6042

IEEE TRANSACTIONS ON NEURAL NETWORKS AND LEARNING SYSTEMS, VOL. 34, NO. 9, SEPTEMBER 2023

Distributed Finite-Time Optimization of
Second-Order Multiagent Systems
With Unknown Velocities
and Disturbances
Xiangyu Wang , Senior Member, IEEE, Wei Xing Zheng , Fellow, IEEE, and Guodong Wang

Abstract— In this article, the distributed finite-time optimization problem is investigated for second-order multiagent systems with unknown velocities, disturbances, and quadratic local
cost functions. To solve this problem, by combining finitetime observers (FTOs), the homogeneous systems theory, and
distributed finite-time estimator techniques together, an output
feedback-based feedforward-feedback composite distributed control scheme is proposed. Specifically, the control scheme consists
of three parts. First, some FTOs are developed for the agents to
estimate their unknown velocities and the disturbances together.
Second, based on the velocity and disturbance estimates, the
homogeneous system theory, and some global information on all
the local cost functions’ gradients, Hessian matrices, and the
velocity estimates, a kind of centralized finite-time optimization
controllers is designed. Third, by designing some distributed
finite-time estimators and using their estimates to replace the
global terms employed in the centralized optimization controllers,
the distributed finite-time optimization controllers are derived.
These controllers achieve the distributed finite-time optimization
goal. Simulations illustrate the effectiveness and superiority of
the proposed control scheme.
Index Terms— Composite control, distributed optimization,
disturbances, finite-time control, multiagent systems, unknown
states.

I. I NTRODUCTION

I

N RECENT years, distributed optimization problems of
multiagent systems have drawn much attention mainly
because of their wide applications in practice, such as distributed optimal coordination of multiple mobile robots [1],
Manuscript received 30 December 2020; revised 8 August 2021; accepted
22 November 2021. Date of publication 10 January 2022; date of current
version 1 September 2023. This work was supported in part by the National
Natural Science Foundation of China under Grant 61873060, Grant 61973081,
and Grant 62025302; in part by the Natural Science Foundation of Jiangsu
Province under Grant BK20190061; in part by the Key Research and Development Plan of Jiangsu Province under Grant BE2020082-4; in part by the
Zhishan Young Scholar Program of Southeast University; and in part by the
Qing Lan Project of the Higher Education Institutions of Jiangsu Province.
(Corresponding authors: Wei Xing Zheng; Xiangyu Wang.)
Xiangyu Wang and Guodong Wang are with the School of Automation,
Southeast University, Nanjing 210096, China, and also with the Key Laboratory of Measurement and Control of Complex Systems of Engineering,
Ministry of Education, Nanjing 210096, China (e-mail: w.x.y@seu.edu.cn;
w.g.d@seu.edu.cn).
Wei Xing Zheng is with the School of Computer, Data and Mathematical
Sciences, Western Sydney University, Sydney, NSW 2751, Australia (e-mail:
w.zheng@westernsydney.edu.au).
Color versions of one or more figures in this article are available at
https://doi.org/10.1109/TNNLS.2021.3132658.
Digital Object Identifier 10.1109/TNNLS.2021.3132658

economic dispatch of smart grids [2], [3], and cooperative
source localization of sensor networks [4]. In a distributed
optimization problem, each agent has a local cost function
only known to itself and the multiagent system has a global
cost function as the sum function of all the local cost
functions. The control objective is to cooperatively minimize
the global cost function by designing proper distributed controllers for the agents. Compared with traditional distributed
cooperative control, such as consensus and containment control, distributed optimization is more complex and challenging, since it requires not only cooperative control but also
an optimization task (i.e., minimization of the global cost
function).
In distributed optimization field, the results are generally
classified into two groups, namely, discrete-time results and
continuous-time results. In [2] and [4]–[9], several distributed
optimization control algorithms were designed in discretetime domain. Since a lot of practical multiagent systems
have continuous-time dynamics (such as multimobile robot
systems, multimanipulator systems, and multiunmanned aerial
vehicle systems), recently many distributed optimization control algorithms were proposed for continuous-time multiagent
systems [1], [3], [10]–[18]. It is worth pointing out that those
algorithms achieve asymptotic optimization and the corresponding closed-loop systems have exponential convergence
rates at best, namely, the minimizer of the global cost function
is not reached by the agents until time goes to infinity. In fact,
most of the practical distributed optimization tasks (such as
those mentioned above) are expected to be accomplished in
finite time. However, since finite-time convergence means
nonsmoothness of the closed-loop systems, the traditional
distributed optimization methods (such as the primal method,
the dual method, or the primal-dual method) cannot realize
distributed finite-time optimization.
To improve the convergence of the closed-loop systems,
some distributed finite-time optimization results have been
presented for first-order [19]–[21] and second-order [22]
multiagent systems. However, for one thing, disturbances
(i.e., system uncertainties or external disturbances) were not
considered in those results. In practice, the disturbances cannot be neglected in control systems, because they usually
bring adverse effects onto system performances [23]–[26].
In [16], [17], and [27]–[29], some distributed optimization

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WANG et al.: DISTRIBUTED FINITE-TIME OPTIMIZATION OF SECOND-ORDER MULTIAGENT SYSTEMS

problems of multiagent systems with disturbances were studied, but the results were asymptotic rather than finite-time
optimization results. In [30], distributed finite-time optimization was achieved for first-order multiagent systems with
disturbances. For another, most of the existing distributed
optimization results were attained based on the condition that
all the agents’ states are measurable and known. In some
cases, control systems may have unmeasured states due to
the absence of corresponding sensors, such as motors’ speeds,
manipulators’ forces, and hydraulic cylinders’ pressures. Such
control design is more difficult than that based on full-state
feedback. Usually, to obtain the speed for control design,
a direct manner is to compute the position difference. However, this operation amplifies measurement noises. Hence,
some other ways are needed to obtain the unknown state
information and an efficient way is to design observers.
In [1], via some linear observers and output feedback control,
a distributed asymptotic optimization algorithm was proposed
for higher-order multiagent systems with unknown states but
without disturbances.
From the above review, distributed finite-time optimization
problems for multiagent systems are important. However, the
results considering unknown states and disturbances have
been rarely reported, since it requires accurate finite-time
estimation of both unknown states and disturbances and an
organic integration of such estimates with the nonsmooth
control and optimization techniques to derive output-feedback
based distributed optimization controllers, which is effortful.
In this article, the distributed finite-time optimization problem
is studied for second-order multiagent systems with unknown
velocities, disturbances, and quadratic local cost functions.
To solve this problem, an output-feedback based composite
distributed control scheme is developed through three phases.
Firstly, some finite-time observers (FTOs) are designed for
the agents to get their unknown velocity and disturbance
information accurately in finite time. Secondly, by using the
velocity and disturbance estimates generated from the FTOs,
the homogeneous systems theory and some global information
(i.e., the sum for all the agents’ local cost functions’ gradients
and the sum for all the products of each agent’s local cost function’s Hessian matrix and the agent’s observed velocity), a kind
of centralized finite-time optimization controllers are designed
in the form of finite-time consensus controllers plus some
global optimization terms. Third, some distributed estimators
are designed, which estimate each agent’s local cost function’s
gradient and the product of each agent’s local cost function’s
Hessian matrix and its observed velocity in finite time. Then
by using these estimates to reconstruct the global optimization
terms employed in the centralized controllers, the agents’
distributed finite-time optimization controllers are proposed.
Under the controllers, all the agents’ outputs converge to the
unique minimizer of the global cost function in finite time,
i.e., the distributed finite-time optimization goal is achieved.
The contribution of this article lies in three aspects. Firstly,
the distributed finite-time optimization problem of secondorder multiagent systems with both unknown velocities and
disturbances is solved. The considered problem is more difficult than the existing distributed finite-time optimization

