=Paper=
{{Paper
|id=Vol-2853/paper22
|storemode=property
|title=Optimal Selection of Membership Functions Types for Fuzzy Control and Decision Making Systems
|pdfUrl=https://ceur-ws.org/Vol-2853/paper22.pdf
|volume=Vol-2853
|authors=Oleksiy Kozlov
|dblpUrl=https://dblp.org/rec/conf/intelitsis/Kozlov21
}}
==Optimal Selection of Membership Functions Types for Fuzzy Control and Decision Making Systems==
https://ceur-ws.org/Vol-2853/paper22.pdf
Optimal Selection of Membership Functions Types for Fuzzy
Control and Decision Making Systems
Oleksiy Kozlova
a
Petro Mohyla Black Sea National University, 10 68th Desantnykiv st., Mykolaiv, 54003, Ukraine
Abstract
This paper proposes the advanced approach for optimal selection of linguistic terms
membership functions (LTMF) types for fuzzy control and decision making systems based on
bioinspired optimization techniques. The developed approach allows to effectively optimize
membership functions for improvement the fuzzy system (FS) performance as well as for
simplification the further procedure of its parametric optimization of linguistic terms by
reducing the number of optimized parameters. In order to study and validate the efficiency of
the presented approach the optimization of the LTMF types for the fuzzy control system of
the clamping device for a mobile robot (MR) of vertical movement is carried out in this
work. The obtained simulation results confirm the high efficiency of the developed approach
for optimal selection of LTMF types, as well as the expediency of its application for
structural optimization of fuzzy systems and devices of various types, configurations and
purposes.
Keywords 1
Fuzzy system, membership functions, intelligent approach, bioinspired optimization
techniques, fuzzy controller, mobile robot.
1. Introduction
Fuzzy logic is a popular and powerful tool for automation of control and decision-making
processes, which has a great potential [1, 2]. Intelligent decision-making systems based on the theory
of fuzzy sets and fuzzy logic are widely used in the following areas: transport logistics, medical and
technical diagnostics, financial management, stock market forecasting, etc. [3-5]. Also, fuzzy
inference devices are successfully developed and applied as controllers, observers, adaptive devices,
identifiers, tactical control units and others in automation systems of complicated nonlinear and/or
nonstationary plants, such as robotic production lines and warehouses, power plants and chemical
reactors, mobile robots and drones, marine floating structures and ships, unmanned aerial and
underwater vehicles, etc. [6-8].
An essential feature of fuzzy systems and devices is a large number of adjustable parameters and
structure elements, which significantly affect their performance [9]. Namely, the following are the
subjects for adjustment: the number of linguistic terms of input and output variables, as well as the
types and parameters (vertices) of their membership functions, the antecedents and consequents of the
rule base (RB), the number of RB fuzzy rules, the input normalizing coefficients, the types of
aggregation, activation and accumulation procedures, as well as the defuzzification method [10, 11].
This allows to implement complex, flexible and most effective strategies of control and decision-
making, however, significantly complicates the design procedures of these systems and devices. So,
in a number of cases, FSs developed on the basis of expert knowledge and assessments do not provide
a significant improvement in indicators compared to similar conventional systems and, at the same
IntelITSIS’2021: 2nd International Workshop on Intelligent Information Technologies and Systems of Information Security, March 24–26,
2021, Khmelnytskyi, Ukraine
EMAIL: kozlov_ov@ukr.net (O. Kozlov)
ORCID: 0000-0003-2069-5578 (O. Kozlov)
© 2021 Copyright for this paper by its authors.
Use permitted under Creative Commons License Attribution 4.0 International (CC BY 4.0).
CEUR Workshop Proceedings (CEUR-WS.org)
time, have a more complex software and hardware implementation [12]. Therefore, in recent years,
one of the leading research directions of the modern theory of fuzzy systems is the development and
approbation of highly efficient approaches, methods and information technologies for their design,
which include certain procedures of structural-parametric optimization [13, 14].
