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

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 𝐻_∞ 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 𝑁 identical linear dynamics with fixed directed communication topologies. Consensus problem with ℒ₂ 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 𝐻_∞ 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 𝐻_∞ performance is transmitted into an 𝐻_∞ control problem of another reduced-order system. It is shown that consensus with 𝐻_∞ performance can be achieved if there exists a common dynamic output feedback controller which can be realized by solving 𝐻_∞ problem for 𝑁−1 linear dynamic systems simultaneously, where 𝑁 is the number of agents. Secondly, in terms of the 𝑁−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 𝐻_∞ control theory. Section 3 formulates the problem and conditions for reaching consensus with 𝐻_∞ 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^𝑛×𝑛 and ℂ^𝑛×𝑛 be the set of 𝑛×𝑛 real matrices and 𝑛×𝑛 complex matrices, respectively. Matrices, if not explicitly stated, are assumed to have compatible dimensions. 𝐼_𝑛 and 0_𝑛×𝑚 are the 𝑛×𝑛 identity matrix and the 𝑛×𝑚 zero matrix, respectively. For a matrix 𝐴, ‖𝐴‖₂ is the induced 2-norm of the vector norm, and ‖𝐴‖=₂^▵𝜎(𝐴). The notation 𝜎(𝐴) is the maximal singular value of matrix 𝐴. Notations 𝐴^𝑇, 𝐴⁻¹, and 𝐴^∗ represent the transpose, the inverse, and the complex conjugate transpose of matrix 𝐴, respectively. Let Im(𝐴) and ker(𝐴) be the image space and kernel of 𝐴. 𝐴<0 (𝐴>0) means that the matrix 𝐴 is negative (positive) definite. Span{𝑣₁,𝑣₂,…𝑣_𝑚} is a subspace of ℝ^𝑛 spanned by {𝑣₁,𝑣₂,…𝑣_𝑚}, where 𝑣_𝑖∈ℝ^𝑛,𝑖=1,2,…,𝑚. The notation ⊗ represents the Kronecker product. For a vector 𝑥∈ℂ^𝑛,|𝑥|=^▵√𝑥^∗𝑥 is the Euclidean norm. 𝟏_𝑁 denotes the 𝑁×1 column vector whose elements are all ones. The space of piecewise continuous functions in ℝ^𝑚 that are square integrable over [0,+∞) is denoted by ℒ^𝑚₂[0,∞), for any 𝑣(𝑡)∈ℒ^𝑚₂[0,∞), and its normalized energy is defined by ‖𝑣(𝑡)‖_ℒ2=(∫^∞₀|𝑣(𝑡)|²𝑑𝑡)^1/2. Let 𝐴𝐵𝐶𝐷 be a state space realization of 𝐺(𝑠)=𝐶(𝑠𝐼−𝐴)⁻¹𝐵+𝐷.

2. Preliminaries

2.1. Graph Theory

Directed graphs are used to model the information interaction among agents. Let 𝒢_𝑁=(𝒱_𝑁,ℰ_𝑁,𝒜_𝑁) be a directed weighted graph, where 𝒱_𝑁={1,2,…,𝑁} is the node set, ℰ_𝑁⊆𝒱_𝑁×𝒱_𝑁 is the edge set, and 𝒜_𝑁∈ℝ^𝑁×𝑁 is a weighted adjacency matrix with nonnegative elements 𝑎_𝑖𝑗. An edge of 𝒢_𝑁 is denoted by (𝑖,𝑗) which means that agent 𝑗 can directly get information from agent 𝑖. (𝑗,𝑖)∈ℰ_𝑁 if and only if 𝑎_𝑖𝑗>0, otherwise 𝑎_𝑖𝑗=0. If (𝑖,𝑗)∈ℰ_𝑁⇔(𝑗,𝑖)∈ℰ_𝑁, then 𝒢_𝑁 is said to be an undirected graph. In this paper, we assume that there are no self-cycles in 𝒢_𝑁; that is, 𝑎_𝑖𝑖=0, 𝑖=1,2,…,𝑁. The in-degree and out-degree of the 𝑖th agent are, respectively, defined as 𝑑_in(𝑖)=∑^𝑁_𝑗=1𝑎_𝑖𝑗 and 𝑑_out(𝑖)=∑^𝑁_𝑗=1𝑎_𝑗𝑖. Let 𝑑_max=max_𝑖{𝑑_in(𝑖)}. Correspondingly, the Laplacian matrix of graph 𝒢_𝑁 is denoted by 𝐿_𝑁=Δ_𝑁−𝒜_𝑁∈ℝ^𝑁×𝑁, where Δ_𝑁=[Δ_𝑖𝑗] is a diagonal matrix with Δ_𝑖𝑖=𝑑_in(𝑖).

A sequence (𝑖₁,𝑖₂),(𝑖₂,𝑖₃),…,(𝑖_𝑘−1,𝑖_𝑘) of edges is called a directed path from node 𝑖₁ to node 𝑖_𝑘. 𝒢_𝑁 is called a strongly connected digraph if for any 𝑖,𝑗∈𝒱_𝑁, there is a directed path from 𝑖 to 𝑗. 𝒢_𝑁 has a directed spanning tree if there exists a node 𝑟∈𝒱_𝑁 (a root) such that all other nodes can be linked to 𝑟 via a directed path. A directed graph is called balanced if ∑_𝑗≠𝑖𝑎_𝑖𝑗=∑_𝑖≠𝑗𝑎_𝑗𝑖 for all 𝑖∈𝒱.

Below are well-known results for the Laplacian matrix.

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

Lemma 2 (see [3]). The Laplacian matrix 𝐿_𝑁 of a directed graph has a simple zero eigenvalue with an associated eigenvector 𝟏_𝑁, 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 𝐿_𝑁 be the Laplacian matrix of a directed graph 𝒢_𝑁, then there exists an orthogonal matrix 𝑈=[𝟏_𝑁/√𝑁𝑈]∈ℝ^𝑁×𝑁 such that 𝑈^𝑇𝐿_𝑁0𝟏𝑈=√^𝑇𝑁𝑁𝐿𝑈0𝑈^𝑇.𝐿𝑈 (1) Furthermore, if 𝒢 is a balanced graph, then 𝑈^𝑇𝐿_𝑁𝑈=000𝑈^𝑇𝐿_𝑁𝑈. (2)

