We consider an extension of the probability space [Ω;A;P] by the dimension of fuzziness, i.e., by introducing a membership scale. This enables the consideration of imprecise observations as fuzzy realizations
X x x
x( )=(~, ,~n)⊆
~ω 1 … of each elementary event ω∈Ω.
For constructing probability measures on fuzzy sets, an appropriate metric is needed. We consider the well-known Hausdorff metric dH, on the class Kc(ℜ) of nonempty compact intervals
{ K K K K }
k k k
k K
K
dH k Kk K k K k K
− ′
− ′
=
− ′ − ′
′ =
′ ∈
′∈
∈ ′
∈ ′
sup sup , inf inf max
|
| inf sup
|,
| inf sup max ) ,
( (4)
A fuzzy random variable X~
is the fuzzy result of the uncertain mapping )
(
~:
ℜ
→ Ω Fc
X (5)
such that for each α∈[0,1] the α -level mapping )
( :Ω→Kc ℜ
Xα , with Xα(ω)=[inf(X(ω))α,sup(X(ω))α], ∀ω∈Ω (6) is a (compact convex) random set, (i.e., it is Borel-measurable w.r.t. the Borel
σ -field generated by the topology associated with dH).
The fuzzy probability distribution function ~( ) x
F of X~
is the set of probability distribution functions of all originals Xj of X~
with the membership values à(F(x)). The cuantification of fuzziness by fuzzy parameters leads to the description of the fuzzy probability distribution function F~(x) of X~
as a function of the fuzzy bunch parameter ~ . s
)
~, ( )
~(x F s x
F = (7)
For the purposes of numerical evaluation, α-discretization is advantageously applied.
{
}
] 1 , 0 [ , )) ( (
)], ( ), ( [ ) (
| )) ( ( );
( )
~, (
∈
∀
=
=
=
α α à
à
α
α α α
α α
x F
x F x F x F x F x F x s
F (8)
with Fα(x)=inf{F(s,x)|s∈sα}, Fα(x)=sup{F(s,x)|s∈sα}.
Figure 3. Fuzzy probability density function and fuzzy cumulative distribution function.
With the aid of α-discretization a fuzzy random function may be formulated as a set of α-level sets of ordinary random functions
) x ( F0 i
) x ( F0 i
( ) [ ] {
( ()) , [0,1] }
, ) ( ), ( ) (
| ) ( );
( )
~(
∈
∀
=
=
=
α α à
à
α
α α α
α α
t X
t X t X t X t X t X t
X (9)
3.2. Hybrid approaches to propagating randomness and fuzziness in risk assessment
The idea is to find the output of a model g(X Xn X X~m)
,
~ , , ,
, 1
1 … … that has
both random variables X1,…,Xn, given by probabilistic distributions, and fuzzy variables X X~m
,
~ ,
1 … , for the inputs. To estimate the output of this generalized model, most researchers attempt to eliminate or transform one type of uncertainty to another before performing a simulation (e.g. possibility to probability transformation). Guyonnet et al. (2003) first proposed a “hybrid approach” with both fuzzy and random types of uncertainty without transforming one type to another. They calculated the Inf and Sup values of the model g considering all the values that are located within the α−cuts of the input fuzzy sets and suggested that minimization and maximization algorithm can be used for finding Inf and Sup values of a general model. However, in their application, the model was a simple monotonic function, and the Inf and Sup values were identified directly without using minimization or maximization algorithms.
A more tractable way to propagating both randomness and fuzziness is based of a fuzzy generalization of the Monte Carlo (FMC) simulation framework, which integrates fuzzy arithmetic method with Monte Carlo simulation to find the output of a model with both fuzzy and probabilistic inputs.
0.4
0.45 0.5
0.55 0.6
0 0.2 0.4 0.6 0.8 1
0 0.5 1
C X
m u at ul e iv isd ibtr io ut n
Alpha
Figure 4. 3D view of fuzzy CDF resulting from the output of FMC by aggregating α-CDF bounds.
Since in FMC, fuzzy arithmetic (in α-cut form) is performing for each sample set, the output of FMC is represented as a number of fuzzy sets with random variation. This randomness results from random sampling of random input parameters. The fuzzy CDF is used for finding the fuzzy probability of not exceeding a given threshold and a fuzzy quantile corresponding to a given probability.