6043

problems without unknown states or disturbances. Thus, the
obtained result enlarges the research scope of distributed finitetime optimization problems. Secondly, the proposed control
scheme in this article builds a bridge from finite-time consensus design to distributed finite-time optimization design.
By adding some appropriate optimization terms into the finitetime consensus controllers, the derived controllers realize
distributed finite-time optimization. Since neither the design of
finite-time consensus controllers nor the design of optimization
terms is specific, the proposed control scheme has good versatility in solving the distributed finite-time optimization problems. Last but not least, the agents’ disturbances are handled
by feedforward-feedback composite control. Compared with
the existing results based on feedback control, the proposed
composite control scheme can handle the disturbances’ effects
more directly and promptly. Moreover, the agents’ unknown
velocities and disturbances are observed together by designing
some FTOs, which play important roles in the control design.
The remainder of this article is organized as follows.
In Section II, some preliminaries and problem formulation are
introduced. In Section III, the distributed finite-time optimization results are presented. In Section IV, numerical simulations
are made to illustrate the advantages of the proposed control
scheme. In Section V, the conclusions are drawn.
II. P RELIMINARIES AND P ROBLEM F ORMULATION
A. Notations
Let 0n×m denote the n × m null matrix and Im as the
m × m identity matrix, and 1m = [1, . . . , 1]T ∈ Rm , 0m =
[0, . . . , 0]T ∈ Rm . For a vector x = [x 1 , . . . , x m ]T ∈
Rm and α ∈ R, denote [x]k = x k , k = 1, . . . , m and
x α = [x 1α , . . . , x mα ]T , sigα (x) = [sigα (x 1 ), . . . , sigα (x m )]T ,
where sigα (z) = |z|α sgn(z), ∀z, α ∈ R and sgn(·) is
the standard sign function. Especially, for x ∈ Rm ,
sgn(x) = [sgn(x 1 ), . . . , sgn(x m )]T . For a matrix B = [bi j ] ∈
Rn×m , [B]i, j represents its (i, j )th element
m and sgn(B) =
[sgn(bi j )] ∈ Rn×m . Denote x1 =
i=1 |x i |, x2 =
(x T x)1/2 , and x∞ = maxi=1,...,m {|x i |} as the 1-norm,
Euclidean norm, and infinity norm of vector x √∈ Rm ,
respectively. A basic property is that x2 ≤ x1 ≤ mx2 ,
Denote B2 = (λmax (B T B))1/2 and B∞ =
∀x ∈ Rm . 
m
maxi=1,...,n { l=1
|bil |} as the Euclidean norm and infinity
norm of matrix B ∈ Rn×m , respectively. For a symmetric
matrix P ∈ Rm×m , its eigenvalues are denoted as λmin (P) =
λ1 (P) ≤ λ2 (P) ≤ · · · ≤ λm (P) = λmax (P) in a nondecreasing order. For any two matrices A ∈ Rm×n and B ∈
R p×q , A⊗B ∈ Rmp×nq denotes their Kronecker product, where
m, n, p, q are positive integers.
B. Lemmas and Definitions
A few lemmas and definitions are given for later
muse. q
,
.
.
.
,
x
≥
0.
Then
(
Lemma
1
([31]):
Let
x
1
m
i=1
m
xmi ) ≤
m q
q
q
q−1
x
,
if
0
<
q
≤
1
and
(
x
)
≤
m
i
i=1 i
i=1
i=1 x i ,
if 1 < q < +∞.
Lemma 2 ([32]): If c > 0, d > 0, and γ (x, y) > 0
is a real-valued function in x, y ∈ R, then |x|c |y|d ≤
(cγ (x, y)|x|c+d )/(c + d) + (dγ −c/d (x, y)|y|c+d )/(c + d).

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6044

IEEE TRANSACTIONS ON NEURAL NETWORKS AND LEARNING SYSTEMS, VOL. 34, NO. 9, SEPTEMBER 2023

Consider the following autonomous system

system is written as
1

ẋ = f (x), x = [x 1 , . . . , x m ] ∈ R , f (0m ) = 0m
T

m

(1)

where f : D → Rm is the continuous function on an open
neighborhood D ⊆ Rm of the origin. Denote the initial state
as x 0 = x(0).
Definition 1 ([33]): The origin is a finite-time convergent
equilibrium of system (1) if there is an open neighborhood
U ⊆ D of the origin and a function Tx : U \{0m } → (0, ∞),
such that every solution trajectory x(t, x 0 ) of system (1)
starting from the initial point x 0 ∈ U \{0m } is well-defined for
t ∈ [0, Tx (x 0 )), and limt→Tx (x0 ) x(t, x 0 ) = 0m . Tx (x 0 ) is called
as the settling-time function (with respect to x 0 ). The origin
is said to be a finite-time stable equilibrium if it is finite-time
convergent and Lyapunov stable. If U = D = Rm , then the
origin is said to be a globally finite-time stable equilibrium.
Lemma 3 ([33]): For system (1), if there exist a positivedefinite continuous function V (x): D → R, c > 0 and α ∈
(0, 1), and an open neighborhood U0 ⊂ D of the origin such
that V̇ (x) + cV α (x) ≤ 0, ∀x ∈ U0 \{0}, then the origin is a
finite-time stable equilibrium of system (1). If U0 = D = Rm
and V (x) is proper, then the origin is a globally finite-time
stable equilibrium. The finite convergence time Tx (x 0 ) satisfies
Tx (x 0 ) ≤ (V 1−α (x 0 )/c(1 − α)).
Definition 2 ([33]): Consider system (1), where f : U →
Rm is continuous on a neighborhood U of the origin x = 0m
in Rm . Denote [r1 , . . . , rm ] with rk > 0, k = 1, . . . , m and
f (x) = [ f 1 (x), . . . , f m (x)]T as a continuous vector field.
f (x) is said to be homogeneous of degree τ ∈ R with
respect to the dilation [r1 , . . . , rm ], if ∀ > 0, x ∈ Rm ,
f k ( r1 x 1 , . . . ,  rm x m ) =  τ +rk f k (x), k = 1, . . . , m, where
τ ≥ − min{rk , k = 1, . . . , m}. System (1) is said to be
homogeneous if f (x) is homogeneous.
Lemma 4 ([33]): If the origin is a globally asymptotically
stable equilibrium of system (1) and f (x) is homogenous
of a negative degree, then it is a globally finite-time stable
equilibrium.
Consider the system ẋ = g(x) + d(t), x ∈ R, where x is
the state, g(x) ∈ R, and d(t) is the disturbance. Assume that
d(t) is p − 1 times differentiable with p ≥ 1, and d ( p−1) (t)
has a Lipschitz constant L > 0. For this system, a nonlinear
observer is designed to estimate d(t) and its derivatives in the
following:
1

p

ż 0 = ψ0 + g(x), ψ0 = −λ0 L p+1 sig p+1 (z 0 − x) + z 1
p− 

1
ż  = ψ , ψ = −λ L p+1− sig p+1− z  − ψ−1 + z +1
 = 1, . . . , p − 1


ż p = −λ p Lsgn z p − ψ p−1

(2)

where λ0 , . . . , λ p are positive gains, and z 0 = x̂, z  =
(−1) , 
= 1, . . . , p are the estimates of x, d (−1) ,
d
respectively. By denoting the observation errors w0 = x 1 − z 0 ,
w = d (−1) − z  ,  = 1, . . . , p, the observation error

p

ẇ0 = −λ0 L p+1 sig p+1 (w0 ) + w1
p− 

1
ẇ = −λ L p+1− sig p+1− w − ẇ−1 + w+1
 = 1, . . . , p − 1


ẇ p ∈ −λ p Lsgn w p − ẇ p−1 + [−L, L].

(3)

Lemma 5 ([34]): If the gains λ0 , . . . , λ p are appropriately
chosen, then observer (2) is finite-time convergent.
C. Convex Functions
Some fundamental concepts and principles on convex functions are now introduced [35]. A set D ⊆ Rm is convex if
ϑ x + (1 − ϑ)y ∈ D, ∀x, y ∈ D and 0 ≤ ϑ ≤ 1. The
whole m-dimensional space Rm is a convex set. For a twice
differentiable function f : Rm → R, its gradient and Hessian
matrix (or second-order derivative) are denoted as ∇ f (x) and
∇ 2 f (x), respectively. A function f : Rm → R is convex,
if its domain is a convex set D and f (ϑ x + (1 − ϑ)y) ≤
ϑ f (x) + (1 − ϑ) f (y), ∀x, y ∈ D with 0 ≤ ϑ ≤ 1. A twice
continuously differentiable convex function f : Rm → R is
θ -strongly convex with θ > 0, if and only if ∇ 2 f (x) ≥ θ Im ,
∀x ∈ D. For a θ -strongly convex function f (x), it has a
unique minimizer x ∗  arg min x∈Rm f (x) satisfying f (x ∗ ) =
minx∈Rm f (x) and ∇ f (x ∗ ) = 0m .
D. Graph Theory Notions
Let G = (V, E, A) be a directed graph. Here
V = {1, . . . , N} is the node set, E ⊆ V × V is the edge set,
and A = [ai j ] ∈ R N ×N is the weighted adjacency matrix.
For the edge weights, ai j > 0 if ( j, i ) ∈ E while ai j = 0
otherwise, and aii = 0, ∀i ∈ V. The neighbor set of node i is
Ni = { j ∈ V | ( j, i ) ∈ E}. G is an undirected graph if and
only if ( j, i ) ∈ E ⇔ (i, j ) ∈ E. In an undirected graph G,
N ×N
,
ai j = a j i . The
 Laplacian matrix of G is L = [li j ] ∈ R
where lii =
j ∈Ni ai j , l i j = −ai j for i  = j . In a directed
graph, a directed path is an edge sequence (k1 , k2 ), (k2 , k3 ), . . .
with ki ∈ V. An undirected path in an undirected graph is
defined analogously. An undirected graph is connected if there
is an undirected path between any two nodes. A directed tree
is a graph, where every node has exactly a parent except for a
node (called the root) which has no parent, and the root has a
directed path to every other node. A directed spanning tree of
a graph is a directed tree formed by graph edges that connect
all the nodes of the graph. A graph has a directed spanning
tree if there exists at least one node having a directed path to
all the other nodes.
Lemma 6 ([36]): For a connected undirected graph G =
(V, E, A), its Laplacian matrix L is positive semi-definite. 0 is
a simple eigenvalue of L and 1 N is the associated eigenvector.
The eigenvalues of L satisfy 0 < λ2 (L) ≤ · · · ≤ λ N (L) =
λmax (L). Moreover, if 1TN x = 0 for x ∈ R N , then x T Lx ≥
λ2 (L)x T x.
Lemma 7 ([37]): For a connected undirected graph G =
(V, E, A) with the Laplacian matrix L, if there is a
nonnegative-definite diagonal matrix B = diag{b1 , . . . , b N }
with at least one positive diagonal element, then the matrix
L̄ = L + B is positive definite.

