# Optimal distributed control with stability guarantees
by training a network of neural closed-loop maps

Danilo Saccani, Leonardo Massai, Luca Furieri and Giancarlo Ferrari-Trecate
This research has been supported by the Swiss National Science Foundation
under the NCCR Automation (grant agreement 51NF40_180545).The authors are with
the Institute of Mechanical Engineering, Ecole Polytechnique Fédérale de
Lausanne (EPFL), CH-1015 Lausanne, Switzerland. (email:
{l.massai,danilo.saccani,luca.furieri<EMAIL_ADDRESS>
###### Abstract

This paper proposes a novel approach to improve the performance of distributed
nonlinear control systems while preserving stability by leveraging Deep Neural
Networks (DNNs). We build upon the Neural System Level Synthesis (Neur-SLS)
framework and introduce a method to parameterize stabilizing control policies
that are distributed across a network topology. A distinctive feature is that
we iteratively minimize an arbitrary control cost function through an
unconstrained optimization algorithm, all while preserving the stability of
the overall network architecture by design. This is achieved through two key
steps. First, we establish a method to parameterize interconnected Recurrent
Equilibrium Networks (RENs) that guarantees a bounded $\mathcal{L}_{2}$ gain
at the network level. This ensures stability. Second, we demonstrate how the
information flow within the network is preserved, enabling a fully distributed
implementation where each subsystem only communicates with its neighbors. To
showcase the effectiveness of our approach, we present a simulation of a
distributed formation control problem for a fleet of vehicles. The simulation
demonstrates how the proposed neural controller enables the vehicles to
maintain a desired formation while navigating obstacles and avoiding
collisions, all while guaranteeing network stability.

##  I Introduction

Controlling interconnected systems with non-linear dynamics is a crucial task
across various fields ranging from robotics, and power grids, to social
sciences [1, 2, 3, 4]. However, achieving networked control becomes
significantly more challenging when the system dynamics are non-linear and a
convex function does not accurately capture the desired performance metric.

Indeed, several Optimal Distributed Control (ODC) approaches exist for the
case of linear systems with convex performance metrics [5, 6, 7, 8, 9];
however, for non-convex problems, solutions are scarce. Dissipativity-based
approaches, like passivity control [10, 11] and control by interconnection
[12], offer ways to design stabilizing distributed control policies without
minimizing a specific cost. However, these methods become prohibitive when it
comes to minimizing a user-defined cost function. Nonlinear Distributed Model
Predictive Control (DMPC) [13, 14] gives an alternative, but computing crucial
terminal ingredients in a distributed manner remains unsolved for non-linear
systems. Additionally, suitable distributed optimization algorithms for online
implementation are limited, often relying on the Alternating Direction Method
of Multipliers (ADMM) [15], whose potential lack of convergence poses
limitations in real-time applications.

Recently, there has been a surge of interest in using Deep Neural Networks
(DNNs) to parameterize highly non-linear distributed controllers. Approaches
like Graph Neural Networks (GNNs) have shown promise in designing such
policies [16, 17] in a scalable way. Nevertheless, stability guarantees are
often limited by restrictive assumptions. For instance, some methods require a
system to be initially linear and open-loop stable, with the trained network
maintaining small Lipschitz constants across layers [17]. These conditions
might not hold during exploration, potentially leading to system failures
before a safe policy is learned [18].

An alternative approach using deep Hamiltonian control policies offers built-
in closed-loop stability guarantees for specific systems [19]. However, these
policies only apply to a narrow class of systems with a particular skew-
symmetric interconnection structure.

#### Contributions

With this work, we provide two main contributions. First, we propose a novel
distributed state-feedback control scheme leveraging the Neural System Level
Synthesis (Neur-SLS) framework and dissipativity theory. The scheme is
composed of a parametrized family of nonlinear sub-operators, interconnected
according to a prescribed network structure that mimics the sparsity pattern
of the distributed system itself. This means that communication between sub-
operators only occurs between interconnected agents in the physical system. We
prove that closed-loop stability is guaranteed as long as the parameters
satisfy a Linear Matrix Inequality (LMI). Secondly, we present a free
parametrization of this networked ensemble of nonlinear sub-operators, with
each sub-operator modeled as a Recurrent Equilibrium Network (REN). The
parameterization is free in the sense that the resulting distributed
controller guarantees the closed-loop stability for any value of the
parameters. This eliminates the need to enforce LMIs, thereby enabling the
training of a distributed controller using unconstrained learning algorithms
akin to neural networks. This sparse neural controller structure facilitates
distributed implementation across individual subsystems while preserving
communication topology among the subsystems.

## Notation

Throughout the paper, we denote with $\mathbb{N}$ the set of non-negative
integers. $I_{a}$ ($0_{a}$) is the identity (zero) square matrix with
dimensions $a\times a$, $0_{a\times b}$ is the zero matrix with dimension
$a\times b$ and $\mathds{1}$ is the vector of all ones, where the dimension is
clear from the context. We denote with diag$(v)$ a square diagonal matrix with
the elements of vector $v$ on the main diagonal, while blkdiag$(A,B)$ denotes
a block diagonal matrix created by aligning the matrices $A$ and $B$ along the
diagonal. Positive semidefinite matrices $A$ are denoted as $A\succeq 0$. The
set of all sequences $\mathbf{v}=(v_{0},v_{1},v_{2},\dots)$, where
$v_{t}\in\mathbb{R}^{n}$ for all $t\in\mathbb{N}$, is denoted as $l^{n}$.
Moreover, $\mathbf{v}$ belongs to $l^{n}_{2}\subset l^{n}$ if
$\|\mathbf{v}\|_{2}=\left(\sum_{t=0}^{\infty}|v_{t}|^{2}\right)^{\frac{1}{2}}<\infty$.
We refer to $\mathbf{v_{T}}$ to denote a truncation of $\mathbf{v}$ with $t$
ranging from $0$ to $T$. An operator $\mathbf{A}:l^{n}\rightarrow l^{m}$ is
said to be causal if
$\mathbf{A}(\mathbf{x})=(A_{0}(\mathbf{x_{0}}),A_{1}(\mathbf{x_{1}}),\dots,A_{t}(\mathbf{x_{t}}),\dots)$.
If $A_{t}(\mathbf{x_{t}})=A_{t}(0,\mathbf{x_{t-1}}),$ then $\mathbf{A}$ is
said to be strictly causal. An operator $\mathbf{A}:l^{n}\rightarrow l^{m}$ is
said to be $\mathcal{L}_{2}$ stable if it is causal and
$\mathbf{A}(\mathbf{a})\in l^{m}_{2}$ for all $\mathbf{a}\in l^{n}_{2}$.
Equivalently, we write $\mathbf{A}\in\mathcal{L}_{2}$. Given an undirected
graph $\mathcal{G}=(\mathcal{V},\mathcal{E})$ described by the set of nodes
$\mathcal{V}$ and the set of edges
$\mathcal{E}\subset\mathcal{V}\times\mathcal{V}$, we denote set of neighbors
of node $i$, including $i$ itself by $\mathcal{N}_{i}=\\{i\\}\cup\\{j\ |\
\\{i,j\\}\in\mathcal{E}\\}\subseteq\mathcal{V}$. We denote with
col${}_{j\in\mathcal{V}}(v^{[j]})$ a vector which consists of the stacked
subvectors $v^{[j]}$ from $j=1$ to $j=|\mathcal{V}|$ and with
$v^{[\mathcal{N}_{i}]}$ a vector composed by the stacked subvectors $v^{[i]}$,
i.e. $v^{[\mathcal{N}_{i}]}=col_{i\in\mathcal{N}_{i}}(v^{[i]})$.

