# Enforcing contraction via data

Zhongjie Hu, Claudio De Persis, Pietro Tesi Zhongjie Hu and Claudio De Persis
are with the Engineering and Technology Institute, University of Groningen,
9747AG, The Netherlands (e-mail<EMAIL_ADDRESS>c.de.persis@rug.nl).
Pietro Tesi is with DINFO, University of Florence, 50139 Florence, Italy
(e-mail: pietro.tesi@unifi.it). *This publication is part of the project
Digital Twin with project number P18-03 of the research programme TTW
Perspective which is (partly) financed by the Dutch Research Council (NWO).
The first author is supported by the China Scholarship Council.

###### Abstract

We present data-based conditions for enforcing contractivity via feedback
control and obtain desired asymptotic properties of the closed-loop system. We
focus on unknown nonlinear control systems whose vector fields are expressible
via a dictionary of functions and derive data-dependent semidefinite programs
whose solution returns the controller that guarantees contractivity. When data
are perturbed by disturbances that are linear combination of sinusoids of
known frequencies (but unknown amplitude and phase) and constants, we
remarkably obtain conditions for contractivity that do not depend on the
magnitude of the disturbances, with imaginable positive consequences for the
synthesis of the controller. Finally, we show how to design from data an
integral controller for nonlinear systems that achieves constant reference
tracking and constant disturbance rejection.

##  I Introduction

There is widespread attention in the control community towards new ways to
utilize data collected from dynamical systems in the process of designing
feedback control. The interest is motivated by the shared belief that machine
learning is revolutionising many branches of science and engineering,
including automatic control, and learning to design control from data is a
first step towards a thorough cross-fertilization of machine learning and
control theory.

Albeit the use of data has always represented the backbone of control design
in areas such as adaptive control, recently a new wave of solutions have
focused on the so-called direct data-driven control design, which produces
control policies from data collected in offline experiments without explicitly
attempting to identify the system [1, 2]. Among the various proposed
solutions, the one that relates control design to data-dependent convex
programs is one of the approaches that has caught on the most in recent times.
Such approach has mainly focused on linear control systems [3, 4, 5, 6].

In this paper, we are interested to data-driven control for nonlinear systems.
Designing feedback control from data is inevitably harder due to the varied
complexity of nonlinear dynamical systems. Many of the most successful model-
based nonlinear control design techniques have exploited special structures of
nonlinear systems [7], which are difficult to work with when the systems to
control are hardly known. To date, contributions to data-driven control design
for nonlinear systems are sparse. In [4], by taking into account the impact of
the remainder as perturbations corrupting the data, a feedback controller that
stabilizes an equilibrium of nonlinear systems by the first approximation was
designed from data. As for linear systems, the design was based on the
solution of semidefinite programs and prompted the observation that this
approach might be fruitful for those classes of nonlinear systems whose
control design relies on convex programs. These include bilinear systems [8,
9] and polynomial systems for which [10] and [11] independently proposed
distinct approaches. Later, [12] remarked that the results of [11] could be
extended to rational systems. By Taylor’s series expansion, the results on
data-driven control for linear and polynomials systems can be used for general
nonlinear systems by a careful analysis of the remainder [13, 14, 15]. Another
way to deal with general nonlinear systems is to express the system’s vector
fields via dictionaries of nonlinear functions and then design controllers
that makes the closed-loop system dominantly linear [16]. When coupled with
coordinates transformations, these techniques do mimic a data-dependent
version of nonlinear feedback linearization [16, Section VII.B]. Data-
dependent nonlinear feedback linearization has been also investigated in the
context of predictive control [17], and using Gaussian process regression
[18]. Other approaches consist of looking at the nonlinear system as a linear
one with state dependent matrices and recursively designing the feedback
controller at each time step via linear control techniques [19] or carrying
out an LPV embedding of an incremental representation of the nonlinear system
followed by a data-driven control synthesis [20].

Contribution and related works. For autonomous discrete-time dynamical systems
whose vector field is a contraction mapping, convergence to a fixed point is a
consequence of the contraction mapping theorem. More in general and for both
discrete- and continuous-time systems, concepts such as contraction analysis
[21], incremental stability [22] and convergent systems [23] (whose relations
have been studied in [24], [25]) have established deep connections between
asymptotic properties of trajectories with respect to each other and Lyapunov
analysis. When specialized to systems that have an equilibrium, establishing
contractivity does not require the explicit knowledge of the equilibrium
itself, a feature which is useful when dealing with unknown systems. More
importantly, contractivity is instrumental in solving various problems such as
observer design and output regulation.

The purpose of this paper is to present data-based conditions for enforcing
contractivity via feedback control and obtain desired asymptotic properties of
the closed-loop system. As in [16], we focus on unknown nonlinear control
systems whose vector fields are expressible via a dictionary of functions and
aim at deriving data-dependent semidefinite programs whose solution returns
the controller that guarantees contractivity. Using semidefinite programs to
guarantee contractivity via feedback has been used in the case the system’s
model is known [26, 27, 28], but the use of data to offset the uncertainty on
the model makes the presented results distinctively novel. The presented
approach also carries over to the case of data that are perturbed by unknown
but bounded process disturbances. Even more, if the disturbances are linear
combinations of constant and sinusoidal signals of known frequencies (but
unknown amplitude and phase), we present a new result where the data-based
conditions for contractivity do not depend on the magnitude of the disturbance
affecting the data and remarkably bear similarities with the ones obtainable
in the case of noise-free data, with imaginable positive consequences for the
synthesis of the controller. As our final contribution, in order to underscore
the relevance of contractivity in data-driven control problems different from
mere stabilization, we show how to design from data an integral controller for
nonlinear systems that achieves constant reference tracking and constant
disturbance rejection. For this problem, we follow the idea in [29], [30] of
rendering the given system, extended with the integral controller and without
forcing inputs, contractive and then analyzing the impact of the reference and
the disturbance by relying on the established contractivity. This idea does
not rely on special forms of the system and is therefore well suited for data-
based control design.

Being a powerful concept to deal with nonlinear systems analysis and control,
contraction theory has been already employed in learning-based control. Some
existing results have been surveyed in [31], which uses estimation techniques
based on neural networks ([31]), while [32, Section 4] uses Gaussian process
regression to estimate a closed-loop system and to give a condition for
incremental exponential stability in the second moment.

Outline. The framework and problem formulation are discussed in Section II.
The main result is given in Section III, which is then expanded to deal with
noisy data (Section V). The special case of disturbances given by linear
combination of sinusoids and constants is studied in Section VI. The paper
ends with the application of the previous results to the design of integral
controllers (Section VII). To make the paper self-contained, we added an
Appendix that contains a few basic results about contractive and convergent
systems that are used in the paper.

Notation. Throughout the paper, $\mathbb{R}$ denotes the set of real numbers,
while $\mathbb{R}_{\geq c}$ denotes the set of real numbers greater than or
equal to the real number $c$. Given a symmetric matrix $M$, $M\succ 0$
($M\succeq 0$) indicates that $M$ is positive definite (positive
semidefinite), while $M\prec 0$ ($M\preceq 0$) indicates that $M$ is negative
definite (negative semidefinite). $\mathbb{S}^{n\times n}$ denotes the set of
real-valued symmetric matrices. We abbreviate a symmetric matrix
$\left[\begin{smallmatrix}M&N\\\ N^{\top}&P\end{smallmatrix}\right]$ as
$\left[\begin{smallmatrix}M&\star\\\ N^{\top}&P\end{smallmatrix}\right]$.
Finally, we let $|v|$ be the 2-norm of a vector $v$, and $\lVert M\rVert$ be
the induced 2-norm of a matrix $M$. Other, less standard, notions are
introduced throughout the paper.

##  II Set-up and problem formulation

We consider a continuous-time system of the form

$\dot{x}=f(x)+Bu$ (1)

where $x\in\mathbb{R}^{n}$ is the state, $u\in\mathbb{R}^{m}$ is the control
input. The vector field $f(x)$ and the input matrix $B$ are unknown. To ease
the analysis we focus on the case of a constant input matrix $B$. We are
interested in the problem of making the system contractive via feedback. We
will introduce later a problem that motivates the interest in this property.

Without further priors about the vector field $f$, any control design is
challenging. Hence, we introduce the following:

###### Assumption 1

We know a continuously differentiable vector-valued function
$Z:\mathbb{R}^{n}\rightarrow\mathbb{R}^{s}$ such that $f(x)=AZ(x)$ for some
matrix $A\in\mathbb{R}^{n\times s}$. $\Box$

Under Assumption 1, system (1) can be equivalently written as

$\dot{x}=AZ(x)+Bu$ (2)

with $A,B$ unknown.

For control design purposes, it is advantageous to have in $Z(x)$ both the
state vector $x$ and the vector of nonlinear functions $Q(x)$, namely we
consider $Z(x)$ of the form

$Z(x)=\begin{bmatrix}x\\\ Q(x)\end{bmatrix}\,,$ (3)

with $Q:\mathbb{R}^{n}\rightarrow\mathbb{R}^{s-n}$ containing only nonlinear
functions.

_Dataset._ The lack of knowledge about $A,B$ are offset by data collected in
an experiment, which returns the following dataset to the designer:

$\mathbb{D}:=\left\\{(x_{i},u_{i},\dot{x}_{i})\right\\}_{i=0}^{T-1}$ (4)

where $T>0$ is an integer equal to the number of samples of the dataset,
$x_{i}:=x(t_{i}),u_{i}:=u(t_{i}),\dot{x}_{i}:=\dot{x}(t_{i})$ and $0\leq
t_{0}\leq t_{1}\leq\ldots\leq t_{T-1}$ are the sampling times. No requirements
are imposed on the sampling times. By construction, the samples satisfy the
relation $\dot{x}_{i}=AZ(x_{i})+Bu_{i}$ for $i=0,\ldots,T-1$, $T>0$. Note
that, $Z(x_{i})$ can be computed from $x_{i}$ and $Z(x)$, since the latter is
known. We also assume that the time derivative of the state $\dot{x}_{i}$ can
be measured. While this is an unrealistic assumption, there are two ways to
relax it. One is to consider approximate estimates of the derivatives, regard
the estimation error as a perturbation on the data and design the control
accordingly (see Section V on how to establish contractivity with noisy data).
The other is to use a different data collection scheme that considers integral
versions of the relation (2) (see [33, Appendix A] for details). Finally, we
begin to study the scenario in which data are not perturbed by disturbances or
noise.

_Problem._ We are interested in the design of a feedback control of the form
$u=KZ(x)$, which makes the closed-loop system

$\dot{x}=(A+BK)Z(x)$ (5)

contractive on a set $\mathcal{X}\subseteq\mathbb{R}^{n}$. We recall below
such a property in a form that is suitable for our purposes:

###### Definition 1

Consider a set $\mathcal{X}\subseteq\mathbb{R}^{n}$. System (5) is
exponentially contractive on $\mathcal{X}$ if

$\begin{array}[]{l}\exists P_{1}\succ 0,\beta>0\textrm{ such that }\forall
x\in\mathcal{X}\\\ ((A+BK)\displaystyle\frac{\partial Z}{\partial
x})^{\top}P_{1}^{-1}+P_{1}^{-1}(A+BK)\displaystyle\frac{\partial Z}{\partial
x}\preceq-\beta P_{1}^{-1}.\end{array}$ (6)

The set $\mathcal{X}$ is the contraction region with respect to the metric
$P_{1}^{-1}$. $\Box$

This is a property instrumental in several estimation and control problems,
whose use in the control literature has been popularized by [21]. In the
original definition therein, the metric $P_{1}^{-1}$ is taken as state- and
time-dependent. Since the search for a state- or time-dependent metric is
difficult, here we concentrate our efforts on a constant metric. In fact, to
complicate the situation here is the lack of knowledge of the matrices $A,B$
in (2) and the presence of the design variable $K$. Hence, beside $P_{1}$ and
$\beta$, one has to simultaneously seek a $K$ that enforces (6). Finally, in
Section VII we will consider a tracking problem with constant reference and
disturbance signal of arbitrary magnitude, for which considering a constant
metric results in no loss of generality [34, Theorem 2].

##  III Control design

###  III-A A data-dependent representation of the closed-loop system

A first step we take toward designing the controller is to obtain a data-
dependent representation of the closed-loop system. To this end, we arrange
the dataset in the following matrices:

$\displaystyle
U_{0}:=\begin{bmatrix}u_{0}&u_{1}&\cdots&u_{T-1}\end{bmatrix}\in\mathbb{R}^{m\times
T}\,,$ (7a) $\displaystyle
X_{0}:=\begin{bmatrix}x_{0}&x_{1}&\cdots&x_{T-1}\end{bmatrix}\in\mathbb{R}^{n\times
T}\,,$ (7b) $\displaystyle
X_{1}:=\begin{bmatrix}\dot{x}_{0}&\dot{x}_{1}&\cdots&\dot{x}_{T-1}\end{bmatrix}\in\mathbb{R}^{n\times
T}\,,$ (7c) $\displaystyle Z_{0}:=\begin{bmatrix}x_{0}&x_{1}&\cdots&x_{T-1}\\\
Q(x_{0})&Q(x_{1})&\cdots&Q(x_{T-1})\end{bmatrix}\in\mathbb{R}^{s\times T}.$
(7d)

Note that these matrices satisfy the identity

$X_{1}=AZ_{0}+BU_{0}.$

The following result, established in [16, Lemma 1] and recalled here without
proof, returns the data-dependent representation of the closed-loop system we
are interested in.

###### Lemma 1

Consider any matrices $K\in\mathbb{R}^{m\times s}$, $G\in\mathbb{R}^{T\times
s}$ such that

$\begin{bmatrix}K\\\ I_{s}\end{bmatrix}=\begin{bmatrix}U_{0}\\\
Z_{0}\end{bmatrix}G\,.$ (8)

Let $G$ be partitioned as $G=\begin{bmatrix}G_{1}&G_{2}\end{bmatrix}$, where
$G_{1}\in\mathbb{R}^{T\times n}$ and $G_{2}\in\mathbb{R}^{T\times(s-n)}$. Let
Assumption 1 hold. Then system (1) under the control law $u=KZ(x)$ results in
the closed-loop dynamics

$\dot{x}=(A+BK)Z(x)=Mx+NQ(x)$ (9)

where $M:=X_{1}G_{1}$ and $N:=X_{1}G_{2}$. $\Box$

_Proof._ See [16, Lemma 1]. $\blacksquare$

We note that the condition (8) and the right-hand side of (9) only depend on
the matrices of data $U_{0},Z_{0},X_{1}$ and the decision variable $G$. Hence,
we will design $G$ that enforces the desired contractivity property and then
obtain the control gain $K$ via (8).

###  III-B Contractivity from data

The first statement gives data-dependent conditions under which there exists
$P_{1},K$ for which (6) holds.

