Abstract
This 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 different 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, sufficient 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 effectiveness of the proposed stability analysis framework for Markov jump systems.
1. 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 [1–3]. Before the system data are transmitted through the network, they must be quantized and coded. Therefore, the impact of quantization errors on the system performance must be considered. Moreover, the network may suffer a malicious attack initiated by an attacker, which will seriously affect the safe operation of the system [4–6]. 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. Thus, the system’s performance is seriously reduced. Considering that the event-triggered transmission scheme can effectively 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 influence of deception attacks.
For the Markov jump systems suffered 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-to-state practical stability of the system. By using a novel dynamic-memory event-triggered protocol, a memory-based sliding mode control for singular semi-Markov jump systems is provided in [10] to ensure the mean-square 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 specified /passive performance.
As we can see, the event-triggered scheme is a useful method to deal with the effect of deception attacks. Different from the existing results, the triggered scheme proposed in this paper does not rely on the triggered error, which can effectively 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 filter connections. The problem of finite-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 affine systems, a compensation scheme is adopted to deal with the quantized measurement output loss intermittently, and sufficient conditions are provided such that the filtering 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 sufficient criteria are established such that the closed-loop system is stochastic mean-square exponential ultimately bounded.
Different 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 first design the time-varying quantization radius and quantization center for different 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, sufficient 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 effectively 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 sufficient conditions are obtained to ensure exponential convergence in the mean sense of Markov jump systems under deception attacks.
The 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 different cases. The 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
The sets of nonnegative integers and nonnegative real numbers are denoted by and . Let . The signal represents -dimensional Euclidean space. The -norm is adopted by unless otherwise specified. and denote the smallest and the largest eigenvalues of a symmetric matrix, respectively.
2. Problem Formulation
The system configuration studied in this paper is shown in Figure 1. The signal flow is as follows: at time , the sensor collects and transmits the system state and the system mode to the event trigger. If the trigger condition is satisfied (as shown in Section 2.2), the state and mode are transmitted to the quantizer at the triggered instant . The quantizer quantizes the state to (the specific quantization rules are provided in Section 3). Under the role of deception attacks, if the network is attack-free, and are received by the observer. Otherwise, the tempered signals and are adopted to update the observer state. Then, the controller designs the control algorithm according to and the observer state .
2.1. Markov Jump Systems
The plant shown in Figure 1 can be modeled by the following Markovian jump systems:where is the system state and is the control input. The switch signal indicates the system mode at instant . and are known matrices corresponding to different subsystems. The switching signal satisfying the Markov jump process with dwell time is described as follows: assuming that the -th subsystem is activated at , then no switching occurs for any ), where is called as the dwell time. For , the switching occurs according to the transition probability matrix , in which denotes the probability of the system transforming from mode to mode , i.e., for and for with and .
Lemma 1. (see [19]). Let be the switching number of on the interval (), then we havewhere and .
Assumption 2. (see [20]) (stabilizability). For each , the subsystem is stabilizable, i.e., there exists a state feedback gain matrix such that is Hurwitz, i.e., all eigenvalues of have negative real parts.
2.2. Event-Triggered Scheme
The triggered instants are determined by the following scheme:where and are the end instants of deception attacks defined in Section 2.4.
The first condition is used to ensure that there is at most one switch occurring within each triggered interval. The 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 transmitted if with [21–23], 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 , i.e., (73) cannot be guaranteed, which makes it difficult 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 , which is similar to the role of error-based triggered.
2.3. Quantization Rules
At a general triggered instant , it is supposed thatwhere is the quantization center and is the half length of the quantization area. Then, we can divide the hypercube into equal hypercubic sub-boxes, per each dimension. Let the center of the sub-box containing be the quantized value which is transmitted to the controller side along with the system mode .
Obviously, it holds the following:and
Assumption 4. (see [20]) (data rate). We assume that is large enough such that .
2.4. Deception Attacks
For any , let [) with and indicating the -th time interval of deception attacks. Meanwhile, [) indicates the -th time interval of normal communication. Obviously, for any interval [), represents the total time interval of deception attacks on the system. Thus, in order to limit deception attacks in terms of frequency and duration, the following assumption is proposed.
