A Zero-Sum Game Framework for Optimal Sensor Placement in Uncertain Networked Control Systems under Cyber-Attacks Anh Tung Nguyen1 Sribalaji C. Anand2 and Andr e M. H. Teixeira1

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A Zero-Sum Game Framework for Optimal Sensor Placement in

Uncertain Networked Control Systems under Cyber-Attacks

Anh Tung Nguyen1, Sribalaji C. Anand2, and Andr´

e M. H. Teixeira1

Abstract— This paper proposes a game-theoretic approach

to address the problem of optimal sensor placement against

an adversary in uncertain networked control systems. The

problem is formulated as a zero-sum game with two players,

namely a malicious adversary and a detector. Given a protected

performance vertex, we consider a detector, with uncertain

system knowledge, that selects another vertex on which to place

a sensor and monitors its output with the aim of detecting the

presence of the adversary. On the other hand, the adversary,

also with uncertain system knowledge, chooses a single vertex

and conducts a cyber-attack on its input. The purpose of the

adversary is to drive the attack vertex as to maximally disrupt

the protected performance vertex while remaining undetected

by the detector. As our ﬁrst contribution, the game payoff of

the above-deﬁned zero-sum game is formulated in terms of the

Value-at-Risk of the adversary’s impact. However, this game

payoff corresponds to an intractable optimization problem.

To tackle the problem, we adopt the scenario approach to

approximately compute the game payoff. Then, the optimal

monitor selection is determined by analyzing the equilibrium

of the zero-sum game. The proposed approach is illustrated via

a numerical example of a 10-vertex networked control system.

I. INTRODUCTION

Networked control systems have been playing a crucial

role in modeling, analysis, and operation of real-world

large-scale interconnected systems such as power systems,

transportation networks, and water distribution networks.

Those systems consist of multiple interconnected subsystems

which generally communicate with each other via insecure

communication channels to share their information. This

insecure protocol may leave the networked control systems

vulnerable to cyber-attacks such as denial-of-service and

false-data injection attacks [1], inﬂicting serious ﬁnancial

loss and civil damages. Reports on actual damages such as

Stuxnet [2] and Industroyer [3] have described the catas-

trophic consequences of such cyber-attacks for an Iranian

nuclear program and a Ukrainian power grid, respectively.

Motivated by the above observation, cyber-physical security

has increasingly received much attention from control society

in recent years.

*This work is supported by the Swedish Research Council under the

grants 2018-04396 and 2021-06316 and by the Swedish Foundation for

Strategic Research.

1Anh Tung Nguyen and Andr´

e M. H. Teixeira are with

the Department of Information Technology, Uppsala University, PO

Box 337, SE-75105, Uppsala, Sweden. {anh.tung.nguyen,

andre.teixeira}@it.uu.se

2Sribalaji C. Anand is with the Department of Electrical Engi-

neering, Uppsala University, PO Box 65, SE-75103, Uppsala, Sweden.

sribalaji.anand@angstrom.uu.se

One of the most popular security metrics is the game-

theoretic approach that has been successfully applied to deal

with the problem of robustness, security, and resilience of

networked control systems [4]. This approach affords us to

address the robustness and security of networked control

systems within the common well-deﬁned framework of H∞

robust control design. Further, many other concepts of games

considering networked systems subjected to cyber-attacks

such as dynamic [5], stochastic [6], network monitoring [7],

[8], and zero-sum games [9] have been recently studied.

Although the above games were successful in studying

control systems subjected to cyber-attacks such as denial-

of-service and stealthy data injection attacks, the full system

model knowledge was assumed to be available to both the

malicious adversary and the detector. This assumption might

be restrictive when it comes to large-scale interconnected

systems which can consist of a huge number of subsystems.

This can be explained by a variety of facts such as (i)limited

availability of computational resources for modeling, (ii)

limited availability of modeling data, and (iii)modeling er-

rors. Thus, the adversary and the detector might have limited

system knowledge instead of accurate system parameters,

which will be addressed throughout this paper.

In this paper, we deal with the problem of optimal sensor

placement against an adversary in an uncertain networked

control system which is represented by interconnected ver-

tices. Given a protected performance vertex, the detector

monitors the system by selecting a single monitor vertex and

placing a sensor to measure its output with the purpose of

detecting cyber-attacks. Meanwhile, the adversary chooses a

single vertex to attack and directly injects attack signals into

its input via the wireless network. The aim of the adversary

is to steer the attack vertex as to maximally disrupt the

protected performance vertex while remaining undetected by

the detector. The contributions of this paper are the following

1) The problem of optimal sensor placement against the

adversary is formulated as a zero-sum game between

two strategic players, i.e., the adversary and the detec-

tor, with the same uncertain system knowledge.

2) Due to the uncertainty, the game payoff of the zero-

sum game, which is a min-max optimization problem,

is computationally intractable [10]. To deal with the

problem, we adopt the scenario approach [11] to ap-

proximately compute the above game payoff.

3) We show that the existence of a ﬁnite solution to the

problem is related to the system-theoretic properties of

the dynamical system, namely its invariant zeros and

arXiv:2210.04091v1 [eess.SY] 8 Oct 2022

Fig. 1. Visualization of a zero-sum game between a detector and an

adversary in a networked control system.

relative degrees.

4) The solutions to the problem of the optimal sensor

placement are provided by investigating the pure and

the mixed-strategy equilibrium of the zero-sum game

in a numerical example.

