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Order reduction of complex systems described by TSK

fuzzy models based on singular perturbations method
a a a
Anouar Bouazza , Anis Sakly & Mohamed Benrejeb
a
Unité de recherche LARA Automatique, Ecole Nationale d’Ingénieurs de Tunis , BP 37 Le
Belvédère , Tunis 1002 , Tunisia
Published online: 28 Jul 2011.

To cite this article: Anouar Bouazza , Anis Sakly & Mohamed Benrejeb (2013) Order reduction of complex systems described
by TSK fuzzy models based on singular perturbations method, International Journal of Systems Science, 44:3, 442-449, DOI:
10.1080/00207721.2011.602480

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 International Journal of Systems Science
Vol. 44, No. 3, March 2013, 442–449

Order reduction of complex systems described by TSK fuzzy models based on singular
perturbations method
Anouar Bouazza*, Anis Sakly and Mohamed Benrejeb
Unite´ de recherche LARA Automatique, Ecole Nationale d’Inge´nieurs de Tunis, BP 37 Le Belve´de`re,
Tunis 1002, Tunisia
(Received 13 November 2008; final version received 22 June 2011)

This article deals with the concept of order reduction of linear complex systems described by TSK fuzzy models.
The use of singular perturbations technique for process modelling and the choice of Benrejeb arrow form
characteristic matrix provide the decoupling of dynamics of linear systems in the continuous and discrete case.
An original contribution is to apply nonconventional TSK fuzzy approach on a direct current motor in order, on
Downloaded by [Florida State University] at 00:29 29 December 2014

one hand, to highlight the order reduction of this system presented locally in the form of linear models and, on
the other hand, to show the efficiency of the proposed approaches.
Keywords: order reduction technique; method of singular perturbations; two-time scale system; discretisation of
continuous systems; TSK fuzzy models; Benrejeb arrow form matrix

1. Introduction This study is interested in the approach of singular

The study of real systems often presents a number of perturbations for order reduction of a direct current
complexities, including the large size of the (DC) motor (Benrejeb and Abdelkrim 2003), described
process which presents difficulties for the synthesis of by Takagi–Sugeno–Kang (TSK) fuzzy models
control and the determination of observers or (Taniguchi and Tanaka 2000).
filters. To circumvent the drawback of high order,
many works suggest the application of reduction
techniques (Satakshi, Mukherjee, and Mittal 2005;
Sampaio and Soize 2006; Alsmadi and Abdalla 2007; 2. Singular perturbation method
Coluccio, Eisinberg, and Fedele 2007; Frangos and In mathematics, more precisely in perturbation theory,
Jaimoukha 2007; Kherraz 2007; Bellamine and a singular perturbation problem is a problem contain-
Elkamel 2008). ing a small parameter that cannot be approximated by
Therefore, many methods are used to lead to order setting the parameter value to zero. This is in contrast
reduction or to process decomposition and ensure the to regular perturbation problems, for which an
fidelity of the reduced system towards the original approximation can be obtained by simply setting the
system. small parameter to zero.
Using such a reduced system it should be simple to The perturbation methods (Shafrir 2004) are
allow the resolutionof the problems related to the commonly used by mathematicians to approach the
analysis and synthesis of the process, with a limited equations of many systems. They provide solutions
quantity of calculations, and, in particular, the approached by switching to the limit close to the real
calculation of control laws. system.
The concept of simplification is very wide since it The approach by the method of multi-time scale
includes the structural simplifications in nature, the systems, or singular perturbation (Lubuma and
decompositions of the initial system in subsystems and Patidar 2009), enables the supply of simplified
the order of decrease of a model without structure models by decoupling slow and fast dynamics of the
modification. global system and guarantees the fact that each model
The reduced model obtained must maintain eigen- takes into account the dynamics of the other.
values of the global system and therefore, the real The discrete systems deal with the properties of a
physical meaning. two-time scale (Benrejeb, Soudani, Sakly, and

*Corresponding author. Email: bouazza.anouar@yahoo.fr

ISSN 0020–7721 print/ISSN 1464–5319 online

ß 2013 Taylor & Francis
http://dx.doi.org/10.1080/00207721.2011.602480
http://www.tandfonline.com
 International Journal of Systems Science 443

