INT J COMPUT COMMUN, ISSN 1841-9836
8(3):395-406, June, 2013.

An Approach to Fuzzy Modeling of Electromagnetic Actuated
Clutch Systems

C.-A. Dragoş, R.-E. Precup, M.L. Tomescu, S. Preitl, E.M. Petriu, M.-B. Rădac

Claudia-Adina Dragoş, Radu-Emil Precup,
Stefan Preitl, Mircea-Bogdan Rădac
"Politehnica" University of Timişoara
Department of Automation and Applied Informatics
Bd. V. Parvan 2, 300223 Timisoara, Romania
claudia.dragos@aut.upt.ro, radu.precup@aut.upt.ro,
stefan.preitl@aut.upt.ro, mircea.radac@aut.upt.ro

Marius L. Tomescu
Aurel Vlaicu University of Arad
Romania, 310330 Arad, Elena Dragoi, 2
tom_uav@yahoo.com

Emil M. Petriu
University of Ottawa
School of Electrical Engineering and Computer Science
800 King Edward, Ottawa, ON, K1N 6N5 Canada
petriu@eecs.uottawa.ca

Abstract: This paper proposes an approach to fuzzy modeling of a nonlinear servo
system application represented by an electromagnetic actuated clutch system. The
nonlinear model of the process is simplified and linearized around several operating
points of the input-output static map of the process. Discrete-time Takagi-Sugeno
(T-S) fuzzy models of the processes are derived on the basis of the modal equivalence
principle; the rule consequents of these T-S fuzzy models contain the state-space
models of the process. Three discrete-time T-S fuzzy models are suggested, compared
and validated by simulation results.
Keywords: Discrete-time Takagi-Sugeno fuzzy models, electromagnetic actuated
clutch system, linearization, operating points, simulation results.

1 Introduction

The process taken into consideration and modeled in this paper is an electromagnetic actu-
ated clutch system as a representative nonlinear system application. Therefore the derivation
of accurate models is a challenging problem. Several approaches to fuzzy modeling of nonlinear
servo systems are given in the literature. They belong to the general framework of nonlinear
process models [1], [2], [3], [4], [5]. A parallel distributed compensation scheme is proposed in [6]
with focus on fuzzy reference models; the linear matrix inequalities are formulated and solved in
order to linearize the errors between the feedback system and the nonlinear reference model. The
nonlinear system behavior is modeled in [7] by the division of the phase plane into sub-regions and
a linear model represented either in state-space or ARX model form is assigned for each regions;
the linear models are next expressed as fuzzy models. A DSP-based fuzzy-linear-model robust
tracking control is developed in [8] for a piezoelectric servo system with dominant hysteresis in
terms of the weighted combination of N fuzzy linear pulse transfer functions; the fuzzy model is
included in a dead-beat control system. An ANFIS-based neuro-fuzzy model for a low inertia
servomotor is suggested in [9], and several comparisons between the performance of the system
with the standard motor model and its neuro-fuzzy model are carried out in the framework of

Copyright c⃝ 2006-2013 by CCC Publications



396 C.-A. Dragoş, R.-E. Precup, M.L. Tomescu, S. Preitl, E.M. Petriu, M.-B. Rădac

adaptive control. Fuzzy feedback linearization and fuzzy sliding mode control applications are
given in [10] and [11].
This paper offers discrete-time dynamic Takagi-Sugeno (T-S) fuzzy model of an electromagnetic
actuated clutch system. The computation of the T-S fuzzy models starts with the derivation
of the continuous-time models which are obtained on the basis of the local linearization of the
process models at five operating points (o.p.s). The local models are next discretized accepting
a zero-order hold, and these local models are placed in the rule consequents of the T-S fuzzy
model of the process.

Our approach is advantageous because it is relatively simple and it can be incorporated in
many fuzzy control structures [12], [13], [14], [15], [16], [17], [18], [19]. Three fuzzy models are
offered and compared using simulation results.
The paper is organized as follows: Section 2 is dedicated to the mathematical modeling of the
process, the computation of T-S fuzzy models is synthesized in Section 3. Simulation results
are presented in Section 4 to validate the new T-S fuzzy models. The concluding remarks are
highlighted in Section 5.

