the Brunovský form equivalent to $\displaystyle\bar{{\mathcal{H}}}_{G}$ is
$\displaystyle\langle 0,1\rangle$, then we can deduce that the signature of an
explicit solution $\displaystyle s$ of $\displaystyle{\mathcal{H}}_{G}$ must
be $\displaystyle\nu=\langle 0,1,1\rangle$. The dimension of the associated
jet space is $\displaystyle N_{\nu}=8>7=\dim(J^{\langle 1,1\rangle}\times G)$.
As such, we confirm again (this time via Proposition 6.2) that the system (50)
is not SFL; however, if we perform a prolongation of
$\displaystyle\bm{\gamma}^{G}$ along the $\displaystyle w$ jet coordinate by
adjoining the 1-form $\displaystyle dw_{2}-w_{3}\,dt$ to
$\displaystyle\bm{\gamma}^{G}$ (Definition 6.1) in equation (55), we get
$\displaystyle\bm{{\mathchoice{{\ooalign{\hbox{\raise
6.25pt\hbox{\scalebox{1.0}[-1.0]{\lower
6.25pt\hbox{$\displaystyle\displaystyle\widehat{\vrule
width=0.0pt,height=4.30554pt\vrule
height=0.0pt,width=5.1773pt}$}}}}\cr\hbox{$\displaystyle\displaystyle\bm{\gamma}$}}}}{{\ooalign{\hbox{\raise
6.25pt\hbox{\scalebox{1.0}[-1.0]{\lower
6.25pt\hbox{$\displaystyle\textstyle\widehat{\vrule
width=0.0pt,height=4.30554pt\vrule
height=0.0pt,width=5.1773pt}$}}}}\cr\hbox{$\displaystyle\textstyle\bm{\gamma}$}}}}{{\ooalign{\hbox{\raise
5.81944pt\hbox{\scalebox{1.0}[-1.0]{\lower
5.81944pt\hbox{$\displaystyle\scriptstyle\widehat{\vrule
width=0.0pt,height=3.01389pt\vrule
height=0.0pt,width=3.62411pt}$}}}}\cr\hbox{$\displaystyle\scriptstyle\bm{\gamma}$}}}}{{\ooalign{\hbox{\raise
5.53241pt\hbox{\scalebox{1.0}[-1.0]{\lower
5.53241pt\hbox{$\displaystyle\scriptscriptstyle\widehat{\vrule
width=0.0pt,height=2.15277pt\vrule
height=0.0pt,width=2.58865pt}$}}}}\cr\hbox{$\displaystyle\scriptscriptstyle\bm{\gamma}$}}}}}}^{G}:=\bm{\gamma}^{G}\oplus\\{dw_{2}-w_{3}\,dt\\}$,
which turns out to be a DFL of $\displaystyle\bm{\gamma}^{G}$, since
$\displaystyle\bm{{\mathchoice{{\ooalign{\hbox{\raise
6.25pt\hbox{\scalebox{1.0}[-1.0]{\lower
6.25pt\hbox{$\displaystyle\displaystyle\widehat{\vrule
width=0.0pt,height=4.30554pt\vrule
height=0.0pt,width=5.1773pt}$}}}}\cr\hbox{$\displaystyle\displaystyle\bm{\gamma}$}}}}{{\ooalign{\hbox{\raise
6.25pt\hbox{\scalebox{1.0}[-1.0]{\lower
6.25pt\hbox{$\displaystyle\textstyle\widehat{\vrule
width=0.0pt,height=4.30554pt\vrule
height=0.0pt,width=5.1773pt}$}}}}\cr\hbox{$\displaystyle\textstyle\bm{\gamma}$}}}}{{\ooalign{\hbox{\raise
5.81944pt\hbox{\scalebox{1.0}[-1.0]{\lower
5.81944pt\hbox{$\displaystyle\scriptstyle\widehat{\vrule
width=0.0pt,height=3.01389pt\vrule
height=0.0pt,width=3.62411pt}$}}}}\cr\hbox{$\displaystyle\scriptstyle\bm{\gamma}$}}}}{{\ooalign{\hbox{\raise
5.53241pt\hbox{\scalebox{1.0}[-1.0]{\lower
5.53241pt\hbox{$\displaystyle\scriptscriptstyle\widehat{\vrule
width=0.0pt,height=2.15277pt\vrule
height=0.0pt,width=2.58865pt}$}}}}\cr\hbox{$\displaystyle\scriptscriptstyle\bm{\gamma}$}}}}}}^{G}$
is SFL by Proposition 6.2.

This example is an instance of a general result concerning prolongation of the
jet coordinates involved in the partial contact curve reduction. In fact, the
order counting technique demonstrated above can be generalized to any contact
sub-connection and is the subject of Theorem 7.2 below. To prove this we first
introduce some helpful notation and establish a technical lemma concerning the
fundamental functions of a reduced contact sub-connection.

######  Notation 7.1.

Given a signature

$\nu=\langle\rho_{1},\ldots,\rho_{k}\rangle,$

we will use subscripts to denote an increase in order of all jets so that

$\nu_{\ell}=\langle\underbrace{0,\,\ldots,\,0}_{\ell\text{ entries
}},\rho_{1},\ldots,\rho_{k}\rangle,\quad\ell\geq 0,$

and $\displaystyle\nu_{\ell}=\nu$ if $\displaystyle\ell<0$.

######  Notation 7.2.

Consider a collection of signatures $\displaystyle
S=\\{\nu_{\ell}\\}_{\ell=1}^{k}$ and their corresponding jet spaces
$\displaystyle J^{\nu_{\ell}}(\mathbb{R},\mathbb{R}^{m_{\nu_{\ell}}})$, where
$\displaystyle m_{\nu_{\ell}}=m_{\nu}$ for all $\displaystyle 1\leq\ell\leq k$
and $\displaystyle m_{\nu}$ is the sum of the entries of $\displaystyle\nu$.
We denote by $\displaystyle\nu^{*}_{S}$ the signature of the smallest jet
space $\displaystyle J^{\nu^{*}_{S}}(\mathbb{R},\mathbb{R}^{m_{\nu}})$ for
which the collection of jet spaces
$\displaystyle\\{J^{\nu_{\ell}}(\mathbb{R},\mathbb{R}^{m_{\nu_{\ell}}})\\}_{\ell=1}^{k}$
may be embedded into $\displaystyle
J^{\nu^{*}_{S}}(\mathbb{R},\mathbb{R}^{m_{\nu}})$.

For example, if $\displaystyle S=\\{\nu_{0},\nu_{1}\\}$ where
$\displaystyle\nu_{0}=\langle 1,0,1,1\rangle$ and
$\displaystyle\nu_{1}=\langle 0,3\rangle$ then
$\displaystyle\nu^{*}_{S}=\langle 0,1,1,1\rangle$ and $\displaystyle
J^{\nu_{\ell}}\subset J^{\nu^{*}_{S}},\ell=0,1$; $\displaystyle
J^{\nu^{*}_{S}}$ is the smallest such jet space.

######  Definition 7.1.

If $\displaystyle J^{\nu_{1}}$ and $\displaystyle J^{\nu_{2}}$ are jet spaces,
we say that $\displaystyle\nu_{1}$ is less than or equal to
$\displaystyle\nu_{2}$ if $\displaystyle J^{\nu_{1}}\subseteq J^{\nu_{2}}$ and
we write $\displaystyle\nu_{1}\leq\nu_{2}$.

######  Lemma 7.1.

Let
$\displaystyle{\mathcal{H}}_{G}=\ker\left(\bm{\beta}^{\nu^{\perp}}\oplus\bm{\beta}^{\nu}\oplus\Theta^{G}\right)$
be the contact sub-connection of a $\displaystyle G$-invariant control system
$\displaystyle\bm{\omega}$, with $\displaystyle G$ being control admissible.
Assume further that $\displaystyle\bar{{\mathcal{H}}}_{G}$ is an SFL contact
curve reduction of $\displaystyle{\mathcal{H}}_{G}$, where the family of
partial contact curves $\displaystyle\bm{c}^{\nu}_{f}$ are induced by integral
manifolds of $\displaystyle\bm{\beta}^{\nu}$ with
$\displaystyle\nu=\langle\rho_{1},...,\rho_{k_{\nu}}\rangle$ and
$\displaystyle\bm{\beta}^{\kappa}=\bm{\beta}^{\nu^{\perp}}\oplus\bm{\beta}^{\nu}$.
Each order $\displaystyle\ell$ fundamental function
$\displaystyle\phi^{a_{\ell},\ell}$, $\displaystyle 1\leq
a_{\ell}\leq\rho_{\ell}$, $\displaystyle 1\leq\ell\leq\bar{k}$ of
$\displaystyle\bar{{\mathcal{H}}}_{G}$ defining its static feedback
linearization depends on derivatives of $\displaystyle\bm{f}(t)$ having
signature no larger than $\displaystyle\nu_{\ell-1}$, where
$\displaystyle\nu_{\ell-1}$ is the signature of the order
$\displaystyle\ell-1$ time derivative of $\displaystyle
j^{\nu}\\!\\!\bm{f}(t)$ and $\displaystyle\bar{k}$ is the derived length of
$\displaystyle\bar{{\mathcal{H}}}_{G}$.

###### Proof.

Let $\displaystyle\bar{Z}\in\bar{{\mathcal{H}}}_{G}$ denote the vector field

$\displaystyle\bar{Z}=\mathbf{\bar{D}}_{t}+\sum_{a=1}^{r}\lambda^{a}(\bm{z},j^{\nu}\\!\\!\bm{f}(t))R_{a},$

where the $\displaystyle\\{R_{a}\\}_{a=1}^{r}$ form a basis of right invariant
vector fields on $\displaystyle G$ and $\displaystyle\mathbf{\bar{D}}_{t}$ is
the total derivative operator on $\displaystyle J^{\nu^{\perp}}$. Notice that,
depending on the functions $\displaystyle\lambda^{a}$, the vector field
$\displaystyle\bar{Z}$ may depend on at most as many derivatives of
$\displaystyle\bm{f}(t)$ as are defined by the Brunovský normal form
$\displaystyle\bm{\beta}^{\nu}$. By the procedure contact, there exist
fundamental functions $\displaystyle\phi^{a_{\ell},\ell}$ for each
$\displaystyle\ell<\bar{k}-1$ such that the quotient bundle
$\displaystyle\mathrm{Char}\,\bar{{\mathcal{H}}}_{G}^{(\ell)}/\mathrm{Char}\,(\bar{{\mathcal{H}}}_{G})^{(\ell)}_{\ell-1}$
is nontrivial. In particular, the fundamental functions are the invariants of
the integrable quotient bundle
$\displaystyle\bar{\Xi}^{(\ell)}_{\ell-1}/\bar{\Xi}^{(\ell)}$ where
$\displaystyle\bar{\Xi}^{(\ell)}=\mathrm{ann}\,\left(\mathrm{Char}\,\bar{{\mathcal{H}}}_{G}^{(\ell)}\right)$
and
$\displaystyle\bar{\Xi}^{(\ell)}_{\ell-1}=\mathrm{ann}\,\left(\mathrm{Char}\,(\bar{{\mathcal{H}}}_{G})^{(\ell)}_{\ell-1}\right)$.
Each Cauchy bundle
$\displaystyle\mathrm{Char}\,\bar{{\mathcal{H}}}_{G}^{(\ell)}$ satisfies

$\mathrm{Char}\,(\bar{{\mathcal{H}}}_{G})^{(\ell)}_{\ell-1}\subseteq\mathrm{Char}\,\bar{{\mathcal{H}}}_{G}^{(\ell)}\subseteq\mathrm{Char}\,(\bar{{\mathcal{H}}}_{G})^{(\ell+1)}_{\ell},$

by definition of the intersection bundle; see (18). Furthermore, as per
Equation (25),

$\displaystyle\mathrm{Char}\,(\bar{{\mathcal{H}}}_{G})^{(\ell)}_{\ell-1}$
$\displaystyle=\mathrm{span}\\{\bar{C}_{0},\mathrm{ad}(Z)\bar{C}_{0},\ldots,\mathrm{ad}(Z)^{\ell-1}\bar{C}_{0}\\},$
$\displaystyle\mathrm{Char}\,(\bar{{\mathcal{H}}}_{G})^{(\ell+1)}_{\ell}$
$\displaystyle=\mathrm{span}\\{\bar{C}_{0},\mathrm{ad}(Z)\bar{C}_{0},\ldots,\mathrm{ad}(Z)^{\ell}\bar{C}_{0}\\},$
$\displaystyle\bar{C}_{0}$
$\displaystyle:=\mathrm{Char}\,(\bar{{\mathcal{H}}}_{G})^{(1)}_{0}.$

In particular, we notice that for each $\displaystyle 1\leq i\leq\ell$ the
coefficients of the vector fields in
$\displaystyle\mathrm{ad}(Z)^{i}\bar{C}_{0}$ will have at most $\displaystyle
i-1$ time derivatives of $\displaystyle j^{\nu}\\!\\!\bm{f}(t)$. As such, with
$\displaystyle i=\ell$, the first integrals
$\displaystyle\phi^{a_{\ell},\ell}$ of the non-trivial quotient bundles
$\displaystyle\bar{\Xi}^{(\ell)}_{\ell-1}/\bar{\Xi}^{(\ell)}$ have at most
dependence on $\displaystyle\ell-1$ time derivatives of $\displaystyle
j^{\nu}\\!\\!\bm{f}(t)$, which we denote by $\displaystyle
j^{\nu_{\ell-1}}\\!\\!\bm{f}(t)$. Finally, in case
$\displaystyle\bar{\Delta}_{\bar{k}}=1$ the associated fundamental bundle
$\displaystyle\bar{\Pi}^{\bar{k}}$ as given by equation (25) allows us to make
the same conclusion when $\displaystyle\ell=\bar{k}$. In case
$\displaystyle\bar{\Delta}_{\bar{k}}>1$, a similar argument applies. ∎

######  Theorem 7.2 (Explicit solution via symmetry).

Suppose the control system $\displaystyle(M,\bm{\omega})$ of derived length
$\displaystyle k$ is cascade feedback linearizable with respect to a Lie group
$\displaystyle G$. Let $\displaystyle{\mathcal{H}}_{G}$ be its contact sub-
connection on $\displaystyle J^{\kappa}\times G$, where $\displaystyle
J^{\kappa}$ is decomposed as $\displaystyle(J^{\nu}\times
J^{\nu^{\perp}})/\sim$ and $\displaystyle\bar{{\mathcal{H}}}_{G}$ is the
static feedback linearizable reduced contact sub-connection with respect to a
generic family of partial contact curves $\displaystyle\bm{c}^{\nu}_{f}$.
Finally, let $\displaystyle\mathrm{pr}{\mathcal{H}}_{G}$ be the partial
prolongation of $\displaystyle{\mathcal{H}}_{G}$ along its $\displaystyle
J^{\nu}$ components with the resulting signature of
$\displaystyle\mathrm{pr}{\mathcal{H}}_{G}$ given by
$\displaystyle\kappa^{\prime}=\nu^{\perp}+\nu^{\prime}$ where

$\nu^{\prime}=\nu^{*}_{S},$

with $\displaystyle S=\\{\nu^{a_{\ell},\ell}_{\ell}\colon a_{\ell}\neq
0,\,1\leq\ell\leq\bar{k}\\}$ where $\displaystyle\nu^{a_{\ell},\ell}$ denotes
the signature of the derivatives of $\displaystyle\bm{f}(t)$ appearing in each
fundamental function $\displaystyle\phi^{a_{\ell},\ell}$. Then
$\displaystyle\mathrm{pr}{\mathcal{H}}_{G}$ is SFL.

###### Proof.

Let us decompose the contact coordinates on $\displaystyle
J^{\kappa}=(J^{\nu^{\perp}}\times J^{\nu})/\sim$ such that $\displaystyle
J^{\nu^{\perp}}$ has coordinates $\displaystyle(t,\bm{z})$ and $\displaystyle
J^{\nu}$ has coordinates $\displaystyle(t,\bm{w})$. Once we restrict
$\displaystyle{\mathcal{H}}_{G}$ to a family of partial contact curves
$\displaystyle\bm{c}^{\nu}_{\bm{f}}$ we will have

$\bar{{\mathcal{H}}}_{G}=\mathrm{span}\big{\\{}\mathbf{\bar{D}}_{t}+\sum_{a=1}^{r}\lambda^{a}(\bm{z},j^{\nu}\\!\\!\bm{f}(t))R_{a},\
\ \partial_{z^{a_{j}}_{j}}\Big{\\}},$ (61)

where $\displaystyle z^{a_{j}}_{j}$ are the highest order contact coordinates
on $\displaystyle J^{\nu^{\perp}}$. By hypothesis,
$\displaystyle\bar{{\mathcal{H}}}_{G}$ is static feedback linearizable of some
signature $\displaystyle\bar{\kappa}$. Consequently, the static feedback
transformation that implements the linearization is determined by the
procedure contact, [46]. According to this, one computes the fundamental
functions $\displaystyle\phi^{a_{\ell},\ell}$ of all orders, after which the
contact coordinates are determined by Lie differentiation by the total
differential operator $\displaystyle\bar{Z}$, which in this case is given by
the first vector field in (61).888This is because we assume that
$\displaystyle\bar{{\mathcal{H}}}_{G}$ has passed the test of being static
feedback linearizable. Each fundamental function
$\displaystyle\phi^{a_{\ell},\ell}$ depends upon the contact coordinates
$\displaystyle\bm{z}$, the local group coordinates
$\displaystyle\bm{\epsilon}$ and the jet $\displaystyle
j^{\nu^{a_{\ell},\ell}}\\!\\!\bm{f}(t)$ of the family of functions
$\displaystyle\bm{f}(t)$ that define the contact curve reduction. Let
$\displaystyle\bm{\bar{z}}$ denote the contact coordinates for the static
feedback linearization of $\displaystyle\bar{{\mathcal{H}}}_{G}$. Then,

$\bm{\bar{z}}^{a_{\ell},\ell}_{i_{\ell}}={\EuScript
L}^{i_{\ell}}_{\bar{Z}}\phi^{a_{\ell},\ell}(\bm{z},j^{\nu^{a_{\ell},\ell}}\\!\\!\bm{f}(t),\bm{\varepsilon}),\
\ \ 0\leq i_{\ell}\leq\ell,\ \ \ 1\leq\ell\leq\bar{k}.$

As such, each highest order contact coordinate
$\displaystyle\bm{\bar{z}}^{a_{\ell},\ell}_{\ell}$ of the linearizing static
feedback transformation depends upon, at most, the
$\displaystyle\nu^{a_{\ell},\ell}_{\ell}$ jet of $\displaystyle\bm{f}(t)$.
That is,

$\phi^{a_{\ell},\ell}_{\ell}=\phi^{a_{\ell},\ell}_{\ell}(\bm{z},j^{\nu^{a_{\ell},\ell}_{\ell}}\bm{f}(t),\bm{\varepsilon}),\quad
0\leq l\leq\bar{k}.$ (62)

Let $\displaystyle\bar{\kappa}$ is the signature of
$\displaystyle\bar{{\mathcal{H}}}_{G}$ as a SFL control system, that is,
$\displaystyle\bar{{\mathcal{H}}}_{G}\cong\mathrm{ann}\,\bm{\beta}^{\bar{\kappa}}$.
Note the list of non-negative integers $\displaystyle\bar{\kappa}$ does not
contain information about the jet orders of $\displaystyle\bm{f}(t)$ that
occur in the fundamental functions $\displaystyle\phi^{a_{\ell},\ell}$. Let

$\nu^{\prime}=\nu^{*}_{S},\quad S=\\{\nu^{a_{\ell},\ell}_{\ell}\colon
a_{\ell}\neq 0,\,1\leq\ell\leq\bar{k}\\}.$

Then solving for $\displaystyle\bm{z}$ and $\displaystyle\varepsilon$ in (62)
obtains the explicit solution of $\displaystyle\bar{{\mathcal{H}}}_{G}$ in the
form
$\displaystyle(\bm{z},\varepsilon)=\zeta(j^{\bar{\kappa}}\\!\bm{g},j^{\nu^{\prime}}\\!\\!\\!\bm{f})$,
for an arbitrary, smooth function $\displaystyle\bm{g}(t)$, valued in
$\displaystyle\mathbb{R}^{\bar{m}}$, where $\displaystyle\bar{m}$ is the
number of inputs in $\displaystyle\bar{{\mathcal{H}}}_{G}$. Combining this
with the partial contact curve reduction in which
$\displaystyle\bm{w}=j^{\nu}\\!\\!{\bm{f}}$, obtains the explicit solution of
$\displaystyle{\mathcal{H}}_{G}$. Thus we see that the signature of the
explicit solution of $\displaystyle{\mathcal{H}}_{G}$ \- say
$\displaystyle\kappa^{\prime}$ \- is given by,

$\kappa^{\prime}=\bar{\kappa}+\nu^{\prime}.$

We note in passing that $\displaystyle\kappa^{\prime}$ is generally not the
signature of $\displaystyle{\mathcal{H}}_{G}$ itself. Only if
$\displaystyle{\mathcal{H}}_{G}$ is static feedback linearizable will this be
so.

