# Pinning-Like Adaptive Consensus for Networked Mobile Agents with Heterogeneous Nonlinear Dynamics

Authors: Chengjie Xu; Yanwei Wang; Hong Zhang; Xiaoqi Zhou
Journal: Mathematical Problems in Engineering
 (2014)

---

## Abstract

This paper investigates the adaptive consensus for networked mobile agents with heterogeneous nonlinear dynamics. Using tools from matrix, graph, and Lyapunov stability theories, sufficient consensus conditions are obtained under adaptive control protocols for both first-order and second-order cases. We design an adaptive strategy on the coupling strengths, which can guarantee that the consensus conditions do not require any global information except a connection assumption. The obtained results are also extended to networked mobile agents with identical nonlinear dynamics via adaptive pinning control. Finally, numerical simulations are presented to illustrate the theoretical findings.


---

## Body

## 1. Introduction

In recent years, distributed cooperative control for multiagent systems has been intensively investigated by researchers from various disciplines. This is due to its broad potential applications in sensor networks, combat intelligence, surveillance, and so forth [1, 2]. Various scenarios about distributed cooperative control are studied, such as leaderless consensus [3–5], leader-following consensus [6–8], containment [9–13], and flocking [14, 15].Consensus is one of the most fundamental problems in distributed cooperative control, which means that the states of the agents reach an agreement on a common physical quantity of interest by implementing an appropriate consensus protocol based on the information from local neighbors. Numerous interesting results about consensus algorithm were presented in the past decade. The earlier studies on consensus problem are mainly about multiagent systems with first-order dynamics [16–18]. And consensus for multiagent systems with second-order dynamics has been investigated recently [19–21], which is more challenging than the first-order case. Meanwhile, consensus for linear multiagent systems is discussed in [22, 23], which is considered more general than first-order and second-order multiagent systems.However, in reality, mobile agents may be governed by more complicated intrinsic dynamics, so second-order consensus problems for multiagent with nonlinear dynamics have been investigated [24–33], in which the authors assume that all the agents have identical nonlinear dynamics. Recently, second-order consensus of multiagent systems with heterogeneous nonlinear dynamics and time-varying delay was investigated by introducing novel decentralized adaptive control in [28]. In [29], a distributed cooperative tracking problem was studied for a group of second-order multiagents networked system with nonidentical nonlinear dynamics and bounded external disturbances. Impulsive consensus for first-order multiagent with heterogeneous nonlinear dynamics was discussed [30], in which the information of the leader is known by every following agent. This assumption is also adopted in [31]. Motivated by [28–33], we investigate the second-order consensus for networked mobile agents with heterogeneous nonlinear dynamics without assuming that all the nonlinear functions have a common equilibrium solution. A distributed adaptive strategy and a pinning-like control is introduced in the consensus protocol. The protocol is composed of two parts. One is designed to compensate the errors of the nonlinear dynamics and the other one is designed like a pinning controller.The rest of the paper is organized as follows. In Section2, we state the model considered in the paper and give some basic definitions, lemmas, and assumptions. In Section 3, new results on the second-order consensus problem are addressed. And numerical example is given in Section 4. Finally, we conclude the paper in Section 5.


## 2. Preliminaries and Model Description

In this section, some notations and preliminaries are introduced. The following notations are used throughout this paper.I
n denotes the n
×
n identity matrix. For a matrix A (or a vector x), A
T (or x
T) represents the transpose of A (or x).LetG
=
(
V
,
E
,
A
) be a directed graph with a nonempty set of nodes V
=
(
υ
1
,
υ
2
,
…
,
υ
N
+
M
), a set of edges E
⊆
V
×
V, and a weighted adjacent matrix A
=
[
a
i
j
]. An edge is denoted by (
υ
i
,
υ
j
) in a directed graph which means that vertex j can obtain information from vertex i but not necessarily vice versa. a
i
j represents the weight of the edge (
υ
j
,
υ
i
) and a
i
j
>
0
⇔
(
υ
j
,
υ
i
)
∈
E. Node v
j is called the parent node, node v
i is the child node, and v
j is a neighbor of v
i. The neighbor set N
i
=
{
υ
j
∣
(
υ
j
,
υ
i
)
∈
E
}. A graph is undirected if (
υ
j
,
υ
i
)
∈
E implies (
υ
i
,
υ
j
)
∈
E. The elements l
i
j of Laplace matrix L is defined as l
i
j
=
-
a
i
j,  i
≠
j, and l
i
i
=
∑
j
≠
i
,
j
=
1
N
a
i
j.Before moving on, some assumptions and Lemmas are introduced.Assumption 1.
Suppose that the undirected graphG is connected.Lemma 2 (see [33]).
IfL
=
(
l
i
j
)
∈
R
N
×
N is a symmetric irreducible matrix with l
i
i
=
-
∑
j
=
1
,
j
≠
i
N
l
i
j, l
i
j
=
l
j
i
≤
0  (
i
≠
j
), then L is semipositive definite and then for any matrix E
=
diag
⁡
(
e
,
0
,
…
,
0
) with e
>
0, all eigenvalues of the matrix (
L
+
E
) are positive.Lemma 3 (see [34]).
The matrixA of an undirected graph G is irreducible if and only if the undirected graph is connected.Lemma 4 (see [35]).
If a scalar functionV
(
x
,
t
) satisfies the following conditions, then V
˙
(
x
,
t
)
→
0, as t
→
∞. (a)   V
(
x
,
t
) is lower bounded; (b)   V
˙
(
x
,
t
) is negative semidefinite; (c)   V
˙
(
x
,
t
) is uniformly continuous in t.


## 3. Main Results

In this section, consensus conditions for both first-order and second-order multiagent systems with heterogeneous nonlinear dynamics are obtained by designing pinning-like adaptive control protocols. Pinning-like adaptive consensus for first-order multiagent systems with heterogeneous nonlinear dynamics is investigated in the first subsection and the second-order case is investigated in the second subsection. For the first-order case, theith node can be described as
(1)
x
˙
i
=
f
i
(
t
,
x
i
(
t
)
)
+
u
i
,
where i
=
1,2
,
…
,
N, t
∈
R
+,  x
i
=
(
x
i
1
,
x
i
2
,
…
,
x
i
n
)
T
∈
R
n is the state vector of the ith node. f
i
:
R
×
R
n
→
R
n is a continuous vector-value function, which describes the local dynamics of the nodes in the ith node.For the second-order case, theith node can be described as
(2)
x
˙
i
(
t
)
=
v
i
(
t
)
,
1111111111111111
v
˙
i
(
t
)
=
f
i
(
t
,
x
i
(
t
)
,
v
i
(
t
)
)
+
u
i
(
t
)
,
i
=
1
,
…
,
N
,
where t
∈
R
+, x
i
,
v
i
,
u
i
∈
R
n are the position, velocity, and input vector of the ith follower, respectively. f
i
:
R
×
R
n
×
R
n
→
R
n is a continuous vector-value function, which describes the local dynamics of the nodes in the ith node.