2 Process Modeling

The mathematical modeling of the electromagnetic actuator as part of electric drive clutches
is based on the schematic structure of a magnetically actuated mass-spring-damper system pre-
sented in Figure 1 [20]. The state-space model of the nonlinear servo system is:

ẋ1 = x2,

ẋ2 = − kmx1 −
c
m
x2 +

kax
2
3

m(kb+d−x1)2
,

ẋ3 = −
R(kb+d−x1)

2ka
x2x3 +

1
kb+d−x1

x2x3 + [(kb + d − x1)/2ka]V,
y = 1000x1,

(1)

where x1 is the position, i.e., the mass position, x2 is the mass speed, x3 is the current, V is
the control signal, y is the measured position (output), k is the stiffness of the spring, c is the
coefficient of the damper, R is the electromagnetic coil resistance, and ka, kb are the constants
in the relation between the magnetic flux and the current. The numerical values of the process
parameters are listed in [21].

Figure 1: Schematic structure of a magnetically actuated mass-spring-damper system [20].

The linearization of the nonlinear servo system model (1) at five o.p.s Aj(x10, x20, x30, x40)
(with j-the index of the o.p. j = 1, 5, and 0-the index of the coordinates of the o.p.s, i.e., the
state variables) leads to the linearized state-space models:



An Approach to Fuzzy Modeling of Electromagnetic Actuated Clutch Systems 397

ẋ(t) = Ax(t) + b∆V (t),

∆y(t) = cT x(t),

x = [ x1 = x x2 = ẋ x3 = i]
T ,

A =




0 1 0

− k
m

+
2kax

2
30

m(kb+d−x10)
3 − cm

2kax30
m(kb+d−x10)

2

Rx30−V0
2ka

− x20x30
(kb+d−x10)

2 − x30kb+d−x10 −
x20

kb+d−x10
− R(kb+d−x10)

2ka


 ,

b =


 00

kb+d−x10
2ka


 , cT = [ 1000 0 0 ].

(2)

where x(t) is the system state vector, A, b and cT are the linearized system matrices, and
t is the continuous time variable. The matrices of the discrete-time systems developed from (2)
will be presented in the sequel.

3 Approach to Takagi-Sugeno Fuzzy Modeling

In order to capture both the static nonlinearity and the linear dynamics of the process, the
derivation of a discrete-time dynamic T-S fuzzy model of the process is presented as follows.
Figure 2 illustrates the structure of the T-S fuzzy model identification process.

Figure 2: Structure of the discrete-time dynamic Takagi-Sugeno fuzzy model identification pro-
cess.

The steps of our modeling approach are:

Step I. The definition of the membership functions of the input variables x1 and x3.

Step II. The choice of the settling time and the discretization of the continuous-time state-
space models of the process which result in the discrete-time state-space models with the
matrices Ad,i, Bd,i and Cd,i and Cd,i, i = 1, 5.

Step III. The derivation of the T-S fuzzy model of the process, which has the state variables
x1 and x3 as input variables, and the discrete-time state-space models of the process in the
rule consequents.

The step I starts with the setting of the largest domains of variation of the two state variables
used in all electromagnetic actuated clutch system operating regimes:

0 6 x1 6 0.004, 0 6 x3 6 10. (3)



398 C.-A. Dragoş, R.-E. Precup, M.L. Tomescu, S. Preitl, E.M. Petriu, M.-B. Rădac

The fuzzification part of the T-S fuzzy model consists of the linguistic terms assigned to the
input variables and defined as follows.

Three cases were considered for the input variable x1 . The first two cases employ five lin-
guistic terms, LTx1,j, j = 1, 5 , with trapezoidal membership functions defined and referred
to as LTx1,1, with the universe of discourse [0.0019, 0.0023], LTx1,2, with the universe of dis-
course [0.0021, 0.0027] , LTx1,3, with the universe of discourse [0.0023, 0.003], LTx1,4, with the
universe of discourse [0.0027, 0.0033] and LTx1,5, with the universe of discourse [0.003, 0.004] .
The expressions of these trapezoidal membership functions are:

µTLx3,j (x) =




0, x < ax1,j

1 +
x−bx1,j

bx1,j−ax1,j
, x ∈ [ax1,j, bx1,j)
1, x ∈ [bx1,j, cx1,j)

1 − x−cx1,j
dx1,j−cx1,j

, x ∈ [cx1,j, dx1,j)
0, x > dx1,j

, ax1,j < bx1,j 6 cx1,j < dx1,j, j = 1.5 (4)

The modal values of the membership functions are the parameters ax1,j, j = 1, 5, bx1,j, j =
1, 5, cx1,j, j = 1, 5 and dx1,j, j = 1, 5. The values of these parameters are given in Table 1 for
the first case and in Table 2 for the second case.

