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

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 H ∞ performance are presented. Numerical examples are given to illustrate the theoretical results.


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][2][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][6][7][8][9][10][11][12][13][14][15][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 L 2 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 leaderfollower 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][25][26][27][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 reducedorder 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 −1 linear dynamic systems simultaneously, where  is the number of agents.Secondly, in terms of the − 1 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 R × and C × be the set of  ×  real matrices and  ×  complex matrices, respectively.Matrices, if not explicitly stated, are assumed to have compatible dimensions.  and 0 × are the  ×  identity matrix and the  ×  zero matrix, respectively.For a matrix , ‖‖ 2 is the induced 2norm of the vector norm, and ‖‖ 2  = ().The notation () is the maximal singular value of matrix .Notations   ,  −1 , and  * represent the transpose, the inverse, and the complex conjugate transpose of matrix , respectively.Let Im() and ker() be the image space and kernel of . < 0 ( > 0) means that the matrix  is negative (positive) definite.Span{V 1 , V 2 , . . .
The notation ⊗ represents the Kronecker product.For a vector  ∈ C  , ||  = √  *  is the Euclidean norm. 1  denotes the  × 1 column vector whose elements are all ones.The space of piecewise continuous functions in R  that are square integrable over [0, +∞) is denoted by L  2 [0, ∞), for any V() ∈ L  2 [0, ∞), and its normalized energy is defined by . Let [     ] be a state space realization of () = ( − ) −1  + .

Preliminaries
A sequence ( 1 ,  2 ), ( 2 ,  3 ), . . ., ( −1 ,   ) of edges is called a directed path from node  1 to node   .G  is called a strongly connected digraph if for any ,  ∈ V  , there is a directed path from  to .G  has a directed spanning tree if there exists a node  ∈ V  (a root) such that all other nodes can be linked to  via a directed path.A directed graph is called balanced if ∑  ̸ =   = ∑  ̸ =   for all  ∈ V. 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 1  .
Lemma 2 (see [3]).The Laplacian matrix   of a directed graph has a simple zero eigenvalue with an associated eigenvector 1  , 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 G  , then there exists an orthogonal matrix  = [1  / √  ] ∈ R × such that Furthermore, if G is a balanced graph, then where () ∈ R  is the state, () ∈ L  2 [0, ∞) is the external disturbance, () ∈ R  is the control input, () ∈ R  is the regulated output, () ∈ R  is the measured output, and ,   ,   , and   , for ,  = 1, 2, are known real constant matrices of appropriate dimensions.Without loss of generality, we assume that  22 = 0, (,  2 ) is stabilizable and ( 2 , ) is detectable.
The   th-order dynamic output feedback (DOF) controller is described as follows: where   () ∈ R   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  > 0.
To facilitate the consensus protocol design and stability analysis, several results of the  ∞ problem are recalled as follows.
Lemma 4 (see [29]).Given  > 0, there exists a DOF controller (4) which can solve the  ∞ problem for the LTI system (3) if and only if there exist symmetric matrices  > 0 and  > 0 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.
The solution  provides the state space realization for a feasible controller (4) which can solve  ∞ problem for system (3).

𝐻 ∞ Consensus Problem
Consider a multiagent system consisting of  identical agents with linear dynamics described by where   () ∈ R  is the state,   () ∈ R  is the control input,   () ∈ R  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 lim for any initial states.Assume that the communication topology among the  agents is represented by a fixed directed graph G  = (V  , E  , A  ).Based on the relative output information between the agents, the following dynamic output feedback (DOF) control protocol is used by [18]: where V  () ∈ R   ,   is a preassigned dimension of the coordinating law, and   is the element of the corresponding adjacency matrix A  .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, ,  = 1, 2, . . ., , 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   (),   () ∈ R. Let ξ  () =   ().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).
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   ∈ L  2 [0, +∞) 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 G  has a spanning tree, and according to Lemma 2, the Laplacian matrix   of graph G  has a simple zero, and all of the other eigenvalues are in the right half-plane.Let  1 = (1/ √ )1  .By Lemma 3, there exists an orthogonal matrix 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   1  = 0,    =  −1 , and   =   − (1/)1  1   according to [ 1 ] being an orthogonal matrix.
Let  0 = [  0].Define an output vector z() as where where  0 () = (1/) ∑  =1   (), which means that z() can measure the difference between the state of each agent and the average state of all agents.
From the fact that the null-space of matrix   ⊗   is Span {1  ⊗   }, we know that lim if and only if there exist  0 () ∈ R  such that lim  → +∞   () =  0 (), which implies that consensus of the multiagent system (12) can be achieved asymptotically.However, it is obvious that z() cannot approach zero as  tending to infinity due to the existence of communication error d(), 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 d() on the output z() of system (26).Notice that the latter can be quantitatively measured by the  ∞ norm of the transfer function matrix   () from d() to z(), which is defined by ‖  ()‖ ∞ = sup ∈R {‖  ()‖ 2 } = sup ∈R {(  ())}, that results in the following definition.
Definition 7. Given a scalar  > 0. The system (12) with the DOF control protocol ( 19) is called to achieve consensus with  ∞ performance if the following conditions hold.
A sufficient condition is given in the following theorem to ensure that the multiagent system (12) where  0 =  −1 max  and   is the nonzero eigenvalue of Laplacian matrix   ,  = 2, 3, . . ., .
Proof.It is known that the system (12) where Ã, B, and C are defined by (26).
Notice that  12) with the DOF control protocol (19), which is the result shown by [18].

