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IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS: SYSTEMS 1

Adaptive Secure Control for Uncertain

Cyber–Physical Systems With Markov Switching
Against Both Sensor and Actuator Attacks
Zhen Liu , Junye Zhang, and Quanmin Zhu

Abstract—In this article, adaptive secure controller synthesis [8], [9]. It is thus of great significance, from the point of view
for uncertain cyber–physical systems with Markov switching of long-term operation, to deal with the security problem of
(CPSMSs), both sensor and actuator stealthy attacks as well as CPSs with in-depth exploration [10].
generally unknown transition rates (GUTRs), is under consider-
ation via neural sliding mode control (SMC) technique. In order Of special relevance for this work are a class of hybrid
to resist unknown attack signals from both sensor and actuator dynamic systems called Markov jump systems (MJSs), by
channels, a novel neural network (NN)-based SMC design is which physical plants with random switching mechanism
performed, which could not only guarantee the boundedness of could be emulated, such as chemical reactor, DC motor, TD
relevant adaptive data but also force the actual state trajectories jump circuit, wheeled mobile manipulators, and aircraft engine
to arrive at the proposed sliding mode surface (SMS) with
limited moments almost surely. Then, a fresh stochastically stable system [11], [12], [13], [14], [15]. Note that the primary
criterion for the resultant plant is provided in spite of hidden matter of concern for analysis and synthesis of MJSs is
cyber attacks, GUTRs, and structural uncertainty, relying on the relevant to so-called transition rates (TRs), and research under
arrival of the SMS and stochastic stability theory. Finally, an partly unknown TRs reduces conservatism of the investigated
F-404 aircraft engine model with performance comparisons is model [16], in comparison to the case of completely known
offered to confirm the feasibleness of the theoretical result.
TRs. However, given the difficulty in obtaining the TRs and
Index Terms—Cyber–physical systems (CPSs), Markov switch- unknown uncertainties in practical situations, comprehensive
ing parameter, neural network (NN), sensor and actuator attacks, studies on control of MJSs under generally unknown TRs
sliding mode control (SMC).
(GUTRs) may be more realistic, and some remarkable control
strategies have been reported in this direction [7].
As a typical robust control technique, sliding mode control
I. I NTRODUCTION (SMC) has been extensively focused around control commu-
nity and engineering applications in view of its simplicity
YBER–PHYSICAL systems (CPSs), in view of the
C advancement of computers and information interaction
technology, have been extensively found and studied in indus-
in algorithm, robustness to parametric uncertainties and dis-
turbances [17], [18], [19], [20]. It is noteworthy that some
progresses under SMC strategy have been absorbed into the
trial field around the heterogeneous cyber layers and physical
synthesis of uncertain MJSs [12], [13], [15], [17], [20], [21],
plants [1], [2], [3], [4]. Nevertheless, upon the background
[22]. To cite a few, an event-triggered SMC for discrete-
of continuous development and increasing complexity of
time MJSs was deliberated more in [20]; the stabilization for
networked embedded technique, the fragility of CPSs from
MJSs subject to actuator faults was investigated based upon
unknown attackers is escalated. Malicious attacks will degrade
adaptive SMC in [23]. It is also notable that, security control
system’s control precision by fabricating the sensor’s measure-
problems of CPSs have been probed further [24], [25], [26],
ment and/or impeding the actuator’s command, which would
e.g., adaptive control methods against stealthy attacks were
cause unpredictable damage to physical systems [5], [6], [7],
proposed for nonlinear T–S fuzzy systems and discrete-time
Received 15 August 2024; revised 28 December 2024; accepted 26 systems in [24] and [25], respectively.
February 2025. This work was supported in part by the National Natural Due to practical background of hybrid systems and signal
Science Foundation of China under Grant 61803217 and Grant 62003231;
in part by the Natural Science Foundation of Shandong Province under
communications between plant devices and cyber layers,
Grant ZR2023MF029; in part by the Team Plan for Youth Innovation of recent years have witnessed the research progress of CPSs with
Universities in Shandong Province under Grant 2022KJ142; and in part by Markov switching (CPSMSs). Apart from the GUTRs, another
the Taishan Scholar Special Project Fund under Grant TSON202408163.
(Corresponding author: Zhen Liu.)
crucial matter in resilient control of CPSMSs is the malicious
Zhen Liu and Junye Zhang are with the School of Automation cyber attacks, and some control strategies for CPSMSs against
and the Shandong Key Laboratory of Industrial Control Technology, injected attacks were investigated [7], [17], [27], [28]. Notably,
Qingdao University, Qingdao 266071, China (e-mail: zhenliuzz@hotmail.com;
junyezhang1@163.com).
the above researches focused on the actuator attacks from
Quanmin Zhu is with the School of Engineering, University of the West of cyber layers only, while ignoring the general case malicious
England, BS16 1QY Bristol, U.K. (e-mail: quan.zhu@uwe.ac.uk). attackers pose unknown attacks to both the actuator and
Color versions of one or more figures in this article are available at
https://doi.org/10.1109/TSMC.2025.3547865.
sensor’s channels concurrently. In fact, the output signals
Digital Object Identifier 10.1109/TSMC.2025.3547865 may be contaminated due to injected sudden sensor attacks,
2168-2216 
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2 IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS: SYSTEMS

which will be inaccurate for the sliding surface design and D(rt ) and E(rt ) are known matrices, Y(t) is an uncertain
controller synthesis, then the control signal becomes invalid matrix and Y T (t)Y(t) ≤ I. The right-continuous Markov
and the security of the system is not guaranteed, leading process denoted by {rt , t ≥ 0}, give values in the set J =
to the instability of the whole system. Just recently, an {1, 2, . . . , J}.
adaptive resilient control method for CPSMSs under sensor In what follows, attack scenarios are focused in the begin-
and actuator attacks was studied in [29], whereas the previous ning, from which both sensor measurement and control input
control results were discussed in the light of norm-bounded signal may be corrupted. The sensor attack depending on the
assumption made in system attacks. To the authors’ knowl- state is defined by ∂s (t) as
edge, the above-addressed issues or limitations regarding the
analysis and synthesis of CPSMSs still have not been fully ∂s (t) = p(t)x(t), t ≥ 0 (2)
inspected and solved so far. Moreover, neural network (NN), where p(t) denotes the time-varying unknown nonlinear
as a valid approximation technology, has been employed to weight function which satisfies 0 ≤ p(t) ≤ ¯ < 1 and |ṗ(t)| ≤
identify unknown nonlinear signals, which is beneficial for ς̄ < 1, t ≥ 0, ¯ and ς̄ are two unknown scalars. Then, the
the controller design to a great extent [30], however, related actual state information xs (t), stemming from the compromised
reports on NN-based SMC strategy of CPSMSs are relatively sensor channel for feedback, is expressed as
scarce, which generates the motivation of the work finally.
In response to the above observations, this article attempts xs (t) = x(t) + ∂s (t), t ≥ 0. (3)
to provide an NN-based secure control scheme for uncertain
In addition, the actual control signal ua (t) received by the
CPSMSs against structural uncertainty, GUTRs, both sensor
actuator is depicted by
and actuator attacks under a novel SMC framework. The main
innovations and tasks are delivered as follows. ua (t) = u(t) + ∂a (x(t), t), t ≥ 0 (4)
1) In contrast to [17] and [28], a simplified linear-type
sliding mode surface (SMS) is established drawing upon in which u(t) is the devised controller, ∂a (x(t), t) represents
the actual sensor measurement, from which an updated the actuator attack signal estimated by NN in Section II-B.
adaptive SMC framework for uncertain CPSMSs is con- Designate  = [ηij ]J×J (i, j ∈ J ) as TR matrix with mode
ducted, and a novel sufficient criterion for the resultant transition probabilities shown by
system to be stochastically stable is derived relying
Pij = Pr(rt+t = j|rt = i)
on the attainability of the special SMS and stochastic 
ηij t + o(t), if i = j;
stability theory. =
2) In comparison to [17], [27], [28], and [29], a novel 1 + ηii t + o(t), if i = j
NN-based SMC strategy is exhibited to guarantee both
the finite-time reachability of the proposed SMS and where ηij = η̂ij + ηij represents TR from mode i at moment t
security performance with robustness against sensor and to mode j at moment t + t, η̂ij denotes the TR’s deterministic
actuator attacks as well as the GUTRs, where previous term, and ηij stands for uncertain term satisfying |ηij | ≤

