# Distributed Consensus of Stochastic Multi-agent Systems with Prescribed
Performance Constraints

Pushpak Jagtap1 and Dimos V. Dimarogonas2 This work was partially supported by
the Wallenberg AI, Autonomous Systems and Software Program (WASP) funded by
the Knut and Alice Wallenberg Foundation, the Swedish Foundation for Strategic
Research (SSF), the Swedish Research Council (VR), the ERC CoG LEAFHOUND
project, and Robert Bosch Center for Cyber-Physical Systems.1Pushpak Jagtap is
with Robert Bosch Center for Cyber-Physical Systems, Indian Institute of
Science, Bangalore 560012, India<EMAIL_ADDRESS>Dimos V. Dimarogonas is
with the Division of Decision and Control Systems, KTH Royal Institute of
technology, SE-100 44 Stockholm, Sweden<EMAIL_ADDRESS>
###### Abstract

This paper focuses on the problem of distributed consensus control of multi-
agent systems while considering two main practical concerns $(i)$ stochastic
noise in the agent dynamics and $(ii)$ predefined performance constraints over
evolutions of multi-agent systems. In particular, we consider that each agent
is driven by a stochastic differential equation with state-dependent noise
which makes the considered problem more challenging compare to non-stochastic
agents. The work provides sufficient conditions under which the proposed time-
varying distributed control laws ensure consensus in expectation and almost
sure consensus of stochastic multi-agent systems while satisfying prescribed
performance constraints over evolutions of the systems in the sense of the
$q$th moment. Finally, we demonstrate the effectiveness of the proposed
results with a numerical example.

##  I Introduction

The past decade has witnessed an ever-growing interest in the study of multi-
agent systems (MAS) due to their extensive applications in both science and
engineering, see [1, 2], and references therein for examples. Among many
interesting problems, the synchronization or consensus problem of multi-agent
system is an active research topic in the past years [3, 4], whose objective
is to design a distributed consensus algorithm (or protocol) using only
limited neighborhood information to ensure that all the agents achieve some
common control objective, such as convergence to a common state.

In practice, stochastic disturbances, such as thermal noise, channel fading,
quantization effect during encoding and decoding, are inevitable and cannot be
avoided in real-world systems. Therefore, the consensus problems of stochastic
multi-agent systems (SMAS) have attracted much attention. Since the
traditional consensus definition is not applicable in a stochastic setting,
several researchers have proposed several consensus conceptions in different
probabilistic senses such as mean-square consensus, consensus in $q$th moment,
consensus in probability, and almost sure consensus. Examples of few results
include the mean-consensus protocol for SMAS [5], consensus in probability for
discrete-time SMAS [6], mean-square consensus control for time-varying SMAS
[7], exponential consensus of SMAS with delay [8], exponential leader-follower
consensus [9], approximate consensus of discrete-time SMAS [10], and average
consensus of SMAS in $q$th moment [11].

It is worth mentioning that the quality of performance, such as maximum
overshoot, rate of convergence, and steady-state error, is usually required to
be satisfied in practical systems. By considering such performance
constraints, authors in [12] proposed _prescribed performance control_ (PPC)
to ensure stability while respecting those constraints. Very recently many
researchers adapted the PPC approach to guarantee prescribed performance
constraints while solving various problems of multi-agent systems with
deterministic agents [13, 14, 15, 16]. On the other hand, there are a very few
works available on utilizing PPC for stochastic control systems [17, 18].
However, as far as we know, there is no work available in the literature on
the consensus of stochastic multi-agent systems while considering prescribed
performance constraints.

To the best of our knowledge, this paper is the first to address the consensus
control of stochastic multi-agent systems with prescribed performance
constraints. In this paper, we consider that the evolution of each agent is
given by a stochastic differential equation with state-dependent noise and is
driven by a conventional first-order consensus protocol [4] with an external
control input. Further, for a given communication graph topology and
predefined performance constraints, the paper proposes a time-varying
distributed control law that guarantees consensus in considered SMAS along
with the sufficient conditions over system and prescribed performance function
parameters. In particular, we proposed results considering two well-know
stochastic consensus notions: (i) consensus in expectation and (ii) almost
sure consensus.

The remainder of this paper is structured as follows. In Section II, we
introduce stochastic multi-agent systems and prescribed performance
constraints. Then, we formally define the problem considered in this paper.
Section III provides the sufficient conditions under which the proposed
distributed control law ensures stochastic consensus of SMAS while
guaranteeing prescribed performance constraints. Section IV demonstrates the
effectiveness of the results using a numerical example. Finally, Section V
concludes the paper.

##  II Preliminaries and Problem Statement

###  II-A Notations

Let the triplet $(\Omega,\mathcal{F},\mathds{P})$ denote a probability space
with a sample space $\Omega$, a filtration $\mathcal{F}$, and the probability
measure $\mathds{P}$. The filtration $\mathds{F}=(\mathcal{F}_{s})_{s\geq 0}$
satisfies the usual conditions of right continuity and completeness [19]. Let
$(W_{s})_{s\geq 0}$ be a $\mathds{F}$-Brownian motion. We use
$\mathds{E}[\cdot]$ to denote the expectation operator. The symbols
${\mathbb{N}}$, ${\mathbb{N}}_{0}$, ${\mathbb{R}}$, ${\mathbb{R}}^{+},$ and
${\mathbb{R}}_{0}^{+}$ denote the set of natural, nonnegative integer, real,
positive, and nonnegative real numbers, respectively. We use
${\mathbb{R}}^{n\times m}$ to denote a vector space of real matrices with $n$
rows and $m$ columns. We use $\|\cdot\|$ to represent Euclidean norm. For
$a\in{\mathbb{R}}$, we denote absolute value of $a$ by $|a|$. For
$a,b\in{\mathbb{R}}$ and $a<b$, we use $(a,b)$ to represent an open interval
in ${\mathbb{R}}$. For $a,b\in{\mathbb{N}}$ and $a\leq b$, we use $[a;b]$ to
denote a close interval in ${\mathbb{N}}$. We use $I_{n}$ and $0_{n}$ to
denote identity matrix and zero matrix in ${\mathbb{R}}^{n\times n}$,
respectively. A diagonal matrix in ${\mathbb{R}}^{n\times n}$ with diagonal
entries $d_{1},\ldots,d_{n}$ is denoted by $diag\\{d_{1},\ldots,d_{n}\\}$.
Given a matrix $M\in{\mathbb{R}}^{n\times m}$, $M^{T}$ represents transpose of
matrix $M$. Given a matrix $P\in{\mathbb{R}}^{n\times n}$,
$\operatorname{Tr}(P)$ represents the trace of matrix $P$, and $P>0$ and
$P\geq 0$ denote positive definite and semi-definite matrices, respectively.
Given a set $A$, we use $|A|$ to represent the cardinality of the set $A$. We
use notations $\mathcal{K}$ and $\mathcal{K}_{\infty}$ to denote different
classes of comparison functions, as follows:
$\mathcal{K}=\\{\psi:\mathbb{R}_{0}^{+}\rightarrow\mathbb{R}_{0}^{+}|$ $\psi$
is continuous, strictly increasing, and $\psi(0)=0\\}$;
$\mathcal{K}_{\infty}=\\{\psi\in\mathcal{K}|$
$\lim\limits_{r\rightarrow\infty}\psi(r)=\infty\\}$.