To finish the proof, we invoke Proposition 6.1 by constructing a
$\displaystyle\kappa^{\prime}$-prolongation as in Definition 6.1. Toward this
goal, we need some dimension counts:

$\displaystyle\mathrm{dim}\,J^{\kappa}\times G$
$\displaystyle=r+1+\sum_{\rho_{i}\in\kappa}(1+i)\rho_{i},$
$\displaystyle=r+1+\sum_{\rho_{i}\in\nu^{\perp}}(1+i)\rho_{i}+\sum_{\rho_{i}\in\nu}(1+i)\rho_{i},$
$\displaystyle\mathrm{dim}\,J^{\nu^{\perp}}\times G$
$\displaystyle=r+1+\sum_{\rho_{i}\in\nu^{\perp}}(1+i)\rho_{i}$
$\displaystyle\mathrm{dim}\,J^{\kappa^{\prime}}$
$\displaystyle=1+\sum_{\rho_{i}\in\bar{\kappa}}(1+i)\rho_{i}+\sum_{\rho_{i}\in\nu^{\prime}}(1+i)\rho_{i},$

where $\displaystyle\mathrm{dim}\,J^{\nu^{\perp}}\times
G=\mathrm{dim}\,J^{\bar{\kappa}}$. Let $\displaystyle
N_{\nu^{\prime}-\nu}:=\mathrm{dim}\,J^{\nu^{\prime}}-\mathrm{dim}\,J^{\nu}$.
Then it follows that

$\mathrm{dim}\,J^{\kappa^{\prime}}=N_{\nu^{\prime}-\nu}+\mathrm{dim}\,J^{\kappa}\times
G.$

Let us then partially prolong the contact sub-connection
$\displaystyle\bm{\gamma}^{G}=\bm{\beta}^{\kappa}\oplus\Theta^{G}$ so that the
$\displaystyle\bm{\beta}^{\nu}$ Brunovský form summand in
$\displaystyle\bm{\beta}^{\kappa}$ is prolonged to
$\displaystyle\bm{\beta}^{\nu^{\prime}}$. In that case the Pfaffian system
$\displaystyle\bm{\gamma}^{G}$ thus prolonged will be denoted by
$\displaystyle\bm{{\mathchoice{{\ooalign{\hbox{\raise
6.25pt\hbox{\scalebox{1.0}[-1.0]{\lower
6.25pt\hbox{$\displaystyle\displaystyle\widehat{\vrule
width=0.0pt,height=4.30554pt\vrule
height=0.0pt,width=5.1773pt}$}}}}\cr\hbox{$\displaystyle\displaystyle\bm{\gamma}$}}}}{{\ooalign{\hbox{\raise
6.25pt\hbox{\scalebox{1.0}[-1.0]{\lower
6.25pt\hbox{$\displaystyle\textstyle\widehat{\vrule
width=0.0pt,height=4.30554pt\vrule
height=0.0pt,width=5.1773pt}$}}}}\cr\hbox{$\displaystyle\textstyle\bm{\gamma}$}}}}{{\ooalign{\hbox{\raise
5.81944pt\hbox{\scalebox{1.0}[-1.0]{\lower
5.81944pt\hbox{$\displaystyle\scriptstyle\widehat{\vrule
width=0.0pt,height=3.01389pt\vrule
height=0.0pt,width=3.62411pt}$}}}}\cr\hbox{$\displaystyle\scriptstyle\bm{\gamma}$}}}}{{\ooalign{\hbox{\raise
5.53241pt\hbox{\scalebox{1.0}[-1.0]{\lower
5.53241pt\hbox{$\displaystyle\scriptscriptstyle\widehat{\vrule
width=0.0pt,height=2.15277pt\vrule
height=0.0pt,width=2.58865pt}$}}}}\cr\hbox{$\displaystyle\scriptscriptstyle\bm{\gamma}$}}}}}}^{G}$
and we let
$\displaystyle\mathrm{pr}{\mathcal{H}}_{G}:=\ker\bm{{\mathchoice{{\ooalign{\hbox{\raise
6.25pt\hbox{\scalebox{1.0}[-1.0]{\lower
6.25pt\hbox{$\displaystyle\displaystyle\widehat{\vrule
width=0.0pt,height=4.30554pt\vrule
height=0.0pt,width=5.1773pt}$}}}}\cr\hbox{$\displaystyle\displaystyle\bm{\gamma}$}}}}{{\ooalign{\hbox{\raise
6.25pt\hbox{\scalebox{1.0}[-1.0]{\lower
6.25pt\hbox{$\displaystyle\textstyle\widehat{\vrule
width=0.0pt,height=4.30554pt\vrule
height=0.0pt,width=5.1773pt}$}}}}\cr\hbox{$\displaystyle\textstyle\bm{\gamma}$}}}}{{\ooalign{\hbox{\raise
5.81944pt\hbox{\scalebox{1.0}[-1.0]{\lower
5.81944pt\hbox{$\displaystyle\scriptstyle\widehat{\vrule
width=0.0pt,height=3.01389pt\vrule
height=0.0pt,width=3.62411pt}$}}}}\cr\hbox{$\displaystyle\scriptstyle\bm{\gamma}$}}}}{{\ooalign{\hbox{\raise
5.53241pt\hbox{\scalebox{1.0}[-1.0]{\lower
5.53241pt\hbox{$\displaystyle\scriptscriptstyle\widehat{\vrule
width=0.0pt,height=2.15277pt\vrule
height=0.0pt,width=2.58865pt}$}}}}\cr\hbox{$\displaystyle\scriptscriptstyle\bm{\gamma}$}}}}}}^{G}$.
Matters being so, this partial prolongation is exactly the
$\displaystyle\kappa^{\prime}$-prolongation of Definition 6.1, and therefore
by Proposition 6.2 we conclude that
$\displaystyle\mathrm{pr}{\mathcal{H}}_{G}$ is SFL with signature
$\displaystyle\kappa^{\prime}$. ∎

We point out that Theorem 7.2 can also be expressed by the diagram in Fig. 3.

$\displaystyle{{(J^{\nu^{\prime}+\nu^{\perp}}\times
G,\bm{\beta}^{\nu^{\prime}}\oplus\bm{\beta}^{\nu^{\perp}}\oplus\Theta^{G})}}$$\displaystyle{{(J^{\bar{\kappa}+\nu^{\prime}},\bm{\beta}^{\bar{\kappa}+\nu^{\prime}})}}$$\displaystyle{{(J^{\kappa}\times
G,\bm{\gamma}^{G})=(J^{\nu+\nu^{\perp}}\times
G,\bm{\gamma}^{G})}}$$\displaystyle{\varphi^{\prime}}$$\displaystyle{\tilde{\pi}^{\prime}}$
Figure 3: Here $\displaystyle\varphi^{\prime}$ is the static feedback map that
linearizes the prolonged contact sub-connection.

The number of partial prolongations of $\displaystyle{\mathcal{H}}_{G}$
described in Theorem 7.2 to achieve the static feedback linearization
$\displaystyle\text{pr}{\mathcal{H}}_{G}$ is exact. However, as far as
possible we want to avoid the task of computing the fundamental functions of
$\displaystyle\bar{{\mathcal{H}}}_{G}$ explicitly by deducing the maximum
number of prolongations of $\displaystyle{\mathcal{H}}_{G}$ required to
achieve a static feedback linearization. Such a maximum is provided by
Corollary 7.4, below. Before that we need to state the following Lemma.

######  Lemma 7.3.

Let $\displaystyle(J^{\nu^{\perp}+\nu^{\prime}}\times
G,\mathrm{pr}{\mathcal{H}}_{G})$ be the minimally prolonged static feedback
linearizable contact connection provided by Theorem 7.2. Performing any
partial prolongation along any of the components corresponding to the nonzero
entries in $\displaystyle\nu^{\prime}$ preserves the static feedback
linearizability of $\displaystyle\mathrm{pr}{\mathcal{H}}_{G}$.

###### Proof.

This follows easily, for instance by using the Gardner-Shadwick test for
static feedback linearizability [17]. ∎

######  Corollary 7.4.

Let $\displaystyle{\mathcal{H}}_{G}$ be as in the hypotheses of Theorem 7.2
and let $\displaystyle\bar{k}$ be the derived length of
$\displaystyle\bar{{\mathcal{H}}}_{G}$. Then, it is enough to prolong all the
$\displaystyle J^{\nu}$ components of $\displaystyle{\mathcal{H}}_{G}$ by
$\displaystyle 2\bar{k}-1$, so as to achieve a SFL control system
$\displaystyle\mathrm{pr}\,{\mathcal{H}}_{G}$.

###### Proof.

Recall that to compute an optimal set of contact coordinates for
$\displaystyle\text{pr}{\mathcal{H}}_{G}$ we examined the fundamental
functions $\displaystyle\phi^{a_{\ell},\ell}$ of
$\displaystyle\bar{{\mathcal{H}}}_{G}$ which must be differentiated
$\displaystyle\ell$-times and then we minimized over the set of signatures
$\displaystyle S=\\{\nu^{a_{\ell},\ell}_{\ell}\\}_{\ell=1}^{\bar{k}}$. By
Lemma 7.1, the $\displaystyle\phi^{a_{\ell},\ell}$ depend upon $\displaystyle
j^{\lambda}\\!\\!\bm{f}$, where $\displaystyle\lambda\leq\nu_{\ell-1}$ and
hence the $\displaystyle\ell^{\text{th}}$ derivatives of
$\displaystyle\phi^{a_{\ell},\ell}$ depend upon $\displaystyle
j^{\mu}\\!\\!\bm{f}$, where $\displaystyle\mu\leq\nu_{2\ell-1}$. Replacing
$\displaystyle S$ by $\displaystyle
S^{\prime}=\\{\nu_{2\ell-1}\\}_{\ell=1}^{\bar{k}}$ we have
$\displaystyle\nu^{*}_{S^{\prime}}=\nu_{2\bar{k}-1}$. By Lemma 7.3, we
conclude that since the partial prolongation of the summand
$\displaystyle\beta^{\nu}$ in $\displaystyle\bm{\gamma}^{G}$ prolonged to
$\displaystyle\beta^{\nu_{S}^{*}}$ leads to an SFL partial prolongation
$\displaystyle\bm{{\mathchoice{{\ooalign{\hbox{\raise
6.25pt\hbox{\scalebox{1.0}[-1.0]{\lower
6.25pt\hbox{$\displaystyle\displaystyle\widehat{\vrule
width=0.0pt,height=4.30554pt\vrule
height=0.0pt,width=5.1773pt}$}}}}\cr\hbox{$\displaystyle\displaystyle\bm{\gamma}$}}}}{{\ooalign{\hbox{\raise
6.25pt\hbox{\scalebox{1.0}[-1.0]{\lower
6.25pt\hbox{$\displaystyle\textstyle\widehat{\vrule
width=0.0pt,height=4.30554pt\vrule
height=0.0pt,width=5.1773pt}$}}}}\cr\hbox{$\displaystyle\textstyle\bm{\gamma}$}}}}{{\ooalign{\hbox{\raise
5.81944pt\hbox{\scalebox{1.0}[-1.0]{\lower
5.81944pt\hbox{$\displaystyle\scriptstyle\widehat{\vrule
width=0.0pt,height=3.01389pt\vrule
height=0.0pt,width=3.62411pt}$}}}}\cr\hbox{$\displaystyle\scriptstyle\bm{\gamma}$}}}}{{\ooalign{\hbox{\raise
5.53241pt\hbox{\scalebox{1.0}[-1.0]{\lower
5.53241pt\hbox{$\displaystyle\scriptscriptstyle\widehat{\vrule
width=0.0pt,height=2.15277pt\vrule
height=0.0pt,width=2.58865pt}$}}}}\cr\hbox{$\displaystyle\scriptscriptstyle\bm{\gamma}$}}}}}}^{G}$,
the partial prolongation of $\displaystyle\beta^{\nu}$ to
$\displaystyle\beta^{\nu_{2\bar{k}-1}}$ instead, will ensure that this partial
prolongation of $\displaystyle\bm{\gamma}^{G}$ will also be SFL. ∎

Thus, the only data we require from $\displaystyle\bar{{\mathcal{H}}}_{G}$ to
obtain an SFL prolonged contact connection
$\displaystyle\text{pr}{\mathcal{H}}_{G}$ is its derived length which is
algorithmically obtained. The number of prolongations described in Corollary
7.4 agrees with the well-known sharp Sluis-Tilbury bound [42] in the case of
2-input systems. In fact, we can immediately find an entire class of examples
that satisfy the sharp Sluis-Tilbury bound of $\displaystyle 2n-3$
prolongations of a 2-input system where $\displaystyle n$ is the number of
states.

Indeed, consider any contact sub-connection $\displaystyle{\mathcal{H}}_{G}$
with 2 controls whose Brunovský component has signature $\displaystyle\langle
1,0,\ldots 0,1\rangle$, where the final ’1’ is in position $\displaystyle k$
for some $\displaystyle k>1$. If the group $\displaystyle G$ has dimension
$\displaystyle r$ then the contact sub-connection, as a control system, has
$\displaystyle 1+k+r$ state variables. If a static feedback linearizable
partial contact curve reduction is performed along the contact coordinate of
order 1 then the signature of the Brunovský component of the reduced sub-
connection $\displaystyle\bar{{\mathcal{H}}}_{G}$ will have signature
$\displaystyle\langle 0,0,\ldots 0,1\rangle$, where the final ’1’ continues to
be in position $\displaystyle k$. This means that
$\displaystyle\bar{{\mathcal{H}}}_{G}$ will be a rank 2 Goursat bundle and its
derived length will be $\displaystyle\bar{k}=k+r$, where upon Corollary 7.4
tells us that $\displaystyle{\mathcal{H}}_{G}$ requires at most

$\displaystyle 2\bar{k}-1=2(k+r)-1=2(1+k+r)-3=2n-3$

prolongations to achieve an SFL system.

Actually, the bound in Corollary 7.4 is far more efficient than the best known
bounds [15] on the required number of partial prolongations; there they are
polynomial and/or exponential in the number of states and controls, in
contrast to the linear bounds provided by Corollary 7.4. However, in [15] the
authors permit themselves no additional information on the control systems in
question. In particular, unlike in the present paper no assumption on admitted
symmetry is made.

######  Example 7.2.

Consider the following 4-input contact sub-connection on $\displaystyle
J^{\langle 1,2,1\rangle}\times G$, where $\displaystyle G$ is an abelian Lie
group with local coordinates
$\displaystyle(\varepsilon^{1},\varepsilon^{2},\varepsilon^{3})$,

${\mathcal{H}}_{G}=\mathrm{span}\\{\mathbf{D}_{t}+w^{1}_{2}z^{1}_{3}\partial_{\varepsilon^{1}}+w^{1}_{2}z^{2}_{1}\partial_{\varepsilon^{2}}+w^{2}_{2}z^{2}_{1}\partial_{\varepsilon^{3}},\partial_{z^{1}_{3}},\partial_{z^{2}_{1}},\partial_{w^{1}_{2}},\partial_{w^{2}_{2}}\\},$

where $\displaystyle\mathbf{D}_{t}$ is the total derivative operator on
$\displaystyle J^{\langle 1,2,1\rangle}$. Furthermore, the coordinates of
$\displaystyle J^{\kappa}=J^{\langle 1,2,1\rangle}$ are such that if we
decompose $\displaystyle\kappa=\nu^{\perp}+\nu$ such that
$\displaystyle\nu^{\perp}=\langle 1,0,1\rangle$ and $\displaystyle\nu=\langle
0,2,0\rangle$ then the $\displaystyle\bm{z}$ coordinates correspond to
$\displaystyle J^{\nu^{\perp}}$ and the $\displaystyle\bm{w}$ coordinates
correspond to $\displaystyle J^{\nu}$. The refined derived type of
$\displaystyle{\mathcal{H}}_{G}$ is

$[[5,0],[9,0,0],[15,8,13],[16,16]],$

which is clearly not that of a Brunovský form and hence this system is not
SFL. However, let us illustrate Theorem 7.2 by reducing
$\displaystyle{\mathcal{H}}_{G}$ by a family of partial contact curves
$\displaystyle\bm{c}^{\nu}_{\bm{f}}$ that entail integral manifolds of
$\displaystyle\bm{\beta}^{\nu}$. Indeed, let
$\displaystyle\bm{f}(t)=(f_{1}(t),f_{2}(t))$ such that
$\displaystyle\bm{c}^{\nu}_{\bm{f}}$ imposes
$\displaystyle(w^{i}_{0},w^{i}_{1},w^{i}_{2})=(f_{i}(t),\dot{f}_{i}(t),\ddot{f}_{i}(t)),\,i=1,2$.
The reduced system is therefore

$\bar{{\mathcal{H}}}_{G}=\mathrm{span}\\{\mathbf{\bar{D}}_{t}+z^{1}_{3}\ddot{f}_{1}(t)\partial_{\varepsilon^{1}}+z^{2}_{1}\ddot{f}_{1}(t)\partial_{\varepsilon^{2}}+z^{2}_{1}\ddot{f}_{2}(t)\partial_{\varepsilon^{3}},\partial_{z^{1}_{3}},\partial_{z^{2}_{1}}\\},$

where $\displaystyle\mathbf{\bar{D}}_{t}$ is the total derivative operator on
$\displaystyle J^{\nu^{\perp}}$. By calculation we find that the reduced
system $\displaystyle\bar{{\mathcal{H}}}_{G}$ is SFL with signature
$\displaystyle\bar{\kappa}=\langle 0,0,1,1\rangle$. The fundamental functions
of $\displaystyle\bar{{\mathcal{H}}}_{G}$ are easy to compute,

$\displaystyle\phi^{1,3}$
$\displaystyle=\frac{\dddot{f}_{2}(t)}{\dddot{f}_{1}(t)}(\ddot{f}_{1}(t)z^{2}_{0}-\varepsilon^{2})+(\varepsilon^{3}-\ddot{f}_{2}(t)z^{2}_{0}),$
$\displaystyle\phi^{1,4}$
$\displaystyle=\varepsilon^{1}-\ddddot{f}_{1}(t)z^{1}_{0}+\dddot{f}_{1}(t)z^{1}_{1}-\ddot{f}_{2}(t)z^{1}_{2}.$

Note that $\displaystyle j^{\nu}\\!\\!\bm{f}(t)$ has signature
$\displaystyle\nu=\langle 0,2\rangle$ and the jet $\displaystyle
j^{\nu^{1,3}}\\!\\!\bm{f}(t)$ appearing in $\displaystyle\phi^{1,3}\,(\ell=3)$
is of signature $\displaystyle\nu^{1,3}=\langle 0,0,2\rangle$, which is
predicted by the maximal bound of Lemma 7.1. For
$\displaystyle\phi^{1,4}(\ell=4)$ the signature of the jet of
$\displaystyle\bm{f}(t)$ that appears has signature
$\displaystyle\nu^{1,4}=\langle 0,1,0,1\rangle$, which is not the maximum
signature predicted by Lemma 7.1, since the lemma says the signature is no
greater than $\displaystyle\nu_{3}=\langle 0,0,0,0,2\rangle$. We have
$\displaystyle\nu^{1,3}_{3}=\langle 0,0,0,0,0,2\rangle$ and
$\displaystyle\nu^{1,4}_{4}=\langle 0,0,0,0,0,1,0,1\rangle$, so by Theorem 7.2
we set $\displaystyle S=\\{\nu^{1,3}_{3},\nu^{1,4}_{4}\\}$ and obtain
$\displaystyle\nu^{\prime}=\nu^{*}_{S}=\langle 0,0,0,0,0,1,0,1\rangle$.
Finally, recalling that our partial contact curve reduction occurred along the
two second order jet coordinates $\displaystyle w^{1},w^{2}$, we prolong
$\displaystyle\bm{\beta}^{\nu}$ to $\displaystyle\bm{\beta}^{\nu^{\prime}}$ so
that

$\displaystyle\nu^{\perp}+\nu=\langle 1,2,1\rangle=\langle
1,0,1\rangle+\langle 0,2,0\rangle\mapsto\langle 1,0,1\rangle+\langle
0,0,0,0,0,1,0,1\rangle=\nu^{\perp}+\nu^{\prime}$

and thus we manage to construct $\displaystyle\mathrm{pr}{\mathcal{H}}_{G}$ on
$\displaystyle J^{\langle 1,0,1,0,0,1,0,1\rangle}\times G$ as

$\mathrm{pr}{\mathcal{H}}_{G}=\mathrm{span}\\{\mathrm{pr}\,\mathbf{D}_{t}+w^{1}_{2}z^{1}_{3}\partial_{\varepsilon^{1}}+w^{1}_{2}z^{2}_{1}\partial_{\varepsilon^{2}}+w^{2}_{2}z^{2}_{1}\partial_{\varepsilon^{3}},\partial_{z^{1}_{3}},\partial_{z^{2}_{1}},\partial_{w^{1}_{8}},\partial_{w^{2}_{6}}\\},$

where $\displaystyle\mathrm{pr}\,\mathbf{D}_{t}$ is the total derivative
operator on $\displaystyle J^{\langle 1,0,1,0,0,1,0,1\rangle}$. One can then
apply Theorem 3.3 or the GS algorithm to discover that
$\displaystyle\mathrm{pr}{\mathcal{H}}_{G}$ is indeed SFL. The signature of
$\displaystyle\mathrm{pr}{\mathcal{H}}_{G}$ turns out to be
$\displaystyle\langle 0,0,1,1,0,1,0,1\rangle$. The GS algorithm or contact can
now be used to construct an explicit solution of
$\displaystyle\mathrm{pr}{\mathcal{H}}_{G}$ and hence that of
$\displaystyle{\mathcal{H}}_{G}$. This verifies Theorem 7.2 in that the
signature of $\displaystyle\text{pr}\,{\mathcal{H}}_{G}$ is
$\displaystyle\langle 0,0,1,1,0,1,0,1\rangle=\bar{\kappa}+\nu^{\prime}$, this
being the signature of the explicit solution of
$\displaystyle{\mathcal{H}}_{G}$.

Finally, if we did not have the expressions for the fundamental functions we
could nevertheless obtain an upper bound for the number of prolongations using
Corollary 7.4. Since in this case $\displaystyle\bar{k}=4$, we would prolong
$\displaystyle\nu=\langle 0,2\rangle$ by $\displaystyle 2\bar{k}-1=7$
derivatives giving $\displaystyle\nu_{S^{\prime}}^{*}=\langle
0,0,0,0,0,0,0,0,2\rangle$ instead of $\displaystyle\nu^{\prime}=\langle
0,0,0,0,0,\allowbreak 1,0,1\rangle$. Lemma 7.3 says that a prolongation of
$\displaystyle\nu$ to $\displaystyle\nu_{S^{\prime}}^{*}$ instead of
$\displaystyle\nu^{\prime}$ ensures that the resulting prolonged sub-
connection $\displaystyle\text{pr}\,{\mathcal{H}}_{G}$ is also SFL.

