topic scaled consensus for multiagent systems under denial of service attacks and exogenous disturbance

Scaled consensus for multiagent systems underdenial-of-service attacks and exogenousBilal J.. MahmoudPublished online: 11 June 2021Article views: 44 Abstract In this paper, the scaled gr

HANOI UNIVERSITY OF SCIENCE AND TECHNOLOGY School of Electrical and Electronic Engineering

Department of Automation Engineering

Final Exam 20231

EE1024E Introduction to Electrical Engineering

Topic: “Scaled consensus for multiagent systems under denial-of-serviceattacks and exogenous disturbance”

Lê Đặng Công Vinh - 20232557Nhữ Thanh Tùng - 20232554Trần Minh Vương - 20232561

Thái Duy Tùng - 20232555

Hanoi, 1/2024

Scaled consensus for multiagent systems underdenial-of-service attacks and exogenous

Bilal J Karaki & Magdi S Mahmoud

Published online: 11 June 2021Article views: 44

Abstract

In this paper, the scaled group consensus problem of high-order multi-agent systems is investigated over directed graphs In terms of an appropriate Lyapunov function, sufficient conditions are derived to guarantee scaled consensus for high-order continuous-time systems for strongly connected networks For the case where the agents are subject to exogenous inputs, the scaled consensus is achieved with a guaranteed H2/H∞ performance Moreover, the scaled-consensus problem for multiagents subject Denial-of-Service (DoS) attack is investigated The considered attack model allows the adversaries to compromise agents independently Decay rates for each attack mode are obtainedbased on a set of linear matrix inequalities Sufficient conditions in terms of decay rates are derived to guarantee secure scaled consensus Simulation results are also presented to illustrate the effectiveness of the proposed theoretical results.

I Introduction

Recently, the cooperative control of multi-agent systems has received considerable attention In cooperative approaches, every agent communicates with its neighbours over a communication network in order to implement its owncontrol law such that a common global goal is achieved The cooperative control approach distributes the responsibilities among all agents Consensus problem, asa typical cooperative behaviour in multi-agent systems, means that the states of the agents reach a common value of interests based on local information exchange It plays a central role in many fields, including robotics, computer science, biology and economics Different models of multiagent systems have been studied to achieve consensus such as first-order models which contains onlythe dynamics of the position, agents with second-order dynamics where the position and velocity are considered.

All aforementioned results considered common consensus such that all statesconverge to a common value Recently, various new consensus problems have emerged, such as group consensus and scaled consensus Scaled consensus is considered one of the important coordination problems Broadly speaking, the scaled-consensus problem is dedicated to designing distributed protocols such that the states of the agents reach a prescribed proportion rather than a common

value In other words, the states of different agents with the same scale values

will reach the same common value and the agents with different scales will gradually converge to state values with different ratios It is worth mentioning that, by appropriately selecting the scale-value of each agent, the scaled-consensus protocol solves the bipartite consensus problem On the other hand, the usual consensus can be solved by adjusting the scales such that they have the same value For the research of scaled consensus, an enormous number of results have been established, just to name a few.

Multiagent systems deeply integrate communication and control/computation with the agents’ plants Since the agents always

communicate through a network, multiagent systems become more vulnerable to cyber-attacks Denial of service attacks (DoS) aims to disrupt the communicationtopology or jam an agent’s state from reaching the neighbouring agents It is worth mentioning that adversaries use DoS attacks against multiagent systems because defining all channels is almost impossible and it is considered to be simple to launch Recently, some interesting results on the secure control policieshave been developed against DoS attacks An observer-based secure consensus protocol was developed for multiagent systems with lossy sensors DoS attacks, where the attack was modelled by a Bernoulli distribution In De Persis and Tesi (2015), De Persis and Tesi provided a scheduling scheme to guarantee the stability of the cyber-physical systems subject to DoS attacks However, the workwas devoted to systems with a single channel only On the other hand, based on switching techniques, secure control was investigated for systems with multiple transmission channels under DoS.

