Fast Finite-Time Secure Control for Nonlinear Systems under Dynamic Event-triggered Mechanism

This paper addresses the fast finite-time secure control problem for the nonlinear system subjected to deception attacks. To enhance the resilience of the nonlinear system with faster convergence rate, a fast finite-time secure controller (FFSC) with an adaptive compensation law is designed, which deals with the damage effects from the deception attacks. Moreover, for the purpose of alleviating the network burden, a dynamic event-triggered mechanism is established, which is regulated by the internal variable generator contains the error norms with the fractional power. Furthermore, the fast finite-time stability of the closed-loop system and the Zeno behavior elimination of the proposed DETM are rigorously proved, respectively. Consequently, the simulation results demonstrate the effectiveness of the proposed method.


Introduction
Convergence rate is a key performance indicator for the nonlinear systems, such as the flexible air-breathing hypersonic vehicles [1], satellite group formation flying [2], microgrids [3] and unmanned aerial vehicle containment [4], and so on.It is therefore highly necessary to investigate the finite-time control, which was initially studied in [5,6] and then becomes an increasingly interesting topic in the last decades.Different from the asymptotic stability that achieves the control objects in infinite-time, the finite-time control strategy captures plenty of remarkable merits.Many fruitful researches show the superior performances of the finite-time control, such as faster convergence rate [7], more robust against the uncertainty [8] and better anti-interference performance [9], etc.
It is noted that cyber security is a fundamental requirement under the networking trend in many fields, (etc.the internet of things [10], intelligent transportation [11], etc.).Aiming at destroying the cyber security, there exist two main attack patterns, namely the denial of service (DoS) attacks [12] and deception attacks [13], which operate by blocking the communication channels and tampering the data packages, respectively, leading to the system performance degradation or even collapse.Especially, by the deception attack, the systems may wrongly operate toward the hacker's malicious intentions.Therefore, it is of the utmost importance to focus on the secure control strategy of the system due to the misleading and harmful deception attacks.Up to now, lots of secure control strategies against the deception attacks have been obtained.In [14], a neural-network-based controller is proposed for the nonlinear cyber-physical systems to deal with the deception attacks occurring on the actuator side and the sensor side.In the framework of input-to-state stable in probability, Ding et al. [15] studied the security in probability for the stochastic nonlinear systems attacked by the deception attacks.In [16], the security object is guaranteed in the meaning of the almost sure stability, where the deception signals are composed of the randomly impulsive sequence.It should be emphasized that, the aforementioned secure control approaches ensure the closed-loop system to be stable at the infinite time.Since the engineering systems desire high demand of the convergence rate, it is highly significant to study the attack-tolerate performance of the nonlinear systems in the meaning of the finite-time.
Very recently, lots of pioneering works have been proposed to protect the system with stronger resilience and timelier secure actions based on the finite-time technology.To name a few, in [17], with the aid of the finite-time converging extended state observer, a composite resilient controller is formulated, by which the finite-time stability of the high-order nonlinear cyber-physical systems can be ensured.To eliminate the damaging effect from the injected data, Song et al. [18] designed a finite-time fractional order command-filtered backstepping method, which also reduces the computation burden greatly.In [19], the time-varying gain problem caused by the state-dependent sensor attacks is handled by the finite-time Nussbaum based backstepping technology.Furthermore, in order to improve the transient performance, the definition of "fast finite-time stable" (FFTS) was presented in [20], the original intention of which is to enhance the convergence rate of the "finite-time stable" when the system states are far from the equilibrium point.To realize this purpose, the core idea is adding a fractional power term of the Lyapunov function to the asymptotic stability condition.What's more, when taking the cyber-attacks into consideration, how to design the secure control strategy in the sense of the fast finite-time stability is of vital importance and needs further investigation.
Up to now, the event-triggered mechanism (ETM) has attracted extensive attention of many scholars, especially in the limited energy or bandwidth situations.Over the last decades, plentiful ETM schemes have been designed from distinguished prospects, such as the sampled-data-based ETM [21], adaptive ETM [22], etc.Furthermore, in [23] three ETMs were designed for the nonlinear systems, which have been widely applied afterwards (see [24], [25] and the references therein).It is worth pointing out that, differing from the above static ETMs, a novel dynamic event-triggered mechanism (DETM) was firstly established in [26], which cuts down the transmission frequency effectively by adopting an internal dynamic variable generator.Recently, the research on DETM for the nonlinear system has become a popular topic.Such as, in [27] a DETM with a dynamically varying threshold is designed, by which the network resource burden is greatly reduced.Shu et al. [28] studied the DETM for the p-norm nonlinear system, the aperiodically sampled controller is capable of guaranteeing the system signals remain in a small region.However, the above mentioned DETM strategies are not suitable to the finite-time controller design.Therefore, it's challenging and desirable to design a DETM that guarantees the fast finite-time stability of the system.
To sum up, the state art of the secure control strategies and the ETMs have been recalled in the above observations.Inspired by which, this paper aims to design a fast finite-time secure controller with the dynamic event-driven approach.Specifically, to speed up the convergence rate in the presence of the deception attacks, the FFSC is equipped with a fast adaptive control law.Besides, compared with the existing DETM, the proposed DETM in this paper guarantees the system to be fast finite-time stable.The main contributions of this paper are summarized as follows.
1) Different from the existing dynamic ETMs in [27], [28], the designed dynamic event-triggered mechanism (DETM) guarantees the FFTS of the nonlinear system.
2) To enhance the system resilience, especially in aspect of the recovery time, the proposed secure controller against the deception attacks is capable of achieving the fast finite-time stability, which is distinct from stability in the sense of infinite-time [14] or the finite-time [18].
3) Compared with the static event-triggered mechanism, the designed internal variable generator of DETM is formulated.With the utilization of the fractional power term of the triggering errors, the conservation of the triggering condition is relaxed, which attributes to alleviating the network burden.
The rest of this paper is organized as follows.Section 2 formulates the secure control problem of the nonlinear system and presents the preliminary knowledge.Dynamic event -triggered mechanism and the fast finite-time secure controller are designed in the Section 3. A numerical example that verifies the effectiveness of the proposed method is given in Section 4. Consequently, the conclusions and the future works are presented in Section 5.