######  Remark 7.1.

Theorem 7.2 and Corollary 7.4 do more than provide a bound on the number of
partial prolongations required to produce a SFL control system
$\displaystyle\mathrm{pr}{\mathcal{H}}_{G}$. They also tells us which controls
need to be prolonged. Moreover, as is by now clear, cascade linearization
gives a general canonical procedure for expressing a control system that is
not DFL by differentiation of given inputs to one that is.

######  Example 7.3 (Example 6.2 continued).

This makes use of the contact connection worked out in (59) and (60).

We now check the cascade feedback linearization of (58). We can perform a
contact curve reduction along $\displaystyle m$ or $\displaystyle n$. For this
example we fix a family of contact curve reductions by defining the map

$\displaystyle\mathbf{c}_{f}^{\langle 0,1\rangle}:J^{\langle 1\rangle}\times
G\to J^{\langle 1,1\rangle}\times G$

given by

$\displaystyle\mathbf{c}_{f}^{\langle
0,1\rangle}(t,n,n_{1},\varepsilon^{1},\varepsilon^{2},\varepsilon^{3})=$
$\displaystyle(t,n,n_{1},\varepsilon^{1},\varepsilon^{2},\varepsilon^{3},m,m_{1},m_{2})$
$\displaystyle=$
$\displaystyle(t,n,n_{1},\varepsilon^{1},\varepsilon^{2},\varepsilon^{3},f(t),\dot{f}(t),\ddot{f}(t)).$

Then

$\displaystyle\bm{\bar{\gamma}}^{G}$
$\displaystyle=\left(\mathbf{c}_{f}^{\langle
0,1\rangle}\right)^{*}\bm{\gamma}^{G}=\\{dn-n_{1}dt\\}\oplus$
$\displaystyle\Big{\\{}d\varepsilon^{1}+\frac{(n_{1}+1)\sin f-h\ddot{f}}{\cos
f}dt,\ \ d\varepsilon^{2}-t\frac{(n_{1}+1)\sin f-h\ddot{f}}{\cos f}dt,\ \
d\varepsilon^{3}-ndt\Big{\\}}.$

It can be checked that $\displaystyle\bm{\bar{\gamma}}^{G}$ is static feedback
linearizable of signature $\displaystyle\langle 0,0,0,1\rangle$ and therefore
static feedback equivalent to the Brunovský normal form with that signature.
Moreover, by Theorem 7.2, we perform a 7 ($\displaystyle=2\bar{k}-1$) -fold
prolongation in $\displaystyle m$, in order that
$\displaystyle\text{pr}{\mathcal{H}}_{G}$ is static feedback
linearizablegiving a dynamic feedback linearization of
$\displaystyle{\mathcal{H}}_{G}$. In fact it turns out that it is sufficient
to prolong to order 6 in $\displaystyle m$. That is,

$\displaystyle\text{pr}{\mathcal{H}}_{G}$
$\displaystyle=\text{span}\Big{\\{}\partial_{t}+m_{1}\partial_{m}+m_{2}\partial_{m_{1}}+m_{3}\partial_{m_{2}}+m_{4}\partial_{m_{3}}+m_{5}\partial_{m_{4}}+m_{6}\partial_{m_{5}}+n_{1}\partial_{n}$
$\displaystyle\frac{(n_{1}+1)\sin m-hm_{2}}{\cos
m}(t\partial_{\varepsilon^{2}}-\partial_{\varepsilon^{1}})+n\partial_{\varepsilon^{3}},\
\ \ \partial_{m_{6}},\ \ \partial_{n_{1}}\Big{\\}}.$

is static feedback linearizable of signature $\displaystyle\langle
0,0,0,1,0,1\rangle$, which is also the signature of the explicit solution.

###  7.2 From cascade to dynamic feedback linearization

So far we have shown how to construct the explicit solution of a smooth,
invariant control system (1) via the contact sub-connection,
$\displaystyle{\mathcal{H}}_{G}$. The goal of this subsection is to give the
precise relationship between cascade feedback linearization and dynamic
feedback linearization and show how to construct a dynamic feedback
linearization of an invariant control system (1) in local coordinates, as
expressed by Definition 2.4. The basic idea is that the map
$\displaystyle\widetilde{\varphi}$ should lift to a map
$\displaystyle\widetilde{\widetilde{{\varphi}}}$ between
$\displaystyle(J^{\nu^{\prime}+\nu^{\perp}}\times
G,\bm{\beta}^{\nu^{\prime}+\nu^{\perp}}\oplus\Theta^{G})$ and some manifold
and Pfaffian system $\displaystyle({\mathchoice{{\ooalign{\hbox{\raise
7.09259pt\hbox{\scalebox{1.0}[-1.0]{\lower
7.09259pt\hbox{$\displaystyle\displaystyle\widehat{\vrule
width=0.0pt,height=6.83331pt\vrule
height=0.0pt,width=10.79166pt}$}}}}\cr\hbox{$\displaystyle\displaystyle
M$}}}}{{\ooalign{\hbox{\raise 7.09259pt\hbox{\scalebox{1.0}[-1.0]{\lower
7.09259pt\hbox{$\displaystyle\textstyle\widehat{\vrule
width=0.0pt,height=6.83331pt\vrule
height=0.0pt,width=10.79166pt}$}}}}\cr\hbox{$\displaystyle\textstyle
M$}}}}{{\ooalign{\hbox{\raise 6.40926pt\hbox{\scalebox{1.0}[-1.0]{\lower
6.40926pt\hbox{$\displaystyle\scriptstyle\widehat{\vrule
width=0.0pt,height=4.78333pt\vrule
height=0.0pt,width=7.55415pt}$}}}}\cr\hbox{$\displaystyle\scriptstyle
M$}}}}{{\ooalign{\hbox{\raise 5.9537pt\hbox{\scalebox{1.0}[-1.0]{\lower
5.9537pt\hbox{$\displaystyle\scriptscriptstyle\widehat{\vrule
width=0.0pt,height=3.41666pt\vrule
height=0.0pt,width=5.39581pt}$}}}}\cr\hbox{$\displaystyle\scriptscriptstyle
M$}}}}},{\mathchoice{{\ooalign{\hbox{\raise
6.25pt\hbox{\scalebox{1.0}[-1.0]{\lower
6.25pt\hbox{$\displaystyle\displaystyle\widehat{\vrule
width=0.0pt,height=4.30554pt\vrule
height=0.0pt,width=6.22453pt}$}}}}\cr\hbox{$\displaystyle\displaystyle\bm{\omega}$}}}}{{\ooalign{\hbox{\raise
6.25pt\hbox{\scalebox{1.0}[-1.0]{\lower
6.25pt\hbox{$\displaystyle\textstyle\widehat{\vrule
width=0.0pt,height=4.30554pt\vrule
height=0.0pt,width=6.22453pt}$}}}}\cr\hbox{$\displaystyle\textstyle\bm{\omega}$}}}}{{\ooalign{\hbox{\raise
5.81944pt\hbox{\scalebox{1.0}[-1.0]{\lower
5.81944pt\hbox{$\displaystyle\scriptstyle\widehat{\vrule
width=0.0pt,height=3.01389pt\vrule
height=0.0pt,width=4.35718pt}$}}}}\cr\hbox{$\displaystyle\scriptstyle\bm{\omega}$}}}}{{\ooalign{\hbox{\raise
5.53241pt\hbox{\scalebox{1.0}[-1.0]{\lower
5.53241pt\hbox{$\displaystyle\scriptscriptstyle\widehat{\vrule
width=0.0pt,height=2.15277pt\vrule
height=0.0pt,width=3.11226pt}$}}}}\cr\hbox{$\displaystyle\scriptscriptstyle\bm{\omega}$}}}}})$
with  $\displaystyle\displaystyle\widehat{\vrule
width=0.0pt,height=4.30554pt\vrule height=0.0pt,width=6.22453pt}$
$\displaystyle\displaystyle\bm{\omega}$ encoding a dynamic extension of
$\displaystyle\bm{\omega}$.

—- Constructing a dynamic feedback linearization —-

  1. 1.

Construct $\displaystyle\text{pr}{\mathcal{H}}_{G}$ and
$\displaystyle\bm{{\mathchoice{{\ooalign{\hbox{\raise
6.25pt\hbox{\scalebox{1.0}[-1.0]{\lower
6.25pt\hbox{$\displaystyle\displaystyle\widehat{\vrule
width=0.0pt,height=4.30554pt\vrule
height=0.0pt,width=5.1773pt}$}}}}\cr\hbox{$\displaystyle\displaystyle\bm{\gamma}$}}}}{{\ooalign{\hbox{\raise
6.25pt\hbox{\scalebox{1.0}[-1.0]{\lower
6.25pt\hbox{$\displaystyle\textstyle\widehat{\vrule
width=0.0pt,height=4.30554pt\vrule
height=0.0pt,width=5.1773pt}$}}}}\cr\hbox{$\displaystyle\textstyle\bm{\gamma}$}}}}{{\ooalign{\hbox{\raise
5.81944pt\hbox{\scalebox{1.0}[-1.0]{\lower
5.81944pt\hbox{$\displaystyle\scriptstyle\widehat{\vrule
width=0.0pt,height=3.01389pt\vrule
height=0.0pt,width=3.62411pt}$}}}}\cr\hbox{$\displaystyle\scriptstyle\bm{\gamma}$}}}}{{\ooalign{\hbox{\raise
5.53241pt\hbox{\scalebox{1.0}[-1.0]{\lower
5.53241pt\hbox{$\displaystyle\scriptscriptstyle\widehat{\vrule
width=0.0pt,height=2.15277pt\vrule
height=0.0pt,width=2.58865pt}$}}}}\cr\hbox{$\displaystyle\scriptscriptstyle\bm{\gamma}$}}}}}}^{G}:=\text{ann}\,(\text{pr}{\mathcal{H}}_{G})$
as prescribed in Theorem 7.2,

  2. 2.

Compute the Goursat bundle
$\displaystyle\theta=\left(\widetilde{\widetilde{\varphi}}\right)^{*}\bm{{\mathchoice{{\ooalign{\hbox{\raise
6.25pt\hbox{\scalebox{1.0}[-1.0]{\lower
6.25pt\hbox{$\displaystyle\displaystyle\widehat{\vrule
width=0.0pt,height=4.30554pt\vrule
height=0.0pt,width=5.1773pt}$}}}}\cr\hbox{$\displaystyle\displaystyle\bm{\gamma}$}}}}{{\ooalign{\hbox{\raise
6.25pt\hbox{\scalebox{1.0}[-1.0]{\lower
6.25pt\hbox{$\displaystyle\textstyle\widehat{\vrule
width=0.0pt,height=4.30554pt\vrule
height=0.0pt,width=5.1773pt}$}}}}\cr\hbox{$\displaystyle\textstyle\bm{\gamma}$}}}}{{\ooalign{\hbox{\raise
5.81944pt\hbox{\scalebox{1.0}[-1.0]{\lower
5.81944pt\hbox{$\displaystyle\scriptstyle\widehat{\vrule
width=0.0pt,height=3.01389pt\vrule
height=0.0pt,width=3.62411pt}$}}}}\cr\hbox{$\displaystyle\scriptstyle\bm{\gamma}$}}}}{{\ooalign{\hbox{\raise
5.53241pt\hbox{\scalebox{1.0}[-1.0]{\lower
5.53241pt\hbox{$\displaystyle\scriptscriptstyle\widehat{\vrule
width=0.0pt,height=2.15277pt\vrule
height=0.0pt,width=2.58865pt}$}}}}\cr\hbox{$\displaystyle\scriptscriptstyle\bm{\gamma}$}}}}}}^{G}$
and notice that the state and control variables are not well-defined.

  3. 3.

For $\displaystyle\mathcal{T}=\mathrm{ann}\,\theta$, choose any $\displaystyle
t$-preserving diffeomorphism
$\displaystyle\chi:{\mathchoice{{\ooalign{\hbox{\raise
7.09259pt\hbox{\scalebox{1.0}[-1.0]{\lower
7.09259pt\hbox{$\displaystyle\displaystyle\widehat{\vrule
width=0.0pt,height=6.83331pt\vrule
height=0.0pt,width=10.79166pt}$}}}}\cr\hbox{$\displaystyle\displaystyle
M$}}}}{{\ooalign{\hbox{\raise 7.09259pt\hbox{\scalebox{1.0}[-1.0]{\lower
7.09259pt\hbox{$\displaystyle\textstyle\widehat{\vrule
width=0.0pt,height=6.83331pt\vrule
height=0.0pt,width=10.79166pt}$}}}}\cr\hbox{$\displaystyle\textstyle
M$}}}}{{\ooalign{\hbox{\raise 6.40926pt\hbox{\scalebox{1.0}[-1.0]{\lower
6.40926pt\hbox{$\displaystyle\scriptstyle\widehat{\vrule
width=0.0pt,height=4.78333pt\vrule
height=0.0pt,width=7.55415pt}$}}}}\cr\hbox{$\displaystyle\scriptstyle
M$}}}}{{\ooalign{\hbox{\raise 5.9537pt\hbox{\scalebox{1.0}[-1.0]{\lower
5.9537pt\hbox{$\displaystyle\scriptscriptstyle\widehat{\vrule
width=0.0pt,height=3.41666pt\vrule
height=0.0pt,width=5.39581pt}$}}}}\cr\hbox{$\displaystyle\scriptscriptstyle
M$}}}}}\to{\mathchoice{{\ooalign{\hbox{\raise
7.09259pt\hbox{\scalebox{1.0}[-1.0]{\lower
7.09259pt\hbox{$\displaystyle\displaystyle\widehat{\vrule
width=0.0pt,height=6.83331pt\vrule
height=0.0pt,width=10.79166pt}$}}}}\cr\hbox{$\displaystyle\displaystyle
M$}}}}{{\ooalign{\hbox{\raise 7.09259pt\hbox{\scalebox{1.0}[-1.0]{\lower
7.09259pt\hbox{$\displaystyle\textstyle\widehat{\vrule
width=0.0pt,height=6.83331pt\vrule
height=0.0pt,width=10.79166pt}$}}}}\cr\hbox{$\displaystyle\textstyle
M$}}}}{{\ooalign{\hbox{\raise 6.40926pt\hbox{\scalebox{1.0}[-1.0]{\lower
6.40926pt\hbox{$\displaystyle\scriptstyle\widehat{\vrule
width=0.0pt,height=4.78333pt\vrule
height=0.0pt,width=7.55415pt}$}}}}\cr\hbox{$\displaystyle\scriptstyle
M$}}}}{{\ooalign{\hbox{\raise 5.9537pt\hbox{\scalebox{1.0}[-1.0]{\lower
5.9537pt\hbox{$\displaystyle\scriptscriptstyle\widehat{\vrule
width=0.0pt,height=3.41666pt\vrule
height=0.0pt,width=5.39581pt}$}}}}\cr\hbox{$\displaystyle\scriptscriptstyle
M$}}}}}$ such that the integrable bundle
$\displaystyle\mathrm{Char}\,\mathcal{T}^{(1)}_{0}$ is spanned by coordinate
vector fields $\displaystyle\\{\partial_{W^{1}},\ldots,\partial_{W^{m}}\\}$
according to the Frobenius theorem. The diffeomorphism $\displaystyle\chi$
defines the dynamic compensator $\displaystyle u=\beta(t,x,y,W)$ as in Defn.
2.4 and thus defines the new control variables $\displaystyle W$ and new
additional state variables $\displaystyle y$ for the SFL control system
$\displaystyle{\mathchoice{{\ooalign{\hbox{\raise
7.09259pt\hbox{\scalebox{1.0}[-1.0]{\lower
7.09259pt\hbox{$\displaystyle\displaystyle\widehat{\vrule
width=0.0pt,height=6.83331pt\vrule
height=0.0pt,width=7.50002pt}$}}}}\cr\hbox{$\displaystyle\displaystyle\mathcal{V}$}}}}{{\ooalign{\hbox{\raise
7.09259pt\hbox{\scalebox{1.0}[-1.0]{\lower
7.09259pt\hbox{$\displaystyle\textstyle\widehat{\vrule
width=0.0pt,height=6.83331pt\vrule
height=0.0pt,width=7.50002pt}$}}}}\cr\hbox{$\displaystyle\textstyle\mathcal{V}$}}}}{{\ooalign{\hbox{\raise
6.40926pt\hbox{\scalebox{1.0}[-1.0]{\lower
6.40926pt\hbox{$\displaystyle\scriptstyle\widehat{\vrule
width=0.0pt,height=4.78333pt\vrule
height=0.0pt,width=5.25pt}$}}}}\cr\hbox{$\displaystyle\scriptstyle\mathcal{V}$}}}}{{\ooalign{\hbox{\raise
5.9537pt\hbox{\scalebox{1.0}[-1.0]{\lower
5.9537pt\hbox{$\displaystyle\scriptscriptstyle\widehat{\vrule
width=0.0pt,height=3.41666pt\vrule
height=0.0pt,width=3.75pt}$}}}}\cr\hbox{$\displaystyle\scriptscriptstyle\mathcal{V}$}}}}}:=\vartheta^{-1}_{*}\left(\mathrm{pr}{\mathcal{H}}_{G}\right)$
where
$\displaystyle\vartheta=\widetilde{\widetilde{\bm{\varphi}}}\circ\chi^{-1}$.

$\displaystyle{{({\mathchoice{{\ooalign{\hbox{\raise
7.09259pt\hbox{\scalebox{1.0}[-1.0]{\lower
7.09259pt\hbox{$\displaystyle\displaystyle\widehat{\vrule
width=0.0pt,height=6.83331pt\vrule
height=0.0pt,width=10.79166pt}$}}}}\cr\hbox{$\displaystyle\displaystyle
M$}}}}{{\ooalign{\hbox{\raise 7.09259pt\hbox{\scalebox{1.0}[-1.0]{\lower
7.09259pt\hbox{$\displaystyle\textstyle\widehat{\vrule
width=0.0pt,height=6.83331pt\vrule
height=0.0pt,width=10.79166pt}$}}}}\cr\hbox{$\displaystyle\textstyle
M$}}}}{{\ooalign{\hbox{\raise 6.40926pt\hbox{\scalebox{1.0}[-1.0]{\lower
6.40926pt\hbox{$\displaystyle\scriptstyle\widehat{\vrule
width=0.0pt,height=4.78333pt\vrule
height=0.0pt,width=7.55415pt}$}}}}\cr\hbox{$\displaystyle\scriptstyle
M$}}}}{{\ooalign{\hbox{\raise 5.9537pt\hbox{\scalebox{1.0}[-1.0]{\lower
5.9537pt\hbox{$\displaystyle\scriptscriptstyle\widehat{\vrule
width=0.0pt,height=3.41666pt\vrule
height=0.0pt,width=5.39581pt}$}}}}\cr\hbox{$\displaystyle\scriptscriptstyle
M$}}}}},{\mathchoice{{\ooalign{\hbox{\raise
6.25pt\hbox{\scalebox{1.0}[-1.0]{\lower
6.25pt\hbox{$\displaystyle\displaystyle\widehat{\vrule
width=0.0pt,height=4.30554pt\vrule
height=0.0pt,width=6.22453pt}$}}}}\cr\hbox{$\displaystyle\displaystyle\bm{\omega}$}}}}{{\ooalign{\hbox{\raise
6.25pt\hbox{\scalebox{1.0}[-1.0]{\lower
6.25pt\hbox{$\displaystyle\textstyle\widehat{\vrule
width=0.0pt,height=4.30554pt\vrule
height=0.0pt,width=6.22453pt}$}}}}\cr\hbox{$\displaystyle\textstyle\bm{\omega}$}}}}{{\ooalign{\hbox{\raise
5.81944pt\hbox{\scalebox{1.0}[-1.0]{\lower
5.81944pt\hbox{$\displaystyle\scriptstyle\widehat{\vrule
width=0.0pt,height=3.01389pt\vrule
height=0.0pt,width=4.35718pt}$}}}}\cr\hbox{$\displaystyle\scriptstyle\bm{\omega}$}}}}{{\ooalign{\hbox{\raise
5.53241pt\hbox{\scalebox{1.0}[-1.0]{\lower
5.53241pt\hbox{$\displaystyle\scriptscriptstyle\widehat{\vrule
width=0.0pt,height=2.15277pt\vrule
height=0.0pt,width=3.11226pt}$}}}}\cr\hbox{$\displaystyle\scriptscriptstyle\bm{\omega}$}}}}})}}$$\displaystyle{{({\mathchoice{{\ooalign{\hbox{\raise
7.09259pt\hbox{\scalebox{1.0}[-1.0]{\lower
7.09259pt\hbox{$\displaystyle\displaystyle\widehat{\vrule
width=0.0pt,height=6.83331pt\vrule
height=0.0pt,width=10.79166pt}$}}}}\cr\hbox{$\displaystyle\displaystyle
M$}}}}{{\ooalign{\hbox{\raise 7.09259pt\hbox{\scalebox{1.0}[-1.0]{\lower
7.09259pt\hbox{$\displaystyle\textstyle\widehat{\vrule
width=0.0pt,height=6.83331pt\vrule
height=0.0pt,width=10.79166pt}$}}}}\cr\hbox{$\displaystyle\textstyle
M$}}}}{{\ooalign{\hbox{\raise 6.40926pt\hbox{\scalebox{1.0}[-1.0]{\lower
6.40926pt\hbox{$\displaystyle\scriptstyle\widehat{\vrule
width=0.0pt,height=4.78333pt\vrule
height=0.0pt,width=7.55415pt}$}}}}\cr\hbox{$\displaystyle\scriptstyle
M$}}}}{{\ooalign{\hbox{\raise 5.9537pt\hbox{\scalebox{1.0}[-1.0]{\lower
5.9537pt\hbox{$\displaystyle\scriptscriptstyle\widehat{\vrule
width=0.0pt,height=3.41666pt\vrule
height=0.0pt,width=5.39581pt}$}}}}\cr\hbox{$\displaystyle\scriptscriptstyle
M$}}}}},{\mathchoice{{\ooalign{\hbox{\raise
6.25pt\hbox{\scalebox{1.0}[-1.0]{\lower
6.25pt\hbox{$\displaystyle\displaystyle\widehat{\vrule
width=0.0pt,height=4.30554pt\vrule
height=0.0pt,width=6.22453pt}$}}}}\cr\hbox{$\displaystyle\displaystyle\bm{\omega}$}}}}{{\ooalign{\hbox{\raise
6.25pt\hbox{\scalebox{1.0}[-1.0]{\lower
6.25pt\hbox{$\displaystyle\textstyle\widehat{\vrule
width=0.0pt,height=4.30554pt\vrule
height=0.0pt,width=6.22453pt}$}}}}\cr\hbox{$\displaystyle\textstyle\bm{\omega}$}}}}{{\ooalign{\hbox{\raise
5.81944pt\hbox{\scalebox{1.0}[-1.0]{\lower
5.81944pt\hbox{$\displaystyle\scriptstyle\widehat{\vrule
width=0.0pt,height=3.01389pt\vrule
height=0.0pt,width=4.35718pt}$}}}}\cr\hbox{$\displaystyle\scriptstyle\bm{\omega}$}}}}{{\ooalign{\hbox{\raise
5.53241pt\hbox{\scalebox{1.0}[-1.0]{\lower
5.53241pt\hbox{$\displaystyle\scriptscriptstyle\widehat{\vrule
width=0.0pt,height=2.15277pt\vrule
height=0.0pt,width=3.11226pt}$}}}}\cr\hbox{$\displaystyle\scriptscriptstyle\bm{\omega}$}}}}})}}$$\displaystyle{{(J^{\nu^{\perp}+\nu^{\prime}}\times
G,{\mathchoice{{\ooalign{\hbox{\raise 6.25pt\hbox{\scalebox{1.0}[-1.0]{\lower
6.25pt\hbox{$\displaystyle\displaystyle\widehat{\vrule
width=0.0pt,height=4.30554pt\vrule
height=0.0pt,width=5.1773pt}$}}}}\cr\hbox{$\displaystyle\displaystyle\bm{\gamma}$}}}}{{\ooalign{\hbox{\raise
6.25pt\hbox{\scalebox{1.0}[-1.0]{\lower
6.25pt\hbox{$\displaystyle\textstyle\widehat{\vrule
width=0.0pt,height=4.30554pt\vrule
height=0.0pt,width=5.1773pt}$}}}}\cr\hbox{$\displaystyle\textstyle\bm{\gamma}$}}}}{{\ooalign{\hbox{\raise
5.81944pt\hbox{\scalebox{1.0}[-1.0]{\lower
5.81944pt\hbox{$\displaystyle\scriptstyle\widehat{\vrule
width=0.0pt,height=3.01389pt\vrule
height=0.0pt,width=3.62411pt}$}}}}\cr\hbox{$\displaystyle\scriptstyle\bm{\gamma}$}}}}{{\ooalign{\hbox{\raise
5.53241pt\hbox{\scalebox{1.0}[-1.0]{\lower
5.53241pt\hbox{$\displaystyle\scriptscriptstyle\widehat{\vrule
width=0.0pt,height=2.15277pt\vrule
height=0.0pt,width=2.58865pt}$}}}}\cr\hbox{$\displaystyle\scriptscriptstyle\bm{\gamma}$}}}}}^{G})}}$$\displaystyle{\chi^{-1}}$$\displaystyle{\vartheta}$$\displaystyle{\widetilde{\widetilde{\bm{\varphi}}}}$
Figure 4: Deformation of $\displaystyle\widetilde{\widetilde{\varphi}}$ to a
static feedback transformation $\displaystyle\vartheta$.

