Cooperative Control for Uncertain Multiagent Systems via Distributed Output Regulation

Abstract

The distributed robust output regulation problem for multiagent systems is considered. For heterogeneous uncertain linear systems and a linear exosystem, the controlling aim is to stabilize the closed-loop system and meanwhile let the regulated outputs converge to the origin asymptotically, by the help of local interaction. The communication topology considered is directed acyclic graphs, which means directed graphs without loops. With distributed dynamic state feedback controller and output feedback controller, respectively, the solvability of the problem and the algorithm of controller design are both investigated. The solvability conditions are given in terms of linear matrix inequalities (LMIs). It is shown that, for polytopic uncertainties, the distributed controllers constructed by solving LMIs can satisfy the requirements of output regulation property.

1. Introduction

Recently, there are amounts of researches on cooperative control for multiagent systems (MASs) because of broad applications. An MAS is a practical model to describe dynamic agents which can exchange information by communication, such as unmanned air vehicles and sensor networks. According to different control objectives, problems of consensus, tracking, formation, flocking and the rest have been widely studied.

Among those cooperative control problems, the consensus problem and the tracking problem share some common characteristics. The consensus problem requires the MAS to reach an agreement by protocols based on local information. In [1, 2], the consensus problem is primarily studied, and the basic problem framework is formed. The consensus problem has been investigated for different kinds of agents, such as first-order integrators in [3], second-order integrators in [4, 5], linear systems in [6–8], and nonlinear systems in [9–12]. Recently, the output consensus problem for heterogeneous systems also attracted researchers. The dynamics and even dimensions of the agents are possibly different, so it is desirable to focus on the synchronization of outputs. In [13], the consensus of a class of second-order integrators with unknown nonlinear dynamics is considered. As for high-order systems, the frequency domain approach is used to discuss the consensus of heterogenous linear systems in [14, 15]. Uncertain minimum-phase linear MASs are studied in [16], by a low-gain approach. In [7], general uncertain linear MASs are considered, and a sufficient and necessary condition for the solvability of the output consensus problem is proposed. It is admirable that Wieland et al. introduce an important concept of internal model to cooperative control, which is also fundamental in output regulation theory.

For the leader-follower consensus problem, also called consensus tracking problem, it involves one or multiple leaders and several followers. A leader is usually the target to be tracked, or the agent that directly receives the information of the target. Distributed controllers are designed to help all the agents to track one or multiple leaders by cooperation. The tracking problem for MASs has been studied in a lot of papers, such as [17–22]. Note that for heterogenous MASs, the outputs of all the agents are required to be synchronized in the issue of both consensus problem and tracking problem. To consider the two kinds of problems under a unified framework is one of the motivations to introduce distributed output regulation (DOR) problem.

According to [23], output regulation problem involves an exosystem and a regulated output defined by a combination of the measurement output and the output of the exosystem. Controllers are designed to stabilize the closed-loop system and modulate the regulated outputs to the origin. So, it has an attracting performance on solving a tracking problem in the presence of disturbances. The classic output regulation theory cannot directly be applied to MASs nevertheless. Actually the controllers obtained are probably not in a distributed form. So, a framework of DOR for MASs is introduced in [24–26]. What is mainly different from the classic theory is that the controllers have to be distributed, and only local information is available. In [24], homogenous linear MASs and a directed topology are considered, and dynamic state/output feedback controllers are designed. In [25], heterogenous MASs are challenged, and effective controllers are obtained under the directed acyclic topology. Different from the two works above, the limits on topology are dispelled in [26] by reconstructing the form of controllers. The communication of relative states of controllers replaces the communication of relative outputs.

This paper is basically motivated by [25]. In [25], the distributed controllers are robust to uncertain dynamics with sufficiently small uncertainties. However, more analyses based on information of the uncertainties are not involved in [25]. What we focus on in our paper is that how to design robust controllers if the uncertainty is structured. We suppose that the uncertainties are in a polytopic form. For the distributed robust output regulation problem, we give sufficient conditions of the solvability in terms of linear matrix inequalities (LMIs) and present an approach of controllers design.

An outline of this paper is as follows. In Section 2, some preliminaries and the problem statement are given. In Section 3, for distributed output regulation problem with polytopic uncertainties, the sufficient conditions of solvability and algorithms of both state and output feedback controllers design are proposed. In Section 4, a practical example is taken to show the control effect of our approach, which is compared with that of the algorithm given in [25]. In Section 5, a conclusion is given.

The following notations will be used throughout this paper. ℝ is the set of real numbers. 𝐼_𝑛 is the 𝑛-dimensional identity matrix. 0_𝑚×𝑛 is the zero matrix with 𝑚 rows and 𝑛 columns, and 0 is the zero matrix with appropriate dimensions. For a symmetric matrix 𝑀∈ℝ^𝑛×𝑛, 𝑀>0 which means that 𝑀 is positive definite. 𝑀^𝑇 is the transposition of the matrix 𝑀. diagblock(𝑀₁,…,𝑀_𝑝) means a block diagonal matrix with 𝑀₁,…,𝑀_𝑝 as the diagonal entries. ⊗ denotes the Kronecker product.

2. Problem Statement

Let us begin with notations in graph theory [27]. A graph is denoted by 𝒢=(𝒱,ℰ), where 𝒱={𝑣₁,𝑣₂,…,𝑣_𝑁} is the set of 𝑁 nodes and ℰ⊆𝒱×𝒱 is the set of edges. An edge from node 𝑣_𝑗 to node 𝑣_𝑖 is denoted by (𝑖,𝑗)∈ℰ. A subset of ℰ{(𝑖₁,𝑖₂),(𝑖₂,𝑖₃),…,(𝑖_𝑘,𝑖_𝑘+1)} is called a path from 𝑣_𝑖𝑘+1 to 𝑣_𝑖1. If 𝑖₁=𝑖_𝑘+1, the path is called a loop. 𝑁_𝑖 denotes the neighbor set {𝑗∶(𝑖,𝑗)∈ℰ,1≤𝑗≤𝑁}, whose cardinality is |𝑁_𝑖|. A constant matrix 𝐴_𝑑=[𝑎_𝑖𝑗]∈ℝ^𝑁×𝑁 is called the adjacency matrix of graph 𝒢 if 𝑎_𝑖𝑗=1/|𝑁_𝑖| when (𝑖,𝑗)∈ℰ and 𝑎_𝑖𝑗=0 when (𝑖,𝑗)∉ℰ. And a constant matrix 𝐿=[𝑙_𝑖𝑗]=𝐼_𝑁−𝐴_𝑑∈ℝ^𝑁×𝑁 is called the Laplacian matrix of graph 𝒢. A graph is called an undirected graph if for all 1≤𝑖,𝑗≤𝑁, 𝑎_𝑖𝑗=𝑎_𝑗𝑖. Or else, it is called a directed graph. If a directed graph does not contain a loop, it is called a directed acyclic graph.

Consider an exosystem with 𝜔∈ℝ^𝑞 as the state, whose dynamics can be described as follows: 𝜔̇(𝑡)=𝑆𝜔(𝑡), (1) where 𝑆∈ℝ^𝑞×𝑞 is a constant matrix, satisfying the following assumption as that in [23, 25].

