Aperiodic Sampled-Data Control of Distributed Networked Control Systems Under Stochastic Cyber-Attacks

Kritika Bansal and Pankaj Mukhija

Abstract—This paper examines the stabilization problem of a distributed networked control system under the effect of cyberattacks by employing a hybrid aperiodic triggering mechanism.The cyber-attack considered in the paper is a stochastic deception attack at the sensor-controller end. The probability of the occurrence of attack on a subsystem is represented using a random variable. A decentralized hybrid sampled-data strategy is introduced to save energy consumption and reduce the transmission load of the network. In the proposed decentralized strategy,each subsystem can decide independently whether its state should be transmitted to the controller or not.The scheme of the hybrid triggering mechanism for each subsystem composed of two stages: In the first stage, the next sampling instant is computed using a self-triggering strategy.Subsequently, in the second stage,an event-triggering condition is checked at these sampling instants and the control signal is computed only if the event-triggering condition is violated. The self-triggering condition used in the first stage is dependent on the selection of eventtriggering condition of the second stage. Finally,a comparison of the proposed approach with other triggering mechanisms existing in the literature is presented in terms of the sampling instants,transmission frequency and performance measures through simulation examples.

I.Introduction

DISTRIBUTED networked control system (DNCS)consists of geographically distributed subsystems that are connected together via a common communication network.The controller of each subsystem shares its state information with the controller of its neighbouring subsystems in a DNCS.Such systems are found throughout the infrastructure of society,such as smart grids, transportation networks,and robotics[1].These interconnections among the subsystems increase the system complexity as the dynamics of each subsystem are governed not only by its own state but also by the states of its neighbouring subsystems.Furthermore, the presence of a network in a DNCS gives rise to several issues such as network induced delay,data loss[2],and cyber security[3].The consideration of these network-induced effects in an interconnected system makes the control of these systems more challenging.

The problem of cyber-attacks such as denial-of-service(DoS)attacks,deception attacks,replay attacks on networked control systems(NCS)has recently attracted the attention of several researchers[4]–[12].These cyber-attacks risk the security and safety of DNCS and at the least degrade the performance of the system.Thus,it becomes essential to analyze the effect of cyber-attacks on the stability and control of a system.The stability of networked systems in the presence of DoS attacks is ensured in[4] by determining the suitable scheduling of the transmission times for the sensor and controller.In[5],a solution to the problem of optimal location switching data injection attacks is presented for NCS with distributed actuators.The system under dynamic sensor attack is considered in[6]and an index is provided giving the minimum number of sensors to be attacked for the attack to remain undetected.The problem of stability and stabilization of the system under cyber-attacks in the sensor-controller channel is considered in [7],[8].In [7],stochastic cyber-attack with delay in the channel is considered whereas[8]considers the false data injection attacks whereby the attacker can access and manipulate the controller gain causing gain fluctuations.A distributed attack detection method for sensor networks under deception attacks is presented in[9].The systems considered in[4]–[9]for the analysis of cyber-attacks are isolated in nature in the sense that no interconnection with other systems is considered.The problem of cyber-attacks on interconnected subsystems is examined in[10]–[12].Attack on the subsystem dynamics and communicated data between subsystems is considered in[10].An unknown input observer scheme is proposed to detect,locate,and identify the nature of the attack.In[11],an optimization problem from the outlook of a DoS attacker that maximizes the attacking effect under energy constraint is investigated.The consensus control of multi-agent systems with connectivity maintained and connectivity broken attacks on the edges is investigated in[12].However,to the best of authors’knowledge,the effect of cyber-attacks on the DNCS in which different subsystems may suffer attacks of different intensities has not been explored.