######  Theorem 7.5.

If a control system $\displaystyle(M,\bm{\omega})$ is cascade feedback
linearizable with respect to a control admissible symmetry group
$\displaystyle G$, then there exists a dynamic compensator
$\displaystyle\bm{u}=\bm{\beta}(t,\bm{x},\bm{y},\bm{W})$ together with linear
dynamics in new state variables $\displaystyle\bm{y}$ such that the dynamic
augmentation is static feedback linearizable.

###### Proof.

Let  $\displaystyle\displaystyle\widehat{\vrule
width=0.0pt,height=6.83331pt\vrule height=0.0pt,width=10.79166pt}$
$\displaystyle\displaystyle M$ be a manifold that locally has the form of an
open subset of $\displaystyle P\times M$ where $\displaystyle P$ is some
manifold with $\displaystyle\dim P=\dim J^{\nu^{\prime}}-\dim J^{\nu}$ and
local coordinates denoted by $\displaystyle(\bm{p})$. Moreover, let
$\displaystyle\widetilde{\pi}:{\mathchoice{{\ooalign{\hbox{\raise
7.09259pt\hbox{\scalebox{1.0}[-1.0]{\lower
7.09259pt\hbox{$\displaystyle\displaystyle\widehat{\vrule
width=0.0pt,height=6.83331pt\vrule
height=0.0pt,width=10.79166pt}$}}}}\cr\hbox{$\displaystyle\displaystyle
M$}}}}{{\ooalign{\hbox{\raise 7.09259pt\hbox{\scalebox{1.0}[-1.0]{\lower
7.09259pt\hbox{$\displaystyle\textstyle\widehat{\vrule
width=0.0pt,height=6.83331pt\vrule
height=0.0pt,width=10.79166pt}$}}}}\cr\hbox{$\displaystyle\textstyle
M$}}}}{{\ooalign{\hbox{\raise 6.40926pt\hbox{\scalebox{1.0}[-1.0]{\lower
6.40926pt\hbox{$\displaystyle\scriptstyle\widehat{\vrule
width=0.0pt,height=4.78333pt\vrule
height=0.0pt,width=7.55415pt}$}}}}\cr\hbox{$\displaystyle\scriptstyle
M$}}}}{{\ooalign{\hbox{\raise 5.9537pt\hbox{\scalebox{1.0}[-1.0]{\lower
5.9537pt\hbox{$\displaystyle\scriptscriptstyle\widehat{\vrule
width=0.0pt,height=3.41666pt\vrule
height=0.0pt,width=5.39581pt}$}}}}\cr\hbox{$\displaystyle\scriptscriptstyle
M$}}}}}\to M$ be the natural projection with typical fiber given by
$\displaystyle P$. Let $\displaystyle(t,\bm{z},\bm{w},\bm{w}^{\prime})$ be
local coordinates on $\displaystyle J^{\nu^{\prime}+\nu^{\perp}}$ with
$\displaystyle\bm{w}^{\prime}$ denoting the coordinates along the fibers of
$\displaystyle J^{\nu^{\prime}+\nu^{\perp}}\to J^{\nu+\nu^{\perp}}$. Then we
define the map
$\displaystyle\widetilde{\widetilde{\varphi}}:{\mathchoice{{\ooalign{\hbox{\raise
7.09259pt\hbox{\scalebox{1.0}[-1.0]{\lower
7.09259pt\hbox{$\displaystyle\displaystyle\widehat{\vrule
width=0.0pt,height=6.83331pt\vrule
height=0.0pt,width=10.79166pt}$}}}}\cr\hbox{$\displaystyle\displaystyle
M$}}}}{{\ooalign{\hbox{\raise 7.09259pt\hbox{\scalebox{1.0}[-1.0]{\lower
7.09259pt\hbox{$\displaystyle\textstyle\widehat{\vrule
width=0.0pt,height=6.83331pt\vrule
height=0.0pt,width=10.79166pt}$}}}}\cr\hbox{$\displaystyle\textstyle
M$}}}}{{\ooalign{\hbox{\raise 6.40926pt\hbox{\scalebox{1.0}[-1.0]{\lower
6.40926pt\hbox{$\displaystyle\scriptstyle\widehat{\vrule
width=0.0pt,height=4.78333pt\vrule
height=0.0pt,width=7.55415pt}$}}}}\cr\hbox{$\displaystyle\scriptstyle
M$}}}}{{\ooalign{\hbox{\raise 5.9537pt\hbox{\scalebox{1.0}[-1.0]{\lower
5.9537pt\hbox{$\displaystyle\scriptscriptstyle\widehat{\vrule
width=0.0pt,height=3.41666pt\vrule
height=0.0pt,width=5.39581pt}$}}}}\cr\hbox{$\displaystyle\scriptscriptstyle
M$}}}}}\to J^{\nu^{\prime}+\nu^{\perp}}\times G$ by

$\widetilde{\widetilde{\varphi}}=\widetilde{\varphi}\circ\widetilde{\pi}\times\\{\bm{w}^{\prime}=\bm{p}\\}.$

Furthermore, we will define the Pfaffian system and dual distribution by

$\displaystyle\bm{\theta}$
$\displaystyle=\left(\widetilde{\widetilde{\varphi}}\right)^{*}\,{\mathchoice{{\ooalign{\hbox{\raise
6.25pt\hbox{\scalebox{1.0}[-1.0]{\lower
6.25pt\hbox{$\displaystyle\displaystyle\widehat{\vrule
width=0.0pt,height=4.30554pt\vrule
height=0.0pt,width=5.1773pt}$}}}}\cr\hbox{$\displaystyle\displaystyle\bm{\gamma}$}}}}{{\ooalign{\hbox{\raise
6.25pt\hbox{\scalebox{1.0}[-1.0]{\lower
6.25pt\hbox{$\displaystyle\textstyle\widehat{\vrule
width=0.0pt,height=4.30554pt\vrule
height=0.0pt,width=5.1773pt}$}}}}\cr\hbox{$\displaystyle\textstyle\bm{\gamma}$}}}}{{\ooalign{\hbox{\raise
5.81944pt\hbox{\scalebox{1.0}[-1.0]{\lower
5.81944pt\hbox{$\displaystyle\scriptstyle\widehat{\vrule
width=0.0pt,height=3.01389pt\vrule
height=0.0pt,width=3.62411pt}$}}}}\cr\hbox{$\displaystyle\scriptstyle\bm{\gamma}$}}}}{{\ooalign{\hbox{\raise
5.53241pt\hbox{\scalebox{1.0}[-1.0]{\lower
5.53241pt\hbox{$\displaystyle\scriptscriptstyle\widehat{\vrule
width=0.0pt,height=2.15277pt\vrule
height=0.0pt,width=2.58865pt}$}}}}\cr\hbox{$\displaystyle\scriptscriptstyle\bm{\gamma}$}}}}}^{G},$
$\displaystyle\mathcal{T}$ $\displaystyle=\mathrm{ann}\,\bm{\theta}.$

Notice that $\displaystyle\widetilde{\widetilde{\varphi}}$ is not a static
feedback transformation since the state and control variables for
$\displaystyle\mathcal{T}$ are not yet well-defined; however it does leave
$\displaystyle t$ unchanged. Additionally, and again because the state and
control variables are not well defined, it is not obvious that
$\displaystyle\mathcal{T}$ is static feedback equivalent to
$\displaystyle\mathrm{pr}\,{\mathcal{H}}_{G}$ and hence static feedback
linearizable. However, since $\displaystyle\widetilde{\widetilde{\varphi}}$ is
a diffeomorphism preserving the independent variable $\displaystyle t$, and
since the refined derived type is a diffeomorphism invariant, we do have that
$\displaystyle\mathcal{T}$ is still a Goursat bundle whose integral curves are
parameterized by $\displaystyle t$. The intersection bundle
$\displaystyle\mathrm{Char}\,\mathcal{T}^{(1)}_{0}$ will allow us to make a
well defined and canonical choice of state and control variables via a
diffeomorphism, as we now demonstrate.

Let $\displaystyle{\mathchoice{{\ooalign{\hbox{\raise
7.09259pt\hbox{\scalebox{1.0}[-1.0]{\lower
7.09259pt\hbox{$\displaystyle\displaystyle\widehat{\vrule
width=0.0pt,height=6.83331pt\vrule
height=0.0pt,width=6.66669pt}$}}}}\cr\hbox{$\displaystyle\displaystyle\Xi$}}}}{{\ooalign{\hbox{\raise
7.09259pt\hbox{\scalebox{1.0}[-1.0]{\lower
7.09259pt\hbox{$\displaystyle\textstyle\widehat{\vrule
width=0.0pt,height=6.83331pt\vrule
height=0.0pt,width=6.66669pt}$}}}}\cr\hbox{$\displaystyle\textstyle\Xi$}}}}{{\ooalign{\hbox{\raise
6.40926pt\hbox{\scalebox{1.0}[-1.0]{\lower
6.40926pt\hbox{$\displaystyle\scriptstyle\widehat{\vrule
width=0.0pt,height=4.78333pt\vrule
height=0.0pt,width=4.66667pt}$}}}}\cr\hbox{$\displaystyle\scriptstyle\Xi$}}}}{{\ooalign{\hbox{\raise
5.9537pt\hbox{\scalebox{1.0}[-1.0]{\lower
5.9537pt\hbox{$\displaystyle\scriptscriptstyle\widehat{\vrule
width=0.0pt,height=3.41666pt\vrule
height=0.0pt,width=3.33334pt}$}}}}\cr\hbox{$\displaystyle\scriptscriptstyle\Xi$}}}}}^{(1)}_{0}$
be the annihilator of
$\displaystyle\mathrm{Char}\,\,\left(\mathrm{pr}\,{\mathcal{H}}_{G}\right)^{(1)}_{0}$.
Then,

${\mathchoice{{\ooalign{\hbox{\raise
7.09259pt\hbox{\scalebox{1.0}[-1.0]{\lower
7.09259pt\hbox{$\displaystyle\displaystyle\widehat{\vrule
width=0.0pt,height=6.83331pt\vrule
height=0.0pt,width=6.66669pt}$}}}}\cr\hbox{$\displaystyle\displaystyle\Xi$}}}}{{\ooalign{\hbox{\raise
7.09259pt\hbox{\scalebox{1.0}[-1.0]{\lower
7.09259pt\hbox{$\displaystyle\textstyle\widehat{\vrule
width=0.0pt,height=6.83331pt\vrule
height=0.0pt,width=6.66669pt}$}}}}\cr\hbox{$\displaystyle\textstyle\Xi$}}}}{{\ooalign{\hbox{\raise
6.40926pt\hbox{\scalebox{1.0}[-1.0]{\lower
6.40926pt\hbox{$\displaystyle\scriptstyle\widehat{\vrule
width=0.0pt,height=4.78333pt\vrule
height=0.0pt,width=4.66667pt}$}}}}\cr\hbox{$\displaystyle\scriptstyle\Xi$}}}}{{\ooalign{\hbox{\raise
5.9537pt\hbox{\scalebox{1.0}[-1.0]{\lower
5.9537pt\hbox{$\displaystyle\scriptscriptstyle\widehat{\vrule
width=0.0pt,height=3.41666pt\vrule
height=0.0pt,width=3.33334pt}$}}}}\cr\hbox{$\displaystyle\scriptscriptstyle\Xi$}}}}}^{(1)}_{0}=\mathrm{span}\left\\{dt,d\varepsilon,d\bm{z}^{\left\lfloor{\nu^{\perp}}\right\rfloor},d\bm{w}^{\left\lfloor{\nu}\right\rfloor},d\bm{w}^{\left\lceil{\nu}\right\rceil},d\bm{w}^{\left\lfloor{\nu^{\prime}-\nu}\right\rfloor}\right\\},$

where $\displaystyle\bm{w}^{\left\lfloor{\nu^{\prime}-\nu}\right\rfloor}$
denotes all the $\displaystyle\bm{w^{\prime}}$ variables of order strictly
smaller than the highest order $\displaystyle\bm{w^{\prime}}$ variables. The
first integrals of this bundle are precisely all the state variables for the
control system $\displaystyle\mathrm{pr}\,{\mathcal{H}}_{G}$ and time
$\displaystyle t$. Additionally, the flow box coordinates for
$\displaystyle\mathrm{Char}\,\,\left(\mathrm{pr}\,{\mathcal{H}}_{G}\right)^{(1)}_{0}$
provide the control variables, which are precisely
$\displaystyle\left(\bm{z}^{\left\lceil{\nu^{\perp}}\right\rceil},\bm{w}^{{\left\lceil{\nu^{\prime}}\right\rceil}}\right)$
for $\displaystyle\mathrm{pr}\,{\mathcal{H}}_{G}$. With this in mind, we now
find the pullback of the bundle
$\displaystyle{\mathchoice{{\ooalign{\hbox{\raise
7.09259pt\hbox{\scalebox{1.0}[-1.0]{\lower
7.09259pt\hbox{$\displaystyle\displaystyle\widehat{\vrule
width=0.0pt,height=6.83331pt\vrule
height=0.0pt,width=6.66669pt}$}}}}\cr\hbox{$\displaystyle\displaystyle\Xi$}}}}{{\ooalign{\hbox{\raise
7.09259pt\hbox{\scalebox{1.0}[-1.0]{\lower
7.09259pt\hbox{$\displaystyle\textstyle\widehat{\vrule
width=0.0pt,height=6.83331pt\vrule
height=0.0pt,width=6.66669pt}$}}}}\cr\hbox{$\displaystyle\textstyle\Xi$}}}}{{\ooalign{\hbox{\raise
6.40926pt\hbox{\scalebox{1.0}[-1.0]{\lower
6.40926pt\hbox{$\displaystyle\scriptstyle\widehat{\vrule
width=0.0pt,height=4.78333pt\vrule
height=0.0pt,width=4.66667pt}$}}}}\cr\hbox{$\displaystyle\scriptstyle\Xi$}}}}{{\ooalign{\hbox{\raise
5.9537pt\hbox{\scalebox{1.0}[-1.0]{\lower
5.9537pt\hbox{$\displaystyle\scriptscriptstyle\widehat{\vrule
width=0.0pt,height=3.41666pt\vrule
height=0.0pt,width=3.33334pt}$}}}}\cr\hbox{$\displaystyle\scriptscriptstyle\Xi$}}}}}^{(1)}_{0}$
by $\displaystyle\widetilde{\widetilde{\varphi}}$ to be

$\displaystyle\left(\widetilde{\widetilde{\varphi}}\right)^{*}{\mathchoice{{\ooalign{\hbox{\raise
7.09259pt\hbox{\scalebox{1.0}[-1.0]{\lower
7.09259pt\hbox{$\displaystyle\displaystyle\widehat{\vrule
width=0.0pt,height=6.83331pt\vrule
height=0.0pt,width=6.66669pt}$}}}}\cr\hbox{$\displaystyle\displaystyle\Xi$}}}}{{\ooalign{\hbox{\raise
7.09259pt\hbox{\scalebox{1.0}[-1.0]{\lower
7.09259pt\hbox{$\displaystyle\textstyle\widehat{\vrule
width=0.0pt,height=6.83331pt\vrule
height=0.0pt,width=6.66669pt}$}}}}\cr\hbox{$\displaystyle\textstyle\Xi$}}}}{{\ooalign{\hbox{\raise
6.40926pt\hbox{\scalebox{1.0}[-1.0]{\lower
6.40926pt\hbox{$\displaystyle\scriptstyle\widehat{\vrule
width=0.0pt,height=4.78333pt\vrule
height=0.0pt,width=4.66667pt}$}}}}\cr\hbox{$\displaystyle\scriptstyle\Xi$}}}}{{\ooalign{\hbox{\raise
5.9537pt\hbox{\scalebox{1.0}[-1.0]{\lower
5.9537pt\hbox{$\displaystyle\scriptscriptstyle\widehat{\vrule
width=0.0pt,height=3.41666pt\vrule
height=0.0pt,width=3.33334pt}$}}}}\cr\hbox{$\displaystyle\scriptscriptstyle\Xi$}}}}}^{(1)}_{0}=\mathrm{span}\left\\{dt,d\bm{x},d\left(\bm{w}^{\left\lceil{\nu}\right\rceil}(t,\bm{x},\bm{u})\right),d\bm{p}^{\left\lfloor{\nu^{\prime}-\nu}\right\rfloor}\right\\},$

since

$\left(\widetilde{\widetilde{\varphi}}\right)^{*}\mathrm{span}\left\\{dt,d\varepsilon,d\bm{z}^{\left\lfloor{\nu^{\perp}}\right\rfloor},d\bm{w}^{\left\lfloor{\nu}\right\rfloor}\right\\}=\mathrm{span}\\{dt,d\bm{x}\\},$

by construction of $\displaystyle\widetilde{\widetilde{\varphi}}$ from
$\displaystyle\widetilde{\varphi}$. Without loss of generality, let us relabel
the indices of the control variables $\displaystyle(u^{1},\ldots,u^{m})$ of
$\displaystyle(M,\bm{\omega})$ so that of the $\displaystyle
m=m_{\nu^{\perp}}+m_{\nu}$ controls, the last $\displaystyle m_{\nu}$ control
variables $\displaystyle\left(u^{m_{\nu^{\perp}}+1},\ldots,u^{m_{\nu}}\right)$
may be determined from the equations
$\displaystyle\bm{y}=\left(\bm{w}^{\left\lceil{\nu}\right\rceil}(t,\bm{x},\bm{u}),\bm{p}^{\left\lfloor{\nu^{\prime}-\nu}\right\rfloor}\right)$
in terms of $\displaystyle\bm{x},\,\bm{y}$, and
$\displaystyle(u^{1},\ldots,u^{m_{\nu^{\perp}}})$. Let
$\displaystyle\chi:{\mathchoice{{\ooalign{\hbox{\raise
7.09259pt\hbox{\scalebox{1.0}[-1.0]{\lower
7.09259pt\hbox{$\displaystyle\displaystyle\widehat{\vrule
width=0.0pt,height=6.83331pt\vrule
height=0.0pt,width=10.79166pt}$}}}}\cr\hbox{$\displaystyle\displaystyle
M$}}}}{{\ooalign{\hbox{\raise 7.09259pt\hbox{\scalebox{1.0}[-1.0]{\lower
7.09259pt\hbox{$\displaystyle\textstyle\widehat{\vrule
width=0.0pt,height=6.83331pt\vrule
height=0.0pt,width=10.79166pt}$}}}}\cr\hbox{$\displaystyle\textstyle
M$}}}}{{\ooalign{\hbox{\raise 6.40926pt\hbox{\scalebox{1.0}[-1.0]{\lower
6.40926pt\hbox{$\displaystyle\scriptstyle\widehat{\vrule
width=0.0pt,height=4.78333pt\vrule
height=0.0pt,width=7.55415pt}$}}}}\cr\hbox{$\displaystyle\scriptstyle
M$}}}}{{\ooalign{\hbox{\raise 5.9537pt\hbox{\scalebox{1.0}[-1.0]{\lower
5.9537pt\hbox{$\displaystyle\scriptscriptstyle\widehat{\vrule
width=0.0pt,height=3.41666pt\vrule
height=0.0pt,width=5.39581pt}$}}}}\cr\hbox{$\displaystyle\scriptscriptstyle
M$}}}}}\to{\mathchoice{{\ooalign{\hbox{\raise
7.09259pt\hbox{\scalebox{1.0}[-1.0]{\lower
7.09259pt\hbox{$\displaystyle\displaystyle\widehat{\vrule
width=0.0pt,height=6.83331pt\vrule
height=0.0pt,width=10.79166pt}$}}}}\cr\hbox{$\displaystyle\displaystyle
M$}}}}{{\ooalign{\hbox{\raise 7.09259pt\hbox{\scalebox{1.0}[-1.0]{\lower
7.09259pt\hbox{$\displaystyle\textstyle\widehat{\vrule
width=0.0pt,height=6.83331pt\vrule
height=0.0pt,width=10.79166pt}$}}}}\cr\hbox{$\displaystyle\textstyle
M$}}}}{{\ooalign{\hbox{\raise 6.40926pt\hbox{\scalebox{1.0}[-1.0]{\lower
6.40926pt\hbox{$\displaystyle\scriptstyle\widehat{\vrule
width=0.0pt,height=4.78333pt\vrule
height=0.0pt,width=7.55415pt}$}}}}\cr\hbox{$\displaystyle\scriptstyle
M$}}}}{{\ooalign{\hbox{\raise 5.9537pt\hbox{\scalebox{1.0}[-1.0]{\lower
5.9537pt\hbox{$\displaystyle\scriptscriptstyle\widehat{\vrule
width=0.0pt,height=3.41666pt\vrule
height=0.0pt,width=5.39581pt}$}}}}\cr\hbox{$\displaystyle\scriptscriptstyle
M$}}}}}$ be the diffeomorphism whose inverse is given by

$(t,\bm{x},\bm{u},\bm{p})\mapsto(t,\bm{x},\bm{y},\bm{W}),$

where

$\displaystyle\bm{y}$
$\displaystyle=\left(\bm{w}^{\left\lceil{\nu}\right\rceil}(t,\bm{x},\bm{u}),\bm{p}^{\left\lfloor{\nu^{\prime}-\nu}\right\rfloor}\right),$
(63) $\displaystyle\bm{W}$
$\displaystyle=\left(u^{1},\ldots,u^{m_{\nu^{\perp}}},\bm{p}^{\left\lceil{\nu^{\prime}}\right\rceil}\right).$

Using this diffeomorphism, we find that

$\left(\widetilde{\widetilde{\varphi}}\circ\chi^{-1}\right)^{*}{\mathchoice{{\ooalign{\hbox{\raise
7.09259pt\hbox{\scalebox{1.0}[-1.0]{\lower
7.09259pt\hbox{$\displaystyle\displaystyle\widehat{\vrule
width=0.0pt,height=6.83331pt\vrule
height=0.0pt,width=6.66669pt}$}}}}\cr\hbox{$\displaystyle\displaystyle\Xi$}}}}{{\ooalign{\hbox{\raise
7.09259pt\hbox{\scalebox{1.0}[-1.0]{\lower
7.09259pt\hbox{$\displaystyle\textstyle\widehat{\vrule
width=0.0pt,height=6.83331pt\vrule
height=0.0pt,width=6.66669pt}$}}}}\cr\hbox{$\displaystyle\textstyle\Xi$}}}}{{\ooalign{\hbox{\raise
6.40926pt\hbox{\scalebox{1.0}[-1.0]{\lower
6.40926pt\hbox{$\displaystyle\scriptstyle\widehat{\vrule
width=0.0pt,height=4.78333pt\vrule
height=0.0pt,width=4.66667pt}$}}}}\cr\hbox{$\displaystyle\scriptstyle\Xi$}}}}{{\ooalign{\hbox{\raise
5.9537pt\hbox{\scalebox{1.0}[-1.0]{\lower
5.9537pt\hbox{$\displaystyle\scriptscriptstyle\widehat{\vrule
width=0.0pt,height=3.41666pt\vrule
height=0.0pt,width=3.33334pt}$}}}}\cr\hbox{$\displaystyle\scriptscriptstyle\Xi$}}}}}^{(1)}_{0}=\mathrm{span}\\{dt,d\bm{x},d\bm{y}\\}.$

It follows that if $\displaystyle X$ is a vector field on $\displaystyle
J^{\nu^{\perp}+\nu^{\prime}}\times G$ such that
$\displaystyle\big{(}\widetilde{\widetilde{\varphi}}\circ\chi^{-1}\big{)}_{*}X\in\text{ann}\,{\mathchoice{{\ooalign{\hbox{\raise
7.09259pt\hbox{\scalebox{1.0}[-1.0]{\lower
7.09259pt\hbox{$\displaystyle\displaystyle\widehat{\vrule
width=0.0pt,height=6.83331pt\vrule
height=0.0pt,width=6.66669pt}$}}}}\cr\hbox{$\displaystyle\displaystyle\Xi$}}}}{{\ooalign{\hbox{\raise
7.09259pt\hbox{\scalebox{1.0}[-1.0]{\lower
7.09259pt\hbox{$\displaystyle\textstyle\widehat{\vrule
width=0.0pt,height=6.83331pt\vrule
height=0.0pt,width=6.66669pt}$}}}}\cr\hbox{$\displaystyle\textstyle\Xi$}}}}{{\ooalign{\hbox{\raise
6.40926pt\hbox{\scalebox{1.0}[-1.0]{\lower
6.40926pt\hbox{$\displaystyle\scriptstyle\widehat{\vrule
width=0.0pt,height=4.78333pt\vrule
height=0.0pt,width=4.66667pt}$}}}}\cr\hbox{$\displaystyle\scriptstyle\Xi$}}}}{{\ooalign{\hbox{\raise
5.9537pt\hbox{\scalebox{1.0}[-1.0]{\lower
5.9537pt\hbox{$\displaystyle\scriptscriptstyle\widehat{\vrule
width=0.0pt,height=3.41666pt\vrule
height=0.0pt,width=3.33334pt}$}}}}\cr\hbox{$\displaystyle\scriptscriptstyle\Xi$}}}}}^{(1)}_{0}=\mathrm{Char}\,\left(\text{pr}{\mathcal{H}}_{G}\right)^{(1)}_{0}$
then
$\displaystyle\chi^{-1}_{*}X\in\chi^{-1}_{*}\mathrm{Char}\,\left(\text{pr}{\mathcal{H}}_{G}\right)^{(1)}_{0}=\mathrm{Char}\,{\mathcal{T}}^{(1)}_{0}$.
Therefore, $\displaystyle\chi_{*}\mathrm{Char}\,{\mathcal{T}}^{(1)}_{0}$ is
annihilated by $\displaystyle\mathrm{span}\\{dt,d\bm{x},d\bm{y}\\}$, from
which we have

$\chi_{*}\Big{(}\mathrm{Char}\,\mathcal{T}^{(1)}_{0}\Big{)}=\mathrm{span}\\{\partial_{W^{1}},\ldots,\partial_{W^{m}}\\}.$

Now we have that
$\displaystyle\chi_{*}{\mathcal{T}}=\chi_{*}\widetilde{\widetilde{\varphi}}^{-1}_{*}\left(\text{pr}{\mathcal{H}}_{G}\right):=\psi_{*}\left(\text{pr}{\mathcal{H}}_{G}\right)$
and since $\displaystyle\text{pr}{\mathcal{H}}_{G}$ is SFL and
$\displaystyle\psi=\chi\circ\widetilde{\widetilde{\varphi}}^{-1}$ is a
diffeomorphism we have that $\displaystyle\chi_{*}{\mathcal{T}}$ is a Goursat
bundle and so the latter has a Cauchy bundle
$\displaystyle\mathrm{Char}\,\big{(}\chi_{*}{\mathcal{T}}^{(k-1)}\big{)}$ with
an annihilator which we will denote by
$\displaystyle\Lambda^{(k^{\prime}-1)}$, where $\displaystyle k^{\prime}$ is
the derived is the derived length of $\displaystyle{\mathcal{T}}$; therefore
we have that
$\displaystyle\Lambda^{(k^{\prime}-1)}=(\psi^{-1})^{*}{\mathchoice{{\ooalign{\hbox{\raise
7.09259pt\hbox{\scalebox{1.0}[-1.0]{\lower
7.09259pt\hbox{$\displaystyle\displaystyle\widehat{\vrule
width=0.0pt,height=6.83331pt\vrule
height=0.0pt,width=6.66669pt}$}}}}\cr\hbox{$\displaystyle\displaystyle\Xi$}}}}{{\ooalign{\hbox{\raise
7.09259pt\hbox{\scalebox{1.0}[-1.0]{\lower
7.09259pt\hbox{$\displaystyle\textstyle\widehat{\vrule
width=0.0pt,height=6.83331pt\vrule
height=0.0pt,width=6.66669pt}$}}}}\cr\hbox{$\displaystyle\textstyle\Xi$}}}}{{\ooalign{\hbox{\raise
6.40926pt\hbox{\scalebox{1.0}[-1.0]{\lower
6.40926pt\hbox{$\displaystyle\scriptstyle\widehat{\vrule
width=0.0pt,height=4.78333pt\vrule
height=0.0pt,width=4.66667pt}$}}}}\cr\hbox{$\displaystyle\scriptstyle\Xi$}}}}{{\ooalign{\hbox{\raise
5.9537pt\hbox{\scalebox{1.0}[-1.0]{\lower
5.9537pt\hbox{$\displaystyle\scriptscriptstyle\widehat{\vrule
width=0.0pt,height=3.41666pt\vrule
height=0.0pt,width=3.33334pt}$}}}}\cr\hbox{$\displaystyle\scriptscriptstyle\Xi$}}}}}^{(k^{\prime}-1)}$.
In the case $\displaystyle\Delta_{k^{\prime}}=1$, we have that $\displaystyle
dt\in{\mathchoice{{\ooalign{\hbox{\raise
7.09259pt\hbox{\scalebox{1.0}[-1.0]{\lower
7.09259pt\hbox{$\displaystyle\displaystyle\widehat{\vrule
width=0.0pt,height=6.83331pt\vrule
height=0.0pt,width=6.66669pt}$}}}}\cr\hbox{$\displaystyle\displaystyle\Xi$}}}}{{\ooalign{\hbox{\raise
7.09259pt\hbox{\scalebox{1.0}[-1.0]{\lower
7.09259pt\hbox{$\displaystyle\textstyle\widehat{\vrule
width=0.0pt,height=6.83331pt\vrule
height=0.0pt,width=6.66669pt}$}}}}\cr\hbox{$\displaystyle\textstyle\Xi$}}}}{{\ooalign{\hbox{\raise
6.40926pt\hbox{\scalebox{1.0}[-1.0]{\lower
6.40926pt\hbox{$\displaystyle\scriptstyle\widehat{\vrule
width=0.0pt,height=4.78333pt\vrule
height=0.0pt,width=4.66667pt}$}}}}\cr\hbox{$\displaystyle\scriptstyle\Xi$}}}}{{\ooalign{\hbox{\raise
5.9537pt\hbox{\scalebox{1.0}[-1.0]{\lower
5.9537pt\hbox{$\displaystyle\scriptscriptstyle\widehat{\vrule
width=0.0pt,height=3.41666pt\vrule
height=0.0pt,width=3.33334pt}$}}}}\cr\hbox{$\displaystyle\scriptscriptstyle\Xi$}}}}}^{(k^{\prime}-1)}$
and hence since $\displaystyle\psi$ preserves $\displaystyle t$
$\displaystyle(t\circ\psi=t)$, we see that $\displaystyle
dt\in\Lambda^{(k^{\prime}-1)}$. Therefore in the case
$\displaystyle\Delta_{k^{\prime}}=1$, the hypotheses of Theorem 3.3 are
satisfied and we can conclude that $\displaystyle\chi_{*}{\mathcal{T}}$ is
SFL. A similar argument holds in the case
$\displaystyle\Delta_{k^{\prime}}>1$.

Let $\displaystyle{\mathchoice{{\ooalign{\hbox{\raise
7.09259pt\hbox{\scalebox{1.0}[-1.0]{\lower
7.09259pt\hbox{$\displaystyle\displaystyle\widehat{\vrule
width=0.0pt,height=6.83331pt\vrule
height=0.0pt,width=7.50002pt}$}}}}\cr\hbox{$\displaystyle\displaystyle{\mathcal{V}}$}}}}{{\ooalign{\hbox{\raise
7.09259pt\hbox{\scalebox{1.0}[-1.0]{\lower
7.09259pt\hbox{$\displaystyle\textstyle\widehat{\vrule
width=0.0pt,height=6.83331pt\vrule
height=0.0pt,width=7.50002pt}$}}}}\cr\hbox{$\displaystyle\textstyle{\mathcal{V}}$}}}}{{\ooalign{\hbox{\raise
6.40926pt\hbox{\scalebox{1.0}[-1.0]{\lower
6.40926pt\hbox{$\displaystyle\scriptstyle\widehat{\vrule
width=0.0pt,height=4.78333pt\vrule
height=0.0pt,width=5.25pt}$}}}}\cr\hbox{$\displaystyle\scriptstyle{\mathcal{V}}$}}}}{{\ooalign{\hbox{\raise
5.9537pt\hbox{\scalebox{1.0}[-1.0]{\lower
5.9537pt\hbox{$\displaystyle\scriptscriptstyle\widehat{\vrule
width=0.0pt,height=3.41666pt\vrule
height=0.0pt,width=3.75pt}$}}}}\cr\hbox{$\displaystyle\scriptscriptstyle{\mathcal{V}}$}}}}}:=\chi_{*}\mathcal{T}$
and let $\displaystyle{\mathchoice{{\ooalign{\hbox{\raise
6.25pt\hbox{\scalebox{1.0}[-1.0]{\lower
6.25pt\hbox{$\displaystyle\displaystyle\widehat{\vrule
width=0.0pt,height=4.30554pt\vrule
height=0.0pt,width=6.22453pt}$}}}}\cr\hbox{$\displaystyle\displaystyle\bm{\omega}$}}}}{{\ooalign{\hbox{\raise
6.25pt\hbox{\scalebox{1.0}[-1.0]{\lower
6.25pt\hbox{$\displaystyle\textstyle\widehat{\vrule
width=0.0pt,height=4.30554pt\vrule
height=0.0pt,width=6.22453pt}$}}}}\cr\hbox{$\displaystyle\textstyle\bm{\omega}$}}}}{{\ooalign{\hbox{\raise
5.81944pt\hbox{\scalebox{1.0}[-1.0]{\lower
5.81944pt\hbox{$\displaystyle\scriptstyle\widehat{\vrule
width=0.0pt,height=3.01389pt\vrule
height=0.0pt,width=4.35718pt}$}}}}\cr\hbox{$\displaystyle\scriptstyle\bm{\omega}$}}}}{{\ooalign{\hbox{\raise
5.53241pt\hbox{\scalebox{1.0}[-1.0]{\lower
5.53241pt\hbox{$\displaystyle\scriptscriptstyle\widehat{\vrule
width=0.0pt,height=2.15277pt\vrule
height=0.0pt,width=3.11226pt}$}}}}\cr\hbox{$\displaystyle\scriptscriptstyle\bm{\omega}$}}}}}:=\mathrm{ann}\,{\mathchoice{{\ooalign{\hbox{\raise
7.09259pt\hbox{\scalebox{1.0}[-1.0]{\lower
7.09259pt\hbox{$\displaystyle\displaystyle\widehat{\vrule
width=0.0pt,height=6.83331pt\vrule
height=0.0pt,width=7.50002pt}$}}}}\cr\hbox{$\displaystyle\displaystyle{\mathcal{V}}$}}}}{{\ooalign{\hbox{\raise
7.09259pt\hbox{\scalebox{1.0}[-1.0]{\lower
7.09259pt\hbox{$\displaystyle\textstyle\widehat{\vrule
width=0.0pt,height=6.83331pt\vrule
height=0.0pt,width=7.50002pt}$}}}}\cr\hbox{$\displaystyle\textstyle{\mathcal{V}}$}}}}{{\ooalign{\hbox{\raise
6.40926pt\hbox{\scalebox{1.0}[-1.0]{\lower
6.40926pt\hbox{$\displaystyle\scriptstyle\widehat{\vrule
width=0.0pt,height=4.78333pt\vrule
height=0.0pt,width=5.25pt}$}}}}\cr\hbox{$\displaystyle\scriptstyle{\mathcal{V}}$}}}}{{\ooalign{\hbox{\raise
5.9537pt\hbox{\scalebox{1.0}[-1.0]{\lower
5.9537pt\hbox{$\displaystyle\scriptscriptstyle\widehat{\vrule
width=0.0pt,height=3.41666pt\vrule
height=0.0pt,width=3.75pt}$}}}}\cr\hbox{$\displaystyle\scriptscriptstyle{\mathcal{V}}$}}}}}$.
Then,  $\displaystyle\displaystyle\widehat{\vrule
width=0.0pt,height=4.30554pt\vrule height=0.0pt,width=6.22453pt}$
$\displaystyle\displaystyle\bm{\omega}$ is precisely the dynamic feedback
augmentation of $\displaystyle\bm{\omega}$ with dynamic compensator
$\displaystyle\bm{u}=\bm{\beta}(t,\bm{x},\bm{y},\bm{W})$ given by solving
equations (63) for $\displaystyle\bm{u}$. We have that the map
$\displaystyle\vartheta:=\widetilde{\widetilde{\varphi}}\circ\chi^{-1}:{\mathchoice{{\ooalign{\hbox{\raise
7.09259pt\hbox{\scalebox{1.0}[-1.0]{\lower
7.09259pt\hbox{$\displaystyle\displaystyle\widehat{\vrule
width=0.0pt,height=6.83331pt\vrule
height=0.0pt,width=10.79166pt}$}}}}\cr\hbox{$\displaystyle\displaystyle
M$}}}}{{\ooalign{\hbox{\raise 7.09259pt\hbox{\scalebox{1.0}[-1.0]{\lower
7.09259pt\hbox{$\displaystyle\textstyle\widehat{\vrule
width=0.0pt,height=6.83331pt\vrule
height=0.0pt,width=10.79166pt}$}}}}\cr\hbox{$\displaystyle\textstyle
M$}}}}{{\ooalign{\hbox{\raise 6.40926pt\hbox{\scalebox{1.0}[-1.0]{\lower
6.40926pt\hbox{$\displaystyle\scriptstyle\widehat{\vrule
width=0.0pt,height=4.78333pt\vrule
height=0.0pt,width=7.55415pt}$}}}}\cr\hbox{$\displaystyle\scriptstyle
M$}}}}{{\ooalign{\hbox{\raise 5.9537pt\hbox{\scalebox{1.0}[-1.0]{\lower
5.9537pt\hbox{$\displaystyle\scriptscriptstyle\widehat{\vrule
width=0.0pt,height=3.41666pt\vrule
height=0.0pt,width=5.39581pt}$}}}}\cr\hbox{$\displaystyle\scriptscriptstyle
M$}}}}}\to J^{\nu^{\perp}+\nu^{\prime}}\times G$ is a static feedback
transformation, and since $\displaystyle{\mathchoice{{\ooalign{\hbox{\raise
6.25pt\hbox{\scalebox{1.0}[-1.0]{\lower
6.25pt\hbox{$\displaystyle\displaystyle\widehat{\vrule
width=0.0pt,height=4.30554pt\vrule
height=0.0pt,width=5.1773pt}$}}}}\cr\hbox{$\displaystyle\displaystyle\bm{\gamma}$}}}}{{\ooalign{\hbox{\raise
6.25pt\hbox{\scalebox{1.0}[-1.0]{\lower
6.25pt\hbox{$\displaystyle\textstyle\widehat{\vrule
width=0.0pt,height=4.30554pt\vrule
height=0.0pt,width=5.1773pt}$}}}}\cr\hbox{$\displaystyle\textstyle\bm{\gamma}$}}}}{{\ooalign{\hbox{\raise
5.81944pt\hbox{\scalebox{1.0}[-1.0]{\lower
5.81944pt\hbox{$\displaystyle\scriptstyle\widehat{\vrule
width=0.0pt,height=3.01389pt\vrule
height=0.0pt,width=3.62411pt}$}}}}\cr\hbox{$\displaystyle\scriptstyle\bm{\gamma}$}}}}{{\ooalign{\hbox{\raise
5.53241pt\hbox{\scalebox{1.0}[-1.0]{\lower
5.53241pt\hbox{$\displaystyle\scriptscriptstyle\widehat{\vrule
width=0.0pt,height=2.15277pt\vrule
height=0.0pt,width=2.58865pt}$}}}}\cr\hbox{$\displaystyle\scriptscriptstyle\bm{\gamma}$}}}}}^{G}$
is SFL, then  $\displaystyle\displaystyle\widehat{\vrule
width=0.0pt,height=4.30554pt\vrule height=0.0pt,width=6.22453pt}$
$\displaystyle\displaystyle\bm{\omega}$ must also be SFL. Lastly, we show that
the associated dynamics for $\displaystyle\bm{y}$ are linear. Denote the
$\displaystyle\nu^{\prime}$ prolongation of $\displaystyle\bm{\gamma}^{G}$ by

$\displaystyle\alpha=\mathrm{span}\left\\{d\bm{w}^{\left\lceil{\nu}\right\rceil}-\bm{w}_{1}^{\left\lfloor{\nu^{\prime}-\nu}\right\rfloor}\,dt,\ldots,d\bm{w}_{l_{\nu^{\prime}}-1}^{\left\lfloor{\nu-\nu^{\prime}}\right\rfloor}-\bm{w}^{\left\lceil{\nu^{\prime}}\right\rceil}dt\right\\},$

where $\displaystyle l_{\nu^{\prime}}$ is a place holder for the highest order
jet variable determined by Theorem 7.2. Then we merely observe that a pullback
by $\displaystyle\widetilde{\widetilde{\varphi}}$ of $\displaystyle\alpha$
results in

$\displaystyle\left(\widetilde{\widetilde{\varphi}}\right)^{*}\alpha=\mathrm{span}\left\\{d\left(\bm{w}^{\left\lceil{\nu}\right\rceil}(t,\bm{x},\bm{u})\right)-\bm{p}_{1}^{\left\lfloor{\nu^{\prime}-\nu}\right\rfloor}\,dt,\ldots,d\bm{p}_{l_{\nu^{\prime}}-1}^{\left\lfloor{\nu-\nu^{\prime}}\right\rfloor}-\bm{p}^{\left\lceil{\nu^{\prime}}\right\rceil}dt\right\\}.$

Thus, one final pullback by $\displaystyle\chi^{-1}$ results in

$\displaystyle\left(\widetilde{\widetilde{\varphi}}\circ\chi^{-1}\right)^{*}\alpha=\mathrm{span}\left\\{d\bm{y}_{0}-\bm{y}_{1}\,dt,\ldots,d\bm{y}_{l_{\nu}^{\prime}-1}-\bm{W}^{\left\lceil{\nu^{\prime}}\right\rceil}\,dt\right\\},$

where the subscripts on $\displaystyle\bm{y}_{i}$ for $\displaystyle 1\leq
i\leq l_{\nu}$ are used to keep track of how the $\displaystyle\bm{y}$
variables are assigned to the $\displaystyle\bm{p}$ variables and
$\displaystyle\bm{W}^{\left\lceil{\nu^{\prime}}\right\rceil}$ denotes those
control variables arising from the highest order $\displaystyle\bm{p}$
variables. Thus, we can see that the above Pfaffian system is that of a linear
system. ∎

In Example 7.3, the coordinate change $\displaystyle\chi$ in Step 3 turns out
to be the identity transformation, as we demonstrate at the end of this
section. This can always be arranged for systems which are dynamic feedback
linearizable by differentiation of the given inputs. We note also that
sometimes $\displaystyle\mathrm{Char}\,\mathcal{T}^{(1)}_{0}$ is already in a
basis of coordinate vector fields guaranteed by the Frobenius theorem, which
defines the control variables for a dynamic feedback linearization of
$\displaystyle{\mathcal{V}}$ without the need for the map $\displaystyle\chi$
(which can be chosen to be the identity). This choice can be made at the cost
of having nonlinear dynamics for the variables $\displaystyle\bm{p}$; see §9
for an example. We also have the following Corollary concerning prolongation
by differentiation.

