Observer-Based Distributed Fault Detection for Heterogeneous Multi-Agent Systems

: This paper solves the distributed fault detection (FD) problem for heterogeneous multi-agent systems (MAS). For a heterogeneous MAS, we adopt a distributed control law to realise cooperative output regulation (COR) when no fault occurs in the MAS, and propose a state-feedback-based FD scheme, where the adopted distributed control law and proposed FD scheme all utilise state information. Furthermore, we consider the condition that state information is unmeasurable, the output-feedback-based distributed FD scheme is proposed, and the adopted distributed control law also utilises measurement output. Finally, two numerical examples are utilised to verify that the proposed distributed FD schemes could locate and remove the faulty agent in time.


Introduction
Recently, a large amount of literature on cooperative control of multi-agent systems (MAS) has emerged, which investigate this problem from different aspects, such as event-triggered and finite-time cooperative control [1,2]. Besides homogeneous MAS, research on heterogeneous MAS is also significant. As an effective method to realise cooperative control of heterogeneous MAS, for instance, a network of unmanned aerial vehicles (UAVs) with different dynamics, cooperative output regulation (COR) has attracted intensive research attention during the past decade [3][4][5][6], which was based on output regulation theory, where the influence of mismatched disturbance generated by an exosystem could be completely rejected via converting mismatched forms into matched forms [7]. Specifically, H. Basu and S. Y. Yoon considered the condition that only partial information of an exosystem matrix was accessible to each agent, where a distinct estimator network was proposed to cooperatively estimate the value of the exosystem state [8]. As some agents may destroy other healthy agents due to the influence of unexpected faults, security operation of heterogeneous MAS has attracted some researchers' attention.
Existing security operation schemes of heterogeneous MAS are passive, where the fault tolerant COR is ensured by the designed passive fault tolerant controllers in each agent. In [9], Deng et al. designed a distributed adaptive fault tolerant control law to attenuate partial loss of actuator effectiveness faults. Furthermore, they considered the condition that actuators suffered from both partial loss of effectiveness faults and stuck faults [10,11]. Besides designing passive fault tolerant control laws for each agent, detecting and removing faulty agents is the other effective method to ensure the security operation of MAS. Hence, some literature on FD schemes for MAS has emerged during the past decade. The basic idea is to design an additional FD algorithm for MAS, and run the FD and control algorithms simultaneously; the FD algorithm will locate and remove faulty agents in and 3.2. Section 4 gives two simulation examples to verify that the proposed FD schemes are effective. Finally, Section 5 gives conclusions and future directions. Notation: Some standard notation will be adopted in this paper. C and R n denote the set of complex numbers and n-dimensional Euclidean space, respectively. I N and I denote an identity matrix with dimension N and appropriate dimension, respectively. ⊗ denotes the Kronecker product. e i represents a column with only one nonzero entry '1', which locates in the i-th row. Re(ζ) represents the real part of ζ, where ζ ∈ C. diag(A 1 , A 2 , · · · , A N ) represents a block-diagonal matrix with matrices A i , i = 1, 2, · · · , N. λ i (A) denotes the i-th eigenvalue of A. A > 0 means that A is positive definite. '!' denotes the factorial of a non-negative integer, and C b a = a! (a−b)!b! , where a, b are non-negative integers, and b ≤ a. The superscript 'T' represents the transpose of a matrix.

Graph Theory
. What is more, define the in-degree of each agent as d i = ∑ N j=0 a ij ≥ 1, and define the Laplacian matrix associated with For the convenience of analysis, denote L s as the Laplacian matrix associated with G s , and define A 0 = diag(a 10 , a 20 , · · · , a N0 ) and H = L s + A 0 . Furthermore, define N i = {V j ∈ V (G s ) : (i, j) ∈ E (G s ), i = j} as the neighbour set of node V i ∈ V (G s ) in G s , and defineN i = {i} ∪ N i , where |N i | is the cardinality ofN i , and {ī 1 , · · · ,ī |N i | } are sequence numbers of nodes inN i from small to large. Lemma 1 ([26]). If the subgraph G s is undirected, and each follower agent has paths to the leader in the graph G, H is positive definite.

