# Data-driven stabilization of nonlinear polynomial systems with noisy data

Meichen Guo, Claudio De Persis, and Pietro Tesi Meichen Guo and Claudio De
Persis are with ENTEG, Faculty of Science and Engineering, University of
Groningen, 9747 AG Groningen, The Netherlands. Email<EMAIL_ADDRESS><EMAIL_ADDRESS>Pietro Tesi is with DINFO, University of Florence, 50139
Florence, Italy. E-mail<EMAIL_ADDRESS>
###### Abstract

In a recent paper we have shown how to learn controllers for unknown linear
systems using finite-sized noisy data by solving linear matrix inequalities.
In this note we extend this approach to deal with unknown nonlinear polynomial
systems by formulating stability certificates in the form of data-dependent
sum of squares programs, whose solution directly provides a stabilizing
controller and a Lyapunov function. We then derive variations of this result
that lead to more advantageous controller designs. The results also reveal
connections to the problem of designing a controller starting from a least-
square estimate of the polynomial system.

##  I Introduction

The idea of designing control laws from measured data alone for a system with
unknown dynamics is of high relevance, both theoretically and practically.
Following a rough classification, there are two approaches to concretize this
idea. The first one is based on identifying a model of the system and then
designing a controller that deals with the uncertainty resulting from the
identification process. The second one devises methods to directly synthesize
controllers from data without explicitly undertaking a model identification,
hence the name of direct data-driven control.

Both approaches are viable and appealing but the focus of this paper is on
direct data-driven control, which will lead us to provide neat compact
conditions for the synthesis of nonlinear stabilizing controllers and
corresponding Lyapunov functions. Various results on direct data-driven
control have been proposed and much attention has been devoted to the specific
study of nonlinear systems, which remains an open and challenging problem.
Among the contributions on direct data-driven control of nonlinear systems, we
recall works such as the virtual reference feedback tuning (VRFT) [1] and
iterative feedback tuning [2], the on-line data-driven control of [3], the so-
called intelligent PID [4, 5], and reinforcement learning [6, 7].

Related work. Recent works inspired by Willems et al.’s lemma [8] have
revealed a new approach to accomplish direct data-driven control.
Specifically, for linear systems, by Willems et al.’s lemma, any input-output
trajectory of a linear system can be expressed as a linear combination of
finite number of previously measured input-output data. Using this lemma to
parameterize a closed-loop linear system controlled by a state-feedback
controller, the feedback gain can be directly found via data-based linear
matrix inequalities (LMIs) for various control problems [9]. A nonlinear
extension of the linear stabilization result was also considered in [9], where
the nonlinear remainder after linearization is treated as disturbances and
robust data-driven control method is applied to obtain local stabilization
result. Recently, [10] investigated the stabilization of bilinear systems with
a characterization of the basin of attraction. Polynomial systems lend
themselves to a similar analysis as the one in [9] once one expresses the
system’s polynomial vector fields in the span of a suitable basis of
polynomial functions and pursues a simultaneous design of controllers and
Lyapunov functions based on the second Lyapunov theorem [11]. Independently,
and taking a different route, [12] used the dual stability theory and Farkas’
lemma for direct data-driven design of rational state-feedback controllers for
nonlinear systems. A basis of polynomials to express the nonlinearity was
considered earlier in [13] and used to perform a right-inversion of the system
dynamics based on which a reference tracking controller is derived. Model-
inversion errors are then compensated by an outer-loop linear controller
designed according to the VRFT technique.

A reasonable concern in data-driven control is that the measured data can be
affected by unknown noise. For linear systems, [9] posed a signal-to-noise
ratio assumption on the measurement noise and presented sufficient conditions
for data-driven stabilizers. This method was further investigated and analyzed
in [14] for data-driven design of linear quadratic regulators. The paper [15]
has remarked that the condition on the noise introduced in [9] can be
interpreted as an instance of a quadratic matrix inequality and analyzed via a
full-block S-procedure. Another work [16] presented a matrix-valued S-lemma
based on the classical S-lemma and apply it to data-driven control of linear
systems. Other notable works dealing with direct or indirect data-driven
robust control include, but are not limited to, [17, 12].

In this paper, we investigate global data-driven stabilization of continuous-
time nonlinear polynomial systems using noisy data. The focus is on polynomial
systems because, first, polynomials systems are widely used to model processes
in engineering applications such as fluid dynamics [18] and robotics [19]. In
fact, polynomial control systems, and the supporting technical developments in
sum of squares (SOS) optimization, have attracted considerable attention over
the last twenty years [20, 21, 22, 23, 24, 25, 26]. Second, as shown in the
model-based control [27], nonlinear polynomial systems can be written into a
linear-like form and controlled using the Lyapunov method. This inspires us to
adopt the framework of [9] to establish a direct data-driven control synthesis
for the nonlinear polynomial systems. Our previous work [11] has presented
some preliminary results on data-driven stabilization of polynomial systems
with state-independent input vector field and noise-free data. In this paper,
we enhance the results to handle state-dependent input vector field and noisy
data. Besides being of interest in its own right, the study on polynomial
systems will help us better understand and gain more insights in direct data-
driven control of nonlinear systems.

Contribution. The main contribution of this paper is developing state feedback
stabilizers for nonlinear polynomial systems using noisy input-state data
alone. It is assumed that the noise is unknown and it satisfies a quadratic
constraint. Using a variation of the data-based closed-loop representation in
our work [11], we improve the previous results so that high-order polynomial
systems can now be handled more effectively. Following the framework of [9],
the first result of this paper shows that the data-driven design with
measurement noise for linear systems can be extended to nonlinear polynomial
systems using the adjusted closed-loop representation and matrix-valued
Young’s inequality. Then, by adopting other forms of the data-based closed-
loop representation, more advantageous stabilization results are attained
which require less assumptions or have improved computational efficiency. A
connection is also established with the problem of designing controllers based
on least-square estimates of the system’s dynamics. As the data-driven
stabilizing conditions involve semi-positive-definiteness of matrix
polynomials, we use SOS relaxations to make the computation tractable. We also
show that the results can be interpreted as SOS relaxations of pointwise
necessary and sufficient conditions for global robust stabilizability of
polynomial systems.