Preliminaries and problem formulation
In this section, the nonlinear system in the presence of the actuator deception attacks is formulated first.Subsequently, some basic definitions and lemmas are presented.
Consider the following nonlinear system () = ( () ) () where, , n x uR  are the system states and manipulated control input, respectively, (() ) f xt denotes the system nonlinearity.
In this paper, for the system (1), the vicious attack targeting at the controller-to-actuator (CA) channel is considered, which is modeled as follows: where, () ut is the ideal control input, (() ,) , then it follows that: (ii) The setting time T  in (i) can be given as . Lemma 2 ([30]) Given real scalars k, l and 12  rr / r , where 12 , rr are two positive odd integers, define   kkl , if 01  r , then it can be deduced that   where,     .Lemma 3 ([31]) Given a real number  , for any scalar , the following inequality holds: where, 1, 2, iN   .Assumption 1 The vector field nonlinear function (() ) f xt in (1) satisfies the quadratic-condition (QUAD-condition), i.e. there exists a scalar  such that where, , n x yR  .

Remark 1:
The Assumption 1 is reasonable, for one thing the QUAD condition is milder and weaker than the Lipschitz condition.In addition, the QUAD condition is widely adopted for numerous theoretical and engineering systems, such as the Lorenz systems [32] and Chua's circuit system [33].
In this paper, we are aiming to design a fast finite-time secure control strategy for the attacked system (1) with a dynamic event-triggered mechanism.By which, the communication resource will be saved and the fast finite-time stability of the closed-loop system will be guaranteed.

Main results
In this section, first, a dynamic event-triggered mechanism (DETM) is given, in which, a fractional power term of the internal variable generator is added; Furthermore, with the involve of the proposed DETM, the fast finite-time secure controller is designed.Based on which, the fast finite-time stability analysis of the closed loop system and the elimination of the Zeno phenomenon are performed.