Unknown Input Observer
Consider the following system with unknown input: where ξ ∈ R n is the state. u ∈ R r and w ∈ R s are the known and unknown inputs, and W and Y represent their input channels, respectively. y ∈ R m is the measurement output, and H represents the measurement matrix. In order to estimate the actual state of System (1), the following observer is designed.
whereξ ∈ R n and z ∈ R n are the estimated state and observer's state, respectively. Parameter matrices of observer (2) need to be designed to make the state estimation error be not influenced by unknown input w, where the design method is shown as follows: then the state estimation error dynamics is shown as follows: where e = ξ −ξ. If the designed R 1 makes G Hurwitz stable, e will converge to zero asymptotically. Observer (2) is usually called unknown input observer (UIO). Lemma 2 gives the existence conditions of a UIO.

Remark 1.
The above two conditions guarantee existence of H and R 1 in (3), respectively, where R 1 makes G Hurwitz stable.

Problem Formulation
The considered heterogeneous MAS consists of the following N agents: where x i ∈ R n i , e i ∈ R p i , y mi ∈ R p m , u i ∈ R m i and f i ∈ R l i are the state, tracking error, measurement output, control input and fault signal of the i-th agent, respectively. Columns of B f i are linearly independent, and (A i , B i , C mi ) are stabilisable and detectable. v ∈ R q is the reference input to be tracked or the disturbance to be rejected, which is assumed to be generated by the following exosystem: In this paper, the leader labelled as 0 could represent the exosystem. System (5) could be seen as follower agents, where only some follower agents of the MAS could utilise v directly, i.e., the leader is their neighbour in the graph G, and the remaining follower agents just have paths to the leader. The above two facts mean that the leader has paths to all follower agents in the graph G.
System (5) also needs to satisfy the following assumptions.
Assumption 2. The following equations have solution pairs (X i , U i ), respectively.
For System (5), a state-feedback-based distributed control law is proposed in [1].
where η i is a dynamic compensator, c is a positive scalar to design, K 1i and K 2i are parameter matrices to design. Let K 2i be as follows: are Hurwitz stable, c > 0, and the leader has paths to all follower agents, control law (7) will realise COR when there exists no faulty agent [3], i.e., tracking errors e i converge to zero asymptotically. However, the given exosystem under Assumption 1 just has an unforced purely oscillatory solution, which cannot include some kinds of solutions such as those of a damped differential system or the solution of a forced differential system or dynamic system, i.e., some kinds of practical signals cannot be generated by the given exosystem, where the application prospect is limited. This is due to the fact that the given exosystem needs to ensure that matrix Equation (6) has solutions, and the following Theorems 2 and 3 hold. Furthermore, COR will not realise if some agent suffers from fault signals. Hence, there exists a need to detect the faulty agent.
This paper aims at adopting a distributed control law for the MAS, and designing distributed observers in some agents to detect the possibly faulty agent. Running the observers and control law simultaneously, the observers could detect the faulty agent if there occurs a fault, and the distributed control law could also realise COR if no fault occurs. It is worth indicating that if the solution pair (X i , U i ) is not unique, we just need to select one of them to design the control law (7), and design FD observers (15) and (29), where the parameters of FD observers contain the selected solution pairs (X i , U i ). What is more, the designed control law (7) with any chosen solution pair (X i , U i ) will realise consensus control when there exists no faulty agent, and existence of the designed FD observers will also hold under any chosen solution pair (X i , U i ).

