Event-Triggered Quantized Stabilization of Markov Jump Systems under Deception Attacks

. Tis paper focuses on the event-triggered quantized control for Markov jump systems with deception attacks. First, we design an event-triggered scheme relying on dwell time and end instants of attacks. It can limit the number of switches within the triggered intervals and the lower bound of triggered instants. Second, the quantization rules and the increasing/decreasing rate of Lyapunov function are obtained for diferent cases. Next, combined with the increasing/decreasing rate, the lower bound of triggered instants, and the probability of switches occurring, the upper bound of Lyapunov function at the triggered instants is provided. On this basis, sufcient conditions ensuring the exponential convergence in the mean sense of the closed-loop system are given. Finally, atwo-tank system is provided to verify the efectiveness of the proposed stability analysis framework for Markov jump systems.


Introduction
In recent years, due to the powerful modeling ability of Markov process, Markov jump systems have received extensive attention in aircraft control systems and robot systems [1][2][3]. Before the system data are transmitted through the network, they must be quantized and coded. Terefore, the impact of quantization errors on the system performance must be considered. Moreover, the network may sufer a malicious attack initiated by an attacker, which will seriously afect the safe operation of the system [4][5][6]. As a common attack method, deception attacks disrupt the system's performance by tampering with the transmitted data. Especially, in a Markov jump system, the tampered mode will result in a mode mismatch between the system side and the controller side even if the system does not switch. Tus, the system's performance is seriously reduced. Considering that the event-triggered transmission scheme can efectively reduce the amount of data transmission [7,8], this paper will design an event-triggered quantized control strategy to guarantee the stability operation of the Markov jump systems under the infuence of deception attacks.
For the Markov jump systems sufered by deception attacks, some control algorithms have been proposed to guarantee the system stability. Literature [9] designs an event-triggered scheme similar to switching according to triggered errors, switching signals, and deception attack instants to ensure the mean-square exponential input-tostate practical stability of the system. By using a novel dynamic-memory event-triggered protocol, a memorybased sliding mode control for singular semi-Markov jump systems is provided in [10] to ensure the meansquare exponential stability of the system. For Markov jump neural networks subjected to cyber-attacks, which include deception attacks and denial of service attacks, a static output feedback strategy regardless of whether hybrid cyber-attacks occur is designed in reference [11] to guarantee the specifed H ∞ /passive performance.
As we can see, the event-triggered scheme is a useful method to deal with the efect of deception attacks. Diferent from the existing results, the triggered scheme proposed in this paper does not rely on the triggered error, which can efectively avoid the Zeno behavior. Meanwhile, such scheme guarantees the existence of the lower bound of triggered instants, which is the key point to pursue the upper bound of Lyapunov function.
Quantized control for Markov jump systems also gains fruitful results. In reference [12], a quantized iterative learning control scheme is studied by quantizing the tracking error signal based on a logarithmic quantizer. A time-triggered quantized control method is adopted in [13] to ensure the system stability. Literature [14] provides a novel switching delay quantizer with flter connections. Te problem of fnite-time control by using logarithmic quantizer is mainly studied in [15]. For the Markov jump system with data quantization and delay, the authors in [16] designed a hybrid-triggered mechanism and control algorithm to guarantee the asymptotic stability of the system.
If data quantization and deceptive attacks occur in a Markov jump system simultaneously, the quantized state/ output and the system mode may be tampered. It is crucial to ensure the healthy operation of the system under unreliable and inaccurate data transmission. In reference [17], for uncertain fuzzy Markov switched afne systems, a compensation scheme is adopted to deal with the quantized measurement output loss intermittently, and sufcient conditions are provided such that the fltering error system is mean-square exponentially stable. In reference [18], the nonstationary quantized controller design for the Markov jump singularly perturbed systems with deception attacks is studied, and sufcient criteria are established such that the closed-loop system is stochastic mean-square exponential ultimately bounded.
Diferent from the existing results with logarithmic quantizer, we will adopt a time-varying uniform quantizer. Due to the fact that a uniform quantizer does not always assume that the quantizer is unsaturated such as a logarithmic quantizer, we frst design the time-varying quantization radius and quantization center for diferent cases to guarantee the unsaturation of the quantizer. Second, a Lyapunov function is designed based on the quantization radius and quantization center, and the upper bound of which is obtained by using the lower bound of the triggered instants and the probability of the switches occurring. On this basis, sufcient conditions are given to ensure the exponential convergence in the mean sense of the closed-loop system.
Summarized above, the innovations of this paper mainly include the following three aspects: (i) an event-triggered mechanism which is independent of triggered errors is designed, which efectively avoids the Zeno behavior and meanwhile guarantees the existence of the lower bound of triggered instants; (ii) quantization rules are designed for four cases, and the upper bound of Lyapunov function at the triggered instants is obtained by combining the lower bound of triggered instants and the probability of switches occurring; and (iii) some sufcient conditions are obtained to ensure exponential convergence in the mean sense of Markov jump systems under deception attacks.
Te structure of this paper is as follows: Section 2 elaborates on the problem formula, which provides a detailed description of Markov jump systems, event-triggered scheme, quantization rules, deception attacks, control rules and closed-loop systems, and the main purpose of this paper. Section 3 mainly outlines the design of quantization rules for diferent cases. Te increasing/decreasing rate of Lyapunov function is analyzed in Section 4. On this basis, the stability analysis of the system is carried out in Section 5. Simulation and conclusions are provided in Sections 6 and 7, respectively.
1.1. Notations. Te sets of nonnegative integers and nonnegative real numbers are denoted by N and R ≥0 . Let Z � N ∪ 0 { }. Te signal R n represents n-dimensional Euclidean space. Te ∞-norm is adopted by ‖ · ‖ unless otherwise specifed. λ(·) and λ(·) denote the smallest and the largest eigenvalues of a symmetric matrix, respectively.