2. Related works
At the moment, a fairly large number of works have been published on various aspects of the
structural-parametric optimization of different types FSs [15-17]. Particularly, designing methods and
information technologies based of parametric optimization of the LTMF and weight coefficients of
RB are given in [18, 19]. Techniques of FSs structural optimization including the RB reduction and
interpolation, as well as optimal choice of defuzzification procedures are presented in papers [20, 21].
In turn, the most recent studies show that bioinspired intelligent methods and information
technologies are very promising for performing FSs design and optimization and have a number of
advantages compared to conventional optimization techniques [22, 23]. Among them are: methods of
ant colony optimization [24], biogeography based optimization [25], particle swarm optimization
[26], genetic methods [27] and evolutionary strategies [28], methods of artificial immune systems
[29], etc. The given bioinspired methods can be also effectively applied for solving the challenge of
LTMF optimization, in particular optimal selection of their types, that will allow not only to improve
the efficiency of the FS itself, but also to simplify the procedure for its parametric optimization by
reducing the number of optimized parameters of terms.
Thus, the purpose of this paper is development and research of an advanced intelligent approach
for optimal selection of membership functions types for fuzzy systems based on bioinspired
optimization techniques.
3. Intelligent approach for optimal selection of LTMF types for fuzzy systems
It is advisable to implement optimization of the LTMF types for improvement the FS performance
(quality indicators) and to simplify the subsequent optimization procedures of linguistic terms
parameters. In turn, the proposed intelligent approach consists of the following stages.
Stage 1. Setting of operating ranges of input and output variables changing of the developed fuzzy
system. At this stage, for each i-th (i = 1, 2, …, n) input and j-th (j = 1, 2, …, m) output FS variables,
the operating ranges are set, within which these variables can change. For example, if the input
variables are fed to the FS inputs in relative units from their maximum value, then it is advisable to set
their operating ranges from –1 to 1, or from 0 to 1 [2].
Stage 2. Formation of a set of membership functions used in the optimization process. At this
stage, a set of alternative LTMFs SMF is created, at which the search for optimal membership
functions will be carried out for all linguistic terms of each i-th input and j-th output variables of the
developed FS. It is advisable to include in this set the most frequently used membership functions:
triangular TrFN, trapezoidal TrpFN, bell-shaped GbFN, Gaussian 1st Gs1FN and 2nd Gs2FN types, π-
shaped PiFN, S-shaped SFN, Z-shaped ZFN , sigmoid SgFN, double sigmoid difference function
DsgFN, and product of two sigmoid functions PsgFN [4, 30]. The number of adjustable parameters
kMF [10, 11] of these membership functions are shown in Table 1.
Table 1
Number of adjustable parameters kMF for LTMF
SMF TrFN TrpFN GbFN Gs1FN Gs2FN PiFN SFN ZFN SgFN DsgFN PsgFN
kMF 3 4 3 2 4 4 2 2 2 4 4
Stage 3. Selection of the number of linguistic terms for the FS input and output. At this stage, the
number of linguistic terms is selected τi (i = 1, 2, …, n) and τj (j = 1, 2, …, m) for each i-th input and
j-th output variables of the FS. In real FS, the number of linguistic terms for input variables should be
set in the range from 2 to 7 (τi = 2…7) [10, 16], and for output variables – form 3 to 9 (τj = 3…9) [18].
Stage 4. Setting of parameters initial values of the LTMF for the developed fuzzy system. At this
stage, for all LTMFs included in the set formed at Stage 2 (Table 1), the values of their parameters are
pre-set. In most cases, at the beginning of the optimization process, it is advisable to set the
parameters of the membership functions in such a way, that linguistic terms for all input and output
variables, depending on their number τi and τj, selected at Stage 3, are evenly distributed over their
working ranges previously set at Stage 1.