##  II Preliminaries

###  II-A Distributed nonlinear systems

We consider a network of $N$ interconnected nonlinear subsystems. The coupling
network among the subsystems is defined as an undirected communication graph
$\mathcal{G}=(\mathcal{V},\mathcal{E})$, with $\mathcal{V}=\\{1,\dots,N\\}$
representing the subsystems in the network, and the set of edges $\mathcal{E}$
containing the pairs of subsystems $\\{i,j\\}$, which can communicate with
each other. Each subsystem is of the form

$x_{t}^{[i]}=f^{[i]}(x^{[\mathcal{N}_{i}]}_{t-1},u^{[i]}_{t-1})+w^{[i]}_{t},\
\ \ \forall t\geq 1,$ (1)

where state and input of each subsystem $i\in\mathcal{V}$ are denoted by
$x^{[i]}\in\mathbb{R}^{n_{i}}$ and $u^{[i]}\in\mathbb{R}^{m_{i}}$
respectively. Moreover, $w^{[i]}\in\mathbb{R}^{n_{i}}$ is an unknown process
noise with $w^{[i]}_{0}=x^{[i]}_{0}$.
In operator form, we can express the subsystem (1) as:

$\mathbf{x}^{[i]}=\mathbf{F}^{[i]}(\mathbf{x}^{[\mathcal{N}_{i}]},\mathbf{u}^{[i]})+\mathbf{w}^{[i]},$
(2)

where $\mathbf{F}^{[i]}:l^{n_{\mathcal{N}_{i}}}\times l^{m_{i}}\rightarrow
l^{n_{i}}$ is a strictly causal operator defining the subsystem dynamics such
that
$\mathbf{F^{[i]}}(\mathbf{x^{[\mathcal{N}_{i}]},u^{[i]}})=\left(0,f^{[i]}\left(x^{[\mathcal{N}_{i}]}_{0},u_{0}^{[i]}\right),\dots,f^{[i]}\left(x^{[\mathcal{N}_{i}]}_{t},u^{[i]}_{t}\right),\dots\right)$.
By combining the local system dynamics in (1), the dynamics of the global
system result in

$x_{t}=f(x_{t-1},u_{t-1})+w_{t},\ \ \ \forall t\geq 0,$ (3)

where $x=col_{i\in\mathcal{V}}(x^{[i]})\in\mathbb{R}^{n}$,
$u=col_{i\in\mathcal{V}}(u^{[i]})\in\mathbb{R}^{m}$,$w=col_{i\in\mathcal{V}}(w^{[i]})\in\mathbb{R}^{n}$.
Similarly to the subsystems, we can rewrite system (3) in operator form as:

$\mathbf{x}=\mathbf{F}(\mathbf{x,u})+\mathbf{w},$ (4)

We make the following Assumption on the system to be controlled:

###### Assumption 1

We assume that the causal operator $\mathbf{F}$ is such that the map
$(\mathbf{w},\mathbf{u})\mapsto\mathbf{x}$ lies in $\mathcal{L}_{2}$, and that
the process noise $w(t)\sim\mathcal{D}(\mu,\sigma)$ is distributed according
to an unknown distribution $\mathcal{D}$, and that $\mathbf{w}$ belongs to
$l^{n}_{2}$.

The Assumption above means that the interconnection is stable or locally
controlled to achieve $\mathcal{L}_{2}$ stability. This is typically true in
practical applications where systems are either pre-stabilized or can be
stabilized using simple controllers. While these controllers ensure stability,
they often result in suboptimal closed-loop performance. Our goal is to
enhance performance while maintaining stability.

To control the system, we consider nonlinear state-feedback causal policies
parametrized as follows

$\mathbf{u}=\mathbf{K(x,\mathbf{u})}=(K_{0}(x_{0}),\dots,K_{t}(\mathbf{x}_{\mathbf{t}},\mathbf{u}_{\mathbf{t-1}}),\dots).$
(5)

Thanks to causality, each sequence of disturbances, denoted as $\mathbf{w}\in
l^{n}_{2}$, results in unique trajectories for the closed-loop system (4)-
(5). This implies that the mappings from disturbances to state and control,
denoted as $\mathbf{\Phi^{x}}[\mathbf{F,K}]$ and
$\mathbf{\Phi^{u}}[\mathbf{F,K}]$ respectively, are well-defined for any given
system $\mathbf{F}$ and controller $\mathbf{K}$. Thus, we express
$\mathbf{x=\Phi^{x}}[\mathbf{F,K}]\mathbf{(w)}$ and
$\mathbf{u=\Phi^{u}}[\mathbf{F,K}]\mathbf{(w)}$ for all $\mathbf{w}\in
l_{2}^{n}$.

###  II-B Problem formulation

Our goal is to address the following challenges in designing a control policy
$\mathbf{K}(\mathbf{x},\mathbf{u})$:

  1. bx-1

Ensuring the stability of the closed-loop mappings
$\mathbf{\Phi^{x}}[\mathbf{F,K}]$ and $\mathbf{\Phi^{u}}[\mathbf{F,K}]$ in the
$\mathcal{L}_{2}$ sense.

  2. bx-2

Enhancing the performance of the closed-loop system by minimizing a loss
function defined as:

$J=\mathbb{E}_{\mathbf{w}\sim\mathcal{D}}\left[\mathbf{l}(\mathbf{x_{T},u_{T}})\right]\>,$
(6)

where $\mathbf{l}$ is differentiable and $\mathbf{l}(\mathbf{x_{T},u_{T}})\geq
0$ for all $T\in\mathbb{N}$ and all $(x,u)\in\mathbb{R}^{n+m}$.

  3. bx-3

Ensuring that the resulting policy $\mathbf{K}(\mathbf{x},\mathbf{u})$ can be
implemented in a distributed manner exploiting a neighbor-to-neighbor
communication. Specifically, we require that

$\mathbf{u}^{[i]}=\mathbf{K}^{[i]}(\mathbf{x}^{[\mathcal{N}_{i}]},\mathbf{u}^{[\mathcal{N}_{i}]}),\quad\forall
i\in\mathcal{V}\,.$ (7)

###### Remark 1

Unlike most optimal control and dynamic programming problems that rely on a
sum of stage costs over time, our approach is more flexible. We only require
the loss function to be differentiable in its arguments. This allows us to
incorporate complex objectives typical of many robotics and reinforcement
learning tasks, such as logical specifications formulated using Signal
Temporal Logic (STL) [20].

