathbb{N}^m$ and a locally defined $C^\infty$ function $t \mapsto \xi(t) \in \mathbb{R}^m$,

* $\bar{\xi}^{(k)}$ denotes the vector $(\xi_1, \dot{\xi}_1, \dots, \xi_1^{(k_1)}, \dots, \xi_m, \dot{\xi}_m, \dots, \xi_m^{(k_m)})$ of dimension $m + |k|$, with $|k| \triangleq \sum_{i=1}^m k_i$ and $\xi_i^{(j)} \triangleq \frac{d^j \xi_i}{dt^j}$, $j = 1, \dots, k_i$, $i = 1, \dots, m$;

* $\bar{\xi}$ denotes the infinite sequence $(\xi, \dot{\xi}, \ddot{\xi}, \dots) \triangleq (\xi_i^{(k)}; i = 1, \dots, m; k \ge 0) \in \mathbb{R}_\infty^m$, where $\mathbb{R}_\infty^m \triangleq \mathbb{R}^m \times \mathbb{R}^m \times \dots$ is the product of an infinite number of copies of $\mathbb{R}^m$.

We also denote the minimum of two arbitrary integers $k$ and $l$ by $k \vee l \triangleq \min\{k, l\}$, and their maximum by $k \wedge l \triangleq \max(k, l)$. Also, for every $j \triangleq (j_1, \dots, j_m) \in \mathbb{N}^m$ and $k \in \mathbb{N}$, we introduce the *componentwise minimum* notation

$$ j \vee k \triangleq (j_1 \vee k, \dots, j_m \vee k) = (\min(j_1, k), \dots, \min(j_m, k)) \qquad (2.2) $$
and, more generally, for every $\mathbf{k} \triangleq (k_1, \dots, k_m) \in \mathbb{N}^m$,

$$ \mathbf{j} \vee \mathbf{k} \triangleq (j_1 \vee k_1, \dots, j_m \vee k_m) = (\min(j_1, k_1), \dots, \min(j_m, k_m)). \quad (2.3) $$

Accordingly, the componentwise maximum is denoted by

$$ \mathbf{j} \wedge \mathbf{k} \triangleq (j_1 \wedge k_1, \dots, j_m \wedge k_m) = (\max(j_1, k_1), \dots, \max(j_m, k_m)) \quad (2.4) $$

with the notation $\mathbf{j} \wedge \mathbf{k}$ if $k_1 = \dots = k_m = k$.

## 2.1 Recalls on Lie-Bäcklund Isomorphisms

Roughly speaking<fnref n="1" />, system (2.1) is said to be differentially flat at $(x_0, \bar{u}_0)$ [8, 19, 9, 17] if, and only if, there exists an $m$-dimensional vector $y = (y_1, \dots, y_m)$ such that the following statements hold:

* $y$ and its successive derivatives $\dot{y}, \ddot{y}, \dots$ are locally independent,

* $y$ is a function of $x$, $u$ and successive derivatives of $u$ up to a finite order $\mathbf{s} \in \mathbb{N}^m$:

$$ y = \Psi(x, \bar{u}^{(\mathbf{s})}), \quad (2.5) $$

$\Psi$ being defined and $C^\infty$ in a suitable neighborhood of $(x_0, \bar{u}_0)$,

* $x$ and $u$ are functions of $y$ and its derivatives up to a finite order $\mathbf{r} \in \mathbb{N}^m$:

$$ x = \Phi_{-1}(\bar{y}^{(\mathbf{r})}), \quad u = \Phi_0(\bar{y}^{(\mathbf{r}+1)}), \quad (2.6) $$

with $\mathbf{r}+1 \triangleq (r_1+1, \dots, r_m+1)$, and where<fnref n="2" /> $\bar{\Phi} \triangleq (\Phi_{-1}, \Phi_0, \Phi_1, \dots, \Phi_l, \dots)$ is defined and $C^\infty$ in a suitable neighborhood of $\bar{y}_0 = \bar{\Psi}(x_0, \bar{u}_0)$, again with the notation $\bar{\Psi} \triangleq (\Psi, \dot{\Psi}, \dots, \Psi^{(l)}, \dots)$, where $\Psi^{(l)}, l \in \mathbb{N}$, stands for the total $l$-th order derivative of $\Psi$, i.e. $\Psi^{(l)}(x, \bar{u}^{(\mathbf{s}+1)}) = y^{(l)}$, again with the notation $\mathbf{s}+1 \triangleq (s_1+1, \dots, s_m+1)$.

* Moreover, $\Phi$ is such that equation (2.1) is identically satisfied, i.e.

$$ \frac{d}{dt} (\Phi_{-1}(\bar{y}^{(\mathbf{r})})) = f(\Phi_{-1}(\bar{y}^{(\mathbf{r})}), \Phi_0(\bar{y}^{(\mathbf{r}+1)})) $$

for every $C^\infty$ function $t \mapsto y(t)$ such that $\bar{y}(t)$ remains in the above mentioned neighborhood of $\bar{y}_0$.

Then the vector $y$ is called a flat output.

This definition can be made rigorous by viewing $\bar{\Phi}$ and $\bar{\Psi}$ as functions defined on infinite order jet spaces (see e.g. [9, 24, 17, 18, 25]). They are then called Lie-Bäcklund isomorphisms and are inverse of one another. Let us now recall the definitions of system prolongation and Lie-Bäcklund isomorphism for the sake of completeness.

<footnote n="1">¹See section 2.2 of this article for a precise definition.</footnote>

<footnote n="2">²the components of $\Phi$ are indexed from $-1$ to $\infty$, to let the component $u^{(k)}$ of $\bar{u}$ be $\Phi_k$ for all $k \ge 0$.</footnote>
DIFFERENTIAL FLATNESS BY PURE PROLONGATION

**Definition 1.** The infinite order jet space prolongation of system (2.1) is given by the pair $(X \times \mathbb{R}_{\infty}^{m}, C_f)$, where $X \times \mathbb{R}_{\infty}^{m}$ is the product of X with an infinite number of copies of $\mathbb{R}^{m}$, with coordinates $(x, \bar{u})$, endowed with the Cartan field

$$C_f = f(x, u) \frac{\partial}{\partial x} + \sum_{j \ge 0} \sum_{i=1}^{m} u_i^{(j+1)} \frac{\partial}{\partial u_i^{(j)}}, \qquad (2.7)$$

that lifts the Lie derivative operator along f, defined on TX, to the Lie derivative along $C_f$ on $TX \times \mathbb{R}_{\infty}^{m}$, the tangent bundle of $X \times \mathbb{R}_{\infty}^{m}$.

**Definition 2.** Consider two systems:

$$\dot{x} = g(x, u) \quad \text{and} \quad \dot{y} = \gamma(y, v) \qquad (2.8)$$

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

$$C_g = g(x, u) \frac{\partial}{\partial x} + \sum_{j \ge 0} \sum_{i=1}^{m} u_i^{(j+1)} \frac{\partial}{\partial u_i^{(j)}}, \qquad (2.9)$$

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

$$C_{\gamma} = \gamma(y, v) \frac{\partial}{\partial y} + \sum_{j \ge 0} \sum_{i=1}^{\mu} v_i^{(j+1)} \frac{\partial}{\partial v_i^{(j)}}, \qquad (2.10)$$

We say that they are Lie-Bäcklund equivalent at a pair of points $(x_0, \bar{u}_0)$ and $(y_0, \bar{v}_0)$ if, and only if, there exists neighborhoods $N_{x_0, \bar{u}_0} \subset X \times \mathbb{R}_{\infty}^{m}$ and $N_{y_0, \bar{v}_0} \subset Y \times \mathbb{R}_{\infty}^{\mu}$ and a $C^{\infty}$ isomorphism$^3 \Phi : N_{y_0, \bar{v}_0} \to N_{x_0, \bar{u}_0}$ satisfying $\Phi(y_0, \bar{v}_0) = (x_0, \bar{u}_0)$, with $C^{\infty}$ inverse $\Psi$, such that the respective Cartan fields are $\Phi$ and $\Psi$ related, i.e. $\Phi_* C_{\gamma} = C_g$ in $N_{x_0, \bar{u}_0}$ and $\Psi_* C_g = C_{\gamma}$ in $N_{y_0, \bar{v}_0}$.

In other words, the two systems are Lie-Bäcklund equivalent at the points $(x_0, \bar{u}_0)$ and $(y_0, \bar{v}_0)$ 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.

We recall, without proof, a most important result from [19] (see also [8, 9, 17]) giving an interpretation of the Lie-Bäcklund equivalence in terms of diffeomorphism in finite dimension and endogenous dynamic feedback, that will be useful in the next sections.

**Theorem 1** (Martin [19]). If the systems

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

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

<footnote>$^3$Recall 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 $(y, \bar{v})$ (resp. $(x, \bar{u})$) (see e.g. [15, 29, 17]).</footnote>
DIFFERENTIAL FLATNESS BY PURE PROLONGATION

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

(ii) there exist

- an endogenous dynamic feedback<fnref n="4" />

$$u = \alpha(x, z, w), \quad \dot{z} = \beta(x, z, w), \qquad (2.11)$$

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

- a multi-integer<fnref n="5" /> $\mathbf{r} \triangleq (r_1, \dots, r_m)$,

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

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

$$\dot{y} = \gamma(y, v), \quad v(\mathbf{r}) = w \qquad (2.12)$$

and the closed-loop one

$$\dot{x} = g(x, \alpha(x, z, w)), \quad \dot{z} = \beta(x, z, w) \qquad (2.13)$$

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

$$(x, z) = \chi(y, v, \dot{v}, \dots, v^{(\mathbf{r}-1)}), \quad (y, v, \dot{v}, \dots, v^{(\mathbf{r}-1)}) = \chi^{-1}(x, z) \qquad (2.14)$$

and

$$\hat{g} = \chi_* \hat{\gamma}, \quad \hat{\gamma} = \chi_*^{-1} \hat{g} \qquad (2.15)$$

where we have denoted

$$\begin{aligned} \hat{g}(x, z, w) &\triangleq g(x, \alpha(x, z, w)) \frac{\partial}{\partial x} + \beta(x, z, w) \frac{\partial}{\partial z} \\ \hat{\gamma}(y, v, \dot{v}, \dots, v^{(\mathbf{r}-1)}, w) &\triangleq \gamma(y, v) \frac{\partial}{\partial y} + \sum_{i=1}^{m} \sum_{j=0}^{r_i-1} v_i^{(j+1)} \frac{\partial}{\partial v_i^{(j)}} + w_i \frac{\partial}{\partial v_i^{(r_i)}}. \end{aligned}$$

## 2.2 Recalls on Differential Flatness

We say that system (2.1) is differentially flat (or, more shortly, flat) at the pair of points $(x_0, \bar{u}_0)$ and $\bar{y}_0 \in \mathbb{R}_\infty^m$ if and only if, it is Lie-Bäcklund equivalent to the trivial system of $\mathbb{R}_\infty^m$ endowed with the trivial Cartan field

$$\tau \triangleq \sum_{j \ge 0} \sum_{i=1}^{m} y_i^{(j+1)} \frac{\partial}{\partial y_i^{(j)}} \qquad (2.16)$$

<footnote n="4">⁴A 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 [19, 8, 9, 17]).</footnote>

<footnote n="5">⁵Recall that we denote by $v^{(\mathbf{r})} \triangleq (v_1^{(r_1)}, \dots, v_m^{(r_m)}) \triangleq (\frac{d^{r_1} v_1}{dt^{r_1}}, \dots, \frac{d^{r_m} v_m}{dt^{r_m}})$.</footnote>
DIFFERENTIAL FLATNESS BY PURE PROLONGATION

at the considered points.

Otherwise stated, the locally defined flat output $y = \Psi(x, \bar{u})$ is such that $(x, \bar{u}) = \Phi(\bar{y}) = (\Phi_{-1}(\bar{y}), \Phi_0(\bar{y}), \Phi_1(\bar{y}), \dots)$, with $\frac{d}{dt}\Phi_{-1}(\bar{y}) \equiv f(\Phi_{-1}(\bar{y}), \Phi_0(\bar{y}))$ 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 $y = \Psi(x, \bar{u})$ of dimension $m$, depending at most on a finite number of derivatives of $u$, with independent derivatives of all orders, such that $x$ and $\bar{u}$ can be expressed in terms of $y$ and a finite number of its successive derivatives, i.e. $(x, \bar{u}) = \Phi(\bar{y})$, and such that the system equation $\frac{d}{dt}\Phi_{-1}(\bar{y}) = f \circ \Phi(\bar{y})$ is identically satisfied for all sufficiently differentiable $y: \mathbb{R} \to \mathbb{R}^m$. We thus have recovered, in a rigorous mathematical framework, the definition of section 2.1 (see e.g. [17, § 5.2] for a related discussion).

For a flat system, with the notations of theorem 1, the vector field $\gamma$, or $\hat{\gamma}$ indifferently, corresponds to the linear system in Brunovský canonical form

$$y_i^{(r_i+1)} = w_i, \quad i = 1, \dots, m, \qquad (2.17)$$

and $C_\gamma$, defined by (2.10), satisfies $C_\gamma = \tau$, with $\tau$ given by (2.16).

Theorem 1 reads:

**Corollary 1.** If system (2.1) is flat at a given point, there exists an endogenous dynamic feedback of the form (2.11) and a finite dimensional local diffeomorphism $\chi$ such that the systems (2.17) and (2.13), with $f$ in place of $g$, are $\chi$-related for all $w \in \mathbb{R}^m$.

## 2.3 Recalls on Feedback Linearization and Lie brackets

Feedback linearizable systems [14, 12] (see also [13, 23, 16]) constitute a subclass of differentially flat systems. We recall their definition.

**Definition 3.** System (2.1) is said static feedback linearizable or, shortly, feedback linearizable if the context allows, if, and only if, there exist:

* 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 $0 \in \mathbb{R}^n$,

* and a static feedback $u = \alpha(x, v)$, $v \in \mathbb{R}^m$ being the new control input and $\alpha$ being invertible with respect to $v$ for all $x$ in the above mentioned neighborhood of the origin, i.e. $\text{rank}(\frac{\partial \alpha}{\partial v})(x, v) = m$ for all $x$ and $v$ as above,

such that the image by $\varphi$ of the closed-loop vector field, namely $\varphi_*(f \circ \alpha)$, is equal to the vector field associated to the Brunovsky controllability canonical form

$$y_i^{(r_i)} = v_i, \quad i = 1, \dots, m, \qquad (2.18)$$

where the multi-integer $\mathbf{r} \triangleq (r_1, \dots, r_m)$, whose components $r_i$ are called the controllability indices, satisfies $|\mathbf{r}| \triangleq \sum_{i=1}^m r_i = n = \text{dim } X$.
DIFFERENTIAL FLATNESS BY PURE PROLONGATION

Indeed, since the n-dimensional vectors $\bar{y}^{(r-1)}$ and $x$ are diffeomorphic (again we have noted $\mathbf{r} - \mathbf{1} \triangleq (r_1 - 1, \dots, r_m - 1)$), it is immediate to verify that $(y_1, \dots, y_m)$ is a flat output and that a feedback linearizable system is flat.

