Observer-Based Dynamic Event-Triggered Robust H∞ Control of Networked Control Systems Under DoS Attacks

This paper designs an observer-based controller with a dynamic event-triggered strategy for networked control systems with external disturbance, system uncertainty, and unknown periodic Denial-of-Service (DoS) attacks. First, the system under DoS attacks is modeled as active subsystem and dormant subsystem by using the switching system method. The controller is built based on dynamic event-triggered sequence and observer with the hold-input method used as the control signal for DoS attacks active subsystem. An augmented system is constructed with the time delay method, and the <inline-formula> <tex-math notation="LaTeX">$H_{\infty }$ </tex-math></inline-formula> performance index of the closed-loop switched system is given. Then, using Lyapunov functional theory and some technical lemmas, the sufficient conditions for the stability of the system are given without the disturbance, and with disturbance, the sufficient conditions for the closed-loop system with <inline-formula> <tex-math notation="LaTeX">$H_{\infty }$ </tex-math></inline-formula> performance index are deduced. Finally, a numerical example is given to verify the effectiveness of the theory. Through the minimum optimization problem, the <inline-formula> <tex-math notation="LaTeX">$H_{\infty }$ </tex-math></inline-formula> performance parameter is optimized to increase the disturbance suppression degree of the system.

keeps the latest value. Compared with the traditional timetriggered strategy, the event-triggered strategy saves the bandwidth resource of the networked control systems in [15]. Research into event triggering should pay attention to the fact that the minimum interval time between events must be strictly greater than zero to avoid Zeno behavior. One way to overcome this problem is to employ an event-triggered strategy based on sampled data [16]. To further save bandwidth resources, [18] proposes a dynamic event-triggered mechanism. By introducing an additional variable, the eventtriggered number is reduced while holding the stability of the system. A new dynamic event-triggered mechanism is designed in [19], and by setting the threshold value of the dynamic parameter, the parameter neither decays prematurely nor increases excessively, which generates fewer transmissions. The analysis and synthesis of networked control systems based on dynamic event triggering have also been discussed in [20]- [23]. Under the same control performance as the static counterpart, the dynamic event-triggered strategies generate fewer events. Besides, [19] researching dynamic event-triggered mechanism based on sampling instant can also avoid the Zeno behavior.
There are growing focus on event triggering and security control of networked control systems. In [24], using the active period and dormant period of DoS attacks, the resilient event-triggered strategy based on output is designed, and the strategy and DoS attacks are integrated into a unified framework by switching system theory. In [25], considering the practical engineering applications, and the state of the system is hard to get all, the networked control systems with periodic DoS attacks based on state observer are studied for resilient event-triggered control, and the observer and controller of the system are designed collaboratively. Although some static event-triggered methods have been developed with the DoS attacks, only a few results considered dynamic event-triggered strategy into account. In [26], resilient dynamic event-triggered strategy is designed to study the control problem of power systems under DoS attacks and transmission delay. Based on [25], to reduce the control update frequency, the parameters of dynamic event triggering, observer, and controller are designed collaboratively in [27]. Considering that the information transmission between the sensor and the controller may be subject to asynchronous DoS attacks, [28] proposes a resilient control method for dynamic event triggering, and the designed dynamic event-triggered strategy generates fewer events than static event-triggered strategy.
To sum up, this paper mainly assumes that DoS attacks energy are limited. Compared with [25], this paper discusses more common attacks with the unknown period, while the above resilient control strategy mainly adopts the zero-input method to compensate for the impact of DoS attacks on the system in [25]- [28]. Considering that the performance of the hold-input strategy may be better for the compensation of the unstable system, this paper takes the hold-input method to design the controller and studies the secure dynamic event-triggered control of the networked control systems with uncertainty and external disturbance. The novelty of this paper lies in two aspects: 1) Different from [25], [28], [29], considering a more complex networked control systems with DoS attacks, external disturbance, and uncertainty, sufficient conditions are given for robust asymptotic stability of the system without disturbance, and with disturbance, the sufficient condition for the closed-loop system with H ∞ performance index is deduced. 2) Comparing with results in [28], by introducing an integral inequality to deal with the integrals with derivatives in Lyapunov-Krasovskii functional, a low conservatism condition is obtained with an additional parameter, using the hold-input strategy to realize faster and more smoothly the stability of the system. Notation: R n denotes the n-dimensional Euclidean space and R n×m is the set of n × m real matrices; the superscript T stands for matrix transposition; the symbol * represents each of the symmetric blocks; L 2 [0, ∞) denotes the space of square-integrable vector functions defined on [0, ∞); I and 0 represent identify matrix and zero matrix with proper dimensions, respectively.

