# Cyberattack Detection for Nonlinear Leader-Following Multi-Agent Systems
Using Set-Membership Fuzzy Filtering

Mahshid Rahimifard, Amir M. Moradi Sizkouhi, and Rastko R. Selmic M.
Rahimifard, A. M. Moradi Sizkouhi, and R. R. Selmic are with with the
Department of Electrical and Computer Engineering, Concordia University,
Montreal, QC, Canada<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

This paper is concerned with cyberattack detection in discrete-time, leader-
following, nonlinear, multi-agent systems subject to unknown but bounded (UBB)
system noises. The Takagi–Sugeno (T-S) fuzzy model is employed to approximate
the nonlinear systems over the true value of the state. A distributed
cyberattack detection method, based on a new fuzzy set-membership filtering
method, which consists of two steps, namely a prediction step and a
measurement update step, is developed for each agent to identify two types of
cyberattacks at the time of their occurrence. The attacks are replay attacks
and false data injection attacks affecting the leader-following consensus. We
calculate an estimation ellipsoid set by updating the prediction ellipsoid set
with the current sensor measurement data. Two criteria are provided to detect
cyberattacks based on the intersection between the ellipsoid sets. If there is
no intersection between the prediction set and the estimation set of an agent
at the current time instant, a cyberattack on its sensors is declared. Control
signal or communication signal data of an agent are under a cyberattack if its
prediction set has no intersection with the estimation set updated at the
previous time instant. Recursive algorithms for solving the consensus protocol
and calculating the two ellipsoid sets for detecting attacks are proposed.
Simulation results are provided to demonstrate the effectiveness of the
proposed method.

##  I INTRODUCTION

Multi-Agent Systems (MAS) have wide range of applications such as internet of
things (IoT), electrical grids, water distribution systems, transportation
systems, autonomous vehicles and Unmanned Aerial Vehicles (UAVs) [1]. Reaching
consensus in a distributed manner is a fundamental problem in MAS. Some
distributed and decentralized methods for attack detection have been proposed
[2, 3, 4, 5]. The agents transmit their data to neighboring agents through
communication channels in distributed consensus protocols, and these channels
are vulnerable to cyberattacks.

In [2] a distributed method to detect attacks in the communication network for
the distributed control of interconnected systems has been proposed. In this
study, only the local knowledge of the system is needed. However, through this
method, stealthy attacks cannot be detected. In [6, 7, 8, 9, 10], secure
designs against Denial of Service (DoS) attacks in a centralized setting have
been studied. In [11], an investigation on a distributed event-triggered
secure cooperative control of linear multi-agent systems under DoS attacks has
been conducted. This paper studies how to achieve average consensus in the
presence of DoS attacks and does not detect the attacks.

The problem of distributed simultaneous fault detection and leader-following
consensus control for multi-agent systems has been investigated in [12]. In
this study, a single module is used that conducts both tasks of fault
detection and control objectives, simultaneously. Also, the proposed fault
isolation method detect the faulty agent as well as the kind of fault.
However, this method is restricted to the actuator and sensor faults to the
linear systems with undirected topology. The authors of [13] have studied the
leader-following consensus problem for heterogeneous multi-agent systems
subject to both sensor and actuator attacks. In this study, the system is
linear and there have been considered some restricting conditions.

In [14] a state-dependent event-triggered control strategy for time-varying
MASs over a finite horizon has been designed for the first time. The only
attack considered in this study is the false data injection attack to the
linear multi-agent system with undirected topology.

Most of attack detection approaches, which are based on the state estimation,
necessitate systems noises to be in a stochastic framework, and this leads to
a probabilistic state estimation. For many real-world applications, accuracy
in the state estimation is crucial. However, estimation based on probabilistic
approach, such as Kalman filtering method, necessitates the use of mean and
variance to describe the state distributions modelled as random variables
(usually white and Gaussian perturbations). Consequently, considering unknown
but bounded (UBB) noises is a much more appropriate approach to modeling state
distributions.

Additionally, a common attack detection method, called the performance index
test ($\chi^{2}$-detector), uses a residual signal to determine if the
estimated behavior differs from that predicted by a model. Due to the nature
of the Kalman filtering technique, the estimated and predicted states are
single vectors and as a result, they cannot guarantee that a state is included
in some region. Also, as the resulting UBB noises are sub-optimal for Kalman-
type filtering, the reliability of attack detection is decreased. As a result
of the need for set-valued estimation, the ellipsoidal state estimation
technique was developed [15]. This method, known as the set-membership or set-
valued state estimation filtering approach, has been extensively studied in
filtering problems [16, 17, 18, 19, 20, 21, 22] and provides a set of state
estimates in state space that contains the system’s true state [23, 24]. By
using convex optimization approaches, an optimal ellipsoid with minimal size
can be determined for set-membership estimation, improving state estimation
and detection performance.

The authors of [25] have studied a cyberattack detection method for the linear
networked control systems through which for the first time, simultaneously,
using the set-membership filtering for the purpose of the attack detection and
distinguishing attacks on control signals from attacks on measurement outputs
have been considered. However, they only considered the attack detection
problem and there is no approach to the control of the system and the system
is a single agent.

Except a few publications [26, 27], most research on set-membership filtering
considers linear systems [28, 29, 30, 25]. Linearization should best fit the
nonlinear functions over a state estimate set rather than a state estimate
point when we use the set-membership framework. The authors of [26],
linearized the nonlinear dynamics around the current estimate, then bounded
the remaining terms by using interval mathematics and finally incorporated the
remaining bounds as additions to the process or measurement noise bounds. Due
to linearization around the estimated value of the state rather than the true
value, the above approximations, bring a base point error [31].

There are few works on the detection of replay attacks and to the best of our
knowledge, all the existing works have been only done on linear systems.
Therefore, detection of these attacks for the nonlinear systems are of prime
importance since real-world systems are mostly nonlinear. The fuzzy model of
Takagi-Sugeno (T-S) is an effective and universal approximator for a certain
class of nonlinear dynamic systems.

Therefore, in this paper, we use it to approximate nonlinear systems [32, 33].
We linearize the nonlinear systems over the true value of state and eliminate
the base point error. Our objective is to design a simultaneous distributed
attack detection strategy and leader-following consensus control based on a
new two-step fuzzy set-membership filtering approach in a distributed
framework. By utilizing the fuzzy modeling approach and the S-procedure
technique [34], we determine bounding ellipsoidal sets for each agent by a
recursive algorithm in state-space which guarantee the always enclosing of the
system’s true state [23, 24], regardless of UBB noises, assuming no attacks
are being made on the agent.

Each agent has a prediction and a measurement update step in its state
estimation algorithm. The following two criteria are then used to detect
cyberattacks:

  1. 1.

When a cyberattack violates the control signal of any agent, the prediction
ellipsoid set of that agent and its estimation ellipsoid set, updated with the
previous measurement output, do not intersect.

  2. 2.

When a cyberattack violates the sensor signal of any agent, the prediction
ellipsoid set of that agent and its estimation ellipsoid set, updated at the
current time instant, do not intersect.

Comparing with the previous works, the contributions of our work are as
follows:

  * •

To the best of our knowledge, we studied the attack detection problem of
nonlinear multi-agent systems subject to replay attacks for the first time in
the literature.

  * •

We developed the fuzzy set-membership filtering approach for detection of the
attacks.

  * •

We considered false data injection attacks on the control signal and
communication networks, as well as replay attacks on the sensor measurement
data.

  * •

We can distinguish attacks on control signals from attacks on measurement
outputs.

  * •

We also are able to mitigate the effects of the attacks and recover the system
performance.

  * •

Moreover, we ensure that we achieve the control goal which is achieving all
the agent states to the leader-following consensus.

##  II Problem Formulation

Interaction and communication is modelled as a connected directed graph
$\mathcal{G}=\\{\mathcal{V},\mathcal{E},\mathcal{A}\\},\mathcal{V}=\\{1,2,...,N\\},\mathcal{E}=\\{i,j,\left(i,j\right)\in\mathcal{V}\\}$
and $\mathcal{A}=\left(a_{ij}\right)\in\mathbb{R}^{N\times N}$which are the
vertex set, the directed edge set and the weighted adjacency matrix of
$\mathcal{G}$, respectively. The weights are defined as $a_{ij}>0$, if
$(j,i)\in\mathcal{E}$ and $a_{ij}=0$, otherwise. A node from which an edge
goes to node $i$ is a neighbor of node $i$. The set of the neighbors of node
$i$ are indicated by $N_{i}$, where
$N_{i}=\\{j|\left(j,i\right)\in\mathcal{E}$. Moreover, the Laplacian matrix
$\mathcal{L}=\left(l_{i,j}\right)\in\mathbb{R}^{N\times N}$ is defined as
$\mathcal{L}=\mathcal{D}-\mathcal{A}$ and
$\mathcal{D}=\operatorname{diag}_{N}^{i}\left\\{d_{i}\right\\}$, with
$d_{i}=\sum_{j=1}^{N}a_{ij}$.

Consider a discrete-time nonlinear multi-agent system with $N$ agents, and the
dynamics of agent, $i,i\in\\{1,...,N\\}$ is given as

$\begin{cases}{x}_{i}(k+1)=f_{i}(x_{i}(k))+G_{i}u_{i}(k)+I_{i}(x_{i})\omega_{i}(k)\\\
y_{i}(k)=h_{i}(x_{i}(k))+F_{i}(x_{i})v_{i}(k),\end{cases}$ (1)

where $x_{i}(k)\in\mathbb{R}^{n_{x}},u_{i}(k)\in\mathbb{R}^{n_{u}}$ and
$y_{i}(k)\in\mathbb{R}^{n_{y}}$ represent state variables, control inputs and
measurable output, respectively. The functions $f_{i}\left(x_{i}(k)\right)$,
$I_{i}\left(x_{i}(k)\right),h_{i}\left(x_{i}(k)\right)$, and
$F_{i}\left(x_{i}(k)\right)$ are the functions of $x_{i}(k)$ with
$f_{i}(0)=0$, $I_{i}(0)=0,h_{i}(0)=0$, and $F_{i}(0)=0$ and $G_{i}$’s are
known matrces. A process uncertainty is denoted by
$\omega_{i}(k)\in\mathbb{R}^{n_{\omega}}$, and $v_{i}(k)\in\mathbb{R}^{n_{v}}$
as a measurement noise which are assumed to be confined to specified
ellipsoidal sets.

######  Definition 1.

An ellipsoidal set has the form $\mathcal{X}\triangleq\\{\zeta:\zeta=$ $c+\Xi
z,\|z\|\leq 1\\}$, where $c\in\mathbb{R}^{n_{x}}$ is the center and
$\Xi\in\mathbb{R}^{n_{x}\times m}$ with $\operatorname{rank}(\Xi)=m\leq n_{x}$
is its shape matrix. Assume that $\Xi$ is a lower triangular matrix whose
diagonal elements all are positive. According to a Cholesky factorization, it
can be seen that $P=\Xi\Xi^{\mathrm{T}}>0$ and
$z^{\mathrm{T}}z=(\zeta-c)^{\mathrm{T}}P^{-1}(\zeta-c)\leq 1$. Consequently,
the ellipsoidal set can also be represented as
$\mathcal{X}\triangleq\left\\{\zeta:(\zeta-c)^{\mathrm{T}}P^{-1}(\zeta-c)\leq
1\right\\}$. The size of the ellipsoid is dependent on the squares shape
matrix $P$ and can be calculated as $\operatorname{Tr}(P)$, which is the sum
of the squared semiaxes lengths [30].

