# A Dual Geometric Test for Forward-Flatness

Bernd Kolarcor<EMAIL_ADDRESS>Johannes Schrotshamer
<EMAIL_ADDRESS>Markus Schöberl<EMAIL_ADDRESS>Institute
of Automatic Control and Control Systems Technology, Johannes Kepler
University, Linz, Austria

###### Abstract

Forward-flatness is a generalization of static feedback linearizability and a
special case of a more general flatness concept for discrete-time systems.
Recently, it has been shown that this practically quite relevant property can
be checked by computing a unique sequence of involutive distributions which
generalizes the well-known static feedback linearization test. In this paper,
a dual test for forward-flatness based on a unique sequence of integrable
codistributions is derived. Since the main mathematical operations for
determining this sequence are the intersection of codistributions and the
calculation of Lie derivatives of 1-forms, it is computationally quite
efficient. Furthermore, the formulation with codistributions also facilitates
a comparison with the existing discrete-time literature regarding the closely
related topic of dynamic feedback linearization, which is mostly formulated in
terms of 1-forms rather than vector fields. The presented results are
illustrated by two examples.

###### keywords:

Difference flatness; Differential-geometric methods; Discrete-time systems;
Feedback linearization; Nonlinear control systems; Normal forms.

, ,

[cor]Corresponding author.

##  1 Introduction

The concept of flatness has been introduced by Fliess, Lévine, Martin and
Rouchon in the 1990s for nonlinear continuous-time systems, see e.g. [9],
[10], or [11]. Flat continuous-time systems possess the characteristic feature
that all system variables can be expressed by a flat output and its time
derivatives, which allows elegant solutions for trajectory planning and
trajectory tracking problems. Because of its high practical relevance with a
wide variety of applications, flatness has attracted a lot of attention in the
control systems community. Nevertheless, checking the flatness of nonlinear
multi-input systems is a highly nontrivial problem, for which still no
complete systematic solution in the form of verifiable necessary and
sufficient conditions exists (see e.g. [24], [25], or [14]).

In view of the inherent discrete-time nature of digital control circuits,
applying flatness-based methods to discrete-time systems is not only
interesting from a theoretical but also from a practical perspective.
Furthermore, a discrete-time approach is also favorable for a combination with
data-based methods, see e.g. [1] or [8]. However, it should be noted that
regarding the flatness of nonlinear discrete-time systems there exist two
approaches. The first one is to replace the time derivatives of the
continuous-time definition by forward-shifts as e.g. in [17], [29], or [19].
This point of view is also consistent with the notion of endogenous and
exogenous dynamic feedback as it is defined in [3]. The second approach
considers flatness as the existence of a one-to-one correspondence between the
system trajectories and the trajectories of a trivial system. It is a
generalization of the first approach, since here the flat output may also
depend on backward-shifts of the system variables, see [7] or [15]. To ensure
a clear distinction, we refer to the first approach, which we consider in the
present paper, as forward-flatness.

In [20] it has been shown that forward-flat systems can be decomposed by
state- and input transformations into a subsystem and an endogenous dynamic
feedback, such that the complete system is forward-flat if and only if the
subsystem is forward-flat. Based on this result, it has been shown in [18]
that forward-flatness can be checked by computing a unique sequence of
involutive distributions, which generalizes the sequence of distributions from
the static feedback linearization test of [12]. The existence of such a
systematic test is surprising since there does not exist a counterpart in the
continuous-time case. The purpose of the present paper is now to derive a dual
test for forward-flatness, which is based on a unique sequence of integrable
codistributions. Our motivation is twofold: First, deriving a dual approach is
of interest in its own right, like the dual version of the well-known static
feedback linearization test for continuous-time systems (e.g. [30], [32],
[27]). Second, the paper is motivated by the fact that the majority of the
discrete-time literature regarding the closely related topics of dynamic
feedback linearization and controllability uses an algebraic framework based
on 1-forms, see e.g. [13], [2], or [3]. Even though we use differential-
geometric methods, the codistributions of the proposed sequence are also
spanned by 1-forms. Thus, the dual approach should be accessible to a wider
audience and also facilitate a comparison with the existing literature. A
rather recent paper which uses an algebraic framework based on 1-forms is
[16], where necessary and sufficient conditions for flatness of discrete-time
systems in the more general sense including backward-shifts are proposed.
However, as mentioned there, even when applied to the special case of forward-
flat systems the computational complexity is higher since partial differential
equations have to be solved, whereas in the present contribution only the
intersection of codistributions and computing Lie derivatives of 1-forms will
be needed. Another important aspect of our approach is that for a given
discrete-time system the proposed sequence of codistributions is uniquely
defined, i.e., like in [18] there occur no degrees of freedom during the
computations.

The paper is organized as follows: First, in Section 2 we recapitulate the
concept of forward-flatness for discrete-time systems. Subsequently, Section 3
addresses invariant codistributions and Cauchy characteristics, which form the
mathematical foundation upon which the paper is based. In Section 4 we then
introduce the sequence of codistributions, and show how it is related to
system decompositions and forward-flatness. Finally, Section 5 illustrates our
results by two examples.

#### Notation

Throughout the paper we make use of basic differential-geometric concepts. The
Lie derivative of a differential $k$-form $\alpha$ along a vector field $v$ is
denoted by $L_{v}\alpha$, the exterior derivative is denoted by
$\mathrm{d}\alpha$, and the wedge product of differential forms $\alpha$ and
$\beta$ is written as $\alpha\wedge\beta$. The contraction (interior product)
of a vector field with a differential form is denoted by $v\rfloor\alpha$, and
the Lie bracket of two vector fields $v$ and $w$ by $[v,w]$. If $P$ is a
$d$-dimensional codistribution on an $n$-dimensional manifold $\mathcal{M}$,
then $P^{\perp}$ denotes its annihilator, i.e., the $(n-d)$-dimensional
distribution consisting of all vector fields $v$ that meet $v\rfloor\omega=0$
for all 1-forms $\omega\in P$. Likewise, $D^{\perp}$ denotes the annihilator
of a distribution $D$, consisting of all 1-forms $\omega$ that meet
$v\rfloor\omega=0$ for all vector fields $v\in D$. To keep formulas short and
readable, we also use the Einstein summation convention. For instance, if we
have local coordinates $x^{1},\ldots,x^{n}$ on $\mathcal{M}$, then we write a
vector field $v$ as $v^{i}(x)\partial_{x^{i}}$ and a 1-form $\omega$ as
$\omega_{i}(x)\mathrm{d}x^{i}$ without a summation symbol since the index
range is clear from the context. Furthermore, if we write e.g.
$\mathrm{span}\\{\omega^{1},\omega^{2}\\}$ with 1-forms
$\omega^{1},\omega^{2}$ defined on a manifold $\mathcal{M}$, then we mean
$\mathrm{span}_{C^{\infty}(\mathcal{M})}\\{\omega^{1},\omega^{2}\\}$, i.e.,
the span over the ring of smooth functions $C^{\infty}(\mathcal{M})$. Finally,
the symbols $\subset$ and $\supset$ are used in the sense that they also
include equality. For an introduction to differential-geometric concepts with
a focus on nonlinear control applications we refer e.g. to [26] or [27], and
for a rather general introduction to differential geometry e.g. to [4] or
[23].

