International Journal of Computers, Communications & Control
Vol. I (2006), No. 4, pp. 9-19

New Discrete Tanaka Sugeno Kang Fuzzy Systems Characterization and
Stability Domain

Mohamed Benrejeb, Dhaou Soudani, Anis Sakly, Pierre Borne

Abstract: In this paper, an analytical approach to characterize discrete Tanaka
Sugeno Kang (TSK) fuzzy systems is presented. This characterization concerns the
choice of the adequate conjunctive operator between input variables of discrete TSK
fuzzy models, t-norm, and its impact on stability domain estimation. This new ap-
proach is based on stability conditions issued from vector norms corresponding to a
vector-Lyapunov function. In particular, second order discrete TSK models are con-
sidered and this work concludes that Zadeh’s t-norm, logic product min, gives the
largest estimation of stability domain.
Keywords: Discrete nonlinear systems, discrete TSK fuzzy model, t-norm, stability
domain, vector norm, arrow form matrix.

1 Introduction

Fuzzy control of systems presents a major interest in several applications including industrial ones.
However, closed loop system properties are not easily understood and the design of the fuzzy system is
generally based on intuitive approaches.

Different fuzzy control strategies exist. In particular, TSK fuzzy approach permits the description
and the control of a system by defining different models related via a rule base.

The stability of TSK systems has been one of the central issues and is subject of many works either
in the continuous case or in the discrete one. To do this, different approaches are considered mainly
based on Lyapunov functions [5, 10, 12]. In particular, the Linear Matrix Inequality (LMI) formulation
is used [11] and, according to the considered system, permits the stability problem resolution. In our
previous work [1], the used approach is led through the convergence of a regular vector norm [3]. The
vector norm approach, based on the comparison and overevaluating principle, has a major advantage that
it deals with a very large class of systems, since no restrictive assumption is made on the matrices of
state equations. So, in [1], the approach estimates the stability domain of continuous TSK fuzzy systems
and its dependence on the choice of the conjunctive operator between inputs.

In this way, many authors have presented and analyzed several of these operators defined by t-norms
and so said t-operators [4, 6, 7]. In particular, in [6], six t-norms are used to represent this connective
operator in the inverse pendulum control. In another work [9], are used the same t-norms to study their
impact on fuzzy control performances of a second order process by defining a performance criterion
based on error measurement of the closed loop system. However, then exist other operators to represent
this connective such as mean operators developed in [14].

Except [1], all these studies don’t treat the influence of the choice of the conjunctive operator be-
tween inputs on the stability conditions obtained and often only prod or min operators are used without
theoretical argumentation and just for simplification reasons.

Therefore, in this way the study will be considered and constitutes a generalization of [1] in the
discrete case seeing the soft implementation nature of this control strategy.

By exploiting conditions or hypothesis obtained from stability analysis of discrete TSK fuzzy systems
based on vector norms approach, the influence of the choice of t-norm is studied. Similarly to [1], only
second order TSK fuzzy models are considered for simplification reason.

In the next section, is presented the structure of discrete TSK fuzzy models. In section 3, several
definitions dealing with t-norm notion as well as its properties are given. In section 4, sufficient stability
conditions of these TSK systems based on vector norms approach are presented. Then, the study of the

Copyright c© 2006 by CCC Publications



10 Mohamed Benrejeb, Dhaou Soudani, Anis Sakly, Pierre Borne

impact of the choice of t-norm on the proposed stability conditions will be considered in section 5. In
particular, second order TSK fuzzy models are considered. A DC motor is studied in section 6 as an
application example validating this study. Finally, some concluding remarks are pointed out.

2 TSK discrete models description

An r-order TSK discrete fuzzy model of an n-order nonlinear system to be controlled is given by a
rule base where the ith rule is in the form:

Ri : IF x1 is G
i
1 AND x2 is G

i
2 AND···AND xn is Gin

T HEN

{
x(k + 1) = Aix(k) + Biu(k)
y(k) = Cix(k)

i = 1, 2, ..., r
(1)

where Gij, j=1,2,...,n, is the i
th fuzzy set of the state vector x j.

The state vector x ∈ ℜn, the control input u ∈ ℜ and the matrices Ai, Bi and Ci are of appropriate
dimensions.

According to the Parallel Distributed Compensation (PDC) concept [13], the rule base R′i of the fuzzy
controller stabilizing the former system is in the form:

R′i : IF x1 is G
i
1 AND x2 is G

i
2 AND···AND xn is Gin

T HEN u(k) = −Kix(k) i = 1, 2, ..., r
(2)

with Ki = [k1i ,k
2
i ,...,k

n
i ].