We conclude this section by providing the notations which

are used throughout this paper. The problem description

is given in Section II. Thereafter, Section III formulates

the problem of optimal sensor placement as a zero-sum

game with the game payoff based on a risk metric. The

evaluation of the game payoff is carried out in Section IV.

Section V presents a numerical example of the zero-sum

game between an adversary and a detector and computes the

optimal monitor selection based on the mixed-strategy Nash

equilibrium. Concluding remarks are provided in Section VI.

Notation: the set of real positive numbers is denoted as

R+;Rnand Rn×mstand for sets of real n-dimensional

vectors and n-row m-column matrices, respectively. Let us

deﬁne ei∈Rnwith all zero elements except the i-th element

that is set as 1. A continuous-time system with the state-

space model ˙x(t) = Ax(t) + Bu(t), y(t) = Cx(t) +

Du(t)is denoted as Σ,(A, B, C, D). Consider the norm

kxk2

L2[0,T ],RT

0kx(t)k2

2dt. The space of square-integrable

functions is deﬁned as L2,f:R+→R| kfkL2[0,∞]<

∞and the extended space is deﬁned as L2e,f:

R+→R| kfkL2[0,T ]<∞,∀0< T < ∞. We

denote IA(x)as an indicator function such that IA(x)=1

if x∈ A, otherwise IA(x)=0. The probability of X

is denoted as P(X). For x∈R,dxerepresents a value

rounded to the nearest integer greater than or equal to x. Let

G,(V,E, A, Θ) be an undirected weighted digraph with

the set of Nvertices V={v1, v2, ..., vN}, the set of edges

E ⊆ V ×V, the weighted adjacency matrix A,[aij ], and the

weighted self-loop matrix Θ. For any (vi, vj)∈ E, i 6=j,

the element of the weighted matrix aij is positive, and with

(vi, vj)/∈ E or i=j,aij = 0. The degree of vertex

viis denoted as di=Pn

j=1 aij and the degree matrix

of the graph Gis deﬁned as D=diagd1, d2, . . . , dN,

where diag stands for a diagonal matrix. For each vertex

vi, it has a positive weighted self-loop gain θi>0. The

weighted self-loop matrix of the graph Gis deﬁned as Θ =

diagθ1, θ2, . . . , θN. The Laplacian matrix, representing

the graph G, is deﬁned as L= [`ij ] = D−A+ Θ.

Further, Gis called an undirected graph if Ais symmetric.

An edge of an undirected graph Gis denoted by a pair

(vi, vj)∈ E. An undirected graph is connected if for any

pair of vertices there exists at least one path between two

vertices. The set of all neighbours of vertex viis denoted as

Ni={vj∈ V : (vi, vj)∈ E}.

II. PROBLEM DESCRIPTION

This section ﬁrstly presents the description of a networked

control system. Then, we introduce a malicious adversary

who with limited system knowledge conducts a cyber-attack

to maliciously affect the system performance.

A. Networked control system description

Consider a networked control system associated with a

connected undirected graph G,(V,E, A, Θ) with Nver-

tices, the state-space model of each one-dimensional vertex

vi, i ∈1,2, . . . , N, is described as

˙x∆

i(t) = X

vj∈Ni

`∆

ij x∆

i(t)−x∆

j(t)+ ˜ui(t),(1)

y∆

τ(t) = x∆

τ(t),(2)

where x∆

i(t),˜ui(t)∈Rare the state of vertex viand its

control input received from its controller over the wireless

network (see Fig. 1), respectively. The performance of the

networked control system (1) is measured via the state

of a given vertex vτ∈ V in (2). The weight parame-

ters `∆

ij ,∀(vi, vj)∈ E,are uncertain and assumed to be

structured as `∆

ij ,¯

`ij +δij , where ¯

`ij and δij are the

nominal value and the bounded probabilistic uncertainty of

`∆

ij , respectively.

First, we consider the wireless network healthy, i.e., the

absence of cyber-attacks. Thus, the received control input

˜ui(t)of vertex vi, i ∈1,2, . . . , N, is the same as the

control input sent by its controller:

˜ui(t) = ui(t) = −θix∆

i(t),(3)

where ui(t)is the control input designed and sent by the

controller of vertex vi.θi∈R+is an adjustable self-loop

control gain of vertex vi.

For convenience, let us denote x∆(t),

x∆

1(t), x∆

2(t), . . . , x∆

N(t)>as the state of the networked

control system. The dynamics of the networked control

system (1) under the control law (3) can be rewritten as

˙x∆(t) = −L∆x∆(t),(4)

where the uncertain matrix L∆is deﬁned as: L∆,¯

L+ ∆,

∆∈Ω, where Ωis a closed and bounded set, ¯

L,[¯

`ij ]

and ∆,[δij ]are nominal value and bounded uncertainty

of L∆, respectively. Next, let us make use of the following

assumptions.

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AZero-SumGameFrameworkforOptimalSensorPlacementinUncertainNetworkedControlSystemsunderCyber-AttacksAnhTungNguyen1,SribalajiC.Anand2,andAndr´eM.H.Teixeira1AbstractThispaperproposesagame-theoreticapproachtoaddresstheproblemofoptimalsensorplacementagainstanadversaryinuncertainnetworkedcontrolsystems.T...

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A Zero-Sum Game Framework for Optimal Sensor Placement in Uncertain Networked Control Systems under Cyber-Attacks Anh Tung Nguyen1 Sribalaji C. Anand2 and Andr e M. H. Teixeira1

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