### 3.1. Pinning-Like Adaptive Consensus for First-Order Multiagent Systems with Heterogeneous Nonlinear Dynamics

In this subsection, we investigate consensus criteria for first-order multiagent systems with heterogeneous nonlinear dynamics via distributed adaptive pinning control.As mentioned in many literatures, consensus or synchronization for networked systems with nonidentical nodes cannot be realized without control if the nonidentical dynamic functions do not have a common solution. Thereby, a distributed adaptive pinning-like control protocol is proposed, under which leader-following consensus for networked system (1) can be achieved; that is, there exists a desired state s, such that
(3)
lim
⁡
t
→
∞
∥
x
i
(
t
)
-
s
(
t
)
∥
=
0
,
i
=
1,2
,
…
,
N
,
where s satisfies s
˙
=
f
(
t
,
s
(
t
)
), and s can be an equilibrium, a limit cycle, or even a chaotic attractor.The following assumption is necessary for our main results.Assumption 5.
For arbitraryx
,
y
∈
R
n and every i
∈
{
1,2
,
…
,
N
}, there exists a constant ω
>
0 such that
(4)
(
x
-
y
)
T
(
f
i
(
t
,
x
)
-
f
i
(
t
,
y
)
)
≤
ω
(
x
-
y
)
T
(
x
-
y
)
.
We choose the control protocol as
(5)
u
i
(
t
)
=
s
˙
(
t
)
-
f
i
(
t
,
s
(
t
)
)
+
∑
j
∈
N
i
‍
c
i
j
a
i
j
(
x
j
(
t
)
-
x
i
(
t
)
)
+
h
i
c
i
(
t
)
(
s
(
t
)
-
x
i
(
t
)
)
.
In the controller (4), h
i
=
1 for some i, then the ith node is named as pinning-like node and h
i
=
0 for others. c
i
>
0 and c
i
j
>
0 are determined by the following equations:
(6)
c
˙
i
(
t
)
=
h
i
k
i
(
s
(
t
)
-
x
i
(
t
)
)
T
(
s
(
t
)
-
x
i
(
t
)
)
,
111
c
˙
i
j
(
t
)
=
a
i
j
k
i
j
(
x
j
(
t
)
-
x
i
(
t
)
)
T
(
x
j
(
t
)
-
x
i
(
t
)
)
,
i
≠
j
,
i
,
j
=
1,2
,
…
,
N
,
where k
i
j
=
k
j
i
>
0, k
i
>
0 are the weights of adaptive laws for parameters c
i
j
(
t
) and c
i
(
t
). c
i
j
>
0 denotes the coupling strength between node i and node j
,
i
,
j
=
1
,
…
,
N. We define a new matrix, named adaptive weighted Laplace matrix of the network topology, as the following:
(7)
L
~
=
[
l
~
11
l
~
12
⋯
l
~
1
N
l
~
21
l
~
22
⋯
l
~
2
N
⋮
⋮
⋮
⋮
l
~
N
1
l
~
N
2
⋯
l
~
N
N
]
.
The elements l
~
i
j is defined as l
~
i
j
=
c
i
j
l
i
j,  i
≠
j,  i
,
j
=
1,2
,
…
,
N, and c
i
i is defined as l
~
i
i
=
c
i
i
l
i
i
=
∑
j
≠
i
,
j
=
1
N
c
i
j
a
i
j, so L
~ is zero-row-sum. The initial values of the elements of L
~ should be chosen symmetrically for the symmetry of L
~.Remark 6.
From Lemmas2 and 3, if the undirected topology graph G is connected, the adaptive weighted Laplace matrix is semipositive definite.The controlled network for system (1) under protocol (5) with adaptive strategy (6) can be rewritten as
(8)
x
˙
i
=
f
i
(
t
,
x
i
(
t
)
)
+
∑
j
∈
N
i
c
i
j
a
i
j
(
x
j
(
t
)
-
x
i
(
t
)
)
+
s
˙
(
t
)
-
f
i
(
t
,
s
(
t
)
)
+
h
i
c
i
(
t
)
(
s
(
t
)
-
x
i
(
t
)
)
.
Let error state e
i be defined as
(9)
e
i
=
x
i
-
s
,
i
=
1,2
,
…
,
N
.
From (1) and ((8)-(9)), the error systems can be described as
(10)
e
˙
i
=
f
i
(
t
,
x
i
(
t
)
)
-
f
i
(
t
,
s
(
t
)
)
+
∑
j
∈
N
i
c
i
j
a
i
j
(
x
j
(
t
)
-
x
i
(
t
)
)
-
h
i
c
i
(
t
)
e
i
(
t
)
.
Noting that L
~ is zero-row-sum, (10) can be rewritten as
(11)
e
˙
i
=
f
i
(
t
,
x
i
(
t
)
)
-
f
i
(
t
,
s
(
t
)
)
-
∑
j
=
1
N
l
~
i
j
x
j
-
h
i
c
i
(
t
)
e
i
.Theorem 7.
Suppose that Assumptions1 and 5 hold. Then consensus for multiagent system (1) can be achieved under the control protocol (5) with adaptive strategy (6) if there exists at least one pinned-like node in the network.Proof.
LetH
=
diag
⁡
(
h
1
,
h
2
,
…
,
h
N
), according to Lemma 2, L
+
H is positive definite. Consider the following Lyapunov function candidate:
(12)
V
(
e
i
,
c
i
j
,
c
i
)
=
1
2
∑
i
=
1
N
e
i
T
e
i
+
1
2
∑
i
=
1
N
[
∑
j
=
1
,
j
≠
i
N
(
c
i
j
-
m
)
2
2
k
i
j
+
(
c
i
-
m
)
2
k
i
]
,
where m
>
ω
/
λ
1
(
L
+
H
), and λ
1
(
L
+
H
) denotes the minimum eigenvalue of L
+
H.
DifferentiatingV
(
t
) with respect to t along (11), we have
(13)
V
˙
(
t
)
=
∑
i
=
1
N
‍
e
i
T
(
f
i
(
t
,
x
i
(
t
)
)
-
f
i
(
t
,
s
(
t
)
)
)
-
∑
i
=
1
N
‍
e
i
T
[
∑
j
=
1
N
‍
l
~
i
j
x
j
(
t
)
+
h
i
c
i
(
t
)
e
i
(
t
)
]
+
1
2
∑
i
=
1
N
∑
j
=
1
,
j
≠
i
N
(
c
i
j
-
m
)
a
i
j
(
x
j
-
x
i
)
T
(
x
j
-
x
i
)
+
∑
i
=
1
N
‍
(
c
i
-
m
)
h
i
e
i
T
e
i
≤
∑
i
=
1
N
‍
[
ω
e
i
T
e
i
-
m
h
i
e
i
T
e
i
-
∑
j
=
1
N
‍
l
~
i
j
e
i
T
x
j
(
t
)
1111111
+
1
2
∑
j
=
1
,
j
≠
i
N
‍
(
c
i
j
-
m
)
a
i
j
(
x
j
-
x
i
)
T
(
x
j
-
x
i
)
]
.
Noting that L
~ is symmetric and zero-row-sum, we have
(14)
∑
i
=
1
N
∑
j
≠
i
N
(
c
i
j
-
m
)
a
i
j
e
j
T
e
j
=
∑
i
=
1
N
∑
j
≠
i
N
(
c
i
j
-
m
)
a
i
j
e
i
T
e
i
=
∑
i
=
1
N
‍
(
c
i
i
-
m
)
l
i
i
e
i
T
e
i
.
According to (14), we have
(15)
∑
i
=
1
N
∑
j
=
1
,
j
≠
i
N
(
c
i
j
-
m
)
a
i
j
(
x
j
-
x
i
)
T
(
x
j
-
x
i
)
=
∑
i
=
1
N
∑
j
≠
i
N
(
c
i
j
-
m
)
a
i
j
(
e
j
-
e
i
)
T
(
e
j
-
e
i
)
=
∑
i
=
1
N
∑
j
≠
i
N
(
c
i
j
-
m
)
a
i
j
(
e
j
T
e
j
+
e
i
T
e
i
-
2
e
i
T
e
j
)
=
2
∑
i
=
1
N
(
c
i
i
-
m
)
l
i
i
e
i
T
e
i
+
2
∑
i
=
1
N
∑
j
≠
i
N
(
c
i
j
-
m
)
l
i
j
e
i
T
e
j
=
2
∑
i
=
1
N
∑
j
=
1
N
(
c
i
j
-
m
)
l
i
j
e
i
T
e
j
.
So we have
(16)
∑
i
=
1
N
∑
j
=
1
N
(
c
i
j
-
m
)
l
i
j
e
i
T
e
j
=
1
2
∑
i
=
1
N
∑
j
≠
i
N
(
c
i
j
-
m
)
a
i
j
(
e
j
-
e
i
)
T
(
e
j
-
e
i
)
.
From ((14)–(16)), one can easily conclude the following:
(17)
∑
i
=
1
N
∑
j
=
1
N
l
~
i
j
e
i
T
e
j
=
1
2
∑
i
=
1
N
∑
j
≠
i
N
c
i
j
a
i
j
(
e
j
-
e
i
)
T
(
e
j
-
e
i
)
.
From (13), (16), and (17), we have
(18)
V
˙
(
t
)
≤
∑
i
=
1
N
ω
e
i
T
e
i
-
∑
i
=
1
N
∑
j
=
1
N
l
~
i
j
e
i
T
e
j
+
1
2
∑
i
=
1
N
∑
j
=
1
,
j
≠
i
N
(
c
i
j
-
m
)
a
i
j
(
e
j
-
e
i
)
T
(
e
j
-
e
i
)
-
m
∑
i
=
1
N
h
i
e
i
T
e
i
=
∑
i
=
1
N
ω
e
i
T
e
i
-
∑
i
=
1
N
∑
j
=
1
N
m
l
i
j
e
i
T
e
j
-
m
∑
i
=
1
N
h
i
e
i
T
e
i
=
e
T
[
ω
I
-
m
(
L
+
H
)
]
e
.
Since m
>
ω
/
λ
1
(
L
+
H
), ω
I
-
m
(
L
+
H
) is negative definite, which means V
˙
(
t
)
≤
0. This implies that V
(
e
(
t
)
,
c
i
j
(
t
)
,
c
i
(
t
)
)
≤
V
(
e
(
0
)
,
c
i
j
(
0
)
,
c
i
(
0
)
), for all t
>
0. Therefore, e, c
i
j, and c
i are bounded. Noting that c
i
j and c
i are dependent on e and e is bounded, then, from Assumption 1, (13), and (18), one can know that V
¨ is bounded. This means that V
˙ is uniformly continuous in t. Hence, according to Lemma 4, V
˙
(
e
(
t
)
,
c
i
j
(
t
)
,
c
i
(
t
)
)
→
0 as t
→
∞. Again noting that ω
I
-
m
(
L
+
H
) is negative definite, one can conclude that e
(
t
)
→
0, as t
→
∞, which means that the consensus of networked system (1) is achieved.Remark 8.
From Theorem7, we can know all the nodes in network (1) will asymptotically converge to the desired node s. When all the nonlinear dynamic f
i have a common solution, that is, there exists a vector y
∈
R
n, satisfied y
˙
=
f
i
(
t
,
y
), we can choose y as the desired node. And if all the nonlinear dynamic f
i do not have a common solution, we can choose x
-
=
∑
i
=
1
N
x
i
/
N as the desired node. And we also can obtain the distributed consensus protocol for nonlinear multiagent systems with identical nodes. Considering the network with identical nodes, the ith node of which is described as
(19)
x
˙
i
=
f
(
t
,
x
i
(
t
)
)
.
We choose the control protocol as
(20)
u
i
(
t
)
=
∑
j
∈
N
i
‍
c
i
j
a
i
j
(
x
j
-
x
i
)
+
h
i
c
i
(
s
-
x
i
)
,
where the desired state s satisfies s
˙
=
f
(
t
,
s
) and the adaptive strategy is designed as (6).According to Theorem7, we have the following corollary.Corollary 9.
Suppose that the topology is undirected and connected andf satisfies Lipschitz condition with a Lipschitz constant γ
>
0. Then, under the control protocol (20) with adaptive strategy (6), consensus for multiagent system (19) can be achieved if there exists at least one pinned node in the network.