By substituting u in the equation (1) above, it comes:

x(k + 1) =
r

∑
i=1

r

∑
j=1

hih j(Ai −BiK j)x(k) (3)

with:

hi = wi/
r

∑
i=1

wi (4)

wi = T (G
i
1(x1), G

i
2(x2), ..., G

i
n(xn)) (5)

and T a t-norm.
When linear models of the system to be controlled are considered in the controllable form given by:

Ai =




0 1 ··· 0
...

. . .
. . .

...
0 ··· 0 1
−a1i ··· −an−1i −ani


 and Bi = B =




0
...
0
1


 (6)

relation (3) becomes:

x(k + 1) =
r
∑

i=1

r
∑
j=1

hih j(Ai −BiK j)x(k)

=
r
∑

i=1

r
∑
j=1

hih j(Ai −BK j)x(k)

=
r
∑

i=1

r
∑
j=1

hih jAix(k)−
r
∑

i=1

r
∑
j=1

hih jBK jx(k)

=
r
∑

i=1
hi

r
∑
j=1

h jAix(k)−
r
∑
j=1

h j
r
∑

i=1
hiBK jx(k)

=
r
∑

i=1
hiAix(k)−

r
∑
j=1

h jBK jx(k)



New Discrete Tanaka Sugeno Kang Fuzzy Systems Characterization and Stability Domain 11

then finally:

x(k + 1) =
r

∑
i=1

hi(Ai −BKi)x(k) (7)

3 T-norms

An important task to be performed in the design of TSK fuzzy systems is the choice of the conjunctive
operator materializing the connective AND between input variables in the rule base and so corresponding
to the intersection operation between fuzzy subsets relatively to different inputs.

Often, this operator is defined by a t-norm T , whose definition and proprieties are presented as
follows.

Definition 1. A triangular norm (t-norm) is a function T :[0,1]×[0,1] → [0,1] verifying for every u, v, w
and t of [0,1] :

i) T (u,v) = T (v,u) (commutativity)
ii) T (u,T (v,w)) = T (T (u,v),w) (associativity)
iii) T (u,v) ≤ T (w,t) si u ≤ w et v ≤ t (monotonicity)
iv) T (u, 1) = u (one identity)

(8)

min operator is the largest of all possible t-norms, T (u,v) ≤ min (u,v).
The intersection operation between fuzzy subsets is defined by a t-norm T such as:

C = A∩B, then ∀ x ∈ X µC(x) = T (µA(x),µB(x)) (9)
The commonly encountered t-norms are given in table 1.

Table 1: Main t-norms

t-norm Name
min(u, v) Zadeh (logical operators)
uv Bandler (probabilistic operators)
max(u + v−1, 0) Lukasiewicz (bounded operators)





u if v = 1
v if u = 1
0 otherwise

Weber (drastic operators)

Definition 2. A t-norm T is said archimedean if and only if T (u, v) is continuous and T (u, u) < u for
each u of ]0,1[. Moreover, T is said strict archimedean if and only if T (u, v) < T (w,t) as soon as u < w
and v < t.

Any function T :[0,1] × [0,1] → [0,1] is an archimedean t-norm if and only if it exists a continuous
decreasing function f : [0,1]→[0,+∞[ such that f (1)=0 satisfying :

T (u,v) = f −1∗( f (u) + f (v)) (10)

where f −1∗ is the pseudo-inverse of f such as f −1∗ =
(

f −1(w) if w ∈ [0, f (0)[
0 if w ∈ [ f (0), + ∞[ .



12 Mohamed Benrejeb, Dhaou Soudani, Anis Sakly, Pierre Borne

Besides, T is strict if and only if f (0)= +∞.
f is said the additive generator of the t-norm T .

The additive generators associated to main parameterized archimedean t-norms are presented in table
2 [14].