We can now define the finite horizon ODC problem we aim to solve:

$\displaystyle\min_{\mathbf{K(\cdot)}}\quad$
$\displaystyle\mathbb{E}_{\mathbf{w}\sim\mathcal{D}}\left[\mathbf{l}(\mathbf{x_{T},u_{T}})\right]$
(8a) $\displaystyle\operatorname{subject~{}to}\quad$ $\displaystyle
w_{0}=x_{0},\quad\eqref{eq:nonlinsyslocal},~{}\eqref{eq:policy},~{}\eqref{eq:distributed_constraint},~{}\forall
i\in\mathcal{V},$ (8b)
$\displaystyle(\mathbf{\Phi^{x}[F,K],\Phi^{u}[F,K]})\in\mathcal{L}_{2}\>,$
(8c)

where (5), (7) impose the causality and sparsity structure of the obtained
policy while ensuring the satisfaction of (8c) guarantees the closed-loop
stability of the distributed system at a global network level.

###### Remark 2

Under Assumption 1, the problem (8) is always feasible (consider for example
$\mathbf{K=0})$.

###### Remark 3

Critically, (8c) enforces strict closed-loop stability at the network level
for all optimized distributed policies. However, in this paper we consider a
stronger requirement since we are interested in iteratively solving the
optimization problem (8) while ensuring fail-safe learning. This means
remaining within the feasible region defined by (8b) and (8c) at every
iteration, while improving the obtained policy by minimizing the loss function
(8a).

##  III Main result: Free parametrization of distributed stable neural closed-
loop maps

${\textstyle\mathbfcal{R}^{[1]}}(v^{[1]})$$\mathbfcal{R}^{[N]}(v^{[N]})$$\ddots$$M_{vz}$$\mathbf{z}$$\mathbf{v}$$\bm{u}$$\widehat{\mathbf{w}}$$M_{uz}$$\mathbfcal{R}$$M_{vw}$
Figure 1: Interconnection of $N$ operators $\mathbf{\mathbfcal{R}}^{[i]}$.

In this section, we will address the three main objectives we aim to achieve.
Firstly, we will begin by recalling the process of parametrizing stabilizing
control policies for the closed-loop system to establish a framework capable
of satisfying C-1 with a distributed structure C-3. Then, we will demonstrate
how a proper parametrization of the controller enables us to fulfill C-2 in an
unconstrained fashion.

###  III-A Networked controller

To achieve a distributed controller ensuring closed-loop stability, we first
recall the findings introduced in [21] for centralized control. Using insights
from internal model control [22], the main result of [21] is that under
Assumption 1, all and only stabilizing control policies for the system
$\mathbf{F}$ can be parameterized in terms of one stable operator
$\mathbfcal{R}\in\mathcal{L}_{2}$ by implementing the control input
$\mathbf{u}=\mathbfcal{R}(\widehat{\mathbf{w}})$ where
$\widehat{\mathbf{w}}=\mathbf{x}-\mathbf{F(x,u)}$ reconstructs the true
disturbances $\mathbf{w}$. The resulting closed loop system is given by

$\displaystyle\mathbf{x}=\mathbf{F(x,u)}+\mathbf{w},$
$\displaystyle\mathbf{u}=\mathbfcal{R}(\mathbf{x}-\mathbf{F(x,u)})\,,\quad\mathbfcal{R}\in\mathcal{L}_{2}\,,$
(9)

where $\mathbfcal{R}$ defines an overall control policy $\mathbf{K}$ such that
$\mathbf{u}=\mathbf{K}(\mathbf{x},\mathbf{u})=\mathbfcal{R}(\widehat{\mathbf{w}})$.

Beyond achieving closed-loop stability, the information constraints (7) of the
ODC problem (8) bring in additional challenges. Indeed, implementing a
controller $\mathbf{u}=\mathbfcal{R}(\widehat{\mathbf{w}})$ requires, at each
node $i\in\mathcal{V}$, information about other
$\widehat{\mathbf{w}}^{[i]}$’s, which in turn are reconstructed based on the
neighboring states $x^{[\mathcal{N}_{i}]}$’s and $u^{[i]}$’s, that is,
$\mathbf{w}^{[i]}=\mathbf{x}^{[i]}-\mathbf{F}^{[i]}(\mathbf{x}^{[\mathcal{N}_{i}]},\mathbf{u}^{[i]})$.
This raises the question of how the constraints in (7) impact the design of
$\mathbfcal{R}$.

Our solution is based on designing $N$ networked $\mathcal{L}_{2}$ sub-
operators $\mathbfcal{R}^{[i]}:l^{q_{i}}\rightarrow l^{r_{i}}$ whose
interconnection structure is shown in Fig. 1 and resulting in the overall
local controller shown in Fig. 2. Specifically, we aim to design sub-operators
with input $v^{[i]}\in\mathbb{R}^{q_{i}}$ and output
$z^{[i]}\in\mathbb{R}^{r_{i}}$ with $q_{i}\geq n_{i}$ and $r_{i}\geq m_{i}$,
$\forall i\in\mathcal{V}$ whose state-space definition is as follows:

$\displaystyle\xi^{[i]}_{t+1}=$
$\displaystyle\rho^{[i]}\left(\xi^{[i]}_{t},v^{[i]}_{t}\right)\,$
$\displaystyle z^{[i]}_{t}=$
$\displaystyle\chi^{[i]}\left(\xi^{[i]}_{t},v^{[i]}_{t}\right)\>,\quad\forall
i\in\mathcal{V}\,$ $\displaystyle\forall t\geq 0,$ (10)

where $\xi^{[i]}_{t}\in\mathbb{R}^{c}$ is the internal state. We assume that
each sub-operator has a finite $\mathcal{L}_{2}$-gain not grater than
$\gamma^{[i]}$, meaning that there exists a non-negative local storage
function $V^{[i]}$ such that for all $t\in\mathbb{N}$

$V^{[i]}(\xi^{[i]}_{t+1})-V^{[i]}(\xi^{[i]}_{t})\leq
s^{[i]}(v^{[i]}_{t},z^{[i]}_{t})\,,$ (11)

where $s^{[i]}(v^{[i]},z^{[i]})$ is the quadratic supply rate

$s^{[i]}(v^{[i]},z^{[i]})=\left[\begin{array}[]{l}v^{[i]}\\\
z^{[i]}\end{array}\right]^{\top}\underbrace{\left[\begin{array}[]{ll}\gamma^{[i]^{2}}I_{q_{i}}&0_{q_{i}\times
r_{i}}\\\ 0_{r_{i}\times
q_{i}}&-I_{r_{i}}\end{array}\right]}_{X^{[i]}}\left[\begin{array}[]{l}v^{[i]}\\\
z^{[i]}\end{array}\right].$ (12)

It is worth mentioning that (11) is the definition of dissipativity for the
system (10) with respect to the generic supply rate
$s^{[i]}(v^{[i]},z^{[i]})$. For the specific supply given in (12), we retrieve
the well-known concept of finite $\mathcal{L}_{2}$-gain property, which is
indeed a particular case of dissipativity (see [23]).

