Li, C., Liu, Y., Gao, M., & Sheng, L. Fault-tolerant formation consensus control for time-varying multi-agent systems with stochastic communication protocol. International Journal of Network Dynamics and Intelligence. 2024, 3(1), 100004. doi:


Fault-tolerant formation consensus control for time-varying multi-agent systems with stochastic communication protocol

Chunyu Li, Yifan Liu, Ming Gao, and Li Sheng *

1 College of Control Science and Engineering, China University of Petroleum (East China), Qingdao 266580, China

* Correspondence: E-mail:


Received: 4 September 2023

Accepted: 8 November 2023

Published: 26 March 2024


Abstract: This paper is concerned with the problem of fault-tolerant formation consensus control for linear time-varying (LTV) multi-agent systems (MASs) with stochastic communication protocol (SCP). The SCP is introduced to schedule the signal transmission, and only one neighbouring agent is allowed to transmit data at one instant. The purpose of this work is to design a fault-tolerant controller for each agent, so that, for all probabilistic scheduling behaviors, MASs can achieve the formation consensus performance. The state and fault are augmented into a new vector, meanwhile, each agent system is written as a singular one and a state observer is designed. By utilizing the estimated information of states and faults, the designed time-varying compensation term can reduce the impacts of unknown external disturbances and faults. Then, a sufficient condition is obtained to guarantee the performance constraint over the finite horizon for closed-loop systems. The parameters of observers and controllers are derived by solving coupled backward recursive Riccati difference equations. Finally, a numerical example is given to validate the effectiveness of the proposed fault-tolerant control scheme.


fault-tolerant formation consensus control multi-agent systems backward recursive Riccati difference equations stochastic communication protocol

1. Introduction

Formation control is one of the core problems in the research field of MASs. In many tasks, multi-agents need to maintain a predetermined geometrical shape, that is, formation, with each other in the process of moving toward a specific target or direction, such as effective search, patrol and exploration [1–4]. MASs have the characteristics of distributed composition and parallel execution of tasks, so multi-agent formation offers better fault tolerance and mission efficiency than a single agent and can acquire surrounding environment information as well as save fuel or energy in some vehicle systems, such as unmanned aerial vehicles (UAVs) and autonomous underwater vehicles (AUVs) [5,6]. In addition, formation control has a good prospect in industry, military and aviation [7], thus it has exerted much fascination on control communities.

Similar to sensor networks, the structure of MASs is more fragile than general centralized systems [8,9], and coupled with the harsh working environment, agents are easily to prone to the faults or cyberattacks [10], which may propagate to the healthy individuals through the special communication topology [11]. Therefore, to ensure the reliability and safety of MASs, it is essential that a robust and reliable fault-tolerant control (FTC) system completes its operation within an acceptable time window after fault occurrence in the presence of disturbances in the system [12]. The overall goal of FTC systems is to accommodate faults in the system components during operation and maintain stability with little or acceptable degradation in the performance levels [13]. Fault-tolerant formation control is usually considered as a special case of FTC, which is mainly divided into two categories characterized by whether there exists the fault detection and isolation (FDI) module. The active FTC system has its advantages in dealing with all kinds of faults, and the system can achieve the best FTC performance by reconstructing the controller [14]. The passive FTC system draws on the idea of robust control, and the designed controller has strong robustness for the fault of a certain severity [15].

By now, the fault-tolerant formation control over MASs has been widely studied and a rich body of algorithms have been proposed, where the observer approach has been extensively applied in the problem of FTC [16–22]. On the basis of the estimated state, fault and uncertainty information, fault-tolerant controllers were developed to achieve the desired formation objectives. For instance, in [16], two kinds of distributed finite-time observers were presented for each follower, thus eliminating the assumption that the leader’s information is derived by all agents directly. The problem of robust relative motion control in a multirobotic system was solved by sliding mode control technique in [17], where the parameters of fault were estimated using a residual-based synchronous fault-detection scheme. In [18], by virtue of introducing fault estimators, the developed controller was effective to compensate for actuator faults, sensor faults and unknown nonlinearity simultaneously. An adaptive fault-tolerant  output feedback control scheme was developed based on unknown input observers in [19,20]. Moreover, the fault-tolerant time-varying formation control problem has caused considerable attention from a variety of communities and a lot of distributed algorithms have been developed in [23–26]. However, only certain types of faults considered in the design stage can be treated by the passive FTC system. In general, from the perspective of performance, active FTC is superior to passive FTC scheme.

