On the Internal Stability of Diffusively Coupled Multi-Agent Systems and the Dangers of Cancel Culture

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arXiv:2210.06149v2 [eess.SY] 9 May 2023

On the Internal Stability of Diﬀusively Coupled Multi-Agent Systems

and the Dangers of Cancel Culture★

Gal Barkaia, Leonid Mirkina, Daniel Zelazob

aFaculty of Mechanical Engineering, Technion—IIT, Haifa 3200003, Israel

bFaculty of Aerospace Engineering, Technion—IIT, Haifa 3200003, Israel

Abstract

We study internal stability in the context of diﬀusively-coupled control architectures, common in multi-agent systems (i.e. the

celebrated consensus protocol), for linear time-invariant agents. We derive a condition under which the system can not be stabilized

by any controller from that class. In the ﬁnite-dimensional case the condition states that diﬀusive controllers cannot stabilize agents

that share common unstable dynamics, directions included. This class always contains the group of homogeneous unstable agents,

like integrators. We argue that the underlying reason is intrinsic cancellations of unstable agent dynamics by such controllers, even

static ones, where directional properties play a key role. The intrinsic lack of internal stability explains the notorious behavior of

some distributed control protocols when aﬀected by measurement noise or exogenous disturbances.

Keywords: Multi-agent systems, controller constraints and structure, stability.

1. Introduction

A multi-agent system (MAS) is a collection of independent

systems (agents) coupled via the pursuit of a common goal.

In large-scale MASs the information exchange between agents

might be costly. As such, it is commonly limited to a subset

of agents, known as neighbors. Control laws that use only

information from neighboring agents are called distributed.

This work studies a class of distributed control laws, where

only relative measurements are exchanged between neighbors.

In other words, each agent has access only to the diﬀerence

between its output and that of each of its neighbours. Such

control laws are called diﬀusive and systems controlled by them

are known as diﬀusively coupled. Diﬀusive control laws are

common in the MAS literature. Relative sensing appears nat-

urally in MAS tasks, where absolute measurements are hard

to obtain, such as space and aerial exploration and sensor lo-

calization, see (Smith and Hadaegh, 2005; Khan et al., 2009;

Zelazo and Mesbahi, 2011b) and the references therein. The

consensus and synchronization problems are well-known exam-

ples of diﬀusively coupled systems (Olfati-Saber et al., 2007;

Wieland et al., 2011).

However, diﬀusively-coupled systems behave poorly when

aﬀected by disturbances and noise. Measurement noise rapidly

deteriorates performance (Zelazo and Mesbahi, 2011a, §III-A)

and even dynamic controllers can hardly attenuate disturbances

(Ding, 2015). To cope with the diﬃculties, diﬀerent relax-

ing assumptions are assumed. Some allow for non-relative

★Supported by the Israel Science Foundation (grants 3177/21 and 2285/20)

and Sakranut Graydah at the Technion.

Email addresses: galbarkai@campus.technion.ac.il (Gal Barkai),

mirkin@technion.ac.il (Leonid Mirkin), dzelazo@technion.ac.il

(Daniel Zelazo)

state (Yucelen and Egerstedt, 2012) or output (Mo and Guo,

2019) measurements, while others employ an undisturbed leader

(Ding, 2015) or impose limitations even on bounded distur-

bances (Bürger and De Persis, 2015, Prop. 5). Despite these

diﬀerent assumptions, if they fail, the resulting trajectories ex-

hibit certain common traits that can be associated with insta-

bility. These traits can be illustrated by the classical consensus

protocol, considered below for a set of integrator agents and

with a static interaction network.

1.1. Motivating example

Reaching agreement between autonomous agents is a

fundamental building block in multi-agent coordination

(Ren and Beard, 2008). In its simplest form it studies a group of

independent integrator agents ¤𝑥𝑖(𝑡)=𝑢𝑖(𝑡), where 𝑥𝑖and 𝑢𝑖are

their states and control inputs, respectively. The goal is to reach

asymptotic agreement between all agents, in the sense that

lim

𝑡→∞ 𝑥𝑖(𝑡) − 𝑥𝑗(𝑡)=0,∀𝑖, 𝑗, (1)

under the constraint that the 𝑖th agent has access only to states

of its neighbors, whose indices belong to a set N𝑖. This

problem can be solved by the celebrated consensus protocol

(Olfati-Saber et al., 2007), which is a diﬀusive state-feedback

of the form

𝑢𝑖(𝑡)=−Õ

𝑗∈N𝑖𝑥𝑖(𝑡) − 𝑥𝑗(𝑡),∀𝑖. (2)

If certain connectivity conditions on the communication topol-

ogy hold (i.e. connectedness), then the control law (2) drives the

agents to agreement exponentially fast (Mesbahi and Egerstedt,

2010, Ch. 3). The state trajectories of four agents controlled

Preprint submitted to Elsevier May 11, 2023

(a) The states response. (b) The control signals response.

Figure 1: Simulation of protocol (2) perturbed by a step at 𝑡=𝑡𝑑.

by (2) are shown in Fig. 1 in the time interval [0, 𝑡𝑑]. Observe

that on this time interval the states converge exponentially to

the average of their initial conditions and the control signals all

asymptotically vanish.

This might no longer be the case if the agents are aﬀected by

load disturbances 𝑑𝑖, viz.

¤𝑥𝑖(𝑡)=𝑢𝑖(𝑡) + 𝑑𝑖(𝑡).(3)

An example of what happens in such situations is also shown

in Fig. 1. At the time instance 𝑡=𝑡𝑑one agent is aﬀected by a

unit step disturbance. As a result, all states cease to agree and

start to diverge when 𝑡 > 𝑡𝑑, whereas the control signals reach

non-zero steady-state values.

