3088 IEEE TRANSACTIONS ON CYBERNETICS, VOL. 46, NO. 12, DECEMBER 2016
Lemma 2 [3]: ||
i
|| ≤ (||e
i
||/(σ (L + B))), i = 1,...,n,
where σ
(L +B) is a minimum singular value of matrix L +B,
where e
i
= [e
1,i
,...,e
N,i
]
T
.
C. Cooperative Synchronization Error Dynamics
and Sliding Mode Error
Define Notations: e
i
= [e
1,i
,...,e
N,i
]
T
∈ R
N
, i = 1,...,n,
f
0
= [f
0
,...,f
0
]
T
∈ R
N
. Similar to (2), the above synchro-
nization error can be rewritten as
˙e
i
= e
i+1
˙e
n
=−(L +B)
f + u + g − f
0
(4)
where the property L1
= 0 is used.
Define the sliding mode error s
k
for follower k as follows:
s
k
=
n−1
i=1
c
k,i
e
k,i
+ e
k,n
, k = 1, 2,...,n (5)
where c
k,i
= C
i−1
n−1
α
n−i
k
, i = 1,...,n − 1,α
k
> 0 denotes a
designed parameter, C
i−1
n−1
= ((n −1)(n − 2) ···(n −i +1))/
((i − 1)(i −2) ···1).
Let ¯e
k
= [e
k,1
,...,e
k,n
]
T
.
Lemma 3 [19]: Let s
k
be defined by (5). We then have:
1) if s
k
= 0, then lim
t→∞
e
k,1
(t) = 0;
2) if |s
k
|≤w
k
, ¯e
k
(0) ∈
w
k
, then ¯e
k
(t) ∈
w
k
, ∀t ≥ 0;
3) if |s
k
|≤w
k
, ¯e
k
(0)/∈
w
k
, then ∃T
k
= (n − 1)/λ
k
,
∀t ≥ T
k
, ¯e
k
(t) ∈
w
k
, where
w
k
={¯e
k
(t)
|
|e
k,i
|
≤ 2
(i−1)
λ
i−n
k
w
k
, i = 1,...,n, k = 1, 2,...N, w
k
is a
design parameter.
For simplification, let c
1,i
= ··· = c
N,i
= λ
i
,λ
n
= 1,
i = 1,...,n, then s
k
= λ
1
e
k,1
+···+λ
n
e
k,n
. Define the global
sliding mode error s = [s
1
,...,s
N
]
T
, then s = λ
1
e
1
+···+
λ
n
e
n
. Further, one has
˙s = λ
1
˙e
1
+···+λ
n
˙e
n
=
n−1
i=1
λ
i
e
i+1
+˙e
n
= γ − (L +B)
f + u + g − f
0
where γ =
n−1
i=1
λ
i
e
i+1
. It is necessary to point out that, in
this paper, it is assumed that only the outputs of each follower
and its neighbors are measurable, the above synchronization
errors and sliding mode errors cannot used in the controller
design. Hence, the first task is to design state observers for the
followers in the graph to estimate the states of them and their
neighbors, whose detailed design will be given in the next
section. Let ˆx
k,i
denotes the estimate of x
k,i
, k = 1,...,N,
i = 1,...,n. Consequently, using the estimated states, the
above synchronization errors, sliding mode error, and tracking
error are modified as follows:
⎧
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎩
ˆe
k,i
=
j∈N
k
a
kj
ˆx
j,i
−ˆx
k,i
+ b
k
ˆx
0,i
−ˆx
k,i
ˆs
k
=
n−1
i=1
c
k,i
ˆe
k,i
+ˆe
k,n
ˆ
k,i
=ˆx
k,i
−ˆx
0,i
ˆ
i
=ˆx
i
−ˆx
0
(6)
which will be actually used in the controllers, where
ˆx
0,i
= [ˆx
0,i
,...,ˆx
0,i
]
T
∈ R
N
, ˆx
i
= [ˆx
1,i
,...,ˆx
N,i
]
T
, ˆ
i
=
[ ˆ
1,i
,..., ˆ
N,i
]
T
. Next, using the estimated states, modified
synchronization errors, sliding mode error, and tracking error
in (6), distributed adaptive sliding mode controllers will be
designed to obtain the above control objective.
