Mathematical Problems in Engineering

Volume 2013 (2013), Article ID 979087, 12 pages

http://dx.doi.org/10.1155/2013/979087

Research Article

Distributed Robust Consensus Control of Multiagent Systems with Communication Errors Using Dynamic Output Feedback Protocol

Xi Yang and Jinzhi Wang

State Key Laboratory for Turbulence and Complex System, Department of Mechanics and Aerospace Engineering, College of Engineering, Peking University, Beijing 100871, China

Received 17 April 2013; Accepted 8 May 2013

Academic Editor: Guanghui Wen

Copyright © 2013 Xi Yang and Jinzhi Wang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

This paper studies robust consensus problem for multiagent systems modeled by an identical linear time-invariant system under a fixed communication topology. Communication errors in the transferred data are considered, and only the relative output information between each agent and its neighbors is available. A distributed dynamic output feedback protocol is proposed, and sufficient conditions for reaching consensus with a prescribed performance are presented. Numerical examples are given to illustrate the theoretical results.

1. Introduction

Consensus problem of multiagent systems has been a popular subject in system and control theory due to its widespread applications such as satellite formation flying, cooperative unmanned air vehicles, and mobile robots [1–3]. The study of consensus problem focuses on designing a distributed protocol using information which can only be obtained and shared locally to ensure that the resulting closed-loop system has the desired characteristics. A number of solutions that are based on relative states between each agent and its neighbors to the consensus problem have been proposed up to now. The theoretical framework of solving consensus for multiagent system was suggested by [4], providing the convergence analysis of a consensus protocol for a network of single integrators with directed fixed/switching topologies. Later, under different cases of communication topologies such as fixed, switching, and with communication delays, many different types of protocols have been proposed for different types of agent dynamics to reach global asymptotical consensus [2, 3, 5–16].

Recently, solving consensus problem for the multiagent systems by using output information has attracted particular attention due to its theoretical significance and wide applications. Reference [17] constructed a dynamic output feedback protocol based on a observer for the synchronization of a network of identical linear state space models under a possibly time-varying and directed interconnection, where each agent needs to obtain all the observer’s state information of its neighbors. Based on the low gain approach, [18] proposed a consensus protocol which only used the relative outputs for identical linear dynamics with fixed directed communication topologies. Consensus problem with external disturbance under switching undirected communication topologies was studied by [19], where a dynamic output feedback protocol was proposed for subjecting the external disturbances. Reference [20] studied the output consensus problem for a class of heterogeneous uncertain linear SISO multiagent systems, where each agent’s output information and the relative outputs with its neighbors were used to design the controller. Reference [21] designed robust static output feedback controllers to achieve consensus for undirected networks of heterogeneous agents modeled as nonlinear systems of relative degree two.

It can be seen that there is a common assumption in the literatures mentioned above that each agent can receive accurate measurements of relative states or outputs between its neighbors and itself all the time. However, in some practical situations, agents cannot perfectly sense their neighbors due to the existence of sensor failures or some other communication constraints. In view of this, we consider the consensus problems for the multiagent systems with communication errors. It is required to point out that the measurement for communication errors we considered is limited to some errors in the transferred data not including loss of communication. The robustness analysis of first-/second-order leader-follower consensus with communication errors is studied by [22, 23]. Some robustness issues for systems with external disturbances or model uncertainties are investigated by some other researchers [19, 24–28], which are different from the robust consensus problem stated in this paper.

Motivated by the above-mentioned works, we study the consensus problem for linear multiagent systems to attenuate the communication errors by using dynamic output feedback controller. The agent dynamics considered here are general stabilizable and detectable linear systems, and a dynamic consensus protocol is proposed which uses only the relative output information between each agent and its neighbors. The main contributions of this paper can be summarized as two aspects. Firstly, in order to describe the effects of communication errors on consensus, a concept called consensus with performance is introduced which can characterize the effects of communication errors on the difference between the state of each agent and the average of states of all agents. The problem of consensus with performance is transmitted into an control problem of another reduced-order system. It is shown that consensus with performance can be achieved if there exists a common dynamic output feedback controller which can be realized by solving problem for linear dynamic systems simultaneously, where is the number of agents. Secondly, in terms of the linear systems, a sufficient condition based on linear matrix inequalities for the existence of the controller is provided, and the approach to construct the corresponding controller is given.

