# A Triangular Normal Form for $x$-Flat Control-Affine Two-Input Systems
††thanks: 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.

1st Conrad Gstöttner  Institute of Automatic Control
and Control Systems Technology
JKU Linz, Austria
https://orcid.org/0000-0003-2107-3009  2nd Bernd Kolar  Institute of Automatic
Control
and Control Systems Technology
JKU Linz, Austria
https://orcid.org/0000-0001-9710-8445  3rd Markus Schöberl  Institute of
Automatic Control
and Control Systems Technology
JKU Linz, Austria
https://orcid.org/0000-0001-5539-7015

###### Abstract

This paper is devoted to normal forms for $x$-flat control-affine systems with
two inputs. We propose a general triangular normal form which contains several
other normal forms discussed in the literature as special cases. We derive
conditions under which a system with given $x$-flat output can be transformed
into the proposed triangular form. Based on the triangular form we motivate a
simple algorithm for identifying candidates for flat outputs.

###### Index Terms:

flatness, normal forms, nonlinear control systems, geometric methods

##  I Introduction

The concept of differential flatness has been introduced by Fliess, Lévine,
Martin and Rouchon in [1, 2], and has attracted a lot of interest in the
control systems community. The property of a system to be differentially flat
(or just “flat” for short) allows for a systematic solution of feed-forward
and feedback control problems, see e.g. [3, 4, 5, 6]. Roughly speaking, a
nonlinear control system of the form

$\displaystyle\begin{aligned} \dot{x}&=f(x,u)\end{aligned}$ (1)

with $\mathrm{dim}(x)=n$ states and $\mathrm{dim}(u)=m$ inputs is flat if
there exists an $m$-dimensional (fictitious) output

$\displaystyle\begin{aligned}
y&=\varphi(x,u,\dot{u},\ldots,u^{(q)})\,,\end{aligned}$ (2)

$u^{(q)}$ denoting the $q$-th time derivative of the input, such that the
state and the input of the system can locally be expressed as functions of
this output and a finite number of its time derivatives, i.e.,

$\displaystyle\begin{aligned} x&=F_{x}(y,\dot{y},\ldots,y^{(r-1)})\\\
u&=F_{u}(y,\dot{y},\ldots,y^{(r)})\,.\end{aligned}$

Such a (fictitious) output (2) is called a flat output of the system (1). The
computation of flat outputs is known to be a difficult problem. There do not
exist easily verifiable necessary and sufficient conditions, except for
certain classes of systems, including two-input drift-less systems, see [7],
systems which are linearizable by a one-fold prolongation of a suitably chosen
control, see [8], two-input systems which are linearizable by a two-fold
prolongation of a suitably chosen control, see [9, 10, 11]. Necessary and
sufficient conditions for $(x,u)$-flatness of control-affine systems with two
inputs and four states can be found in [12]. In the references [13, 14, 15,
16, 17], structurally flat triangular forms have been used for computing flat
outputs. Such triangular forms have proven to be very useful in this problem.
E.g. in [13, 14, 15, 16] triangular forms have been geometrically
characterized, which provides easily applicable sufficient conditions for
flatness. In [17] an implicit triangular form together with a constructive
scheme for successively transforming a system into the proposed triangular
form is presented.

In this paper, we propose a very general structurally flat triangular form for
a particular class of flat systems, namely control-affine two input systems

$\displaystyle\begin{aligned}
\dot{x}&=f(x)+g_{1}(x)u^{1}+g_{2}(x)u^{2}\end{aligned}$ (3)

which are $x$-flat, i.e., which possess a flat output
$y=(\varphi^{1}(x),\varphi^{2}(x))$ that depends on the system’s state only.
We derive conditions under which a system (3) with given $x$-flat output can
be transformed into the proposed triangular form. Although deriving a complete
characterization of this triangular form does not seem to be a tractable
problem, it is still useful for the problem of computing flat outputs. Indeed,
based on the proposed triangular form we motivate a simple algorithm for
identifying candidates for flat outputs and the triangular form may be used as
a basis for formulating more sophisticated algorithms as well.

The paper is structured as follows. In Section II we introduce some notation
and provide a definition for flatness. The normal form and its relation to
some normal forms discussed in the literature are presented in Section III. A
simple algorithm for identifying candidates for flat outputs is presented in
Section IV. In Section V the results are illustrated by means of an example
and proofs are provided in Section VI.

##  II Notation and Flatness

We make use of the Einstein summation convention. The $k$-fold Lie derivative
of a function $\varphi$ along a vector field $v$ is denoted by
$\mathrm{L}_{v}^{k}\varphi$ and we define the Lie derivative of a
codistribution $P=\mathrm{span}\\{\omega^{1},\ldots,\omega^{p}\\}$ by
$\mathrm{L}_{v}P=\mathrm{span}\\{\mathrm{L}_{v}\omega^{1},\ldots,\mathrm{L}_{v}\omega^{p}\\}$.111Note
that $\mathrm{L}_{v}P$ on its own is meaningless as it depends on the
particular choice of generators $\omega^{l}$, but e.g. $P+\mathrm{L}_{v}P$ is
a well defined codistribution. The Lie bracket of two vector fields $v$ and
$w$ is denoted by $[v,w]$. Given a distribution $D$, its involutive closure is
denoted by $\overline{D}$ and we write $\mathcal{C}(D)$ for its characteristic
distribution, which is spanned by all vector fields $v\in D$ that satisfy
$[v,D]\subset D$. Characteristic distributions are always involutive. By
$\partial_{x}h$ we denote the $m\times n$ Jacobian matrix of
$h=(h^{1},\ldots,h^{m})$ with respect to $x=(x^{1},\ldots,x^{n})$. The symbols
$\subset$ and $\supset$ are used in the sense that they also include equality.
We write $h_{[\alpha]}$ for the $\alpha$-th time derivative of a function $h$.
When $h$ consists of several components, i.e., $h=(h^{1},\ldots,h^{m})$, then
$h_{[\alpha]}=(h^{1}_{[\alpha]},\ldots,h^{m}_{[\alpha]})$. To keep expressions
involving time derivatives of different orders short we use multi-indices. Let
$A=(a_{1},\ldots,a_{m})$ be a multi-index. Then
$h_{[A]}=(h^{1}_{[a_{1}]},\ldots,h^{m}_{[a_{m}]})$ and
$h_{[0,A]}=(h^{1}_{[0,a_{1}]},\ldots,h^{m}_{[0,a_{m}]})$ where
$h^{j}_{[0,a_{j}]}=(h^{j}_{[0]},\ldots,h^{j}_{[a_{j}]})$. Addition and
subtraction of a multi-index with an integer $c$ is defined by $A\pm
c=(a_{1}\pm c,\ldots,a_{m}\pm c)$. Furthermore, we define
$\\#A=\sum_{j=1}^{m}a_{j}$. Throughout, all functions and (co)vector fields
are assumed to be smooth and all (co)distributions are assumed to have locally
constant rank, we consider generic points only. We call two systems
$\dot{x}=f(x,u)$ and $\dot{\bar{x}}=\bar{f}(\bar{x},\bar{u})$ static feedback
equivalent if they are locally equivalent via a state transformation
$\bar{x}=\Phi_{x}(x)$ and a static feedback transformation
$\bar{u}=\Phi_{u}(x,u)$, i.e.,
$(f^{l}(x,u)\partial_{x^{l}}\Phi^{i}_{x}(x))\circ\hat{\Phi}(\bar{x},\bar{u})=\bar{f}^{i}(\bar{x},\bar{u})$
where $\hat{\Phi}$ denotes the inverse of the transformation
$(\bar{x},\bar{u})=(\Phi_{x}(x),\Phi_{u}(x,u))$.

Like e.g. in [18], we use a finite-dimensional differential-geometric
framework with an extended state and input manifold
$\mathcal{X}\times\mathcal{U}_{[0,l_{u}]}$ with coordinates
$(x,u,u_{[1]},\ldots,u_{[l_{u}]})$, where $u_{[\alpha]}$ denotes the
$\alpha$-th time derivative of the input $u$ and $l_{u}$ is some sufficiently
large but finite integer. The time derivative $h_{[1]}$ of a function
$h(x,u,u_{[1]},\ldots,u_{[l_{u}-1]})$ along trajectories of (1) is then given
by the Lie derivative $\mathrm{L}_{f_{u}}h$ along the vector field

$\displaystyle\begin{aligned}
f_{u}&=f^{i}(x,u)\partial_{x^{i}}+\sum_{\alpha=0}^{l_{u}-1}u^{j}_{[\alpha+1]}\partial_{u^{j}_{[\alpha]}}\,.\end{aligned}$
(4)

Throughout, we assume that $l_{u}$ is chosen large enough such that $f_{u}$
acts as time derivative on all functions considered. Within this framework,
flatness can be defined as follows.