2.2. Robust 𝐻_∞ Control Theory

Consider that the 𝑛th-order linear time-invariant (LTI) system is described as follows: 𝑥̇(𝑡)=𝐴𝑥(𝑡)+𝐵₁𝜔(𝑡)+𝐵₂𝑢(𝑡),𝑧(𝑡)=𝐶₁𝑥(𝑡)+𝐷₁₁𝜔(𝑡)+𝐷₁₂𝑢(𝑡),𝑦(𝑡)=𝐶₂𝑥(𝑡)+𝐷₂₁𝜔(𝑡)+𝐷₂₂𝑢(𝑡), (3) where 𝑥(𝑡)∈ℝ^𝑛 is the state, 𝜔(𝑡)∈ℒ^𝑞₂[0,∞) is the external disturbance, 𝑢(𝑡)∈ℝ^𝑚 is the control input, 𝑧(𝑡)∈ℝ^𝑝 is the regulated output, 𝑦(𝑡)∈ℝ^𝑙 is the measured output, and 𝐴, 𝐵_𝑖, 𝐶_𝑖, and 𝐷_𝑖𝑗, for 𝑖,𝑗=1,2, are known real constant matrices of appropriate dimensions. Without loss of generality, we assume that 𝐷₂₂=0, (𝐴,𝐵₂) is stabilizable and (𝐶₂,𝐴) is detectable.

The 𝑛_𝑐th-order dynamic output feedback (DOF) controller is described as follows: 𝑥̇_𝑐(𝑡)=𝐴_𝑘𝑥_𝑐(𝑡)+𝐵_𝑘𝑦(𝑡),𝑢(𝑡)=𝐶_𝑘𝑥_𝑐(𝑡)+𝐷_𝑘𝑦(𝑡), (4) where 𝑥_𝑐(𝑡)∈ℝ^𝑛𝑐 is the controller state and 𝐴_𝑘, 𝐵_𝑘, 𝐶_𝑘, and 𝐷_𝑘 are constant matrices with appropriate dimensions.

Let 𝑆_𝑝𝑐𝐴(𝑠)=^𝑐𝐵_𝑐𝐶_𝑐𝐷_𝑐 be the transfer function from 𝜔(𝑡) to 𝑧(𝑡) of the closed-loop system obtained from (3) and (4), where 𝐴_𝑐=𝐴₀+𝐵𝐾𝐶,𝐵_𝑐=𝐵₀+𝐵𝐾𝐷₂₁,𝐶_𝑐=𝐶₀+𝐷₁₂𝐾𝐶,𝐷_𝑐=𝐷₁₁+𝐷₁₂𝐾𝐷₂₁,𝐴₀=𝐴000,𝐵₀=𝐵₁0,𝐶₀=𝐶₁0,𝐵=0𝐵₂,𝐼0𝐶𝐶=0𝐼₂0,𝐷₁₂=0𝐷₁₂,𝐷₂₁=0𝐷₂₁𝐴,𝐾=^𝑘𝐵_𝑘𝐶_𝑘𝐷_𝑘. (5) The 𝐻_∞ problem for the given LTI system (3) is to find a DOF controller (4) such that the closed-loop system is internally stable and ‖𝑆_𝑝𝑐(𝑠)‖_∞<𝛾 for some constant 𝛾>0.

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

Lemma 4 (see [29]). Given 𝛾>0, there exists a DOF controller (4) which can solve the 𝐻_∞ problem for the LTI system (3) if and only if there exist symmetric matrices 𝑋>0 and 𝑌>0 such that (a)𝑁_𝑋00𝐼^∗𝑋𝐴+𝐴^∗𝑋𝑋𝐵₁𝐶^∗₁𝐵^∗₁𝑋−𝛾𝐼𝐷^∗₁₁𝐶₁𝐷₁₁𝑁−𝛾𝐼^𝑋0(0𝐼<0,b)𝑁_𝑌00𝐼^∗𝑌𝐴^∗+𝐴𝑌𝑌𝐶^∗₁𝐵₁𝐶₁𝑌−𝛾𝐼𝐷₁₁𝐵^∗₁𝐷^∗₁₁𝑁−𝛾𝐼^𝑌0(0𝐼<0,c)𝑋𝐼𝐼𝑌≥0, (6) where 𝑁_𝑋 and 𝑁_𝑌 are full-rank matrices whose images satisfy Im𝑁_𝑋=ker𝐶₂𝐷₂₁,Im𝑁_𝑌=ker𝐵^∗₂𝐷^∗₁₂. (7)

As the results shown by [29], the DOF controller (4) solving the 𝐻_∞ problem for the LTI system (3) can be constructed as follows.(i)Find 𝑋>0 and 𝑌>0 which satisfy Lemma 4.(ii)Let 𝑋_𝑐=^𝑋𝑋2_𝑋^∗_2𝐼, where 𝑋₂∈ℝ^{𝑛×𝑛𝑐} satisfying 𝑋−𝑌⁻¹=𝑋₂𝑋^∗₂.(iii)Solve the the following inequality: 𝐻_𝑋𝑐+𝑄^∗𝐾^∗𝑃_𝑋𝑐+𝑃^∗_𝑋𝑐𝐾𝑄<0. (8) For a feasible solution 𝐾, where 𝑃_𝑋𝑐=𝐵^∗𝑋_𝑐0𝐷^∗₁₂,𝑄=𝐶𝐷₂₁0,𝐻_𝑋𝑐=𝑋_𝑐𝐴₀+𝐴^∗₀𝑋_𝑐𝑋_𝑐𝐵₀𝐶^∗₀𝐵^∗₀𝑋_𝑐−𝛾₀𝐼𝐷^∗₁₁𝐶₀𝐷₁₁−𝛾₀𝐼, (9) and 𝐴₀,𝐵₀,𝐶₀,𝐵,𝐶,𝐷₁₂,𝐷₂₁ are defined by (5). The solution 𝐾 provides the state space realization for a feasible controller (4) which can solve 𝐻_∞ problem for system (3).