Compared with time-invariant MASs, the convergence analysis on the FTC problem for time-varying MASs is more challenging due to the complexity caused by time variance [27]. Consensus problem is a fundamental problem in cooperative control of MASs, to mention a few, the leader-following consensus for time-varying MASs has been adequately discussed in [27–32]. Besides, in [33–36], many distributed algorithms have been developed for consensus problems of nonlinear time-varying MASs models expressed via various representations. However, it should be pointed out that studies on fault-tolerant formation control problem for time-varying MASs has not been properly studied so far.

In a typical network with limited bandwidth, multiple simultaneous transmissions on the network will lead to inevitable data conflicts. One of the effective ways to prevent data conflicts is to arrange the transmission of signals according to specific communication protocols. Communication protocols are extensively applied in MASs including SCP [37,38], try-once-discard (TOD) protocol [39], the round-robin (RR) protocol [40,41], and the random-access (RA), etc. Moreover, the use of communication protocols makes the system information incomplete [42], and hence the development of new control algorithms is of great significance to deal with the incomplete information.

Invoked by the above literature review, in this paper, we aim to deal with the problem of distributed fault-tolerant formation consensus control for a class of LTV MASs. This problem is by no means trivial due to the following identified technical challenges:

1) For a LTV MAS with unknown disturbances and faults, the faults will propagate along the communication topology, resulting in a significant reduction in formation consensus performance. How to design a FTC scheme to improve the system robustness to disturbances and faults?

2) For a LTV MAS under SCP scheduling, how to deal with the difficulty in the analysis of fault-tolerant formation consensus control problem caused by SCP scheduling?

3) How to obtain the state and fault information of the system simultaneously by the observer technique?

In light of those three questions, the main contributions of this paper are highlighted as follows:

1) A novel fault-tolerant formation consensus control scheme is, for the first time, proposed for a LTV MAS subjected to unknown external disturbances, system and sensor faults, directed topologies as well as SCP.

2) Under the premise of allowing a small sacrifice of formation consensus performance, a time-varying fault-tolerant controller is designed for each agent based on estimation information of observers, which can tolerate system and sensor faults without making any assumptions on faults.

3) A sufficient condition for the existence of fault-tolerant  formation consensus controllers is given, and a novel coupled backward recursive RDE method is proposed for time-varying systems with the stochastic parameter matrices.

The rest of this article is organized as follows. In Section II, some basic knowledge of graph theory, the discrete LTV MAS as well as the structures of observers and fault-tolerant formation consensus controllers are introduced, and the problem under the consideration is formulated. In Section III, the observer and controller parameters are obtained by solving backward recursive RDEs. Furthermore, a numerical illustrative example is given in Section IV to show the feasibility and effectiveness of the proposed controller design scheme. Finally, the conclusion is drawn in Section V.

Notations.  is the space of all real numbers.  and  respectively represent the -dimensional Euclidean space and the set of all  real matrices.  denotes the transpose of a matrix  denotes the operation of Kronecker product.  is the Hadamard product of matrices.  is the identity matrix with appropriate dimensions,  denotes the -dimensional identity matrix, and  denotes a zero matrix with appropriate dimensions.  denotes an  dimensional column vector with all ones. The superscript  denotes Moore-Penrose pseudo inverse. diag  stands for a block-diagonal matrix whose diagonal elements are . The notation , where  and  are real symmetric matrices, means that  is positive definite. Prob  refers to the occurrence probability of the event  denotes the mathematical expectation of  represents the Euclidean norm of , and  denotes the Frobenius norm of  denotes the spectral radius of the square matrix  represents the Kronecker delta function with  (if ) and  (otherwise).