######  Assumption 1.

The process noise $w_{i}(k)$ is UBB, which is assumed to belong to the
following specified ellipsoidal sets:

$\displaystyle\mathcal{W}_{i}(k)\triangleq\left\\{w_{i}(k):{w_{i}}(k)^{\mathrm{T}}Q_{i}(k)^{-1}w_{i}(k)\leq
1\right\\}$ (2)
$\displaystyle\mathcal{V}_{i}(k)\triangleq\left\\{v_{i}(k):{v_{i}^{\mathrm{T}}(k)}R_{i}^{-1}(k)v_{i}(k)\leq
1\right\\},$

where $Q_{i}(k)=Q_{i}(k)^{T}>0$ and $R_{i}(k)=R_{i}(k)^{T}>0$ are known
matrices with compatible dimensions.

The system model for the $i$th agent is presented by fuzzy IF-THEN rules.

Plant Rule $l_{i}:$ IF $\theta_{i,1}(k)$ is $\mu_{l_{i},1}$ and
$\theta_{i,2}(k)$ is $\mu_{l_{i},2}\ldots$ and $\theta_{i,q}(k)$ is
$\mu_{l_{i},q}$, THEN

$\begin{cases}x_{i}(k+1)=A_{l_{i}}x_{i}(k)+B_{l_{i}}u_{i}(k)+M_{l_{i}}\omega_{i}(k)\\\
y_{i}(k)=C_{l_{i}}x_{i}(k)+D_{l{i}}v_{i}(k),\end{cases}$ (3)

where $l_{i}=1,\ldots,r$ ($r$ stands for the total number of plant IF-THEN
rules), $\mu_{l_{i},1},\ldots,\mu_{l_{i},q}$ are fuzzy sets, $\theta_{i}(k)=$
$\left[\theta_{i,1}^{T}(k)\theta_{i,2}^{T}(k)\cdots\theta_{i,q}^{T}(t)\right]^{T}$
denotes the premise variable, $A_{l_{i}},B_{l_{i}},M_{l_{i}},C_{l_{i}}$ and
$D_{l_{i}}$ are the system matrices with appropriate dimensions. The above-
mentioned system can be inferred as follows:

$\small{\left\\{\begin{aligned}
{x}_{i}(k+1)=&\sum_{l_{i}=1}^{r}g_{l_{i}}\left(\theta_{i}(k)\right)A_{l_{i}}x_{i}(k)\\\
&+\sum_{l_{i}=1}^{r}g_{l_{i}}\left(\theta_{i}(k)\right)B_{l_{i}}u_{i}(k)\\\
&+\sum_{l_{i}=1}^{r}g_{l_{i}}\left(\theta_{i}(k)\right)M_{l{i}}\omega_{i}(k)\\\
y_{i}(k)=&\sum_{l_{i}=1}^{r}g_{l_{i}}\left(\theta_{i}(k)\right)C{l_{i}}x_{i}(k)\\\
&+\sum_{l_{i}=1}^{r}g_{l_{i}}\left(\theta_{i}(k)\right)D_{l{i}}v_{i}(k)\end{aligned}\right.,}$
(4)

where
$g_{l_{i}}\left(\theta_{i}(k)\right)=\psi_{l_{i}}\left(\theta_{i}(k)\right)/\sum_{l_{i}=1}^{r}\psi_{l_{i}}\left(\theta_{i}(k)\right)$
is the normalized weight for each rule with
$\psi_{l_{i}}\left(\theta_{i}(k)\right)=$
$\Pi_{v=1}^{q}\mu_{l_{i}v}\left(\theta_{iv}(k)\right)\geqslant 0$ and
$\sum_{l_{i}=1}^{r}g_{l_{i}}\left(\theta_{i}(k)\right)=1$, where
$\mu_{l_{i}v}\left(\theta_{iv}(k)\right)$ is the grade of membership of
$\theta_{iv}(k)$ in $\mu_{l_{i}q}$.

By considering fuzzy model as an interpolation of $r$ linear systems through
the membership function $g_{l_{i}}\left(\theta_{i}(k)\right)$, we can
approximate the nonlinear system. Therefore, the nonlinear multi-agent system
can be described as

$\small{\left\\{\begin{aligned}
x_{i}(k+1)=&f_{i}(x_{i}(k))+G_{i}u_{i}(k)+I_{i}(x_{i})\omega_{i}(k)\\\
=&\sum_{l_{i}=1}^{r}g_{l_{i}}\left(\theta_{i}(k)\right)A_{l_{i}}x_{i}(k)+\Delta
f_{i}\left(x_{i}(k)\right)\\\
&+\sum_{l_{i}=1}^{r}g_{l_{i}}\left(\theta_{i}(k)\right)B_{l_{i}}u_{i}(k)\\\
&+\sum_{l_{i}=1}^{r}g_{l_{i}}\left(\theta_{i}(k)\right)M_{l{i}}\omega_{i}(k)\\\
&+\Delta I_{i}\left(x_{i}(k)\right)\omega_{i}(k)\\\
y_{i}(k)=&\sum_{l_{i}=1}^{r}g_{l_{i}}\left(\theta_{i}(k)\right)C_{l_{i}}x_{i}(k)+\Delta
h_{i}\left(x_{i}(k)\right)\\\
&+\sum_{l_{i}=1}^{r}g_{l_{i}}\left(\theta_{i}(k)\right)D_{l{i}}v_{i}(k)\\\
&+\Delta F_{i}\left(x_{i}(k)\right)v_{i}(k),\end{aligned}\right.}$ (5)

where

$\small{\left\\{\begin{aligned} \Delta
f_{i}\left(x_{i}(k)\right)=&f_{i}\left(x_{i}(k)\right)-\sum_{l_{i}=1}^{r}g_{l_{i}}\left(\theta_{i}(k)\right)A_{l_{i}}x_{i}(k)\\\
\Delta
I_{i}\left(x_{i}(k)\right)=&I_{i}\left(x_{i}(k)\right)-\sum_{l_{i}=1}^{r}g_{l_{i}}\left(\theta_{i}(k)\right)M_{l_{i}}\\\
\Delta
h_{i}\left(x_{i}(k)\right)=&h_{i}\left(x_{i}(k)\right)-\sum_{l_{i}=1}^{r}g_{l_{i}}\left(\theta_{i}(k)\right)C_{l_{i}}x_{i}(k)\\\
\Delta
F_{i}\left(x_{i}(k)\right)=&F_{i}\left(x_{i}(k)\right)-\sum_{l_{i}=1}^{r}g_{l_{i}}\left(\theta_{i}(k)\right)D_{l_{i}}\\\
\end{aligned}\right.}$ (6)

denote the approximation (or interpolation) errors between the nonlinear
system and the fuzzy model.

######  Assumption 2.

According to [27], we assume

$\left\\{\begin{aligned} &\Delta
f_{i}\left(x_{i}(k)\right)=H_{i,1}\Delta_{i,1}E_{i,1}x_{i}(k)\\\ &\Delta
I_{i}\left(x_{i}(k)\right)=H_{i,2}\Delta_{i,2}E_{i,2}\\\ &\Delta
h_{i}\left(x_{i}(k)\right)=H_{i,3}\Delta_{i,3}E_{i,3}x_{i}(k)\\\ &\Delta
F_{i}\left(x_{i}(k)\right)=H_{i,4}\Delta_{i,4}E_{i,4},\end{aligned}\right.$
(7)

where $H_{i}$ and $E_{i}$ are known matrices, and $\Delta_{i}$ is unknown but
bounded with $\left\|\Delta_{i}\right\|\leq 1$.

We are interested in constructing the fuzzy-based leader following consensus
protocol, which utilizes the estimated state instead of the full system state.
First, consider the leader agent’s dynamic by the following IF-THEN rules.
Plant Rule $l_{i}:$ IF $\theta_{i,1}(k)$ is $\mu_{l_{i},1}$ and
$\theta_{i,2}(k)$ is $\mu_{l_{i},2}\ldots$ and $\theta_{i,q}(k)$ is
$\mu_{l_{i},q}$, THEN

$x^{l}(k+1)=A_{l_{i}}^{l}x^{l}(k),$ (8)

where $x^{l}(k)\in\mathbb{R}^{n_{x}}$ is the state of the leader, and
$A_{l_{i}}^{l}$ are the system matrices with appropriate dimensions. It is
assumed that the leader’s dynamics are not subject to UBB process noise. The
abovementioned system can be inferred as follows:

$x^{l}(k+1)=\sum_{l_{i}=1}^{r}g_{l_{i}}\left(\theta_{i}(k)\right)A_{l_{i}}^{l}x^{l}(k).$
(9)

######  Assumption 3.

The initial states $x_{i}(0)$ and $x^{l}(0)$ are assumed to belong to a given
ellipsoid

$\displaystyle\mathcal{X}_{i}(0\mid 0)\triangleq\left\\{x\right._{i}(0):$
$\displaystyle\left(x_{i}(0)-\hat{x}_{i}(0\mid
0)\right)^{\mathrm{T}}P_{i}(0\mid 0)^{-1}$ (10)
$\displaystyle\times\left.\left(x_{i}(0)-\hat{x}_{i}(0\mid 0)\right)\leq
1\right\\}$
$\displaystyle\mathcal{U}_{i}(0)\triangleq\left\\{x\right._{i}(0):$
$\displaystyle\left(x_{i}(0)-x^{l}(0)\right)^{\mathrm{T}}U_{i}(0)^{-1}$
$\displaystyle\times\left.\left(x_{i}(0)-x^{l}(0)\right)\leq 1\right\\},$

where $\hat{x}_{i}(0\mid 0)$ is the given estimate of $x_{i}(0)$, and
$P_{i}(0\mid 0)=P_{i}(0\mid 0)^{\mathrm{T}}\succ 0$ and $U_{i}(0\mid
0)=U_{i}(0\mid 0)^{\mathrm{T}}\succ 0$ are known matrices.

In this paper, we consider two kinds of attacks on the system.

###  II-A False Data Injection Attacks

The original data packets are replaced by false ones when they are transferred
from controllers to actuators or from another agent via communication
channels.

$\begin{cases}u_{i}^{c}(k)=u_{i}(k)+u_{i}^{a}(k)\\\
\bar{x}_{j}^{c}(k)=\bar{x}_{j}(k)+\phi_{j}^{a}(k)\bar{x}_{j}^{a}(k),\end{cases}$
(11)

where $u_{i}(k)\in\mathbb{R}^{m_{i}},u_{i}^{a}(k)$ and $u_{i}^{c}(k)$ are the
uncompromised control input, unknown false data injected to the actuator of
agent $i$, and the compromised input available to agent $i$. Moreover,
$\bar{x}_{j}^{c}$ is the corrupted neighbouring data, and in the presence of
an attack on neighbouring channel $\phi_{j}^{a}$ is 1”, otherwise it is ”0”.

###  II-B Replay Attacks

A successful replay attack does not need a priori knowledge of the system
components. It is assumed that the attacker can record sensor’s measurement
data from $k_{i}$ untill $k_{r}$ with the window size $\tau=k_{r}-k_{i}$ in
the first phase. Then, in the second phase, the attacker replays the recorded
data to the system from $k=k_{r}+d$ untill the end of the attack at $k=k_{f}$,
where $d$ is the delay between the recording time and replaying time. We model
this attack according to [25] as

$a^{y_{i}}(k)=y_{i}(k-\tau)-y_{i}(k).$ (12)

Thus, the sensor’s data affected by the attack is

$\tilde{y}_{i}(k)=y_{i}(k)+a^{y_{i}}(k).$ (13)

We propose a distributed attack detector to detect the aforementioned types of
attacks. The modules are tasked to detect attacks as well as ensure that the
desired control specifications are satisfied. Also, the method can recover the
system performance and mitigate the effects of the attacks. The structure of
the system with the detector is shown in Fig. 1.

Figure 1: The structure of a leader-following MAS with a fuzzy-based set-
membership filtering detection method.