Organization. The structure of the paper is specified as follows. In Section
II, we formulate the data-driven control problem. Section III contains the
main results where stabilizers are designed using data corrupted by
measurement noise. Theorem 1 can be regarded as an extension of [9, Theorem 5]
from linear systems to nonlinear polynomial systems. Next, by slightly
modifying the closed-loop system representation, a result in the same spirit
of Theorem 1 is proposed that requires less assumptions on the controlled
dynamics. A more computationally efficient corollary then follows, where a
decision variable is redefined such that its size is independent of the size
of the data. In Section IV, the simulation results on the Van der Pol
oscillators are presented to show the effectiveness of the proposed design
methods. Finally, we summarize the paper and draw the conclusions in Section
V.

Notation. The following notations are adopted throughout the paper:

  * –

$A\succeq 0$: matrix $A$ is positive semi-definite;

  * –

$A\succ 0$: matrix $A$ is positive definite;

  * –

$A\succeq B$: matrix $A-B$ is positive semi-definite;

  * –

$\mathbb{N}_{>0}$: the set of natural numbers excluding $0$;

  * –

$\mathbb{R}$: the set of real numbers;

  * –

$\mathbb{R}_{>0}$: the set of positive real numbers;

  * –

$\mathcal{P}$: the set of polynomials;

  * –

$\Sigma$: the set of SOS polynomials;

  * –

$\mathcal{P}^{r\times s}$: the set of $s\times r$ matrix polynomials;

  * –

$\Sigma^{r}$: the set of $r\times r$ SOS matrix polynomials.

##  II Problem Formulation

Consider the polynomial system

$\displaystyle\dot{x}=f(x)+g(x)u$ (1)

where $x\in\mathbb{R}^{n}$ is the state, $u\in\mathbb{R}^{m}$ is the control
input, $f(x)$ and $g(x)$ are polynomial vector fields of sizes $n\times 1$ and
$n\times m$, respectively. The specific expressions of $f(x)$ and $g(x)$ are
unknown. The polynomial system (1) can be written into the linear-like form

$\displaystyle\dot{x}=AZ(x)+BW(x)u$ (2)

where $A\in\mathbb{R}^{n\times N}$ and $B\in\mathbb{R}^{n\times q}$ are
unknown constant matrices. The $N\times 1$ vector $Z(x)$ is a collection of
distinct monomials in $x$ that may appear in $f(x)$, and the $q\times m$
matrix $W(x)$ contains monomials that may appear in $g(x)$.

As the dynamics of (1) is unknown, to write the linear-like form (2), we need
the vector $Z(x)$ to contain all distinct monomials that may appear in $f(x)$.
For nonlinear systems having high order, $Z(x)$ can have notably large size
and high order, which may cause issues in the controller design. As pointed
out in [27] for the model-based control of polynomial systems, the choice of
$Z(x)$ affects the success of the design method. A big vector $Z(x)$
containing high order monomials can cause computational issues in the SOS
program and fail to give a solution. To overcome this issue, we use another
vector $\hat{Z}(x)$ having smaller size and lower degree than $Z(x)$ for the
controller design and stability analysis. Specifically, the $p\times 1$ vector
$\hat{Z}(x)$ satisfies that

$\displaystyle Z(x)=H(x)\hat{Z}(x)$ (3)

with matrix polynomial $H(x)\in\mathcal{P}^{N\times p}$. Our controller design
and stability analysis will be based on the Lyapunov function
$V(x)=\hat{Z}(x)^{\top}P^{-1}\hat{Z}(x)$, with $P$ a positive definite matrix.
Since in this paper we restrict ourselves to consider a global stabilization
problem, we have that $\hat{Z}(x)=0$ if and only if $x=0$, and $\hat{Z}(x)$ is
radially unbounded. A straightforward choice satisfying these conditions is a
$\hat{Z}(x)$ whose first $n$ components coincide with $x$. This choice will be
used in the subsequent sections of this paper. Note that $\hat{Z}(x)=0$ if and
only if $x=0$ implies that $Z(x)=0$ if $x=0$, _i.e._ , the autonomous system
$\dot{x}=f(x)$ must have an equilibrium at the origin. The latter results in
no loss of generality because, in the case $f(x)$ has no equilibrium at the
origin, one reduces the analysis to the case $f(0)=0$ by a change of state
variables. On the other hand, even though the matrix $W(x)$ having larger size
also increases computational burden in the SOS program, it does not directly
affect the design of the Lyapunov function or the sizes of the decision
variables. Hence, the choice of $W(x)$ has much less impact on the success of
the synthesis and will not be a focus of this work.

To design a data-driven controller, we first run an off-line experiment to
collect the input-state data. The experiment is conducted over the time
interval $[t_{0},t_{0}+(T-1)\tau]$ where $T\in\mathbb{N}_{>0}$ is the number
of sampled data and $\tau\in\mathbb{R}_{>0}$ is the sampling time. The Hankel
matrices of the sampled input-state data are defined as

$\displaystyle U_{0}$
$\displaystyle:=\begin{bmatrix}u(t_{0})\\!&\\!u(t_{0}\\!+\\!\tau)\\!&\\!\cdots\\!&\\!u(t_{0}+(T\\!-\\!1)\tau)\end{bmatrix},$
$\displaystyle X_{0}$
$\displaystyle:=\begin{bmatrix}x(t_{0})\\!&\\!x(t_{0}\\!+\\!\tau)\\!&\\!\cdots\\!&\\!x(t_{0}+(T\\!-\\!1)\tau)\end{bmatrix},$
$\displaystyle X_{1}$
$\displaystyle:=\begin{bmatrix}\dot{x}(t_{0})\\!&\\!\dot{x}(t_{0}\\!+\\!\tau)\\!&\\!\cdots\\!&\\!\dot{x}(t_{0}+(T\\!-\\!1)\tau)\end{bmatrix}.$

Then, we can compute the data

$\displaystyle Z_{0}$
$\displaystyle:=\\!\big{[}Z(x(t_{0}))~{}~{}Z(x(t_{0}\\!+\\!\tau))~{}~{}\cdots~{}~{}Z(x(t_{0}\\!+\\!(T\\!-\\!1)\tau))\big{]},$
$\displaystyle\overline{U}_{0}$
$\displaystyle:=\\!\big{[}W(x(t_{0}))u(t_{0})~{}~{}W(x(t_{0}\\!+\\!\tau))u(t_{0}\\!+\\!\tau)$
$\displaystyle\quad\quad\quad\quad~{}~{}\cdots~{}~{}W(x(t_{0}+(T\\!-\\!1)\tau))u(t_{0}+(T\\!-\\!1)\tau)\big{]}.$

We consider experiments affected by measurement noises $D_{0}$ defined as

$\displaystyle
D_{0}:=\begin{bmatrix}d(t_{0})\\!&\\!d(t_{0}\\!+\\!\tau)\\!&\\!\cdots\\!&\\!d(t_{0}+(T\\!-\\!1)\tau)\end{bmatrix}$

where $d(t)\in\mathbb{R}^{n}$. The specific nature of the noise is discussed
in the next section. Affected by the noise, the measured derivatives $X_{1}$
satisfies

$\displaystyle X_{1}=AZ_{0}+B\overline{U}_{0}+D_{0}.$ (4)

The data-driven stabilization problem of (1) is to design a state-dependent
control gain $F(x)$ using the noisy experimental input-state data alone, such
that under the state-feedback controller

$\displaystyle u=F(x)\hat{Z}(x)$ (5)

the closed-loop system is globally asymptotically stable at the origin.