Borne 2006) and can be written in the following form: Then, the matrix P is considered
       2 3
x_ 1 ðtÞ A11 A12 x1 ðtÞ B1 0 1  1 0
¼ þ uðtÞ ð1Þ 6 1
x_ 2 ðtÞ A21 A22 x2 ðtÞ B2 6 2    n1 07 7
6 2 .. 7
  P¼6
6 1 22    2n1 .77 ð5Þ
  x1 ðtÞ 6 .. .. .. 7
yðtÞ ¼ C1 
C2 ð2Þ 4 . . .  15
x2 ðtÞ
n1
1 n1
2    n1
n1 1
with
The transformation
x1 ðtÞ 2 Rn1 slow vectors of the system
x1 ðtÞ 2 Rn2 fast vectors of the system z ¼ P1 x ð6Þ
deals up to the new description:
Aij C2 B2
Aij ¼ ; C2 ¼ ; B2 ¼ ; n1 þ n2 ¼ n ~
   z_ ¼ AðÞz ð7Þ
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Finally, the Benrejeb arrow form characteristic matrix

 separability degree defined below. is given by
The main difficulty of a singular perturbation method 2 3
is to use the small  parameter to describe the system 1    a~ 1n
in the standard form. The separation of states into slow 6 .. 7
6 . 1   a~ 2n 7
6 7
and fast is generally non-trivial. A permutation and/or 6 7
scaling of states is required to obtain the separable ~ ¼6
AðÞ 6
.. .. .. .. 7
7 ð8Þ
6 . . .  . 7
model. 6 .. .. 7
6 . .  n1 a~n1,n 7
Some analytical methods of identification of 4 5
dynamics and iterative techniques allow us, in most a~n1 ðÞ a~n2 ðÞ  a~ n,n1 ðÞ nn ðÞ
cases, to get around this difficulty.
The verification of two-time scale conditions with
(Benrejeb and Abdelkrim 2003), as it will be shown,
requires a packaging of the characteristic matrix’s 1
a~ i ¼ Qn1 i ¼ 1, . . . , n  1 ð9Þ
system. The arrangement often requires changes such i¼1,iþj ði  j Þ
as permutation and calibration.
The Benrejeb arrow form characteristic matrix  
a~ nj ðÞ ¼ P j ,  j ¼ 1, . . . , n  1 ð10Þ
(Benrejeb and Abdelkrim 2003) is proving to be very
interesting for analysis and synthesis of systems as well
X
n1
as for the order reduction of the process. a~nn ðÞ ¼ ann ðÞ  j ð11Þ
In this sense, for the following Companion matrix, j¼1

2 3
0 1 0  0
6 0 0 1  0 7
6 .. .. .. .. .. 7 3. Reduced TSK fuzzy models
AðÞ ¼ 6
6 . . . . . 7 ð3Þ
7
4 5 The TSK fuzzy approach is a multi-model technique
0 0 0  1
a0 ðÞ a1 ðÞ a2 ðÞ    an1 ðÞ used for modelling and control of complex systems.
The idea is to decompose the process in linear
i ðÞ is a coefficient of the instantaneous characteristic submodels in space state correlated by membership
polynomial of the matrix AðÞ such that functions. The TSK fuzzy approach is a very interest-
ing approach for the description of nonlinear systems.
X
n1
The TSK fuzzy system under duress the rule, Ri, for
Pð, Þ ¼ n þ i ðÞi ð4Þ
i ¼ 1, 2, . . . , r, is given by Benrejeb et al. (2006) and
i¼0
Benrejeb, Sakly, Ben Othman, and Borne (2008)
with 1 , 2 , . . . , n1 : ðn  1Þ being distinct real or
complex numbers. If x1 ðtÞ is Mi1 and . . . and xn ðtÞ is Min
(
The suitable choice of the vectors _ ¼ Ai xðtÞ þ Bi uðtÞ
xðtÞ ð12Þ
 ¼ ½1 , 2 , . . . , n1  has high consequences for mod- Then
elling and analysis of the reduced order TSK model. yðtÞ ¼ Ci xðtÞ
 444 A. Bouazza et al.