##  III Consensus Protocol and Fuzzy-Based Two-Step Set-Membership Estimation
Method

###  III-A Prediction Step

First, the prediction filter is considered in the form of
Plant Rule $l_{i}:$ IF ${\hat{\theta}}_{i,1}(k)$ is $\mu_{l_{i},1}$ and
$\hat{\theta}_{i,2}(k)$ is $\mu_{l_{i},2}\ldots$ and $\hat{\theta}_{i,q}(k)$
is $\mu_{l_{i},q}$, THEN

$\hat{x}_{i}(k+1\mid k)=\hat{A}_{l_{i}}\hat{x}_{i}(k\mid k),$ (14)

where $\hat{x}_{i}(k\mid k)$ is the estimation of the state $x_{i}(k)$,
$\hat{A}_{l_{i}}$ is the fuzzy filter parameter to be determined and
$\hat{\theta}_{i}(k)=\left\\{\hat{\theta}_{i,1}(k),\hat{\theta}_{i,2}(k),\ldots,\hat{\theta}_{i,q}(k)\right\\}$
are premise variables, which maybe functions of the state estimates. The
overall fuzzy filter can be written from (14) as [35], [36]

$\displaystyle\hat{x}_{i}(k+1\mid k)=$
$\displaystyle\sum_{l_{i}=1}^{r}g_{l_{i}}\left(\hat{\theta}_{i}(k)\right)\hat{A}_{l_{i}}\hat{x}_{i}(k\mid
k).$ (15)

For the given state estimation ellipsoid set $\mathcal{X}_{i}(k\mid k)$ with
the center $\hat{x}_{i}(k\mid k)$ and the shape matrix $\Xi_{i}(k\mid k)$, the
real state $x_{i}(k)$ can be described by

$x_{i}(k)=\hat{x}_{i}(k\mid k)+\Xi_{i}(k\mid k)z_{i}.$ (16)

Then, our goal is to obtain the prediction ellipsoid set

$\displaystyle\mathcal{X}_{i}(k+1\mid k)\triangleq\left\\{x_{i}(k+1):\right.$
(17) $\displaystyle\left.\left(x_{i}(k+1)-\hat{x}_{i}(k+1\mid
k)\right)^{\mathrm{T}}P_{i}^{-1}(k+1\mid k)\right.$
$\displaystyle\times\left.\left(x_{i}(k+1)-\hat{x}_{i}(k+1\mid k)\right)\leq
1\right\\}.$

Note that the state $x_{i}(k+1)$ belongs to such an ellipsoid set for any
value of the system noises in their specified sets.

###  III-B Measurement Update Step

The update based on the current measurement is considered for the system (5),
which is in the form of
Plant Rule $l_{i}:$ IF $\hat{\theta}_{i,1}(k)$ is $\mu_{l_{i},1}$ and
$\hat{\theta}_{i,2}(k)$ is $\mu_{l_{i},2}\ldots$ and $\hat{\theta}_{i,q}(k)$
is $\mu_{l_{i},q}$, THEN

$\begin{aligned} \hat{x}_{i}(k+1\mid k+1)=&\hat{x}_{i}(k+1\mid
k)+L_{l_{i}}\left(y_{i}(k+1)\right.\left.-\hat{y}_{i}(k+1\mid
k)\right),\end{aligned}$

(18)

where $L_{l_{i}}$ is the filter parameter to be determined. The overall fuzzy
update can be written from (18) as

$\displaystyle\hat{x}_{i}(k+1\mid k+1)=$ $\displaystyle\hat{x}_{i}(k+1\mid
k)+\sum_{l_{i}=1}^{r}g_{l_{i}}\left(\hat{\theta}_{i}(k)\right)L_{l_{i}}$ (19)
$\displaystyle\times\left(y_{i}(k+1)-\hat{y}_{i}(k+1\mid k)\right).$

According to the prediction ellipsoid set $\mathcal{X}_{i}(k+1\mid k)$ given
by (17), the state $x_{i}(k+1)$ can be written as

$x_{i}(k+1)=\hat{x}_{i}(k+1\mid k)+\Xi_{i}(k+1\mid k)z_{i}.$ (20)

Our objective is to update this prediction set with the one yielding from the
current measurement $y_{i}(k+1)$. In other words, we look for an updated
ellipsoid set $\mathcal{X}_{i}(k+1\mid k+1)$ with the center
$\hat{x}_{i}(k+1\mid k+1)$ and the shape matrix $\Xi_{i}(k+1\mid k+1)$ for the
state $x_{i}(k+1)$, given by the current measurement information at the time
instant $k+1$. Thus, the updated ellipsoid set should satisfy the condition

$\displaystyle(x_{i}(k+1)$ $\displaystyle-\hat{x}_{i}(k+1\mid
k+1))^{\mathrm{T}}P_{i}^{-1}(k+1\mid k+1)$ (21)
$\displaystyle\times\left(x_{i}(k+1)-\hat{x}_{i}(k+1\mid k+1)\right)\leq 1,$

whenever the output constraint

$\displaystyle y_{i}(k+1)=$
$\displaystyle\sum_{l_{i}=1}^{r}g_{l_{i}}\left({\theta}_{i}(k)\right)C_{l_{i}}\left(\hat{x}_{i}(k+1\mid
k)+\Xi_{i}(k+1\mid k)z_{i}\right)$ (22)
$\displaystyle+H_{i,3}\Delta_{i,3}E_{i,3}\left(\hat{x}_{i}(k+1\mid
k)+\Xi_{i}(k+1\mid k)z_{i}\right)$
$\displaystyle+\left(\sum_{l_{i}=1}^{r}g_{l_{i}}\left({\theta}_{i}(k)\right)D_{l_{i}}+H_{i,4}\Delta_{i,4}E_{i,4}\right)v_{i}(k+1)$

holds for some $\|z_{i}\|\leq 1$.

###  III-C Leader Following Consensus Protocol

The distributed observer-based leader following consensus protocol [37] is

$\displaystyle u_{i}(k)=K_{l_{i}}\Bigg{(}\sum_{j\in\mathcal{N}_{i}}$
$\displaystyle a_{ij}\left(\hat{x}_{i}(k\mid k)-\hat{x}_{j}(k\mid k)\right)$
(23) $\displaystyle+\lambda_{i}\left(\hat{x}_{i}(k\mid
k)-x^{l}(k)\right)\Bigg{)},$

where $K_{l_{i}}$ are constant matrices to be designed, $a_{ij}$ is a
nonnegative element of the weighted adjacency matrix
$\mathcal{A}=\left[a_{ij}\right]\in\mathbb{R}^{N\times N}$. The adjacency
matrix of the topology is selected as a binary matrix, where $a_{ij}=1$ if
follower $i$ can receive information from follower $j$, otherwise $a_{ij}=0$.

The leader-following multi-agent system (1), (8) achieves set-membership
leader-following consensus under protocol (23) and two-step filter (15), (19),
if the existence of desired gain sequences $K_{l_{i}},\hat{A}_{l_{i}}$, and
$L_{l_{i}}$ can guarantee that the one step ahead states $x_{i}(k+1),\forall
i\in\mathcal{\nu}$ for all the followers reside in a leader state ellipsoid
$\mathcal{U}_{i}(k+1)$ always enclosing all the followers’ true states, where

$\displaystyle\mathcal{U}_{i}(k+1)\triangleq\left\\{x_{i}(k+1):\right.$ (24)
$\displaystyle\left(x_{i}(k+1)-x^{l}(k+1)\right)^{T}U^{-1}(k+1)$
$\displaystyle\left.\times\left(x_{i}(k+1)-x^{l}(k+1)\right)\leq 1\right\\}$

For the given leader ellipsoid set $\mathcal{U}_{i}(k)$ with the center
$x^{l}(k)$ and the shape matrix $\xi_{i}(k)$, the state $x_{i}(k)$ can be
described by

$x_{i}(k)=x^{l}(k)+\xi_{i}(k)z_{i}.$ (25)

##  IV Attack Detection Using Set-Membership Fuzzy Filtering

The proposed cyberattack detection problem is addressed in this section by
developing a set-membership filter. First, we develop the prediction
elipsoidal sets based on the leader following consensus protocol (23) and then
update the prediction ellipsoid set with the current measurement. Also, we
develop the leader elipsoidal set based on the leader following consensus
protocol (23). Finally, convex optimization problems and one algorithm are
provided to expose the cyberattack diagnosis scheme.

###  IV-A The Prediction Ellipsoid Set Design Based on Leader Following
Consensus

From the system model (5) and (7), and the filter (15) and (16), the
prediction error $x_{i}(k+1)-\hat{x}_{i}(k+1\mid k)$ can be written as

$\displaystyle x_{i}(k$ $\displaystyle+1)-\hat{x}_{i}(k+1\mid k)$ (26)
$\displaystyle=$
$\displaystyle\left(\sum_{l_{i}=1}^{r}g_{l_{i}}\left(\theta_{i}(k)\right)A_{l_{i}}-\sum_{j_{i}=1}^{r}g_{j_{i}}\left(\hat{\theta}_{i}(k)\right)\hat{A}_{j_{i}}\right)\hat{x}_{i}(k\mid
k)$
$\displaystyle+\sum_{l_{i}=1}^{r}g_{l_{i}}\left(\theta_{i}(k)\right)A_{l_{i}}\Xi_{i}(k\mid
k)z_{i}+\sum_{l_{i}=1}^{r}g_{l_{i}}\left(\theta_{i}(k)\right)B_{l_{i}}K_{l_{i}}$
$\displaystyle\times\left(\sum_{j\in\mathcal{N}_{i}}a_{ij}\left(\hat{x}_{i}(k\mid
k)-\hat{x}_{j}(k\mid k)\right)+\lambda_{i}\left(\hat{x}_{i}(k\mid
k)-x^{l}(k)\right)\right)$
$\displaystyle+\sum_{l_{i}=1}^{r}g_{l_{i}}\left(\theta_{i}(k)\right)M_{l{i}}\omega_{i}(k)+H_{i,1}q_{i,1}+H_{i,1}q_{i,2}+H_{i,2}q_{i,3},$

where

$\displaystyle q_{i,1}=\Delta_{i,1}E_{i,1}\hat{x}_{i}(k\mid k)$ (27)
$\displaystyle q_{i,2}=\Delta_{i,1}E_{i,1}\Xi_{i}(k\mid k)z_{i}$
$\displaystyle q_{i,3}=\Delta_{i,2}E_{i,2}\omega_{i}(k).$

Denoting

$\eta_{i,1}(k)=\left[\begin{array}[]{llllll}1&z_{i}&\omega_{i}(k)&q_{i,1}&q_{i,2}&q_{i,3}\end{array}\right]^{T},$
(28)

and considering the fact that
$\sum_{l_{i}=1}^{r}g_{l_{i}}\left(\theta_{i}(k)\right)=1$, we can write (26)
in a compact form as

$\displaystyle x_{i}(k+1)$ $\displaystyle-\hat{x}_{i}(k+1\mid k)$ (29)
$\displaystyle=$
$\displaystyle\sum_{l_{i}=1}^{r}g_{l_{i}}\left(\theta_{i}(k)\right)\sum_{j_{i}=1}^{r}g_{j_{i}}\left(\hat{\theta}_{i}(k)\right)$
$\displaystyle\times\Gamma_{i,1,l_{i}j_{i}}\eta_{i,1}(k).$

By denoting

$\displaystyle P_{i,1,l_{i},j_{i}}=$
$\displaystyle\left(A_{l_{i}}-\hat{A}_{j_{i}}\right)\hat{x}_{i}(k\mid
k)-B_{l_{i}}K_{l_{i}}\lambda_{i}x^{l}(k)$ (30)
$\displaystyle+B_{l_{i}}K_{l_{i}}\sum_{j=1}^{N}\tilde{l}_{ij}\hat{x}_{j}(k\mid
k),$

we have

$\displaystyle\Gamma_{i,1,l_{i}j_{i}}=\left[P_{i,1,l_{i},j_{i}}\quad
A_{l_{i}}\Xi_{i}(k\mid k)\quad M_{l_{i}}\quad H_{i,1}\quad H_{i,1}\quad
H_{i,2}\right],$ (31)

where
$\tilde{\mathcal{L}}=\mathcal{L}+\Lambda=\left[\tilde{l}_{ij}\right]_{N\times
N}$ and
$\Lambda=\operatorname{diag}\left\\{\lambda_{1},\lambda_{2},\ldots,\lambda_{N}\right\\}$.