State-Feedback-Based Distributed FD
Define y i as the observer feedback information that agent i's observers for FD could utilise, which is designed to contain state x i and compensator state η i of agent i, as well as agent i's neighbours', where agent i's neighbours' compensator state η i are also utilised in control law (7).
Substitute (7) into (5), we obtaiṅ and according to Assumption 2, we could obatiṅ Remark 3. Obviously, it is impossible to design observers for System (10) decoupled from e ηi (t) if agent i could not obtain exogenous signal v(t) directly. What is more, the evolution of e ηi (t) is influenced by e ηj (t), j ∈ N i , i.e., ||e ηi (t)|| may not be monotonic for t ≥ 0, therefore, the bound of ||e ηi (t)|| is unknown based only on local information, which means that isolation thresholds cannot be selected [28]. Therefore, observers for FD need to be designed for the following closed-loop System (12), where utilisation of exogenous signal v is avoided. Denote we obtain System (12) for agent i.
and (8) can be rewritten as follows: Combine (12) and (13), we obtain |N i | systems as follows: R 1 iī k is a matrix to make G iī k Hurwitz stable.
Then, construct |N i | residual generators r iī k as follows: Theorem 1. Suppose that observers (15) exist for each agent inN i , and Assumption 3 is satisfied.
If lim t→∞ r iī k (t) = 0, and lim t→∞ r ij (t) = 0, where j ∈N i and j =ī k , there occurs a fault in agent i k .
Proof of Theorem 1. The UIO error (residual) dynamics are shown as follows: where e iī k = ψ i −ψ iī k = T iī k ψ i − z iī k . Therefore, residual generators will only be influenced by f −ik (t). Assumption 3 indicates thatf −iī k (t) = 0 if f¯i k (t) = 0, therefore, lim t→∞ r iī k (t) = 0 and lim t→∞ r ij (t) = 0, j =ī k indicates that agentī k is faulty. Remark 4. Assumption 3 could be relaxed if the number of faulty agents is known, and all the faulty agents locate in the same setN i . Assume the number of faulty agents as h < |N i |, then construct C h |N i | systems as follows: combinations, as well asb f ik .B f −ik is the surplus ofB f i after deletingb f ik . Design C h |N i | observers and residual generators in agent i for above systems following the same steps. Finally, the only residual generator containing all the faults converges to zero. For h = |N i | − 1, if all the residual generators will not converge to zero, there exists no healthy agent inN i .

Theorem 2.
If Assumption 1 is satisfied, c > 0, and each follower agent has paths to the leader in the graph G, there exists an observer (15) for anyī k ∈N i .
Proof of Theorem 2. Forī k ∈N i , Lemma 2 indicates that there exists an observer (15) if rank(C ib f ik ) = rank(b f ik ), and M iī k = sI − A ib f ik C i 0 is of full column rank for ∀s ∈ C, Re(s) ≥ 0.
Firstly, we verify the first condition, where It is easy to verify that coulmns of C ib f ik are linearly independent, where the first condition holds. Then, we need to verify that M iī k is of full column rank for ∀s ∈ C, Re(s) ≥ 0, which is shown as follows: Let H is positive definite according to Lemma 1, then, there exists a Y 1 to make Y 1 HY −1 where Λ 2 is an upper triangular matrix with diagonal elements {λ 1 (S), · · · , λ q (S)} [29]. Then, we obtain eigenvelus of I N ⊗ S − cH ⊗ I q via pre-multiplying by which are shown as follows: According to Assumption 1, we have Re[λ i (S)] = 0, i.e., Re[λ i (I N ⊗ S − cH ⊗ I q )] < 0. Hence, columns of sI − (I N ⊗ S − cH ⊗ I q ) are linearly independent for ∀s ∈ C, Re(s) ≥ 0, i.e., ν 2 = 0, as well as ν 1 . As columns of B fī k are linearly independent, we have ν 3 = 0, therefore, (19) is of full column rank for ∀s ∈ C, Re(s) ≥ 0. Above conditions guarantee existence of observers (15) for any node in N i .

Remark 5.
If compensator state η¯i 1 , η¯i 2 , · · · , η¯i |N i | in y i are replaced by exosystem state v, existence of the observers for FD is still ensured. In practical operation, only partial follower agents could obtain exosystem state v, but each follower agent has access to itself and its neighbours' compensator state, i.e., the estimate of v, therefore, feedback information y i is designed as (8).