These systems have been first characterized by [14, 11, 12] 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 easily be proven that systems of the form (2.1) are feedback linearizable if, and only if, the control-affine prolongation

$$ 
\begin{aligned}
\dot{x} &= f(x, u), \\
\dot{u}_i &= u_i^{(1)}, \quad i = 1, \dots, m
\end{aligned} 
\qquad (2.19) 
$$

with state $\bar{x}^{(0)} \triangleq (x, u) = (x_1, \dots, x_n, u_1, \dots, u_m) \in X^{(0)} \triangleq X \times \mathbb{R}^m$ and control vector $u^{(1)} \triangleq (u_1^{(1)}, \dots, u_m^{(1)}) \in \mathbb{R}^m$, is feedback linearizable (see e.g. [5, 26, 27, 16]).

Indeed, in the local coordinates<fnref n="6" /> $\bar{x}^{(0)}$ of $X^{(0)}$, denoting the associated vector fields by

$$ g_0^{(0)}(\bar{x}^{(0)}) \triangleq \sum_{i=1}^{n} f_i(x, u) \frac{\partial}{\partial x_i}, \quad g_i^{(0)}(\bar{x}^{(0)}) \triangleq \frac{\partial}{\partial u_i}, \quad i = 1, \dots, m, \qquad (2.20) $$

defined on the tangent bundle $TX^{(0)} = TX \times \mathbb{R}^m$, system (2.19) reads

$$ \dot{\bar{x}}^{(0)} = g_0^{(0)}(\bar{x}^{(0)}) + \sum_{i=1}^{m} u_i^{(1)} g_i^{(0)}(\bar{x}^{(0)}) \qquad (2.21) $$

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 (2.21), even if the vector-field $f$ is already given in control-affine form. For the sake of coherence, we set $u = (u_1, \dots, u_m) \triangleq u^{(0)} = (u_1^{(0)}, \dots, u_m^{(0)})$, so that $\dot{u}^{(0)} = u^{(1)}$.

We recall that the Lie bracket $[\eta, \gamma]$ of two arbitrary vector fields $\eta$ and $\gamma$ of $TX^{(0)}$ is given, in the $\bar{x}^{(0)}$-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 $\bar{x}^{(0)} = (x, u^{(0)}) \triangleq (x_1^{(0)}, \dots, x_{n+m}^{(0)})$. 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 $k \ge 1$, with the convention that $\mathrm{ad}_{\eta}^0 \gamma = \gamma$. In addition, if $\Gamma$ is an arbitrary distribution of vector fields on $TX^{(0)}$, 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 $\bar{\Gamma} = \Gamma$, where $\bar{\Gamma}$ denotes the involutive closure of $\Gamma$, i.e. the smallest involutive distribution containing $\Gamma$.

<footnote n="6">⁶We introduce the superscript (0) from now on to get ready to work with higher order prolongations (see section 3).</footnote>
DIFFERENTIAL FLATNESS BY PURE PROLONGATION

If the distribution $\Gamma$ is locally generated by $p$ vector fields $\gamma_1, \dots, \gamma_p$, with $p$ arbitrary, we write $\Gamma \triangleq \{\gamma_1, \dots, \gamma_p\}$. We also denote by $\Gamma(\xi) \triangleq \{\gamma_1(\xi), \dots, \gamma_p(\xi)\}$ the vector space generated by the vectors $\gamma_1(\xi), \dots, \gamma_p(\xi)$ at a point $\xi \in X^{(0)}$.

Consider the (0th order or non prolonged) filtration of distributions built on the vector fields (2.20)<fnref n="7" />:

$$G_0^{(0)} \triangleq \{g_1^{(0)}, \dots, g_m^{(0)}\}, \quad G_{k+1}^{(0)} \triangleq G_k^{(0)} + \mathrm{ad}_{g_0^{(0)}} G_k^{(0)}, \quad \forall k \ge 0, \quad (2.22)$$

indeed satisfying $G_0^{(0)} \subset \dots \subset G_k^{(0)} \subset G_{k+1}^{(0)} \subset \dots \subset TX^{(0)}$.

**Theorem 2** ([14, 12]). System (2.1), or equivalently system (2.19), is feedback linearizable in a neighborhood of the origin of $X^{(0)}$ if, and only if, in this neighborhood:

(i) $G_k^{(0)}$ is involutive with constant dimension for all $k \ge 0$,

(ii) there exists an integer $k_\star^{(0)} \le n$ such that $G_k^{(0)} = G_{k_\star^{(0)}}^{(0)} = TX^{(0)}$ for all $k \ge k_\star^{(0)}$.

Note that, according to (2.20), $G_0^{(0)}$ is involutive with constant dimension, equal to $m$, by construction.

Theorem 2 provides a construction of flat outputs via Frobenius theorem (see e.g. [7]) and the list of so-called *Brunovský's controllability indices* [3] as follows:

**Definition 4.** Consider the sequence of integers

$$\rho_k^{(0)} \triangleq \dim G_k^{(0)} / G_{k-1}^{(0)} \quad \forall k \ge 1, \quad \rho_0^{(0)} \triangleq \dim G_0^{(0)} = m.$$

The *Brunovský controllability indices* $\kappa_k^{(0)}$'s are defined by

$$\kappa_k^{(0)} \triangleq \#\{l \mid \rho_l^{(0)} \ge k\}, \quad k = 1, \dots, m,$$

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

It can be proven (see e.g. [14, 12, 13, 23, 16]) that, for a feedback linearizable nonlinear system (2.1), or (2.19), we have:

* $\rho_k^{(0)}$'s and $\kappa_k^{(0)}$'s are non increasing sequences,

* $\rho_k^{(0)} \le m$ for all $k$, $\rho_k^{(0)} = 0$, for all $k \ge k_\star^{(0)} + 1$,

* $\kappa_1^{(0)} = k_\star^{(0)} + 1$, $\kappa_m^{(0)} \ge 1$,

<footnote n="7">⁷As before, the superscript (0) is used to indicate that the distributions $G_k^{(0)}$ and the related indices $\rho_k^{(0)}$ and $\kappa_k^{(0)}$ are built on the non prolonged vector fields (2.20) and to distinguish them from the prolonged distributions $G_k^{(j)}$ of arbitrary jth order, $j \in \mathbb{N}^m$, and related indices, $\rho_k^{(j)}$ and $\kappa_k^{(j)}$, introduced in sections 3 and 4.</footnote>
DIFFERENTIAL FLATNESS BY PURE PROLONGATION

$$ \bullet \sum_{k=0}^{k_{\star}^{(0)}} \rho_k^{(0)} = \sum_{k=1}^{m} \kappa_k^{(0)} = \dim G_{k_{\star}^{(0)}}^{(0)} = n+m. $$

The list $\kappa_1^{(0)}, \dots, \kappa_m^{(0)}$ 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 (2.18) with $r_i = \kappa_i^{(0)}$, $i = 1, \dots, m$.

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

$$ G_k^{(0)} = \bigoplus_{j=1}^{m} \{ \mathrm{ad}_{g_0^{(0)}}^l g_j^{(0)} \mid l = 0, \dots, k \lor (\kappa_j^{(0)} - 1) \}, \quad G_{\kappa_1^{(0)}-1}^{(0)} = G_{k_{\star}^{(0)}}^{(0)} = \mathrm{TX}^{(0)}. $$

Then, flat outputs $(y_1, \dots, y_m)$ are locally non trivial solutions of the system of PDE's

$$ L_{\mathrm{ad}^k_{g_0^{(0)}} g_j^{(0)}} y_i = 0, \quad k = 0, \dots, \kappa_i^{(0)} - 2, \quad j = 1, \dots, m, \quad \text{with} \quad L_{\mathrm{ad}^{\kappa_i^{(0)}-1}_{g_0^{(0)}} g_i^{(0)}} y_i \neq 0, \qquad (2.23) $$

for $i = 1, \dots, m$, 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

$$ \bar{x}^{(0)} \mapsto (y_1, \dots, y_1^{(\kappa_1^{(0)}-1)}, \dots, y_m, \dots, y_m^{(\kappa_m^{(0)}-1)}) $$

is a local diffeomorphism.

# 3 System Pure Prolongation

## 3.1 Purely Prolonged distributions

We now introduce higher order prolongations of the vector fields defined by (2.20), called pure prolongations<fnref n="8" /> after [6] (see also [27, 28, 1, 10]).

Given a multi-integer $\mathbf{j} \triangleq (j_1, \dots, j_m) \in \mathbb{N}^m$, we note, as before, $|\mathbf{j}| \triangleq \sum_{i=1}^m j_i$ and the prolonged state:

$$ \bar{x}^{(\mathbf{j})} \triangleq (x, \bar{u}^{(\mathbf{j})}) \triangleq (x_1, \dots, x_n, u_1^{(0)}, \dots, u_1^{(j_1)}, \dots, u_m^{(0)}, \dots, u_m^{(j_m)}), $$

with the convention $u_i^{(0)} = u_i$, $i = 1, \dots, m$.

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

<footnote n="8">⁸They are called *pure dynamic extensions* in [27, 28], a terminology used for feedback design, whereas *prolongation* here refers to the jet space coordinates in the spirit of É. Cartan [4] who used the French word “prolongement”.</footnote>
The pure prolongation of order **j** of system (2.19), or otherwise said, of the vector
fields (2.20), in the tangent bundle $TX^{(j)} = TX \times \mathbb{R}^{m+|j|}$, is defined by

$$
\begin{align}
g_0^{(j)}(\bar{x}^{(j)}) &= \sum_{i=1}^{n} f_i(x, u) \frac{\partial}{\partial x_i} + \sum_{i=1}^{m} \sum_{k=0}^{j_i-1} u_i^{(k+1)} \frac{\partial}{\partial u_i^{(k)}} \tag{3.1} \\
g_i^{(j)}(\bar{x}^{(j)}) &\triangleq g_i^{(j_i)}(\bar{x}^{(j)}) \triangleq \frac{\partial}{\partial u_i^{(j_i)}}, \quad i = 1, \dots, m,
\end{align}
$$

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 **j** pure integrators to $u = u^{(0)}$ in (2.19) (with the same usual abuse of notations as in (2.21)):

$$
\dot{\bar{x}}^{(j)} = g_0^{(j)}(\bar{x}^{(j)}) + \sum_{i=1}^{m} u_i^{(j_i+1)} g_i^{(j_i)}(\bar{x}^{(j)}) \quad (3.2)
$$

or

$$
\dot{x} = f(x, u), \quad \dot{u}_i^{(k)} = u_i^{(k+1)}, \quad k = 0, \dots, j_i, \quad i = 1, \dots, m,
$$

$u^{(j+1)} = (u_1^{(j_1+1)}, \dots, u_m^{(j_m+1)})$ being the new control vector of this *purely prolonged system*, whose state is $(x, \bar{u}^{(j)}) = \bar{x}^{(j)}$.

**Remark 1.** Note that the state of the **j**-th prolonged system, $\bar{x}^{(j)}$, coincides with the image of $\bar{x}$ by the projection $p_j : \bar{x} \in X \times \mathbb{R}_\infty^m \mapsto p_j(\bar{x}) = \bar{x}^{(j)} \in X^{(j)}$ for all **j**. In addition, the family of projections $p_{i,j} : \bar{x}^{(i)} \in X^{(i)} \mapsto \bar{x}^{(j)} = p_{i,j}(\bar{x}^{(i)}) \in X^{(j)}$ for all **i**, **j** such that $i_k \ge j_k$ for all $k = 1, \dots, m$, that we note $\mathbf{i} \succeq \mathbf{j}$, indeed satisfies $p_{i,j} \circ p_{j,k} = p_{i,k}$ for all $\mathbf{i} \succeq \mathbf{j} \succeq \mathbf{k}$ and thus allows us to identify the manifold $X \times \mathbb{R}_\infty^m$ with the projective limit of the family $(X^{(i)}, p_{i,j})$ for all $\mathbf{i}$ and all $\mathbf{j}$ such that $\mathbf{i} \succeq \mathbf{j}$, i.e. $X \times \mathbb{R}_\infty^m \simeq \lim_{\leftarrow} X^{(i)}$ (see e.g. [2, Chap. I, §10]). A similar identification trivially holds for the associated tangent bundles, i.e. $TX \times \mathbb{R}_\infty^m \simeq \lim_{\leftarrow} TX^{(i)}$ relatively to the family $Tp_{i,j}$ of tangent projections, hence the identification of the Cartan field $C_f$, defined by (2.7), with $\lim_{\leftarrow} g_0^{(j)}$, the projective limit of the vector fields $g_0^{(j)}$. Nevertheless, this property does not hold for the control vector fields $g_i^{(j)}$ since $Tp_{j,k}(g_i^{(j)})$ is not equal to $g_i^{(k)}$, for $i = 1, \dots, m$ 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 system transformation range.

**Remark 2.** Given an arbitrary point $\bar{x}_0 \triangleq (x_0, \bar{u}_0)$ around which system (2.19) is defined, it is convenient to consider the shift $\theta : (x, \bar{u}) \in X \times \mathbb{R}_\infty^m \mapsto \theta(x, \bar{u}) = (x - x_0, (\bar{u} - u_0)) \triangleq (z, \bar{v}) \in X \times \mathbb{R}_\infty^m$ such that $\bar{x}_0$ is mapped to the origin of $TX \times \mathbb{R}_\infty^m$, denoted by $\bar{0}$, thus inducing the shift of vector fields:

$$
\theta_{\star}(g_{i}^{(j)})(z, \bar{v}) \triangleq g_{i}^{(j)}(z + x_{0}, \overline{(v + u_{0})}), \quad i = 0, \dots, m, \qquad (3.3)
$$
DIFFERENTIAL FLATNESS BY PURE PROLONGATION

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

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

$$G_0^{(j)} \triangleq \{g_1^{(j_1)}, \dots, g_m^{(j_m)}\}, \quad G_{k+1}^{(j)} \triangleq G_k^{(j)} + \mathrm{ad}_{g_0^{(j)}} G_k^{(j)}, \quad \forall k \ge 0 \quad (3.4)$$

Indeed, for $\mathbf{j} = \mathbf{0} = (0, \dots, 0)$, i.e. $j_1 = \dots = j_m = 0$, this filtration coincides with the $\mathbf{0}$th order one given by (2.22). Similarly to the $\mathbf{0}$th order case, $G_0^{(j)}$ is involutive with constant dimension, equal to $m$, by construction.

Moreover, since every $G_k^{(j)} \subset TX^{(j)}$, with $\dim TX^{(j)} = n + m + |\mathbf{j}|$, we have

**Proposition 1.** There exists a finite integer $k_{\star}^{(j)}$ such that $G_k^{(j)} = G_{k_{\star}^{(j)}}^{(j)}$ for all $k \ge k_{\star}^{(j)}$ and

$$k_{\star}^{(j)} \le n + |\mathbf{j}|. \quad (3.5)$$

*Proof.* Since

$$n + m + |\mathbf{j}| \ge \dim G_{k_{\star}^{(j)}}^{(j)} = \sum_{k=1}^{k_{\star}^{(j)}} \dim G_k^{(j)} / G_{k-1}^{(j)} + \dim G_0^{(j)} \ge k_{\star}^{(j)} + m,$$

we get (3.5).

<figure><img src="image_3.png" /></figure>

**Remark 3.** In full generality, $k_{\star}^{(j)}$ depends on the point where it is evaluated. However, if $\dim G_k^{(j)}$ is constant in an open dense subset of $X^{(j)}$ for all large enough $k$, so is $k_{\star}^{(j)}$.

