Automatica 141 (2022) 110259

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Automatica
journal homepage: www.elsevier.com/locate/automatica

Brief paper

Exponential convergence of distributed optimization for
heterogeneous linear multi-agent systems over unbalanced digraphs✩
∗

Li Li a,b,c , Yang Yu a , Xiuxian Li a,c,d , , Lihua Xie e
a

Department of Control Science and Engineering, College of Electronics and Information Engineering, Tongji University, Shanghai 201804, China
Shanghai Institute of Intelligent Science and Technology, Tongji University, Shanghai 201804, China
c
Shanghai Research Institute for Intelligent Autonomous Systems, Shanghai, 201210, China
d
Institute for Advanced Study, Tongji University, Shanghai, 200092, China
e
School of Electrical and Electronic Engineering, Nanyang Technological University, Singapore 639798, Singapore
b

article

info

Article history:
Received 11 January 2021
Received in revised form 30 December 2021
Accepted 30 January 2022
Available online 26 April 2022
Keywords:
Distributed optimization
Multi-agent systems
Event-triggered communication
Sampling control

a b s t r a c t
In this work we study a distributed optimal output consensus problem for heterogeneous linear
multi-agent systems over unbalanced directed networks where the agents aim to reach consensus
with the purpose of minimizing the sum of private smooth costs. Based on output feedback, a
distributed continuous time control law is proposed by using the proportional–integral (PI) control
technique. Under the assumption that the global cost function satisfies the restricted secant inequality
condition, the designed controller can achieve convergence exponentially in an unbalanced and
strongly connected network. Furthermore, to remove the requirement of continuous communications,
a sampling-based event-triggered algorithm with a lower bound of the communication interval is
provided, which also converges exponentially. Two simulation examples are given to verify the
proposed control algorithms.
© 2022 Elsevier Ltd. All rights reserved.

1. Introduction
In recent few decades, distributed optimization has been attracting more and more research interests because of its wide
applications in multi-agent systems, smart grids, machine learning and so on. Specifically, the purpose of each node in a network
is to minimize the sum of private costs under constraints only
by exchanging local information with neighbors. In the discretetime setting, the distributed subgradient method is a seminal
work proposed in Nedic and Ozdaglar (2009); to achieve faster
convergence rate, several fixed step-size algorithms are proposed
such as EXTRA (Shi, Ling, Wu, & Yin, 2015), DIGing (Nedic, Olshevsky, & Shi, 2017), Newton–Raphson algorithm (Varagnolo,
Zanella, Cenedese, Pillonetto, & Schenato, 2016) and so on. In the
continuous-time setting, a distributed PI algorithm (Kia, Cortes,
& Martinez, 2015; Wang & Elia, 2011) is proposed by using
first-order gradient information, and a Zero-Gradient-Sum algorithm (Lu & Tang, 2012) is proposed by using second-order
✩ The material in this paper was not presented at any conference. This paper
was recommended for publication in revised form by Associate Editor Luca
Schenato under the direction of Editor Christos G. Cassandras.
∗ Corresponding author at: Department of Control Science and Engineering,
College of Electronics and Information Engineering, Tongji University, Shanghai
201804, China.
E-mail addresses: lili@tongji.edu.cn (L. Li), 1910639@tongji.edu.cn (Y. Yu),
xli@tongji.edu.cn (X. Li), elhxie@ntu.edu.sg (L. Xie).
https://doi.org/10.1016/j.automatica.2022.110259
0005-1098/© 2022 Elsevier Ltd. All rights reserved.

Hessian information. For more details, please refer to the surveys (Nedic, Olshevsky, & Rabbat, 2018; Yang et al., 2019).
So far, most existing distributed optimization algorithms can
be regarded as distributed optimal coordination algorithms for
single-integrator multi-agent systems. But in practical physical
systems such as AGVs and UAVs, the implementation of distributed strategies must consider the complicated dynamics of
each agent. Therefore, in recent years, interest has been attracted
increasingly in distributed optimization combined with physical
systems. This problem requires each of a group of continuoustime physical systems to achieve the best performance. In Zhang,
Deng, and Hong (2017) where the system dynamics are Euler–
Lagrange systems, two distributed algorithms are developed for
the case without parametric uncertainties and the case with
parametric uncertainties respectively, but the control parameters
are dependent on some global information. To overcome this
shortcoming, Zou, Meng, and Hong (2020) propose an adaptive
fully distributed algorithm for a group of Euler–Lagrange systems. For high-order multi-agent systems, Tang, Deng, and Hong
(2019) make use of a virtual first-order optimizer to generate an
optimal signal and then uses an underlying controller to make
the system track this signal, which has been proved to converge to the optimal solution exponentially under the strong
convexity condition of local cost functions. Wang, Wang, and
Li (2020) construct a nonsmooth embedded control framework
covering first-order, second-order, and higher order types, which

L. Li, Y. Yu, X. Li et al.

Automatica 141 (2022) 110259

solves the distributed optimization in finite time. For heterogeneous uncertain nonlinear multi-agent systems, Tang and Wang
(2021) use a similar idea to Tang et al. (2019) to achieve optimal
output consensus. Zhao, Liu, Wen, and Chen (2017) study the
optimal consensus problem of linear systems, but it requires
the local objective function to be of a certain special structure,
which only works in limited situations. Xie and Lin (2020)
study the distributed optimization problem for a group of weakly
nonminimum phase linear systems and achieve suboptimal output consensus. For heterogeneous linear multi-agent systems, an
asymptotically stable and fully distributed controllers is designed
in Li, Wu, Li, and Ding (2020). Note that all the above algorithms
assume that the communication topology is balanced. When the
communication topology is unbalanced, Touri and Gharesifard
(2019) and Zhu, Yu, Wen, and Ren (2018) design distributed optimization algorithms for the first-order integrator systems, and
obtain the result of asymptotic convergence. For homogeneous
linear multi-agent systems, an exponential stable controller is
designed in Zhang, Liu, and Ji (2020) on the premise that agents
know the network imbalance in advance.
On the other hand, in continuous-time distributed optimization algorithms, it is necessary for agents to exchange information
continuously, which is unrealistic in actual physical systems.
In order to avoid continuous communication and reduce communication overhead, event-triggered mechanisms (Ding, Han,
Ge, & Zhang, 2018; Ge, Han, Ding, Wang, & Zhang, 2020) are
introduced for first-order integrator systems (Kia et al., 2015;
Wu, Li, Ding, & Li, 2020), second-order integrator systems (Tran,
Wang, Liu, Xiao, & Lei, 2019), Euler–Lagrange systems (Deng &
Hong, 2016), nonlinear systems (Ma, Yang, & Chen, 2021), linear
multi-agent systems (Li, Chen, & Su, 2019; Li et al., 2020; Yu
& Chen, 2021; Zhang, Papachristodoulou, & Li, 2018) and so
on. Agents communicate with their neighbors only after reaching certain trigger conditions, which need to be continuously
monitored by agents. However, in many control applications,
the controller is implemented on a digital platform with finite
working frequency. To overcome this shortcoming, a samplingbased event-triggered method, using a certain sampling period
to detect event triggering conditions, is studied in the multiagent consensus problems (Duan, Xiao, & Wang, 2018; Nowzari
& Cortes, 2016; Shi, Lin, Yang, & Wang, 2021), which discretizes
both communication and detection. Furthermore, if there exists
a lower bound for interval samplings, Zeno’s behavior, meaning
that an infinite number of events occur in a finite time, is naturally avoided. However, the application of this method in the
distributed optimization of complex systems remains an open
problem.
Aiming at the optimal output consensus problem of heterogeneous linear multi-agent systems over unbalanced directed
graphs, this paper designs a proportional–integral (PI) controller
to solve the problem, in which the proportional term drives all
the agents to the consensus space, the integral term eliminates
the consensus errors (Qiu, Xie, & You, 2019), and the adaptive
parameters are added to offset the imbalance of the network. The
main contributions of this paper are as follows.

