基于SVR学习的非线性空间分布式系统模糊逻辑控制器
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"这篇文章是关于使用支持向量机回归（SVR）学习设计3维模糊逻辑控制器（3-D FLC）的方法，针对非线性空间分布动态系统的研究论文。"
在"IEEETRANSACTIONSONNEURALNETWORKSANDLEARNINGSYSTEMS"的2013年10月刊中，作者Xian-Xia Zhang, Ye Jiang, Han-Xiong Li和Shao-Yuan Li提出了一种基于SVR学习的3-D模糊逻辑控制器设计方法。该方法主要针对非线性空间分布动态系统，通过将数据驱动的策略应用于3-D模糊逻辑控制。
首先，他们整合了3-D FLC的空间信息表达和处理以及模糊语言表达和规则推理，形成一种称为空间模糊基函数（SFBFs）。然后，3-D FLC被表示为一个由三层网络结构组成的模型。关键创新在于，通过直接将SFBFs与SVR的空间核函数关联，建立了3-D FLC与SVR之间的等价关系。这意味着可以利用SVR的学习能力来设计3-D FLC。
为了便于实际应用，作者进一步构建了一个系统性的、基于SVR学习的3-D FLC设计框架。这个框架强调了控制器的普适性和适应性，这对于解决非线性空间分布动态系统的复杂控制问题至关重要。通过SVR学习，3-D FLC能够适应不断变化的环境和系统特性，提供更精确的控制输出。
此外，文章还可能探讨了SVR学习的优势，如其在处理非线性问题时的高效性和鲁棒性，以及如何通过优化算法来调整控制器参数，以达到最佳性能。 SVR的学习过程可能涉及到选择合适的核函数、确定惩罚参数C和核宽度γ，这些都会直接影响到控制器的性能和稳定性。
这项研究为非线性空间分布动态系统的控制提供了一个新的视角，即结合了模糊逻辑的灵活性和SVR的强大的学习能力，为实际工程问题的解决提供了理论基础和实用工具。这不仅有助于提高控制系统的性能，还有可能促进相关领域的技术进步，比如环境监测、能源管理、自动化生产等领域。
ZHANG et al.: SVR LEARNING-BASED SPATIOTEMPORAL FUZZY LOGIC CONTROLLER 1637
the form of 3-D fuzzy rules. As the research on the 3-D FLC
is just at the beginning stage, control rule extraction from a
spatiotemporal dataset is still a challenging and open problem
for spatially distributed dynamic systems.
Data-driven design methods for the traditional FLC were
developed in past three decades. Generally speaking, the data-
driven FLC design consists of two parts: 1) rule generation;
and 2) system optimization (including structure optimization
and parameter optimization) [15]. For instance, clustering
techniques [16] can be used to generate rules automatically;
fusing similar fuzzy sets [17] can be applied to reduce the rule
number and realize the structure optimization; gradient decent
approaches [13] can be adopted for fine-tuning the member-
ship functions and realizing the parameter optimization. In
addition, an FLC can be designed directly through support
vector machine (SVM) learning [18], [19], i.e., the optimal
rule generation and parameter design of an FLC come from
the learning results of a SVM. For instance, a λ-fuzzy rule-
based system was constructed from a trained SVM in [20] and
additive fuzzy systems can be directly derived from the SVM
in [21] and [22].
SVM, as a new learning algorithm with the superior
learning power and the good generalization ability, brings
forward a promising solution to the data-driven-based design
of the traditional fuzzy system. However, the current SVM
has no inherent ability to express and cope with spatial
information. In particular, the SVM may not work well for
a spatiotemporal input–output dataset from a spatiotemporal
system.