Problem Formulation
Te system confguration studied in this paper is shown in Figure 1. Te signal fow is as follows: at time t ∈ R ≥0 , the sensor collects and transmits the system state x(t) and the system mode r(t) to the event trigger. If the trigger condition is satisfed (as shown in Section 2.2), the state x(t k ) and mode r(t k ) are transmitted to the quantizer at the triggered instant t k . Te quantizer quantizes the state x(t k ) to c k (the specifc quantization rules are provided in Section 3). Under the role of deception attacks, if the network is attack-free, c k and r(t k ) are received by the observer. Otherwise, the tempered signals c k and σ(t k ) are adopted to update the observer state. Ten, the controller designs the control algorithm according to r(t k )/σ(t k ) and the observer state x(t). Figure 1 can be modeled by the following Markovian jump systems:

Markov Jump Systems. Te plant shown in
where x(t) ∈ R n is the system state and u(t) ∈ R m is the control input. Te switch signal r(t) ∈ M � 1, . . . , s { } indicates the system mode at instant t. A r(t) and B r(t) are known matrices corresponding to diferent subsystems. Te switching signal r satisfying the Markov jump process with dwell time is described as follows: assuming that the p-th subsystem is activated at p i , then no switching occurs for any t ∈p i , p i + h), where h ≥ 0 is called as the dwell time. For t ≥ p i + h, the switching occurs according to the transition probability matrix Π � [π pq ], in which π pq denotes the probability of the system transforming from mode p ∈ M to mode q ∈ M, i.e., P( Lemma 1 (see [19]). Let N r (t) be the switching number of r(t) on the interval (0, t), then we have where t ≥ t k and a m + b m are the end instants of deception attacks defned in Section 2.4. Te frst condition is used to ensure that there is at most one switch occurring within each triggered interval. Te second one ensures that transmission occurs as soon as the attack ends, which is to minimize the impact of attacks on the system's performance.
Remark 3. Although many papers have used triggered errors to design event-triggered scheme, i.e., the data are [21][22][23], this paper does not adopt such condition for two reasons. On the one hand, if the error-based triggered condition is adopted, then the lower bound of t k , i.e., (73) cannot be guaranteed, which makes it difcult to obtain the upper bound of the Lyapunov function. On the other hand, as shown in (23), the quantization rules designed in Section 2.3 have actually limited the range of x(t), which is similar to the role of error-based triggered.

Quantization Rules. At a general triggered instant
t k k∈N , it is supposed that where x * k is the quantization center and E k is the half length of the quantization area. Ten, we can divide the hypercube S k : � x ∈ R n : ‖x − x * k ‖ ≤ E k into N n equal hypercubic sub-boxes, N per each dimension. Let the center of the subbox containing x(t k ) be the quantized value c k which is transmitted to the controller side along with the system mode r(t k ).
Obviously, it holds the following: and Assumption 4 (see [20]) (data rate). We assume that N is large enough such that Λ p ≔ e ‖A p ‖h < N, ∀p ∈ M.