###  II-B Graph Theory

An undirected graph [3] is defined as
${\mathcal{G}}=({\mathcal{V}},{\mathcal{E}})$ with the vertices set
${\mathcal{V}}=\\{1,2,\ldots,n\\}$ and the edges set
${\mathcal{E}}=\\{(i,j)\in{\mathcal{V}}\times{\mathcal{V}}\mid
j\in\mathcal{N}_{i}\\}$, where $\mathcal{N}_{i}$ denotes the set of
neighbouring agents of agent $i$ that can communicate with agent $i$. Let us
index edges in set ${\mathcal{E}}$ as $e_{1},e_{2},\ldots,e_{m}$, where
$m=|{\mathcal{E}}|$ is the number of edges in the graph. A _path_ is a
sequence of edges that connects two different vertices. A graph is _connected_
if and only if there exist a path between any pair of vertices. A graph is a
_tree_ if and only if there exist exactly one path between any pair of
vertices. By assigning an arbitrary orientation to each edge of
${\mathcal{G}}$, we define the _incidence matrix_ , $D\in{\mathbb{R}}^{n\times
m}$, with the rows of $D$ being indexed by the vertices and columns being
indexed by edges; and the element $d_{ij}=1$ if the vertex $i$ is the head of
the edge $(i,j)$, $d_{ij}=-1$ if the vertex $i$ is the tail of the edge
$(i,j)$, and $d_{ij}=0$ otherwise. The _graph Laplacian_ of ${\mathcal{G}}$ is
described as $L=DD^{T}$. In addition, $L_{e}=D^{T}D$ is the so-called _edge
Laplacian_.

###  II-C System Description

In this work, we consider a multi-agent system with $n$ first-order stochastic
agents modeled by following stochastic differential equation:

$\displaystyle\operatorname{d}x_{i}=u_{i}\operatorname{d}t+g(x_{i})\operatorname{d}{W}_{t},\quad
i\in[1;n],$ (1)

where $x_{i}\in{\mathbb{R}}$, $u_{i}\in{\mathbb{R}}$ are the position and
control input of the $i$th agent, respectively and
$g:{\mathbb{R}}\rightarrow{\mathbb{R}}$ is a Lipschitz continuous diffusion
function with Lipschitz constant $\mathsf{k}_{g}\in{\mathbb{R}}_{0}^{+}$ such
that: $\|g(x)-g(x^{\prime})\|\leq\mathsf{k}_{g}\|x-x^{\prime}\|$ for all
$x,x^{\prime}\in{\mathbb{R}}$. Note that for the sake of simplicity, here we
consider one-dimensional agents and the result can be extended to higher
dimensions with the appropriate use of Kronecker product.

Let us consider that agents represent the vertices
${\mathcal{V}}=\\{1,2,\ldots,n\\}$ of an undirected graph ${\mathcal{G}}$. We
assume that the communication graph is static, i.e., neighbouring agents
$\mathcal{N}_{i}$ of agent $i\in[1;n]$ do not vary over time and each agent is
driven by a first-order consensus protocol with an external input
$v_{i}\in{\mathbb{R}}$, i.e.,
$u_{i}=-\sum_{j\in\mathcal{N}_{i}}(x_{i}-x_{j})+v_{i}$, with corresponding
stochastic differential equation given as:

$\displaystyle\operatorname{d}x_{i}=\Big{(}-\sum_{j\in\mathcal{N}_{i}}(x_{i}-x_{j})+v_{i}\Big{)}\operatorname{d}t+g(x_{i})\operatorname{d}{W}_{t},\quad
i\in{\mathcal{V}}.$ (2)

Let $x=[x_{1},\ldots,x_{n}]^{T}\in{\mathbb{R}}^{n}$ and
$v=[v_{1},\ldots,v_{n}]^{T}\in{\mathbb{R}}^{n}$ be stack vectors of absolute
positions and external inputs of all agents, respectively. Denote
$\overline{x}=[\overline{x}_{1},\ldots\overline{x}_{m}]^{T}\in{\mathbb{R}}^{m}$
the stack vector of relative positions between the pair of communicating
agents $(i,j)=e_{k}\in{\mathcal{E}}$, where $\overline{x}_{k}\triangleq
x_{ij}=x_{i}-x_{j}$ such that $(i,j)=e_{k}\in{\mathcal{E}},k\in[1;m]$. We also
mention some interesting properties which are useful in the paper:
$Lx=D\overline{x}$, $\overline{x}=D^{T}x$, and if $\overline{x}=0$, we have
$Lx=0$. By stacking agents in (2), the dynamics of the stochastic multi-agent
system is rewritten as:

$\displaystyle\operatorname{d}x=(-Lx+v)\operatorname{d}t+G(x)\operatorname{d}W_{t},$
(3)

where $L$ is the graph Laplacian and $G(x):=[g(x_{1}),\ldots,g(x_{n})]^{T}$.