Remark 6. Establishment of Theorem 1 depends on Assumption 3 and existence of observers for FD,
where existence of the observers for FD is proved in Theorem 2. Then, Theorems 1 and 2 together guarantee that the faulty agent will be detected.

Remark 7.
Existence of FD observers just requires c > 0, which is the same as realisation conditions of COR, i.e., control law (7) and observers (15) are able to run simultaneously. What is more, if the remaining follower agents still have paths to the leader after removing the faulty agent, the remaining follower agents could still realise COR.
According to residual dynamics (17), residual generators will not converge to zero until time approaches infinity, therefore, appropriate isolation thresholds need to be set [21]. Then, the following location algorithm is given to locate the possibly faulty agent inN i .

Remark 8.
A simple selection method of isolation thresholds Θ ik is shown as follows. Assume that the initial error e iī k (0) ≤ is bounded, i.e., ||e iī k (0)|| ≤ . Then, one can compute the threshold by considering an upper estimate of the error expression (17): , which can be obtained by Jordan decomposition, is such that ||e What is more, selection of isolation thresholds is also related with trade-offs between false alarm and misdetection rate, among others [30], where more details could be found in [24] and references there-in.

Output-Feedback-Based Distributed FD
As state information is difficult to obtain, an output-feedback-based control law is designed here, as well as an output-feedback-based distributed FD scheme, where matrices A i , B f i and C mi need to satisfy the following assumption. Inspired by Reference [2], we design a dynamic output-feedback-based distributed control law: where c and K 1i , K 2i and L i are the scalar and parameter matrices to design, respectively. Let K 2i be designed as follows: X i and U i are still determined by the matrix equations introduced in Assumption 2.

Remark 9.
Control law (23) consists of two observers, i.e., the compensator η i and state observerx i , which aim at estimating exosystem and system states, respectively. y i is designed as follows: which contains measurement output and state estimate of agents inN i .
Substitute (23) into (5), we havė and according to Assumption 2, we could obatin (24): Denote then we obtain the following MAS closed-loop system: and (24) could be changed to the following form: Combine (26) and (27), we obtain |N i | systems as follows: Follow the same steps in Section 2.1, |N i | observers are designed in agent i.
where the design of parameter matrices is the same as (16). Next we just need to prove existence of observers for FD.

Remark 10.
In comparison with y i in (8), (22) contains measurement output and state estimate of one agent's and its neighbours'. In the following, Theorem 3 will prove that existence of the above observers is still ensured, where y i does not contain state information. The objective proposed in Section 1 will realise under the condition that state information is unmeasurable.
Theorem 3. If Assumptions 1 and 4 are satisfied, c > 0, A¯i k + B¯i k K 1ī k are Hurwitz stable, k = 1, · · · , |N i |, and each follower agent has paths to the leader in the graph G, there exists an observer (29) for any agentī k ∈N i .
Proof of Theorem 3. Forī k ∈N i , Lemma 2 indicates that there exists an observer (29) Firstly, we verify the first condition, for k = 1, we have where C mī 1 B fī 1 is of full column rank according to Assumption 4. Furthermore, for k = 2, · · · , |N i |, columns of C ib f iī k are always linearly independent, where the first condition holds.
Then, we need to verify that M ik is of full column rank for ∀s ∈ C, Re(s) ≥ 0, which is shown as follows:  It has been proved in Theorem 2 that columns of sI − (I N ⊗ S − cH ⊗ I q ) are linearly independent for ∀s ∈ C, Re(s) ≥ 0, we have ν 3 = 0. Therefore, ν 2 = ν 4 = 0 if (33) is of full column rank for ∀s ∈ C, Re(s) ≥ 0, where (33) could be transformed to (34) through appropriate row and column transformation, the rank of which is equal to (33),   sI − diag(A¯i 1 , · · · , A¯i according to Assumption 4, (34) is of full column rank for ∀s ∈ C, Re(s) ≥ 0, i.e., ν 2 = ν 4 = 0.