###### Definition 1

The system (1) is called flat if there exists an $m$-tuple of smooth functions

$\displaystyle\varphi^{j}(x,u,u_{[1]},\ldots,u_{[q]}),\quad j=1,\ldots,m$ (5)

defined on $\mathcal{X}\times\mathcal{U}_{[0,l_{u}]}$ and smooth functions
$F_{x}^{i}$, $F_{u}^{j}$ such that locally the conditions

$\displaystyle\begin{aligned} x^{i}&=F_{x}^{i}(\varphi_{[0,R-1]}),\quad
i=1,\ldots,n\\\ u^{j}&=F_{u}^{j}(\varphi_{[0,R]}),\quad
j=1,\ldots,m\end{aligned}$ (6)

with some multi-index $R=(r_{1},\ldots,r_{m})$ hold. The $m$-tuple (5) is
called a flat output.

Definition 1 implies that the differentials
$\mathrm{d}\varphi,\mathrm{d}\varphi_{[1]},\ldots,\mathrm{d}\varphi_{[\beta]}$
up to an arbitrary order $\beta$ are linearly independent, and based on that,
it can be shown that locally there exist a unique minimal multi-index $R$ and
unique maps $F_{x}$ and $F_{u}$ such that (6) holds, see e.g. [18] for further
details. The map
$(F_{x},F_{u}):\mathbb{R}^{\\#R+m}\rightarrow\mathbb{R}^{n+m}$ is a submersion
(it degenerates to a diffeomorphism for linearizing outputs in the sense of
static feedback linearization). We denote the difference of the dimensions of
the domain and the codomain of this map by $d$, i.e., $d=\\#R+m-(n+m)=\\#R-n$,
and refer to it as the differential difference of the corresponding flat
output. In [19, 8], the number $\\#R+m$ is called the differential weight of
the flat output. (The differential weight of a flat output with differential
difference $d$ is thus given by $n+m+d$.) A flat output is called minimal if
its differential difference is the smallest among all flat outputs of the
system.

##  III Triangular Normal Form

In the following, we consider nonlinear two-input control-affine systems

$\displaystyle\begin{aligned}
\dot{x}&=f(x)+g_{1}(x)u^{1}+g_{2}(x)u^{2}\end{aligned}$ (7)

with $\mathrm{dim}(x)=n$. Let

$\displaystyle y$ $\displaystyle=(\varphi^{1}(x),\varphi^{2}(x))$ (8)

be an $x$-flat output of (7) and let $K=(k_{1},k_{2})$ be the relative degrees
of the flat output components, which are defined via

$\displaystyle\begin{aligned}
\mathrm{L}_{f_{u}}^{k_{j}-1}\varphi^{j}&=\varphi^{j}_{[k_{j}-1]}(x)\,,&&&\mathrm{L}_{f_{u}}^{k_{j}}\varphi^{j}&=\varphi^{j}_{[k_{j}]}(x,u)\,.\end{aligned}$

Since we are dealing with a control-affine system, the vector field $f_{u}$ is
of the form

$\displaystyle\begin{aligned}
f_{u}&=\left(f^{i}(x)+u^{1}g_{1}^{i}(x)+u^{2}g_{2}^{i}(x)\right)\partial_{x^{i}}+\sum_{\alpha=0}^{l_{u}-1}u^{j}_{[\alpha+1]}\partial_{u^{j}_{[\alpha]}}\end{aligned}$

and $\varphi^{j}_{[k_{j}]}(x,u)$ are actually affine functions of $u$. Now
consider the static feedback transformation
$\bar{u}^{1}=\varphi^{1}_{[k_{1}]}(x,u)$, $\bar{u}^{2}=u^{2}$ (permute $u^{1}$
and $u^{2}$ if necessary).222The static feedback
$\bar{u}^{1}=\varphi^{1}_{[k_{1}]}(x,u)$, $\bar{u}^{2}=u^{2}$ implies the
transformation  $\displaystyle\begin{aligned}
\bar{u}^{1}_{[1]}&=\varphi^{1}_{[k_{1}+1]}(x,u,u_{[1]})&&&\bar{u}^{2}_{[1]}&=u^{2}_{[1]}\\\\[-4.30554pt]
&\mathmakebox[\widthof{{}={}}][c]{\vdots}&&&&\mathmakebox[\widthof{{}={}}][c]{\vdots}\\\\[-4.30554pt]
\bar{u}^{1}_{[l_{u}]}&=\varphi^{1}_{[k_{1}+l_{u}]}(x,u,u_{[1]},\ldots,u_{[l_{u}]})&&&\bar{u}^{2}_{[l_{u}]}&=u^{2}_{[l_{u}]}\end{aligned}$
for the derivatives of the inputs. In the new coordinates the vector field
$f_{u}$ reads  $\displaystyle\begin{aligned}
f_{u}&=\underbrace{\bar{f}^{i}(x,\bar{u})\partial_{x^{i}}+\sum_{\alpha=0}^{l_{u}-1}\bar{u}^{j}_{[\alpha+1]}\partial_{\bar{u}^{j}_{[\alpha]}}}_{\bar{f}_{u}}+\ldots\partial_{\bar{u}^{1}_{[l_{u}]}}+\ldots\partial_{\bar{u}^{2}_{[l_{u}]}}\,,\\\\[-4.30554pt]
\end{aligned}$ where $\bar{f}=f\circ\hat{\Phi}_{u}$ and $\hat{\Phi}_{u}$
denotes the inverse of the transformation
$\bar{u}^{1}=\varphi^{1}_{[k_{1}]}(x,u)$, $\bar{u}^{2}=u^{2}$ (which depends
affinely on $\bar{u}$ when $\varphi^{1}_{[k_{1}]}(x,u)$ depends affinely on
$u$). The in general non-zero components in the
$\partial{\bar{u}^{j}_{[l_{u}]}}$-directions can be ignored as long as $l_{u}$
is chosen large enough, i.e., after applying a transformation, we can solely
work with $\bar{f}_{u}$, which is of the same structure as $f_{u}$ in the
original coordinates. (An additional state transformation
$\bar{x}=\Phi_{x}(x)$ would only affect the $\partial_{x}$-components and
result in the components
$\bar{f}^{i}(\bar{x},\bar{u})=(f^{l}(x,u)\partial_{x^{l}}\Phi^{i}_{x}(x))\circ\hat{\Phi}(\bar{x},\bar{u})$
with respect to $\partial_{\bar{x}^{i}}$.) Such transformations have also been
used in [20] and [21] where it has been shown that by applying such a
transformation, the flat output and its derivatives up to the orders
$R=(r_{1},r_{2})$ take the form333The control-affine two-input $x$-flat
systems studied here belong to the broader class of general nonlinear two-
input systems $\dot{x}=f(x,u^{1},u^{2})$ with $(x,u)$-flat outputs, i.e., flat
outputs $\varphi(x,u)$ which may depend on the state $x$ and also the input
$u$, which are studied in [20] and [21]. The results therein are thus directly
applicable here.

$\displaystyle\begin{aligned}
\varphi^{1}&=\varphi^{1}(x)&&&\varphi^{2}&=\varphi^{2}(x)\\\\[-4.30554pt]
&\mathmakebox[\widthof{{}={}}][c]{\vdots}&&&&\mathmakebox[\widthof{{}={}}][c]{\vdots}\\\\[-4.30554pt]
\varphi^{1}_{[k_{1}-1]}&=\varphi^{1}_{[k_{1}-1]}(x)&&&\varphi^{2}_{[k_{2}-1]}&=\varphi^{2}_{[k_{2}-1]}(x)\\\
\varphi^{1}_{[k_{1}]}&=\bar{u}^{1}&&&\varphi^{2}_{[k_{2}]}&=\varphi^{2}_{[k_{2}]}(x,\bar{u}^{1})\\\
\varphi^{1}_{[k_{1}+1]}&=\bar{u}^{1}_{[1]}&&&\varphi^{2}_{[k_{2}+1]}&=\varphi^{2}_{[k_{2}+1]}(x,\bar{u}^{1},\bar{u}^{1}_{[1]})\\\\[-4.30554pt]
&\mathmakebox[\widthof{{}={}}][c]{\vdots}&&&&\mathmakebox[\widthof{{}={}}][c]{\vdots}\\\\[-4.30554pt]
\varphi^{1}_{[r_{1}]}&=\bar{u}^{1}_{[r_{1}-k_{1}]}&&&\varphi^{2}_{[r_{2}]}&=\varphi^{2}_{[r_{2}]}(x,\bar{u}^{1},\bar{u}^{1}_{[1]},\ldots,\\\
&&&&&\hskip
40.00006pt\bar{u}^{1}_{[r_{2}-k_{2}]},\bar{u}^{2})\,.\\\\[-4.30554pt]
\end{aligned}$ (9)