Secure cooperative protocols should not only consider DoS attacks, but also various sources of performance degradations such as exogenous disturbances In the presence of external signals, the guaranteed performance should be introduced to reduce the impact of exogenous disturbances on the disagreement behaviour It is worth mentioning that the exogenous disturbance of an agent could adversely infect its neighbouring agents This motivates researchers to investigate the H∞ performance for multiagent systems with exogenous signals Li and coworkers transformed the H∞ consensus problem into H∞ control problem for undirected topologies Robust H∞ consensus problem was solved foragents with uncertain dynamics and exogenous disturbance Besides, Zhao et al (2012) investigated the problem based on output-based measurements and an observer-based methodology It is worth pointing out that most of the

aforementioned results are mainly devoted to solving the H∞ consensus problem under undirected topologies Nevertheless, the communication topology subject to cyber-attacks or network imperfection becomes directed For instance, when an adversary compromises communication links in one direction, an undirected graph switches into a directed communication topology immediately Besides, the network topologies are usually directed and undirected connection can be considered as a special case Therefore, the H∞ consensus problem with directed topologies has been the subject of a renewed surge of interest In Saboori and

Khorasani (2012), H∞ consensus was solved based on the algebraic Riccati equation for strongly connected graph topologies.

It is still a challenging problem to develop effective secure consensus protocols for multiagent systems with directed graphs under DoS attacks and external disturbances Despite the considerable progress, most of the existing results including those mentioned above does not focus on scaled consensus in the presence of cyber-attacks Motivated by the above observation, the major contributions of this paper are reflected as follows:

(1) This article investigates the secure consensus protocol for multiagent systems subject to a severe class of DoS attacks The considered DoS attack model is different from the on/off model with a single adversary, where the rule generated by adversaries can compromise different channels with different intensities.

(2) The protocol is further investigated to guarantee scaled consensus with external-disturbance rejection via H∞ performance Moreover, another theorem is provided to solve the scaled consensus with guaranteed H2 performance Based on the DoS intensities on the set of communication channels, sufficient conditions in terms of linear matrix inequalities are derived based on Lyapunov and directed graph theory.

(3) The proposed secure consensus protocol can be applied to multiagent systems with switching topologies Besides, the proposed theoretical results are proposed for higher order dynamics compared with the existing results that focus on single- or second-order dynamics without external disturbances.

The organisation of this paper is as follows In Section 2, problem formulation and preliminary results are presented, and the secure scaled-consensus protocol isdesigned Consensus results of the proposed techniques are provided in Section 3 A simulation example to verify the proposed theoretical results is presented in Section 4 Finally, Section 5 concludes the paper.

Notation: Define G = (V, E, A) as a directed graph with a set of vertexes V ={v1, , vN}, a set of edges E = V × V, and A is the adjacency matrix.

II Problem formulation

Consider a group of N identical linear agents whose dynamics are described

using the following differential equations:

x˙i(t) = Axi(t) + Bui(t) + Dwi(t)

where xi(t) Rn represents the state of each agent and ui(t) Rm is the control ∈ ∈law of the ith agent wi(t) Rp represents an external disturbance A, B and D ∈are known constant matrices with appropriate dimensions.

Assumption 2.1: The directed graph G is assumed to be strongly connected Assumption 2.2: The pair (A, B) is stabilisable.

Remark 2.1: It is worth pointing out that no assumptions are imposed on locations of the eigenvalues of the state matrix A Assumption 2.2 is very mild inthe investigations of distributed coordination problems of linear multiagent systems Besides, this assumption usually is not used in the next analysis, but it isa necessary condition to find a feasible solution for the LMI conditions Lemma 2.1: Let G be a strongly connected graph Then, the following key properties hold.

(1) The vector 1 = [1, 1, , 1]T is the right eigenvector of the Laplacian matrix L associated with the simple zero eigenvalue, i.e L1 = 0.

(2) There exists a positive left eigenvector ξ = [ξ1ξ2, , ξN] T of the Laplacian matrix L associated with the simple zero eigenvalue, i.e ξTL = 0 such that ξi ≥ 0 for all i = 1, , N.

(3) There exists a positive scalar μ, the generalised algebraic connectivity Definition 2.2: Consider multiagent system (1).

(1) It is said to accomplish the scaled consensus with the nonzero scales (α1, , αN), if the states of multiagent system (1) satisfy:

(2) Under the zero initial conditions, the distributed protocol solves the scaled consensus if dynamics (5) is asymptotically stable and there exists a positive scalar γ such that the following holds:

(3) The distributed protocol solves the H2-scaled consensus if dynamics

is asymptotically stable.