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WANG et al.: DISTRIBUTED FINITE-TIME OPTIMIZATION OF SECOND-ORDER MULTIAGENT SYSTEMS

E. Problem Formulation
Consider the leaderless second-order multiagent system
ẋ i = v i , v̇ i = u i + di , i ∈ V

(4)

where x i = [x i,1 , . . . , x i,m ]T ∈ Rm is the position of agent
i , v i = [v i,1 , . . . , v i,m ]T ∈ Rm is the unknown velocity, yi =
[yi,1 , . . . , yi,m ]T = x i is the output, di = [di,1 , . . . , di,m ]T ∈
Rm is the disturbance, and u i = [u i,1 , . . . , u i,m ]T ∈ Rm is
the control input. The system’s communication topology is
described by an undirected graph G = (V, E, A). Agent i (i ∈
V) has a local cost function f i (yi ) ∈ R only known
 Nto itself.
The global cost function is defined as f (y) =
i=1 f i (y),
y = [y 1 , . . . , y m ]T ∈ Rm .
Assumption 1: For agent i (i ∈ V), its cost function f i (yi )
has a quadratic form as f i (yi ) = (1/2)yiT Ai yi + Bi yi + ci ,
T
m×m
where
, Bi ∈ R1×m , ci ∈ R, i ∈ V, and
 N Ai = Ai ∈ R
i=1 A i is positive definite.
Remark 1: A similar assumption was made in [12]. In practice, the quadratic cost functions satisfying Assumption 1 are
widely used, such as quadratic aggregate distance functions
in optimal rendezvous of multimobile robot systems [1],
quadratic cost functions in economic dispatch of smart
grids [2], [3], and quadratic distance functions to the disks
in source localization of wireless sensor networks [4]. With
Assumption
1, the global
functionis given by 
f (y) =
cost
N
N
N
N
f i (y) = (1/2)y T ( i=1
Ai )y + ( i=1
Bi )y + i=1
ci .
i=1 
N
Since i=1 A
is
positive
definite,
f
(y)
is
-strongly
convex
i
N
with
≥
i=1 λmin (A i ) > 0. Thus, f (y) has a unique
minimizer y ∗ = [(y 1 )∗ , . . . , (y m )∗ ]T ∈ Rm satisfying f (y ∗ ) =
min y∈Rm f (y) and ∇ f (y ∗ ) = 0m .
This article aims to achieve distributed finite-time optimization (i.e., yi = x i → y ∗ , ẏi = v i → 0m , i ∈ V in finite time)
for multiagent system (4) with both unknown velocities and
disturbances by designing proper distributed controllers u i .
III. D ISTRIBUTED F INITE -T IME O PTIMIZATION
C ONTROL D ESIGN
In this section, output feedback-based composite distributed
control scheme is proposed with a three-part design. Firstly,
some FTOs are designed to estimate the agents’ unknown
velocities and disturbances. Secondly, based on the homogeneous systems theory and velocity and disturbance estimates
generated from the FTOs, a kind of centralized finite-time
optimization controllers are designed in the form of finitetime consensus controllers plus some global terms on the local
cost functions. Thirdly, based on the centralized controllers
and by developing some distributed finite-time estimators, the
distributed finite-time optimization controllers are derived by
replacing the global terms with estimates generated from the
distributed estimators.
A. FTOs Design
To begin with, the following assumption is introduced.
Assumption 2: The disturbance di,l (t), l = 1, . . . , m, i ∈ V
of system (4) is Mi,l − 1 (Mi,l ≥ 1) times differentiable and
(M −1)
di,l i,l (t) has a Lipschitz constant L i,l .

6045

Remark 2: On one hand, several kinds of practical disturbances satisfy Assumption 2, such as constant, ramp, parabolic, high-order, sinusoidal disturbances, and their mixtures.
On the other hand, it is hard for estimators to estimate
very fast switching disturbances. From these two aspects,
Assumption 2 is reasonable, under which some FTOs can be
designed to estimate the unknown velocities and disturbances
simultaneously.
To estimate the agents’ unknown velocities v i and disturbances di , i ∈ V in system (4), some observers are
designed
0
0
= ψi,l
ż i,l

Mi,l +1 
  1

0
0
1
ψi,l
= −λ0i,l L i,l Mi,l +2 sig Mi,l +2 z i,l
− x i,1 + z i,l

1
1
ż i,l
= ψi,l
+ u i,l
Mi,l 
  1

1
1
0
2
+ z i,l
ψi,l
= −λ1i,l L i,l Mi,l +1 sig Mi,l +1 z i,l
− ψi,l
ρ

ρ

ż i,l = ψi,l


Mi,l +1−ρ 
 1
ρ
ρ 
ρ
ρ−1
ρ+1
+ z i,l
ψi,l = −λi,l L i,l Mi,l +2−ρ sig Mi,l +2−ρ z i,l − ψi,l
ρ = 2, . . . , Mi,l , l = 1, . . . , m,


Mi,l +1
M +1
M +1
M
= −λi,li,l L i,l sgn z i,li,l − ψi,l i,l
ż i,l

i ∈V
(5)


ρ
(ρ−2)
0
1
where z i,l
= x

, ρ =
i,l , z i,l = v
i,l , and z i,l = di,l
2, . . . , Mi,l + 1 are, respectively, the estimates of x i,l , v i,l ,
M +1
(ρ−2)
and di,l , and λ0i,l , . . . , λi,li,l
> 0 are the observer
T
T
gains. Denote xi = [x
i = [v
i,1 , . . . , x i,m ] , v
i,1 , . . . , v i,m ] ,
T


di = [di,1 , . . . , di,m ] , i ∈ V. Denote the observation
ρ
(ρ−2)
0
1
errors wi,l
= x i,l − x

−
i,l , wi,l = v i,l − v
i,l , wi,l = di,l

(ρ−2)
di,l , ρ = 2, . . . , Mi,l + 1, l = 1, . . . , m, i ∈ V, and
0
0 T
1
1 T
, . . . , wi,m
] , wi1 = [wi,1
, . . . , wi,m
] , wi2 =
wi0 = [wi,1
2
2 T
[wi,1 , . . . , wi,m ] . From (4) and (5), the observation error
systems are given by
Mi,l +1 
  1

0
0
1
+ wi,l
= −λ0i,l L i,l Mi,l +2 sig Mi,l +2 wi,l
ẇi,l
M
i,l
  1
 1

1
0
2
+ wi,l
ẇi,l
= −λ1i,l L i,l Mi,l +1 sig Mi,l −1 wi,l
− ẇi,l


M
+1−ρ
i,l
 1
ρ
ρ 
ρ
ρ−1
ρ+1
+ wi,l
ẇi,l = −λi,l L i,l Mi,l +2−ρ sig Mi,l +2−ρ wi,l − ẇi,l

ρ = 2, . . . , Mi,l , l = 1, . . . , m, i ∈ V


Mi,l +1
M +1
M +1
M
ẇi,l
∈ −λi,li,l L i,l sgn wi,li,l − ẇi,li,l +

−L i,l , L i,l . (6)

Proposition 1: By choosing appropriate positive observer
M +1
gains λ0i,l , . . . , λi,li,l , l = 1, . . . , m, i ∈ V, observers (5) are
finite-time convergent.
Proof:
By Lemma 5 and observation error system (6), the result follows immediately and no further proof
is needed.
Remark 3: By this proposition, there is finite convergence
0
(t) =
time Tob (related to the initial conditions) such that wi,l
Mi,l +1
0, . . . , wi,l
(t) = 0, ∀t ≥ Tob . For agent i (i ∈ V), the
key parameters for implementation of an observer (5) are the
M +1
order parameter Mi,l and the gains λ0i,l , . . . , λi,li,l . Mi,l is
determined by the order of the disturbance di,l . According
to Remark 2, Mi,l can usually be obtained by disturbance
modeling. If Mi,l cannot be accurately obtained, then one can

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initially select a relatively higher order for the observer to
guarantee the validity and next gradually reduce the order
until the observation performance significantly deteriorates.
After several times of trial and error, a suitable order for the
M +1
can be
observer can be attained. The gains λ0i,l , . . . , λi,li,l
0
chosen recursively and only λi,l needs to be set ahead. For
instance, the gains can be chosen relatively larger at first and
then be reduced gradually according to the desired observation
performance (generally, the observer dynamics should be set
much faster than the controller dynamics). More details on
gain tuning can be found in [34] and the references therein.