### 3.2. Pinning-Like Adaptive Consensus for Second-Order Multiagent Systems with Heterogeneous Nonlinear Dynamics

In this subsection, we investigate second-order consensus of networked nonlinear multiagent system (2) via pinning-like adaptive control.Assumption 10.
For everyf
i in (2) and arbitrary u
,
v
,
x
,
y
∈
R
n, there exist corresponding constants θ
>
0 such that
(21)
∥
f
i
(
t
,
u
,
v
)
-
f
i
(
t
,
x
,
y
)
∥
≤
θ
(
∥
u
-
x
∥
+
∥
v
-
y
∥
)
.Remark 11.
Choosingθ
=
max
⁡
{
ρ
1
,
ρ
2
} in [24], we can get the well-definition of Assumption 10. In fact, this condition can be satisfied by many systems. Examples include the pendulum system with a control torque, car-like robots, the Chuas circuit, the Lorenz system, and the Chen system and so forth [27].
Let(22)
x
~
i
=
x
i
-
x
0
,
v
~
i
=
v
i
-
v
0
,
i
=
1,2
,
…
,
N
.
Synchronization or consensus for network with nonidentical nodes cannot be realized without control if the nonidentical dynamic functions do not have a common equilibrium. The purpose of this paper is to design distributed tracking consensus algorithm for networked system (1). Therefore, a distributed adaptive control protocol is proposed, under which leader-following second-order consensus for network (1) can be achieved; that is, there exists a desired state s
=
(
x
0
T
,
v
0
T
)
T, satisfying x
˙
i
(
t
)
=
v
0
(
t
), v
˙
0
(
t
)
=
f
0
(
t
,
x
0
(
t
)
,
v
0
(
t
)
), such that
(23)
lim
⁡
t
→
∞
∥
x
i
(
t
)
-
x
0
(
t
)
∥
=
0
,
lim
⁡
t
→
∞
∥
v
i
(
t
)
-
v
0
(
t
)
∥
=
0
,
11111111
i
=
1,2
,
…
,
N
.
The following consensus protocol is designed for networked system (2):
(24)
u
i
(
t
)
=
v
˙
0
(
t
)
-
f
i
(
t
,
x
0
(
t
)
,
v
0
(
t
)
)
+
∑
j
=
1
,
j
≠
i
N
c
i
j
a
i
j
(
x
j
-
x
i
)
+
h
i
c
i
(
t
)
(
x
0
(
t
)
-
x
i
(
t
)
)
+
∑
j
=
1
,
j
≠
i
N
c
i
j
a
i
j
(
v
j
-
v
i
)
+
h
i
c
i
(
t
)
(
v
0
(
t
)
-
v
i
(
t
)
)
,
1111111111111111111111111111
i
=
1,2
,
…
,
N
.
In the controller (3), h
i
=
1 for some i, then the ith node is named as pinning-like node and h
i
=
0 for others. The adaptive weights c
i
j
>
0, c
i
>
0, i
,
j
=
1,2
,
…
,
N are determined by the following equations:
(25)
c
˙
i
j
(
t
)
=
h
i
j
a
i
j
[
(
x
j
(
t
)
-
x
i
(
t
)
)
T
(
x
j
(
t
)
-
x
i
(
t
)
)
1111111
+
(
v
j
(
t
)
-
v
i
(
t
)
)
T
(
v
j
(
t
)
-
v
i
(
t
)
)
]
,
1111111111111111
i
≠
j
,
i
,
j
=
1,2
,
…
,
N
,
c
˙
i
(
t
)
=
h
i
d
i
[
(
x
0
(
t
)
-
x
i
(
t
)
)
T
(
x
0
(
t
)
-
x
i
(
t
)
)
111111
+
(
v
0
(
t
)
-
v
i
(
t
)
)
T
(
v
0
(
t
)
-
v
i
(
t
)
)
]
,
1111111111111,111111111
i
=
1,2
,
…
,
N
,
where h
i
j
=
h
j
i
>
0, d
i
>
0 are the weights of adaptive laws for parameters c
i
j
(
t
) and c
i
(
t
).
Under consensus protocol (24), the error systems can be rewritten as
(26)
x
~
˙
i
(
t
)
=
v
~
i
(
t
)
,
v
~
˙
i
(
t
)
=
f
i
(
t
,
x
i
(
t
)
,
v
i
(
t
)
)
-
f
i
(
t
,
x
0
(
t
)
,
v
0
(
t
)
)
v
~
˙
i
(
t
)
=
+
∑
j
=
1
,
j
≠
i
N
c
i
j
(
t
)
a
i
j
(
x
~
j
(
t
)
-
x
~
i
(
t
)
)
-
h
i
c
i
(
t
)
x
~
i
(
t
)
v
~
˙
i
(
t
)
=
+
∑
j
=
1
,
j
≠
i
N
c
i
j
(
t
)
a
i
j
(
v
~
j
(
t
)
-
v
~
i
(
t
)
)
-
h
i
c
i
(
t
)
v
~
i
(
t
)
,
1111111111111111111111111111111111
i
=
1
,
…
,
N
.
The following theorem addresses a sufficient condition for the consensus of networked system (2).Theorem 12.
Suppose that Assumptions1 and 5 hold. Then consensus of heterogeneous nonlinear networked system (2) can be achieved under the control protocol (24) with adaptive strategy (25) if there exists at least one pinned-like node in the network.Proof.
LetQ
=
[
2
L
~
+
2
Δ
+
I
N
I
N
I
N
I
N
], Δ
=
diag
⁡
(
c
1
h
1
,
…
,
c
N
h
N
). According to [19, Theorem 6.2
.
24], for an undirected graph G, the Laplace matrix L is irreducible if and only if G is connected. Noting that the corresponding graph of L
~ has the same connectivity with L, L
~ is irreducible under Assumption 5. Then one can conclude that L
~ is semipositive definite according to Lemma 2. And then by virtue ofSchur Complement Lemma, we can know that Q is positive definite.