Assumption 1. 𝑆has no eigenvalues with negative real parts.

A MAS consists of 𝑁 nonidentical dynamic agents which can exchange information among neighborhood. For 𝑖=1,…,𝑁, the 𝑖th agent can be expressed by 𝑥̇_𝑖(𝑡)=𝐴_𝑖𝑥_𝑖(𝑡)+𝐵_𝑖𝑢_𝑖(𝑡)+𝐸_𝑖𝜔(𝑡),𝑦_𝑖(𝑡)=𝐶_𝑖𝑥_𝑖(𝑡),𝑒_𝑖(𝑡)=𝐶_𝑖𝑥_𝑖(𝑡)+𝐹𝜔(𝑡), (2) where 𝑥_𝑖∈ℝ^𝑝𝑖, 𝑢_𝑖∈ℝ^𝑚𝑖, 𝑦_𝑖∈ℝ^𝑛, and 𝑒_𝑖∈ℝ^𝑛 are, respectively, the state, the control input, the measurement output, and the regulated output of the 𝑖th agent. 𝐹∈ℝ^𝑛×𝑞 is a certain constant matrix, while those matrices 𝐴_𝑖∈ℝ^{𝑝𝑖×𝑝𝑖}, 𝐵_𝑖∈ℝ^{𝑝𝑖×𝑚𝑖}, and 𝐶_𝑖∈ℝ^{𝑛×𝑝𝑖} are uncertain matrices represented as 𝐴_𝑖=^𝑔𝑖_𝑘=1𝜆_𝑘𝑖𝐴_𝑘𝑖,𝐵_𝑖=^𝑔𝑖_𝑘=1𝜆_𝑘𝑖𝐵_𝑘𝑖,𝐶_𝑖=^𝑔𝑖_𝑘=1𝜆_𝑘𝑖𝐶_𝑘𝑖, (3) where 𝐴_𝑘𝑖,𝐵_𝑘𝑖,and𝐶_𝑘𝑖 are known constant matrices, and 𝜆_𝑘𝑖 are nonnegative constants satisfying that ∑^𝑔𝑖_𝑘=1𝜆_𝑘𝑖=1, for 𝑖=1,…,𝑁,𝑘=1,…,𝑔_𝑖. 𝐸_𝑖∈ℝ^{𝑝𝑖×𝑞} is an arbitrary constant matrix.

The controlling aim is to stabilize the closed-loop system and also to regulate 𝑒₁,…,𝑒_𝑁 to the origin. For 𝑖=1,…,𝑁, if 𝑒_𝑖 is available to the 𝑖th agent, the output regulation of the 𝑖th agent is simple to be achieved by classic output regulation theory. However, only some of the agents can get information of their own regulated outputs, which are called leader nodes. The set of their serial numbers is denoted by ℒ, while other agents utilize the relative output among neighbors to accomplish the output regulation property. To ensure that all the agents can receive the information of the exosystem by local interaction, the communication graph of the MAS satisfies the following assumption.

Assumption 2. Graph 𝒢 is a directed acyclic graph. And for each nonleader node 𝑣_𝑖, there exists a leader node 𝑣_𝑗 such that a path from node 𝑣_𝑗 to node 𝑣_𝑖 exists.

In this note, two kinds of distributed dynamic feedback controllers are considered.(I) Distributed dynamic state feedback controller: 𝑧̇_𝑖(𝑡)=𝐺₁𝑧_𝑖(𝑡)+𝐺₂𝑒_𝑖𝜔(𝑡),𝑢_𝑖(𝑡)=𝐾_1𝑖𝑥_𝑖(𝑡)+𝐾_2𝑖𝑧_𝑖(𝑡),𝑧_𝑖(𝑡)∈ℝ^𝑐,𝑞≤𝑐,𝑖=1,…,𝑁, (4) where 𝑒_𝑖𝜔(𝑡)=𝑒_𝑖(𝑡)1𝑖∈ℒ𝑁_𝑖𝑗∈∑_𝑁𝑖𝑦_𝑖(𝑡)−𝑦_𝑗(𝑡)𝑖∉ℒ. (5)𝐾_1𝑖,  𝐾_2𝑖, 𝐺₁, and 𝐺₂ are designed matrices with appropriate dimensions.(II) Distributed dynamic output feedback controller: 𝑧̇_𝑖(𝑡)=𝐺_1𝑖𝑧_𝑖(𝑡)+𝐺_2𝑖𝑒_𝑖𝜔(𝑡),𝑢_𝑖(𝑡)=𝐾_𝑖𝑧_𝑖(𝑡),𝑧_𝑖(𝑡)∈ℝ^{𝑐+𝑝𝑖},𝑞≤𝑐,𝑖=1,…,𝑁. (6)𝐾_𝑖, 𝐺_1𝑖, and 𝐺_2𝑖 are designed matrices with appropriate dimensions.

The variables 𝑒_𝑖𝜔(𝑡) in distributed feedback control laws (4) and (6) are measurable relative outputs. For 𝑖∈ℒ, 𝑒_𝑖𝜔(𝑡) is the regulated output 𝑒_𝑖(𝑡). For 𝑖∉ℒ, 𝑒_𝑖𝜔(𝑡) is the average of relative output errors between 𝑖th agent and its neighbors.

In the sequel, we rewrite the closed-loop system into a composite form. Let 1_𝑁 be a column vector with all the elements as 1. 𝐿_𝑖 and 𝑙_𝑖, respectively, denote the 𝑖th row of 𝐿 and 𝐿1_𝑁. For 𝑖∈ℒ, 𝑙_𝑖=1, while for 𝑖∉ℒ, 𝑙_𝑖=0. Let 𝜒=(𝑥^𝑇₁,𝑧^𝑇₁,…,𝑥^𝑇_𝑁,𝑧^𝑇_𝑁)^𝑇,𝑒=(𝑒^𝑇₁,…,𝑒^𝑇_𝑁)^𝑇, 𝒜₁=diagblock𝒜₁₁,…,𝒜_1𝑁,𝒜₂=diagblock𝒜₂₁,…,𝒜_2𝑁,𝒞₁=diagblock𝐶₁0_𝑛×𝑐,…,𝐶_𝑁0_𝑛×𝑐,𝒞₂=diagblock𝐶₁0_{𝑛×(𝑐+𝑝1)},…,𝐶_𝑁0_{𝑛×(𝑐+𝑝𝑁)},𝒢₁=𝐿₁⊗𝐺^′₂₁⋮𝐿_𝑁⊗𝐺^′_2𝑁,𝒢₂=𝐿₁⊗𝐺^′₂₁⋮𝐿_𝑁⊗𝐺^′_2𝑁,ℰ₁=𝐸₁𝑙₁𝐺₂𝐹⋮𝐸_𝑁𝑙_𝑁𝐺₂𝐹,ℰ₂=𝐸₁𝑙₁𝐺₂₁𝐹⋮𝐸_𝑁𝑙_𝑁𝐺_2𝑁𝐹, (7) where 𝒜_1𝑖=^{𝐴𝑖+𝐵𝑖𝐾1𝑖𝐵𝑖𝐾2𝑖}_0𝐺1,  𝒜_2𝑖=^{𝐴𝑖𝐵𝑖𝐾𝑖}_0𝐺1𝑖,  𝐺^′_2𝑖=^{0𝑖𝑛×𝑝}_𝐺2, and 𝐺^′_2𝑖=^{0𝑖𝑛×𝑝}_𝐺2𝑖, for 𝑖=1,…,𝑁. Then, the closed-loop system (1), (2) with controllers (4) or (6) can be rewritten as 𝜔̇(𝑡)(𝑡),𝜒̇(𝑡)=𝑆𝜔=𝐴_𝑐𝜒(𝑡)+𝐵_𝑐𝜔(𝑡),𝑒(𝑡)=𝐶_𝑐𝜒(𝑡)+𝐹_𝑐𝜔(𝑡). (8) With the controller (4), 𝐴_𝑐=𝒜₁+𝒢₁𝒞₁,𝐵_𝑐=ℰ₁,𝐶_𝑐=𝒞₁,𝐹_𝑐=1_𝑁⊗𝐹, (9) while with the controller (6), 𝐴_𝑐=𝒜₂+𝒢₂𝒞₂,𝐵_𝑐=ℰ₂,𝐶_𝑐=𝒞₂,𝐹_𝑐=1_𝑁⊗𝐹. (10) As a result, the distributed output regulation problem studied in this paper is given as follows.

