IEEE Systems, Man and Cybernetics Magazine - April 2021 - 41

Observer-Based Methods
Observer-based approaches, such as high-gain observers
and Luenberger observers, normally apply a deterministic
state-space mathematical model of a system to estimate
states. Pajic et al. [101] design an attack-resilient state estimator for systems with dynamic uncertainty and noise. An
L 0 -norm-based state observer is developed to estimate
states during attacks. The authors obtain a bound for the
SE error caused by modeling uncertainty and noise and
consider these bounds in the design procedure. Simulation
results on a UGV show the capability of the proposed estimator. Secure SE and the control of dynamic systems during cyberattacks are examined in [102]. The authors
investigate an observer during sensor attacks and obtain
	

the maximum number of attacks that it can tolerate so
that the state vector can still be reconstructed via a function of system and measurement models, i.e., knowledge of
matrices A and C. The authors show that if more than half
the sensors are simultaneously attacked, the estimator
becomes unstable. Then, they develop an output feedback
controller that is resilient against attacks. They prove that
if the output control law can stabilize the system during
sensor attacks, the state error covariance matrix converges, and the estate variable can be reconstructed.
Figure 5 conveys the probability of detecting attacks
with the proposed model-based approach in an IEEE
14-bus power system experiencing sensor attacks. Note
that the proposed method can estimate the state when
fewer than four sensors is under attack. On the other hand,

Detector

0.4
0.3
0.2
0.1
0

5

10

15
20
Time (s)
(a)

25

30

5

10

15
20
Time (s)
(b)

25

30

Detector

4
3
2
1
0

Figure 4. An | 2 detector with (a) a nonoptimal

authentical signal and (b) an optimal authentication
signal [96].

1
Probability of Detection

estimate the system state and detect attacks. An upper
bound to filter the error covariance is provided based on
solving two sets of coupled algebraic Riccati-like equations. The proposed upper band proves to be exponentially
bounded. A new secure control approach using a distributed KF is introduced in [97] for a CPS that has bandwidth
constraints. Local KFs are developed through sink nodes,
and their estimations are transmitted to a fusion center.
A fusion estimator from distributed KFs is obtained to
minimize the linear variance. The CPS is examined during
several replay attacks and under various bandwidth conditions, and the approach performs well.
In another work, Chen et al. [98] develop a fusion
strategy for CPSs experiencing DoS attacks. They model
resource constraints, including energy and bandwidth,
and design distributed KFs to detect DoS events. System
stability is also investigated, and the abounded mean
square error is guaranteed for the distributed KF estimators. Secure estimation using a Kalman filtering
approach is developed in [99] for multichannel CPSs. A
distributed Kalman filtering method is created by using
two wireless sensors to detect DoS attacks. The stability
condition for the distributed estimators is proved, and
optimal scenarios to maximize the filters' expected estimation errors are discussed.
A coding-based secure control method is introduced in
[100] for CPSs. The main idea is to apply estimation and
control methodologies with coding information. Sensor
measurements are coded before they are transmitted into
channels and decoded in estimators. The method applies a
coding matrix in a way that increases the state estimation
(SE) error during sensor attacks yet keeps the estimation
error/residual small under normal conditions. An algorithm
is proposed to determine the coding matrix, which is randomly renewed to lessen the chance of hackers finding it. If
a high-fidelity mathematical model of a CPS is available,
the KF can provide somewhat reasonable estimates. However, its performance declines if a good model is not available and when exogenous disturbances are acting on a
system. Moreover, its estimate quality depends on the initial conditions of the state and the covariance matrices.

0.8
0.6
0.4
0.2
0

5

10
15
20
25
30
Number of Attacks on Sensors

Figure 5. The probability of detecting CPS attacks

with a model-based method [102].

Ap ri l 2021

IEEE SYSTEMS, MAN, & CYBERNETICS MAGAZINE	

41



IEEE Systems, Man and Cybernetics Magazine - April 2021

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