##  III Data-driven stabilization with noisy data

In this section, we present conditions for the stabilization of unknown
polynomial systems with data affected by noise. It is unrealistic to expect to
stabilize the system using data that are affected by arbitrary noise. Hence,
we introduce the following assumption on $D_{0}$ :

######  Assumption 1.

The matrix $D_{0}$ satisfies $D_{0}D_{0}^{\top}\preceq R_{D}R_{D}^{\top}$ for
some known $R_{D}\in\mathbb{R}^{n\times T}$.

Assumption 1 is in a similar form as [9, Assumption 2] where a signal-to-noise
ratio assumption is posed on the noise $D_{0}$ to solve the data-driven
control problems. In fact, if we set $R_{D}=\gamma X_{1}$ for some constant
$\gamma>0$, Assumption 1 becomes the same as [9, Assumption 2]. Interestingly,
the noisy data result in [9, Theorem 5] can be extended to nonlinear
polynomial systems, which will be presented in Subsection III-A. Then, in
Subsection III-B and III-C, by slightly changing the data-based closed-loop
representation and the decision variable, we can derive the result under more
advantageous conditions.

###  III-A Data-driven stabilization with noisy data

In the result below, inspired by the results of [9], we will introduce a
controller of the form (a) $u=F(x)\hat{Z}(x)$ where $F(x)=U_{0}Y(x)P^{-1}$ and
$Y(x),P\succ 0$ are matrices that satisfy the condition (b) $Z_{0}Y(x)=H(x)P$.
This choice allows us to express the closed-loop dynamics as

$\begin{array}[]{rcl}\dot{x}&=&AZ(x)+BW(x)u\\\
&\stackrel{{\scriptstyle{\rm(a)}}}{{=}}&\big{(}AH(x)PP^{-1}+BW(x)U_{0}Y(x)P^{-1}\big{)}\hat{Z}(x)\\\
&\stackrel{{\scriptstyle{\rm(b)}}}{{=}}&\big{(}AZ_{0}Y(x)P^{-1}+BW(x)U_{0}Y(x)P^{-1}\big{)}\hat{Z}(x)\\\
&=&(AZ_{0}+BW(x)U_{0})Y(x)P^{-1}\hat{Z}(x)\\\
&\stackrel{{\scriptstyle\eqref{data-
equation}}}{{=}}&(X_{1}-D_{0}-B\overline{U}_{0}+BW(x)U_{0})Y(x)P^{-1}\hat{Z}(x)\\\
&=&(X_{1}-E\hat{U}_{0}(x))Y(x)P^{-1}\hat{Z}(x)\end{array}$ (6)

having introduced the symbols

$\displaystyle
E:=[B~{}~{}D_{0}],\quad\widehat{U}_{0}(x):=\begin{bmatrix}\,\overline{U}_{0}-W(x)U_{0}\\\
I_{T}\end{bmatrix}$

to obtain the last identity.

To design the stabilizing gain $F(x)$, we need another assumption on the
unknown matrix $B$.

######  Assumption 2.

The matrix $B$ satisfies $BB^{\top}\preceq R_{B}R_{B}^{\top}$ for some known
$R_{B}\in\mathbb{R}^{n\times q}$.

Under Assumptions 1 and 2,

$EE^{\top}=BB^{\top}+D_{0}D_{0}^{\top}\preceq
R_{B}R_{B}^{\top}+R_{D}R_{D}^{\top}.$

Define $R_{E}:=[R_{B}~{}~{}R_{D}]$. Then, it holds that

$EE^{\top}\preceq R_{E}R_{E}^{\top}.$ (7)

Obtaining stabilization results requires to impose positive conditions on
matrix polynomials, which is often computationally intractable. To resolve
this issue, we use SOS relaxations in the subsequent results.

######  Theorem 1.

For the polynomial system (1), under Assumptions 1 and 2, if there exist a
positive definite matrix $P\in\mathbb{R}^{p\times p}$, matrix polynomial
$Y(x)\in\mathcal{P}^{T\times p}$, $\epsilon_{1}(x)\in\Sigma$, and
$\epsilon_{2}>0$ such that

$\displaystyle Z_{0}Y(x)=H(x)P,$ (8)
$\displaystyle\begin{bmatrix}\Upsilon_{E}(x)-\epsilon_{1}(x)I_{p}&Y(x)^{\top}\widehat{U}_{0}(x)^{\top}\\\
\widehat{U}_{0}(x)Y(x)&\epsilon_{2}I_{q+T}\end{bmatrix}\in\Sigma^{p+q+T},$ (9)

where

$\displaystyle\Upsilon_{E}(x)$
$\displaystyle:=-\frac{\partial\hat{Z}(x)}{\partial
x}X_{1}Y(x)-Y(x)^{\top}X_{1}^{\top}\frac{\partial\hat{Z}(x)}{\partial
x}^{\top}$
$\displaystyle\quad~{}-\epsilon_{2}\frac{\partial\hat{Z}(x)}{\partial
x}R_{E}R_{E}^{\top}\frac{\partial\hat{Z}(x)}{\partial x}^{\top},$

then the state-feedback controller

$\displaystyle u=U_{0}Y(x)P^{-1}\hat{Z}(x)$ (10)

stabilizes the continuous-time polynomial system (1). Moreover, if
$\epsilon_{1}(x)>0$ for all $x\neq 0$, the closed-loop system is globally
asymptotically stable.