Table 1
Parameters of input membership functions in the first case

Linguistic terms, Trapezoidal membership functions
LTx1,j, j = {1, 5} ax1,j, j = 1, 5 bx1,j, j = 1, 5 cx1,j, j = 1, 5 dx1,j, j = 1, 5

LTx1,1 0.0019 0.0019 0.0021 0.0023
LTx1,2 0.0019 0.0021 0.0023 0.0027
LTx1,3 0.0021 0.0023 0.0027 0.003
LTx1,4 0.0023 0.0027 0.003 0.00384
LTx1,5 0.003 0.00384 0.004 0.004

Table 2
Parameters of input membership functions in the second case

Linguistic terms, Trapezoidal membership functions
LTx1,j, j = {1, 5} ax1,j, j = 1, 5 bx1,j, j = 1, 5 cx1,j, j = 1, 5 dx1,j, j = 1, 5

LTx1,1 0.0019 0.0019 0.0021 0.0023
LTx1,2 0.0021 0.0023 0.0025 0.0027
LTx1,3 0.0025 0.0027 0.003 0.0033
LTx1,4 0.003 0.0033 0.0035 0.00384
LTx1,5 0.0035 0.00384 0.004 0.004

Five linguistic terms, LTx1,j, j = 1, 5, with trapezoidal and triangular membership functions
are defined and employed in the third case, and referred to as LTx1,1, with the universe of
discourse [0.0019, 0.0023], LTx1,2, with the universe of discourse [0.0021, 0.0027], LTx1,3, with
the universe of discourse [0.0023, 0.003], LTx1,4, with the universe of discourse [0.0027, 0.0033],
and LTx1,5, with the universe of discourse [0.003, 0.004]. The modal values of the trapezoidal
membership functions are the parameters ax1,j, j ∈ {1, 5}, bx1,j, j ∈ {1, 5}, cx1,j, j ∈ {1, 5} and
dx1,j, j ∈ {1, 5} given in Table 3.



An Approach to Fuzzy Modeling of Electromagnetic Actuated Clutch Systems 399

Table 3
Parameters of trapezoidal input membership functions in the third case

Linguistic terms, Trapezoidal membership functions
LTx1,j, j = {1, 5} ax1,j, j ∈ {1, 5} bx1,j, j ∈ {1, 5} cx1,j, j ∈ {1, 5} dx1,j, j ∈ {1, 5}

LTx1,1 0.0019 0.0019 0.0021 0.0023
LTx1,5 0.0033 0.00384 0.004 0.004

The expressions of the triangular membership functions are:

µTLx1,j (x) =




0, x < ax1,j

1 +
x−bx1,j

bx1,j−ax1,j
, x ∈ [ax1,j, bx1,j)

1 − x−bx1,j
cx1,j−bx1,j

, x ∈ [bx1,j, cx1,j)
0, x > cx1,j

, ax1,j < bx1,j < cx1,j, j = 2, 4 (5)

where the modal values of the membership functions are the parameters ax1,j, bx1,j, and cx1,j,
j = 2, 4 presented in Table 4.

Table 4
Modal values of linguistic terms in the third case

Linguistic terms, Trapezoidal membership functions
LTx1,j, j = 2, 4 ax1,j bx1,j cx1,j

LTx1,1 0.0021 0.0023 0.0027
LTx1,3 0.0023 0.0027 0.003
LTx1,4 0.0027 0.003 0.0033

Figure 3 shows the membership functions of x1 in these three cases: the first case in Figure
3 (a), the second case in Figure 3 (b) and the third case in Figure 3 (c).

Figure 3: Membership functions of the input variable x1 in the first case (a), in the second case
(b) and in the third case (c).

