Scale-free linear protocol design for global regulated state synchronization of discrete-time double-integrator multi-agent systems subject to

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Scale-free linear protocol design for global

regulated state synchronization of discrete-time

double-integrator multi-agent systems subject to

actuator saturation

Zhenwei Liu

College of Information

Science and Engineering,

Northeastern University

Shenyang, China

liuzhenwei@ise.neu.edu.cn

Ali Saberi

School of Electrical Engineering

and Computer Science,

Washington State University

Pullman, WA, USA

saberi@wsu.edu

Anton A. Stoorvogel

Department of Electrical Engineering,

Mathematics and Computer Science,

University of Twente

Enschede, The Netherlands

A.A.Stoorvogel@utwente.nl

Abstract—This paper studies global regulated state synchro-

nization of discrete-time double-integrator multi-agent systems

subject to actuator saturation by utilizing localized information

exchange. We propose a scale-free linear protocol that achieves

global regulated state synchronization for any network with arbi-

trary number of agents and arbitrarily directed communication

graph that has a path between each agent and exosystem which

generates the reference trajectory.

Index Terms—Discrete-time; double-integrator; multi-agent

systems; actuator saturation; global regulated state synchroniza-

tion; scale-free

I. Introduction

The synchronization or consensus problem of multi-agent

systems (MAS) has drawn a lot of investigation in recent years,

because of its developing applications in a few ﬁelds, like

automotive vehicle control, satellites/robots formation, sensor

networks, and so forth. The work can be seen, for example

in the books [1], [2], [4], [12], [13], [14], [17] and references

therein.

MAS subjects to actuator saturation has been studied in both

semi-global and global frameworks. Some researchers have

worked on (semi) global state and output synchronizations for

both continuous- and discrete-time MAS subject to actuator

saturation. We can summarized the current existing literature

on MAS with linear agents subject to actuator saturation as

follows:

1) The semi-global synchronization has been studied in

[16] for full-state coupling. For partial state coupling,

we have [15], [22] which are based on the extra com-

munication. Meanwhile, the result without the extra

communication is developed in [23]. The static protocols

via partial state coupling is designed in [9] for G-passive

agents and G-passiﬁable via input feedforward agents.

2) Global synchronization for the case of full-state coupling

and an undirected network has been studied by [18]

for neutrally stable and double-integrator discrete-time

agents. For special case, the result dealing with networks

that has a directed spanning tree are provided in [5] for

single integrator agents. Then, global synchronization

results for partial-state coupling has been studied in [9]

for both continuous- and discrete-time G-passive and G-

passiﬁable via input feedforward agent models by a static

design, which require the graph is strongly connected

and detailed balanced.

Recently, we have initiated a research eﬀort on developing

scale-free protocols design for MAS, in which the agent model

can be continuous- or discrete-time homogeneous and het-

erogeneous, and includes external disturbances, input delays,

and communication delays ([6]). The scale-free framework

for protocol design means that the design does not depend

on communication network and the size of network or the

number of agents. In particular, for the case of linear agents

subject to actuator saturation, we have designed the scale-free

nonlinear and linear protocols to achieve the global regulated

state synchronization for both continuous- or discrete-time

agents, see [11], [7], [8]. The nonlinear design is focusing on

at most weakly unstable agents, i.e., eigenvalues of agents are

in the closed left half plane for continuous-time systems and

in the closed unit disc for discrete-time systems. The linear

design focused only on the neutrally stable agents.

We continue the research eﬀort on extending the scale-free

linear protocol design to global regulated synchronization of

discrete-time double-integrator MAS in this paper. We would

like to point out that the double-integrator is polynomially

unstable system. It is also well known that the only signiﬁcant

class of MAS with polynomially unstable agents subject to

actuator saturation for which global synchronization can be

achievable is double-integrator MAS subject to actuator satu-

ration. This class is important due to the practical signiﬁcance.

We propose a family of the linear protocols that are scalable,

and any number of this family of protocols can achieve global

arXiv:2210.14453v1 [eess.SY] 26 Oct 2022

regulated state synchronization for MAS with any communi-

cation graph with arbitrary number of agents as long as the

communication graph has a path between the exosystem which

generates reference trajectory and each agent.

Graphs

Aweighted graph Gis deﬁned by a triple (V,E,A) where

V={1, . . . , 𝑁}is a node set, Eis a set of pairs of nodes

indicating connections among nodes, and A=[𝑎𝑖 𝑗 ] ∈ R𝑁×𝑁

is the adjacency matrix. Moreover, 𝑎𝑖 𝑗 =0if there is no edge

from node 𝑗to node 𝑖. We assume there are no self-loops, i.e.

we have 𝑎𝑖𝑖 =0. A path from node 𝑖1to 𝑖𝑘is a sequence of

nodes {𝑖1, . . . , 𝑖𝑘}such that (𝑖𝑗, 𝑖 𝑗+1) ∈ E for 𝑗=1, . . . , 𝑘 −

1. A directed tree with root 𝑟is a subgraph of the graph

Gin which there exists a unique path from node 𝑟to each

node in this subgraph. A directed spanning tree is a directed

tree containing all the nodes of the graph. The weighted in-

degree of node 𝑖is given by 𝑑in (𝑖)=Í𝑁

𝑗=1𝑎𝑖 𝑗 . See [14]. For

a weighted graph G, the matrix 𝐿=[ℓ𝑖 𝑗 ]with

ℓ𝑖 𝑗 =Í𝑁

𝑘=1𝑎𝑖𝑘 , 𝑖 =𝑗,

−𝑎𝑖 𝑗 , 𝑖 ≠𝑗,

is called the Laplacian matrix associated with the graph G.

