Distributed Unknown Input Observer-Based Global Fault-Tolerant Average Consensus Control for Linear Multi-Agent Systems

Authors: ximing  yang
tieshan  li
yue  long
hanqing  yang
Venue: IEEE_org_ICIST_2024_Conference
Published: 2024-08-12 04:02:36

Abstract: The paper mainly investigates the distributed unknown input observer-based global fault-tolerant average consensus control problem for multi-agent systems (MASs). First, a distributed unknown input observer based on relative estimation error is proposed, which can effectively reduce the impact of external disturbances and achieve accurate estimation of the agent states and the faults they suffered. Then, based on the obtained accurate estimations and using the relative estimation error, a global fault-tolerant average consensus controller is proposed. The proposed controller can compensate for the effects of faults and enable the MASs to achieve global average consensus. Finally, simulations are given to verify the effectiveness of the proposed scheme.

Main paper: 
 i. introduction: 
 In the past decades, the study of multi-agent systems (MASs) has been highly emphasized. Due to their extensive civilian and military applications, MASs are subject to stringent performance requirements, such as adaptability, flexibility, and robustness [1] . To meet these requirements, considerable attention has been given to coordination issues in MASs, such as consensus [2] , containment control [3] , and formation control [4] . These coordination mechanisms have been utilized in a wide range of applications such as intelligent transportation systems [5] , drone formation [6] , and smart grids [7] . However, the scalability and complexity of MASs render traditional centralized control schemes insufficient to meet these requirements. Therefore, the exploration of distributed control schemes for MASs is of significant importance.Compared with centralized control schemes, distributed control schemes are more suitable for the coordinated control of autonomous agents in MASs [8] . Currently, the existing control schemes can be categorized into two types based on the structure of MASs: leaderless and leader-follower. The goal of control in leaderless MASs is to reach the consensus among the agents [9] . In contrast, the control objective of leaderfollower MASs is for the follower agents to track the state of the leader [10] . A formation control scheme based on dynamic output feedback was proposed for cases where velocity cannot be measured, ensuring that the agents converge to the desired formation pattern within a finite time [11] . In [12] , an adaptive control strategy with a fully distributed neural network was proposed to ensure that all followers track the leader's state and that the synchronization error remains within a specified range. A formation control method based on constructing a direction alignment law and formation control law using the displacement between agents was proposed to address the direction misalignment issue caused by local reference frames [13] . Overall, distributed control has emerged as a popular research direction, attracting considerable research efforts and yielding abundant results. However, many research outcomes focus solely on the control methods design and consider relatively idealized cases, assuming precise knowledge of system states and the absence of system faults, which diminishes their engineering feasibility.In practical applications, MASs consist of numerous agents distributed across a spatial area, with each agent facing distinct environmental challenges. Agents may encounter uncertainties, such as actuator faults, which can incapacitate the entire control system [14] . To enhance the reliability and safety of the system, it is necessary to implement measures to compensate the adverse influences of faults on the system. In this context, fault-tolerant consensus control has attracted widespread attention as an effective method to compensate for the impact of faults [15] . A virtual actuator framework-based adaptive fault-tolerant control method was proposed to achieve leaderfollower consensus control under time-varying actuator faults [16] . Based on an observer framework, a reliable consensus control design method under stochastic actuator failures was proposed to achieve multi-agent consensus [17] . A distributed fault-tolerant consensus protocol based on a distributed intermediate observer was proposed to achieve finite-time faulttolerant consensus control with enhanced dissipation rate [18] . However, although [18] has addressed the consensus problem of MASs under faults, they have not considered the impact of external disturbances present in practical environments on estimation performance. Fortunately, The unknown input observer, as an effective method based on disturbance decoupling technology to handle external disturbances in estimation error systems, has been widely applied [19] - [20] . Depending on [19] , to address the problem of distributed secure control in MASs, a decentralized unknown input observer-based distributed secure control scheme was proposed [21] .Based on these observations, a distributed unknown input observer and a fault-tolerant average consensus controller based on relative estimation error are proposed in this paper. Major contributions of this work are summarized below:(1) Compared with reference [18] , a control scheme utilizing disturbance decoupling technology to handle external disturbances is proposed. This scheme effectively reduces the adverse influence of disturbances on estimation performance and achieves global average consensus for MASs.