with Mij , j ¼ 1, 2, . . . , n, the ith fuzzy subset relating to The inferred model of the reduced TSK fuzzy models
the state Vector xi, x 2 Rn , u 2 R. can be described by Benrejeb and Abdelkrim (2003)
The final outputs of the fuzzy model are inferred as Xr
follows: A~ ¼ hi A~ i ð22Þ
i¼1
X r
_ ¼
xðtÞ hi ðxðtÞÞðAi xðtÞ þ Bi uðtÞÞ ð13Þ
A~ i ¼ A22,i  A21,i A1
11,i A12,i i ¼ 1, 2, . . . , r ð23Þ
i¼1

X
r
yðtÞ ¼ hi ðxðtÞÞCi xðtÞ ð14Þ 4. Discretisation of the continuous systems
i¼1
The discretisation concerns the process of transferring
wi ðxðtÞÞ continuous models and equations into discrete coun-
hi ðxðtÞÞ ¼ Pr ð15Þ
i¼1 wi ðxðtÞÞ terparts. This process is usually carried out as a first
The hi verifies the property of a sum convex, which is step towards making them suitable for numerical
to say that evaluation and implementation on digital computers.
Xr Among the methods of discretisation, the homo-
hj  0; h ¼1 ð16Þ
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j¼1 j graphic transformation is considered here; this techni-

que is also called the method of trapeze (Zhang and
Given the concept of the Parallel Distributed
Li 2005).
Compensation (PDC) control law (Wang, Tanaka,
This transformation allows us to associate, rigor-
and Griffin 1996), for each fuzzy submodel, and based
ously, to a continuous system or, conversely, to a
on the same fuzzy rules of the system to be controlled,
discrete system, and is thus an ideal way to transpose
it will be
the continuous optimal problem to a discrete
Xr X r   optimal one.
_ ¼
xðtÞ hi hi Ai  Bi Kj xðtÞ ð17Þ
i¼1 j¼1
If the initial continuous system is described by the
equation
where the characteristic matrix of the linear models of
the system to be controlled is considered in the _ ¼ AxðtÞ þ BuðtÞ
xðtÞ
ð24Þ
Benrejeb arrow form (8). yðtÞ ¼ CxðtÞ
As a result, the determination of the reduced and
it is possible to deduct the representation of a recurrent
the inferred model of the reduced TSK fuzzy models is
solution describing in the space state the evolution of
deduced as follows.
the vector x(k) obtained by discretisation of x(t).
. If the global model verifies the two-time scale T is the sampling time.
8
separability conditions (Benrejeb and Abdelkrim > T
>
< x þ x_ ¼ xðk þ 1Þ
2003), 2
 1  1     ð25Þ
>
A   A22  A21 A1 A12  þ kA21 kA1 A12  1
11 11 11 : x  T x_ ¼ xðkÞ
>
3 2
i ¼ 1, 2, ...,r ð18Þ The triplet (A, B, C ) relating to the continuous system
A reduced model of the TSK fuzzy system is described match, through the homographic transformation, the
by Taniguchi and Tanaka (2000) triplet (F, G, H ) concerning the discrete system is
~_ ¼ A~ xðtÞ defined as follows:
xðtÞ ~ ð19Þ 8
>
> AT 1 AT
with >
> F¼ I Iþ
< 2 2
A~ ¼ A22  A21 A1
11 A12 ð20Þ AT 1 ð26Þ
>
> G¼ I BT
>
> 2
:
C¼C
. If the local TSK fuzzy system verifies the two-
time scale separability conditions (Benrejeb and This leads to the following discrete state
Abdelkrim 2003), representation:
xðk þ 1Þ ¼ FxðkÞ þ GvðkÞ
    ð27Þ
 1  1   yðxÞ ¼ CxðkÞ
A11,i   A22,i  A21,i A1 11,i A12,i 
3
 

 1
 1
þ A21,i A1
11,i A 12,i  i ¼ 1, 2, ...,r ð21Þ vðkÞ ¼ ð28Þ
2½uðk þ 1Þ þ uðkÞ
 International Journal of Systems Science 445

u 1 y By application of singular perturbation method,

s(s + d)(s + cid) a reduced second order system is obtained and
described by the following equation (Bouazza
Figure 1. Modelling of DC Motor. et al. 2008):

x_~ ¼ A~ i x~ þ Bu
~ i ¼ ð1, 2Þ ð35Þ
5. Exploitation of the TKS fuzzy approach for order with
reduction of DC motor
A~ i ¼ A22,i  A21,i A1
11,i A12,i i ¼ ð1, 2Þ ð36Þ
The singular perturbation method presupposes the
knowledge of both slow and fast state vectors. such as
Generally, the partitioning is based on some physical  
properties of the process. ~ 0:1321 0:0770
A1 ¼ ð37Þ
An example of a very important application is that 0:0103 0:0066
of electrical machinery rotating.
 