Consider the following Lyapunov function candidate:(27)
V
(
e
i
,
c
i
j
,
c
i
)
=
1
2
(
x
~
T
v
~
T
)
Q
(
x
~
v
~
)
+
1
2
∑
i
=
1
N
[
∑
j
=
1
,
j
≠
i
N
(
c
i
j
-
α
)
2
2
h
i
j
+
(
c
i
-
α
)
2
d
i
]
,
where α
>
(
2
θ
+
1
)
/
λ
1
(
L
+
H
), H
=
diag
⁡
(
h
1
,
…
,
h
N
), and λ
1
(
L
+
H
) denotes the minimum eigenvalue of (
L
+
H
).
DifferentiatingV
(
t
) with respect to t along (28), we have
(28)
V
˙
(
t
)
=
x
~
T
(
2
L
~
+
2
Δ
+
I
N
)
v
~
+
x
~
T
v
~
˙
+
v
~
T
v
~
+
v
~
T
v
~
˙
+
∑
i
=
1
N
[
∑
j
=
1
,
j
≠
i
N
(
c
i
j
-
α
)
c
˙
i
j
2
h
i
j
+
(
c
i
-
α
)
c
˙
i
d
i
]
=
x
~
T
(
2
L
~
+
2
Δ
+
I
N
)
v
~
+
v
~
T
v
~
+
∑
i
=
1
N
(
x
~
i
+
v
~
i
)
T
(
f
i
(
t
,
x
i
,
v
i
)
-
f
i
(
t
,
x
0
,
v
0
)
)
+
∑
i
=
1
N
‍
(
x
~
i
+
v
~
i
)
T
(
∑
j
=
1
,
j
≠
i
N
c
i
j
a
i
j
(
x
j
-
x
i
)
1111111111111111
+
∑
j
=
1
,
j
≠
i
N
c
i
j
a
i
j
(
v
j
-
v
i
)
111111111111111111111111
-
c
i
h
i
x
~
i
-
c
i
h
i
v
~
i
∑
j
=
1
,
j
≠
i
N
‍
)
+
1
2
∑
i
=
1
N
∑
j
=
1
,
j
≠
i
N
[
(
c
i
j
-
α
)
a
i
j
(
x
j
-
x
i
)
T
(
x
j
-
x
i
)
111111111111111
+
(
c
i
j
-
α
)
a
i
j
(
v
j
-
v
i
)
T
111111111111111
×
(
v
j
-
v
i
)
(
x
j
-
x
i
)
T
]
+
∑
i
=
1
N
‍
(
c
i
-
α
)
h
i
x
~
i
T
x
~
i
+
∑
i
=
1
N
‍
(
c
i
-
α
)
h
i
v
~
i
T
v
~
i
.
Noting that L
~ is symmetric and zero-row-sum, similarly as (16), we have
(29)
∑
i
=
1
N
∑
j
=
1
,
j
≠
i
N
(
c
i
j
-
α
)
a
i
j
(
x
~
j
-
x
~
i
)
T
(
x
~
j
-
x
~
i
)
=
2
∑
i
=
1
N
∑
j
=
1
N
(
l
~
i
j
-
α
l
i
j
)
x
~
i
T
x
~
j
,
∑
i
=
1
N
∑
j
=
1
,
j
≠
i
N
(
c
i
j
-
α
)
a
i
j
(
v
~
j
-
v
~
i
)
T
(
v
~
j
-
v
~
i
)
=
2
∑
i
=
1
N
∑
j
=
1
N
(
l
~
i
j
-
α
l
i
j
)
v
~
i
T
v
~
j
.
From ((11)–(13)), we have
(30)
V
˙
(
t
)
=
x
~
T
(
2
L
~
+
2
Δ
+
I
N
)
v
~
+
v
~
T
v
~
+
∑
i
=
1
N
(
x
~
i
+
v
~
i
)
T
(
f
i
(
t
,
x
i
,
v
i
)
-
f
i
(
t
,
x
0
,
v
0
)
)
+
∑
i
=
1
N
(
x
~
i
+
v
~
i
)
T
[
∑
j
=
1
N
(
-
l
~
i
j
(
x
~
j
+
v
~
j
)
)
-
c
i
h
i
(
x
~
i
+
v
~
i
)
]
+
∑
i
=
1
N
∑
j
=
1
,
j
≠
i
N
[
(
l
~
i
j
-
α
l
i
j
)
x
~
i
T
x
j
+
(
l
~
i
j
-
α
l
i
j
)
v
~
i
T
v
~
i
]
+
∑
i
=
1
N
‍
(
c
i
-
α
)
h
i
x
~
i
T
x
~
i
+
∑
i
=
1
N
‍
(
c
i
-
α
)
h
i
v
~
i
T
v
~
i
≤
(
2
θ
+
1
)
(
x
~
T
x
~
+
v
~
T
v
~
)
-
α
x
~
T
(
L
+
H
)
x
~
-
α
v
~
T
(
L
+
H
)
v
~
=
x
~
T
[
(
2
θ
+
1
)
I
-
α
(
L
+
H
)
]
x
~
+
v
~
T
[
(
2
θ
+
1
)
I
-
α
(
L
+
H
)
]
v
~
.
According to Lemma 2, (
L
+
H
) is positive definite. Since α
>
(
2
θ
+
1
)
/
λ
1
(
L
+
H
), (
2
θ
+
1
)
I
-
α
(
L
+
H
) is negative definite, which means V
˙
(
t
)
≤
0. The same to Theorem 7, by in virtue of Lemma 4, one can conclude that x
~
i
→
0 and v
~
i
→
0, as t
→
∞, which means that the consensus of networked system (2) is achieved.Remark 13.
From Theorem12, we can know that all the nodes in network (2) will asymptotically converge to the desired node s. When all the nonlinear dynamic f
i have a common solution, we can choose it as the desired node. And if all the nonlinear dynamics f
i do not have a common solution, we can choose average state as the desired node. And we also can obtain the pinning adaptive consensus control protocol for nonlinear mobile agents networked system with identical nonlinear dynamics.
When all the nonlinear functionf
i, i
=
1
,
…
,
N are identical, that is, the ith node in system (1) can be described as
(31)
x
˙
i
(
t
)
=
v
i
(
t
)
,
v
˙
i
(
t
)
=
f
(
t
,
x
i
(
t
)
,
v
i
(
t
)
)
+
u
i
(
t
)
,
11111111111111111
i
=
1
,
…
,
N
.
We choose the control protocol as
(32)
u
i
(
t
)
=
∑
j
=
1
,
j
≠
i
N
c
i
j
a
i
j
(
x
j
-
x
i
)
+
h
i
c
i
(
t
)
(
x
0
-
x
i
)
+
∑
j
=
1
,
j
≠
i
N
c
i
j
a
i
j
(
v
j
-
v
i
)
+
h
i
c
i
(
t
)
(
v
0
-
v
i
)
,
11111111111111111111111
i
=
1,2
,
…
,
N
.
In the protocol, the adaptive weights c
i
j
>
0, c
i
>
0, i
,
j
=
1,2
,
…
,
N, are determined by (25). Then, from Theorem 12, one can easily obtain the following corollary.Corollary 14.
Suppose that Assumptions1 and 5 hold. Then consensus for nonlinear mobile multiagent networked system (31) can be achieved under the control protocol (32) with adaptive strategy (25) if there exists at least one pinned node in the network.


