Fault Tolerant Consensus of Multi-Agent Systems with Linear Dynamics

This paper deals with the consensus problem of linear multi-agent systems with actuator faults. A fault estimator based consensus protocol is provided, together with a convergence analysis. It is shown that the consensus errors of all agents will converge to a small set around the origin, if parameters in the consensus protocol are properly chosen. A numerical example is given to illustrate the effectiveness of the proposed protocol.


Introduction
Recently, distributed consensus problems for multi-agent systems have become a hot research area in the control theory community [1][2][3][4][5][6][7][8][9][10][11][12].This is partly because of their widespread applications in areas such as robots, flocking, unmanned air vehicles, sensor fusion, and microgrids (see [13][14][15][16][17][18] and the references therein).For multi-agent systems, consensus means the group of agents asymptotically agree on certain quantities of interest that depends on the states of all agents [19].In the research of multi-agent systems, the main challenge is how to design simple control rule for simple agents to achieve a prescribed group behavior.
Fault detection and fault accommodation are very important problems for control systems [20][21][22].The fault detection and the fault tolerant consensus problems of multi-agent systems have attracted attention of researchers in the last few years.In [23], the fault detection problem was considered for discrete-time multi-agent systems with first-order dynamics, while the continuous-time second-order multi-agent systems were considered in [24].The fault tolerant consensus problem for first-order multi-agent systems was investigated in [25], under the assumption that the faults are detected in time.The resilient consensus problem was considered in [26], for multi-agent systems with adversary agents.A consensus protocol is provided; under which consensus can be achieved, if the number of adversary agents satisfies a certain condition related to the degree of the communication graph.
The fault tolerant consensus problem for high-order linear multi-agent systems has not been considered in the literature, which motivated the work in this paper.In this paper, consensus problems will be considered for linear multi-agent systems with actuator faults.A fault estimator is provided, based on which a consensus protocol is derived.It is proved that consensus error can converge to a small set around the origin, if parameters in the fault estimator and the consensus protocol are properly chosen.The rest of the paper is organized as follows.Section 2 formulates the fault tolerant consensus problem of the multi-agent systems with linear dynamics.The main results are presented in Section 3. A numerical example is given in Section 4 to illustrate the proposed results and the paper is concluded in Section 5.
Notations.Throughout this paper, matrix  > 0 means that  is symmetric positive definite.Consider

Problem Formulation
Consider a team of  agents with the following linear dynamics: where   () ∈ R  is the state of agent ,   () ∈ R  is the control input, and   () ∈ R  is the output.  () ∈ R  denotes the actuator fault of agent .If   () ≡ 0, then no actuator fault occurs at agent .In this paper, we have the following assumptions.
Remark 3. Assumption 2 means that the state of the agent can be constructed from the output, and there is no subsystem decoupled from the faults in the linear description [20].Under Assumption 2, there exist a symmetric positive definite  and matrices  and  such that The communication topology among the  agents can be represented by an undirected graph G = (V, E), where V = {1, . . ., } is the node set and E ∈ V × V is the edge set.An edge (, ) ∈ E if agent  and agent  can access information from each other.An undirected path is a sequence of undirected edges of the form ( 1 ,  2 ), ( 2 ,  3 ), . .., where   ∈ V.An undirected graph is connected if for any ,  ∈ V there exists a path between them.The neighbor set N  of agent  is defined as N  ≜ { | (, ) ∈ E}.The adjacency matrix A ≜ [  ] ∈ R × is defined as   =   > 0 if (, ) ∈ E and   = 0 otherwise.The Laplacian matrix L is defined as L ≜  − , where  = diag{∑  =1  1 , . . ., ∑  =1   }.It is well known that if the communication topology is connected, then L has a simple zero eigenvalue and  − 1 nonnegative eigenvalues  2 ≤  3 ≤ ⋅ ⋅ ⋅ ≤   .Definition 4. We say algorithm   () asymptotically solves the consensus problem if   () − (∑  =1   ())/ → 0 as  → ∞, for any  = 1, . . ., .
The objective of this paper is to derive a consensus protocol   () under which consensus can be achieved even if some agents are subject to actuator faults.

Main Results
This section gives the fault tolerant consensus protocol for multi-agent systems described in the last section.Before giving the consensus protocol, we give the fault estimator first.
Lemma 5. Define   () ≜   () − f ().Under Assumptions 1 and 2, estimator (4) guarantees that (  ,   ) converge exponentially to the following set: where Proof.The proof is similar to the proof of Theorem 1 in [20] and hence is omitted here.

𝑇
. Equation ( 11) can be written in a compact form as where It can be seen that  =   − 1  1  /,  2 = , 1    = 0   , 1  = 0  , and L = L = L. Before giving the main results of this paper, the following lemma is needed.
Lemma 7 (see [10]).Let  and L be matrices previously defined; the following statements hold.
(1) The eigenvalues of  are 1 with multiplicity  − 1 and 0 with multiplicity 1.The vectors 1   and 1  are the left and right eigenvectors of  associated with zero eigenvalue, respectively.
(2) There exists an orthogonal matrix  ∈ R × with last column 1  /√, such that Next, we give the main results of this paper.
Theorem 8. Suppose the undirected graph G is connected, and the nonzero eigenvalues of L are  2 ≤ ⋅ ⋅ ⋅ ≤   .Using protocol (6), with the fault estimator (4), consensus errors {  () :  = 1, . . ., } will converge to a small ball around the origin if there exists a symmetric positive definite matrix  such that the following LMI holds: and  is chosen as  = −   −1 .
Remark 9. Theorem 8 shows that   () can converge to a small ball surrounding the origin.From the proof, we know that this set is determined by  and Γ.Since  and Γ can be selected freely, this ball can be chosen arbitrarily small.However, if  is chosen too small,   () may converge very slowly.To overcome this problem, dynamically changing  and Γ can be used in the practice.This is out of the scope of this paper and will be considered in our future research.

A Numerical Example
Consider a multi-agent system consisting of 4 agents with and parameters in the estimator (4) can be chosen as  = 1, Γ = 1, and Figures 2 and 3 show, respectively, the position and velocity responses of nodes 1-4.It can be seen that consensus can be achieved in this case.

Conclusions and Future Work
The fault-tolerant consensus problem for multi-agent systems with actuator faults was considered.A fault estimator based consensus protocol is provided, together with a sufficient condition under which the consensus can be achieved.It is proved that consensus errors of all agents can converge to a small set around the origin.The numerical example confirmed the proposed theoretical results.In practice, many systems have stochastic Markovian jumping dynamics [27][28][29][30][31][32].
Future research efforts will be devoted to the fault tolerant consensus problem of multi-agent systems with stochastic Markovian jumping dynamics.