2. Problem Formulation and Preliminaries

2.1. Graph Theory

The communication topology among the  agents is described by a weighted directed graph , where  represents the collection of nodes, and  represents the -th node of the graph . In the description of MASs, the nodes represent the location of agents. A directed edge  means agent  can receive information from agent , and  means the set of directed edges, where  denotes the weighted adjacency matrix, which consists of the non-negative elements  satisfying . In this paper, self-edges are not allowed, which means  and  for any  means the neighborhood of agent  is on behalf of the degree matrix with . The Laplacian matrix  consists of the elements , and  for all .

2.2. Problem Formulation

Considering a MAS composed of  identical agents, in order to prevent data collisions, we assume that for agent , only one neighboring agent  is allowed to transmit data to agent  at each transmission instant. Let  represent the neighboring agent which gets the opportunity to transmit data to the agent  at time instant . As discussed in [43],  could be regarded as a sequence of random variables to represent the scheduling behavior of the SCP. All the random variables  are mutually independent, and the probability of  is

where  represents that agent  transmits data to agent  at time instant , and there has been a common assumption that , and .

Remark 1: As discussed in extensive existing literature, the SCP is generally employed to determine which node has the priority of accessing to the communication networks at each transmission instant, which can effectively reduce the communication burden among sensors [44]. The probabilistic accesses of SCP have been identified as a set of random variables affecting the system performance [45]. For the considered SCP,  is a stochastic process, which can randomly generate the number from the set  (e.g. the neighborhood of agent ) subject to the transition probability determined in (1).

Consider the following discrete LTV MAS defined on the finite horizon  and composed of  agents, whose topology is a directed graph, and the dynamics of the -th agent is described by

where  and  represent the system state, the measurement output, the control input, the fault and the control output respectively for the -th agent.  and  are external disturbances, respectively.  satisfies , where  is a non-negative constant.  and  are known and time-varying matrices with appropriate dimensions.

To facilitate the analysis of the problem, the following definitions are needed. Define the reference formation as a time-varying matrix , which is assumed to be known. Subsequently, define formation error and formation consensus error for agent  as  and , respectively.

The premise of achieving fault-tolerant formation consensus control is to obtain the state and the fault information of the system. For the sake of the subsequent analysis, by defining , system (2) can be rewritten as the following singular system:

where . Assume that  is row full rank and  is observable.

In this article, the problem of finite-horizon formation consensus control is studied for the LTV MAS under SCP. In the objective system, exogenous disturbances, sensor and system faults are considered simultaneously. To deal with this problem, an observer is designed for each agent to obtain the system state and fault information, and a new distributed active FTC method is developed via adding a compensation term to alleviate the effects of system faults, time-varying parameters and external disturbances on the formation consensus among the agents. The following introduces the definition of achieving  formation consensus for MAS (2).

Definition 1: Let a disturbance attenuation level  and a positive definite matrix  be given. Considering MAS (2) with a connected directed topology, if

holds, then the MAS is said to satisfy the  consensus performance constraint over the finite horizon , where , and  represents the influence of faults, estimation errors and external disturbances on the formation consensus error system, and its definition will be given later.

2.3. State Observers Design

Assumption 1: At each transmission instant  in the finite horizon , there exists  that would make  holds.

In order to estimate the faults and system states simultaneously, a decentralized state observer is designed for system (3):

where  and  respectively denote the observer state and the estimation of augmented vector  are the observer parameters that need to be computed, and there exists a sequence of matrices  satisfying the following time-varying equations:

where Assumption 1 guarantees that equation (6) must have a non-zero solution .

In this paper, the idea of active fault-tolerant scheme is employed to deal with formation consensus control problem, and the FTC scheme is shown in Figure 1. By utilizing the state and fault information obtained by the estimator, the fault-tolerant formation consensus controllers would be construct for the MAS to compensate for the influence of faults and exogenous disturbances.

Figure 1. Structure of fault-tolerant distributed formation consensus control scheme.

2.4. Formation Consensus Controllers Design

The core idea of the fault-tolerant controller design is to attenuate the influence of external disturbances and faults by adding the compensation term to the formation consensus controller. Under the SCP scheduling, the fault-tolerant formation consensus controller is designed as follows:

where  are the gain matrices to be determined, , and  is the element in the -th row and the -th column of the weighted adjacency matrix .