DoS aims to disrupt the communication topology or jam the agent’s state from reaching the neighbouring agents Broadly speaking, DoS attack models aredifferent from packet dropouts modelled by stochastic models The DoS attack model changes its targets and strategies such that the consequences of the risk areconstantly changing Therefore, to develop consensus protocols for risk-sensitive multiagent systems, we pay attention to the case when the DoS attacks cannot be modelled as a stochastic process such as the Bernoulli process No assumptions have been imposed on the DoS attacks except for attack duration As mentioned in Feng and Hu (2017), DoS requires to terminate its activation and turn into a stop-mode to store energy for the future attack instant Similar to De Persis and

Tesi (2015) and Lu and Yang (2018), attack intensity/duration for each communication edge (i, j) E should be restricted as follows.∈ Assumption 2.3: There exist ij ϑ ∈ R+ and σij < 1 satisfying:

where σij represents the DoS attack intensity, and D(i,j)(τ , t) represents the unionof jamming periods of the link (i, j) E during the interval [τ , t), τ < t.∈

Remark 2.2: It is worth mentioning that larger σij indicates a more intensive

DoS attack on channel (i, j) In De Persis and Tesi (2015) and Feng and Hu (2017), two attack modes are considered (under DoS or not) Nevertheless, we consider various attack modes: all or partial links can be attacked with different attack intensities: partial or all links Besides, it should be pointed out that while Lu and Yang (2018) assumes channels (i, j) and (j, i) as one link, in this paper, we consider directed graphs such that (i, j) and (j, i) are considered as different channels.

III Main result

Theorem 3.1: Suppose that the scales αi are given and the communication

graph G is strongly connected Multiagent system (1) solves the scaled consensus with the proposed protocol (2), if there exists a positive definite matrixX > 0 and positive-scalars γ > 0 and c>0 satisfying:

H∞-where , and μ is the generalised algebraic connectivity Moreover, the consensus gain is given by

Remark 3.1: It is worth mentioning that the Laplacian matrix L is not

symmetric Consequently, employing the unitary transformation used in Li et al (2010, 2012) is not possible Theorem 3.1 transforms the global scaled consensusof the multiagent systems in (1) to a feasibility problem of a low-dimensional LMI Compared with the results in P Lin and Jia (2010) where the agents are modelled by second-order dynamics, the conditions in Theorem 3.1 are applicable to any high-order multiagent systems.

It is worth mentioning that scaled consensus with H∞-performance increasesthe robustness of the multiagent system to the worst exogenous disturbances On the other hand, H2-performance indicates that the multiagent system can reject bounded disturbances The following theorem provides a set of sufficient conditions to achieve scaled consensus with guaranteed H2-index.

Theorem 3.2: Suppose that the scales αi are given and the communication graph G is strongly connected Multiagent system (1) achieves H2-scaled

consensus with via distributed protocol (2), if there exists a positive definite matrix X > 0 and positive-scalars ν > 0 and c>0 satisfying:

where , and μ is the generalised algebraic connectivity Moreover, the consensus gain is given by

Theorem 3.3: For a strongly connected directed graph G with multiagent

dynamics (1), if there exist positivedefinite matrix X and positive scalars ς,c, φ1 and φ2 such that

< 0 (38)

Remark 3.2: It is noted thatμis not always larger than μ In light of the LMIcondition (38), if μ>μ the feasibility of the inequality is feasible with a negative scalar κ < 0 On the other hand, κ could become positive to guarantee the feasibility of LMI condition (38) for some severe DoS modes, particularly when μ>μ However, inequalities (39) and (40) distribute the effects and compensate for the growth of energy due to the positivity of based on the intensities of the attack modes.

Remark 3.3: It is worth mentioning that the number of decision variables in LMI-conditions is independent of the number of agents of the group However, some conservativeness appears due to the attack intestines which may prevent obtaining a feasible solution from the LMIs One could consider multiple Lyapunov functions can obtain less conservative conditions This may increase the analysis difficulty and may lead to conditions with a large number of decisionvariables which require more computing resources.

4 Simulation results

The system to be controlled is not asymptotically stable and is characterisedby the following dynamics: