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### 基于内部模型原理的非同构线性代理强输出同步研究 #### 概述本文讨论了通过内部模型原理实现非同构线性代理的强输出同步问题。研究针对受参数扰动影响的异质代理网络，利用输出调节理论解决了这一问题。主要成果包括构建参考发生器和稳健控制器，确保非同构代理的输出能够跟踪领导者的输出。设计方法分为两个步骤：使代理网络中局部相同参考发生器的输出渐近地跟随领导者输出；在个体控制器的作用下，每个代理的输出被强制跟踪对应局部参考发生器的输出。 #### 研究背景与意义近年来，多代理系统的同步问题受到了系统与控制领域的广泛关注。这类问题的应用范围广泛，包括传感器网络、交汇、队形控制以及群集控制等。代理之间通过时变图或时不变图进行信息交换。网络中单个代理的动力学可以是相同的，也可以是非同构的。通过控制律可以修改代理的性能，使其表现出集体行为，例如实现关于感兴趣变量的同步。在非同构代理组成的异质网络中，可以通过输出调节理论实现所有代理间的输出同步。 #### 内部模型原理简介内部模型原理是一种重要的理论基础，它指出若一个控制系统想要稳定地跟踪或抑制某种类型的输入信号（如阶跃信号、正弦信号等），则该系统必须包含一个与输入信号结构相匹配的子系统。这通常意味着控制器的设计需要内嵌一个“内部模型”，用于模拟外部扰动或期望轨迹的动态特性。在本研究中，该原理被应用于设计参考发生器和稳健控制器，从而实现输出同步。 #### 研究方法与成果 1. **参考发生器的设计**：研究者首先设计了一个网络中的局部参考发生器集群，使得这些发生器的输出能够渐近地跟随领导者输出。这一步骤是通过构造适当的内部模型来实现的，以确保参考信号能够准确地反映领导者的动态特征。 2. **个体控制器的设计**：接下来，为每个代理设计了个体控制器，这些控制器基于输出调节理论，确保代理的输出能够跟踪对应的局部参考发生器输出。这种设计不仅考虑了代理之间的非同构性，还能够应对可能存在的参数扰动，提高了整个系统的鲁棒性和稳定性。 #### 分离原则的应用所提出的两步设计方法体现了分离原则的思想。第一步中，参考发生器的输出被强制跟随领导者输出；第二步中，则确保每个代理跟踪其对应的参考发生器输出。这种分离设计不仅可以简化控制器的设计过程，还能提高系统的整体性能。 #### 结论与展望通过对非同构线性代理的输出同步问题的研究，本文提出了一种基于内部模型原理的有效解决方案。通过设计合适的参考发生器和个体控制器，实现了代理间输出的强同步，并且这种方法对参数扰动具有较强的鲁棒性。未来的工作可以进一步探索更复杂的异质网络结构，以及如何将这一理论应用于实际工程系统中，比如分布式传感器网络、无人驾驶车辆编队等场景。此外，还可以考虑引入更多的智能算法和技术，如机器学习和优化技术，以增强系统的自适应能力和智能水平。

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IET Control Theory & Applications

Research Article

Robust output synchronisation of

non-identical linear agents via

internal model principle

ISSN 1751-8644

Received on 6th November 2014

Revised on 16th January 2015

Accepted on 12th March 2015

doi: 10.1049/iet-cta.2014.1192

www.ietdl.org

Yuanqing Wu, Zhengguang Wu, Hongye Su

National Laboratory of Industrial Control Technology, Institute of Cyber-Systems and Control, Zhejiang Uni versity, Yuquan Campus,

Hangzhou Zhejiang, 310027, People’s Republic of China

E-mail: hysu@iipc.zju.edu.cn

Abstract: This study investigates the output synchronisation of a heterogeneous network of agents affected by parameter

perturbations. The authors use the output regulation theory to solve this problem. The main result is the construction

of the reference generators and the robust regulators, which can guarantee the outputs of the non-identical agents track

the output of the leader. The designed method includes two steps, which can be seen as a kind of separation principle.

Firstly, the outputs of a network of local identical reference generators are forced to asymptotically follow the output of

the leader. Secondly, under the control of an individual regulator, the output of each agent is forced to track the output of

the corresponding local reference generator, via robust output regulation theory.

1 Introduction

Recently, the synchronisation problem of multiagent systems has

attracted considerable attention in systems and control commu-

nity, because of its application to a wide range of problems, such

as sensor networks, rendezvous, formation control and ﬂocking

control [1–14]. The agents exchange information through a com-

munication graph, which is a time-varying graph [15–18]ora

time-invariant graph [19–22]. The dynamics of individual agents

in the network can be identical [23–28] or non-identical [29–31].