norm boundedness assumption for unknown attacks is no ϑij , ϑij > 0, when i = j; and ηii = − Jj=1,j=i ηij . For
longer required, which generalizes the result in previous convenience, the following two index sets are introduced:
works.  
k  j:η̂ij is accessible for j ∈ J
i
3) The resiliency of the developed control mechanism  
uk  j:η̂ij is inaccessible for j ∈ J
i
against pernicious actuator and sensor attacks is verified (5)
through simulation analysis, and the efficiency of the
proposed theoretical result is justified by performance with the TRs divided by four categories: 1) i ∈ k i and j ∈
uk  = ; 2) i ∈ uk i and k i = ; 3) uk i = J and
i
comparisons with specified indicators.
Notation: sym{P} represents P + PT . In symmetric block j ∈ k for some j = i; and 4) uk i = J and j ∈
j / k j for any
matrices, “∗” is employed to denote abbreviations of symmetry j ∈ J . Let rt = i ∈ J , A(rt ), A(rt , t), B(rt ), D(rt ), E(rt ) and
terms. Mathematical expectation is expressed by E{·}, and ·F Y(t) can be shortened by Ai , Ai (t), Bi , Di , Ei , and Y.
represents the Frobenius-norm. Consequently, (1) is reorganized as
ẋ(t) = (Ai + ΔAi (t))x(t)
II. M ODEL D ESCRIPTION AND P REPARATIONS + Bi [u(t) + ∂a (x(t), t)]. (6)
A. System Description
The following class of uncertain CPSMSs are considered: B. Neural Network Architecture
ẋ(t) = (A(rt ) + ΔA(rt , t))x(t) + B(rt )ua (t) (1) As stated in [30], a three-layer feedforward NN is used
to design an adaptive neural controller in this research (see
where x(t) = [x1 (t), x2 (t), . . . , xn (t)]T ∈ Rn and ua (t) ∈ Rm Fig. 1), and the net with the input x and output y is given
stand for state vector and control signal subject to certain as y(x) = W T ϕ(V T x), where W and V denote the connection
actuator attacks, respectively. A(rt ) ∈ Rn×n and B(rt ) ∈ weights between neurons with corresponding dimensions, and
Rn×m represent system matrices, A(rt , t) is the structural ϕ(·) represents the activation function in the hidden layer
uncertainty that fulfills A(rt , t) = D(rt )Y(t)E(rt ), where neurons.

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LIU et al.: ADAPTIVE SECURE CONTROL FOR UNCERTAIN CPSMSs 3

hidden layer of NN related to the input signal, l = 1, 2, . . . , r,

ϕ̂  [ϕ1 (Ô1T X̄ ), ϕ2 (Ô2T X̄ ), . . . , ϕr (ÔrT X̄ )]T ∈ Rr×1 , and o(·)
represents the infinitesimal.
Lemma 2 [32]: The norm of v(xs (t)) satisfies v(xs (t)) <
α T ω, in which the vector α ∈ R4 is unknown, and ω is denoted
by ω  (1, X̄ , X̄ ŴF , X̄ ÔF )T . Herein, denote α̂
and α̃ as the estimation vector and error of α, respectively,
and α̃ = α̂ − α.
Lemma 3 [7]: For a matrix T > 0, given any real scalar 
and a matrix Q, one has (Q + QT ) ≤  2 T + QT −1 QT .

III. M AIN R ESULTS

Fig. 1. NN structure.
A. Reachability Analysis of SMS
This part is focused on a simplified SMS design first, and
the reachability of the SMS is confirmed with the developed
In view of the approximation capability of the three-layer NN-based adaptive controller then. In this position, the SMS
NN for continuous functions [30], [31], [32], one has is proposed as
 