(2) To reduce the communication overhead, this paper further
introduces event-triggered communication mechanisms for
the above-proposed algorithm, which is proven to guarantee
exponential convergence in the case of aperiodic discrete
communication. Besides the established exponential rate
here, compared with the gradually decreasing communication interval and continuous measurement of triggering
conditions in Li et al. (2020), the proposed algorithm can
clearly give a lower bound of the communication interval
by verifying the event-triggering condition at a discrete time
sampling sequence.
The rest of the paper is organized as follows. Preliminaries are
given in Section 2. In Section 3, the heterogeneous multi-agent
systems under investigation are described mathematically, the
optimal output consensus problem is defined and some useful
lemmas are given. Following that, the control laws with continuous and sampling-based event-triggered communication are
proposed respectively and their exponential convergence is established in Section 4. Then two simulation examples are provided to verify the effectiveness of the algorithms in Section 5.
Finally, conclusions and future works are discussed in Section 6.
2. Preliminaries
2.1. Notations
Let N, R, Rn , Rm×n be the sets of natural numbers, real numbers, real vectors of dimension n and real matrices of
dimension m × n, respectively. In denotes the n-dimensional
identity matrix. 1n and 0n denote n-dimensional all-one and
all-zero column vectors, respectively. For a matrix A ∈ Rm×n ,
A⊤ is its transpose, and diag(A1 , . . . , An ) = blkdiag(A1 ; . . . ; An )
denotes a block diagonal matrix with diagonal blocks of A1 , . . . ,
⊤ ⊤
An . col(x1 , . . . , xn ) = (x⊤
1 , . . . , xn ) is a column vector by stacking
vectors x1 , . . . , xn . ∥A∥ and ∥x∥ are the induced 2-norm of matrix
A and the Euclidean norm of vector x respectively. A⊗B represents
the Kronecker product of matrices A and B.
2.2. Graph theory
A communication network of N agents is modeled by a digraph
G = (V , E , A), where V = {v1 , . . . , vN } is a node set and E ∈ V × V
is an edge set. If agent i can receive information from agent j,
then (vj , vi ) ∈ E with aij > 0 denoting its weight, and otherwise
aij = 0. A = [aij ] ∈ RN ×N is the adjacency matrix. If there
exists a path from any node to any other node in V , then G is
called strongly connected, otherwise
∑N disconnected. The in-degree
in
of node vi is denoted by din
i =
j=1 aij . Denote L = D − A as
in
in
the Laplacian matrix of G , where D = diag(d1 , . . . , din
N ) is the
in-degree matrix of G .
Lemma 1 (Godsil & Royle, 2001; Yu, Chen, & Cao, 2011). If G is
strongly connected with the Laplacian matrix L ∈ RN ×N , then

(1) Through the feedback combination of an agent’s own state
and neighbor output information, a PI based control law is
designed, which is shown to have an exponential convergence rate over unbalanced directed graphs if the global cost
function satisfies the restricted secant inequality condition.
In comparison, the related work (Li et al., 2020; Yu & Chen,
2021) only gives the result of asymptotic convergence over
undirected graphs; the work in Zhang et al. (2020) is for
homogeneous systems and agents need to know the imbalance of the network in advance. Furthermore, all of them are
based on stronger assumptions on cost functions and system
structures than those in this paper.

(1) L1N = 0N ;
(2) there is a positive left eigenvector ξ = col(ξ1 , . . . , ξ∑
N ) associN
ated with eigenvalue 0 such that ξ ⊤ L = 0⊤
N and
i=1 ξi =
1;
(3) L = (Ξ L + L⊤ Ξ )/2 is a valid Laplacian matrix for a connected
⊤
and balanced graph and λ2 (L) = min1⊤ x=0 xx⊤Lxx , where Ξ :=
diag {ξ1 , . . . , ξN } and λ2 (L) is its second smallest eigenvalue.
Denote ξ := min{ξ1 , . . . , ξN };
(4) the general algebraic connectivity of G satisfies λL
⊤ Lx
minx⊤ ξ =0 xx⊤ Ξ
.
x
2

=

L. Li, Y. Yu, X. Li et al.

Automatica 141 (2022) 110259

3. Problem formulation
Consider a multi-agent system with N heterogeneous agents,
and the ith agent has the linear dynamics:
ẋi = Ai xi + Bi ui ,

(1)

yi = Ci xi ,

where xi ∈ Rni , ui ∈ Rpi and yi ∈ Rq are the state, input
and output variables, respectively. Ai ∈ Rni ×ni , Bi ∈ Rni ×pi and
Ci ∈ Rq×ni are the state, input and output matrices, respectively,
which are constant.
Each agent has a private
cost function fi (y) : Rq → R.
∑local
N
˜
The sum of them f (y) :=
i=1 fi (y) is the global cost function.
The objective of this paper is to design a controller ui (t) for each
agent by using only local interaction and information such that
all agents cooperatively reach an optimal output y∗ that solves
the following convex optimization problem:
min f˜ (y).

Fig. 1. Proportional–Integral controller to solve (3).

(2)

y∈Rq

Lemma 2.

∗

Define Y as the optimal set of the optimization problem (2)
and Y ∗ := {1N ⊗ y∗ |y∗ ∈ Y ∗ }.
Since each agent in the network has its own output variable yi ,
by defining y := col(y1 , . . . , yN ) ∈ RNq , problem (2) is equivalent
to
min f (y) = min

yi ∈Rq

y ∈RNq

Assumption 1.
nected.

N
∑

(3)

Ci Bi Ψi = Iq ,

(5b)

rank [Ci Bi

(

Ci Ai ] = q = rank(Ci Bi ),

)

(

Iq ] = q = rank(Ci Bi ).

i=1

)

rank [Ci Bi

The communication network G is strongly con-

Thus (5a) and (5b) have solutions Υi , Ψi .
Remark 2. The controllability in Assumption 5 is quite standard
in dealing with the problem for linear systems. And the requirement (4) is employed to guarantee the solvability of matrix
Eqs.
] is strictly weaker than the assumption (i.e., rank
[ (5), which
Ci Bi
0
= ni + q, i = 1, . . . , N) employed in Li et al. (2020),
−Ai Bi Bi
Yu and Chen (2021) and Zhang et al. (2020).

Denote w := max{w1 , . . . , wN }.
The optimal set Y ∗ is nonempty and convex.

4. Main results

Assumption 4. The global cost function f˜ satisfies the restricted
secant inequality condition (Yi, Zhang, Yang, Chai, & Johansson,
2020; Yi, Zhang, Yang, Johansson, & Chai, 2019) with constant
m > 0:

4.1. Continuous communication
A PI controller for the ith agent is proposed as

)
)⊤ (
∇ f˜ (a) −∇ f˜ (PY∗ (a)) ≥ m∥a − PY∗ (a)∥2 , ∀a ∈ Rq ,

∗
a − PY
(a)

ui = −Υi xi + Ψi (−α

∗
where PY
(a) is the projection of a onto the set Y ∗ , which satisfies
∗
∇ f˜ (PY (a)) = 0.