The rest of this paper is organized as follows. Section 2 introduces basic notations and reviews some useful results on graph theory and robust control theory. Section 3 formulates the problem and conditions for reaching consensus with performances that are derived. The existence for a dynamic output feedback protocol and a method to construct such controller are proposed in Section 4. Numerical simulations are provided in Section 5. Section 6 concludes the paper.

Notations. Let and be the set of real matrices and complex matrices, respectively. Matrices, if not explicitly stated, are assumed to have compatible dimensions. and are the identity matrix and the zero matrix, respectively. For a matrix , is the induced 2-norm of the vector norm, and . The notation is the maximal singular value of matrix . Notations , , and represent the transpose, the inverse, and the complex conjugate transpose of matrix , respectively. Let and be the image space and kernel of . () means that the matrix is negative (positive) definite. is a subspace of spanned by , where . The notation represents the Kronecker product. For a vector is the Euclidean norm. denotes the column vector whose elements are all ones. The space of piecewise continuous functions in that are square integrable over is denoted by , for any , and its normalized energy is defined by . Let be a state space realization of .

2. Preliminaries

2.1. Graph Theory

Directed graphs are used to model the information interaction among agents. Let be a directed weighted graph, where is the node set, is the edge set, and is a weighted adjacency matrix with nonnegative elements . An edge of is denoted by which means that agent can directly get information from agent . if and only if , otherwise . If , then is said to be an undirected graph. In this paper, we assume that there are no self-cycles in ; that is, , . The in-degree and out-degree of the th agent are, respectively, defined as and . Let . Correspondingly, the Laplacian matrix of graph is denoted by , where is a diagonal matrix with .

A sequence of edges is called a directed path from node to node . is called a strongly connected digraph if for any , there is a directed path from to . has a directed spanning tree if there exists a node (a root) such that all other nodes can be linked to via a directed path. A directed graph is called balanced if for all .

Below are well-known results for the Laplacian matrix.

Lemma 1 (see [3]). The Laplacian matrix of a directed graph has at least one zero eigenvalue with an associated eigenvector .

Lemma 2 (see [3]). The Laplacian matrix of a directed graph has a simple zero eigenvalue with an associated eigenvector , and all of the other eigenvalues have positive real parts if and only if the directed graph has a directed spanning tree.

Lemma 3 (see [24]). Let be the Laplacian matrix of a directed graph , then there exists an orthogonal matrix such that

Furthermore, if is a balanced graph, then

2.2. Robust Control Theory

Consider that the th-order linear time-invariant (LTI) system is described as follows:

where is the state, is the external disturbance, is the control input, is the regulated output, is the measured output, and , , , and , for , are known real constant matrices of appropriate dimensions. Without loss of generality, we assume that , is stabilizable and is detectable.

The th-order dynamic output feedback (DOF) controller is described as follows:

where is the controller state and , , , and are constant matrices with appropriate dimensions.

Let be the transfer function from to of the closed-loop system obtained from (3) and (4), where

The problem for the given LTI system (3) is to find a DOF controller (4) such that the closed-loop system is internally stable and for some constant .

To facilitate the consensus protocol design and stability analysis, several results of the problem are recalled as follows.

Lemma 4 (see [29]). Given , there exists a DOF controller (4) which can solve the problem for the LTI system (3) if and only if there exist symmetric matrices and such that

where and are full-rank matrices whose images satisfy

As the results shown by [29], the DOF controller (4) solving the problem for the LTI system (3) can be constructed as follows.

(i)Find and which satisfy Lemma 4.(ii)Let , where satisfying .(iii)Solve the the following inequality:

For a feasible solution , where

and are defined by (5). The solution provides the state space realization for a feasible controller (4) which can solve problem for system (3).