Defining estimation errors of augmented states , other variables

yields the following compact form of fault-tolerant controller (10):


Remark 2: Obviously,  is a stochastic parameter matrix whose value depends on the sequences . It can be seen that  indicates the -th row-sum of . Bring the SCP in mind, if  is regarded as the element of the new adjacency matrix, then  acts as the adjacency matrix, and  indicates the degree matrix in this case.

To facilitate the subsequent analysis, a relationship between  and one random sequence  is developed by mapping technique.

Lemma 1: The random variable sequences   representing the scheduling behavior of SCP can be mapped to sequence  :

Moreover, if  is given, then the value of  can be obtained by :

Proof: By noting (12), it readily follows that, for sequence  calculated by (12),  holds. Below we continue the proof on that  can be acquired by (13). Since  holds for all , for any given , it follows

It can be concluded that there is a one-to-one correspondence between the variable  and the vector . Hence, the peculiarity of  is determined by the random sequences .

Lemma 2: For any , the occurrence probability of  is

Proof: Because  holds for all , and  is a random variables with independent distribution according to the character of SCP, it yields (14) by noticing (1), where continued multiplication indicates intersection, that is  holds simultaneously for all .

2.5. Closed-Loop System

The following lemma is presented before moving to next part.

Lemma 3: [46] For arbitrary matrices  and , the row-sums of  are the diagonal elements of , i.e.,

For the sake of presentation, let us introduce some notations:

Then, the MAS (3) and the observers (5) can be respectively written in the following compact forms:


where .

Subsequently, the estimation error system can be obtained by (10)

Substituting (6) into (17) and considering (3), it follows

Implementing (5), (7) and (8) into (18), we have

By defining variables , and , the following estimation error system is derived from (19):

One of the research objectives of this paper is to design the observer parameters  so that the estimation error  of the controlled output  satisfies the  performance constraint defined below.

Definition 2: Let a disturbance attenuation level  and a positive definite matrix  be given. Considering LTV MAS (2), for any disturbance sequences , if the following inequality is satisfied, then the estimation error system (20) satisfies the  consensus performance over the finite horizon :

where it can be known from the definition of  that .

By denoting formation consensus error , it can be inferred that

where . Subsequently, it readily follows that the formation consensus error system is

Substituting the controller (11) into (22) yields

where .

Then, (23) can be further written into

In terms of Lemma 3, the following equation can be obtained:

which means all the row-sums of  equal to 0. Therefore, it can be proved that the equation  holds.

On the basis of (24), it can be inferred that

where .

Subject to the distribution of  given by Lemma 2, one obtains

Define , then (26) can also be equivalently expressed as


Summarizing the aforementioned discussion, in this paper, we aim to deal with the problem of designing the controller gains  and  which can ensure that the  performance is achieved for formation consensus error . The core idea is to attenuate the effect of unknown factors including estimation errors, faults, external disturbances and changing formation control targets on achieving formation consensus of MASs. Recalling the form of formation consensus error system (28), we treat  and  as unknown disturbances, and then give the following definition:

Definition 3: For all possible realizations of the random sequence , if inequality

holds over the finite horizon , then  formation consensus is achieved for time-varying MAS (2), where . In virtue of the definition of  in (4), we can get that , where .

3. Main Results

In this section, the design methods of observer and controller parameters are proposed.

3.1. Observer Parameters Design

Lemma 4: [47] Let , and  be known nonzero matrices with appropriate dimensions. The solution  to  is .

Theorem 1: Consider time-varying MAS (3) with the decentralized state observer (5), and let the disturbance attenuation level  and the positive definite matrix  be given. Then, for any disturbance sequence , the augmented state estimation error system (20) satisfies the  performance defined in (21) if there exist solutions  (with condition ) satisfying the recursive RDE:

subject to

Proof: Define , then defining  and considering (20) yields

Adding the zero term  to the right side of (33), one obtains

By applying the completing squares method, (34) can be converted into the following form:

where .

Substituting (30) into (35) and summing up it from 0 to  with respect to  yield

Therefore, recalling (17), one obtains

Noticing condition (31), apparently, the augmented state estimation error system (20) satisfies the  performance in Definition 2.

Subsequently, to develop an approach to solve the observer parameter sequence , we consider the worst situation, that is , where . Then, (20) can be converted into

where . Meanwhile, the following cost functional is constructed to represent the estimation effect of the observer:

where  is typically a positive scalar.