It is immediate that the derivatives of $\varphi^{1}$ from order $k_{1}$ on
are just derivatives of the new input $\bar{u}^{1}$. The form of the
derivatives of $\varphi^{2}$ can be derived by exploiting the fact that by
assumption $R=(r_{1},r_{2})$ are the minimal integers such that (6) holds,
from which it additionally follows that $r_{1}-k_{1}=r_{2}-k_{2}$, see e.g.
[20] for details. Furthermore, it follows that $k_{1}+r_{2}=n$ since it must
be possible to construct exactly $n$ independent functions of $x$ only from
the functions (9). In conclusion, the relative degrees $K=(k_{1},k_{2})$, the
state dimension $n$, the orders $R=(r_{1},r_{2})$ and the differential
difference $d$ are related as follows (see also [22], Proposition 3.19)

$\displaystyle\begin{aligned} k_{1}+r_{2}&=n\,,\qquad k_{2}+r_{1}=n\,,\qquad
n-k_{1}-k_{2}=d\,.\end{aligned}$

From these properties, it also immediately follows that $R=K+d$.

With any $x$-flat output (8), we associate the following sequence of
codistributions

$\displaystyle\begin{aligned}
P_{K-1}&=\mathrm{span}\\{\mathrm{d}\varphi_{[0,K-1]}\\}\\\
P_{K}&=\mathrm{span}\\{\mathrm{d}\varphi_{[0,K]}\\}\\\\[-4.30554pt]
&\mathmakebox[\widthof{{}={}}][c]{\vdots}\\\\[-4.30554pt]
P_{K+d-1}&=P_{R-1}=\mathrm{span}\\{\mathrm{d}\varphi_{[0,R-1]}\\}\\\
P_{K+d}&=P_{R}=\mathrm{span}\\{\mathrm{d}\varphi_{[0,R]}\\}\,,\end{aligned}$
(10)

as well as the following intersections with $\mathrm{span}\\{\mathrm{d}x\\}$

$\displaystyle\begin{aligned}
Q_{K-1}&=P_{K-1}\cap\mathrm{span}\\{\mathrm{d}x\\}\\\
Q_{K}&=P_{K}\cap\mathrm{span}\\{\mathrm{d}x\\}\\\\[-4.30554pt]
&\mathmakebox[\widthof{{}={}}][c]{\vdots}\\\\[-4.30554pt]
Q_{R-1}&=P_{R-1}\cap\mathrm{span}\\{\mathrm{d}x\\}\,.\end{aligned}$ (11)

Due to the linear independence of
$\mathrm{d}\varphi,\mathrm{d}\varphi_{[1]},\ldots,\mathrm{d}\varphi_{[\beta]}$
up to an arbitrary order $\beta$ (see below Definition 1), we obviously have

$\displaystyle\begin{aligned}
P_{K-1}\underset{2}{\subset}P_{K}\underset{2}{\subset}\ldots\underset{2}{\subset}P_{R-1}\underset{2}{\subset}P_{R}\,.\end{aligned}$

By the definition of the relative degree, we furthermore have
$Q_{K-1}=P_{K-1}$ and from (9) it is also immediate that

$\displaystyle\begin{aligned}
Q_{K-1}\underset{1}{\subset}Q_{K}\underset{1}{\subset}\ldots\underset{1}{\subset}Q_{R-1}=\mathrm{span}\\{\mathrm{d}x\\}\,.\end{aligned}$

Except for $Q_{K-1}$ and $Q_{R-1}$ the remaining codistributions in (11) are
not necessarily completely integrable. In case that all of them are indeed
completely integrable, the system can be transformed into a triangular form
compatible with the corresponding (not necessarily minimal) $x$-flat output
$\varphi(x)$, as the following theorem asserts.

###### Theorem 2

The system (7) with $x$-flat output $\varphi=(\varphi^{1}(x),\varphi^{2}(x))$
is static feedback equivalent to the triangular form

$\displaystyle\begin{aligned} \dot{z}^{1}&=z^{2}\\\\[-4.30554pt]
&\mathmakebox[\widthof{{}={}}][c]{\vdots}\\\\[-4.30554pt]
\dot{z}^{k_{1}-1}&=z^{k_{1}}\\\
\dot{z}^{k_{1}}&=v^{1}\end{aligned}\quad\begin{aligned}
\dot{z}^{k_{1}+1}&=z^{k_{1}+2}\\\\[-4.30554pt]
&\mathmakebox[\widthof{{}={}}][c]{\vdots}\\\\[-4.30554pt]
\dot{z}^{k_{1}+k_{2}-1}&=z^{k_{1}+k_{2}}\\\
\dot{z}^{k_{1}+k_{2}}&=a^{k_{1}+k_{2}}(z^{1},\ldots,z^{k_{1}+k_{2}+1})+\\\
&\hskip
25.83325ptb^{k_{1}+k_{2}}(z^{1},\ldots,z^{k_{1}+k_{2}+1})v^{1}\\\\[6.45831pt]
\dot{z}^{k_{1}+k_{2}+1}&=a^{k_{1}+k_{2}+1}(z^{1},\ldots,z^{k_{1}+k_{2}+2})+\\\
&\hskip
25.83325ptb^{k_{1}+k_{2}+1}(z^{1},\ldots,z^{k_{1}+k_{2}+2})v^{1}\\\\[-4.30554pt]
&\mathmakebox[\widthof{{}={}}][c]{\vdots}\\\\[-4.30554pt]
\dot{z}^{n-1}&=a^{n-1}(z)+b^{n-1}(z)v^{1}\\\
\dot{z}^{n}&=v^{2}\\\\[-12.91663pt] \end{aligned}$ (12)

with $\varphi=(z^{1},z^{k_{1}+1})$ and $b^{k_{1}+k_{2}}\neq 0$, as well as
$\partial_{z^{l+1}}a^{l}\neq 0$ or $\partial_{z^{l+1}}a^{l}\neq 0$ for
$l=k_{1}+k_{2},\ldots,n-1$, if and only if all the corresponding
codistributions (11) are completely integrable.

A proof of this theorem is provided in Section VI. Due to the property
$\partial_{z^{l+1}}a^{l}\neq 0$ or $\partial_{z^{l+1}}b^{l}\neq 0$ for
$l=k_{1}+k_{2},\ldots,n-1$, one of the functions $a^{l}$ or $b^{l}$ in each
equation can always be normalized to $z^{l+1}$. Such normalizations preserve
the triangular structure. The normal form (12) contains several normal forms
discussed in the literature as special cases:

  * •

For $k_{1}+k_{2}=n$, i.e., if the sum of the relative degrees of the flat
ouput components coincides with the state dimension, the form (12) degenerates
to the Brunovský normal form and $\varphi=(z^{1},z^{k_{1}+1})$ is a
linearizing output in the sense of static feedback linearization, see [23, 24,
25, 26, 27].

  * •

When $k_{1}=k_{2}=1$ and all $a^{l}=0$, (12) reduces (after normalizing all
$b^{l}$) to the chained form

$\displaystyle\begin{aligned}
\dot{z}^{1}&=v^{1}\end{aligned}\quad\begin{aligned}
\dot{z}^{2}&=z^{3}v^{1}\\\\[-4.30554pt]
&\mathmakebox[\widthof{{}={}}][c]{\vdots}\\\\[-4.30554pt]
\dot{z}^{n-1}&=z^{n}v^{1}\\\ \dot{z}^{n}&=v^{2}\,,\\\\[-8.61108pt]
\end{aligned}$

see e.g. [7, 28, 29, 30, 31, 32].

  * •

The case $k_{1}=k_{2}=1$ with $a^{l}$ not necessarily zero and
$\partial_{z^{l+1}}b^{l}\neq 0$ leads (after normalizing all $b^{l}$) to so-
called extended chained systems, see [33, 34, 14, 35].

  * •

In [36], normal forms for $x$-flat two-input control-affine systems in
dimension five are presented. All except for one of these normal forms are
special cases of (12) (the one exception is discussed below).