Next, we introduce stochastic consensus notions for stochastic multi-agent
systems which are adapted from [20].

######  Definition II.1 (Consensus in expectation)

The agents in the stochastic multi-agent system (3) are said to reach
consensus in expectation if the following holds

$\displaystyle\lim_{t\rightarrow\infty}\mathbb{E}[\overline{x}(t)^{T}\overline{x}(t)]=0,$
(4)

where $\overline{x}$ is the stack vector of relative positions.

######  Definition II.2 (Almost sure consensus)

The agents in the stochastic multi-agent system (3) are said to reach almost
sure (a.s.) consensus if the following holds
$\lim_{t\rightarrow\infty}\overline{x}(t)=0,$ almost surely (i.e.,
$\mathbb{P}[\lim_{t\rightarrow\infty}\overline{x}(t)=0]=1$).

For later use, we recall the infinitesimal generator (denoted by the operator
$\mathcal{L}$) for a stochastic system
$S:\operatorname{d}x=f(x)\operatorname{d}t+g(x)\operatorname{d}W_{t}$,
$x\in{\mathbb{R}}^{n}$ using Itô’s differentiation [19]. Let
$V:{\mathbb{R}}^{n}\rightarrow{\mathbb{R}}_{0}^{+}$ be a twice differentiable
continuous function. The infinitesimal generator of $V$ associated with a
stochastic system $S$ is an operator, denoted by $\mathcal{L}V$, and given by

$\displaystyle\mathcal{L}V(x)=\frac{\partial V}{\partial
x}f(x)+\frac{1}{2}\operatorname{Tr}\Big{(}g(x)^{T}\frac{\partial^{2}V}{\partial
x^{2}}g(x)\Big{)},$ (5)

for all $x\in{\mathbb{R}}^{n}$.

######  Lemma II.3 ([19])

Consider a stochastic system
$S:\operatorname{d}x=f(x)\operatorname{d}t+g(x)\operatorname{d}W_{t}$, a twice
differentiable continuous function
$V:{\mathbb{R}}^{n}\rightarrow{\mathbb{R}}_{0}^{+}$, and a constant
$q\in{\mathbb{R}}^{+}$. If there exists a constant
$\kappa\in{\mathbb{R}}^{+}$, a function
$\overline{\psi}\in\mathcal{K}_{\infty}$, and a convex function
$\underline{\psi}\in\mathcal{K}_{\infty}$ such that for all
$x\in{\mathbb{R}}^{n}$ the following hold

$\displaystyle\underline{\psi}(\|x\|^{q})\leq
V(x)\leq\overline{\psi}(\|x\|^{q}),$ (6)
$\displaystyle\mathcal{L}V(x)\leq-\kappa V(x),$ (7)

then the solution of $S$ satisfies

$\displaystyle\mathds{E}[\|x(t)\|^{q}]\leq\underline{\psi}^{-1}(\overline{\psi}(\|x(0)\|^{q})\mathsf{e}^{-\kappa
t})$ (8)

for all $t\in{\mathbb{R}}_{0}^{+}$.

######  Lemma II.4 ([21])

For a stochastic system
$S:\operatorname{d}x=f(x)\operatorname{d}t+g(x)\operatorname{d}W_{t}$, if
there exist a twice differentiable continuous function
$V:{\mathbb{R}}^{n}\rightarrow{\mathbb{R}}_{0}^{+}$ satisfying (6) and

$\mathcal{L}V(x)\leq-\beta(x),$

where $\beta:{\mathbb{R}}^{n}\rightarrow{\mathbb{R}}_{0}^{+}$ is continuous
and nonnegative. Then, for each $x_{0}\in{\mathbb{R}}^{n}$,
$\lim\limits_{t\rightarrow\infty}\beta(x(t))=0$ a.s.

###  II-D Prescribed Performance

This subsection provides preliminary knowledge on the prescribed performance
control (PPC) [12]. The aim of PPC is to prescribe the evolution of the
relative position $\overline{x}_{k}(t)$ within some predefined region which
can be expressed in the form of the following inequality

$\displaystyle-\rho_{k}(t)<\overline{x}_{k}(t)<\rho_{k}(t)$ (9)

for all $t\in{\mathbb{R}}_{0}^{+}$, where
$\rho_{k}:{\mathbb{R}}_{0}^{+}\rightarrow{\mathbb{R}}^{+}$, $k\in[1;m]$ are
positive, smooth, and strictly decreasing performance functions that introduce
the desired predefined bounds for the relative positions. In this work, we
consider the following performance function

$\displaystyle\rho_{k}(t)=(\rho_{k0}-\rho_{k\infty})\mathsf{e}^{-\epsilon_{k}t}+\rho_{k\infty},$
(10)

where $\rho_{k0}$, $\rho_{k\infty}$, and $\epsilon_{k}$ are positive constants
with $\rho_{k0}>\rho_{k\infty}$ and
$\rho_{k\infty}=\lim_{t\rightarrow\infty}\rho_{k}(t)$ represents relative
positions at steady state. Now by normalizing $\overline{x}_{k}$ with respect
to the performance function $\rho_{k}$, we define the modulating error as
$\hat{x}_{k}(t)=\frac{\overline{x}_{k}(t)}{\rho_{k}(t)}$ and the corresponding
prescribed performance region
$\hat{\mathcal{D}}_{k}:=\\{\hat{x}_{k}\mid\hat{x}_{k}\in(-1,1)\\}$. Then the
modulated error is transformed through a transformation function
$T_{k}:\hat{\mathcal{D}}_{k}\rightarrow{\mathbb{R}}$ such that $T_{k}(0)=0$
and is chosen as

$\displaystyle
T_{k}(\hat{x}_{k})=\ln\Big{(}\frac{1+\hat{x}_{k}}{1-\hat{x}_{k}}\Big{)}.$ (11)

The transformed error is then defined as $\xi_{k}=T_{k}(\hat{x}_{k})$. By
differentiating $\xi_{k}$ with respect to time, we obtain transformed error
dynamics as