II. PROBLEM FORMULATION AND MODELING
Consider a class of networked control systems with uncertainty and disturbance: where x(t) ∈ R n is the plant state, y(t) ∈ R q is the plant output, u(t) ∈ R m is the control input, w(t) ∈ R p denotes the disturbance input belonging to L 2 [0, ∞), A, B 1 , B, C are the known parameter matrices with suitable dimensions, and A, B are the unknown time-variant matrices representing norm-bounded parameter uncertainties, which are assumed to be of the form A B = HF(t) E 1 E 2 , where H , E 1 and E 2 are known constant matrices with suitable dimensions, and F(t) is an unknown time-varying matrix function, which satisfies the inequality condition F T (t)F(t) ≤ I for all t.
In Fig.1, the unknown periodic DoS attacks can be modeled as the following switching signals where [g n , g n + h n ) denotes the sleep interval of the nth DoS attacks with normal transmission, and [g n + h n , g n+1 ) denotes the active interval of the nth DoS attacks with interrupted transmission. {h n } represents the time instants when DoS attacks are sleeping, and {g n + h n } represents the active start time of the nth DoS attacks which end at g n+1 . In order to simplify the derivation, let H 1,n = [g n , g n + h n ), H 2,n = [g n + h n , g n+1 ), t 1,n = g n , t 2,n = g n + h n . Assumption 1: Same as [29], n(t) is the number of the positive edge triggering of DoS attacks in the current period [t, ∞). There are two real scalars ν ≥ 0 and > 0 for all t ≥ 0 that satisfy Assumption 2: Same as [29], (t) is the total duration of DoS attacks within n(t) of the current positive edge triggering. There are two real scalars ω ≥ 0 and D > 0 for all t ≥ 0 that satisfy (4) Considering the following full-dimensional state observer for system (1) wherex ∈ R n is the observer state,ŷ ∈ R q is the observer output, and L(t) ∈ R n×q is the observer gain to be designed as From Fig.1, h > 0 is the sampling period, t j h is the sampling time, and t k h is the latest trigger time, t j , t k ∈ N. During the nth dormant DoS attacks, the jth sampling is defined as t n j , n, t n j , j ∈ N, and t n j h is the current sampling time. If t n j h satisfies (6) as follows the data will be transmitted and the trigger time will be undated.
To compensate for the impact of the system from the active DoS attacks, we designed the data transmitted at g n+1 which represents the end of the nth DoS attacks. Therefore, the transmission instant of observer statex is determined by dynamic event generator which is as follows t k,n h ∈ t n j h satisfying (6) | t n j h ∈ H 1,n {g n+1 }. (7) In (6), θ > 0, σ ∈ (0, 1), is a positive definite symmetric matrix to be designed, and the dynamic parameter η(t) satisfieṡ where t ∈ [t k,n h, t k+1,n h), is a positive definite weighted matrix to be designed, β is a function of class K ∞ and satisfies Lipschitz continuity, andη is an adjustable parameter according to the actual condition.
Remark 1: In order to avoid Zeno behavior, the designed trigger condition is based on the sampling time of the system, so the minimum event-triggered interval of the system is an integer multiple of the sampling period.