1) Distributed Finite-Time Estimator Design: Denote matrices Bk = diag{bk1 , . . . , bki , . . . , bk N } ∈ R N ×N , k ∈ V, where
bkk > 0 and bki = 0, ∀i = k ∈ V. By Lemma 7, matrices
, k ∈ V are positive definite. To reconstruct the
L̄k = L + Bk
N
global term k=1
∇ f k (x k ) for each agent, ∇ f k (x k ), k ∈ V
are distributedly estimated. On the cost function’s gradient
∇ f k (x k ) of agent k (k ∈ V), a distributed estimator is developed to estimate it such that all the other agents get it in finite
time (i ∈ V)
⎡
⎤




j
φ̇ki =−l1,k sgn⎣
ai j φki − φk + bki φki − ∇ f k (x k ) ⎦ (8)
j ∈Ni

B. Centralized Finite-Time Optimization Controller Design
By using the homogeneous systems theory and employing the velocity and disturbance estimates generated from
FTOs (5), the following nonsmooth centralized optimization
controllers are designed for multiagent system (4):




ai j sigα1 x i − x j − k2
ai j sigα2 vi − vj
u i = −k1
j ∈Ni

−γ1 sigβ1

j ∈Ni



N

∇ f k (x k )
k=1
N

−γ2 sigβ2

i
i
where φki = [φk,1
, . . . , φk,m
]T ∈ Rm is the estimate of ∇ f k (x k )
for agent i, i ∈ 
V, and l1,k is a positive gain. To reconstruct
N
the global term k=1
∇ 2 f k (x k )
v k for each agent, ∇ 2 fk (x k )
vk ,
v k of
k ∈ V are distributedly estimated. On the term ∇ 2 f k (x k )
agent k (k ∈ V), a distributed estimator is designed to estimate
it such that all the other agents get it in finite time (i ∈ V)
⎤
⎡




j
χ̇ki = −l2,k sgn⎣ ai j χki −χk +bki χki −∇ 2 f k (x k )
v k ⎦ (9)
j ∈Ni


∇ 2 f k (x k )
v k − di , i ∈ V

χki

(7)

k=1

where k1 , k2 , γ1 , γ2 > 0, 0 < α1 < 1, α2 = (2α1 )/(1 + α1 ),
0 < β1 < 1, β2 = (2β1 )/(1 + β1 ), and vi and di are estimates
of v i and di generated from FTOs (5), respectively.
Theorem 1: For multiagent system (4), if the graph G
is connected and Assumptions 1 and 2 hold, then under
optimization controllers (7), yi = x i → y ∗ (i.e., the unique
minimizer of the global cost function f (y)) and ẏi = v i → 0m ,
i ∈ V in finite time.
Proof: See Appendix A.
Remark 4: For agent i (i ∈ V), controller (7) is a
nonsmooth controller and has a 
feedforward-feedback comα1
posite
form.
Specifically,
−k
1
j ∈Ni ai j sig (x i − x j ) −

α2
k2 j ∈Ni ai j sig (
v i − vj ) are finite-time consensus terms in
the form of a finite-time consensus controller (for which the
details are given in Part 2 “Finite-time consensus”
of the proof
β1  N
sig
(
of Theorem
1
in
Appendix
A),
and
−γ
1
k=1 ∇ f k (x k ))−
N
γ2 sigβ2 ( k=1
∇ 2 f k (x k )
v k ) are finite-time optimization terms,
and −d̂i is the disturbance compensation
term. In conN
N
2
trollers (7),
∇
f
(x
)
and
∇
f
(x
v k are global
k k
k k )
k=1
k=1
terms on the local cost functions. So controllers (7) are actually centralized finite-time optimization controllers. To overcome this problem, in the next subsection, some distributed
finite-time estimators are developed to estimate ∇ f k (x k ) and
∇ 2 f k (x k )
v k , k ∈ V. Then these global terms are replaced by
using the estimates generated from the distributed finite-time
estimators and finally the distributed finite-time optimization
controllers are derived.
C. Distributed Finite-Time Optimization Controller Design
The design comprises devising distributed finite-time estimators and distributed finite-time optimization controllers.

i
i
where
= [χk,1
, . . . , χk,m
]T ∈ Rm is the estimate of
2
∇ f k (x k )
v k for agent i, i ∈ V, and l2,k is a positive gain.
Proposition 2: For multiagent system (4), if the graph G is
connected and Assumption 1 holds, and the estimator gains
l1,k and l2,k , k ∈ V are properly chosen, then distributed
estimators (8) and (9) are both finite-time convergent.
Proof: Take distributed estimator (8) for instance. For
vk +
one thing, for i, k ∈ V, denote k = 1 N ⊗ (∇ 2 f k (x k )(
i
T
i ,...,e i ]
=
φ
−
∇
f
(x
),
e
wk1 )), eφki = [eφk,1
k k
φk =
φk,m
k
T
T T
T
1
N
[(eφk ) , . . . , (eφk ) ] , and gk = [gk,1 , . . . , gk,N m ] = (L̄k ⊗
v k + wk1 )∞ . Set an energy
Im )eφk . Then k ∞ = ∇ 2 f k (x k )(
T
function Veφk = eφk (L̄k ⊗ Im )eφk /2 = eφTk gk /2. Since the matrix
L̄k = L + Bk is positive definite, there hold λmax (L̄k ) ≥
λmin (L̄k ) > 0 and Veφk ≤ λmax (L̄k )eφk 22 /2. Then V̇eφk
along (8) satisfies


V̇eφk = gkT −l1,k sgn(gk ) − k = −l1,k gk 1 − gkT k . (10)

For another thing, since |gkT k | ≤ gk 1 k ∞ , gk =
(L̄k ⊗ Im )eφk and gk 1 ≥ gk 2 ≥ λmin (L̄k )eφk 2 , it
follows from (10) that V̇eφk ≤ −(l1,k − k ∞ )λmin (L̄k )eφk 2 .
Moreover, ∇ 2 f k (x k ) = Ak and k (t)∞ ≤ Ak vk (t)∞ +
Ak wk1 (t)∞ . Set l1,k (t) ≥ Ak vk (t)∞ + τ1,k , ∀t ≥ 0, where
τ1,k is a positive constant. Then it is verified that
 1
 

1
τ1,k − Ak wk1 (t)∞ 2 2 λmin L̄k
Ve2φk
(11)
V̇eφk ≤ −
1


2
L̄k
λmax
which is followed by V̇eφk ≤ (supt Ak wk1 (t)2∞ λ2min (L̄k ))/
(λmax (L̄k ))Veφk + (1/2). Since Ak and the estimation error
wk1 are both bounded, Veφk and eφk are bounded in any
finite time interval covering [0, Tob ] with Tob being the finite
convergence time of FTOs (5). When t ≥ Tob , wk1 (t) =
0m . By (11), Veφk converges to zero in finite time and
eφki = φki − ∇ f k (x k ) → 0m in finite time, i, k ∈ V.
For distributed estimator (9), denote the estimation errors as

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WANG et al.: DISTRIBUTED FINITE-TIME OPTIMIZATION OF SECOND-ORDER MULTIAGENT SYSTEMS