∂a (x(t), t) = WdT ϕ OdT X¯d + φd (x), x(t) ∈ (7) S(xs (t))  Bi T Pi xs (t) = 0 (11)
where Pi > 0 will be computed later.
where Wd ∈ Rr×m and Od ∈ R(n+1)×r are the
Remark 1: In contrast to [17] and [28], a linear SMS func-
optimization weights of hidden-to-output layer and input-
tion is designed for the CPSMSs, which is computed with the
to-hidden layer with r neurons, Od  [Od1 , Od2 , . . . , Odr ],
actual measurement xs (t) from the impaired sensor channel.
contains the threshold; X¯d = [xT (t), −1]T represents the
Theorem 1: If a NN-based SMC law is developed as
neural input vector with input bias −1, ϕ(OdT X¯d )   
[ϕ1 (Od1 T X¯ ), ϕ (O T X¯ ), . . . , ϕ (O T X¯ )]T ,
d 2 d2 d r dr d in which u(t) = −Ŵ T ϕ ÔT X̄ − Fi (t) (12)
ϕl (Odl X¯d ) = [1/1 + e−Odl X̄d ] is a sigmoid function, l =
T
T
where Fi (t) = (Bi T Pi Bi )−1 [ζ ϒ(α̂ T ω + β̂(t) + σ ) + Bi T Pi 
1, . . . , r, is a compact set; φd (x) indicates the estimation
(Ai +Di Ei )xs (t) sgn(S(xs (t))), and the adaptive rules
error vector meeting φd (x) ≤ φ̄d , φ̄d > 0 is an unknown
constant [30]. However, considering that the original state
are shown ⎧ by   T 
⎪ −1
⎨ 1 ζ ϕ̂ − ϕ̂ Ô X̄ S (xs (t)) Bi Pi Bi , if Λi > 0
⎪ T T
signal x(t) may not be acquired due to the sensor attack, we ˙
redefine the actuator attack by NN as follows: Ŵ = 0,  if Λi = 0
⎪
⎪  
  ⎩ 1 −1 ζ ϕ̂ − ϕ̂ ÔT X̄ ST (xs (t)) BT Pi Bi , if Λi < 0
i
∂a (x(t), t) = W T ϕ OT X̄ + φ(xs (t)) ⎧ −1  T  T
⎨ 2 ζ (X̄ S (xs (t)) Bi Pi Bi Ŵ ϕ̂ ),
T if Λi > 0
 ∂a (xs (t), t), xs (t) ∈ (8) ˙
Ô = 0, if Λi = 0
⎩ −1  
in which X̄ = (xsT (t), −1)T ; similarly, W ∈ Rr×m and O ∈ 2 ζ (X̄ ST (xs (t)) BTi Pi Bi Ŵ T ϕ̂ ), if Λi > 0
R(n+1)×r are the optimization weights, and the approximation α̂˙ =  −1 ζ ϒS(x (t))ω; β̂(t) ˙ =  −1 ζ ϒS(x (t)), where
3 s 4 s
error vector is described by φ(xs (t)) with φ(xs (t) ) ≤ φ̄, Λi = ST (xs (t))(Bi T Pi Bi )(Ŵ T ϕ̂ ÕT X̄ + W̃ T (ϕ̂ − ϕ̂ ÔT X̄ )),
φ̄ > 0. ϒ = maxi∈J (Bi T Pi Bi ), 1 , 2 , 3 and 4 represent the
Lemma 1 [30], [31], [32]: The attack signal ∂a (xs (t), t) is coefficients of the adaptive rules, β̂(0) ≥ 0 denotes the
estimated by ∂ˆa (xs (t), t) = Ŵ T ϕ(ÔT X̄ ) based on NN. Then, initial condition of β̂(t) by the  designer, σ is a positive

the approximated error ∂ˆa (xs (t), t) − ∂a (xs (t), t) shown by constant, ζ  max 1/(1 − q(t)) , ζ  min 1/(1 − q(t)) , and
∂˜a (xs (t), t), is rebuilt as q(t) = p(t)/(p(t) + 1), then the finite-time reachability of the
 proposed SMS can be guaranteed almost surely.
∂˜a (xs (t), t) = W̃ T ϕ̂ − ϕ̂ ÔT X̄ Proof:
+ Ŵ T ϕ̂ ÕT X̄ + v(xs (t)) (9) Step 1: The following Lyapunov function is constructed as:
1 T
where the estimation error matrices of the weights are provided V1 (S(xs (t)), i) = S (xs (t))S(xs (t))
2
by W̃ = Ŵ − W, and Õ = Ô − O, in which Ŵ ∈ Rr×m and 1 1 1
Ô ∈ R(n+1)×r indicate the hidden-output and input-hidden + tr{W̃ T 1 W̃} + tr{ÕT 2 Õ} + α̃ T 3 α̃.
2 2 2
weight matrices, and r refers to the number of hidden neurons. (13)
Moreover, the residual term v(xs (t)) is described as
 Note that the weight function q(t) is provided as
v(xs (t)) = W̃ T ϕ̂ OT X̄ + W T o ÕT X̄ + φ(xs (t)) (10) q(t) = p(t)/(p(t) + 1). Then, the SMS function can be rewrit-
ten as
where ϕ̂  diag{ϕ1 (Ô1T X̄ ), ϕ2 (Ô2T X̄ ), . . . , ϕr (ÔrT X̄ )} ∈ S(x(t))
S(xs (t)) = Bi T Pi xs (t) = (14)
Rr×r , ϕl (ÔT X̄ ) means the derivative of the lth neuron in the 1 − q(t)

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4 IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS: SYSTEMS

in which S(x(t))  Bi T Pi x(t). Thus, the following can be In view of β̂(t) ˙ ≥0, one
˙ =  −1 ζ ϒST (x (t))≥ 0 and β̂(0)
4 s
derived: also gives
ẋ(t) (1 − q̇(t))x(t) R1,i ≤ S(xs (t))Bi T Pi (Ai + ΔAi (t))xs (t)
Ṡ(xs (t)) = Bi T Pi [ − ]. (15)  
1 − q(t) (1 − q(t))2 − S(xs (t))Bi T Pi (Ai  + Di Ei )xs (t)
Then, the infinitesimal operator L on V1 (S(xs (t)), i) yields Λi
+ ζ ϒS(xs (t))α T ω −
1 − q(t)
LV1 (S(xs (t) ), i) = R1,i + R2,i (16) − ζ ϒS(xs (t))(α̂ T ω + σ )
   
where R1,i = (1/(1 − q(t)))ST (xs (t))Bi T Pi ẋ(t) + + tr W̃ T 1 W̃˙ + tr ÕT  Õ˙ + α̃ T  α̃˙
2 3
 T ˙  T ˙
tr W̃ 1 W̃ + tr Õ 2 Õ + α̃ T 3 α̃, ˙ R2,i = −((1 − Λi
q̇(t))/((1 − q(t)) ))S (xs (t))Bi Pi x(t). Further, R1,i is
2 T T ≤ ζ ϒS(xs (t))α T ω −
1 − q(t)
converted into
− ζ ϒS(x(t))(α̂ T ω + σ ) + α̃ T 3 α̃˙
   
ST (xs (t)) T ˙ + tr ÕT  Õ˙ .
+ tr W̃ T  W̃ (21)
R1,i = Bi Pi (Ai + ΔAi (t))x(t) 1 2
1 − q(t)
In the position, for the case Λi > 0, it follows that:
ST (xs (t)) T
+ (Bi Pi Bi )(u(t) + ∂a (xs (t), t)) R1,i ≤ ζ ϒS(xs (t))(α T ω − α̂ T ω) − ζ ϒS(xs (t))σ
1 − q(t)    
˙ + tr{ÕT  Õ} ˙ + α̃ T  α̃.
˙ ˙ + tr ÕT  Õ˙ + α̃ T  α̃˙
− ζ Λi + tr W̃ T 1 W̃
+ tr{W̃ T 1 W̃} 2 3 2 3
 T 
(17) ≤ ζ ϒS(xs (t)) α ω − α̂ ω T

