SEKE 2012 Proceedings - Knowledge Systems Institute

Modeling and Analysis of Switched Fuzzy Systems
Zuohua Ding, Jiaying Ma
Lab of Scientific Computing and Software Engineering
Zhejiang Sci-Tech University
Hangzhou, Zhejiang 310018, P.R. China
Abstract
Switched fuzzy systems can be used to describe the hybrid
systems with fuzziness. However, the languages to describe
the switching logic and the fuzzy subsystems are in
general different, and this difference makes the system analysis
hard. In this paper we use Differential Petri Net (DPN)
as a unified model to represent both the discrete logic and
fuzzy dynamic processes. We then apply model checking
technique to DPN to check the correctness of the requirements.
1 Introduction
Switched systems have been widely used in a variety
of industrial processes. If a system is too complex or illdefined,
i.e. the system contains fuzzyness, switched fuzzy
system has been adopted to model such systems[1].
Compared with conventional switched system modeling,
switched fuzzy system modeling is essentially a multimodel
approach in which simple sub-models (typically linear
models) are fuzzily combined to describe the global behavior
of a nonlinear system. A typical sub-model is Takagi
and Sugeno (T-S) fuzzy model, which consists of a set of
If-Then rules and a set of ordinary differential equations[5].
This model raises some issues in the software engineering.
For example, in the T-S fuzzy model, the languages to
describe the switching logic and the fuzzy subsystems are in
general different, and this difference makes the system analysis
and implementation hard. So far, the efforts to address
this issue are only for the deterministic switched systems.
It is our attempt to solve this issue for switched fuzzy
systems. We use Differential Petri net defined by Demongodin
and Koussoulas[2] as a unified behavior model to represent
switched fuzzy systems. We then use model checking
tool HyTech to check the DPN, and use an enhanced
version of the tool Visual Object Net++ to simulate DPN.
We employ the the Differential-Drive Two-Wheeled Mobile
Robots as the running example to illustrate our method.
2 Switched Fuzzy Systems
A Switched Fuzzy System (SFS) consists of a family
of T-S fuzzy models and a set of rules that orchestrates
the switching among them. Assume that the Switched
Fuzzy System has m subsystems, each is described by a
Takagi-Sugeno fuzzy model[5], and σ : R + → M =
{1, 2,...,m} is a piecewise constant function that represents
the switching signal.
2.1 Takagi-Sugeno Fuzzy Model
Let N σ(t) be the number of inference rules. Then the T-S
fuzzy model is described as follows:
[Local Plant Rule]
Rσ(t) l : if ξ 1 is Mσ(t)1 l ∧···∧ξ p is Mσ(t)p l ,
then x ′ (t) =A σ(t)l x(t)+B σ(t)l u σ(t) (t),
l =1, 2,...,N σ(t) .
In this model, R l σ(t)
denotes the l th inference
rule, u σ(t) is the input variable, vector x(t) =
[x 1 (t),x 2 (t),...,x n (t)] T ∈ R n represents the state
variables, vector ξ =[ξ 1 ,ξ 2 ,...,ξ p ] represents the vector
of rule antecedents (premises) variables, and matrices
A σ(t)l ∈ R n×n and B σ(t)l ∈ R n×p .
Hence, the i th subsystem can be represented as follows:
⎧
⎪⎨
subsystem i :
⎪⎩
R 1 i :ifξ 1 is M 1 i1 ∧···∧ξ p is M 1 ip
then x ′ (t) =A i1 x(t)+B i1 u i (t)
R 2 i :ifξ 1 is M 2 i1 ∧···∧ξ p is M 2 ip
then x ′ (t) =A i2 x(t)+B i2 u i (t)
.
R N i
i
:ifξ 1 is M N i
i1 ∧···∧ξ p is M N i
ip
then x ′ (t) =A iNi x(t)+B iNi u i (t)
By using the center of gravity method for defuzzification,
the global model of the i th fuzzy subsystem can be de-
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