######  Corollary 7.6.

Let $\displaystyle(M,\bm{\omega})$ be a CFL control system with respect to a
state-space symmetry group $\displaystyle G$. Then
$\displaystyle(M,\bm{\omega})$ is DFL by differentiation of the given inputs.

###### Proof.

The proof of Theorem 7.5 essentially shows that the dynamic compensator
consists of components of the inverse of the map
$\displaystyle\widetilde{\widetilde{\varphi}}$ which, in turn, is constructed
from $\displaystyle\widetilde{\varphi}$. The inverse of
$\displaystyle\widetilde{\varphi}$ may be written in terms of the inverse of
$\displaystyle\varphi$ which gives expressions for the control variables
$\displaystyle\bm{v}$ on $\displaystyle M/G$. That is,
$\displaystyle\varphi^{-1}$ acts as the identity on $\displaystyle\bm{v}$.
However, if $\displaystyle G$ is only a state-space symmetry group, then the
controls of $\displaystyle(M,\bm{\omega})$ descend to
$\displaystyle(M/G,\bm{\omega}/G)$ since they are invariant functions of the
action of $\displaystyle G$ i.e. $\displaystyle\bm{v}=\bm{u}$. As a result, we
have
$\displaystyle\bm{u}=\big{(}\bm{W},\bm{w}^{\left\lceil{\nu}\right\rceil}\big{)}$
which is a dynamic compensator by partial prolongation. ∎

The need for - and the construction of - a nontrivial $\displaystyle\chi$ in
the general case will be illustrated by the example in §8.

Finally, we summarize the relationship between CFL control systems and DFL
control systems by the diagram in Fig. 5.

$\displaystyle{{({\mathchoice{{\ooalign{\hbox{\raise
7.09259pt\hbox{\scalebox{1.0}[-1.0]{\lower
7.09259pt\hbox{$\displaystyle\displaystyle\widehat{\vrule
width=0.0pt,height=6.83331pt\vrule
height=0.0pt,width=10.79166pt}$}}}}\cr\hbox{$\displaystyle\displaystyle
M$}}}}{{\ooalign{\hbox{\raise 7.09259pt\hbox{\scalebox{1.0}[-1.0]{\lower
7.09259pt\hbox{$\displaystyle\textstyle\widehat{\vrule
width=0.0pt,height=6.83331pt\vrule
height=0.0pt,width=10.79166pt}$}}}}\cr\hbox{$\displaystyle\textstyle
M$}}}}{{\ooalign{\hbox{\raise 6.40926pt\hbox{\scalebox{1.0}[-1.0]{\lower
6.40926pt\hbox{$\displaystyle\scriptstyle\widehat{\vrule
width=0.0pt,height=4.78333pt\vrule
height=0.0pt,width=7.55415pt}$}}}}\cr\hbox{$\displaystyle\scriptstyle
M$}}}}{{\ooalign{\hbox{\raise 5.9537pt\hbox{\scalebox{1.0}[-1.0]{\lower
5.9537pt\hbox{$\displaystyle\scriptscriptstyle\widehat{\vrule
width=0.0pt,height=3.41666pt\vrule
height=0.0pt,width=5.39581pt}$}}}}\cr\hbox{$\displaystyle\scriptscriptstyle
M$}}}}},{\mathchoice{{\ooalign{\hbox{\raise
6.25pt\hbox{\scalebox{1.0}[-1.0]{\lower
6.25pt\hbox{$\displaystyle\displaystyle\widehat{\vrule
width=0.0pt,height=4.30554pt\vrule
height=0.0pt,width=6.22453pt}$}}}}\cr\hbox{$\displaystyle\displaystyle\bm{\omega}$}}}}{{\ooalign{\hbox{\raise
6.25pt\hbox{\scalebox{1.0}[-1.0]{\lower
6.25pt\hbox{$\displaystyle\textstyle\widehat{\vrule
width=0.0pt,height=4.30554pt\vrule
height=0.0pt,width=6.22453pt}$}}}}\cr\hbox{$\displaystyle\textstyle\bm{\omega}$}}}}{{\ooalign{\hbox{\raise
5.81944pt\hbox{\scalebox{1.0}[-1.0]{\lower
5.81944pt\hbox{$\displaystyle\scriptstyle\widehat{\vrule
width=0.0pt,height=3.01389pt\vrule
height=0.0pt,width=4.35718pt}$}}}}\cr\hbox{$\displaystyle\scriptstyle\bm{\omega}$}}}}{{\ooalign{\hbox{\raise
5.53241pt\hbox{\scalebox{1.0}[-1.0]{\lower
5.53241pt\hbox{$\displaystyle\scriptscriptstyle\widehat{\vrule
width=0.0pt,height=2.15277pt\vrule
height=0.0pt,width=3.11226pt}$}}}}\cr\hbox{$\displaystyle\scriptscriptstyle\bm{\omega}$}}}}})}}$$\displaystyle{{({\mathchoice{{\ooalign{\hbox{\raise
7.09259pt\hbox{\scalebox{1.0}[-1.0]{\lower
7.09259pt\hbox{$\displaystyle\displaystyle\widehat{\vrule
width=0.0pt,height=6.83331pt\vrule
height=0.0pt,width=10.79166pt}$}}}}\cr\hbox{$\displaystyle\displaystyle
M$}}}}{{\ooalign{\hbox{\raise 7.09259pt\hbox{\scalebox{1.0}[-1.0]{\lower
7.09259pt\hbox{$\displaystyle\textstyle\widehat{\vrule
width=0.0pt,height=6.83331pt\vrule
height=0.0pt,width=10.79166pt}$}}}}\cr\hbox{$\displaystyle\textstyle
M$}}}}{{\ooalign{\hbox{\raise 6.40926pt\hbox{\scalebox{1.0}[-1.0]{\lower
6.40926pt\hbox{$\displaystyle\scriptstyle\widehat{\vrule
width=0.0pt,height=4.78333pt\vrule
height=0.0pt,width=7.55415pt}$}}}}\cr\hbox{$\displaystyle\scriptstyle
M$}}}}{{\ooalign{\hbox{\raise 5.9537pt\hbox{\scalebox{1.0}[-1.0]{\lower
5.9537pt\hbox{$\displaystyle\scriptscriptstyle\widehat{\vrule
width=0.0pt,height=3.41666pt\vrule
height=0.0pt,width=5.39581pt}$}}}}\cr\hbox{$\displaystyle\scriptscriptstyle
M$}}}}},{\mathchoice{{\ooalign{\hbox{\raise
6.25pt\hbox{\scalebox{1.0}[-1.0]{\lower
6.25pt\hbox{$\displaystyle\displaystyle\widehat{\vrule
width=0.0pt,height=4.30554pt\vrule
height=0.0pt,width=6.22453pt}$}}}}\cr\hbox{$\displaystyle\displaystyle\bm{\omega}$}}}}{{\ooalign{\hbox{\raise
6.25pt\hbox{\scalebox{1.0}[-1.0]{\lower
6.25pt\hbox{$\displaystyle\textstyle\widehat{\vrule
width=0.0pt,height=4.30554pt\vrule
height=0.0pt,width=6.22453pt}$}}}}\cr\hbox{$\displaystyle\textstyle\bm{\omega}$}}}}{{\ooalign{\hbox{\raise
5.81944pt\hbox{\scalebox{1.0}[-1.0]{\lower
5.81944pt\hbox{$\displaystyle\scriptstyle\widehat{\vrule
width=0.0pt,height=3.01389pt\vrule
height=0.0pt,width=4.35718pt}$}}}}\cr\hbox{$\displaystyle\scriptstyle\bm{\omega}$}}}}{{\ooalign{\hbox{\raise
5.53241pt\hbox{\scalebox{1.0}[-1.0]{\lower
5.53241pt\hbox{$\displaystyle\scriptscriptstyle\widehat{\vrule
width=0.0pt,height=2.15277pt\vrule
height=0.0pt,width=3.11226pt}$}}}}\cr\hbox{$\displaystyle\scriptscriptstyle\bm{\omega}$}}}}})}}$$\displaystyle{{(J^{\nu^{\perp}+\nu^{\prime}}\times
G,{\mathchoice{{\ooalign{\hbox{\raise 6.25pt\hbox{\scalebox{1.0}[-1.0]{\lower
6.25pt\hbox{$\displaystyle\displaystyle\widehat{\vrule
width=0.0pt,height=4.30554pt\vrule
height=0.0pt,width=5.1773pt}$}}}}\cr\hbox{$\displaystyle\displaystyle\bm{\gamma}$}}}}{{\ooalign{\hbox{\raise
6.25pt\hbox{\scalebox{1.0}[-1.0]{\lower
6.25pt\hbox{$\displaystyle\textstyle\widehat{\vrule
width=0.0pt,height=4.30554pt\vrule
height=0.0pt,width=5.1773pt}$}}}}\cr\hbox{$\displaystyle\textstyle\bm{\gamma}$}}}}{{\ooalign{\hbox{\raise
5.81944pt\hbox{\scalebox{1.0}[-1.0]{\lower
5.81944pt\hbox{$\displaystyle\scriptstyle\widehat{\vrule
width=0.0pt,height=3.01389pt\vrule
height=0.0pt,width=3.62411pt}$}}}}\cr\hbox{$\displaystyle\scriptstyle\bm{\gamma}$}}}}{{\ooalign{\hbox{\raise
5.53241pt\hbox{\scalebox{1.0}[-1.0]{\lower
5.53241pt\hbox{$\displaystyle\scriptscriptstyle\widehat{\vrule
width=0.0pt,height=2.15277pt\vrule
height=0.0pt,width=2.58865pt}$}}}}\cr\hbox{$\displaystyle\scriptscriptstyle\bm{\gamma}$}}}}}^{G})}}$$\displaystyle{{(J^{\bar{\kappa}+\nu^{\prime}},\bm{\beta}^{\bar{\kappa}+\nu^{\prime}})}}$$\displaystyle{{(M,\bm{\omega})}}$$\displaystyle{{(J^{\kappa}\times
G,\bm{\gamma}^{G})}}$$\displaystyle{{(J^{\nu^{\perp}}\times
G,\bm{\bar{\gamma}}^{G})}}$$\displaystyle{{(J^{\bar{\kappa}},\bm{\beta}^{\bar{\kappa}})}}$$\displaystyle{{(M/G,\bm{\omega}/G)}}$$\displaystyle{{\,(J^{\kappa},\bm{\beta}^{\kappa})}}$$\displaystyle{\chi^{-1}}$$\displaystyle{\vartheta}$$\displaystyle{\widetilde{\pi}}$$\displaystyle{\widetilde{\widetilde{\varphi}}}$$\displaystyle{\widetilde{\pi}^{\prime}}$$\displaystyle{\varphi^{\prime}}$$\displaystyle{\widetilde{\varphi}}$$\displaystyle{\pi}$$\displaystyle{G}$$\displaystyle{\pi^{\prime}}$$\displaystyle{\bm{c}^{\nu}_{\bm{f}}}$$\displaystyle{\bar{\varphi}}$$\displaystyle{(2)}$$\displaystyle{\varphi}$$\displaystyle{(1)}$
Figure 5: In this diagram the signature $\displaystyle\kappa$ decomposes as
$\displaystyle\kappa=\nu+\nu^{\perp}$. Additionally, the maps
$\displaystyle\varphi,\bar{\varphi},\widetilde{\varphi}$, and
$\displaystyle\vartheta$ are all SFTs.

######  Example 7.4.

We continue with Example 7.3. According to our method, we now compute
$\displaystyle\widetilde{\widetilde{\varphi}}$ given by

$\displaystyle t=t,\ m=\theta,\ m_{1}=\theta_{1},\ m_{2}=v^{2},\ n=z_{1},\
n_{1}=v^{1}\cos\theta+hv^{2}\sin\theta-1,$
$\displaystyle\varepsilon^{1}=x_{1},\ \varepsilon^{2}=x-tx_{1},\
\varepsilon^{3}=z,\ m_{3}=p_{1},\ m_{4}=p_{2},\ m_{5}=p_{3},\ m_{6}=p_{4}.$

We then find that

$\displaystyle\mathcal{T}=\left(\widetilde{\widetilde{\varphi}}\right)^{-1}_{*}\text{pr}{\mathcal{H}}_{G}=\Big{\\{}\partial_{t}+x_{1}\partial_{x}-(u^{1}\sin\theta-
hu^{2}\cos\theta)\partial_{x_{1}}+z_{1}\partial_{z}$
$\displaystyle+(u^{1}\cos\theta+hu^{2}\sin\theta-1)\partial_{z_{1}}+\theta_{1}\partial_{\theta}+u^{2}\partial_{\theta_{1}}+p_{1}\partial_{u^{2}}+p_{2}\partial_{p_{1}}+p_{3}\partial_{p_{2}}+p_{4}\partial_{p_{3}},$
$\displaystyle\hskip 200.0pt\partial_{u^{1}},\ \partial_{p_{4}}\Big{\\}}.$

This is precisely a dynamic feedback linearization by differentiation of the
PVTOL by a four-fold prolongation of the control $\displaystyle u^{2}$, which
can be compared to the bound of $\displaystyle 2\bar{k}-1=7$ established in
Corollary 7.4. Here $\displaystyle\chi$ is the identity transformation, but we
still use the fact that

$\displaystyle\mathrm{Char}\,\mathcal{T}^{(1)}_{0}=\mathrm{span}\\{\partial_{u^{1}},\partial_{p^{4}}\\}$

to guarantee that the new control variables are $\displaystyle u^{1}$ and
$\displaystyle p^{4}$, respectively. Thus
$\displaystyle\vartheta=\widetilde{\widetilde{\varphi}}\circ\chi^{-1}=\widetilde{\widetilde{\varphi}}$
is the static feedback transformation

$\displaystyle\vartheta:{\mathchoice{{\ooalign{\hbox{\raise
7.09259pt\hbox{\scalebox{1.0}[-1.0]{\lower
7.09259pt\hbox{$\displaystyle\displaystyle\widehat{\vrule
width=0.0pt,height=6.83331pt\vrule
height=0.0pt,width=10.79166pt}$}}}}\cr\hbox{$\displaystyle\displaystyle
M$}}}}{{\ooalign{\hbox{\raise 7.09259pt\hbox{\scalebox{1.0}[-1.0]{\lower
7.09259pt\hbox{$\displaystyle\textstyle\widehat{\vrule
width=0.0pt,height=6.83331pt\vrule
height=0.0pt,width=10.79166pt}$}}}}\cr\hbox{$\displaystyle\textstyle
M$}}}}{{\ooalign{\hbox{\raise 6.40926pt\hbox{\scalebox{1.0}[-1.0]{\lower
6.40926pt\hbox{$\displaystyle\scriptstyle\widehat{\vrule
width=0.0pt,height=4.78333pt\vrule
height=0.0pt,width=7.55415pt}$}}}}\cr\hbox{$\displaystyle\scriptstyle
M$}}}}{{\ooalign{\hbox{\raise 5.9537pt\hbox{\scalebox{1.0}[-1.0]{\lower
5.9537pt\hbox{$\displaystyle\scriptscriptstyle\widehat{\vrule
width=0.0pt,height=3.41666pt\vrule
height=0.0pt,width=5.39581pt}$}}}}\cr\hbox{$\displaystyle\scriptscriptstyle
M$}}}}}\to J^{\kappa^{\prime}}\times G$

where $\displaystyle\kappa^{\prime}=\langle 1,0,0,0,0,1\rangle$. The
fundamental functions of $\displaystyle{\mathcal{T}}$ are flat outputs of the
PVTOL system (58) with respect to the Galilean group generated by
$\displaystyle\bm{\Gamma}$ in Example 7.3.

######  Remark 7.2.

Example 7.4 highlights the fact that an invariant flat control system will
generally have distinct sets of flat outputs depending upon the Lie subgroup
$\displaystyle G$ being considered. In each case the flat outputs are
generated by the fundamental functions of the corresponding dynamic
linearization of  $\displaystyle\displaystyle\widehat{\vrule
width=0.0pt,height=6.83331pt\vrule height=0.0pt,width=7.50002pt}$
$\displaystyle\displaystyle{\mathcal{V}}$ . In section 9 we will derive the
well-known flat outputs of (58) using a different subgroup of the control
symmetry group for this system.

##  8  A Non- ’Integrator Chain’ Dynamic Feedback
Linearization Example

In this section we illustrate the dynamic feedback linearization framework
developed in this paper via an elementary example. From [42] we deduce that
this system does not possess a linearization by the differentiation of the
given inputs. The goal of this section is to systematically derive a dynamic
feedback linearization in local coordinates using only the framework
established in this paper.

While the system in question cannot be linearized by integrator chains, it can
nevertheless be transformed to such a system by a preliminary feedback
transformation which can, in this case, be guessed by inspection. However, the
theory developed in this paper does not rely on such ad hoc steps. In the
search for a dynamic linearization, a preliminary feedback transformation, if
one is required, arises canonically from our general theory, as will be
demonstrated in this case.

The control system to be studied is

$\displaystyle\dot{x}_{1}=x^{2},\ \ \
\dot{x}_{2}=-x^{2}u^{1}(x^{1}+x^{5})+u^{2},\ \ \
\dot{x}_{3}=(x^{3})^{2}x^{4},$ (64)
$\displaystyle\dot{x}_{4}=-x^{5}(x^{1}x^{2}u^{1}-u^{2})+x^{2}u^{1}-1,\ \ \
\dot{x}_{5}=x^{6},\ \ \ \dot{x}_{6}=u^{2}-x^{1}x^{2}u^{1},$

in 6 states and 2 controls. We denote the ambient manifold of the system by
$\displaystyle M$ and calculate that the abelian Lie algebra spanned by

$\displaystyle\Gamma=\mathrm{span}\Big{\\{}$ $\displaystyle
Y_{1}=t\partial_{x^{1}}+\partial_{x^{2}}-\frac{u^{1}}{x^{2}}\partial_{u^{1}}+tx^{2}u^{1}\partial_{u^{2}},\
Y_{2}=\partial_{x^{1}}+x^{2}u^{1}\partial_{u_{2}},\
Y_{3}=(x^{3})^{2}\partial_{x^{3}}\Big{\\}}$

generates a control addmissible symmetry group. Let
$\displaystyle{\mathcal{V}}\subset TM$ denote the given control system for
(64). Away from the submanifold $\displaystyle x^{2}=0$ the functions

$\displaystyle t,\ q_{1}=x^{4},\ q_{2}=x^{5},\ q_{3}=x^{6},\
v^{1}=x^{2}u^{1},\ v^{2}=u^{2}-x^{1}x^{2}u^{1}$

are $\displaystyle G$-invariant, serve as local coordinates on $\displaystyle
M/G$ and determine the projection $\displaystyle\pi$ in local coordinates.