##  2 Discrete-time Systems and Forward-flatness

In this contribution, we consider nonlinear time-invariant discrete-time
systems

$x^{i,+}=f^{i}(x,u)\,,\quad i=1,\ldots,n$ (1)

with $\dim(x)=n$, $\dim(u)=m$, and smooth functions $f(x,u)$ that satisfy the
submersivity condition

$\mathrm{rank}(\partial_{(x,u)}f)=n\,.$ (2)

The assumption (2) is quite common in the discrete-time literature and
necessary for accessibility (see e.g. [13]). The superscript $+$ on the left-
hand side of the system equations (1) denotes the forward-shift of the
corresponding variables. For indicating also higher-order forward-shifts, we
use subscripts in brackets. For instance, the $\alpha$-th forward-shift of a
component $u^{j}$ of the input with some $\alpha\in\mathbb{N}$ is denoted by
$u_{[\alpha]}^{j}$, and
$u_{[\alpha]}=(u_{[\alpha]}^{1},\ldots,u_{[\alpha]}^{m})$. In order to
summarize the concept of forward-flatness, we make use of a space with
coordinates $(x,u,u_{[1]},u_{[2]},\ldots)$. If $g$ is a smooth function
defined on this space, then its future values can be determined by a repeated
application of the forward-shift operator, which is defined according to the
rule

$\delta(g(x,u,u_{[1]},u_{[2]},\ldots))=g(f(x,u),u_{[1]},u_{[2]},u_{[3]},\ldots)\,.$

To define a backward-shift operator $\delta^{-1}$, in general it would be
necessary to extend the system map (1) such that it becomes invertible.
However, in this contribution we only need backward-shifts of functions of the
form $h(f(x,u))$, and in this case the backward-shift is obviously given by

$\delta^{-1}(h(f(x,u))=h(x)\,.$

The shift operators can also be applied to 1-forms by shifting both their
coefficients and differentials. In particular, we will need the backward-shift
of 1-forms of the form $\omega_{i}(f(x,u))\mathrm{d}f^{i}$, which is given by

$\delta^{-1}(\omega_{i}(f(x,u))\mathrm{d}f^{i})=\omega_{i}(x)\mathrm{d}x^{i}\,.$
(3)

For codistributions which are spanned by 1-forms of the form
$\omega_{i}(f(x,u))\mathrm{d}f^{i}$, the backward-shift is defined
accordingly. It is also important to emphasize that all our results are local
since we apply the inverse- and the implicit function theorem as well as the
Frobenius theorem, which allow only local results. Thus, like the discrete-
time static feedback linearization problem, we consider discrete-time flatness
in a suitable neighborhood of an equilibrium $(x_{0},u_{0})$ of the system
(1). Since the map (1) is continuous, it can then be guaranteed that the
system trajectories do not leave the regions of validity of the coordinate
transformations derived by the above-mentioned theorems (see e.g. [12], [26],
or [22]). Forward-flatness can now be defined as follows.

######  Definition 1.

The system (1) is said to be forward-flat around an equilibrium
$(x_{0},u_{0})$, if the $n+m$ coordinate functions $x$ and $u$ can be
expressed locally by an $m$-tuple of functions

$y^{j}=\varphi^{j}(x,u,u_{[1]},\ldots,u_{[q]})\,,\quad j=1,\ldots,m$ (4)

and their forward-shifts
$y_{[1]}=\delta(\varphi(x,u,u_{[1]},\ldots,u_{[q]}))$,
$y_{[2]}=\delta^{2}(\varphi(x,u,u_{[1]},\ldots,u_{[q]}))$, $\ldots$ up to some
finite order. The $m$-tuple (4) is called a flat output.

The representation of $x$ and $u$ by the flat output and its forward-shifts is
unique, and has the form111The multi-index $R=(r_{1},\ldots,r_{m})$ contains
the number of forward-shifts of the individual components of the flat output
which is needed to express $x$ and $u$, and $y_{[0,R]}$ is an abbreviation for
$y$ and its forward-shifts up to order $R$.

$\begin{array}[]{cclcl}x^{i}&=&F_{x}^{i}(y_{[0,R-1]})\,,&&i=1,\ldots,n\\\
u^{j}&=&F_{u}^{j}(y_{[0,R]})\,,&&j=1,\ldots,m\,.\end{array}$ (5)

The term forward-flatness refers to the fact that both in the flat output (4)
as well as in the corresponding parameterization of the system variables (5)
there occur forward-shifts but no backward-shifts like in the more general
case discussed in [7] or [15]. As proven in [20], forward-flat systems can
always be transformed into a certain triangular form.

######  Theorem 2.

A forward-flat system (1) can be transformed by a state- and input
transformation

$\displaystyle(\bar{x}_{1},\bar{x}_{2})=\>\>$ $\displaystyle\Phi_{x}(x)$ (6a)
$\displaystyle(\bar{u}_{1},\bar{u}_{2})=\>\>$ $\displaystyle\Phi_{u}(x,u)$
(6b)

into the form

$\displaystyle\bar{x}_{2}^{+}$
$\displaystyle=f_{2}(\bar{x}_{2},\bar{x}_{1},\bar{u}_{2})$ (7a)
$\displaystyle\bar{x}_{1}^{+}$
$\displaystyle=f_{1}(\bar{x}_{2},\bar{x}_{1},\bar{u}_{2},\bar{u}_{1})$ (7b)

with $\dim(\bar{x}_{1})\geq 1$ and
$\mathrm{rank}(\partial_{\bar{u}_{1}}f_{1})=\dim(\bar{x}_{1})$.

######  Proof 2.1.

For systems (1) with $\mathrm{rank}(\partial_{u}f)=m$, i.e., systems without
redundant inputs, this is shown in Theorem 6 of [20]. In the case
$\mathrm{rank}(\partial_{u}f)<m$, redundant inputs can always be eliminated by
an input transformation without affecting the forward-flatness of the system,
see e.g. Lemma 9 of [20]. Thus, combining Theorem 6 and Lemma 9 of [20]
completes the proof for general systems (1) with
$\mathrm{rank}(\partial_{u}f)\leq m$.

The importance of the triangular form (7) is due to the following property.

######  Lemma 3.

A system of the form (7) with
$\mathrm{rank}(\partial_{\bar{u}_{1}}f_{1})=\dim(\bar{x}_{1})$ is forward-flat
if and only if the subsystem (7a) with the inputs $(\bar{x}_{1},\bar{u}_{2})$
is forward-flat. The flat outputs are related as follows:

  1. (i)

If $\dim(\bar{u}_{1})=\dim(\bar{x}_{1})$, then every flat output $y_{2}$ of
(7a) is also a flat output $y$ of (7).

  2. (ii)

If $\dim(\bar{u}_{1})>\dim(\bar{x}_{1})$, then the components of $\bar{u}_{1}$
can be split in the form $\bar{u}_{1}=(\hat{u}_{1},\tilde{u}_{1})$ such that
$\mathrm{rank}(\partial_{\hat{u}_{1}}f_{1})=\dim(\bar{x}_{1})$, and a flat
output of (7) is given by $y=(y_{2},\tilde{u}_{1})$.

######  Proof 2.2.

Because of $\mathrm{rank}(\partial_{\bar{u}_{1}}f_{1})=\dim(\bar{x}_{1})$, it
is always possible to choose the input transformation (6b) such that with
$\bar{u}_{1}=(\hat{u}_{1},\tilde{u}_{1})$ the equations (7b) have the form
$\bar{x}_{1}^{+}=\hat{u}_{1}$. Since (7b) is then only a simple prolongation
of the input variables $\bar{x}_{1}$ of the subsystem (7a), the claims follow
immediately from Definition 1.

######  Remark 4.

In the case $\dim(\bar{u}_{1})=\dim(\bar{x}_{1})$, the equations (7b) can be
considered as an endogenous dynamic feedback for the subsystem (7a), since a
solution of the subsystem uniquely determines a solution of the complete
system (7). In the case $\dim(\bar{u}_{1})>\dim(\bar{x}_{1})$ the trajectory
of the input variables $\tilde{u}_{1}$ can still be chosen arbitrarily,
reflecting the fact that a flat output of (7) is given by
$y=(y_{2},\tilde{u}_{1})$.

######  Remark 5.

A special case of forward-flat systems are systems which are linearizable by a
static feedback. Such systems allow (repeated) decompositions of the form (7)
with $\bar{u}_{2}$ empty, see [26].

##  3 Invariant Codistributions and Cauchy Characteristics

The test for forward-flatness which we propose in Section 4 is based on the
concept of invariant codistributions. The notion of invariant distributions
and codistributions is used quite frequently in control theory, see e.g. [26].
A typical field of application is the analysis of the controllability and
observability properties of nonlinear continuous-time systems. In this
section, we recapitulate basic facts and formulate some technical results
which we need in the main part of the paper. Throughout the paper, we assume
that all considered distributions and codistributions have locally constant
dimension.