D. Mathematical Description of Fuzzy Logic Systems
An FLS consists of four parts: 1) the knowledge base; 2) the
fuzzifier; 3) the fuzzy inference engine working on fuzzy rules;
and 4) the defuzzifier. The knowledge base for FLS comprises
a collection of fuzzy if-then rules of the following form:
R
l
:ifx
1
is A
l
1
and x
2
is A
l
2
··· and x
n
is A
l
n
,
then y is B
l
, l = 1, 2,...,M
where x
=
[
x
1
,...,x
n
]
T
⊂ R
n
and y are the FLS input and
output, respectively. Fuzzy sets A
l
i
and B
l
are associated with
the fuzzy functions μ
A
l
i
(x
i
) = exp(−(((x
i
− a
l
i
)/b
l
i
))
2
) and
μ
B
l
(y
l
) = 1, respectively. M is the rules number. Through
singleton function, center average defuzzification and product
inference, the FLS can be expressed as
y(x) =
M
l=1
¯y
l
n
i=1
μ
A
l
i
(
x
i
)
M
l=1
n
i=1
μ
A
l
i
(
x
i
)
where ¯y
l
= max
y∈R
μ
B
l
. Define the fuzzy basis functions as
ξ
l
(x) =
n
i=1
μ
A
l
i
(
x
i
)
M
l=1
n
i=1
μ
A
l
i
(
x
i
)
and define θ
T
= [¯y
1
, ¯y
2
,...,¯y
M
] = [θ
1
,θ
2
,...,θ
M
] and
ξ(x) = [ξ
1
(x),...,ξ
M
(x)]
T
, then the above FLS can be
rewritten as: y(x) = θ
T
ξ(x).
Lemma4[20]: Let f (x) be a continuous function defined
on a compact set . Then for any constant ε>0, there exists
an FLS such as sup
x∈
|f (x) − θ
T
ξ(x)|≤ε.
By Lemma 4, we know, an FLS can approximate any
smooth function on a compact space. Due to this approxi-
mation capability, we can assume that the nonlinear term f(x)
can be approximated as: f (x,θ) = θ
T
ξ(x). Define the optimal
parameter vector θ
∗
as: θ
∗
= arg min
θ∈
[sup
x∈U
|f (x) − f (x,θ
∗
)|],
where and U are compact regions for θ and x, respectively.
Also the FLS minimum approximation error is defined as: ε =
f (x)−θ
∗T
ξ(x). In this paper, we use the above FLS to approx-
imate the unknown functions f
k
(¯x
k
), k = 1,...,N, namely,
there exist optimal vector θ
∗
k
and minimum approximation
error ε
k
such that
f
k
(¯x
k
) = θ
∗T
k
ξ
k
(
¯x
k
)
+ ε
k
.
Define the actual approximate error as
δ
k
= f
k
(¯x
k
) −
ˆ
θ
T
k
ξ
k
ˆ
¯x
k
where
ˆ
¯x
k
= [ˆx
k,1
,...,ˆx
k,n
]
T
is the estimate of ¯x
k
, and
ˆ
θ
k
is
the estimate of θ
∗
k
.
From Lemma 4, the following assumption is made.
剩余11页未读，继续阅读
weixin_38673237
- 粉丝: 2
- 资源: 843
我的内容管理
展开
最新资源
- WebLogic集群配置与管理实战指南
- AIX5.3上安装Weblogic 9.2详细步骤
- 面向对象编程模拟试题详解与解析
- Flex+FMS2.0中文教程：开发流媒体应用的实践指南
- PID调节深入解析：从入门到精通
- 数字水印技术：保护版权的新防线
- 8位数码管显示24小时制数字电子钟程序设计
- Mhdd免费版详细使用教程：硬盘检测与坏道屏蔽
- 操作系统期末复习指南：进程、线程与系统调用详解
- Cognos8性能优化指南：软件参数与报表设计调优
- Cognos8开发入门：从Transformer到ReportStudio
- Cisco 6509交换机配置全面指南
- C#入门：XML基础教程与实例解析
- Matlab振动分析详解：从单自由度到6自由度模型
- Eclipse JDT中的ASTParser详解与核心类介绍
- Java程序员必备资源网站大全
资源上传下载、课程学习等过程中有任何疑问或建议，欢迎提出宝贵意见哦~我们会及时处理！
点击此处反馈