## 3.1. Pinning-Like Adaptive Consensus for First-Order Multiagent Systems with Heterogeneous Nonlinear Dynamics

In this subsection, we investigate consensus criteria for first-order multiagent systems with heterogeneous nonlinear dynamics via distributed adaptive pinning control.As mentioned in many literatures, consensus or synchronization for networked systems with nonidentical nodes cannot be realized without control if the nonidentical dynamic functions do not have a common solution. Thereby, a distributed adaptive pinning-like control protocol is proposed, under which leader-following consensus for networked system (1) can be achieved; that is, there exists a desired state s, such that
(3)
lim
⁡
t
→
∞
∥
x
i
(
t
)
-
s
(
t
)
∥
=
0
,
i
=
1,2
,
…
,
N
,
where s satisfies s
˙
=
f
(
t
,
s
(
t
)
), and s can be an equilibrium, a limit cycle, or even a chaotic attractor.The following assumption is necessary for our main results.Assumption 5.
For arbitraryx
,
y
∈
R
n and every i
∈
{
1,2
,
…
,
N
}, there exists a constant ω
>
0 such that
(4)
(
x
-
y
)
T
(
f
i
(
t
,
x
)
-
f
i
(
t
,
y
)
)
≤
ω
(
x
-
y
)
T
(
x
-
y
)
.
We choose the control protocol as
(5)
u
i
(
t
)
=
s
˙
(
t
)
-
f
i
(
t
,
s
(
t
)
)
+
∑
j
∈
N
i
‍
c
i
j
a
i
j
(
x
j
(
t
)
-
x
i
(
t
)
)
+
h
i
c
i
(
t
)
(
s
(
t
)
-
x
i
(
t
)
)
.
In the controller (4), h
i
=
1 for some i, then the ith node is named as pinning-like node and h
i
=
0 for others. c
i
>
0 and c
i
j
>
0 are determined by the following equations:
(6)
c
˙
i
(
t
)
=
h
i
k
i
(
s
(
t
)
-
x
i
(
t
)
)
T
(
s
(
t
)
-
x
i
(
t
)
)
,
111
c
˙
i
j
(
t
)
=
a
i
j
k
i
j
(
x
j
(
t
)
-
x
i
(
t
)
)
T
(
x
j
(
t
)
-
x
i
(
t
)
)
,
i
≠
j
,
i
,
j
=
1,2
,
…
,
N
,
where k
i
j
=
k
j
i
>
0, k
i
>
0 are the weights of adaptive laws for parameters c
i
j
(
t
) and c
i
(
t
). c
i
j
>
0 denotes the coupling strength between node i and node j
,
i
,
j
=
1
,
…
,
N. We define a new matrix, named adaptive weighted Laplace matrix of the network topology, as the following:
(7)
L
~
=
[
l
~
11
l
~
12
⋯
l
~
1
N
l
~
21
l
~
22
⋯
l
~
2
N
⋮
⋮
⋮
⋮
l
~
N
1
l
~
N
2
⋯
l
~
N
N
]
.
The elements l
~
i
j is defined as l
~
i
j
=
c
i
j
l
i
j,  i
≠
j,  i
,
j
=
1,2
,
…
,
N, and c
i
i is defined as l
~
i
i
=
c
i
i
l
i
i
=
∑
j
≠
i
,
j
=
1
N
c
i
j
a
i
j, so L
~ is zero-row-sum. The initial values of the elements of L
~ should be chosen symmetrically for the symmetry of L
~.Remark 6.
From Lemmas2 and 3, if the undirected topology graph G is connected, the adaptive weighted Laplace matrix is semipositive definite.The controlled network for system (1) under protocol (5) with adaptive strategy (6) can be rewritten as
(8)
x
˙
i
=
f
i
(
t
,
x
i
(
t
)
)
+
∑
j
∈
N
i
c
i
j
a
i
j
(
x
j
(
t
)
-
x
i
(
t
)
)
+
s
˙
(
t
)
-
f
i
(
t
,
s
(
t
)
)
+
h
i
c
i
(
t
)
(
s
(
t
)
-
x
i
(
t
)
)
.
Let error state e
i be defined as
(9)
e
i
=
x
i
-
s
,
i
=
1,2
,
…
,
N
.
From (1) and ((8)-(9)), the error systems can be described as
(10)
e
˙
i
=
f
i
(
t
,
x
i
(
t
)
)
-
f
i
(
t
,
s
(
t
)
)
+
∑
j
∈
N
i
c
i
j
a
i
j
(
x
j
(
t
)
-
x
i
(
t
)
)
-
h
i
c
i
(
t
)
e
i
(
t
)
.
Noting that L
~ is zero-row-sum, (10) can be rewritten as
(11)
e
˙
i
=
f
i
(
t
,
x
i
(
t
)
)
-
f
i
(
t
,
s
(
t
)
)
-
∑
j
=
1
N
l
~
i
j
x
j
-
h
i
c
i
(
t
)
e
i
.Theorem 7.
Suppose that Assumptions1 and 5 hold. Then consensus for multiagent system (1) can be achieved under the control protocol (5) with adaptive strategy (6) if there exists at least one pinned-like node in the network.Proof.
LetH
=
diag
⁡
(
h
1
,
h
2
,
…
,
h
N
), according to Lemma 2, L
+
H is positive definite. Consider the following Lyapunov function candidate:
(12)
V
(
e
i
,
c
i
j
,
c
i
)
=
1
2
∑
i
=
1
N
e
i
T
e
i
+
1
2
∑
i
=
1
N
[
∑
j
=
1
,
j
≠
i
N
(
c
i
j
-
m
)
2
2
k
i
j
+
(
c
i
-
m
)
2
k
i
]
,
where m
>
ω
/
λ
1
(
L
+
H
), and λ
1
(
L
+
H
) denotes the minimum eigenvalue of L
+
H.
DifferentiatingV
(
t
) with respect to t along (11), we have
(13)
V
˙
(
t
)
=
∑
i
=
1
N
‍
e
i
T
(
f
i
(
t
,
x
i
(
t
)
)
-
f
i
(
t
,
s
(
t
)
)
)
-
∑
i
=
1
N
‍
e
i
T
[
∑
j
=
1
N
‍
l
~
i
j
x
j
(
t
)
+
h
i
c
i
(
t
)
e
i
(
t
)
]
+
1
2
∑
i
=
1
N
∑
j
=
1
,
j
≠
i
N
(
c
i
j
-
m
)
a
i
j
(
x
j
-
x
i
)
T
(
x
j
-
x
i
)
+
∑
i
=
1
N
‍
(
c
i
-
m
)
h
i
e
i
T
e
i
≤
∑
i
=
1
N
‍
[
ω
e
i
T
e
i
-
m
h
i
e
i
T
e
i
-
∑
j
=
1
N
‍
l
~
i
j
e
i
T
x
j
(
t
)
1111111
+
1
2
∑
j
=
1
,
j
≠
i
N
‍
(
c
i
j
-
m
)
a
i
j
(
x
j
-
x
i
)
T
(
x
j
-
x
i
)
]
.
Noting that L
~ is symmetric and zero-row-sum, we have
(14)
∑
i
=
1
N
∑
j
≠
i
N
(
c
i
j
-
m
)
a
i
j
e
j
T
e
j
=
∑
i
=
1
N
∑
j
≠
i
N
(
c
i
j
-
m
)
a
i
j
e
i
T
e
i
=
∑
i
=
1
N
‍
(
c
i
i
-
m
)
l
i
i
e
i
T
e
i
.
According to (14), we have
(15)
∑
i
=
1
N
∑
j
=
1
,
j
≠
i
N
(
c
i
j
-
m
)
a
i
j
(
x
j
-
x
i
)
T
(
x
j
-
x
i
)
=
∑
i
=
1
N
∑
j
≠
i
N
(
c
i
j
-
m
)
a
i
j
(
e
j
-
e
i
)
T
(
e
j
-
e
i
)
=
∑
i
=
1
N
∑
j
≠
i
N
(
c
i
j
-
m
)
a
i
j
(
e
j
T
e
j
+
e
i
T
e
i
-
2
e
i
T
e
j
)
=
2
∑
i
=
1
N
(
c
i
i
-
m
)
l
i
i
e
i
T
e
i
+
2
∑
i
=
1
N
∑
j
≠
i
N
(
c
i
j
-
m
)
l
i
j
e
i
T
e
j
=
2
∑
i
=
1
N
∑
j
=
1
N
(
c
i
j
-
m
)
l
i
j
e
i
T
e
j
.
So we have
(16)
∑
i
=
1
N
∑
j
=
1
N
(
c
i
j
-
m
)
l
i
j
e
i
T
e
j
=
1
2
∑
i
=
1
N
∑
j
≠
i
N
(
c
i
j
-
m
)
a
i
j
(
e
j
-
e
i
)
T
(
e
j
-
e
i
)
.
From ((14)–(16)), one can easily conclude the following:
(17)
∑
i
=
1
N
∑
j
=
1
N
l
~
i
j
e
i
T
e
j
=
1
2
∑
i
=
1
N
∑
j
≠
i
N
c
i
j
a
i
j
(
e
j
-
e
i
)
T
(
e
j
-
e
i
)
.
From (13), (16), and (17), we have
(18)
V
˙
(
t
)
≤
∑
i
=
1
N
ω
e
i
T
e
i
-
∑
i
=
1
N
∑
j
=
1
N
l
~
i
j
e
i
T
e
j
+
1
2
∑
i
=
1
N
∑
j
=
1
,
j
≠
i
N
(
c
i
j
-
m
)
a
i
j
(
e
j
-
e
i
)
T
(
e
j
-
e
i
)
-
m
∑
i
=
1
N
h
i
e
i
T
e
i
=
∑
i
=
1
N
ω
e
i
T
e
i
-
∑
i
=
1
N
∑
j
=
1
N
m
l
i
j
e
i
T
e
j
-
m
∑
i
=
1
N
h
i
e
i
T
e
i
=
e
T
[
ω
I
-
m
(
L
+
H
)
]
e
.
Since m
>
ω
/
λ
1
(
L
+
H
), ω
I
-
m
(
L
+
H
) is negative definite, which means V
˙
(
t
)
≤
0. This implies that V
(
e
(
t
)
,
c
i
j
(
t
)
,
c
i
(
t
)
)
≤
V
(
e
(
0
)
,
c
i
j
(
0
)
,
c
i
(
0
)
), for all t
>
0. Therefore, e, c
i
j, and c
i are bounded. Noting that c
i
j and c
i are dependent on e and e is bounded, then, from Assumption 1, (13), and (18), one can know that V
¨ is bounded. This means that V
˙ is uniformly continuous in t. Hence, according to Lemma 4, V
˙
(
e
(
t
)
,
c
i
j
(
t
)
,
c
i
(
t
)
)
→
0 as t
→
∞. Again noting that ω
I
-
m
(
L
+
H
) is negative definite, one can conclude that e
(
t
)
→
0, as t
→
∞, which means that the consensus of networked system (1) is achieved.Remark 8.
From Theorem7, we can know all the nodes in network (1) will asymptotically converge to the desired node s. When all the nonlinear dynamic f
i have a common solution, that is, there exists a vector y
∈
R
n, satisfied y
˙
=
f
i
(
t
,
y
), we can choose y as the desired node. And if all the nonlinear dynamic f
i do not have a common solution, we can choose x
-
=
∑
i
=
1
N
x
i
/
N as the desired node. And we also can obtain the distributed consensus protocol for nonlinear multiagent systems with identical nodes. Considering the network with identical nodes, the ith node of which is described as
(19)
x
˙
i
=
f
(
t
,
x
i
(
t
)
)
.
We choose the control protocol as
(20)
u
i
(
t
)
=
∑
j
∈
N
i
‍
c
i
j
a
i
j
(
x
j
-
x
i
)
+
h
i
c
i
(
s
-
x
i
)
,
where the desired state s satisfies s
˙
=
f
(
t
,
s
) and the adaptive strategy is designed as (6).According to Theorem7, we have the following corollary.Corollary 9.
Suppose that the topology is undirected and connected andf satisfies Lipschitz condition with a Lipschitz constant γ
>
0. Then, under the control protocol (20) with adaptive strategy (6), consensus for multiagent system (19) can be achieved if there exists at least one pinned node in the network.