According to (29), we can write

$\displaystyle\left(x_{i}(k+1)\right.$
$\displaystyle\left.-\hat{x}_{i}(k+1\mid k)\right)^{T}P_{i}^{-1}(k+1\mid k)$
(32) $\displaystyle\times\left(x_{i}(k+1)-\hat{x}_{i}(k+1\mid k)\right)$
$\displaystyle=$
$\displaystyle\sum_{l_{i}=1}^{r}g_{l_{i}}\left(\theta_{i}(k)\right)\sum_{j_{i}=1}^{r}g_{j_{i}}\left(\hat{\theta}_{i}(k)\right)$
$\displaystyle\times\sum_{m_{i}=1}^{r}g_{m_{i}}\left(\theta_{i}(k)\right)\sum_{n_{i}=1}^{r}g_{n_{i}}\left(\hat{\theta}_{i}(k)\right)$
$\displaystyle\times\eta_{i,1}^{T}(k)\Gamma_{i,1,l_{i}j_{i}}^{T}P_{i}^{-1}(k+1\mid
k)$ $\displaystyle\times\Gamma_{i,1,m_{i}n_{i}}\eta_{i,1}(k).$

Therefore, we can achieve

$\displaystyle\left(x_{i}(k+1)\right.$
$\displaystyle\left.-\hat{x}_{i}(k+1\mid k)\right)^{T}P_{i}^{-1}(k+1\mid k)$
(33) $\displaystyle\times\left(x_{i}(k+1)-\hat{x}_{i}(k+1\mid k)\right)$
$\displaystyle\leq$
$\displaystyle\sum_{l_{i}=1}^{r}g_{l_{i}}\left(\theta_{i}(k)\right)\sum_{j_{i}=1}^{r}g_{j_{i}}\left(\hat{\theta}_{i}(k)\right)$
$\displaystyle\times\eta_{i,1}^{T}(k)\Gamma_{i,1,l_{i}j_{i}}^{T}P_{i}^{-1}(k+1\mid
k)$ $\displaystyle\times\Gamma_{i,1,l_{i}j_{i}}\eta_{i,1}(k).$

The condition in (17) can be written as

$\displaystyle\eta_{i,1}^{T}(k)\Bigg{[}$
$\displaystyle\sum_{l_{i}=1}^{r}g_{l_{i}}\left(\theta_{i}(k)\right)\sum_{j_{i}=1}^{r}g_{j_{i}}\left(\hat{\theta}_{i}(k)\right)$
(34) $\displaystyle\times\Gamma_{i,1,l_{i}j_{i}}^{T}P_{i}^{-1}(k+1\mid
k)\Gamma_{i,1,l_{i}j_{i}}$
$\displaystyle-\operatorname{diag}\\{1,0,0,0,0,0\\}\Bigg{]}\eta_{i,1}(k)\leq
0.$

With $\left\|\Delta_{i}\right\|\leq 1$, we can infer from (27) that

$\left\\{\begin{array}[]{l}q_{i,1}^{T}q_{i,1}-\hat{x}_{i}^{T}(k\mid
k)E_{i,1}^{T}E_{i,1}\hat{x}_{i}(k\mid k)\leq 0\\\
q_{i,2}^{T}q_{i,2}-z_{i}^{T}\Xi_{i}^{T}(k\mid
k)E_{i,1}^{T}E_{i,1}\Xi_{i}(k\mid k)z_{i}\leq 0\\\
q_{i,3}^{T}q_{i,3}-\omega_{i}^{T}(k)E_{i,2}^{T}E_{i,2}\omega_{i}(k)\leq
0\end{array}\right..$ (35)

From (2), (16) and (35), the unknown variables
$z_{i},\omega_{i}(k),q_{i,1},q_{i,2}$ and $q_{i,3}$ satisfy the following
constraints:

$\displaystyle\|z_{i}\|\leq 1$ (36) $\displaystyle
w_{i}^{T}(k)Q_{i}^{-1}(k)w_{i}(k)\leq 1$ $\displaystyle
q_{i,1}^{T}q_{i,1}-\hat{x}_{i}^{T}(k\mid k)E_{i,1}^{T}E_{i,1}\hat{x}_{i}(k\mid
k)\leq 0$ $\displaystyle q_{i,2}^{T}q_{i,2}-z_{i}^{T}\Xi_{i}^{T}(k\mid
k)E_{i,1}^{T}E_{i,1}\Xi_{i}(k\mid k)z_{i}\leq 0$ $\displaystyle
q_{i,3}^{T}q_{i,3}-\omega_{i}^{T}(k)E_{i,2}^{T}E_{i,2}\omega_{i}(k)\leq 0,$

which can be written in $\eta_{i,1}(k)$ as

$\displaystyle\eta_{i,1}^{T}(k)\operatorname{diag}\\{-1,I,0,0,0,0\\}\eta_{i,1}(k)\leq
0$ (37)
$\displaystyle\eta_{i,1}^{T}(k)\operatorname{diag}\\{-1,0,Q_{i}^{-1}(k),0,0,0\\}\eta_{i,1}(k)\leq
0$ $\displaystyle\eta_{i,1}^{T}(k)\operatorname{diag}\\{-\hat{x}_{i}^{T}(k\mid
k)E_{i,1}^{T}E_{i,1}\hat{x}_{i}(k\mid k),$
$\displaystyle\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad
0,0,I,0,0\\}\eta_{i,1}(k)\leq 0$
$\displaystyle\eta_{i,1}^{T}(k)\operatorname{diag}\\{0,-\Xi_{i}^{T}(k\mid
k)E_{i,1}^{T}E_{i,1}\Xi_{i}(k\mid k),$
$\displaystyle\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad
0,0,I,0\\}\eta_{i,1}(k)\leq 0.$
$\displaystyle\eta_{i,1}^{T}(k)\operatorname{diag}\\{0,0,-E_{i,2}^{T}E_{i,2},0,0,I\\}\eta_{i,1}(k)\leq
0.$

Applying S-procedure [34] to (34) and (37), we can conclude that the
inequality (34) holds if there exist nonnegative scalars $\tau_{i,1}(k)$,
$\tau_{i,2}(k)$, $\tau_{i,3}(k),\tau_{i,4}(k)$ and $\tau_{i,5}(k)$ such that

$\displaystyle\Gamma_{i,1,l_{i}j_{i}}^{T}P_{i}^{-1}(k+1\mid
k)\Gamma_{i,1,l_{i}j_{i}}-\operatorname{diag}\\{1,0,0,0,0,0\\}$ (38)
$\displaystyle-\tau_{i,1}(k)\operatorname{diag}\\{-1,I,0,0,0,0\\}$
$\displaystyle-\tau_{i,2}(k)\operatorname{diag}\\{-1,0,Q_{i}^{-1}(k),0,0,0\\}$
$\displaystyle-\tau_{i,3}(k)$
$\displaystyle\quad\times\operatorname{diag}\\{-\hat{x}_{i}^{T}(k\mid
k)E_{i,1}^{T}E_{i,1}\hat{x}_{i}(k\mid k),0,0,I,0,0\\}$
$\displaystyle-\tau_{i,4}(k)$
$\displaystyle\quad\times\operatorname{diag}\\{0,-\Xi_{i}^{T}(k\mid
k)E_{i,1}^{T}E_{i,1}\Xi_{i}(k\mid k),0,0,I,0\\}$
$\displaystyle-\tau_{i,5}(k)\operatorname{diag}\\{0,0,-E_{i,2}^{T}E_{i,2},0,0,I\\}\leq
0.$

Inequality (38) can be written in the following compact form

$\displaystyle\Gamma_{i,1,l_{i}j_{i}}^{T}$ $\displaystyle P_{i}^{-1}(k+1\mid
k)\Gamma_{i,1,l_{i}j_{i}}-\operatorname{diag}\\{1-\tau_{i,1}(k)$ (39)
$\displaystyle-\tau_{i,2}(k)-\tau_{i,3}(k)\hat{x}_{i}^{T}(k\mid
k)E_{i,1}^{T}E_{i,1}\hat{x}_{i}(k\mid k),$
$\displaystyle\tau_{i,1}(k)I-\tau_{i,4}(k)\Xi_{i}^{T}(k\mid
k)E_{i,1}^{T}E_{i,1}\Xi_{i}(k\mid k),$
$\displaystyle\tau_{i,2}(k)Q_{i}^{-1}(k)-\tau_{i,5}(k)E_{i,2}^{T}E_{i,2},$
$\displaystyle\tau_{i,3}(k)I,\tau_{i,4}(k)I,\tau_{i,5}(k)I\\}\leq 0.$

Finally, denoting

$\displaystyle\Theta_{i,1}$ $\displaystyle(k)$ (40) $\displaystyle=$
$\displaystyle\operatorname{diag}\\{1-\tau_{i,1}(k)-\tau_{i,2}(k)$
$\displaystyle-\tau_{i,3}(k)\hat{x}_{i}^{T}(k\mid
k)E_{i,1}^{T}E_{i,1}\hat{x}_{i}(k\mid k),\tau_{i,1}(k)I$
$\displaystyle-\tau_{i,4}(k)\Xi_{i}^{T}(k\mid
k)E_{i,1}^{T}E_{i,1}\Xi_{i}(k\mid k),$
$\displaystyle\tau_{i,2}(k)Q_{i}^{-1}(k)-\tau_{i,5}(k)E_{i,2}^{T}E_{i,2},\tau_{i,3}(k)I,$
$\displaystyle\tau_{i,4}(k)I,\tau_{i,5}(k)I\\},$

we can write (39) as

$\displaystyle\Gamma_{i,1,l_{i}j_{i}}^{T}P_{i}^{-1}(k+1\mid
k)\Gamma_{i,1,l_{i}j_{i}}-\Theta_{i,1}(k)\leq 0.$ (41)

By using Schur complements, (41) is equivalent to

$\left[\begin{array}[]{cc}-P_{i}(k+1\mid k)&\Gamma_{i,1,l_{i}j_{i}}\\\
\Gamma_{i,1,l_{i}j_{i}}^{\mathrm{T}}&-\Theta_{i,1}(k)\end{array}\right]\leq
0.$ (42)

Moreover, from the system model (5), (8), and (25), and by considering (27)
the error $x_{i}(k+1)-x^{l}(k+1\mid k)$ can be written as

$\displaystyle x_{i}(k$ $\displaystyle+1)-x^{l}(k+1)$ (43) $\displaystyle=$
$\displaystyle\sum_{l_{i}=1}^{r}g_{l_{i}}\left(\theta_{i}(k)\right)A_{l_{i}}\left(x^{l}(k)+\xi_{i}(k)z_{i}\right)+\sum_{l_{i}=1}^{r}g_{l_{i}}\left(\theta_{i}(k)\right)B_{l_{i}}K_{l_{i}}$
$\displaystyle\times\left(\sum_{j\in\mathcal{N}_{i}}a_{ij}\left(\hat{x}_{i}(k\mid
k)-\hat{x}_{j}(k\mid k)\right)+\lambda_{i}\left(\hat{x}_{i}(k\mid
k)-x^{l}(k)\right)\right)$
$\displaystyle+\sum_{l_{i}=1}^{r}g_{l_{i}}\left(\theta_{i}(k)\right)M_{l{i}}\omega_{i}(k)-\sum_{l_{i}=1}^{r}g_{l_{i}}\left(\theta_{i}(k)\right)A_{l_{i}}^{l}x^{l}(k)$
$\displaystyle+H_{i,1}q_{i,1}+H_{i,1}q_{i,2}+H_{i,2}q_{i,3}.$