Stage 5. Formation of the vector S structure, which determines the set of LTMF for the developed
FS. The LTMF vector S for fuzzy systems of Mamdani [9] and Takagi-Sugeno [14] types can be
represented by equations (1) and (2), respectively:
= S {S ini (q ), S=j
out
=
(k )},q (1,2,..., =
τi ), k (1,2,..., =
τ j ), i (1,2,..., n ), j (1,2,..., m ),
S ini (q )∈ {TrFN ,TrpFN ,GbFN ,Gs 1FN ,Gs 2FN , PiFN , SFN , ZFN , SgFN , DsgFN , PsgFN }, (1)
j
S out (k )∈ {TrFN ,TrpFN ,GbFN ,Gs 1FN ,Gs 2FN , PiFN , SFN , ZFN , SgFN , DsgFN , PsgFN },
i
=S {S=
in
=
(q )},q (1,2,..., τi ), i (1,2,..., n ),
(2)
S ini (q )∈ {TrFN ,TrpFN ,GbFN ,Gs 1FN ,Gs 2FN , PiFN , SFN , ZFN , SgFN , DsgFN , PsgFN },
where Sini (q ) , Soutj
(k ) are variables that determine the types of membership functions of the q-th
linguistic term of the i-th input variable and the k-th term of the j-th output variable, respectively.
Moreover, vector S may have certain restrictions. For instance, S-shaped SFN and sigmoid SgFN
membership functions can be used only for rightmost linguistic terms (q = τi, k = τj), and Z-shaped
ZFN – only for leftmost terms (q = 1, k = 1) for all FS variables.
Stage 6. Selection of the initial hypothesis of the vector S, which determines the set of LTMF for
the developed FS. At this stage, in accordance with the number of terms selected at Stage 3 τi (i = 1,
2, …, n) and τj (j = 1, 2, …, m) the initial values of the components of the vector S are selected. These
initial values can be selected on the basis of an expert approach or randomly. For example, if at the
beginning of the optimization process in a MISO Mamdani-type FS (with parameters n = 2, τi = {7,
5}, m = 1, τj = {5}), Gaussian membership functions of the 2st type Gs2FN are selected for all
linguistic terms of input variables, and triangular functions TrFN are selected for all terms of output
variable, then the initial value of the vector S0 is determined by expression (3)
i j
= S0 {S= in
(q ), S out (k )} {S in1 (1), S in1 (2), S in1 (3), S in1 (4), S in1 (5), S in1 (6), S in1 (7),
S in2 (1), S in2 (2), S in2 (3), S in2 (4), S in2 (5), S out
1
(1), S out
1
(2), S out
1
(3), S out
1
(4), S out
1
(5)} =
(3)
= {Gs 2FN ,Gs 2FN ,Gs 2FN ,Gs 2FN ,Gs 2FN ,Gs 2FN ,Gs 2FN ,
Gs 2FN ,Gs 2FN ,Gs 2FN ,Gs 2FN ,Gs 2FN ,TrFN ,TrFN ,TrFN ,TrFN ,TrFN }.
Stage 7. Formation of a complex objective function JC for evaluating the effectiveness of the
developed fuzzy system. At this stage, the type, parameters and optimal value of the complex objective
function JC, used to find the optimal types of membership functions, are determined. Since, the total
number of linguistic terms parameters, that will be optimized at subsequent stages of FS design,
depends on the LTMF types, then in the process of vector S optimization it is advisable to use
criterion J1, which evaluates the FS performance itself, and criterion J2, which takes into account the
degree of complexity of further parametric optimization of the system being developed. Therefore, the
current value of the complex objective function JС should be calculated based on the equation (4)
J=
C
J1 + k J 2J 2 , (4)
where kJ2 is the scaling factor for J2, which determines its importance in the process of computational
search.
For example, when designing a MISO fuzzy automatic control system, the criterion J1 can be
presented as the generalized integral deviation of the real transient response YR(t, S) from the desired
transient response of the reference model YD(t) [9, 31, 32]:
t max
1
J 1 (t =
,S) (E )2 + k 11 (EY )2 + k 12 (EY )2 dt ,
t max ∫0 Y (5)
where EY is the deviation of YR(t, S) from YD(t), EY = YD(t) – YR(t, S); tmax is the system’s total transient
time; k11, k12 are the corresponding weights for the components ( EY ) 2 and ( EY ) 2 .