η̇i = αβ

Remark 1. Compared with Li et al. (2020), Yi et al. (2020) and
Yu and Chen (2021) which require the network to be undirected,
Assumption 1 allows the graph to be unbalanced. Assumption 2
does not require the convexity of the local cost functions. Assumptions 3–4 do not even need f˜ to be convex, which are
weaker than the strong convexity (Li, Su, & Liu, 2021; Li et al.,
2020; Tang et al., 2019; Tang & Wang, 2021; Yu & Chen, 2021;
Zhu, Ren, Yu, & Wen, 2021; Zhu et al., 2018; Zou et al., 2020) and
quadratic gradient growth (Meng & Li, 2020; Necoara, Nesterov,
& Glineur, 2019).
Assumption 5.

(5a)

Proof. From (4), we can get
fi (yi ), s.t . y1 = · · · = yN .

∥∇ fi (a) − ∇ fi (b)∥ ≤ wi ∥a − b∥, ∀a, b ∈ Rq , wi > 0.

(

Ci Bi Υi = Ci Ai ,
exist solutions Υi , Ψi .

Assumption 2. The local objective function fi is differentiable
and its gradient is wi -Lipschitz in Rq :

Assumption 3.

Under Assumption 5, the matrix equations

zii

−β

N
∑

aij (yi − yj ) − δηi ),

(6a)

j=1

aij (yi − yj ),

(6b)

aij (zi − zj ),

(6c)

j=1

żi = −

N
∑
j=1

j

where ηi ∈ Rq , zi ∈ RN , zi is the jth component of zi , j =
1, . . . , N, Υi , Ψi are feedback matrices defined in Lemma 2, α, β, δ
are positive parameters, and aij is the weight corresponding to the
edge (j, i).
As shown in Fig. 1, the compact form of the closed-loop system
is

(Ai , Bi ) is controllable, and

rank(Ci Bi ) = q, i = 1, . . . , N .

N
∑

∇ fi (yi )

ẋ = (A − BΥ )x + BΨ −α (ZN−1 ⊗ Iq )∇ f (y) − β (L ⊗ Iq )y − δη ,

(

(4)

)

(7a)
3

L. Li, Y. Yu, X. Li et al.

Automatica 141 (2022) 110259

η̇ = αβ (L ⊗ Iq )y ,
y = C x,

(7c)

The system (11) can be regarded as a perturbed system of the
nominal system

ż = −(L ⊗ IN )z ,

(7d)

ṗ = f1 (p),

(7b)

(12)

where x = col(x1 , . . . , xN ), η = col(η1 , . . . , ηN ), y = col(y1 , . . . ,
yN ), z = col(z1 , . . . , zN ), A = diag(A1 , . . . , AN ), B = diag(B1 , . . . ,
BN ), C = diag(C1 , . . . , CN ), Υ = diag(Υ1 , . . . , ΥN ), Ψ = diag(Ψ1 ,
. . . , ΨN ), ZN = diag(z11 , . . . , zNN ), and ∇ f (y) = col(∇ f (y1 ), . . . ,
∇ f (yN )).

with the perturbation item g1 (t , p).
First consider the stability of nominal system (12). Select a
Lyapunov candidate as

Remark 3. The introduction of η is to eliminate errors in the
consensus process with η(0) = 0Nq . The introduction of z is to
counteract the imbalance of the communication topology, which
is based on the methods in Zhu et al. (2021, 2018). The initial
j
value z(0) satisfies zii = 1 for i = 1, . . . , N and zi = 0 for all i ̸ = j,
−1
which ensures that ZN is well-defined (Zhu et al., 2018).

for which it is easy to verify that c1 ∥p∥2 ≤ V1 ≤ c2 ∥p∥2 and
∥ ∂∂Vp1 ∥ ≤ c3 ∥p∥ for some positive constants c1 , c2 and c3 . Its
derivative along (12) is

2

2

2N w 2
m

(6), problem (3) is solved and yi (t) converges to the optimal set Y ∗
exponentially as t → ∞ for i = 1, . . . , N.

ż = −(L ⊗ IN )z .

Define y := PY ∗ (y) = 1N ⊗ y . From (8b), the corresponding

η∗ satisfies

q

Nq .

σ ),

(13)

)

(14)

such that ȳ ∈ S and y ⊥ ∈ S ⊥ , where S = {1N ⊗ y : y ∈ Rq } and
S ⊥ is the orthogonal complement space of S. For convenience, let
ȳ be the component element of ȳ, that is, ȳ = 1N ⊗ ȳ. Moreover,
∗
it holds that PY ∗ (ȳ) = PY ∗ (y) = y ∗ , i.e., PY
(ȳ) = y∗ .
Because the gradients of local cost functions are Lipschitz, one
has

∗

η˙∗ = αβ (L ⊗ I )y ∗ = 0

1

α

y = ȳ + y ⊥ ,

(8c)

∗

σ )⊤ (Ξ ⊗ Iq )(ρ +

For any y ∈ RNq , y can be decomposed as

Proof. First, we discuss the relationship between the equilibrium
point of (7) and the optimal solution of (3). Pre-multiplying (7a)
by C , we get

(8b)

α

β ⊤
δ
σ (Ξ L ⊗ Iq )ρ − σ ⊤ (Ξ ⊗ Iq )σ
α
α
(
)
β
β
+ ρ ⊤ (Ξ L + L⊤ Ξ ) ⊗ Iq ρ + σ ⊤ (Ξ L ⊗ Iq )ρ
2
α
= − 2αρ ⊤ h − βρ ⊤ (L ⊗ Iq )ρ − 2δρ ⊤ (Ξ ⊗ Iq )σ
δ
− σ ⊤ h − σ ⊤ (Ξ ⊗ Iq )σ .
α

+ w+

η̇ = αβ (L ⊗ Iq )y ,

1

2

− σ ⊤h −

δ + 2N . For linear multi-agent system (1) with control protocol

(8a)

1

ρ ⊤ (Ξ ⊗ Iq )ρ + (ρ +

(

)
m

ẏ = −α (ZN−1 ⊗ Iq )∇ f (y) − β (L ⊗ Iq )y − δη,

2

V̇1 = − 2αρ ⊤ h − βρ ⊤ (Ξ L + L⊤ Ξ ) ⊗ Iq ρ − 2δρ ⊤ (Ξ ⊗ Iq )σ

Theorem 1. Suppose that Assumptions 1–5 hold and( the parameters α, δ, β satisfy α > 0, δ > Nmwξ and β > λ 2(αL)

1

V1 =

− σ ⊤h ≤

(9)

It follows from Zhu et al. (2021) that limt →∞ z = limt →∞
e−(L⊗IN )t z(0) = 1N ⊗ ξ , i.e., limt →∞ ZN−1 = Ξ −1 . Because η(0) =
⊤
0Nq and ξ ⊤ L = 0⊤
⊗ Iq )η(t) = 0, ∀t > 0.
N , we can get (ξ
Pre-multiply (8a) by (ξ ⊤ ⊗ Iq ), one has

N w2
2α mξ

σ ⊤Ξ σ +

αm
2N

ρ ⊤ ρ,

(15)

(y − ȳ)⊤ (∇ f (y) − ∇ f (ȳ)) ≥ −w∥y − ȳ ∥2 = −w∥y ⊥ ∥2 ,

(16)

(ȳ − y ∗ )⊤ (∇ f (y) − ∇ f (ȳ)) ≥ −w∥y ⊥ ∥ ∥ȳ − y ∗ ∥,

(17)

(y − ȳ)⊤ (∇ f (ȳ) − ∇ f (y ∗ )) ≥ −w∥ȳ − y ∗ ∥ ∥y ⊥ ∥.