The apparent instability of the whole system,manifested in the

unboundedness of the states, can be explained by the well-known

fact that the consensus protocol has a closed-loop eigenvalue at

the origin (Olfati-Saber et al., 2007). Nevertheless, the bound-

edness of the control signals under such conditions is intriguing.

Situations wherein some signals in the closed-loop system are

bounded while some others are not normally indicate unstable

pole-zero cancellations in the feedback loop (Zhou et al., 1996,

Sec. 5.3). However, controller (2) is static and thus has no zeros.

1.2. Contribution

The example above suggests that a deeper inspection of the

internal stability property could oﬀer insight into the behavior

of diﬀusively-coupled systems. The internal stability of any

feedback interconnection requires the stability of all possible

input / output relations in the system, see (Zhou et al., 1996;

Skogestad and Postlethwaite, 2005). However, to the best of

our knowledge, internal stability has not been explicitly studied

in the context of diﬀusively-coupled architectures of MASs yet.

In this paper we show that diﬀusively-coupled systems of

LTI (linear time-invariant) agents might not be internally sta-

bilizable. Loosely speaking, this happens if the agents share

common unstable dynamics, directions counting. This, for ex-

ample, is always the case in a group of homogeneous unstable

agents, like those discussed in §1.1.

When restricting the result to ﬁnite-dimensional agents, we

also explain the mechanism behind the shown internal instabil-

ity. It is caused by unstable cancellations in the cascade of the

aggregate plant and a diﬀusive controller. Important is that these

cancellations are caused not by controller zeros, but rather by an

intrinsic spatial deﬁciency of the diﬀusively-coupled conﬁgura-

tion. These cancellations are intrinsic to the diﬀusive structure

and cannot be aﬀected by controller dynamics. Consequently,

the internal stability of feedback systems utilizing only relative

measurements depends solely on the agent dynamics.

In addition to providing a rigorous analysis of the internal

stability of diﬀusively-coupled systems, we show how the anal-

ysis is readily applied to common extensions found in the lit-

erature. In particular, we discuss more general symmetrically

coupled multi-agent systems (i.e. not restricted to only diﬀu-

sive coupling), asymmetric coupling (i.e. MASs over directed

graphs), unstable systems with no closed right-half plane poles,

and MASs over time-varying networks. Numerous examples

are also provided along the way to illustrate the main results.

The paper is organized as follows. The problem is set up in

Section 2 and the main result is presented in Section 3, with sev-

eral generalizations discussed in §3.2. Section 4 addresses the

case of ﬁnite-dimensional agents, reformulating the main result

in a more transparent form and revealing the underlying reason

for the reported behavior. Concluding remarks are provided in

Section 5. Two appendices collect deﬁnitions and technical re-

sults about coprime factorizations over 𝐻∞and poles and zero

directions of multivariable real-rational transfer functions.

Notation

The sets of integer, real, and complex numbers are ℤ,ℝ,

and ℂ, respectively, with subsets ℕ𝜈≔{𝑖∈ℤ|1≤𝑖≤𝜈},

ℂ0≔{𝑠∈ℂ|Re 𝑠 > 0}, and ¯

ℂ0≔{𝑠∈ℂ|Re 𝑠≥0}. By 𝐼𝜈

and 𝟙𝜈we denote the 𝜈×𝜈identity matrix and 𝜈-dimensional

vector of ones, respectively. When the dimension is immaterial

or clear from context, we use 𝐼and 𝟙. The complex-conjugate

transpose of a matrix 𝐴is denoted by 𝐴⊤, the set of all its

eigenvalues by spec(𝐴), and its minimal singular value by 𝜎(𝐴).

The notation diag{𝐴𝑖}stands for a block-diagonal matrix with

diagonal elements 𝐴𝑖. The image (range) and kernel (null)

spaces of a matrix 𝐴are notated Im 𝐴and ker 𝐴, respectively.

Given two matrices 𝐴and 𝐵,𝐴⊗𝐵denotes their Kronecker

product.

By the stability of a system 𝐺we understand its 𝐿2-stability.

It is known (Curtain and Zwart, 2020, §A.6.3) that a 𝑝×𝑚LTI

system is causal and stable iﬀ its transfer function 𝐺(𝑠)be-

longs to 𝐻𝑝×𝑚

∞, which is the space of holomorphic and bounded

functions ℂ0→ℂ𝑝×𝑚(we write 𝐻∞when the dimensions are

clear). Given a real-rational transfer function 𝐺(𝑠), its McMil-

lan degree is denoted by deg(𝐺). By nrank 𝐺(𝑠)we understand

the normal rank of a function 𝐺(𝑠).

A digraph G=(V,E) consists of a vertex set Vand an

edge set E ⊂ V × V, see (Godsil and Royle, 2001) for more

details. The (oriented) incidence matrix of Gis denoted by 𝐸G

or simply 𝐸when the association with a concrete graph is clear.

It is a |V | × |E| matrix, whose (𝑖, 𝑗)entry is

[𝐸G]𝑖 𝑗 =









1 if vertex 𝑖is the head of edge 𝑗

−1 if vertex 𝑖is the tail of edge 𝑗

0 if vertex 𝑖does not belong to edge 𝑗

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arXiv:2210.06149v2[eess.SY]9May2023OntheInternalStabilityofDiﬀusivelyCoupledMulti-AgentSystemsandtheDangersofCancelCulture★GalBarkaia,LeonidMirkina,DanielZelazobaFacultyofMechanicalEngineering,Technion—IIT,Haifa3200003,IsraelbFacultyofAerospaceEngineering,Technion—IIT,Haifa3200003,IsraelAbstractWest...

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