3.1Dynamic Event-triggered Mechanism
In order to save communication frequency, a dynamic event-triggered mechanism is formulated in this subsection.The positivity of the designed internal variable is presented, which lays a foundation of the fast finite-time stability analysis hereinafter.
The dynamic event-triggered mechanism in this paper is considered to be located in the sensor-to-controller side.
, in which j t denotes the jth triggering instant, where,  will be defined in the triggering condition (5).
To judge the transmission of sensor data, the triggering condition of the DETM is given as:   , the proposed DETM belongs to the same type of the classic DETM, such as the one utilized in [34].
To avoid the singular problem of the DETM (especially the Zeno problem), the positivity of the internal variable generator (6) must be guaranteed.To tackle this problem, the following Lemma 4 is presented.
Lemma 4 For   0, t   , the following conditions hold: Proof: During the operating period, by the triggering condition of the DETM that presented in (5), the above condition a) holds naturally.Now, we are at the stage of proving the condition b).
Based on the condition a), it is obvious that the inequality with the internal variable generator (6), one arrives at Construct the auxiliary ordinary differential equation Suggest that at the instant ' tt  , the trajectory of the differential equation ( 8  .This contrasts with the above assumption. Thus, the solution of (8) stays nonnegative all the time.
Then the condition b) holds.This completes the proof.■ With the aid of the internal variable, the current value and the dynamics of the sampled error are characterized and further utilized in the triggering condition of the DETM (5), by which the redundant transmissions can be cut down.

Fast finite-time secure controller
In this subsection, a fast finite-time secure controller is formulated to address the deception attack problem described in (2).With the involve of the DETM (5), only the triggered states are available for the controller design.
The fast finite-time secure controller (FFSC) is formed as follows: where, k   , 0 G  ,  and  are the adaptive attack compensation law, given as follows.
  , ,,R  ssnn .Remark 3: For the designed FFSC (9), i) The FFSC only makes use of the triggered sensor data   j x t to combat with deception attacks, which is aware of network resource saving.
ii) The utilization of the fractional power " 21   " and the sign function in (9) are expected to enhance the resilience of the nonlinear system and achieve the goal of fast finite-time stability.Accordingly, if 0 G  , ˆ0   , ˆ0   , the FFSC will reduce to the continuous controller that processes the asymptotic behavior.
Nonlinear system: Eq. ( 1) Fast finite-time secure controller: Eq. ( 9) Internal variable generator: Eq. ( 6) DETM: Eq. ( 5) Adaptive compensate law: Eq. ( 10) Deception actuator attack: Eq. ( 2) Fig. 1.The diagram of the FFSC with the DETM (The dotted line implies the time-triggered transmission, the dash-dotted line represents the event-triggered transmission, the solid line stands for the local data exchange) Till now, the formulation of the dynamic event-triggered fast finite-time secure controller for the nonlinear system is completed.To clearly capture the main outline of this article, the diagram of the proposed dynamic event-triggered secure control scheme is illustrated in the Fig. 1.

Stability analysis
The main purpose of this subsection is to analyze the fast finite-time stability of the closed-loop system, which ensures that the proposed FFSC is capable of enhancing the security of the nonlinear system in the presence of the deception actuator attack.
Theorem 1 Consider the system (1) under the DETM (5) with the deception actuator attacks, if there exist positive scalars  being two odd integers, and k   , then by the designed FFSC ( 9) and the adaptive law (10), the FFTS of the closed loop system can be guaranteed.
Proof: Consider the following Lyapunov candidate function The derivative of 1 V is calculated as Subscribing the adaptive compensate law (10) in to (18) With the aid of Young's inequality and the Lemma 2, one has Recall the definition of   in (6), based on the Lemma 3, (21) can be deduced as   where, kk  , Thus, based on the Lemma 1, (22) implies that there exists finite-time such that the Lyapunov function V will converge into the set Hence, the fast finite-time stability of the closed-loop system can be achieved.This completes the proof.■