The interconnection structure of sub-operators shown in Fig. 1 is formalized
as follows:

$\left[\begin{array}[]{l}v\\\
u\end{array}\right]=\bar{M}\left[\begin{array}[]{l}z\\\
\widehat{w}\end{array}\right]=\left[\begin{array}[]{ll}M_{vz}&M_{vw}\\\
M_{uz}&0_{m\times n}\end{array}\right]\left[\begin{array}[]{l}z\\\
\widehat{w}\end{array}\right],$ (13)

where $v\in\mathbb{R}^{q}$ are the inputs to local controllers and
$z\in\mathbb{R}^{r}$ the outputs. The matrix $\bar{M}$ describes how each sub-
operator input $v^{[i]}$ is coupled with each sub-operator output $z^{[i]}$ as
well as with the exogenous input $w^{[i]}$ and output $u^{[i]}$. Specifically,
the matrix $M_{vz}$ describes how the input of each sub-operator is linked to
the outputs of the others. In practice, $M_{vz}$ is defined coherently with
the network structure defined by the graph $\mathcal{G}$ so that $v_{i}$
depends on $z_{\mathcal{N}_{i}}$.

To obtain an overall operator $\mathbfcal{R}$ with a finite $\mathcal{L}_{2}$
gain $\gamma_{R}>0$ between $\widehat{w}$ and $u$, we need to guarantee the
dissipativity of the interconnected operator $\mathbfcal{R}$, with respect to
the quadratic supply rate:

$\left[\begin{array}[]{l}\widehat{w}\\\
u\end{array}\right]^{\top}\underbrace{\left[\begin{array}[]{cc}\gamma_{R}^{2}I_{n}&0_{n\times
m}\\\ 0_{m\times
n}&-I_{m}\end{array}\right]}_{S}\left[\begin{array}[]{l}\widehat{w}\\\
u\end{array}\right].$ (14)

We are now in the position to state the following proposition guaranteeing the
$\mathcal{L}_{2}$ stability of the closed-loop (9).

###### Proposition 1

Consider the closed-loop system given by (9) where the controller operator
$\mathbfcal{R}(\widehat{\mathbf{w}})$ is composed of sub-operators
interconnected according to (13). Let Assumption 1 hold, then the system (9)
is $\mathcal{L}_{2}$ stable, i.e.,
$(\mathbf{\Phi^{x}[F,K],\Phi^{u}[F,K]})\in\mathcal{L}_{2}$ if there exist
scalars $\alpha^{[i]}\geq 0,i\in\mathcal{V}$ such that the following
inequality holds:

$\left[\begin{array}[]{ll}M_{vz}&M_{vw}\\\ I&0\\\ 0&I\\\
M_{uz}&0\end{array}\right]^{\top}\left[\begin{array}[]{cc}\mathbf{X}(\alpha^{[i]}X^{[i]})&0\\\
0&-S\end{array}\right]\left[\begin{array}[]{ll}M_{vz}&M_{vw}\\\ I&0\\\ 0&I\\\
M_{uz}&0\end{array}\right]\preceq 0\>,$

(15)

where $\mathbf{X}(\alpha^{[i]}X^{[i]})=\sum_{i=1}^{N}\alpha^{[i]}X^{[i]}$, and
the matrices $M_{vz}$, $M_{vw}$, $M_{uz}$ and $S$ are defined in (13) and
(15).

###### Proof:

Thanks to Assumption 1, from [21, Thm. 2] we have that
$(\mathbf{\Phi^{x}[F,K],\Phi^{u}[F,K]})\in\mathcal{L}_{2}$ for any
$\mathbfcal{R}\in\mathcal{L}_{2}$. Now we have to prove that the operator
$\mathbfcal{R}$ composed by interconnecting $\mathcal{L}_{2}$ stable operators
$\mathbfcal{R}^{[i]}$ as in (13) has a finite $\mathcal{L}_{2}$ gain if the
inequality (15) is satisfied. Following [11], we consider as candidate storage
function the weighted sum of the local storage functions for the whole
interconnection
$V(\xi)=\alpha^{[1]}V^{[1]}\left(\xi^{[1]}\right)+\cdots+\alpha^{[N]}V^{[N]}\left(\xi^{[N]}\right)$,
with $\alpha^{[i]}\geq 0$ and $\xi=col_{i\in\mathcal{V}}(\xi^{[i]})$. Thus, to
obtain a finite $\mathcal{L}_{2}$ gain between $\widehat{w}$ and $u$ we want
that the interconnection is dissipative with respect to the weighted sum of
the supply rates (12) and that this term is dominated by the performance
supply rate (14), i.e.,

$\sum_{i=1}^{N}\alpha^{[i]}\left[V^{[i]}(\xi^{[i]}_{t+1})-V^{[i]}(\xi^{[i]}_{t})\right]\leq\\\
\sum_{i=1}^{N}\alpha^{[i]}\left[\begin{array}[]{l}v^{[i]}\\\
z^{[i]}\end{array}\right]^{\top}X^{[i]}\left[\begin{array}[]{l}v^{[i]}\\\
z^{[i]}\end{array}\right]\leq\\\ \left[\begin{array}[]{l}\widehat{w}\\\
u\end{array}\right]^{\top}\underbrace{\left[\begin{array}[]{cc}\gamma_{R}^{2}I_{n}&0_{n\times
m}\\\ 0_{m\times
n}&-I_{m}\end{array}\right]}_{S}\left[\begin{array}[]{l}\widehat{w}\\\
u\end{array}\right]\leq 0.$ (16)

By rearranging the right-hand side of (16) we can rewrite it as:

$\left[\begin{array}[]{l}v\\\ z\\\ \widehat{w}\\\
u\end{array}\right]^{\top}\left[\begin{array}[]{cc}\mathbf{X}(\alpha^{[i]}X^{[i]})&0\\\
0&-S\end{array}\right]\left[\begin{array}[]{l}v\\\ z\\\ \widehat{w}\\\
u\end{array}\right]\leq 0\>.$ (17)

By substituting the interconnection (13), we obtain (15).

∎

Proposition 1 allows us to fulfill C-1 and C-3 by providing a condition that
ensures the closed-loop stability at the network level
$\eqref{eq:closedloopglobal}$ with a sparse and causal controller. We will
address C-2 in the next section.

$\mathbf{F}^{[i]}\left(\mathbf{x}^{[\mathcal{N}_{i}]},\mathbf{u}^{[i]}\right)$$\mathbfcal{R}^{[i]}(\cdot)$$\mathbf{w}^{[i]}$$\widehat{\mathbf{w}}^{[i]}$$\mathbf{x}^{[i]}$$+$$+$$\mathbf{x}^{[\mathcal{N}_{i}\backslash
i]}$$+$$-$Subsystem Local controller $\mathbf{u}^{[i]}$ Neighbour-to-neighbour
communication
$\mathbf{F}^{[i]}\left(\mathbf{x}^{[\mathcal{N}_{i}]},\mathbf{u}^{[i]}\right)$$\mathbf{x}^{[\mathcal{N}_{i}]}$$\mathbf{z}^{[\mathcal{N}_{i}\backslash
i]}$ Figure 2: Proposed local controller parametrizing stabilizing controllers
in terms of freely chosen sub-operator
$\mathbfcal{R}^{[i]}\in\mathcal{L}_{2}$.