By Theorem 3.8, to determine the structure of the quotient of
$\displaystyle{\mathcal{V}}$ by $\displaystyle G$ we need only study the
refined derived type of the distribution
$\displaystyle\widehat{{\mathcal{V}}}={\mathcal{V}}\oplus\bm{\Gamma}$. We find
that $\displaystyle\widehat{{\mathcal{V}}}$ is a static feedback relative
Goursat bundle (Definition 3.15) of signature $\displaystyle\langle
1,1\rangle$. By Theorem 3.8, it follows that $\displaystyle{\mathcal{V}}/G$ is
SFL with signature, $\displaystyle\langle 1,1\rangle$; that is,
$\displaystyle(M/G,{\mathcal{V}}/G)$ is static feedback equivalent to
Brunovský normal form $\displaystyle\left(J^{\langle
1,1\rangle},{\mathcal{B}}_{\langle 1,1\rangle}\right)$. An easy calculation
shows that the quotient control system $\displaystyle{\mathcal{V}}/G$ has the
local form

${\mathcal{V}}/G=\mathrm{span}\left\\{\partial_{t}+(v^{2}q_{2}+v^{1}-1)\partial_{q_{1}}+q_{3}\partial_{q_{2}}+v^{2}\partial_{q_{3}},\
\partial_{v^{1}},\ \partial_{v^{2}}\right\\}$ (65)

where $\displaystyle v^{1},v^{2}$ are the controls on $\displaystyle M/G$, and
a further calculation verifies the above mentioned properties of the quotient.

Indeed, applying procedure contact [46] to (65) determines the local
diffeomorphism $\displaystyle\varphi:M/G\to J^{\langle 1,1\rangle}$ given by

$\varphi=(t=t,\ z=q_{1},\ z_{1}=v^{2}q_{2}+v^{1}-1,\ w=q_{2},\ w_{1}=q_{3},\
w_{2}=v^{2}),$

where $\displaystyle(t,z,z_{1},w,w_{1},w_{2})$ are the standard contact
coordinates on $\displaystyle J^{\langle 1,1\rangle}$. Next we consider the
static feedback linearization $\displaystyle\widetilde{\varphi}$ constructed
as per the discussion following Equation (51). The transformation group
$\displaystyle G$ generated by $\displaystyle\bm{\Gamma}$ consists of

$\displaystyle\bar{x}_{1}=$ $\displaystyle
x^{1}+t\varepsilon^{1}+\varepsilon^{2},\ \ \bar{x}_{2}=x^{2}+\varepsilon^{1},\
\ \bar{x}_{3}=\frac{x^{3}}{1-x^{3}\varepsilon^{3}},$
$\displaystyle\bar{u}^{1}=\frac{x^{2}u^{1}}{x^{2}+\varepsilon^{1}},\ \
\bar{u}^{2}=u^{2}+(t\varepsilon^{1}+\varepsilon^{2})x^{2}u^{1},$

with the remaining variables fixed by $\displaystyle G$. Taking the section

$\displaystyle\varphi:(t,q^{1},q^{2},q^{3},v^{1},v^{2})\mapsto(t,0,1,0,x^{4},x^{5},x^{6},u^{1},u^{2})$

yields

$\displaystyle\varepsilon^{1}=x^{2}-1,\ \varepsilon^{2}=x^{1}-t(x^{2}-1),\
\varepsilon^{3}=1-\frac{1}{x^{3}}$

and so

$\displaystyle\widetilde{\varphi}=\Big{(}$ $\displaystyle
t=t,z=x^{4},z_{1}=x^{5}(u^{2}-x^{1}x^{2}u^{1})+x^{2}u^{1}-1,w=x^{5},w_{1}=x^{6}$
$\displaystyle w_{2}=u^{2}-x^{1}x^{2}u^{1},\varepsilon^{1}=x^{2}-1,\
\varepsilon^{2}=x^{1}-t(x^{2}-1),\ \varepsilon^{3}=1-\frac{1}{x^{3}}\Big{)}.$

In particular, we compute the contact sub-connection
$\displaystyle{\mathcal{H}}_{G}=\widetilde{\varphi}_{*}{\mathcal{V}}$ in this
case to be

$\displaystyle{\mathcal{H}}_{G}=\mathrm{span}\big{\\{}\partial_{t}+$
$\displaystyle z_{1}\partial_{z}+w_{1}\partial_{w}+w_{2}\partial_{w_{1}}+$
(66)
$\displaystyle\lambda\partial_{\varepsilon^{1}}+(1-t\lambda)\partial_{\varepsilon^{2}}+z\partial_{\varepsilon^{3}},\partial_{z_{1}},\partial_{w_{2}}\Big{\\}},$

where $\displaystyle\lambda=w_{2}(w^{2}+1)-w(z_{1}+1)$.

To determine the existence of a dynamic feedback linearization of the system
(64), we must carry out a partial contact curve reduction (§5, Definition
5.1). As there are 2 inputs, we can do this along either the order 1 variable
$\displaystyle z$ or the order 2 variable $\displaystyle w$. In this case we
choose $\displaystyle w$ and form the map $\displaystyle\mathbf{c}^{\langle
0,1\rangle}_{f}=j^{\langle 0,1\rangle}\\!\\!f\times\text{Id}_{J^{\langle
1\rangle}}\times\text{Id}_{G\to G}:J^{\langle 1\rangle}\times G\to J^{\langle
1,1\rangle}\times G$, where $\displaystyle j^{\langle 0,1\rangle}\\!\\!f=(t,\
f(t),\ \dot{f}(t),\ \ddot{f}(t))$ and $\displaystyle f$ is an arbitrary,
smooth real-valued function of $\displaystyle t$. As in §5, we form the
Pfaffian system

$\displaystyle\left(\mathbf{c}^{\langle
0,1\rangle}_{f}\right)^{*}\bm{\gamma}^{G}=:\bm{\bar{\gamma}}^{G}.$

Then

$\displaystyle\bar{{\mathcal{H}}}_{G}=\mathrm{span}\left\\{\partial_{t}+z_{1}\partial_{z}+\bar{\lambda}\partial_{\varepsilon^{1}}+(1-t\bar{\lambda})\partial_{\varepsilon^{2}}+z\partial_{\varepsilon^{3}},\
\partial_{z_{1}}\right\\},$

where $\displaystyle\bar{{\mathcal{H}}}_{G}$ is the kernel of
$\displaystyle\bm{\bar{\gamma}}^{G}$ on $\displaystyle J^{\langle
1\rangle}\times G$ and
$\displaystyle\bar{\lambda}=\ddot{f}(f^{2}+1)-f(z_{1}+1)$. By the results in
§3.1 and §3.2, we find that $\displaystyle\bar{{\mathcal{H}}}_{G}$ is static
feedback linearizable with signature $\displaystyle\langle 0,0,0,1\rangle$.
Thus, the derived length of $\displaystyle\bar{{\mathcal{H}}}_{G}$ is
$\displaystyle\bar{k}=4$ and by Corollary 7.4, a maximum 7-fold partial
prolongation of $\displaystyle{\mathcal{H}}_{G}$ along the $\displaystyle
w$-series of contact coordinates is sufficient for a static feedback
linearizable control system. In fact, in this case it is enough to perform a
4-fold partial prolongation to obtain,

$\displaystyle\text{pr}{\mathcal{H}}_{G}=\mathrm{span}$
$\displaystyle\big{\\{}\partial_{t}+z_{1}\partial_{z}+w_{1}\partial_{w}+w_{2}\partial_{w_{1}}+w_{3}\partial_{w_{2}}+w_{4}\partial_{w_{3}}+w_{5}\partial_{w_{4}}+w_{6}\partial_{w_{5}}$
$\displaystyle+\lambda\partial_{\varepsilon^{1}}+(1-t\lambda)\partial_{\varepsilon^{2}}+z\partial_{\varepsilon^{3}},\partial_{z_{1}},\partial_{w_{6}}\Big{\\}},$

on $\displaystyle J^{\langle 1,0,0,0,0,1\rangle}\times G$, and a further
calculation shows that

$\displaystyle\text{pr}{\mathcal{H}}_{G}\simeq_{\text{SF}}{\mathcal{B}}_{\langle
0,0,0,1,0,1\rangle}.$

We have therefore proven that (64) is cascade feedback linearizable and hence
dynamic feedback linearizable with a dynamic extension of signature
$\displaystyle\langle 0,0,0,1,0,1\rangle$. Our next goal is to derive the
linearizable dynamic extension of (64), according to Theorem 7.5.

For this we apply the results of §7.5 by computing
$\displaystyle\widetilde{\widetilde{\varphi}}=\widetilde{\varphi}\circ\widetilde{\pi}\times\\{w_{3}=p_{1}^{1},w_{4}=p_{2}^{1},w_{5}=p_{3}^{1},w_{6}=p_{4}^{1}\\}$.
Indeed,

$\displaystyle\widetilde{\widetilde{\varphi}}=\Big{(}$ $\displaystyle
t=t,z=x^{4},z_{1}=x^{5}(u^{2}-x^{1}x^{2}u^{1})+x^{2}u^{1}-1,w=x^{5},w_{1}=x^{6}$
$\displaystyle w_{2}=u^{2}-x^{1}x^{2}u^{1},w_{3}=p^{1}_{1},\ w_{4}=p^{1}_{2},\
w_{5}=p^{1}_{3},\ w_{6}=p^{1}_{4},$ $\displaystyle\hskip
70.0pt\varepsilon^{1}=x^{2}-1,\ \varepsilon^{2}=x^{1}-t(x^{2}-1),\
\varepsilon^{3}=1-\frac{1}{x^{3}}\Big{)}.$

Calculating
$\displaystyle\mathcal{T}:=\widetilde{\widetilde{\varphi}}^{-1}_{*}\text{pr}{\mathcal{H}}_{G}$
shows that this distribution has the form

$\mathcal{T}=\mathrm{span}\big{\\{}Y,\partial_{u^{1}}+x^{1}x^{2}\partial_{u^{2}},\partial_{p^{1}_{4}}\Big{\\}},$

where $\displaystyle Y$ is the image of the first vector field in
$\displaystyle\text{pr}{\mathcal{H}}_{G}$; in particular,

$\mathrm{Char}\,\mathcal{T}^{(1)}_{0}=\mathrm{span}\,\\{\partial_{u^{1}}+x^{1}x^{2}\partial_{u^{2}},\partial_{p^{1}_{4}}\\}.$
(67)

Because of the first vector field in (67), $\displaystyle\mathcal{T}$ does not
yield a well-defined control system in its current form. Therefore we compose
$\displaystyle\widetilde{\widetilde{\varphi}}$ with a map $\displaystyle\chi$
as in Fig. 4, embodying the change of variable $\displaystyle
u=\beta(t,x,y,W)$ of Definition 2.4, chosen so that the flowbox coordinates of
$\displaystyle\mathrm{Char}\,\mathcal{T}^{(1)}_{0}$ are the new controls.

In this case it is easy to see that (67) is transformed by

$\displaystyle\chi^{-1}=$
$\displaystyle(t=t,\,x^{i}=x^{i},\,y^{1}=u^{2}-x^{1}x^{2}u^{1},\,y^{2}=p^{1}_{1},$
$\displaystyle
y^{3}=p^{1}_{2},\,y^{4}=p^{1}_{3},\,W^{1}=u^{1},\,W^{2}=p^{1}_{4}),$

where $\displaystyle i=1,\ldots,6.$ We see that the new controls are
$\displaystyle W^{1}=u^{1},\ W^{2}=p^{1}_{4}$ with new additional states given
by $\displaystyle
y^{1}=u^{2}-x^{1}x^{2}u^{1},\,y^{2}=p^{1}_{1},\,y^{3}=p^{1}_{2},,y^{4}=p^{1}_{3}$.
Thus the map $\displaystyle u=\beta(t,x,y,W)$ is given by

$u^{1}=W^{1},\ \ u^{2}=y^{1}+x^{1}x^{2}W^{1}.$ (68)

Hence the dynamic feedback linearization of (64), written symbolically in the
form $\displaystyle\dot{x}=f(x,u^{1},u^{2})$, is given by

$\displaystyle\dot{x}=f\left(\bm{x},W^{1},\ \ y^{1}+x^{1}x^{2}W^{1}\right)$
(69) $\displaystyle\dot{y_{i}}=y_{i+1},\,i=1,\ldots\,3,$
$\displaystyle\dot{y_{4}}=W^{2}.$

It can be checked directly that this control system is SFL. Its trajectories
uniquely determine those of (64). If $\displaystyle s^{\prime}$ is a solution
of  $\displaystyle\displaystyle\widehat{\vrule
width=0.0pt,height=6.83331pt\vrule height=0.0pt,width=7.50002pt}$
$\displaystyle\displaystyle{\mathcal{V}}$ then
$\displaystyle\widetilde{\pi}\circ\chi\circ
s^{\prime}:{\mathrm{I}}\subseteq\mathbb{R}\to M$ solves
$\displaystyle{\mathcal{V}}$. The distribution
$\displaystyle\displaystyle\widehat{\vrule width=0.0pt,height=6.83331pt\vrule
height=0.0pt,width=7.50002pt}$ $\displaystyle\displaystyle{\mathcal{V}}$ is
defined by (69).

It remains only to identify the fundamental functions which must be Lie
differentiated by a vector field $\displaystyle
Z\in{\mathchoice{{\ooalign{\hbox{\raise
7.09259pt\hbox{\scalebox{1.0}[-1.0]{\lower
7.09259pt\hbox{$\displaystyle\displaystyle\widehat{\vrule
width=0.0pt,height=6.83331pt\vrule
height=0.0pt,width=7.50002pt}$}}}}\cr\hbox{$\displaystyle\displaystyle{\mathcal{V}}$}}}}{{\ooalign{\hbox{\raise
7.09259pt\hbox{\scalebox{1.0}[-1.0]{\lower
7.09259pt\hbox{$\displaystyle\textstyle\widehat{\vrule
width=0.0pt,height=6.83331pt\vrule
height=0.0pt,width=7.50002pt}$}}}}\cr\hbox{$\displaystyle\textstyle{\mathcal{V}}$}}}}{{\ooalign{\hbox{\raise
6.40926pt\hbox{\scalebox{1.0}[-1.0]{\lower
6.40926pt\hbox{$\displaystyle\scriptstyle\widehat{\vrule
width=0.0pt,height=4.78333pt\vrule
height=0.0pt,width=5.25pt}$}}}}\cr\hbox{$\displaystyle\scriptstyle{\mathcal{V}}$}}}}{{\ooalign{\hbox{\raise
5.9537pt\hbox{\scalebox{1.0}[-1.0]{\lower
5.9537pt\hbox{$\displaystyle\scriptscriptstyle\widehat{\vrule
width=0.0pt,height=3.41666pt\vrule
height=0.0pt,width=3.75pt}$}}}}\cr\hbox{$\displaystyle\scriptscriptstyle{\mathcal{V}}$}}}}}$
satisfying $\displaystyle{\EuScript L}_{Z}(t)=1$. By procedure contact
(§Procedure Contact) we deduce that there is one fundamental function
$\displaystyle\phi^{4}$ of order 4 and one function $\displaystyle\phi^{6}$ of
order 6. Indeed,

$\displaystyle\phi^{6}=x^{5},\ \ \ \phi^{4}=x^{1}.$

If $\displaystyle(t,\alpha_{a},\ \beta_{b})$, $\displaystyle 0\leq a\leq 4$,
$\displaystyle 0\leq b\leq 6$ are the contact coordinates on $\displaystyle
J^{\langle 0,0,0,1,0,1\rangle}$, then the static feedback transformation
$\displaystyle\psi:P\times\pi^{-1}(\mathrm{U})\to J^{\langle
0,0,0,1,0,1\rangle}$ that identifies
$\displaystyle\displaystyle\widehat{\vrule width=0.0pt,height=6.83331pt\vrule
height=0.0pt,width=7.50002pt}$ $\displaystyle\displaystyle{\mathcal{V}}$ with
its Brunovský normal form $\displaystyle{\mathcal{B}}_{\langle
0,0,0,1,0,1\rangle}$ is given by

$\displaystyle\alpha_{a}(t,x,y,W)=\left({\EuScript L}_{Z}\right)^{a}\phi^{4},\
\ 0\leq a\leq 4,\ \ \ \ \ \beta_{b}(t,x,y,W)=\left({\EuScript
L}_{Z}\right)^{b}\phi^{6},\ \ 0\leq b\leq 6,$

from which the explicit solution is readily deduced, but too lengthy to record
here.

Note that the functions $\displaystyle\phi^{4}$ and $\displaystyle\phi^{6}$
are flat outputs for (64) and that they arise canonically as the fundamental
functions of the dynamic linearization (69). This is a consequence of the
general theory. Thereby cascade feedback linearization provides a geometric
characterization of flat outputs in the case of invariant flat control
systems; see also §9.

##  9  Yet Another Look at the PVTOL

In [29] the authors made a study of the well known PVTOL control system (58)
by writing down a complete set of flat outputs and thereby derived a dynamic
linearization feedback equivalent to the Brunovský form of signature
$\displaystyle\langle 0,0,0,2\rangle$. According to their account the authors
achieved this by inspired guesswork. In this section, we show how these flat
outputs can be derived systematically using the theory of the present paper.

We denote the ambient manifold of the control system (58) by $\displaystyle M$
and calculate that the abelian Lie algebra spanned by

$\bm{\Gamma}=\text{span}\\{X_{1},X_{2}\\}$ (70)

where

$\displaystyle X_{1}=$
$\displaystyle\,h\sin^{2}(\theta)\cos(\theta)\partial_{x}+h\theta_{1}(3\cos^{2}(\theta)-1)\sin(\theta)\partial_{x_{1}}+(x-h\sin(\theta)\cos^{2}(\theta))\partial_{z}$
$\displaystyle+(x_{1}+2h\theta_{1}\cos(\theta)-3h\theta_{1}\cos^{3}(\theta))\partial_{z_{1}}+\sin^{2}(\theta)\partial_{\theta}+\theta_{1}\sin(2\theta)\partial_{\theta_{1}}$
$\displaystyle+\cos(\theta)\sin(\theta)(5h\theta_{1}^{2}-u_{1})\partial_{u_{1}}+(2\theta_{1}^{2}\cos(2\theta)+u_{2}\sin(2\theta))\partial_{u_{2}}$
$\displaystyle X_{2}=$ $\displaystyle\partial_{z}$

generates a control admissible symmetry group acting on an open subset of
$\displaystyle M$. The action of the control admissible symmetry group
$\displaystyle G$ generated by $\displaystyle\bm{\Gamma}$ can be found in the
appendix. For the present purposes we need not record further details about
this here and instead focus on its application to the dynamic feedback
linearization issue at hand.

Letting $\displaystyle{\mathcal{V}}$ denote the distribution for system (58),
we have

$\displaystyle{\mathcal{V}}=\text{span}\big{\\{}\partial_{t}+x_{1}\partial_{x}-$
$\displaystyle(u^{1}\sin\theta-
hu^{2}\cos\theta)\partial_{x_{1}}+z_{1}\partial_{z}+(u^{1}\cos\theta+hu^{2}\sin\theta-1)\partial_{z_{1}}$
$\displaystyle\hskip
120.0pt+\theta_{1}\partial_{\theta}+u^{2}\partial_{\theta_{1}},\
\partial_{u^{1}},\ \partial_{u^{2}}\big{\\}}.$

We find that
$\displaystyle\widehat{{\mathcal{V}}}:={\mathcal{V}}\oplus\bm{\Gamma}$ is a
static feedback relative Goursat bundle of signature $\displaystyle\langle
0,2\rangle$, and hence $\displaystyle{\mathcal{V}}$ has a contact sub-
connection
$\displaystyle\widetilde{\varphi}_{*}{\mathcal{V}}={\mathcal{H}}_{G}$, which
we calculate to be of the form 999See the appendix for the local
trivialization $\displaystyle\widetilde{\varphi}$, in this case.