(2) Distinguished from [21] , a novel distributed unknown input observer utilizing relative estimation error is proposed to obtain the estimations of the state and the fault experienced by each agent. Specifically, it uses relative estimation error to determine fault estimation, incorporating output estimates rather than just the outputs themselves into the distributed algorithm.The structure is given as follows: Section II presents the problem formulation and give some useful assumptions. In Section III, the main results including distributed unknown input observer-based global fault-tolerant average consensus control scheme and stability analysis are given. Simulations are given in Section IV. Finally, the conclusion of this work is presented in Section V.An undirected graph g is defined as a pair (υ, ϵ, A), where υ = {υ 1 , . . . , υ N } represents a nonempty finite set of nodes, and ϵ ⊆ υ × υ represents a set of edges. An edge (υ i , υ j ) denotes a pair of nodes υ i and υ j . The adjacency matrix, denoted as A = [a ij ] ∈ R N ×N , has elements a ij representing the weight coefficient of the edge (υ i , υ j ), with a ii = 0 and a ij = 1 if (υ i , υ j ) ∈ ϵ. The Laplacian matrix, denoted asL = D -A, is constructed where D = [d ii ] is a diagonal matrix with d ii = N j=1 a ij .Considering a MASs with N agents (i ∈ {1, . . . , N }), and the dynamics of ith agent with actuator faults are denoted as follows:EQUATIONwhere x i (t) ∈ R n , u i (t) ∈ R m , y i (t) ∈ R p representEQUATIONTo facilitate subsequent analysis, the following useful assumptions and lemma are given:EQUATIONAssumption 2. [22] The actuator fault f i (t) is differentiable with respect to time, and its time derivative ḟi (t) belongs to L 2 [0, ∞). Similarly, the external disturbance ω i (t) is bounded and also belongs to L 2 [0, ∞).Lemma 1. [21] For the undirected and connected graph g, one has LM = ML = L.To reconstruct the state and actuator fault of the agent, the relative estimation error-based distributed unknown input observer for agent i is proposed:EQUATIONwhere m i (t), xi (t), fi (t), and ŷi denote the state of unknown input observer, state estimation, actuator fault estimation, and output estimation for agent i, respectively. And η i = y i (t) -ŷi (t) denotes output estimation error, η i -η j denotes the relative estimation error. In addition, the global fault-tolerant average consensus controller for agent i is proposed:EQUATIONThen, for agent i, the state estimation error system can be denoted as below:EQUATIONThe following condition for the matrices Υ and Θ can be obtained based on Assumption 1:Υ Θ I D C 0 = I 0which could be re-written as followsEQUATIONThen, based on the above conditions, one hasEQUATIONand the fault estimation error system can be denoted as:EQUATIONDenote vector e i (t) = [e T xi (t), e T f i (t)], the augmented estimation error system can be obtained:EQUATIONwhereÃ = ΥA ΥB 0 0 , L = L 1 L 2 , C = C 0 , Î = 0 I .ḟ (t) = ḟ1 (t) . . . ḟN (t) T , e(t) = e T 1 (t) . . . e T N (t) T .Then, the estimation error system can be rewritten as:EQUATIONIn addition, for agent i:, the closed-loop system can be denoted as:EQUATIONwhere B = [-BE B].To achieve global average consensus, recall the global average consensus error (2) for agent i, defining vectorx(t) = xT 1 (t) . . . xT N (t) T , x(t) = x T 1 (t) . . . x T N (t) T , ω(t) = ω T 1 (t) . . . ω T N (t) T .Then, the closed-loop system can be rewritten as:EQUATIONSo, for the global average consensus errorEQUATIONwhere M = I N -EQUATIONTheorem 1. For given scalar α > 0, matrices Υ, Θ, L, K, controller feedback gain matrix E, Laplacian matrix L, matrix M, if there exist matrices Q = Q T > 0, P = P T > 0 with appropriate dimensions, such that the following condition holdsEQUATION)whereΦ 1 = He{I N ⊗ (QA + QBE)} + αI N ⊗ Q, Φ 2 = M⊗Q B+L⊗QBK C, Φ 3 = M⊗QD, Φ 4 = He{I N ⊗P Ã- L⊗P L C}+αI N ⊗P , Φ 5 = I N ⊗P Î, Φ 6 = -I N ⊗I nω , Φ 7 = -I N ⊗ I n f ,then the all the signals of the estimation error system (11) and the global average consensus error system (15) are bounded.Proof. The Lyapunov function can be chosen as below:EQUATIONwhere V 1 (t) = xT (t) Qx(t), V 2 (t) = e T (t) P e(t), P = I N ⊗ P , Q = I N ⊗ Q. Take the derivative of the above function, the following can be obtained:EQUATIONAccording to the properties of the Kronecker product, we can get:V (t) ≤e T (t)He{I N ⊗ P Ã -L ⊗ P L C}e(t) + xT (t)He{I N ⊗ (QA + QBE)}x(t) + 2x T (t)(M ⊗ Q B + L ⊗ QBK C)e(t) + 2x T (t)(M ⊗ QD)ω(t) + 2e T (t)(I N ⊗ P Î) ḟ (t).Define ξ(t) = [x T (t), e T (t), ω T (t), ḟ T (t)], if the following linear matrix inequality holdsEQUATIONwhereΦ 1 =He{I N ⊗ (QA + QBE)} + αI N ⊗ Q, Φ 2 =M ⊗ Q B + L ⊗ QBK C, Φ 3 =M ⊗ QD, Φ 4 =He{I N ⊗ P Ã -L ⊗ P L C} + αI N ⊗ P, Φ 5 =I N ⊗ P Î, Φ 6 = -I N ⊗ I nω , Φ 7 = -I N ⊗ I n f ,we haveEQUATIONAs can be seen from the above conclusion, the global average consensus of MASs (1) and the boundedness of the estimation error system (11) can be guaranteed. The proof is completed.Without loss of generality, the gain matrices L, K can be solved by some algebraic operations, and the theorem is given as follows.Theorem 2. For