Lemma 5 (see [30]). Let 𝛾>0, 𝐴𝐺(𝑠)=𝐵𝐶𝐷 with 𝐴 is Hurwitz stable, and 𝐻≜𝐴+𝐵𝑅⁻¹𝐷^∗𝐶𝐵𝑅⁻¹𝐵^∗−𝐶^∗𝐼+𝐷𝑅⁻¹𝐷^∗𝐶−𝐴+𝐵𝑅⁻¹𝐷^∗𝐶^∗∗, (10) where 𝑅=𝛾²𝐼−𝐷^∗𝐷. Then, the following conditions are equivalent:(1)‖𝐺(𝑠)‖_∞<𝛾,(2)𝜎(𝐷)<𝛾 and 𝐻 have no eigenvalues on the imaginary axis,(3)there exists a matrix 𝑋=𝑋^∗>0 such that 𝑋𝐴+𝐴^∗𝑋𝑋𝐵𝐶^∗𝐵^∗𝑋−𝛾𝐼𝐷^∗𝐶𝐷−𝛾𝐼<0. (11)

3. 𝐻_∞ Consensus Problem

Consider a multiagent system consisting of 𝑁 identical agents with linear dynamics described by 𝑥̇_𝑖(𝑡)=𝐴𝑥_𝑖(𝑡)+𝐵𝑢_𝑖(𝑡)𝑦_𝑖(𝑡)=𝐶𝑥_𝑖(𝑡)𝑖=1,…,𝑁, (12) where 𝑥_𝑖(𝑡)∈ℝ^𝑛 is the state, 𝑢_𝑖(𝑡)∈ℝ^𝑚 is the control input, 𝑦_𝑖(𝑡)∈ℝ^𝑝 is the measured output, and 𝐴, 𝐵, and 𝐶 are constant matrices with compatible dimensions. It is assumed that (𝐴,𝐵) is stabilizable and (𝐶,𝐴) is detectable, and without loss of generality, 𝐵 is of full column rank. We say that the control input 𝑢_𝑖(𝑡) solves the consensus problem for the multiagent system (12) if the states of the agents satisfy lim_𝑡→+∞𝑥_𝑖(𝑡)−𝑥_𝑗(𝑡)=0,∀𝑖,𝑗∈𝒱_𝑁 (13) for any initial states.

Assume that the communication topology among the 𝑁 agents is represented by a fixed directed graph 𝒢_𝑁=(𝒱_𝑁,ℰ_𝑁,𝒜_𝑁). Based on the relative output information between the agents, the following dynamic output feedback (DOF) control protocol is used by [18]: 𝑣̇_𝑖(𝑡)=𝐴_𝑘𝑣_𝑖(𝑡)+𝐵∑_𝑘^𝑁_𝑗=0𝑎_𝑖𝑗𝑦_𝑗(𝑡)−𝑦_𝑖(𝑡𝑢_𝑖)(𝑡)=𝐶_𝑘𝑣_𝑖(𝑡)+𝐷∑_𝑘^𝑁_𝑗=0𝑎_𝑖𝑗𝑦_𝑗(𝑡)−𝑦_𝑖(𝑡)𝑖=1,…,𝑁, (14) where 𝑣_𝑖(𝑡)∈ℝ^𝑛𝑐, 𝑛_𝑐 is a preassigned dimension of the coordinating law, and 𝑎_𝑖𝑗 is the element of the corresponding adjacency matrix 𝒜_𝑁. The system matrix 𝐴𝐾=^𝑘𝐵_𝑘𝐶_𝑘𝐷_𝑘 (15) 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 𝐾 was presented by [18].

However, if there exist communication errors between the 𝑖th agent and the 𝑗th agent, 𝑖,𝑗=1,2,…,𝑁, 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 𝜉̈_𝑖(𝑡)=𝑢_𝑖(𝑡),𝑦_𝑖(𝑡)=𝜉_𝑖(𝑡),𝑖=1,2,…,𝑁, (16) where 𝜉_𝑖(𝑡),𝑦_𝑖(𝑡)∈ℝ. Let 𝜉̇_𝑖(𝑡)=𝜁_𝑖(𝑡). Then, the above system can be rewritten as the form of (12) with 𝑥_𝑖(𝑡)=𝜉_𝑖(𝑡)𝜁_𝑖(𝑡)01,.,𝐴=0100,𝐵=𝐶=10 (17) 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 𝐴_𝑘=−10−1−50.1039−0.485,𝐵_𝑘=,𝐶1050_𝑘=0.10390.485,𝐷_𝑘=0. (18) 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 𝑥𝑖 and 𝑥1 without/with communication errors, 𝑖=2,3,…,6.

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: 𝑣̇_𝑖(𝑡)=𝐴_𝑘𝑣_𝑖(𝑡)+𝐵_𝑘^𝑁_𝑗=0𝑎_𝑖𝑗𝑦_𝑗(𝑡)−𝑦_𝑖(𝑡)+𝜔_𝑖𝑗(𝑡,𝑢_𝑖)(𝑡)=𝐶_𝑘𝑣_𝑖(𝑡)+𝐷_𝑘^𝑁_𝑗=0𝑎_𝑖𝑗𝑦_𝑗(𝑡)−𝑦_𝑖(𝑡)+𝜔_𝑖𝑗(𝑡)𝑖=1,…,𝑁, (19) where 𝜔_𝑖𝑗∈ℒ^𝑝₂[0,+∞) represents the communication error when the 𝑖th agent gets information from the 𝑗th agent. For convenience, denote 𝐱(𝑡)=𝐱^𝑇₁(𝑡),…,𝐱^𝑇_𝑁(𝑡)^𝑇∈ℝ^{𝑁(𝑛+𝑛𝑐)},𝐱_𝑖(𝑡)=𝑥^𝑇_𝑖(𝑡),𝑣^𝑇_𝑖(𝑡)^𝑇∈ℝ^{𝑛+𝑛𝑐}𝐲(𝑡)=𝑦,𝑖=1,2,…,𝑁,_𝑇1(𝑡),𝑦^𝑇₂(𝑡),…,𝑦^𝑇_𝑁(𝑡)^𝑇∈ℝ^𝑁𝑝,𝐝(𝑡)=𝐝^𝑇₁(𝑡),𝐝^𝑇₂(𝑡),…,𝐝^𝑇_𝑁(𝑡)^𝑇∈ℝ^{𝑁(𝑁−1)𝑝},𝐝_𝑖(𝑡)=𝜔^𝑇_𝑖1(𝑡),…,𝜔^𝑇_{𝑖()𝑖−1}(𝑡),𝜔^𝑇_𝑖()𝑖+1(𝑡),…,𝜔^𝑇_𝑖𝑁^𝑇(𝑡)∈ℝ^{(𝑁−1)𝑝},𝑖=1,2,…,𝑁. (20) Then, the overall dynamics result in the system (12) with the DOF control protocol (19) can be written as 𝐱̇(𝑡)=𝐼_𝑁⊗𝐴−𝐿_𝑁⊗𝐵𝐶𝐱(𝑡)+𝐷_𝑁⊗𝐵𝐝(𝑡), (21) where 𝐴=𝐴𝐵𝐶_𝑘0𝐴_𝑘,𝐵=𝐵𝐷_𝑘𝐵_𝑘,,𝐷𝐶=𝐶0_𝑁=𝛿₁0_1×𝑁−1⋯0_1×𝑁−10_1×𝑁−1𝛿₂⋯0_1×𝑁−10⋮⋮⋱⋮_1×𝑁−10_1×𝑁−1⋯𝛿_𝑁∈ℝ^{𝑁×𝑁(𝑁−1)},𝛿_𝑖=𝑙_𝑖1⋯𝑙_{𝑖(𝑖−1)}𝑙_𝑖(𝑖+1)⋯𝑙_𝑖𝑁,𝑖=1,…,𝑁, (22) and 𝑙_𝑖𝑗 is the element of the Laplacian matrix 𝐿_𝑁.