Considering the fact that
$\sum_{l_{i}=1}^{r}g_{l_{i}}\left(\theta_{i}(k)\right)=1$, we can write (43)
in a compact form as

$\displaystyle x_{i}(k+1)$
$\displaystyle-x^{l}(k+1)=\sum_{l_{i}=1}^{r}g_{l_{i}}\left(\theta_{i}(k)\right)\sum_{j_{i}=1}^{r}\Gamma_{i,2,l_{i}}\eta_{i,1}(k).$
(44)

By denoting

$\displaystyle P_{i,2,l_{i}}=$
$\displaystyle\left(A_{l_{i}}+A_{l{i}}^{l}-B_{l_{i}}K_{l_{i}}\lambda_{i}\right)x^{l}(k)$
(45)
$\displaystyle+B_{l_{i}}K_{l_{i}}\sum_{j=1}^{N}\tilde{l}_{ij}\hat{x}_{j}(k\mid
k),$

we have

$\displaystyle\Gamma_{i,2,l_{i}}=\left[P_{i,2,l_{i}}\quad
A_{l_{i}}\xi_{i}(k)\quad M_{l_{i}}\quad H_{i,1}\quad H_{i,1}\quad
H_{i,2}\right].$ (46)

According to (44), we can write

$\displaystyle\left(x_{i}(k+1)\right.$
$\displaystyle\left.-x^{l}(k+1)\right)^{T}U_{i}^{-1}(k+1)$ (47)
$\displaystyle\times\left(x_{i}(k+1)-x^{l}(k+1)\right)$ $\displaystyle=$
$\displaystyle\sum_{l_{i}=1}^{r}g_{l_{i}}\left(\theta_{i}(k)\right)\sum_{j_{i}=1}^{r}g_{j_{i}}\left(\theta_{i}(k)\right)$
$\displaystyle\times\eta_{i,1}^{T}(k)\Gamma_{i,2,l_{i}}^{T}U_{i}^{-1}(k+1)$
$\displaystyle\times\Gamma_{i,2,j_{i}}\eta_{i,1}(k).$

Therefore, we can achieve

$\displaystyle\left(x_{i}(k+1)\right.$
$\displaystyle\left.-x^{l}(k+1)\right)^{T}U_{i}^{-1}(k+1)$ (48)
$\displaystyle\times\left(x_{i}(k+1)-x^{l}(k+1)\right)$ $\displaystyle\leq$
$\displaystyle\sum_{l_{i}=1}^{r}g_{l_{i}}\left(\theta_{i}(k)\right)\eta_{i,1}^{T}(k)\Gamma_{i,2,l_{i}}^{T}U_{i}^{-1}(k+1)$
$\displaystyle\times\Gamma_{i,2,l_{i}}\eta_{i,1}(k).$

Therefore, the condition in (24) can be written as

$\displaystyle\eta_{i,1}^{T}(k)\Bigg{[}$
$\displaystyle\sum_{l_{i}=1}^{r}g_{l_{i}}\left(\theta_{i}(k)\right)\Gamma_{i,2,l_{i}}^{T}U_{i}^{-1}(k+1)\Gamma_{i,2,l_{i}}$
(49)
$\displaystyle-\operatorname{diag}\\{1,0,0,0,0,0\\}\Bigg{]}\eta_{i,1}(k)\leq
0.$

Applying S-procedure to (37) and (49), we can conclude that the inequality
(49) holds if there exist nonnegative scalars $\tau_{i,6}(k)$,
$\tau_{i,7}(k)$, $\tau_{i,8}(k),\tau_{i,9}(k)$ and $\tau_{i,10}(k)$ such that

$\displaystyle\Gamma_{i,2,l_{i}}^{T}U_{i}^{-1}(k+1)\Gamma_{i,2,l_{i}}-\operatorname{diag}\\{1,0,0,0,0,0\\}$
(50) $\displaystyle-\tau_{i,6}(k)\operatorname{diag}\\{-1,I,0,0,0,0\\}$
$\displaystyle-\tau_{i,7}(k)\operatorname{diag}\\{-1,0,Q_{i}^{-1}(k),0,0,0\\}$
$\displaystyle-\tau_{i,8}(k)\operatorname{diag}\\{-\hat{x}_{i}^{T}(k\mid
k)E_{i,1}^{T}E_{i,1}\hat{x}_{i}(k\mid k),0,0,I,0,0\\}$
$\displaystyle-\tau_{i,9}(k)\operatorname{diag}\\{0,-\Xi_{i}^{T}(k\mid
k)E_{i,1}^{T}E_{i,1}\Xi_{i}(k\mid k),0,0,I,0\\}$
$\displaystyle-\tau_{i,10}(k)\operatorname{diag}\\{0,0,-E_{i,2}^{T}E_{i,2},0,0,I\\}\leq
0.$

Inequality (50) can be written in the following compact form

$\displaystyle\Gamma_{i,2,l_{i}}^{T}$ $\displaystyle
U_{i}^{-1}(k+1)\Gamma_{i,2,l_{i}}-\operatorname{diag}\\{1-\tau_{i,6}(k)-\tau_{i,7}(k)$
(51) $\displaystyle-\tau_{i,8}(k)\hat{x}_{i}^{T}(k\mid
k)E_{i,1}^{T}E_{i,1}\hat{x}_{i}(k\mid k),\tau_{i,6}(k)I$
$\displaystyle-\tau_{i,9}(k)\Xi_{i}^{T}(k\mid
k)E_{i,1}^{T}E_{i,1}\Xi_{i}(k\mid k),\tau_{i,7}(k)Q_{i}^{-1}(k)$
$\displaystyle-\tau_{i,10}(k)E{i,2}^{T}E_{i,2},\tau_{i,8}(k)I,\tau_{i,9}(k)I,$
$\displaystyle\tau_{i,10}(k)I\\}\leq 0.$

Finally, denoting

$\displaystyle\Theta_{i,2}$ $\displaystyle(k)$ (52) $\displaystyle=$
$\displaystyle\operatorname{diag}\\{1-\tau_{i,6}(k)-\tau_{i,7}(k)$
$\displaystyle-\tau_{i,8}(k)\hat{x}_{i}^{T}(k\mid
k)E_{i,1}^{T}E_{i,1}\hat{x}_{i}(k\mid k),\tau_{i,6}(k)I$
$\displaystyle-\tau_{i,9}(k)\Xi_{i}^{T}(k\mid
k)E_{i,1}^{T}E_{i,1}\Xi_{i}(k\mid k),\tau_{i,7}(k)Q_{i}^{-1}(k)$
$\displaystyle-\tau_{i,10}(k)E_{i,2}^{T}E_{i,2},\tau_{i,8}(k)I,\tau_{i,9}(k)I,\tau_{i,10}(k)I\\},$

we can write (51) as

$\displaystyle\Gamma_{i,2,l_{i}}^{T}U_{i}^{-1}(k+1)\Gamma_{i,2,l_{i}}-\Theta_{i,2}(k)\leq
0.$ (53)

By using Schur complements, (53) is equivalent to

$\left[\begin{array}[]{cc}-U_{i}(k+1)&\Gamma_{i,2,l_{i}}\\\
\Gamma_{i,2,l_{i}}^{\mathrm{T}}&-\Theta_{i,2}(k)\end{array}\right]\leq 0.$
(54)

######  Theorem 1.

Consider the leader following multi-agent system (1), (8) that satisfies
Assumption 2, Assumption 7 and Assumption 10. Suppose that the state
$x_{i}(k)$ belongs to its state estimation ellipsoid
$\left(x_{i}(k)-\hat{x}_{i}(k\mid k)\right)^{\mathrm{T}}P_{i}^{-1}(k\mid
k)\left(x_{i}(k)-\hat{x}_{i}(k\mid k)\right)\leq 1$ and leader state ellipsoid
$\left(x_{i}(k)-x^{l}(k)\right)^{\mathrm{T}}U_{i}^{-1}(k)\left(x_{i}(k)-x^{l}(k)\right)\leq
1$, then the one-step ahead state $x_{i}(k+1)$ will reside in its state
prediction ellipsoid $\left(x_{i}(k+1)-\hat{x}_{i}(k+1\mid
k)\right)^{\mathrm{T}}P_{i}^{-1}(k+1\mid
k)\left(x_{i}(k+1)-\hat{x}_{i}(k+1\mid k)\right)\leq 1$ as well as leader
state ellipsoid
$\left(x_{i}(k+1)-x^{l}(k+1)\right)^{\mathrm{T}}U_{i}^{-1}(k+1)\left(x_{i}(k+1)-x^{l}(k+1)\right)\leq
1$, if there exist $P_{i}(k+1\mid
k)>0,U_{i}(k+1)>0,\hat{A}_{l_{i}},K_{l{i}},\tau_{i,m}(k)\geq 0,$ for
$m=1,\ldots,10,$ such that the linear matrix inequalities (LMI) (42) and (54)
hold for all $l_{i},j_{i}=1,\dots,r$.

###### Proof.

According to the above discussion, if there exist $P_{i}(k+1\mid
k)>0,U_{i}(k+1)>0,\hat{A}_{l_{i}},K_{l{i}},\tau_{i,m}(k)\geq 0,$ for
$m=1,\ldots,10,$ such that (42) and (54) hold for all $l_{i},j_{i}=1,\dots,r$,
then we have

$\displaystyle\sum_{l_{i}=1}^{r}$ $\displaystyle
g_{l_{i}}\left(\theta_{i}(k)\right)\sum_{j_{i}=1}^{r}g_{j_{i}}\left(\hat{\theta}_{i}(k)\right)$
(55)
$\displaystyle\times\eta_{i,1}^{T}(k)\Gamma_{i,1,l_{i}j_{i}}^{T}P_{i}^{-1}(k+1\mid
k)\Gamma_{i,1,l_{i}j_{i}}\eta_{i,1}(k)\leq 1$

and

$\displaystyle\sum_{l_{i}=1}^{r}g_{l_{i}}\left(\theta_{i}(k)\right)\eta_{i,1}^{T}(k)\Gamma_{i,2,l_{i}}^{T}U_{i}^{-1}(k+1)\Gamma_{i,2,l_{i}}\eta_{i,1}(k)\leq
1.$ (56)

From (33) and (48), we obtain

$\begin{aligned} \left(x_{i}(k+1)-\hat{x}_{i}(k+1\mid
k)\right)^{\mathrm{T}}P_{i}^{-1}(k+1\mid
k)\left(x_{i}(k+1)-\hat{x}_{i}(k+1\mid k)\right)\leq 1\end{aligned}$

(57)

and

$\left(x_{i}(k+1)-x^{l}(k+1)\right)^{\mathrm{T}}U_{i}^{-1}(k+1)\left(x_{i}(k+1)-x^{l}(k+1)\right)\leq
1,$ (58)

which complete the proof. ∎

According to the Theorem 1 and in order to find the optimal state prediction
ellipsoid containing $x_{i}(k+1)$, the convex optimization is performed as

$\displaystyle\underset{\begin{subarray}{c}P_{i}(k+1\mid
k),U_{i}(k+1),\hat{A}_{l{i}}(k),K_{l{i}},\\\
\tau_{i,1}(k),\tau_{i,2}(k),\tau_{i,3}(k),\tau_{i,4}(k),\tau_{i,5}(k),\\\
\tau_{i,6}(k),\tau_{i,7}(k),\tau_{i,8}(k),\tau_{i,9}(k),\tau_{i,10}(k)\\\
\end{subarray}}{\operatorname{min}}\operatorname{Tr}\left(T_{i}(k+1\mid
k)\right)$ (59)

subject to (42) for all $l_{i},j_{i}=1,\dots,r$ in which the trace of
$T_{i}(k+1\mid k)=\operatorname{diag}\\{U_{i}(k+1),P_{i}(k+1\mid k)\\}$ is
optimized at each time step in order to find the prediction ellipsoid set with
minimal size.