Remark 11.
In comparison with the output-feedback-based COR problem in Reference [4], where the authors design two control laws, which correspond to two kinds of follower agents in the MAS, the first kind are called informed agents, i.e., measurement output y mi contains exogenous signal v, then v is detectable from y mi . The second kind are called uninformed agents, where measurement output y mi does not contain exogenous signal v. As Section 2 just considers that all the follower agents are uninformed, i.e., a special case of literature [2], a single control law (21) will realise COR if no fault occurs, where c > 0, and the designed K 1i and L i make A i + B i K 1i and A i + L i C mi Hurwitz stable, which is not contradicted with the existence conditions of observers (29), i.e., control law (21) and observers (29) could run simultaneously.

Remark 12.
The faulty agent location algorithm for the output-feedback-based distributed FD scheme is the same as Algorithm 1 and omitted here.

Simulation Example
In this section, we will provide an example to illustrate the effectiveness of the two proposed distributed FD schemes.
Consider the following agent dynamics in Reference [31]: Assume the communication graph G among all the follower agents and the exosystem can be described by Figure 1, where node 0 represents the exosystem and the other nodes represent four follower agents, where agent 2 is assumed as faulty, it can be observed that only agent 3 can access the exosystem state v, v is generated by the exosystemv = Sv, where Then, it can be verified that Assumptions 1-3 are satisfied, (A i , B i ) are stabilisable, (A i , C mi ) are detectable, and four follower agents have paths to the leader in the graph G. What is more, which satisfy Assumption 4. The solutions of (6) are given by Choose the coupling gain coefficient c, parameter matrices of control laws as Theorems 2 and 3, which correspond to the state-feedback-based and output-feedback-based distributed FD schemes, respectively, as well as parameter matrices of the observers for FD.
For the condition that state information is measurable, run control law (7) and observers (15) for FD.
The fault is assumed to be a constant and occur in the first element of x 2 , i.e., f 2 = 3.5, which occurs after 25 s. Residual generators r 11 , r 12 and r 14 in agent 1 are shown as Figure 2, which are represented by 2-norm type. According to Remark 8, we could choose a positive scalar , which is larger than ||e 21 (0)||, ||e 22 (0)|| and ||e 24 (0)||, and e σt , which is larger than ||e G 21 t ||, ||e G 22 t || and ||e G 24 t ||, combined with ||C 2 || = 1.64. Then, the isolation threshold could be set as 1.64 · e σt .
It is shown that above residuals converge to little enough values before the fault occurs, then r 11 and r 14 fluctuate when the fault occurs at the time of 25 s. However, r 12 does not fluctuate as r 11 and r 14 when the fault occurs, then according to Algorithm 1, the fault occurs in agent 2. For the condition that state information is unmeasurable, run control law (21) and observers (29) for FD.
The fault signal is the same as the state feedback case, as well as the isolation threshold selection method. Simulation results of residual generators r 11 , r 12 and r 14 in agent 1 are shown as Figure 3, and still represented by 2-norm type. It is shown that r 12 remains converging after the fault occurs, where r 11 and r 14 fluctuate; therefore, the fault occurs in agent 2.

Conclusions
Two distributed FD schemes for heterogeneous MAS are proposed in this paper. A state-feedback-based distributed control law is adopted to realise COR when there does not occur any fault, where state-feedback-based observers for FD are designed, and existence of the designed observers is also proved. Furthermore, we consider the condition that state information is unmeasurable, an output-feedback-based distributed control law is designed, as well as output-feedback-based observers for FD, where existence of the above observers are ensured through appropriately designed feedback information. Finally, two simulation examples verify the effectiveness of the proposed FD schemes. However, the above distributed control laws and FD schemes require one agent to obtain information from its neighbours, such as state, compensator state, measurement output and state estimate, which will exert a heavy burden on communication networks.
Possible future work includes considering the condition that agents suffer from disturbances and faults simultaneously, as well as reducing communication burden.