## 3.2. Pinning-Like Adaptive Consensus for Second-Order Multiagent Systems with Heterogeneous Nonlinear Dynamics

In this subsection, we investigate second-order consensus of networked nonlinear multiagent system (2) via pinning-like adaptive control.Assumption 10.
For everyf
i in (2) and arbitrary u
,
v
,
x
,
y
∈
R
n, there exist corresponding constants θ
>
0 such that
(21)
∥
f
i
(
t
,
u
,
v
)
-
f
i
(
t
,
x
,
y
)
∥
≤
θ
(
∥
u
-
x
∥
+
∥
v
-
y
∥
)
.Remark 11.
Choosingθ
=
max
⁡
{
ρ
1
,
ρ
2
} in [24], we can get the well-definition of Assumption 10. In fact, this condition can be satisfied by many systems. Examples include the pendulum system with a control torque, car-like robots, the Chuas circuit, the Lorenz system, and the Chen system and so forth [27].
Let(22)
x
~
i
=
x
i
-
x
0
,
v
~
i
=
v
i
-
v
0
,
i
=
1,2
,
…
,
N
.
Synchronization or consensus for network with nonidentical nodes cannot be realized without control if the nonidentical dynamic functions do not have a common equilibrium. The purpose of this paper is to design distributed tracking consensus algorithm for networked system (1). Therefore, a distributed adaptive control protocol is proposed, under which leader-following second-order consensus for network (1) can be achieved; that is, there exists a desired state s
=
(
x
0
T
,
v
0
T
)
T, satisfying x
˙
i
(
t
)
=
v
0
(
t
), v
˙
0
(
t
)
=
f
0
(
t
,
x
0
(
t
)
,
v
0
(
t
)
), such that
(23)
lim
⁡
t
→
∞
∥
x
i
(
t
)
-
x
0
(
t
)
∥
=
0
,
lim
⁡
t
→
∞
∥
v
i
(
t
)
-
v
0
(
t
)
∥
=
0
,
11111111
i
=
1,2
,
…
,
N
.
The following consensus protocol is designed for networked system (2):
(24)
u
i
(
t
)
=
v
˙
0
(
t
)
-
f
i
(
t
,
x
0
(
t
)
,
v
0
(
t
)
)
+
∑
j
=
1
,
j
≠
i
N
c
i
j
a
i
j
(
x
j
-
x
i
)
+
h
i
c
i
(
t
)
(
x
0
(
t
)
-
x
i
(
t
)
)
+
∑
j
=
1
,
j
≠
i
N
c
i
j
a
i
j
(
v
j
-
v
i
)
+
h
i
c
i
(
t
)
(
v
0
(
t
)
-
v
i
(
t
)
)
,
1111111111111111111111111111
i
=
1,2
,
…
,
N
.
In the controller (3), h
i
=
1 for some i, then the ith node is named as pinning-like node and h
i
=
0 for others. The adaptive weights c
i
j
>
0, c
i
>
0, i
,
j
=
1,2
,
…
,
N are determined by the following equations:
(25)
c
˙
i
j
(
t
)
=
h
i
j
a
i
j
[
(
x
j
(
t
)
-
x
i
(
t
)
)
T
(
x
j
(
t
)
-
x
i
(
t
)
)
1111111
+
(
v
j
(
t
)
-
v
i
(
t
)
)
T
(
v
j
(
t
)
-
v
i
(
t
)
)
]
,
1111111111111111
i
≠
j
,
i
,
j
=
1,2
,
…
,
N
,
c
˙
i
(
t
)
=
h
i
d
i
[
(
x
0
(
t
)
-
x
i
(
t
)
)
T
(
x
0
(
t
)
-
x
i
(
t
)
)
111111
+
(
v
0
(
t
)
-
v
i
(
t
)
)
T
(
v
0
(
t
)
-
v
i
(
t
)
)
]
,
1111111111111,111111111
i
=
1,2
,
…
,
N
,
where h
i
j
=
h
j
i
>
0, d
i
>
0 are the weights of adaptive laws for parameters c
i
j
(
t
) and c
i
(
t
).
Under consensus protocol (24), the error systems can be rewritten as
(26)
x
~
˙
i
(
t
)
=
v
~
i
(
t
)
,
v
~
˙
i
(
t
)
=
f
i
(
t
,
x
i
(
t
)
,
v
i
(
t
)
)
-
f
i
(
t
,
x
0
(
t
)
,
v
0
(
t
)
)
v
~
˙
i
(
t
)
=
+
∑
j
=
1
,
j
≠
i
N
c
i
j
(
t
)
a
i
j
(
x
~
j
(
t
)
-
x
~
i
(
t
)
)
-
h
i
c
i
(
t
)
x
~
i
(
t
)
v
~
˙
i
(
t
)
=
+
∑
j
=
1
,
j
≠
i
N
c
i
j
(
t
)
a
i
j
(
v
~
j
(
t
)
-
v
~
i
(
t
)
)
-
h
i
c
i
(
t
)
v
~
i
(
t
)
,
1111111111111111111111111111111111
i
=
1
,
…
,
N
.
The following theorem addresses a sufficient condition for the consensus of networked system (2).Theorem 12.
Suppose that Assumptions1 and 5 hold. Then consensus of heterogeneous nonlinear networked system (2) can be achieved under the control protocol (24) with adaptive strategy (25) if there exists at least one pinned-like node in the network.Proof.
LetQ
=
[
2
L
~
+
2
Δ
+
I
N
I
N
I
N
I
N
], Δ
=
diag
⁡
(
c
1
h
1
,
…
,
c
N
h
N
). According to [19, Theorem 6.2
.
24], for an undirected graph G, the Laplace matrix L is irreducible if and only if G is connected. Noting that the corresponding graph of L
~ has the same connectivity with L, L
~ is irreducible under Assumption 5. Then one can conclude that L
~ is semipositive definite according to Lemma 2. And then by virtue ofSchur Complement Lemma, we can know that Q is positive definite.