Feasibility Analysis
To make the DETM applicable, one key issue is to avoid the infinite triggering during a finite period, which is named as the Zeno behavior.Hence, in this part, the feasibility of proposed the DETM is proved, which eliminates the Zeno behavior.
Theorem 2 For the DETM (5), based on the Theorem 1, given the triggering parameters and the controller parameters as the in ( 5), ( 6), ( 9) and ( 10), then for any the two successive triggering instants j t , 1 j t  , then there exists a positive scalar  such that, 1 jj tt    .
Proof: For the sampling error Note that   x t  is a function of   x t and adaptive variable  , which have been proved to be bound in the Theorem 1. Then, As at every triggering instant   0 j et  , based on the comparison lemma ( [35]), one can obtain Recall the triggering condition (5), which implies that the triggered states won't be updated before   Thus, the Zeno phenomenon of the DETM ( 5) is eliminated, which proves the feasibility of the presented DETM.■ From the abovementioned analyses, it can be seen that by the FFSC (9), the security in the sense of fast finite-time stability against deception actuator attacks (2) can be guaranteed.Meanwhile, the transmission frequency will be cut down by the designed DETM.

Results
The Chua's circuit system is utilized to illustrate the validity of the dynamic event-triggered fast finite-time secure control (DET-FFSC) strategy, meanwhile, the comparisons analysis of the static event-triggered fast finite-time secure control (SET-FFSC) and the time-triggered state feedback control (TT-SFC) are also carried out.
The structure of the NCCS is shown in Fig. 9, the dynamic model of which is described as where, 1 v and 2 v denote the voltage across the capacitors 1 C , 2 C , respectively.L i ,   1 g v are the current through the inductor L and nonlinear resistor g , respectively.i u are the control input signals, R is the linear resistor.

. Nonlinear Chua's circuit with control inputs
The nonlinear term   Trajectories the adaptive law  From Fig. 3, it is obvious that by the proposed FFSC the system states can maintain in a small region of the equilibrium points with faster convergence rate (within 0.1s ).However, by the TT-SFC, the system states deviate from the equilibrium points and keep oscillating.Thus, the effectiveness of the FFSC at suppressing the cyber-attacks is testified.
Fig. 4-6 show the triggering performance of the proposed DETM.Specifically, the Fig. 4. shows that less data is transmitted from the sensor to the controller.It is calculated that the triggering number of the DETM and SETM are 128 and 220, respectively.Accordingly, the judgment for the triggering condition are presented in Fig. 5.The dynamic trajectory of the internal variable   t  is given in Fig. 6, which enhances adaptive performance of the triggering condition, saving more network resource.Fig. 7-9 demonstrate the efforts of the control strategy against the attacks.Besides, Fig. 7 conforms that longer time intervals can be obtained by the proposed method.The Fig. 8 and Fig. 9 show the response of the adaptive laws designed in (10), addressing the parameterized state attack     1 , x tt  and the uniformly bounded attack 2 (() ,) x tt  effectively.From the above analysis, it can be seen that the fast finite-time stability of the closed-loop system that subjected to the deception attack can be achieved.Also, the effectiveness and the feasibility of the developed DETM is validated.

5.conclusions
In this paper, we proposed a fast finite-time secure control strategy via a dynamic event driven approach to deal with the cyber-attack problem.A fast finite-time secure controller has been deployed to cope with the wicked effect from the cyber-attacks, an adaptive term is designed for the attack compensation.To relieve the communication burden between the sensor to the controller, a dynamic event-triggered mechanism is established, which has been further proved to be Zeno-free and capable of maintaining the fast finite-time stability of the closed-loop system.Consequently, a comparative simulation has been exploited to demonstrate the effectiveness of the proposed DET-FFSC (faster convergence rate, more resilience to the deception attacks and larger triggering intervals).In the future, the optimal fixed time secure control for the interconnected nonlinear system subjected to the stealthy deception attacks will be considered.

12 VVV
ee  , together with the Lemma 2, it follows that sampling errors bring difficulties to the system analysis.Note that the positivity of the internal variable   t  has been proved in Lemma 4. Define the following Lyapunov candidate function , the derivation of which is given as

:
Through this paper, tt