given scalar α > 0, matrices Υ, Θ, controller feedback gain matrix E, Laplacian matrix L, matrix M, if there exist symmetric positive definite matrices S, P , matrices K, P L with appropriate dimensions, such that the following condition holdsEQUATIONwhereΨ 1 = He{I N ⊗ (AS + BES)} + αI N ⊗ S, Ψ 2 = M ⊗ B + L ⊗ BK C, Ψ 3 = M ⊗ D, Ψ 4 = He{I N ⊗ P Ã - L ⊗ P L C} + αI N ⊗ P , Ψ 5 = I N ⊗ P Î, Ψ 6 = -I N ⊗ I nω , Ψ 7 = -I N ⊗ I n f , S = Q -1, then the all the signals of the estimation error system (11) and the global average consensus error system (15) are bounded, and gain matrix L = P -1 P L .Proof. Post-and pre-multiplying (19) by diag{I N ⊗ Q -1 , I N ⊗ I nx+n f , I N ⊗ I nω , I N ⊗ I n f }, the linear matrix inequality ( 21) can be deduced. This proof is completed.In this example, a group of five agents is considered. And the dynamics of the agents are in the form of ẋi (t) =Ax i (t) + B(u i (t) + f i (t)) + Dω i (t)EQUATIONwhich are borrowed from [23] , and parameter matrices are given as belowA = 0 1 0.2 -2 , B = 0 1 , C = 0 1 1 0 , D = 0.1 0.1 .The communication graph considered in this paper is shown below: From Fig. 1 , one hasL =       2 0 -1 -1 0 0 2 0 -1 -1 -1 0 2 -1 0 -1 -1 -1 3 0 0 -1 0 0 1      .To obtain the pre-design unknown input observer gain matrices, the matrix M κ can be selected as follows: As can be seen from Figs. 2-6, the proposed scheme (4) can effectively reduce the influence of external disturbance ω i (t) on the estimation performance and realize accurate estimations of the agent state and fault. Based on the accurate estimations obtained by scheme (4) and the relative estimation error η i -η j , the proposed global fault-tolerant average consensus controller (5) can make the global average consensus errors xi (t) approach zero, as shown in Fig. 7 .EQUATIONIn this paper, the distributed unknown input observer-based global fault-tolerant average consensus control problem for linear MASs has been investigated. First, a distributed unknown input observer based on relative estimation error has been proposed, which can mitigate the impact of external disturbances on estimation performance, thereby achieving accurate estimations of state and fault. Then, based on the obtained estimations and the relative estimation error, a global fault-tolerant average consensus controller has been developed. The proposed scheme can compensate for fault impacts while ensuring global average consensus of the MASs. Finally, simulation experiments have been given to validate the effectiveness of the proposed control scheme.This work was supported in part by the National Natural Science Foundation of China under Grant 51939001, Grant 62273072, and Grant 62203088, in part by the Natural Science Foundation of Sichuan Province under Grant 2022NSFSC0903. (Corresponding author: Tieshan Li.) 
 
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Reviews: review: title: title: Distributed Unknown Input Observer-Based Global Fault-Tolerant Average Consensus Control for Linear Multi-Agent Systems
main_review: main_review: The obtained result is valuable and can be accepted if the following problems can be clarified.
1. The paper should include comparisons against the existing literature to demonstrate its advantages.
2. The paper does not analyze the transient performance of the control system, which is highly influenced by the overshoots introduced by the nonlinear observer.
3. In simulation, this paper should give a result of the trajectories of leaders and followers in the multi-agent systems.
title: title: This article is quite fascinating and of high quality.
main_review: main_review: The paper titled "Distributed Unknown Input Observer-Based Global Fault-Tolerant Average Consensus Control for Linear Multi-Agent Systems" investigates the distributed unknown input observer-based global fault-tolerant average consensus control problem for multi-agent systems. Firstly, a distributed unknown input observer based on relative estimation errors is proposed. This observer reduces the impact of external disturbances on estimation performance, enabling accurate estimation of states and faults. A global fault-tolerant average consensus controller is designed based on the obtained estimates and relative estimation errors. Finally, the effectiveness of the proposed control scheme is validated through simulation experiments. My specific feedback is as follows: 1) Contribution Point 2 is compared with other methods, but it does not explain the advantages of the author's method. 2) What role does disturbance decoupling play in control schemes dealing with external disturbances?
title: title: Accept
main_review: main_review: In this paper, " Distributed Unknown Input Observer-Based Global Fault-Tolerant Average Consensus Control for Linear Multi-Agent Systems", a global fault-tolerant average consistent control method is proposed. Firstly, a distributed unknown input observer based on relative estimation error is proposed. Then, using the relative estimation error, a global fault-tolerant mean consistent controller is proposed. The article has clear logic and organization, but there are still some problems. My specific feedback is as follows :1) What are the advantages of relative estimation error compared with estimation error? 2) What are the advantages of distributed unknown input observers compared to distributed observers?