(18)

∗
(ξ ⊤ ⊗ Iq )y˙∗ = −α (1⊤
N ⊗ Iq )∇ f (y ) = 0q ,

By Assumption 4, we have
(∇ f (ȳ) − ∇ f (y ∗ ))⊤ (ȳ − y ∗ )

i.e.,
y˙∗ = (ξ ⊤ 1N ) ⊗ (Iq y˙∗ ) = (ξ ⊤ ⊗ Iq )y˙∗ = 0q .

=

So one has y˙∗ = 1N ⊗ y˙∗ = 0Nq , which together with (9) ensures
that (y ∗ , η∗ ) is an equilibrium point of system (8).
To proceed, taking ρ = y − y ∗ , σ = η − η∗ , one has

(∇ fi (ȳ) − ∇ fi (y∗ ))⊤ (ȳ − y∗ )

i=1

=(∇ f˜ (ȳ) − ∇ f˜ (y∗ ))⊤ (ȳ − y∗ )
(
)⊤ (
)
∗
ȳ − PY
(ȳ)
= ∇ f˜ (ȳ) − ∇ f˜ (PY∗ (ȳ))

(
)
ρ̇ = −α (ZN−1 ⊗ Iq )∇ f (y) − (Ξ −1 ⊗ Iq )∇ f (y ∗ ) − β (L ⊗ Iq )ρ − δσ ,
(10a)

σ̇ = αβ (L ⊗ Iq )ρ.

N
∑

≥m∥ȳ − PY∗ (ȳ)∥2 =

(10b)

m
N

∥ȳ − y ∗ ∥2 .

(19)

Using (16)–(19), it follows

Next we only need to discuss the convergence of (10).
For simplicity, set p := col(ρ, σ ) and h := ∇ f (y) − ∇ f (y ∗ ).
Then, it can be concluded from (10) that p satisfies

= − (y − y ∗ )⊤ (∇ f (y) − ∇ f (y ∗ ))

ṗ = f1 (p) + g1 (t , p),

= − (y − ȳ + ȳ − y ∗ )⊤ (∇ f (y) − ∇ f (ȳ) + ∇ f (ȳ) − ∇ f (y ∗ ))

− ρ⊤h

(11)

= − (y − ȳ)⊤ (∇ f (y) − ∇ f (ȳ)) − (y − ȳ)⊤ (∇ f (ȳ) − ∇ f (y ∗ ))

where

− (ȳ − y ∗ )⊤ (∇ f (y) − ∇ f (ȳ)) − (∇ f (ȳ) − ∇ f (y ∗ ))⊤ (ȳ − y ∗ )

(
)
−α (Ξ −1 ⊗ Iq )h − β (L ⊗ Iq )ρ − δσ
,
αβ (L ⊗ Iq )ρ
)
)
( (( −1
)
α Ξ − ZN−1 ⊗ Iq ∇ f (y)
g1 (t , p) =
.
f1 (p) =

≤−
≤−

0

4

m

∥ȳ − y ∗ ∥2 + 2w∥y ⊥ ∥ ∥ȳ − y ∗ ∥ + w∥y ⊥ ∥2

N
m

2N

∥ȳ − y ∗ ∥2 + (

2N w 2
m

+ w)∥y ⊥ ∥2 .

(20)

L. Li, Y. Yu, X. Li et al.

Automatica 141 (2022) 110259

By the fact that (ξ ⊤ ⊗ Iq )η(t) = 0, one has

4.2. Sampling-based event-triggered communication

− 2δρ (Ξ ⊗ Iq )σ
⊤

The above continuous time algorithm requires each agent to
know the real-time outputs of neighbors, which is impossible in
practice due to limited communication bandwidths. Therefore,
in order to avoid continuous communication and reduce communication overhead, a sampling-based event-triggered control
algorithm is proposed with a minimum monitoring and communication interval.
l
Suppose that ti 1 is the l1 th communication instant of zi for the
l
l
l +1
ith agent, and denote ẑi (t) := zi (ti 1 ), ∀t ∈ [ti 1 , ti 1 ) as the latest
known zi of agent i ∈ V transmitted to its neighbors. Similarly let
l
ti 2 denote the l2 th communication instant of yi for the ith agent
l
l
l +1
and ŷi (t) := yi (ti 2 ), ∀t ∈ [ti 2 , ti 2 ). The communication instant
l
l
sequences {ti1 , . . . , ti 1 , . . . } and {ti1 , . . . , ti 2 , . . . } corresponding to
zi and yi , respectively, of the ith agent will be determined later.
By defining measurement error e1,i := ẑi (t) − zi (t) and e2,i :=
l
ŷi (t) − yi (t), it is clear that e1,i = 0 at any instant ti 1 and e2,i = 0
l2
at any instant ti .
Consider the next implementation of algorithm (6) with
discrete-time communication,

= − 2δ (ȳ + y ⊥ − y ∗ )⊤ (Ξ ⊗ Iq )σ
⊤

= − 2δ (y ⊥ ) (Ξ ⊗ Iq )σ
δ ⊤
σ (Ξ ⊗ Iq )σ .
≤2αδ∥y ⊥ ∥2 +
2α

(21)

According to Lemma 1 and L1N = 0, it can be concluded that
ρ ⊤ (L ⊗ Iq )ρ = (y ⊥ )⊤ (L ⊗ Iq )y ⊥ ≥ λ2 (L)∥y ⊥ ∥2 . Substituting (15),
(20) and (21) into (14), it is simplified as
V̇1 ≤ −

αm
N

(
∥ȳ − y ∥ −

αm

+

2N

∗ 2

N w2
δ
−
2α
2α mξ

)
σ ⊤ ( Ξ ⊗ Iq ) σ

ρ⊤ρ

)
)
(
2N w 2
+ w − 2αδ ∥y ⊥ ∥2
βλ2 (L) − 2α
m
(
)
δξ
αm
N w2
2
≤−
∥ρ∥ −
−
∥σ ∥2
2N
2α
2α m
(
(
))
m
2N w 2
− βλ2 (L) − 2α
+w+δ+
∥y ⊥ ∥2 .
(

−

m

⎛
ui = −Υi xi + Ψi ⎝−α

2N

According to the parameters selection in Theorem 1, it can be
2
δξ
obtained that V̇1 ≤ −c4 ∥p∥2 with c4 := min{ α2Nm , 2α − N2αwm } > 0,
which means the exponential stability of system (12) by Theorem
4.10 in Khalil and Grizzle (2002).
Next, consider the effect of the disturbance term g1 (t , p) on
nominal system (12). From Zhu et al. (2021), it follows that
max |(zii )−1 − ξi−1 | ≤ ϵ1 e−ι1 t for some positive constants ϵ1 , ι1 .
So

η̇i = αβ

żi = −

2

c

− 2c4 t

∫ t − c4 (t −τ )
0

e

2c2

(22a)

aij (ŷi − ŷj ),

(22b)

N
∑

aij (ẑi − ẑj ).

