滑模观测器驱动的非线性多主体系统分布式自适应模糊控制稳定性研究
39 浏览量
更新于2024-08-26
收藏 844KB PDF 举报
本文主要探讨了在有向图上具有固定拓扑结构的高阶不确定非线性多主体系统中的分布式自适应模糊控制问题。针对这类系统，研究的核心关注点在于每个跟随者仅能获取自身输出及其邻居的信息，如何在这种受限条件下设计控制器以实现有效的协同控制。
作者Shen、Shi和Yan Shi提出了一种新颖的方法，即为每个跟随者设计等效输出注入滑模观测器（Sliding Mode Observer, SMO），这些观测器旨在实时估计每个跟随者及其邻节点的状态。滑模技术在此被巧妙地运用，以处理非线性和不确定性，其优势在于能够保证系统状态的快速收敛和鲁棒性。
通过模糊逻辑系统（Fuzzy Logic Systems, FLS），未知函数被近似处理，这使得控制器能够在有限的适应能力下处理复杂动态环境。在设计过程中，采用了基于Lyapunov理论的分析框架，并利用Filippov变分原理，确保了整个闭合控制系统在处理非线性动态和不完全信息时的稳定性。
该论文的关键贡献在于提出了一个基于观测器的分布式自适应控制策略，它不仅能够使所有跟随者渐近同步于一个领导者，而且跟踪误差可以达到半全局统一最终有界，这对于多主体系统的协调和控制性能具有重要意义。最后，作者通过数值仿真验证了所提方法的有效性和稳健性，展示了在实际应用中的可行性。
这篇研究论文深入探讨了非线性多主体系统中的分布式控制问题，融合了滑模观测器、自适应模糊控制以及图论和Lyapunov稳定理论，为解决此类复杂系统中的协同控制难题提供了理论基础和实用策略。
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
点击了解资源详情
点击了解资源详情
点击了解资源详情
2023-02-23 上传
2021-04-06 上传
2022-12-22 上传
2021-08-10 上传
2022-11-03 上传
127 浏览量
weixin_38673237
- 粉丝: 2
- 资源: 843
我的内容管理
展开
最新资源
- 深入浅出：自定义 Grunt 任务的实践指南
- 网络物理突变工具的多点路径规划实现与分析
- multifeed: 实现多作者间的超核心共享与同步技术
- C++商品交易系统实习项目详细要求
- macOS系统Python模块whl包安装教程
- 掌握fullstackJS：构建React框架与快速开发应用
- React-Purify: 实现React组件纯净方法的工具介绍
- deck.js：构建现代HTML演示的JavaScript库
- nunn：现代C++17实现的机器学习库开源项目
- Python安装包 Acquisition-4.12-cp35-cp35m-win_amd64.whl.zip 使用说明
- Amaranthus-tuberculatus基因组分析脚本集
- Ubuntu 12.04下Realtek RTL8821AE驱动的向后移植指南
- 掌握Jest环境下的最新jsdom功能
- CAGI Toolkit：开源Asterisk PBX的AGI应用开发
- MyDropDemo: 体验QGraphicsView的拖放功能
- 远程FPGA平台上的Quartus II17.1 LCD色块闪烁现象解析