Lemma 1 [19]: Suppose β is a K ∞ class function and satisfies Lipschitz continuity, let σ ∈ (0, 1), Considering the switching signals of DoS attacks, the controller of the system (1) is designed as follows where K 1 , K 2 are the controller gain matrices, the sequence t k,n h is determined by iteration of the trigger condition (7), t 0,n = g n , h > 0, k ∈ {0, 1, · · · , k(n)} = K (n), k(n) = sup k ∈ N | t k,n ≤ g n + h n , t k(n)+1,n h > g n + h n . From the above definition, it can be known that t k(n),n h is the k(n)th normal triggering within the time of the nth dormant DoS attacks.
In order to simplify the derivation, similar to [25], let R k,n = t k,n h, t k+1,n h . Substitute (9) into (1), we can obtain R k,n is divided into inter cell according to sampling interval as follows where k ∈ K (n), n ∈ N , ρ k,n = inf{m ∈ N | t k,n h + mh ≥ t k+1,n h}.
145628 VOLUME 9, 2021 For k ∈ K (n), n ∈ N , same as [29], two piecewise functions are defined, The error state is defined as e(t) =x(t)−x(t), and the augmented state is ξ T (t) = x T (t) e T (t) , then the augmented system is derived as follows: where In this paper, we design a robust dynamic event-triggered controller (9) based on the observer (5) for the networked control systems (1) with unknown periodic DoS attacks (2) so that the closed-loop switched system (14) satisfies the following two conditions: 1) When the external disturbance w(t) = 0, the closedloop switched system (14) is robust asymptotically stable. 2) When the external disturbance w(t) = 0, the system in its initial state x(t) = 0 satisfies the following H ∞ performance In order to facilitate, the lemmas are given for the treatment of uncertainty and integral item in Lyapunov functional.
Lemma 2 [30]: If there are two known matrices D and E of proper dimensions and F(t) satisfies F T (t)F(t) ≤ I , there is a scalar λ > 0 such that the following inequality holds Lemma 3 [31]: → R n such that the integration in the following inequality is well defined, then it holds that

Lemma 4:
Given fixed DoS attacks parameters ν ≥ 0, ≥ 0, ω ≥ 0, D ≥ 0, the feedback control gain matrices K 1 , K 2 , the observer gain matrices L 1 , L 2 , and the parameters α 1 > 0, 2 > 0 and matrices T 1 , T 2 of appropriate dimensions, such that the following matrix inequalitieŝ wherê it can be obtained Proof: An candidate Lyapunov-Krasovskii functional for the closed-loop switched system (14) is constructed as follows η(t) is defined by (8), β(η(t)) = −2α 1 η(t), and V i (t) is constructed as where by taking the derivative of V 1 (t) along the trajectory of the system (14) at t ∈ R k,n ∩ H 1,n , applying Lemma 3 to the integral term where Substituting (24) for (23), we can geṫ According to (8) and (25) we can obtaiṅ To apply Schur's Complement Lemma [30], we can obtain where F 1 = A 1B1 0B 1 and To eliminate the uncertainty in 1 , we obtain 1 ≤ˆ 1 , and the form ofˆ 1 is shown in (17) of Lemma 4.
Obviously, ifˆ 1 < 0 then 1 < 0,Ẇ 1 (t) < −2α 1 W 1 (t), when t ∈ R k,n ∩ H 1,n , we can obtain When i = 2, w(t) = 0, in the same way by taking the derivative of V 2 (t) along the trajectory of the system (14) at t ∈ R k(n),n ∩ H 2,n , we can obtaiṅ taking Schur complement of 2 < 0, and it can be obtained to where F 2 = A 2B2 0B 2 , and Similarly, the uncertainty of (28) is eliminated to obtain 2 ≤ˆ 2 , and the form ofˆ 2 is shown in (18) of Lemma 4. Obviously, ifˆ 2 < 0 then 2 < 0,Ẇ 2 (t) < 2α 2 W 2 (t), when t ∈ R k(n),n ∩ H 2,n , we can obtain The proof is now completed. Remark 2: Since the closed-loop switched system has time delay, Lyapunov-Krasovskii functional with the integral term is designed to deal with the delay effect of the system. In [25], the integral term is processed by the free weight matrix, and an additional constraint is introduced. In this paper, the integral term is processed by Lemma 3 in [31], and no redundant constraint is introduced, thus providing a low conservative condition.