T
i ,...,e i ]
eχki = [eχk,1
= χki − ∇ 2 f k (x k )
v k , eχ k =
χk,m
˙k (t)∞ +τ2,k , ∀t ≥ 0,
[(eχk1 )T , . . . , (eχkN )T ]T . Set l2,k (t) > Ak v
where τ2,k is a positive constant. Similarly, it can be proved
that eχki = χki − ∇ 2 f k (x k )
v k → 0m in finite time, i, k ∈ V. This
completes the proof.
Remark 5: From Proposition 2, there is the finite time Tde
(related to the initial conditions) such that φki → ∇ f k (x k ) and
v k , i, k ∈ V within Tde . For node k (k ∈ V) of
χki → ∇ 2 f k (x k )
the undirected communication topology graph G, by adding a
virtual edge from it to itself with a weight bkk , node k becomes
a virtual leader node. At the same time, the communication
topology graph becomes a directed graph Ḡ k with such a
virtual leader node. Ḡ k has at least one directed spanning tree
and an invertible Laplacian matrix L̄k = Lk + Bk , where Bk
is a “virtual leader adjacency” matrix actually. Based on Ḡ k ,
distributed finite-time estimators (8) and (9) are developed to
vk .
estimate ∇ f k (x k ) and ∇ 2 f k (x k )
2) Distributed Finite-Time Optimization Controller Design:
By using the estimates φki and χki , i, k ∈ V of ∇ f k (x k ) and
∇ 2 f k (x k )
v k generated respectively from distributed
 N estimators 
(8) and (9) to replace the global terms
k=1 ∇ f k (x k )
N
and k=1
∇ 2 f k (x k )
v k used in centralized optimization controllers (7), the following nonsmooth distributed optimization
controllers are derived (i ∈ V):




ai j sigα1 x i − x j − k2
ai j sigα2 vi − vj
u i = −k1
j ∈Ni



N

−γ1 sig

β1

φki
k=1

j ∈Ni



N

− γ2 sig

β2

χki

− di

(12)

k=1

where the parameters are the same as those in controllers (7).
The main result of this article is now presented as follows.
Theorem 2: For multiagent system (4), if the graph G
is connected and Assumptions 1 and 2 hold, then under
distributed optimization controllers (12), the distributed finitetime optimization goal is achieved, i.e., yi = x i → y ∗
(i.e., the minimizer of the global cost function f (y)) and
ẏi = v i → 0m , i ∈ V in finite time.
Proof: See Appendix B.
Remark 6: The work principle of the proposed distributed
finite-time optimization scheme is as follows: FTOs (5) first
generate accurate velocity and disturbance estimates vi and
di , i ∈ V for the agents in finite time, and then distributed
estimators (8) and (9) produce accurate estimates of ∇ fk (x k )
and ∇ 2 f k (x k )
v k , k ∈ V in finite time. With these estimates,
under distributed optimization controllers (12), all the agents’
outputs yi , i ∈ V reach finite-time consensus and converge
jointly to the minimizer y ∗ of the global cost function f (y)
in finite time. Accordingly, the dynamics of FTOs (5) and
distributed estimators (8) and (9) should be regulated faster
than the controllers dynamics. For FTOs (5), the dynamics
can be regulated by adjusting their gains and parameters
(see Remark 3). For distributed estimators (8) and (9), the
dynamics can be regulated by adjusting their gains l1,k and
l2,k , k ∈ V. For distributed optimization controllers (12),
by making gains k1 , k2 , γ1 , γ2 larger, the closed-loop system
convergence rate is accelerated. The closed-loop system
convergence time has a nonlinear relationship with the power

6047

parameters α1 , α2 , β1 , β2 ∈ (0, 1). If α1 → 1 or β1 → 1, then
the smoothness of controllers (12) (and also the closed-loop
system) becomes stronger, but the convergence time of the
closed-loop system becomes longer or even tends to +∞.
If α1 → 0 or β1 → 0, then controllers (12) tend to be
discontinuous controllers which have undesired chattering.
So there is a compromise between the controller smoothness
and dynamic performances (e.g., the convergence time) of
the closed-loop system. In practice, α1 , α2 , β1 , β2 are usually
set between 0.5 and 1 such that the closed-loop system is
relatively “smooth”. In this article, the connected undirected
communication graphs are considered. A parallel extension
would be to connected balanced directed communication
graphs, and the extensions to more kinds of directed
communication graphs deserve further research. Moreover,
for calculation of the finite optimization settling time range,
a Lyapunov function construction and analysis method can
be used. For example, for system (A.14), the Lyapunov
function can be taken as Vz = l z [γ1 (sigβ1 (z 1 ))T z 1 +
(1 + β1 )/(2)z 2T (∇ 2 f (x))−1 z 2 ](3+β1 )/(2(1+β1 ))
+
z 1T (∇ 2 f (x))−1 z 2 , where l z is a sufficiently large positive
constant related to γ1 , γ2 , β1 . Then a pair of K z > 0
β
and 0 < βz < 1 can be found such that V̇z ≤ −K z Vz z .
By Lemma 3, the finite settling time Tz of system (A.14)
1−β
satisfies Tz ≤ (Vz z (0))/(K z (1 − βz )). The similar
Lyapunov function construction and analysis methods were
also introduced in [38] and [39].
Remark 7: If there are no disturbances, then to achieve
distributed finite-time optimization for multiagent system (4),
some FTOs are still needed to estimate the unknown velocities
v i , i ∈ V. By Lemma 5, the observers can be designed as
 1 1  0

0
0
0
1
z̄˙ i,l
= ψ̄i,l
, ψ̄i,l
= −λ̄0i,l L̄ i,l 2 sig 2 z̄ i,l
− x i,1 + z̄ i,l


1
1
1
1
0
z̄˙ i,l
= ψ̄i,l
+ u i,l , ψ̄i,l
= −λ̄1i,l L̄ i,l sgn z̄ i,l
− ψ̄i,l
l = 1, . . . , m, i ∈ V

(13)

0
1
where z̄ i,l
= x

i,l and z̄ i,l = v
i,l are, respectively, the estimates
0
of x i,l and v i,l , λ̄i,l and λ̄1i,l > 0 are the observer gains,
and L̄ i,l are arbitrary positive constants. Still denote 
xi =
T
T
[x
,
.
.
.
,
x
]
and
v

=
[
v

,
.
.
.
,
v
]
,
i
∈
V.
Denote
the
i,1
i,m
i
i,1
i,m
0
1
= x i,l − x
,
w̄
=
v
−
v

,
i
∈
V,
observation errors w̄i,l
i,l
i,l
i,l
i,l
0
0 T
1
1 T
and w̄i0 = [w̄i,1
, . . . , w̄i,m
] , w̄i1 = [w̄i,1
, . . . , w̄i,m
] . By (4)
and (13), the observation error systems are given by
 1 1  0 
0
1
+ w̄i,l
w̄˙ i,l
= −λ̄0i,l L̄ i,l 2 sig 2 w̄i,l


1
1
0
w̄˙ i,l
+ − L̄ i,l , L̄ i,l
∈ −λ̄1i,l L̄ i,l sgn w̄i,l
− w̄˙ i,l

l = 1, . . . , m, i ∈ V.

(14)

By Lemma 5 and observation error system (6), if proper
positive observer gains λ̄0i,l and λ̄1i,l , l = 1, . . . , m, i ∈ V are
chosen, then observers (13) are finite-time convergent. The
corresponding reduced distributed optimization controllers are




ai j sigα1 x i − x j − k2
ai j sigα2 vi − vj
u i = −k1
j ∈Ni
β1

φki

−γ1 sig

k=1

j ∈Ni



N



N

− γ2 sig

β2

χki

, i ∈V

k=1

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(15)

6048

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where φki and χki , i, k ∈ V are estimates of ∇ f k (x k ) and
∇ 2 f k (x k )
v k generated respectively from distributed estimators (8) and (9), and other parameters are the same as those in
controllers (12). In the absence of disturbances, by a proof similar to that of Theorem 2, distributed finite-time optimization
is achieved under reduced controllers (15). However, in the
presence of disturbances, neither finite-time consensus nor
distributed finite-time optimization can be realized by reduced
controllers (15) due to the absence of disturbance compensations. This reflects the main superiority of the proposed
composite distributed finite-time optimization control scheme
in Theorem 2, i.e., handling the effects of both unknown states
and disturbances in finite time simultaneously.
Remark 8: Consider the following second-order multiagent
system with unknown velocities and mismatched disturbances:
ẋ i = v i + dix , v̇ i = u i + div , i ∈ V

(16)

x
x T
where dix = [di,1
, . . . , di,m
]
∈ Rm and div =
v
v T
m
[di,1 , . . . , di,m ] ∈ R are the mismatched disturbance and
matched disturbance of agent i , respectively, and the other
variables are defined like those of multiagent system (4). Since
both v i and dix are unknown and in the same bridge, they
cannot be estimated separately. Instead, a feasible way is to
estimate them together. By denoting v̄ i = v i + dix , it follows
from (16) that:

ẋ i = v̄ i , v̄˙i = u i + d̄i , i ∈ V

TABLE I
A GENTS ’ I NITIAL P OSITIONS AND L OCAL C OST F UNCTIONS

Fig. 1.

Communication topology (V = {1, 2, 3, 4, 5, 6}).

be adopted, e.g., the embedded control technique (used in
[1] and [37]) that decomposes the distributed optimization
controller design into a virtual optimal signal generator design
for the agents and the agents’ tracking controller design. For
different tasks, different control methods may be used.