###### Theorem 1

Consider the nonlinear system (1). Let Assumption 1 hold and $Z(x)$ be of the
form (3). Let $R_{Q}\in\mathbb{R}^{n\times r}$ be a known matrix and
$\mathcal{X}\subseteq\mathbb{R}^{n}$ a set such that

$\frac{\partial Q}{\partial x}(x)^{\top}\frac{\partial Q}{\partial
x}(x)\preceq R_{Q}R_{Q}^{\top}\text{ for any $x\in\mathcal{X}$}.$ (10)

Consider the following semidefinite program (SDP) in the decision variables
$P_{1}\in\mathbb{S}^{n\times n}$, $Y_{1}\in\mathbb{R}^{T\times n}$,
$G_{2}\in\mathbb{R}^{T\times(s-n)}$ and $\alpha\in\mathbb{R}_{>0}$:

$\displaystyle P_{1}\succ 0,$ (11a) $\displaystyle
Z_{0}Y_{1}=\begin{bmatrix}P_{1}\\\ 0_{(s-n)\times n}\end{bmatrix}\,,$ (11b)
$\displaystyle\begin{bmatrix}X_{1}Y_{1}+(X_{1}Y_{1})^{\top}+\alpha
I_{n}&X_{1}G_{2}&P_{1}R_{Q}\\\ (X_{1}G_{2})^{\top}&-I_{s-n}&0\\\
(P_{1}R_{Q})^{\top}&0&-I_{r}\end{bmatrix}\preceq 0\,,$ (11c) $\displaystyle
Z_{0}G_{2}=\begin{bmatrix}0_{n\times(s-n)}\\\ I_{s-n}\end{bmatrix}\,.$ (11d)

If the program is feasible then the control law

$u=KZ(x)$ (12)

with

$\displaystyle K=U_{0}\begin{bmatrix}Y_{1}P_{1}^{-1}&G_{2}\end{bmatrix}$ (13)

is such that the closed-loop dynamics (1), (12) is exponentially contractive
on $\mathcal{X}$, _i.e._ , (6) holds. $\Box$

_Proof._ If (11) is feasible, then we can define $G_{1}:=Y_{1}P_{1}^{-1}$ and
note that (11b), (11d) and the definition of $K$ in (13), give (8). Hence, by
Lemma 1, the closed-loop dynamics $\dot{x}=(A+BK)Z(x)$ can be written as
$\dot{x}=Mx+NQ(x)$, with $M=X_{1}G_{1}$ and $N=X_{1}G_{2}$. By Schur
complement, (11c) gives

$\begin{bmatrix}X_{1}Y_{1}+(X_{1}Y_{1})^{\top}+P_{1}R_{Q}R_{Q}^{\top}P_{1}+\alpha
I_{n}&X_{1}G_{2}\\\ (X_{1}G_{2})^{\top}&-I_{s-n}\end{bmatrix}\preceq 0\,.$

Another Schur complement returns

$X_{1}Y_{1}+(X_{1}Y_{1})^{\top}+\alpha
I_{n}+P_{1}R_{Q}R_{Q}^{\top}P_{1}+X_{1}G_{2}(X_{1}G_{2})^{\top}\preceq 0\,.$

Left- and right-multiply by $P_{1}^{-1}$ and recall that
$G_{1}:=Y_{1}P_{1}^{-1}$. Then

$\begin{array}[]{l}P_{1}^{-1}X_{1}G_{1}+(X_{1}G_{1})^{\top}P_{1}^{-1}+\alpha
P_{1}^{-2}+R_{Q}R_{Q}^{\top}\\\\[2.84526pt] \hskip
99.58464pt+P_{1}^{-1}X_{1}G_{2}(X_{1}G_{2})^{\top}P_{1}^{-1}\preceq
0\,.\end{array}$ (14)

This shows that there exist a real number $\lambda=1$ and matrices

$\begin{array}[]{rl}H:=&P_{1}^{-1}X_{1}G_{1}+(X_{1}G_{1})^{\top}P_{1}^{-1}+\alpha
P_{1}^{-2}\\\ J:=&P_{1}^{-1}X_{1}G_{2}\\\ L:=&I_{n}\\\
O:=&R_{Q}R_{Q}^{\top}\end{array}$

that satisfy $H+\lambda JJ^{\top}+\lambda^{-1}L^{\top}OL\preceq 0$. If the
solution of (11) is such that $J=0$, then $(A+BK)\frac{\partial Z}{\partial
x}=X_{1}G_{1}$ and (14) implies (6) with
$\beta:=\alpha\lambda_{\min}(P_{1}^{-1})$, where $\lambda_{\min}(P_{1}^{-1})$
is the minimum eigenvalue of $P_{1}^{-1}$, which ends the proof. On the other
hand, if the solution of (11) is such that $J\neq 0$, then by the nonstrict
Petersen’s lemma [35, Fact 2], we have that $H+\lambda
JJ^{\top}+\lambda^{-1}L^{\top}OL\preceq 0$ implies
$H+JR^{\top}L+L^{\top}RJ^{\top}\preceq 0$ for all $R\in\mathcal{R}:=\\{R\colon
RR^{\top}\preceq O\\}$, _i.e._ ,

$\begin{array}[]{l}P_{1}^{-1}X_{1}G_{1}+(X_{1}G_{1})^{\top}P_{1}^{-1}+\alpha
P_{1}^{-2}\\\ \hskip
85.35826pt+P_{1}^{-1}X_{1}G_{2}R^{\top}+RG_{2}^{\top}X_{1}^{\top}P_{1}^{-1}\preceq
0,\\\\[5.69054pt] \text{ for all }R\in\mathcal{R}:=\\{R\colon RR^{\top}\preceq
R_{Q}R_{Q}^{\top}\\}.\end{array}$

By (10), for any $x\in\mathcal{X}$, the inequality above gives

$\begin{array}[]{l}P_{1}^{-1}X_{1}G_{1}+G_{1}^{\top}X_{1}^{\top}P_{1}^{-1}+\alpha
P_{1}^{-2}+P_{1}^{-1}X_{1}G_{2}\displaystyle\frac{\partial Q}{\partial
x}(x)\\\ \hskip 85.35826pt+\displaystyle{\frac{\partial Q}{\partial
x}(x)}^{\top}(X_{1}G_{2})^{\top}P_{1}^{-1}\preceq 0,\end{array}$ (15)

that is, recalling that $\beta:=\alpha\lambda_{\min}(P_{1}^{-1})$,

$P_{1}^{-1}X_{1}G\frac{\partial Z}{\partial x}+\frac{\partial Z}{\partial
x}^{\top}G^{\top}X_{1}^{\top}P_{1}^{-1}\preceq-\beta P_{1}^{-1}\text{ for all
$x\in\mathcal{X}$},$

which ends the proof. $\blacksquare$

Having enforced the contraction property on the closed-loop system, we state
its asymptotic properties, provided an equilibrium exists.

###### Corollary 1

Let the conditions of Theorem 1 hold and assume additionally that $Z(x_{*})=0$
and $\mathcal{X}$ is a convex set containing $x_{*}$ in its interior. Then
$x_{*}$ is the unique111In fact, it is the unique solution defined and bounded
for all $t\in(-\infty,+\infty)$ and contained in $\mathcal{X}$. equilibrium
point in $\mathcal{X}$, it is uniformly exponentially stable for
$\dot{x}=(A+BK)Z(x)$ and any solution initialized in the set
$\mathcal{V}:=\left\\{x\in\mathbb{R}^{n}\colon(x-x_{*})^{\top}P_{1}^{-1}(x-x_{*})\leq\delta\right\\}$,
with $\delta>0$ such that $\mathcal{V}$ is contained in $\mathcal{X}$,
converges to $x_{*}$.

_Proof._ It is a consequence of Corollary 7 in the Appendix. It can also be
directly derived by the following standard arguments. Note that

$\begin{array}[]{rl}&\frac{\partial V}{\partial x}(A+BK)Z(x)\\\
\stackrel{{\scriptstyle(a)}}{{=}}&\\!\\!\\!\\!\\!2(x-x_{*})^{\top}P_{1}^{-1}(A+BK)(Z(x)-Z(x_{*}))\\\\[5.69054pt]
\stackrel{{\scriptstyle(b)}}{{=}}&\\!\\!\\!\\!\\!2(x-x_{*})^{\top}P_{1}^{-1}(A+BK)\left[\frac{\partial
Z}{\partial x}(x_{*}+\lambda(x-x_{*}))\right]_{\lambda\in(0,1)}\cdot\\\
&\hskip 170.71652pt\cdot(x-x_{*})\\\\[2.84526pt]
\stackrel{{\scriptstyle(c)}}{{\leq}}&-\beta(x-x_{*})^{\top}P_{1}^{-1}(x-x_{*})\end{array}$

where (a) holds thanks to $Z(x_{*})=0$, (b) by the mean value theorem for
vector valued functions and thanks to the convexity of the set $\mathcal{X}$,
(c) by (6). The other claims are immediate. $\blacksquare$

The assumption $Z(x_{*})=0$ necessarily implies that $x_{*}=0$. The assumption
can be lifted under a more stringent condition on $\mathcal{X}$.

###### Corollary 2

Let the conditions of Theorem 1 hold where $\mathcal{X}$ is either (i) a
convex, closed, forward invariant set for $\dot{x}=(A+BK)Z(x)$ and
$\dot{x}=(A+BK)Z(x)$ is forward complete on $\mathcal{X}$ or (ii)
$\mathcal{X}=\mathbb{R}^{n}$ and there exists $\overline{x}\in\mathbb{R}^{n}$
such that $|Z(\overline{x})|<\infty$. Then there exists a unique equilibrium
point $x_{*}$ in $\mathcal{X}$ for $\dot{x}=(A+BK)Z(x)$. Moreover, $x_{*}$ is
uniformly exponentially stable for $\dot{x}=(A+BK)Z(x)$ and any solution of
$\dot{x}=(A+BK)Z(x)$ initialized in $\mathcal{X}$ uniformly exponentially
converges to $x_{*}$.

_Proof._ Under the conditions of Theorem 1, (6) holds. If $\mathcal{X}$
satisfies (i), then the thesis follows from Theorem 5 in the Appendix. If
$\mathcal{X}$ satisfies (ii), then the thesis follows from Theorem 6 in the
Appendix. $\blacksquare$

A few remarks are in order.

(On condition (10)) Theorem 1 holds under the local Lipschitz property (10).
The result also holds for nonlinearities satisfying more general conditions
such as

$\begin{array}[]{l}\displaystyle\frac{\partial Q}{\partial
x}(x)^{\top}R\frac{\partial Q}{\partial x}(x)+S\frac{\partial Q}{\partial
x}(x)\\\\[5.69054pt] \hskip 71.13188pt\displaystyle+\frac{\partial Q}{\partial
x}(x)^{\top}S^{\top}\preceq W\text{ for any $x\in\mathcal{X}$}\end{array}$
(16)

where $W=W^{\top}\succeq 0$, $R=R^{\top}\succeq 0$, $S$ are known matrices,
provided that the LMI (11c) is replaced by

$\begin{bmatrix}X_{1}Y_{1}+(X_{1}Y_{1})^{\top}+\alpha
I_{n}&X_{1}G_{2}-P_{1}S&P_{1}W^{1/2}\\\ (X_{1}G_{2}-P_{1}S)^{\top}&-R&0\\\
(P_{1}W^{1/2})^{\top}&0&-I_{r}\end{bmatrix}\preceq 0$

In fact, the Schur complement applied to the inequality above returns

$\begin{bmatrix}X_{1}Y_{1}+(X_{1}Y_{1})^{\top}+P_{1}WP_{1}+\alpha
I_{n}&X_{1}G_{2}-P_{1}S\\\ (X_{1}G_{2}-P_{1}S)^{\top}&-R\end{bmatrix}\preceq
0\,.$

By left- and right-multiplying by ${\rm block.diag}(P_{1}^{-1},I_{s-n})$ and
setting $G_{1}:=Y_{1}P_{1}^{-1}$, the inequality above is rewritten as

$\\!\\!\begin{bmatrix}P_{1}^{-1}X_{1}G_{1}+(P_{1}^{-1}X_{1}G_{1})^{\top}+W+\alpha
P_{1}^{-2}&\\!\\!\\!\\!\star\\\
(P_{1}^{-1}X_{1}G_{2}-S)^{\top}&-R\end{bmatrix}\preceq 0,$

Left-multiplying the last inequality by $\left[\begin{smallmatrix}I\\\
\frac{\partial Q}{\partial x}\end{smallmatrix}\right]^{\top}$, right-
multiplying it by $\left[\begin{smallmatrix}I\\\ \frac{\partial Q}{\partial
x}\end{smallmatrix}\right]$, and bearing in mind (16), it follows that

$\left[\begin{matrix}I\\\ \frac{\partial Q}{\partial
x}\end{matrix}\right]^{\top}\\!\\!\begin{bmatrix}P_{1}^{-1}X_{1}G_{1}+(P_{1}^{-1}X_{1}G_{1})^{\top}+\alpha
P_{1}^{-2}&\star\\\
(P_{1}^{-1}X_{1}G_{2})^{\top}&0\end{bmatrix}\\!\\!\left[\begin{matrix}I\\\
\frac{\partial Q}{\partial x}\end{matrix}\right]\\!\\!\preceq\\!\\!0$

for all $x\in\mathcal{X}$, which is (6), as claimed. Examples of
nonlinearities that satisfy (16) include those that satisfy incremental sector
bound conditions [36, Lemma 4.4]. For other nonlinearities covered by this
analysis see [37, 30].

(On condition (11)) Contractivity and the asymptotic properties given above
are guaranteed by (11). The conditions (11b), (11d) are used to obtain the
data-dependent representation (9) of the closed-loop system:
$\dot{x}=Mx+NQ(x)$, where $M=X_{1}G_{1}$ and $N=X_{1}G_{2}$. On the other
hand, to enforce contractivity on $\mathcal{X}$, i.e., to enforce the
inequality (cf. (14))

$\begin{array}[]{l}P_{1}^{-1}X_{1}G_{1}+(P_{1}^{-1}X_{1}G_{1})^{\top}+\alpha
P_{1}^{-2}+P_{1}^{-1}X_{1}G_{2}\displaystyle\frac{\partial Q}{\partial
x}(x)\\\ \hskip 99.58464pt+\displaystyle{\frac{\partial Q}{\partial
x}(x)}^{\top}(X_{1}G_{2})^{\top}P_{1}^{-1}\preceq 0,\end{array}$

the key property is (11c), which, written as

$\begin{array}[]{l}P_{1}^{-1}X_{1}G_{1}+(X_{1}G_{1})^{\top}P_{1}^{-1}+\alpha
P_{1}^{-2}+R_{Q}R_{Q}^{\top}\\\ \hskip
99.58464pt+P_{1}^{-1}X_{1}G_{2}(X_{1}G_{2})^{\top}P_{1}^{-1}\preceq
0\,,\end{array}$

reveals the aim of our control design. In fact, the possibility of fulfilling
this condition is related to the existence of a matrix $P_{1}\succ 0$ and for
the linear part of the controller
$P_{1}^{-1}X_{1}G_{1}+(X_{1}G_{1})^{\top}P_{1}^{-1}$ to dominate both
$P_{1}^{-1}X_{1}G_{2}(X_{1}G_{2})^{\top}P_{1}^{-1}$ and the contribution of
the Jacobian $\frac{\partial Q}{\partial x}(x)$, which appears through the
term ${R}_{Q}{R}_{Q}^{\top}$. Controllability of the pair $(\overline{A},B)$,
where $A=\begin{bmatrix}\overline{A}&\hat{A}\end{bmatrix}$,
$\overline{A}\in\mathbb{R}^{n\times n}$, also plays a role in achieving this
goal.

###  III-C Comparison with other methods and a numerical example

It is useful to compare this approach with other methods on controller design
that are based on convex programming, see [16, 38, 39]. Starting from the
data-based representation of the closed-loop dynamics
$\dot{x}=X_{1}G_{1}x+X_{1}G_{2}Q(x)$, and by assuming that the term $Q(x)$
acts as a “remainder” term, _i.e._ , $\lim_{x\rightarrow
0}\frac{|Q(x)|}{|x|}=0$, 222This implies that $\overline{x}=0$ is a known
equilibrium of the system. a simple approach is to search for matrices
$P_{1}\in\mathbb{S}^{n\times n},Y_{1}\in\mathbb{R}^{T\times n}$, and a
positive scalar $\alpha$ satisfying the following conditions:

$\displaystyle\text{\eqref{eq:SDP0}, \eqref{eq:SDP1}, \eqref{eq:SDP4}},$ (17a)
$\displaystyle X_{1}Y_{1}+(X_{1}Y_{1})^{\top}+\alpha I_{n}\preceq 0.$ (17b)

If (17) is feasible then $u=Kx$ with $K=U_{0}Y_{1}P_{1}^{-1}$ locally
stabilizes the origin of the closed-loop system. This approach is inspired by
Lyapunov’s indirect method, which guarantees stability of the nonlinear
dynamics by stabilizing the linearized dynamics around the desired equilibrium
point.