In general, not all of the codistributions (11) are completely integrable. An
example is the following system in dimension five with $x$-flat output
$(x^{1},x^{2})$

$\displaystyle\begin{aligned}
\dot{x}^{1}&=u^{1}\end{aligned}\quad\begin{aligned}
\dot{x}^{2}&=x^{3}+x^{4}u^{1}\\\
\dot{x}^{3}&=a(x^{1},\ldots,x^{4})+(-x^{5}+b(x^{1},\ldots,x^{4}))u^{1}\\\
\dot{x}^{4}&=x^{5}+c(x^{1},\ldots,x^{4})u^{1}\\\
\dot{x}^{5}&=u^{2}\\\\[-4.30554pt] \end{aligned}$

taken from [36], see also [34]. For the flat output $(x^{1},x^{2})$ the
relative degrees are $K=(1,1)$ and the codistributions (11) follow as

$\displaystyle Q_{(0,0)}$
$\displaystyle=\mathrm{span}\\{\mathrm{d}x^{1},\mathrm{d}x^{2}\\}$
$\displaystyle Q_{(1,1)}$
$\displaystyle=\mathrm{span}\\{\mathrm{d}x^{1},\mathrm{d}x^{2},\mathrm{d}x^{3}+u^{1}\mathrm{d}x^{4}\\}$
$\displaystyle Q_{(2,2)}$
$\displaystyle=\mathrm{span}\\{\mathrm{d}x^{1},\mathrm{d}x^{2},\mathrm{d}x^{3},\mathrm{d}x^{4}\\}$
$\displaystyle Q_{(3,3)}$
$\displaystyle=\mathrm{span}\\{\mathrm{d}x^{1},\mathrm{d}x^{2},\mathrm{d}x^{3},\mathrm{d}x^{4},\mathrm{d}x^{5}\\}\,,$

with $Q_{(1,1)}$ being not completely integrable.

##  IV Identifying Candidates for Flat Outputs

Deriving verifiable necessary and sufficient conditions for static feedback
equivalence to (12), as it has been done for most of the specializations
listed below Theorem 2 in the cited references, does not seem to be a
tractable problem. However, candidates for flat outputs, i.e., candidates for
the top variables $(z^{1},z^{k_{1}+1})$ in (12), may be identified by simple
algorithms like the one we will propose below. In fact it suffices to identify
only one potential component. When one component of a flat output of a two-
input system is known, a second component can be computed using e.g. the
results in [37]. (In this reference, the problem of computing the missing
component of a flat output of a system with $m$ inputs when $m-1$ components
are known, resp. have been guessed, is addressed.) As an alternative to [37],
for two-input systems (7) for which one component of an $x$-flat output is
known, a second component can also be computed as follows:

  1. 1.

Let $\varphi^{1}(x)$ be a candidate for a component of an $x$-flat output.

  2. 2.

Introduce $\varphi^{1}_{[k_{1}]}(x,u)$ as a new input via the static feedback
$v^{1}=\varphi^{1}_{[k_{1}]}(x,u)$, $v^{2}=u^{2}$ (permute $u^{1}$ and $u^{2}$
if necessary, note that this is actually an affine feedback).

  3. 3.

Let $\dot{x}=\bar{f}(x)+\bar{g}_{1}(x)v^{1}+\bar{g}_{2}(x)v^{2}$ be the system
obtained by applying this feedback. Prolong $(n-1)$-fold the new input
$v^{1}$, which leads to the prolonged system

$\displaystyle\begin{aligned}
\dot{x}&=\bar{f}(x)+\bar{g}_{1}(x)v^{1}+\bar{g}_{2}(x)v^{2}\\\
\dot{v}^{1}&=v^{1}_{[1]}\\\\[-4.30554pt]
&\mathmakebox[\widthof{{}={}}][c]{\vdots}\\\\[-4.30554pt]
\dot{v}^{1}_{[n-2]}&=v^{1}_{[n-1]}\end{aligned}$ (13)

with input $(v^{1}_{[n-1]},v^{2})$ and state
$(x,v^{1},v^{1}_{[1]},\ldots,v^{1}_{[n-2]})$.

  4. 4.

Let $f_{p}$, $g_{1,p}$, $g_{2,p}$ be the drift and control vector fields of
the prolonged system (13). Apply the well-known test for static feedback
linearizability to the prolonged system (13), i.e., compute the
linearizability distributions $D_{1}=\mathrm{span}\\{g_{1,p},g_{2,p}\\}$,
$D_{i}=D_{i-1}+[f_{p},D_{i-1}]$ for $i\geq 2$ and check them for involutivity
and their ranks, see e.g. [26].

  5. 5.

If the prolonged system (13) is indeed static feedback linearizable, then the
linearizing outputs of the prolonged system are flat outputs of the original
system.

  6. 6.

If the prolonged system (13) is not static feedback linearizable, then it can
be concluded that the candidate $\varphi^{1}(x)$ is not part of an $x$-flat
output of the system and the procedure may be repeated with a different
candidate.

The claims made in 5) and 6) can be proven based on the structure (9). See
also [20], where a similar result for $(x,u)$-flat two-input systems
$\dot{x}=f(x,u^{1},u^{2})$ is derived.

With these preliminaries at hand, let us now state a simple procedure for
identifying flat output candidates, resp. candidates for one component of a
flat output. Let $f=f^{i}(x)\partial_{x^{i}}$,
$g_{1}=g_{1}^{i}(x)\partial_{x^{i}}$ and $g_{2}=g_{2}^{i}(x)\partial_{x^{i}}$
be the drift and control vector fields of system (7). Starting from the
distribution $D_{1}=\mathrm{span}\\{g_{1},g_{2}\\}$, a sequence

$\displaystyle\begin{aligned} D_{1}\subset D_{2}\subset\ldots\subset
D_{i}\subset\ldots\subset\mathcal{T}(\mathcal{X})\end{aligned}$ (14)

of distributions is computed as follows:

  * •

If $D_{i}$ is involutive, then $D_{i+1}=D_{i}+[f,D_{i}]$.

  * •

If $D_{i}$ is non-involutive, then $D_{i+1}=D_{i}+[D_{i},D_{i}]$.

These distributions are feedback invariant and we can assume that
$D_{s}=\mathcal{T}(\mathcal{X})$ for some integer $s$, otherwise the system is
not accessible and thus certainly not flat. Let $s$ be the smallest integer
such that $D_{s}=\mathcal{T}(\mathcal{X})$, then
$D_{s-1}\underset{p}{\subset}\mathcal{T}(\mathcal{X})$ with $p\geq 1$ and
$D_{s-1}$ can either be involutive or non-involutive.

  1. (a)

If $D_{s-1}$ is involutive, then there exist $p$ functions
$h^{1}(x),\ldots,h^{p}(x)$ such that
$D_{s-1}^{\perp}=\mathrm{span}\\{\mathrm{d}h^{1},\ldots,\mathrm{d}h^{p}\\}$.
Choose a function $\psi:\mathbb{R}^{p}\rightarrow\mathbb{R}$ and take
$\varphi^{1}(x)=\psi(h^{1}(x),\ldots,h^{p}(x))$ as candidate for a flat output
component.

  2. (b)

If $D_{s-1}$ is non-involutive, then replace $D_{s-1}$ by its characteristic
distribution $\mathcal{C}(D_{s-1})$ and proceed as in (a).

In item (a), when $p=1$, only one candidate has to be checked, but for $p\geq
2$, there are in fact infinitely many ways to combine the $p$ functions, i.e.,
there are infinitely many non-equivalent choices for $\psi$. The proposed
procedure is motivated by the triangular form (12) based on the following
considerations.

  * •

In case that in (12) we have $\partial_{z^{l+1}}b^{l}\neq 0$ for all
$l=k_{1}+k_{2},\ldots,n-1$, the procedure certainly succeeds:

    * –

When in this case additionally $k_{1}=k_{2}=1$ holds, then the sequence (14)
ends with $D_{s}=\overline{D}_{1}=\mathcal{T}(\mathcal{X})$ and
$\mathcal{C}(D_{s-1})\underset{3}{\subset}\mathcal{T}(\mathcal{X})$. A flat
output component is then indeed any function $\varphi^{1}$ such that
$\mathrm{d}\varphi^{1}\neq 0$ and
$\mathrm{d}\varphi^{1}\perp\mathcal{C}(D_{s-1})$. This follows from the
properties of (extended) chained systems, see [32, 35] for details.

    * –

For $k_{1}\geq 2$ and/or $k_{2}\geq 2$, it follows that $D_{s-1}$ is
involutive and its annihilator is spanned by $\mathrm{d}z^{1}$ and/or
$\mathrm{d}z^{k_{1}+1}$.