Assumption 5. (see [9]). There exist , , and satisfyingandwhere and represent the number and duration of deception attack in [), respectively. The inverses of and provide the upper bounds of the average number and the average duration per unit time of deception attacks, respectively. Under the effect of deception attack, the transmitted and may be tampered. To facilitate the following analysis, a binary process is adopted to characterize the attacked situations of the network at the triggered instant . Specially, indicates that the transmission is normal, and means that the network is under deception attacks.
2.5. Control Rule and Closed-Loop System
Let be the mode received by the controller, then the control rule is designed as follows:where is the feedback matrix given in Assumption 2 and is the observer state satisfyingwith
It is assumed that the tampered quantization value is also a center of sub-box to reduce the attack detection rate. It is obvious that if , then meetsand
Note that the system mode transmitted from the system side may also be tampered. At the triggered instant , we denote the tampered mode as . Then,
Hence, the control rule can be rewritten asand the closed-loop system is written as
2.6. Main Objective
Similar to the exponential convergence defined in [20], the property of exponential convergence in the mean sense is defined as follows.
Definition 6. The closed-loop system (16) is exponential convergence in the mean sense that if there exist constants and and a function: such that
The control objective of this paper is designing the suitable quantization rules and a controller with the feedback matrix defined in Assumption 2 such that the closed-loop system (16) is the exponential convergence in the mean sense.
3. The Design of Quantization Rules
From Section 2.3, we should guarantee that (4) always holds for any triggered instant . To achieve this purpose, we first pursue an initial instant and an initial hypercubic box with such that .
Let , then system (1) is operated in an open-loop. For any given constants and , it defines an increasing function as follows:
Due to that grows fast to dominate the growth rate of the open-loop dynamics. It must be a finite time such that , i.e., . Denote as the initial triggered instant, and turn the system (1) to the closed-loop form for any .
Next, we will give an iterative design method for quantization rules. Assuming that (4) holds, and will be designed such thatis satisfied for different cases.
3.1. Triggered Interval with No Switch
To facilitate the following analysis, for any system mode and the controller mode , we define the matrix as follows:
3.1.1. No Attack Occurs at
If and , the error satisfies [). Due to by recalling (5) and (11), one haswith defined in Assumption 4. To ensure (19), we can let
3.1.2. An Attack Occurs at
For , we denote the system mode as and the tampered mode as . On the triggered interval [), the close-loop dynamics are
Let, (23) can be rewritten as
By (11) and (12), we can easily get
By introducing an auxiliary systemwe know that
Moreover, is designed by projecting onto the component
3.2. Triggered Interval with a Switch
3.2.1. No Attack Occurs at
Suppose that and . Similar to the analysis in Section 4.2 of [20], and can be designed as follows:andwhere and are any given constants belonging to .
3.2.2. An Attack Occurs at
Let , , and . Obviously, there is an unknown instant such that [) and [).
(a) Analysis before the Switch. On [), similar to the analysis of (23)–(27), we have
For any , it is easy to see that
By recalling (13), the triangle inequality, it obtains
(b) Analysis after the Switch. On the interval [), 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 , we pick a . Then, it yields
Combined with the above inequalities, one hasby using the properties and . Moreover, can be defined as follows:
4. Increasing/Decreasing Rate of Lyapunov Function
Let denote (). According to Assumption 2, there exist positive-definite matrices and such that , with . Definingwhere and , there must exist a large enough positive constant such thatby recalling Assumption 5. Obviously, such defined can eliminate the dependency of on . However, the matrix that satisfies solved by linear matrix inequality always changes with the value of . Then, we define Lyapunov function as
This section will provide the increasing/decreasing rate of such Lyapunov function for different cases, which is the basis of stability analysis.
4.1. Triggered Interval with No Switch
4.1.1. No Attack Occurs at
If and , one has and by recalling (21) and (22).
Similar to the analysis in [20], one gets the following:wherewith .
4.1.2. An Attack Occurs at
If , let and . It follows from (28) that with defined by the following:
It gives thatwith . Moreover, we know from (27) that
Since , where denotes the Euclidean norm of the vector , it yieldswherewith .