Building on (17), an alternative approach is to stabilize the linearized
dynamics while simultaneously reducing the impact of the nonlinearities in
closed-loop. This amounts to searching for matrices
$P_{1}\in\mathbb{S}^{n\times n},Y_{1}\in\mathbb{R}^{T\times n}$,
$G_{2}\in\mathbb{R}^{T\times(s-n)}$ and a positive scalar $\alpha$ satisfying
the following program:

minimize $\displaystyle\|X_{1}G_{2}\|$ (18a) subject to
$\displaystyle\text{\eqref{eq:SDP0}, \eqref{eq:SDP1},
\eqref{eq:SDP4}},\eqref{SDP-taylor3}.$ (18b)

If the minimum is attained at zero, then $(A+BK)\frac{\partial Z}{\partial
x}=X_{1}G_{1}$ and condition (6) holds with $\mathcal{X}=\mathbb{R}^{n}$. In
particular, in this case the closed-loop dynamics becomes linear.

Compared with the method proposed in this paper, (17) and (18) entail milder
feasibility conditions since condition (17b) is less demanding than (11c).
Furthermore, Lyapunov’s indirect method allows to treat the term
$X_{1}G_{2}Q(x)$ that appears in the expression
$\dot{x}=X_{1}G_{1}x+X_{1}G_{2}Q(x)$ as an _unknown_ quantity, see [38] for
details. This is somewhat harder with the method proposed in this paper since
the control strategy relies on the full nonlinear behaviour to enforce a
stable (contracting) behaviour. On the other hand, the proposed method
possesses some nice features, notably:

  1. (1)

_Stability properties._ The proposed method permits to have _global_ stability
properties, as long as condition (6) holds with $\mathcal{X}=\mathbb{R}^{n}$.
This is instead not achievable (at least not by design) with Lyapunov’s
indirect method, and requires exact nonlineary cancellation, _i.e._ ,
$X_{1}G_{2}=0$, with (18), which is possible only when the system to control
has a specific structure, see [16, 40].

  2. (2)

_Knowledge of the equilibrium_. As detailed in Corollary 2, there are case of
practical interest in which the proposed method does not require the knowledge
of the equilibrium point, which follows because contractive systems have the
incremental stability property that all the trajectories converge to a unique
solution, see Theorems 5 and 6 in the Appendix. In contrast, neither (17) nor
(18) possess this property. For these methods, _exact_ knowledge of the
equilibrium point is needed to transform the problem into the standard form of
a zero-regulation problem. (This is not surprising considering that both (17)
and (18) rest on Lyapunov’s indirect method.)

We recall the definition of region of attraction used in the following
numerical results.

###### Definition 2

([16, Def. 1]) Let $\overline{x}$ be an asymptotically stable equilibrium
point for the system $\dot{x}=f(x)$. A set $\mathcal{R}$ defines a region of
attraction (ROA) for the system relative to $\overline{x}$ if for every
$x(0)\in\mathcal{R}$ we have $\lim_{t\rightarrow\infty}x(t)=\overline{x}$.

We illustrate the foregoing considerations via a numerical example. 333The
code to reproduce the examples is available at
https://github.com/zjhurug/codes-for-contraction-paper.git

###### Example 1

Consider the dynamics of a one-link robot arm [7, Section 4.10]

$\displaystyle\dot{x}_{1}=x_{2}$ (19a)
$\displaystyle\dot{x}_{2}=-\frac{K_{c}}{J_{2}}x_{1}-\frac{F_{2}}{J_{2}}x_{2}+\frac{K_{c}}{J_{2}N_{c}}x_{3}-\frac{mgd}{J_{2}}\cos
x_{1}$ (19b) $\displaystyle\dot{x}_{3}=x_{4}$ (19c)
$\displaystyle\dot{x}_{4}=-\frac{K_{c}}{J_{1}N_{c}}x_{1}+\frac{K_{c}}{J_{1}N_{c}^{2}}x_{3}-\frac{F_{1}}{J_{1}}x_{4}+\frac{1}{J_{1}}u.$
(19d)

where $x_{1}$, $x_{3}$ denote the angular positions of the link and of the
actuator shaft, respectively, and $u$ is the torque produced at the actuator
axis. We will collect data from the system setting the parameters as
$K_{c}=0.4$, $F_{2}=0.15$, $J_{2}=0.2$, $N_{c}=2$, $F_{1}=0.1$, $J_{1}=0.15$,
$m=0.4$, $g=9.8$ and $d=0.1$.

We introduce $Q(x)=\cos x_{1}$. Note that

$\frac{\partial Q}{\partial x}(x)^{\top}\frac{\partial Q}{\partial
x}(x)=\begin{bmatrix}\sin(x_{1})^{2}&0&0&0\\\ 0&0&0&0\\\ 0&0&0&0\\\
0&0&0&0\end{bmatrix}.$

If we set $\mathcal{X}=\mathbb{R}^{4}$, then (10) is satisfied with

$R_{Q}=R_{Q}^{\top}=\begin{bmatrix}1&0&0&0\\\ 0&0&0&0\\\ 0&0&0&0\\\
0&0&0&0\end{bmatrix}.$ (20)

We collect $T=10$ samples by running an experiment with input uniformly
distributed in $[-0.1,0.1]$, and with an initial state within the same
interval. The SDP (11) is feasible and returns the controller $K$ and the
closed-loop dynamics in (21).

$\displaystyle K$
$\displaystyle=\begin{bmatrix}-3.1639&-4.6751&-3.9299&-0.7614&0.0106\end{bmatrix}$
(21a) $\displaystyle\dot{x}$
$\displaystyle={\scriptsize\begin{bmatrix}0.0000&1.0000&0.0000&0.0000&0.0000\\\
-2.0000&-0.7500&1.0000&0.0000&-1.9600\\\ 0.0000&0.0000&0.0000&1.0000&0.0000\\\
-22.4261&-31.1671&-25.5328&-5.7430&0.0708\end{bmatrix}}Z(x)$ (21b)

Since $\mathcal{X}=\mathbb{R}^{4}$ and $|Z(\overline{x})|<\infty$ for any
$\overline{x}\in\mathbb{R}^{n}$, by Corollary 2, the equilibrium point
$x_{*}=\begin{bmatrix}-0.5718&0&0.5046&0\end{bmatrix}^{\top}$ obtained from
the simulation experiment or analytically from (21b) is unique in
$\mathbb{R}^{4}$ for (21b). Moreover, $x_{*}$ is globally uniformly
exponentially stable for (21b).

$\displaystyle
K=\begin{bmatrix}-280.8884&-257.1662&-91.3493&-8.2326&0.1591&-0.0032&-0.0030\end{bmatrix}$
(22a)
$\displaystyle\dot{x}=\begin{bmatrix}0.0000&1.0000&0.0000&0.0000&0.0000&-0.0000&-0.0000\\\
-2.0000&-0.7500&1.0000&0.0000&-1.9600&0.0000&0.0000\\\
0.0000&0.0000&0.0000&1.0000&-0.0000&0.0000&0.0000\\\
-1873.9225&-1714.4416&-608.3286&-55.5510&1.0603&-0.0213&-0.0201\end{bmatrix}Z(x)$
(22b)

Next, in order to test the effectiveness of Theorem 1 for other choices of
$Q(x)$, we add more nonlinearities in $Q(x)$ and let $Q(x)=\begin{bmatrix}\cos
x_{1}&x_{1}^{2}&\sin x_{2}\end{bmatrix}^{\top}$. Note that

$\frac{\partial Q}{\partial x}(x)^{\top}\frac{\partial Q}{\partial
x}(x)=\begin{bmatrix}\sin(x_{1})^{2}+4x_{1}^{2}&0&0&0\\\
0&\cos(x_{2})^{2}&0&0\\\ 0&0&0&0\\\ 0&0&0&0\end{bmatrix}.$

If we set $\mathcal{X}=[-w,w]\times\mathbb{R}^{3}$, where
$w\in\mathbb{R}_{>0}$, then (10) is satisfied with

$R_{Q}=R_{Q}^{\top}=\begin{bmatrix}\sqrt{4w^{2}+1}&0&0&0\\\ 0&1&0&0\\\
0&0&0&0\\\ 0&0&0&0\end{bmatrix}.$

Here, we set $w=1$. We collect $T=10$ samples under the same experiment setup.
The SDP (11) is feasible and returns the controller $K$ and the closed-loop
dynamics in (22). The unique equilibrium point obtained from the simulation
experiment or analytically from (22) is
$x_{*}=\begin{bmatrix}-0.3607&0&1.1126&0\end{bmatrix}^{\top}$. Note that
$x_{*}\in\text{int}(\mathcal{X})$. By Corollary 7 in the Appendix, any
solution of (22b) initialized in any sub-level set of
$V(x)=(x-x_{*})^{\top}P_{1}^{-1}(x-x_{*})$ contained in $\mathcal{X}$
uniformly exponentially converges to $x_{*}$. We observe the following: (i)
from numerical results, the solutions of (22b) initialized at the points
outside the largest Lyapunov sublevel set contained in $\mathcal{X}$ still
converge to $x_{*}$, which means the exact ROA is much larger and the obtained
controller (22b) is possibly a global controller. (ii) (11) remains feasible
up to $w=100$, and as $w$ increases, the magnitude of the coefficients for the
linear part of the obtained controller tends to increase, which is consistent
with the comments about condition (11). With $w=100$, the magnitude is of
order $10^{5}$, which results in a high-gain controller.

We do not compare (11) with (18), since the lack of a specific structure of
the systems does not allow (18) to exactly cancel the nonlinearity. It is more
interesting to compare (11) with the Lyapunov’s indirect method [38, Theorem
2]. Denote system (19) as $\dot{x}=f(x,u)$ and assume that the desired
equilibrium point $(x_{*},u_{*})$ is known, i.e.
$x_{*}=\begin{bmatrix}-0.3607&0&1.1126&0\end{bmatrix}^{\top}$ and
$u_{*}=-0.1834$. Linearize system (19) at $(x_{*},u_{*})$ which yields

$\dot{\tilde{x}}=A{\tilde{x}}+B{\tilde{u}}+r(\tilde{x},\tilde{u})$

where $\tilde{x}=x-x_{*}$, $\tilde{u}=u-u_{*}$ are the shifted state and input
variables,

$A=\left.\displaystyle\frac{\partial
f}{\partial{x}}\right|_{(x,u)=(x_{*},u_{*})},\quad
B=\left.\displaystyle\frac{\partial
f}{\partial{u}}\right|_{(x,u)=(x_{*},u_{*})}$

are unknown constant matrices and
$r\colon\mathbb{R}^{4}\times\mathbb{R}\to\mathbb{R}^{4}$ denotes the unknown
remainder. Collect $\\{(x_{i},u_{i},\dot{x}_{i})\\}_{i=0}^{T-1}$ from system
(19) and compute the samples
$\\{(\tilde{x}_{i},\tilde{u}_{i},\dot{\tilde{x}}_{i})\\}_{i=0}^{T-1}$, which
are used to define the data matrices

$\displaystyle\tilde{U}_{0}:=\begin{bmatrix}\tilde{u}_{0}&\tilde{u}_{1}&\ldots&\tilde{u}_{T-1}\end{bmatrix}\in\mathbb{R}^{m\times
T}\,,$ (23a)
$\displaystyle\tilde{X}_{0}:=\begin{bmatrix}\tilde{x}_{0}&\tilde{x}_{1}&\ldots&\tilde{x}_{T-1}\end{bmatrix}\in\mathbb{R}^{n\times
T}\,,$ (23b)
$\displaystyle\tilde{X}_{1}:=\begin{bmatrix}\dot{\tilde{x}}_{0}&\dot{\tilde{x}}_{1}&\ldots&\dot{\tilde{x}}_{T-1}\end{bmatrix}\in\mathbb{R}^{n\times
T}\,,$ (23c)
$\displaystyle\tilde{Z}_{0}:=\begin{bmatrix}\tilde{x}_{0}&\tilde{x}_{1}&\ldots&\tilde{x}_{T-1}\\\\[1.42262pt]
Q(\tilde{x}_{0})&Q(\tilde{x}_{1})&\ldots&Q(\tilde{x}_{T-1})\end{bmatrix}\in\mathbb{R}^{(n+2)\times
T}.$ (23d)

These matrices satisfy the identity
$\tilde{X}_{1}=A\tilde{X}_{0}+B\tilde{U}_{0}+R_{0}$, where
$R_{0}:=\begin{bmatrix}r_{0}&r_{1}&\cdots&r_{T-1}\end{bmatrix}\in\mathbb{R}^{n\times
T}$ is the (unknown) data matrix of samples of the reminder $r$. Under the
control law $\tilde{u}=K\tilde{x}$, the data-based representation of the
closed-loop dynamics is

$\dot{\tilde{x}}=(\tilde{X}_{1}-R_{0})G{\tilde{x}}+r(\tilde{x},\tilde{u}).$
(24)

Let $\mathcal{R}:=\\{R\in\mathbb{R}^{n\times T}\colon
RR^{\top}\preceq\Delta\Delta^{\top}\\}$ where $\Delta\in\mathbb{R}^{n\times
p}$ is chosen by the designer, and assume $R_{0}\in\mathcal{R}$. Similar to
(17), a simple approach to obtain a stabilizing controller $K$ is to search
for matrices $P_{1}\in\mathbb{S}^{n\times n},Y_{1}\in\mathbb{R}^{T\times n}$,
and positive scalars $\alpha$ and $\mu$ satisfying the following program [38]:

$\displaystyle\text{\eqref{eq:SDP0}, \eqref{eq:SDP1}, \eqref{eq:SDP4}},$ (25a)
$\displaystyle\begin{bmatrix}X_{1}Y_{1}+(X_{1}Y_{1})^{\top}+\alpha
I_{n}+\mu\Delta\Delta^{\top}&Y_{1}^{\top}\\\ Y_{1}&-\mu
I_{T}\end{bmatrix}\preceq 0.$ (25b)

If SDP (25) with $X_{1}$ replaced by $\tilde{X}_{1}$ and $Z_{0}$ replaced by
$\tilde{Z}_{0}$ is feasible then $\tilde{u}=K\tilde{x}$ with
$K=\tilde{U}_{0}Y_{1}P_{1}^{-1}$ locally stabilizes the origin of the closed-
loop system (24). We collect $T=10$ samples from system (19). In this example,
the reminder term is
$r(\tilde{x},\tilde{u})=\begin{bmatrix}0&1&0&0\end{bmatrix}^{\top}\overline{r}(\tilde{x})$
with
$\overline{r}(\tilde{x}):=-\frac{mgd\cos{x_{*}}_{1}}{J_{2}}(\cos{\tilde{x}}_{1}-1)+\frac{mgd\sin{x_{*}}_{1}}{J_{2}}(\sin{\tilde{x}}_{1}-{\tilde{x}}_{1})$.
Let
$\delta(\tilde{x}):=4|\cos{\tilde{x}}_{1}-1|+2|\sin{\tilde{x}}_{1}-{\tilde{x}}_{1}|$,
which over-approximates $\overline{r}(\tilde{x})$ by more than 100%. Then, we
set $\Delta=\sqrt{T}\text{diag}(0,c,0,0)$, with $c$ equal to the maximum of
$\delta(\tilde{x})$ over the experimental data, such that
$R_{0}\in\mathcal{R}$ is satisfied. Finally, we set
$\Delta=\text{diag}(0,0.0145,0,0)$. Then the SDP (25) is feasible and returns
the controller
$K=\begin{bmatrix}2.3662&-0.2030&-10.7730&-1.4698\end{bmatrix}$. For this
controller, we numerically determine the set
$\mathcal{H}=\\{\tilde{x}:\dot{V}(\tilde{x})<0\\}$, with
$\dot{V}(\tilde{x}):=2\tilde{x}^{\top}P_{1}^{-1}\dot{\tilde{x}}$, over which
the Lyapunov function $V(\tilde{x})=\tilde{x}^{\top}P_{1}^{-1}\tilde{x}$
decreases and any sub-level set
$\mathcal{R}_{\gamma}:=\\{\tilde{x}:V(\tilde{x})\leq\gamma\\}$ of $V$ with
$\gamma>0$ contained in $\mathcal{H}\cup\\{0\\}$ gives an estimate of the ROA
for the closed-loop system. Planar projections of the set $\mathcal{H}$ and of
a sub-level set of $V$ are displayed in Figure 1. Hence, the controller
$u=K(x-x_{*})+u_{*}$ renders $(x_{*},u_{*})$ a locally asymptotically stable
equilibrium point for system (19).