$\displaystyle\dot{\xi}_{k}=\phi_{k}(\hat{x}_{k},t)[\dot{\overline{x}}_{k}+\alpha_{k}(t)\overline{x}_{k}],$
(12)

where
$\phi_{k}(\hat{x}_{k},t):=\frac{1}{\rho_{k}(t)}\frac{2}{1-\hat{x}_{k}^{2}}>0$
for all $\hat{x}_{k}\in(1,-1)$ and
$\alpha_{k}(t):=-\frac{\dot{\rho}_{k}(t)}{\rho_{k}(t)}>0$ for all
$t\in{\mathbb{R}}_{0}^{+}$ are the normalized Jacobian of the transformation
function $T_{k}$ and the normalized derivative of the performance function
$\rho_{k}$, respectively. Note that, since the evolution of $\overline{x}_{k}$
is stochastic, the transform error dynamics can also be written as a
stochastic differential equation and the corresponding incremental form will
be given in (14). It can be verified that if the transformed error is bounded,
then the modulated error $\hat{x}_{k}$ is constrained within the region
$\hat{\mathcal{D}}_{k}$. This further implies that the error
$\overline{x}_{k}$ evolves within the predefined performance bound (9). Since
we are dealing with stochastic systems, we ensure the satisfaction of
considered performance constraints in the sense of $q$th moment by showing the
boundedness of $\mathbb{E}[\|\xi_{k}\|^{q}]$ (i.e. boundedness of $\xi_{k}$ in
$q$th moment).

###  II-E Problem Statement

In this paper, we are interested in designing a consensus control law for
stochastic multi-agent systems (3) such that they achieve consensus as defined
in Definition II.1 (or II.2) and the evolution of the relative positions
between neighboring agents should satisfy some prescribed performance bounds
in the sense of $q$th moment. Next, we formally define the problem.

###### Problem II.5

Given a multi-agent system defined by (3) with the communication graph
${\mathcal{G}}=({\mathcal{V}},{\mathcal{E}})$ and the prescribed performance
functions $\rho_{k}$, $k\in[1;m]$, as in (10), derive a distributed control
strategy such that the controlled multi-agent system achieves consensus in
expectation (or almost sure consensus) while satisfying prescribed performance
constraints (9) in the sense of $q$th moment.

##  III Consensus Control with Prescribed Performance Guarantees

In this section, we design the control law for the system (3) that guarantees
consensus while satisfying prescribed performance constraints in the sense of
$q$th moment. Here, we assume that the communicating agents can share
information about their performance functions $\rho_{k}$ and transformation
functions $T_{k}$, $k\in[1;m]$. This means the communication between agents is
bidirectional and the graph ${\mathcal{G}}$ is assumed to be undirected.

We first rewrite the dynamics of the considered multi-agent system in edge
space by multiplying (3) with $D^{T}$ on both sides as follows:

$\displaystyle\operatorname{d}\overline{x}=(-L_{e}\overline{x}+D^{T}v)\operatorname{d}t+D^{T}G(x)\operatorname{d}W_{t},$
(13)

where $L_{e}$ is the edge Laplacian which is positive definite if the graph is
a tree [22].

From (13), the transformed error dynamics in an incremental form is written as
follows:

$\displaystyle\operatorname{d}\xi=\big{(}\Phi_{t}(-L_{e}\overline{x}+D^{T}v+\alpha_{t}\overline{x})\big{)}\operatorname{d}t+\Phi_{t}D^{T}G(x)\operatorname{d}W_{t},$
(14)

where $\xi=[\xi_{1},\ldots,\xi_{m}]^{T}$,
$\Phi_{t}=diag\\{\phi_{1}(\hat{x}_{1},t),\ldots,$
$\phi_{m}(\hat{x}_{m},t)\\}$, and
$\alpha_{t}=diag\\{\alpha_{1}(t),\ldots\alpha_{m}(t)\\}$. Now by considering
augmented state-space $\eta=[\overline{x},\xi]^{T}$, we define an augmented
dynamics as:

$\displaystyle\operatorname{d}\eta=$
$\displaystyle\Big{(}\begin{bmatrix}-L_{e}&0_{m}\\\
-\phi_{t}(L_{e}+\alpha_{t})&0_{m}\end{bmatrix}\eta+\begin{bmatrix}I_{m}\\\
\phi_{t}\end{bmatrix}D^{T}v\Big{)}\operatorname{d}t$
$\displaystyle+\begin{bmatrix}I_{m}\\\
\phi_{t}\end{bmatrix}D^{T}G(x)\operatorname{d}W_{t}.$ (15)

In the next theorem, we provide a distributed control law and sufficient
conditions over system parameters under which we have a solution to Problem
II.5 for consensus in expectation.

###### Theorem III.1

Consider the stochastic multi-agent system (3) with the communication graph
being a tree, the predefined performance functions
$\rho_{k}:{\mathbb{R}}_{0}^{+}\rightarrow{\mathbb{R}}^{+},k\in[1;m]$ with
decay rates $\epsilon_{k}$ as in (10), the transformation functions
$T_{k},k\in[1;m]$ as in (11) with
$\overline{\epsilon}:=\max\limits_{k\in[1;m]}\epsilon_{k}$ satisfying

$\displaystyle
L_{e}(I_{m}+\gamma\Phi_{t}^{-1}+\Phi_{t}^{2})-\overline{\epsilon}I_{m}$
$\displaystyle\geq 0$ (16)

for all $t\in{\mathbb{R}}_{0}^{+}$ and a time-varying distributed control law

$\displaystyle
v_{i}(t)=-\sum_{k\in\Omega_{i}}\big{(}(\phi_{k}(\hat{x}_{k},t))^{2}\overline{x}_{k}+\xi_{k}\big{)},i\in{\mathcal{V}},t\in{\mathbb{R}}_{0}^{+},$
(17)

where $\Omega_{i}=\\{k\mid(i,j)=e_{k},j\in\mathcal{N}_{i}\\}$, i.e., the set
of all the edges that include agent $i\in{\mathcal{V}}$ as a node. If there
exist constants $\kappa,\gamma\in{\mathbb{R}}^{+}$ such that