###  III-B Free parametrization of networked RENs

Recently, finite-dimensional DNN approximations of nonlinear $\mathcal{L}_{p}$
operators [24, 25, 21] have been proposed. Specifically, as suggested in [21]
an operator $\mathbfcal{R}:l^{n}\rightarrow l^{m}$ is a REN if
$\mathbf{u}=\mathbf{\mathbfcal{R}(\widehat{w})}$ is generated by the following
dynamical system with input $\widehat{w}\in\mathbb{R}^{n}$ and output
$u\in\mathbb{R}^{m}$:

$\left[\begin{array}[]{l}\xi_{t}\\\ \nu_{t}\\\
u_{t}\end{array}\right]\underbrace{\left[\begin{array}[]{lll}A_{1}&B_{1}&B_{2}\\\
C_{1}&D_{11}&D_{12}\\\
C_{2}&D_{21}&D_{22}\end{array}\right]}_{Z(\theta,\gamma)}\left[\begin{array}[]{l}\xi_{t-1}\\\
\sigma(\nu_{t})\\\ \widehat{w}_{t}\end{array}\right],\ \ \ \xi_{-1}=0,$ (18)

where $\xi\in\mathbb{R}^{c}$ and $\nu\in\mathbb{R}^{s}$ are the hidden states
and the input of neurons of the REN, while
$\sigma:\mathbb{R}\rightarrow\mathbb{R}$ is the activation function applied
element-wise. Further, $\sigma(\cdot)$ must be piecewise differentiable and
with first derivatives restricted to the interval $[0,1]$. As highlighted in
[24, 25], RENs are es expressive as various existing DNN architectures.
However, when picking any random value for their parameters, the operator
$\mathbfcal{R}$ defined by (18) might not lie in the space $\mathcal{L}_{2}$.
In [24, 25] the authors introduce a smooth function $Z$ that takes
unconstrained training parameters $\theta\in\mathbb{R}^{n_{\theta}}$, for a
suitable $n_{\theta}$, and a prescribed $\mathcal{L}_{2}$ gain $\gamma>0$ map
them to a matrix $Z(\theta,\gamma)$, which defines (18). Notably, they ensure
that the resulting operator $\mathbfcal{R}[\theta,\gamma]$ is in
$\mathcal{L}_{2}$ and has $\mathcal{L}_{2}$-gain at most $\gamma$ for any
value of the parameters $\theta$. For this reason the parametrization induced
by $Z$ is termed free.

Since our focus is on using a networked operator as explained in Section
III-A, we consider an interconnection of RENs that replicates the sparsity of
the system. To meet requirement C-2 without dealing with a large-dimensional
constrained optimization problem, we aim to develop a free parametrization of
the networked operator. This means finding a parametrization that allows any
parameter value while still meeting the constraint. To achieve this, we
consider the following dynamics of the interconnected sub-operators
$\mathbfcal{R}^{[i]}$:

$\left[\begin{array}[]{l}\xi^{[i]}_{t}\\\ \nu^{[i]}_{t}\\\
z^{[i]}_{t}\end{array}\right]\underbrace{\left[\begin{array}[]{lll}A_{1}^{[i]}&B_{1}^{[i]}&B_{2}^{[i]}\\\
C_{1}^{[i]}&D_{11}^{[i]}&D_{12}^{[i]}\\\
C_{2}^{[i]}&D_{21}^{[i]}&D_{22}^{[i]}\end{array}\right]}_{Z^{[i]}(\theta^{[i]},\gamma^{[i]})}\left[\begin{array}[]{l}\xi^{[i]}_{t-1}\\\
\sigma(\nu^{[i]}_{t})\\\ v^{[i]}_{t}\end{array}\right],$ (19)

where $\xi^{[i]}_{-1}=0$, $\forall i\in\mathcal{V}$, and
$Z^{[i]}(\theta^{[i]},\gamma^{[i]})$ defining the free parametrization of the
local REN model (19). Before entering the details of the proposed
parametrization, we consider the following Assumption on the chosen
interconnection for the network of RENs.

###### Assumption 2

We assume a particular structure for the matrices $M_{vw}$ and $M_{uz}$, more
in detail:

  1. (a)

we assume that there exists a permutation matrix $C\in[0,1]^{q\times q}$ such
that

$\displaystyle CM_{vw}=\begin{bmatrix}I_{n}\\\ 0_{(q-n)\times
n}\end{bmatrix}\>.$ (20)

Notice that this also implies that $M_{vw}$ is a positive-valued semi-
orthogonal matrix satisfying $M_{vw}^{\top}M_{vw}=I_{n}$.

  2. (b)

We assume that $M_{uz}$ is semi-orthogonal, i.e., it satisfies

$\displaystyle M_{uz}^{\top}M_{uz}=H,\quad
H=\operatorname{diag}(h),\>\>h\in\mathbb{R}^{r}\>.$ (21)

###### Remark 4

A few comments on Assumption 1 are in order. With (a) we allow the input of
one or more sub-operators to depend on the exogenous input. Although the
structure of $M_{vw}$ is seemingly restrictive, it is general enough to
capture the way exogenous inputs affect most physical systems. More in detail,
the assumption covers all cases where each sub-system can have inputs
depending on different exogenous inputs. What it does not cover, is the case
in which different sub-systems have inputs depending on the combination of
_common_ exogenous inputs. We aim to relax this assumption in future work.
Condition (b) states that the columns of $M_{uz}$ are mutually orthogonal,
which in turn implies that the output of the operator $\mathbfcal{R}$ is
comprised of linearly independent combinations of the sub-operators output
$z$. This is fairly general and gives one enough leeway on how the output $u$
is chosen. As a particular case, we have $M_{uz}=I_{r}$ where the output of
the operator coincides with the output of all sub-operators.

Before presenting our main result, it is useful to define the set of such sub-
operators that takes at least one exogenous input, which will be called _out-
connected_. First of all, we define

$\displaystyle\mathcal{A}_{u_{i}}=\mbox{set of indices of $u$ associated with
$i\in\mathcal{V}$},$ (22) $\displaystyle\mathcal{A}_{z_{i}}=\mbox{set of
indices of $z$ associated with $i\in\mathcal{V}$}.$ (23)

With this, we can define

$\displaystyle\mathcal{A}_{u_{i}}^{1}$
$\displaystyle=\left\\{k\in\mathcal{A}_{u_{i}}\>|\>(M_{vw}\mathds{1})_{k}=1\right\\},\quad
i\in\mathcal{V},$ (24) $\displaystyle\mathcal{A}_{u_{i}}^{0}$
$\displaystyle=\left\\{k\in\mathcal{A}_{u_{i}}\>|\>(M_{vw}\mathds{1})_{k}=0\right\\},\quad
i\in\mathcal{V}.$ (25)

These are those indices of $u_{i}$ that are associated with the exogenous
inputs and those that are not respectively. If $\mathcal{A}_{u_{i}}^{1}$ is
not empty, we say that the sub-operator $i$ is out-connected. The set of out-
connected operators is

$\displaystyle\mathcal{V}^{1}=\left\\{i\in\mathcal{V}\>|\>\mathcal{A}_{u_{i}}^{1}\neq\emptyset\right\\}\>.$
(26)

We are now able to present our main result concerning a free parametrization
of distributed $\mathcal{L}_{2}$-stable operators $\mathbfcal{R}$ composed by
a network of RENs.