###### Proof.

Design the Lyapunov function as $V(x)=\hat{Z}(x)^{\top}P^{-1}\hat{Z}(x)$ whose
derivative along the dynamics (6) is

$\displaystyle\dot{V}(x)=-\hat{Z}$
$\displaystyle(x)^{\top}P^{-1}\Big{[}-\frac{\partial\hat{Z}(x)}{\partial
x}\big{(}X_{1}\\!-\\!E\widehat{U}_{0}(x)\big{)}Y(x)$
$\displaystyle-Y(x)^{\top}\big{(}X_{1}\\!-\\!E\widehat{U}_{0}(x)\big{)}^{\top}\frac{\partial\hat{Z}(x)}{\partial
x}^{\top}\Big{]}P^{-1}\hat{Z}(x).$ (11)

The SOS condition (9) guarantees the non-negativity of the matrix, which, by
Schur complement, is equivalent to

$\displaystyle\Upsilon_{E}(x)-\epsilon_{2}^{-1}Y(x)^{\top}\widehat{U}_{0}(x)^{\top}\widehat{U}_{0}(x)Y(x)\succeq\epsilon_{1}(x)I_{p}$

for all $x\in\mathbb{R}^{n}$. Under Assumptions 1 and 2, the left-hand side of
the inequality satisfies

$\displaystyle\quad\Upsilon_{E}(x)-\epsilon_{2}^{-1}Y(x)^{\top}\widehat{U}_{0}(x)^{\top}\widehat{U}_{0}(x)Y(x)$
$\displaystyle\preceq-\frac{\partial\hat{Z}(x)}{\partial
x}X_{1}Y(x)-Y(x)^{\top}X_{1}^{\top}\frac{\partial\hat{Z}(x)}{\partial
x}^{\top}$
$\displaystyle\quad~{}-\epsilon_{2}\frac{\partial\hat{Z}(x)}{\partial
x}EE^{\top}\frac{\partial\hat{Z}(x)}{\partial x}^{\top}$
$\displaystyle\quad~{}-\epsilon_{2}^{-1}Y(x)^{\top}\widehat{U}_{0}(x)^{\top}\widehat{U}_{0}(x)Y(x)$
$\displaystyle\preceq-\frac{\partial\hat{Z}(x)}{\partial
x}X_{1}Y(x)-Y(x)^{\top}X_{1}^{\top}\frac{\partial\hat{Z}(x)}{\partial
x}^{\top}$ $\displaystyle\quad~{}+\frac{\partial\hat{Z}(x)}{\partial
x}E\widehat{U}_{0}(x)Y(x)+Y(x)^{\top}\widehat{U}_{0}(x)^{\top}E^{\top}\frac{\partial\hat{Z}(x)}{\partial
x}^{\top}$ $\displaystyle=-\frac{\partial\hat{Z}(x)}{\partial
x}\big{(}X_{1}-E\widehat{U}_{0}(x)\big{)}Y(x)$
$\displaystyle\quad~{}-Y(x)^{\top}\big{(}X_{1}-E\widehat{U}_{0}(x)\big{)}^{\top}\frac{\partial\hat{Z}(x)}{\partial
x}^{\top},$

where the last inequality follows from the matrix-valued Young’s inequality
$\epsilon_{2}L(x)L(x)^{\top}+\epsilon_{2}^{-1}M(x)M(x)^{\top}\succeq-L(x)M(x)^{\top}-M(x)L(x)^{\top}$.
Thus, it holds that

$\displaystyle-\\!\frac{\partial\hat{Z}(x)}{\partial
x}\big{(}X_{1}\\!-\\!E\widehat{U}_{0}(x)\big{)}Y(x)$
$\displaystyle\quad~{}-Y(x)^{\top}\big{(}X_{1}\\!-\\!E\widehat{U}_{0}(x)\big{)}^{\top}\frac{\partial\hat{Z}(x)}{\partial
x}^{\top}\succeq\epsilon_{1}(x)I_{p}$

for all $x\in\mathbb{R}^{n}$. Then, the derivative of $V(x)$ satisfies that

$\displaystyle\dot{V}(x)\leq-\epsilon_{1}(x)\hat{Z}(x)^{\top}P^{-1}\cdot
P^{-1}\hat{Z}(x)$ (12)

for all $x\in\mathbb{R}^{n}$. Therefore, the closed-loop system is stable.
Moreover, if $\epsilon_{1}(x)>0$ for all $x\neq 0$, then $\dot{V}(x)<0$ for
all $x\neq 0$, and thus the closed-loop system is globally asymptotically
stable at the origin since $V(x)$ is radially unbounded (recall that
$\hat{Z}(x)$ contains $x$ as a sub-vector). ∎

To find $P$ and $Y(x)$ in Theorem 1, one can use MATLAB based tools such as
the SOSTOOLS [28]. The decision variables of the SOS program include the
constant matrix $P$ and the coefficients of $Y(x)$. The coefficients of
$\epsilon_{1}(x)$ and $\epsilon_{2}$ can also be set as decision variables.

######  Remark 1.

_(Input vector field independent of $x$)_ In the case where the input vector
field is independent of $x$, Theorem 1 can achieve data-driven stabilization
without Assumption 2. In this case, the polynomial system can be written into
the linear-like form

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

where $B\in\mathbb{R}^{n\times m}$, and the corresponding closed-loop dynamics
can be expressed using data as

$\displaystyle\dot{x}=(X_{1}-D_{0})Y(x)P^{-1}\hat{Z}(x).$

Then, a stabilization result in the same fashion as Theorem 1 can be derived
by changing (9) into

$\displaystyle\begin{bmatrix}\Upsilon(x)-\epsilon_{1}(x)I_{p}&Y(x)^{\top}\\\
Y(x)&\epsilon_{2}(x)I_{T}\end{bmatrix}\in\Sigma^{p+T}$

where

$\displaystyle\Upsilon(x)$ $\displaystyle:=-\frac{\partial\hat{Z}(x)}{\partial
x}X_{1}Y(x)-Y(x)^{\top}X_{1}^{\top}\frac{\partial\hat{Z}(x)}{\partial
x}^{\top}$
$\displaystyle\quad~{}-\epsilon_{2}(x)\frac{\partial\hat{Z}(x)}{\partial
x}R_{D}R_{D}^{\top}\frac{\partial\hat{Z}(x)}{\partial x}^{\top}.$

######  Remark 2.

_(Rank condition on $Z_{0}$)_ The feasibility of condition (8) implies a rank
condition of $Z_{0}$. As $P\succ 0$,
$\text{rank}\big{(}H(x)P\big{)}=\text{rank}\big{(}H(x)\big{)}$. Based on the
assumptions on $Z(x)$ and $\hat{Z}(x)$, the rank of
$H(x)\in\mathcal{P}^{N\times p}$ should satisfy
$1\leq\text{rank}\big{(}H(x)\big{)}\leq p$. This means that $Z_{0}Y(x)$ should
also have the same rank between $1$ and $p$. Hence, the rank condition needed
for $Z_{0}$ is
$\text{rank}\big{(}Z_{0}\big{)}\geq\text{rank}\big{(}H(x)\big{)}$. Depending
on $H(x)$, in many practical cases $Z_{0}$ must have full row rank in order to
fulfil condition (8). These conditions on $Z_{0}$ imply that the number $T$ of
sampling data must be sufficiently large, since $Z_{0}\in\mathbb{R}^{N\times
T}$.