$\displaystyle\Phi_{t}L_{e}$ $\displaystyle\geq 2\kappa I_{m},$ (18a)
$\displaystyle
L_{e}(I_{m}+\Phi_{t}^{2})-\frac{\mathsf{k}_{g}^{2}}{2}(I_{m}+\frac{1}{\gamma}\Phi_{t}^{2})$
$\displaystyle\geq 2\kappa I_{m},$ (18b)

for all $t\in{\mathbb{R}}_{0}^{+}$, where $\mathsf{k}_{g}$ is the Lipschitz
constant of the diffusion term $g(\cdot)$ in (1), then the controller in (17)
achieves consensus in expectation for any initial relative position
$\overline{x}_{k}(0)\in(-\rho_{k}(0),\rho_{k}(0)),\ k\in[1;m]$ while
guaranteeing prescribed performance in (9) in the sense of $2nd$ moment.

###### Proof:

Consider a Lyapunov-like function
$V:{\mathbb{R}}^{2m}\rightarrow{\mathbb{R}}_{0}^{+}$ as

$\displaystyle
V(\xi,\overline{x})=\frac{1}{2}\xi^{T}\xi+\frac{\gamma}{2}\overline{x}^{T}\overline{x},$

where $\gamma\in{\mathbb{R}}_{0}^{+}$ is a constant satisfying (18). One can
readily verify that the function $V$ satisfies condition (6) in Lemma II.3
with functions $\underline{\psi}(s):=\frac{1}{2}\min\\{1,\gamma\\}s$ and
$\overline{\psi}(s):=\frac{1}{2}\max\\{1,\gamma\\}s$ for all
$s\in{\mathbb{R}}_{0}^{+}$. The corresponding infinitesimal generator as
defined in (5) along the multi-agent system in edge space (13) and transformed
error dynamics (14) is given by

$\displaystyle\mathcal{L}V(\xi,\overline{x})$
$\displaystyle=\xi^{T}\Phi_{t}(-L_{e}\overline{x}+D^{T}v+\alpha_{t}\overline{x})$
$\displaystyle+\gamma\overline{x}^{T}(-L_{e}\overline{x}+D^{T}v)+\frac{1}{2}G(x)^{T}D\Phi_{t}^{T}\Phi_{t}D^{T}G(x)$
$\displaystyle+\frac{\gamma}{2}G(x)^{T}DD^{T}G(x).$

The stack vector of external inputs (17) is written as

$\displaystyle v(t)=-D\Phi_{t}^{2}\overline{x}-D\xi,\
t\in{\mathbb{R}}_{0}^{+}.$ (19)

By substituting control law (19) and using the Lipschitz continuity of
$g(x_{i})$, we get

$\displaystyle\mathcal{L}$ $\displaystyle
V(\xi,\overline{x})\leq-\xi^{T}\Phi_{t}L_{e}\overline{x}-\xi^{T}\Phi_{t}D^{T}D\Phi_{t}^{2}\overline{x}-\xi^{T}\Phi_{t}D^{T}D\xi$
$\displaystyle+\xi^{T}\Phi_{t}\alpha_{t}\overline{x}-\gamma\overline{x}^{T}L_{e}\overline{x}-\gamma\overline{x}^{T}D^{T}D\Phi_{t}^{2}\overline{x}$
$\displaystyle-\gamma\overline{x}^{T}D^{T}D\xi+\frac{1}{2}\mathsf{k}_{g}^{2}\overline{x}^{T}\Phi_{t}^{2}\overline{x}+\frac{\gamma}{2}\mathsf{k}_{g}^{2}(\overline{x}^{T}\overline{x})$
(20)

Since the considered graph is a tree, we know that the edge Laplacian
$L_{e}=D^{T}D$ is positive definite. We also know from the fact that
$\alpha_{k}(t)=-\frac{\dot{\rho_{k}}(t)}{\rho_{k}(t)}>0$ and (10) that
$\alpha_{k}(t)<\epsilon_{k}$ for all $t\in{\mathbb{R}}_{0}^{+}$ and hence
$\alpha_{t}<\overline{\epsilon}:=\max\limits_{k\in[1;m]}\epsilon_{k}$. With
the aforementioned facts, inequality (III) reduces to

$\displaystyle\mathcal{L}$ $\displaystyle V(\xi,\overline{x})\leq\
\xi^{T}\big{(}\Phi_{t}(-L_{e}(I_{m}+\gamma\Phi_{t}^{-1}+\Phi_{t}^{2})+\overline{\epsilon}I_{m})\big{)}\overline{x}$
$\displaystyle-\gamma\overline{x}^{T}\big{(}L_{e}(I_{m}+\Phi_{t}^{2})-\frac{\mathsf{k}_{g}^{2}}{2}(I_{m}+\frac{1}{\gamma}\Phi_{t}^{2})\big{)}\overline{x}-\xi^{T}\Phi_{t}L_{e}\xi.$

Note that $\Phi_{t}$ is a positive definite matrix and
$\xi_{k}(\overline{x}_{k}/\rho_{k})$ is strictly increasing with
$\xi_{k}(0)=0$ which implies that
$\xi_{k}(\overline{x}_{k}/\rho_{k})\overline{x}_{k}\geq 0$. Thus, by using
condition (16), we can readily verify that the first term is non-positive.
Further, by following conditions in (18), one obtains

$\mathcal{L}V(\xi,\overline{x})\leq-\kappa V(\xi,\overline{x})$

with a constant $\kappa\in{\mathbb{R}}^{+}$ satisfying (18). Now by following
the result of Lemma II.3, one ensures the consensus in expectation; and since
$\overline{x}_{k}(0)\in(-\rho_{k}(0),\rho_{k}(0)),\ k\in[1;m]$ and
$\mathds{E}[\|\xi(t)\|^{2}]$ is bounded for all $t\in{\mathbb{R}}_{0}^{+}$, we
ensure that the $\overline{x}$ satisfies prescribed transient constraints (9)
in the sense of $2nd$ moment. This concludes the proof. ∎

###### Remark III.2

Note that the proposed time-varying distributed control law (17) applied to
agents is the composition of the term based on the prescribed performance and
the relative positions of the neighbours.