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_𝑁 has a spanning tree, and according to Lemma 2, the Laplacian matrix 𝐿_𝑁 of graph 𝒢_𝑁 has a simple zero, and all of the other eigenvalues are in the right half-plane. Let 𝑒₁=(1/√𝑁)𝟏_𝑁. By Lemma 3, there exists an orthogonal matrix [𝑒₁𝑈]∈ℝ^𝑁×𝑁 such that 𝑒^𝑇₁𝑈^𝑇𝐿_𝑁𝑒₁𝑈=0𝑒^𝑇₁𝐿_𝑁𝑈0𝐿_𝑁, (23) where 𝐿_𝑁=𝑈^𝑇𝐿_𝑁𝑈. It is obvious that the eigenvalues of 𝐿_𝑁 are equal to the nonzero eigenvalues of 𝐿_𝑁, which means that all of the eigenvalues of 𝐿_𝑁 are in the right half-plane. Here, the matrix 𝑈 satisfies 𝑈^𝑇𝟏_𝑁=0,𝑈^𝑇𝑈=𝐼_𝑁−1, and 𝑈𝑈^𝑇=𝐼_𝑁−(1/𝑁)𝟏_𝑁𝟏^𝑇_𝑁 according to [𝑒₁𝑈] being an orthogonal matrix.

Let 𝐶₀=[𝐼_𝑛0]. Define an output vector 𝐳(𝑡) as 𝐳(𝑡)=𝑈^𝑇⊗𝐶₀𝐱(𝑡)=𝑈^𝑇⊗𝐼_𝑛𝑥(𝑡), (24) where 𝐳(𝑡)=[𝐳^𝑇₁(𝑡),…,𝐳^𝑇_𝑁−1(𝑡)]^𝑇∈ℝ^{(𝑁−1)𝑛}, 𝐳_𝑖(𝑡)∈ℝ^𝑛, and 𝑥(𝑡)=[𝑥^𝑇₁(𝑡),𝑥^𝑇₂(𝑡),…,𝑥^𝑇_𝑁(𝑡)]^𝑇. Then, |𝐳(𝑡)|²=𝑈^𝑇⊗𝐶₀𝐱(𝑡)²=𝑥(𝑡)−𝟏_𝑁⊗𝑥₀(𝑡)², (25) where 𝑥₀(𝑡)=(1/𝑁)∑^𝑁_𝑖=1𝑥_𝑖(𝑡), which means that 𝐳(𝑡) can measure the difference between the state of each agent and the average state of all agents.

Let 𝐱(𝑡)=(𝑈^𝑇⊗𝐼_{𝑛+𝑛𝑐})𝐱(𝑡). Using the DOF control protocol (19), the system dynamics with the output 𝐳(𝑡) can be represented as 𝐱̇(𝑡)=𝐴𝐱(𝑡)+(𝑡),𝐳(𝑡)=𝐶𝐱(𝑡),𝐵𝐝 (26) where 𝐴=𝐼_𝑁−1⊗𝐴−𝐿_𝑁⊗𝐵𝐶, 𝐵=𝑈^𝑇𝐷_𝑁⊗𝐵, and 𝐶=𝐼_𝑁−1⊗𝐶₀.

From the fact that the null-space of matrix 𝑈^𝑇⊗𝐼_𝑛 is Span{𝟏_𝑁⊗𝐼_𝑛}, we know that lim_𝑡→+∞𝐳(𝑡)=lim_𝑡→+∞𝑈^𝑇⊗𝐼_𝑛𝑥(𝑡)=0 (27) if and only if there exist 𝑥₀(𝑡)∈ℝ^𝑛 such that lim_𝑡→+∞𝑥_𝑖(𝑡)=𝑥₀(𝑡), which implies that consensus of the multiagent system (12) can be achieved asymptotically. However, it is obvious that 𝐳(𝑡) cannot approach zero as 𝑡 tending to infinity due to the existence of communication error 𝐝(𝑡), 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(𝑡) on the output 𝐳(𝑡) of system (26). Notice that the latter can be quantitatively measured by the 𝐻_∞ norm of the transfer function matrix 𝐺_𝑑𝑧(𝑠) from 𝐝(𝑡) to 𝐳(𝑡), which is defined by ‖𝐺_𝑑𝑧(𝑠)‖_∞=sup_𝜔∈ℝ{‖𝐺_𝑑𝑧(𝑗𝜔)‖₂}=sup_𝜔∈ℝ{𝜎(𝐺_𝑑𝑧(𝑗𝜔))}, that results in the following definition.

Definition 7. Given a scalar 𝛾>0. The system (12) with the DOF control protocol (19) is called to achieve consensus with 𝐻_∞ performance if the following conditions hold.(1)It can reach consensus when 𝐝(𝑡)=0; (2)‖𝐺_𝑑𝑧(𝑠)‖_∞<𝛾, where 𝐺_𝑑𝑧(𝑠) is the transfer function matrix of system (26) from 𝐝(𝑡) to 𝐳(𝑡) and the output 𝐳(𝑡) 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 𝐻_∞ performance.

Theorem 8. Given a scalar 𝛾>0. Assume that the fixed communication topology 𝒢_𝑁 has a spanning tree. The system (12) with the DOF control protocol (19) achieves consensus with 𝐻_∞ performance if there exists a matrix 𝑋_𝑐=𝑋^∗_𝑐>0 such that 𝑋_𝑐𝐴−𝜆_𝑖𝐵𝐶+𝐴−𝜆_𝑖𝐵𝐶^∗𝑋_𝑐𝑋_𝑐𝐵𝐶^∗₀𝐵^∗𝑋_𝑐−𝛾₀𝐶𝐼0₀0−𝛾₀𝐼<0, (28) where 𝛾₀=𝑑⁻¹_max𝛾 and 𝜆_𝑖 is the nonzero eigenvalue of Laplacian matrix 𝐿_𝑁, 𝑖=2,3,…,𝑁.