###  III-B Data-driven stabilization without bounds on the input matrix $B$

By defining $S:=[B~{}~{}A]$ and
$W_{0}(x):=[U_{0}^{\top}W(x)^{\top}~{}~{}Z_{0}^{\top}]^{\top}$, the chain of
equalities in (6) has highlighted the identity

$\displaystyle X_{1}-E\hat{U}_{0}(x)=SW_{0}(x).$ (13)

We can expect then that the analysis carried out in the previous subsection
for the representation $\dot{x}=(X_{1}-E\hat{U}_{0}(x))Y(x)P^{-1}\hat{Z}(x)$
can be repeated for the representation

$\displaystyle\dot{x}=SW_{0}(x)Y(x)P^{-1}\hat{Z}(x)$ (14)

and the same stability result can be established. In doing so, we obtain a few
advantages compared to Theorem 1, including the relaxation of Assumption 2 on
the unknown matrix $B$.

As a first remark, standard calculations show that the derivative of the
Lyapunov function $V(x)=\hat{Z}(x)^{\top}P^{-1}\hat{Z}(x)$ along the closed-
loop system $\dot{x}=SW_{0}(x)Y(x)P^{-1}\hat{Z}(x)$ is

$\displaystyle\dot{V}(x)$
$\displaystyle=-\hat{Z}(x)^{\top}P^{-1}\begin{bmatrix}I_{p}\\\
S^{\top}\frac{\partial\hat{Z}(x)}{\partial x}^{\top}\end{bmatrix}^{\top}$
$\displaystyle\quad\cdot\begin{bmatrix}0_{p\times
p}&-Y(x)^{\top}W_{0}(x)^{\top}\\\ -W_{0}(x)Y(x)&0_{(q+N)\times
p}\end{bmatrix}$ $\displaystyle\quad\cdot\begin{bmatrix}I_{p}\\\
S^{\top}\frac{\partial\hat{Z}(x)}{\partial
x}^{\top}\end{bmatrix}P^{-1}\hat{Z}(x).$ (15)

On the other hand, since we choose a $\hat{Z}(x)$ whose first $n$ components
coincide with $x$, $\frac{\partial\hat{Z}(x)}{\partial x}$ has full column
rank. Hence, the quadratic constraint in Assumption 1 can be equivalently
written as

$\displaystyle\frac{\partial\hat{Z}(x)}{\partial
x}D_{0}D_{0}^{\top}\frac{\partial\hat{Z}(x)}{\partial
x}^{\top}-\frac{\partial\hat{Z}(x)}{\partial
x}R_{D}R_{D}^{\top}\frac{\partial\hat{Z}(x)}{\partial x}^{\top}\preceq 0$

and then expressed as

$\begin{array}[]{l}\begin{bmatrix}I_{p}\\\
D_{0}^{\top}\frac{\partial\hat{Z}(x)}{\partial
x}^{\top}\end{bmatrix}^{\top}\\\
\cdot\begin{bmatrix}-\frac{\partial\hat{Z}(x)}{\partial
x}R_{D}R_{D}^{\top}\frac{\partial\hat{Z}(x)}{\partial x}^{\top}&0_{N\times
T}\\\ 0_{T\times N}&I_{T}\end{bmatrix}\begin{bmatrix}I_{p}\\\
D_{0}^{\top}\frac{\partial\hat{Z}(x)}{\partial x}^{\top}\end{bmatrix}\preceq
0.\end{array}$ (16)

Using the equation $D_{0}=X_{1}-S\overline{W}_{0}$ which follows from (4) with

$\displaystyle\overline{W}_{0}:=\begin{bmatrix}\,\overline{U}_{0}\\\
\,Z_{0}\end{bmatrix}\in\mathbb{R}^{(q+N)\times T},$ (17)

we can derive the following condition equivalent to (16)

$\displaystyle\begin{bmatrix}I_{p}\\\
S^{\top}\frac{\partial\hat{Z}(x)}{\partial x}^{\top}\end{bmatrix}^{\top}$
$\displaystyle\quad\cdot\begin{bmatrix}\frac{\partial\hat{Z}(x)}{\partial
x}\big{(}X_{1}X_{1}^{\top}-R_{D}R_{D}^{\top}\big{)}\frac{\partial\hat{Z}(x)}{\partial
x}^{\top}&-\frac{\partial\hat{Z}(x)}{\partial
x}X_{1}\overline{W}_{0}^{\top}\\\
-\overline{W}_{0}X_{1}^{\top}\frac{\partial\hat{Z}(x)}{\partial
x}^{\top}&\overline{W}_{0}\overline{W}_{0}^{\top}\end{bmatrix}$
$\displaystyle\quad\cdot\begin{bmatrix}I_{p}\\\
S^{\top}\frac{\partial\hat{Z}(x)}{\partial x}^{\top}\end{bmatrix}\preceq 0.$
(18)

Using these conditions, we present a data-driven stabilizing design free of
Assumption 2 as follows:

######  Theorem 2.

For the polynomial system (1), under Assumption 1, if there exist a positive
definite matrix $P\in\mathbb{R}^{p\times p}$, matrix polynomial
$Y(x)\in\mathcal{P}^{T\times p}$, $\epsilon_{1}(x)\in\Sigma$, and
$\epsilon_{2}(x)\in\Sigma$ such that

$\displaystyle Z_{0}Y(x)=H(x)P,$ (19)
$\displaystyle\begin{bmatrix}\Upsilon_{D}(x)\\!-\epsilon_{1}(x)I_{p}&-Y(x)\\!^{\top}\\!W_{0}(x)\\!^{\top}\\!-\epsilon_{2}(x)\frac{\partial\hat{Z}(x)}{\partial
x}X_{1}\overline{W}_{0}^{\top}\\\
\star&\epsilon_{2}(x)\overline{W}_{0}\overline{W}_{0}^{\top}\end{bmatrix}$
$\displaystyle\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\in\Sigma^{p+q+N},$
(20)

where

$\displaystyle\Upsilon_{D}(x)=\epsilon_{2}(x)\frac{\partial\hat{Z}(x)}{\partial
x}\big{(}X_{1}X_{1}^{\top}-R_{D}R_{D}^{\top}\big{)}\frac{\partial\hat{Z}(x)}{\partial
x}^{\top},$ (21)

then the state-feedback controller (10) stabilizes the polynomial system.
Moreover, if $\epsilon_{1}(x)>0$ for all $x\neq 0$, the closed-loop system is
globally asymptotically stable.