Substituting (12) into (17), it yields − ζ ϒS(xs (t))σ + α̃ T 3 α̃˙

= −δ1 S(xs (t)) < 0 (22)
ST (xs (t)) T
R1,i ≤ Bi Pi (Ai . + ΔAi (t))x(t) where δ1 = −ζ σ ϒ. Then, for the case that Λi is less than 0,
1 − q(t)
the following inequality can be obtained:
ST (xs (t)) T
+ (Bi Pi Bi )(∂a (xs (t), t) R1,i ≤ ζ ϒS(xs (t))(α T ω − α̂ T ω) − ζ Λi
1 − q(t)  
˙ ˙
− ζ ϒS(xs (t))σ + tr W̃ T 1 W̃
− Ŵ T ϕ(ÔT X̄ ) − Fi (t)) + tr{W̃ T 1 W̃}
˙ + α̃ T  α̃.  
+ tr{ÕT 2 Õ} 3
˙ (18) + tr ÕT 2 Õ˙ + α̃ T 3 α̃˙
 
Considering the actuator attack in (8) and Lemma 1, (18) can ≤ ζ ϒS(xs (t)) α T ω − α̂ T ω (23)
be derived as − ζ ϒS(xs (t))σ + α̃ T 3 α̃˙
ST (xs (t)) T = −δ1 S(xs (t)) < 0.
R1,i ≤ Bi Pi (Ai + ΔAi (t))x(t)
1 − q(t) In like manner, if Λi = 0, then the inequality R1,i ≤
ST (xs (t)) T −δ1 S(xs (t)) < 0 still holds. Moreover, R2,i can be calcu-
+ (Bi Pi Bi )(−v(xs (t))
1 − q(t) lated as
− W̃ T (ϕ̂ − ϕ̂ ÔT X̄ ) − Ŵ T ϕ̂ ÕT X̄ − Fi (t)) 1 − q̇(t)
R2,i ≤ − S(xs (t))2 < 0. (24)
˙ + tr{ÕT  Õ}
+ tr{W̃ T 1 W̃} ˙ + α̃ T  α̃.
˙ (19) (1 − q(t))2
2 3
From the above discussion, we have LV1 (S(xs (t)), i) < 0.
Noting that all variables have finite dimensions, xs (t) = Then, it can be obtained that S(xs (t)), W̃, Õ and α̃ are
x(t)/(1 − q(t)) and · ≤ ·1 , (19) is further shown as bounded, which gives the boundedness of ∂˜a (xs (t), t) further,
i.e., ∂˜a (xs (t), t) ≤ β, where β > 0 is an unknown scalar,
R1,i ≤ S(xs (t))Bi T Pi (Ai + ΔAi (t))xs (t) and its estimation is defined by β̂(t).
− S(xs (t))1 Bi T Pi (Ai  + Di Ei )xs (t) Step 2: The arrival of the SMS can be justified further with
Λi a selected Lyapunov function as follows:
+ ζ S(xs (t))Bi T Pi Bi α T ω −
1 − q(t) 1 4
V2 (S(xs (t)), i) = ST (xs (t))S(xs (t)) + β̃ 2 (t) (25)
− ζ ϒS(xs (t))1 (σ + β̂(t) + α̂ T ω) 2 2
˙ + tr{ÕT  Õ}
+ tr{W̃ T 1 W̃} ˙ + α̃ T  α̃˙ in which β̃(t) = β̂(t) − β. The infinitesimal operator
2 3
≤ S(xs (t))Bi T Pi (Ai + ΔAi (t))xs (t) LV2 (S(xs (t)), i) can be redefined as
− S(xs (t))Bi T Pi (Ai  + Di Ei )xs (t) LV2 (S(xs (t)), i) = R3,i (26)
Λi ˙
+ ζ S(xs (t))Bi T Pi Bi α T ω − where R3,i = ST (xs (t))Bi T Pi [ẋ(t)/1 − q(t)] + 4 β̃(t)β̃(t).
1 − q(t) Substituting (12) into (26), one has
− ζ ϒS(xs (t))(α̂ T ω + β̂(t) + σ ) ST (xs (t)) T
˙ + tr{ÕT  Õ}˙ + α̃ T  α̃.
˙ R3,i ≤ Bi Pi (Ai +ΔAi (t))x(t)
+ tr{W̃ T 1 W̃} 2 3 (20) 1 − q(t)

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LIU et al.: ADAPTIVE SECURE CONTROL FOR UNCERTAIN CPSMSs 5

ST (xs (t))  T  B. Performance Analysis

+ Bi Pi Bi ∂a (xs (t), t)
1 − q(t) In this part, a new stochastically stable criterion of the
   ˙
− Ŵ T ϕ ÔT X̄ − Fi (t) + 4 β̃(t)β̃(t) (27) resultant CPSMSs subject to GUTRs on the sliding motion is
ST (xs (t)) T put forward.
≤ Bi Pi (Ai + ΔAi (t))x(t) Theorem 2: The system (6) is stochastically stable during
1 − q(t)
the sliding phase if there exist matrices Pi > 0, Ug,i > 0,
ST (xs (t))  T 
+ Bi Pi Bi − ∂˜a (xs (t), t) Vg,i > 0, Wi > 0 and Xi,jα > 0, Ki and a positive scalar ι
1 − q(t) such that the conditions in the following are held.
 ˙
− F (t) +  β̃(t)β̃(t)
i 4 Case 1: i ∈ k i and g ∈ uk i
⎡ ⎤
Meanwhile, in view of  ·  ≤  · 1 , one has i,1 + 1 (Pi Di )T ψ1
  ⎣ ∗ −ιI 0 ⎦ < 0. (34)
R3,i ≤ S(x(t))Bi T Pi (Ai + ΔAi (t))xs (t) ∗ ∗ −Ug,i
− S(xs (t))Bi T Pi (Ai  + Di Ei ) Case 2: i ∈ i ,g ∈ uk i and ik =
uk
· xs (t) − ζ ϒS(xs (t))(α̂ T ω + β̂(t) + σ ) ⎧⎡ ⎤
  ˙ ⎪
⎪ i,2 + 2 (Pi Di )T ψ2
+ ζ ϒS(xs (t))∂˜a (xs (t), t) + 4 β̃(t)β̃(t) ⎨⎣
∗ −ιI 0 ⎦<0
(35)
≤ −ζ ϒS(xs (t))(α̂ T ω + σ ). (28) ⎪
⎪ ∗ ∗ −Vg,i
⎩
˙ there Pi − Pj ≥ 0.
It is noteworthy that, based upon the adaptive rule α̂,
∗ ∗
exists an instant T > 0 such that α̂ ω > 0 for t ≥ T . Then, uk = J , g ∈ uk and j ∈ k for some j  = i
T Case 3: i i j