In this paper, we will expand the learning ability of the SVM
to spatiotemporal datasets and develop a new data-driven 3-D
FLC design method based on the SVM learning. As SVM can
be classified into SVM classification [23] and SVM regression
(also called support vector regression (SVR)] [24]–[26], we
herein focus on SVR with an ε-insensitive loss function,
i.e., ε-SVR. Initially, the spatial information expression and
processing as well as the fuzzy linguistic expression and rule
inference of a 3-D FLC are integrated into spatial fuzzy basis
functions (SFBFs), and then the 3-D FLC can be described
by a three-layer network structure. Through relating SFBFs
of the 3-D FLC directly to spatial kernel functions (SKFs)
of an SVR, an equivalence relationship of the 3-D FLC and
the SVR is established. With the SKFs, the learning ability
of the SVR can be enhanced for spatiotemporal datasets; the
3-D FLC can be designed with the help of the SVR learning.
For an easy implementation, a systematic SVR learning-based
3-D FLC design scheme is formulated in four steps as follows:
1) data collection; 2) spatial support-vector learning; 3) 3-D
fuzzy rule construction; and 4) 3-D fuzzy controller integra-
tion. In addition, the universal approximation capability of the
proposed 3-D FLC is presented.
This paper is organized as follows. In Section II, pre-
liminaries about 3-D FLC are addressed. In Section III, an
SVR learning-based 3-D FLC design methodology is pre-
sented, including the design theory and the systematic design
framework, and the universal approximation capability of
the proposed 3-D FLC is given. In Section IV, a catalytic
packed-bed reactor is presented as an example to illustrate the
proposed 3-D FLC and validate its effectiveness, and finally,
Section V summarizes the conclusion.
II. P
RELIMINARIES
A. 3-D FLC
A 3-D FLC is a controller based on a linguistic rule
base, which can be represented and computed using linguistic
words; on the other hand, it is a nonlinear mapping from
an input space to an output space, which can be expressed
in a precise mathematical formula. Here, we will present the
mathematical description of a 3-D FLC as a nonlinear mapping
and then describe its network structure based on SFBFs.
1) Nonlinear Mapping: The basic structure of a 3-D FLC is
composed of 3-D fuzzifier, 3-D rule inference, and defuzzifier.
Because of its unique 3-D nature, some detailed operations
of a 3-D FLC are different from a traditional one for spatial
information expression, processing, and compression. For their
detailed operations, we can refer to [9]. Once each component
of a 3-D FLC is set, the nonlinear mathematical description of
the 3-D FLC can be derived (see Appendix I for a brief deriva-
tion). Assume we employ 3-D singleton fuzzifier, Gaussian
type 3-D MF, 3-D fuzzy rules as shown in (A.1) of Appendix I,
product t-norm and weighted aggregation dimension reduction
[9] in the 3-D rule inference, singleton fuzzy sets for the
output variable, and linear defuzzifier [27], the 3-D FLC can
be mathematically expressed as follows:
u(x
z
) =
N
l=1
ζ
l
p
j=1
a
j
s
i=1
μ
¯
C
l
i
(x
i
(z
j
))
=
N
l=1
ζ
l
p
j=1
a
j
s
i=1
exp
−((x
i
(z
j
) − c
l
ij
)/σ
j
)
2
=
N
l=1
ζ
l
p
j=1
a
j
exp
−||x(z
j
) − c
l
(z
j
)||
2
/σ
2
j
(1)
where σ
j
is set as 10% of the length of the input interval of
x(z
j
) [28] denoted by
σ
j
= max
1≤i≤s
(x
max
i
(z
j
) − x
min
i
(z
j
))/10
(2)
where x
max
i
(z
j
) and x
min
i
(z
j
) are the maximum and the mini-
mum bound value of the ith spatial input variable, respectively,
at the jth sensing location.
Equation (1) shows that the 3-D FLC is a nonlinear mapping
from the input space x
z
∈ ⊂ R
p×s
to the output space
u(x
z
) ∈ U ⊂ R. It provides a way to understand and analyze
the 3-D FLC from the point of view of function approximation.
2) SFBFs and Three-Layer Network Structure: In (1), let
l
(x
z
) =
p
j=1
a
j
s
i=1
μ
¯
C
l
i
(x
i
(z
j
)) (3)
then (1) can be rewritten as follows:
u(x
z
) =
N
l=1
ζ
l
l
(x
z
).
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