  * •

When however $\partial_{z^{l+1}}b^{l}\neq 0$ does not hold for all
$l=k_{1}+k_{2},\ldots,n-1$, then besides obtaining new directions via
$D_{i+1}=D_{i}+[D_{i},D_{i}]$, there are also steps in which new directions
are obtained by taking Lie brackets with the drift vector field, i.e.,
$D_{i+1}=D_{i}+[f,D_{i}]$. It follows that as long as at least one of the flat
output components has a sufficiently large relative degree $k_{j}$, then this
component can still be found via the proposed procedure.

Finally, it should be noted that a system (7) can of course have many
different $x$-flat outputs for which the condition of Theorem 2 is met, and
these flat outputs of course need not have the same differential difference.
It is thus not guaranteed that flat output components found via the proposed
procedure belong to minimal flat outputs. The procedure only aims at finding a
component that belongs to one of the system’s flat outputs which is compatible
with (12).

##  V Example

Consider the following model of an induction motor (already simplified by a
static feedback), taken from [38], see also [19]

$\displaystyle\begin{aligned} \dot{\theta}&=\omega\\\
\dot{\omega}&=\mu\psi_{d}I_{q}-\tfrac{\tau_{L}}{J}\\\
\dot{\psi}_{d}&=-\eta\psi_{d}+\eta MI_{d}\end{aligned}$
$\displaystyle\qquad\begin{aligned} \dot{\rho}&=n_{p}\omega+\tfrac{\eta
MI_{q}}{\psi_{d}}\\\ \dot{I}_{d}&=v_{d}\\\
\dot{I}_{q}&=v_{q}\,.\\\\[-4.30554pt] \end{aligned}$

The inputs are $v_{d},v_{q}$. The drift and input vector fields are given by
$f=\omega\partial_{\theta}+(\mu\psi_{d}I_{q}-\tfrac{\tau_{L}}{J})\partial_{\omega}+(-\eta\psi_{d}+\eta
MI_{d})\partial_{\psi_{d}}+(n_{p}\omega+\tfrac{\eta
MI_{q}}{\psi_{d}})\partial_{\rho}$, $g_{1}=\partial_{I_{d}}$,
$g_{2}=\partial_{I_{q}}$. Applying the procedure for identifying candidates
for flat output components from the previous section, for the distributions
(14), we obtain

$\displaystyle\begin{aligned}
D_{1}&=\mathrm{span}\\{g_{1},g_{2}\\}=\mathrm{span}\\{\partial_{I_{d}},\partial_{I_{q}}\\}\\\
D_{2}&=D_{1}+[f,D_{1}]=\mathrm{span}\\{\partial_{I_{d}},\partial_{I_{q}},\partial_{\psi_{d}},\psi_{d}^{2}\mu\partial_{\omega}+M\eta\partial_{\rho}\\}\\\
D_{3}&=D_{2}+[D_{2},D_{2}]=\mathrm{span}\\{\partial_{I_{d}},\partial_{I_{q}},\partial_{\psi_{d}},\partial_{\omega},\partial_{\rho}\\}=\overline{D}_{2}\\\
D_{4}&=D_{3}+[f,D_{3}]=\mathcal{T}(\mathcal{X})\end{aligned}$

and since $D_{s-1}=D_{3}$ is involutive, item (a) applies. We have
$D_{3}^{\perp}=\mathrm{span}\\{\mathrm{d}\theta\\}$ and thus
$\varphi^{1}=\theta$ as candidate for a component of an $x$-flat output. It
follows that $\varphi^{1}=\theta$ is indeed a flat output component, and a
suitable second component is given by $\varphi^{2}=\rho$. It can be computed
using e.g. the 6 step procedure from the previous section. Following the
sufficiency part of the proof of Theorem 2, the system can be represented in
the form (12). This is achieved by the state transformation
$z^{1}=\varphi^{1}=\theta$, $z^{2}=\varphi^{1}_{[1]}=\omega$,
$z^{3}=\varphi^{1}_{[2]}=\mu\psi_{d}I_{q}-\tfrac{\tau_{L}}{J}$,
$z^{4}=\varphi^{2}=\rho$, $z^{5}=\varphi^{2}_{[1]}=n_{p}\omega+\tfrac{\eta
MI_{q}}{\psi_{d}}$, $z^{6}=I_{d}$ and the static feedback
$v^{1}=\varphi^{1}_{[3]}$, $v^{2}=u_{d}$, which leads to

$\displaystyle\begin{aligned} \dot{z}^{1}&=z^{2}\\\ \dot{z}^{2}&=z^{3}\\\
\dot{z}^{3}&=v^{1}\end{aligned}\qquad\begin{aligned} \dot{z}^{4}&=z^{5}\\\
\dot{z}^{5}&=a^{5}(z^{2},z^{3},z^{5},z^{6})+b^{5}(z^{2},z^{3},z^{5})v^{1}\\\
\dot{z}^{6}&=v^{2}\,.\end{aligned}$

##  VI Proof of Theorem 2

Necessity. It is straightforward to verify that for a system in the form (12)
all the codistributions (11) computed for $\varphi=(z^{1},z^{k_{1}+1})$ are
indeed completely integrable.

Sufficiency. We have to show that a system (7) with $x$-flat output
$(\varphi^{1}(x),\varphi^{2}(x))$ for which the codistributions (11) are all
completely integrable can be put into the form (12) by means of a state
transformation and a static feedback.For that, let us first show that the
required state transformation is given by

$\displaystyle\begin{aligned} z^{1}&=\varphi^{1}(x)\\\
z^{2}&=\varphi^{1}_{[1]}(x)\\\\[-4.30554pt]
&\mathmakebox[\widthof{{}={}}][c]{\vdots}\\\\[-4.30554pt]
z^{k_{1}}&=\varphi^{1}_{[k_{1}-1]}(x)\end{aligned}\qquad\begin{aligned}
z^{k_{1}+1}&=\varphi^{2}(x)\\\
z^{k_{1}+2}&=\varphi^{2}_{[1]}(x)\\\\[-4.30554pt]
&\mathmakebox[\widthof{{}={}}][c]{\vdots}\\\\[-4.30554pt]
z^{k_{1}+k_{2}}&=\varphi^{2}_{[k_{2}-1]}(x)\\\
z^{k_{1}+k_{2}+1}&=g^{k_{1}+k_{2}+1}(x)\\\\[-4.30554pt]
&\mathmakebox[\widthof{{}={}}][c]{\vdots}\\\\[-4.30554pt]
z^{n}&=g^{n}(x)\,,\\\\[-4.30554pt] \end{aligned}$

where the functions $g$ can be chosen as any function such that
$Q_{K-1+l}=Q_{K-1}+\mathrm{span}\\{\mathrm{d}g^{k_{1}+k_{2}+1},\ldots,\mathrm{d}g^{k_{1}+k_{2}+l}\\}$,
$l=1,\ldots,n-k_{1}-k_{2}$. The existence of such functions $g$ is assured
since by assumption all the codistributions (11) are completely integrable. In
other words, we introduce new coordinates $z$ in which all the codistributions
(11) are straightened out simultaneously and use $\varphi_{[0,K-1]}(x)$ as a
part of these new coordinates. In these new coordinates we by construction
have

$\displaystyle Q_{K-1}$
$\displaystyle=\mathrm{span}\\{\mathrm{d}z^{1},\ldots,\mathrm{d}z^{k_{1}+k_{2}}\\}$
$\displaystyle Q_{K}$
$\displaystyle=\mathrm{span}\\{\mathrm{d}z^{1},\ldots,\mathrm{d}z^{k_{1}+k_{2}+1}\\}$
$\displaystyle\mathmakebox[\widthof{{}={}}][c]{\vdots}$ $\displaystyle
Q_{R-1}$
$\displaystyle=\mathrm{span}\\{\mathrm{d}z^{1},\ldots,\mathrm{d}z^{n}\\}\,.$

Next, introduce $v^{1}=\varphi^{1}_{[k_{1}]}(x,u)$ by means of a static
feedback (and keep $u^{2}$ as the other input, which may require permuting
$u^{1}$ and $u^{2}$). Note that this is actually an affine feedback. For the
codistributions (10) we then have