Figure 1: The grey set represents the projection of the set $\mathcal{H}$ onto
the plane $\\{\tilde{x}:\tilde{x}_{3}=\tilde{x}_{4}=0\\}$. The blue set is the
projection onto the same plane of a Lyapunov sub-level set
$\mathcal{R}_{\gamma}$ contained in $\mathcal{H}$; here,
$P_{1}^{-1}=\left[\begin{smallmatrix}4.2206&0.2028&-2.7615&-0.1858\\\
0.2028&1.1915&0.9613&0.0219\\\ -2.7615&0.9613&11.4795&0.6800\\\
-0.1858&0.0219&0.6800&0.1742\\\ \end{smallmatrix}\right]$ and $\gamma=16.3$.

##  IV  An extension to general nonlinear systems

The results of the paper can be extended to a more general class of nonlinear
systems, which contains as a special case not only the system (2) but also the
important class of input-affine systems. In fact, here we consider systems of
the form

$\dot{x}=f(x,u)$ (26)

We introduce the stacked vector of states and inputs
$\xi:=[\begin{smallmatrix}x\\\ u\end{smallmatrix}]$, and consistently with the
previous section we assume the following:

###### Assumption 2

A continuously differentiable function
$\mathcal{Z}:\mathbb{R}^{n+m}\rightarrow\mathbb{R}^{R}$ is known such that
$f(x,u)=A\mathcal{Z}(\xi)$ for some matrix $A\in\mathbb{R}^{n\times R}$.
$\Box$

Under this assumption the system dynamics (26) is written as

$\dot{x}=A\mathcal{Z}(\xi)$ (27)

We partition $\mathcal{Z}(\xi)$ into subvectors $\xi$ and
$\mathcal{Q}:\mathbb{R}^{n+m}\to\mathbb{R}^{R-n-m}$, where $\mathcal{Q}(\xi)$
collects all the entries of $\mathcal{Z}(\xi)$ that are nonlinear functions of
the extended state $\xi$, namely we consider

$\mathcal{Z}(\xi)=\begin{bmatrix}\xi\\\ \mathcal{Q}(\xi)\end{bmatrix}.$ (28)

Finally, we regard $\xi$ as the new state variable and extend the dynamics
(27) with the integral control $\dot{u}=v$, where $v\in\mathbb{R}^{m}$ is a
new control input. Other uses of the integral control will be considered in
Section VII. We obtain the overall system

$\dot{\xi}=\mathcal{A}\mathcal{Z}(\xi)+\mathcal{B}v,$ (29)

where

$\mathcal{A}:=\begin{bmatrix}\,\overline{A}&\hat{A}\\\
{0}_{m\times(n+m)}&{0}_{m\times(R-n-m)}\\\
\end{bmatrix},\quad\mathcal{B}:=\begin{bmatrix}0_{n\times m}\\\
I_{m}\end{bmatrix}$ (30)

having partitioned $A$ as
$A=\begin{bmatrix}\,\overline{A}&\hat{A}\end{bmatrix}$ with
$\overline{A}\in\mathbb{R}^{n\times(n+m)}$. We then aim at the design of the
feedback control

$v=\mathcal{K}\mathcal{Z}(\xi)$

where
$\mathcal{K}=\begin{bmatrix}\,\overline{\mathcal{K}}&\hat{\mathcal{K}}\end{bmatrix}$
is to enforce desired properties on
$\dot{\xi}=(\mathcal{A}+\mathcal{B}\mathcal{K})\mathcal{Z}(\xi)$. The design
can be pursued as in the previous section mutatis mutandis. Below we concisely
present the main result.

Collect the dataset $\\{x(k),u(k),v(k)\\}_{k=0}^{T}$ from the system (29) and
define the data matrices

$\begin{array}[]{rl}V_{0}:=&\begin{bmatrix}v(0)&v(1)&\ldots&v(T-1)\end{bmatrix}\in\mathbb{R}^{m\times
T}\\\\[5.69046pt]
\Xi_{0}:=&\begin{bmatrix}\xi(0)&\xi(1)&\ldots&\xi(T-1)\end{bmatrix}\in\mathbb{R}^{(n+m)\times
T}\\\\[5.69046pt]
\Xi_{1}:=&\begin{bmatrix}\xi(1)&\xi(2)&\ldots&\xi(T)\end{bmatrix}\in\mathbb{R}^{(n+m)\times
T}\\\\[5.69046pt]
{\footnotesize\mathcal{Z}_{0}:=}&\begin{bmatrix}\xi(0)&\xi(1)&\ldots&\xi(T-1)\\\\[1.42262pt]
\mathcal{Q}(\xi(0))&\mathcal{Q}(\xi(1))&\ldots&\mathcal{Q}(\xi(T-1))\end{bmatrix}\in\mathbb{R}^{R\times
T}\end{array}$

which satisfy the identity
$\Xi_{1}=\mathcal{A}\mathcal{Z}_{0}+\mathcal{B}V_{0}$.

###### Corollary 3

Consider the nonlinear system (26). Let Assumption 2 hold and
$\mathcal{Z}(\xi)$ be of the form (28), with $\xi=[\begin{smallmatrix}x\\\
u\end{smallmatrix}]$. Let $\mathcal{R}_{Q}\in\mathbb{R}^{(n+m)\times r}$ be a
known matrix and $\mathcal{W}\subseteq\mathbb{R}^{n}\times\mathbb{R}^{m}$ a
set such that

$\frac{\partial\mathcal{Q}}{\partial\xi}(\xi)^{\top}\frac{\partial\mathcal{Q}}{\partial\xi}(\xi)\preceq\mathcal{R}_{\mathcal{Q}}\mathcal{R}_{\mathcal{Q}}^{\top}\text{
for any $\xi\in\mathcal{W}$}$ (31)

Consider the following SDP in the decision variables
$\mathcal{P}_{1}\in\mathbb{S}^{(n+m)\times(n+m)}$,
$\mathcal{Y}_{1}\in\mathbb{R}^{T\times(n+m)}$,
$\mathcal{G}_{2}\in\mathbb{R}^{T\times(R-n-m)}$ and
$\alpha\in\mathbb{R}_{>0}$:

$\displaystyle\mathcal{P}_{1}\succ 0,$ (32a)
$\displaystyle\mathcal{Z}_{0}\mathcal{Y}_{1}=\begin{bmatrix}\mathcal{P}_{1}\\\
0_{(R-n-m)\times(n+m)}\end{bmatrix}\,,$ (32b)
$\displaystyle\begin{bmatrix}\Xi_{1}\mathcal{Y}_{1}+(\Xi_{1}\mathcal{Y}_{1})^{\top}+\alpha
I_{n+m}&\Xi_{1}\mathcal{G}_{2}&\mathcal{P}_{1}\mathcal{R}_{\mathcal{Q}}\\\
(\Xi_{1}\mathcal{G}_{2})^{\top}&-I_{R-n-m}&0\\\
\mathcal{R}_{\mathcal{Q}}^{\top}\mathcal{P}_{1}^{\top}&0&-I_{r}\end{bmatrix}\preceq
0\,,$ (32c)
$\displaystyle\mathcal{Z}_{0}\mathcal{G}_{2}=\begin{bmatrix}0_{(n+m)\times(R-n-m)}\\\
I_{R-n-m}\end{bmatrix}\,.$ (32d)

If the program is feasible then the dynamic feedback control

$\dot{u}=\mathcal{K}\mathcal{Z}(\xi)$ (33)

with

$\displaystyle\mathcal{K}=V_{0}\begin{bmatrix}\mathcal{Y}_{1}\mathcal{P}_{1}^{-1}&\mathcal{G}_{2}\end{bmatrix}$
(34)

is such that the closed-loop dynamics (1), (12) is exponentially contractive
on $\mathcal{W}$.

Proof. It is similar to the proof of Theorem 1 and is omitted. $\blacksquare$

The asymptotic properties of the closed-loop system can be established
similarly to Corollaries 1 and 2. For the sake of brevity, we do not formally
state them.

###### Example 2

Consider the dynamics of a continuous stirred tank reactor [41]

$\displaystyle\dot{x}_{1}=4.25x_{1}+x_{2}-0.25u-x_{1}u$ (35a)
$\displaystyle\dot{x}_{2}=-6.25x_{1}-2x_{2}$ (35b)

where $x_{1}$ and $x_{2}$ are the deviations from the steady-state output
temperature and concentration, respectively, and the control input $u$ is the
effect of coolant flow on the chemical reaction. Note that the open-loop
system is unstable. The control objective is to maintain the output
temperature and concentration close to their steady-state values by regulating
the flow of coolant fluid.

We introduce $\xi:=[\begin{smallmatrix}x\\\ u\end{smallmatrix}]$ and
$Q(\xi)=\xi_{1}\xi_{3}$, and then the overall system is given as

$\dot{\xi}=\mathcal{A}\mathcal{Z}(\xi)+\mathcal{B}v,$ (36)

where

$\mathcal{A}:=\begin{bmatrix}4.25&1&-0.25&-1\\\ -6.25&-2&0&0\\\
0&0&0&0\end{bmatrix},\quad\mathcal{B}:=\begin{bmatrix}0\\\ 0\\\
1\end{bmatrix}.$ (37)

Note that $\frac{\partial Q}{\partial\xi}(\xi)^{\top}\frac{\partial
Q}{\partial\xi}(\xi)=\begin{bmatrix}\xi_{3}^{2}&0&\xi_{1}\xi_{3}\\\ 0&0&0\\\
\xi_{1}\xi_{3}&0&\xi_{1}^{2}\end{bmatrix}$. Let
$\mathcal{W}=[-w,w]\times\mathbb{R}\times[-w,w]$, where $w=0.05$ and let
$\mathcal{R}_{\mathcal{Q}}=\mathcal{R}_{\mathcal{Q}}^{\top}=\begin{bmatrix}0.1&0&0\\\
0&0&0\\\ 0&0&0.1\end{bmatrix}$ such that (31) is satisfied.

We collect $T=10$ samples by running an experiment on system (36) with input
uniformly distributed in $[-0.1,0.1]$, and with an initial state within the
same interval. The SDP (32) is feasible and returns the following controller
$\mathcal{K}$ and closed-loop dynamics

$\displaystyle\mathcal{K}=\begin{bmatrix}851.2634&169.3755&-27.9383&-0.9400\end{bmatrix}$
(38a)
$\displaystyle\dot{\xi}={\scriptsize\begin{bmatrix}4.2500&1.0000&-0.2500&-1.0000\\\
-6.2500&-2.0000&0.0000&0.0000\\\
851.2634&169.3755&-27.9383&-0.9400\end{bmatrix}}\mathcal{Z}(\xi)$ (38b)

and by Corollary 3, (38b) is exponentially contractive on $\mathcal{W}$.

Figure 2: The grey set represents the projection of the set $\mathcal{H}$ onto
the plane $\\{\xi:\xi_{2}=0\\}$. The blue set is the projection onto the same
plane of a Lyapunov sub-level set $\mathcal{R}_{\gamma}$ contained in
$\mathcal{H}$; here,
$\mathcal{P}_{1}^{-1}=\left[\begin{smallmatrix}32.9160&6.5218&-1.0322\\\
6.5218&1.3114&-0.2042\\\ -1.0322&-0.2042&0.0376\end{smallmatrix}\right]$ and
$\gamma=0.06$.

Note that $\mathcal{Z}(0)=0$ and $\mathcal{W}$ is a convex set containing the
origin. By Corollary 1, any solution of (38b) initialized in any sub-level set
of $V(\xi)=\xi^{\top}\mathcal{P}_{1}^{-1}\xi$ contained in $\mathcal{W}$
uniformly exponentially converges to the origin. For this controller, we
numerically determine the set $\mathcal{H}=\\{\xi:\dot{V}(\xi)<0\\}$, with
$\dot{V}(\xi):=2\xi^{\top}\mathcal{P}_{1}^{-1}\dot{\xi}$, over which the
Lyapunov function $V(\xi)=\xi^{\top}\mathcal{P}_{1}^{-1}\xi$ computed along
the solutions of (38b) decreases and any sub-level set
$\mathcal{R}_{\gamma}:=\\{\xi:V(\xi)\leq\gamma\\}$ of $V$ with $\gamma>0$
contained in $\mathcal{H}\cup\\{0\\}$ gives an estimate of the ROA for the
closed-loop system (38b). Planar projections of the set $\mathcal{H}$ and of a
sub-level set of $V$ are displayed in Figure 2. Note that $\mathcal{W}$ is
determined by $w$, i.e., the range of $\xi_{1}$ and $\xi_{3}$, and thus we
project the set $\mathcal{H}$ onto the plane $\\{\xi:\xi_{2}=0\\}$ to observe
the Lyapunov sub-level set contained in $\mathcal{W}$. Moreover, we observe
that (32) remains feasible up to $w=0.1$, and as $w$ increases, the magnitude
of the coefficients for the linear part of the obtained controller tends to
increase. With $w=0.1$ and
$\mathcal{R}_{\mathcal{Q}}=\mathcal{R}_{\mathcal{Q}}^{\top}=\begin{bmatrix}0.2&0&0\\\
0&0&0\\\ 0&0&0.2\end{bmatrix}$ such that (31) is satisfied, the magnitude is
of order $10^{3}$ in this case.

##  V Establishing contractivity with noisy data

In this section we extend the previous results to the setup where an additive
process disturbance affects the dynamics both during the data acquisition
phase and the execution of the control task. Beside the interest per se,
establishing contractivity in the presence of disturbance and from noisy data
have interesting implications in the solution of other control tasks than
stabilization, as we will investigate later.

In the presence of process disturbances, system (1) becomes

$\dot{x}=f(x)+Bu+Ed$ (39)

which, under Assumption 1, can be written as

$\dot{x}=AZ(x)+Bu+Ed$ (40)

where $d\in\mathbb{R}^{q}$ is an unknown signal representing a process
disturbance and $E\in\mathbb{R}^{n\times q}$ is a known matrix that indicates
which parts of the dynamics are affected by $d$.

Similar to the noiseless case, we perform an experiment to collect the dataset
$\mathbb{D}$ in (4) and arrange the data into the matrices
$U_{0},X_{0},X_{1},Z_{0}$ defined in (7). Note that now these matrices are
related by the equation $X_{1}=AZ_{0}+BU_{0}+ED_{0}$, where

$D_{0}:=\begin{bmatrix}d_{0}&d_{1}&\cdots&d_{T-1}\end{bmatrix}\in\mathbb{R}^{q\times
T}$ (41)

is an unknown matrix.

A data-dependent representation of system (39) in closed-loop with the
controller $u=KZ(x)$ can also be given in the presence of disturbances. In
fact, the following holds:

###### Lemma 2

Consider any matrices $K\in\mathbb{R}^{m\times s}$, $G\in\mathbb{R}^{T\times
s}$ such that (8) holds. Let $G$ be partitioned as
$G=\begin{bmatrix}G_{1}&G_{2}\end{bmatrix}$, where
$G_{1}\in\mathbb{R}^{T\times n}$ and $G_{2}\in\mathbb{R}^{T\times(s-n)}$. Let
Assumption 1 hold. Then system (39) under the control law (12) results in the
closed-loop dynamics

$\\!\\!\\!\begin{array}[]{l}\dot{x}=(A+BK)Z(x)+Ed\\\
=(X_{1}-ED_{0})G_{1}x+(X_{1}-ED_{0})G_{2}Q(x)+Ed.~{}{\tiny\text{$\Box$}}\end{array}$
(42)

_Proof._ See [16, Lemma 2]. $\blacksquare$

To further the design and the analysis, we impose some requirements on
$D_{0}$, namely we introduce the following assumption:

###### Assumption 3

For a given $\Delta$, $D_{0}\in\mathcal{D}=\\{D\in\mathbb{R}^{q\times T}\colon
DD^{\top}\preceq\Delta\Delta^{\top}\\}$. $\Box$

This assumption on the disturbance has been shown to be flexible enough to
capture various classes of disturbances, such as disturbances satisfying
energy and pointwise bounds as well as stochastic disturbances. We refer the
reader to [4, 35, 42, 16] for details.