Problem 3. Distributed output regulation problem: design controllers in the form of (4) or (6) such that the closed-loop system (8) has the following properties. (i)It is exponentially stable at the origin with 𝜔=0.(ii)For all initial values 𝜒(0) and 𝜔(0), lim_𝑡→∞𝑒(𝑡)=0.

3. Main Results

In this section, we give two theorems about the solvability of Problem 3 and the approach of controller design. First of all, we introduce the concept of quadratic stability and related lemmas, which will be used later.

Definition 4 (Amato [28]). Consider a parametric uncertain linear system given by 𝑥̇(𝑡)=𝐴𝑝₀𝑥(𝑡),𝑡∈[0,+∞), (11) where 𝑥(𝑡)∈ℝ^𝑛0, 𝑝₀∈𝐻⊂ℝ^𝑚0 is the vector of uncertain parameters, where 𝐻 is a hyperbox, and 𝐴(⋅) is continuous. This system is said to be quadratically stable (QS) in 𝐻 if and only if there exists a symmetric positive definite matrix 𝑃∈ℝ^𝑛0×𝑛0 such that for all 𝑝₀∈𝐻, 𝐴^𝑇(𝑝₀)𝑃+𝑃𝐴(𝑝₀)<0.

Lemma 5 (Amato [28]). Assume that the system (11) is QS. Then, for any function 𝑝₀(⋅) that is piecewise continuous on ℝ⁺ and valued on 𝑊, the linear time-varying system 𝑥̇(𝑡)=𝐴(𝑝₀(𝑡))𝑥(𝑡) is exponentially stable.

Lemma 6 (Amato [28]). Assume that the uncertain system is in a polytopic form; that is, 𝑥̇(𝑡)=𝐴(𝑡)𝑥(𝑡),𝐴(𝑡)∈conv𝐴₁,𝐴₂,…,𝐴_𝑔, (12) where 𝐴₁,…,𝐴_𝑔∈ℝ^𝑛0×𝑛0 and conv{⋅} represents the convex hull of the following matrices. It is QS if and only if there exists a positive definite symmetric matrix 𝑃 such that for 𝑘=1,…,𝑔,  𝐴^𝑇_𝑘𝑃+𝑃𝐴_𝑘<0.

Second, we need to recall the concept of internal model and its property.

Definition 7 (Huang [23]). Given any square matrix 𝑆, a pair of matrices (𝒢₁,𝒢₂) is said to incorporate a 𝑝-copy internal model of the matrix 𝑆 if the pair satisfies that 𝒢₁𝑆=𝑇¹𝑆₂0𝐺₁𝑇⁻¹,𝒢₂𝑆=𝑇³𝐺₂, (13) where 𝑆₁, 𝑆₂, and 𝑆₃ are arbitrary constant matrices of appropriate dimensions, 𝑇 is any nonsingular matrix with the same dimension as 𝒢₁, and 𝐺₁, 𝐺₂ are described as follows: 𝐺₁=blockdiag𝛽₁,…,𝛽_𝑝,𝐺₂=blockdiag𝜎₁,…,𝜎_𝑝, (14) where for 𝑗=1,…,𝑝, 𝛽_𝑗 is a constant square matrix of dimension 𝑑_𝑗 for some integer 𝑑_𝑗 and 𝜎_𝑗 is a constant column vector of dimension 𝑑_𝑗 such that (i)𝛽_𝑖 and 𝜎_𝑖 are controllable,(ii)the minimal polynomial of 𝑆 divides the characteristic polynomial of 𝛽_𝑖.

Lemma 8 (Huang [23]). Under Assumption 1, assume that (𝒢₁,𝒢₂) incorporates a 𝑝-copy internal model of 𝑆. Let 𝐴𝐵𝒢₂𝐶𝒢₁ (15) be exponentially stable, where 𝐴,𝐵, and 𝐶 are any matrices with appropriate dimensions. Then, for any matrices 𝐸 and 𝐹 of appropriate dimensions, the following matrix equations 𝑋𝑆=𝐴𝑋+𝐵𝑍+𝐸,𝑍𝑆=𝒢₁𝑍+𝒢₂𝐶𝑋+𝐹 (16) have a unique solution 𝑋 and 𝑍. Moreover, 𝑋 satisfies 𝐶𝑋+𝐹=0.

Based on these preparations, theorems on solvability of Problem 3 with state/output feedback controllers are given as follows.

Theorem 9. Suppose that Assumptions 1 and 2 hold and the pair (𝐺₁,𝐺₂) is a 𝑝-copy internal model of the exosystem (1). For 𝑖=1,…,𝑁, if there exist a matrix 𝑌_𝑖 and a positive definite matrix 𝑃_𝑖 satisfying that 𝐴_𝑘𝑖𝑃_𝑖+𝑃_𝑖𝐴^𝑇_𝑘𝑖+𝐵_𝑘𝑖𝑌_𝑖+𝑌^𝑇_𝑖𝐵^𝑇_𝑘𝑖<0,𝑘=1,…,𝑔_𝑖, (17) where 𝐴_𝑘𝑖=𝐴_𝑘𝑖0𝐺₂𝐶_𝑘𝑖𝐺₁,𝐵_𝑘𝑖=𝐵_𝑘𝑖0, (18) Then, Problem 3 is solvable by a dynamic state feedback controller (4), where (𝐾_1𝑖,𝐾_2𝑖)=𝐾_𝑖=𝑌_𝑖𝑃_𝑖⁻¹.