Lemma 5 (see [30]). Let , with is Hurwitz stable, and

where . Then, the following conditions are equivalent:

(1),(2) and have no eigenvalues on the imaginary axis,(3)there exists a matrix such that

3. Consensus Problem

Consider a multiagent system consisting of identical agents with linear dynamics described by

where is the state, is the control input, is the measured output, and , , and are constant matrices with compatible dimensions. It is assumed that is stabilizable and is detectable, and without loss of generality, is of full column rank. We say that the control input solves the consensus problem for the multiagent system (12) if the states of the agents satisfy

for any initial states.

Assume that the communication topology among the agents is represented by a fixed directed graph . Based on the relative output information between the agents, the following dynamic output feedback (DOF) control protocol is used by [18]:

where , is a preassigned dimension of the coordinating law, and is the element of the corresponding adjacency matrix . The system matrix

of the DOF control protocol (14) need to be designed to make the multiagent system (12) achieve consensus. A general method for constructing the system matrix was presented by [18].

However, if there exist communication errors between the th agent and the th agent, , then the performance of consensus will be affect by these errors, as illustrated by the example given below.

Example 6. We consider double-integrator systems given by

where . Let . Then, the above system can be rewritten as the form of (12) with

The weighted communication topology with 6 agents is shown in Figure 1. Using the results presented in [18], the DOF control protocol (14) can be constructed with

It is known that the consensus is asymptotically achieved when there are no communication errors with the designed protocol (see Figure 2(a)). However, communication errors are inevitable. Assume that a 1% error appears in all of the communication channels. Simulation results show that, under the same protocol, the system diverges in the sense that the position state of each agent is far away from the position state of leader (node 1) as can be seen in Figure 2(b).

Figure 1: Communication topology.

(a)

(b)

Figure 2: The disagreement states between and without/with communication errors, .

Example 6 implies that, under the influence of communication errors, consensus cannot be achieved for each agent with the given control protocol. This provides motivation to design an appropriate DOF control protocol to attenuate the effects of communication errors on the consensus performance. In this paper, we assume that there exist communication errors in the transferred data; that is, the DOF control protocol takes the following form:

where represents the communication error when the th agent gets information from the th agent. For convenience, denote

Then, the overall dynamics result in the system (12) with the DOF control protocol (19) can be written as

where

and is the element of the Laplacian matrix .

In order to characterize the effects of the communication errors on consensus performance, we need to define a controlled output for the multiagent system (12) as follows.

Assume that the fixed directed communication graph has a spanning tree, and according to Lemma 2, the Laplacian matrix of graph has a simple zero, and all of the other eigenvalues are in the right half-plane. Let . By Lemma 3, there exists an orthogonal matrix such that

where . It is obvious that the eigenvalues of are equal to the nonzero eigenvalues of , which means that all of the eigenvalues of are in the right half-plane. Here, the matrix satisfies , and according to being an orthogonal matrix.

Let . Define an output vector as

where , , and . Then,

where , which means that can measure the difference between the state of each agent and the average state of all agents.

Let . Using the DOF control protocol (19), the system dynamics with the output can be represented as

where , , and .

From the fact that the null-space of matrix is , we know that

if and only if there exist such that , which implies that consensus of the multiagent system (12) can be achieved asymptotically. However, it is obvious that cannot approach zero as tending to infinity due to the existence of communication error , which indicates that consensus cannot be achieved for the system (12) with the DOF control protocol (19). Inspired by the analysis above, it is reasonable to evaluate the effects of communication error on consensus of the system (12) with the DOF control protocol (19) by using the effects of communication error on the output of system (26). Notice that the latter can be quantitatively measured by the norm of the transfer function matrix from to , which is defined by , that results in the following definition.

Definition 7. Given a scalar . The system (12) with the DOF control protocol (19) is called to achieve consensus with performance if the following conditions hold.

(1)It can reach consensus when ; (2), where is the transfer function matrix of system (26) from to and the output is defined by (24).

A sufficient condition is given in the following theorem to ensure that the multiagent system (12) with the DOF control protocol (19) can achieve consensus with performance.