(22c)

)
m
+ w + δ + 2N
. For linear multi-agent system (1) with

2N w 2
m

control protocol (22), problem (3) is solved and yi (t) converges to
the optimal set Y ∗ exponentially as t → ∞ for i = 1, . . . , N, with
triggering conditions

−ι1

(zii )−1

(

4α
λ2 (L)

f (y∗)∥ −ι1 t
θ (τ ) dτ = αϵ1c∥∇
(e
−
4
2c2

aij (ŷi − ŷj ) − δηi ⎠ ,

j=1

Theorem 2.
Suppose that Assumptions 1–5 hold and the pa2
α2 m
and β >
rameters α, δ, β satisfy α > 0, δ > Nmwξ + 28N
ξ

l +1

⏐
:= inf {kτ ⏐∥e1,i (kτ )∥2

ti 1

e 2 ). Because e
is an upper bound about max |
− ξi |,
we can relax it to make the inequality always satisfied. From
Lemma 9.4 in Khalil and Grizzle (2002) or following its proofs,
it can be concluded that 0 is an exponentially stable equilibrium
point for perturbed system (11), i.e., every yi converges exponentially to the optimal set Y ∗ . This completes the proof. ■
−ι1 t

−β

⎞

j=1

where γ (t) = αϵ1 w e−ι1 t and∫ θ (t) = αϵ1 ∥∇∫f (y ∗ )∥e−ι1 t .
t
∞
It is easily verified that 0 γ (τ ) dτ < 0 γ (τ ) dτ = αϵ1 ι1 w .
c

N
∑

zii

N
∑

j=1

∥g1 (t , p)∥ ≤ max |(zii )−1 − ξi−1 | · α∥∇ f (y)∥
≤γ (t)∥p∥ + θ (t),

When 2c4 ̸ = ι1 , one has

∇ fi (yi )

−1

l

kτ >ti 1

≥

1

N
∑

2(2din
i + κ1 )

j=1

aij ∥ẑi − ẑj ∥2 , k ∈ N},

(23a)

l +1

⏐
:= inf {kτ ⏐∥e2,i (kτ )∥2

ti 2

Remark 4. In comparison, an exponential convergence rate is
established over unbalanced directed graphs, while the most
closely related work (Li et al., 2020) only provides an asymptotic convergence over undirected graphs without analysis of the
convergence speed building upon a stronger assumption than
Assumptions 3–5 here. Meanwhile, compared with Zhang et al.
(2020), where an exponential convergence rate is obtained for homogeneous linear multi-agent systems over unbalanced graphs,
the exponential convergence is established here for heterogeneous linear multi-agent systems based on a strictly weaker
Assumptions 3–5, including homogeneous linear multi-agent systems as a special case. Moreover, the agents in Zhang et al. (2020)
need to know the imbalance of the network ξ in advance, which
is not required in this paper. The exponential convergence result
is obtained in Zhu et al. (2021) for single-integrator multi-agent
systems, but it requires the local cost functions to be strongly
convex.

l

kτ >ti 2

≥

1

N
∑

2(2din
i +βκ2)

j=1

aij ∥ŷi − ŷj ∥2 , k ∈ N},

(23b)

where the sampling period τ satisfies

{
τ < min{τ1 , τ2 } = min
1

ς4

ln

ς1 − ς0
,
∥L∥(1 + ς0 )(1 + ς1 )

}
ς5 (ς2 + 1)(ς3 + 1) + ς4 (ς3 + 1)
,
ς4 + ς5 + ς3 ς5

with the parameter κ1 > max{ ξ2 ,
N ∥L∥2
2α mξ

, β N 4∥αL2∥βξ+m82α m }, ς0 =

ς3 =

2 2α m
,
β N ∥L∥

2

√

2

4

2

2

∥L∥2
2 ξ 2 λL

,

4∥L∥4 +2ξ 2 λ2L

+ δ and ς5 =
ς4 = αw
ξ

√

2β∥L∥.

Proof. It can be found in the appendix.
5

2
}, κ > max{ βξ
,

2
ξ 3 λ2L
ξ λL
2∥L∥
√
, ς1 = √
, ς2 =
2∥L∥
κ1 ξ −2

√

(24)

√

√ 2∥L∥ ,
βκ2 ξ −2

L. Li, Y. Yu, X. Li et al.

Automatica 141 (2022) 110259

Fig. 2. Communication network among six agents.

Remark 5. Note that the proposed algorithms require the knowledge of global parameters ξ , λ2 (L), λL , ∥L∥, m, and w . To ensure that this condition can be determined in advance by the
designer, the agents in network can execute distributed initialization procedures, such as Charalambous, Rabbat, Johansson, and
Hadjicostis (2016), Li, Chen, Su, and Li (2017) and Ren and Beard
(2008), which can calculate these parameters in finite time.

Fig. 3. The outputs error

∑6

i=1

∥yi (t) − y∗ ∥2 .

5. Simulation
Example 1.

Consider a network
where A1,2 =
[ of six agents,
]
[
]
[ ]
1 1 0
1 0
0
1
−2
, A3,4 =
, A5,6 = 0 1 1 , B1,2 =
, B3,4 =
1 1
−2 1
1
1 0 2
[ ]
[ ]
1
[
]
[
]
1
0 , C1,2 =
1 1 , C3,4 =
3 1 ,
, B5.6 =
−1
[
]2
C5,6 = 0.5 −1 0 . Agent i has the following local cost function:

[

]

Fig. 4. Triggering instants of six agents.

⎧
c1,i y2 ,
y ≤ 0,
⎪
√
⎪
√
⎪
⎪
⎨1 − (1 − y2 ) + c2,i y2 ,
0 < y ≤ 22 ,
√
√
√
√
fi (y) =
2
< y ≤ 1,
⎪ 1 − (x − 2)2 − 2 + 1 + c2,i y2 ,
⎪
2
√√
⎪
√√
√
⎪
⎩1
(x − 1 + 22−1 )2 + 2 2 − 2 + 5−54 2 + c2,i y2, y > 1,
2
where c1 = col(c1,1 , . . . , c1,6 ) = [0.2; 0.5; 0.5; 0.1; 0.3; 0.4] and
c2 = col(c2,1 , . . . , c1,6 ) = [−0.2; 0.3; 0.1; −0.2; 0.2; −0.2]. These
fi (y), modified from Example 2 in Zhang and Cheng (2015), are
differentiable,
smooth but non-convex. The global cost function
∑N
f˜ (y) =
i=1 fi (y) is non-convex and satisfies the restricted secant
inequality condition with the optimal set {0}, i.e., y∗ = 0.
The communication network among these agents is unbalanced and depicted as Fig. 2 with all the edge weights as 1.

Fig. 5. The outputs error

[

∑6

i=1

∥yi (t) − y∗ ∥2 .