Theorem 2: Consider singular system (3) with observer (5), and let the disturbance attenuation level , scalar  and the positive definite matrix  be given. The state estimation error system (20) satisfies the  performance requirement (14) if there exist solutions  (with condition ) satisfying the recursive RDEs (30) and the following one:

where .

Moreover, the observer parameters  can be calculated by

Proof: Based on Theorem 1, define the function as  . Combining with (38), it is easy to deduce that

Then, add zero term  to the right side of (42), and it can be further written into

It readily follows that, based on completing squares method, (43) is equivalent to

where .

Substituting (40) into (44) and summing up it from 0 to , it follows

Noticing the cost function (39), it follows that

Recalling the definitions of  and , one obtains

Apparently, the observer gain  should make the cost function  as small as possible. A method of calculating  is given below.

Denoting  and considering (41), it can be found that

In light of Lemma 4,  can be obtained from the following equation:

which is the solution of the optimization problem (41). Recalling Theorem 1, (20) satisfies the  performance (14) under the influence of designed , and the proof is complete.

So far, we have designed the observer (5) for the augmented singular system (3), and given the conditions to ensure that the augmented estimation error achieves the  performance given in Definition 2, thus the simultaneous estimation of state and fault vectors is realized. Considering the SCP, based on the above work, the following will deal with the problem of designing controller parameters for the fault-tolerant formation consensus controller as shown in (48).

3.2. Controller Parameters Design

Theorem 3: Consider MAS (3) with the observer (5) and the fault-tolerant controller (10), and let the disturbance attenuation level , the scalar  and the positive definite matrix  be given. The time-varying system (28) satisfies the  performance requirement (29) if there exist solutions  (with condition ) satisfying (30) and the following recursive RDE:

subject to


Proof: Define  and calculate the expectation of  as follows:

Then (55) can be further written into

Adding the zero term  to (56), it follows

Applying the completing squares method, one obtains


Noticing (49), it follows from (29) that

Noticing the definition of , it can be inferred that

Owing to the condition (54), we have , which is equivalent to . Since , it readily follows that , which indicates that . Therefore, under the condition of (53), the  performance constraint defined by (29) is satisfied over the finite horizon . A sufficient condition is established for time-varying MAS (2) achieving formation consensus, and the proof is complete.

Remark 3: Both  and  are  performance constraints over the finite horizon , the difference being that they are designed for different systems. Specifically, if , then the augmented state estimation error system (20) satisfies the  performance constraint. If , the formation consensus error system (28) satisfies the  performance constraint, which is equivalent to MAS achieving fault-tolerant  formation consensus control. Therefore,  is used to prove Theorem 1 for solving the time-varying parameter matrices  of the state observer (16). And  is used to prove Theorem 3 for solving  of the fault-tolerant controller (11).

In Theorem 3, we discuss the condition that the formation consensus error system (28) satisfies the  performance by resorting to backward recursive RDEs. In order to give a solution to the controller gain , rewrite (28) as follows:

where . Consider the particular case, that is, specifying  has a special form , where .

Subsequently, from (61), the formation consensus error system in this case can be obtained:

where .

Define the following cost functional:

where  represents the control performance,  stands for the control effort, and scalar  is selected according to special needs.

Theorem 4: Consider MAS (3) with the observer (5) and controller (10), and let the disturbance attenuation level , the scalar  and the positive definite matrix  be given. The MAS achieves formation consensus if there exist solutions  [with condition ] satisfying (30), (49) and the following recursive RDE:

subject to


Besides, the controller gain can be obtained by the following equation:

Proof: Similar to the proof process of Theorem 2, we define   based on Theorem 3, and recalling (62) yields

Noticing the definitions of  and , add the zero term   to (67), then it can be equivalently expressed as

where .

Subsequently, it follows from (64) and the cost functional (63) that

According to the definition of cost functional in (63), the controller gain  obtained by (66) is the best choice to suppress . Considering that  has a special structure, similar to the calculation method of observer gain , we introduce the notation

It readily follows from Lemma 4 that  can be obtained by

Recalling Theorem 3, it can be seen that under the effect of fault-tolerant controller (11), the MAS can achieve the formation consensus defined in Definition 3.