Theorem 8. Given a scalar . Assume that the fixed communication topology has a spanning tree. The system (12) with the DOF control protocol (19) achieves consensus with performance if there exists a matrix such that

where and is the nonzero eigenvalue of Laplacian matrix , .

Proof. It is known that the system (12) with the DOF control protocol (19) achieves consensus with performance if and only if is Hurwitz and

where , , and are defined by (26).

According to Lemma 5 and (28), we have is Hurwitz stable, and

has no eigenvalues on the imaginary axis; that is, for any ,

if and only if and , where .

For matrix , there are two unitary matrices and such that

where and are upper triangular, with diagonal entries and , respectively, . Now, suppose that is an eigenvalue of

Then, there exists a vector , where , such that

where . When , from (31), it is easy to get that . Then, when , (34) can be rewritten as

which implies that . Similarly, it can be known that for all . This contradicts our assumption, and hence matrix has no eigenvalues on the imaginary axis.

Notice that

which means that matrix has no pure imaginary eigenvalues. Moreover, matrices are Hurwitz stable, , which implies that matrix is Hurwitz stable. Noting that and are unitary matrices, then there must exist two matrices and such that , and . Thus, according to Lemma 5, it can be obtained that

In addition, it is easy to know that

Then, we have

This completes the proof.

Remark 9. If there are no communication errors, that is, , then from Theorem 8, it is only required that is Hurwitz for all nonzero eigenvalues of the Laplacian matrix . In this case, consensus can be achieved asymptotically for the multiagent system (12) with the DOF control protocol (19), which is the result shown by [18].

Remark 10. In fact, from Lemma 5, there exists such that the inequality (28) holds, , if and only if there exists a common th-order DOF controller

solving problem with performance for th-order LTI systems

where , , and are the state space matrices for system (12), , and is the nonzero eigenvalue of Laplacian matrix .

4. Dynamic Output Feedback Design for Consensus

In this section, we determine the system matrix of the DOF control protocol (19) for the multiagent system (12) to achieve consensus with performance. According to Remark 10, it is required to design a common DOF controller (40) to solve problem for LTI systems. Notice that the common DOF controller is difficult to obtain; thus, we firstly consider the problem for the systems , .

Let and be full-rank matrices whose images satisfy

Denote , where and have and rows, respectively.

Lemma 11. If there exist matrices and such that

where , then there exists a DOF controller which can solve the problem with a given performance for the LTI system given by (41), .

Proof. According to Lemma 4, we have that there exists a DOF controller to solve the problem for the LTI system with a given index if and only if there exist matrices and such that

where spans the kernel of . Notice the following facts:

and, for all nonzero ,

Thus, we can choose . Let for all . Then, it is easy to obtain that

and (b), (c) naturally hold from (44), (45), respectively. This completes the proof.

Remark 12. Lemma 11 gives a sufficient condition for the existence of the controller which can solve the problem for the system , . As the results stated in Section 2.2, if there exist and satisfying Lemma 11, then the DOF controller can be obtained by solving the following inequality:

for , where

Obviously, if there exists a common that makes the inequality (50) hold for all , then there exists a common th-order DOF controller which can solve the problem for the LTI systems . Thus, we have the following result.

Theorem 13. Given a scalar , and let . Assume that the fixed communication topology has a spanning tree. Then, there exists a DOF control protocol (19) for the system (12) achieving consensus with performance if

(i)there exist matrices and satisfying (43), (44), and (45), (ii)there exists a matrix satisfying the following LMIs:

for all nonzero eigenvalues of Laplacian matrix , where , , and are given in (51) and is defined by (53).

Proof. From the analysis above, the conditions (i) and (ii) hold which implies that there exist matrices and such that (28) holds due to the fact that (54) is exactly (28). The proof is completed by using Theorem 8 directly.

Remark 14. If we want controllers of order less than , it is only required to add the additional constraint

to (i) of Theorem 13, which can be obtained by using Corollary 7.8 given by [29] and Theorem 13 directly.