A $p$-dimensional codistribution
$P=\mathrm{span}\\{\omega^{1},\ldots,\omega^{p}\\}$, defined on some
$n$-dimensional manifold $\mathcal{M}$ with local coordinates
$(x^{1},\ldots,x^{n})$, is called invariant w.r.t. a vector field $v$ if
$L_{v}\omega\in P$ for all 1-forms $\omega\in P$. This condition is often
abbreviated as $L_{v}P\subset P$, cf. [26]. Accordingly, $P$ is called
invariant w.r.t. a $d$-dimensional distribution
$D=\mathrm{span}\\{v_{1},\ldots,v_{d}\\}$ if $L_{v}\omega\in P$ for all
1-forms $\omega\in P$ and vector fields $v\in D$. It is straightforward to
verify that this is the case if and only if

$L_{v_{i}}\omega^{j}\in P\,,\quad i=1,\ldots,d\,,\,j=1,\ldots,p$

for arbitrary bases $\\{v_{1},\ldots,v_{d}\\}$ of $D$ and
$\\{\omega^{1},\ldots,\omega^{p}\\}$ of $P$. Furthermore, it is an immediate
consequence of the definition of invariance that also all higher order
(repeated and mixed) Lie derivatives of 1-forms $\omega\in P$ w.r.t. vector
fields $v\in D$ are contained in $P$.

Given a vector field $v$ and a codistribution $P$ which is not invariant
w.r.t. $v$, one can pose the question how to extend $P$ such that it becomes
invariant. Indeed, there always exists a unique smallest invariant
codistribution which contains $P$. Computing the smallest invariant
codistribution which contains a given codistribution is used in the control
literature e.g. for checking the observability of nonlinear continuous-time
systems, see [26]. In the following, we consider the general case of
invariance w.r.t. a distribution, and show that the invariant extension can be
constructed by adding suitable (also higher-order) Lie derivatives of the
1-forms of an arbitrary basis of $P$.

######  Proposition 6.

Given a codistribution $P$ and a distribution $D$, there exists a unique
codistribution $\hat{P}$ of minimal dimension which contains $P$ and is
invariant w.r.t. $D$.

######  Proof 3.1.

The statement can be proven by giving a procedure for the construction of a
basis of $\hat{P}$. According to the definition of invariance, $\hat{P}$ must
contain all (also higher order, repeated and mixed) Lie derivatives of the
1-forms $\omega^{1},\ldots,\omega^{p}$ of $P$ w.r.t. all vector fields
$v_{1},\ldots,v_{d}$ of $D$. Thus, in a first step, the basis
{$\omega^{1},\ldots,\omega^{p}\\}$ of $P$ is extended by first-order Lie
derivatives $L_{v_{i}}\omega^{j}$ until a maximal set of linearly independent
1-forms is obtained. In a second step, Lie derivatives of the 1-forms which
were added in the first step are added until again a maximal set of linearly
independent 1-forms is obtained. All other second-order Lie derivatives do not
need to be considered, since they are linearly dependent by construction. For
this reason, the number of new linearly independent 1-forms either remains
constant or decreases with every further step. Since the dimension of a
codistribution cannot exceed the dimension of the underlying manifold, the
procedure terminates after a finite number of steps. The constructed basis for
$\hat{P}$ consists only of the 1-forms $\omega^{1},\ldots,\omega^{p}$ and
their Lie derivatives, which, by the definition of invariance, must be
contained in an invariant extension of $P$ anyway. Thus, the constructed
codistribution $\hat{P}$ is the unique smallest invariant codistribution which
contains $P$.

If the distribution $D$ is involutive and meets $D\rfloor P=0$, which means
that $v\rfloor\omega=0$ for all $v\in D$ and $\omega\in P$, then the following
result for $\hat{P}$ can be shown.

######  Proposition 7.

Consider a codistribution $P$ and a distribution $D$, and let $\hat{P}$ denote
the smallest codistribution which contains $P$ and is invariant w.r.t. $D$. If
$D\rfloor P=0$ and $D$ is involutive, then also $D\rfloor\hat{P}=0$.

######  Proof 3.2.

The proof is based on the fact that a basis for $\hat{P}$ can be constructed
by simply adding suitable Lie derivatives to a basis of $P$ as shown above.
According to Cartan’s magic formula, the Lie derivative of a 1-form $\omega$
along a vector field $v$ is given by

$L_{v}\omega=v\rfloor\mathrm{d}\omega+\mathrm{d}(v\rfloor\omega)\,.$ (8)

Since $v\rfloor\omega=0$ holds for all 1-forms $\omega$ of $P$ and vector
fields $v$ of $D$, the first-order Lie derivatives are of the form
$L_{v}\omega=v\rfloor\mathrm{d}\omega$. With $w$ denoting another arbitrary
vector field of $D$ and using the identity222See e.g. [23].

$w\rfloor\left(v\rfloor\mathrm{d}\omega\right)=L_{v}\left(w\rfloor\omega\right)-L_{w}\left(v\rfloor\omega\right)-\left[v,w\right]\rfloor\omega\,,$

we get

$w\rfloor
L_{v}\omega=L_{v}\left(w\rfloor\omega\right)-L_{w}\left(v\rfloor\omega\right)-\left[v,w\right]\rfloor\omega\,.$

Because of $D\rfloor P=0$ and the involutivity of $D$, which implies
$\left[v,w\right]\in D$, all terms on the right-hand side vanish and hence
$w\rfloor L_{v}\omega=0$. Consequently, since $w$ is an arbitrary vector field
of $D$, the first-order Lie derivatives meet $D\rfloor L_{v}\omega=0$.
Repeating this argumentation shows that this is also true for all higher-order
Lie derivatives, and hence $D\rfloor\hat{P}=0$.

With the additional property $D\rfloor P=0$, invariance is closely related to
the notion of Cauchy-characteristic vector fields and distributions, which is
discussed in detail e.g. in [28], [31], or [5].

######  Definition 8.

([28]) A vector field $v$ is called a Cauchy-characteristic vector field of a
codistribution $P$ if

$v\rfloor P=0\quad\text{and}\quad v\rfloor\mathrm{d}P\subset P\,.$ (9)

The set of all Cauchy-characteristic vector fields forms the Cauchy-
characteristic distribution $\mathcal{C}(P)$, which is involutive.

The importance of Cauchy-characteristic vector fields lies in the existence of
coordinate transformations such that $P$ can be represented by a reduced
number of coordinates. More precisely, if an arbitrary vector field
$v\in\mathcal{C}(P)$ is straightened out by the flow-box theorem using a
coordinate transformation $\bar{x}=\Phi(x)$ such that
$v=\partial_{\bar{x}^{1}}$, then there exists a basis for $P$ which is
independent of $\bar{x}^{1}$. Since $\mathcal{C}(P)$ is involutive, the
Frobenius theorem allows to straighten out even the whole distribution. In
such coordinates, there exists a basis for $P$ which is independent of the
corresponding $\dim(\mathcal{C}(P))$ coordinates (see [28], [31], or [5]). In
the main part of the paper, we make use of the following straightforward
observation.

######  Proposition 9.

If a codistribution $P$ is invariant w.r.t. a distribution $D$ which meets
$D\rfloor P=0$, then $D$ is a subdistribution of the Cauchy-characteristic
distribution $\mathcal{C}(P)$ of $P$.

######  Proof 3.3.

The Lie derivative of a 1-form $\omega$ along a vector field $v$ is given by
(8). Thus, every vector field $v$ with $v\rfloor P=0$ which meets the
invariance condition $L_{v}P\subset P$ also meets the condition (9) for a
Cauchy-characteristic vector field of $P$. Consequently,
$D\subset\mathcal{C}(P)$.

In contrast to the Cauchy-characteristic distribution, the distribution $D$ of
Proposition 9 is not necessarily involutive. However, if $D$ is involutive, it
can also be straightened out by the Frobenius theorem, and then because of
$D\subset\mathcal{C}(P)$ there exists a basis for $P$ which is independent of
the corresponding $\dim(D)$ coordinates. The following corollary summarizes
and combines the content of this section such that we can directly apply it in
the remainder of the paper.