it further follows: ⎡ ⎤
i,3 + 3 (Pi Di )T Pi − Pg
R3,i ≤ −δ1 S(xs (t)) < 0, for t ≥ T ∗ . (29) ⎣ ∗ −ιI 0 ⎦ < 0. (36)
∗ ∗ −Wi
Moreover, introduce the function V3 (S(xs (t)), i) =
˙
[1/2]ST (xs (t))S(xs (t)). Considering the case that β̃(t) = Case 4: uk
i = J, g ∈
uk and j ∈
i / k j for any j ∈ J
˙ ⎧⎡ ⎤
β̂(t) > 0, there exists a finite instant T > 0 such that ⎪
⎪  + 4 (Pi Di )T Pjα − Pj
˙ ⎨ ⎣ i,4
4 β̃(t)β̃(t) ≥ 0 for t ≥ Tf = max{T ∗ , T }. In view of (29), ∗ −ιI 0 ⎦<0
(37)
one can get ⎪
⎪ ∗ ∗ −Xi,jα
⎩
Pi − Pj ≥ 0.
LV3 (S(xs (t)), i) ≤ −δ1 S(xs (t)) < 0 ∀t ≥ Tf .
where the matrix Ki is set for Ai − Bi Ki is Hurwitz, and
(30)
i, = sym{Pi (Ai − Bi Ki )} + ιEi T Ei ,  ∈ {1, 2, 3, 4}
Further, one has    1  2
 1 = η̂ij Pj − Pg + ϑij Ug,i
LV3 (S(xs (t)), i) ≤ −ζ V3 (S(xs (t)), i)∀t ≥ Tf (31) i
4
j∈
√ k
where ζ = 2δ1 . By Itô’s formula, it yields    1  2
 2 = η̂ij Pj − Pg + ϑij Vg,i
LS(xs (t)) = L V3 (S(xs (t)), i) ≤ −ζ /2 (32) i
4
j∈ k

and hence   1  2
3 = αi η̂jj Pi − Pg + ϑjj Wi ,
   4
ES(xs (t)) ≤ ES xs Tf  − (ζ /2)(t − Tf ) (33)   1  2
4 = η̂jjα Pjα − Pg + ϑjjα Xi,jα
which concludes there exists an instant tf = Tf + 2m0 /ζ    4   
satisfying ES(xs (t)) = 0 for all t ≥ tf , where m0 = ψ1 = Pj,1 − Pg , Pj,2 − Pg , . . . , Pj1 − Pg
     
ES(xs (Tf )) < ∞. Thus, the arrival of the devised SMS can ψ2 = Pj,1 − Pg , Pj,2 − Pg , . . . , Pj2 − Pg .
be determined almost surely.
Proof: The Lyapunov function candidate is selected as
Remark 2: In contrast to the existing control strat-
egy [17], [19] that guaranteed the uniform boundedness of V4 (x, i) = xT (t)Pi x(t). (38)
sliding variable only, the proposed NN-based adaptive SMC
The infinitesimal generator of V4 (x, i) along system (6) gives
design can not only satisfy the boundedness of related data
but also admit the state onto the proposed SMS S(xs (t)) = 0 
J
in finite time almost surely. LV4 (x, i) = 2xT (t)Pi ẋ(t) + ηij xT (t)Pj x(t)
Remark 3: Differing from the projection operator approach j=1
to handle the attack signals in [29], the NN-based control = x (t) · sym{Pi (Ai − Bi Ki )}x(t)
T

method is introduced herein. It is worth mentioning that the + 2xT (t)Pi ΔAi x(t) + 2xT (t)Pi Bi
norm-bounded assumption is not required in this work, which
indicates the control design in (12) generalizes the result    J
· ua (t) + Ki x(t) + ηij xT (t)Pj x(t).
proposed in [29]. j=1

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6 IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS: SYSTEMS

    ηij
Besides, the following inequality is held: = ηij Pj + ηii Pi − ηii + λi,k Pj
i i ,j =i
−ηii − λi,k
2xT (t)Pi ΔAi x(t) ≤ ι−1 xT (t)Pi Di Di T Pi x(t) j∈ k j∈ uk

+ ιxT (t)Ei T Ei x(t) (39) where j∈ i ,j =i (ηij /−ηii − λi,k ) = 1, 0 ≤ ηij /(−ηii − λi,k )
uk

for a positive scalar ι. By taking reachability of the SMS into ≤ 1 and g ∈ i .

uk For any j ∈ i ,
uk one gives
account, we get ST (xs (t)) = xT (t)Pi Bi /(1 − q(t)) = 0, yielding 
J
xT (t)Pi Bi = 0. Thus, it follows that: i,2 + ηij Pj
LV4 (x, i) ≤ x (t)Hi x(t) T
(40) 
j=1
ηij  
= [i,2 + diag{ηii Pi − Pg (46)
J Hi = sym{Pi (Ai − Bi Ki )} +
where + + ι−1 Pi Di Di T Pi ιEi T Ei −ηii − λi,k
g∈ uk ,j=i
i
j=1 ηij Pj .         
Case 1 (i ∈ k ): Denote λi,k 
i
i ηij . Due to + ηij Pj − Pg , ηii Pi − Pg + ηij Pj − Pg }].
i = J i∈ k
k , it gives λi,k < 0. Note that j=1 ηij Pj could be
j∈ i
k j∈ i
k
expressed as 
Since 0 ≤ ηij ≤ −ηii − λi,k , the inequality i,2 + Jj=1 ηij Pj

J   < 0 is equivalent to
ηij Pj = + ηij Pj     
j=1 j∈ k
i j∈ uk
i i,2 + ηii Pi − Pg + ηij Pj − Pg < 0. (47)
  ηij j∈ i
= ηij Pj − λi,k Pj (41) k
−λi,k
j∈ k
i j∈ uk
i From ηii < 0, (47) holds if
 
in which j∈ i (ηij /−λi,k ) = 1 and 0 < (ηij /−λi,k ) < 1, j ∈ Pi − Pg≥ 0,  
uk
i,2 + j∈ i ηij Pj − Pg < 0. (48)
uk . Hence, for any g ∈ uk , one also gives
i i
k
  
i,1 + ηij Pj − Pg Moreover, for any Vi,g > 0, by Lemma 3, one has
j∈ i   
ηij Pj − Pg
ηij  
k
  
= i,1 + ηij Pj − Pg . (42) j∈ i
−λi,k k
g∈ i
uk
i j∈ k
    1  2
= η̂ij Pj − Pg + [ ϑij Vg,i (49)
Evidently, one has i i
4
  
j∈ k j∈ k
ηij Pj − Pg    T
+ Pj − Pg Vg,i −1 Pj − Pg ].
j∈ i
  
k
   Recalling the similar line and condition (35), one also has
= η̂ij Pj − Pg + ηij Pj − Pg . (43) LV4 (x, i) < 0.
j∈ i
k j∈ i
k Case 3 ( uk i = J and j ∈ k j for Some j = i): In such
In addition, for any Ug,i > 0, it yields by Lemma 3 that case, ηii is estimated by αi ηjj . Denote λi,k  ηii . Therefore,
J
   j=1 ηij Pj is changed into
ηij Pj − Pg
j∈ i 
J 
k
ηij Pj = ηii Pi + ηij Pj
 1    
= Δηij Pj − Pg + Pj − Pg j=1 j∈ uk
i
2  ηij
= ηii Pi − λi,k Pj .
j∈ i
k (50)
  1  2    T  i
−λi,k
≤ ϑij Ug,i + Pj − Pg Ug,i −1 Pj − Pg . (44) j∈ uk
4 
i j∈ k
Since j∈ uk
i ηij = −ηii = −λi,k > 0, it follows that for
∀g ∈ i
Via Schur complement and (34), it gives LV4 (x, i) < 0, which uk

shows system (6) during the sliding motion is stochastically 

J
stable. i,3 + ηij Pj

Case 2 (i ∈ iuk and ik = ): Denote λi,k  j∈ i ηij . j=1
k 
ηig   
Since i
uk = , one has λi,k > 0 and = i,3 + ηii Pi − Pg
−λ i,k
g∈ uk i