Based on the above analysis, the finite-horizon fault-tolerant formation consensus control design algorithm can be summarized.

4. Illustrative Example

In this section, a numerical example is provided to validate the effectiveness of the developed scheme of distributed  fault-tolerant formation consensus controller design. Consider a time-varying MAS that consists of five agents, and the dynamics of agents are modelled as (2) with following parameters:

In the simulation, it is assumed that the topology of MAS is denoted by a directed graph  shown in Figure 2 with all the edge weight as 1. Assume that the following faults occur simultaneously on agents 1 and 3:

The disturbance input  are respectively selected as  and , where  are sequences which obey uniform distribution over . In this simulation example, we set the time horizon , scalars , and positive definite matrices , and the initial values  is chosen as . Specify the reference formation matrix by . Assume that the probability  of agent  transmitting data to agent  at the discrete time  is shown in Table 1. In accordance with Algorithm 1, the observer and controller parameters  are respectively listed in Table 2.

Figure 2. Directed communication graph.

Table 1. The probability of transmitting data between agents

0 0.4 0.6 0 0
0.5 0 0.5 0 0
0 0.4 0 0.3 0.3
0.4 0 0.3 0 0.3
0.2 0.3 0 0.5 0


Table 2. Observer and controller parameters

1 2 3 4 5


The simulation results are presented in Figures 3-8, where the red dash dotted lines indicate the discrete time when faults occur, i.e., . The state estimation errors  are shown in Figure 3, where  and  represent two first-order components of , respectively. And Figure 4 depicts the faults , and the fault estimations . In order to demonstrate the effectiveness of the proposed fault-tolerant formation consensus controller (11), we compare it with the formation consensus controller without the time-varying fault compensation term, i.e.,

Figure 3. State estimation error(upside) and(downside),.

and the state trajectories  of agents by using the developed controller  given by (11) and  are shown in Figures 5 and 6.

Obviously, when , both  and  can make MAS achieve the formation consensus control well. Moreover, the corresponding formation consensus errors  are depicted by Figures 7 and 8, which clearly reveal that, the proposed fault-tolerant controller (11), compared with the one without time-varying fault compensation term, can significantly suppress the effect of the faults on  formation consensus. It can be seen that the simulation results verify the validity and feasibility of the proposed fault-tolerant  control scheme.

Figure 4. Fault, and its estimation.

Figure 5. State trajectories, with(upside) and(downside).

Figure 6. State trajectories, with(upside) and(downside).

Figure 7. Formation consensus error, with(upside) and(downside).

Figure 8. Formation consensus error, with(upside) and(downside).

5. Conclusions

In this paper, the problem of finite-horizon fault-tolerant distributed formation consensus control has been addressed for a LTV MAS with SCP. To this end, a singular system has been constructed by augmenting the state and fault of the system into a new vector, and a decentralized observer has been designed. A sufficient condition for the existence of observers has been given to ensure that the state estimation error system satisfies the given  performance constraint. By resorting to describing the scheduling behavior of the SCP by a stochastic variable sequence, the closed-loop MAS has been modeled as a time-varying system with stochastic parameter matrices. Utilizing the obtained state and fault information, fault-tolerant distributed formation consensus controllers have been constructed, which can suppress the adverse effects mainly induced by faults and external disturbances. In addition, a sufficient condition has been given to ensure that the MAS achieves formation consensus, and the parameters of the controller have been obtained by solving two coupled backward recursive RDEs. Finally, an illustrative example has been presented to validate the feasibility and effectiveness of the developed methods.

Author Contributions: Chunyu Li: Draft writing, conceptualization, methodology, simulation. Yifan Liu: Investi-gation, simulation. Ming Gao: Supervision, review & editing. Li Sheng: Funding acquisition, review & editing, supervision.

Funding: This work was supported in part by the National Natural Science Foundation of China under Grant 62073339, Grant 62173343, and Grant 62033008; in part by the Natural Science Foundation of Shandong Province of China under Grant ZR2022ZD34; and in part by the Research Fund for the Taishan Scholar Project of Shandong Province of China.

Data Availability Statement: Not applicable.

Conflicts of Interest: The authors declare no conflict of interest.


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