###### Remark III.3

Since the term $\phi_{k}(\hat{x}_{k},t)$ is lower bounded by
$\min\limits_{t\in{\mathbb{R}}_{0}^{+},\hat{x}_{k}\in(-1,1)}\frac{1}{\rho_{k}(t)}\frac{2}{1-\hat{x}_{k}^{2}}=\frac{2}{\rho_{k0}}$,
$k\in[1;m]$ for all $t\in{\mathbb{R}}_{0}^{+}$, one can find a upper bound for
$\kappa$ satisfying inequalities (18) for appropriate values of $\gamma$. From
(18b), we would like to emphasize that smaller values of $\kappa$ imply that
one can handle stronger noise, i.e., bigger value of $\mathsf{k}_{g}$.
Moreover, since one can always find a constant $\gamma$ for any value of
$\overline{\epsilon}$ satisfying (16), we can use the proposed controller to
have a result for prescribed performance functions with any decay rate.

The next theorem provides result for ensuring almost sure consensus while
considering performance constraints.

###### Theorem III.4

Consider the stochastic multi-agent system (3) with the communication graph
being a tree, the predefined performance functions
$\rho_{k}:{\mathbb{R}}_{0}^{+}\rightarrow{\mathbb{R}}^{+},k\in[1;m]$ as in
(10), and a time-varying distributed control law

$\displaystyle
v_{i}(t)=-\sum_{k\in\Omega_{i}}(\phi_{k}(\hat{x}_{k},t))^{2}\overline{x}_{k},i\in{\mathcal{V}},t\in{\mathbb{R}}_{0}^{+}.$
(21)

If

$\displaystyle L_{e}-\frac{1}{2}k_{g}^{2}I_{m}>0,$ (22)

where $\mathsf{k}_{g}$ is the Lipschitz constant of the diffusion term
$g(\cdot)$ in (1), then the controller in (21) achieves almost sure consensus
for any initial relative position
$\overline{x}_{k}(0)\in(-\rho_{k}(0),\rho_{k}(0)),\ k\in[1;m]$ while
guaranteeing prescribed performance in (9) in the sense of $q$th moment, where
$q\in\\{1,2\\}$.

###### Proof:

Consider a Lyapunov-like function
$V:{\mathbb{R}}^{2m}\rightarrow{\mathbb{R}}_{0}^{+}$ as

$\displaystyle V(\eta)=\Big{(}\frac{1}{q}\eta^{T}\eta\Big{)}^{\frac{q}{2}},$

where $q\in\\{1,2\\}$. One can readily verify that the function $V$ satisfies
condition (6) with functions
$\underline{\psi}(s):=(\frac{1}{q})^{\frac{q}{2}}s$ and
$\overline{\psi}(s):=(\frac{2m}{q})^{\frac{q}{2}}s$ for all
$s\in{\mathbb{R}}_{0}^{+}$. The corresponding infinitesimal generator as
defined in (5) along the augmented dynamics (15) of the multi-agent system in
edge space (13) and transformed error dynamics (14) is given by

$\displaystyle\mathcal{L}V(\eta)=\eta^{T}\Big{(}\frac{1}{q}\eta^{T}\eta\Big{)}^{\frac{q}{2}-1}\Big{(}\begin{bmatrix}-L_{e}&0_{m}\\\
-\phi_{t}(L_{e}+\alpha_{t})&0_{m}\end{bmatrix}\eta$
$\displaystyle+\begin{bmatrix}I_{m}\\\
\phi_{t}\end{bmatrix}D^{T}v\Big{)}+\frac{1}{2}G(x)^{T}D\begin{bmatrix}I_{m}\\\
\Phi_{t}\end{bmatrix}^{T}\Bigg{(}\Big{(}\frac{1}{q}\eta^{T}\eta\Big{)}^{\frac{q}{2}-1}I_{2m}$
$\displaystyle+\frac{q-2}{q}\eta\eta^{T}\Big{(}\frac{1}{q}\eta^{T}\eta\Big{)}^{\frac{q}{2}-2}\Bigg{)}\begin{bmatrix}I_{m}\\\
\phi_{t}\end{bmatrix}D^{T}G(x).$

The stack vector of external inputs is written as

$\displaystyle v(t)=-D\Phi_{t}^{2}\overline{x},\ t\in{\mathbb{R}}_{0}^{+}.$
(23)

By substituting control law (23) and using the Lipschitz continuity of
$g(x_{i})$ and the fact that $q\in\\{1,2\\}$, the $\mathcal{L}V(\eta)$ is
rewritten as