$\displaystyle\begin{aligned}
P_{K-1}&=Q_{K-1}=\mathrm{span}\\{\mathrm{d}z^{1},\ldots,\mathrm{d}z^{k_{1}}\\!,\mathrm{d}z^{k_{1}+1},\ldots,\mathrm{d}z^{k_{1}+k_{2}}\\}\\\\[2.58334pt]
P_{K}&=\mathrm{span}\\{\mathrm{d}z^{1},\ldots,\mathrm{d}z^{k_{1}},\mathrm{d}v^{1},\\\
&\hskip
38.74988pt\mathrm{d}z^{k_{1}+1},\ldots,\mathrm{d}z^{k_{1}+k_{2}},\mathrm{d}\bar{\varphi}^{2}_{[k_{2}]}\\}\\\\[2.58334pt]
P_{K+1}&=\mathrm{span}\\{\mathrm{d}z^{1},\ldots,\mathrm{d}z^{k_{1}},\mathrm{d}v^{1},\mathrm{d}v^{1}_{[1]},\\\
&\hskip
38.74988pt\mathrm{d}z^{k_{1}+1},\ldots,\mathrm{d}z^{k_{1}+1},\mathrm{d}\bar{\varphi}^{2}_{[k_{2}]},\mathrm{d}\bar{\varphi}^{2}_{[k_{2}+1]}\\}\\\\[-8.61108pt]
&\mathmakebox[\widthof{{}={}}][c]{\vdots}\\\\[-2.15277pt] \end{aligned}$

where $\bar{\varphi}^{2}_{[\alpha]}$ denotes $\varphi^{2}_{[\alpha]}$
expressed in the new coordinates. Because of $Q_{A}\subset P_{A}$ and
$\mathrm{d}v^{1},\mathrm{d}v^{1}_{[1]},\ldots\notin Q_{A}$, we can construct
new bases for the codistributions $P_{A}$, namely

$\displaystyle\begin{aligned}
P_{K-1}&=Q_{K-1}=\mathrm{span}\\{\mathrm{d}z^{1},\ldots,\mathrm{d}z^{k_{1}}\\!,\mathrm{d}z^{k_{1}+1},\ldots,\mathrm{d}z^{k_{1}+k_{2}}\\}\\\\[2.58334pt]
P_{K}&=\mathrm{span}\\{\mathrm{d}z^{1},\ldots,\mathrm{d}z^{k_{1}},\mathrm{d}v^{1},\\\
&\hskip
38.74988pt\mathrm{d}z^{k_{1}+1},\ldots,\mathrm{d}z^{k_{1}+k_{2}},\mathrm{d}z^{k_{1}+k_{2}+1}\\}\\\\[2.58334pt]
P_{K+1}&=\mathrm{span}\\{\mathrm{d}z^{1},\ldots,\mathrm{d}z^{k_{1}},\mathrm{d}v^{1},\mathrm{d}v^{1}_{[1]},\\\
&\hskip
38.74988pt\mathrm{d}z^{k_{1}+1},\ldots,\mathrm{d}z^{k_{1}+k_{2}},\mathrm{d}z^{k_{1}+k_{2}+1},\mathrm{d}z^{k_{1}+k_{2}+2}\\}\\\\[-4.30554pt]
&\mathmakebox[\widthof{{}={}}][c]{\vdots}\\\
P_{R-1}&=\mathrm{span}\\{\mathrm{d}z^{1},\ldots,\mathrm{d}z^{k_{1}},\mathrm{d}v^{1},\mathrm{d}v^{1}_{[1]},\ldots,\mathrm{d}v^{1}_{[n-k_{1}-k_{2}-1]},\\\
&\hskip
38.74988pt\mathrm{d}z^{k_{1}+1},\ldots,\mathrm{d}z^{n}\\}\\\\[2.58334pt]
P_{R}&=\mathrm{span}\\{\mathrm{d}z^{1},\ldots,\mathrm{d}z^{k_{1}},\mathrm{d}v^{1},\mathrm{d}v^{1}_{[1]},\ldots,\mathrm{d}v^{1}_{[n-k_{1}-k_{2}]},\\\
&\hskip
38.74988pt\mathrm{d}z^{k_{1}+1},\ldots,\mathrm{d}z^{n},\mathrm{d}u^{2}\\}\,,\end{aligned}$

where the differentials
$\mathrm{d}\bar{\varphi}^{2}_{[k_{2}]},\mathrm{d}\bar{\varphi}^{2}_{[k_{2}+1]},\ldots,\mathrm{d}\bar{\varphi}^{2}_{[r_{2}-1]}$
got replaced by
$\mathrm{d}z^{k_{1}+k_{2}+1},\mathrm{d}z^{k_{1}+k_{2}+2},\ldots,\mathrm{d}z^{n}$,
and $\mathrm{d}\bar{\varphi}^{2}_{[r_{2}]}$ by $\mathrm{d}u^{2}$

(which can be done because flatness obviously implies that
$\mathrm{span}\\{\mathrm{d}z,\mathrm{d}v^{1},\mathrm{d}u^{2}\\}\subset
P_{R}$). Using these bases of the codistributions $P_{A}$, the triangular
structure of the system equations in the $(z,v)$-coordinates can be deduced as
follows. Because of $P_{A+1}=P_{A}+\mathrm{L}_{\bar{f}_{u}}P_{A}$, with
$\bar{f}_{u}$ denoting (4) in the newly introduced coordinates (see also
footnote 2), it follows that for $i=k_{1}+k_{2},\ldots,n-1$ the differential
$\mathrm{d}\mathrm{L}_{\bar{f}_{u}}z^{i}$ can be expressed as a linear
combination of the differentials
$\mathrm{d}z^{1},\ldots,\mathrm{d}z^{i+1},\mathrm{d}v^{1}$ and this linear
combination necessarily involves $\mathrm{d}z^{i+1}$, but it does not involve
differentials of $v^{1}_{[\alpha]}$, $\alpha>0$ due to the structure of the
vector field $\bar{f}_{u}$. This in turn is equivalent to the property that
for $i=k_{1}+k_{2},\ldots,n-1$ the time derivative
$\mathrm{L}_{\bar{f}_{u}}z^{i}$ of $z^{i}$ depends on
$z^{1},\ldots,z^{i+1},v^{1}$ only and explicitly depends on $z^{i+1}$.
Furthermore, due to the structure of $\bar{f}_{u}$, the derivative
$\mathrm{L}_{\bar{f}_{u}}z^{i}$ depends affinely on $v^{1}$. Hence, in
conclusion, there exist functions $a^{i}$, $b^{i}$ such that
$\mathrm{L}_{\bar{f}_{u}}z^{i}=a^{i}(z^{1},\ldots,z^{i+1})+b^{i}(z^{1},\ldots,z^{i+1})v^{1}$
and at least $\partial_{z^{i+1}}a^{i}\neq 0$ or $\partial_{z^{i+1}}b^{i}\neq
0$. Analogously, it follows that $\mathrm{L}_{\bar{f}_{u}}z^{n}$ explicitly
depends affinely on $u^{2}$ and can thus be normalized to $v^{2}$ by an affine
input transformation. Together with the triangular structure of the equations
for $\dot{z}^{i}$ for $i=1,\ldots,k_{1}+k_{2}-1$, which is an immediate
consequence of the fact that we have chosen $z^{i}$ for
$i=1,\ldots,k_{1}+k_{2}$ as successive time derivatives of flat output
components, we indeed obtain the triangular form (12). $\square$

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Mode-locked ultrashort pulses from an 8 µm wavelength semiconductor laser

Johannes Hillbrand1,2,∗, Nikola Opačak1, Marco Piccardo2,3, Harald Schneider4,
Gottfried Strasser1, Federico Capasso2, Benedikt Schwarz1,2,†

1Institute of Solid State Electronics, TU Wien, Gußhausstraße 25, 1040 Vienna,
Austria
2Harvard John A. Paulson School of Engineering and Applied Sciences, Harvard
University, Cambridge, USA
3CNST – Fondazione Istituto Italiano di Tecnologia, Via Pascoli 70/3, 20133
Milano, Italy
4Institute of Ion Beam Physics and Materials Research, Helmholtz-Zentrum
Dresden-Rossendorf, Germany