Let us inductively define the $n$-dimensional vector functions $\gamma_{k,j,i}$, for $k \ge 1$, $i = 1, \dots, m$, and arbitrary $\mathbf{j} = (j_1, \dots, j_m)$ as follows:

$$\gamma_{k+1,j,i} \triangleq L_{g_0^{(j)}} \gamma_{k,j,i} - \gamma_{k,j,i} \frac{\partial f}{\partial x} = L_{g_0^{(0)}} \gamma_{k,j,i} + \sum_{p=1}^{m} \sum_{l=0}^{j_p-1} u_p^{(l+1)} \frac{\partial \gamma_{k,j,i}}{\partial u_p^{(l)}} - \gamma_{k,j,i} \frac{\partial f}{\partial x} \quad (3.6)$$

with

$$\gamma_{1,j,i} = (-1)^{(j_i+1)} \frac{\partial f}{\partial u_i^{(0)}}. \quad (3.7)$$

For an arbitrary $\mathbf{j}$ and given $i = 1, \dots, m$, thanks to (3.7), it is readily seen that $\gamma_{1,j,i}$ depends at most of $\bar{x}^{(0)}$ and thus, if $k \le j_i - 1$, thanks to (3.6), $\gamma_{k+1,j,i}$ depends at most of $\bar{x}^{(j \lor k)}$.
## 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 $\pi_{\mathbf{r}}: X^{(\mathbf{r})} \to \mathbb{R}^{m+|\mathbf{r}|}$, with $\pi_{\mathbf{r}}(\bar{x}^{(\mathbf{r})}) = \bar{u}^{(\mathbf{r})}$, its vertical space at $\bar{x}^{(\mathbf{r})}$, denoted by $V_{\bar{x}^{(\mathbf{r})}} X^{(\mathbf{r})}$, is the tangent space $T_{\bar{x}}X$. Its vertical bundle, denoted by $VX^{(\mathbf{r})}$, is the vector bundle made of the vertical spaces at each $\bar{x}^{(\mathbf{r})}$, i.e. the set of linear combinations $\sum_{i=1}^n \alpha_i(\bar{x}^{(\mathbf{r})}) \frac{\partial}{\partial x_i}$ whose coefficients $\alpha_i$ are smooth functions that depend at most on $\bar{x}^{(\mathbf{r})}$ and where $(x_1, \dots, x_n)$ are local coordinates of $X$.

The same definition indeed holds for the vertical bundle $V(X \times \mathbb{R}_{\infty}^m)$ associated to the fiber bundle $\pi: X \times \mathbb{R}_{\infty}^m \to \mathbb{R}_{\infty}^m$, i.e. the set of linear combinations of $\frac{\partial}{\partial x_1}, \dots, \frac{\partial}{\partial x_n}$ whose coefficients are smooth functions of $\bar{x}$.

We now establish comparison formulae between Lie brackets of the vector fields of the finitely purely prolonged system and those of the original (non prolonged) one.

**Lemma 1** (Comparison formulae). For all $\mathbf{j} = (j_1, \dots, j_m) \in \mathbb{N}^m$ satisfying $0 \le j_1 \le \dots \le j_m$, with $j_m$ finite, for all $k \le j_i$ and $i = 1, \dots, m$, we have:

$$ \mathrm{ad}_{g_0^{(j)}}^k g_i^{(j_i)} = (-1)^k \frac{\partial}{\partial u_i^{(j_i-k)}} = (-1)^k g_i^{(j_i-k)} \quad (3.8) $$

and for all $k \ge 1$:

$$ \mathrm{ad}_{g_0^{(j)}}^{j_i+k} g_i^{(j_i)} = (-1)^{j_i} \mathrm{ad}_{g_0^{(j)}}^k g_i^{(0)} = \gamma_{k,j,i} \frac{\partial}{\partial x} \in VX(j \lor (k-1)), \quad (3.9) $$

Moreover, we have

$$ [g_p^{(j_p-k)}, \mathrm{ad}_{g_0^{(j)}}^{l-j_q} g_q^{(0)}] = 0, \quad \forall k < j_p, \forall l \ge j_q \text{ s.t. } k+l < j_p+j_q+1. \quad (3.10) $$

*Proof.* It is immediately seen that

$$ \mathrm{ad}_{g_0^{(j)}} g_i^{(j_i)} = \left[ f \frac{\partial}{\partial x} + \sum_{k=1}^{m} \sum_{l \ge 0} u_k^{(l+1)} \frac{\partial}{\partial u_k^{(l)}}, \frac{\partial}{\partial u_i^{(j_i)}} \right] = - \frac{\partial}{\partial u_i^{(j_i-1)}} = -g_i^{(j_i-1)}. $$

Iterating this computation up to $k = j_i$ yields (3.8):

$$ \mathrm{ad}_{g_0^{(j)}}^{j_i} g_i^{(j_i)} = (-1)^{j_i} \frac{\partial}{\partial u_i^{(0)}} = (-1)^{j_i} g_i^{(0)}. $$

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 $l \ge 0$, we have:

$$ \mathrm{ad}_{g_0^{(j)}}^{j_i+1} g_i^{(j_i)} = (-1)^{j_i} \left[ f \frac{\partial}{\partial x} + \sum_{k=1}^{m} \sum_{l=0}^{j_k-1} u_k^{(l+1)} \frac{\partial}{\partial u_k^{(l)}}, \frac{\partial}{\partial u_i^{(0)}} \right] = (-1)^{j_i} \mathrm{ad}_{g_0^{(0)}} g_i^{(0)}. $$
DIFFERENTIAL FLATNESS BY PURE PROLONGATION

Moreover, a direct calculation shows that

$$(-1)^{j_i} \mathrm{ad}_{g_0^{(0)}} g_i^{(0)} = (-1)^{j_i} \left[ f \frac{\partial}{\partial x}, \frac{\partial}{\partial u_i^{(0)}} \right] = (-1)^{(j_i+1)} \frac{\partial f}{\partial u_i^{(0)}} (\bar{x}^{(0)}) \frac{\partial}{\partial x}, \quad (3.11)$$

which proves that $\mathrm{ad}_{g_0^{(j)}}^{j_i+1} g_i^{(j_i)} \in \mathrm{VX}^{(0)}$. Thus (3.9) holds at the order $k=1$.

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

$$\mathrm{ad}_{g_0^{(j)}}^{j_i+\nu+1} g_i^{(j)} = (-1)^{j_i} \left[ g_0^{(j)}, \mathrm{ad}_{g_0^{(j)}}^{\nu} g_i^{(0)} \right] = \left( L_{g_0^{(j)}} \gamma_{\nu,j,i} - \gamma_{\nu,j,i} \frac{\partial f}{\partial x} \right) \frac{\partial}{\partial x} = \gamma_{\nu+1,j,i} \frac{\partial}{\partial x},$$

and, according to (3.6)-(3.7), for all $j_p + 1 \le \nu < j_{p+1}$, $p = 1, \dots, m-1$, since we differentiate $\gamma_{\nu,j,i}$ 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 $\gamma_{\nu+1,j,i}$ depends at most on

$$\bar{x}^{(j \vee \nu)} = (x, u_1^{(0)}, \dots, u_1^{(j_1)}, \dots, u_p^{(0)}, \dots, u_p^{(j_p)}, u_{p+1}^{(0)}, \dots, u_{p+1}^{(\nu)}, \dots, u_m^{(0)}, \dots, u_m^{(\nu)})$$

and on $\bar{x}^{(j)} = \bar{x}^{(j \vee \nu)}$ if $\nu \ge j_m$, hence (3.9).

Concerning (3.10), since $\mathrm{ad}_{g_0^{(j)}}^{l-j_q} g_q^{(0)} = \pm \gamma_{l-j_q,j,q} \frac{\partial}{\partial x}$ depends only on $\bar{x}^{(j \vee (l-j_q-1))}$, the derivative of $\gamma_{l-j_q,j,q}$ with respect to $g_p^{(j_p-k)}$ is indeed null if $j_p - k > l - j_q - 1$, hence the result. $\square$

Let us assume, without loss of generality, that the control components have been reordered in such a way that $j_1 \le j_2 \le \dots \le j_m$. Moreover, we may suppose that $j_1 = 0$, as shown to be sufficient in the next section.

We now introduce two new filtrations of $\mathrm{TX}^{(j)}$, noted $\Gamma_k^{(j)}$ and $\Delta_k^{(j)}$, for $k \ge 0$, as follows

$$\Gamma_k^{(j)} \triangleq \bigoplus_{p=2}^{m} \{g_p^{((j_p-1))} \mid l = 0, \dots, k \lor (j_p - 1)\}, \quad (3.12)$$

$$\Delta_k^{(j)} \triangleq \sum_{p=1}^{m} \{ \mathrm{ad}_{g_0^{(j)}}^{l-j_p} g_p^{(0)} \mid l = j_p, \dots, k \}. \quad (3.13)$$

with the convention that $\mathrm{ad}_{g_0^{(j)}}^{k-j_p} g_p^{(0)} = 0$ if $k < j_p$, $p = 1, \dots, m$.

We indeed have $\Gamma_k^{(j)} = \Gamma_{j_{m-1}}^{(j)} = \bigoplus_{p=2}^m \{g_p^{(j_p)}, \dots, g_p^{(1)}\}$ for all $k \ge j_m - 1$. Thus, $\dim \Gamma_k^{(j)} \le |\mathbf{j}|$ for all $k$.

The definitions (3.12)-(3.13) and Lemma 1 readily yield

$$\begin{align}
\overline{\Gamma_k^{(j)}} &= \Gamma_k^{(j)}, \quad [\Gamma_k^{(j)}, \Gamma_l^{(j)}] \subset \Gamma_{k \wedge l}^{(j)}, \tag{3.14} \\
\Delta_l^{(j)} &\subset \mathrm{V}(X \times \mathbb{R}_\infty^m), \quad [\Gamma_k^{(j)}, \Delta_l^{(j)}] \subset \mathrm{V}(X \times \mathbb{R}_\infty^m). \nonumber
\end{align}$$
DIFFERENTIAL FLATNESS BY PURE PROLONGATION

**Remark 4.** Note that, in contrast to the increasing filtration {$G_k^{(j)}$}$_{0\le k\le k_\star^{(j)}}$, the mapping $k \mapsto \dim \Delta_k^{(j)}$ is not non-decreasing in general.

**Remark 5.** In our definition of $\Delta_k^{(j)}$, we consider Lie brackets of the form $\mathrm{ad}_{g_0}^k g_p^{(0)}$, as opposed to the Lie brackets of the form $\mathrm{ad}_{g_0}^k g_p^{(0)}$ used in [1].

**Proposition 2.** For all $\mathbf{j}$ such that $0 = j_1 \le \dots \le j_{m-1} \le j_m$, with $j_m$ finite, if $\mathbf{j} = (j_1, \dots, j_i, j_{i+1}, \dots, j_m)$ and $\mathbf{j}' = (j_1, \dots, j_i, j'_{i+1}, \dots, j'_m)$ for some $(j'_{i+1}, \dots, j'_m)$, we have $\Delta_k^{(j)} = \Delta_k^{(j')}$ for all $k = 0, \dots, (j_{i+1} \lor j'_{i+1}) - 1$. Moreover, for all $k \ge 0$,

$$ \dim \Gamma_k^{(j)} = \begin{cases} \sum_{p=1}^{i} j_p + (k+1)(m-i) & \text{if } j_i \le k < j_{i+1}, i = 1, \dots, m-1 \\ |\mathbf{j}| & \text{if } k \ge j_m \end{cases} \tag{3.15} $$

$$ \dim \Delta_k^{(j)} \le \begin{cases} ((k+1)i - \sum_{p=1}^i j_p) \lor (n+i) & \text{if } j_i \le k < j_{i+1}, i = 1, \dots, m-1 \\ ((k+1)m - |\mathbf{j}|) \lor (n+m) & \text{if } k \ge j_m \end{cases} \tag{3.16} $$

and we have

$$ G_k^{(j)} = \Gamma_k^{(j)} \oplus \Delta_k^{(j)}, \quad \forall k \ge 0. \tag{3.17} $$

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

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

$$ n + |\mathbf{j}| \ge k_{\star}^{(j)} \ge j_m \land \frac{n + |\mathbf{j}|}{m}. \tag{3.18} $$

*Proof.* By definition, the generators of $\Gamma_k^{(j)}$ are independent for all $k$ and thus their number is equal to $\dim \Gamma_k^{(j)}$, hence (3.15). The dimension of $\Delta_k^{(j)}$, in turn, is lesser than, or equal to, the number of its generators, in number $(k+1)+\dots+(k-j_i+1) = (k+1)i - \sum_{p=1}^i j_p$, if $j_i \le k < j_{i+1}$ (respectively $(k+1)+\dots+(k-j_m+1) = (k+1)m - |\mathbf{j}|$, if $k \ge j_m$), and, since, according to (3.9) of Lemma 1, $\Delta_k^{(j)}$ is contained in $\{\frac{\partial}{\partial u_1^{(0)}}, \dots, \frac{\partial}{\partial u_i^{(0)}}, \frac{\partial}{\partial x_1}, \dots, \frac{\partial}{\partial x_n}\}$ if $j_i \le k < j_{i+1}$, $i = 1, \dots, m-1$ (respectively in $\{\frac{\partial}{\partial u_1^{(0)}}, \dots, \frac{\partial}{\partial u_i^{(0)}}, \frac{\partial}{\partial x_1}, \dots, \frac{\partial}{\partial x_n}\}$ if $k \ge j_m$), 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 $G_0^{(j)} = \{g_1^{(0)}, g_2^{(j_2)}, \dots, g_m^{(j_m)}\} = \{g_1^{(0)}\} \oplus \{g_2^{(j_2)}, \dots, g_m^{(j_m)}\} = \Delta_0^{(j)} \oplus \Gamma_0^{(j)}$. Thus, (3.17) is valid at the order 0.
DIFFERENTIAL FLATNESS BY PURE PROLONGATION

Assume now that (3.17) holds true up to the order $\nu > 0$ with $j_r \le \nu < j_{r+1}$ for some $r \in \{1, \dots, m\}$.

Note that 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 $j_r \le \nu+1 < j_{r+1}$ or $\nu+1 = j_{r+1}$. In the first case, using Lemma 1, we get:

$$ 
\begin{aligned}
G_{\nu+1}^{(j)} &= G_{\nu}^{(j)} + ad_{g_0^{(j)}} G_{\nu}^{(j)} = \Gamma_{\nu}^{(j)} \oplus \Delta_{\nu}^{(j)} + ad_{g_0^{(j)}} \Gamma_{\nu}^{(j)} + ad_{g_0^{(j)}} \Delta_{\nu}^{(j)} \\
&= \Gamma_{\nu}^{(j)} \oplus \Delta_{\nu}^{(j)} + \left\{ ad_{g_0^{(j)}}^{\nu+1} g_1^{(0)}, \dots, ad_{g_0^{(j)}}^{\nu+1-j_r} g_r^{(0)}, g_{r+1}^{(j_{r+1}-\nu-1)}, \dots, g_m^{(j_m-\nu-1)} \right\} \\
&= \Gamma_{\nu+1}^{(j)} \oplus \Delta_{\nu+1}^{(j)}.
\end{aligned}
$$

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

$$ 
\begin{aligned}
G_{\nu+1}^{(j)} &= \Gamma_{\nu}^{(j)} \oplus \Delta_{\nu}^{(j)} + \left\{ ad_{g_0^{(j)}}^{\nu+1} g_1^{(0)}, \dots, ad_{g_0^{(j)}}^{\nu+1-j_r} g_r^{(0)}, g_{r+1}^{(0)}, g_{r+2}^{(j_{r+2}-\nu-1)}, \dots, g_m^{(j_m-\nu-1)} \right\} \\
&= \Gamma_{\nu+1}^{(j)} \oplus \Delta_{\nu+1}^{(j)}.
\end{aligned}
$$

The case $j_m \le \nu + 1$ follows the same lines:

$$ G_{\nu+1}^{(j)} = \Gamma_{\nu}^{(j)} \oplus \Delta_{\nu}^{(j)} + \left\{ ad_{g_0^{(j)}}^{\nu+1} g_1^{(0)}, \dots, ad_{g_0^{(j)}}^{\nu+1-j_m} g_m^{(0)} \right\} = \Gamma_{\nu+1}^{(j)} \oplus \Delta_{\nu+1}^{(j)} $$

hence (3.17) is proven and the property of the number of iterations $k_*^{(j)}$ to simultaneously saturate the dimensions of $\Gamma_k^{(j)}$ and $\Delta_k^{(j)}$ immediately follows.