Deception Attacks. For any
) represents the total time interval of deception attacks on the system. Tus, in order to limit deception attacks in terms of frequency and duration, the following assumption is proposed.
Assumption 5 (see [9]). Tere exist α ≥ 0, β ≥ 0, ζ ≥ 0, and T ≥ 1 satisfying and where n(a, b) and |Ξ(a, b)| represent the number and duration of deception attack in [a, b), respectively. Te inverses of β and T provide the upper bounds of the average number and the average duration per unit time of deception attacks, respectively. Under the efect of deception attack, the transmitted c k and r(t k ) may be tampered. To facilitate the following analysis, a binary process ϕ(t k ) ∈ 0, 1 { } is adopted to characterize the attacked situations of the network at the triggered instant t k . Specially, ϕ(t k ) � 0 indicates that the transmission is normal, and ϕ(t k ) � 1 means that the network is under deception attacks.

Control Rule and Closed-Loop System.
Let σ c (t k ) be the mode received by the controller, then the control rule is designed as follows: where K σ c (t k ) is the feedback matrix given in Assumption 2 and x(t) is the observer state satisfying  It is assumed that the tampered quantization value c k is also a center of sub-box to reduce the attack detection rate. It is obvious that if x(t k ) ∈ S k , then c k meets and Note that the system mode r(t k ) transmitted from the system side may also be tampered. At the triggered instant t k , we denote the tampered mode as σ(t k ). Ten, Hence, the control rule can be rewritten as and the closed-loop system is written as 2.6. Main Objective. Similar to the exponential convergence defned in [20], the property of exponential convergence in the mean sense is defned as follows.
Defnition 6. Te closed-loop system (16) is exponential convergence in the mean sense that if there exist constants η > 0 and ω > 0 and a function: Te control objective of this paper is designing the suitable quantization rules and a controller with the feedback matrix defned in Assumption 2 such that the closedloop system (16) is the exponential convergence in the mean sense.

The Design of Quantization Rules
From Section 2.3, we should guarantee that (4) always holds for any triggered instant t k k∈N . To achieve this purpose, we frst pursue an initial instant t 0 and an initial hypercubic box Let u(t) � 0, then system (1) is operated in an open-loop. For any given constants E 0 > 0 and δ > 0, it defnes an increasing function as follows: Due to that E(t) grows fast to dominate the growth rate of the open-loop dynamics. It must be a fnite time t 0 such that ‖x(t 0 )‖ ≤ E(t 0 ), i.e., x(t 0 ) ∈ S 0 . Denote t 0 as the initial triggered instant, and turn the system (1) to the closed-loop form for any t ≥ t 0 .
Next, we will give an iterative design method for quantization rules. Assuming that (4) holds, E k+1 and x * k+1 will be designed such that is satisfed for diferent cases.

Triggered Interval with No Switch.
To facilitate the following analysis, for any system mode r(t) � p∈ M and the controller mode σ c (t k ) � q ∈ M, we defne the matrix A qp as follows: (5) and (11), one has with Λ p defned in Assumption 4. To ensure (19), we can let

An Attack Occurs at t k .
For ϕ(t k ) � 1, we denote the system mode as r(t k ) � r(t k+1 ) � p ∈ M and the tampered mode as Lety ≔ x x , (23) can be rewritten as By (11) and (12), we can easily get By introducing an auxiliary system 4 Journal of Control Science and Engineering we know that and x * k+1 ≔ I n * n 0 n * n e A pq t ″ I n * n I n * n e A p +B p K p t ′ c k , where t ′ and t ″ are any given constants belonging to

An Attack Occurs at
(a) Analysis before the Switch. On [t k , t k + t), similar to the analysis of (23)-(27), we have For any t ′ ∈ [0, t k+1 − t k ], it is easy to see that By recalling (13), the triangle inequality, it obtains (b) Analysis after the Switch. On the interval [t k + t, t k+1 ), the closed-loop dynamic is as follows: Considering the second auxiliary system as follows: one can see that To eliminate the dependence of the quantization center on the unknown time t, we pick a t ″ ∈ [0, t k+1 − t k ]. Ten, it yields Combined with the above inequalities, one has Journal of Control Science and Engineering by using the properties ‖M − I‖ ≤ ‖M‖ + 1 and ‖e As ‖ ≤ e ‖A‖ |s|. Moreover, x * k+1 can be defned as follows: x * k+1 ≔ I n * n 0 n * n y t ″ � I n * n 0 n * n e A mq t ″ e A mp t ′ c k . (39)