###  IV-B Update on Prediction Ellipsoid Set With Current Measurement

We develop here a scheme to determine the shape matrix $E_{i}(k+1\mid k+1)$
and the filter gain $L_{i}(k+1)$ with the output constraint (22).

From the system (1), the prediction ellipsoid set (20), and the filter based
on the current measurement (20), the current estimation error
$x_{i}(k+1)-\hat{x}_{i}(k+1\mid k+1)$ can be written as

$\displaystyle x_{i}(k+1)$ $\displaystyle-\hat{x}_{i}(k+1\mid k+1)$ (60)
$\displaystyle=$
$\displaystyle\sum_{l_{i}=1}^{r}g_{l_{i}}\left(\theta_{i}(k)\right)\sum_{j_{i}=1}^{r}g_{j_{i}}\left(\hat{\theta}_{i}(k)\right)$
$\displaystyle\times\left[\left(I-L_{j_{i}}C_{l{i}}\right)\Xi_{i}(k+1\mid
k)z_{i}\right.$ $\displaystyle\quad\quad-
D_{l_{i}}v_{i}(k+1)-L_{j_{i}}H_{i,3}q_{i,4}$
$\displaystyle\quad\quad\left.-L_{j_{i}}H_{i,3}q_{i,5}-L_{j_{i}}H_{i,4}q_{i,6}\right],$

where

$\displaystyle q_{i,4}=\Delta_{i,3}E_{i,3}\hat{x}_{i}(k+1\mid k)$ (61)
$\displaystyle q_{i,5}=\Delta_{i,3}E_{i,3}\Xi_{i}(k+1\mid k)z_{i}$
$\displaystyle q_{i,6}=\Delta_{i,4}E_{i,4}v_{i}(k+1).$

Therefore, we can define

$\eta_{i,2}(k+1)=\left[\begin{array}[]{llllll}1&z_{i}&v_{i}(k+1)&q_{i,4}&q_{i,5}&q_{i,6}\end{array}\right]^{T}.$
(62)

Thus, the above estimation error dynamics can be written in a compact form

$\displaystyle x_{i}(k+1)$ $\displaystyle-\hat{x}_{i}(k+1\mid k+1)$ (63)
$\displaystyle=$
$\displaystyle\sum_{l_{i}=1}^{r}g_{l_{i}}\left(\theta_{i}(k)\right)\sum_{j_{i}=1}^{r}g_{j_{i}}\left(\hat{\theta}_{i}(k)\right)$
$\displaystyle\times\Gamma_{i,2,l_{i}j_{i}}\eta_{i,2}(k+1).$

By denoting

$\displaystyle
P_{i,2,l_{i},j_{i}}=\left(I-L_{j_{i}}C_{l{i}}\right)\Xi_{i}(k+1\mid k),$ (64)

we have

$\begin{aligned} \Gamma_{i,3,l{i}j_{i}}=\left[0\quad P_{i,2,l_{i},j_{i}}\quad-
L_{j_{i}}D_{l{i}}\quad-L_{j_{i}}H_{i,3}\quad-L_{j_{i}}H_{i,3}\quad-
L_{j_{i}}H_{i,4}\right].\end{aligned}$

(65)

Taking (63) into account, we can write

$\displaystyle(x_{i}(k+1)$ $\displaystyle-\hat{x}_{i}(k+1\mid
k+1))^{\mathrm{T}}P_{i}^{-1}(k+1\mid k+1)$ (66)
$\displaystyle\times\left(x_{i}(k+1)-\hat{x}_{i}(k+1\mid k+1)\right)$
$\displaystyle=$
$\displaystyle\sum_{l_{i}=1}^{r}g_{l_{i}}\left(\theta_{i}(k)\right)\sum_{j_{i}=1}^{r}g_{j_{i}}\left(\hat{\theta}_{i}(k)\right)$
$\displaystyle\sum_{m_{i}=1}^{r}g_{m_{i}}\left(\theta_{i}(k)\right)\sum_{n_{i}=1}^{r}g_{n_{i}}\left(\hat{\theta}_{i}(k)\right)$
$\displaystyle\eta_{i,2}^{T}(k+1)\Gamma_{i,3,l{i}j_{i}}^{T}\Gamma_{i,3,m{i}n_{i}}\eta_{i,2}(k+1)$

Therefore, we can achieve

$\displaystyle\left(x_{i}(k+1)\right.$
$\displaystyle\left.-\hat{x}_{i}(k+1\mid
k+1)\right)^{\mathrm{T}}P_{i}^{-1}(k+1\mid k+1)$ (67)
$\displaystyle\times\left(x_{i}(k+1)-\hat{x}_{i}(k+1\mid k+1)\right)$
$\displaystyle\leq$
$\displaystyle\sum_{l_{i}=1}^{r}g_{l_{i}}\left(\theta_{i}(k)\right)\sum_{j_{i}=1}^{r}g_{j_{i}}\left(\hat{\theta}_{i}(k)\right)$
$\displaystyle\eta_{i,2}^{T}(k+1)\Gamma_{i,3,l{i}j_{i}}^{T}\Gamma_{i,3,l{i}j_{i}}\eta_{i,2}(k+1)$

Therefore, the condition (21) in Section III-B can be described as

$\displaystyle\eta_{i,2}^{\mathrm{T}}(k+1)\Bigg{[}$
$\displaystyle\sum_{l_{i}=1}^{r}g_{l_{i}}\left(\theta_{i}(k)\right)\sum_{j_{i}=1}^{r}g_{j_{i}}\left(\hat{\theta}_{i}(k)\right)\Gamma_{i,3,l_{i}j_{i}}^{\mathrm{T}}$
(68) $\displaystyle\times P_{i}^{-1}(k+1\mid k+1)\Gamma_{i,3,l_{i}j_{i}}$
$\displaystyle-\operatorname{diag}\\{1,0,0,0,0,0\\}\Bigg{]}\eta_{i,2}(k+1)\leq
0$

On the other hand, from (2), (20), and (61) the unknown variables
$z_{i},v_{i}(k+1),q_{i,4},q_{i,5},$ and $q_{i,6}$ satisfy the following
constraints:

$\displaystyle\|z_{i}\|\leq 1$ (69) $\displaystyle
v_{i}^{T}(k+1)R_{i}^{-1}(k+1)v_{i}(k+1)\leq 1$ $\displaystyle
q_{i,4}^{T}q_{i,4}-\hat{x}_{i}^{T}(k+1\mid
k)E_{i,3}^{T}E_{i,3}\hat{x}_{i}(k+1\mid k)\leq 0$ $\displaystyle
q_{i,5}^{T}q_{i,5}-z_{i}^{T}\Xi_{i}^{T}(k+1\mid k)$
$\displaystyle\quad\quad\quad\quad\quad\quad\quad\quad\times
E_{i,3}^{T}E_{i,3}\Xi_{i}(k+1\mid k)z_{i}\leq 0$ $\displaystyle
q_{i,6}^{T}q_{i,6}-v_{i}^{T}(k+1)E_{i,4}^{T}E_{i,4}v_{i}(k+1)\leq 0$

which can be written in $\eta_{i,2}(k+1)$ as

$\displaystyle\eta_{i,2}^{\mathrm{T}}(k+1)\operatorname{diag}\\{-1,I,0,0,0,0\\}\eta_{i,2}(k+1)\leq
0$ (70)
$\displaystyle\eta_{i,2}^{\mathrm{T}}(k+1)\operatorname{diag}\\{-1,0,R_{i}^{-1}(k+1),0,0,0\\}$
$\displaystyle\quad\quad\quad\quad\quad\quad\quad\quad\quad\times\eta_{i,2}(k+1)\leq
1$
$\displaystyle\eta_{i,2}^{\mathrm{T}}(k+1)\operatorname{diag}\\{-\hat{x}_{i}^{T}(k+1\mid
k)E_{i,3}^{T}E_{i,3}\hat{x}_{i}(k+1\mid k),$
$\displaystyle\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad
0,0,I,0,0\\}\eta_{i,2}(k+1)\leq 1$
$\displaystyle\eta_{i,2}^{\mathrm{T}}(k+1)\operatorname{diag}\\{0,-\Xi_{i}^{T}(k+1\mid
k)E_{i,3}^{T}E_{i,3}$
$\displaystyle\quad\quad\quad\quad\quad\quad\quad\quad\quad\times\Xi_{i}(k+1\mid
k),0,0,I,0\\}$
$\displaystyle\quad\quad\quad\quad\quad\quad\quad\quad\quad\times\eta_{i,2}(k+1)\leq
1$
$\displaystyle\eta_{i,2}^{\mathrm{T}}(k+1)\operatorname{diag}\\{0,0,-E_{i,4}^{T}E_{i,4},0,0,,I\\}$
$\displaystyle\quad\quad\quad\quad\quad\quad\quad\quad\quad\times\eta_{i,2}(k+1)\leq
1$

By applying S-procedure to (68) and (70), we can conclude that the inequality
(68) holds if there exist nonnegative scalars
$\tau_{i,11}(k),\tau_{i,12}(k),\tau_{i,13}(k),\tau_{i,14}(k),$ and
$\tau_{i,15}(k)$ such that

$\displaystyle\Gamma_{i,3,l_{i}j_{i}}^{T}P_{i}^{-1}(k+1\mid
k+1)\Gamma_{i,3,l_{i}j_{i}}$ (71)
$\displaystyle-\operatorname{diag}\\{1,0,0,0,0,0\\}-\tau_{i,11}(k)\operatorname{diag}\\{-1,I,0,0,0,0\\}$
$\displaystyle-\tau_{i,12}(k)\operatorname{diag}\\{-1,0,R_{i}^{-1}(k+1),0,0,0\\}$
$\displaystyle-\tau_{i,13}(k)\operatorname{diag}\\{-\hat{x}_{i}^{T}(k+1\mid
k)E_{i,3}^{T}E_{i,3}\hat{x}_{i}(k+1\mid k),$
$\displaystyle\quad\quad\quad\quad\quad\quad\quad\quad\quad 0,0,I,0,0\\}$
$\displaystyle-\tau_{i,14}(k)\operatorname{diag}\\{0,-\Xi_{i}^{T}(k+1\mid
k)E_{i,3}^{T}E_{i,3}$
$\displaystyle\quad\quad\quad\quad\quad\quad\quad\quad\quad\times\Xi_{i}(k+1\mid
k),0,0,I,0\\}$
$\displaystyle-\tau_{i,15}(k)\operatorname{diag}\\{0,0,-E_{i,4}^{T}E_{i,4},0,0,,I\\}\leq
0.$

Inequality (71) can be written in the following compact form

$\displaystyle\Gamma_{i,3,l_{i}j_{i}}^{T}P$ $\displaystyle{}_{i}^{-1}(k+1\mid
k+1)\Gamma_{i,3,l_{i}j_{i}}-\operatorname{diag}\\{1$ (72)
$\displaystyle-\tau_{i,11}(k)-\tau_{i,12}(k)-\tau_{i,13}(k)$
$\displaystyle\times\hat{x}_{i}^{T}(k+1\mid
k)E_{i,3}^{T}E_{i,3}\hat{x}_{i}(k+1\mid k),$
$\displaystyle\tau_{i,11}(k)I-\tau_{i,14}(k)\Xi_{i}^{T}(k+1\mid
k)E_{i,3}^{T}E_{i,3}$ $\displaystyle\times\Xi_{i}(k+1\mid
k),\tau_{i,12}(k)R_{i}^{-1}(k+1)$
$\displaystyle-\tau_{i,15}(k)E{i,4}^{T}E_{i,4},\tau_{i,13}(k)I,\tau_{i,14}(k)I,$
$\displaystyle\tau_{i,15}(k)I\\}\leq 0.$