Proof. It is known that the system (12) with the DOF control protocol (19) achieves consensus with 𝐻_∞ performance if and only if 𝐴 is Hurwitz and 𝐺_𝑑𝑧(𝑠)_∞=𝐶𝑠𝐼−𝐴⁻¹𝐵_∞<𝛾, (29) where 𝐴, 𝐵, and 𝐶 are defined by (26).
According to Lemma 5 and (28), we have 𝐴−𝜆_𝑖𝐵𝐶 is Hurwitz stable, and 𝐻_𝑖≜𝐴−𝜆_𝑖𝐵𝐶𝛾₀⁻²𝐵𝐵^∗−𝐶^∗₀𝐶₀−𝐴−𝜆_𝑖𝐵𝐶^∗ (30) has no eigenvalues on the imaginary axis; that is, for any 𝜔∈ℝ, 𝑗𝜔𝐼−𝐴−𝜆_𝑖𝐵𝐶𝜂₁−𝛾₀⁻²𝐵𝐵^∗𝜂₂𝐶=0,_∗0𝐶₀𝜂₁+𝑗𝜔𝐼+𝐴−𝜆_𝑖𝐵𝐶^∗𝜂₂=0, (31) if and only if 𝜂₁=0 and 𝜂₂=0, where 𝜂₁,𝜂₂∈ℂ^{𝑛+𝑛𝑐}.
For matrix 𝐿_𝑁, there are two unitary matrices 𝑉₁ and 𝑉₂ such that 𝑉^∗₁𝐿_𝑁𝑉₁=𝑇=𝑡_𝑖𝑗∈ℝ^{(𝑁−1)×(𝑁−1)},𝑉^∗₂𝐿^∗_𝑁𝑉₂=𝑆=𝑠_𝑖𝑗∈ℝ^{(𝑁−1)×(𝑁−1)}, (32) where 𝑇 and 𝑆 are upper triangular, with diagonal entries 𝑡_𝑖𝑖=𝜆_𝑖+1 and 𝑠_𝑖𝑖=𝜆^∗_𝑖+1, respectively, 𝑖=1,2,…,𝑁−1. Now, suppose that 𝑗𝜔₀ is an eigenvalue of 𝐼𝐻≜^𝑁−1⊗𝐴−𝑇⊗𝐵𝐶𝛾₀⁻²𝐼_𝑁−1⊗𝐵𝐵^∗−𝐼_𝑁−1⊗𝐶^∗₀𝐶₀−𝐼_𝑁−1⊗𝐴−𝑆^∗⊗𝐵𝐶^∗. (33) Then, there exists a vector 𝜂=[𝜂^𝑇_1,1,𝜂^𝑇_1,2,…,𝜂^𝑇_1,(𝑁−1),𝜂^𝑇_2,1,𝜂^𝑇_2,2,…,𝜂^𝑇_2,(𝑁−1)]^𝑇≠0, where 𝜂_𝑘,𝑙∈ℂ^{𝑛+𝑛𝑐},𝑘=1,2,𝑙=1,2,…,𝑁−1, such that 𝑗𝜔₀𝐼−𝐴−𝜆_𝑖+1𝐵𝐶𝜂_1,𝑖+^𝑁−1_𝑗=𝑖+1𝑡_𝑖𝑗𝐵𝐶𝜂_1,𝑗−𝛾₀⁻²𝐵𝐵^∗𝜂_2,𝑖𝐶=0,_∗0𝐶₀𝜂_1,𝑖+𝑗𝜔₀𝐼+𝐴−𝜆_𝑖+1𝐵𝐶^∗𝜂_2,𝑖−^𝑁−1_𝑗=𝑖+1𝑠_𝑖𝑗𝐵𝐶^∗𝜂_2,𝑗=0, (34) where 𝑖=1,2,…,𝑁−1. When 𝑖=𝑁−1, from (31), it is easy to get that 𝜂_1,𝑁−1=𝜂_2,𝑁−1=0. Then, when 𝑖=𝑁−2, (34) can be rewritten as 𝑗𝜔₀𝐼−𝐴−𝜆_𝑁−1𝐵𝐶𝜂_1,𝑁−2−𝛾₀⁻²𝐵𝐵^∗𝜂_2,𝑁−2𝐶=0,_∗0𝐶₀𝜂_1,𝑁−2+𝑗𝜔₀𝐼+𝐴−𝜆_𝑁−1𝐵𝐶^∗𝜂_2,𝑁−2=0, (35) which implies that 𝜂_1,𝑁−2=𝜂_2,𝑁−2=0. Similarly, it can be known that 𝜂_1,𝑖=𝜂_2,𝑖=0 for all 𝑖=1,2,…,𝑁−1. This contradicts our assumption, and hence matrix 𝐻 has no eigenvalues on the imaginary axis.
Notice that 𝐼𝐻≜^𝑁−1⊗𝐴−𝐿_𝑁⊗𝐵𝐶𝛾₀⁻²𝑉₁𝑉^∗₂⊗𝐵𝐵^∗−𝑉₂𝑉^∗₁⊗𝐶^∗₀𝐶₀−𝐼_𝑁−1⊗𝐴−𝐿_𝑁⊗𝐵𝐶^∗=𝑉₁⊗𝐼_{𝑛+𝑛𝑐}00𝑉₂⊗𝐼_{𝑛+𝑛𝑐}𝐻𝑉^∗₁⊗𝐼_{𝑛+𝑛𝑐}00𝑉^∗₂⊗𝐼_{𝑛+𝑛𝑐}, (36) which means that matrix 𝐻 has no pure imaginary eigenvalues. Moreover, matrices 𝐴−𝜆_𝑖𝐵𝐶 are Hurwitz stable, 𝑖=2,3,…,𝑁, which implies that matrix 𝐴=𝐼_𝑁−1⊗𝐴−𝐿_𝑁⊗𝐵𝐶 is Hurwitz stable. Noting that 𝑉₁ and 𝑉₂ are unitary matrices, then there must exist two matrices 𝑌₁ and 𝑌₂ such that 𝑌₁𝑌^∗₁=𝑉₁𝑉^∗₂,𝑌^∗₂𝑌₂=𝑉₂𝑉^∗₁, and ‖𝑌₁‖₂=‖𝑌₂‖₂=1. Thus, according to Lemma 5, it can be obtained that 𝑌₂⊗𝐶₀𝑠𝐼−𝐼_𝑁−1⊗𝐴−𝐿_𝑁⊗𝐵𝐶⁻¹𝑌₁⊗𝐵_∞=𝐶𝑠𝐼−𝐴⁻¹𝐼_𝑁−1⊗𝐵_∞<𝛾₀. (37)
In addition, it is easy to know that 𝑈^𝑇𝐷_𝑁2≤𝐷_𝑁2=max_𝑖𝛿_𝑖𝛿^𝑇_𝑖≤𝑑_max. (38) Then, we have 𝐺_𝑑𝑧(𝑠)_∞=𝐶𝑠𝐼−𝐴⁻¹𝐵_∞<𝑑_max𝐶𝑠𝐼−𝐴⁻¹𝐼_𝑁−1⊗𝐵_∞<𝛾. (39) This completes the proof.