Indeed, they are modelled by two types of 0:1321 0:0770
~
A2 ¼ ð38Þ
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equations such as electrical and mechanical. In 0:0982 0:0572

addition, it is well known that the electrical variables
are faster than the mechanical ones.  
~ ~ ~ 0
Therefore, it is appropriate to pick the mechanical B1 ¼ B2 ¼ B ¼ ð39Þ
1
variables as slow and electrical as fast ones.
A DC motor can be described like a dynamic, The smallest time constant of continuous system is its
electromechanical, nonlinear and third class process. electrical one (30), equal to 0.08 s, therefore, it is
In this sense, a dynamic, electromechanical, non- appropriate to choose a sampling period T  0:08 s.
linear, third order system (Benrejeb and Abdelkrim It is wise to choose
2003; Bouazza, Sakly, and Benrejeb 2008), is consid-
ered and defined by its two time constants (Figure 1). T ¼ 0:05 s:
The pairs (Ai, Bi) and ðA~ i , B~ i Þ, for i ¼ 1, 2, correspond,
Slow mechanical time constant:
via the homographic transformation or the method of
1 trapeze, to the pairs (Fi, Gi) and ðF~i , G~ i Þ relating to
ðci  1, i ¼ 1, 2Þ ¼ f8s, 13:33sg ð29Þ
ci  discrete global and reduced models and are defined as
follows (Benrejeb et al. 2008):
1 2 3
Fast electrical time constant: ¼ 0:08s ð30Þ 0:5 0 0:0029

F1 ¼ 4 0:0078 0:9934 0:0039 5 ð40Þ
This system can be in the form of two linear submodels 4:0628 0:0005 1:0241
and is described locally, around an operating point, by
its state equation 2 3
0:5 0 0:0029
_ ¼ Ai xðtÞ þ BuðtÞ i ¼ ð1, 2Þ
xðtÞ ð31Þ F2 ¼ 4 0:0078 0:9934 0:0039 5 ð41Þ
4:0852 0:0049 1:0268
The characteristic matrix Ai, for i ¼ 1, 2, of the process
is represented in the Benrejeb arrow form as 3 2
0
2 3 G1 ¼ G2 ¼ G ¼ 4 0 5 ð42Þ
13:125 0 0:077
A1 ¼ 4 0 0:132 0:077 5 ð32Þ 0:05
106:634 0:010 0:631    
0:9934 0:0038 0:9934 0:0038
2 3 F~1 ¼ F~2 ¼
0:0005 1:0003 0:0049 1:0029
13:125 0 0:077
A2 ¼ 4 0 0:132 0:077 5 ð33Þ ð43Þ
107:076 0:098 0:685  
0
2 3 G~ 1 ¼ G~ 2 ¼ G~ ¼ ð44Þ
0 0:05
B1 ¼ B 2 ¼ B ¼ 4 0 5 ð34Þ A fuzzy controller is designed for the global TSK fuzzy
1 models (32), (33). Reduced models (37), (38) are
 446 A. Bouazza et al.

defined and, now, a fuzzy controller is designed for If ðx1 is M2 Þ and ðx2 is M2 Þ and ðx3 is M2 Þ
them. ð50Þ
Then u ¼ K2 x
For the global third order system, the following
fuzzy rules are defined. with Ki ði ¼ 1, 2Þ: Vectors of gains, defined from the
quadratic optimal control relating to the global system
. For the open loop’s system considered in
described locally in the form of two linear submodels.
autonomous regime:
. For the global system:
 
If ðx1 is M1 Þ and ðx2 is M1 Þ and ðx3 is M1 Þ K1 ¼ 10:9898 0:3419 1:5019 ð51Þ
ð45Þ
_ ¼ F1 xðtÞ
Then xðtÞ  
K2 ¼ 11:4055 0:2593 1:5260 ð52Þ
If ðx1 is M2 Þ and ðx2 is M2 Þ and ðx3 is M2 Þ
ð46Þ
_ ¼ F2 xðtÞ
Then xðtÞ . For the reduced system:
 