Finally denoting

$\displaystyle\Theta_{i,3}(k)\quad=$
$\displaystyle\operatorname{diag}\\{1-\tau_{i,11}(k)-\tau_{i,12}(k)-\tau_{i,13}(k)$
(73) $\displaystyle\times\hat{x}_{i}^{T}(k+1\mid
k)E_{i,3}^{T}E_{i,3}\hat{x}_{i}(k+1\mid k),$
$\displaystyle\tau_{i,11}(k)I-\tau_{i,14}(k)\Xi_{i}^{T}(k+1\mid
k)E_{i,3}^{T}E_{i,3}$ $\displaystyle\times\Xi_{i}(k+1\mid
k),\tau_{i,12}(k)R_{i}^{-1}(k+1)$
$\displaystyle-\tau_{i,15}(k)E{i,4}^{T}E_{i,4},\tau_{i,13}(k)I,\tau_{i,14}(k)I,$
$\displaystyle\tau_{i,15}(k)I\\},$

we can write (72) as

$\displaystyle\Gamma_{i,3,l_{i}j_{i}}^{T}P_{i}^{-1}(k+1\mid
k+1)\Gamma_{i,3,l_{i}j_{i}}-\Theta_{i,3}(k)\leq 0.$ (74)

Now, we deal with the output constraint (22) in Section III-B. First, it can
be described by

$\Gamma_{l_{i}y_{i}}\left(\hat{x}_{i}(k+1\mid k)\right)\eta_{i,2}(k+1)=0.$
(75)

By denoting

$\displaystyle P_{l_{i}y_{i},1}=C_{l_{i}}\hat{x}_{i}(k+1\mid k)-y_{i}(k+1)$
(76) $\displaystyle P_{l_{i}y_{i},2}=C_{l_{i}}\Xi_{i}(k+1\mid k),$

we have

$\displaystyle\Gamma_{l_{i}y_{i}}(\hat{x}_{i}(k+1\mid
k))=\left[P_{l_{i}y_{i},1}\quad P_{l_{i}y_{i},2}\quad D_{l_{i}}\quad
H_{i,3}\quad H_{i,3}\quad H_{i,4}\right].$ (77)

By virtue of Finsler’s lemma [38], the inequality (68) under constraint (75)
holds if there exists a $Z_{i}(k+1)$ such that

$\displaystyle\Gamma_{i,3,l_{i}j_{i}}^{T}$ $\displaystyle P_{i}^{-1}(k+1\mid
k+1)\Gamma_{i,3,l_{i}j_{i}}-\Theta_{i,3}(k)$ (78)
$\displaystyle+Z_{i}^{\mathrm{T}}(k+1)\Gamma_{l_{i}y_{i}}\left(\hat{x}_{i}(k+1\mid
k)\right)$
$\displaystyle+\Gamma_{l_{i}y_{i}}^{\mathrm{T}}\left(\hat{x}_{i}(k+1\mid
k)\right)Z_{i}(k+1)\leq 0.$

For the purpose of simplicity, denote

$\displaystyle\Theta_{i,4}(k)=$
$\displaystyle\Theta_{i,3}(k)-Z_{i}^{\mathrm{T}}(k+1)\Gamma_{l_{i}y_{i}}\left(\hat{x}_{i}(k+1\mid
k)\right)$ (79)
$\displaystyle-\Gamma_{l_{i}y_{i}}^{\mathrm{T}}\left(\hat{x}_{i}(k+1\mid
k)\right)Z_{i}(k+1).$

Then, by using Schur complements, (78) is equivalent to

$\left[\begin{array}[]{cc}-P_{i}(k+1\mid k+1)&\Gamma_{i,3,l_{i}j_{i}}\\\
\Gamma_{i,3,l_{i}j_{i}}^{\mathrm{T}}&-\Theta_{i,4}(k)\end{array}\right]\leq
0.$ (80)

######  Theorem 2.

Consider the leader following multi-agent system (1), (8) that satisfies
Assumption 2, Assumption 7 and Assumption 10. If the state $x_{i}(k+1)$
belongs to its state prediction ellipsoid
$\left(x_{i}(k+1)-\hat{x}_{i}(k+1\mid k)\right)^{\mathrm{T}}P_{i}^{-1}(k+1\mid
k)\left(x_{i}(k+1)-\hat{x}_{i}(k+1\mid k)\right)\leq 1$, then such a state
also resides in its updated state estimation ellipsoid
$\left(x_{i}(k+1)-\hat{x}_{i}^{\mathrm{T}}(k+1\mid
k+1)\right)P_{i}^{-1}(k+1\mid k+1)\left(x_{i}(k+1)-\hat{x}_{i}(k+1\mid
k+1)\right)\leq 1$ with the center determined by (18), where $P_{i}(k+1\mid
k+1)>0$ satisfies matrix inequality (80) with other decision variables
$L_{l_{i}}(k+1)$, $N_{i}(k+1),$ and $\tau_{i,m}(k)\geq 0$ for $m=11,\ldots,15$
for all $l_{i},j_{i}=1,\dots,r$.

###### Proof.

According to the above discussion, if there exist $P_{i}(k+1\mid
k+1)>0,L_{l_{i}}(k+1)$, $N_{i}(k+1),$ and $\tau_{i,m}(k)\geq 0$ for
$m=11,\ldots,15$ such that (80) holds for all $l_{i},j_{i}=1,\dots,r$, then we
have

$\displaystyle\sum_{l_{i}=1}^{r}$ $\displaystyle
g_{l_{i}}\left(\theta_{i}(k)\right)\sum_{j_{i}=1}^{r}g_{j_{i}}\left(\hat{\theta}_{i}(k)\right)$
(81)
$\displaystyle\eta_{i,2}^{\mathrm{T}}(k+1)\Gamma_{i,3,l_{i}j_{i}}^{\mathrm{T}}P_{i}^{-1}(k+1\mid
k+1)\Gamma_{i,3,l_{i}j_{i}}\eta_{i,2}(k+1)\leq 1.$

From (67), we obtain

$\displaystyle\left(x_{i}(k+1)\right.$
$\displaystyle\left.-\hat{x}_{i}(k+1\mid
k+1)\right)^{\mathrm{T}}P_{i}^{-1}(k+1\mid k+1)$ (82)
$\displaystyle\times\left(x_{i}(k+1)-\hat{x}_{i}(k+1\mid k+1)\right)\leq 1,$

which completes the proof. ∎

Now, the convex optimization approach is applied to determine an optimal
ellipsoid with the minimal size. Therefore, $P_{i}(k+1\mid k+1)$ is obtained
by solving the following optimization problem:

$\displaystyle\underset{\begin{subarray}{c}P_{i}(k+1\mid
k+1),L_{l{i}}(k+1),\\\ \tau_{i,11}(k),\tau_{i,12}(k),\tau_{i,13}(k),\\\
\tau_{i,14}(k),\tau_{i,15}(k),Z_{i}(k+1)\\\
\end{subarray}}{\operatorname{min}}\operatorname{Tr}\left(P_{i}(k+1\mid
k+1)\right)$ (83)

subject to (80).

###  IV-C Recursive Algorithm for Attack Diagnosis

The recursive algorithm based on the set-membership filtering to compute the
state ellipsoids so that a cyberattack can be detected is summarized below.

Algorithm 1 recursively computes the prediction ellipsoid
$\mathcal{X}_{i}(k+1\mid k)$ and its update $\mathcal{X}_{i}(k+1\mid k+1)$
with the current measurement $y_{i}(k+1)$. Steps 3 and 6 of the algorithm are
proposed to detect cyberattacks that affect control signals, communication
channels and sensor measurements.

Algorithm 1 Recursive State Estimation
---
1\. Initialization:
Given an initial ellipsoid $\mathcal{X}_{i}(0\mid 0),\mathcal{U}_{i}(0)$,
recursive times $T_{N}$, and set $k=0$. Let
$x_{i}(k)=x_{i}(0),\hat{x}_{i}(k\mid k)=\hat{x}_{i}(0\mid 0),\Xi_{i}(k\mid
k)=\Xi_{i}(0\mid 0),x^{l}(k)=x^{l}(0),$ and $\xi_{i}(k)=\xi_{i}(0)$.
2\. Prediction:
1) Calculate $P_{i}(k+1\mid k),U_{i}(k+1),\hat{A}_{l_{i}}(k),K_{l_{i}}$ by
solving the optimization problem (59).
2) Obtain the matrix $\Xi_{i}(k+1\mid k),$ and $\xi_{i}(k+1\mid k)$ according
to $P_{i}(k+1\mid k)=\Xi_{i}(k+1\mid k)\Xi_{i}^{\mathrm{T}}(k+1\mid k),$ and
$U_{i}(k+1)=\xi_{i}(k+1)\xi_{i}^{\mathrm{T}}(k+1)$.
3) Calculate the centre of the prediction ellipsoid $\hat{x}_{i}(k+1\mid k)$
by (14).
3\. Attack Detection: Control Signal Data or Communication Signal Data Cyber
Attack Diagnosis
1) If $\mathcal{X}_{i}(k\mid k)\bigcap\mathcal{X}_{i}(k+1\mid
k)\neq\varnothing$, there is no attack and go to step 5.
2) If $\mathcal{X}_{i}(k\mid k)\cap\mathcal{X}_{i}(k+1\mid k)=\varnothing$,
data is subject to attack and go to step 4.
4\. Recovery Step and Attack Mitigation:
Set $\mathcal{X}_{i}(k+1\mid k)\leftarrow\mathcal{X}_{i}(k\mid k),$
$\mathcal{U}_{i}(k+1)\leftarrow\mathcal{U}_{i}(k),u_{i}(k)\leftarrow
u_{i}(k-1),$ and go to step 5.
5\. Measurement Update:
1) Calculate $P_{i}(k+1\mid k+1)$ and $L_{l_{i}}(k+1)$ by solving the
optimization problem (83).
2) Obtain the new $\Xi_{i}(k+1\mid k+1)$ according to $P_{i}(k+1\mid
k+1)=\Xi_{i}(k+1\mid k+1)\Xi_{i}^{\mathrm{T}}(k+1\mid k+1)$.
3) Calculate the centre of the updated estimation ellipsoid
$\hat{x}_{i}(k+1\mid k+1)$ by (18).
6\. Attack Detection: Sensor Measurement Data Cyber Attack Diagnosis
1) If $\mathcal{X}_{i}(k+1\mid k+1)\bigcap{\mathcal{X}_{i}(k+1\mid
k)}\neq\varnothing$, there is no attack and go to step 8.
2) If $\mathcal{X}_{i}(k+1\mid k+1)\bigcap\mathcal{X}_{i}(k+1\mid
k)=\varnothing$, data is subject to attack and go to step 7.
7\. Recovery Step and Attack Mitigation:
Set $\mathcal{X}_{i}(k+1\mid k+1)\leftarrow\mathcal{X}_{i}(k+1\mid
k),y_{i}(k+1)\leftarrow y_{i}(k)$ and go to step 8.
8\. Loop
If $k==T_{N}$ then Exit, Else $k\leftarrow k+1$ and go to step 2.