Moreover, if $\dim \Delta_{k_*}^{(j)} = m + n$, we must have $j_m \le k_*^{(j)}$ since otherwise, using definition (3.13) for $k_*^{(j)} < j_m$, $\Delta_{k_*}^{(j)}$ 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 $\dim \Delta_{k_*}^{(j)} = m + n$, we get $(k_*^{(j)} + 1)m - |\mathbf{j}| \ge m + n$, which, combined with (3.5), immediately yields (3.18). $\square$

**Remark 6.** The inequality (3.18) reads $k_*^{(j)} - \frac{|\mathbf{j}|}{m} \ge \frac{n}{m}$ and may thus be interpreted as an estimate of the gap between $k_*^{(j)}$ and the average value $\frac{|\mathbf{j}|}{m}$ of the prolongation lengths $j_1, j_2, \dots, j_m$, provided that the prolonged system satisfies the strong accessibility rank condition $\dim \Delta_{k_*}^{(j)} = m + n$.

# 4 Flatness by Pure Prolongation

## 4.1 Necessary and Sufficient Conditions

**Definition 5.** We say that a system is flat by pure prolongation at a point $\bar{x}_0 \in X \times \mathbb{R}_\infty^m$ if, and only if, there exists a pure prolongation of finite order $\mathbf{j}$ such that the prolonged system is feedback linearizable at $\bar{x}_0$.
DIFFERENTIAL FLATNESS BY PURE PROLONGATION

The feedback linearizability of the prolonged system of order **j** indeed implies that this prolonged system is flat and, consequently, the original system too, which justifies qualifying this property as flatness by pure prolongation. The corresponding class of systems is thus clearly a strict subclass of differentially flat systems.

The following Proposition is a straightforward adaptation of Theorem 2 for an arbitrary order **j**. Note that, at this stage, nothing is said about a possible choice of **j**, a question that will be dealt with in subsection 4.2, theorem 4.

**Proposition 3.** *The prolonged system of order j is feedback linearizable at $\bar{0}$ if, and only if, $G_k^{(j)}$ is involutive with locally constant dimension for all k and such that $G_{k_*}^{(j)} = TX^{(j)}$.*

Again, flat outputs can be computed via Frobenius theorem, once established the list of Brunovský's controllability indices of order **j**, as follows:

**Definition 6.** Consider the sequence of integers

$$ \rho_k^{(j)} \triangleq \dim G_k^{(j)} / G_{k-1}^{(j)} \quad \forall k \ge 1, \quad \rho_0^{(j)} \triangleq \dim G_0^{(j)} = m. $$

The Brunovský controllability indices of order **j** are defined by

$$ \kappa_k^{(j)} \triangleq \#\{l \mid \rho_l^{(j)} \ge k\}, \quad k = 1, \dots, m, $$

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

* $\rho_k^{(j)}$'s and $\kappa_k^{(j)}$'s are non increasing sequences,

* $\rho_k^{(j)} \le m$ for all $k$ and $\rho_k^{(j)} = 0$ for all $k \ge k_*^{(j)} + 1$,

* $\kappa_1^{(j)} = k_*^{(j)} + 1$, $\kappa_m^{(j)} \ge 1$,

* $\sum_{k=0}^{k_*^{(j)}} \rho_k^{(j)} = \sum_{k=1}^{m} \kappa_k^{(j)} = \dim G_{k_*^{(j)}}^{(j)} = n + m + |\mathbf{j}|$.

The list $\kappa_1^{(j)}, \dots, \kappa_m^{(j)}$ 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 (2.18) with $r_i = \kappa_i^{(j)}$, $i = 1, \dots, m$.

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

$$ G_k^{(j)} = \bigoplus_{p=1}^{m} \left\{ \mathrm{ad}_{g_0^{(j)}}^{l} g_p^{(j)} \mid l = 0, \dots, k \lor (\kappa_p^{(j)} - 1) \right\}, \quad G_{\kappa_1^{(j)}-1}^{(j)} = G_{k_*^{(j)}}^{(j)} = TX^{(j)} $$
DIFFERENTIAL FLATNESS BY PURE PROLONGATION

and flat outputs $(y_1, \dots, y_m)$ are locally non trivial solutions of the system of PDE's

$$ \langle G_k^{(j)}, dy_i \rangle = 0, \ k = 0, \dots, \kappa_i^{(j)} - 2, \ \text{with} \ \langle G_{\kappa_i^{(j)} - 1}^{(j)}, dy_i \rangle \neq 0, \ i = 1, \dots, m. \tag{4.1} $$

Finally, the mapping

$$ \bar{x}^{(j)} \mapsto (y_1, \dots, y_1^{(\kappa_1^{(j)} - 1)}, \dots, y_m, \dots, y_m^{(\kappa_m^{(j)} - 1)}) $$

is a local diffeomorphism.

The next Lemma extends a well-known result (see e.g. [5, 27]) to the context of jets of infinite order (see also [10]).

**Lemma 2.** We consider system (3.2), denoted by $\Sigma_j$, with $j = (j_1, \dots, j_m)$, assuming, without loss of generality, that $0 \le j_1 \le \dots \le j_m$, possibly up to input renumbering. We denote by $j' = j - j_1 = (0, j_2 - j_1, \dots, j_m - j_1)$, and by $\Sigma_{j'}$ the corresponding system. Then $\Sigma_j$ is flat at a given point $(x_0, \bar{u}_0)$ if, and only if, $\Sigma_{j'}$ is also flat at this point. Moreover, every flat output of $\Sigma_j$ at $(x_0, \bar{u}_0)$ is a flat output of $\Sigma_{j'}$ at the same point, and conversely.

*Proof.* It is immediately seen that $\Sigma_j$ and $\Sigma_{j'}$ are Lie-Bäcklund equivalent. Hence, if $\Sigma_j$ is Lie-Bäcklund equivalent to the trivial system, the same property holds for $\Sigma_{j'}$ and conversely. Moreover, $\Sigma_j$'s flatness implies the existence of $y$ such that $(x, \bar{u}^{(j-1)}, \bar{v}) = (x, \bar{u}) = \Phi(\bar{y})$, with $v = u^{(j)}$ and $\bar{y} = \Psi(x, \bar{u}) = \Psi(x, \bar{u}^{(j-1)}, \bar{v})$, and thus, setting $w = u^{(j')}$, we immediately get that $(x, \bar{u}^{(j'-1)}, \bar{w}) = \Phi(\bar{y})$, with $\bar{y} = \Psi(x, \bar{u}^{(j'-1)}, \bar{w})$. The converse is obvious and left to the reader. $\square$

In virtue of Lemma 2, it suffices to restrict our analysis to prolongations of order $j = (j_1, \dots, j_m)$ such that $0 = j_1 \le \dots \le j_m$.

**Theorem 3.** A necessary and sufficient condition for flatness by pure prolongation at $\bar{0}$ is that there exists $j = (j_1, \dots, j_m) \in \mathbb{N}^m$, $0 = j_1 \le \dots \le j_m < +\infty$, such that

(i) $\overline{\Delta_k^{(j)}} = \Delta_k^{(j)}$ with $\dim \Delta_k^{(j)}$ locally constant for all $k \ge 0$ (involutivity with locally constant dimension of all $\Delta_k^{(j)}$'s),

(ii) $[\Gamma_k^{(j)}, \Delta_k^{(j)}] \subset \Delta_k^{(j)}$ for all $k \ge 0$ (invariance of $\Delta_k^{(j)}$ by $\Gamma_k^{(j)}$),

(iii) The integer $k_*^{(j)}$ is such that $\Delta_k^{(j)} = \mathrm{TX} \times \mathrm{TR}^m$ and $\Gamma_k^{(j)} = \mathrm{TR}^{|j|}$ for all $k \ge k_*^{(j)}$ (strong controllability).

*Proof.* By (3.17) of Proposition 2, we have $G_k^{(j)} = \Gamma_k^{(j)} \oplus \Delta_k^{(j)}$ for all $k \ge 0$. Then, $G_k^{(j)} = \overline{G_k^{(j)}}$ implies that $[\Gamma_k^{(j)} \oplus \Delta_k^{(j)}, \Gamma_k^{(j)} \oplus \Delta_k^{(j)}] \subset \Gamma_k^{(j)} \oplus \Delta_k^{(j)}$. Since $\Gamma_k^{(j)} = \overline{\Gamma_k^{(j)}}$ for all $k$, and since $[\Gamma_k^{(j)}, \Delta_k^{(j)}] \cap \Gamma_k^{(j)} = \{0\}$ by Lemma 1, we deduce that $[\Gamma_k^{(j)}, \Delta_k^{(j)}] =$
$[\Delta_k^{(j)}, \Delta_k^{(j)}] \subset \Delta_k^{(j)}$, hence $[\Gamma_k^{(j)}, \Delta_k^{(j)}] \subset \Delta_k^{(j)}$ and $[\Delta_k^{(j)}, \Delta_k^{(j)}] \subset \Delta_k^{(j)}$, i.e. $\Delta_k^{(j)} = \overline{\Delta_k^{(j)}}$, for all $k \ge 0$.

Conversely, $[\Gamma_k^{(j)}, \Delta_k^{(j)}] \subset \Delta_k^{(j)}$ and $\Delta_k^{(j)} = \overline{\Delta_k^{(j)}}$ for all $k \ge 0$ trivially implies that $G_k^{(j)} = \overline{G_k^{(j)}}$ for all $k \ge 0$.

Moreover, since dim $\Gamma_k^{(j)}$ is constant by construction, the fact that $G_k^{(j)}$ has locally constant dimension is equivalent to the fact that $\Delta_k^{(j)}$ has locally constant dimension too for all $k \ge 0$, hence (i).

Finally, (iii) is an immediate consequence of the condition that $G_k^{(j)} = TX \times \mathbb{R}^{m+|\mathbf{l}|}$ for all $k \ge k_*(^j)$, and the theorem is proven. $\square$

## 4.2 The Pure Prolongation Algorithm

Let us now assume that conditions (i)-(iii) of theorem 3 are satisfied for some $\mathbf{j} = (j_1, \dots, j_{i-1})$, $0 = j_1 \le \dots \le j_{i-1}$, for some $i = 2, \dots, m$. We denote by $(\mathbf{j}, \mathbf{l}) = (j_1, \dots, j_{i-1}, l_i, \dots, l_m)$ for some $\mathbf{l} \triangleq (l_i, \dots, l_m) \in \mathbb{N}^{m-i+1}$, with $j_{i-1} \le l_i \le \dots \le l_m$.

For $l_i - 1 \ge k \ge j_{i-1}$, (3.12) and (3.13) read:

$$ 
\begin{aligned}
\Gamma_k^{(\mathbf{j},\mathbf{l})} &= \{g_2^{(j_2)}, \dots, g_2^{(1)}, \dots, g_{i-1}^{(j_{i-1})}, \dots, g_{i-1}^{(1)}, g_i^{(l_i)}, \dots, g_i^{(l_i-k)}, \dots, g_m^{(l_m)}, \dots, g_m^{(l_m-k)}\} \\
\Delta_k^{(\mathbf{j},\mathbf{l})} &= \{g_1^{(0)}, \dots, \mathrm{ad}_{g_0^{(j,1)}}^k g_1^{(0)}, \dots, g_{i-1}^{(0)}, \dots, \mathrm{ad}_{g_0^{(j,1)}}^{k-j_{i-1}} g_{i-1}^{(0)}\}
\end{aligned}
\quad (4.2) $$

and for $l_{i+1} - 1 \ge k \ge l_i \ge j_{i-1}$:

$$ 
\begin{aligned}
\Gamma_k^{(\mathbf{j},\mathbf{l})} &= \{g_2^{(j_2)}, \dots, g_2^{(1)}, \dots, g_i^{(l_i)}, \dots, g_i^{(1)}, g_{i+1}^{(l_{i+1})}, \dots, g_{i+1}^{(l_{i+1}-k)}, \dots, g_m^{(l_m)}, \dots, g_m^{(l_m-k)}\} \\
\Delta_k^{(\mathbf{j},\mathbf{l})} &= \{g_1^{(0)}, \dots, \mathrm{ad}_{g_0^{(j,1)}}^k g_1^{(0)}, \dots, g_{i-1}^{(0)}, \dots, \mathrm{ad}_{g_0^{(j,1)}}^{k-j_{i-1}} g_{i-1}^{(0)}, g_i^{(0)}, \dots, \mathrm{ad}_{g_0^{(j,1)}}^{k-l_i} g_i^{(0)}\}
\end{aligned}
\quad (4.3) $$

Note that in (4.2), according to (3.9), $\Delta_k^{(\mathbf{j},\mathbf{l})}$ is contained in $\mathrm{VX}^{(k-1)}$ and thus does not depend on $\mathbf{l}$ at all, and in (4.3), if $k \ge l_i \ge j_{i-1}$, $\Delta_k^{(\mathbf{j},\mathbf{l})}$, depends at most on $l_i$ only.

We deduce that $\Delta_k^{(\mathbf{j},\mathbf{l})}$ must be involutive for all $k \ge j_{i-1}$ and all $\mathbf{l}$ such that $l_i - 1 \ge k$, therefore

**Proposition 4.** If conditions (i)-(iii) of theorem 3 are satisfied for some $\mathbf{j} = (j_1, \dots, j_{i-1})$, $0 = j_1 \le \dots \le j_{i-1}$, and some $i = 2, \dots, m$, then, if there exists an $l_i > j_{i-1}$ such that $\Delta_k^{(\mathbf{j},\mathbf{l})}$ is not involutive for at least one $k$ such that $l_i - 1 \ge k \ge j_{i-1}$ and such that dim $\Delta_k^{(\mathbf{j},\mathbf{l})} < n+m$, we cannot conclude on the system flatness by pure prolongation with non prolonged input $u_1^{(0)}$. Moreover, if this property holds for every choice of non prolonged input, then the system is not flat by pure prolongation.

In the case $k + 1 \ge l_i \ge j_{i-1}$, let us introduce the number

$$ \sigma_{\Delta}(k) \triangleq \min\{l_i \mid k + 1 \ge l_i \ge j_{i-1}, \overline{\Delta_k^{(\mathbf{j},\mathbf{l})}} = \Delta_k^{(\mathbf{j},\mathbf{l})}\}. \quad (4.4) $$
DIFFERENTIAL FLATNESS BY PURE PROLONGATION

We indeed have

$$j_{i-1} \le \sigma_{\Delta}(k) \le k+1 \quad \text{for } k \ge j_{i-1} \qquad (4.5)$$

and $\sigma_{\Delta}(k)$ is independent of $(l_{i+1}, \dots, l_m)$.