(19) is a sufficient condition that will be used to prove the asymptotic stability of (14) in Theorem 1.
When i = 1, the alternative Lyapunov-Krasovskii functional form same as Lemma 4 is selected. Define the augmented stateζ T (t) = ζ T (t) w T (t) , the same proof process as Lemma 4 can be obtained Similarly, when i = 2, we can obtain By repeatedly deriving the stability theory of the augmented system in Theorem 1, the conditions (29) -(31) are obtained. Multiply J 1 by the left and right sides of (45), and J 1 = diag{X 1 , J 3 , J 3 , X 11 , X 1 , I , I , I , I }, where J 3 = diag{X 11 , X 11 }, multiply X 1 by the left and right sides of (29), let X 1 = P −1

IV. ILLUSTRATIVE EXAMPLES AND SIMULATION
Consider an unstable batch reactor system [32], whose system parameters are as follows  In order to verify the effectiveness of Theorem 2, the closed-loop switched system (14) is simulated as shown in Fig. 2 -Fig. 5. As shown in Fig. 2 and Fig. 3, when there is no disturbance, the state of the closed-loop batch reactor system with unknown periodic DoS attacks converges to zero in finite time. When there is a disturbance, the states of the closed-loop system converge to zero in finite time as well as the system still can effectively restrain disturbance with H ∞   performance. According to Fig. 4, the observation errors converge to zero before the state of the closed-loop system is stable, which verifies the effectiveness of the full-dimensional state observer designed in this paper.
In addition, instants and intervals of dynamic event triggering are shown in Fig. 5, and we can derive that the average triggering interval is 0.14s, which is much longer than the sampling period 0.001s. Thus, system resources such as networked bandwidth can be saved. The minimum triggering  interval equals to the sampling period 0.001s, which excludes Zeno behavior. These observations above confirm Remark 1. Significantly, by comparing Fig. 2 and Fig. 3, the system can suppress the disturbance with short-term DoS attacks, such as the 5s to 15s. However, the disturbance suppression ability of the system is reduced with long-term DoS attacks, such as the 30s to 40s. It can be known that high energy DoS attacks will worsen the adverse effects of disturbance on the system and interfere with the normal operation of the system.
Here we compare the results of the proposed method with [26]. The responses of state show in Fig.6 employ the two different control strategies without disturbance and uncertainty. Apparently, Theorem 1 can achieve satisfactory control performance faster and more smoothly. This result is mainly because the hold-input strategy adopted in this paper compensates for the impact of DoS attacks on the system. In contrast, the zero-input strategy is adopted in [28] to compensate for the impact of DoS attacks on the system.
To further optimize the H ∞ performance of the system, we use MATLAB to solve the following optimization The optimal value γ = 0.36 of H ∞ performance parameter is obtained through mincx solver of the LMI toolbox, and the simulation result is shown in Fig. 7. By comparing Fig. 3 and Fig. 7, it can be seen that the disturbance suppression degree of the closed-loop system increases after the H ∞ performance parameter γ is optimized. According to (15), the H ∞ performance parameter γ determines the degree of disturbance suppression, and the smaller γ is, the greater the degree of disturbance suppression. However, if the given γ is smaller, the LMIs in Theorem 2 will be more challenging to solve. In this article, the control parameters of the closed-loop system and minimum γ can be effectively found through solving optimization problem (47).

V. CONCLUSION
This paper studied an observer-based dynamic eventtriggered control for networked control systems with unknown periodic DoS attacks, external disturbance, and uncertainty. Firstly, we have analyzed the case without the disturbance and obtained sufficient conditions for the robust asymptotic stability of the closed-loop switched system by using Lyapunov functional theory, improved Jensen's inequality, Schur's Complement Lemma, etc. Secondly, we have considered the case with the disturbance. By introducing the H ∞ performance index, we obtained the sufficient conditions that the closed-loop switched system has the H ∞ performance with the disturbance. Finally, an example of batch reactor has been used to verify the correctness of the theory studied in this paper, and H ∞ performance parameter optimization has been realized to increase the disturbance suppression level of the system.