Proof. Let 𝛾_𝑖(𝑡)=(𝑥^𝑇_𝑖,𝑧^𝑇_𝑖)^𝑇. Then, the subsystems of (8) with controller (4) can be written as follows: for 𝑖∈ℒ, 𝛾̇_𝑖(𝑡)=𝐴_𝑖+𝐵_𝑖𝐾_1𝑖𝐵_𝑖𝐾_2𝑖𝐺₂𝐶_𝑖𝐺₁𝛾_𝑖(𝑡)+𝐸_𝑖𝐺₂𝐹𝑤(𝑡), (19) and for 𝑖∉ℒ, 𝛾̇_𝑖(𝑡)=𝐴_𝑖+𝐵_𝑖𝐾_1𝑖𝐵_𝑖𝐾_2𝑖𝐺₂𝐶_𝑖𝐺₁𝛾_𝑖(𝑡)+𝐸_𝑖0𝑤(𝑡)+1𝑁_{𝑖𝑗∈𝑁𝑖}00−𝐺₂𝐶_𝑗0𝛾_𝑗(𝑡). (20) According to [29], by relabeling the nodes, a directed acyclic graph could be put into an ordered form. That is to say, for all edges (𝑖,𝑗)∈ℰ, 𝑖>𝑗 holds. Notice that matrix 𝐴_𝑐 is consequently a block lower triangular matrix and system (8) with 𝑤=0 is asymptotically stable if and only if all the subsystems below are asymptotically stable 𝛾̇_𝑖(𝑡)=𝐴_𝑖+𝐵_𝑖𝐾_1𝑖𝐵_𝑖𝐾_2𝑖𝐺₂𝐶_𝑖𝐺₁𝛾_𝑖(𝑡)=^𝑔𝑖_𝑘=1𝜆_𝑘𝑖𝐴_𝑘𝑖+𝐵_𝑘𝑖𝐾_1𝑖𝐵_𝑘𝑖𝐾_2𝑖𝐺₂𝐶_𝑘𝑖𝐺₁𝛾_𝑖(𝑡)=^𝑔𝑖_𝑘=1𝜆_𝑘𝑖𝐴_𝑘𝑖+𝐵_𝑘𝑖𝐾_𝑖𝛾_𝑖(𝑡). (21) Consider LMI (17). When it holds for a symmetric positive definite matrix 𝑃_𝑖 and a matrix 𝑌_𝑖, let 𝑃_𝑖=𝑃_𝑖⁻¹, and let 𝐾_𝑖=𝑌_𝑖𝑃_𝑖. Pre- and postmultiplied by 𝑃_𝑖, (17) is equivalent to the following inequality: 𝐴_𝑘𝑖+𝐵_𝑘𝑖𝐾_𝑖^𝑇𝑃_𝑖+𝑃_𝑖𝐴_𝑘𝑖+𝐵_𝑘𝑖𝐾_𝑖<0. (22) According to Lemma 6, when it holds, the subsystems (21) are all QS. And according to Lemma 5, for any polytopic uncertainties, the subsystems are exponentially stable. The system (8) with 𝑤=0 is consequently exponentially stable. The condition (i) of Problem 3 has been satisfied. In the following, the error 𝑒(𝑡) is proved to converge to zero.
Since the matrix 𝐴_𝑖+𝐵_𝑖𝐾_1𝑖𝐵_𝑖𝐾_2𝑖𝐺₂𝐶_𝑖𝐺₁ is Hurwitz, according to Lemma 8, for any 𝐸_𝑖 and 𝐹_𝑖, the following matrix equations 𝑋_𝑖𝑆=𝐴_𝑖+𝐵_𝑖𝐾_1𝑖𝑋_𝑖+𝐵_𝑖𝐾_2𝑖𝑍_𝑖+𝐸_𝑖,𝑍_𝑖𝑆=𝐺₁𝑍_𝑖+𝐺₂𝐶_𝑖𝑋_𝑖+𝐹_𝑖 (23) have a unique solution 𝑋_𝑖 and 𝑍_𝑖, and at the same time, 𝐶_𝑖𝑋_𝑖+𝐹_𝑖=0. For 𝑖∈ℒ, 𝐹_𝑖=𝐹, otherwise, 𝐹_𝑖=(−1/|𝑁_𝑖|)∑_{𝑗∈𝑁𝑖}𝐶_𝑗𝑋_𝑗. Therefore, the 𝑁 coupled matrix equations have a unique solution (𝑋₁,𝑍₁,…,𝑋_𝑁,𝑍_𝑁).
The proof will be given by induction. As mentioned earlier, each agent can only receive information from the agents with smaller labels, after appropriately relabeling the directed acyclic graph. That is to say, for 𝑖=1,2,…,𝑁, 𝐹_𝑖=𝐹 or 𝐹_𝑖=(−1/|𝑁_𝑖|)∑^𝑖−1_{𝑗=1,𝑗∈𝑁𝑖}𝐶_𝑗𝑋_𝑗. For 𝑖=1, the first agent does not communicate with any other agents but the exosystem, so 𝐹₁=𝐹. And we can obtain a unique pair of (𝑋₁,𝑍₁) that satisfies (23) and 𝐶₁𝑋₁+𝐹₁=0. For 𝑖=2, if it is a leader, then 𝐹₂=𝐹; or else, it can only communicate with the first agent, which means that 𝐹₂=−𝐶₁𝑋₁. In both cases 𝐹₂ is a certain and known matrix, so there exists a unique solution (𝑋₂,𝑍₂) of (23) and 𝐶₂𝑋₂+𝐹₂=0. Suppose that for 𝑖=1,2,…,𝑚, the solution (𝑋_𝑖,𝑍_𝑖) has already been obtained. And then for 𝑖=𝑚+1, either 𝐹_𝑖=𝐹 or 𝐹_𝑖=(−1/|𝑁_𝑖|)∑^𝑖−1_{𝑗=1,𝑗∈𝑁𝑖}𝐶_𝑗𝑋_𝑗 is a certain and known matrix. So, the unique solution (𝑋_𝑚+1,𝑍_𝑚+1) also exists. By induction, we can obtain the unique solution (𝑋₁,𝑍₁,…,𝑋_𝑁,𝑍_𝑁), and at the same time, 𝐶₁𝑋₁+𝐹₁=⋯=𝐶_𝑁𝑋_𝑁+𝐹_𝑁=0.
Take notations Γ_𝑖=^𝑋𝑖_𝑍𝑖 and 𝛾_𝑖=𝛾_𝑖−Γ_𝑖𝜔, and substitute (23) into (19) and (20). It is obtained that ̇𝛾_𝑖(𝑡)=𝛾̇_𝑖(𝑡)−Γ_𝑖(𝑡)=𝑆𝜔𝐴_𝑖+𝐵_𝑖𝐾_1𝑖𝐵_𝑖𝐾_2𝑖𝐺₂𝐶_𝑖𝐺₁𝛾_𝑖(𝑡)̇𝛾,𝑖∈ℒ,_𝑖(𝑡)=𝛾̇_𝑖(𝑡)−Γ_𝑖(𝑡)=𝑆𝜔𝐴_𝑖+𝐵_𝑖𝐾_1𝑖𝐵_𝑖𝐾_2𝑖𝐺₂𝐶_𝑖𝐺₁𝛾_𝑖(𝑡)−1𝑁_{𝑖𝑗∈𝑁𝑖}𝐺00₂𝐶_𝑗0𝛾_𝑗(𝑡),𝑖∉ℒ. (24) By the first part of the proof, the system above is exponentially stable; that is, lim_𝑡→∞𝛾_𝑖(𝑡)=0,𝑖=1,…,𝑁. (25) Since 𝑒_𝑖𝑤(𝑡)=𝐶_𝑖𝑥_𝑖(𝑡)(𝑡)=+𝐹𝜔𝐶_𝑖0𝛾_𝑖(𝑡)+𝐶_𝑖𝑋_𝑖𝜔(𝑡)𝑒+𝐹,𝑖∈ℒ,_𝑖𝜔(𝑡)=𝐶_𝑖𝑥_𝑖(𝑡)−1𝑁_{𝑖𝑗∈𝑁𝑖}𝐶_𝑗𝑥_𝑗(𝑡)=𝐶_𝑖0𝛾_𝑖(𝑡)−1𝑁_{𝑖𝑗∈𝑁𝑖}𝐶_𝑗0𝛾_𝑗(𝑡)+𝐶_𝑖𝑋_𝑖+𝐹_𝑖𝜔(𝑡),𝑖∉ℒ, (26) it is obvious that lim_𝑡→∞𝑒_𝑖𝜔(𝑡)=0,𝑖=1,…,𝑁. (27) Under Assumption 2, the statement (27) is equivalent to lim_𝑡→∞𝑒(𝑡)=0. This completes the proof.