(17)

where d̄i = [d̄i,1 , . . . , d̄i,m ]T = ḋix +div . Multiagent system (17)
has the same form as multiagent system (4). Similarly, if d̄i =
ḋix + div (i ∈ V) satisfy Assumption 2, then by using the proposed control method, distributed finite-time optimization can
be achieved for multiagent system (17) [also (16)]. Actually,
Assumption 2 is a sufficient condition that may be unnecessary. For the cases that some di (also the derivative ḋix of
the mismatched disturbance dix ) do not satisfy Assumption 2,
the distributed finite-time optimization method for multiagent
system (16) needs further research.
Remark 9: In this article, distributed finite-time optimization has been studied for second-order multiagent system (4)
with unknown velocities and disturbances. The proposed control method builds a bridge from finite-time consensus design
to distributed finite-time optimization design. By adding some
appropriate optimization terms into the finite-time consensus
controllers, the derived controllers can achieve distributed
finite-time optimization. Since neither the design of finite-time
consensus controllers nor the design of optimization terms
is specific, the proposed control method has good versatility
in solving the distributed finite-time optimization problems.
Actually, the proposed control method can also be used for
dealing with distributed finite-time optimization of higherorder multiagent systems with unknown states and disturbances. In such cases, some higher-order FTOs are needed
to estimate the unknown states and disturbances, additional
distributed finite-time estimators are needed to estimate the
local functions’ higher-order time derivatives, and new finitetime optimization controllers for higher-order systems also
need to be designed, all of which will bring new challenges. Alternatively, some other control methods may also

IV. N UMERICAL S IMULATIONS
In this section, a simulation example is presented. The
effectiveness and advantages of the proposed distributed
finite-time optimization control scheme in Theorem 2 are
illustrated by making comparisons between the closed-loop
system performances under the proposed distributed composite controllers (12) and those under the reduced distributed
controllers (15). The communication topology of the agents is
shown in Fig. 1.
The agents are assumed to be initially static, and their
initial positions and local cost functions are given in Table I.
Then the global cost function is f (y) = 9.5(y 1 − 1.3789)2 +
9(y 2 − 0.4056)2 + 15.0305, which is -strongly convex with
= 18. Accordingly, the unique minimizer of f (y) is y ∗ =
[(y 1 )∗ , (y 2 )∗ ]T = [1.3789, 0.4056]T with f (y ∗ ) = 15.0305.
When t ≤ 8 s, there are no disturbances and the disturbances
are suddenly imposed on the agents when t = 8 s with
the forms listed in Table II. To make a fair comparison,
the amplitudes of both the proposed composite distributed
controllers (12) and the reduced distributed controllers (15) are
limited within ±20. Considering this constraint and according
to conditions given in Theorem 2 and Remark 7, considerable
time and efforts have been spent on regulating the parameters
of both kinds of controllers. Parameter settings of FTOs (5) are
given in Table II and parameters of the reduced observers (13)
are set the same as those used in the first two orders of
FTOs (5). For both FTOs (5) and observers (13), their initial
ρ
values are set as zero, i.e., z i,l (0) = 0, ρ = 0, . . . , Mi,l + 1,
0
1
z̄ i,l (0) = 0, z̄ i,l (0) = 0, l = 1, . . . , m, i ∈ V. The diagonal
matrices Bk of distributed finite-time estimators (8) and (9)
are chosen as b11 = 1.5, b22 = 1.2, b33 = 1.4, b44 = 1.8,

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WANG et al.: DISTRIBUTED FINITE-TIME OPTIMIZATION OF SECOND-ORDER MULTIAGENT SYSTEMS

6049

TABLE II
D ISTURBANCES AND FTO S ’ PARAMETERS

Fig. 2. Response curves of the agents’ positions (i.e., outputs) and velocities. (a), (c), and (e) Under composite controllers (12). (b), (d), and (f) Under
reduced controllers (15).

b55 = 1.6, b66 = 1.3. For both kinds of controllers, the control
parameters are set as k1 = 4.8, k2 = 5, α1 = 0.72, γ1 = 14.6,
γ2 = 15, β1 = 0.7.

The simulation results are displayed in Figs. 2–4. From
Fig. 3, the amplitudes of both kinds of controllers (12) and (15)
are within ±20. As shown in Fig. 4, the agents’ unknown

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6050

Fig. 3.

IEEE TRANSACTIONS ON NEURAL NETWORKS AND LEARNING SYSTEMS, VOL. 34, NO. 9, SEPTEMBER 2023

Time histories of the control inputs. (a) Composite controllers (12). (b) Reduced controllers (15).

Fig. 4. Response curves for the agents’ velocity and disturbance estimation errors of FTOs (5). (a) Agent 1. (b) Agent 2. (c) Agent 3. (d) Agent 4. (e) Agent 5.
(f) Agent 6.

velocities and disturbances are all accurately estimated by
the designed FTOs (5) in finite time. From Fig. 2, in the
absence of disturbances, the agents’ outputs under both kinds
of controllers track the minimizer y ∗ = [(y 1 )∗ , (y 2 )∗ ]T =
[1.3789, 0.4056]T of f (y) in finite time. However, only composite controllers (12) overcome the disturbances’ effects and
the agents’ outputs return to the minimizer fast after a short
fluctuation when the disturbances are suddenly imposed on
the system. In contrast, affected by the disturbances, the

agents’ outputs under reduced controllers (15) escape from
the minimizer and do not return any more. These simulation
results illustrate the effectiveness and advantages of the proposed distributed composite controllers (12) and validate the
comparison analysis given in Remark 7.
V. C ONCLUSION
This article has provided a solution to the distributed
finite-time optimization problem of second-order multiagent

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WANG et al.: DISTRIBUTED FINITE-TIME OPTIMIZATION OF SECOND-ORDER MULTIAGENT SYSTEMS

systems with unknown velocities, disturbances, and quadratic
local cost functions. Based on the combination of FTOs,
the homogeneous systems theory and the distributed finitetime estimator techniques, output feedback-based composite
distributed control scheme has been proposed. Under the
proposed control scheme, all the agents’ outputs converge
to the minimizer of the global cost function in finite time.
Simulations have validated the proposed control scheme.

6051

errors wi1 , i ∈ V are bounded due to finite-time convergence
of FTOs (5). Notice that the edge weights ai j , i, j ∈ V
are bounded. Then by Lemma 2, there exist finite positive
constants δ2 and δ3 such that



k2 ai j v iT sigα2 v i − v j − wi1 − w1j

N

−
i=1 j ∈Ni

N

v i 22 + δ3 .

≤ δ2
A PPENDIX A
P ROOF OF T HEOREM 1
Proof of Theorem 1: Denote ηi j = x i − x j , ∀i = j ∈ V.
By (4), (5), (7) and ηi j , the resultant closed-loop system is
(∀i = j ∈ V)
η̇i j = v i − v j
v̇ i = −k1

 
ai j sigα1 ηi j −k2

j ∈Ni




ai j sigα2 v i −v j − wi1 −w1j

j ∈Ni



N

− γ1 sigβ1

∇ f k (x k )
k=1
N

− γ2 sigβ2

By Assumption 1, ∇ f k (x k ) = Ak x k + Bk , k ∈ V. With
0 < β1 < 1, by Lemmas 1 and 2, and
there
 (A.4),
N
N
β1
(1−β1 )/(2) 
holds sig ( k=1 ∇ f k (x k ))2 ≤ m
k=1 (A k x k +
β
N
β1
β
β
Bk )2 1 ≤ m (1−β1 )/(2) k=1
(λmax
(Ak )x k 2 1 + Bk 2 1 ) ≤

β
N
1
m (1−β1 )/(2) β1 k=1 λmax
(Ak )x k 2
+
m (1−β1 )/(2) (1
−
 N β1
N
β1
(1−β1 )/(2)
β1 ) k=1 λmax (Ak ) + m
k=1 Bk 2 . Since λmax (A k )
and Bk 2 are bounded, by Lemma 2, there are finite δ4 > 0,
δ5 > 0 and δ6 > 0 satisfying

N
N
γ1 v iT sigβ1

−





∇ 2 f k (x k ) v k − wk1 + wi2 .

∇ fk (x k )

i=1

k=1
N

(A.1)

The proof has three parts, i.e., finite-time boundedness of the
states ηi j , x i , v i , ∀i = j ∈ V, finite-time consensus of x i , v i
(i.e., ηi j , η̇i j → 0m , ∀i = j ∈ V in finite time), and finite-time
convergence of yi = x i to y ∗ and ẏi = v i to 0m , i ∈ V.
Part 1 (Finite-Time State
Set an energy
 N Boundedness):
N
2
=
(1/2)
x

+
(k
/2(1
+
α
))
function
V
b
i
1
1
2
i=1
i=1
N

α1
T
2
j ∈Ni ai j (sig (ηi j )) ηi j + (1/2)
i=1 v i 2 , which is positive definite. From (4), (7) and (A.1), we have
N




k2 ai j v iT sigα2 v i − v j − wi1 − w1j

N

x iT v i −
i=1 j ∈Ni

i=1
N

N

+
i=1
N

i=1
N

i=1

N

i=1

N

v iT wi2 ≤
i=1

k=1
N

v i 22 + δ8 .