Five linguistic terms, LTx3,j, j = 1, 5, are defined for the input variable x3 . The first and the
fifth one are modeled by trapezoidal membership functions, and the second, the third and the
fourth one are modeled by trapezoidal membership functions. The universes of discourse of the



400 C.-A. Dragoş, R.-E. Precup, M.L. Tomescu, S. Preitl, E.M. Petriu, M.-B. Rădac

membership functions of these linguistic terms are: [4, 8] for LTx3,1, [5, 9] for LTx3,2, for [6, 10],
LTx3,3, [7, 11] for LTx3,4, and [8, 12] for LTx3,5. The expressions of the trapezoidal membership
functions are:

µTLx3,j (x) =




0, x < ax3,j

1 +
x−bx3,j

bx3,j−ax3,j
, x ∈ [ax3,j, bx3,j)
1, x ∈ [bx3,j, cx3,j)

1 − x−cx3,j
dx3,j−cx3,j

, x ∈ [cx3,j, dx3,j)
0, x > dx3,j

, ax3,j < bx3,j 6 cx3,j < dx3,j, j ∈ {1, 5} (6)

The modal values of the membership functions are the parameters ax3,j, j ∈ {1, 5}, bx3,j, j ∈
{1, 5}, cx3,j, j ∈ {1, 5}, and dx3,j, j ∈ {1, 5}, given in Table 5.

Table 5
Parameters of trapezoidal linguistic terms

Linguistic terms, Trapezoidal membership functions
LTx3,j, j = {1, 5} ax3,j, j ∈ {1, 5} bx3,j, j ∈ {1, 5} cx3,j, j ∈ {1, 5} dx3,j, j ∈ {1, 5}

LTx3,1 4 4 6 8
LTx3,5 8 10 12 12

The expressions of the triangular membership functions are:

µTLx1,j (x) =




0, x < ax1,j

1 +
x−bx1,j

bx1,j−ax1,j
, x ∈ [ax1,j, bx1,j)

1 − x−bx1,j
cx1,j−bx1,j

, x ∈ [bx1,j, cx1,j)
0, x > cx1,j

, ax1,j < bx1,j < cx1,j, j = 2, 4, (7)

where the modal values of the membership functions are the parameters ax3,j, bx3,j, and cx3,j,
j = 2, 4, given in Table 6.

Table 6
Modal values of linguistic terms

Trapezoidal
Linguistic terms, membership functions
LTx3,j, j = 2, 4 ax3,j bx3,j cx3,j

LTx3,1 5 7 9
LTx3,3 6 8 10
LTx3,4 7 9 11

Figure 4 shows the membership functions of the input x3.
The rule consequents of the T-S fuzzy models correspond to the discrete-time state-space

models characterized by the matrices Ad,i, and Bd,i, Cd,i, i = 1, 5, detailed in Table 7. These
models are obtained by discretization of the continuous-time state-space linearized models (1)
using the sampling period Ts = 0.001 s.



An Approach to Fuzzy Modeling of Electromagnetic Actuated Clutch Systems 401

Figure 4: Membership functions of the input variable x3.

Table 7
Numerical values of matrices of discrete-time state-space models

O.p.s Numerical values of the matrices

1
Ad1 =


 0.9864 0.0007 0.0000062−24.3234 0.4847 0.011

3.018 −0.1579 0.9816


 , Bd1 =


 0.0000000310.000088

0.0142


 ,

Cd1 = [1000 0 0]

2
Ad1 =


 0.9869 0.0007 0.0000073−23.352 0.4847 0.013

3.4059 −0.1856 0.9811


 , Bd1 =


 0.0000000360.000088

0.0142


 ,

Cd1 = [1000 0 0]

3
Ad1 =


 0.9876 0.0007 0.0000086−22.0872 0.4847 0.0153

3.7379 −0.2153 0.9807


 , Bd1 =


 0.0000000420.00012

0.0139


 ,

Cd1 = [1000 0 0]

4
Ad1 =


 0.9885 0.0007 0.0000101−20.4072 0.4847 0.018

3.9765 −0.2479 0.9801


 , Bd1 =


 0.0000000490.000139

0.0135


 ,

Cd1 = [1000 0 0]

5
Ad1 =


 0.9897 0.0007 0.0000101−18.3864 0.4847 0.0209

4.1934 −0.2814 0.9793


 , Bd1 =


 0.0000000550.000158

0.0132


 ,

Cd1 = [1000 0 0]

The modal equivalence principle guarantees the equivalence between the fuzzy models and
the nonlinear state-space models. That is the reason to express the rule base of the discrete-time
dynamic T-S fuzzy models in the following general form:

Ri : IFx1,kISLTx1,jANDx3,kISLTx3,j THEN

{
xk+1 = Ad,ixk + Bd,iuk

yk,m = Cd,ixk
,

i = 1, nR, j = 1, nLT,

(8)

where k is the index of the current sampling interval, i is the index of the current rule, j is the
index of the current linguistic term, nR is the number of rules, nLT is the number of linguistic
terms, nR = nLT = 5 in our discrete-time dynamic T-S fuzzy models.