###### Proof.

Under condition (20), it holds that

$\displaystyle\begin{bmatrix}\Upsilon_{D}(x)&-Y(x)^{\top}W_{0}(x)^{\top}\\!-\epsilon_{2}(x)\frac{\partial\hat{Z}(x)}{\partial
x}X_{1}\overline{W}_{0}^{\top}\\\
\star&\epsilon_{2}(x)\overline{W}_{0}\overline{W}_{0}^{\top}\end{bmatrix}$
$\displaystyle\succeq$
$\displaystyle\begin{bmatrix}\epsilon_{1}(x)I_{p}&0_{p\times(q+T)}\\\
0_{(q+T)\times p}&0_{(q+T)\times(q+T)}\end{bmatrix}.$

Recall the expression of $\dot{V}$ (III-B) along the closed-loop system and
the expression (III-B) of Assumption 1. Then, for any $\epsilon_{2}(x)\geq 0$
$\forall x\in\mathbb{R}^{n}$, it holds that

$\displaystyle\dot{V}(x)$
$\displaystyle\leq-\hat{Z}(x)^{\top}P^{-1}\begin{bmatrix}I_{p}\\\
S^{\top}\frac{\partial\hat{Z}(x)}{\partial x}^{\top}\end{bmatrix}^{\top}$
$\displaystyle\quad\cdot\begin{bmatrix}\Upsilon_{D}(x)&-Y(x)^{\top}W_{0}(x)^{\top}\\!-\\!\epsilon_{2}(x)\frac{\partial\hat{Z}(x)}{\partial
x}X_{1}\overline{W}_{0}^{\top}\\\
\star&\epsilon_{2}(x)\overline{W}_{0}\overline{W}_{0}^{\top}\end{bmatrix}$
$\displaystyle\quad\cdot\begin{bmatrix}I_{p}\\\
S^{\top}\frac{\partial\hat{Z}(x)}{\partial
x}^{\top}\end{bmatrix}P^{-1}\hat{Z}(x).$ (22)

Using this inequality and condition (20), we can conclude that

$\displaystyle\dot{V}(x)\leq-\epsilon_{1}(x)\hat{Z}(x)^{\top}P^{-1}\cdot
P^{-1}\hat{Z}(x)~{}~{}\forall x\in\mathbb{R}^{n},$

and the thesis follows. ∎

Theorem 1 and Theorem 2 use the same stabilizing approach but different data
parameterizations of the closed-loop system. Comparing these two results,
Theorem 2 is more advantageous in a few aspects. First, by pulling out the
unknown dynamics $A$ and $B$ in one matrix $S$, Theorem 2 no longer requires
Assumption 2. Second, the SOS condition (20) can be made more computationally
efficient by a change of decision variable, as we discuss in the next
subsection. Nonetheless, Theorem 1 is important because it provides insights
on how to arrive at Theorem 2 while establishing connections with the robust
stabilization results of [9].

######  Remark 3.

(A least-square-based design) Identity (4) can be used to establish a
variation of Theorem 1 that has a similar feature as Theorem 2 but is based on
a system’s representation that uses a least-square estimate of the system’s
matrices. By (4), we can adopt an estimate of the model $S$ that minimizes the
Frobenius norm of $X_{1}-S\overline{W}_{0}=D_{0}$. This choice returns the
least-square estimate $S_{*}:=X_{1}\overline{W}_{0}^{\dagger}$ of $S$, with
$\overline{W}_{0}^{\dagger}$ the pseudo-inverse of $\overline{W}_{0}$, and
entails an estimation error $S_{*}-S$. We note that the estimate $S_{*}$ can
be used for a model-based control design in which the actual model $S$ is
replaced by the estimate $S_{*}$, with the expectation that for smaller
$S_{*}-S$, the result will be increasingly accurate. To show this, let us
consider the case of a full-row rank matrix $\overline{W}_{0}$, so that
$S_{*}-S$ is given by $D_{0}\overline{W}_{0}^{\dagger}$. Then, we obtain the
representation

$\displaystyle
X_{1}-E\hat{U}_{0}(x)=SW_{0}(x)=S_{*}W_{0}(x)-D_{0}\overline{W}_{0}^{\dagger}W_{0}(x),$

Based on this representation and on the same arguments of the proof of Theorem
1, one realizes that, under Assumption 1 and without Assumption 2, an
analogous of Theorem 1 holds provided that in condition (9) the matrices
$X_{1},\hat{U}_{0}(x)$ are replaced by
$S_{*}W_{0}(x),\overline{W}_{0}^{\dagger}W_{0}(x)$ respectively.

###  III-C Computationally more efficient stabilization conditions

Recalling that $F(x)=U_{0}Y(x)P^{-1}$, we can obtain $F(x)P=U_{0}Y(x)$.
Defining $K(x):=F(x)P$, we can write the term $W_{0}(x)Y(x)$ in (III-B) as

$\displaystyle\quad~{}W_{0}(x)Y(x)$
$\displaystyle=\begin{bmatrix}W(x)U_{0}Y(x)\\\
Z_{0}Y(x)\end{bmatrix}=\begin{bmatrix}W(x)F(x)P\\\
H(x)P\end{bmatrix}=\begin{bmatrix}W(x)K(x)\\\ H(x)P\end{bmatrix},$

which suggests to adopt $K(x)$ as a new decision variable, modify condition
(20) and obtain the following result:

######  Corollary 1.

For the polynomial system (1), under Assumption 1, if there exist a positive
definite matrix $P\in\mathbb{R}^{p\times p}$, matrix polynomial
$K(x)\in\mathcal{P}^{m\times p}$, $\epsilon_{1}(x)\in\Sigma$, and
$\epsilon_{2}(x)\in\Sigma$ such that

$\displaystyle\begin{bmatrix}\Upsilon_{D}\\!(x)\\!-\\!\epsilon_{1}(x)I_{p}&-\\!\begin{bmatrix}W\\!(x)K(x)\\\
H(x)P\end{bmatrix}^{\top}\\!-\\!\epsilon_{2}(x)\frac{\partial\hat{Z}(x)}{\partial
x}X_{1}\\!\overline{W}_{0}^{\top}\\\
\star&\epsilon_{2}(x)\overline{W}_{0}\overline{W}_{0}^{\top}\end{bmatrix}$
$\displaystyle\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\in\Sigma^{p+q+N},$
(23)

where $\Upsilon_{D}(x)$ is as in (21), then the state-feedback controller

$\displaystyle u=K(x)P^{-1}\hat{Z}(x)$ (24)

stabilizes the polynomial system. Moreover, if $\epsilon_{1}(x)>0$ for all
$x\neq 0$, the closed-loop system is globally asymptotically stable.