Note also, again since $\Delta_k^{(j,1)}$ is contained in $VX^{(k-1)}$, that $[g_p^{(l_p-r)}, \Delta_k^{(j,1)}] = 0$ for all $r = 0, \dots, k$ and $l$ such that $l_i - k > k - 1$ with $k \ge j_{i-1} \wedge 2$. Thus, by assumption (ii) of theorem 3 for $j = (j_1, \dots, j_{i-1})$, $0 = j_1 \le \dots \le j_{i-1}$, we get that $[\Gamma_k^{(j,1)}, \Delta_k^{(j,1)}] \subset \Delta_k^{(j,1)}$ for all $l_i > 2k - 1$. Thus, introducing the number

$$\sigma_{\Gamma, \Delta}(k) \triangleq \min\{l_i \mid l_i - 1 \ge j_{i-1}, [\Gamma_k^{(j,1)}, \Delta_k^{(j,1)}] \subset \Delta_k^{(j,1)}\} \quad \text{for } k \ge j_{i-1}, \qquad (4.6)$$

we have

$$j_{i-1} \le \sigma_{\Gamma, \Delta}(k) \le 2k - 2 \quad \text{for all } k \ge j_{i-1} \qquad (4.7)$$

and $\sigma_{\Gamma, \Delta}(k)$ is independent of $(l_{i+1}, \dots, l_m)$.

We thus have proven the following alternative

**Theorem 4.** Assume that conditions (i)-(iii) of theorem 3 are satisfied for some $j = (j_1, \dots, j_{i-1})$, $0 = j_1 \le \dots \le j_{i-1}$, and some $i = 2, \dots, m$. Then

1. if, for every choice of non prolonged input $u_1^{(0)}$, there exists a $k \ge j_{i-1}$ such that $\Delta_k^{(j,1)}$ is not involutive for all $l_i > j_{i-1}$ or if $\max_k \dim \Delta_k^{(j,1)} < n+m$ for all $l$, then the system is not flat by pure prolongation;

2. otherwise, $j_i$ is given by

$$j_i = \max_{k \ge j_{i-1}} (\sigma_{\Gamma, \Delta}(k) \wedge \sigma_{\Delta}(k)) \qquad (4.8)$$

and $j_i \le k_{\star}^{(j)}$ for all $i = 1, \dots, m$, i.e. every $j_i$ is such that $\dim \Delta_{j_i}^{(j,1)} < n+m$.

*Proof.* In view of what precedes, it only remains to be proven that $j_i$ is such that $\dim \Delta_{j_i}^{(j,1)} < n+m$, which indeed implies that $j_i \le k_{\star}^{(j)}$. By (4.8), we clearly have that $[\Gamma_k^{(j,1)}, \Delta_k^{(j,1)}] \subset \Delta_k^{(j,1)}$ and $\overline{\Delta_k^{(j,1)}} = \Delta_k^{(j,1)}$ for all $k \ge j_{i-1}$. Moreover, by (iii) and (3.18), we have $j_i \le j_m \le k_{\star}^{(j)}$, hence the boundedness of every $j_i$, $i = 1, \dots, m$. $\square$

We thus immediately deduce the following

**Algorithm 1. flatness by pure prolongation**

**Input:** the vector fields $g_0^{(0)}, g_1^{(0)}, \dots, g_m^{(0)}$ (see (2.20))

**output:** the minimal lengths $0 = j_1 \le \dots \le j_m$ and $k_{\star}^{(j)}$ or FAIL if the system is not flat by pure prolongation.

**Procedure: Initialization.** Choose $l = (l_1, \dots, l_m)$, associated to the prolongations $u_i^{(l_i)}$, with $l_1 = 0$ and $l_i \ge 0$, $i = 2, \dots, m$ arbitrary, and compute $\Gamma_0^{(l)}$ and $\Delta_0^{(l)}$ (at this step, $\Delta_0^{(l)}$ depends only on the choice of $u_1^{(0)}$). Note that for every choice of $l_1$, we have $\overline{\Delta_0^{(l)}} = \Delta_0^{(l)}$ and $[\Gamma_0^{(j,1)}, \Delta_0^{(l)}] \subset \Delta_0^{(l)}$.
**Step 1.** For $k \ge 1$, compute $\Delta_k^{(1)}$ for all $1 \le l_2 \le k+1$, $\sigma_\Delta(k)$ and $\sigma_{\Gamma,\Delta}(k)$. Continue up to the first $k_1$ such that $\sigma_{\Gamma,\Delta}(k_1) \wedge \sigma_\Delta(k_1)$ is maximal. Then set $j_2 = \sigma_{\Gamma,\Delta}(k_1) \wedge \sigma_\Delta(k_1)$. If $\Delta_k^{(1)}$ is non involutive for some $k \le k_1$, permute the inputs to change $u_1^{(0)}$ and restart.

**Step i, 1 $\le i \le m$.** Same as step 1 with $k \ge j_{i-1}$ to obtain $k_{i-1}$ and $j_i$. If $\Delta_k^{(1)}$ is non involutive for some $k \le k_{i-1}$, permute the inputs to change $u_1^{(0)}$ and restart. If $\Delta_k^{(1)}$ is non involutive for all such permutations, then FAIL.

**Step i $\ge m$.** Determine $k_*^{(j)}$. If $\dim \Delta_{k_*}^{(j)} = n+m$, STOP. Otherwise, if for every permutation of non prolonged input, there is an $i = 2, \dots, m$, a $k \ge j_{i-1}$ and if for all $l_i \ge k+1$, $\Delta_k^{(1)}$ is non involutive, or if $\dim \Delta_k^{(j)} < n+m$ for all $k$, then FAIL.

# 5 Examples

All the examples of this paper concern two input systems, i.e. $m = 2$, with prolongation index $j = (j_1, j_2)$. Such prolongations are supposed to be such that $0 = j_1 \le j_2$, up to a suitable input permutation. However, for the sake of readability, we will keep the original input numbering unchanged and thus consider that $j = (0, j_2)$ or $(j_1, 0)$ depending on the context. At the exception of this modification, we strictly apply algorithm 1 in all the examples.

The first example gives a detailed presentation of the application of algorithm 1, in particular the role played by $\sigma_{\Gamma,\Delta}(k)$ and $\sigma_\Delta(k)$. The second one shows the importance of the number $\sigma_\Delta(k)$ alone to determine the prolongation length, and the third one, borrowed from [6], and taken up again in [1, Section 3.1], is reported here to compare our approach with the one of [1]. Finally, the pendulum example is presented to show that non flat systems by pure prolongation can be detected in a finite number of steps.

## 5.1 Chained System [20]

$$ 
\begin{aligned}
x_1^{(3)} &= u_1 \\
\ddot{x}_2 &= u_2 \\
\dot{x}_3 &= u_1 u_2
\end{aligned} 
\qquad (5.1) 
$$

This system has been proven to be flat in [20, section 3.1.1] with the following flat output

$$ y_1 = x_3 - \ddot{x}_1 u_2 + \dot{x}_1 \dot{u}_2 - x_1 \ddot{u}_2, \quad y_2 = x_2. \qquad (5.2) $$
### 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 $X \triangleq (x_{1,1}, x_{1,2}, x_{1,3}, x_{2,1}, x_{2,2}, x_3, u_1^{(0)}, u_2^{(0)})$ ($n = 6$ 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

$$ 
\begin{aligned}
g_0^{(0)} &\triangleq x_{1,2} \frac{\partial}{\partial x_{1,1}} + x_{1,3} \frac{\partial}{\partial x_{1,2}} + u_1^{(0)} \frac{\partial}{\partial x_{1,3}} + x_{2,2} \frac{\partial}{\partial x_{2,1}} + u_2^{(0)} \frac{\partial}{\partial x_{2,2}} + u_1^{(0)} u_2^{(0)} \frac{\partial}{\partial x_3} \\
g_1^{(0)} &\triangleq \frac{\partial}{\partial u_1^{(0)}}, \quad g_2^{(0)} \triangleq \frac{\partial}{\partial u_2^{(0)}}
\end{aligned} 
\quad (5.3) $$

One can verify that

$$ \mathrm{ad}_{g_0^{(0)}} g_1^{(0)} = -\frac{\partial}{\partial x_{1,3}} - u_2^{(0)} \frac{\partial}{\partial x_3}, \quad \mathrm{ad}_{g_0^{(0)}}^2 g_1^{(0)} = \frac{\partial}{\partial x_{1,2}}, \quad \mathrm{ad}_{g_0^{(0)}}^3 g_1^{(0)} = -\frac{\partial}{\partial x_{1,1}}, \quad \mathrm{ad}_{g_0^{(0)}}^4 g_1^{(0)} = 0 $$

and

$$ \mathrm{ad}_{g_0^{(0)}} g_2^{(0)} = -\frac{\partial}{\partial x_{2,2}} - u_1^{(0)} \frac{\partial}{\partial x_3}, \quad \mathrm{ad}_{g_0^{(0)}}^2 g_2^{(0)} = \frac{\partial}{\partial x_{2,1}}, \quad \mathrm{ad}_{g_0^{(0)}}^3 g_2^{(0)} = 0. $$

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., $[g_2^{(0)}, \mathrm{ad}_{g_0^{(0)}} g_1^{(0)}] = -\frac{\partial}{\partial x_3} \notin G_1^{(0)}$, 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 $k \ge 3$. Moreover, $\dim G_3^{(0)} = 7 < \dim \overline{G_3^{(0)}} = n + m = 8$. We conclude that the system is not feedback linearizable.

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

Let us now apply theorem 3 and algorithm 1 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 $j_1 \ge 1$. Recall that we have set $g_0^{(l_1,0)} = g_0^{(0)} + \sum_{p=0}^{l_1-1} u_1^{(p+1)} \frac{\partial}{\partial u_1^{(p)}}$.
DIFFERENTIAL FLATNESS BY PURE PROLONGATION

• **k = 0.** Setting $l_1 \ge 1$, we have $\Gamma_0^{(l_1,0)} = \{g_1^{(l_1)}\} = \left\{\frac{\partial}{\partial u_1^{(l_1)}}\right\}$, $\Delta_0^{(l_1,0)} = \{g_2^{(0)}\} = \left\{\frac{\partial}{\partial u_2^{(0)}}\right\} = \overline{\Delta_0^{(l_1,0)}}$ and $[\Gamma_0^{(l_1,0)}, \Delta_0^{(l_1,0)}] \subset \Delta_0^{(l_1,0)}$ for all $l_1 \ge 1$.

• **k = 1.** If $l_1 \ge 2$,

$$ \Gamma_1^{(l_1,0)} = \{g_1^{(l_1)}, g_1^{(l_1-1)}\} = \left\{ \frac{\partial}{\partial u_1^{(l_1)}}, \frac{\partial}{\partial u_1^{(l_1-1)}} \right\} $$

and

$$ \Delta_1^{(l_1,0)} = \{g_2^{(0)}, \mathrm{ad}_{g_0^{(l_1,0)}} g_2^{(0)}\} = \left\{ \frac{\partial}{\partial u_2^{(0)}}, \frac{\partial}{\partial x_{2,2}} + u_1^{(0)} \frac{\partial}{\partial x_3} \right\} = \overline{\Delta_1^{(l_1,0)}}. $$

Moreover, it is readily seen that $[\Gamma_1^{(l_1,0)}, \Delta_1^{(l_1,0)}] = \{0\} \subset \Delta_1^{(l_1,0)}$.

Now, for $l_1 = 1$, we have $\Gamma_1^{(1,0)} = \{g_1^{(1)}\} = \left\{\frac{\partial}{\partial u_1^{(1)}}\right\}$ and $\Delta_1^{(1,0)} = \{g_1^{(0)}, g_2^{(0)}, \mathrm{ad}_{g_0^{(1,0)}} g_2^{(0)}\} = \left\{\frac{\partial}{\partial u_1^{(0)}}, \frac{\partial}{\partial u_2^{(0)}}, \frac{\partial}{\partial x_{2,2}} + u_1^{(0)} \frac{\partial}{\partial x_3}\right\}$ which is not involutive.

Thus $\sigma_{\Delta}(1) = 2$ and $\sigma_{\Gamma,\Delta}(1) = 0$ which implies that $j_1 = \max_{k \ge 0} \sigma_{\Delta}(k) \wedge \sigma_{\Gamma,\Delta}(k) \ge \sigma_{\Delta}(1) \wedge \sigma_{\Gamma,\Delta}(1) = 2$.

• **k = 2.** If $l_1 \ge 3$, we have

$$ \Gamma_2^{(l_1,0)} = \{g_1^{(l_1)}, g_1^{(l_1-1)}, g_1^{(l_1-2)}\} = \left\{ \frac{\partial}{\partial u_1^{(l_1)}}, \frac{\partial}{\partial u_1^{(l_1-1)}}, \frac{\partial}{\partial u_1^{(l_1-2)}} \right\} $$

and

$$ \begin{aligned} \Delta_2^{(l_1,0)} &= \{g_2^{(0)}, \mathrm{ad}_{g_0^{(l_1,0)}} g_2^{(0)}, \mathrm{ad}_{g_0^{(l_1,0)}}^2 g_2^{(0)}\} \\ &= \left\{ \frac{\partial}{\partial u_2^{(0)}}, \frac{\partial}{\partial x_{2,2}} + u_1^{(0)} \frac{\partial}{\partial x_3}, \frac{\partial}{\partial x_{2,1}} - u_1^{(1)} \frac{\partial}{\partial x_3} \right\} = \overline{\Delta_2^{(l_1,0)}}. \end{aligned} $$

Moreover, it is readily verified that $[\Gamma_2^{(l_1,0)}, \Delta_2^{(l_1,0)}] \subset \Delta_2^{(l_1,0)}$ only if $l_1 \ge 4$, condition (ii) of theorem 3 being violated if $l_1 = 3$ and we have $\sigma_{\Delta}(2) = 0$ and $\sigma_{\Gamma,\Delta}(2) = 4$ which implies that $j_1 = \max_{k \ge 0} \sigma_{\Delta}(k) \wedge \sigma_{\Gamma,\Delta}(k) \ge \max_{r=1,2} \sigma_{\Delta}(r) \wedge \sigma_{\Gamma,\Delta}(r) \ge 4$.

• **k = 3.** Again, if $l_1 \ge 4$, we have:

$$ \Gamma_3^{(l_1,0)} = \{g_1^{(l_1)}, g_1^{(l_1-1)}, g_1^{(l_1-2)}, g_1^{(l_1-3)}\} = \left\{ \frac{\partial}{\partial u_1^{(l_1)}}, \frac{\partial}{\partial u_1^{(l_1-1)}}, \frac{\partial}{\partial u_1^{(l_1-2)}}, \frac{\partial}{\partial u_1^{(l_1-3)}} \right\} $$
and

$$
\begin{align*}
\Delta_3^{(l_1, 0)} &= \left\{ g_2^{(0)}, \operatorname{ad}_{g_0^{(l_1, 0)}} g_2^{(0)}, \operatorname{ad}_{g_0^{(l_1, 0)}}^2 g_2^{(0)}, \operatorname{ad}_{g_0^{(l_1, 0)}}^3 g_2^{(0)} \right\} \\
&= \left\{ \frac{\partial}{\partial u_2^{(0)}}, \frac{\partial}{\partial x_{2,2}} + u_1^{(0)} \frac{\partial}{\partial x_3}, \frac{\partial}{\partial x_{2,1}} - u_1^{(1)} \frac{\partial}{\partial x_3}, u_1^{(2)} \frac{\partial}{\partial x_3} \right\} \\
&= \left\{ \frac{\partial}{\partial u_2^{(0)}}, \frac{\partial}{\partial x_{2,2}}, \frac{\partial}{\partial x_{2,1}}, \frac{\partial}{\partial x_3} \right\} = \overline{\Delta_3^{(l_1, 0)}}
\end{align*}
$$

provided that $u_1^{(2)} \neq 0$. We also indeed have $[\Gamma_3^{(l_1,0)}, \Delta_3^{(l_1,0)}] \subset \Delta_3^{(l_1,0)}$ for $l_1 \ge 4$
hence $\sigma_{\Gamma,\Delta}(3) \le 4$ and $j_1 \ge \max_{r=1,2,3} \sigma_{\Delta}(r) \wedge \sigma_{\Gamma,\Delta}(r) \ge 4$.