$\displaystyle\mathcal{L}V(\eta)\leq\eta^{T}\Big{(}\frac{1}{q}\eta^{T}\eta\Big{)}^{\frac{q}{2}-1}\Big{(}\begin{bmatrix}-L_{e}&0_{m}\\\
-\Phi_{t}(L_{e}+\alpha_{t})&0_{m}\end{bmatrix}\eta$ $\displaystyle\
-\begin{bmatrix}I_{m}\\\
\Phi_{t}\end{bmatrix}D^{T}D\Phi_{t}^{2}\overline{x}\Big{)}+\frac{1}{2}k_{g}^{2}\overline{x}^{T}\Big{(}\frac{1}{q}\eta^{T}\eta\Big{)}^{\frac{q}{2}-1}\begin{bmatrix}I_{m}\\\
\Phi_{t}\end{bmatrix}^{T}\begin{bmatrix}I_{m}\\\
\Phi_{t}\end{bmatrix}\overline{x}.$
$\displaystyle=\eta^{T}\Big{(}\frac{1}{q}\eta^{T}\eta\Big{)}^{\frac{q}{2}-1}\Big{(}\begin{bmatrix}-L_{e}&0_{m}\\\
-\Phi_{t}(L_{e}+\alpha_{t})&0_{m}\end{bmatrix}\eta-\begin{bmatrix}I_{m}\\\
\Phi_{t}\end{bmatrix}L_{e}\begin{bmatrix}\Phi_{t}^{2}\\\
0_{m}\end{bmatrix}\eta\Big{)}$ $\displaystyle\
+\frac{1}{2}k_{g}^{2}\Big{(}\frac{1}{q}\eta^{T}\eta\Big{)}^{\frac{q}{2}-1}\eta^{T}\begin{bmatrix}\begin{bmatrix}I_{m}\\\
\Phi_{t}\end{bmatrix}^{T}\begin{bmatrix}I_{m}\\\
\phi_{t}\end{bmatrix}&0_{m}\\\ 0_{m}&0_{m}\end{bmatrix}\eta$
$\displaystyle=\eta^{T}\Big{(}\frac{1}{q}\eta^{T}\eta\Big{)}^{\frac{q}{2}-1}\begin{bmatrix}-L_{e}(I_{m}+\Phi_{t}^{2})+\frac{1}{2}k_{g}^{2}(I_{m}+\Phi_{t}^{2})&0_{m}\\\
-\Phi_{t}(L_{e}+\alpha_{t}+L_{e}\Phi_{t}^{2})&0_{m}\end{bmatrix}\eta$
$\displaystyle=-\eta^{T}\Big{(}\frac{1}{q}\eta^{T}\eta\Big{)}^{\frac{q}{2}-1}\begin{bmatrix}(L_{e}-\frac{1}{2}k_{g}^{2}I_{m})(I_{m}+\Phi_{t}^{2})&0_{m}\\\
\Phi_{t}(L_{e}+\alpha_{t}+L_{e}\Phi_{t}^{2})&0_{m}\end{bmatrix}\eta$
$\displaystyle=-\beta(\eta).$

Note that $\Phi_{t}$ and $L_{e}$ is a positive definite matrices. Hence, if
$L_{e}-\frac{1}{2}k_{g}^{2}I_{m}>0$, we get $\mathcal{L}V(\eta)\leq 0$. This
implies that $\mathbb{E}[\|\eta(t)\|^{q}]$ is bounded and hence
$\mathbb{E}[\|\xi(t)\|^{q}]$ is bounded for all $t\in{\mathbb{R}}_{0}^{+}$ and
all $\overline{x}_{k}(0)\in(-\rho_{k}(0),\rho_{k}(0))$, $k\in[1;m]$. This
ensures that the $\overline{x}$ satisfies prescribed transient constraints (9)
in the sense of $q$th moment, where $q\in\\{1,2\\}$. Now, since $\beta(x)$ is
continuous and nonnegative, by utilizing stochastic Barbalat’s lemma (Lemma
II.4), we have $\lim\limits_{t\rightarrow\infty}\beta(\eta(t))=0$ almost
surely. This implies almost sure consensus (i.e.
$\lim\limits_{t\rightarrow\infty}\overline{x}(t)=0$ a.s.). ∎

Figure 1: Communication graph with tree topology Figure 2: Several
realizations of relative positions of stochastic multi-agent system (24)
starting from $\overline{x}(0)=[-4.9,1,-3,-1.5,4.5]^{T}$ $(a)$ without
prescribed performance control law; $(b)$ with the proposed control law (21);
$(c)$ and $(d)$ are with the proposed control law (17) and decay rates of
prescribed performance functions $\epsilon_{k}=1.5$ and $\epsilon_{k}=10$
$\forall k\in[1;5]$, respectively. The dashed black lines represents
prescribed performance constraints.

##  IV Numerical Examples

Figure 3: Mean-squared values of relative position starting from
$\overline{x}(0)=[-4.9,1,-3,-1.5,4.5]^{T}$ under proposed control laws (17)
[blue] and (21) [red]. The dashed black line shows squared values of
performance function $\rho(t)=[\rho_{1}(t),\ldots,\rho_{m}(t)]^{T}$,
$\rho_{k}(t)=4.9\mathsf{e}^{-1.5t}+0.1$ for all $k\in[1;5]$.

In order to demonstrate the effectiveness of the proposed results, we consider
a simulation example of a stochastic multi-agent system consist of six agents
given by stochastic differential equations as

$\displaystyle\operatorname{d}x_{i}=u_{i}\operatorname{d}t+\mathsf{e}^{-0.1|x_{i}|}\sin(x_{i})\operatorname{d}W_{t},\
i\in[1;6],$ (24)

where $x_{i}\in{\mathbb{R}}$ and $u_{i}\in{\mathbb{R}}$ are the absolute
position and the control input of the $i$th agent, respectively, $W_{t}$ is
the standard Brownian motion, and the diffusion term
$g(x_{i})=\mathsf{e}^{-0.1|x_{i}|}\sin(x_{i})$ with the corresponding
Lipschitz constant $\mathsf{k}_{g}=1$. The undirected communication graph with
${\mathcal{V}}=\\{1,2,3,4,5,6\\}$ and
${\mathcal{E}}=\\{e_{1},e_{2},e_{3},e_{4},e_{5}\\}$ (i.e., $n=6$ and $m=5$) is
shown in Figure 1. We consider the prescribed performance functions
$\rho_{k}(t)=4.5\mathsf{e}^{-\epsilon t}+0.1$ for all $k\in[1;m]$ and we
consider two cases: $\epsilon=1.5$ and $\epsilon=10$. The performance bounds
$-\rho_{k}(t)$ and $\rho_{k}(t)$ are depicted with black dashed lines in
Figure 2. We obtain $\gamma=4$ and $\kappa=0.39$ satisfying conditions (18a),
and (18b) in both the cases. Figure 2 shows several realizations of relative
positions of the controlled stochastic multi-agent systems starting from
$\overline{x}(0)=[-4.9,1,-3,-1.5,4.5]^{T}$ without an external input (Figure
2(a)) and with the proposed distributed controller (21) for decay rate
$\epsilon=1.5$ (Figure 2(b)), and with the proposed distributed controller
(17) with different values of decay rates of performance bounds (Figure 2(c)
and (d)). The mean-squared value of relative positions of the controlled
stochastic multi-agent system $\mathds{E}[\|\overline{x}(t)\|^{2}]$ computed
for 1000 realizations is shown in Figure 3. From Figures 2 and 3, one can
readily verify that the proposed control laws achieve almost sure consensus
and consensus in expectation, respectively, while respecting prescribed
performance constraints in $2nd$ moment.

