Distributed fixed-time optimization for multi-agent systems 777
stability theory. By using improved fixed-time sta-
bility theory [27], a high-precision estimation of the
settling-time is obtained.
3. In order to reduce the communication burden, two
event-triggered protocols are designed by using
sign function and saturation function. For the event-
triggered protocol with sign function, although the
optimal value can be achieved theoretically, there
exists a high frequency chattering. To overcome this
disadvantage, the saturation function is employed
and an improved event-triggered protocol is pre-
sented. It is found that the optimal value also can
be reached in a fixed-time interval and the high-
frequency chattering can be eliminated.
The rest of this paper is organized as follows. Some
preliminaries including graph theory, fixed-time stabil-
ity theory, and problem statement are given in Sect. 2.
The main results are presented in Sect. 3. In Sect. 4,
an example is given to illustrate the correctness of the
results. Section 5 gives a conclusion of t his paper.
Notations In this paper, R and R
n
denote real num-
ber set and n-dimensional Euclidean space, respec-
tively. For a vector y (or matrix A), let y
T
(A
T
) rep-
resents its transpose. Let y =[y
1
, y
2
,...,y
n
]
T
be an
n-dimensional column vector, and F(y) is a twice dif-
ferentiable function. Then,
∂ F(y)
∂y
i
denotes the partial
derivative of F(y) with respect to y
i
(i = 1, 2,...,n).
∇F(y) and ∇
2
F(y) represent its gradient and Hes-
sian matrix, respectively. y represents the 2-norm of
y.Letsig(y)
α
=[sig(y
1
)
α
, sig(y
2
)
α
,...,sig(y
n
)
α
]
T
with sig(y
i
)
α
= sign(y
i
)|y
i
|
α
(i = 1, 2,...,n),in
which sign(·) represents the sign function. N denotes
the natural number set. diag(·) represents the diagonal
matrix.
2 Preliminaries
2.1 Graph theory
Let G = (V, E, A) represent a directed graph, in which
V ={v
1
,...,v
N
} denotes the nodes set, E ⊆ V × V
denotes the edges set, and A =[a
ij
]
N ×N
denotes the
weighted adjacency matrix. An edge is described by
an ordered pair of distinct nodes. For an edge e
ij
=
(v
i
,v
j
), we refer to v
i
and v
j
as the tail node and head
node, respectively. The weighted adjacency matrix A =
[a
ij
]
N ×N
is defined by a
ii
= 0, a
ij
> 0ife
ji
∈ E
and a
ij
= 0, otherwise. The neighbors of node v
i
are
denoted by N
i
={v
j
∈ V:(v
j
,v
i
) ∈ E }. The graph G
is called strongly connected if there exists an directed
path between any different nodes. The Laplacian matrix
L =[l
ij
]
N ×N
of G is defined as l
ii
=
N
j=1, j=i
a
ij
and l
ij
=−a
ij
for i = j.
2.2 Fixed-time stability theory
Consider the following differential equation
˙x(t) = f (x(t)), x(0) = x
0
, (1)
where x(t) ∈ R
n
denotes the state variable and
f : R
n
→ R
n
is a nonlinear function. Assume that the
origin is the equilibrium point of system (1).
Definition 1 [26].Theoriginofsystem(1)issaidtobe
globally uniformly finite-time stable if for any solution
x(t, t
0
, x
0
) with x
0
∈ R
n
, there exists a positive number
T (x
0
) such that x(t, t
0
, x
0
) = 0 for all t ≥ t
0
+T (x
0
).
The positive number T (x
0
) is called the settling time.
Moreover, if the positive number T (x
0
) independent of
initial value x
0
, the origin is said to be globally fixed-
time stable.
Lemma 1 [27] Consider the differential equation (1).
If there exists a regular, positive definite and radially
unbounded function V (x): R
n
→ R such that any solu-
tion of (1) satisfies the inequality
˙
V (x(t)) ≤−
aV
p
(x(t)) + bV
q
(x(t))
k
,
x(t) ∈ R
n
\0,
where a, b, p, k > 0,q ≥ 0 and pk > 1,qk < 1,
then the origin of system (1) is fixed-time stable, and
the settling time T (x
0
) is estimated by
T (x
0
) ≤
1
b
k
b
a
1−qk
p−q
1
1 − qk
+
1
pk − 1
.
Lemma 2 [29] Let θ
1
,θ
2
,...θ
n
≥ 0. Then,
n
i=1
θ
a
i
≥
n
i=1
θ
i
a
, if 0 < a ≤ 1
n
i=1
θ
a
i
≥ n
1−a
n
i=1
θ
i
a
, if 1 < a ≤∞.
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