Remark 10. When there is no uncertainty in the system (2), 𝐴_𝑖=𝐴_1𝑖, 𝐵_𝑖=𝐵_1𝑖, and 𝐶_𝑖=𝐶_1𝑖, 𝑖=1,…,𝑁. Then, the conclusion of Theorem 9 still holds if the solvability of LMI (17) is replaced by the statement that the pair (𝐴_1𝑖,𝐵_1𝑖) is stabilizable. In fact, according to [23], from the two statements,(i)the pair (𝐴_1𝑖,𝐵_1𝑖) is stabilizable,(ii)Assumption 1 holds, and the pair (𝐺₁,𝐺₂) is an internal model of 𝑆; it is followed that the pair 𝐴_1𝑖0𝐺₂𝐶_1𝑖𝐺₁,𝐵_1𝑖0 (28)is stabilizable. It is equivalent to LMI (17) that holds with 𝑔_𝑖=1, 𝑖=1,…,𝑁. Specially, if for 𝑖=1,…,𝑁, 𝐴_1𝑖=𝐴, 𝐵_2𝑖=𝐵, and 𝐶_1𝑖=𝐶, 𝑔_𝑖=1, then in Theorem 9, the solvability of LMI (17) can be replaced by the statement that the pair (𝐴,𝐵) is stabilizable.

Theorem 11. Suppose that Assumptions 1 and 2 hold and the pair (𝐺₁,𝐺₂) is an internal model of the exosystem (1) if the following conditions are satisfied: (i)For 𝑖=1,…,𝑁, there exist a matrix 𝑌_𝑖 and a symmetric positive definite matrix 𝑃_𝑖 satisfying LMI (17),(ii)For 𝑖=1,…,𝑁, and the matrices 𝑌_𝑖, 𝑃_𝑖 obtained in (i), let 𝑃_𝑖=𝑃_𝑖⁻¹, and let 𝐾_𝑖=(𝐾_1𝑖,𝐾_2𝑖)=𝑌_𝑖𝑃_𝑖, and there exist matrices 𝑀_1𝑖, 𝑀_2𝑖, 𝑀_3𝑖 and a symmetric positive definite matrix 𝑄_𝑖 satisfying the following LMI: Π_11,𝑘𝑖Π_12,𝑘𝑖Π^𝑇_12,𝑘𝑖Π_22,𝑘𝑖<0, (29)where Π_11,𝑘𝑖=𝑃_𝑖𝐴_𝑘𝑖+𝐴^𝑇_𝑘𝑖𝑃_𝑖+𝐵_𝑘𝑖𝐾_𝑖𝑃_𝑖+𝑃_𝑖𝐾^𝑇_𝑖𝐵^𝑇_𝑘𝑖,Π_12,𝑘𝑖=𝑃_𝑖𝐵_𝑘𝑖𝐾_1𝑖0+−𝐴^𝑇_𝑘𝑖𝑄_𝑖−𝐾^𝑇_1𝑖𝐵_𝑘𝑖𝑄_𝑖+𝑀^𝑇_1𝑖+𝐶^𝑇_𝑘𝑖𝑀^𝑇_3𝑖−𝐾^𝑇_2𝑖𝐵^𝑇_𝑘𝑖𝑄_𝑖+𝑀^𝑇_2𝑖,Π_22,𝑘𝑖=−𝑄_𝑖𝐵_𝑘𝑖𝐾_1𝑖−𝐾^𝑇_1𝑖𝐵^𝑇_𝑘𝑖𝑄_𝑖+𝑀_1𝑖+𝑀^𝑇_1𝑖,𝑘=1,…,𝑔_𝑖. (30) Then, Problem 3 is solvable by a dynamic output feedback controller (6) with 𝐺_1𝑖=𝑇_1𝑖𝑇_2𝑖0𝐺₁,𝐺_2𝑖=𝑇_3𝑖𝐺₂, (31) where 𝑇_1𝑖=𝑄_𝑖⁻¹𝑀_1𝑖, 𝑇_2𝑖=𝑄_𝑖⁻¹𝑀_2𝑖, and 𝑇_3𝑖=𝑄_𝑖⁻¹𝑀_3𝑖.