Remark 9. If there are no communication errors, that is, 𝐝(𝑡)=0, then from Theorem 8, it is only required that 𝐴−𝜆_𝑖𝐵𝐶 is Hurwitz for all nonzero eigenvalues 𝜆_𝑖 of the Laplacian matrix 𝐿_𝑁. 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 𝑋_𝑐>0 such that the inequality (28) holds, 𝑖=2,3,…,𝑁, if and only if there exists a common 𝑛_𝑐th-order DOF controller 𝒦(𝑠)=𝐴_𝑘𝐵_𝑘𝐶_𝑘𝐷_𝑘 (40) solving 𝐻_∞ problem with performance 𝛾₀ for 𝑁−1𝑛th-order LTI systems Σ_𝑖(𝑠)=𝐴0𝐵𝐼_𝑛00−𝜆_𝑖𝐶𝐷₂₁0, (41) where 𝐴, 𝐵, and 𝐶 are the state space matrices for system (12), 𝐷₂₁=𝐼_𝑝, and 𝜆_𝑖 is the nonzero eigenvalue of Laplacian matrix 𝐿_𝑁,𝑖=2,3,…,𝑁.

4. Dynamic Output Feedback Design for 𝐻_∞ Consensus

In this section, we determine the system matrix 𝐾 of the DOF control protocol (19) for the multiagent system (12) to achieve consensus with 𝐻_∞ performance. According to Remark 10, it is required to design a common DOF controller (40) to solve 𝐻_∞ problem for 𝑁−1 LTI systems. Notice that the common DOF controller is difficult to obtain; thus, we firstly consider the 𝐻_∞ problem for the systems Σ_𝑖(𝑠), 𝑖=2,3,…,𝑁.

Let 𝑁_𝑋 and 𝑁_𝑌 be full-rank matrices whose images satisfy Im𝑁_𝑋=ker𝐶𝐼_𝑝,Im𝑁_𝑌=ker𝐵^∗0. (42) Denote 𝑁_𝑋=[𝑁^∗_𝑐𝑁^∗_𝐼]^∗, where 𝑁_𝑐 and 𝑁_𝐼 have 𝑛 and 𝑝 rows, respectively.

Lemma 11. If there exist matrices 𝑋=𝑋^∗>0 and 𝑌=𝑌^∗>0 such that (i)𝑁_𝑐00𝐼^∗𝑋𝐴+𝐴^∗𝑋𝐼_𝑛𝐼_𝑛−𝛾₀𝐼𝑁_𝑐0−𝛾0𝐼_𝑚𝑁^∗_𝐼𝑁_𝐼000<0,(ii)𝑁^∗_𝑌𝑌𝐴^∗+𝐴𝑌𝑌𝑌−𝛾₀𝐼𝑁_𝑌<0,(iii)𝑋𝐼𝐼𝑌≥0, (43)(44)(45) where 𝛾_𝑚=𝛾₀{min_𝑖|𝜆_𝑖|²}, then there exists a DOF controller 𝒦_𝑖𝐴(𝑠)=^𝑖𝑘𝐵^𝑖_𝑘𝐶^𝑖_𝑘𝐷^𝑖_𝑘 which can solve the 𝐻_∞ problem with a given performance 𝛾₀>0 for the LTI system Σ_𝑖(𝑠) given by (41), 𝑖=2,3,…,𝑁.

Proof. According to Lemma 4, we have that there exists a DOF controller 𝒦_𝑖(𝑠) to solve the 𝐻_∞ problem for the LTI system Σ_𝑖(𝑠) with a given index 𝛾₀>0 if and only if there exist matrices 𝑋_𝑖=𝑋^∗_𝑖>0 and 𝑌=𝑌^∗>0 such that (a)𝑁^𝑖_𝑋00𝐼^∗𝑋_𝑖𝐴+𝐴^∗𝑋_𝑖0𝐼_𝑛0−𝛾₀𝐼𝐼0_𝑛0−𝛾₀𝐼𝑁^𝑖_𝑋0(0𝐼<0,b)𝑁_𝑌00𝐼^∗𝑌𝐴^∗+𝐴𝑌𝑌0𝑌−𝛾₀𝐼000−𝛾₀𝐼𝑁_𝑌0(0𝐼<0,c)𝑋_𝑖𝐼𝐼𝑌≥0, (46) where 𝑁^𝑖_𝑋 spans the kernel of [−𝜆_𝑖𝐶𝐼_𝑝]. Notice the following facts: −𝜆_𝑖𝐶𝐼_𝑝𝑁_𝑐−𝜆_𝑖𝑁_𝐼=−𝜆_𝑖𝐶𝑁_𝑐+𝐼_𝑝𝑁_𝐼=0, (47) and, for all nonzero 𝜆_𝑖, 𝑁rank^𝑐−𝜆_𝑖𝑁_𝐼𝑁=rank^𝑐𝑁_𝐼. (48) Thus, we can choose 𝑁^𝑖_𝑋=^𝑁𝑐_{−𝜆𝑖𝑁𝐼}. Let 𝑋_𝑖=𝑋 for all 𝑖=2,3,…,𝑁. Then, it is easy to obtain that 𝑁^𝑖_𝑋00𝐼^∗𝑋_𝑖𝐴+𝐴^∗𝑋_𝑖0𝐼_𝑛0−𝛾₀𝐼𝐼0_𝑛0−𝛾₀𝐼𝑁^𝑖_𝑋0=𝑁0𝐼_𝑐00𝐼^∗𝑋𝐴+𝐴^∗𝑋𝐼_𝑛𝐼_𝑛−𝛾₀𝐼𝑁_𝑐0−𝛾0𝐼₀𝜆_𝑖²𝑁^∗_𝐼𝑁_𝐼0≤𝑁00_𝑐00𝐼^∗𝑋𝐴+𝐴^∗𝑋𝐼_𝑛𝐼_𝑛−𝛾₀𝐼𝑁_𝑐0−𝛾0𝐼_𝑚𝑁^∗_𝐼𝑁_𝐼000<0 (49) 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_𝑖(𝑠) which can solve the 𝐻_∞ problem for the system Σ_𝑖(𝑠), 𝑖=2,3,…,𝑁. As the results stated in Section 2.2, if there exist 𝑋=𝑋^∗>0 and 𝑌=𝑌^∗>0 satisfying Lemma 11, then the DOF controller 𝒦_𝑖(𝑠) can be obtained by solving the following inequality: 𝐻_𝑋𝑐+𝑄𝜆_𝑖^∗𝐾^∗_𝑖𝑃_𝑋𝑐+𝑃^∗_𝑋𝑐𝐾_𝑖𝑄𝜆_𝑖<0, (50) for 𝐾_𝑖,𝑖=2,3,…,𝑁, where 𝐻_𝑋𝑐=𝑋_𝑐𝐴₀+𝐴^∗₀𝑋_𝑐0𝐶^∗₀0−𝛾₀𝐶𝐼0₀0−𝛾₀𝐼,𝑃_𝑋𝑐=𝐵^∗𝑋_𝑐00,𝑄𝜆_𝑖=𝐶𝜆_𝑖𝐷0,𝐴₀=𝐴000,𝐶₀=,,𝐶𝜆𝐼0𝐵=0𝐵𝐼0_𝑖=0𝐼−𝜆_𝑖,0𝐼,𝐶0𝐷=𝐾_𝑖=𝐴^𝑖_𝑘𝐵^𝑖_𝑘𝐶^𝑖_𝑘𝐷^𝑖_𝑘,𝑋_𝑐=𝑋𝑋₂𝑋^∗₂𝐼,𝑋₂𝑋^∗₂=𝑋−𝑌⁻¹. (51)(52)(53) Obviously, if there exists a common 𝐾 that makes the inequality (50) hold for all 𝑖=2,3,…,𝑁, then there exists a common 𝑛_𝑐th-order DOF controller 𝒦(𝑠) which can solve the 𝐻_∞ problem for the LTI systems Σ_𝑖,𝑖=2,3,…,𝑁. Thus, we have the following result.