###### Proof.

Under the control law (24), the closed-loop system can be expressed as

$\displaystyle\dot{x}=S\begin{bmatrix}W(x)K(x)P^{-1}\\\
H(x)\end{bmatrix}\hat{Z}(x).$

Then, the time derivative of the Lyapunov function
$V(x)=\hat{Z}(x)^{\top}P^{-1}\hat{Z}(x)$ is

$\displaystyle\dot{V}(x)$
$\displaystyle=\hat{Z}(x)^{\top}P^{-1}\Bigg{(}\frac{\partial\hat{Z}(x)}{\partial
x}S\begin{bmatrix}W(x)K(x)\\\ H(x)P\end{bmatrix}$
$\displaystyle\quad\quad+\begin{bmatrix}W(x)K(x)\\\
H(x)P\end{bmatrix}^{\top}S^{\top}\frac{\partial\hat{Z}(x)}{\partial
x}^{\top}\Bigg{)}P^{-1}\hat{Z}(x)$
$\displaystyle=-\hat{Z}(x)^{\top}P^{-1}\begin{bmatrix}I_{p}\\\
S^{\top}\frac{\partial\hat{Z}(x)}{\partial x}^{\top}\end{bmatrix}^{\top}$
$\displaystyle\quad\cdot\begin{bmatrix}0_{p\times
p}&-K(x)^{\top}W(x)^{\top}&-PH(x)^{\top}\\\ -W(x)K(x)&0_{q\times q}&0_{q\times
N}\\\ -H(x)P&0_{N\times q}&0_{N\times N}\end{bmatrix}$
$\displaystyle\quad\cdot\begin{bmatrix}I_{p}\\\
S^{\top}\frac{\partial\hat{Z}(x)}{\partial
x}^{\top}\end{bmatrix}P^{-1}\hat{Z}(x).$

Using the expression (III-B) of Assumption 1 and the condition (23), we can
conclude that

$\displaystyle\dot{V}(x)\leq-\epsilon_{1}(x)\hat{Z}(x)^{\top}P^{-1}P^{-1}\hat{Z}(x)~{}~{}\forall
x\in\mathbb{R}^{n}.$

The thesis follows. ∎

In Theorem 2, the size of decision variable $Y(x)$ is dependent on the number
of sampled data $T$. For systems having higher order and degree, $T$ can be
large which will result in heavy computational burden in the SOS programming.
Hence, by changing the decision variable $Y(x)$ used in Theorem 2, the SOS
condition (20) becomes (23), which is independent of $T$, and the stabilizer
design is more computationally efficient.

######  Remark 4.

(An SOS relaxation of a pointwise necessary and sufficient condition) Observe
that investigating the stabilization problem via the Lyapunov function
$V(x)=\hat{Z}(x)^{\top}P^{-1}\hat{Z}(x)$ and the feedback controller
$u=K(x)P^{-1}\hat{Z}(x)$, the problem becomes one of finding the matrix
$P\succ 0$ and the matrix $K(x)$ such that for all
$x\in\mathbb{R}^{n}\setminus\\{0\\}$, the matrix in the proof of Corollary 1
satisfies

$\begin{bmatrix}0_{p\times p}&-K(x)^{\top}W(x)^{\top}&-PH(x)^{\top}\\\
-W(x)K(x)&0_{q\times q}&0_{q\times N}\\\ -H(x)P&0_{N\times q}&0_{N\times
N}\end{bmatrix}\succ 0$

for all $S$ such that (III-B) holds. For each fixed
$x\in\mathbb{R}^{n}\setminus\\{0\\}$, one can pointwise apply the matrix
S-lemma of [16] to this formulation. Under the technical condition that for
each $x\in\mathbb{R}^{n}\setminus\\{0\\}$, there exists a matrix
$\overline{S}$ for which (III-B) with $S$ replaced by $\overline{S}$ holds,
the stabilization problem is solvable if and only if there exist a matrix
$P\succ 0$ and a matrix $K(x)$ such that for all
$x\in\mathbb{R}^{n}\setminus\\{0\\}$ there exist $\epsilon_{1}(x)>0$ and
$\epsilon_{2}(x)\geq 0$ for which the matrix (23) is positive semi-definite.
Having this condition satisfied leads to an infinite dimensional problem. A
natural way to overcome this obstacle is to relax the positive semi-
definiteness condition by requiring $K(x),\epsilon_{1}(x),\epsilon_{2}(x)$ to
be matrix polynomials and the resulting matrix polynomial (23) to be an SOS
matrix, which is the condition in Corollary 1.

######  Remark 5.

(Another convex relaxation) The stabilization problem of polynomial systems
from data has been tackled in [12] using density functions and polyhedral
constraints on the uncertainties, which leads to Quadratically Constrained
Quadratic Program whose convex relaxation is solved via moments-based
techniques. Stemming from the results in [9, 11], our approach uses Lyapunov
second stability theorem and quadratic constraints, leading to SOS programs
whose solutions directly provide stabilizing controllers and Lyapunov
functions.

##  IV Example

In this section, we apply the data-driven control method to the stabilization
of the Van der Pol oscillator, which is a popular benchmark to test data-
driven control results. The simulations are conducted using the SOSTOOLS in
MATLAB.

Consider the controlled Van der Pol oscillator

$\displaystyle\dot{x}_{1}$ $\displaystyle=x_{2},$ $\displaystyle\dot{x}_{2}$
$\displaystyle=-x_{1}+(1-x_{1}^{2})x_{2}+u.$ (25)

Let $Z(x)$ be the power vector that contains all monomials of $x$ having
degrees $1$ to $3$. Choose $\hat{Z}(x)=[x_{1}~{}~{}x_{2}]^{\top}$, then

$\displaystyle
H(x)=\begin{bmatrix}1&0&x_{2}&x_{1}&0&x_{1}x_{2}&0&x_{1}^{2}&0\\\
0&1&0&0&x_{2}&0&x_{1}x_{2}&0&x_{2}\end{bmatrix}^{\top}.$

Run an experiment with initial condition $x_{0}=[-0.1~{}~{}0.1]^{\top}$ and
input $u=\sin t$ from $t=0$ to $t=10$. The sampling period is $0.5$.