The Laplacian matrix 𝐿has all its eigenvalues in the closed

right half plane and at least one eigenvalue at zero associated

with right eigenvector 1. See [3].

II. System description and problem formulation

Consider a MAS consisting of 𝑁identical discrete-time

double-integrator with actuator saturation:

(𝑥𝑖(𝑡+1)=𝐴𝑥𝑖(𝑡) + 𝐵𝜎(𝑢𝑖(𝑡)),

𝑦𝑖(𝑡)=𝐶𝑥𝑖(𝑡)(1)

where 𝑥𝑖(𝑡) ∈ R2𝑛,𝑦𝑖(𝑡) ∈ R𝑛and 𝑢𝑖(𝑡) ∈ R𝑛are the state,

output, and input of agent 𝑖, respectively, for 𝑖=1, . . . , 𝑁. And

𝐴=𝐼 𝐼

0𝐼, 𝐵 =0

𝐼, 𝐶 =𝐼0.

Meanwhile,

𝜎(𝑣)=©«

sat(𝑣1)

sat(𝑣𝑚)ª®®¬

,where 𝑣=©«

𝑣1

𝑣𝑚ª®®¬

∈R𝑚

and sat(𝑤)is standard saturation function satisfying sat(𝑤)=

sgn(𝑤)min(1,|𝑤|).

The communication network provides agent 𝑖with the

following information,

𝜁𝑖(𝑡)=

𝑁



𝑗=1,𝑖≠𝑗

𝑎𝑖 𝑗 (𝑦𝑖(𝑡) − 𝑦𝑗(𝑡)),(2)

where 𝑎𝑖 𝑗 >0and 𝑎𝑖𝑖 =0. The network topology can be

described by a weighted graph Gassociated with (2), where

the 𝑎𝑖 𝑗 are the coeﬃcients of adjacency matrix A.𝜁𝑖(𝑡)can

be rewritten using associated Laplacian matrix as

𝜁𝑖(𝑡)=

𝑁



𝑗=1

ℓ𝑖 𝑗 𝑦𝑗(𝑡).

Now, we consider regulated state synchronization prob-

lem. We need a reference trajectory, which is generated by the

following exosystem

𝑥𝑟(𝑡+1)=𝐴𝑥𝑟(𝑡),(3)

𝑦𝑟(𝑡)=𝐶𝑥𝑟(𝑡).(4)

The objective of regulated state synchronization is

lim

𝑡→∞(𝑥𝑖(𝑡) − 𝑥𝑟(𝑡)) =0(5)

for all 𝑖∈1, ..., 𝑁.

Clearly, we need some level of communication between the

reference trajectory and the agents. Thus, we assume that a

nonempty subset S. The agents in this set have access to their

own output relative to the reference trajectory 𝑦𝑟(𝑡) ∈ R𝑝. It

means that each agent has access to the quantity

𝜓𝑖(𝑡)=𝜄𝑖(𝑦𝑖(𝑡) − 𝑦𝑟(𝑡)), 𝜄𝑖=1, 𝑖 ∈ S,

0, 𝑖 ∉S.(6)

Then, the information available for each discrete-time agent

can be expressed by

𝜁𝑖(𝑡)=1

2+¯

𝐷in (𝑖)

𝑁



𝑗=1

𝑎𝑖 𝑗 (𝑦𝑖(𝑡)−𝑦𝑗(𝑡))+𝜄𝑖(𝑦𝑖(𝑡)−𝑦𝑟(𝑡)),(7)

where ¯

𝐷in (𝑖)is the upper bound of 𝑑in (𝑖)=Í𝑁

𝑗=1𝑎𝑖 𝑗 for 𝑖=

1,· · · , 𝑁.

Remark 1 As the explanations in [8, Remark 2], ¯

𝐷in (𝑖)is still

a local information.

Thus, we use the node set Sas root set. Then, we deﬁne

an expanded Laplacian matrix as

𝐿=𝐿+diag{𝜄𝑖}=[¯

ℓ𝑖 𝑗 ]𝑁×𝑁(8)

for any graph with the Laplacian matrix 𝐿. Since the sum of

its rows does not need to be zero, ¯

𝐿is not a regular Laplacian

matrix associated to the graph. Furthermore, it should be

emphasized that ¯

ℓ𝑖 𝑗 =ℓ𝑖 𝑗 for 𝑖≠𝑗in ¯

𝐿. And (7) can be

rewritten as

𝜁𝑖(𝑡)=1

2+¯

𝐷in (𝑖)

𝑁



𝑗=1

ℓ𝑖 𝑗 (𝑦𝑗(𝑡) − 𝑦𝑟(𝑡)).(9)

To guarantee that each agent can achieve the required regu-

lation, we need to make sure that there exists a path to each

node starting with node from the set S. Therefore, we denote

the following set of graphs.

Deﬁnition 1 Given a node set S, we denote by G𝑁

Sthe set

of all directed graphs with 𝑁nodes containing the node set

S, such that every node of the network graph G∈G𝑁

Sis a

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Scale-freelinearprotocoldesignforglobalregulatedstatesynchronizationofdiscrete-timedouble-integratormulti-agentsystemssubjecttoactuatorsaturationZhenweiLiuCollegeofInformationScienceandEngineering,NortheasternUniversityShenyang,Chinaliuzhenwei@ise.neu.edu.cnAliSaberiSchoolofElectricalEngineeringandC...

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Scale-free linear protocol design for global regulated state synchronization of discrete-time double-integrator multi-agent systems subject to

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