Theorem 13. Given a scalar 𝛾>0, and let 𝛾₀=𝑑⁻¹_max𝛾. Assume that the fixed communication topology 𝒢_𝑁 has a spanning tree. Then, there exists a DOF control protocol (19) for the system (12) achieving consensus with 𝐻_∞ performance if(i)there exist matrices 𝑋>0 and 𝑌>0 satisfying (43), (44), and (45), (ii)there exists a matrix 𝐾 satisfying the following LMIs: 𝐻_𝑋𝑐+𝑄𝜆_𝑖^∗𝐾^∗𝑃_𝑋𝑐+𝑃^∗_𝑋𝑐𝐾𝑄𝜆_𝑖<0, (54) for all nonzero eigenvalues 𝜆_𝑖 of Laplacian matrix 𝐿_𝑁, where 𝐻_𝑋𝑐, 𝑃_𝑋𝑐, and 𝑄(𝜆_𝑖) are given in (51) and 𝑋_𝑐 is defined by (53).

Proof. From the analysis above, the conditions (i) and (ii) hold which implies that there exist matrices 𝐾 and 𝑋_𝑐>0 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 𝑛_𝑐 less than 𝑛, it is only required to add the additional constraint rank𝑋𝐼𝐼𝑌≤𝑛+𝑛_𝑐, (55) 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 𝐻_∞ 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: 𝑋>0 and 𝑌>0. Step 2: Construct 𝑋_𝑐>0 as (53). Step 3: Solve the 𝑁−1 LMIs (54) for a common feasible solution 𝐾.

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(𝑠) defined by (40) for the LTI systems (41) with 𝐷₂₁=0. Using Theorem 5.8 given by [29] and the fact that the kernels of 𝜆_𝑖𝐶 and 𝐶 are exactly equal for all nonzero 𝜆_𝑖, reproducing the steps of the proof of Theorem 13, we have the following results. Assume that the fixed communication topology 𝒢_𝑁 has a spanning tree, then there exists a DOF control protocol (14) with order 𝑛_𝑐 for the system (12) achieving consensus if (1) there exist matrices 𝑋=𝑋^∗>0 and 𝑌=𝑌^∗>0 such that 𝑁^∗_𝑋𝐴^∗𝑋+𝑋𝐴𝑁_𝑋<0,𝑁^∗_𝑌𝐴𝑌+𝑌𝐴^∗𝑁_𝑌<0,𝑋𝐼𝐼𝑌≥0,rank𝑋𝐼𝐼𝑌≤𝑛+𝑛_𝑐, (56) where 𝑁_𝑋 and 𝑁_𝑌 span the kernels of 𝐶 and 𝐵^∗, respectively,(2) there exists a matrix 𝐾 satisfying the following LMIs: 𝐴^∗₀𝑃+𝑃𝐴₀+𝐶𝜆_𝑖^∗𝐾^∗𝐵^∗𝑃+𝑃𝐵𝐾𝐶𝜆_𝑖<0, (57) for all nonzero eigenvalues 𝜆_𝑖 of Laplacian matrix 𝐿_𝑁, where 𝑃=𝑋𝑋₂𝑋^∗₂𝐼 (58) and 𝑋₂ satisfies 𝑋₂𝑋^∗₂=𝑋−𝑌⁻¹ and 𝐴₀, 𝐵, and 𝐶(𝜆_𝑖) 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 𝑁−1 LMIs after constructing 𝑋_𝑐, which increase the difficulty of the numerical calculation if the size of the multiagent system 𝑁 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 𝛼_𝑖 and 𝛽_𝑖 are the real part and imaginary part of 𝜆_𝑖, respectively, where 𝜆_𝑖 is the nonzero eigenvalue of Laplacian matrix 𝐿_𝑁,𝑖=2,3,…,𝑁. Let 𝛼₀=min_𝑖{𝛼_𝑖},𝛼₀=max_𝑖{𝛼_𝑖}, 𝛽₀=min_𝑖{𝛽_𝑖}, and 𝛽₀=max_𝑖{𝛽_𝑖}=−𝛽₀.

Theorem 17. Given a scalar 𝛾>0, and let 𝛾₀=𝑑⁻¹_max𝛾. Assume that the fixed communication topology 𝒢_𝑁 has a spanning tree. Then, there exists a DOF control protocol (19) for the system (12) achieving consensus with 𝐻_∞ performance if (i)there exist matrices 𝑋>0 and 𝑌>0 satisfying (43), (44), and (45);(ii)there exists a matrix 𝐾 such that 𝐻_𝑋𝑐+𝑄𝛼_𝑘+𝑗𝛽_𝑘^∗𝐾^∗𝑃_𝑋𝑐+𝑃^∗_𝑋𝑐𝐾𝑄𝛼_𝑘+𝑗𝛽_𝑘<0 (59) for (𝛼_𝑘,𝛽_𝑘)∈{(𝛼₀,𝛽₀),(𝛼₀,𝛽₀),(𝛼₀,𝛽₀),(𝛼₀,𝛽₀)}, where 𝐻_𝑋𝑐, 𝑃_𝑋𝑐, and 𝑄(𝛼_𝑘+𝑗𝛽_𝑘) are given in (51) and 𝑋_𝑐 is defined by (53).