Remark 15. Theorem 13 gives the sufficient conditions under which there exists a DOF control protocol such that the multiagent system (12) achieve consensus with a given performance. When the conditions are satisfied, the procedure to construct the DOF control protocol is presented as follows.

 Sept 1: Solve LMIs (43), (44), and (45) for getting a solution: and . Step 2: Construct as (53). Step 3: Solve the LMIs (54) for a common feasible solution .

Remark 16. Assume that there is no communication error in the system. As shown in Remark 9, in this case, the given problem is to design a stabilizing controller defined by (40) for the LTI systems (41) with . Using Theorem 5.8 given by [29] and the fact that the kernels of and are exactly equal for all nonzero , reproducing the steps of the proof of Theorem 13, we have the following results. Assume that the fixed communication topology has a spanning tree, then there exists a DOF control protocol (14) with order for the system (12) achieving consensus if

(1) there exist matrices and such that

where and span the kernels of and , respectively,(2) there exists a matrix satisfying the following LMIs:

for all nonzero eigenvalues of Laplacian matrix , where

and satisfies and , , and are given in (51).

Moreover, using the method similar to that stated in Remark 15, we can construct the DOF controller for the multiagent system (12) reaching consensus.

Notice that condition (ii) in Theorem 13 implies that we need to solve LMIs after constructing , which increase the difficulty of the numerical calculation if the size of the multiagent system is large. We give the following conditions, which can reduce the computational complexity for getting the DOF control protocol by solving four LMIs.

Denote that and are the real part and imaginary part of , respectively, where is the nonzero eigenvalue of Laplacian matrix . Let , , and .

Theorem 17. Given a scalar , and let . Assume that the fixed communication topology has a spanning tree. Then, there exists a DOF control protocol (19) for the system (12) achieving consensus with performance if

(i)there exist matrices and satisfying (43), (44), and (45);(ii)there exists a matrix such that

for , where , , and are given in (51) and is defined by (53).

Proof. To complete the proof, we only need to show that if LMI (59) holds for , then LMI (54) holds for all nonzero eigenvalues of Laplacian matrix .

Notice that

which, in virtue of the Schur Complement Lemma, is equivalent to

where , , and are defined by (22). For convenience, we denote

where , , and .

In fact, there must exist such that . When ,

Similarly, when , we have

This completes the proof.

5. Numerical Example

An example is shown to verify the results obtained in the above section. The agent dynamics and the communication topology are given in Example 6, and the performance index is chosen as . According to the results presented in Section 4, we have

Then, by solving the LMIs (54), we can get a feasible controller (40) with

and the norm of system (26) is . With the designed DOF control protocol, the disagreement states between and without communication error are shown in Figure 3, , which implies that the consensus can be reached when . The communication error is supposed to be

where . Under zero initial condition, the state trajectories of the six agents are depicted in Figure 4, and the corresponding energy trajectories of and are given in Figure 5. It is noted obviously that . Thus, the multiagent system with the given DOF controller can achieve the consensus with the given performance, which validates the effectiveness of the proposed protocol and demonstrates the correctness of the obtained theoretical results.

(a) Position states of the 6 agents

(b) Velocity states of the 6 agents

Figure 3: The disagreement states between and without communication error , .

(a) Position states of the 6 agents

(b) Velocity states of the 6 agents

Figure 4: The state trajectories of the multiagent system with communication error .

Figure 5: Energy trajectories of the controlled output and the communication error .

6. Conclusions

This paper is devoted to the consensus problem for multiagent systems molded by linear time-invariant systems under fixed directed communication topologies and subject to communication errors in the transferred data. A dynamic output feedback control algorithm is proposed. The theoretical analysis shows that if there exists a common dynamic output feedback controller which can solve problem for linear time-invariant systems of order , then the consensus with a desired level can be reached. By using theory, a sufficient condition in terms of linear matrix inequalities is given to ensure the existence for such a controller. A procedure for the controller design is presented.

Acknowledgment

This work is supported by National Natural Science Foundation of China under Grants 90916003 and 61074026.

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