]

C = 1 1 . Agent i has the following strongly convex local cost
function

It can be verified that Assumptions 1–5 hold. The parameters
of each [agent can]be selected
[ by the
] proposed
[ algorithms,
] where
−
2
−
1
−
1
2
1
−
1
2
Υ
=
,
Υ
=
,
Υ
=
, Ψ1,2 =
1
,
2
3
,
4
5
,
6
[ ]
[ ]
[ ]
−1 , Ψ3,4 = 0.5 , and Ψ5,6 = 2 . The initial values xi (0) are
randomly selected in [−10, 10].
∑6
Fig. 3 depicts the optimization errors i=1 ∥yi (t) − y∗ ∥2 under
the continuous time control law and event-triggered control law.
It can be seen that the outputs of all agents converge to the optimal value y∗ exponentially. Fig. 4 shows the triggering instants of
six agents with sampling-based event-triggered communication
control laws, from which we can observe that the communication
among six agents is discrete and has a lower bound.

fi (y) = c3,i y2 + c4,i y + c5,i ,

(25)

where c3 = col(c3,1 , . . . , c3,6 ) = [0.2, 0.5, 0.7, 0.1, 0.2, 0.3], c4 =
col(c4,1 , . . . , c4,6 ) = [2, 1, −2, −3, 1, −3] and c5 = col(c5,1 , . . . ,
c5,6 ) = [3, −4, 2, 2, −1, −2]. These fi (y) are differentiable and
∑N
strongly convex. The global cost function f˜ (y) =
i=1 fi (y) is
∗
strongly convex with the optimal point y = 1. The communication network among these agents is unbalanced and depicted
as Fig. 2 with all the edge weights as 1. The initial values xi (0) are
randomly selected in [−10, 10].
Fig. 5 shows that the algorithms in Li et al. (2020) and Yu
and Chen (2021) cannot converge to the optimal point in the
unbalanced network. The algorithm (6) proposed in this paper
has a convergence speed similar to Zhang et al. (2020), and both
converge to the optimal point exponentially. However, Zhang
et al. (2020) need agents to know the exact imbalance eigenvector
ξ of the network in advance, while only a lower bound on ξ is
enough in algorithm (6).

Example 2. In order to verify the convergence speed of our algorithm, we compare it with some related works (Li et al., 2020; Yu
& Chen, 2021; Zhang et al., 2020) with the centralized algorithm
as the baseline. Consider a system composed
of
[
] 6 homogeneous
[
]
0 1
0
1
agents, the dynamics are given as A =
,B =
,
0 0
1 −2
6

L. Li, Y. Yu, X. Li et al.

Automatica 141 (2022) 110259

6. Conclusion

and then its derivative along (27) is
V̇2 = −ϱ⊤ (L ⊗ IN )ϱ − ϱ⊤ (Ξ L ⊗ IN )e1 .

This paper has investigated the optimal output consensus
problem for heterogeneous linear multi-agent systems over unbalanced directed networks. A proportional–integral (PI) control
law has been proposed, which can converge to the optimal set
exponentially if the global cost function satisfies the restricted
secant inequality condition. Then, in order to avoid continuous
communication among agents, the algorithm has been extended
to event-triggered communication schemes with a lower bound
of the communication interval, which also guarantees exponential convergence. The detection of triggering conditions occurs
on discrete time series through sampling technology, thereby
reducing the computing cost.
Future works can be placed on eliminating the global parameters of the algorithms and considering the impact of time
delay.

(29)

By Lemma 1 and the fact that ϱ (ξ ⊗ IN ) = 0, one has
⊤

ϱ (L ⊗ IN )ϱ ≥ ξ λL ∥ϱ∥2 .
⊤

(30)

From the definition of ϱ̂, one has

ϱ⊤ (Ξ L ⊗ Iq )ϱ + 2ϱ⊤ (Ξ L ⊗ Iq )e1
⊤
=ϱ̂⊤ (Ξ L ⊗ Iq )ϱ̂+ϱ⊤ (Ξ L ⊗ Iq )e1 − e⊤
1 (Ξ L ⊗ Iq )ϱ− e1 (Ξ L ⊗ Iq )e1
∥ L∥ 2
≥ϱ̂⊤ (Ξ L ⊗ Iq )ϱ̂ −
(31)
∥ϱ∥2 − κ1 ξ ∥e1 ∥2 − e⊤
1 (Ξ L ⊗ Iq )e1 ,
2κ1 ξ

with a positive parameter κ1 .
Substituting (30) and (31) into (29), it is simplified as

(
V̇2 ≤ −

Acknowledgments

ξ λL

∥L∥2
−
2
4κ1 ξ

)

1

∥ϱ∥2 − s1 ,
2

where s1 = ϱ̂ (Ξ L ⊗ Iq )ϱ̂ − κ1 ξ ∥e1 ∥2 − e⊤
1 (Ξ L ⊗ Iq )e1 .
Because L = Din −A and Din +A ≥ 0, we can get e⊤
1 (L⊗Iq )e1 ≤
∑N
in
⊤
in
ξ
d
e
e
.
2e⊤
(
D
⊗
I
)e
=
2
i
1
,
i
1
,
i
q
1
i
1
i=1
⊤
By using 1⊤
N L = 0N , L1N = 0N and ϱ̂ = M ẑ, we get
⊤
ϱ̂⊤ (L ⊗ Iq )ϱ̂ = ẑ (L ⊗ IN )ẑ. Together with the relationship
⊤

The authors are grateful to the Editor, the Associate Editor and
the anonymous reviewers for their insightful suggestions.
This research was supported by the National Natural Science
Foundation of China under Grant 62003243, 72171172, Basic
Science Centre Program by National Natural Science Foundation
of China under grant 62088101, Ministry of Education of Republic of Singapore under Grant AcRF TIER 1-2019-T1-001-088
(RG72/19), the Shanghai Municipal Commission of Science and
Technology, China No. 19511132100, 19511132101, the Shanghai Municipal Science and Technology Major Project, China, No.
2021SHZDZX0100, and National Key R&D Program of China, No.
2018YFE0105000, 2018YFB1305304.

⊤

ẑ (L ⊗ IN )ẑ
N

=

N

2

1 ∑∑
(ξi aij + ξj aji ) ẑi − ẑj 
2
i=1 j=1

≥

N
N
1 ∑∑

2

2

ξi aij ẑi − ẑj  ,

i=1 j=1

it can be obtained that

Appendix

s1 ≥

Before proving Theorem 2, it is helpful to notice that triggering condition (23a) ensures that zi can still estimate the left
eigenvalue ξ under discrete communication, as shown in the
subsequent Lemma.

)
N (
N
∑
1 ∑
2
ξi
aij ∥ẑi − ẑj ∥2 − (2ξi din
+
κ
ξ
)
∥
e
∥
1
1,i
i
2

i=1

)
N
∑( 1 ∑
2
≥
.
ξi
aij ∥ẑi − ẑj ∥2 − (2ξi din
+
κ
ξ
)
∥
e
∥
1
i
1
,
i
i
2

i=1

Lemma 3. In the subsystem (22c), with the initial condition in
Remark 3 and triggering condition (23a), zi converges to ξ exponentially for i = 1, . . . , N.

(
V̇2 ≤ −

∥L∥2
−
2
4κ1 ξ

)
∥ϱ∥2 .

(32)
ξ λL
2

2

− 4∥κL∥ξ > 0.
1

By using Young’s inequality, one has
2∥L∥2 ϱ(kτ )⊤ ϱ(kτ ) − (κ1 ξ − 2)e1 (kτ )⊤ e1 (kτ ) ≥ s1 (kτ ) ≥ 0,

(26)

(33)

i.e.,

∑N

⊤
Let z :=
⊗ IN )z and define ϱ := z − 1N ⊗ z̄ =
i=1 ξi zi = (ξ
⊤
Mz, where M = (IN − 1N ξ ) ⊗ IN . It can be confirmed that LM = L
and M ⊤ Ξ L = Ξ L hold. Furthermore define ϱ̂ := M ẑ = ϱ + Me1 .
Then (26) can be further written as

ϱ(kτ )⊤ ϱ(kτ ) − ς0−2 e1 (kτ )⊤ e1 (kτ ) ≥ 0,

(34)

√

where ς0 = √

2∥L∥

κ1 ξ −2

.