###### Theorem 1

Consider a set $\mathcal{V}=\\{1,\dots,N\\}$ of sub-operators
$\mathbfcal{R}^{[i]}[\theta^{[i]},\gamma^{[i]}],\>i\in\mathcal{V}$, each
dissipative with respect to the supply (12), and interconnected according to
(13). By choosing the $\mathcal{L}_{2}$ gain $\gamma^{[i]}$ of each sub-
operator $i\in\mathcal{V}$ as follows:

(27)

where

$\displaystyle\alpha^{[i]}$
$\displaystyle=h^{[i]}+\max_{j\in\mathcal{A}_{z_{i}}}\left(\sum_{k}|m_{kj}|\right)+b^{[i]^{2}},\>\forall
i\in\mathcal{V},$ (28)

the interconnected system composed by the sub-operators
$\mathbfcal{R}^{[i]}[\theta^{[i]},\eta(b^{[i]})]$ satisfy condition (15) for
all $\theta^{[i]},b^{[i]}$, and the resulting operator
$\mathbfcal{R}[\theta,\gamma_{R}]$ lies in $\mathcal{L}_{2}$.

The proof of Theorem 1 is moved to the Appendix for the sake of readability.
Theorem 1, together with Proposition 1, allows us to fulfill C-2 by enabling
an unconstrained parametrization of $\mathcal{L}_{2}$-stable distributed
operators, effectively removing the need to enforce the hard constraint 8c
which, thanks to Proposition 1, led to the inequality (15). This is important
as it allows the use of parametrized families of highly non-linear operators
such as RENs, which can be optimized very efficiently by solving an
unconstrained optimization problem over the free parameters
$\theta^{[i]},b^{[i]}$ with standard backpropagation and unconstrained
gradient descent methods.

###### Remark 5

Notice that Theorem 1 generalizes [26, Thm. 1]. In that paper, we considered
the special case where $M_{uz}^{\top}M_{uz}=I_{r}$ and $M_{vw}=I_{q}$. The
latter condition implies that $w$ can only be an additive term in each sub-
operator’s input $v$ (note that this also implies that $w$ is constrained to
have the same dimension as the input $v$). In such a special case, we have
$\mathcal{V}^{1}=\mathcal{V},\mathcal{A}_{u_{i}}^{1}=\mathcal{A}_{u_{i}}$ and
$\mathcal{A}_{u_{i}}^{0}=\emptyset\>\forall i\in\mathcal{V}$, therefore (27)
reduces to

$\displaystyle\eta(b^{[i]})=\sqrt{\frac{1}{\alpha^{[i]}}\frac{\gamma_{R}^{2}}{\max_{j\in\mathcal{A}_{u_{i}}}\left(\sum_{k}|m_{jk}|\right)\gamma_{R}^{2}+1}},\>\>\forall
i\in\mathcal{V}\>,$ (29)

which is the parametrization originally proposed in [26].

A detailed description of how Theorem 1 can be leveraged for the training of a
network of $\mathcal{L}_{2}$-bounded RENs, is provided by Algorithm 1.

Algorithm 1 Training of networked $\mathcal{L}_{2}$ stable RENs

Input: Input data: initial conditions $\\{w_{0}^{j}\\}_{j=1}^{n_{exp}}$
sampled from $\mathcal{D}$, system model $f(x,u)$, learning rate $\eta$,
number of epochs $E$.

Initialize randomly the parameters of each subsystem: $\theta^{[i]}$,
$b^{[i]}$, $i\in\mathcal{V}$ and $\gamma_{R}$ .

for $e=1$ to $E$ do

for $j=1$ to $n_{exp}$ do

Initialize hidden state $\xi^{[i]},\>\>\forall i\in\mathcal{V}$.

Initialize control action $u^{[i]},\>\>\forall i\in\mathcal{V}$.

Compute $\gamma^{[i]}=\eta(b^{[i]}),\>\>\forall i\in\mathcal{V}$ using Eq.
(27).

for $t=0$ to $T$ do

Simulate system (3)

Reconstruct the noise $\widehat{w}$

Compute the inputs:

$u^{[i]}_{t}=M^{[i]}_{uz}\left(C^{[i]}_{2}\xi^{[i]}_{t-1}+D^{[i]}_{21}\sigma(\nu^{[i]}_{t})+D_{22}^{[i]}\widehat{w}^{[i]}_{t}\right)$

end for

end for

Compute loss
$\mathbb{E}_{\mathbf{w}\sim\mathcal{D}}\left[\mathbf{l}(\mathbf{x_{T},u_{T}})\right]$.

Backpropagate gradients through time (see [27]) to update parameters via
gradient descent with learning rate $\eta$.

end for

Output: Trained model parameters $\theta_{i},b^{[i]}$, $i\in\mathcal{V}$.

###### Remark 6

There is usually a “price to pay” for transforming the problem (8) into an
unconstrained one via a free parametrization like the one we derived with
Theorem 1. Mainly, this stems from the fact that the map $\eta$ will only
output a subset of all possible $\mathcal{L}_{2}$-gains such that the overall
operator $\mathbfcal{R}$ lies in $\mathcal{L}_{2}$, which in turn means that
we are restricting ourselves to optimize over a subset of
$\mathcal{L}_{2}$-bounded operators. In particular, this comes from the use of
sufficient but not necessary conditions, like Gershgorin Theorem, to ensure
that (15) is satisfied and the reader can take a look at the proof in the
Appendix for the details. Finding less conservative methods for guaranteeing
the satisfaction of (15) is still a matter of ongoing research. Despite this,
as it is shown in [24], the advantages of having a free parametrization
greatly outweigh such a drawback.

###### Remark 7

Notice that, due to the assumption made regarding the structure of the
matrices $M_{vz}$ and $M_{vw}$, the computation of each $\gamma^{[i]}$
according to (1) only necessitates knowledge of the coupling matrices $M_{uz}$
and $M_{vz}$. Therefore, by assuming suitable separability properties of the
costs among agents, the proposed algorithm could be adapted to a distributed
implementation. The distributed design of local controllers is not within the
scope of this paper and will be addressed in future research.