Quantum cascade lasers (QCL) have revolutionized the generation of mid-
infrared light. Yet, the ultrafast carrier transport in mid-infrared QCLs has
so far constituted a seemingly insurmountable obstacle for the formation of
ultrashort light pulses. Here, we demonstrate that careful quantum design of
the gain medium and control over the intermode beat synchronization enable
transform-limited picosecond pulses from QCL frequency combs. Both an
interferometric radio-frequency technique and second-order autocorrelation
shed light on the pulse dynamics and confirm that mode-locked operation is
achieved from threshold to rollover current. Being electrically pumped and
compact, mode-locked QCLs pave the way towards monolithically integrated non-
linear photonics in the molecular fingerprint region beyond 6 µm wavelength.
<EMAIL_ADDRESS><EMAIL_ADDRESS>
The discovery of ultrashort light pulses has led to numerous breakthroughs in
science and technology, including frequency combs1, high-speed optical
telecommunication2 and refractive surgery in ophthalmology3. Nowadays, optical
pulses are routinely generated in mode-locked lasers operating in the visible
or near-infrared range4, 5. Currently, large efforts are aimed at bringing
ultrafast laser science in the mid-infrared (MIR) region to a similarly high
degree of maturity6. Due to the lack of suitable gain media, methods for the
generation of pulses in the molecular fingerprint region beyond 5 µm
wavelength have so far relied on non-linear downconversion of near-infrared
pulses 7. Established techniques such as optical parametric oscillators8 or
difference frequency generation9, 10 either require sophisticated optical
setups with tabletop dimensions or are restricted to mW-level of output power.
Quantum cascade lasers11 (QCL) have matured to become the dominant MIR laser
source. While being microchip-sized and electrically pumped, they are capable
of producing Watt-level average power12, 13. Quantum engineering of the active
region allows to tailor the emission wavelength throughout the entire mid-
infrared region. Hence, harnessing high-performance QCL technology for the
generation of MIR pulses represents a long-sought milestone in ultrafast laser
science. Mode-locked QCLs could serve as monolithic pump lasers for
microresonators and resonant supercontinuum generation14, paving the way
towards broadband and high-brightness frequency combs. So far, the sub-
picosecond carrier transport in QCL active regions has constituted a seemingly
insurmountable obstacle for the formation of short light pulses15, 16, 17. To
date, the only successful attempt of mode-locking in monolithic MIR QCLs was
observed using a specially designed active region with strongly enhanced
upper-state lifetime of the lasing transition17. However, the necessary design
modifications limited mode-locked operation to cryogenic temperatures and peak
power below 10 mW, thus impeding their practical use.

Figure 1: Bi-functional quantum cascade lasers for mode-locking. a: Scanning
electron microscope image of three adjacent laser ridges. Each laser consists
of a roughly 3 mm long gain section and a shorter (320-480 µm) modulation
section. b: Simulated gain and loss spectrum in a standard active region
design18 depending on the applied bias. Upon decreasing the bias, the
structure becomes almost transparent at the lasing wavelength $\lambda_{L}$,
limiting the maximally achievable modulation depth. c: Simulated gain and loss
spectrum in a bi-functional active region design12, allowing to tune the gain
at 10 V continuously to absorption (shown as negative gain) at 0 V. d:
Measured light-current-voltage (L-I-V) characteristics of an epi-up mounted
bi-functional QCL at 15$\,{}^{\circ}$C. e: Illustration of a system of coupled
oscillators. This system shows an in-phase and anti-phase synchronization
state, which oscillate at different frequencies depending on the coupling.
Without external stimulation, the anti-phase state is more favorable due to
the damped coupling. However, both synchronization frequencies can be probed
by exerting mechanical force on the platform coupling the oscillators. In the
QCL, the oscillators are represented by the intermode beatings, which tend to
synchronize in anti-phase due to gain damping19, 20. Both synchronization
frequencies are probed by applying modulation to the laser. f: Average optical
power depending on the modulation frequency and power. Two synchronization
states at $f_{\mathrm{rep}}^{0}$ and 60 MHz above are observed. g: Signal of a
2-QWIP sensitive to peak power as function of modulation frequency and power.
The strongly increased signal of the lobe at $f_{\mathrm{rep}}^{0}$+60 MHz
indicates in-phase synchronization.

In this work, we demonstrate the generation of transform limited picosecond
pulses in high-performance 8 µm wavelength QCLs at room temperature both
experimentally and theoretically. Mode-locking is achieved by electrically
modulating the intracavity loss using a short modulation section designed for
efficient radio-frequency (RF) injection (Fig. 1a). In order to achieve the
large modulation depth required for stable mode-locking, close attention has
to be paid to the band structure of the QCL active region. This effect is
illustrated in Figs. 1a,b. As the bias applied to a standard QCL structure is
decreased, it does not switch to absorption at the lasing wavelength, but
becomes nearly transparent for the intracavity light due to a bias-dependent
shift of the electronic levels, known as Stark effect (Fig. 1b). Hence, the
modulation depth is severely limited in standard QCL designs. For this reason,
we employ a bi-functional active region whose lasing wavelength and absorption
wavelength at zero bias were matched to each other 21, 12 (Fig. 1c). This
strategy allows to overcome the aforementioned limitations of the modulation
depth caused by the Stark shift. Most importantly, the bi-functional design
shows excellent overall performance, which is competitive with other state-of-
the-art designs. A 3.5 mm long device mounted epitaxial-side up emits more
than 130 mW average power in continuous wave at room temperature (Fig. 2d).

As a first step towards pulse generation, it is essential to determine the
optimal modulation frequency. For this purpose, mode-locking can be seen as
synchronization of coupled oscillators20 (Fig. 1e). Each pair of neighboring
cavity modes creates a beating at their difference frequency, which is equal
to the cavity roundtrip frequency. These beatings can be seen as oscillators
coupled by the non-linearity of the gain medium. Thanks to this coupling, the
cavity modes of a free-running QCL can be locked together without modulation,
thus giving rise to a self-starting frequency comb 19. Yet, this kind of
frequency comb does not emit isolated pulses, but rather a quasicontinuous
wave accompanied by a strong linear frequency chirp22. This corresponds to
anti-phase synchronization and will be called frequency modulated (FM) comb in
the following. In contrast, in-phase synchronization of the intermode beatings
leads to the formation of short pulses. It is well known from coupled
oscillators that the in-phase and anti-phase states synchronize at different
frequencies depending on the coupling. As a consequence, while the cavity
roundtrip frequency of the FM QCL comb $f_{\mathrm{rep}}^{0}$ may seem like a
reasonable choice, we expect the optimal modulation frequency for generating
pulses to differ from $f_{\mathrm{rep}}^{0}$.

In order to investigate these two synchronization states experimentally, we
start by operating the laser well above its threshold current. Subsequently,
the DC bias of the modulation section is decreased to 2.8 V, where the large
absorption caused by the bifunctional design (Fig. 1c) brings the QCL just
slightly below lasing threshold. In these conditions, modulation at the right
frequencies can provide enough additional gain to reach threshold. Fig. 1f
shows the laser power depending on modulation frequency and power. At 33 dBm
modulation power, the QCL reaches threshold when modulating close to
$f_{\mathrm{rep}}^{0}$. Strikingly, a second modulation frequency where lasing
occurs is observed almost 60 MHz higher than $f_{\mathrm{rep}}^{0}$, as
predicted by the picture of synchronized oscillators. Both the range around
the two synchronization frequencies, where lasing is observed, as well as the
optical power grow upon increasing the modulation power.

Figure 2: Mode-locked pulses from an 8 µm wavelength QCL. a: SWIFTS
characterization of the QCL operated close to lasing threshold. The laser is
modulated at the in-phase synchronization frequency and at 37 dBm power level.
b: Reconstructed time-domain signal of the QCL, showing a train of transform
limited pulses with 6.5 ps FWHM. c: Simulation of the QCL using the coherent
Master equation described in supp. section 1. d: Interferometric
autocorrelation (IAC) of the QCL pulses close to threshold. Red dots: envelope
of the IAC reconstructed using SWIFTS. e: IAC at higher current. f: IAC at the
rollover current, still displaying the 8:1 ratio. The second burst at a delay
equal to the cavity roundtrip time is due to the interference of subsequently
emitted pulses. Its peak value of 8 provides another proof for the coherence
of the pulses because phase-decoherence would smear out the fringes of the IAC
and thus decrease the peak value to smaller than 8. Inset: zoom on the
interferometric fringes.

Even more insight is provided by using a two-photon quantum well infrared
photodetector (2-QWIP) to detect the emitted light. The signal of the 2-QWIP
is proportional to the square of the intensity. This allows to identify which
modulation frequency leads to in-phase and which to anti-phase synchronization
(Fig. 1g). Again, two lobes appear around $f_{\mathrm{rep}}^{0}$ and 60 MHz
above. Yet, the 2-QWIP signal is more than an order of magnitude larger in the
lobe at higher $f_{\mathrm{mod}}$. At this frequency, the laser operates in
the in-phase synchronization regime and emits intense pulses, which leads to a
strongly increased 2-QWIP signal.