0.4 0.45 0.5 0.55 0.6
0 0.2 0.4 0.6 0.8 1
X
Cumulative distribution
Fuzzy Probability
t 00.4 0.45 0.5 0.55 0.6
0.2 0.4 0.6 0.8 1
X
Cumulative distribution
Fuzzy quantile
(a) (b)
Figure 5. (a) The fuzzy probability of not exceeding a specific threshold t∈X (b) A fuzzy quantile corresponding to a given probability.
References
1. C. Baudrit, D. Guyonnet and D. Dubois, “Postprocessing the Hybrid Method for Addressing Uncertainty in Risk Assessments.” Journal of Environmental Engineering 131(12), 1750-1754, (2005).
2. EPA’s guidance: “Risk Assessment Guidance for Superfund (RAGS) Volume III – Part A: Process for Conducting Probabilistic Risk Assessment”, 2001, www.epa.gov/oswer/riskassessment/rags3adt/index.htm 3. S. Ferson and W.T. Tucker, “Probability Bounds Analysis in Environmental
Risk Assessments” (Technical report), Applied Biomathematics, Setauket, New York (2003). Available at www.ramas.com/pbawhite.pdf.
4. S. Ferson, L, Ginzburg, V. Kreinovich, H.T.Nguyen and S.A. Starks,
“Uncertainty in risk analysis: towards a general second-order approach combining interval, probabilistic, and fuzzy techniques”, Proceedings of the 2002 IEEE Int. Conference on Fuzzy Systems, pp. 1342-1347. (2002) 5. D. Guyonnet, B. Bourgigne, D. Dubois, H. Fargier, B. Côme and J. Chilès,
“Hybrid approach for addressing uncertainty in risk assessments.” Journal of environmental engineering 129(1), 68-78 (2003).
6. B. Mửller and M. Beer, Fuzzy Randomness – Uncertainty Computational Mechanics, Springer, Berlin and New York, (2004).
7. P. Terán, “Probabilistic foundations for measurement modeling with fuzzy random variables”, Fuzzy Sets and Systems 158(9), 973-986, (2007).
STRUCTURAL OPTIMIZATION OF LINGUISTIC KNOWLEDGE BASE OF FUZZY CONTROLLERS
YURIY P. KONDRATENKO
Intelligent Information Systems Department, Petro Mohyla Black Sea State University 10, 68th Desantnykiv Str., Nikolaev, 54003, Ukraine, y_kondrat2002@yahoo.com
LEONID P. KLYMENKO
Ecology Department, Petro Mohyla Black Sea State University 10, 68th Desantnykiv Str., Nikolaev, 54003, Ukraine, rector@kma.mk.ua
EYAD YASIN MUSTAFA AL ZU’BI
College of Science and Arts, King Saud University, p/o Box 30 Shagra,11961, Kingdom of Saudi Arabia, eyad19762000@yahoo.com The paper considers the problem of developing effective methods and algorithms for optimization of fuzzy rules bases of Sugeno-type fuzzy controllers that can be applied to control of dynamic objects, including objects with non-stationary parameters. Proposed approach based on calculating the coefficient of influence of each of the rules on the formation of control signals for different types of input signals provides optimization of a linguistic rules database by using exclusion mechanism for rules with negligible influence.
1. Introduction
Fuzzy sets theory and fuzzy logic have been widely introduced into research and design practice recently. From the first study of fuzzy sets [16]
researches received the theoretical foundation introducing fuzzy sets for successful solving of various problems in uncertainty [2,3,4]. Especially it is important and effective for control of such objects with non-stationary functioning conditions as ships, underwater robots, manipulated systems with moving base and others. Special attention should be paid to structural-parameter optimization of fuzzy systems for their applications in engineering where fuzzy controllers are components of embedded computer systems [7,8]. A number of methods of fuzzy controllers synthesis [1,6,9] have been developed to date. They provide the desired quality control in fuzzy systems by optimizing parameters of
membership functions of linguistic terms. As an objective function in the implementation of these methods an integral quadratic criterion or standard deviation of the real transition from desirable are usually used. At the same time rules, that determine the control strategy, are based on expert assessments for all possible combinations of linguistic terms. The described above approach does not take into account the redundancy of complete database rules, which makes the structure of fuzzy controllers too complicated and does not use the ability of fuzzy logic systems for extrapolation. Thus, the development of algorithms to optimize the structure of fuzzy controllers is quite necessary [5,10,12,13,15], as it will simultaneously improve the performance of fuzzy control devices by decreasing the number of computer operations and reduce the complexity of the synthesis process of fuzzy controllers by decreasing the number of optimization parameters in the tasks of nonlinear programming.