Dynamic Output Feedback Design for 𝐻 ∞ 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  − 1 LTI systems.Notice that the common DOF controller is difficult to obtain; thus, we firstly consider the  ∞ problem for the systems Σ  (),  = 2, 3, . . ., .
Proof.According to Lemma 4, we have that there exists a DOF controller K  () to solve the  ∞ problem for the LTI system Σ  () with a given index  0 > 0 if and only if there exist matrices   =  *  > 0 and  =  * > 0 such that where    spans the kernel of [−     ].Notice the following facts: and, for all nonzero   , Thus, we can choose Let   =  for all  = 2, 3, . . ., .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 K  () which can solve the  ∞ problem for the system Σ  (),  = 2, 3, . . ., .As the results stated in Section 2.2, if there exist  =  * > 0 and  =  * > 0 satisfying Lemma 11, then the DOF controller K  () can be obtained by solving the following inequality: for   ,  = 2, 3, . . ., , where Obviously, if there exists a common  that makes the inequality (50) hold for all  = 2, 3, . . ., , then there exists a common   th-order DOF controller K() which can solve the  ∞ problem for the LTI systems Σ  ,  = 2, 3, . . ., .Thus, we have the following result.
Theorem 13.Given a scalar  > 0, and let  0 =  −1 max .Assume that the fixed communication topology G  has a spanning tree.
Proof.From the analysis above, the conditions (i) and (ii) hold which implies that there exist matrices  and   > 0 such that (28) holds due to the fact that (54) is exactly (28).
The proof is completed by using Theorem 8 directly. Remark 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.
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 K() defined by (40) for the LTI systems (41) with  21 = 0. 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 G  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  =  * > 0 and  =  * > 0 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  2 satisfies  2  * 2 = − −1 and  0 , B, and Ĉ(  ) 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  − 1 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.
Theorem 17.Given a scalar  > 0, and let  0 =  −1 max .Assume that the fixed communication topology G  has a spanning tree.Then, there exists a DOF control protocol (19)    where , , and  are defined by (22).For convenience, we denote     It is noted obviously that ‖z()‖ L 2 ≤ ‖d()‖ L 2 .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.

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  − 1 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.
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  = 10.According to the results presented in Section 4, we have solving the LMIs (54), we can get a feasible controller (the  ∞ norm of system (26) is ‖  ()‖ ∞ = 3.0707.With the designed DOF control protocol, the disagreement states between   and  1 without communication error d() are shown in Figure3,  = 2, 3, . . ., 6, which implies that the consensus can be reached whend() = 0.The communication error d() ∈ L 2 [0, ∞) is supposed to be d () = {  sin () ,  ∈ [0, 40] , 0,  > 40,(67)where  = [1.110.58 1.48 1.59 2.12 2.53 0.42 1.31 4.0 0.14]  .Under zero initial condition, the state trajectories of the six agents are depicted in Figure4, and the corresponding energy trajectories of d() and z() are given in Figure5.

Figure 5 :
Figure 5: Energy trajectories of the controlled output z() and the communication error d().
Directed graphs are used to model the information interaction among agents.Let G  = (V  , E  , A  ) be a directed weighted graph, where V  = {1, 2, . . ., } is the node set, E  ⊆ V  × V  is the edge set, and A  ∈ R × is a weighted adjacency matrix with nonnegative elements   .An edge of G  is denoted by (, ) which means that agent  can directly get information from agent .(, ) ∈ E  if and only if   > 0, otherwise   = 0.If (, ) ∈ E  ⇔ (, ) ∈ E  , then G  is said to be an undirected graph.In this paper, we assume that there are no self-cycles in G  ; that is,   = 0,  = 1, 2, . . ., .The in-degree and out-degree of the th agent are, respectively, defined as  in () = ∑  =1   and  out () = ∑  =1   .Let  max = max  { in ()}.Correspondingly, the Laplacian matrix of graph G  is denoted by 2.1.Graph Theory.