Proof. To complete the proof, we only need to show that if LMI (59) holds for (𝛼_𝑘,𝛽_𝑘)∈{(𝛼₀,𝛽₀),(𝛼₀,𝛽₀),(𝛼₀,𝛽₀),(𝛼₀,𝛽₀)}, then LMI (54) holds for all nonzero eigenvalues 𝜆_𝑖=𝛼_𝑖+𝑗𝛽_𝑖 of Laplacian matrix 𝐿_𝑁,𝑖=2,3,…,𝑁.
Notice that 𝐻_𝑋𝑐+𝑄𝛼_𝑖+𝑗𝛽_𝑖^∗𝐾^∗𝑃_𝑋𝑐+𝑃^∗_𝑋𝑐𝐾𝑄𝛼_𝑖+𝑗𝛽_𝑖=^{𝑋𝑐𝛼𝐴−𝑖+𝑗𝛽𝑖𝐵𝐶+𝛼𝐴−𝑖+𝑗𝛽𝑖𝐵𝐶∗𝑋𝑐𝑋𝑐𝐵𝐶∗0𝐵∗𝑋}_𝑐^−𝛾_0𝐶^𝐼0_{00−𝛾0𝐼}<0 (60) which, in virtue of the Schur Complement Lemma, is equivalent to 𝑋_𝑐𝐴−𝛼_𝑖+𝑗𝛽_𝑖𝐵𝐶+𝐴−𝛼_𝑖+𝑗𝛽_𝑖𝐵𝐶^∗𝑋_𝑐+𝛾₀⁻¹𝑋_𝑐𝐵𝐵^∗𝑋_𝑐+𝐶^∗₀𝐶₀<0, (61) where 𝐴, 𝐵, and 𝐶 are defined by (22). For convenience, we denote ℋ𝛼_𝑖,𝛽_𝑖=𝑋_𝑐𝐴−𝛼_𝑖+𝑗𝛽_𝑖𝐵𝐶+𝐴−𝛼_𝑖+𝑗𝛽_𝑖𝐵𝐶^∗𝑋_𝑐+𝛾₀⁻¹𝑋_𝑐𝐵𝐵^∗𝑋_𝑐+𝐶₀𝐶^∗₀=𝐻₀+𝛼_𝑖𝐻₁+𝑗𝛽_𝑖𝐻₂, (62) where 𝐻₀=𝑋_𝑐𝐴+𝐴^∗𝑋_𝑐+𝛾₀⁻¹𝑋_𝑐𝐵𝐵^∗𝑋_𝑐+𝛾₀⁻¹𝐶₀𝐶^∗₀, 𝐻₁=−𝑋_𝑐𝐵𝐶−𝐶^∗𝐵^∗𝑋_𝑐, and 𝐻₂=−𝑋_𝑐𝐵𝐶+𝐶^∗𝐵^∗𝑋_𝑐.
In fact, there must exist 𝑠∈[0,1],𝑟∈[0,1] such that 𝛼_𝑖=𝑠𝛼₀+(1−𝑠)𝛼₀,𝛽_𝑖=𝑟𝛽₀+(1−𝑟)𝛽₀. When 𝑟≥𝑠, ℋ𝛼_𝑖,𝛽_𝑖=𝐻₀+𝑠𝛼₀+(1−𝑠)𝛼₀𝐻₁+𝑗𝑟𝛽₀+(1−𝑟)𝛽₀𝐻₂=𝑠ℋ𝛼₀,𝛽₀+(𝑟−𝑠)ℋ𝛼₀,𝛽₀+(1−𝑟)ℋ𝛼₀,𝛽₀<0. (63) Similarly, when 𝑠≥𝑟, we have ℋ𝛼_𝑖,𝛽_𝑖=𝑟ℋ𝛼₀,𝛽₀+(𝑠−𝑟)ℋ𝛼₀,𝛽₀+(1−𝑠)ℋ𝛼₀,𝛽₀<0. (64) 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 𝐻_∞ performance index is chosen as 𝛾=10. According to the results presented in Section 4, we have ,𝑋𝑋=1.8218−1.2978−1.29783.6435,𝑌=1.2145−0.9109−0.91093.6435_𝑐=.1.8218−1.2978−0.86880.2312−1.29783.64351.81480.1107−0.86881.81481.000000.23120.110701.0000 (65) Then, by solving the LMIs (54), we can get a feasible controller (40) with 𝐴_𝑘=−7.0377−5.6811−0.6005−5.2415,𝐵_𝑘=,𝐶−9.5464−0.5110_𝑘=3.45202.9461,𝐷_𝑘=4.9523, (66) and the 𝐻_∞ norm of system (26) is ‖𝐺_𝑑𝑧(𝑠)‖_∞=3.0707. With the designed DOF control protocol, the disagreement states between 𝑥_𝑖 and 𝑥₁ without communication error 𝐝(𝑡) are shown in Figure 3, 𝑖=2,3,…,6, which implies that the consensus can be reached when 𝐝(𝑡)=0. The communication error 𝐝(𝑡)∈ℒ₂[0,∞) is supposed to be 𝐝(𝑡)=𝜃sin(𝑡),𝑡∈[0,40],0,𝑡>40, (67) where 𝜃=[1.110.581.481.592.122.530.421.314.00.14]^𝑇. Under zero initial condition, the state trajectories of the six agents are depicted in Figure 4, and the corresponding energy trajectories of 𝐝(𝑡) and 𝐳(𝑡) are given in Figure 5. It is noted obviously that ‖𝐳(𝑡)‖_ℒ2≤‖𝐝(𝑡)‖_ℒ2. Thus, the multiagent system with the given DOF controller can achieve the consensus with the given 𝐻_∞ 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 𝑥𝑖 and 𝑥1 without communication error 𝐝(𝑡), 𝑖=2,3,…,6.

(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 𝐝(𝑡).

Figure 5 

Energy trajectories of the controlled output 𝐳(𝑡) and the communication error 𝐝(𝑡).

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 𝐻_∞ problem for 𝑁−1 linear time-invariant systems of order 𝑛, then the consensus with a desired 𝐻_∞ level can be reached. By using 𝐻_∞ 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.