(A.7)

i=1

(A.2)

k=1

x iT v i +

(A.6)

i=1

With ∇ 2 f k (x k ) = Ak , k ∈ V and 0 < β2 < 1,
by Lemmas
1 and 2, and (A.4), there holds
N
∇ 2 f k (x k )(v k − wk1 ))2
≤
m (1−β2 )/(2) 
sigβ2 ( k=1

N
β2
β2
N
1 β2
(1−β2 )/(2)
k=1 A k (v k − wk )2 ≤m
k=1 λmax (A k )(v k 2 +
β2
N
1 β2
(1−β2 )/(2)
(1−β2 )/(2)
wk 2 ) ≤ m
β2 k=1 λmax (Ak )v k 2 +m
(1−
 N β2
 N β2
β
β2 ) k=1
λmax (Ak ) + m (1−β2 )/(2) k=1
λmax (Ak )wk1 2 2 . Since
λmax (Ak ) and wk1 are bounded, by Lemma 2, there are finite
δ7 > and δ8 > 0 such that

N
N


T
β2
2
1
−
γ2 v i sig
∇ fk (x k ) v k − wk





∇ 2 f k (x k ) v k − wk1 .

v i 22 + δ6 .

≤ δ7

By Proposition 1, observation errors wi2 , i ∈ V are bounded
due to finite-time convergence of FTOs (5) By Lemma 2,
N

i=1

i=1

∇ f k (x k )
k=1

γ2 v iT sigβ2

−



N

γ1 v iT sigβ1

v iT wi2 −

N

x i 22 + δ5

≤ δ4

k=1

V̇b =

(A.5)

i=1

N

1
x i 22 +
v i 22 + δ1 (A.3)
2 i=1
i=1

where δ1 is a finite positive constant. By Lemma 1, ∀z =
[z 1 , . . . , z m ]T ∈Rm and 0 < α < 1, there holds that
2/α
m
sigα (z)2 = ( l=1
z l2α )1/α ≤ m 1/α−1 z22 . Then
 α 
α
sig (z) ≤ m 1−α
2 z .
(A.4)
2
2
(A.4), it follows
Since 0 < α2 < 1, by Lemmas 1 and
 2, and
1
1 
(1−α2 )/(2) 
−
v
−
(w
−
w
))
≤
m
that sigα2 (v
vi − v j −
j
i
j
2
αi
(wi1 − w1j )2 2 ≤ m (1−α2 )/(2) (v i α2 2 + v j α2 2 + wi1 α2 2 +
w1j α2 2 ) ≤ m (1−α2 /2) α2 (v i 2 + v j 2 ) + 2m (1−α)/(2) (1 − α2 ) +
m (1−α2 )/(2) (wi1 α2 2 + w1j α2 2 ). By Proposition 1, observation

Substituting
(A.3), (A.5), (A.6), (A.7) into (A.2) yields V̇b ≤
N
(x i 22 +v i 22 )+δ10 ≤ 2δ9 Vb +δ10 , where δ9 , δ10 > 0.
δ9 i=1
So Vb and states x i , ηi j , v i , ∀i = j ∈ V are bounded in any
finite time interval covering [0, Tob ], where Tob is the finite
convergence time of FTOs (5).
Part 2 (Finite-Time Consensus): For t ≥ Tob , by Proposition 1, observation errors wi1 (t) = 0m and wi2 (t) = 0m hold,
∀i ∈ V. Then system (A.1) becomes (∀i = j ∈ V)
η̇i j = v i − v j
v̇ i = −k1

 
ai j sigα1 ηi j − k2

j ∈Ni



N

−γ1 sigβ1

j ∈Ni

∇ f k (x k )
k=1
N

−γ2 sig



ai j sigα2 v i − v j

β2


∇ f k (x k )v k .
2

k=1

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(A.8)

6052

IEEE TRANSACTIONS ON NEURAL NETWORKS AND LEARNING SYSTEMS, VOL. 34, NO. 9, SEPTEMBER 2023

Consider the following reduced system (A.8):
η̇i j = v i − v j

 
ai j sigα1 ηi j − k2

v̇ i = −k1
j ∈Ni



ai j sigα2 v i −v j . (A.9)
j ∈Ni

If system (A.9) is globally asymptotically stable, then from
the equilibrium conditions η̇i, j ≡ 0m , v̇ i ≡ 0m , ∀i, j ∈ V, it is
verified that v i (+∞) = v j (+∞) = v̄ with v̄ being the agents’
common steady-state constant velocity. Denote v̄ i = v i − v̄,
i ∈ V. For system (A.9), set 
a positive-definite
Lyapunov funcN 
α1
a
(sig
(ηi j ))T ηi j +
tion Vc = (k1 /2(1 + α1 )) i=1
i
j
j ∈Ni
N
2
(1/2) i=1 v̄ i 2 . Then V̇c satisfies


T 

ai j sigα2 v̄ i − v̄ j
v̄ i − v̄ j ≤ 0 (A.10)

N

V̇c = −

k2
2 i=1

j ∈Ni

which means that V̇c is negative semi-definite. On one hand,
define the invariant set ϒ = {(ηi j , v̄ i ) | V̇c ≡ 0}. V̇c ≡ 0
means that v̄ i − v̄ j = v i − v j ≡ 0m , ∀i = j ∈
and then
V,
N
v̇ i −v̇ j ≡ 0m . Besides, it follows from (A.9)
that
i=1 v̇ i ≡ 0m .

α1
Thus, v̇ i ≡ 0m , i ∈ V, and
a
sig
(ηi j ) ≡
i
j
 N  N j ∈Ni T α1
0m , i
∈
V and
a
η
sig
(η
=
ij)
i=1
j ∈Ni i j i j
N T N
α1
2 i=1 x i
≡ 0. Since the graph
j ∈Ni ai j sig (ηi j )
G is connected, there holds ηi j ≡ 0m , ∀i = j ∈ V.
By LaSalle’s invariant principle [40], system (A.9) is globally
asymptotically stable, i.e., ηi j → 0m , v̄ i = v i − v̄ → 0m and
v i − v j → 0m , ∀i = j ∈ V globally and asymptotically.
On the other hand, system (A.9) has a negative homogeneous
degree −(1 − α1 )/(1 + α1 ) with respect to the dilation
(where
the
[(2)/(1 + α1 ), . . . , (2)/(1 + α1 ), 1, . . . , 1]
numbers of (2)/(1 + α1 ) and 1 are N(N − 1)m and
Nm, respectively). By Lemma 4, system (A.9) is finite-time
stable, i.e., ηi j = x i − x j → 0m and v i − v j → 0m ,
∀i = j ∈ V in finite time. Denote the consensus error vectors
Ex = (L ⊗ Im )[x 1T , . . . , x mT ]T and Ev = (L ⊗ Im )[v 1T , . . . , v mT ]T .
Then the consensus error system of system (A.8) is given by

both consensus error system (A.11) and system (A.8) are
finite-time stable. This means that finite-time consensus is
achieved, namely, ηi j = x i −x j → 0m and η̇i j = v i −v j → 0m ,
∀i = j ∈ V in finite time Tc ≥ Tob .
Part 3 (Finite-Time Convergence): For t ≥ Tc , x i (t) = x j (t)
and v i (t) = v j (t), ∀i = j ∈ V. Define x = x i and v = v i ,
∀i ∈ V. It follows from (A.8) that:
ẋ = v


v̇ = −γ1 sigβ1 (∇ f (x)) − γ2 sigβ2 ∇ 2 f (x)v . (A.13)
Denote
z 1 = ∇ f (x) and z 2 = ∇ 2 f (x)v. Note that ∇ 2 f (x) =
N
i=1 A i . Then system (A.13) becomes
z1 = z2
ż 2 = −γ1 ∇ 2 f (x)sigβ1 (z 1 ) − γ2 ∇ 2 f (x)sigβ2 (z 2 ). (A.14)
For one thing, for system (A.14), take the Lyapunov function
Vd = (γ1 )/(1 + β1 )(sigβ1 (z 1 ))T z 1 + (1/2)z 2T (∇ 2 f (x))−1 z 2 ,
which is positive definite in z 1 and z 2 . V̇d along system (A.14)
satisfies
T

(A.15)
V̇d = −γ2 sigβ2 (z 2 ) z 2 ≤ 0
which is negative semi-definite. Denote the invariant set  =
{[z 1T , z 2T ]T | V̇d ≡ 0}. Since V̇d ≡ 0 ⇔ z 2 ≡ 0m , there
holds ż 2 ≡ 0m . From (A.14) it follows that z 1 ≡ 0m .
By LaSalle’s invariant principe [40], system (A.14) is globally asymptotically stable. For another thing, with the dilation [(2)/(1 + β1 ), . . . , (2)/(1 + β1 ), 1, . . . , 1] (the numbers
of (2)/(1 + β1 ) and 1 are both m), system (A.14) has a negative homogeneous degree −(1 − β1 )/(1 + β1 ). By Lemma 4,
system (A.14) is finite-time stable, i.e., z 1 = ∇ f (x) → 0m
and z 2 = ∇ 2 f (x)v → 0m in finite time. So y = x → y ∗ and
ẏ = v → 0m in finite time.
Based on Parts 1–3, for multiagent system (4) under controllers (7), there hold yi = x i → y ∗ and ẏi = v i → 0m , i ∈ V
in finite time. This completes the proof.
A PPENDIX B
P ROOF OF T HEOREM 2