##  V Simulation Results

Consider the following multi-agent, discrete-time, nonlinear system:

$\displaystyle x_{1,1}(k+1)=$ $\displaystyle
0.2x_{1,1}(k)-0.3\left(x_{1,2}(k)-x_{1,1}^{2}(k)\right)$ (84)
$\displaystyle+u_{1}(k)+w_{k}$ $\displaystyle x_{1,2}(k+1)=$ $\displaystyle
0.3x_{1,1}(k)+0.2\left(x_{1,2}(k)-x_{1,1}^{2}(k)\right)$
$\displaystyle+0.3u_{1,1}(k)+0.9u_{1,2}(k)+\omega_{k}$ $\displaystyle y_{k}=$
$\displaystyle x_{1,1}(k)+0.1x_{1,1}^{2}(k)+x_{1,2}(k)+v_{k}$ $\displaystyle
x_{2,1}(k+1)=$ $\displaystyle
0.5x_{2,1}(k)-0.1\left(x_{2,2}(k)-x_{2,1}^{2}(k)\right)$
$\displaystyle+0.9u_{2,1}(k)+0.2u_{2,2}(k)+w_{k}$ $\displaystyle
x_{2,2}(k+1)=$ $\displaystyle
0.9x_{2,1}(k)+0.5\left(x_{2,2}(k)-x_{2,1}^{2}(k)\right)$
$\displaystyle+u_{2}(k)+\omega_{k}$ $\displaystyle y_{k}=$ $\displaystyle
x_{2,1}(k)+0.1x_{2,1}^{2}(k)+x_{2,2}(k)+v_{k}$

where the state
$x_{i}(k)=\left[\begin{array}[]{ll}x_{i,1}(k)&x_{i,2}(k)\end{array}\right]^{T}$.
Now, we construct the following fuzzy models to approximate the above
nonlinear multi-agent system for each agent:
Agent 1:
\- Rule 1: IF $x_{1,1}(k)$ is about 1,THEN

$\displaystyle
x_{1}(k+1)=A_{1,1}x_{1}(k)+B_{1,1}u_{1}(k)+M_{1,1}\omega_{1}(k),$ (85)
$\displaystyle y_{1}(k)=C_{1,1}x_{1}(k)+D_{1,1}v_{1}(k)$

\- Rule 2: IF $x_{1,1}(k)$ is about 0, THEN

$\displaystyle x_{1}(k+1)=A_{1,2}x_{1}(k)+B_{1,2}\omega_{1}(k)$ (86)
$\displaystyle y_{1}(k)=C_{1,2}x_{1}(k)+D_{1,2}v_{1}(k)$

where

$\begin{array}[]{rlr}A_{1,1}&=\left[\begin{array}[]{cc}0.5&-0.3\\\
0.1&0.2\end{array}\right]\quad B_{1,1}=\left[\begin{array}[]{l}1\\\
1\end{array}\right]\\\
C_{1,1}&=\left[\begin{array}[]{lll}1.1&1.1\end{array}\right]\quad D_{1,1}=1\\\
A_{1,2}&=\left[\begin{array}[]{cc}0.2&-0.3\\\ 0.3&0.2\end{array}\right]\quad
B_{1,2}=\left[\begin{array}[]{l}1\\\ 1\end{array}\right]\\\
C_{1,2}&=\left[\begin{array}[]{lll}1.0&1.0\end{array}\right]\quad
D_{1,2}=1.\end{array}$

Agent 2:
\- Rule 1: IF $x_{2,1}(k)$ is about 1,THEN

$\displaystyle
x_{2}(k+1)=A_{2,1}x_{2}(k)+B_{2,1}u_{2}(k)+M_{2,1}\omega_{1}(k),$ (87)
$\displaystyle y_{2}(k)=C_{2,1}x_{2}(k)+D_{2,1}v_{2}(k)$

\- Rule 2: IF $x_{2,1}(k)$ is about 0, THEN

$\displaystyle x_{2}(k+1)=A_{2,2}x_{2}(k)+B_{2,2}\omega_{2}(k)$ (88)
$\displaystyle y_{2}(k)=C_{2,2}x_{2}(k)+D_{2,2}v_{2}(k)$

where

$\begin{array}[]{rlr}A_{2,1}&=\left[\begin{array}[]{cc}0.6&-0.1\\\
0.4&0.5\end{array}\right]\quad B_{2,1}=\left[\begin{array}[]{l}1\\\
1\end{array}\right]\\\
C_{2,1}&=\left[\begin{array}[]{lll}1.1&1.1\end{array}\right]\quad D_{2,1}=1\\\
A_{2,2}&=\left[\begin{array}[]{cc}0.5&-0.1\\\ 0.9&0.5\end{array}\right]\quad
B_{2,2}=\left[\begin{array}[]{l}1\\\ 1\end{array}\right]\\\
C_{2,2}&=\left[\begin{array}[]{lll}1.0&1.0\end{array}\right]\quad
D_{2,2}=1.\end{array}$ (89)

For the convenience of simulation, triangular membership functions are used
for Rule 1 and Rule 2 in this example.

In the above fuzzy models, the approximation errors between the nonlinear
system and the fuzzy models are assumed to satisfy (7), where

$\displaystyle\begin{array}[]{ll}H_{1,1}=\left[\begin{array}[]{c}0.1\\\
0.1\end{array}\right]&H_{2,1}=\left[\begin{array}[]{c}0.3\\\
0.3\end{array}\right]\\\
E_{1,1}=\left[\begin{array}[]{ll}0&0.5\end{array}\right]&E_{2,1}=\left[\begin{array}[]{ll}0&0.6\end{array}\right]\\\
H_{i,2}=\left[\begin{array}[]{l}0\\\ 0\end{array}\right]&E_{i,2}=0\\\
H_{i,3}=0.1&E_{i,3}=\left[\begin{array}[]{ll}0&0.5\end{array}\right]\\\
H_{i,4}=0&E_{i,4}=0\end{array}$ (90)

The leader matrices described in (9) is defined as follow:

$\begin{array}[]{ll}A_{1}^{l}=\left[\begin{array}[]{cc}0.5&0.2\\\
-0.6&0.7\end{array}\right]&A_{2}^{l}=\left[\begin{array}[]{cc}0.5&0.2\\\
-0.4&0.7\end{array}\right]\end{array}$ (91)

In the simulation, $\omega_{i}(k)$ and $v_{i}(k)$ are chosen as $0.5\sin(2k)$
and $0.5\sin(20k)$, respectively. The initial state is set as
$x_{i}(0)=\left[\begin{array}[]{ll}0&0\end{array}\right]^{T}$, which belongs
to the ellipsoids $\left(x_{i}(0\mid
0)-\hat{x}_{i}(0)\right)^{T}P_{i}^{-1}(0\mid
0)\left(x_{i}(0)-\hat{x}_{i}(0\mid 0)\right)\leq 1$ and
$\left(x_{i}(0)-{x}^{l}(0)\right)^{T}U_{i}^{-1}(0)\left(x_{i}(0)-{x}_{l}(0)\right)\leq
1$, where
$\hat{x}_{i}(0)={x}^{l}(0)=\left[\begin{array}[]{ll}1&1\end{array}\right]^{T}$,
and $P_{i}(0\mid 0)=U_{i}(0)=\left[\begin{array}[]{cc}100&0\\\
0&100\end{array}\right],Q_{i}(k)=1-k/50$, and $R_{i}(k)=1-k/50$. The
communication between the agents and the leader is modeled as Fig. 2

Agent 1Agent 2Leader Figure 2: Multi-agent system with a leader.

We obtained the simulation results under MATLAB 9.8 with YALMIP and SDPT3. We
considered the following scenarios in 50 sampling steps.

###  V-A Attack Free System

In this case, the prediction ellipsoid set and the updated estimation
ellipsoid set must always have the intersection. Fig. 3 (a) and Fig. 3 (b)
show the existence of the intersection between these sets for agent 1 and
agent 2, respectively.

---
(a) Agent 1
(b) Agent 2
Figure 3: Prediction ellipsoid $\mathcal{X}_{i}(k+1\mid k)$ (pink) and the
updated estimation ellipsoid $\mathcal{X}_{i}(k+1\mid k+1)$ (green) at k=22.

###  V-B Replay Attacks on Sensor Data

We consider adding the replay attack on the sensor measurement data based on
the definition in the Subsection II-B. Therefore, we assume that the attack
records the data from k=5 to k=10 and replaces the sensor data at k=20 to k=25
with them. Therefore, Fig. 4 (a) and Fig. 4 (b) confirm that the prediction
ellipsoid set and the updated measurement set for the next iteration do not
have the intersection during this attack period.

---
(a) Agent 1
(b) Agent 2
Figure 4: Prediction ellipsoid $\mathcal{X}_{i}(k+1\mid k)$ (pink) and the
updated estimation ellipsoid $\mathcal{X}_{i}(k+1\mid k+1)$ (green) at k=23.

###  V-C False Data Injection Attacks on Control Signal

In this case, the original data packets are replaced by false ones when they
are transferred from controllers to actuators via communication channels.
Therefore, we consider that the attacker replaces the control signal via
targeting the communication channel between the controller and actuator. In
the simulation, the attack vector in (11) is modeled as $u_{i}^{a}(k)=[4\quad
4]^{T}$ for both agents from step k = 20 to k = 25. Therefore, as it is shown
in Fig. 5 (a) and Fig. 5 (b), when the attack on the control signal occurs,
the prediction ellipsoid set and the estimation ellipsoid set updated with the
previous time instant has no intersection.

---
(a) Agent 1
(b) Agent 2
Figure 5: Prediction ellipsoid $\mathcal{X}_{i}(k+1\mid k)$ (pink) and the
previous updated estimation ellipsoid $\mathcal{X}_{i}(k\mid k)$ (green) at
k=24.

###  V-D False Data Injection Attacks on Communication Channel

In this case, the original data packets were replaced by false ones when they
were transferred between two agents via communication channels. Therefore, we
consider that the attacker replace the signal transferring from agent 2 to
agent 1 via communication channel. In the simulation, the attack vector in
(11) is modeled as $\bar{x}_{2}^{c}(k)=[5\quad 5]^{T}$ from step k=20 to k=25.
Therefore, as it is shown in Fig. 6, for agent 1, the prediction ellipsoid set
and the estimation ellipsoid set updated with the previous time instant has no
intersection. Also, as there is no attack on the communication channel from
agent 1 to agent 2, the prediction ellipsoid set and the estimation ellipsoid
set updated with the previous time instant for agent 2 has an intersection.

---
(a) Agent 1
(b) Agent 2
Figure 6: Prediction ellipsoid $\mathcal{X}_{i}(k+1\mid k)$ (pink) and the
previous updated estimation ellipsoid $\mathcal{X}_{i}(k\mid k)$ (green) at
k=23.

Finally, Fig. 7 illustrates that we can guarantee the leader following
consensus in the attack free system and in the presence of the attacks for all
the above scenarios.

---
(a) Attack free system
(b) Replay attack on sensor data
(c) False data injection attack on control signal
(d) False data injection attack on communication channel
Figure 7: Leader-following consensus achievements of the agents in all
scenarios.

##  VI Conclusion

This paper deals with the problem of cyberattack detection in discrete-time
leader-following nonlinear multi-agent systems subject to unknown but bounded
system noises. For the approximation of the nonlinear systems over the true
value of the state, the T-S fuzzy model has been employed. A new fuzzy set-
membership filtering method consisting of two steps has been developed for
each agent to detect two types of cyberattacks at the time of their
occurrence. It has considered the detection of replay attacks and false data
injection attacks affecting the leader-following consensus. We proposed
recursive algorithms for achieving the consensus protocol and finding the two
ellipsoid sets for detecting attacks. The cyberattacks have been detected
based on two criteria about the intersections between the ellipsoid sets.
Finally, simulation results have been provided to demonstrate the
effectiveness of the proposed method.

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