The performances of the agents can be modiﬁed by a control law,

to perform collective behaviors, such as to reach synchronisation

about relevant variables of interest. In a heterogeneous network of

non-identical agents, output synchronisation among all agents can

be achieved via the theory of output regulation. As shown by the

internal model principle, if the outputs of the agents in the hetero-

geneous network track a trajectory, this trajectory needs to be the

output of autonomous system. Inspired by the authors [19, 20, 29],

we consider the leader-following consensus problem. Under the

dynamical control law, the outputs of the followers exponentially

track the output of a prescribed leader.

In the leader-following consensus problem, the leader’s motion

is independent of all the other agents and followed by them. We

solve this consensus problem in two stages, which can be treated

as a kind of separation principle. In the ﬁrst stage, a homoge-

neous network of N identical reference generators which track

the dynamic of the leader is considered. The reference genera-

tors exchange information through a time-varying communication

graph. Under the distributed control law which depends on relative

information, the outputs of these local reference generators can

exponential synchronise to the output of the leader. The method

in this stage is to synchronise a set of networked identical linear

reference generators to reach a common steady state. In the second

stage, the kth of the reference generators designed in ﬁrst stage is

to be treated as the exosystem for the kth agent. So the problem is

approached in the framework of the classical output regulation the-

ory. We design a decentralised dynamical control law for individual

non-identical agent in such a way that its output tracks the refer-

ence output of its corresponding generator. The individual regulator

includes a stabiliser and an internal model, if the stabiliser can

stabilise the augmented system, the output regulation problem is

solved. The method used in this stage has the internal model prop-

erty with respect to the local identical linear exosystem in steady

state. These two stages guarantee the outputs of heterogeneous

networked linear systems can achieve output consensus with the

output of the leader.

This design philosophy was originally proposed in [22, 29]. In

our approach, we extend the results of such works in two ways:

we require only exchange of information at the level of controlled

output (as opposite to exchange of full state information) and we

handle the cases in which the models of the followers are affected

by parameter perturbations. In this way, we take full advantage of

the potentials of the theory of robust output regulation.

The rest of this paper is organised as follows. In the following

section, we give the description of the communication graph, for-

mulate the investigated problem and specify the structure of the

control law. Section 3 presents the consensus of the diffusively-

coupled reference generators for the ﬁrst stage. In Section 4,we

address the output consensus of the heterogeneous network for the

second stage, by means of internal model-based regulators. Some

simulation results will be given in Section 5 to present the effective

of the obtained method. Finally, Section 6 concludes this paper.

Notation: The notations used throughout this paper are fairly

standard. R

and R

m×n

denote the n-dimensional Euclidean space

and the set of all m × n real matrices, respectively. The notation

X > Y (X  Y ), where X and Y are symmetric matrices, means

that X − Y is positive deﬁnite (positive semi-deﬁnite). I and 0

represent the identity matrix and a zero matrix, respectively. The

superscript ‘T’ represents the transpose, and diag{···} stands for a

block-diagonal matrix. ||·|| denotes the Euclidean norm of a vec-

tor and its induced norm of a matrix. λ

max

(Q) (λ

min

(Q)) denotes

the maximum (minimum) of the eigenvalue of a real symmetric

matrix Q. Matrices, if their dimensions are not explicitly stated, are

assumed to have compatible dimensions for algebraic operations.

The vector 1

denotes a column vector with all entries equal to 1.

A ⊗ B denotes the Kronecker product of the matrices A and B.

2 Preliminaries

2.1 Communication graphs

A digraph G(V, E, A(t)) contains a set of N vertices V =

, v

, ..., v

}, an edge set E ⊆ V × V and an adjacency

matrix A(t) =[a

(t)]

N ×N

. For simplicity, digraph G(V, E, A(t))

is denoted by G(t). If an edge (v

, v

) ∈ E, then the information

IET Control Theory Appl., 2015, Vol. 9, Iss. 12, pp. 1755–1765

is transmitted from parent node v

to child node v

.IfE contains

a subset of the form {(v

, v

), (v

, v

), ..., (v

, v

)}, this sequence

of edges is called a path from v

to v

. The elements of the

weighted adjacency matrix A(t) are deﬁned as a

(t) = 0, a

(t)>

0if(v

, v

) ∈ E; otherwise, a

(t) = 0, (i, j = 1, ..., N ). The set of

neighbours of the node v

is the set N

(t) ={v

∈ V : a

(t) = 0}.

The Laplacian matrix L =[l

(t)]

N ×N

of G(t) is deﬁned as l

(t) =



j=1,j=i

(t), l

(t) =−a

(t), ∀i = j, which guarantee L1

= 0.

The digraph G(t) is said to be balanced at time t if and only if the

in-degree of each node is equal to out-degree of the node, that is,

L = 0[19, 23]. For a node, the number of head endpoints adja-

cent to a node is called the in-degree of the node and the number

of tail endpoints adjacent to a node is its out-degree. Denote G(t)

as the mirror graph of G(t), which is an undirected graph with

the same set of nodes as G(t), the symmetric adjacency matrix

A (t) with elements ˆa

(t) =ˆa

(t) =[{a

(t) + a

(t)}/2]≥0, and

the set of edges

E = E



E where

E be the set of reverse edges

of G(t) obtained by reversing the order of nodes of all the pairs

in E [32]. If G(t) is a balanced digraph, then the Laplacian matrix

of G(t) is positive semi-deﬁnite: L(G(t)) =[{L + L

}/2]≥0. The

undirected underlying graph of digraph G(t) can be obtained by

replacing all directed edges of G(t) with undirected edges. If the

undirected underlying graph is a connected graph, then the digraph

G(t) is weakly connected.