J
   
ηij Pj = i,3 + ηii Pi − Pg = i,3 + αi ηjj Pi − Pg . (51)
j=1
  Further, by recalling ηij = η̂ij + Δηij , one has
= ηij Pj + ηii Pi + ηij Pj (45)     
j∈ i j∈ i αi ηjj Pi − Pg = αi η̂jj + Δηjj Pi − Pg . (52)
k uk

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LIU et al.: ADAPTIVE SECURE CONTROL FOR UNCERTAIN CPSMSs 7

Then, by Lemma 3, for any Wi > 0, one has

  1  
Δηjj Pi − Pg = · 2Δηjj Pi − Pg
2
  2 
ϑjj   −1
 T
≤ Wi + Pi − Pg (Wi ) Pi − Pg .
4
(53)
Combining (51), (52) and (53), it also yields LV4 (x, i) < 0 by
Schur complement and condition (36).
Case 4 ( uk i = J and j ∈ / k j for any j ∈ J ): In this case,
denote λi,k  ηija (ja = i), and ηija is estimated by ηjja . Then,
J Fig. 2. Control diagram against both sensor and actuator attacks.
j=1 ηij Pj is changed into


J
ηij Pj IV. S IMULATION VALIDATION
j=1 In this section, a representative example is taken, from

= ηija Pja + ηii Pi + ηij Pj which the viability of the presented control method is substan-
i ,j =i tiated with simulation comparisons.
j∈ uk
   Example 1: As a simulation object, the linearized F-404
ηij Pj
= ηija Pja + ηii Pi − ηii + λi,k (54) aircraft engine plant model in [15] is borrowed, in which the
i ,j =i
−ηii − λi,k relevant state vector with physical data is written as
j∈ uk
⎡ ⎤
thus one has sideslip angle(o /s)

J x(t) = ⎣ roll rate(o /s) ⎦.
i,4 + ηij Pj yaw rate(o /s)
j=1
      The corresponding matrix Ai is described as
= i,4 +ηii Pi − Pg + ηija Pja − Pg . (55) ⎡ ⎤
g∈Ωuk i ,j=i −1.46 0 2.48
Ai = ⎣0.1643 + 0.5τi −0.4 + τi −0.3788⎦
Since 0 ≤ ηij ≤ −ηii −λi,k , the inequality i,4 +ηii (Pi −Pg )+ 0.3107 0 −2.23
ηija (Pja − Pg ) < 0 is equivalent to
    where τi , i = 1, 2, 3, are unknown parameters of the system
i,4 + ηii Pi − Pg + αi ηjja Pja − Pg < 0. (56)
model and set by −1, −2 and −3, respectively, then it follows:
Due to ηii < 0, (56) < 0 holds if ⎡ ⎤ ⎡ ⎤
 −1.46 0 2.428 −0.1
Pi − Pg ≥ 0,  (57) A1 = ⎣−0.3357 −1.4 −0.3788⎦, B1 = ⎣ 0.2 ⎦,
i,4 + αi ηjja Pja − Pg < 0. 0.3107 0 −2.23 −0.2
⎡ ⎤ ⎡ ⎤
Further, by Lemma 3, for any Xijα > 0, it gives −1.46 0 2.428 −0.2
  A2 = ⎣−0.8357 −2.4 −0.3788⎦, B2 = ⎣ 0.1 ⎦,
ηjja Pja − Pg 0.3107 0 −2.23 −0.1
    ⎡ ⎤ ⎡ ⎤
= η̂jja Pja − Pg + Δηjja Pja − Pg −1.46 0 2.428 −0.2
 2
   ϑjja A3 = ⎣−1.3357 −3.4 −0.3788⎦, B3 = ⎣ 0.1 ⎦;
≤ η̂jja Pja − Pg + Xija 0.3107 0 −2.23 −0.2
  −1 
4
T  ⎡ ⎤ ⎡ ⎤
+ Pja − Pg Xija Pja − Pg . (58) 0.2 0.2 0.1 0.2 0.3 0.1
D1 = ⎣ 0 0.2 0 ⎦, E1 = ⎣ 0 0.2 0.1⎦,
From the similar line and condition (37), LV4 (x, i) < 0 still 0 0.2 0.1 0 0.3 0.3
⎡ ⎤ ⎡ ⎤
holds, which implies the stochastic stability of system (6) on 0.2 0.1 0.1 0.1 0.2 0.2
the sliding motion. D2 = ⎣0.1 0.2 0.1⎦, E2 = ⎣ 0 0.2 0.2⎦,
Remark 4: As seen, a novel NN-based secure controller 0.1 0.2 0.1 0.1 0.2 0.3
design is developed for the underlying CPSMSs, where the ⎡ ⎤ ⎡ ⎤
0.1 0.2 0 0.2 0.2 0.2
control diagram against potential sensor and actuator attacks D3 = ⎣0.1 0.2 0 ⎦, E3 = ⎣ 0 0.2 0.1⎦.
is displayed in Fig. 2, and the main components of the whole 0.2 0.2 0.1 0.1 0.2 0.3
CPSMS, i.e., the physical layer, cyber layer, and control layer,
are integrated, in which both the sensor and actuator attack The TR matrix is set as
signals described through the communication channels may ⎡ ⎤
endanger the system security, and the effects of the two −0.45 + Δη11 ? 0.27 + Δη13
malicious cyber attacks can be well suppressed by the current =⎣ ? ? 0.38 + Δη23 ⎦
adaptive control method. ? ? ?