######  Corollary 10.

Consider a codistribution $P$ and an involutive distribution $D$ with
$D\rfloor P=0$. Then the following holds.

  1. (i)

There exists a unique smallest codistribution $\hat{P}$ which contains $P$ and
is invariant w.r.t. $D$.

  2. (ii)

The codistribution $\hat{P}$ meets $D\rfloor\hat{P}=0$.

  3. (iii)

The distribution $D$ is contained in the Cauchy-characteristic distribution of
$\hat{P}$, i.e., $D\subset\mathcal{C}(\hat{P})$.

  4. (iv)

After performing a coordinate transformation $\bar{x}=\Phi(x)$ such that
$D=\mathrm{span}\\{\partial_{\bar{x}^{1}},\ldots,\partial_{\bar{x}^{d}}\\}$,
there exists a basis for $\hat{P}$ which is independent of
$\bar{x}^{1},\ldots,\bar{x}^{d}$.

##  4 Dual Test for Forward-flatness

In this section we introduce a certain sequence of codistributions, and
subsequently show how it is related to system decompositions and forward-
flatness.

###  4.1 Definition of the Sequence of Codistributions

In the following, we define a sequence of codistributions which allows to
efficiently check the forward-flatness of nonlinear discrete-time systems (1).
For a given system the sequence is uniquely determined, i.e., there occur no
degrees of freedom. The main mathematical operations are the intersection of
codistributions, the calculation of Lie derivatives of 1-forms to determine
the smallest invariant extensions of codistributions according to Proposition
6, and backward-shifts of 1-forms. Hence, solving ODEs or even PDEs is not
required. The calculations are performed on the $(n+m)$-dimensional state- and
input manifold $\mathcal{X}\times\mathcal{U}$ with coordinates $(x,u)$. In
order to simplify the calculations as much as possible, it can be convenient
to introduce adapted coordinates

$\begin{array}[]{ccl}\theta^{i}&=&f^{i}(x,u)\,,\quad i=1,\ldots,n\\\
\xi^{j}&=&h^{j}(x,u)\,,\quad j=1,\ldots,m\end{array}$ (10)

instead of $x$ and $u$, since then
$\mathrm{span}\\{\mathrm{d}f\\}=\mathrm{span}\\{\mathrm{d}\theta\\}$ and
$\mathrm{span}\\{\mathrm{d}f\\}^{\perp}=\mathrm{span}\\{\partial_{\xi}\\}$.
Because of the submersivity property (2), there always exist functions
$h^{j}(x,u)$ such that the Jacobian matrix of the right-hand side of (10) is
regular and the transformation hence invertible.

######  Algorithm 11.

Start with $P_{1}=\mathrm{span}\\{\mathrm{d}x\\}$, and repeat the following
steps for $k\geq 1$:

  1. _1._

Compute the intersection $P_{k}\cap\mathrm{span}\\{\mathrm{d}f\\}$.

  2. _2._

Determine the smallest codistribution $P_{k+1}^{+}$ which is invariant w.r.t.
the distribution $\mathrm{span}\\{\mathrm{d}f\\}^{\perp}$ and contains
$P_{k}\cap\mathrm{span}\\{\mathrm{d}f\\}$.333In case
$P_{k}\cap\mathrm{span}\\{\mathrm{d}f\\}$ is already invariant w.r.t.
$\mathrm{span}\\{\mathrm{d}f\\}^{\perp}$, this step is trivial since
$P_{k+1}^{+}=P_{k}\cap\mathrm{span}\\{\mathrm{d}f\\}$.

  3. _3._

Define $P_{k+1}=\delta^{-1}(P_{k+1}^{+})$.

Stop if $P_{\bar{k}+1}=P_{\bar{k}}$ for some $k=\bar{k}$.

According to Proposition 6 with $D=\mathrm{span}\\{\mathrm{d}f\\}^{\perp}$ and
$P=P_{k}\cap\mathrm{span}\\{\mathrm{d}f\\}$, the smallest invariant extension
$P_{k+1}^{+}$ in Step 2 (corresponding to $\hat{P}$ of Proposition 6) is
unique. Because of item (ii) of Corollary 10, like
_$P_{k}\cap\mathrm{span}\\{\mathrm{d}f\\}$_ also the codistribution
$P_{k+1}^{+}$ is contained in $\mathrm{span}\\{\mathrm{d}f\\}$. Moreover,
because of item (iii) and (iv) of Corollary 10, $P_{k+1}^{+}$ possesses even a
basis with 1-forms of the form $\omega_{i}(f(x,u))\mathrm{d}f^{i}$, where also
the coefficients $\omega_{i}$ only depend on the functions $f(x,u)$. This is
obvious in adapted coordinates (10) with
$\mathrm{span}\\{\mathrm{d}f\\}=\mathrm{span}\\{\mathrm{d}\theta\\}$ and
$\mathrm{span}\\{\mathrm{d}f\\}^{\perp}=\mathrm{span}\\{\partial_{\xi}\\}$.
Thus, in Step 3, a basis for $P_{k+1}$ is obtained by applying (3) to these
1-forms.

######  Remark 12.

Note that the computational effort is lower than in [18], where the
calculation of the corresponding sequence of distributions requires the
calculation of largest projectable subdistributions in every step. The
computation of the smallest invariant codistributions in Step 2 of Algorithm
11 by just adding Lie derivatives of 1-forms can be considered here as the
simpler task.

In order to show that the stop condition of Algorithm 11 is reasonable, we
prove now the following.

######  Proposition 13.

The codistributions $P_{1},\ldots,P_{\bar{k}}$ form a nested sequence

$P_{\bar{k}}\subset P_{\bar{k}-1}\subset\ldots\subset P_{1}\,.$ (11)

######  Proof 4.1.

First, let us show that all codistributions $P_{1},\ldots,P_{\bar{k}}$ are
contained in $\mathrm{span}\\{\mathrm{d}x\\}$. Indeed, an application of
Corollary 10 with $D=\mathrm{span}\\{\mathrm{d}f\\}^{\perp}$ and
$P=P_{k}\cap\mathrm{span}\\{\mathrm{d}f\\}$ shows that
_$P_{k+1}^{+}\subset\mathrm{span}\\{\mathrm{d}f\\}$_ for all $k\geq 1$.
Because of $\delta^{-1}(\mathrm{d}f)=\mathrm{d}x$,

$P_{k+1}\subset\mathrm{span}\\{\mathrm{d}x\\}\,,\quad k\geq 1$

follows. Next, we show that the codistributions $P_{1},\ldots,P_{\bar{k}}$
form a nested sequence. For $k=1$, because of
$P_{1}=\mathrm{span}\\{\mathrm{d}x\\}$ this directly implies $P_{2}\subset
P_{1}$. Now assume that for some $k>1$ we have $P_{k}\subset P_{k-1}$. Then
obviously also $P_{k}\cap\mathrm{span}\\{\mathrm{d}f\\}\subset
P_{k-1}\cap\mathrm{span}\\{\mathrm{d}f\\}$ holds, and the invariant extensions
computed in Step 2 of the procedure meet $P_{k+1}^{+}\subset P_{k}^{+}$ (an
invariant codistribution which contains
$P_{k-1}\cap\mathrm{span}\\{\mathrm{d}f\\}$ must also contain
$P_{k}\cap\mathrm{span}\\{\mathrm{d}f\\}\subset
P_{k-1}\cap\mathrm{span}\\{\mathrm{d}f\\}$). Applying the backward-shift
operator to this relation yields $P_{k+1}\subset P_{k}$, and by induction we
finally get (11).

Next, we address the integrability of the codistributions (11).

######  Proposition 14.

The codistributions of the sequence (11) are integrable.