2.2 Problem statement

Consider a heterogeneous network in which the kth agent (k =

1, 2, ..., N ) is a linear system described by equations of the form

˙x

(t) = A

(t) + B

(t)

(t) = C

(t)

(1)

where x

∈ R

is the state, u

∈ R

is the control input and y

∈

is the measured output. A

∈ R

×n

, B

∈ R

×m

, C

∈ R

p×n

are constant matrices.

The leader is an autonomous linear time-invariant system

˙w

(t) = Sw

(t)

(t) = Qw

(t) := θ

(t)

(2)

with state w

∈ R

, output y

∈ R

. S ∈ R

×n

and Q ∈ R

p×n

are constant matrices.

The objective of the control law is to achieve output synchroni-

sation

(t) − y

(t) → 0

as t →∞for each agent.

Inspired by the authors [22, 29], the design method in our paper

consists of two stages. First, we consider the problem of achieving

consensus among the leader and a set of linear systems, each one of

which will be taken as reference generator for a (decentralised) reg-

ulation problem. Second, the theory of output regulation is applied

in a decentralised mode, that is, a dynamic error feedback control

law is designed so as to guarantee that the output of the different

agents robustly track the output generated by the corresponding

reference generator.

Remark 1: In [28, 33], the authors use dynamic output feedback

control law to solve the problem of output regulation. They use

distributed observer in [33] and visual regulated output in [28],

which represent the information exchange in the network, to con-

struct the controllers. In our paper, the structure of the controller

is different from the distributed controllers in [28, 33]. We use the

decentralised controller in the second stage, and the information

exchange through the communication graph among the reference

generators in the ﬁrst stage.

The actual control scheme can be described as follows. Con-

sider a set of N systems (henceforth referred to as ‘local reference

generators’) modelled by

˙w

(t) = Sw

(t) + Gv

(t)

(t) = Qw

(t)

(3)

Each one of them can be seen as a ‘copy’ of (2) which characterise

the dynamics and deﬁne the output of the leader, with a control

injected for synchronisation purposes. Owing to communication

constraints, a full information control law may not be suitable.

Hence it is assumed that v

depends only on the relative values of

the θ

(t)’s, namely that

(t) =



j=1

(t)(θ

(t) − θ

(t)) + d

(t)(θ

(t) − θ

(t)) (4)

where d

(t)>0 denotes the ﬂow of information from the leader

to the agent k at time t, and a

(t) is the (k, j)th entry of the

adjacency matrix A(t) that models the ﬂow of information from

agent j to agent k at time t. The design parameter G in (3)is

chosen in such a way as to induce output consensus among all

such reference generators.

Remark 2: Under the assumption that the communication graph is

uniformly connected, the information exchange considered in [19]

includes the full state of the internal controller and the full state

of an observer. This, in principle, requires the exchange of a large

amount of information. In our paper, the information exchange is

only about the relative of the outputs θ

, a substantially reduced

amount of information.

Set now

(t) = y

(t) − θ

(t) = C

(t) − Qw

(t) (5)

With this in mind, we consider for the kth agent (1) a (decen-

tralised) regulation problem in which the regulated variable is e

and the control u

is provided by an individual regulator.

For brevity, in what follows the argument t will be dropped. The

block diagram for the heterogeneous multiagent system is given in

Fig. 1.

Observe now that the set of interconnected reference generators

appears as a system with no inputs, modelled by

˙w

= Sw

+ GQ

⎡

⎣



j=1

− w

) + d

− w

)

⎤

⎦

Let δ

= w

− w

, k = 1, 2, ..., N .Sow

= w

+ δ

= w

+ E

δ,

in which E

is a matrix of the form



0 ··· I ··· 0



where the identity matrix in the kth place, and δ = col(δ

, δ

, ...,

). In this notation, it is seen that

˙w

= Sw

δ = L(t)δ

(6)

where

L(t) =

[

⊗ S) − (L(t) + D(t)) ⊗ GQ

]

with D(t) = diag{d

(t), ..., d

(t)} denoting the ﬂow of informa-

tion from the leader to the followers, L(t) denoting the Laplacian

matrix of G(t) [23]. It will be shown later that, under suitable con-

nectivity hypotheses, in the ﬁrst stage of the design it is possible to

choose G in such a way that lim

t→∞

δ(t) = 0, that is, such that the

states (and hence the outputs) of the N local reference generators

synchronise with those of the leader.

IET Control Theory Appl., 2015, Vol. 9, Iss. 12, pp. 1755–1765

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