• **k = 4** If $l_1 \ge 5$, we have $\Gamma_4^{(l_1,0)} = \left\{ \frac{\partial}{\partial u_1^{(l_1)}}, \dots, \frac{\partial}{\partial u_1^{(l_1-4)}} \right\}$ and $\Delta_4^{(l_1,0)} = \Delta_3^{(l_1,0)}$ since
$\mathrm{ad}_{g_0^{(l_1,0)}}^4 g_2^{(0)} = -u_1^{(3)} \frac{\partial}{\partial x_3} \in \Delta_3^{(l_1,0)}$.

If now $l_1 = 4$, $\Gamma_4^{(4,0)} = \left\{ \frac{\partial}{\partial u_1^{(4)}}, \dots, \frac{\partial}{\partial u_1^{(1)}} \right\}$ and

$$
\Delta_4^{(4,0)} = \left\{ g_1^{(0)}, g_2^{(0)}, \mathrm{ad}_{g_0^{(4,0)}} g_2^{(0)}, \dots, \mathrm{ad}_{g_0^{(4,0)}}^4 g_2^{(0)} \right\} = \left\{ \frac{\partial}{\partial u_1^{(0)}}, \frac{\partial}{\partial u_2^{(0)}}, \frac{\partial}{\partial x_{2,2}}, \frac{\partial}{\partial x_{2,1}}, \frac{\partial}{\partial x_3} \right\} = \overline{\Delta_4^{(4,0)}}.
$$

We thus immediately get $[\Gamma_4^{(l_1,0)}, \Delta_4^{(l_1,0)}] \subset \Delta_4^{(l_1,0)}$ for all $l_1 \ge 4$.

• **k ≥ 5** Finally, the reader may easily check that $\Gamma_k^{(4,0)} = \Gamma_4^{(4,0)}$ for all $k \ge 5$ and

$$
\Delta_5^{(4,0)} = \Delta_4^{(4,0)} + \left\{ \mathrm{ad}_{g_0^{(4,0)}} g_1^{(0)} \right\} = \left\{ \frac{\partial}{\partial u_1^{(0)}}, \frac{\partial}{\partial x_{1,3}}, \frac{\partial}{\partial u_2^{(0)}}, \frac{\partial}{\partial x_{2,2}}, \frac{\partial}{\partial x_{2,1}}, \frac{\partial}{\partial x_3} \right\} = \overline{\Delta_5^{(4,0)}}
$$

$$
\Delta_6^{(4,0)} = \Delta_5^{(4,0)} + \left\{ \mathrm{ad}_{g_0^{(4,0)}}^2 g_1^{(0)} \right\} = \left\{ \frac{\partial}{\partial u_1^{(0)}}, \frac{\partial}{\partial x_{1,3}}, \frac{\partial}{\partial x_{1,2}}, \frac{\partial}{\partial u_2^{(0)}}, \frac{\partial}{\partial x_{2,2}}, \frac{\partial}{\partial x_{2,1}}, \frac{\partial}{\partial x_3} \right\} = \overline{\Delta_5^{(4,0)}}
$$

$$
\Delta_7^{(4,0)} = \Delta_6^{(4,0)} + \left\{ \mathrm{ad}_{g_0^{(4,0)}}^3 g_1^{(0)} \right\} = \mathrm{TR}^8.
$$

We also indeed have $[\Gamma_k^{(4,0)}, \Delta_k^{(4,0)}] \subset \Delta_k^{(4,0)}$ for all $k \ge 5$.

Hence, for all $k \ge 0$, 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 $k_*^{(4,0)} = 7 \ge \frac{n+j_1}{2} \wedge j_1 = 5$ and $k_*^{(4,0)} = 7 \le n + j_1 = 10$, but they are not tight.

Accordingly, evaluating the bound on the number of integrators needed to linearize the system, proposed by [28] for $m = 2$, we find $2n - 3 = 9$, and the one proposed by [10] gives $2n - \frac{1}{6}(8 + 24 - 14) = 2n - 3 = 9$ as well.
### 5.1.3 Verification.

The prolonged system is now expressed in the state coordinates

$$ \bar{x}^{(4,0)} \triangleq (x_{1,1}, x_{1,2}, x_{1,3}, x_{2,1}, x_{2,2}, x_3, u_1^{(0)}, u_1^{(1)}, u_1^{(2)}, u_1^{(3)}, u_1^{(4)}, u_2^{(0)}) $$

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

$$ g_0^{(4,0)} \triangleq g_0^{(0)} + \sum_{j=0}^{3} u_1^{(j+1)} \frac{\partial}{\partial u_1^{(j)}} $$

$$ g_1^{(4,0)} \triangleq \frac{\partial}{\partial u_1^{(4)}} $$

$$ g_2^{(4,0)} \triangleq g_2^{(0)} = \frac{\partial}{\partial u_2^{(0)}} $$

with $g_0^{(0)}$ and $g_2^{(0)}$ given by (5.3).

We check the linearizability by computing the successive Lie brackets:

$$ \mathrm{ad}_{g_0^{(4,0)}}^k g_1^{(4,0)} = (-1)^k \frac{\partial}{\partial u_1^{(4-k)}} \quad k = 0, \dots, 4, \quad \mathrm{ad}_{g_0^{(4,0)}}^5 g_1^{(4,0)} = \frac{\partial}{\partial x_{1,3}} + u_2^{(0)} \frac{\partial}{\partial x_3}, $$

$$ \mathrm{ad}_{g_0^{(4,0)}}^6 g_1^{(4,0)} = -\frac{\partial}{\partial x_{1,2}} + u_2^{(1)} \frac{\partial}{\partial x_3}, \quad \mathrm{ad}_{g_0^{(4,0)}}^7 g_1^{(4,0)} = u_2^{(2)} \frac{\partial}{\partial x_3} + \frac{\partial}{\partial x_{1,1}}, $$

$$ \mathrm{ad}_{g_0^{(4,0)}}^k g_1^{(4,0)} = u_2^{(k-1)} \frac{\partial}{\partial x_3} \quad \forall k \ge 8 $$

and

$$ \mathrm{ad}_{g_0^{(4,0)}} g_2^{(4,0)} = -\frac{\partial}{\partial x_{2,2}} - u_1^{(0)} \frac{\partial}{\partial x_3}, \quad \mathrm{ad}_{g_0^{(4,0)}}^2 g_2^{(4,0)} = \frac{\partial}{\partial x_{2,1}} - u_1^{(1)} \frac{\partial}{\partial x_3}, $$

$$ \mathrm{ad}_{g_0^{(4,0)}}^k g_2^{(4,0)} = -u_1^{(k-1)} \frac{\partial}{\partial x_3} \quad \forall k \ge 3 $$
Hence,

$$
\begin{align*}
G_0^{(4,0)} &= \left\{ \frac{\partial}{\partial u_1^{(4)}}, \frac{\partial}{\partial u_2^{(0)}} \right\} = \overline{G_0^{(4,0)}}, \\
G_1^{(4,0)} &= G_0^{(4,0)} + \left\{ \frac{\partial}{\partial u_1^{(3)}}, -\frac{\partial}{\partial x_{2,2}} - u_1^{(0)} \frac{\partial}{\partial x_3} \right\} = \overline{G_1^{(4,0)}}, \\
G_2^{(4,0)} &= G_1^{(4,0)} + \left\{ \frac{\partial}{\partial u_1^{(2)}}, \frac{\partial}{\partial x_{2,1}} - u_1^{(1)} \frac{\partial}{\partial x_3} \right\} = \overline{G_2^{(4,0)}}, \\
G_3^{(4,0)} &= G_2^{(4,0)} + \left\{ \frac{\partial}{\partial u_1^{(1)}}, -u_1^{(2)} \frac{\partial}{\partial x_3} \right\} = \overline{G_3^{(4,0)}}, \\
G_4^{(4,0)} &= G_3^{(4,0)} + \left\{ \frac{\partial}{\partial u_1^{(0)}} \right\} = \overline{G_4^{(4,0)}}, \\
G_5^{(4,0)} &= G_4^{(4,0)} + \left\{ \frac{\partial}{\partial x_{1,3}} \right\} = \overline{G_5^{(4,0)}}, \\
G_6^{(4,0)} &= G_5^{(4,0)} + \left\{ \frac{\partial}{\partial x_{1,2}} \right\} = \overline{G_6^{(4,0)}}, \\
G_7^{(4,0)} &= G_6^{(4,0)} + \left\{ \frac{\partial}{\partial x_{1,1}} \right\} = \overline{G_7^{(4,0)}} = \mathbb{TR}^{12}.
\end{align*}
$$

It results that the system (5.1) is flat by pure prolongation in any neighborhood excluding $u_1^{(2)} = 0$, with $k_*^{(4,0)} = 7$. The reader may easily verify that $\rho_k^{(4,0)} = 2$ for $k = 0, \dots, 3$ and $\rho_k^{(4,0)} = 1$ for $k = 4, \dots, 7$, 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

$$
\begin{equation} \tag{5.4}
\begin{aligned}
\langle G_k^{(4,0)}, dy_1 \rangle &= 0, \quad k = 0, \dots, 6, & \langle G_7^{(4,0)}, dy_1 \rangle &\neq 0 \\
\langle G_k^{(4,0)}, dy_2 \rangle &= 0, \quad k = 0, \dots, 2, & \langle G_3^{(4,0)}, dy_2 \rangle &\neq 0
\end{aligned}
\end{equation}
$$

whose solution is

$$
y_1 = x_1, \quad y_2 = x_3 - x_{2,2}u_1^{(0)} + x_{2,1}u_1^{(1)} = x_3 - \dot{x}_2 u_1 + x_2 \dot{u}_1. \qquad (5.5)
$$

**Remark 7.** Another linearization by pure prolongation could have been obtained by interchanging the first and second input. However, we leave to the reader the verification that, by prolonging the second input, one obtains a prolongation of order 6, with flat outputs given by (5.2), thus larger than the one given by (5.5), associated to the first input, of the 4th order.

**Remark 8.** In [20, 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.

$$
r = \min_{Y: \text{flat output}} \max_{i=1,...,m} \{s_i \mid \mathbf{s} = (s_1, \dots, s_m), y = Y(x, \bar{u}^{(s)})\}.
$$
They conjectured $r$ to 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 (5.4), $y_1$ neither can depend on $\bar{u}_1^{(4)}$ 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 [20, 21, 22]

$$ 
\begin{aligned}
\dot{x}_1 &= u_1 \\
\dot{x}_2 &= x_3 u_1 \\
\dot{x}_3 &= x_4 u_1 \\
\dot{x}_4 &= u_2
\end{aligned}
\qquad (5.6) $$

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

$$ y_1 = x_1, \quad y_2 = x_2 \qquad (5.7) $$

(see [21, 22] and [20, 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

$$ g_0^{(0)} = u_1^{(0)} \left( \frac{\partial}{\partial x_1} + x_3 \frac{\partial}{\partial x_2} + x_4 \frac{\partial}{\partial x_3} \right) + u_2^{(0)} \frac{\partial}{\partial x_4}, \quad g_1^{(0)} = \frac{\partial}{\partial u_1^{(0)}}, \quad g_2^{(0)} = \frac{\partial}{\partial u_2^{(0)}}. $$

We compute the distributions $\Gamma_k^{(j)}$ and $\Delta_k^{(j)}$ and the conditions of theorem 3 using algorithm 1 with $\mathbf{j} = (j_1, 0)$, $j_1 \ge 1$.

We thus have

$$ g_0^{(l_1, 0)} = g_0^{(0)} + \sum_{p=0}^{l_1-1} u_1^{(p+1)} \frac{\partial}{\partial u_1^{(p)}}, \quad g_1^{(l_1)} = \frac{\partial}{\partial u_1^{(l_1)}}, \quad g_2^{(0)} = \frac{\partial}{\partial u_2^{(0)}} $$

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 $l_1 \ge 1$.

• $\mathbf{k} = \mathbf{1}$. If $l_1 \ge 2$, we have

$$ \Gamma_1^{l_1, 0} = \left\{ \frac{\partial}{\partial u_1^{(l_1)}}, \frac{\partial}{\partial u_1^{(l_1-1)}} \right\}, \quad \Delta_1^{(l_1, 0)} = \Delta_0^{(l_1, 0)} + \{\mathrm{ad}_{g_0^{(l_1, 0)}} g_2^{(0)}\} = \left\{ \frac{\partial}{\partial u_2^{(0)}}, \frac{\partial}{\partial x_4} \right\} $$
and if $l_1 = 1$,

$$ \Gamma_1^{(1,0)} = \left\{ \frac{\partial}{\partial u_1^{(1)}} \right\}, \quad \Delta_1^{(1,0)} = \Delta_0^{(1,0)} + \{g_1^{(0)}, \mathrm{ad}_{g_0^{(1,0)}} g_2^{(0)}\} = \left\{ \frac{\partial}{\partial u_1^{(0)}}, \frac{\partial}{\partial u_2^{(0)}}, \frac{\partial}{\partial x_4} \right\}. $$

We thus have $\Delta_1^{(l_1,0)} = \overline{\Delta_1^{(l_1,0)}}$ and $[\Gamma_1^{(l_1,0)}, \Delta_1^{(l_1,0)}] = \{0\} \subset \Delta_1^{(l_1,0)}$ for all $l_1 \ge 1$, hence $\sigma_{\Gamma,\Delta}(1) = \sigma_{\Delta}(1) = 0$.

• k = 2. If $l_1 \ge 3$, we have

$$ \Gamma_2^{(l_1,0)} = \left\{ \frac{\partial}{\partial u_1^{(l_1)}}, \frac{\partial}{\partial u_1^{(l_1-1)}}, \frac{\partial}{\partial u_1^{(l_1-2)}} \right\}, \quad \Delta_2^{(l_1,0)} = \left\{ \frac{\partial}{\partial u_2^{(0)}}, \frac{\partial}{\partial x_4}, \frac{\partial}{\partial x_3} \right\}, \quad [\Gamma_2^{(l_1,0)}, \Delta_2^{(l_1,0)}] = \{0\} $$

and if $l_1 = 2$,

$$ \Gamma_2^{(2,0)} = \left\{ \frac{\partial}{\partial u_1^{(2)}}, \frac{\partial}{\partial u_1^{(1)}} \right\}, \quad \Delta_2^{(2,0)} = \left\{ \frac{\partial}{\partial u_1^{(0)}}, \frac{\partial}{\partial u_2^{(0)}}, \frac{\partial}{\partial x_4}, \frac{\partial}{\partial x_3} \right\}, \quad [\Gamma_2^{(2,0)}, \Delta_2^{(2,0)}] = \{0\}. $$

Finally, if $l_1 = 1$,

$$ \Gamma_2^{(1,0)} = \left\{ \frac{\partial}{\partial u_1^{(1)}} \right\}, \quad \Delta_2^{(1,0)} = \left\{ \frac{\partial}{\partial u_1^{(0)}}, -\frac{\partial}{\partial x_1} - x_3 \frac{\partial}{\partial x_2} - x_4 \frac{\partial}{\partial x_3}, \frac{\partial}{\partial u_2^{(0)}}, \frac{\partial}{\partial x_4}, \frac{\partial}{\partial x_3} \right\} \neq \overline{\Delta_2^{(1,0)}}. $$

We thus have $\Delta_2^{(l_1,0)} = \overline{\Delta_2^{(l_1,0)}}$ and $[\Gamma_2^{(l_1,0)}, \Delta_2^{(l_1,0)}] \subset \Delta_2^{(l_1,0)}$ for all $l_1 \ge 2$ since, if $l_1 = 1$, $\Delta_2^{(1,0)}$ is not involutive. Therefore, $\sigma_{\Gamma,\Delta}(2) \le \sigma_{\Delta}(2) = 2$ and our search may be restricted to $j_1 \ge 2$.