######  Proof 4.2.

For $k=1$, the codistribution $P_{1}=\mathrm{span}\\{\mathrm{d}x\\}$ is
clearly integrable. In the following, we prove that for $k\geq 1$ the
integrability of $P_{k}$ implies the integrability of $P_{k+1}$. In fact, we
only need to prove the integrability of _$P_{k+1}^{+}$_ , since the
application of the backward-shift operator in Step 3 of the procedure does not
affect the integrability. For this purpose, let

$P_{k}\cap\mathrm{span}\\{\mathrm{d}f\\}=\mathrm{span}\\{\omega^{1},\ldots,\omega^{d_{1}}\\}$

and

$P_{k}=\mathrm{span}\\{\omega^{1},\ldots,\omega^{d_{1}},\mu^{1},\ldots,\mu^{d_{2}}\\}\,,$

with 1-forms $\mu$ that are not contained in $\mathrm{span}\\{\mathrm{d}f\\}$.
Furthermore, let

$P_{k+1}^{+}=\mathrm{span}\\{\omega^{1},\ldots,\omega^{d_{1}},\rho^{1},\ldots\rho^{d_{3}}\\}\,,$
(12)

with the 1-forms $\rho$ denoting suitable Lie derivatives of the 1-forms
$\omega$ according to the construction of an invariant codistribution
discussed in the proof of Proposition 6. First, it is important to note that

$\omega^{1}\wedge\ldots\wedge\omega^{d_{1}}\wedge\mu^{1}\wedge\ldots\wedge\mu^{d_{2}}\wedge\rho^{1}\wedge\ldots\wedge\rho^{d_{3}}\neq
0$ (13)

since all these 1-forms are linearly independent. The 1-forms
$\omega^{1},\ldots,\omega^{d_{1}},\rho^{1},\ldots\rho^{d_{3}}$ are clearly
linearly independent since they form a basis for
$P_{k+1}^{+}\subset\mathrm{span}\\{\mathrm{d}f\\}$, and after adding
$\mu^{1},\ldots,\mu^{d_{2}}$ the linear independence still holds since there
exists no linear combination of the latter 1-forms which is contained in
$\mathrm{span}\\{\mathrm{d}f\\}$. The codistribution (12) is now integrable if
and only if

$\mathrm{d}\omega^{s}\wedge\omega^{1}\wedge\ldots\wedge\omega^{d_{1}}\wedge\rho^{1}\wedge\ldots\wedge\rho^{d_{3}}=0\,,\quad
s=1,\ldots,d_{1}$ (14)

as well as

$\mathrm{d}\rho^{s}\wedge\omega^{1}\wedge\ldots\wedge\omega^{d_{1}}\wedge\rho^{1}\wedge\ldots\wedge\rho^{d_{3}}=0\,,\quad
s=1,\ldots,d_{3}\,.$ (15)

To prove (14), we make use of the assumption that $P_{k}$ is integrable, which
implies

$\mathrm{d}\omega^{s}\wedge\omega^{1}\wedge\ldots\wedge\omega^{d_{1}}\wedge\mu^{1}\wedge\ldots\wedge\mu^{d_{2}}=0\,,$

and because of (13) also

$\mathrm{d}\omega^{s}\wedge\omega^{1}\wedge\ldots\wedge\omega^{d_{1}}\wedge\mu^{1}\wedge\ldots\wedge\mu^{d_{2}}\wedge\rho^{1}\wedge\ldots\wedge\rho^{d_{3}}=0\,.$
(16)

In the following, it is convenient to introduce adapted coordinates (10) since
then $\mathrm{span}\\{\mathrm{d}f\\}=\mathrm{span}\\{\mathrm{d}\theta\\}$ and
$\mathrm{span}\\{\mathrm{d}f\\}^{\perp}=\mathrm{span}\\{\partial_{\xi}\\}$. In
such coordinates, the 1-forms $\omega$ have the form

$\omega^{s}=\omega_{i}^{s}(\theta,\xi)\mathrm{d}\theta^{i}\,,$

and their exterior derivative can be written as

$\displaystyle\mathrm{d}\omega^{s}$
$\displaystyle=\partial_{\theta^{k}}\omega_{i}^{s}\mathrm{d}\theta^{k}\wedge\mathrm{d}\theta^{i}+\partial_{\xi^{j}}\omega_{i}^{s}\mathrm{d}\xi^{j}\wedge\mathrm{d}\theta^{i}$
$\displaystyle=\partial_{\theta^{k}}\omega_{i}^{s}\mathrm{d}\theta^{k}\wedge\mathrm{d}\theta^{i}-L_{\partial_{\xi^{j}}}(\omega^{s})\wedge\mathrm{d}\xi^{j}$
(17)

with $L_{\partial_{\xi^{j}}}(\omega^{s})$ denoting the Lie derivative of
$\omega^{s}$ along the vector field $\partial_{\xi^{j}}$. Because of
$L_{\partial_{\xi^{j}}}(\omega^{s})\in
P_{k+1}^{+}=\mathrm{span}\\{\omega,\rho\\}$ (invariance of $P_{k+1}^{+}$
w.r.t.
$\mathrm{span}\\{\mathrm{d}f\\}^{\perp}=\mathrm{span}\\{\partial_{\xi}\\}$) we
have

$L_{\partial_{\xi^{j}}}(\omega^{s})\wedge\mathrm{d}\xi^{j}\wedge\omega^{1}\wedge\ldots\wedge\omega^{d_{1}}\wedge\rho^{1}\wedge\ldots\wedge\rho^{d_{3}}=0\,,$
(18)

and hence substituting (17) into (16) yields

$\partial_{\theta^{k}}\omega_{i}^{s}\mathrm{d}\theta^{k}\wedge\mathrm{d}\theta^{i}\wedge\omega^{1}\wedge\ldots\wedge\omega^{d_{1}}\\\
\wedge\mu^{1}\wedge\ldots\wedge\mu^{d_{2}}\wedge\rho^{1}\wedge\ldots\wedge\rho^{d_{3}}=0\,,$
(19)

i.e., the terms with
$L_{\partial_{\xi^{j}}}(\omega^{s})\wedge\mathrm{d}\xi^{j}$ vanish. Since
there exists no linear combination of the 1-forms $\mu$ which is contained in
$\mathrm{span}\\{\mathrm{d}f\\}=\mathrm{span}\\{\mathrm{d}\theta\\}$, (19) is
equivalent to

$\partial_{\theta^{k}}\omega_{i}^{s}\mathrm{d}\theta^{k}\wedge\mathrm{d}\theta^{i}\wedge\omega^{1}\wedge\ldots\wedge\omega^{d_{1}}\wedge\rho^{1}\wedge\ldots\wedge\rho^{d_{3}}=0\,.$
(20)

Subtracting (18) from (20) and replacing
$\partial_{\theta^{k}}\omega_{i}^{s}\mathrm{d}\theta^{k}\wedge\mathrm{d}\theta^{i}-L_{\partial_{\xi^{j}}}(\omega^{s})\wedge\mathrm{d}\xi^{j}$
by $\mathrm{d}\omega^{s}$ according to (17) finally shows that (14) indeed
holds. Next, we also have to prove (15). We do this by induction, using the
fact that the 1-forms $\rho$ are just Lie derivatives (possibly also higher
order) of the 1-forms $\omega$ along vector fields
$v\in\mathrm{span}\\{\mathrm{d}f\\}^{\perp}$. Let us assume that a 1-form
$\bar{\omega}\in P_{k+1}^{+}$ meets

$\mathrm{d}\bar{\omega}\wedge\omega^{1}\wedge\ldots\wedge\omega^{d_{1}}\wedge\rho^{1}\wedge\ldots\wedge\rho^{d_{3}}=0\,,$
(21)

which we have just proven for the 1-forms $\omega^{1},\ldots,\omega^{d_{1}}$,
and consider the expression

$\mathrm{d}\left(L_{v}\bar{\omega}\right)\wedge\omega^{1}\wedge\ldots\wedge\omega^{d_{1}}\wedge\rho^{1}\wedge\ldots\wedge\rho^{d_{3}}\,.$
(22)