Increasing/Decreasing Rate of Lyapunov Function
Let r(t k ) � p denote τ k � t k+1 − t k ∈(0, h). According to Assumption 2, there exist positive-defnite matrices P p (τ k ) and where S p � max τ k ∈(0,h] ‖S p (τ k ) T P p (τ k )S p (τ k )‖ and Q p � min τ k ∈(0,h] λ(Q p (τ k )), there must exist a large enough positive constant ρ p such that by recalling Assumption 5. Obviously, such defned β 1,p can eliminate the dependency of ρ p on τ k . However, the matrix P p that satisfes S p (τ k ) T P p (τ k )S p (τ k ) − P p (τ k ) � − Q p (τ k ) < 0 solved by linear matrix inequality always changes with the value of τ k . Ten, we defne Lyapunov function as Tis section will provide the increasing/decreasing rate of such Lyapunov function for diferent cases, which is the basis of stability analysis.
Similar to the analysis in [20], one gets the following: where with P p � max τ k ∈(0,h] λ(P p (τ k )).

An Attack Occurs at
It follows from (28) that x * k+1 � H mp (τ k )c k with H mp (τ k ) defned by the following: H mp τ k ≔ I n * n 0 n * n e A mp τ k ( ) I n * n I n * n .

No Attack Occurs at t k .
When r(t k ) � p, r(t k+1 ) � q ≠ p and ϕ(t k ) � 0, (29) and (30) yield where with h pq � ‖H pq ‖ and H pq ≔ I n * n 0 n * n e A pq t ″ I n * n I n * n e A p +B p K p t ′ . (52) Similar to the proof of Lemma 7 in [20], one gets with (54)

An Attack Occurs at t k .
Let r(t k ) � p, r(t k+1 ) � q ≠ p, and σ(t k ) � m, it follows from (38) and (39) that where with h mpq � ‖H mpq ‖, H mpq � [I n * n 0 n * n ]e A mq t ″ e A mp t ′ . Ten, one gets the following: where Journal of Control Science and Engineering 7

Stability Analysis
To establish the stability of the closed-loop system (16), we frst pursue the upper bound of Lyapunov function at t k based on the analysis in Section 4. On this basis, the sufcient conditions ensuring exponential convergence in the mean sense of the closed-loop system are provided.

Te Upper Bound of Lyapunov Function at t k
Lemma 7. Let μ � max μ 2 , μ 3 , μ 4 > 1. If dwell time h and β defned in (7) meet and π, π defned in Lemma 1 and α, ζ, T defned in Assumption 5 satisfy then the Lyapunov function follows the following property: where Proof. Assume that there are m time attacks that occur within the interval [t 0 , t k ). Denote a 1 , a 2 , . . . , a m and a 1 + b 1 , a 2 + b 2 , . . . , a m + b m as the beginning and ending instants of these attacks. Let a i , ∀i ∈ 1, 2, . . . , m { } be the frst triggered instant after a i .
Denote F 1 as the increasing/decreasing rate of the Lyapunov function during the interval ∪ m i�1 [a i , a i + b i ) and F 2 as the one corresponding to [t 0 , t k )/ ∪ m i�1 [a i , a i + b i ). It is obvious that the increasing/decreasing rate of Lyapunov function from t 0 to t k , denoted by F, meets F � F 1 F 2 .
First, by recalling Lemma 1, (48), and (57), one has [a i , a i + b i ). Obviously, χ 1 meets χ 1 ≤ |Ξ(t 0 , t k )|. Second, let k 2 represent the number of triggers during [t 0 , t k )/ ∪ m i�1 [a i , a i + b i ) and χ 2 denote the length of [t 0 , t k )/ ∪ m i�1 [a i , a i + b i ). Ten, we have Similarly, one has Combining the above two inequalities, one gets If π ≥ π as shown in (60a), then π − π ≥ 0 and πμ 3 /μ 1 − π ≥ 0. Recalling χ 1 ≤ |Ξ(t 0 , t k )| and (64), one gets where the second inequality is based on μ 1 ≤ μ 3 and (7). Considering that k 2 satisfes the following: we can obtain Because μ 1 ≤ μ, we combined (67) and (69), which yields On the one hand, by recalling 1/h > 1/β according to (59), there exists a large enough T such that On the other hand, based on the event-triggered scheme (3), it is easy to see that i.e., If θ defned in (62) meets θ < 1, one has We summarized (70)-(74), which indicates that Journal of Control Science and Engineering 9 with c, θ, and ω defned in (62). By considering x * 0 � 0, one gets To ensure θ < 1, i.e., T should satisfy Moreover, (60a) is used to guarantee that the denominator of lower bound of T is greater than 0 and (59) can ensure that the upper bound of π is a positive number.
From Lemma 7, we can get the following: and