• k = 3. If $l_1 \ge 4$, 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 $l_1 = 2$,

$$ \Gamma_3^{(2,0)} = \left\{ \frac{\partial}{\partial u_1^{(2)}}, \frac{\partial}{\partial u_1^{(1)}} \right\}, \quad \Delta_3^{(2,0)} = \left\{ \frac{\partial}{\partial u_1^{(0)}}, \frac{\partial}{\partial u_2^{(0)}}, \frac{\partial}{\partial x_4}, \frac{\partial}{\partial x_3}, \frac{\partial}{\partial x_2}, \frac{\partial}{\partial x_1} \right\} = \mathbb{R}^6 $$

Therefore, $\Delta_k^{(2,0)} = \overline{\Delta_k^{(2,0)}}$ and $[\Gamma_k^{(2,0)}, \Delta_k^{(2,0)}] \subset \Delta_k^{(2,0)}$ for all $k \ge 0$ and $\Delta_3^{(2,0)} = \mathbb{R}^6$.

We conclude that the conditions of theorem 3 hold true whenever $j_1 \ge 2$, which proves that system (5.6) with the pure prolongation of order $j = (2,0)$ is feedback linearizable, this prolongation being minimal.
**Verification.** The prolonged state is now $\bar{x}^{(2,0)} \triangleq (x_1, x_2, x_3, x_4, u_1^{(0)}, u_1^{(1)}, u_1^{(2)}, u_2^{(0)})$ of dimension 8, and the new inputs are $(u_1^{(3)}, u_2^{(1)})$.

$$G_0^{(2,0)} = \left\{ \frac{\partial}{\partial u_1^{(2)}}, \frac{\partial}{\partial u_2^{(0)}} \right\} = \overline{G_0^{(2,0)}},$$

$$G_1^{(2,0)} = \left\{ \frac{\partial}{\partial u_1^{(2)}}, \frac{\partial}{\partial u_1^{(1)}}, \frac{\partial}{\partial u_2^{(0)}}, \frac{\partial}{\partial x_4} \right\} = \overline{G_1^{(2,0)}},$$

$$G_2^{(2,0)} = \left\{ \frac{\partial}{\partial u_1^{(2)}}, \frac{\partial}{\partial u_1^{(1)}}, \frac{\partial}{\partial u_1^{(0)}}, \frac{\partial}{\partial u_2^{(0)}}, \frac{\partial}{\partial x_4}, \frac{\partial}{\partial x_3} \right\} = \overline{G_2^{(2,0)}},$$

$$G_3^{(2,0)} = \left\{ \frac{\partial}{\partial u_1^{(2)}}, \frac{\partial}{\partial u_1^{(1)}}, \frac{\partial}{\partial u_1^{(0)}}, \frac{\partial}{\partial x_1} + x_3 \frac{\partial}{\partial x_2} + x_4 \frac{\partial}{\partial x_3}, \frac{\partial}{\partial u_2^{(0)}}, \frac{\partial}{\partial x_4}, \frac{\partial}{\partial x_3}, \frac{\partial}{\partial x_2} \right\} \\ = \overline{G_3^{(2,0)}} = \mathbb{R}^8$$

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 $\kappa_1^{(2,0)} = \kappa_2^{(2,0)} = 4$ (see definition 6).

Flat outputs $(y_1, y_2)$ are locally non trivial solutions of the system (5.8), i.e. :

$$\langle G_k^{(2,0)}, dy_i \rangle = 0, \quad k = 0, 1, 2, \quad \text{with} \quad \langle G_3^{(2,0)}, dy_i \rangle \neq 0, \quad i = 1, 2. \quad (5.8)$$

It is immediate to verify that

$$y_1 = x_1, \quad y_2 = x_2$$

is a solution of (5.8) and that the mapping

$$\bar{x}^{(2,0)} \mapsto (y_1, \dots, y_1^{(3)}, y_2, \dots, y_2^{(3)})$$

is a local diffeomorphism.

**Remark 9.** The reader may easily check that, since $\Delta_2^{(0,l_2)}$ is not involutive for all $l_2 \ge 3$, no pure prolongation on the second channel, $u_2^{(0)}$, leads to the linearizability conditions.
## 5.3 An Example from [6]

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 [6, Example 2] reads:

$$
\begin{align}
\dot{x}_1 &= x_2 + x_3 u_2^{(0)} \nonumber \\
\dot{x}_2 &= x_3 + x_1 u_2^{(0)} \nonumber \\
\dot{x}_3 &= u_1^{(0)} + x_2 u_2^{(0)} \nonumber \\
\dot{x}_4 &= u_2^{(0)} \nonumber \\
\dot{u}_1^{(0)} &= u_1^{(1)} \nonumber \\
\dot{u}_2^{(0)} &= u_2^{(1)}. \tag{5.9}
\end{align}
$$

It is shown in [6] that this system does not satisfy the sufficient, but not necessary, condition for dynamic linearization of Theorem 4.2 of this paper. Nevertheless, it satisfies the algorithm of [1, 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 and thus give the minimal prolongation.

We prolong the second input at an arbitrary order $j_2 \ge 1$ and set $\mathbf{j} = (0, j_2)$. For $l_2 \ge 1$, we denote, as before,

$$
\begin{align*}
g_0^{(0,l_2)} &= (x_2 + x_3 u_2^{(0)}) \frac{\partial}{\partial x_1} + (x_3 + x_1 u_2^{(0)}) \frac{\partial}{\partial x_2} + (u_1^{(0)} + x_2 u_2^{(0)}) \frac{\partial}{\partial x_3} \\
&\quad + u_2^{(0)} \frac{\partial}{\partial x_4} + \sum_{p=0}^{l_2-1} u_2^{(p+1)} \frac{\partial}{\partial u_2^{(p)}} \\
g_1^{(0)} &= \frac{\partial}{\partial u_1^{(0)}}, \quad g_2^{(l_2)} = \frac{\partial}{\partial u_2^{(l_2)}}.
\end{align*}
$$

We have

$$
\Gamma_0^{(0,l_2)} = \left\{ \frac{\partial}{\partial u_2^{(l_2)}} \right\}, \quad \Delta_0^{(0,l_2)} = \left\{ \frac{\partial}{\partial u_1^{(0)}} \right\} = \overline{\Delta_0^{(0,l_2)}}, \quad [\Gamma_0^{(0,l_2)}, \Delta_0^{(0,l_2)}] \subset \Delta_0^{(0,l_2)}, \quad \forall l_2 \ge 1.
$$

• **k = 1.** For all $l_2 \ge 2$, ad$_{g_0} (0, l_2) g_1^{(0)} = -\frac{\partial}{\partial x_3}$ and

$$
\Gamma_1^{(0,l_2)} = \left\{ \frac{\partial}{\partial u_2^{(l_2)}}, \frac{\partial}{\partial u_2^{(l_2-1)}} \right\}, \quad \Delta_1^{(0,l_2)} = \left\{ \frac{\partial}{\partial u_1^{(0)}}, -\frac{\partial}{\partial x_3} \right\} = \overline{\Delta_1^{(0,l_2)}}, \\
[\Gamma_1^{(0,l_2)}, \Delta_1^{(0,l_2)}] \subset \Delta_1^{(0,l_2)}.
$$
DIFFERENTIAL FLATNESS BY PURE PROLONGATION

For $l_2 = 1$:

$$
\Gamma_1^{(0,1)} = \left\{ \frac{\partial}{\partial u_2^{(1)}} \right\}, \quad \Delta_1^{(0,1)} = \left\{ \frac{\partial}{\partial u_2^{(0)}}, \frac{\partial}{\partial u_1^{(0)}}, -\frac{\partial}{\partial x_3} \right\} = \overline{\Delta_1^{(0,1)}}, \\
\left[ \Gamma_1^{(0,1)}, \Delta_1^{(0,1)} \right] \subset \Delta_1^{(0,1)}.
$$

• **k = 2.** For all $l_2 \ge 3$, $\mathrm{ad}_{g_0}^2_{(0,l_2)} g_1^{(0)} = u_2^{(0)} \frac{\partial}{\partial x_1} + \frac{\partial}{\partial x_2}$ and

$$
\Gamma_2^{(0,l_2)} = \left\{ \frac{\partial}{\partial u_2^{(l_2)}}, \frac{\partial}{\partial u_2^{(l_2-1)}}, \frac{\partial}{\partial u_2^{(l_2-2)}} \right\}, \\
\Delta_2^{(0,l_2)} = \left\{ \frac{\partial}{\partial u_1^{(0)}}, -\frac{\partial}{\partial x_3}, u_2^{(0)} \frac{\partial}{\partial x_1} + \frac{\partial}{\partial x_2} \right\} = \overline{\Delta_2^{(0,l_2)}}, \\
\left[ \Gamma_2^{(0,l_2)}, \Delta_2^{(0,l_2)} \right] \subset \Delta_2^{(0,l_2)}.
$$

But for $l_2 = 2$:

$$
\Gamma_2^{(0,2)} = \left\{ \frac{\partial}{\partial u_2^{(2)}}, \frac{\partial}{\partial u_2^{(1)}} \right\}, \quad \Delta_2^{(0,2)} = \left\{ \frac{\partial}{\partial u_2^{(0)}}, \frac{\partial}{\partial u_1^{(0)}}, -\frac{\partial}{\partial x_3}, u_2^{(0)} \frac{\partial}{\partial x_1} + \frac{\partial}{\partial x_2} \right\} \neq \overline{\Delta_2^{(0,2)}} \\
\text{therefore, we must exclude } j_2 = 2.
$$

• **k = 3.** For all $l_2 \ge 4$, $\mathrm{ad}_{g_0}^3_{(0,l_2)} g_1^{(0)} = (u_2^{(1)} - 1) \frac{\partial}{\partial x_1} - (u_2^{(0)})^2 \frac{\partial}{\partial x_2} - u_2^{(0)} \frac{\partial}{\partial x_3}$ and, if we exclude the points where $u_2^{(0)} = 0$ and $u_2^{(1)} = 1$,

$$
\Gamma_3^{(0,l_2)} = \left\{ \frac{\partial}{\partial u_2^{(l_2)}}, \frac{\partial}{\partial u_2^{(l_2-1)}}, \frac{\partial}{\partial u_2^{(l_2-2)}}, \frac{\partial}{\partial u_2^{(l_2-3)}} \right\}, \\
\Delta_3^{(0,l_2)} = \left\{ \frac{\partial}{\partial u_1^{(0)}}, -\frac{\partial}{\partial x_3}, u_2^{(0)} \frac{\partial}{\partial x_1} + \frac{\partial}{\partial x_2}, (u_2^{(1)} - 1) \frac{\partial}{\partial x_1} + (u_2^{(0)})^2 \frac{\partial}{\partial x_2} - u_2^{(0)} \frac{\partial}{\partial x_3} \right\} \\
= \left\{ \frac{\partial}{\partial u_1^{(0)}}, \frac{\partial}{\partial x_3}, \frac{\partial}{\partial x_2}, \frac{\partial}{\partial x_1} \right\} = \overline{\Delta_3^{(0,l_2)}}, \\
\left[ \Gamma_3^{(0,l_2)}, \Delta_3^{(0,l_2)} \right] \subset \Delta_3^{(0,l_2)}.
$$

The reader may then easily check that the same holds for $l_2 = 3$:

$$
\Gamma_3^{(0,3)} = \left\{ \frac{\partial}{\partial u_2^{(3)}}, \frac{\partial}{\partial u_2^{(2)}}, \frac{\partial}{\partial u_2^{(1)}} \right\}, \\
\Delta_3^{(0,3)} = \left\{ \frac{\partial}{\partial u_2^{(0)}}, \frac{\partial}{\partial u_1^{(0)}}, \frac{\partial}{\partial x_3}, \frac{\partial}{\partial x_2}, \frac{\partial}{\partial x_1} \right\} = \overline{\Delta_3^{(0,3)}}.
$$
• For all $k \ge 4$. Since $\mathrm{ad}_{g_0^{(0,l_2)}}^k g_1^{(0)}$ and $\mathrm{ad}_{g_0^{(0,l_2)}}^{k-3} g_2^{(0)}$ 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 $l_2 \ge 4$:

$$ \Delta_k^{(0,l_2)} = \overline{\Delta_k^{(0,l_2)}}, \quad [\Gamma_k^{(0,l_2)}, \Delta_k^{(0,l_2)}] \subset \Delta_k^{(0,l_2)}, $$

and, using the fact that $\mathrm{ad}_{g_0^{(0,l_2)}} g_2^{(0)} = -x_3 \frac{\partial}{\partial x_1} - x_1 \frac{\partial}{\partial x_2} - x_2 \frac{\partial}{\partial x_3} - \frac{\partial}{\partial x_4}$, we have

$$ \Gamma_4^{(0,3)} = \left\{ \frac{\partial}{\partial u_2^{(3)}}, \frac{\partial}{\partial u_2^{(2)}}, \frac{\partial}{\partial u_2^{(1)}} \right\}, $$

$$ \begin{aligned} \Delta_4^{(0,3)} &= \left\{ \frac{\partial}{\partial u_2^{(0)}}, -x_3 \frac{\partial}{\partial x_1} - x_1 \frac{\partial}{\partial x_2} - x_2 \frac{\partial}{\partial x_3} - \frac{\partial}{\partial x_4}, \frac{\partial}{\partial u_1^{(0)}}, \frac{\partial}{\partial x_3}, \frac{\partial}{\partial x_2}, \frac{\partial}{\partial x_1} \right\} \\ &= \left\{ \frac{\partial}{\partial u_2^{(0)}}, \frac{\partial}{\partial u_1^{(0)}}, \frac{\partial}{\partial x_4}, \frac{\partial}{\partial x_3}, \frac{\partial}{\partial x_2}, \frac{\partial}{\partial x_1} \right\} = \overline{\Delta_4^{(0,3)}} = \mathrm{TR}^6 \end{aligned} $$

and

$$ \Delta_k^{(0,3)} = \mathrm{TR}^6 \quad \forall k \ge 4, $$

hence $k_*^{(0,3)} = 4$.

We conclude that the conditions of theorem 3 are satisfied for all $k$ provided that $j_2 \ge 3$ and thus that the minimal prolongation of the second input channel for which the system is feedback linearizable is equal to 3.

Let us finally give the construction of the 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 $9 = n+m+|\mathbf{j}|$.