Because of
$\mathrm{d}\left(L_{v}\bar{\omega}\right)=L_{v}\left(\mathrm{d}\bar{\omega}\right)$
and the property
$L_{v}(\alpha\wedge\beta)=L_{v}(\alpha)\wedge\beta+\alpha\wedge L_{v}(\beta)$
of the Lie derivative for arbitrary (also higher-order) differential forms
$\alpha,\beta$, (22) can be written as

$-\mathrm{d}\bar{\omega}\wedge
L_{v}\left(\omega^{1}\wedge\ldots\wedge\omega^{d_{1}}\wedge\rho^{1}\wedge\ldots\wedge\rho^{d_{3}}\right)\,,$
(23)

where we have already used the assumption (21). Due to the invariance of
$P_{k+1}^{+}=\mathrm{span}\\{\omega,\rho\\}$ w.r.t. vector fields
$v\in\mathrm{span}\\{\mathrm{d}f\\}^{\perp}$, (23) is of the form

$c\,\mathrm{d}\bar{\omega}\wedge\omega^{1}\wedge\ldots\wedge\omega^{d_{1}}\wedge\rho^{1}\wedge\ldots\wedge\rho^{d_{3}}$
(24)

with some smooth function $c\in C^{\infty}(\mathcal{X}\times\mathcal{U})$.
However, because of the assumption (21) the expression (24) vanishes, and
hence we have shown that (21) implies

$\mathrm{d}\left(L_{v}\bar{\omega}\right)\wedge\omega^{1}\wedge\ldots\wedge\omega^{d_{1}}\wedge\rho^{1}\wedge\ldots\wedge\rho^{d_{3}}=0\,.$

Since the 1-forms $\rho$ are Lie derivatives of the 1-forms $\omega$, (15)
follows and the proof is complete.

###  4.2 System Decompositions and Forward-Flatness

Before we prove that the sequence (11) gives rise to necessary and sufficient
conditions for forward-flatness, we show how it is related to the triangular
decomposition (7) of Theorem 2.

######  Theorem 15.

A discrete-time system (1) can be transformed into a triangular form (7) with
$\dim(\bar{x}_{1})\geq 1$ and
$\mathrm{rank}(\partial_{\bar{u}_{1}}f_{1})=\dim(\bar{x}_{1})$ if and only if
the corresponding sequence (11) meets $\dim(P_{2})<\dim(P_{1})$.

######  Proof 4.3.

_Necessity:_ To prove the necessity of $\dim(P_{2})<\dim(P_{1})$, we show that
for a system of the form (7) the condition
$P_{2}\subset\mathrm{span}\\{\mathrm{d}\bar{x}_{2}\\}$ holds.444Note that
$P_{2}$ corresponds to a decomposition (7) with the minimal possible dimension
of the subsystem (7a). There may also exist decompositions (7) of a system (1)
with $\dim(\bar{x}_{2})>\dim(P_{2})$. Because of
$\mathrm{rank}(\partial_{\bar{u}_{1}}f_{1})=\dim(\bar{x}_{1})$ and the fact
that the functions $f_{2}$ are independent of $\bar{u}_{1}$, all linear
combinations of the differentials $\mathrm{d}f$ which are contained in
$P_{1}=\mathrm{span}\\{\mathrm{d}\bar{x}\\}$ are linear combinations of the
differentials $\mathrm{d}f_{2}$ alone. Thus, a system of the form (7) meets

$P_{1}\cap\mathrm{span}\\{\mathrm{d}f\\}=P_{1}\cap\mathrm{span}\\{\mathrm{d}f_{2}\\}\subset\mathrm{span}\\{\mathrm{d}f_{2}\\}\,.$

Since $\mathrm{span}\\{\mathrm{d}f_{2}\\}$ contains
$P_{1}\cap\mathrm{span}\\{\mathrm{d}f\\}$ and is obviously invariant w.r.t.
$\mathrm{span}\\{\mathrm{d}f\\}^{\perp}$, it must also contain $P_{2}^{+}$,
which is by definition the smallest codistribution with these properties. A
backward-shift of the relation
$P_{2}^{+}\subset\mathrm{span}\\{\mathrm{d}f_{2}\\}$ then yields
$P_{2}\subset\mathrm{span}\\{\mathrm{d}\bar{x}_{2}\\}$, and because of
$P_{1}=\mathrm{span}\\{\mathrm{d}\bar{x}_{1},\mathrm{d}\bar{x}_{2}\\}$ with
$\dim(\bar{x}_{1})\geq 1$ we immediately get $\dim(P_{2})<\dim(P_{1})$.
_Sufficiency:_ To prove the sufficiency of $\dim(P_{2})<\dim(P_{1})$, we show
how the coordinate transformation (6) that achieves the decomposition (7) can
be derived. First, since $P_{2}$ is integrable and
$P_{2}\subset\mathrm{span}\\{\mathrm{d}x\\}$, there exists a state
transformation (6a) with

$\dim(\bar{x}_{1})=\dim(P_{1})-\dim(P_{2})\geq 1$

such that $P_{2}=\mathrm{span}\\{\mathrm{d}\bar{x}_{2}\\}$. For the resulting
system

$\displaystyle\bar{x}_{2}^{+}$
$\displaystyle=f_{2}(\bar{x}_{2},\bar{x}_{1},u)$ (25a)
$\displaystyle\bar{x}_{1}^{+}$
$\displaystyle=f_{1}(\bar{x}_{2},\bar{x}_{1},u)\,,$ (25b)

we perform an input transformation (6b) with
$\dim(\bar{u}_{2})=\mathrm{rank}(\partial_{u}f_{2})$ such that
$\dim(\bar{u}_{2})$ of the equations of the subsystem (25a) are simplified to
$\bar{x}_{2}^{i_{2},+}=\bar{u}_{2}^{i_{2}}$ (by just setting
$\bar{u}_{2}^{i_{2}}=f_{2}^{i_{2}}(\bar{x}_{2},\bar{x}_{1},u)$). After this
input transformation the system (25) must have the form (7) with $f_{2}$
independent of $\bar{u}_{1}$, since otherwise
$\mathrm{rank}(\partial_{\bar{u}}f_{2})>\dim(\bar{u}_{2})$. Now let us show
that also the condition
$\mathrm{rank}(\partial_{\bar{u}_{1}}f_{1})=\dim(\bar{x}_{1})$ indeed holds.
In the case $\mathrm{rank}(\partial_{\bar{u}_{1}}f_{1})<\dim(\bar{x}_{1})$
there would exist a nonzero linear combination of the differentials
$\mathrm{d}f_{1}$ which is contained in
$\mathrm{span}\\{\mathrm{d}\bar{x},\mathrm{d}\bar{u}_{2}\\}$, and because of
$\mathrm{rank}(\partial_{\bar{u}_{2}}f_{2})=\dim(\bar{u}_{2})$ the
differentials $\mathrm{d}\bar{u}_{2}$ of this linear combination could be
cancelled out by performing a further linear combination with the
differentials $\mathrm{d}f_{2}$. In other words, there would exist a linear
combination $\omega$ of the differentials $\mathrm{d}f_{1}$ and
$\mathrm{d}f_{2}$ which is contained in
$P_{1}=\mathrm{span}\\{\mathrm{d}\bar{x}\\}$ and involves at least one of the
differentials $\mathrm{d}f_{1}$. That is, $\omega\in
P_{1}\cap\mathrm{span}\\{\mathrm{d}f\\}$ but
$\omega\notin\mathrm{span}\\{\mathrm{d}f_{2}\\}$.555Because of the
submersivity property (2), a linear combination which involves at least one of
the differentials $\mathrm{d}f_{1}$ cannot be contained in
$\mathrm{span}\\{\mathrm{d}f_{2}\\}$. However, according to Step 2 of
Algorithm 11, with $P_{2}^{+}=\mathrm{span}\\{\mathrm{d}f_{2}\\}$ due to the
above state transformation we have
$P_{1}\cap\mathrm{span}\\{\mathrm{d}f\\}\subset\mathrm{span}\\{\mathrm{d}f_{2}\\}$,
which is a contradiction.

With this theorem, we have established a connection between a first
decomposition step (7) for a system (1) and the sequence (11). However, for
checking the forward-flatness of a system (1), in general several
decomposition steps are needed. In order to establish a connection between the
sequence (11) and repeated decompositions of the form (7), we prove the
following.