Exponential Convergence in the Mean Sense of Closed-
Loop System. Tis section will provide a structural proof for the following theorem. To achieve this, we modify the relevant calculations in Section 3.2.2, which is corresponding to the worst case, to derive simpler and more conservative boundaries.
Consider that the closed-loop dynamic is _ y � A mq y during the interval [t k + t, t k+1 ). If we rewrite the second auxiliary system as it is easy to see that holds for any t ∈[t k + t, t k+1 ). Because Hence, we have Projecting onto the x-component, we deduce with α � max m,p,q∈M α mpq , β � max m,p,q∈M β mpq , and Combined (79) and (80) induces that Based on the analysis in Section 3, one knows that the design of E(t 0 ) relies on ‖x(0)‖. Hence, there exists a function g(·) such that E(t 0 ) � g(‖x(0)‖).

Simulation Example
Te two-tank system borrowed from [24] is used to verify the efectiveness of the control strategy, which can be modeled as system (1) with M � 1, 2 { } and where the system states represent the deviations from the nominal reservoir levels. Te fow between two tanks is proportional to the diference of the reservoir levels, and the fow control can be switched arbitrarily.
From Figure 2, it is obvious that t 0 � 6 can be selected as the initial triggered instant. Te mean values of the system states are shown as the red lines in Figure 3, from which we can see that the closed-loop system is exponential convergence in the mean sense under the quantization control algorithm designed in this paper.
Moreover, E k and ‖x(t k ) − x * k ‖ are displayed in Figure 4, which illustrates that the quantizer is unsaturated after t 0 . In such fgure, the blue vertical dotted lines indicate the switching instants, which are randomly generated according to r(0) � 1 and π pq , ∀p, q ∈ M, and the red vertical dotted lines denote the beginning instants of deception attacks, which are assumed to be random variables following the independent and identically distributed variables with a probability of 23%. Obviously, both switches and attacks result in an increase in E k and ‖x(t k ) − x * k ‖. It is worth mentioning that the growth rate of E k is much greater than that of E (x 1 (t)) under this paper E (x 1 (t)) under reference [25] E (x 2 (t)) under this paper ‖x(t k ) − x * k ‖. Tis is because E k is designed from the worst case in order to ensure the unsaturation of the quantizer in all cases. Figure 3, by comparing the state trajectories of this paper and [25], where the triggered mechanism is designed without considering deception attacks, it can be seen that the convergence speed of this paper is slower than that of [25], and the oscillation amplitude of the state trajectories in this paper is greater than that of [25] under the infuence of deception attacks. It means that the deception attacks inevitably reduce the system's performance. However, Figure 5 shows that the number of triggers within 50 seconds is 28 under the triggered mechanism proposed in this paper, but the one under the triggered mechanism in [25] is 31 as shown in Figure 6. Hence, the algorithm proposed in this paper has certain advantages from the perspective of saving network resources.

Conclusions
Te stabilization problem of the Markov jump systems with data quantization and deception attacks has been studied. By designing a suitable event-triggered scheme and quantization coding rules, the unsaturation of the quantizer at the triggered instants has been guaranteed. By analyzing the upper bound of the Lyapunov function, sufcient conditions ensuring the stability of the closed-loop system have been provided.
To simplify the analysis, this paper only considered a single channel. In fact, if the dual channel is executed, it means that the signal transmitted from the controller to the system has also sufered deception attacks, which brings challenges to the quantizer design. Te quantized feedback control under bilateral network sufered deception attacks is one of our future research directions. Moreover, the triggered condition t k+1 � t k + h proposed in this paper is relatively conservative, which may result in more triggered times. How to remove such condition is another research direction in the future.

Data Availability
Te data used to support the fndings of this study are available from the corresponding author upon request.

Conflicts of Interest
Te authors declare that they have no conficts of interest.

Acknowledgments
Te work was supported by the National Natural Science Foundation of China (61773154 and U1804163), the Training Plan of Young Backbone Teachers in Higher