##  IV Numerical example

In this section, we demonstrate the effectiveness of the proposed approach in
addressing a complex control problem. We use PyTorch to build and train our
distributed neural controller, employing stochastic gradient descent method.
The implementation of our approach can be found at
https://github.com/DecodEPFL/Distributed_neurSLS.
The system we are considering consists of four point-mass vehicles (see Fig.
3), each with a position denoted as $p^{[i]}\in\mathbb{R}^{2}$ and velocity
represented as $v^{[i]}\in\mathbb{R}^{2}$. Each vehicle experiences friction
forces, which leads to the discrete-time model:

$\left[\begin{array}[]{l}{p}^{[i]}_{t+1}\\\ {v}^{[i]}_{t+1}\\\
\end{array}\right]=\left[\begin{array}[]{l}{p}^{[i]}_{t}\\\
{v}^{[i]}_{t}\end{array}\right]+T_{s}\left[\begin{array}[]{c}v^{[i]}_{t}\\\
m^{[i]^{-1}}\left(-c^{[i]}|v^{[i]}_{t}|_{2}+F^{[i]}_{t}+u^{[i]}_{t}\right)\end{array}\right],$

(30)

where, $m^{[i]}>0$ is the vehicle mass, $c^{[i]}>0$ is the friction
coefficient, $u^{[i]}$ is the local control input, $T_{s}>0$ represents the
sampling time and $F^{[i]}\in\mathbb{R}^{2}$ stands for the action of a local
base controller whose goal is to maintain a specific formation between the
agents, which is described by a fixed distance $\delta^{[i]}>0$ between them.
More specifically, the base controller stabilizes each agent around a final
reference point $\bar{p}^{[i]}$, by applying the following force:

$F^{[i]}=-\sum_{j=1}^{|\mathcal{N}_{i}\backslash
i|}k^{[ij]}(|p^{[i]}-p^{[j]}|_{2}-\delta^{[i]})-k^{[ir]}|\bar{p}^{[i]}-p^{[i]}|_{2},$
(31)

where $,k^{[ij]},k^{[ir]}>0$ are the gains used in the local controllers. The
overall system obtained thus consists of nonlinear coupled subsystems (30)-
(31). These subsystems can be written as (1), for which the origin is a stable
equilibrium. We chose the loss function in (6) as:

$\displaystyle l(\mathbf{x_{T}},\mathbf{u_{T}})$
$\displaystyle=\sum^{T}_{t=0}l(x_{t},u_{t}),$ (32) $\displaystyle
l(x_{t},u_{t})$
$\displaystyle=l_{traj}(x_{t},u_{t})+l_{ca}(x_{t})+l_{obs}(x_{t})+l_{form}(x_{t}),$

where, $l_{traj}(x,u)=[x^{\top}\ u^{\top}]Q[x^{\top}\ u^{\top}]^{\top}$ with
$Q\succeq 0$ penalizes the distance between agents and their reference, as
well as the magnitude of the control action. The other terms, $l_{ca}(x)$,
$l_{obs}(x)$ and $l_{form}(x_{t})$ instead, are barrier functions that
penalize collisions between agents and obstacles, and incentivize maintaining
the desired formation, respectively (see Fig. 3). The expected value in the
cost function (8a) is approximated by an empirical average (see [21]) over
$10$ randomly sampled initial conditions $\\{w_{0}^{s}\\}_{s=1}^{10}$. The
parameters employed in the simulation are listed in Tab I. Fig. 3 shows the
closed loop trajectories of the controlled system. The goal promoted by the
cost $l(x,u)$ is to coordinate the formation in order to pass through a narrow
valley avoiding collision with each others while keeping the connectivity
required by the formation control. The agents start from a randomly sampled
initial position marked with “$\circ$” and have to reach the target position
marked with “$\star$’".

Figure 3: Closed-loop trajectories after training over randomly sampled initial conditions marked with “$\circ$” (filled transparent “$\circ$” are the sampled $w_{0}$ used for the training). Images captured at time instants $\tau_{1}=0.3$ $s$, $\tau_{2}=8.5$ $s$ and $\tau_{3}=25$ $s$. Colored (gray) lines depict the trajectories within $[0,\tau_{i}]$ ($[\tau_{i},\infty)$). Colored circles (and their radius) represent the agents (and their size for collision avoidance), while colored “$\star$” symbols represent the reference position and green ellipsoids are the obstacles. $m^{[i]}$ | $1\ kg$ | $c^{[i]}$ | $1\ Ns/m$
---|---|---|---
$k^{[ij]}$ | $1\ N/m$ | $k^{[ir]}$ | $1\ N/m$
$\delta^{[1]}=\delta^{[3]}$ | $4\ m$ | $\delta^{[2]}=\delta^{[4]}$ | $1.5\ m$
$T_{s}$ | $0.05\ s$ | $Q$ | blkdiag$(I_{16},0.01\cdot I_{8})$
$\eta$ | $0.001$ | $E$ | $1500$
$\xi^{[i]}$ | $25,\ \forall i\in\mathcal{V}$ | $\nu^{[i]}$ | $25$
TABLE I: List of parameters employed in the simulation $\forall
i,j\in\mathcal{V}$.

##  V Conclusions

This paper introduces a novel distributed control method for interconnected
nonlinear systems. By harnessing the framework of neural system level
synthesis and dissipativity theory, we achieve optimized performance for user-
defined cost functions while guaranteeing stability throughout the training
process at the global network level. A key feature is the ability to train
local control policies using unconstrained optimization algorithms. We hope
that the proposed framework will pave the way for applications in several
domains where achieving both stability and optimal performance has
traditionally been difficult, such as smart-grid control [28], flocking
control for autonomous vehicles [29], and distributed optimization algorithm
design [30]. Further important theoretical developments include the addition
of noisy outputs, unknown system dynamics, and parallelizing the training
across agents.

###  -A Proof of Theorem 1

Firstly we rewrite $\mathbf{X}(\alpha^{[i]}X^{[i]})$ in a more compact form
as:

$\displaystyle\mathbf{X}(\alpha^{[i]}X^{[i]})=$
$\displaystyle\left[\begin{array}[]{ll}\Pi_{N,v}&0\\\
0&-\Pi_{N,z}\end{array}\right]\>,$ (35)

with
$\Pi_{N,v}=\text{blkdiag}\\{\alpha^{[1]}\gamma^{[1]^{2}}I_{q_{1}},\dots,\alpha^{[N]}\gamma^{[N]^{2}}I_{q_{N}}\\}$
and
$\Pi_{N,z}=\text{blkdiag}\\{\alpha^{[1]}I_{r_{1}},\dots,\alpha^{[N]}I_{r_{N}}\\}$.
By Proposition 1, (15) needs to be satisfied, yielding:

$\displaystyle\left[\begin{smallmatrix}-M_{vz}^{\top}\Pi_{N,v}M_{vz}+\Pi_{N,z}-M_{uz}^{\top}M_{uz}&-M_{vz}^{\top}\Pi_{N,v}M_{vw}\\\\[8.61108pt]
-M_{vw}^{\top}\Pi_{N,v}M_{vz}&-M_{vw}^{\top}\Pi_{N,v}M_{vw}+\gamma_{R}^{2}I_{n}\end{smallmatrix}\right]\succ
0\>.$ (36)

By applying the Schur complement, the above condition is equivalent to

$\displaystyle-M_{vz}^{\top}\Pi_{N,v}M_{vz}+\Pi_{N,z}-M_{uz}^{\top}M_{uz}$
$\displaystyle-M_{vz}^{\top}\Pi_{N,v}M_{vw}[-M_{vw}^{\top}\Pi_{N,v}M_{vw}$
(37) $\displaystyle+\gamma_{R}^{2}I_{n}]^{-1}M_{vw}^{\top}\Pi_{N,v}M_{vz}\succ
0$ $\displaystyle-M_{vw}^{\top}\Pi_{N,v}M_{vw}+\gamma_{R}^{2}I_{n}\succ 0\>.$
(38)