######  Lemma 16.

Consider a system (1) with the corresponding sequence (11) as well as a
decomposition (7) such that $P_{2}=\mathrm{span}\\{\mathrm{d}\bar{x}_{2}\\}$.
If $P_{k}^{\prime}$, $k\geq 1$ denotes the sequence (11) computed for the
subsystem (7a) on a smaller-dimensional manifold with coordinates
$(\bar{x}_{2},\bar{x}_{1},\bar{u}_{2})$, then $P_{k}^{\prime}=P_{k+1}$, $k\geq
1$.

######  Proof 4.4.

For $k=1$, because of
$P_{1}^{\prime}=\mathrm{span}\\{\mathrm{d}\bar{x}_{2}\\}$ it is obvious that
$P_{1}^{\prime}=P_{2}$. For $k>1$, the proof can be reduced to the question
whether a computation of the sequence according to Algorithm 11 on a smaller-
dimensional manifold $\mathcal{X}\times\mathcal{U}_{2}$ with coordinates
$(\bar{x}_{2},\bar{x}_{1},\bar{u}_{2})$ and $\mathrm{d}f_{2}$ instead of
$\mathrm{d}f$ yields the same result as a computation on the original manifold
$\mathcal{X}\times\mathcal{U}$ with the additional coordinates $\bar{u}_{1}$
and $\mathrm{d}f$. In Step 1 of Algorithm 11, because of
$\mathrm{rank}(\partial_{\bar{u}_{1}}f_{1})=\dim(\bar{x}_{1})$ and the fact
that the functions $f_{2}$ are independent of $\bar{u}_{1}$, it does not make
a difference whether a codistribution
$P_{k}\subset\mathrm{\mathrm{span}}\\{\mathrm{d}\bar{x}\\}$ is intersected
with $\mathrm{\mathrm{span}}\\{\mathrm{d}f\\}$ or
$\mathrm{\mathrm{span}}\\{\mathrm{d}f_{2}\\}$. Regarding Step 2, assume there
has been performed an additional input transformation
$(\hat{u}_{1},\tilde{u}_{1})=\Phi_{u}(\bar{x},\bar{u})$ which replaces
$\bar{u}_{1}$ such that the equations (7b) are simplified to the form
$\bar{x}_{1}^{i_{1},+}=\hat{u}_{1}^{i_{1}}$,
$i_{1}=1,\ldots,\dim(\bar{x}_{1})$. Because auf
$\mathrm{rank}(\partial_{\bar{u}_{1}}f_{1})=\dim(\bar{x}_{1})$ this is always
possible. Then it can be observed that in contrast to the distribution
$\mathrm{\mathrm{span}}\\{\mathrm{d}f_{2}\\}^{\perp}$ on
$\mathcal{X}\times\mathcal{U}_{2}$, the distribution
$\mathrm{\mathrm{span}}\\{\mathrm{d}f\\}^{\perp}$ on
$\mathcal{X}\times\mathcal{U}$ is larger since it contains the additional
vector fields $\partial_{\tilde{u}_{1}}$ corresponding to the redundant inputs
$\tilde{u}_{1}$ (provided that the system (1) has redundant inputs,
$\dim(\tilde{u}_{1})=m-\mathrm{rank}(\partial_{u}f)$). However, since the
considered codistributions are invariant w.r.t. the vector fields
$\partial_{\tilde{u}_{1}}$ anyway, also Step 2 yields the same result in both
cases. Finally, since Step 3 consists only in a backward-shift, indeed
$P_{k}^{\prime}=P_{k+1}$, $k\geq 1$.

With Lemma 16, we can now prove our main result.

######  Theorem 17.

A system (1) is forward-flat if and only if the sequence (11) terminates with
$P_{\bar{k}}=0$.

######  Proof 4.5.

From Theorem 2 and Lemma 3, it is clear that a system (1) is forward-flat if
and only if it can be decomposed repeatedly until in some step the subsystem
(7a) is trivial with $\dim(\bar{x}_{2})=0$. Because of Theorem 15 and Lemma
16, the existence of these repeated decompositions can be checked by computing
the codistributions of sequence (11). They correspond to the codistributions
spanned by the differentials of the state variables of the successively
computed subsystems (7a).666Assuming that in every step the decomposition is
performed such that $\dim(\bar{x}_{2})$ is minimal. Thus, the last subsystem
(7a) is trivial if and only if $P_{\bar{k}}=0$.

If Theorem 17 confirms the forward-flatness of a system (1), a flat output can
be obtained e.g. by actually performing the repeated decompositions (7), i.e.,
by successively straightening out the integrable codistributions $P_{k}$,
$k\geq 2$ by state transformations (6a) and transforming the resulting systems
(25) into the form (7) by further input transformations (6b) as it is shown in
the sufficiency part of the proof of Theorem 15. A flat output of the last,
trivial subsystem (7a) with $\dim(\bar{x}_{2})=0$ corresponding to
$P_{\bar{k}}=0$ is given by its inputs $(\bar{x}_{1},\bar{u}_{2})$, and by
adding the redundant inputs $\tilde{u}_{1}$ according to item (ii) of Lemma 3
for all decomposition steps, a flat output $y$ of the original system (1) can
be obtained. To get the flat output in original coordinates $(x,u)$, it is of
course necessary to apply the corresponding inverse coordinate
transformations.

######  Remark 18.

Systems which are linearizable by static feedback are contained in the class
of forward-flat systems, and hence the sequence (11) also terminates with
$P_{\bar{k}}=0$. However, Step 2 of Algorithm 11 is always trivial since
$P_{k}\cap\mathrm{span}\\{\mathrm{d}f\\}$ is already invariant and hence
$P_{k+1}^{+}=P_{k}\cap\mathrm{span}\\{\mathrm{d}f\\}$ for all $k\geq 1$, cf.
also Remark 5.

##  5 Examples

In this section, we illustrate our results by two examples.

###  5.1 Academic Example

Consider the system

$\displaystyle x^{1,+}$ $\displaystyle=x^{2}(u^{1}+1)\,,$ $\displaystyle
x^{4,+}$ $\displaystyle=x^{5}+1-\tfrac{x^{1}(u^{1}+1)}{x^{2}+1}$ (26)
$\displaystyle x^{2,+}$ $\displaystyle=u^{1}\,,$ $\displaystyle x^{5,+}$
$\displaystyle=x^{2}+u^{2}$ $\displaystyle x^{3,+}$
$\displaystyle=x^{4}+u^{2}-1\,,$

with the equilibrium $x_{0}=(0,0,0,1,0)$ and $u_{0}=(0,0)$, which is not
static feedback linearizable and has also been studied in [21] in the context
of implicit system decompositions. To check its forward-flatness, we compute
the sequence (11) according to Algorithm 11. The first codistribution of (11)
is given by

$P_{1}=\mathrm{span}\\{\mathrm{d}x\\}=\mathrm{span}\\{\mathrm{d}x^{1},\mathrm{d}x^{2},\mathrm{d}x^{3},\mathrm{d}x^{4},\mathrm{d}x^{5}\\}\,.$

As explained before Algorithm 11, it is convenient to perform the computations
in adapted coordinates

$\begin{array}[]{ccl}\theta^{i}&=&f^{i}(x,u)\,,\quad i=1,\ldots,n\\\
\xi^{1}&=&x^{1}\\\ \xi^{2}&=&x^{3}\,,\end{array}$ (27)

where the functions $h(x,u)$ of (10) have been chosen such that the
transformation is invertible. For $k=1$, the intersection of Step 1 yields