Since (32) is only established at every sampling time kτ , k ∈ N,
we analyze what happens to the Lyapunov function V2 between
sampling instants. For t ∈ (kτ , (k + 1)τ ), by the inequality
2
ξλ
−ϱ⊤ (Ξ L ⊗ IN )e1 ≤ 2 L ϱ⊤ ϱ + 2∥ξL∥λ e1 ⊤ e1 , one has

(27)

It can be seen from Theorem 2 that the detection of triggering
condition (23a) is only performed on a time series with intervals
of τ . Therefore, the proof is also divided into two parts. Let us first
discuss the convergence of the algorithm at the detection time
kτ , k ∈ N, and then discuss the convergence of the algorithm in
the time interval (kτ , (k + 1)τ ).
Select a Lyapunov candidate as
V2 = ϱ⊤ (Ξ ⊗ IN )ϱ,

ξ λL

The choice of κ1 in Theorem 2 guarantees that

For simplicity, let z := col(z1 , . . . , zN ), ẑ := col(ẑ1 , . . . , ẑN ) and
e1 := col(e1,i , . . . , e1,N ). One has ẑ = z + e1 . The subsystem (22c)
can be written in the compact form as

ϱ̇ = −(L ⊗ IN )ϱ̂.

j=1

By the triggering condition (23a), at every sampling time
kτ , k ∈ N one can get s1 ≥ 0, thereby implying

Proof of Lemma 3

ż = −(L ⊗ IN )ẑ .

j=1

N

L

V̇2 ≤ −

=−

ξ λL
2

ξ λL
4

ϱ⊤ ϱ +
ϱ⊤ ϱ −
ξ λL

where ς1 = √

(28)
7

2∥L∥

.

∥L∥

2

2ξ λL

ξ λL
4

e1 ⊤ e1

(ϱ⊤ ϱ − ς1−2 e1 ⊤ e1 ),

(35)

L. Li, Y. Yu, X. Li et al.

Automatica 141 (2022) 110259

Inspired by Kia et al. (2015), let χ1 (t) =
derivative is

∥e1 (t)∥
, and then its
∥ϱ(t)∥

First consider the stability of nominal system (42). Selecting
the Lyapunov candidate as in (13), its derivative along (42) is
V̇1 = − 2αρ ⊤ h − βρ ⊤ (L ⊗ Iq )ρ − βρ ⊤ (Ξ L ⊗ Iq )e2

˙ ∥
˙ ∥
∥e1˙(t)∥∥ϱ(t)∥ − ∥e1 (t)∥∥ϱ(t)
(1 + χ1 (t))∥ϱ(t)
≤
.
2
∥ϱ(t)∥
∥ϱ(t)∥

χ̇1 (t) =

δ ⊤
σ ( Ξ ⊗ Iq ) σ .
α
From the relationship ρ̂ = ρ + e2 , one has
− 2δρ ⊤ (Ξ ⊗ Iq )σ − σ ⊤ h −

˙ ∥ ≤ ∥L∥ ∥ϱ(t)∥ + ∥L∥ ∥e1 (t)∥, one can conclude
Because ∥ϱ(t)
χ̇1 (t) ≤ ∥L∥(1 + χ1 (t))2 .

ρ ⊤ (Ξ L ⊗ Iq )ρ + 2ρ ⊤ (Ξ L ⊗ Iq )e2

By t = kτ , one has χ (kτ ) ≤ ς0 . Choosing the differential equation

µ̇1 = ∥L∥(1 + µ1 )2 ,

⊤
=ρ̂ ⊤ (Ξ L ⊗ Iq )ρ̂ +ρ ⊤ (Ξ L ⊗ Iq )e2 − e⊤
2 (Ξ L ⊗ Iq )ρ − e2 (Ξ L ⊗ Iq )e2
∥L∥2
≥ρ̂ ⊤ (Ξ L ⊗ Iq )ρ̂ −
∥ρ∥2 −βκ2 ξ ∥e∥2 − e⊤
2 (Ξ L ⊗ Iq )e2 , (44)
2βκ2 ξ

(36)

and the solution µ1 (t) of (36) with µ1 (kτ ) = ς0 is

with a positive parameter κ2 .
Similar to the analysis in the proof of Theorem 1, substituting
(15), (20), (21) and (44) into (43), it is simplified as

ς0 + (1 + ς0 )∥L∥(t − kτ )
1
µ1 (t) =
, t ∈ [kτ , kτ +
).
1 − (1 + ς0 )∥L∥(t − kτ )
1 + ς0

)
)
(
δξ
∥L∥2
N w2
2
V̇1 ≤ −
−
−
∥σ ∥2
∥ρ∥ −
2N
4κ2 ξ
2α
2α m
(
(
))
βλ2 (L)
2N w 2
β
m
−
− 2α
+w+δ+
∥y ⊥ ∥2 − s2 ,
(

Based on Lemma 3.4 in Khalil and Grizzle (2002), for t − kτ ≤
ς1 −ς0
τ1 = ∥L∥(1+ς
, it yields that χ1 (t) ≤ µ1 (τ1 ) = ς1 . Therefore,
0 )(1+ς1 )
⊤
it implies that p p − ς1−2 e1 ⊤ e1 ≥ 0.
By choosing the sampling period τ < τ1 , for t ∈ (τ , (k + 1)τ ),

ξ λL
4

ϱ⊤ ϱ.

αm

2

it can be obtained that
V̇2 ≤ −

(43)

m

2N

2

where s2 = ρ̂ ⊤ (L ⊗ Iq )ρ̂ − βκ2 ξ ∥e2 ∥2 − e⊤
2 (L ⊗ Iq )e2 .
Similar to the proof in Lemma 3, at every sampling time
kτ , k ∈ N, one can get

(37)

Combined with (32) and (37), by Theorem 4.10 in Khalil and
Grizzle (2002), we can conclude that ϱ converges to 0 exponentially, i.e., zi converges to ξ exponentially for i = 1, . . . , N. This
completes the proof.

)
N (
N
∑
1 ∑
2
in
2
s2 ≥
ξi
aij ∥ŷi − ŷj ∥ − (2ξi di + βκ2 ξi )∥e2,i ∥
≥ 0,
2

i=1

j=1

by the triggering condition (23b). Thereby implying
V̇1 ≤ −c5 ∥p∥2 ,

Proof of Theorem 2
The compact form of the system with controller (22) is

By using Young’s inequality, one has 2∥L∥2 p(kτ )⊤ p(kτ ) −
(βκ2 ξ − 2)e2 (kτ )⊤ e2 (kτ ) ≥ s2 (kτ ) ≥ 0, i.e., p(kτ )⊤ p(kτ ) −

ẋ = (A − BΥ )x + BΨ (−α (ZN−1 ⊗ Iq )∇ f (y) − β (L ⊗ Iq )ŷ − δη),

√

(38a)

η̇ = αβ (L ⊗ Iq )ŷ ,

(38b)

y = C x,

(38c)

ς2−2 e2 (kτ )⊤ e2 (kτ ) ≥ 0, where ς2 = √ 2∥L∥ .
βκ2 ξ −2

Since (45) is only established at every sampling time kτ , k ∈ N,
we analyze what happens to the Lyapunov function V1 between
sampling instants. For t ∈ (kτ , (k + 1)τ ), by the inequality
2 ∥L∥2
e2 ⊤ e2 , one has
−βρ ⊤ (Ξ L ⊗ IN )e2 ≤ α4Nm ρ ⊤ ρ + N βαm

where ŷ = col(ŷ1 , . . . , ŷN ).
Using the same state transformation as in the previous section
and letting ρ̂ := ŷ − y ∗ , the dynamics (38) can be written as

(
V̇1 ≤ −

)
(
ρ̇ = −α (ZN−1 ⊗ Iq )∇ f (y) − (Ξ −1 ⊗ Iq )∇ f (y ∗ ) − β (L ⊗ Iq )ρ̂ −δσ ,
(39a)

σ̇ = αβ (L ⊗ Iq )ρ̂.