In turn, the values of the criterion J2 can be calculated depending on the number of optimized
parameters of linguistic terms for FS of Mamdani and Takagi-Sugeno types based on the following
dependencies:
n τi m τj
=J2 ∑∑ k ini (q ) + ∑∑ k outj (k );
=i 1=
q 1 =j 1=
k 1
(6)
n τi
J 2 = ∑∑ k ini (q ), (7)
=i 1=
q 1
where kini (q ) and kout
j
(k ) are the numbers of optimized parameters of the q-th linguistic term for the
i-th input variable and the k-th linguistic term for the j-th output variable, depending on their
membership functions types (Table 1).
Stage 8. Conduction of an iterative global search for the optimal vector Sopt of the membership
functions types using bioinspired intelligent algorithms. At this stage, the search for the global
extremum of the objective function J C → min is carried out using one or several bioinspired
algorithms of global optimization. In turn, the following algorithms can be used, which are well
adapted to complex high-dimensional discrete optimization problems: genetic algorithms, ant colony
optimization, biogeography based optimization, evolutionary strategies, algorithms of artificial
immune systems. In this case, vector S is a vector of unknown parameters, the complex objective
function JС is a fitness function, iterative procedures are carried out based on the features of a
specifically selected algorithm.
Stage 9. Analysis of the results obtained and selection of the best variant of the membership
functions types vector. At this stage the obtained optimization results are analyzed and the best variant
of the vector of LTMF types is then selected. After that, the FS parametric optimization and its
software and hardware implementation can be carried out for further application [33, 34].
The efficiency research of the proposed intelligent approach is performed in this paper at
optimization of the LTMF types for a fuzzy control system of the clamping device for a mobile robot
of vertical movement [35, 36].
4. Optimal selection of LTMF types for fuzzy control system of the mobile
robot clamping device
The considered mobile robot moves by means of wheeled or caterpillars propulsors and uses an
electromagnetic clamping device to hold on vertical and inclined ferromagnetic surfaces when
implementing different types technological operations (painting, welding, rust removal, inspection)
[35, 37, 38]. The clamping device must provide a certain value of the clamping force F for safe and
effective movement of the MR on various surfaces under the action of various external disturbances.
The simplified mathematical model of the MR clamping device consists of the following
equations [36]:
Φ 2δ
F= ; (8)
2μ 0s δ
μs I W
Φδ = 0 δ m m ; (9)
δ
dI (10)
Lm m + R mI m =
um ;
dt
du m (11)
TC + um = k Cu F ,
dt
where Φδ is the magnetic flux of the clamping electromagnet; μ0 is the magnetic permeability of the
gap; sδ is the gap cross-sectional area; Im is the current of the clamping electromagnet; Wm is the
number of turns of the electromagnet winding; δ is the gap length; Lm and Rm are the inductance and
resistance of the electromagnet winding; um is the supply voltage of the electromagnet; TC and kC are
the time constant and gain of the power converter; uF is the clamping device control signal.
For automatic control of the MR clamping force it is advisable to use the sliding mode fuzzy
controller that gives the opportunity to effectively automate complex nonlinear plants [39-41]. Since,
the clamping device is a 2nd order nonlinear plant, the sliding surface sF for its control has the form
[39]
s= F
eF + k s e F , (12)
where eF is the clamping force control error; ks is the reference model coefficient.
The conventional sliding mode controller calculate the control signal uF based on the sliding
surface value sF according to the expression (13) [39]
+u Fmax , at s F > 0;
= u F = 0, at s F 0;
(13)
−u
Fmax , at s F < 0,
where uFmax is the maximum value of the clamping device control signal.
In turn, the fuzzy sliding mode controller provides the implementation of the given above control
law using fuzzy inference on the basis of the dependence uFC = fFC(eF; eF ). This allows to obtain high
quality control indicators, as well as to eliminate the chattering problem that is inherent in
conventional sliding mode controllers [39, 40]. In this paper LTMF types optimization is carried out
for the given fuzzy sliding mode controller in accordance with the main stages of the proposed
advanced intelligent approach.