Ėx = Ev
Ėv = (L ⊗ Im )Uo



N
β1

−γ1 (L ⊗ Im )(1 N ⊗ Im )sig

∇ f k (x k )
k=1
N

−γ1 (L ⊗ Im )(1 N ⊗ Im )sigβ2


∇ 2 f k (x k )v k

Proof of Theorem 2: Still denote ηi j = x i − x j , ∀i =
j ∈ V. By (4), (5), (8), (9) and (12), the closed-loop system
is described as (∀i = j ∈ V)
η̇i j = v i − v j
v̇ i = −k1

k=1

(A.11)

 
ai j sigα1 ηi j −k2

j ∈Ni



N
T
T
where
=
[u o,1
,
. . . , u o,N
]T and u o,i
=
 Uo
α1
−k1 j ∈Ni ai j sig (ηi j ) − k2 j ∈Ni ai j sigα2 (v i − v j ), i ∈ V.
Since the graph G is connected, by Lemma 6, there holds
(L⊗ Im )(1 N ⊗ Im ) = L1 N ⊗ Im = 0 N m×m . Then system (A.11)
is equivalent to the system

Ėx = Ev , Ėv = (L ⊗ Im )Uo

(A.12)

which is just the consensus error system associated with the
reduced system (A.9). Since system (A.9) is finite-time stable,
the consensus error system (A.12) is also finite-time stable.
Since the states x i , ηi j , v i , ∀i = j ∈ V are finite-time bounded,

− γ1 sigβ1




ai j sigα2 v i −v j − wi1 −w1j

j ∈Ni

φki
k=1
N

− γ2 sigβ2


χki

+ wi2 .

(B.1)

k=1

The proof contains two parts, i.e., finite-time boundedness of
the states ηi j , x i , v i , ∀i = j ∈ V, and finite-time convergence
of yi = x i to y ∗ and ẏi = v i to 0m , i ∈ V.
Part 1 (Finite-Time State Boundedness): Still
 N take 2the
=
(1/2)
2 +
positive-definiteenergy
function
V
b
i=1 x i
N 
N
α1
T
a
(sig
(η
))
η
+
(1/2)
(k1 /2(1 + α1 )) i=1
ij
ij
i=1
j ∈Ni i j

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WANG et al.: DISTRIBUTED FINITE-TIME OPTIMIZATION OF SECOND-ORDER MULTIAGENT SYSTEMS

v i 22 in x i , ηi j , v i , ∀i = j ∈ V. Then V̇b along system (B.1)
satisfies
N

V̇b =




k2 ai j v iT sigα2 v i − v j − wi1 − w1j

N

x iT v i −
i=1

i=1 j ∈Ni

N

N

γ1 v iT sigβ1

v iT wi2 −

+
i=1
N

i=1



N

γ2 v iT sigβ2

−



N

φki
k=1

χki .

i=1

(B.2)

k=1

Note that φki = eφki + ∇ f k (x k ), ∇ fk (x k ) = Ak x k + Bk , i, k ∈ V.
By Proposition 2, estimation errors eφki , i, k ∈ V are bounded
in t ∈ [0, +∞). Similar to (A.6), it is verified that

N
N
γ1 v iT sigβ1

−
i=1

φki
k=1
N

N

x i 22 + δ12

≤ δ11
i=1

v i 22 + δ13

(B.3)

i=1

where δ11 , δ12 , δ13 are proper finite positive constants. Notice
v k and ∇ 2 f k (x k ) = Ak , i, k ∈ V.
that χki = eχki + ∇ 2 f k (x k )
By Proposition 2, estimation errors eχki , i, k ∈ V are bounded
in t ∈ [0, +∞). Similar to (A.7), it is obtained that

N
N
N
γ2 v iT sigβ2

−
i=1

χki

v i 22 + δ15

≤ δ14

k=1

(B.4)

i=1

where δ14 and δ15 are finite positive constants. Substituting 
(A.3), (A.5), (B.3) and (B.4) into (B.2) yields V̇b ≤
N
(x i 22 + v i 22 ) + δ17 ≤ 2δ16 Vb + δ17 , where δ16
δ16 i=1
and δ17 are finite positive constants. Thus, Vb and states
x i , ηi j , v i , ∀i = j ∈ V are bounded in any finite time
interval covering [0, max{Tob , Tde }], where Tob is the finite
convergence time of FTOs (5) and Tde is the finite convergence
time of distributed estimators (8) and (9).
Part 2 (Finite-Time Convergence): For t ≥ max{Tob , Tde },
observation errors wi1 (t) = 0m and wi2 (t) = 0m , ∀i ∈ V by
Proposition 1, and estimation errors eφki (t) = 0m and eχki (t) =
0m , ∀i, k ∈ V by Proposition 2. So, system (B.1) becomes
system (A.8). Via Parts 2 and 3 of the proof for Theorem 1,
there hold yi = x i → y ∗ and ẏi = v i → 0m , i ∈ V in finite
time. This ends the proof.
ACKNOWLEDGMENT
The authors would like to thank the Associate Editor and the
Reviewers for the insightful comments and helpful suggestions
which have helped to improve this article considerably.
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Xiangyu Wang (Senior Member, IEEE) received the
B.S. degree in mathematics and the Ph.D. degree in
control theory and control engineering from Southeast University, Nanjing, China, in 2009 and 2014,
respectively.
Since 2014, he has been with the School of
Automation, Southeast University, where he is currently an Associate Professor. His research interests
include nonlinear control, anti-disturbance control,
robot control, control and optimization of power
systems, and distributed control and optimization of
multiagent systems.
Dr. Wang serves as an Associate Editor for IET Control Theory and
Applications.

Wei Xing Zheng (Fellow, IEEE) received the B.Sc.
degree in applied mathematics and the M.Sc. and
Ph.D. degrees in electrical engineering from Southeast University, Nanjing, China, in 1982, 1984, and
1989, respectively.
Over the years Dr. Zheng has held various faculty/research/visiting positions with Southeast University; the Imperial College of Science, Technology and Medicine, London, U.K.; the University of
Western Australia, Perth, Australia; the Curtin University of Technology, Perth, Australia; the Munich
University of Technology, Munich, Germany; the University of Virginia,
Charlottesville, VA, USA; and the University of California at Davis, Davis,
CA, USA. He is currently a University Distinguished Professor with Western
Sydney University, Sydney, NSW, Australia.
Dr. Zheng was named a Highly Cited Researcher by Clarivate Analytics
(formerly Thomson Reuters) in 2015, 2016, 2017, 2018, 2019, 2020 and
2021, consecutively. He is a recipient of the 2017 Vice-Chancellor’s Award for
Excellence in Research (Researcher of the Year) at Western Sydney University.
Previously, he served as an Associate Editor for the IEEE T RANSACTIONS
ON C IRCUITS AND S YSTEMS -I: F UNDAMENTAL T HEORY AND A PPLICA TIONS , the IEEE T RANSACTIONS ON AUTOMATIC C ONTROL, the IEEE S IG NAL P ROCESSING L ETTERS , the IEEE T RANSACTIONS ON C IRCUITS AND
S YSTEMS -II: E XPRESS B RIEFS , and the IEEE T RANSACTIONS ON F UZZY
S YSTEMS . He also served as a Guest Editor for the IEEE T RANSACTIONS ON
C IRCUITS AND S YSTEMS -I: R EGULAR PAPERS . He is currently an Associate
Editor for Automatica, the IEEE T RANSACTIONS ON C YBERNETICS , the
IEEE T RANSACTIONS ON N EURAL N ETWORKS AND L EARNING S YSTEMS ,
the IEEE T RANSACTIONS ON C ONTROL OF N ETWORK S YSTEMS , the IEEE
T RANSACTIONS ON C IRCUITS AND S YSTEMS -I: R EGULAR PAPERS , and
other scholarly journals. He served as the Chair for the IEEE Circuits and
Systems Society’s Technical Committee on Neural Systems and Applications
and as the Chair of IEEE Circuits and Systems Society’s Technical Committee
on Blind Signal Processing. He is currently a Distinguished Lecturer of the
IEEE Control Systems Society and the Chair of the IEEE Circuits and Systems
Society’s Technical Committee on Digital Signal Processing.

Guodong Wang received the B.S. degree in automation from the Nanjing University of Aeronautics
and Astronautics, Nanjing, China, in 2017, and the
M.S. degree in control theory and control engineering from Southeast University, Nanjing, in 2019,
where he is currently pursuing Ph.D. degree with
the School of Automation.
His research interests include distributed cooperative control of multiagent systems and antidisturbance control.

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