−

(39b)

(40a)

−

Similar to the proof in Theorem 1, p = col(ρ, σ ) satisfies

where f2 (p)

(α

((

=

)
)
Ξ −1 −ZN−1 ⊗Iq f (y))

αβ (L⊗Iq )(ρ+e2 )

and g2 (t , p)

m
2N

))

∥y ⊥ ∥2

∥ρ∥2

δξ

αm

−

N w2

)

m

2N

2

√

2
Similar to the proof of Lemma 3, let χ2 (t) = ∥p(t)
and choose
∥
the differential equation √
µ̇2 = ς4 (1 + µ2 ) + ς5 (1 + µ2 )2 with
+ δ and ς5 = 2β∥L∥. Based on Lemma 3.4 in Khalil
ς4 = αw
ξ

=

.
0
The system (41) can be regarded as a perturbed system of the
nominal system
ṗ = f2 (p),

m

+w+δ+

where p = col(ρ, σ ) and ς3 = 2β N2∥αLm
.
∥

(41)

(−α(Ξ −1 ⊗Iq )h−β (L⊗Iq )(ρ+e2 )−δσ )

4N

2N w 2

∥ρ∥
8N
(
)
δξ
N w2
αm
αm ⊤
∥σ ∥2 −
−
−
−
(p p −ς3−2 e2 ⊤ e2 ),
2α
2α m
8N
8N

(40b)

ṗ = f2 (p) + g2 (t , p),

αm

(

N β 2 ∥ L∥ 2 ⊤
∥σ ∥2 +
e2 e2
2α
2α m
αm
(
(
))
2N w 2
m
≤ − βλ2 (L) − 2α
+w+δ+
∥y ⊥ ∥2

(
)
ρ̇ = −α (ZN−1 ⊗ Iq )∇f (y) − (Ξ−1 ⊗ Iq )∇ f (y ∗)
−β (L ⊗ Iq )(ρ + e2 ) −δσ,

βλ2 (L) − 2α

(

−

And because e2,i = ŷi (t) − yi (t), we have ρ̂ = ρ + e2 with
e2 = col(e2,1 , . . . , e2,N ). Then (39) is equivalent to

σ̇ = αβ (L ⊗ Iq )(ρ + e2 ).

(45)

2
∥L∥2 δξ
where c5 := min{ 2N − 4κ ξ , 2α − N2αwm } > 0.
2

αm

∥e (t)∥

ς (ς +1)(ς +1)+ς (ς +1)

and Grizzle (2002), for t − kτ ≤ τ2 = ς1 ln 5 2 ς +ς3 +ς ς 4 3 ,
4
4
5
3 5
it yields that χ2 (t) ≤ µ2 (τ2 ) = ς3 . Therefore, it implies that
−2 ⊤
⊤
p p − ς3 e2 e2 ≥ 0.
By choosing the sampling period τ < τ2 , for t ∈ (τ , (k + 1)τ ),
it can be obtained that

(42)

with the perturbation item g2 (t , p).
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L. Li, Y. Yu, X. Li et al.

Automatica 141 (2022) 110259

V̇1 ≤ −c6 ∥p∥2 ,

(46)

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2
δξ
where c6 := min{ 8N , 2α − N2αwm − α8Nm } > 0.
i −1

αm

From Lemma 3, it follows that max |(zi ) − ξi−1 | ≤ ϵ2 e−ι2 t for
some positive constants ϵ2 , ι2 . By following the proof procedures
similar to those given in Theorem 1, it can be concluded that 0
is an exponentially stable equilibrium point for perturbed system
(41), i.e., every yi converges exponentially to the optimal set Y ∗ .
The sampling-based event-triggered mechanism naturally avoids
Zeno’s behavior. This completes the proof.
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Li Li received the B.S. and M.S. degrees in electrical automation from Shengyang Agriculture University,
Shengyang, China, in 1996 and 1999, respectively,
and the Ph.D. degree in mechatronics engineering
from the Shenyang Institute of Automation, Chinese
Academy of Science, Shenyang, in 2003. She then
joined Tongji University, Shanghai, China, where she
is currently a Professor of control science and engineering. She has over 50 publications, including 5
books, over 30 journal papers, and 2 book chapters. Her
current research interests include unmanned systems,
data-driven modeling and optimization, and energy systems.
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L. Li, Y. Yu, X. Li et al.

Automatica 141 (2022) 110259
His research interests include distributed control and (online) optimization, algorithms, operator theory, game theory, and machine learning, with
applications to UAVs and autonomous vehicles, etc.

Yang Yu received the B.E. degree in automation from
Tongji University, Shanghai, China in 2019. He is
currently pursuing the Ph.D. degree in control science and engineering from the College of Electronic
and Information Engineering, Tongji University. His
current research interests include control theory and
distributed optimization.

Lihua Xie (F’07) received the Ph.D. degree in electrical
engineering from the University of Newcastle, Australia,
in 1992. Since 1992, he has been with the School of
Electrical and Electronic Engineering, Nanyang Technological University, Singapore, where he is currently a
professor and Director, Delta-NTU Corporate Laboratory
for Cyber–Physical Systems and Director, Center for
Advanced Robotics Technology Innovation. He served
as the Head of Division of Control and Instrumentation
from July 2011 to June 2014. He held teaching appointments in the Department of Automatic Control, Nanjing
University of Science and Technology from 1986 to 1989.
Dr Xie’s research interests include robust control and estimation, networked
control systems, multi-agent networks, localization and unmanned systems. He
is an Editor-in-Chief for Unmanned Systems and has served as Editor of IET Book
Series in Control and Associate Editor of a number of journals including IEEE
Transactions on Automatic Control, Automatica, IEEE Transactions on Control
Systems Technology, IEEE Transactions on Control of Network Systems, and IEEE
Transactions on Circuits and Systems-II. He was an IEEE Distinguished Lecturer
(Jan. 2012-Dec. 2014). Dr Xie is Fellow of Academy of Engineering Singapore,
IEEE, IFAC, and CAA.

Xiuxian Li (SM’21) received the B.S. degree in mathematics and applied mathematics and the M.S. degree
in pure mathematics from Shandong University, Jinan,
China, in 2009 and 2012, respectively, and the Ph.D.
degree in mechanical engineering from the University
of Hong Kong, Hong Kong, in 2016. From 2016 to
2020, he has been a research fellow with the School
of Electrical and Electronic Engineering, Nanyang Technological University, Singapore, and he has also been
a senior research associate with the Department of
Biomedical Engineering, City University of Hong Kong,
Hong Kong, in 2018. He held a visiting position at King Abdullah University
of Science and Technology, Saudi Arabia, in September 2019. Since 2020,
he has been a research professor with the Department of Control Science
and Engineering, and Shanghai Research Institute for Intelligent Autonomous
Systems, Tongji University, Shanghai, China.

10