$\displaystyle{\mathcal{H}}_{G}=\text{span}\Big{\\{}\partial_{t}+$
$\displaystyle
z_{1}\partial_{z}+z_{2}\partial_{z_{1}}+w_{1}\partial_{w}+w_{2}\partial_{w_{1}}-\frac{1+z_{1}}{w_{1}}\partial_{\varepsilon^{1}}$
$\displaystyle\hskip
30.0pt+\frac{w_{1}(z-w_{1})+w(1+z_{1})}{w_{1}}\partial_{\varepsilon^{2}},\ \
\partial_{z_{2}},\ \ \partial_{w_{2}}\Big{\\}}.$

To determine the existence of a dynamic feedback linearization of the system
(58), we carry out a partial contact curve reduction (§5, Definition 5.1). As
there are 2 inputs, we can do this along either the order 1 variable
$\displaystyle z$ or the order 2 variable $\displaystyle w$. It turns out that
we can choose $\displaystyle w$ and form the map

$\displaystyle\mathbf{c}^{\langle 0,1\rangle}_{f}=j^{\langle
0,1\rangle}\\!\\!f\times\text{Id}_{J^{\langle 0,1\rangle}\times G}:J^{\langle
0,1\rangle}\times G\to J^{\langle 0,2\rangle}\times G,$

where $\displaystyle j^{\langle 0,1\rangle}\\!\\!f=(t,\ f(t),\ \dot{f}(t),\
\ddot{f}(t))$ and $\displaystyle f$ is an arbitrary, smooth, real-valued
function of $\displaystyle t$. As in §5, we form the Pfaffian system

$\displaystyle\left(\mathbf{c}^{\langle
0,1\rangle}_{f}\right)^{*}\bm{\gamma}^{G}=:\bm{\bar{\gamma}}^{G}.$

Then

$\displaystyle\bar{{\mathcal{H}}}_{G}=\text{span}\left\\{\partial_{t}+z_{1}\partial_{z}+z_{2}\partial_{z_{1}}-\frac{1+z_{1}}{\dot{f}}\partial_{\varepsilon^{1}}+\frac{\dot{f}(z-\dot{f})+f(1+z_{1})}{\dot{f}}\partial_{\varepsilon^{2}},\
\partial_{z_{2}}\right\\},$

where $\displaystyle\bar{{\mathcal{H}}}_{G}$ is the kernel of
$\displaystyle\bm{\bar{\gamma}}^{G}$ on $\displaystyle J^{\langle
0,1\rangle}\times G$. By Theorems 3.2 and 3.3, we find that
$\displaystyle\bar{{\mathcal{H}}}_{G}$ is extended static feedback
linearizable with signature $\displaystyle\langle 0,0,0,1\rangle$. Thus, the
derived length of $\displaystyle\bar{{\mathcal{H}}}_{G}$ is
$\displaystyle\bar{k}=4$ and by Corollary 7.4, the 7-fold partial prolongation
of $\displaystyle{\mathcal{H}}_{G}$ along the $\displaystyle w$-series of
contact coordinates is sufficient to achieve a static feedback linearizable
control system. However, this over-estimates the number of prolongations and
in fact, it is easy to discover that only 2 prolongations by differentiation
of $\displaystyle w_{2}$ are required; this can be easily deduced by computing
the fundamental function of $\displaystyle\bar{{\mathcal{H}}}_{G}$. With only
2 prolongations, we get

$\displaystyle\text{pr}\,{\mathcal{H}}_{G}=\text{span}\Big{\\{}\partial_{t}+$
$\displaystyle
z_{1}\partial_{z}+z_{2}\partial_{z_{1}}+w_{1}\partial_{w}+w_{2}\partial_{w_{1}}+w_{3}\partial_{w_{2}}+w_{4}\partial_{w_{3}}-\frac{1+z_{1}}{w_{1}}\partial_{\varepsilon^{1}}$
$\displaystyle\hskip
70.0pt+\frac{w_{1}(z-w_{1})+w(1+z_{1})}{w_{1}}\partial_{\varepsilon^{2}},\ \
\partial_{z_{2}},\ \ \partial_{w_{4}}\Big{\\}}$

on $\displaystyle J^{\langle 0,1,0,1\rangle}\times G$ and a further
calculation shows that
$\displaystyle\text{pr}{\mathcal{H}}_{G}\simeq{\mathcal{B}}_{\langle
0,0,0,2\rangle}$. We have therefore proven that (58) is cascade feedback
linearizable and hence dynamic feedback linearizable with a dynamic
linearization of signature $\displaystyle\langle 0,0,0,2\rangle$. Our next
goal is to derive the linearizable dynamic extension of (58).

The map $\displaystyle\widetilde{\widetilde{\varphi}}$ is given by
$\displaystyle\widetilde{\widetilde{\varphi}}=\widetilde{\varphi}\circ\widetilde{\pi}\times\\{w_{3}=p_{1},\
w_{4}=p_{2}\\}$. We do not record the transformation
$\displaystyle\widetilde{\widetilde{\varphi}}^{-1}$ in this example; however,
the relevant part of $\displaystyle\widetilde{\widetilde{\varphi}}^{-1}$
needed to construct the dynamic feedback is

$w_{2}=\sin(\theta)(h\theta_{1}^{2}-u^{1}).$ (71)

The distribution
$\displaystyle\mathcal{T}:=\widetilde{\widetilde{\varphi}}^{-1}_{*}\text{pr}{\mathcal{H}}_{G}$
has the form

$\displaystyle\mathcal{T}$
$\displaystyle=\text{span}\big{\\{}\partial_{t}+x_{1}\partial_{x}-(u^{1}\sin\theta-
hu^{2}\cos\theta)\partial_{x_{1}}+z_{1}\partial_{z}$
$\displaystyle+(u^{1}\cos\theta+hu^{2}\sin\theta-1)\partial_{z_{1}}+\theta_{1}\partial_{\theta}+u^{2}\partial_{\theta_{1}}$
$\displaystyle+(\theta_{1}(\theta_{1}^{2}h-u^{1})\cot\theta+2hu^{2}\theta_{1}-p_{1})\partial_{u^{1}}+p_{2}\partial_{p_{1}},\
\ \partial_{u^{2}},\ \ \partial_{p_{2}}\big{\\}}.$

We notice that

$\displaystyle\mathrm{Char}\,\mathcal{T}^{(1)}_{0}=\mathrm{span}\\{\partial_{u^{2}},\partial_{p_{2}}\\}$

which ensures that $\displaystyle\mathcal{T}$ is a dynamic extension of the
original PVTOL control system $\displaystyle\mathcal{V}$. Notice, however,
that we can still use equation (71) to construct a dynamic extension as
described in Theorem 7.5. Indeed, let $\displaystyle\chi^{-1}$ be the
diffeomorphism such that

$\displaystyle(y^{1},y^{2})$
$\displaystyle=(\sin(\theta)(h\theta_{1}^{2}-u^{1}),p_{1}),$
$\displaystyle(W^{1},W^{2})$ $\displaystyle=(u^{2},p_{2}),$

and the identity on all other variables. Then the dynamic compensator
$\displaystyle\bm{u}=\bm{\beta}(t,\bm{x},\bm{y},\bm{W})$ is

$\displaystyle u^{1}$ $\displaystyle=h\theta_{1}^{2}-\csc(\theta)y^{1},$
$\displaystyle u^{2}$ $\displaystyle=W^{1}.$

As such, the control system (58) is dynamically extended to

$\displaystyle\dot{x}=x_{1},\ \ \
\dot{x}_{1}=-h\theta_{1}^{2}\sin\theta-y^{1}+hW^{1}\cos\theta,\ \ \
\dot{z}=z_{1},$ (72)
$\displaystyle\dot{z}_{1}=h\theta_{1}^{2}\cos(\theta)-\cot(\theta)y^{1}+hW^{1}\sin\theta-1,\
\ \ \dot{\theta}=\theta_{1},\ \ \ \dot{\theta}_{1}=W^{1},$
$\displaystyle\dot{y^{1}}=y^{2},\ \ \ \dot{y^{2}}=W^{2},$

with the corresponding distribution denoted by
$\displaystyle\displaystyle\widehat{\vrule width=0.0pt,height=6.83331pt\vrule
height=0.0pt,width=7.50002pt}$ $\displaystyle\displaystyle{\mathcal{V}}$ .

To compute the flat outputs of (58), we compute the fundamental functions of
either $\displaystyle\mathcal{T}$ or
$\displaystyle\displaystyle\widehat{\vrule width=0.0pt,height=6.83331pt\vrule
height=0.0pt,width=7.50002pt}$ $\displaystyle\displaystyle{\mathcal{V}}$ in
the usual way. These turn out to be

$\displaystyle x-h\sin\theta\ \ \ \ \ \ \text{and}\ \ \ \ \ z+h\cos\theta,$

the so called centre of oscillation flat outputs of (58) first provided by
Martin, Devasia and Paden in [29] but here obtained canonically from the
fundamental functions of the dynamic linearization (72). The authors in [29]
rather proceed in the opposite direction: the flat outputs are firstly
ingeniously guessed and then verified by deriving the dynamic linearization
(72) from them.

##  10 Conclusion

We have studied generic, smooth control systems $\displaystyle(M,\bm{\omega})$
that are invariant under a Lie group $\displaystyle G$ of control admissible
transformations satisfying the further property that they admit static
feedback linearizable quotients by $\displaystyle G$ (symmetry reductions). An
infinitesimal test for rapidly identifying static feedback linearizable
quotients was given, based on the Lie algebra of infinitesimal generators of
the action of $\displaystyle G$. The requirement that an invariant control
system possesses a static feedback linearizable quotient appears to be very
mild. In practice, it appears to hold quite generally, in contrast to the
static feedback linearizability of a control system itself, which is rare.

We proved that a $\displaystyle C^{\infty}$ control system invariant under a
Lie group $\displaystyle G$ of control admissible transformations and that has
a static feedback linearizable quotient, admits the local normal form
$\displaystyle\bm{\gamma}^{G}=\bm{\beta}^{\kappa}\oplus\Theta$ on a local
trivialization $\displaystyle\pi^{\prime}:J^{\kappa}\times G\to J^{\kappa}$ of
the principal bundle $\displaystyle\pi:(M,\bm{\omega})\to(M/G,\bm{\omega}/G)$
in which $\displaystyle(J^{\kappa},\bm{\beta}^{\kappa})$ is the Brunovský
normal form of $\displaystyle(M/G,\bm{\omega}/G)$ and $\displaystyle\Theta$ is
a differential system for the reconstruction of the trajectories of
$\displaystyle\bm{\omega}$ from those of the quotient
$\displaystyle\bm{\omega}/G$. The present paper built on this result to
construct dynamic feedback linearizations of control systems with symmetry.

A widely applicable procedure was established showing that any cascade
feedback linearizable control system can be dynamically linearized with linear
dynamics in the fiber coordinates. It was proven that this can be achieved by
a prolongation by differentiation of canonically chosen inputs of the
associated contact sub-connection $\displaystyle{\mathcal{H}}_{G}$. Theorem
7.2 and Corollary 7.4 established a bound on the number of differentiations
that must be performed to achieve a static feedback linearizable dynamically
extended control system that appears to outperform the best currently known
bounds [15], being linear in the derived length of the reduced sub-connection
$\displaystyle\bar{{\mathcal{H}}}_{G}$. For instance, the sharp bound of Sluis
and Tilbury [42] in the case of 2-input control systems is achieved by our
general bound. However, we have not, in this paper, made a rigorous general
comparison of efficiency. On the other hand, a procedure was given whereby the
exact number of differentiations can be predicted.

We have given numerous examples to show how the general theory can be applied
in practice. In particular, a symmetry-based construction of the centre of of
oscillation flat outputs of the PVTOL control system was described. These flat
outputs together with the corresponding dynamic linearization were first
presented in [29]. It was also shown that the PVTOL also has a dynamic
linearization by differentiation of a given input by studying the control
system’s invariance under a subgroup of the Galilean transformations in
$\displaystyle x$ and $\displaystyle z$. These are instances of a general
symmetry-based procedure for deriving the flat outputs of flat control systems
with symmetry.

Throughout we have emphasized intrinsic geometric structures underlying
control systems with symmetry to derive coordinate-free, canonical procedures
that inform the local coordinate calculations that lead to their dynamic
feedback linearizations and explicit solutions.

##  11 Appendix: PVTOL Group Action

The local group action of the two dimemsional control admissible symmetry
algebra (70) of Section 9 is given by

$\displaystyle t$ $\displaystyle\mapsto t,$ $\displaystyle\theta$
$\displaystyle\mapsto F(\theta,\epsilon_{1}),$ $\displaystyle\theta_{1}$
$\displaystyle\mapsto\theta_{1}\csc^{2}(\theta)\sin^{2}(F(\theta,\epsilon_{1})),$
$\displaystyle x$ $\displaystyle\mapsto
x+h(\sin(F(\theta,\epsilon_{1}))-\sin(\theta)),$ $\displaystyle x_{1}$
$\displaystyle\mapsto
x_{1}-h\theta_{1}\cos(\theta)+h\theta_{1}\csc^{2}(\theta)\cos(F(\theta,\epsilon_{1}))\sin^{2}(F(\theta,\epsilon_{1})),$
$\displaystyle z$
$\displaystyle\mapsto\epsilon_{1}x+z+\epsilon_{2}+h\cos(\theta)-h\epsilon_{1}\sin(\theta)-h\cos(F(\theta,\epsilon_{1})),$
$\displaystyle z_{1}$ $\displaystyle\mapsto
h\theta_{1}\csc^{2}(\theta)\sin^{3}(F(\theta,\epsilon_{1}))+(x_{1}-h\theta_{1}\cos(\theta))\epsilon_{1}+z_{1}-h\theta_{1}\sin(\theta),$
$\displaystyle u_{1}$
$\displaystyle\mapsto(u_{1}\sin(\theta)-h\theta_{1}^{2}\sin(\theta))\csc(F(\theta,\epsilon_{1}))+h\theta_{1}^{2}\csc^{4}(\theta)\sin^{4}(F(\theta,\epsilon_{1})),$
$\displaystyle u_{2}$
$\displaystyle\mapsto(u_{2}\csc(\theta)-2\theta_{1}^{2}\cos(\theta)\csc^{3}(\theta))\sin^{2}(F(\theta,\epsilon_{1}))$
$\displaystyle\phantom{\mapsto}+2\theta_{1}^{2}\csc^{4}(\theta)\cos(F(\theta,\epsilon_{1}))\sin^{3}(F(\theta,\epsilon_{1})),$

where $\displaystyle
F(\theta,\epsilon_{1})=\mathrm{arccot}(\cot(\theta)-\epsilon_{1})$, and
$\displaystyle\varepsilon_{1}$ and $\displaystyle\varepsilon_{2}$ form the
local coordinates of the control admissible symmetry group $\displaystyle G$.
The local trivialization map $\displaystyle\widetilde{\varphi}$ is given by

$\begin{aligned} t&=t,\\\ w_{0}&=x-h\sin(\theta),\\\
w_{1}&=x_{1}-h\theta_{1}\cos(\theta),\\\
w_{2}&=(h\theta_{1}^{2}-u_{1})\sin(\theta),\\\
\varepsilon_{1}&=1-\cot(\theta),\\\ \end{aligned}\begin{aligned}
z_{0}&=(x_{1}\cos(\theta)-h\theta_{1})\csc(\theta),\\\
z_{1}&=(h\theta_{1}^{2}\cos(\theta)-\theta_{1}x_{1})\csc^{2}(\theta)-1,\\\
z_{2}&=-\csc^{3}(\theta)\theta_{1}^{2}(\theta_{1}h-x_{1}\cos(\theta))+u_{1}\theta_{1}\csc(\theta))\\\
&-\csc^{3}(\theta)(\cos(\theta)\theta_{1}h-x_{1})(\theta_{1}^{2}\cos(\theta)-u_{2}\sin(\theta)),\\\
\varepsilon_{2}&=z-x-\frac{h}{\sqrt{2}}+h\sin(\theta)+x\cot(\theta).\end{aligned}$

## References

  * [1] I. M. Anderson and M. E. Fels, Exterior differential systems with symmetry, Acta Appl. Math., 87 (2005), pp. 3–31.
  * [2] E. Aranda-Bricaire, C. H. Moog, and J.-B. Pomet, A linear algebraic framework for dynamic feedback linearization, IEEE Transactions on Automatic Control, 40 (1995), pp. 127–132.
  * [3] E. Aranda-Bricaire and J.-B. Pomet, Some explicit conditions for a control system to be feedback equivalent to extended goursat form, IFAC Proceedings Volumes, 28 (1995), pp. 489–494.
  * [4] D. Avanessoff and J.-B. Pomet, Flatness and monge parameterization of two-input systems, control-affine with 4 states or general with 3 states, ESAIM: Control, Optimisation and Calculus of Variations, 13 (2007), pp. 237–264.
  * [5] S. Battilotti and C. Califano, A geometric approach to dynamic feedback linearization, in Analysis and design of nonlinear control systems, Springer, Berlin, 2008, pp. 397–411.
  * [6] P. Brunovský, A classification of linear controllable systems, Kybernetika (Prague), 6 (1970), pp. 173–188.
  * [7] R. L. BRYANT, SOME ASPECTS OF THE LOCAL AND GLOBAL THEORY OF PFAFFIAN SYSTEMS., The University of North Carolina at Chapel Hill, 1979.
  * [8] B. Charlet, J. Lévine, and R. Marino, Sufficient conditions for dynamic state feedback linearization, SIAM journal on control and optimization, 29 (1991), pp. 38–57.
  * [9] V. Chetverikov, New flatness conditions for control systems, IFAC Proceedings Volumes, 34 (2001), pp. 191–196.
  * [10] J. Clelland and Y. Hu, On absolute equivalence and linearization i, arXiv preprint arXiv:2005.00643, (2020).
  * [11] J. De Doná, N. Tehseen, and P. J. Vassiliou, Symmetry reduction, contact geometry, and partial feedback linearization, SIAM Journal on Control and Optimization, 56 (2018), pp. 201–230.
  * [12] V. Elkin, Reduction of Nonlinear Control Systems: A Differential Geometric Approach, vol. 472 of Mathematics and its Applications, Springer, Dordrecht, 1999.
  * [13] M. Fliess, J. Lévine, P. Martin, and P. Rouchon, Flatness and defect of non-linear systems: introductory theory and examples, International journal of control, 61 (1995), pp. 1327–1361.
  * [14] , A lie-backlund approach to equivalence and flatness of nonlinear systems, IEEE Transactions on automatic control, 44 (1999), pp. 922–937.
  * [15] J. Franch and E. Fossas, Linearization by prolongations: New bounds on the number of integrators, European journal of control, 11 (2005), pp. 171–179.
  * [16] R. B. Gardner and W. F. Shadwick, An algorithm for feedback linearization, Differential Geom. Appl., 1 (1991), pp. 153–158.
  * [17] , The GS algorithm for exact linearization to Brunovsky normal form, IEEE Trans. Automat. Control, 37 (1992), pp. 224–230.
  * [18] J. Grizzle and S. Marcus, The structure of nonlinear control systems possessing symmetries, IEEE Trans. Automat. Control, 37 (1992), pp. 224–230.
  * [19] M. Guay, P. J. Mclellan, and D. Bacon, A condition for dynamic feedback linearization of control-affine nonlinear systems, International Journal of Control, 68 (1997), pp. 87–106.
  * [20] J. Hauser, S. Sastry, and G. Meyer, Nonlinear control design for slightly non-minimum phase systems: Application to v/stol aircraft, Automatica, 28 (1992), pp. 665–679.
  * [21] L. R. Hunt, R. J. Su, and G. Meyer, Design for multi-input nonlinear systems, in Differential geometric control theory (Houghton, Mich., 1982), vol. 27 of Progr. Math., Birkhäuser, Boston, Mass., 1983, pp. 268–298.
  * [22] B. a. Jakubczyk and W. Respondek, On linearization of control systems, Bull. Acad. Polon. Sci. Sér. Sci. Math., 28 (1980), pp. 517–522.
  * [23] T. J. Klotz, Geometry of Cascade Feedback Linearizable Control Systems, PhD thesis, University of Colorado at Boulder, 2020.
  * [24] S. Kobayashi and K. Nomizu, Foundations of differential geometry, vol. 1, New York, London, 1963.
  * [25] J. Levine, Analysis and control of nonlinear systems: A flatness-based approach, Springer Science & Business Media, 2009.
  * [26] S. Li, F. Nicolau, and W. Respondek, Multi-input control-affine systems static feedback equivalent to a triangular form and their flatness, International Journal of Control, 89 (2016), pp. 1–24.
  * [27] R. Marino, On the largest feedback linearizable subsystem, Systems & Control Letters, 6 (1986), pp. 345–351.
  * [28] R. Marino, W. Boothby, and D. L. Elliott, Geometric properties of linearizable control systems, Mathematical systems theory, 18 (1985), pp. 97–123.
  * [29] P. Martin, S. Devasia, and B. Paden, A different look at output tracking: control of a vtol aircraft, Automatica, 32 (1996), pp. 101 – 107.
  * [30] F. Nicolau and W. Respondek, Flatness of multi-input control-affine systems linearizable via one-fold prolongation, SIAM Journal on Control and Optimization, 55 (2017), pp. 3171–3203.
  * [31] P. J. Olver, Applications of Lie groups to differential equations, vol. 107 of Graduate Texts in Mathematics, Springer-Verlag, New York, second ed., 1993.
  * [32] , Equivalence, invariants, and symmetry, Cambridge University Press, Cambridge, 1995.
  * [33] R. S. Palais, A global formulation of the Lie theory of transformation groups, American Mathematical Society, 1957.
  * [34] W. Pasillas-Lepine and W. Respondek, On the geometry of Goursat structures, ESAIM: Control, Optimisation and Calculus of Variations, 6 (2001), pp. 119–181.
  * [35] P. Pereira da Silva, Some remarks on static feedback linearization for time-varying systems, Automatica, 44 (2008), pp. 3219–3222.
  * [36] J.-B. Pomet, A differential geometric setting for dynamic equivalence and dynamic linearization, PhD thesis, INRIA, 1994.
  * [37] J.-B. Pomet, On dynamic feedback linearization of four-dimensional affine control systems with two inputs, ESAIM: Control, optimisation and calculus of variations, 2 (1997), pp. 151–230.
  * [38] W. Respondek, Partial linearization, decompositions and fibre linear systems, Theory and Applications of Nonlinear Control Systems, (1986), pp. 137–154.
  * [39] W. Respondek, Symmetries and minimal flat outputs of nonlinear control systems, in New trends in nonlinear dynamics and control and their applications, Springer, 2003, pp. 65–86.
  * [40] W. Shadwick, Absolute equivalence and dynamic feedback linearization, Systems & Control Letters, 15 (1990), pp. 35–39.
  * [41] H. B. Silveira, P. S. P. da Silva, and P. Rouchon, A flat triangular form for nonlinear systems with two inputs: Necessary and sufficient conditions, European Journal of Control, 22 (2015), pp. 17–22.
  * [42] W. Sluis and D. Tilbury, A bound on the number of integrators needed to linearize a control system, in Proceedings of 1995 34th IEEE Conference on Decision and Control, vol. 1, IEEE, 1995, pp. 602–607.
  * [43] A. J. van der Schaft, Linearization and input-output decoupling for general nonlinear systems, Systems and Control Letters, 5 (1984), pp. 27–33.
  * [44] M. van Nieuwstadt, M. Rathinam, and R. M. Murray, Differential flatness and absolute equivalence of nonlinear control systems, SIAM Journal on Control and Optimization, 36 (1998), pp. 1225–1239.
  * [45] P. J. Vassiliou, A constructive generalised goursat normal form, Differential Geometry and its Applications, 24 (2006), pp. 332–350.
  * [46] , Efficient construction of contact coordinates for partial prolongations, Foundations of Computational Mathematics, 6 (2006), pp. 269–308.
  * [47] P. J. Vassiliou, Cascade linearization of invariant control systems, J. Dyn. Control Syst., 24 (2018), pp. 593–623.