4.2. Triggered Interval with a Switch
4.2.1. No Attack Occurs at
When and , (29) and (30) yieldwherewith and
Similar to the proof of Lemma 7 in [20], one getswith
4.2.2. An Attack Occurs at
Let , , and , it follows from (38) and (39) thatwherewith
Then, one gets the following:where
5. Stability Analysis
To establish the stability of the closed-loop system (16), we first pursue the upper bound of Lyapunov function at based on the analysis in Section 4. On this basis, the sufficient conditions ensuring exponential convergence in the mean sense of the closed-loop system are provided.
5.1. The Upper Bound of Lyapunov Function at
Lemma 7. Let . If dwell time and defined in (7) meetand defined in Lemma 1 and defined in Assumption 5 satisfythen the Lyapunov function follows the following property:where
Proof. Assume that there are time attacks that occur within the interval [). Denote and as the beginning and ending instants of these attacks. Let be the first triggered instant after .
Denote as the increasing/decreasing rate of the Lyapunov function during the interval [) and as the one corresponding to [)/[). It is obvious that the increasing/decreasing rate of Lyapunov function from to , denoted by , meets .
First, by recalling Lemma 1, (48), and (57), one haswhere represents the number of triggers within the interval [) and denotes the length of [). Obviously, meets .
Second, let represent the number of triggers during [)/ [) and denote the length of [)/ [). Then, we haveSimilarly, one hasCombining the above two inequalities, one getsIf as shown in (60a), then and . Recalling and (64), one getswhere the second inequality is based on and (7).
Considering that satisfies the following:we can obtainBecause , we combined (67) and (69), which yieldsOn the one hand, by recalling according to (59), there exists a large enough such thatOn the other hand, based on the event-triggered scheme (3), it is easy to see thati.e.,If defined in (62) meets , one hasWe summarized (70)–(74), which indicates thatwith , , and defined in (62). By considering , one getsTo ensure , i.e., should satisfyMoreover, (60a) is used to guarantee that the denominator of lower bound of is greater than 0 and (59) can ensure that the upper bound of is a positive number.
From Lemma 7, we can get the following:and
5.2. Exponential Convergence in the Mean Sense of Closed-Loop System
This 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.
Theorem 8. Suppose that Assumptions 2, 4, and 5 hold. If the conditions in Lemma 7 are satisfied, there exists a coding and control strategy such that the closed-loop Markov jump system (16) achieves exponential convergence in the mean sense, i.e.,holds for every initial condition , where , , andwith and defined in (90).
Proof. For all (), by (25), one hasBecause and , it yieldsBy the triangle inequality, one getsSpecially, one has .
Consider that the closed-loop dynamic is during the interval ). If we rewrite the second auxiliary system asit is easy to see thatholds for any [). Because , one getsHence, we haveProjecting onto the -component, we deduce , which implies thatwith , , andCombined (79) and (80) induces thatBased on the analysis in Section 3, one knows that the design of relies on . Hence, there exists a function such that .
By defining and , (92) implies thatThe proof is completed.
6. Simulation Example
The two-tank system borrowed from [24] is used to verify the effectiveness of the control strategy, which can be modeled as system (1) with andwhere the system states represent the deviations from the nominal reservoir levels. The flow between two tanks is proportional to the difference of the reservoir levels, and the flow control can be switched arbitrarily.
Let , , , , , and . Select , , and . To ensure (59) and ((60a) and (60b)), we set , , , and .
From Figure 2, it is obvious that can be selected as the initial triggered instant. The 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, and are displayed in Figure 4, which illustrates that the quantizer is unsaturated after . In such figure, the blue vertical dotted lines indicate the switching instants, which are randomly generated according to and , 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 and . It is worth mentioning that the growth rate of is much greater than that of . This is because is designed from the worst case in order to ensure the unsaturation of the quantizer in all cases.
6.1. Comparison
As shown in 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 influence 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.
7. Conclusions
The 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, sufficient 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 suffered deception attacks, which brings challenges to the quantizer design. The quantized feedback control under bilateral network suffered deception attacks is one of our future research directions. Moreover, the triggered condition 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
The data used to support the findings of this study are available from the corresponding author upon request.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
The work was supported by the National Natural Science Foundation of China (61773154 and U1804163), the Training Plan of Young Backbone Teachers in Higher Schools of Henan Province (2020GGJS085), and the Young Scholars Funding Program of Henan University of Technology (21420079).