Arguments similar to those used to prove Theorem 1 lead to the following:

###### Theorem 2

Consider the nonlinear system (39), let Assumption 1 hold and $Z(x)$ be of the
form (3). Let $R_{Q}\in\mathbb{R}^{n\times r}$ be a known matrix and
$\mathcal{X}\subseteq\mathbb{R}^{n}$ a set such that (10) holds. Let
Assumption 3 hold. Consider the following SDP in the decision variables
$P_{1}\in\mathbb{S}^{n\times n}$, $Y_{1}\in\mathbb{R}^{T\times n}$,
$G_{2}\in\mathbb{R}^{T\times(s-n)}$, $\alpha\in\mathbb{R}_{>0}$,
$\mu\in\mathbb{R}_{>0}$:

$\displaystyle P_{1}\succ 0,$ (43a) $\displaystyle
Z_{0}Y_{1}=\begin{bmatrix}P_{1}\\\ 0_{(s-n)\times n}\end{bmatrix}\,,$ (43b)
$\displaystyle\begin{bmatrix}M(Y_{1},\alpha,\mu)&X_{1}G_{2}&P_{1}R_{Q}&Y_{1}^{\top}\\\\[2.84526pt]
(X_{1}G_{2})^{\top}&-I_{s-n}&0_{n\times r}&G_{2}^{\top}\\\
R_{Q}^{\top}P_{1}^{\top}&0_{r\times n}&-I_{r}&0_{r\times T}\\\\[2.84526pt]
Y_{1}&G_{2}&0_{T\times r}&-\mu I_{T}\end{bmatrix}\preceq 0\,,$ (43c)
$\displaystyle Z_{0}G_{2}=\begin{bmatrix}0_{n\times(s-n)}\\\
I_{s-n}\end{bmatrix}$ (43d)

where $M(Y_{1},\alpha,\mu):=X_{1}Y_{1}+(X_{1}Y_{1})^{\top}+\alpha I_{n}+\mu
E\Delta\Delta^{\top}E^{\top}$. If the program is feasible then the control law
(12) with $K$ as in (13) is such that the closed-loop dynamics (39), (12) is
exponentially contractive on $\mathcal{X}$, _i.e._ , (6) holds. $\Box$

_Proof._ If (43) is feasible, then we can define $G_{1}:=Y_{1}P_{1}^{-1}$, and
(11b), (11d) as well as the definition of $K$ in (13), imply that (8) holds.
Hence, by Lemma 2, the closed-loop dynamics $\dot{x}=(A+BK)Z(x)+Ed$ can be
written as in (42). By Schur complement, (43c) is equivalent to444To avoid
burdening the notation, in the following expressions we omit the size of the
zero matrices.

$\begin{array}[]{l}\begin{bmatrix}X_{1}Y_{1}+(X_{1}Y_{1})^{\top}+\alpha
I_{n}&X_{1}G_{2}&P_{1}R_{Q}\\\\[2.84526pt] (X_{1}G_{2})^{\top}&-I_{s-n}&0\\\
R_{Q}^{\top}P_{1}^{\top}&0&-I_{r}\\\\[2.84526pt] \end{bmatrix}\\\
+\mu\begin{bmatrix}-E\\\ 0\\\
0\end{bmatrix}\Delta\Delta^{\top}\begin{bmatrix}-E\\\ 0\\\
0\end{bmatrix}^{\top}+\mu^{-1}\begin{bmatrix}Y_{1}^{\top}\\\ G_{2}^{\top}\\\
0\end{bmatrix}\begin{bmatrix}Y_{1}^{\top}\\\ G_{2}^{\top}\\\
0\end{bmatrix}^{\top}\preceq 0\end{array}$

which is compactly written as $H+\mu
L^{\top}\Delta\Delta^{\top}L+\mu^{-1}JJ^{\top}\preceq 0$, with obvious
definition of $H,J,L$. Trivially, $L\neq 0$, whereas
$\Delta\Delta^{\top}\succeq 0$. Moreover, $J\neq 0$, for otherwise $Y_{1}=0$,
which would contradict (43b). Hence, by the nonstrict Petersen’s lemma [35,
Fact 2], we have that $H+JD^{\top}L+L^{\top}DJ^{\top}\preceq 0$ for all
$D\in\mathcal{D}$, _i.e._ ,

$\begin{array}[]{l}\begin{bmatrix}X_{1}Y_{1}+(X_{1}Y_{1})^{\top}+\alpha
I_{n}&X_{1}G_{2}&P_{1}R_{Q}\\\\[2.84526pt] (X_{1}G_{2})^{\top}&-I_{s-n}&0\\\
R_{Q}^{\top}P_{1}^{\top}&0&-I_{r}\\\\[2.84526pt] \end{bmatrix}\\\
+\begin{bmatrix}-E\\\ 0\\\
0\end{bmatrix}D\begin{bmatrix}Y_{1}&G_{2}&0\end{bmatrix}\\\\[14.22636pt]
+\begin{bmatrix}Y_{1}^{\top}\\\ G_{2}^{\top}\\\
0\end{bmatrix}D^{\top}\begin{bmatrix}-E^{\top}&0&0\end{bmatrix}\preceq
0,\\!\forall D\in\mathcal{D},\end{array}$

which can be rewritten as

$\\!\\!\\!\\!\begin{array}[]{c}{\scriptsize\begin{bmatrix}(X_{1}-ED)Y_{1}+Y_{1}^{\top}(X_{1}-ED)^{\top}+\alpha
I_{n}&(X_{1}-ED)G_{2}&P_{1}R_{Q}\\\\[2.84526pt]
G_{2}^{\top}(X_{1}-ED)^{\top}&-I_{s-n}&0\\\
R_{Q}^{\top}P_{1}^{\top}&0&-I_{r}\\\\[2.84526pt]
\end{bmatrix}}\\!\\!\preceq\\!0,\\\ \forall D\in\mathcal{D}.\end{array}$

Observe now that this inequality is the same as (11c) provided that
$X_{1}Y_{1}$ and $X_{1}G_{2}$ in the latter are replaced by $(X_{1}-ED)Y_{1}$
and $(X_{1}-ED)G_{2}$. Hence, retracing the arguments of the proof of Theorem
1, one arrives at showing that

$\begin{array}[]{l}\\!\\!\\!P_{1}^{-1}(X_{1}-ED)G\frac{\partial Z}{\partial
x}+\frac{\partial Z}{\partial
x}^{\top}G^{\top}(X_{1}-ED)^{\top}P_{1}^{-1}\preceq-\beta P_{1}^{-1}\\\ \hskip
113.81102pt\text{ for all $x\in\mathcal{X}$ and
$D\in\mathcal{D}$}{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}.}\end{array}$

Since $(X_{1}-ED_{0})G\frac{\partial Z}{\partial x}$ is the Jacobian of the
closed-loop vector field in (42) and $D_{0}\in\mathcal{D}$ by Assumption 3,
the thesis is proven. $\blacksquare$

To state the asymptotic properties of the closed-loop system, we need to
impose assumptions on the entire signal $d$ and not only on the samples
collected during the experiment. Thus, we introduce the following:

###### Assumption 4

$d$ is a continuous function of time that, for some known $\delta>0$ and for
all $t\in\mathbb{R}$, satisfies $|d(t)|\leq\delta$. $\Box$

A disturbance $d$ that satisfies Assumption 4 also satisfies Assumption 3 with
$\Delta:=\delta\sqrt{T}I_{q}$.

The following result underscores that, by enforcing (6), one obtains that the
closed-loop system is convergent (see Definition 3 in the Appendix) and as
such it enjoys certain properties in the presence of forcing signals.

###### Corollary 4

Let the conditions of Theorem 2 hold, with Assumption 3 replaced by Assumption
4 and $\Delta$ in (43c) replaced by $\delta\sqrt{T}I_{q}$, and where
$\mathcal{X}=\mathbb{R}^{n}$ and there exists $\overline{x}\in\mathbb{R}^{n}$
such that $|Z(\overline{x})|<\infty$. Then system $\dot{x}=(A+BK)Z(x)+Ed$ is
convergent. Moreover, if $d(t)$ is periodic of period $T$ then $x_{*}(t)$, the
unique solution that is defined and bounded for all $t\in(-\infty,+\infty)$
and globally uniformly exponentially stable, is periodic of period $T$. If
$d(t)$ is constant, then $x_{*}(t)=x_{*}$.

_Proof._ Under the stated conditions, the Jacobian of the right-hand side of
the closed-loop dynamics $\dot{x}=(A+BK)Z(x)+Ed$, i.e., $(A+BK)\frac{\partial
Z}{\partial x}$, satisfies (6). The thesis is implied by Theorem 6 in the
Appendix. $\blacksquare$

Corollary 4 gives conditions under which all the solutions of the closed-loop
system designed from data converge to a unique solution, which is periodic or
constant depending on the nature of the disturbance. In fact, if the
disturbance is a linear combination of constant and sinusoidal signals of
known frequencies, then Corollary 4 holds with a simpler form of the SDP, as
we show in the next section.

##  VI Data generated under disturbances of known frequencies

In this subsection, we investigate the case in which the disturbance
perturbing the system is a linear combination of constant and sinusoidal
signals of known frequencies. Formally, we replace Assumption 4 with the
following one:

###### Assumption 5

$d$ is a function of time solution of the autonomous system

$\dot{w}=\Psi w,\quad d=\Gamma w$ (44)

where $w={\rm
col}(w_{1},\ldots,w_{\sigma_{1}},w_{\sigma_{1}+1},\ldots,w_{\sigma_{1}+\sigma_{2}})$,

$\Psi={\rm block.diag}(\left[\begin{smallmatrix}0&\psi_{1}\\\
-\psi_{1}&0\end{smallmatrix}\right],\ldots,\left[\begin{smallmatrix}0&\psi_{\sigma_{1}}\\\
-\psi_{\sigma_{1}}&0\end{smallmatrix}\right],\underbrace{0,\ldots,0}_{\text{$\sigma_{2}$
times}}),$

with $\psi_{1},\ldots,\psi_{\sigma_{1}}\in\mathbb{R}_{>0}$ known frequencies,
and $\Gamma\in\mathbb{R}^{q\times(2\sigma_{1}+\sigma_{2})}$ an unknown matrix.
$\Box$

We denote by

$w(0)=\begin{bmatrix}w_{1}(0)^{\top}\\!\\!\\!\\!&\ldots&\\!\\!\\!\\!w_{\sigma_{1}}(0)^{\top}&\\!\\!\\!\\!w_{\sigma_{1}+1}(0)&\\!\\!\ldots&\\!\\!w_{\sigma_{1}+\sigma_{2}}(0)\end{bmatrix}^{\top}$

the initial condition of the system (44) generating the disturbance during the
data collection phase, where
$w_{i}(0)=[w_{i1}(0)\;w_{i2}(0)]^{\top}\in\mathbb{R}^{2}$ for
$i=1,\ldots,\sigma_{1}$ and $w_{i}(0)\in\mathbb{R}$ for
$i=\sigma_{1}+1,\ldots,\sigma_{1}+\sigma_{2}$. Under Assumption 5 the matrix
of disturbance samples $D_{0}$ in (41) becomes

$D_{0}=\Gamma\begin{bmatrix}w_{1}(t_{0})&\ldots&w_{1}(t_{T-1})\\\
\vdots&\ddots&\vdots\\\
w_{\sigma_{1}+\sigma_{2}}(t_{0})&\ldots&w_{\sigma_{1}+\sigma_{2}}(t_{T-1})\\\
\end{bmatrix}$

where, for $i=1,2,\ldots,\sigma_{1}$,

$\begin{array}[]{rl}&\begin{bmatrix}w_{i}(t_{0})&\ldots&w_{i}(t_{T-1})\end{bmatrix}\\\\[8.53581pt]
=&\underbrace{\begin{bmatrix}w_{i1}(0)&w_{i2}(0)\\\
w_{i2}(0)&-w_{i1}(0)\end{bmatrix}}_{=:L_{i}}\underbrace{\begin{bmatrix}\cos(\psi_{i}t_{0})&\ldots&\cos(\psi_{i}t_{T-1})\\\
\sin(\psi_{i}t_{0})&\ldots&\sin(\psi_{i}t_{T-1})\\\
\end{bmatrix}}_{=:M_{i}}\end{array}$

and, for $i=\sigma_{1}+1,\sigma_{1}+2,\ldots,\sigma_{1}+\sigma_{2}$,

$\begin{bmatrix}w_{i}(t_{0})&\ldots&w_{i}(t_{T-1})\end{bmatrix}=\underbrace{w_{i}(0)}_{=:L_{i}}\underbrace{\begin{bmatrix}1&\ldots&1\end{bmatrix}}_{=:M_{i}}$

Set

$\begin{array}[]{rl}L:=&{\rm
block.diag}(L_{1},\ldots,L_{\sigma_{1}+\sigma_{2}})\\\ M:=&{\rm
block.col}(M_{1},\ldots,M_{\sigma_{1}+\sigma_{2}})\\\ N:=&\Gamma L\end{array}$
(45)

and note that $M$ is known while $N$ is unknown. Bearing in mind the system
(40), and the expression of $D_{0}$ given above, namely $D_{0}=\Gamma LM=NM$,
we realize that the matrices of data satisfy

$X_{1}=AZ_{0}+BU_{0}+ENM.$

Hence, we have the following interesting data-dependent representation of the
closed-loop system

###### Lemma 3

Let Assumption 1 and 5 hold. Consider any matrices $K\in\mathbb{R}^{m\times
s}$, $G\in\mathbb{R}^{T\times s}$ such that

$\begin{bmatrix}K\\\ I_{s}\\\ 0_{(2\sigma_{1}+\sigma_{2})\times
s}\end{bmatrix}=\begin{bmatrix}U_{0}\\\ Z_{0}\\\ M\end{bmatrix}G$ (46)

holds. Let $G$ be partitioned as $G=\begin{bmatrix}G_{1}&G_{2}\end{bmatrix}$,
where $G_{1}\in\mathbb{R}^{T\times n}$ and
$G_{2}\in\mathbb{R}^{T\times(s-n)}$. Then system (39) under the control law
(12) results in the closed-loop dynamics

$\\!\\!\\!\\!\\!\\!\begin{array}[]{rl}\dot{x}=&(A+BK)Z(x)+Ed\\\
=&X_{1}G_{1}x+X_{1}G_{2}Q(x)+Ed.\end{array}$ (47)

$\Box$

_Proof._ Left-multiply both sides of (46) by
$\left[\begin{smallmatrix}B&A&EN\end{smallmatrix}\right]$ and obtain

$A+BK=(AZ_{0}+BU_{0}+ENM)G=X_{1}G.$

This ends the proof. $\blacksquare$

The remarkable feature of this result is that, for systems affected by
disturbances generated by (44), a data-dependent representation that is
independent of the unknown matrix $D_{0}$ is derived, provided that the
additional condition $0_{q\times s}=MG$ holds. The result allows us to derive
contractivity and asymptotic properties of the closed-loop system similar to
those obtained in the case of noise-free data. These results are introduced
below.