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8 IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS: SYSTEMS

where “?” indicates the totally unavailable TRs. The paramet-

ric matrices K1 , K2 and K3 are chosen as

K1 = −0.8234 8.6907 −0.4476

K2 = −0.4853 1.4484 7.1411

K3 = −1.2797 −0.3621 −3.2277

and Y is taken as Y = 0.3 cos(100t). From Theorem 2, the

feasible parameters are obtained as follows:
⎡ ⎤
16.3642 4.2610 6.6614
P1 = ⎣ 4.2610 18.2106 8.0298 ⎦
6.6614 8.0298 22.4143
⎡ ⎤
10.4125 3.1423 6.7523 Fig. 3. Jump mode rt and SMS variable S(xs (t)).
P2 = ⎣ 3.1423 14.0692 3.3443 ⎦
6.7523 3.3443 22.4143
⎡ ⎤
15.5899 0.1647 4.9571
P3 = ⎣ 0.1647 12.3043 1.5121 ⎦.
4.9571 1.5121 18.4719

Then, the established sliding surface function is shown by

⎧
⎨ [−2.1165, 1.6101, −3.5430]xs (t), i = 1
S(xs (t)) = [−2.4435, 0.4440, −4.2573]xs (t), i = 2
⎩
[−4.0929, 0.8951, −4.5346]xs (t), i = 3

and the related SMC law is designed as

⎧
⎪
⎪ −Ŵ T ϕ(ÔT X̄ ) − 0.8050[1.0429(α̂ T ω + β̂(t)
⎪
⎪ +1) + 3.5604xs (t)]sgn(S(xs (t))), i = 1
⎪
⎪
⎨
−Ŵ T ϕ(ÔT X̄ ) − 1.0429[1.0429(α̂ T ω + β̂(t)
u(t) = Fig. 4. Norm curves of adaptive parameters Ŵ(t), Ô(t), α̂(t), and β̂(t).
⎪
⎪ +1) + 3.6651xs (t)]sgn(S(xs (t))), i = 2
⎪
⎪
⎪
⎩ Ŵ ϕ(Ô X̄ ) − 0.5510[1.0429(α̂ ω + β̂(t)
− T T T
⎪
+1) + 4.8566xs (t)]sgn(S(xs (t))), i = 3 Then, the devised sliding surface function S(t) in [17] is
exhibited by
where the updating gains are set as 1 = 0.2, 2 = 0.5, ⎧
3 = 0.3 and 4 = 0.1. Besides, the utilized NN consists of ⎪
⎪
⎪  t 0.2, −0.2]xs (t)
[−0.1,
nine neurons, where the relevant initial values of Ŵ, Ô, and ⎪ −
⎪
⎪ 0 [0.5955, −0.2370, −0.1049]xs (κ)dκ, i = 1
⎨ [−0.2,
α̂ are set as [0.1]9×1 , [0.1]4×9 and [0.3]4×1 , respectively. In
S(t) =  t 0.1, −0.1]xs (t)
addition, the values of the learning rate and batch size of the ⎪
⎪ − 0 [0.6453, 0.1494, −0.2239]xs (κ)dκ, i=2
⎪
⎪
NN are chosen as 0.1 and 4, respectively. Simulation exper- ⎪
⎪ [−0.2, 0.1, −0.2]x (t)
⎩ t s
iments with comparisons of specified performance measures − 0 [0.4284, 0.2677, 0.2769]xs (κ)dκ, i=3
are conducted under the current method and the algorithms
of [17], [28] in the following three cases. and the synthesized controller is utilized as same as that
Case 1 (Performance Comparison Under Sensor Attacks): in [17, Th. 3]. For simulation experiment, Fig. 3 displays the
The sensor attack denoted by ∂s (t) = p(t)x(t), is imposed on evolution of jump mode and SMS variable via the current
the communication channel, where the time-varying function method, and norm curves of the adaptive parameters Ŵ(t),
p(t) is given as Ô(t), α̂(t) and β̂(t) are shown in Fig. 4, respectively. The plots
⎧ of the system response and control input under the two control
⎨ 0.5 + 0.25 sin(t), 0 ≤ t ≤ 2 methods are provided in Fig. 5 based on the initial condition
p(t) = 0.5 + 0.25 cos(2t), 2 < t ≤ 6 x(0) = [2 − 1 − 2]T . As observed from Fig. 5, the control
⎩
0.5 + 0.45 cos(5t), 6 < t ≤ 40. signal becomes invalid, and the security of the system can not
be guaranteed due to the potential sensor attack signal under
By the same system settings, the associated gain matrices the method in [17], leading to the instability of the closed-
Kcom,i (i = 1, 2, 3) in [17, Th. 2] are computed as loop system, while the influence of the sensor attack can be
restrained via the developed control algorithm.
Kcom,1 = [6.4313, 0.4782, 2.5813]
Case 2 (Performance Comparison Under Actuator Attacks):
Kcom,2 = [7.7995, 6.4901, 1.2767] As observed from [28], the specific form of actuator attack
Kcom,3 = [3.6903, 6.7520, 3.9376]. signal is shown by ∂a (x(t), t) = δ(t)Q(t)(x(t), t), t ≥ 5s,

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LIU et al.: ADAPTIVE SECURE CONTROL FOR UNCERTAIN CPSMSs 9

Fig. 5. State response and control signal under two methods.

Fig. 6. Jump mode rt and SMS variable S(xs (t)).

where δ(t) stands for the Bernoulli variable, Q(t) denotes an

unknown weight matrix, and (x(t), t) represents the system
information used by the adversaries. In such case, Q(t) and
(x(t), t) are chosen as Q(t) = cos(t) and (x(t), t) =

x22 + x32 + 2. By the same system settings, the matching
matrices Mcom,i (i = 1, 2, 3) in [28] are shown as

Mcom,1 = [−0.8234, 8.6907, −0.4476] (a) (b)

Mcom,2 = [−0.4853, 1.4484, 7.1411]
Mcom,3 = [−1.2797, −0.3621, −3.2277].

In the position, the corresponding sliding surface variable

s(t) in [28] is computed as
⎧
⎪
⎪ [−0.5794,
t 0.4886, −0.8849]xs (t) (c) (d)
⎪
⎪
⎪
⎪ − [0.1330, 2.2068, 0.2326]xs (κ)dκ, i=1
⎨ [−0.6536,
0 Fig. 7. Norm curves of adaptive parameters Ŵ(t), Ô(t), α̂(t), and β̂(t).
t 0.1247, −1.2177]x s (t)
s(t) =
⎪
⎪ − 0 [0.3431, 0.0844, 2.9735]xs (κ)dκ, i=2
⎪
⎪
⎪
⎪ [−1.3144, 0.4353, −1.1282]x (t)
⎩ t s
− 0 [0.3062, −1.6726, −2.5576]xs (κ)dκ, i = 3

while the developed controller is provided as same as that

in [28, Th. 3]. In simulation, Fig. 6 displays evolution of
the jump mode and SMS variable, and norm curves of the
adaptive parameters Ŵ(t), Ô(t), α̂(t) and β̂(t) are shown in
Fig. 7, respectively. The plots of the system response and (a) (b)
control input are provided in Fig. 8 between the current control
method and that in [28] under the same initial condition in
Case 1.
In order to further illustrate the availability of the proposed
result more clearly, data exploration including tracking
precision performance [e.g., integral absolute error (IAE),
integral time-weighted absolute error (ITAE), integral square
error (ISE)] and energy consumption
 20 (EC) takenfrom [33], is (c) (d)
20
carried out, i.e., IAE  0 xs (t)dt, ITAE  0 txs (t)dt
 20  20
and ISE  0 xs (t)2 dt, EC  0 u2 (t)dt are introduced Fig. 8. State response and control signal under two methods.
herein. Comparison results of the four indicators are provided
in Fig. 9. It is observed that the resultant system via the Case 3 (Performance Comparison Under Both Sensor and
proposed control method can be resilient against attacks with Actuator Attacks): In this case, the actuator attack ∂a (x(t), t)
better control accuracy under lower EC. is taken as

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10 IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS: SYSTEMS

(a) (b)

(c) (d)
Fig. 10. Jump mode rt and SMS variable S(xs (t)).
Fig. 9. Comparison of performance measures.