Consider the following Lyapunov function candidate:(27)
V
(
e
i
,
c
i
j
,
c
i
)
=
1
2
(
x
~
T
v
~
T
)
Q
(
x
~
v
~
)
+
1
2
∑
i
=
1
N
[
∑
j
=
1
,
j
≠
i
N
(
c
i
j
-
α
)
2
2
h
i
j
+
(
c
i
-
α
)
2
d
i
]
,
where α
>
(
2
θ
+
1
)
/
λ
1
(
L
+
H
), H
=
diag
⁡
(
h
1
,
…
,
h
N
), and λ
1
(
L
+
H
) denotes the minimum eigenvalue of (
L
+
H
).
DifferentiatingV
(
t
) with respect to t along (28), we have
(28)
V
˙
(
t
)
=
x
~
T
(
2
L
~
+
2
Δ
+
I
N
)
v
~
+
x
~
T
v
~
˙
+
v
~
T
v
~
+
v
~
T
v
~
˙
+
∑
i
=
1
N
[
∑
j
=
1
,
j
≠
i
N
(
c
i
j
-
α
)
c
˙
i
j
2
h
i
j
+
(
c
i
-
α
)
c
˙
i
d
i
]
=
x
~
T
(
2
L
~
+
2
Δ
+
I
N
)
v
~
+
v
~
T
v
~
+
∑
i
=
1
N
(
x
~
i
+
v
~
i
)
T
(
f
i
(
t
,
x
i
,
v
i
)
-
f
i
(
t
,
x
0
,
v
0
)
)
+
∑
i
=
1
N
‍
(
x
~
i
+
v
~
i
)
T
(
∑
j
=
1
,
j
≠
i
N
c
i
j
a
i
j
(
x
j
-
x
i
)
1111111111111111
+
∑
j
=
1
,
j
≠
i
N
c
i
j
a
i
j
(
v
j
-
v
i
)
111111111111111111111111
-
c
i
h
i
x
~
i
-
c
i
h
i
v
~
i
∑
j
=
1
,
j
≠
i
N
‍
)
+
1
2
∑
i
=
1
N
∑
j
=
1
,
j
≠
i
N
[
(
c
i
j
-
α
)
a
i
j
(
x
j
-
x
i
)
T
(
x
j
-
x
i
)
111111111111111
+
(
c
i
j
-
α
)
a
i
j
(
v
j
-
v
i
)
T
111111111111111
×
(
v
j
-
v
i
)
(
x
j
-
x
i
)
T
]
+
∑
i
=
1
N
‍
(
c
i
-
α
)
h
i
x
~
i
T
x
~
i
+
∑
i
=
1
N
‍
(
c
i
-
α
)
h
i
v
~
i
T
v
~
i
.
Noting that L
~ is symmetric and zero-row-sum, similarly as (16), we have
(29)
∑
i
=
1
N
∑
j
=
1
,
j
≠
i
N
(
c
i
j
-
α
)
a
i
j
(
x
~
j
-
x
~
i
)
T
(
x
~
j
-
x
~
i
)
=
2
∑
i
=
1
N
∑
j
=
1
N
(
l
~
i
j
-
α
l
i
j
)
x
~
i
T
x
~
j
,
∑
i
=
1
N
∑
j
=
1
,
j
≠
i
N
(
c
i
j
-
α
)
a
i
j
(
v
~
j
-
v
~
i
)
T
(
v
~
j
-
v
~
i
)
=
2
∑
i
=
1
N
∑
j
=
1
N
(
l
~
i
j
-
α
l
i
j
)
v
~
i
T
v
~
j
.
From ((11)–(13)), we have
(30)
V
˙
(
t
)
=
x
~
T
(
2
L
~
+
2
Δ
+
I
N
)
v
~
+
v
~
T
v
~
+
∑
i
=
1
N
(
x
~
i
+
v
~
i
)
T
(
f
i
(
t
,
x
i
,
v
i
)
-
f
i
(
t
,
x
0
,
v
0
)
)
+
∑
i
=
1
N
(
x
~
i
+
v
~
i
)
T
[
∑
j
=
1
N
(
-
l
~
i
j
(
x
~
j
+
v
~
j
)
)
-
c
i
h
i
(
x
~
i
+
v
~
i
)
]
+
∑
i
=
1
N
∑
j
=
1
,
j
≠
i
N
[
(
l
~
i
j
-
α
l
i
j
)
x
~
i
T
x
j
+
(
l
~
i
j
-
α
l
i
j
)
v
~
i
T
v
~
i
]
+
∑
i
=
1
N
‍
(
c
i
-
α
)
h
i
x
~
i
T
x
~
i
+
∑
i
=
1
N
‍
(
c
i
-
α
)
h
i
v
~
i
T
v
~
i
≤
(
2
θ
+
1
)
(
x
~
T
x
~
+
v
~
T
v
~
)
-
α
x
~
T
(
L
+
H
)
x
~
-
α
v
~
T
(
L
+
H
)
v
~
=
x
~
T
[
(
2
θ
+
1
)
I
-
α
(
L
+
H
)
]
x
~
+
v
~
T
[
(
2
θ
+
1
)
I
-
α
(
L
+
H
)
]
v
~
.
According to Lemma 2, (
L
+
H
) is positive definite. Since α
>
(
2
θ
+
1
)
/
λ
1
(
L
+
H
), (
2
θ
+
1
)
I
-
α
(
L
+
H
) is negative definite, which means V
˙
(
t
)
≤
0. The same to Theorem 7, by in virtue of Lemma 4, one can conclude that x
~
i
→
0 and v
~
i
→
0, as t
→
∞, which means that the consensus of networked system (2) is achieved.Remark 13.
From Theorem12, we can know that all the nodes in network (2) will asymptotically converge to the desired node s. When all the nonlinear dynamic f
i have a common solution, we can choose it as the desired node. And if all the nonlinear dynamics f
i do not have a common solution, we can choose average state as the desired node. And we also can obtain the pinning adaptive consensus control protocol for nonlinear mobile agents networked system with identical nonlinear dynamics.
When all the nonlinear functionf
i, i
=
1
,
…
,
N are identical, that is, the ith node in system (1) can be described as
(31)
x
˙
i
(
t
)
=
v
i
(
t
)
,
v
˙
i
(
t
)
=
f
(
t
,
x
i
(
t
)
,
v
i
(
t
)
)
+
u
i
(
t
)
,
11111111111111111
i
=
1
,
…
,
N
.
We choose the control protocol as
(32)
u
i
(
t
)
=
∑
j
=
1
,
j
≠
i
N
c
i
j
a
i
j
(
x
j
-
x
i
)
+
h
i
c
i
(
t
)
(
x
0
-
x
i
)
+
∑
j
=
1
,
j
≠
i
N
c
i
j
a
i
j
(
v
j
-
v
i
)
+
h
i
c
i
(
t
)
(
v
0
-
v
i
)
,
11111111111111111111111
i
=
1,2
,
…
,
N
.
In the protocol, the adaptive weights c
i
j
>
0, c
i
>
0, i
,
j
=
1,2
,
…
,
N, are determined by (25). Then, from Theorem 12, one can easily obtain the following corollary.Corollary 14.
Suppose that Assumptions1 and 5 hold. Then consensus for nonlinear mobile multiagent networked system (31) can be achieved under the control protocol (32) with adaptive strategy (25) if there exists at least one pinned node in the network.