###### Theorem 3

Consider the nonlinear system (39), let Assumption 1 hold and $Z(x)$ be of the
form (3). Let $R_{Q}\in\mathbb{R}^{n\times r}$ be a known matrix and
$\mathcal{X}\subseteq\mathbb{R}^{n}$ a set such that (10) holds. Let
Assumption 5 hold. Consider the following SDP in the decision variables
$P_{1}\in\mathbb{S}^{n\times n}$, $Y_{1}\in\mathbb{R}^{T\times n}$,
$G_{2}\in\mathbb{R}^{T\times(s-n)}$, $\alpha\in\mathbb{R}_{>0}$:

${\rm\eqref{eq:SDP}},\;0_{(2\sigma_{1}+\sigma_{2})\times
s}=M\begin{bmatrix}Y_{1}&G_{2}\end{bmatrix},$ (48)

where $M$ is the matrix defined in (45). If the program is feasible then the
control law (12) with $K$ as in (13) is such that the closed-loop dynamics
(39), (12) is exponentially contractive on $\mathcal{X}$, _i.e._ , (6) holds.
$\Box$

_Proof._ If (48) is feasible then (11) is feasible. Then we can define
$G_{1}:=Y_{1}P_{1}^{-1}$ and note that (11b), (11d), the definition of $K$ in
(13) and the constraint $0_{q\times
s}=M\begin{bmatrix}Y_{1}&G_{2}\end{bmatrix}$ give (46). Hence, by Lemma 3, the
closed-loop dynamics $\dot{x}=(A+BK)Z(x)+Ed$ can be written as
$\dot{x}=Mx+NQ(x)+Ed$, with $M=X_{1}G_{1}$ and $N=X_{1}G_{2}$. By (11c) and
(10), the proof of Theorem 1 shows that (6) holds. $\blacksquare$

The proof of the following result is identical to the proof of Corollary 4 and
is omitted.

###### Corollary 5

Let the conditions of Theorem 3 with $\mathcal{X}=\mathbb{R}^{n}$ and let
$\overline{x}\in\mathbb{R}^{n}$ be such that $|Z(\overline{x})|<\infty$. Then
system $\dot{x}=(A+BK)Z(x)+Ed$ is convergent. Moreover, if $d(t)$ is periodic
of period $T$ then $x_{*}(t)$ is periodic of period $T$. If $d(t)$ is
constant, then $x_{*}(t)=x_{*}$. $\Box$

The requirement for $d(t)$ to be periodic is needed since the sum of
sinusoidal signals of different frequencies might not be periodic. On the
other hand, if $d(t)$ is a constant disturbance, then the result above
guarantees convergence to the unique equilibrium under conditions that are
simpler than the ones required by Corollary 4.

##  VII Data driven integral control for nonlinear systems

In this section we discuss how establishing contractivity via data paves the
way for the data-driven design of an integral controller for nonlinear systems
of the form

$\begin{array}[]{rl}\dot{x}=&f(x)+Bu+Ed\\\ y=&h(x)\\\ e=&y-r\end{array}$ (49)

where $d\in\mathbb{R}^{q},r\in\mathbb{R}^{p}$ are a constant disturbance and a
constant reference signal, $y\in\mathbb{R}^{p}$ is the regulated output and
$e\in\mathbb{R}^{p}$ is the tracking error. We assume that $y$ is also
available for measurements, in addition to the state $x$. We are interested in
the problem of designing the integral controller

$\begin{array}[]{rl}\dot{\xi}=&e\\\ u=&k(x,\xi)\end{array}$ (50)

such that the solutions $x(t),\xi(t)$ of the closed-loop system

$\begin{array}[]{rl}\dot{x}=&f(x)+Bk(x,\xi)+Ed\\\ \dot{\xi}=&e\\\
e=&h(x)-r\end{array}$ (51)

are bounded for all $t\geq 0$ and satisfy $e(t)\to 0$ as $t\to+\infty$.

The following assumption on the system replaces Assumption 1.

###### Assumption 6

A continuously differentiable vector-valued function
$Z:\mathbb{R}^{n}\rightarrow\mathbb{R}^{s}$ is known such that $f(x)=AZ(x)$
and $h(x)=CZ(x)$ for some matrices $A\in\mathbb{R}^{n\times s}$,
$C\in\mathbb{R}^{p\times s}$. $\Box$

Hence, the system (49) considered in this section takes the form

$\begin{array}[]{rl}\dot{x}=&AZ(x)+Bu+Ed\\\ e=&CZ(x)-r\end{array}$ (52)

We let $Z(x)$ be of the form (3) and partition $A,C$ accordingly as

$\begin{bmatrix}A\\\ C\end{bmatrix}=\begin{bmatrix}\,\overline{A}&\hat{A}\\\
\,\overline{C}&\hat{C}\end{bmatrix}$

After stacking $Z(x)$ and the controller state variable $\xi$ in the vector

$\mathcal{Z}(x,\xi)=\begin{bmatrix}x\\\ \xi\\\ Q(x)\end{bmatrix}$ (53)

the system (49) extended with the integral action $\dot{\xi}=e$ can be written
as

$\begin{array}[]{rl}\begin{bmatrix}\dot{x}\\\
\dot{\xi}\end{bmatrix}=&\mathcal{A}\mathcal{Z}(x,\xi)+\mathcal{B}u+\mathcal{E}d-\mathcal{I}r\end{array}$
(54)

where

$\begin{array}[]{c}\mathcal{A}=\begin{bmatrix}\,\overline{A}&0_{n\times
p}&\hat{A}\\\ \,\overline{C}&0_{p\times
p}&\hat{C}\end{bmatrix},\;\mathcal{B}=\begin{bmatrix}B\\\ 0_{p\times
m}\end{bmatrix}\\\ \mathcal{E}=\begin{bmatrix}E\\\ 0_{p\times
q}\end{bmatrix},\;\mathcal{I}=\begin{bmatrix}0_{n\times p}\\\
I_{p}\end{bmatrix}.\end{array}$

The constant disturbance $d$ is generated by the system (44) with
$\Sigma=0_{\sigma\times\sigma}$, and $\Gamma\in\mathbb{R}^{q\times\sigma}$
unknown.

We collect data

$\mathbb{D}:=\left\\{(x_{i},u_{i},y_{i},\dot{x}_{i},\xi_{i})\right\\}_{i=0}^{T-1}$
(55)

from systems (49), (50) through an experiment, where, as before,
$x_{i}:=x(t_{i})$, etc., and define, in addition to the data matrix $U_{0}$
defined in (7), the matrices

$\begin{array}[]{rl}\mathcal{Z}_{1}:=&\begin{bmatrix}\dot{x}(t_{0})&\dot{x}(t_{1})&\ldots&\dot{x}(t_{T-1})\\\
y(t_{0})&y(t_{1})&\ldots&y(t_{T-1})\\\ \end{bmatrix}\\\\[11.38109pt]
\mathcal{Z}_{0}:=&\begin{bmatrix}x(t_{0})&x(t_{1})&\ldots&x(t_{T-1})\\\
\xi(t_{0})&\xi(t_{1})&\ldots&\xi(t_{T-1})\\\
Q(x(t_{0}))&Q(x(t_{1}))&\ldots&Q(x(t_{T-1}))\\\ \end{bmatrix}\end{array}$

Bearing in mind the analysis of Section VI, the matrix of disturbance samples
$D_{0}$ in (41) is equal to $NM$ where $N=\Gamma L$,

$L={\rm
diag}(w_{1}(0),\ldots,w_{\sigma}(0))\in\mathbb{R}^{\sigma\times\sigma}$ (56)
$M=\left[\begin{smallmatrix}\mathds{1}_{1\times T}\\\ \vdots\\\
\mathds{1}_{1\times T}\end{smallmatrix}\right]\in\mathbb{R}^{\sigma\times T}$
(57)

and $\mathds{1}_{1\times T}$ denotes a $1\times T$ matrix of all ones. The
matrices of data satisfy the identity

$\mathcal{Z}_{1}=\mathcal{A}\mathcal{Z}_{0}+\mathcal{B}U_{0}+\mathcal{E}NM$

Note that to derive this identity, we are using the relation

$y(t_{i})=e(t_{i})+r=CZ(x(t_{i})),\;i=0,1,\ldots,T-1.$

We can give a data-dependent representation of system (54) in feedback with

$\begin{array}[]{rl}u=k(x,\xi):=\begin{bmatrix}\,\overline{K}&\widetilde{K}&\hat{K}\end{bmatrix}\begin{bmatrix}x\\\
\xi\\\ Q(x)\end{bmatrix}=:\mathcal{K}\mathcal{Z}(x,\xi)\end{array}$ (58)

###### Lemma 4

Consider any matrices $\mathcal{K}\in\mathbb{R}^{m\times(s+p)}$,
$\mathcal{G}\in\mathbb{R}^{T\times(s+p)}$ such that

$\begin{bmatrix}\mathcal{K}\\\ I_{s+p}\\\
0_{\sigma\times(s+p)}\end{bmatrix}=\begin{bmatrix}U_{0}\\\ \mathcal{Z}_{0}\\\
M\end{bmatrix}\mathcal{G}$ (59)

holds. Let $\mathcal{G}$ be partitioned as
$\mathcal{G}=\begin{bmatrix}\mathcal{G}_{1}&\mathcal{G}_{2}\end{bmatrix}$,
where $\mathcal{G}_{1}\in\mathbb{R}^{T\times(n+p)}$ and
$\mathcal{G}_{2}\in\mathbb{R}^{T\times(s-n)}$. Then, the dynamics of the
system (49), (50), where $k(x,\xi)$ is as in (58), results in the closed-loop
dynamics

$\begin{array}[]{l}\begin{bmatrix}\dot{x}\\\
\dot{\xi}\end{bmatrix}=(\mathcal{A}+\mathcal{B}\mathcal{K})\mathcal{Z}(x,\xi)+\mathcal{E}d-\mathcal{I}r\\\
=\mathcal{Z}_{1}\mathcal{G}_{1}\begin{bmatrix}x\\\
\xi\end{bmatrix}+\mathcal{Z}_{1}\mathcal{G}_{2}Q(x)+\mathcal{E}d-\mathcal{I}r.\end{array}$
(60)

_Proof._ The proof is similar to the proof of Lemma 3 and is omitted.
$\blacksquare$

The result below shows contractivity of the closed-loop dynamics.

###### Theorem 4

Consider the nonlinear system (49), let Assumption 6 hold and
$\mathcal{Z}(x,\xi)$ be of the form (53). Let $R_{Q}\in\mathbb{R}^{n\times r}$
be a known matrix and $\mathcal{X}\subseteq\mathbb{R}^{n}$ a set such that
(10) holds. Set $\mathcal{R}_{Q}:=\left[\begin{smallmatrix}R_{Q}\\\ 0_{p\times
r}\end{smallmatrix}\right]$. Consider the following SDP in the decision
variables $\mathcal{P}_{1}\in\mathbb{S}^{(n+p)\times(n+p)}$,
$\mathcal{Y}_{1}\in\mathbb{R}^{T\times(n+p)}$,
$\mathcal{G}_{2}\in\mathbb{R}^{T\times(s-n)}$, $\alpha\in\mathbb{R}_{>0}$:

$\displaystyle\mathcal{P}_{1}\succ 0,$ (61a)
$\displaystyle\mathcal{Z}_{0}\mathcal{Y}_{1}=\begin{bmatrix}\mathcal{P}_{1}\\\
0_{(s-n)\times(n+p)}\end{bmatrix}\,,$ (61b)
$\displaystyle\begin{bmatrix}\mathcal{Z}_{1}\mathcal{Y}_{1}+(\mathcal{Z}_{1}\mathcal{Y}_{1})^{\top}+\alpha
I_{n+p}&\mathcal{Z}_{1}\mathcal{G}_{2}&\mathcal{P}_{1}\mathcal{R}_{Q}\\\\[2.84526pt]
(\mathcal{Z}_{1}\mathcal{G}_{2})^{\top}&-I_{s-n}&0\\\
\mathcal{R}_{Q}^{\top}P_{1}^{\top}&0&-I_{r}\end{bmatrix}\preceq 0\,,$ (61c)
$\displaystyle\mathcal{Z}_{0}\mathcal{G}_{2}=\begin{bmatrix}0_{(n+p)\times(s-n)}\\\
I_{s-n}\end{bmatrix},$ (61d) $\displaystyle\mathds{1}_{1\times
T}\begin{bmatrix}\mathcal{Y}_{1}&\mathcal{G}_{2}\end{bmatrix}=0_{1\times(s+p)}\,.$
(61e)

If the program is feasible then the control law (58) with $\mathcal{K}$
defined as

$\displaystyle\mathcal{K}=U_{0}\begin{bmatrix}\mathcal{Y}_{1}\mathcal{P}_{1}^{-1}&\mathcal{G}_{2}\end{bmatrix}$
(62)

is such that the closed-loop dynamics (49), (50) is exponentially contractive
on $\mathcal{X}\times\mathbb{R}^{p}$, _i.e._ ,

$\begin{array}[]{l}\exists\beta>0\text{ such that for all
}(x,\xi)\in\mathcal{X}\times\mathbb{R}^{p}\\\
\left[(\mathcal{A}+\mathcal{B}\mathcal{K})\frac{\partial\mathcal{Z}}{\partial(x,\xi)}\right]^{\top}\\!\\!\\!\mathcal{P}_{1}^{-1}+\mathcal{P}_{1}^{-1}\\!\\!\left[(\mathcal{A}+\mathcal{B}\mathcal{K})\frac{\partial\mathcal{Z}}{\partial(x,\xi)}\right]\\!\\!\preceq\\!-\beta\mathcal{P}_{1}^{-1}.\end{array}$

$\Box$

_P_ roof. If (61) is feasible and we set
$\mathcal{G}_{1}=\mathcal{Y}_{1}\mathcal{P}_{1}^{-1}$, then it holds that

$\begin{array}[]{l}\mathcal{P}_{1}^{-1}\mathcal{Z}_{1}\mathcal{G}_{1}+\mathcal{G}_{1}^{\top}\mathcal{Z}_{1}^{\top}\mathcal{P}_{1}^{-1}+\alpha\mathcal{P}_{1}^{-2}+\mathcal{R}_{Q}\mathcal{R}_{Q}^{\top}\\\\[2.84526pt]
\hskip
14.22636pt+\mathcal{P}_{1}^{-1}\mathcal{Z}_{1}\mathcal{G}_{2}\mathcal{G}_{2}^{\top}\mathcal{Z}_{1}\mathcal{P}_{1}^{-1}\preceq
0\end{array}$ (63)

If $\mathcal{Z}_{1}\mathcal{G}_{2}=0$, then
$(\mathcal{A}+\mathcal{B}\mathcal{K})\frac{\partial\mathcal{Z}}{\partial(x,\xi)}=\mathcal{Z}_{1}\mathcal{G}_{1}\left[\begin{smallmatrix}x\\\
\xi\end{smallmatrix}\right]$, and (63) implies the exponential contractivity
property with $\beta:=\alpha\lambda_{\min}(P_{1}^{-1})$, as claimed. If
$\mathcal{Z}_{1}\mathcal{G}_{2}\neq 0$, (63) implies that

$\begin{array}[]{l}\mathcal{P}_{1}^{-1}\mathcal{Z}_{1}\mathcal{G}_{1}+\mathcal{G}_{1}^{\top}\mathcal{Z}_{1}^{\top}\mathcal{P}_{1}^{-1}+\alpha\mathcal{P}_{1}^{-2}\\\
\hskip
14.22636pt+\mathcal{P}_{1}^{-1}\mathcal{Z}_{1}\mathcal{G}_{2}\mathcal{R}^{\top}+\mathcal{R}\mathcal{G}_{2}^{\top}\mathcal{Z}_{1}^{\top}\mathcal{P}_{1}^{-1}\preceq
0,\\\\[5.69054pt] \text{ for all
}\mathcal{R}\in\\{\mathcal{R}\colon\mathcal{R}\mathcal{R}^{\top}\preceq\mathcal{R}_{Q}\mathcal{R}_{Q}^{\top}\\}.\end{array}$
(64)