The fuzzy controller employs the SUM and PROD operators and the weighted average de-
fuzzification method. Other operators can be used [22], [23], [24], [25], [26], [27].



402 C.-A. Dragoş, R.-E. Precup, M.L. Tomescu, S. Preitl, E.M. Petriu, M.-B. Rădac

4 Experimental Results

The modeling approach presented in the previous sections is applied and exemplified in this
section in order to obtain fuzzy models for the electromagnetic actuated clutch system. Three T-S
fuzzy models were developed for this nonlinear process. Some comparisons were done to illustrate
the difference between them. A part of the results is presented as follows. The simulation results
include the evolutions of the position versus time (in Figure 5), the evolution of the measured
position versus time (in Figure 6), the evolution of the modeling error versus time (in Figure 7),
and the evolution of the current versus time (in Figure 8).
Figure 5 and Figure 6 present the evolution of both position and measured position y in four
cases: nonlinear model (1) of the process, first T-S fuzzy model, second T-S fuzzy model and
third T-S fuzzy model. These evolutions point out in all cases an aperiodical evolution with a
small overshoot, but the fuzzy modeled responses exhibit a delay of 0.1 s and they exceed the
steady-state values of the nonlinear model. The modeling error versus time is highlighted in
Figure 7 to outline the difference between the nonlinear model and the fuzzy models.

Figure 5: Position of nonlinear model (a), of first T-S fuzzy model (b), of second T-S fuzzy model
(c) and of third T-S fuzzy model (d) versus time.

Figure 8 points out the evolution of the current in the same four cases: nonlinear model (1)
of the process, first T-S fuzzy model, second T-S fuzzy model and third T-S fuzzy model. Figure
8 illustrates that the current exhibited by the T-S fuzzy models has a delay, but it reaches the
steady-state value in approximately 0.5 s and with aperiodical response as that of the model (1).
All responses point out a delay of 0.1 s which must be reduced. Moreover, the convergence of
the modeling error to zero can be achieved by the optimization of the parameters of several
parameters of the fuzzy models including input membership functions or parameters in the rule
consequents. Various optimization algorithms can be implemented in this context [28], [29], [30],
[31], [32], [33], [34], [35], [36].



An Approach to Fuzzy Modeling of Electromagnetic Actuated Clutch Systems 403

Figure 6: Measured position of nonlinear model (a), of first T-S fuzzy model (b), of second T-S
fuzzy model (c) and of third T-S fuzzy model (d) versus time.

Figure 7: Modeling error of first T-S fuzzy model (a), of second T-S fuzzy model (b) and of third
T-S fuzzy model (c) versus time.

5 Conclusions

The paper has proposed an approach to the fuzzy modeling of an electromagnetic actuated
clutch system. This approach is important because it is easily applicable with adequate but
not complicated generalizations to a wide category of industrial applications. Other similar T-S
fuzzy models can be obtained in order to be further used in the T-S fuzzy controller design and
tuning.
The future work will be dedicated to separating a part of the parameters of the input member-
ship functions. These parameters will be obtained by different optimization algorithms which



404 C.-A. Dragoş, R.-E. Precup, M.L. Tomescu, S. Preitl, E.M. Petriu, M.-B. Rădac

Figure 8: Current of nonlinear model (a), of first T-S fuzzy model (b), of second T-S fuzzy model
(c) and of third T-S fuzzy model (d) versus time.

will solve the optimization problems with objective functions that depend on the modeling errors.
The reduction of the modeling errors will be thus ensured.

Acknowledgements

This work was supported by a grant in the framework of the Partnerships in priority areas
- PN II program of the Romanian National Authority for Scientific Research ANCS, CNDI -
UEFISCDI, project number PN-II-PT-PCCA-2011-3.2-0732.

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