At the stage 1, the operating ranges for the fuzzy controller inputs eF and eF as well as for output
uFC are set from –1 to 1. Then, the set of alternative LTMFs SMF is created at the stage 2, which
include all membership functions presented in Table 1. At the third stage for the SMFC inputs eF and
eF five linguistic terms are selected for each: BN – big negative; SN – small negative; Z – zero; SP –
small positive; BP – big positive. As for output uFC it has only 3 terms to implement the sliding mode
control strategy: N – negative; Z – zero; P – positive. For all these linguistic terms at stage 4, the
initial parameters (vertices) are set in such a way, that terms are evenly distributed over their working
ranges. Further, at the fifth stage the vector S structure is formed as follows
i j
= S {S= in
(q ), S out (k )} {S in1 (1), S in1 (2), S in1 (3), S in1 (4), S in1 (5),
(14)
S in2 (1), S in2 (2), S in2 (3), S in2 (4), S in2 (5), S out
1
(1), S out
1
(2), S out
1
(3)}.
As the initial hypothesis of the vector S triangular membership functions are selected for all
linguistic terms at the stage 6:
i j
= S {S= (q ), S out (k )} {TrFN ,TrFN ,TrFN ,TrFN ,TrFN ,
in
(15)
TrFN ,TrFN ,TrFN ,TrFN ,TrFN ,TrFN ,TrFN ,TrFN }.
In turn, the rule base for the developed SMFC is composed to implement the sliding mode control
principle and presented in Table 2.
Table 2
SMFC rule base
eF / eF BN SN Z SP BP
BN N N N N Z
SN N N N N Z
Z N N Z P P
SP Z P P P P
BP Z P P P P
At the stage 7 the complex objective function JC is formed that is calculated according to equation
(4). In turn, the criterions J1 and J2 are represented by the expressions (5) and (6) respectively. As the
optimal values of the functions JC, J1 and J2 the following values are selected: JCopt = 7·105; J1opt =
3,5·105; J2opt = 35. The scaling factor kJ2, in this case, is equal to 104. Before implementation of the
LTMF types optimization process, the objective functions JC, J1 and J2 for the SMFC with RB
presented in Table 2 and triangular membership functions (according to initial hypothesis of the
vector S) had the following values: JС = 7,62·105; J1 = 3,72·105; J2 = 39.
At the stage 8 the iterative global search for the optimal vector Sopt of the LTMF types is
conducted by means of the following bioinspired intelligent algorithms (adapted to the specifics of
this task): GA, BBO and AIS. At carrying out the procedure of searching for optimal membership
functions using bioinspired algorithms, the main parameters of GA, BBO and AIS were selected
experimentally for this specific problem. In particular, the initial population of 100 chromosomes was
created for the genetic algorithm. Proportional selection was chosen as the selection operator, single-
point crossover as the crossover operator, and simple mutation as the mutation operator [23, 27].
Moreover, the values of the probabilities of crossover PC and mutation PM are set: PC = 0,25; PM =
0,1. For the BBO algorithm the initial ecosystem of 100 habitats (islands) was created. The
dependences of species migration on the number of species on the islands λ(N) and ν(N) are linear,
wherein λmax = νmax = 1 [25]. Mutation operator coefficient r = 0,1, the maximum possible number of
species on the island corresponding to the optimal value of the habitat suitability index fopt, Nmax = 10.
In turn, the initial population of 100 immune cells was created for the artificial immune systems
algorithm. The uniform clone operator is used as the clone operator [29]. Also, the number of memory
cells Nm = 10, the number of cells in the population with the worst affinity Nw = 50, the clone operator
parameter Nc = 5, the mutation operator parameter r = 2. For all used bioinspired algorithms, 100
iterations were performed. The Fig. 1 shows the curves of changes in the best values of the complex
objective function JС (convergence curves) in the process of searching for the optimal types of
membership functions S at stage 8 based on the considered algorithms (GA, BBO, AIS).
