Propagation Stability Concepts for Network Synchronization Processes Sandip Roy Subir Sarker and Mengran Xue Abstract A notion of disturbance propagation stability is

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Propagation Stability Concepts for Network Synchronization Processes

Sandip Roy, Subir Sarker, and Mengran Xue

Abstract— A notion of disturbance propagation stability is

deﬁned for dynamical network processes, in terms of decres-

cence of an input-output energy metric along cutsets away from

the disturbance source. A characterization of the disturbance

propagation notion is developed for a canonical model for

synchronization of linearly-coupled homogeneous subsystems.

Speciﬁcally, propagation stability is equivalenced with the

frequency response of a certain local closed-loop model, which

is deﬁned from the subsystem model and local network connec-

tions, being sub-unity gain. For the case where the subsystem

is single-input single-output (SISO), a further simpliﬁcation in

terms of the subsystem’s open loop Nyquist plot is obtained.

An extension of the disturbance propagation stability concept

toward imperviousness of subnetworks to disturbances is brieﬂy

developed, and an example focused on networks with planar

subsystems is considered.

I. INTRODUCTION

The synchronization of coupled systems is common in

nature and in the engineered world. For this reason, there has

been a substantial cross-disciplinary research effort to deﬁne

and characterize synchronization phenomena in networks

of coupled systems [1]–[3]. These studies deﬁne synchro-

nization in terms of the internal asymptotic stability of a

manifold, on which each coupled system has an identical

state or output. However, synchronization in a practical sense

requires not only that the coupled systems come to a common

state/output, but also that this equilibrium is impervious to

external disturbances. Based on this recognition, a body of

recent work has considered the disturbance responses of

network synchronization processes [4]–[6]. Broadly, these

efforts model disturbances as impinging on all or a subset

of subsystems within a network, and evaluate their potential

impacts on network-wide synchrony according to a perfor-

mance metric (typically, a H2or H∞gain). The studies in

this direction have largely focused on evaluating the metrics

from a graph-theoretic perspective, and designing controllers

to bound the performance metrics within a threshold.

Engineers working with large-scale built networks often

do not think about coordination in terms of either internal

stability characteristics or global (network-wide) disturbance

response metrics. Rather, they are concerned with the extent

of propagation of local disturbances, whether arising from

exogenous inputs or state deviations. For instance, bulk

power grid operators often distinguish well-damped networks

where oscillatory disturbance responses remain localized

from poorly-damped ones where such disturbances have

network-wide impact [7]. Similar assessments of network

The ﬁrst two authors are with School of Electrical Engineering and Com-

puter Science, Washington State University, Pullman, WA 99164, USA. The

third author is with Raytheon BBN Technologies. Correspondence:

sandip@wsu.edu

performance in terms of disturbance propagation are of

interest in disciplines ranging from air trafﬁc management

to infectious-disease epidemiology and cyber-security [8].

The spatial propagation of disturbances among intercon-

nected systems in cascade or line-topology conﬁgurations

has been extensively researched under the heading of string

stability [9], [10]. The string stability concept has also been

extended to general directional networks in the mesh stability

literature [11]. The recent study [12] has recognized the need

for disturbance-propagation stability notions for general (bi-

directionally connected) dynamical networks, and therefore

has proposed a deﬁnition for network stability in terms

of boundedness of input-to-state or input-to-output gains

which parallels the basic string stability deﬁnition. It has

also introduced an alternate network stability deﬁnition for

tree-like graphs which captures monotonic decresence of the

propagative response away from the disturbance source; this

deﬁnition is a generalization of the strict string stability con-

cept [10]. Other recent studies have also examined stability

notions for linear and nonlinear network processes, which are

concerned with disturbance propagation (e.g., [13]). Addi-

tionally, a number of recent studies have characterized local-

input-to-output properties of dynamical networks (e.g., gains,

zeros), without however explicitly considering propagation

[14], [15]. However, deﬁnitions for propagation stability are

not yet mature, and the formal analysis of propagation is

incomplete even for networks of coupled linear systems.

The purpose of this article is to deﬁne and characterize a

notion of disturbance propagation stability in the context of

a canonical network model for synchronization of coupled

homogeneous linear subsystems. The main contributions of

the study are two-fold:

1) A general deﬁnition is introduced for (strict) propaga-

tion stability, based on decrescence of response norms across

cutsets in the network’s digraph, or equivalently along paths

away from the disturbance source.

2) Propagation stability is characterized in terms of the

closed-loop frequency responses of the subsystem model

with a proportional feedback controller applied. For the case

of single-input single-output subsystems, these conditions are

further simpliﬁed to conditions on the subsystem’s open-loop

frequency response and/or transfer function.

The analysis of propagation stability depends on a new,

local characterization of synchronization models. This char-

acterization is markedly different from the spectral decom-

position that has been exhaustively used to understand both

internal and global external stability [1], [2], and provides

a further assessment/categorization of synchronization pro-

cesses.

arXiv:2210.04370v1 [eess.SY] 9 Oct 2022

The remainder of the article is organized as follows. The

network synchronization model is described in Section II,

and disturbance propagation stability notions are deﬁned

in Section III. The main characterizations of propagation

stability are presented in Section IV. Finally, in Section V, an

example is given which focuses on the impact of damping on

propagation stability when the subsytems are planar devices.

II. MODEL

A network with Nidentical, interconnected devices or

nodes or subsystems, labeled 1, . . . , N, is considered. Each

subsystem i∈1, . . . , N has a state xi∈Rnand output

yi∈Rmwhich are governed by the following linear or

linearized state-space equations:

xi=Axi+B

αX

j6=i

gij (yj−yi) + γiwi

(1)

yi=Cxi.

Here, A,B, and Care a subsystem’s state, input, and output

matrices, respectively; the scalars gij ≥0are coupling

weights; the vector wi∈Rmrepresents an external dis-

turbance input at subsystem i; and αis a global coupling-

strength parameter which allows tuning of the network

connectivity (see [1]). Our focus here is on an external

disturbance impinging on a single source node s∈1, . . . , N,

which is modeled by setting γs= 1 and γi= 0 for i6=s.

The model (1) is a standard representation for the small-

signal dynamics of synchronizing coupled oscillators [1],

[2], with two distinctions. First, a disturbance is applied

at a single subsystem, to allow for analysis of propagative

impacts. Second, the subsystems are modeled as being inter-

connected through commensurately-dimensioned inputs and

outputs, rather than through an explicit inner-coupling term

or alternately through a designable protocol. This format

is used to stress that the network is made up of input-

output devices with ﬁxed connections, but the formulation

encompasses the scenarios with an inner coupling or a

designable protocol.

Analyses of (1) often are phrased in terms of a graph

that represents the network interconnections. For our devel-

opment, a weighted digraph Γis deﬁned with Nvertices

corresponding to the Nsubsystems. A directed edge is

drawn from vertex jto vertex iif gij >0, reﬂecting a

direct inﬂuence of the output of subsystem jon the state

evolution of subsystem i. The edge is assigned a weight of

gij . We use the notation Vfor the set of vertices, and Efor

the set of edges. Additionally, it is convenient to deﬁne an

(asymmetric) Laplacian matrix L= [lij ]∈RN×N. Each off-

diagonal entry lij is given by −gij , while the diagonal entries

are selected so that each row sums to 0(i.e. lii =Pj6=igij ).

For the model (1), the synchronization manifold where the

states x1,...,xnare identical is known to be asympotically

stable under broad conditions on the network graph, the

subsystem model, and the coupling-strength parameter. More

precisely, stability can be related to Hurwitz stability of the

Ncomplex matrices A+λiBC, where 0 = λ1, λ2, . . . , λN

are the eigenvalues of the Laplacian matrix L. From this anal-

ysis, stability can be distilled to a simple test on the Laplacian

matrix’s spectrum via the master stability function construct,

see e.g. [1] for details. A number of other characteristics of

(1), including global disturbance stability and controllability

via external stimulation, can also be related to the matrices

A+λiBC [4], [16].

From here on we refer to the model (1) as the network

synchronization model. The model is approximative of a

number of synchronization phenomena, including the swing

dynamics of the bulk power grid, multi-vehicle formation

ﬂight, and the nonlinear dynamics of electrical oscillator

networks.

III. PROPAGATION STABILITY DEFINITION

A notion of propagation stability is deﬁned based on the

spatial patterns of output energies (squared two norms) at

network subsystems over a time interval [0, T ], when an

exogenous disturbance input ws(t)is applied at a single

node. In our development, the disturbance is assumed to sat-

isfy the Dirichlet conditions (absolute integrability over any

period, ﬁnite number of discontinuities and minima/maxima,

bounded over any interval), but otherwise may be arbi-

trary. In deﬁning stability, the squared two-norm metric

Ei(T) = RT

t=0 yT

i(t)yi(t)dt is considered for each network

subsystem i. Conceptually, the network can be viewed as

propagation stable, if these energies are attenuated away from

the disturbance source with respect to the network graph.

However, since the network’s graph in general has a spatially

inhomogeneous structure, deﬁning attenuation requires some

care.

Fig. 1. Illustration of the propagation stability concept.

One natural way to assess disturbance propagation is

to consider the energy metric Ei(T)for vertex-cutsets in

the network’s graph (i.e., sets of vertices whose removal

partition the graph). If the metric value for at least one cutset

vertex is larger than the metric values for vertices that are

separated from the source by the cutset, then the response

can be viewed as being attenuated away from the source

(see Figure 1). To formalize this notion, let us consider a

set of vertices VC∈ V. We refer to VCas a separating

cutset for the source s, if the remaining vertices V \ VC

can be partitioned into two subsets V1and VBsuch that: 1)

there are no edges from vertices in V1to vertices in VBand

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PropagationStabilityConceptsforNetworkSynchronizationProcessesSandipRoy,SubirSarker,andMengranXueAbstractAnotionofdisturbancepropagationstabilityisdenedfordynamicalnetworkprocesses,intermsofdecres-cenceofaninput-outputenergymetricalongcutsetsawayfromthedisturbancesource.Acharacterizationofthedistu...

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Propagation Stability Concepts for Network Synchronization Processes Sandip Roy Subir Sarker and Mengran Xue Abstract A notion of disturbance propagation stability is

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