## 4. Simulations

In this section, several simulation results are presented to illustrate the previous theoretical results.Example 1.
We choose a network with 4 nodes and the reference node is chosen asv
˙
0
(
t
)
=
sin
t. The nonlinear dynamic functions are chosen as f
i
=
sin
(
i
x
i
+
π
/
i
), i
=
1,2
,
3,4, and nonlinear coupled function g
(
x
)
=
x
+
tanh
x. The topology is described by the following adjacent matrix:
(33)
A
=
(
0
1
0
1
1
0
1
0
0
1
0
1
1
0
1
0
)
.In Figure1, the dash dotted line denotes the state of the reference node. And other lines represent the trajectories of the nodes in the network. The trajectories of the adaptive gains are drawn in Figure 2. According to the simulation, we can know that the consensus for the proposed network can be achieved rapidly, which results in that the adaptive gains are stabilized rapidly.Figure 1
State trajectories of the four nodes with the reference state.Figure 2
Trajectories of the adaptive gains.Example 2.
The topology is chosen the same as Example1 and the reference node is chosen as
(34)
x
˙
0
(
t
)
=
sin
t
,
v
˙
0
(
t
)
=
cos
⁡
t
.The nonlinear dynamic functions are chosen asf
i
=
sin
(
i
x
i
+
π
/
i
), i
=
1,2
,
3,4. We choose the first node as the pinning-like node. In Figures 3 and 4, the dash dotted line denotes the position trajectory and the velocity trajectory of the reference node. And other lines represent the trajectories of the nodes in the network. The trajectories of the adaptive gains are drawn in Figure 5. According to Figures 3 and 4, the consensus can be achieved.Figure 3
Trajectories for position.Figure 4
Trajectories for velocity.Figure 5
Trajectories of the adaptive gains.


## 5. Conclusions

Adaptive consensus for networked mobile agents with heterogeneous nonlinear dynamics was investigated in this paper. Sufficient consensus conditions for both first-order and second-order networked mobile agents with heterogeneous nonlinear dynamics were obtained. By designing an adaptive strategy on the coupling strengths, the consensus can be achieved without requiring any global information except a connection assumption. We also extended the results to the consensus for nonlinear mobile agents with identical nonlinear dynamics. Simulation examples were given to demonstrate the feasibility and effectiveness of the proposed consensus scheme.



---

*Source: 269031-2014-09-25.xml*