Figure 1: Convergence curves in the process of searching for the optimal LTMF types
At the final 9th stage the obtained results of the LTMF types optimization by means of the given
above bioinspired algorithms are analyzed. In turn, the best value of the objective function JС is
attained by means of the BBO algorithm (JСmin = 6,463·105) at the 72nd iteration. The found vector S
that is correspondent to the given best value of the objective function JСmin is considered to be optimal
optimal Sopt for this system and has the following form
Sopt = {ZFN ,TrFN ,Gs 1FN ,TrFN , SFN ,
(16)
Gs 1FN ,TrFN ,TrFN ,Gs 1FN , SgFN ,TrFN ,Gs 1FN ,TrFN }.
Moreover, for this variant, separately taken components of the complex objective function J1 and J2
also have the best values (J1min = 3,263·105, J2min = 32). Also, for this particular problem, the BBO
algorithm showed the best convergence, since the optimal value of the complex objective function
JСopt was achieved with the lowest computational costs (for the least number of iterations – 55). In
turn, genetic and AIS algorithms also showed satisfactory results, since they also made it possible to
achieve the optimal value of the complex objective function JС for an acceptable number of iterations
(63 and 69), which in general confirms the expediency of their application in this approach.
To evaluate the results of optimization of the LTMF types by means of the proposed IIT, transient
graphs for the control system of the MR clamping force are presented in Fig. 2 for two cases: 1 – for
SMFC with triangular types LTMF (according to initial hypothesis S0); 2 – for SMFC with optimal
types LTMF (according optimal vector Sopt). In turn, for both cases, the simulation was carried out at
a set value of clamping force F = 5000 N.
As can be seen from Fig. 2, the clamping force control system with optimized LTMF types of the
SMFC has higher quality indicators in comparison with the same system, that has triangular LTMF of
its SMFC. In particular, it has a lower static error value (ΔF = 0,52% for SMFC with optimal LTMF
types; ΔF = 2,6% for SMFC with triangular LTMF types) with the same rise and transient time.
Figure 2: Transients for the control system of the MR clamping force
In addition to improved quality indicators of the control system, SMFC linguistic terms with
optimal membership functions have fewer parameters that need to be optimized at further parametric
optimization. Namely, the total number of parameters of linguistic terms was reduced by 7
parameters, which will significantly reduce the computational costs for their further parametric
optimization. Also, to find the optimal LTMF vector Sopt for the given fuzzy controller using proposed
intelligent approach did not require significant computational and time costs, which confirms its high
efficiency.
5. Conclusions
The development and research of the advanced intelligent approach for optimal selection of
membership functions types for fuzzy systems based on bioinspired optimization techniques is
presented in this paper. The proposed IIT allows to effectively optimize LTMF types for improvement
the FS performance as well as for simplification the further procedure of its parametric optimization
by reducing the number of optimized parameters.
The efficiency research of the proposed approach is performed in this paper at optimization of the
LTMF types for the sliding mode fuzzy controller of the clamping device control system for a mobile
robot of vertical movement. In particular, at the implementation of this IIT, the iterative global search
for the optimal vector Sopt of the LTMF types for SMFC is conducted by means of several bioinspired
algorithms (GA, BBO and AIS) with further comparison of the obtained results. In this case, the BBO
algorithm showed the best results, as it provided the fastest convergence (JС ≤ JСopt at the 55th
iteration) and the highest SMFC performance (JСmin = 6,463·105 at the 72nd iteration). In turn, genetic
and AIS algorithms also showed satisfactory results, since they also made it possible to attain the
optimal value of the complex objective function JСopt using acceptable computational costs (63 and 69
iterations), which in general confirms the expediency of their application in this intelligent approach.
Moreover, the clamping force control system with optimized LTMF types (obtained by IIT based on
BBO algorithm) of the SMFC has higher quality indicators in comparison with the same system, that
has triangular LTMF of its SMFC. In addition, the SMFC linguistic terms with optimal membership
functions obtained by the presented IIT have fewer parameters that need to be optimized at further
parametric optimization procedures.
Thus, the results obtained in this work confirm the high efficiency of the developed IIT for optimal
selection of LTMF types, as well as the expediency of its application for structural optimization of
fuzzy systems and devices of various types, configurations and purposes.
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