Due to the spatial nature of the system and the sharing of a common network in a DNCS configuration,it becomes practically significant to address the issue of communication load,energy consumption,and computational load arising because of limited resources. A conventional strategy of control over a network is to employ periodic control where sensing and actuation tasks are done at equal intervals of time.This leads to inefficient use of network bandwidth,increase in the computational load,and energy wastage due to redundant data transmission[13].To overcome the above-mentioned limitations,aperiodic sampled-data control,namely,selftriggering(ST)and event-triggering(ET)strategies are proposed in[13],[14].In event-triggered control,the data from sensor to controller is transmitted only if a predefined ET condition for the sensor-controller end is violated.Similarly,the control signal from the controller is transmitted to the actuator only if the ET condition for the controlleractuator end is violated.This reduces the number of transmissions in a control loop and thereby reduces the unnecessary usage of limited resources.ET strategy has been widely used for NCSs(see[15],[16],and references therein)and DNCSs[1],[17],[18].Several researchers have investigated event-triggered control of NCS under cyberattacks[19]–[22].Some discussions on ET strategy and hybrid triggering mechanism combining time-triggered and ET schemes for NCSs under deception attacks are provided in[19],[20],respectively.In[21],a solution to the stability and stabilization problem of NCS under actuator and sensor attacks is presented using an ET observer based control.However,the implementation of ET strategy on a DNCS under cyber-attacks has not been explored.In ET strategy, the continuous and periodic verification of triggering conditions may still lead to wastage of resources at the sensor/controller end.This problem of ET control is addressed by an alternate triggering mechanism,ST control.In ST control, the next sampling instant and the control transmission instant is computed based on the plant dynamics and the previously transmitted data.The ST control has been proposed for NCSs and multi-agent systems considering network effects such as delay and data loss in[23]–[26].In[27],ST control is implemented to mitigate DoS attacks.An ST consensus control is investigated in the presence of DoS attacks in[28].However,the problem of ST control of NCSs and DNCSs considering other types of attacks has not been considered.The ST control is more conservative than ET control in the sense that at some of the sampling instants computed in ST control,the ET condition may not have been actually violated.To overcome the limitations of both ET control and ST control,a combined strategy for an isolated uncertain system is presented in[29].In the triggering mechanism proposed in[29],the ET condition is verified only at the sampling instants computed using ST strategy.This combined mechanism further reduces the number of transmissions as compared to ST strategy and also alleviates the problem of continuous/periodic monitoring required in ET control.However, the problem of implementation of the combined triggering mechanism for the case of interconnected systems is not considered.This problem is addressed in[30]for the interconnected system having DNCS configuration in which a hybrid mechanism combining ST and ET strategies is developed.The ST and ET conditions in[29],[30]are chosen independently. Also, the problem of implementation of the combined triggering mechanism for the case of DNCS suffering from intrusion has not been considered.To ensure the performance of DNCS employing aperiodic triggering mechanisms,it is imperative to investigate it under cyberattacks.

This paper proposes a hybrid aperiodic sampled-data mechanism combining ST and ET strategies for DNCS subjected to stochastic cyber-attacks.The influence of cyberattacks is taken into consideration in the form of deception attacks.These attacks modify the transmitted signal,thus affecting the controller’s decision.The deception attacks reduce the efficiency of the system gradually.In this paper,a random variable is used to represent the occurrence probability of an attack.The proposed aperiodic sampled-data mechanism consists of two steps:First,the states of each subsystem in a DNCS are sampled using the ST strategy.In the next step,the ET condition is checked at these sampling instants,and the control law is computed only if the given condition is violated.The condition for the ST strategy in the first step is derived using the ET condition utilized for computing the control law in the second step.In this hybrid strategy,the system states are sampled in aperiodic manner using ST control and thereby reduce the usage of battery resources at the sensor end.The subsequent checking of ET condition at the sampling instant determined by ST strategy further reduces the computational load and leads to efficient bandwidth utilization.The controller design problem of the hybrid strategy controlled DNCS under cyber-attack has also been addressed.The results of the proposed hybrid strategy have also been applied for an isolated system under deception attack.Also,minimum inter-event time(M IET)is provided for an ET controlled isolated system under attack.The efficacy of the proposed results has been demonstrated through simulation examples of a truck-trailer system and a network of inverted pendulums coupled together.Also,a comparison is drawn between the proposed hybrid strategy and other aperiodic control strategies.