The data $X_{1}$ is corrupted by a noise $D_{0}=0.05X_{1}$. We set $T=12$ and
$R_{D}R_{D}^{\top}=0.1X_{1}X_{1}^{\top}$. SOS programs are then formulated to
search for $P$, $\epsilon_{2}$, and the coefficients of $\epsilon_{1}(x)$ and
$Y(x)$ with both $\epsilon_{1}(x)$ and $Y(x)$ having degree $2$.

First, we apply the special case of Theorem 1 where the input vector field is
independent of $x$ (Remark 1). The SOSTOOLS obtains the solution

$\displaystyle P$ $\displaystyle=\begin{bmatrix}8.67\times
10^{-7}&-2.199\times 10^{-7}\\\ -2.199\times 10^{-7}&1.925\times
10^{-7}\end{bmatrix},$ $\displaystyle\epsilon_{1}(x)$
$\displaystyle=x_{1}(5.203\times 10^{-5}x_{1}+1.627\times 10^{-6}x_{2})$
$\displaystyle\quad+x_{2}(1.627\times 10^{-6}x_{1}+5.39\times 10^{-5}x_{2}),$
$\displaystyle\epsilon_{2}$ $\displaystyle=5.71\times 10^{-7}.$

The stabilizer is calculated as

$\displaystyle u$
$\displaystyle=-x_{1}(9.2x_{1}^{2}+2.2x_{1}x_{2}+x_{1}+8.1x_{2}^{2}+0.24x_{2}+9.7)$
$\displaystyle\quad-
x_{2}(38x_{1}^{2}+6.1x_{1}x_{2}+5.5x_{1}+33x_{2}^{2}+2.3x_{2}+29).$

Using the design conditions in Theorem 2, the solution given by SOSTOOLS is

$\displaystyle P$ $\displaystyle=\begin{bmatrix}1.692\times
10^{-6}&1.311\times 10^{-7}\\\ 1.311\times 10^{-7}&1.104\times
10^{-6}\end{bmatrix},$ $\displaystyle\epsilon_{1}(x)$
$\displaystyle=x_{1}(9.165\times 10^{-6}x_{1}+6.389\times 10^{-10}x_{2})$
$\displaystyle\quad+x_{2}(6.389\times 10^{-10}x_{1}+9.258\times
10^{-6}x_{2}),$ $\displaystyle\epsilon_{2}$ $\displaystyle=1.061\times
10^{-6}.$

The designed stabilizer is

$\displaystyle u$ $\displaystyle=-x_{1}(3.7\times 10^{-4}x_{1}^{2}+5.1\times
10^{-5}x_{1}x_{2}-0.021x_{1}$ $\displaystyle\quad\quad\quad+2.2\times
10^{-4}x_{2}^{2}-0.092x_{2}+0.94)$ $\displaystyle\quad-x_{2}(-1\times
10^{-4}x_{1}^{2}-5.3\times 10^{-3}x_{1}x_{2}-3.1\times 10^{-3}x_{1}$
$\displaystyle\quad\quad\quad+5.5\times 10^{-4}x_{2}^{2}-0.064x_{2}+1.2).$

Alternatively, we can set $K(x)$ having degree $2$ as a decision variable and
utilize Corollary 1 to design the data-driven stabilizer. The solution to the
SOS program is

$\displaystyle P$ $\displaystyle=\begin{bmatrix}6.332\times
10^{-7}&-5.787\times 10^{-8}\\\ -5.787\times 10^{-8}&6.934\times
10^{-7}\end{bmatrix},$ $\displaystyle\epsilon_{1}(x)$
$\displaystyle=x_{1}(5.125\times 10^{-6}x_{1}+1.366\times 10^{-10}x_{2})$
$\displaystyle\quad+x_{2}(1.366\times 10^{-10}x_{1}+5.16\times 10^{-6}x_{2}),$
$\displaystyle\epsilon_{2}$ $\displaystyle=7.184\times 10^{-7}.$

The stabilizer is

$\displaystyle u$ $\displaystyle=-x_{1}(1.9\times 10^{-4}x_{1}^{2}-8.9\times
10^{-5}x_{1}x_{2}-0.026x_{1}$ $\displaystyle\quad\quad\quad+4.2\times
10^{-4}x_{2}^{2}-0.091x_{2}+1.5)$ $\displaystyle\quad-x_{2}(4.8\times
10^{-5}x_{1}^{2}-1.1\times 10^{-4}x_{1}x_{2}-3.6\times 10^{-3}x_{1}$
$\displaystyle\quad\quad\quad-2.4\times 10^{-4}x_{2}^{2}-0.072x_{2}+1.4).$

The phase portraits of the closed-loop systems under the designed controllers
are illustrated in Figures 1 to 3 respectively. The figures show that,
designed using the same data set, the controllers stabilize system (IV) at the
origin with different transient performances. The computational time needed
for formulating and solving the SOS program and then obtaining the control
gain $F(x)$ is $146.5957s$ for Theorem 1, $151.3056s$ for Theorem 2, and
$10.6662s$ for Corollary 1. This verifies that Corollary 1 is more
computational efficient than Theorems 1 and 2.

Figure 1: Phase portrait of the closed-loop system under the controller
designed using Theorem 1.
Figure 2: Phase portrait of the closed-loop system under the controller
designed using Theorem 2.
Figure 3: Phase portrait of the closed-loop system under the controller
designed using Corollary 1.

##  V Conclusion

We have shown how to synthesize stabilizers and Lyapunov functions for unknown
nonlinear polynomial systems starting from noisy data and using Lyapunov
second theorem. We do not assume the unknown noise to have any specific form
as long as it has a known quadratic bound over the experiment. The state-
dependent stabilizing gain is solved via SOS programs that are computationally
tractable. Interestingly, the efficiency of the stabilization design can be
improved by changing the data parameterization of the closed-loop system even
when the same design method is utilized.

Nonlinear data-driven stabilization is a fundamental and important problem
that lays the foundation for our ongoing works on nonlinear polynomial systems
such as local control with guaranteed domain of attraction and optimal control
with quadratic costs. Another interesting topic is looking into the
computational aspects of SOS programming and further improving the efficiency
of the data-driven designs. Our results can play an important role in learning
control policies for those manifold applications where polynomial systems and
SOS optimization have found wide use.

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