⎧
⎪
⎪ M(t) + N1 (t)N2 (t), 0≤t<3
⎨
0, 3≤t<8
∂a (x(t), t) =
⎪
⎪ (1 + 1.2 sin(200t)), 8 ≤ t < 15
⎩
0, 15 ≤ t < 20
where M(t) = 0.4 sin(3t), N1 (t) = 0.2 cos(3t) and N2 (t) = (a) (b)
sin(x2 (t)) cos(x3 (t)), and the time-varying function p(t) of the
sensor attack is chosen as
⎧
⎨ 0, 0≤t<4
p(t) = 0.6 + 0.25 sin(2t), 4 ≤ t < 10
⎩
0.5 + 0.45 sin(2t), 10 ≤ t ≤ 20.
Then, the evolutions of transfer mode rt and SMS variable are (c) (d)
shown by Fig. 10, the norm curves of the adaptive parameters
Fig. 11. Norm curves of adaptive parameters Ŵ(t), Ô(t), α̂(t), and β̂(t).
Ŵ(t), Ô(t), α̂(t) and β̂(t) are provided in Fig. 11, and the
curves of the state response and control signal, based on the
two different methods, are described in Fig. 12.
Moreover, differing from the case in [28] that just takes
actuator attacks into consideration, the proposed method deals
with a more general case that the sensor and actuator are
simultaneously interfered. As seen from Fig. 12, although the
system is obviously affected by both the actuator and sensor
attacks during [0s, 3s] and [8s, 10s], the state response will (a) (b)
return to stable status more robustly and quickly by virtue
of the proposed NN-based SMC than the method in [28],
which shows the former is more efficient in achieving desirable
operation while withstanding complicated threats.
Furthermore, the analysis of the above quantitative data
indicators is carried out, see Fig. 13, which demonstrates the
resultant system using the current method is also resilient (c) (d)
against both attacks with better control accuracy and lower
EC. At last, the statistical results under 30 sets of repeated Fig. 12. State response and control signal under two methods.
experiments have been performed to further validate the
efficiency and advantage of the current control approach with
other peer methods, where the corresponding curves of state the controller and sensor can always be synchronized with the
response are shown in Fig. 14, which indicates the closed-loop system mode. However, some limitations using the proposed
system has better control accuracy with more robustness under methodology may exist in a real system. For instance, in prac-
the proposed algorithm in contrast to the method in [28]. tical engineering, identification errors of system mode, certain
Remark 5: It should be mentioned that the results of this delays in switching signals, etc., may lead to asynchronous
article are obtained based on the premise of that the modes of switching phenomenon among the modes of the controller,

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LIU et al.: ADAPTIVE SECURE CONTROL FOR UNCERTAIN CPSMSs 11

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 This article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of pagination.

12 IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS: SYSTEMS

[23] H. Yang, S. Yin, and O. Kaynak, “Neural network-based adaptive fault- Zhen Liu received the Ph.D. degree in control
tolerant control for Markovian jump systems with nonlinearity and theory and applications from the Ocean University
actuator faults,” IEEE Trans. Syst., Man, Cybern., Syst., vol. 51, no. 6, of China, Qingdao, China, in 2017.
pp. 3687–3698, Jun. 2021. He was a joint Ph.D. candidate with the
[24] J. Song, X.-H. Chang, and Z.-M. Li, “Secure P2P nonfragile sampled- School of Engineering, University of the West
data controller design for nonlinear networked system under sensor of England, Bristol, U.K., and a Visiting Scholar
saturation and DoS attack,” IEEE Trans. Netw. Sci. Eng., vol. 10, no. 3, with the College of Engineering, University of
pp. 1575–1585, May/Jun. 2023. Kentucky, Lexington, KY, USA. He is currently
[25] C. Wu, X. Li, W. Pan, J. Liu, and L. Wu, “Zero-sum game-based optimal a Distinguished Professor with the School of
secure control under actuator attacks,” IEEE Trans. Autom. Control, Automation, Qingdao University, Qingdao. His cur-
vol. 66, no. 8, pp. 3773–3780, Aug. 2021. rent research interests include intelligent control and
[26] Z. Kazemi, A. A. Safavi, M. M. Arefi, and F. Naseri, “Finite-time secure robot, cyber–physical systems, UAV Control, and sliding mode control.
dynamic state estimation for cyber–physical systems under unknown Dr. Liu was the recipient of the Shandong Province Taishan Scholar Special
inputs and sensor attacks,” IEEE Trans. Syst., Man, Cybern., Syst., Project Fund.
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Markov jump cyber-physical systems subject to randomly occurring is currently pursuing the M.Sc. degree in con-
deception attacks,” IEEE Trans. Circuits Syst. II, Exp. Briefs, vol. 69, trol science and engineering with the School of
no. 1, pp. 159–163, Jan. 2022. Automation, Qingdao University, Qingdao, China.
[29] H. Yang, H. Han, S. Yin, H. Han, and P. Wang, “Sliding mode-based His research interests include stochastic systems
adaptive resilient control for Markovian jump cyber–physical systems in and intelligent control.
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Syst., Tokyo, Japan, 1996, pp. 148–153. the Harbin Institute of Technology, Harbin, China,
[33] S. Mobayen and V. J. Majd, “Robust tracking control method based on in 1983, and the Ph.D. degree from the Faculty
composite nonlinear feedback technique for linear systems with time- of Engineering, University of Warwick, Coventry,
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no. 1, pp. 171–180, 2012. He is a Professor of Control Systems with the
[34] Z.-G. Wu, P. Shi, Z. Shu, H. Su, and R. Lu, “Passivity-based asyn- School of Engineering, University of the West of
chronous control for Markov jump systems,” IEEE Trans. Autom. England, Bristol, U.K. His main research interest is
Control, vol. 62, no. 4, pp. 2020–2025, Apr. 2017. in the area of nonlinear system modeling, identifica-
[35] X. Li, C. K. Ahn, W. Zhang, and P. Shi, “Asynchronous event-triggered- tion, and control.
based control for stochastic networked Markovian jump systems with Prof. Zhu is acting as a member of Editorial
FDI attacks,” IEEE Trans. Syst., Man, Cybern., Syst., vol. 53, no. 9, Committee of Chinese Journal of Scientific Instrument, and the President of
pp. 5955–5967, Sep. 2023. International Conference of Modeling, Identification and Control.

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