K~ 1 ¼ 0:3402 1:4392 ð53Þ
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. For the controlled open loop’s system:

 
K~ 2 ¼ 0:2579 1:4865 ð54Þ
If ðx1 is M1 Þ and ðx2 is M1 Þ and ðx3 is M1 Þ
ð47Þ
_ ¼ F1 xðtÞ þ GuðtÞ
Then xðtÞ

If ðx1 is M2 Þ and ðx2 is M2 Þ and ðx3 is M2 Þ

ð48Þ
_ ¼ F2 xðtÞ þ GuðtÞ
Then xðtÞ

. Rules of the fuzzy controller:

If ðx1 is M1 Þ and ðx2 is M1 Þ and ðx3 is M1 Þ

ð49Þ
Then u ¼ K1 x Figure 2. Membership functions M1 and M2.

Figure 3. Responses of the continuous and discrete global system before introducing the TSK fuzzy controller.
 International Journal of Systems Science 447
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Figure 4. Responses of the continuous and discrete global system after introducing the TSK fuzzy controller.

Figure 5. Responses of the continuous and discrete reduced system before introducing the TSK fuzzy controller.

In the same way, the fuzzy rules for the second order According to Figures 4 and 6, the robustness of the
reduced system (37), (38) are defined. proposed models is examined; the responses of
Similarly, vectors of gains and fuzzy rules for the the continuous and discrete global system and the
global (40), (41) and reduced (43) systems are defined responses of the continuous and discrete reduced
in the discrete case (Bouazza et al. 2008). system are similar. The TSK fuzzy approach conserves
The membership functions relating to global and the state variables of the initial system and, therefore,
reduced systems are represented as shown in Figure 2. the real physical meaning.
The simulation of the global system gives the
following responses (Figures 3 and  4). The initial
0:5
conditions are such that xð0Þ ¼ 0:5 . 6. Conclusion
0:5
The simulation of the reduced system gives the This work is interested in TSK fuzzy approach for the
following responses (Figures 5 and 6). The initial modelling and control of nonlinear systems presented
conditions are such that xð0Þ ¼ 0:5
0:5 . locally in the form of linear submodels.
 448 A. Bouazza et al.
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Figure 6. Responses of the continuous and discrete reduced system after introducing the TSK fuzzy controller.

This approach has enabled, rigorously, the applica-

tion of the singular perturbations technique in order to Anis Sakly was born in Tunisia in
1970. He received the Electrical
reduce the order of DC motor. The reduced model, Engineering diploma in 1994 from
simplified and obtained, conserve the state variables of Ecole Nationale d’Ingénieurs de
initial system and, therefore, the real physical meaning. Monastir (ENIM), then the PhD
The accuracy obtained depends of course on the degree in Electrical Engineering in
effective separation of the dynamics of the system. 2005 from Ecole Nationale
d’Ingénieurs de Tunis (ENIT). He is
Moreover, an original contribution, which implements currently Maı̂tre Assistant (Assistant
a nonconventional TSK fuzzy approach, is considered Professor) at ENIM. His research interests are in analysis
to highlight the order reduction of systems presented and synthesis of intelligent control systems, particularly soft
locally in the form of linear models. computing-based control approaches.
Indeed, this study has detailed the TSK fuzzy
models in the continuous and discrete case, within the Mohamed Benrejeb was born in
Tunisia in 1950. He obtained the
framework of PDC control, and specified the advan- diploma of ‘Ingénieur IDN’ in 1973,
tages of the TSK fuzzy controller for the stabilisation the Master degree of Automatic
of dynamic systems. Control in 1974, the PhD in
Some future extensions are possible, particularly Automatic Control from the
the order reduction of nonlinear complex systems. University of Lille in 1976 and the
DSc from the same University in 1980.
Full professor at Ecole Nationale
d’Ingénieurs de Tunis, since 1985 and at Ecole Centrale de
Lille, since 2003. His research interests are in the area of
Notes on contributors
analysis and synthesis of complex systems based on classical
Anouar Bouazza was born in Tunisia and non-conventional approaches and recently in discrete
in 1982. He got his Electrical event systems domain.
Engineering Diploma in 2006 from
the National Engineering School of References
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