By (10), for any $(x,\xi)\in\mathcal{X}\times\mathbb{R}^{p}$, we have

$\begin{array}[]{l}\displaystyle\frac{\partial
Q}{\partial(x,\xi)}^{\top}\frac{\partial
Q}{\partial(x,\xi)}=\begin{bmatrix}\frac{\partial Q}{\partial x}^{\top}\\\
\frac{\partial
Q}{\partial\xi}^{\top}\end{bmatrix}\begin{bmatrix}\frac{\partial Q}{\partial
x}&\frac{\partial Q}{\partial\xi}\end{bmatrix}\preceq\\\ \hskip
99.58464pt\begin{bmatrix}R_{Q}R_{Q}^{\top}&0\\\
0&0\end{bmatrix}=\mathcal{R}_{Q}\mathcal{R}_{Q}^{\top}\end{array}$

that is, $\frac{\partial
Q}{\partial(x,\xi)}^{\top}\in\\{\mathcal{R}\colon\mathcal{R}\mathcal{R}^{\top}\preceq\mathcal{R}_{Q}\mathcal{R}_{Q}^{\top}\\}$.
Therefore

$\begin{array}[]{l}\mathcal{P}_{1}^{-1}\mathcal{Z}_{1}\mathcal{G}_{1}+\mathcal{G}_{1}^{\top}\mathcal{Z}_{1}^{\top}\mathcal{P}_{1}^{-1}+\alpha\mathcal{P}_{1}^{-2}\\\
+\mathcal{P}_{1}^{-1}\mathcal{Z}_{1}\mathcal{G}_{2}\frac{\partial
Q}{\partial(x,\xi)}+\frac{\partial
Q}{\partial(x,\xi)}^{\top}\mathcal{G}_{2}^{\top}\mathcal{Z}_{1}^{\top}\mathcal{P}_{1}^{-1}\preceq
0\end{array}$

that is, recalling that $\beta:=\alpha\lambda_{\min}(P_{1}^{-1})$,

$\begin{array}[]{l}\mathcal{P}_{1}^{-1}\mathcal{Z}_{1}\mathcal{G}\frac{\partial\mathcal{Z}}{\partial(x,\xi)}+\frac{\partial\mathcal{Z}}{\partial(x,\xi)}^{\top}G^{\top}\mathcal{Z}_{1}^{\top}\mathcal{P}_{1}^{-1}\preceq-\beta\mathcal{P}_{1}^{-1}\end{array}$

for all $(x,\xi)\in\mathcal{X}\times\mathbb{R}^{p}$. Bearing in mind that, by
(61a), (61b), (61d), (61e), (62) and Lemma 4,

$(\mathcal{A}+\mathcal{B}\mathcal{K})\mathcal{Z}(x,\xi)=\mathcal{Z}_{1}\mathcal{G}_{1}\begin{bmatrix}x\\\
\xi\end{bmatrix}+\mathcal{Z}_{1}\mathcal{G}_{2}Q(x)$

the thesis is complete. $\blacksquare$

The application to the data-driven integral control problem is immediate.

###### Corollary 6

Let the conditions of Theorem 4 with $\mathcal{X}=\mathbb{R}^{n}$ hold and let
$\overline{x}\in\mathbb{R}^{n}$ be such that $|Z(\overline{x})|<\infty$. Then
the control law (58) with $\mathcal{K}$ defined as in (62) is such that the
solutions of the closed-loop dynamics (49), (50) are bounded for all $t\geq 0$
and satisfy $e(t)\to 0$ as $t\to+\infty$.

Proof. By Theorem 4, the closed-loop dynamics (49), (50), where $k(x,\xi)$ is
as in (58) and $\mathcal{K}$ is as in (62), is exponentially contractive on
$\mathbb{R}^{n}\times\mathbb{R}^{p}$. Moreover, it is time invariant. By
Theorem 6 in the Appendix, all its solutions are defined for all
$t\in\mathbb{R}_{\geq 0}$ and for all
$(x(0),\xi(0))\in\mathbb{R}^{n}\times\mathbb{R}^{p}$. There exists a unique
equilibrium $(x_{\star},\xi_{\star})$, which is globally uniformly
exponentially stable. Hence, all the solutions are bounded. Finally, since
$\dot{\xi}=e$, at the equilibrium $0=h(x_{*})-r$, hence by continuity of $h$,
$e(t)\to 0$ as $t\to+\infty$. $\blacksquare$

The result just stated gives conditions for the regulation of the output of a
nonlinear system to a constant reference. Given the nature of the result, one
would expect some conditions on the feasibility on the problem to be present
in the statement. For instance, given that the constant reference value $r$
can be any vector in $\mathbb{R}^{p}$, it would be natural to require that the
matrix $C$ appearing in the output function $y=h(x)$ has full row rank. Yet,
not such an assumption explicitly appears in the statement. We show below
that, in fact, the condition is implicitly encoded in the SDP (61). To show
this, assume by contradiction that $C$ has not full row rank and that (61) is
feasible. This implies in particular that there exists $\mathcal{Y}_{1}$ such
that
$\mathcal{Z}_{1}\mathcal{Y}_{1}+(\mathcal{Z}_{1}\mathcal{Y}_{1})^{\top}+\alpha
I_{n+p}\preceq 0$. We rewrite the matrix on the left-hand side of the
inequality as

$\begin{bmatrix}X_{1}\\\
C\mathcal{Z}_{0}\end{bmatrix}\mathcal{Y}_{1}+\mathcal{Y}_{1}^{\top}\begin{bmatrix}X_{1}^{\top}\\\
\mathcal{Z}_{0}^{\top}C^{\top}\end{bmatrix}+\alpha I_{n+p}\preceq 0$

where $X_{1}$ is defined in (7c). Let $v\neq 0$ be such that $v^{\top}C=0$ and
multiply the inequality above by
$\left[\begin{smallmatrix}0^{\top}&v^{\top}\end{smallmatrix}\right]$ on the
left and by $\left[\begin{smallmatrix}0\\\ v\end{smallmatrix}\right]$ on the
right. Then

$v^{\top}C\mathcal{Z}_{0}\mathcal{Y}_{1}v+v^{\top}\mathcal{Y}_{1}^{\top}\mathcal{Z}_{0}^{\top}C^{\top}v+\alpha|v|^{2}\leq
0$

which is a contradiction. We conclude that $C$ not fulfilling a necessary
condition for the feasibility of the regulation problem consistently yields
infeasibility of the SDP that is at the basis of the design.

Finally, before considering a numerical example, we observe that, by Corollary
7 in the Appendix, a local version of Corollary 6 can be given where
$\mathcal{X}$ is a convex subset of $\mathbb{R}^{n}$ and the condition that
there exist $\overline{x}\in\mathbb{R}^{n}$ such that
$|Z(\overline{x})|<\infty$ is replaced by the existence of an equilibrium
$(\overline{x},\overline{\xi})$ of the closed-loop system (60) such that
$\overline{x}\in{\rm int}(\mathcal{X})$. The challenge of applying this result
lies in the determination of $\overline{x}$ due to the unknown disturbance
$d$.

###### Example 3

We consider again the one-link robot arm of Example 1, here assuming that the
system is corrupted by the constant disturbance
$d=\begin{bmatrix}0.1&0.2&0.3&0.4\end{bmatrix}^{\top}$ and $E=I_{4}$. The
regulated output is $y=x_{1}$ and the constant reference signal is
$r=\frac{\pi}{3}$. Note that matrix $C$ satisfies the full row rank condition.
The control objective is to regulate the angular position of the link $x_{1}$
to a desired constant reference $r$. We introduce $Q(x)=\cos x_{1}$. If we set
$\mathcal{X}=\mathbb{R}^{4}$, then (10) is satisfied with $R_{Q}$ as in (20).
We collect $T=10$ samples, i.e.
$\mathbb{D}:=\left\\{(x_{i},u_{i},y_{i},\dot{x}_{i},\xi_{i})\right\\}_{i=0}^{9}$,
by running an experiment with input uniformly distributed in $[-0.1,0.1]$, and
with an initial state within the same interval. Note that
$M=\mathds{1}_{1\times 9}$. The SDP (61) is feasible and returns the
controller $K$ and the closed-loop dynamics in (65). The state evolution of
(65b) with initial state uniformly distributed in $[-1,1]$ is displayed in
Figure 3, and the closed-loop trajectories converge to an equilibrium point
where the tracking error is zero.

Figure 3: State response of system (65b) with initial state uniformly
distributed in $[-1,1]$.

Next, in order to test the effectiveness of Theorem 4 for other choices of
$Q(x)$, let $Q(x)=\begin{bmatrix}\cos x_{1}&\sin x_{2}\end{bmatrix}^{\top}$.
Note that

$\frac{\partial Q}{\partial x}(x)^{\top}\frac{\partial Q}{\partial
x}(x)=\begin{bmatrix}\sin(x_{1})^{2}&0&0&0\\\ 0&\cos(x_{2})^{2}&0&0\\\
0&0&0&0\\\ 0&0&0&0\end{bmatrix}.$

If we set $\mathcal{X}=\mathbb{R}^{4}$, then (10) is satisfied with

$R_{Q}=R_{Q}^{\top}=\begin{bmatrix}1&0&0&0\\\ 0&1&0&0\\\ 0&0&0&0\\\
0&0&0&0\end{bmatrix}.$

We collect $T=10$ samples under the same experiment setup. The SDP (61) is
feasible and returns the controller $K$ and the closed-loop dynamics in (66).
The state evolution of (66b) with initial state uniformly distributed in
$[-1,1]$ is displayed in Figure 4, and the closed-loop trajectories converge
to an equilibrium point where the tracking error is zero.

Figure 4: State response of system (66b) with initial state uniformly
distributed in $[-1,1]$. $\displaystyle\mathcal{K}$
$\displaystyle=\begin{bmatrix}-4.1190&-4.9995&-4.3688&-0.8453&-0.9417&0.0030\end{bmatrix}$
(65a) $\displaystyle\begin{bmatrix}\dot{x}\\\ \dot{\xi}\end{bmatrix}$
$\displaystyle=\begin{bmatrix}0.0000&1.0000&0.0000&0.0000&0.0000&-0.0000\\\
-2.0000&-0.7500&1.0000&0.0000&0.0000&-1.9600\\\
-0.0000&-0.0000&-0.0000&1.0000&-0.0000&-0.0000\\\
-28.7936&-33.3298&-28.4588&-6.3019&-6.2778&0.0197\\\
1.0000&-0.0000&-0.0000&-0.0000&-0.0000&0.0000\end{bmatrix}\begin{bmatrix}x\\\
\xi\\\ \cos x_{1}\end{bmatrix}+\begin{bmatrix}0.1\\\ 0.2\\\ 0.3\\\ 0.4\\\
0\end{bmatrix}-\begin{bmatrix}0\\\ 0\\\ 0\\\ 0\\\ \frac{\pi}{3}\end{bmatrix}$
(65b) $\displaystyle\mathcal{K}$
$\displaystyle=\begin{bmatrix}-23.4198&-46.1397&-20.1243&-1.6394&-5.9556&-0.0580&0.0041\end{bmatrix}$
(66a) $\displaystyle\begin{bmatrix}\dot{x}\\\ \dot{\xi}\end{bmatrix}$
$\displaystyle=\begin{bmatrix}-0.0000&1.0000&-0.0000&-0.0000&-0.0000&0.0000&0.0000\\\
-2.0000&-0.7500&1.0000&0.0000&-0.0000&-1.9600&-0.0000\\\
-0.0000&0.0000&0.0000&1.0000&-0.0000&0.0000&-0.0000\\\
-157.4652&-307.5983&-133.4954&-11.5961&-39.7042&-0.3869&0.0274\\\
1.0000&-0.0000&-0.0000&-0.0000&0.0000&0.0000&0.0000\\\
\end{bmatrix}\begin{bmatrix}x\\\ \xi\\\ \cos x_{1}\\\ \sin
x_{2}\end{bmatrix}+\begin{bmatrix}0.1\\\ 0.2\\\ 0.3\\\ 0.4\\\
0\end{bmatrix}-\begin{bmatrix}0\\\ 0\\\ 0\\\ 0\\\ \frac{\pi}{3}\end{bmatrix}$
(66b)

##  VIII Conclusions

We have derived data-based semidefinite programs that once solved return
nonlinear feedback controllers that guarantee contractivity. For data
perturbed by a class of periodic disturbances, these semidefinite programs
have been shown to be independent of the magnitude of the disturbances, a
remarkable feature. We have also studied the design of data-dependent integral
controllers for tracking and disturbance rejection problems. We have chosen to
focus on the data-based representation of closed loop systems introduced in
[4], but a similar analysis can be carried out with other representations,
such as those studied in [35]. Although we have considered continuous-time
systems, results close to the ones presented here can also be derived for
discrete-time systems. There are manifold options for future work, which could
aim at designing controllers for more sophisticated output regulation problems
[43, 44], working with dictionaries of functions $Z(x)$ generated via kernels
[45, 46] establishing connections with Gaussian process regression [32], and
using non-Euclidean norms [47, 37], to name a few.

This appendix collects a few basic results on contraction borrowed from the
existing literature. For the sake of a self-contained exposition, some of the
proofs are also reported.

###  -A Convergence results for periodic and time-invariant contractive
systems

We consider the system

$\dot{x}=f(t,x)$ (67)

defined for $(t,x)\in\mathbb{R}\times\mathcal{X}$, where $\mathcal{X}$ is a
subset of $\mathbb{R}^{n}$. The notation $\phi(t,\tau,\xi)$ indicates the
solution of $\dot{x}=f(t,x)$, $x(\tau)=\xi$. It is assumed that $f(t,x)$ is
differentiable on $x$ and $f(t,x)$ and $\frac{\partial f}{\partial x}(t,x)$
are continuous in $t,x$. It is also assumed that the set $\mathcal{X}$ is
forward invariant, _i.e._ , for any
$(t_{0},x_{0})\in\mathbb{R}\times\mathcal{X}$, the solution
$\phi(t,t_{0},x_{0})$ exists for all $t\in\mathbb{R}_{\geq t_{0}}$ that belong
to the domain of existence of the solution, and system (67) is forward
complete on $\mathcal{X}$,555Forward completeness is guaranteed if
$\mathcal{X}$ is closed and bounded. _i.e._ , for any
$(t_{0},x_{0})\in\mathbb{R}\times\mathcal{X}$, the solution of (67)
initialized at $x(t_{0})=x_{0}$ exists for all $t\in\mathbb{R}_{\geq t_{0}}$.

If in addition to the previous conditions, other properties hold, then the
following results can be stated.

###### Lemma 5

For a given $T\in\mathbb{R}_{>0}$, assume that $f$ is $T$-periodic, _i.e._ ,
$f(t+T,x)=f(t,x)$ for all $(t,x)\in\mathbb{R}\times\mathcal{X}$. Then, for all
integers $k\in\mathbb{Z}_{\geq 0}$, for all $t_{0}\in\mathbb{R}_{\geq 0}$, for
all $t\in\mathbb{R}_{\geq t_{0}}$, for all $\xi\in\mathcal{X}$, the following
holds

  1. 1.

$\phi(t+kT,t_{0}+kT,\xi)=\phi(t,t_{0},\xi)$;

  2. 2.

$\phi(t+kT,t_{0},\xi)=\phi(t+kT,t_{0}+kT,\phi(t_{0}+kT,t_{0},\xi))=\phi(t,t_{0},\phi(t_{0}+kT,t_{0},\xi))$.

_Proof._ 1) See [48, Remark 1]. 2) Recall the semigroup property (e.g., [49,
p. 26]), namely, for any $t>t_{1}>t_{0}$
$\phi(t,t_{0},x_{0})=\phi(t,t_{1},\phi(t_{1},t_{0},x_{0}))$ and apply it to
$\phi(t+kT,t_{0},\xi)$ with $t\to t+kT$, $t_{1}\to t_{0}+kT$, $t_{0}\to t_{0}$
and $x_{0}\to\xi$, to obtain

$\phi(t+kT,t_{0},\xi)=\phi(t+kT,t_{0}+kT,\phi(t_{0}+kT,t_{0},\xi))$

which is the first identity in 2). Then apply 1) to
$\phi(t+kT,t_{0}+kT,\phi(t_{0}+kT,t_{0},\xi))$, to obtain
$\phi(t+kT,t_{0}+kT,\phi(t_{0}+kT,t_{0},\xi))=\phi(t,t_{0},\phi(t_{0}+kT,t_{0},\xi))$,
which is the second identity in 2). $\blacksquare$

The following theorem shows the existence of an attractive solution for
periodic systems that are contractive on a subset of the state space, provided
that the latter is forward invariant with respect to the system’s dynamics and
the system’s dynamics is forward complete on the set. Claims similar to the
one below are given in [26, Property 3] and [50, Theorem 3.8] but here we have
mostly followed [48, Theorem 5].