Figure 3: Synchronization under strong modulation. a: Schematic of a 3-section
QCL comprised of modulation, gain and high-speed detector sections. b: First
three harmonics of the beatnote of the free-running 7 mm long QCL FM comb
(red) compared to the actively mode-locked QCL (AM comb, blue). c: laser
beatnote while free-running (bottom), at
$f_{\mathrm{mod}}{=}f_{\mathrm{rep}}^{0}$ (middle) and at
$f_{\mathrm{mod}}{=}f_{\mathrm{rep}}^{0}{+}33\,$MHz (top). While a broad
pedestal is visible for $f_{\mathrm{mod}}{=}f_{\mathrm{rep}}^{0}$, the
beatnote is perfectly locked for
$f_{\mathrm{mod}}{=}f_{\mathrm{rep}}^{0}{+}33\,$MHz. d: RF spectrum around
$f^{0}_{\mathrm{rep}}{=}$6.196 GHz as the modulation frequency is varied
around $f^{0}_{\mathrm{rep}}$. The phase-noise of the RF spectrum disappears
abruptly at $f_{\mathrm{mod}}{\approx}f_{\mathrm{rep}}^{0}{+}20\,$MHz,
corresponding to in-phase synchronization. Here, the beatnote consists of a
single narrow peak, indicating that the laser is phase-locked to the
modulation. Furthermore, the sharp sidepeaks visible at
$f_{\mathrm{mod}}{=}6.18\,$GHz are attributed to a periodic modulation of the
QCL output, as previously observed in simulations16.

In order to unequivocally prove mode-locking, we employ two independent
methods to characterize the pulse dynamics at three points of operation from
threshold up to the rollover current. Firstly, an interferometric RF technique
called ’Shifted wave interference Fourier transform spectroscopy’ (SWIFTS)23,
24 is used to measure the phases of the QCL spectrum (details in Methods
section). This information not only enables the reconstruction of the temporal
waveform, but also allows to assess the phase-coherence of the pulses and
whether they form a frequency comb. Secondly, we measure the interferometric
autocorrelation (IAC) of the pulses using the 2-QWIP, which constitutes an
additional well-established proof for mode-locking and the pulse width. Fig.
2a shows the SWIFTS characterization of the QCL operated close to threshold.
In contrast to the free-running laser, the intensity spectrum consists of a
single Gaussian-shaped lobe. The SWIFTS spectrum represents the part of the
intensity spectrum which is beating exactly at the modulation frequency. Since
the SWIFTS spectrum has the same shape as the intensity spectrum over its
entire span, the QCL generates a frequency comb whose repetition frequency is
given by the modulation frequency. The intermodal difference phases
$\Delta\phi$, which correspond to the spectral group delay, are synchronized
almost perfectly in-phase. Hence, all parts of the spectrum have the same
group delay and form a pulse. Indeed, the reconstructed waveform in Fig. 2b
shows the emission of 6.5 ps short pulses. The full-width-half-maximum (FWHM)
of the reconstructed pulses is given by the transform limit of the spectrum in
Fig. 2b, indicating that there is negligible chirp in the pulses. In these
conditions, the peak power reaches almost 250 mW, which constitutes an
enhancement of more than 12 compared to the average power of 20 mW. In order
to model the cavity dynamics, we use a fully coherent master equation25 (supp.
section 1). This single equation for the complex field replaces the entire
Maxwell-Bloch system and reliably predicts the spectral shape, phase
relationship and pulse width observed experimentally (Fig. 2c). Furthermore,
it allows experimentally unavailable analyses, e.g. the influence of
dispersion and nonlinearities (supp. Figs. 2,3,7).

The IAC close to threshold (Fig. 2d) shows a prominent peak at zero path
difference caused by constructive interference of the pulses after the
Michelson interferometer. The ratio of this peak to the background at a delay
larger than the pulse width is 8:1, which is generally regarded as the smoking
gun of mode-locked pulses. Encouragingly, the measured IAC is in excellent
agreement with the expected IAC, which was calculated using the pulses
obtained by SWIFTS (red dots in Fig. 2d). This confirms successful mode-
locking and the retrieved pulse width.

The generation of pulses becomes increasingly challenging at higher gain
current. Due to gain saturation, the wings of a pulse experience more gain
than the peak. This effect leads to pulse broadening and can destabilize mode-
locking. Fortunately, the large modulation depth provided by the bi-functional
quantum design enables mode-locking over the entire lasing range from
threshold to rollover. The IAC traces at 3.7 kA/cm2 and at the rollover
current still show the required peak-to-background ratio of 8:1. The pulses at
rollover are slightly broadened to roughly 12 ps, which is attributed
partially to a slight chirp and partially to the gain saturation effect
mentioned above (supp. Fig. 6). Yet, the average power is greatly increased to
62 mW, which results in over 430 mW peak power and 5 pJ pulse energy - more
than an order of magnitude higher than recent reports of comparable mid-
infrared semiconductor lasers emitting at shorter wavelengths 26, 27.

Another fascinating aspect of bi-functional quantum design is the possibility
to monolithically integrate ultrafast photodetectors. While this is
particularly important in applications such as photonic integrated circuits,
it also provides a tool to measure the beatnote with very large signal-to-
noise ratio directly on the chip (Fig. 3a). This provides crucial information
about the type of synchronization state and about its stability. Fig. 3b shows
the first three harmonics of the beatnote in the free-running and the actively
mode-locked regime. In the latter conditions, the beatnote amplitudes increase
by 19 dB due to the much larger amplitude modulation. The zoom on the first
harmonic beatnote (Fig. 3c) allows to assess the phase-coherence and stability
of the frequency comb. The free-running QCL is operating in the anti-phase
state showing a weak beatnote at $f_{\mathrm{rep}}^{0}$. Previous work28, 29,
30 has shown that a weak electrical modulation can be used to lock and
stabilize the beatnote of the anti-phase state. However, the situation is very
different when applying strong modulation at $f_{\mathrm{rep}}^{0}$. In this
case, the modulation enforces an AM waveform, which is contrary to the natural
anti-phase behaviour of the laser. As a result, the beatnote of this waveform
is not phase-locked, as indicated by the pedestal around $f_{\mathrm{mod}}$.
The situation changes completely, when the modulation frequency is tuned to
the synchronization frequency of the in-phase state
($f_{\mathrm{rep}}^{0}{+}33\,$MHz). There, the strong modulation is in
consensus with the natural behavior of the laser, leading to a phase-locked
frequency comb with narrow beatnote. This can also be seen in Fig. 3d, which
shows the laser beatnote while tuning the modulation frequency across the in-
phase and anti-phase synchronization frequencies. While the frequency of the
beatnote is controlled by the modulation over the entire span, a phase-locked
comb is only generated around the in-phase synchronization frequency.

In conclusion, our experiments provide unambiguous proof for the generation of
mode-locked pulses in high-performance QCLs at room temperature - a goal which
remained elusive since the invention of the QCL - and confirm stunning
similarities to synchronization in coupled oscillators. These mode-locked QCLs
constitute the first compact and electrically pumped source for ultrashort
pulses beyond 5 µm wavelength, demonstrating that they are a highly promising
technology for ultrafast laser science in the long-wave infrared region. The
availability of such a source paves the way towards a semiconductor-based
platform for non-linear photonics, potentially enabling broadband mid-infrared
frequency combs and supercontinuum generation.

## Methods

QCLs optimized for RF modulation: The QCLs are processed as buried
heterostructures. The width of the QCL ridges is 12 µm and the facets are left
as cleaved. The area of the top contact of the modulation section is minimized
to decrease its parasitic capacitance, which increases the RF injection
efficiency. Ground contacts for the modulation are provided by etching through
the Fe-doped InP layer next to the laser ridges. The modulation signal is
provided by a HP8341B synthesized sweeper, amplified up to roughly 5 W and
injected via coplanar tips. The insertion loss at 12 GHz is 14 dB including
cables and bias-tee.

SWIFTS and IAC: The light emitted by the QCL is shone through a Bruker Vertex
70v FTIR spectrometer. In order to perform SWIFTS, the light is then detected
by a home-built fast QWIP at the exit of the FTIR. The optical beating
obtained from the QWIP is subsequently amplified and mixed down to roughly 10
MHz using a local oscillator. A Zurich Instruments HF2LI lock-in amplifier and
the trigger of the FTIR are used to record the SWIFTS and intensity
interferograms in rapid scan mode. The IAC is obtained by detecting the pulses
at the exit of the FTIR using the 2-QWIP and recording its photocurrent
depending on the path delay of the FTIR.

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## Acknowledgements

This work was supported by the Austrian Science Fund (FWF) in the framework of
”Building Solids for Function” (Project W1243), the projects ”NanoPlas”
(P28914-N27) and ”NextLite” (F4909-N23).

## Author contributions

J.H. processed the QCLs and carried out the experiments. B.S. and J.H. built
up the SWIFTS and IAC setups. N.O. and B.S. developed the simulation tool.
M.P. carried out the temporal reconstruction using the IAC data. H.S. provided
the 2-QWIP. J.H. wrote the manuscript with editorial input from N.O., B.S.,
G.S and F.C. All authors analysed the results and commented on the paper.