##  V Conclusion

In this work, we studied a consensus problem of stochastic multi-agent systems
with prescribed performance bounds. Under the assumption of a tree graph, a
distributed control law has been proposed for multi-agent systems containing
agents with state-dependent stochastic noise such that the entire system can
achieve consensus in expectation (or almost surely) while satisfying
predefined performance constraints in the sense of $q$th moment. Future work
includes extending the results for more general graphs with cycles and for
heterogeneous agents.

## References

  * [1] A. Dorri, S. S. Kanhere, and R. Jurdak, “Multi-agent systems: A survey,” _IEEE Access_ , vol. 6, pp. 28 573–28 593, 2018.
  * [2] R. N. Darmanin and M. K. Bugeja, “A review on multi-robot systems categorised by application domain,” in _2017 25th Mediterranean Conference on Control and Automation (MED)_. IEEE, 2017, pp. 701–706.
  * [3] M. Mesbahi and M. Egerstedt, _Graph theoretic methods in multiagent networks_. Princeton University Press, 2010, vol. 33.
  * [4] R. Olfati-Saber and R. M. Murray, “Consensus problems in networks of agents with switching topology and time-delays,” _IEEE Transactions on automatic control_ , vol. 49, no. 9, pp. 1520–1533, 2004.
  * [5] X. Wang, N. Xiao, L. Xie, E. Frazzoli, and D. Rus, “Decentralised dynamic games for large population stochastic multi-agent systems,” _IET Control Theory & Applications_, vol. 9, no. 3, pp. 503–510, 2014.
  * [6] D. Ding, Z. Wang, B. Shen, and G. Wei, “Event-triggered consensus control for discrete-time stochastic multi-agent systems: The input-to-state stability in probability,” _Automatica_ , vol. 62, pp. 284–291, 2015.
  * [7] L. Ma, Z. Wang, and H.-K. Lam, “Event-triggered mean-square consensus control for time-varying stochastic multi-agent system with sensor saturations,” _IEEE Transactions on Automatic Control_ , vol. 62, no. 7, pp. 3524–3531, 2016.
  * [8] Y. Tang, H. Gao, W. Zhang, and J. Kurths, “Leader-following consensus of a class of stochastic delayed multi-agent systems with partial mixed impulses,” _Automatica_ , vol. 53, pp. 346–354, 2015.
  * [9] R. Sakthivel, R. Sakthivel, B. Kaviarasan, and F. Alzahrani, “Leader-following exponential consensus of input saturated stochastic multi-agent systems with markov jump parameters,” _Neurocomputing_ , vol. 287, pp. 84–92, 2018.
  * [10] N. Amelina and A. Fradkov, “Approximate consensus in multi-agent nonlinear stochastic systems,” in _2014 European Control Conference (ECC)_. IEEE, 2014, pp. 2833–2838.
  * [11] J. Liu, H. Zhang, X. Liu, and W.-C. Xie, “Distributed stochastic consensus of multi-agent systems with noisy and delayed measurements,” _IET Control Theory & Applications_, vol. 7, no. 10, pp. 1359–1369, 2013.
  * [12] C. P. Bechlioulis and G. A. Rovithakis, “Robust adaptive control of feedback linearizable MIMO nonlinear systems with prescribed performance,” _IEEE Transactions on Automatic Control_ , vol. 53, no. 9, pp. 2090–2099, 2008.
  * [13] L. Macellari, Y. Karayiannidis, and D. V. Dimarogonas, “Multi-agent second order average consensus with prescribed transient behavior,” _IEEE Transactions on Automatic Control_ , vol. 62, no. 10, pp. 5282–5288, 2016.
  * [14] F. Chen and D. V. Dimarogonas, “Consensus control for leader-follower multi-agent systems under prescribed performance guarantees,” in _2019 IEEE 58th Conference on Decision and Control (CDC)_. IEEE, 2019, pp. 4785–4790.
  * [15] C. J. Stamouli, C. P. Bechlioulis, and K. J. Kyriakopoulos, “Multi-agent formation control based on distributed estimation with prescribed performance,” _IEEE Robotics and Automation Letters_ , vol. 5, no. 2, pp. 2929–2934, 2020.
  * [16] F. Mehdifar, C. P. Bechlioulis, F. Hashemzadeh, and M. Baradarannia, “Prescribed performance distance-based formation control of multi-agent systems,” _Automatica_ , vol. 119, p. 109086, 2020.
  * [17] X. Shao and S. Tong, “Adaptive fuzzy prescribed performance control for MIMO stochastic nonlinear systems,” _IEEE Access_ , vol. 6, pp. 76 754–76 767, 2018.
  * [18] S. Sui, C. P. Chen, and S. Tong, “A novel adaptive NN prescribed performance control for stochastic nonlinear systems,” _IEEE Transactions on Neural Networks and Learning Systems_ , 2020.
  * [19] B. Oksendal, _Stochastic differential equations: An introduction with applications_. Springer Science & Business Media, 2013.
  * [20] M. Huang and J. H. Manton, “Stochastic approximation for consensus seeking: Mean square and almost sure convergence,” in _2007 46th IEEE Conference on Decision and Control_. IEEE, 2007, pp. 306–311.
  * [21] Z. Wu, Y. Xia, and X. Xie, “Stochastic barbalat’s lemma and its applications,” _IEEE Transactions on Automatic Control_ , vol. 57, no. 6, pp. 1537–1543, 2011.
  * [22] D. V. Dimarogonas and K. H. Johansson, “Stability analysis for multi-agent systems using the incidence matrix: Quantized communication and formation control,” _Automatica_ , vol. 46, no. 4, pp. 695–700, 2010.