Table 2: Main parametrized t-norms and their additive generators

t-norm Generator f (u) Parameter Name

uv
γ + (1−γ)(u + v−uv) 1

γ
Log

[
γ + (1−γ )u

u

]

Strict Archimedean

γ > 0 Hamacher

1
[(

1
u

)p
+

(
1
v

)p −1
] 1

p

1

(1 + u)
1
p

Strict Archimedean

p > 0 Schweizer & Sklar

1

1 +
[
( 1u −1)λ + ( 1v −1)λ

] 1
λ

1

1 + u
1
λ

Strict Archimedean

λ > 0 Dombi

1−min [1, ((1−u)ω + (1−v)ω ) 1ω ]
(1−u)ω

Archimedean no strict
ω > 0

Yager

uv
max (u,v,α ) No archimedean α ∈ [0,1] Dubois & Prade

4 New stability conditions

In [2], a change of base of (6) under the arrow form give:

x(k + 1) =
r

∑
i=1

hiMix(k) (11)

where Mi is a matrix in the arrow form and P is the corresponding passage matrix:

Mi = P
−1(Ai −BKi)P (12)



New Discrete Tanaka Sugeno Kang Fuzzy Systems Characterization and Stability Domain 13

Mi =




α1 0 ··· 0 β1
0

. . .
. . .

...
...

...
. . .

. . . 0
...

0 ··· 0 αn−1 βn−1
γ 1i ··· ··· γ n−1i γ ni




and P =




1 1 ··· 1 0
α1 α2 ··· αn−1 0
α 21 α

2
2 ··· α 2n−1

...
...

... ··· ... 0
α n−11 α

n−1
2 ··· α n−1n−1 1




(13)

with:

β j =
n−1
∏

q = 1
q 6= j

(α j −αq)−1 ∀ j = 1,2,...,n−1





γ ji = −Pi(α j) ∀ j = 1,2,...,n−1
Pi(λ ) = λ n +

n−1
∑

q=0
(aq+1i + k

q+1
i )λ

q

γ ni = −(ani + kni )−
n−1
∑
j=1

α j

(14)

The application of the classical Borne-Gentina criterion [3] leads to the following theorem.

Theorem 3. The discrete system described by (7) is asymptotically stable if there exist
0 < α j < 1, α j 6= αk, ∀ j 6= k, such as ∀x ∈ S:

i) 1−
∣∣α j

∣∣ > 0 ∀ j = 1, 2, ..., n−1
ii) 1−

∣∣∣∣
r
∑

i=1
hiγ ni

∣∣∣∣−
n−1
∑
j=1

∣∣∣∣
r
∑

i=1
hiγ

j
i β j

∣∣∣∣(1−
∣∣α j

∣∣ )−1 > 0 (15)

If S = ℜn, the stability is global.

Furthermore, if there exist α j, j = 1, 2, ..., n−1, such as :

i) 0 < α j < 1 j = 1, 2, ..., n−1
ii)

r
∑

i=1
hiγ ni > 0

iii)
r
∑

i=1
hiγ

j
i β j > 0 j = 1, 2, ..., n−1

(16)

then the previous theorem can be simplified to the following corollary.

Corollary 4. The discrete system described by (7) is asymptotically stable if there exist 0 < α j < 1,
α j 6= αk, ∀ j 6= k, such as ∀x ∈ S:

BPH < 0 (17)

where matrices B ∈ ℜn×n, P∈ ℜn×r and H∈ ℜr×1 are such as :

B =




β1 0 ··· 0
0

. . .
. . .

...
... ··· βn−1 0
0 ··· 0 −1




, P =




P1 (α1) ··· Pr(α1)
... ··· ...

P1 (αn−1) ··· Pr(αn−1)
P1 (1) ··· Pr(1)


 and H =




h1
...

hr


 (18)

If S = ℜn, the stability is global.

According to the choice of the coefficients α j, different conditions on the parameters hi can be
obtained.



14 Mohamed Benrejeb, Dhaou Soudani, Anis Sakly, Pierre Borne

5 T-norm influence on the proposed stability conditions

Let us consider the following classical fuzzy partition of a second order TSK model given by figure
1, with:

µi = G1i (xi) i = 1, 2, . . ., n (19)

and:

G1i (xi) = min (1, max (0,
Li −xi

2Li
)) and G2i (xi) = 1−G1i (xi) (20)

-

µ

Figure 1: Fuzzy partition of state variables for r=2

Then, the whole fuzzy state space considered is the open subset
Ω =]−L1, L1[×]−L2, L2[×. . .×]−Ln, Ln[. So, the notion of global stability, whenever used in
the literature on fuzzy control, refers to local stability within a such domain [8].