###### Theorem 5

Assume that

  1. 1.

$\mathcal{X}$ is a closed convex subset of $\mathbb{R}^{n}$;

  2. 2.

there exist $P_{1}\succ 0,\beta>0$ such that for all
$(t,x)\in\mathbb{R}\times\mathcal{X}$,

$\displaystyle\frac{\partial f(t,x)^{\top}}{\partial
x}P_{1}^{-1}+P_{1}^{-1}\displaystyle\frac{\partial f(t,x)}{\partial
x}\preceq-\beta P_{1}^{-1};$

  3. 3.

for a given $T\in\mathbb{R}_{>0}$, $f$ is $T$-periodic, _i.e._ ,
$f(t+T,x)=f(t,x)$ for all $(t,x)\in\mathbb{R}\times\mathcal{X}$;

  4. 4.

$c{\rm e}^{-\beta T/2}<1$, where
$c=\left(\frac{\lambda_{\max}(P_{1}^{-1})}{\lambda_{\min}(P_{1}^{-1})}\right)^{1/2}$.

Then there exists a unique periodic solution $x_{*}(t)$ of (67) of period $T$,
which is uniformly exponentially stable. Furthermore, if 3) is replaced by

  * 3’)

$f(t,x)$ is time invariant, _i_.e., $f(t,x)=f(x)$ for all
$(t,x)\in\mathbb{R}\times\mathcal{X}$,

and 4) is removed, then there exists a unique equilibrium $x_{*}$, and every
solution $\xi(t)$ of (67) which is uniformly exponentially stable.

_Proof._ Consider any two solutions $\phi(t,t_{0},\xi),\phi(t,t_{0},\zeta)$ of
(67) from the initial conditions $\xi,\zeta\in\mathcal{X}$. These solutions
exist for all $t\in\mathbb{R}_{\geq t_{0}}$ and belong to $\mathcal{X}$, by
forward completeness and forward invariance. By 1) and 2), they satisfy
$|\phi(t,t_{0},\xi)-\phi(t,t_{0},\zeta)|\leq c{\rm
e}^{-\beta(t-t_{0})/2}|\xi-\zeta|$, for all $t\in\mathbb{R}_{\geq t_{0}}$.
Define $\mathcal{T}(\xi):=\phi(t_{0}+T,t_{0},\xi)$. $\mathcal{T}$ maps
$\mathcal{X}$ into $\mathcal{X}$ because of forward invariance of
$\mathcal{X}$. By 4),
$|\mathcal{T}(\xi)-\mathcal{T}(\zeta)|\leq\rho|\xi-\zeta|$ for all
$\xi,\zeta\in\mathcal{X}$, where $\rho=c{\rm e}^{-\beta T/2}<1$. Consider the
sequence $\xi_{k+1}=\mathcal{T}(\xi_{k})$, which is defined for all666By
definition, $\xi_{1}=T(\xi_{0})=\phi(t_{0}+T,t_{0},\xi_{0})$. Assume now that
$\xi_{k}=\phi(t_{0}+kT,t_{0},\xi_{0})$ for some $k\in\mathbb{Z}_{\geq 1}$.
Then $\xi_{k+1}=\phi(t_{0}+T,t_{0},\phi(t_{0}+kT,t_{0},\xi_{0}))$. Applying
the identity $\phi(t+kT,t_{0},\xi)=\phi(t,t_{0},\phi(t_{0}+kT,t_{0},\xi))$ in
1) of Lemma 5, the expression on the right-hand side becomes
$\phi(t_{0}+T+kT,t_{0},\xi_{0})$. The inductive argument shows that
$\xi_{k}=\phi(t_{0}+kT,t_{0},\xi_{0})$ for all $k\in\mathbb{Z}_{\geq 0}$. Note
that the righ-hand side is defined for all $k\in\mathbb{Z}_{\geq 0}$ by
forward completeness of the solution.  $k\in\mathbb{Z}_{\geq 0}$. Since
$\mathcal{X}$ is a closed subset of $\mathbb{R}^{n}$, $\mathcal{X}$ is a
complete metric space. Hence, by the Contraction Mapping theorem, there exists
a unique $\xi_{*}\in\mathcal{X}$ such that $\xi_{*}=\mathcal{T}(\xi_{*})$ and
$\xi_{k}\to\xi_{*}$ as $k\to\infty$. Define $x_{*}(t)=\phi(t,t_{0},\xi_{*})$.
Then
$x_{*}(t_{0}+T)=\phi(t_{0}+T,t_{0},\xi_{*})=\mathcal{T}(\xi_{*})=\xi_{*}=x_{*}(t_{0})$.
Assume that $x_{*}(t_{0}+jT)=x_{*}(t_{0})$ for all $j=0,1,\ldots,k$, for some
$k\in\mathbb{Z}_{\geq 0}$. Then

$\begin{array}[]{l}x_{*}(t_{0}+(k+1)T)=\phi(t_{0}+(k+1)T,t_{0},\xi_{*})\\\
=\phi(t_{0}+T,t_{0},\phi(t_{0}+kT,t_{0},\xi_{*}))\\\
=\phi(t_{0}+T,t_{0},x_{*}(t_{0}+kT))=\phi(t_{0}+T,t_{0},x_{*}(t_{0}))\\\
=\phi(t_{0}+T,t_{0},\xi_{*})=x_{*}(t_{0}+T)=x_{*}(t_{0})\end{array}$

where the first equation holds by definition of $x_{*}(t)$, the second by 2)
in Lemma 5, the third by the definition of $x_{*}(t)$, the fourth by the
inductive hypothesis, the fifth by the identity $\xi_{*}=x_{*}(t_{0})$, the
sixth by the definition of $x_{*}(t)$ and the last one by the inductive
hypothesis. Hence, $x_{*}(t_{0}+kT)=x_{*}(t_{0})$ for all
$k\in\mathbb{Z}_{\geq 0}$, _i_.e., $x_{*}(t)$ is a periodic solution.
Uniqueness is due to the periodicity of the solution (see [48, p. 223]). It
was shown before that any two solutions
$\phi(t,t_{0},\xi),\phi(t,t_{0},\zeta)$ satisfy
$|\phi(t,t_{0},\xi)-\phi(t,t_{0},\zeta)|\leq c{\rm
e}^{-\beta(t-t_{0})/2}|\xi-\zeta|$, for all $t\in\mathbb{R}_{\geq t_{0}}$. In
particular, for any $\xi\in\mathcal{X}$, the solution $\phi(t,t_{0},\xi)$
satisfies $|\phi(t,t_{0},\xi)-\phi(t,t_{0},\xi_{*})|\leq c{\rm
e}^{-\beta(t-t_{0})/2}|\xi-\xi_{*}|$, for all $t\in\mathbb{R}_{\geq t_{0}}$,
that is, $x_{*}(t)$ is uniformly exponentially stable.

Assume now that $f(t,x)$ is time invariant, _i.e._ $f(t,x)=f(x)$ for all
$(t,x)\in\mathbb{R}\times\mathcal{X}$. $f(x)$ is periodic of any period $T>0$.
By any $T>T_{*}:=2\ln c/\beta$, condition 4) is satisfied. Hence, there exists
a unique solution $x_{*}(t)$ of $\dot{x}=f(x)$ that is periodic of any period
$T>T_{*}$ and every solution $\xi(t)$ of $\dot{x}=f(x)$ converges to it. This
implies that for any $t\in\mathbb{R}$ and any $T>T_{*}$,
$x_{*}(t)=x_{*}(t+T)$. Fixing $t=t_{0}$, this implies that, for any $T>T_{*}$,
$x_{*}(t_{0})=x_{*}(t_{0}+T)$, that is, after time $t_{0}+T_{*}$, $x_{*}(t)$
is constant and equal to $x_{*}(t_{0})$. Take now any
$t\in(t_{0},t_{0}+T_{*}]$ and recall that $x_{*}(t)=x_{*}(t+T)$ for any
$T>T_{*}$. But for any $t\in(t_{0},t_{0}+T_{*}]$ and $T>T_{*}$,
$t+T>t_{0}+T_{*}$, therefore $x_{*}(t)=x_{*}(t+T)=x_{*}(t_{0})$, and we
conclude that $x_{*}(t)=x_{*}(t_{0})$ for all $t\in\mathbb{R}_{\geq t_{0}}$.
That is, $x_{*}(t)$ is a constant. Since by construction $x_{*}$ is a solution
of $\dot{x}=f(x)$ (see the first part of the proof) and it is constant, then
it must be an equilibrium of $\dot{x}=f(x)$. $\blacksquare$

###  -B Convergent systems

We recall here the class of convergent systems, which exhibits a solution
defined and bounded for all time, to which all the solutions uniformly
exponentially converge [23]. Compared to the results in the previous
subsection, here $\mathcal{X}=\mathbb{R}^{n}$.

The system of interest is (67) defined for
$(t,x)\in\mathbb{R}\times\mathbb{R}^{n}$, with $f(t,x)$ differentiable on $x$
and $f(t,x)$ and $\frac{\partial f}{\partial x}(t,x)$ continuous in $t,x$.

###### Definition 3

The system (67) is convergent if

  1. 1.

all its solutions $x(t)$ are defined for all $t\in[t_{0},+\infty)$ and all
initial conditions $t_{0}\in\mathbb{R}$, $x(t_{0})\in\mathbb{R}^{n}$;

  2. 2.

there exists a unique solution $x_{*}(t)$ defined and bounded for all
$t\in(-\infty,+\infty)$;

  3. 3.

the solution $x_{*}(t)$ is globally uniformly exponentially stable.

###### Theorem 6

Assume that

  1. 1.

$\mathcal{X}=\mathbb{R}^{n}$;

  2. 2.

there exist $P_{1}\succ 0,\beta>0$ such that for all
$(t,x)\in\mathbb{R}\times\mathbb{R}^{n}$,

$\displaystyle\frac{\partial f(t,x)^{\top}}{\partial
x}P_{1}^{-1}+P_{1}^{-1}\displaystyle\frac{\partial f(t,x)}{\partial
x}\preceq-\beta P_{1}^{-1};$

  3. 3.

there exists $\overline{x}\in\mathcal{X}$ such that, for all $t\in\mathbb{R}$,
$|f(t,\overline{x})|\leq\bar{f}<+\infty$.

Then the system is convergent. In addition, if, for a given
$T\in\mathbb{R}_{>0}$, $f$ is $T$-periodic, then $x_{*}(t)$ is periodic of
period $T$. If $f(t,x)$ is time-invariant, then $x_{*}(t)=x_{*}$.

_P_ roof. See [23, Theorem 1] and [26, Property 3]. $\blacksquare$

The proof shows that, denoted by $V(x)$ the function
$(x-\overline{x})^{\top}P_{1}^{-1}(x-\overline{x})$, it holds that
$\dot{V}(x):=\frac{\partial V}{\partial x}f(t,x)<0$ for all
$x\in\\{x\in\mathbb{R}^{n}\colon V(x)>\gamma\\}$, where
$\gamma=(2\beta^{-1}\bar{f}\|P_{1}^{-1/2}\|)^{2}$. This implies that the set
$\mathcal{R}_{\gamma}=\\{x\in\mathbb{R}^{n}\colon V(x)\leq\gamma\\}$ is a
closed bounded set of $\mathbb{R}^{n}$ that is forward invariant for
$\dot{x}=f(t,x)$. Hence, it must contain at least one solution $x_{*}(t)$
defined for all $t\in(-\infty,+\infty)$ [23, p. 260] (its uniqueness can be
shown as in [23, p. 261] and uniform exponential stability is
straightforwardly proven).

The possibility of restricting the result to a convex subset $\mathcal{X}$ of
$\mathbb{R}^{n}$, without explicitly asking for the forward invariance of
$\mathcal{X}$ and forward completeness of the system on $\mathcal{X}$, which
are the standing assumptions of Theorem 5, depends on whether or not the set
$\mathcal{R}_{\gamma}$ is contained in $\mathcal{X}$, which cannot be in
general established, as it depends on $\beta,P_{1},\bar{f}$. A special case in
which this is true is when $\overline{x}$ is an equilibrium of the system,
i.e., $\bar{f}=0$, and the set $\mathcal{X}$ contains $\overline{x}$, for in
this case $\bar{f}=0$ implies $\mathcal{R}_{\gamma}=\\{\overline{x}\\}$, and
$x_{*}(t)=\overline{x}$. Hence, the following consequence of Theorem 6 holds:

###### Corollary 7

Assume that

  1. 1.

$\mathcal{X}$ is a convex subset of $\mathbb{R}^{n}$;

  2. 2.

there exist $P_{1}\succ 0,\beta>0$ such that for all
$(t,x)\in\mathbb{R}\times\mathcal{X}$,

$\displaystyle\frac{\partial f(t,x)^{\top}}{\partial
x}P_{1}^{-1}+P_{1}^{-1}\displaystyle\frac{\partial f(t,x)}{\partial
x}\preceq-\beta P_{1}^{-1};$

  3. 3.

there exists $\overline{x}\in\text{int}(\mathcal{X})$ such that, for all
$t\in\mathbb{R}$, $f(t,\overline{x})=0$.

Then, there exists a subset $\mathcal{V}$ of $\mathcal{X}$ containing
$\overline{x}$ such that

  1. 1.

all the solutions $x(t)$ of $\dot{x}=f(t,x)$ are defined for all
$t\in[t_{0},+\infty)$ and all initial conditions $t_{0}\in\mathbb{R}$,
$x(t_{0})\in\mathcal{V}$;

  2. 2.

there exists a unique solution $x_{*}(t)$ defined and bounded for all
$t\in(-\infty,+\infty)$ and contained in $\mathcal{X}$, and it satisfies
$x_{*}(t)=\overline{x}$;

  3. 3.

the solution $x_{*}(t)=\overline{x}$ is uniformly exponentially stable.

Proof. Note that the function
$V(x)=(x-\overline{x})^{\top}P_{1}^{-1}(x-\overline{x})$ satisfies
$\dot{V}(x):=\frac{\partial V}{\partial x}f(t,x)<0$ for all $x\in\mathcal{X}$.
Let $\delta\in\mathbb{R}_{>0}$ and define
$\mathcal{V}:=\\{x\in\mathbb{R}^{n}\colon(x-\overline{x})^{\top}P_{1}^{-1}(x-\overline{x})\leq\delta\\}$.
Fix $\delta$ such that $\mathcal{V}\subseteq\mathcal{X}$. The set
$\mathcal{V}$ is forward invariant with respect to $\dot{x}=f(t,x)$ and
contains a unique solution $x_{*}(t)$ defined and bounded for all
$t\in(-\infty,+\infty)$, which satisfies $x_{*}(t)=\overline{x}$. To prove
uniqueness, we use the argument of [23, p. 261]. Suppose there exists another
solution $x^{\prime}(t)$ defined and bounded for all $t\in(-\infty,+\infty)$
and contained in $\mathcal{X}$. Then such a solution satisfies
$|x^{\prime}(t)-\overline{x}|\leq c{\rm
e}^{-\beta(t-t_{0})/2}|x^{\prime}(t_{0})-\overline{x}|$ for all
$t\in\mathbb{R}_{\geq t_{0}}$. Letting $t_{0}\to-\infty$ yields
$|x^{\prime}(t)-\overline{x}|=0$, which leads to a contradiction by the
arbitrariness of $t$. Finally, the solution $x_{*}(t)=\overline{x}$ is
uniformly exponentially stable as any solution $x(t)$ with initial condition
in $\mathcal{V}$ satisfies $|x(t)-\overline{x}|\leq c{\rm
e}^{-\beta(t-t_{0})/2}|x(t_{0})-\overline{x}|$ for all $t\in\mathbb{R}_{\geq
t_{0}}$. $\blacksquare$

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