Using the structure of $M_{vw}$ given by Assumption 2, easy computations show
that $M_{vw}^{\top}\Pi_{N,v}M_{vw}=P_{1}\in\mathbb{R}^{n\times n}$ is a
diagonal matrix where the diagonal elements are given by
$\alpha^{[i]}\gamma^{[i]^{2}},\>\forall i\in\mathcal{V}^{1}$, each one
repeated as many times as the cardinality of $\mathcal{A}_{u_{i}}^{1}$.
Therefore, condition (38) reads as

$\displaystyle-M_{vw}^{\top}\Pi_{N,v}M_{vw}+\gamma_{R}^{2}I_{n}$
$\displaystyle\succ 0$ (39) $\displaystyle-P_{1}+\gamma_{R}^{2}I_{n}$
$\displaystyle\succ 0$
$\displaystyle\iff-\alpha^{[i]}\gamma^{[i]^{2}}+\gamma_{R}^{2}$
$\displaystyle>0$ $\displaystyle\alpha^{[i]}\gamma^{[i]^{2}}$
$\displaystyle<{\gamma_{R}^{2}},\forall i\in\mathcal{V}^{1}\>.$ (40)

As for condition (37) by collecting $M_{vz}$ we obtain

$M_{vz}^{\top}(-\Pi_{N,v}-\Pi_{N,v}M_{vw}[-M_{vw}^{\top}\Pi_{N,v}M_{vw}\\\
+\gamma_{R}^{2}I_{n}]^{-1}M_{vw}^{\top}\Pi_{N,v})M_{vz}+\Pi_{N,z}-M_{uz}^{\top}M_{uz}\succ
0$ (41)

By applying the Schur complement again this is equivalent to:

$\displaystyle\left[\begin{array}[]{ll}\Pi_{N,z}-H&M_{vz}^{\top}\\\\[8.61108pt]
M_{vz}&-D^{-1}\end{array}\right]$ $\displaystyle\succ 0$ (44) $\displaystyle
D$ $\displaystyle\prec 0\>,$ (45)

where we recall that $M_{uz}^{\top}M_{uz}=H$ is a diagonal matrix by
Assumption 2 and

$D=(-\Pi_{N,v}-\Pi_{N,v}M_{vw}[-M_{vw}^{\top}\Pi_{N,v}M_{vw}\\\
+\gamma_{R}^{2}I_{n}]^{-1}M_{vw}^{\top}\Pi_{N,v})\>.$ (46)

Using again the structure of $M_{vw}$ given in Assumption 2, it is easy to
show that $M_{vw}[-M_{vw}^{\top}\Pi_{N,v}M_{vw}\\\
+\gamma_{R}^{2}I_{n}]^{-1}M_{vw}^{\top}=P_{2}\in\mathbb{R}^{q\times q}$ is a
diagonal matrix such that
$(P_{2})_{jj}=(-\alpha^{[i]}\gamma^{[i]^{2}}+\gamma_{R}^{2})^{-1}$ if
$j\in\mathcal{A}_{u_{i}}^{1},\>i\in\mathcal{V}^{1}$ and $(P_{2})_{jj}=0$
otherwise. From this, we can see that $D$ is a diagonal matrix given by

$\displaystyle(D)_{jj}=\begin{cases}\displaystyle\frac{\alpha^{[i]}\gamma^{[i]^{2}}\gamma_{R}^{2}}{\alpha^{[i]}\gamma^{[i]^{2}}-\gamma_{R}^{2}}\quad&\mbox{for
}j\in\mathcal{A}_{u_{i}}^{1},\>i\in\mathcal{V}^{1}\\\
-\alpha^{[i]}\gamma^{[i]^{2}}\quad&\mbox{for
}j\in\mathcal{A}_{u_{i}}^{0},\>i\in\mathcal{V}\>.\end{cases}$ (47)

Notice that condition (45) is always satisfied when (40) holds. Finally, we
enforce (44) by applying the Gershgorin Theorem [31]. In particular, let us
focus on the first $p$ rows of the matrix (44). We want all the Gershgorin
circles to lie in the non-negative half-plane, i.e.

$\displaystyle\underbrace{\left(\Pi_{N,z}-H\right)_{jj}}_{\mbox{Center}}-\underbrace{\sum_{k}|{\left(M_{vz}^{\top}\right)_{jk}}|}_{\mbox{Radius}}\geq
0,\quad\forall j\in\mathcal{A}_{z_{i}},i\in\mathcal{V}.$ (48)

Notice that the Gershgorin center is $\alpha^{[i]}-1$ for every
$j\in\mathcal{A}_{z_{i}}$, hence condition (48) can be enforced by imposing
$\alpha^{[i]}$ such that:

$\displaystyle\alpha^{[i]}$
$\displaystyle=h^{[i]}+\max_{j\in\mathcal{A}_{z_{i}}}\left(\sum_{k}|m_{kj}|\right)+b^{[i]^{2}},\>\forall
i\in\mathcal{V},$ (49)

where $b^{[i]}\in\mathbb{R}$. Similarly, we can impose the other Gershgorin
circles associated with the remaining rows of the matrix (44) to lie in the
non-negative half of the plane. Using (47), this yields the conditions:

$\displaystyle 0$
$\displaystyle\leq\alpha^{[i]}\gamma^{[i]^{2}}\leq\frac{\gamma_{R}^{2}}{\max_{j\in\mathcal{A}_{u_{i}^{w}}}\left(\sum_{k}|m_{jk}|\right)\gamma_{R}^{2}+1},\>\forall
i\in\mathcal{V}^{1}$ (50) $\displaystyle 0$
$\displaystyle\leq\alpha^{[i]}\gamma^{[i]^{2}}\leq\frac{1}{\max_{j\in\mathcal{A}_{u_{i}^{0}}}\left(\sum_{k}|m_{jk}|\right)},\>\forall
i\in\mathcal{V}\>,$ (51)

notice that (50) ensures that (40) holds true. The conditions above can be
rewritten as

$\displaystyle 0$
$\displaystyle\leq\alpha^{[i]}\gamma^{[i]^{2}}\leq\min\left\\{\frac{\gamma_{R}^{2}}{\max_{j\in\mathcal{A}_{u_{i}}^{1}}\left(\sum_{k}|m_{jk}|\right)\gamma_{R}^{2}+1},\right.$
$\displaystyle\left.\frac{1}{\max_{j\in\mathcal{A}_{u_{i}}^{0}}\left(\sum_{k}|m_{jk}|\right)}\right\\},\>\>\forall
i\in\mathcal{V}^{1}$ (52) $\displaystyle 0$
$\displaystyle\leq\alpha^{[i]}\gamma^{[i]^{2}}\leq\frac{1}{\max_{j\in\mathcal{A}_{u_{i}}^{0}}\left(\sum_{k}|m_{jk}|\right)},\>\forall
i\in\mathcal{V}\setminus\mathcal{V}^{1}\>,$ (53)

where $\alpha^{[i]}$ is given by (49). From this, we readily obtain the
parametrization of the $\mathcal{L}_{2}$ gains given in (27). $\blacksquare$

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