We get

$$ G_0^{(0,3)} = \left\{ \frac{\partial}{\partial u_1^{(0)}}, \frac{\partial}{\partial u_2^{(3)}} \right\} = \overline{G_0^{(0,3)}}, \quad G_1^{(0,3)} = \left\{ \frac{\partial}{\partial x_3}, \frac{\partial}{\partial u_2^{(2)}} \right\} \oplus G_0^{(0,3)} = \overline{G_1^{(0,3)}}, $$

$$ G_2^{(0,3)} = \left\{ u_2^{(0)} \frac{\partial}{\partial x_1} + \frac{\partial}{\partial x_2}, \frac{\partial}{\partial u_2^{(1)}} \right\} \oplus G_1^{(0,3)} = \overline{G_2^{(0,3)}}, $$

$$ G_3^{(0,3)} = \left\{ (u_2^{(1)} - 1) \frac{\partial}{\partial x_1} - u_2^{(0)} \frac{\partial}{\partial x_3}, \frac{\partial}{\partial u_2^{(0)}} \right\} \oplus G_2^{(0,3)} = \overline{G_3^{(0,3)}}, $$

$$ G_4^{(0,3)} \left\{ -x_3 \frac{\partial}{\partial x_1} - x_1 \frac{\partial}{\partial x_2} - x_2 \frac{\partial}{\partial x_3} - \frac{\partial}{\partial x_4} \right\} \oplus G_4^{(0,3)} = \mathrm{TR}^9 $$

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:

$$ \langle G_k^{(0,3)}, dy_1 \rangle = 0, \quad k = 0, \dots, 3, \quad \langle G_4^{(0,3)}, dy_1 \rangle \neq 0 $$

$$ \langle G_k^{(0,3)}, dy_2 \rangle = 0, \quad k = 0, \dots, 2, \quad \langle G_3^{(0,3)}, dy_2 \rangle \neq 0. $$
Its solution is given by $y_1 = x_4$, $y_2 = x_1 - u_2^{(0)} x_2$.

Note, on the contrary, that no pure prolongation of the first channel $u_1$ can linearize the system. Indeed, if we consider $\mathbf{j} = (j_1, 0)$, we may easily check that, for all $l_1 \ge 3$,

$$ \Delta_2^{(l_1, 0)} = \left\{ \frac{\partial}{\partial u_2^{(0)}}, -x_3 \frac{\partial}{\partial x_1} - x_1 \frac{\partial}{\partial x_2} - x_2 \frac{\partial}{\partial x_3} - \frac{\partial}{\partial x_4}, (x_1 - u_1^{(0)}) \frac{\partial}{\partial x_1} - x_3 \frac{\partial}{\partial x_3} \right\} $$

is not involutive, thus contradicting condition (i) of Theorem 3.

## 5.4 The Pendulum Example [9, section II. C]

This model of pendulum in the vertical plane has been studied in [9, section II. C], [17, section 6.2.3],[18, 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 (2.19):

$$
\begin{align*}
\dot{x}_1 &= x_2 \\
\dot{x}_2 &= u_1^{(0)} \\
\dot{y}_1 &= y_2 \\
\dot{y}_2 &= u_2^{(0)} \\
\dot{\theta}_1 &= \theta_2 \tag{5.10} \\
\dot{\theta}_2 &= -\frac{u_1^{(0)}}{\varepsilon} \cos \theta_1 + \frac{u_2^{(0)} + 1}{\varepsilon} \sin \theta_1 \\
\dot{u}_1^{(0)} &= u_1^{(1)} \\
\dot{u}_2^{(0)} &= u_2^{(1)}
\end{align*}
$$

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

$$
\begin{align}
g_0^{(0)} &= x_2 \frac{\partial}{\partial x_1} + y_2 \frac{\partial}{\partial y_1} + \theta_2 \frac{\partial}{\partial \theta_1} + \frac{1}{\varepsilon} \sin \theta_1 \frac{\partial}{\partial \theta_2} \nonumber \\
&\quad + u_1^{(0)} \left( \frac{\partial}{\partial x_2} - \frac{1}{\varepsilon} \cos \theta_1 \frac{\partial}{\partial \theta_2} \right) + u_2^{(0)} \left( \frac{\partial}{\partial y_2} + \frac{1}{\varepsilon} \sin \theta_1 \frac{\partial}{\partial \theta_2} \right) \tag{5.11} \\
g_1^{(0)} &= \frac{\partial}{\partial u_1^{(0)}} \qquad g_2^{(0)} = \frac{\partial}{\partial u_2^{(0)}} \nonumber
\end{align}
$$

Note that the inputs $u_1^{(0)}$ and $u_2^{(0)}$ play a symmetric role and one can choose indifferently one of them as the non prolonged input. This assertion will be clarified later on. We choose $u_1^{(0)}$ as non prolonged input. Thus, the vector fields associated to a
prolongation of length $l_2$ on the second input read:

$$ 
\begin{aligned}
g_0^{(0,l_2)} &= x_2 \frac{\partial}{\partial x_1} + y_2 \frac{\partial}{\partial y_1} + \theta_2 \frac{\partial}{\partial \theta_1} + \frac{1}{\varepsilon} \sin \theta_1 \frac{\partial}{\partial \theta_2} + u_1^{(0)} \left( \frac{\partial}{\partial x_2} - \frac{1}{\varepsilon} \cos \theta_1 \frac{\partial}{\partial \theta_2} \right) \\
&\quad + u_2^{(0)} \left( \frac{\partial}{\partial y_2} + \frac{1}{\varepsilon} \sin \theta_1 \frac{\partial}{\partial \theta_2} \right) + \sum_{p=0}^{l_2-1} u_2^{(p+1)} \frac{\partial}{\partial u_2^{(p)}} \\
g_1^{(0)} &= \frac{\partial}{\partial u_1^{(0)}} \quad g_2^{(l_2)} = \frac{\partial}{\partial u_2^{(l_2)}}.
\end{aligned} 
\quad (5.12) $$

• k = 1. For all $l_2 \ge 2$,

$$ \Gamma_1^{(0,l_2)} = \left\{ \frac{\partial}{\partial u_2^{(l_2)}}, \frac{\partial}{\partial u_2^{(l_2-1)}} \right\}, \quad \Delta_1^{(0,l_2)} = \left\{ \frac{\partial}{\partial u_1^{(0)}}, \frac{\partial}{\partial x_2} - \frac{1}{\varepsilon} \cos \theta_1 \frac{\partial}{\partial \theta_2} \right\}, $$

and if $l_2 = 1$,

$$ \Gamma_1^{(0,1)} = \left\{ \frac{\partial}{\partial u_2^{(1)}} \right\}, \quad \Delta_1^{(0,l_2)} = \left\{ \frac{\partial}{\partial u_1^{(0)}}, \frac{\partial}{\partial x_2} - \frac{1}{\varepsilon} \cos \theta_1 \frac{\partial}{\partial \theta_2}, \frac{\partial}{\partial u_2^{(0)}} \right\}. $$

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 $l_2 \ge 1$.

• k = 2. For all $l_2 \ge 3$,

$$ \Gamma_2^{(0,l_2)} = \left\{ \frac{\partial}{\partial u_2^{(l_2)}}, \frac{\partial}{\partial u_2^{(l_2-1)}}, \frac{\partial}{\partial u_2^{(l_2-2)}} \right\}, $$

$$ \Delta_2^{(0,l_2)} = \left\{ \frac{\partial}{\partial u_1^{(0)}}, \frac{\partial}{\partial x_2} - \frac{1}{\varepsilon} \cos \theta_1 \frac{\partial}{\partial \theta_2}, -\frac{\partial}{\partial x_1} + \frac{1}{\varepsilon} \sin \theta_1 \frac{\partial}{\partial \theta_2} \right\}. $$

If $l_2 = 2$,

$$ \Gamma_2^{(0,2)} = \left\{ \frac{\partial}{\partial u_2^{(2)}}, \frac{\partial}{\partial u_2^{(1)}} \right\}, $$

$$ \Delta_2^{(0,2)} = \left\{ \frac{\partial}{\partial u_1^{(0)}}, \frac{\partial}{\partial x_2} - \frac{1}{\varepsilon} \cos \theta_1 \frac{\partial}{\partial \theta_2}, -\frac{\partial}{\partial x_1} + \frac{1}{\varepsilon} \sin \theta_1 \frac{\partial}{\partial \theta_2}, \frac{\partial}{\partial u_2^{(0)}} \right\}. $$

If $l_2 = 1$,

$$ \Gamma_2^{(0,1)} = \left\{ \frac{\partial}{\partial u_2^{(1)}} \right\}, $$

$$ \Delta_2^{(0,1)} = \left\{ \frac{\partial}{\partial u_1^{(0)}}, \frac{\partial}{\partial x_2} - \frac{1}{\varepsilon} \cos \theta_1 \frac{\partial}{\partial \theta_2}, -\frac{\partial}{\partial x_1} + \frac{1}{\varepsilon} \sin \theta_1 \frac{\partial}{\partial \theta_2}, \frac{\partial}{\partial u_2^{(0)}}, \frac{\partial}{\partial y_2} + \frac{1}{\varepsilon} \sin \theta_1 \frac{\partial}{\partial \theta_2} \right\}. $$

Again, we have $\overline{\Delta_2^{(0,l_2)}} = \Delta_2^{(0,l_2)}$ and $[\Gamma_2^{(0,l_2)}, \Delta_2^{(0,l_2)}] \subset \Delta_2^{(0,l_2)}$ for all $l_2 \ge 1$.
• **k = 3.** For all $l_2 \ge 4$,

$$ \Gamma_3^{(0,l_2)} = \left\{ \frac{\partial}{\partial u_2^{(l_2)}}, \frac{\partial}{\partial u_2^{(l_2-1)}}, \frac{\partial}{\partial u_2^{(l_2-2)}}, \frac{\partial}{\partial u_2^{(l_2-3)}} \right\}, $$

$$ \Delta_3^{(0,l_2)} = \left\{ \begin{aligned} & \frac{\partial}{\partial u_1^{(0)}} \frac{\partial}{\partial x_2} - \frac{1}{\varepsilon} \cos \theta_1 \frac{\partial}{\partial \theta_2} - \frac{\partial}{\partial x_1} + \frac{1}{\varepsilon} \sin \theta_1 \frac{\partial}{\partial \theta_2}, \\ & - \frac{1}{\varepsilon} \theta_2 \sin \theta_1 \frac{\partial}{\partial \theta_1} + \frac{1}{\varepsilon} \left( \left( \theta_2^2 - \frac{1}{\varepsilon} u_1^{(0)} \sin \theta_1 \right) \cos \theta_1 + \frac{1}{\varepsilon} (1 + u_2^{(0)}) \sin^2 \theta_1 \right) \frac{\partial}{\partial \theta_2} \end{aligned} \right\}. $$

The reader may easily verify that the Lie bracket of the last element of $\Delta_3^{(0,l_2)}$ with the second or third one is a combination of $\frac{\partial}{\partial \theta_1}$ and $\frac{\partial}{\partial \theta_2}$ not in $\Delta_3^{(0,l_2)}$. Thus, $\Delta_3^{(0,l_2)}$ is not involutive for all $l_2 \ge 4$. Moreover, exchanging the non prolonged input $u_1^{(0)}$ in $u_2^{(0)}$, a similar calculation, left to the reader, shows that $\Delta_3^{(l_1,0)}$ is not involutive for all $l_1 \ge 4$, which proves, according to the first item of theorem 4, that system (5.10) is not flat by pure prolongation.

## 6 Concluding Remarks

We have established necessary and sufficient conditions for a system to be flat by pure prolongation, extending preliminary results of [5, 6, 27, 28, 1, 10], and drawn up a precise picture on the role played by pure prolongations of vector fields.

These conditions allow the construction of 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 for many fruitful discussions.

## References

[1] S. Battilotti and C. Califano, *A constructive condition for dynamic feedback linearization*, Systems & Control Letters **52** (2004), 329–338.

[2] N. Bourbaki, *Algèbre*, Éléments de Mathématiques, Chapitres 1-3, Hermann, Paris, 1970.

[3] P. Brunovský, *A classification of linear controllable systems*, Kybernetika **6** (1970), 173–178.

[4] 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.
[5] B. Charlet, J. Lévine, and R. Marino, *On dynamic feedback linearization*, Systems & Control Letters **13** (1989), 143–151.

[6] ________, *Sufficient conditions for dynamic state feedback linearization*, SIAM J. on Control and Optimization **29** (1991), no. 1, 38–57.

[7] C. Chevalley, *Theory of Lie groups*, Princeton University Press, 1946.

[8] M. Fliess, J. Lévine, Ph. Martin, and P. Rouchon, *Flatness and defect of non-linear systems: introductory theory and examples*, Int. J. Control **61** (1995), no. 6, 1327–1361.

[9] ________, *A Lie-Bäcklund approach to equivalence and flatness of nonlinear systems*, IEEE Trans. Automat. Contr. **44** (1999), no. 5, 922–937.

[10] J. Franch and E. Fossas, *Linearization by prolongations: New bound on the number of integrators*, European Journal of Control **11** (2005), 171–179.

[11] 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.

[12] ________, *Global transformations of nonlinear systems*, IEEE Trans. Automat. Contr. **28** (1983), 24–31.

[13] A. Isidori, *Nonlinear control systems*, 3rd ed., Springer, New York, 1995.

[14] B. Jakubczyk and W. Respondek, *On linearization of control systems*, Bull. Acad. Pol. Sci. Ser. Sci. Math. **28** (1980), no. 9–10, 517–522.

[15] I. S. Krasil'schik, V. V. Lychagin, and A. M. Vinogradov, *Geometry of jet spaces and nonlinear partial differential equations*, Gordon and Breach, New York, 1986.

[16] 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.

[17] ________, *Analysis and Control of Nonlinear Systems: A Flatness-based Approach*, Mathematical Engineering, Springer, 2009.

[18] ________, *On necessary and sufficient conditions for differential flatness*, Applicable Algebra in Engineering, Communication and Computing **22** (2011), 47–90.

[19] Ph. Martin, *Contribution à l'étude des systèmes différentiellement plats*, Ph.D. thesis, École des Mines de Paris, 1992.
[20] Ph. Martin, R.M. Murray, and P. Rouchon, *Flat systems*, Plenary Lectures and Minicourses, Proc. ECC 97, Brussels (G. Bastin and M. Gevers, eds.), 1997, pp. 211–264.

[21] Ph. Martin and P. Rouchon, *Any (controllable) driftless system with 3 inputs and 5 states is flat*, Systems & Control Letters **25** (1995), 167–173.

[22] __________, *Any (controllable) driftless system with m inputs and m+2 states is flat*, Proc. of the 34th IEEE Conf. on Decision and Control, New Orleans, 1995, pp. 2886–2891.

[23] H. Nijmeijer and A.J. van der Schaft, *Nonlinear dynamical control systems*, Springer, New York, 1990.

[24] J.-B. Pomet, *A differential geometric setting for dynamic equivalence and dynamic linearization*, Geometry in Nonlinear Control and Differential Inclusions (B. Jakubczyk, W. Respondek, and T. Rzeżuchowski, eds.), Banach Center Publications, Warsaw, 1993, pp. 319–339.

[25] __________, *A necessary condition for dynamic equivalence*, SIAM J. on Control and Optimization **48** (2009), 925–940.

[26] A.J. van der Schaft, *Linearization and input-output decoupling for general non-linear systems*, Systems & Control Letters **5** (1984), 27–33.

[27] W.M. Sluis, *A necessary condition for dynamic feedback linearization*, Systems & Control Letters **21** (1993), 277–283.

[28] W.M. Sluis and D.M. Tilbury, *A bound on the number of integrators needed to linearize a control system*, Systems & Control Letters **29** (1996), no. 1, 43–50.

[29] V.V. Zharinov, *Geometrical aspect of partial differential equations*, World Scientific, Singapore, 1992.