$P_{1}\cap\mathrm{span}\\{\mathrm{d}\theta\\}=\mathrm{span}\\{(\theta^{2}+1)\mathrm{d}\theta^{1}-\theta^{1}\mathrm{d}\theta^{2},\mathrm{d}\theta^{3}-\mathrm{d}\theta^{5},\\\
\xi^{1}(\theta^{2}+1)^{2}\mathrm{d}\theta^{1}+\left(\theta^{1}+\theta^{2}+1\right)\theta^{1}\mathrm{d}\theta^{4}\\}\,,$

which is not yet invariant w.r.t. the distribution
$\mathrm{span}\\{\mathrm{d}\theta\\}^{\perp}=\mathrm{span}\\{\partial_{\xi}\\}$.
However, adding in Step 2 the 1-form $(\theta^{2}+1)^{2}\mathrm{d}\theta^{1}$,
which is the Lie derivative of the last 1-form of the above basis of
$P_{1}\cap\mathrm{span}\\{\mathrm{d}\theta\\}$ w.r.t. the vector field
$\partial_{\xi^{1}}$, results in an invariant codistribution $P_{2}^{+}$. A
basis for $P_{2}^{+}$ which is independent of the coordinates $\xi$ is given
by __

$P_{2}^{+}=\mathrm{span}\\{\mathrm{d}\theta^{1},\mathrm{d}\theta^{2},\mathrm{d}\theta^{3}-\mathrm{d}\theta^{5},\mathrm{d}\theta^{4}\\}\,,$

and because of $\theta^{i}=f^{i}(x,u)$, the backward-shift of Step 3 yields
$P_{2}$ in original coordinates as

$P_{2}=\mathrm{span}\\{\mathrm{d}x^{1},\mathrm{d}x^{2},\mathrm{d}x^{3}-\mathrm{d}x^{5},\mathrm{d}x^{4}\\}\,.$

Continuing Algorithm 11 with $k=2$ and using again the adapted coordinates
(27), Step 1 yields

$P_{2}\cap\mathrm{span}\\{\mathrm{d}\theta\\}=\mathrm{span}\\{(\theta^{2}+1)\mathrm{d}\theta^{1}-\theta^{1}\mathrm{d}\theta^{2},\mathrm{d}\theta^{3}-\mathrm{d}\theta^{5}\\}\,,$

which is already invariant w.r.t.
$\mathrm{span}\\{\mathrm{d}\theta\\}^{\perp}=\mathrm{span}\\{\partial_{\xi}\\}$.
Thus, Step 2 is trivial with
$P_{3}^{+}=P_{2}\cap\mathrm{span}\\{\mathrm{d}\theta\\}$, and after the
backward-shift of Step 3 we get

$P_{3}=\mathrm{span}\\{(x^{2}+1)\mathrm{d}x^{1}-x^{1}\mathrm{d}x^{2},\mathrm{d}x^{3}-\mathrm{d}x^{5}\\}$

in original coordinates. Finally, for $k=3$, the intersection of Step 1 yields
$P_{3}\cap\mathrm{span}\\{\mathrm{d}\theta\\}=0$. Hence, $P_{4}=0$ and
according to Theorem 17 the system (26) is forward-flat. A flat output can be
obtained by actually performing the system decompositions (7) induced by the
sequence $P_{4}\subset P_{3}\subset P_{2}\subset P_{1}$. In original
coordinates, it is given by $y=(\frac{x^{1}}{x^{2}+1},x^{3}-x^{5})$.

###  5.2 VTOL Aircraft

As a second example, let us consider the planar VTOL aircraft discussed e.g.
in [27] or [11], which is described by the continuous-time dynamics

$\begin{array}[]{ll}\dot{x}=v_{x}\,,&\dot{v}_{x}=\varepsilon\cos(\theta)u^{2}-\sin(\theta)u^{1}\\\
\dot{z}=v_{z}\,,&\dot{v}_{z}=\cos(\theta)u^{1}+\varepsilon\sin(\theta)u^{2}-1\\\
\dot{\theta}=\omega\,,&\dot{\omega}=u^{2}\,.\end{array}$ (28)

It is well-known that this system is flat with a flat output
$y=(x-\varepsilon\sin(\theta),z+\varepsilon\cos(\theta))$. By combining an
Euler discretization for some sampling time $T_{s}>0$ with a suitable prior
state transformation which transforms (28) into a structurally flat triangular
form as it is shown in [6] for a gantry crane, a flat discrete-time system

$\displaystyle x^{1,+}$ $\displaystyle=x^{1}+T_{s}x^{3}$ (29) $\displaystyle
x^{2,+}$ $\displaystyle=x^{2}+T_{s}x^{4}$ $\displaystyle x^{3,+}$
$\displaystyle=x^{3}+T_{s}\sin(x^{5})(\varepsilon(x^{6})^{2}-u^{1})$
$\displaystyle x^{4,+}$
$\displaystyle=x^{4}+T_{s}\cos(x^{5})(-\varepsilon(x^{6})^{2}+u^{1})-T_{s}$
$\displaystyle x^{5,+}$ $\displaystyle=x^{5}+T_{s}x^{6}$ $\displaystyle
x^{6,+}$ $\displaystyle=x^{6}+T_{s}u^{2}$

with a flat output $y=(x^{1},x^{2})$ can be obtained. Computing the sequence
(11) for this system yields

$\displaystyle P_{1}$
$\displaystyle=\mathrm{span}\\{\mathrm{d}x^{1},\mathrm{d}x^{2},\mathrm{d}x^{3},\mathrm{d}x^{4},\mathrm{d}x^{5},\mathrm{d}x^{6}\\}$
(30) $\displaystyle P_{2}$
$\displaystyle=\mathrm{span}\\{\mathrm{d}x^{1},\mathrm{d}x^{2},\mathrm{d}x^{3},\mathrm{d}x^{4},\mathrm{d}x^{5}\\}$
$\displaystyle P_{3}$
$\displaystyle=\mathrm{span}\\{\mathrm{d}x^{1},\mathrm{d}x^{2},\mathrm{d}x^{3},\mathrm{d}x^{4}\\}$
$\displaystyle P_{4}$
$\displaystyle=\mathrm{span}\\{\mathrm{d}x^{1},\mathrm{d}x^{2}\\}$
$\displaystyle P_{5}$ $\displaystyle=0\,,$

which confirms its forward-flatness. The fact that the codistributions (30)
are already straightened out is due to the structurally flat triangular form
of (29). For $k=1$ and $k=2$, Step 2 of Algorithm 11 is nontrivial, since
$P_{1}\cap\mathrm{span}\\{\mathrm{d}f\\}$ and
$P_{2}\cap\mathrm{span}\\{\mathrm{d}f\\}$ are not invariant w.r.t.
$\mathrm{span}\\{\mathrm{d}f\\}^{\perp}$ and a Lie derivative has to be added.
Thus, the system is not static feedback linearizable.

##  6 Conclusion

We have derived a test for forward-flatness based on a unique sequence of
integrable codistributions (11), which can be considered as dual version of
the test proposed in [18]. The sequence of integrable codistributions gives
rise to repeated system decompositions of the form (7), where the complete
system is forward-flat if and only if the subsystem (7a) is forward-flat. The
dimension of the last codistribution of the sequence corresponds to the
dimension of the last subsystem, for which no further decomposition exists.
Since the existence of such decompositions is a necessary condition for
forward-flatness, the original system (1) is forward-flat if and only if the
last codistribution has dimension zero. The only additional effort compared to
a test for static feedback linearizability consists in the computation of the
smallest invariant codistributions in Step 2 of Algorithm 11, which can be
achieved in a straightforward way by adding suitable Lie derivatives of
1-forms. For static feedback linearizable systems this is trivial, since
already the codistributions of Step 1 are invariant. The computational effort
is also lower than in [18], where compared to a static feedback linearization
test an additional calculation of largest projectable subdistributions is
required. The computation of the smallest invariant codistributions in the
presented dual approach by just adding Lie derivatives of 1-forms can be
considered here as the simpler task. Furthermore, the calculations in [18]
involve two manifolds, whereas here all calculations are performed on only one
manifold. Future research will address extensions of the presented results to
flatness in the more general sense of [7], pursuing similar ideas as in [16]
while trying to keep the computational effort as low as possible.

This research was funded in whole, or in part, by the Austrian Science Fund
(FWF) P36473. For the purpose of open access, the author has applied a CC BY
public copyright licence to any Author Accepted Manuscript version arising
from this submission.

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