Proof. For 𝑖=1,…,𝑁, the subsystems of (8) with (6) and (31) can be written in such a form: for 𝑖∈ℒ, 𝛾̇_𝑖(𝑡)=𝐴_𝑖𝐵_𝑖𝐾_1𝑖𝐵_𝑖𝐾_2𝑖𝑇_3𝑖𝐶_𝑖𝑇_1𝑖𝑇_2𝑖𝐺₂𝐶_𝑖0𝐺₁𝛾_𝑖(𝑡)+𝐸_𝑖𝑇_3𝑖𝐹𝐺₂𝐹𝑤(𝑡), (32) and for 𝑖∉ℒ, 𝛾̇_𝑖(𝑡)=𝐴_𝑖𝐵_𝑖𝐾_1𝑖𝐵_𝑖𝐾_2𝑖𝑇_3𝑖𝐶_𝑖𝑇_1𝑖𝑇_2𝑖𝐺₂𝐶_𝑖0𝐺₁𝛾_𝑖(𝑡)+𝐸_𝑖00𝑤(𝑡)−1𝑁_{𝑖𝑗∈𝑁𝑖}𝑇000_3𝑖𝐶_𝑗𝐺00₂𝐶_𝑗𝛾00^𝑗(𝑡). (33)
Similar to the proof of the previous theorem, the stability of the system (8) with 𝜔=0 is dependent on the following subsystems: 𝛾̇_𝑖(𝑡)=𝐴_𝑖𝐵_𝑖𝐾_1𝑖𝐵_𝑖𝐾_2𝑖𝑇_3𝑖𝐶_𝑖𝑇_1𝑖𝑇_2𝑖𝐺₂𝐶_𝑖0𝐺₁𝛾_𝑖(𝑡)=^𝑔𝑖_𝑘=1𝜆_𝑘𝑖𝐴_𝑘𝑖𝐵_𝑘𝑖𝐾_1𝑖𝐵_𝑘𝑖𝐾_2𝑖𝑇_3𝑖𝐶_𝑘𝑖𝑇_1𝑖𝑇_2𝑖𝐺₂𝐶_𝑘𝑖0𝐺₁𝛾_𝑖(𝑡). (34) Let 𝜉_𝑖(𝑡)=𝐼_𝑝𝑖0000𝐼_𝑐−𝐼_𝑝𝑖𝐼_𝑝𝑖0𝛾_𝑖(𝑡). (35) Then, the subsystem (34) is similar to the following system: 𝜉̇_𝑖(𝑡)=^𝑔𝑖_𝑘=1𝜆_𝑘𝑖𝐴_𝑘𝑖+𝐵_𝑘𝑖𝐾_1𝑖𝐵_𝑘𝑖𝐾_2𝑖𝐵_𝑘𝑖𝐾_1𝑖𝐺₂𝐶_𝑘𝑖𝐺₁0𝑈_𝑘𝑖𝑉_𝑘𝑖𝑊_𝑘𝑖𝜉_𝑖(𝑡), (36) where 𝑈_𝑘𝑖=𝑇_1𝑖+𝑇_3𝑖𝐶_𝑘𝑖−𝐴_𝑘𝑖−𝐵_𝑘𝑖𝐾_1𝑖,𝑉_𝑘𝑖=𝑇_2𝑖−𝐵_𝑘𝑖𝐾_2𝑖,𝑊_𝑘𝑖=𝑇_1𝑖−𝐵_𝑘𝑖𝐾_1𝑖. (37) According to Lemma 6, the system (36) is QS if there exists a symmetric positive definite matrix 𝑅_𝑖>0 such that 𝐴_𝑘𝑖+𝐵_𝑘𝑖𝐾_1𝑖𝐵_𝑘𝑖𝐾_2𝑖𝐵_𝑘𝑖𝐾_1𝑖𝐺₂𝐶_𝑘𝑖𝐺₁0𝑈_𝑘𝑖𝑉_𝑘𝑖𝑊_𝑘𝑖^𝑇𝑅_𝑖+𝑅_𝑖𝐴_𝑘𝑖+𝐵_𝑘𝑖𝐾_1𝑖𝐵_𝑘𝑖𝐾_2𝑖𝐵_𝑘𝑖𝐾_1𝑖𝐺₂𝐶_𝑘𝑖𝐺₁0𝑈_𝑘𝑖𝑉_𝑘𝑖𝑊_𝑘𝑖<0. (38) Suppose that 𝑅_𝑖 is a block diagonal matrix, which means that 𝑅_𝑖=^𝑃𝑖0_0𝑄𝑖, 𝑃_𝑖∈ℝ^{(𝑝𝑖+𝑐)×(𝑝𝑖+𝑐)}, and 𝑄_𝑖∈ℝ^{𝑝𝑖×𝑝𝑖}. Then, the inequality above can be rewritten as Π_11,𝑘𝑖Π_12,𝑘𝑖Π^𝑇_12,𝑘𝑖Π_22,𝑘𝑖<0, (39) where Π_11,𝑘𝑖=𝐴_𝑘𝑖+𝐵_𝑘𝑖𝐾_𝑖^𝑇𝑃_𝑖+𝑃_𝑖𝐴_𝑘𝑖+𝐵_𝑘𝑖𝐾_𝑖,Π_12,𝑘𝑖=𝑃_𝑖𝐵_𝑘𝑖𝐾_1𝑖0+𝑈^𝑇_𝑘𝑖𝑉^𝑇_𝑘𝑖𝑄_𝑖,Π_22,𝑘𝑖=𝑊^𝑇_𝑘𝑖𝑄_𝑖+𝑄_𝑖𝑊_𝑘𝑖. (40)
Notice that Π_11,𝑘𝑖<0 is just the inequality (22), which is equivalent to LMI (17). When Π_11,𝑘𝑖<0 holds, the matrices 𝑃_𝑖, 𝑌_𝑖, 𝑃_𝑖=𝑃_𝑖⁻¹, 𝐾_𝑖=(𝐾_1𝑖,𝐾_2𝑖)=𝑌_𝑖𝑃_𝑖 have already been obtained. Let 𝑄_𝑖𝑇_1𝑖=𝑀_1𝑖, 𝑄_𝑖𝑇_2𝑖=𝑀_2𝑖, and 𝑄_𝑖𝑇_3𝑖=𝑀_3𝑖. We turn the inequality (39) into LMI (29). This implies that the system (8) with 𝜔=0 is asymptotically stable.
The rest part of the proof is similar to that in Theorem 9. Recalling that 𝐴_𝑖𝐵_𝑖𝐾_𝑖𝐺_2𝑖𝐶_𝑖𝐺_1𝑖 is Hurwitz and (𝐺_1𝑖,𝐺_2𝑖) incorporates a 𝑝-copy internal model of 𝑆, we can obtain that for any matrices 𝐸_𝑖 and 𝐹_𝑖, the following matrix equations 𝑋_𝑖𝑆=𝐴_𝑖𝑋_𝑖+𝐵_𝑖𝐾_𝑖𝑍_𝑖+𝐸_𝑖,𝑍_𝑖𝑆=𝐺_1𝑖𝑍_𝑖+𝐺_2𝑖𝐶_𝑖𝑋_𝑖+𝐹_𝑖 (41) have a unique solution 𝑋_𝑖 and 𝑍_𝑖. And meanwhile, 𝐶_𝑖𝑋_𝑖+𝐹_𝑖=0. For 𝑖∈ℒ, 𝐹_𝑖=𝐹, otherwise 𝐹_𝑖=(−1/|𝑁_𝑖|)∑_{𝑗∈𝑁𝑖}𝐶_𝑗𝑋_𝑗.
Take notations Γ_𝑖=^𝑋𝑖_𝑍𝑖 and 𝛾_𝑖(𝑡)=𝛾_𝑖(𝑡)−Γ_𝑖𝜔(𝑡). By calculation, it can be verified that lim_𝑡→∞𝛾_𝑖(𝑡)=0. Consequently, lim_𝑡→∞𝑒_𝑖𝜔(𝑡)=0. When Assumption 2 holds, the above statement is equivalent to lim_𝑡→∞𝑒(𝑡)=0. This completes the proof.