The main contributions of the paper are listed as follows:

1)A hybrid aperiodic sampled-data mechanism for DNCS is introduced to alleviate the problem of computational load,energy consumption and communication load.The proposed strategy combines ST and ET strategies.The state of each subsystem in the DNCS is sampled using ST strategy.Later,ET condition is checked at each self-triggered sampling instant and control law is computed only if this ET condition is violated.

2)Unlike[29],[30],where the ST condition and ET condition are independent of each other,in the proposed hybrid strategy,the condition for ST strategy is derived using the ET condition used in the second step.This as shown in Section V leads to reduced transmissions.

3) A more general attack scenario on a DNCS is considered whereby stochastic deception attacks of different intensities on different subsystems may occur.

4)The implementation of ST strategy alone for DNCS under stochastic deception attacks is also presented(see Corollary 1).

5)As a special case of the proposed results,the analysis of hybrid aperiodic sampled-data strategy for an isolated system is also presented (see Corollary 2). Also,MIET for an isolated system under deception attack is obtained.

The remainder of this paper is organized as follows. After presenting the system description in Section II,conditions for stability and stabilization of DNCSs under attack are derived in Section III.The conditions for aperiodic sampling along with aperiodic control update considering deception attack are presented in Section IV.The obtained results are illustrated using numerical examples in Section V.Finally, the concluding remarks are provided in Section VI.

II.System Description

III.Stability Analysis and Controller Design

IV.Aperiodic Sampled-Data Strategy for DNCS

TABLE V =0.2,0.1,0.3αi=0.1,0.15,0.1 i=1,2,3 Comparison of Strategies for Example 2with and for ,Respectively

TABLE V =0.2,0.1,0.3αi=0.1,0.15,0.1 i=1,2,3 Comparison of Strategies for Example 2with and for ,Respectively

Performance measures Subsystem 1 Subsystem 2 Subsystem 3 Theorem 2 Corollary 1[38]Theorem 2 Corollary 1[38]Theorem 2 Corollary 1[38]ts1(s)2.70 2.97 2.84 4.18 4.52 4.23 2.78 4.18 4.37 ts2(s)2.90 3.16 3.14 3.91 4.27 4.61 3.64 4.73 4.96 M p 3 3 3 2.20 2.43 2.26 1 1.10 1 Samples 314 437–591 864– 391 508–Control updates 131 437 326 92 864 326 118 508 326

TABLE VI Comparison of Strategies for Example 2with =0(without At tack) and α i=0.15,0.2,0.25for i =1,2,3,Respectively

TABLE VI Comparison of Strategies for Example 2with =0(without At tack) and α i=0.15,0.2,0.25for i =1,2,3,Respectively

Subsystem/Strategy ts1(s)ts2(s)Mp Samples Control updates Subsystem 1 Subsystem 2 Subsystem 3 Theorem 2 2.03 2.66 3 80 34 Corollary 1 2.30 2.71 3 226 226[30]2.10 2.67 3 115 40[38]2.14 2.94 3–213 Theorem 2 3.24 3.44 2.13 70 31 Corollary 1 3.33 3.61 2.30 143 143[30]4.61 4.90 2.28 136 38[38]4.01 3.91 2.15–213 Theorem 2 2.34 3.46 1 78 36 Corollary 1 3.02 3.52 1 206 206[30]3.57 3.99 1 183 38[38]3.97 4.12 1–213

VI.Conclusion

This paper presents a hybrid aperiodic sampled-data strategy for linear DNCS under stochastic deception type cyber-attack at the sensor-controller end.Each subsystem is subjected to a deception attack of different intensity.A condition for stability and stabilization of DNCS is derived along with the computation of next sampling instant for each subsystem.The ET condition is verified at these sampling instants to further reduce the transmission of data.The proposed hybrid strategy combines ST and ET schemes in a more coherent manner. Also,a condition for obtaining MIET for an isolated system under deception attack is proposed.Finally,different simulation examples are provided to illustrate the efficiency of the proposed results.The proposed strategy improves the efficiency of the system by reducing the transmission frequency,communication bandwidth, battery consumption and improving the control measures like settling time and overshoot as compared to other aperiodic sampleddata strategies.The proposed strategy can be further developed to address the control problem under other communication constraints arising in various applications involving interconnected systems.