Remark 12. Consider the robust tracking problem for heterogenous MAS in the presence of disturbances. Suppose that an active leader is described as 𝑥̇_𝑘(𝑡)=𝐴_𝑘𝑥_𝑘(𝑡),𝑦_𝑘(𝑡)=𝐶_𝑘𝑥_𝑘(𝑡), (42) and there is an environmental disturbance 𝑑(𝑡) satisfying 𝑑̇(𝑡)=𝐴_𝑑𝑑(𝑡). (43)
The model of the MAS can be given in 𝑥̇_𝑖(𝑡)=𝐴_𝑖𝑥_𝑖(𝑡)+𝐵_𝑖𝑢_𝑖(𝑡)+𝐸_𝑑𝑖𝑑(𝑡),𝑦_𝑖(𝑡)=𝐶_𝑖𝑥_𝑖(𝑡),𝑖=1,…,𝑁, (44) where 𝐴_𝑖,  𝐵_𝑖,  𝐶_𝑖, and 𝐸_𝑑𝑖 are uncertain matrices with polytopic uncertainties. The control target is to design distributed controllers such that the outputs of followers track the output of the leader. That is to say, for 𝑖=1,…,𝑁,  𝑒_𝑖(𝑡)=𝑦_𝑖(𝑡)−𝑦_𝑘(𝑡) converges to zero. This problem is just Problem 3 with 𝜔=(𝑥^𝑇_𝑘,𝑑^𝑇)^𝑇, 𝑆=diagblock(𝐴_𝑘,𝐴_𝑑), 𝐸=0𝐸_𝑑𝑖, and 𝐹=−𝐶_𝑘0. The robust tracking problem for the systems (44), therefore, can be studied under the framework of DOR problem. It can be solved by designing the controller (4) or (6) if the conditions in Theorems 9 or 11 are satisfied.

Remark 13. By Theorems 9 and 11, the sufficient condition of the solvability for Problem 3 depends on the dynamics of each agent. For a heterogenous MAS, 𝑁 LMIs have to be solved to obtain appropriate distributed controllers, which limits scalability. When the agents share some common characteristics, such as nominal parts, the complexity of LMIs will decrease. Specially, if agents are identical in the sense of common nominal parts and common bounds of the uncertainties, the solution to one LMI will construct the required control law.

4. A Numerical Example

In this section, a numerical example of a heterogenous MAS consisting seven agents is given to show the control effect of controllers obtained by Theorems 9 and 11. A comparison is also given between our controllers and that of [25]. In the example, suppose that there is a constant environmental noise and the reference signal (the output of the tracking target) is a sinusoidal signal. Let 𝜔=(𝜔₁,𝜔₂,𝜔₃)^𝑇, where 𝜔₁, 𝜔₂ serve as the reference signals and 𝜔₃ as the exogenous disturbance. Obviously, the differential equations about them are 𝜔̈₁(𝑡)=−𝜔₁(𝑡), 𝜔̈₂(𝑡)=−𝜔₂(𝑡), and 𝜔̈₃(𝑡)=0. Therefore, the exosystem can be described by (1) with 𝑆=010−100000. (45)

According to [23], we can immediately obtain a 1-copy internal model as follows: 𝐺₁=0100010−10,𝐺₂=001. (46)

In the example, three kinds of linear agents with 2, 3, and 4 orders are considered. 2 or 3 basement matrices are randomly valued for each kind. We present the detailed description of basement matrices in the appendix.

And as for 𝐸₁,…,𝐸₇, their first two columns are zeros, noticing that only 𝜔₃ refers to the disturbance, while the other elements are valued randomly. Finally, the regulated output, regarded as the tracking error between 𝑦_𝑖(𝑡) and the sine reference signal 𝜔₁(𝑡), is defined as 𝑒_𝑖(𝑡)=𝑦_𝑖(𝑡)−𝜔₁(𝑡)=𝐶_𝑖𝑥_𝑖(𝑡)+𝐹𝜔(𝑡), so, value 𝐹=−100.

The communication network is described by the directed acyclic graph shown in Figure 1, which satisfies Assumption 2. According to Theorem 11, by solving three kinds of LMIs as (17), we get control gains 𝐾₁=𝐾₂=(−6.3230,28.5507,2.6636,0.9019,6.8977), 𝐾₃=𝐾₄=𝐾₅=(0.9071,−9.9908,10.4217,−1.2503,0.9513,−3.1324), 𝐾₆=𝐾₇=(226.6539, 117.7418, −20.0749, −91.1218, 4.6064, −2.6648, 11.6774). If the states of each agent are measurable, then a candidate distributed controller can be chosen as (4) with the coefficients 𝐾₁,…,𝐾₇. Or else, one more group of LMIs (29) need to be solved to construct the output feedback controllers. In this situation, we obtain 𝑇_1𝑖,𝑇_2𝑖 and 𝑇_3𝑖 for 𝑖=1,…,7, which are also listed in the appendix due to space limitation. And then the distributed dynamic output feedback controllers are given as (6). For simplicity, we denote this control law as controller 𝐶_𝑎.

Figure 1 

Information flow among exosystem and agents.

On the other hand, we translate the polytopic uncertainty into an equivalent form that consists of a nominal part and an uncertain part. In this case, the nominal part 𝐴₁,…,𝐴₇, 𝐵₁,…,𝐵₇, and 𝐶₁,…,𝐶₇ can be valued as the average of the basement matrices. Then, by the algorithm in [25], for 𝑖=1,…,7, the candidate control law can be chosen as (6) with 𝐺_1𝑖=𝐴_𝑖+𝐵_𝑖𝐾_1𝑖+𝐿_𝑖𝐶_𝑖𝐵_𝑖𝐾_2𝑖0𝐺₁,𝐺_2𝑖=−𝐿_𝑖𝐺₂,𝐽_𝑖=𝑄_𝑖⁻¹𝑍_𝑖, (47)

where 𝐾₁=𝐾₂=(−10,0,2,1,−1), 𝐿₁=𝐿₂=(9.5,10)^𝑇, 𝐾₃=𝐾₄=𝐾₅=(−9,−2,−4,3,2,4), 𝐿₃=𝐿₄=𝐿₅=(30,−70,28)^𝑇, 𝐾₆=𝐾₇=(10,10,−2,5,1,2,−2)^𝑇, 𝐿₅=𝐿₆=𝐿₇=(−69,−41,−10,−10)^𝑇. For simplicity, this control law obtained by [25] is denoted by controller 𝐶_𝑏.

Note that both controllers 𝐶_𝑎 and 𝐶_𝑏 are robust controllers against the model uncertainty. To compare the effects, two cases of uncertainties are considered in the example. For one case, the model uncertainties denoted by Δ₁ are rather close to the “nominal part” set earlier. For 𝑖=1,…,7, 𝑘=1,…,𝑔_𝑖, the values of 𝜆_𝑘𝑖 are randomly chosen around 1/𝑔_𝑖, whose values are shown in the appendix. For another case, the model uncertainties denoted by Δ₂ are valued freely with reasonable coefficients, which are relatively farther from the “nominal part.” With the same initial values, the numerical results of the regulated output for each agent are demonstrated in Figure 2 and Figure 3. Our controller 𝐶_𝑎 is effective in the presence of either Δ₁ or Δ₂. However, as for Controller 𝐶_𝑏, when the uncertainty is relatively small, it can help to achieve the output regulation property. While in the case of Δ₂, the regulated errors diverge rapidly as shown in Figure 3(b). As a consequence of the above, for Problem 3 with polytopic uncertainties, Theorems 9 and 11 help construct distributed dynamic state/output feedback controllers of stronger robustness, at the price of increasing the calculation complexity.

(a) For uncertainty Δ1

(b) For uncertainty Δ2

Figure 2 

Regulated output of the MAS with the controller 𝐶𝑎.

(a) For uncertainty Δ1

(b) For uncertainty Δ2

Figure 3 

Regulated output of the MAS with the controller 𝐶𝑏.

5. Conclusions

In this paper, we consider the distributed robust output regulation problem for MASs under the topology of directed acyclic graphs. As an extension of the existing results, we focus on a special class of parametric uncertainty. Assume that the dynamics of heterogeneous agents are in a polytopic form, and we study the solvability of the problem and the design of distributed controllers. Both dynamic state feedback controller and dynamic output feedback controller are under consideration. At last, a practical example is presented to validate our results. By local interaction, a team of heterogenous agents is required to achieve a common output, which is generated by an exosystem. The controller generated by our theorem can realize the property of output regulation and the performance of tracking the reference signal and rejecting the disturbances as a special case.

Appendix

Here are some detailed descriptions of the example in Section 4. Agents 𝑣₁ and 𝑣₂ are systems with 2 orders. And there are three basement matrices for an uncertain system, which means 𝑔₁=𝑔₂=3𝐴₁₁=𝐴₁₂=,𝐴−4.8512−8.9578−8.71087.4003₂₁=𝐴₂₂=,𝐴−4.5841−7.5263−9.31357.0596₃₁=𝐴₃₂=,𝐵−5.7140−8.2750−9.11757.0544₁₁=𝐵₁₂=,𝐵1.6684−0.0861₂₁=𝐵₂₂=1.9897−0.7854,𝐵₃₁=𝐵₃₂=,𝐶1.8509−0.2403₁₁=𝐶₁₂=(),𝐶1.06140.8390₂₁=𝐶₂₂=(),𝐶1.01020.9982₃₁=𝐶₃₂=(0.92830.7936). (A.1) Agents 𝑣₃,  𝑣₄, and 𝑣₅ are systems with 3 orders. And 𝑔₃=𝑔₄=𝑔₅=2𝐴₁₃=𝐴₁₄=𝐴₁₅=,𝐴2.5419−1.23194.70381.9598−1.9843−4.6719−2.6654−2.4159−4.6598₂₃=𝐴₂₄=𝐴₂₅=,𝐵1.1025−2.17013.64880.0927−0.1010−2.6299−3.4064−3.1707−3.2973₁₃=𝐵₁₄=𝐵₁₅=,𝐵3.27382.3681−0.0088₂₃=𝐵₂₄=𝐵₂₅=,𝐶4.35701.9363−0.0489₁₃=𝐶₁₄=𝐶₁₅=(),𝐶−2.9090−1.6423−0.6208₂₃=𝐶₂₄=𝐶₂₅=(−3.5357−1.0246−2.3913). (A.2)(A.3)(A.4) Agents 𝑣₆ and 𝑣₇ are systems with 4 orders. And 𝑔₆=𝑔₇=3𝐴₁₆=𝐴₁₇=,𝐴7.38587.0606−1.9638−6.32181.59412.4411−8.4807−5.20090.9972−2.9810−5.2017−1.6547−7.10090.2650−7.5336−9.0069₂₆=𝐴₂₇=,𝐴5.77507.7097−3.0848−4.7077−0.18510.8409−8.0396−3.72881.0337−2.4579−4.1684−3.4229−7.05791.8202−7.1493−10.8314₃₆=𝐴₃₇=,𝐵7.68678.3454−1.3674−6.1338−0.16680.5027−7.5538−6.0157−0.0637−4.8089−4.6107−0.6759−7.6883−1.0590−7.7299−10.2511₁₆=𝐵₁₇=,𝐵0.06810.08622.54102.3587₂₆=𝐵₂₇=,𝐵−0.22060.32901.91792.2930₃₆=𝐵₃₇=,𝐶−0.23050.96423.29292.4590₁₆=𝐶₁₇=(),𝐶1.4942−2.5319−1.05211.0049₂₆=𝐶₂₇=(),𝐶0.9337−1.6942−1.91090.4535₃₆=𝐶₃₇=(1.4678−2.6602−1.15860.6176). (A.5) Next, these are the matrices obtained according to the solution of LMIs (29) 𝑇₁₁=𝑇₁₂=,𝑇37.5608109.7053−118.3196−149.5498₂₁=𝑇₂₂=,𝑇4.89841.658612.6849−1.7007−0.5759−4.4042₃₁=𝑇₃₂=,𝑇−43.5409−74.382691.9349157.0558₁₃=𝑇₁₄=𝑇₁₅=,𝑇−6.8197−42.193133.91329.9034−14.787519.6377−2.6741−1.6089−3.5333₂₃=𝑇₂₄=𝑇₂₅=,𝑇−4.28953.2637−10.7466−2.30381.7529−5.77180.1139−0.08670.2854₃₃=𝑇₃₄=𝑇₃₅=,𝑇11.15665.92915.2847−7.3616−3.9123−3.4871−0.3312−0.1760−0.1569₁₆=𝑇₁₇=,𝑇−167.8742206.0977254.4031−44.663015.0769147.1908106.7331−62.5135462.5231174.7488−97.4980−165.1735532.2860236.1899−93.3484−210.2411₂₆=𝑇₂₇=,𝑇−1.01620.5879−2.57601.5155−0.87673.84198.8346−5.110822.396110.5625−6.110426.7763₃₆=𝑇₃₇=.123.6493−224.3619−253.059360.056759.3071−107.6129−121.377328.8056−26.789948.610354.8279−13.0119−19.626535.612340.1673−9.5326 (A.6) The coefficients in case of the uncertainty Δ₁ are as follows: 𝜆₁₁=0.33,𝜆₂₁=0.33,𝜆₃₁𝜆=0.34,₁₂=0.4,𝜆₂₂=0.3,𝜆₃₂𝜆=0.3,₁₃=0.6,𝜆₂₃𝜆=0.4,₁₄=0.45,𝜆₂₄𝜆=0.55,₁₅=0.33,𝜆₂₅=0.37,𝜆₃₅𝜆=0.4,₁₆=0.35,𝜆₂₆=0.36,𝜆₃₆𝜆=0.39,₁₇=0.41,𝜆₂₇=0.3,𝜆₃₇=0.39. (A.7) The coefficients in case of the uncertainty Δ₂ are as follows: 𝜆₁₁=0.1,𝜆₂₁=0.2,𝜆₃₁𝜆=0.7,₁₂=0.5,𝜆₂₂=0,𝜆₃₂𝜆=0.5,₁₃=0,𝜆₂₃𝜆=1,₁₄=0.81,𝜆₂₄𝜆=0.19,₁₅=0.3,𝜆₂₅=0.5,𝜆₃₅𝜆=0.2,₁₆=0.02,𝜆₂₆=0.13,𝜆₃₆𝜆=0.85,₁₇=0.3,𝜆₂₇=0.1,𝜆₃₇=0.6. (A.8)

Acknowledgment

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