For second order TSK model, i.e. r=2, then h2 = 1−h1 and:

h1 =
w1

w1 + w2
=

T (µ1,µ2, ..., µn)
T (µ1,µ2,...,µn) + T (1− µ1,1− µ2,...,1− µn)

(21)

where T is a t-norm. Propositions [1]

1. Let µ1, µ2, . . ., µn and µmax ∈ [0, 1] such as µ1 ≤ µmax, µ2 ≤ µmax, ..., and µn ≤ µmax.
Then, we obtain:

h1(µ1,µ2,...,µn) ≤ h1(µmax,µmax,...,µmax) (22)
This proposition allows the determination of the domain S by its characteristic point M in figure 2.

 

x
1

S

x
2

x
1max

0
1

1

0.5

0.5 max

x
2max

L
1

L
2

M

Ω

µ

1

2

µµ

Figure 2: Correspondence between the state space variables domain and the membership values domain



New Discrete Tanaka Sugeno Kang Fuzzy Systems Characterization and Stability Domain 15

2. Let us note ϕn(µ) = T (µ1, µ2, ..., µn) when µ 1=µ 2 =. . . =µ n=µ . ϕ n is an increasing function on
[0, 1] with ϕn(0) = 0 and ϕn(1) = 1.
If the following conditions are satisfied :

i) ϕn is continuous
ii) ∀ µ > 0 , ϕn(µ ) 6= 0
iii) ∀ µ1, µ2 ∈ ]0,1] with µ2 > µ1 , ϕn(µ2)µ2 ≥

ϕn(µ1)
µ1

(23)

then we have: {
h1(µ ,µ ,...,µ ) ≥ µ if µ ≥ 0.5
h1(µ ,µ ,...,µ ) ≤ µ if µ ≤ 0.5

(24)

This proposition means that for a given µ , the smallest coefficient h1 is obtained with the t-norm
of Zadeh, the logic product min; and so for the condition (22) we obtain the largest rectangular
form domain S with this t-norm by taking µ max = c.

3. Conditions (23) of the proposition 2 are satisfied for each strict archimedean t-norm such as ϕn(µ )
is derivable on ]0,1].

The proposed approach is illustrated by considering the following parameterized t-norms:

• the Hamacher one: T (µ1,µ2) = µ1 µ2γ+(1−γ)(µ1+µ2−µ1 µ2) for γ>0,

• the Yager one: T (µ1,µ2) = 1−min (1,((1− µ1)ω + (1− µ2)ω )
1
ω ) for ω>0,

• and the Dubois one: T (µ1,µ2) = µ1 µ2max (µ1,µ2,α) for α ∈ [0,1].

6 Application example

As an example, let consider in figure 3 the case of a DC motor with two linear models G1(p) and
G2(p) with δ 1=0.5 and δ 2=1.

 

)(

1
)(

i

i
pp

pG
δ+

=

u x

i=1,2)(0 pB

Figure 3: DC motor discrete models

B0(p) = 1−e
−Te p
p is a zero order holder and Te=0.2s is the sampling time.

The discrete models of the DC motor are given by the following z-transmittances:

Gi(z) =
Ni(z)

(z−1)(z−ψi)
i = 1, 2

with ψi = e−δiTe and Ni(z) = δiTe+e
−δi Te−1

δ 2i
z + 1−e

−δi Te (1+δiTe)
δ 2i

.

The two discrete DC linear models are considered in the following controllable form:

A1 =
[

0 1
−0.905 1.905

]
, A2 =

[
0 1

−0.819 1.819
]

and B =
[

0
1

]

We suppose that for particular constraints the choice of Ki is imposed such that the pole placement is
different for the two models by taking:



16 Mohamed Benrejeb, Dhaou Soudani, Anis Sakly, Pierre Borne

K1 =
[
−0.7 1.2

]
and K2 =

[
−1.1 1.2

]

According to [3], the minimal overvaluing matrix relatively to the regular vector norm p:

p(x) = [|x1|,|x2|]T

is such as:

M(.) =
[
|α| 1
|γ1| |γ2|

]

with
γ1 = (0.486−0.086α )h1 + α 2 −0.619α −0.281
γ2 = 0.086h1 + 0.619−α

Then, stability conditions deduced from the corollary are:

i) 0 < α < 1
ii) 0.086h1 + 0.619−α > 0
iii) (0.486−0.086α)h1 + α 2 −0.619α −0.281 < 0
iv) 0.4h1 + 0.1 > 0

When i) is satisfied, relations ii) and iv) are too. Then condition iii) leads to the following inequality

h1 < −
α 2 −0.619α −0.281

0.486−0.086α = c

When 0.5 < c < 1 , let S a neighborhood of the equilibrium point 0, included in Ω,
Ω =]−1, 1[×]−1, 1[, that verifies a such condition. S is an overvaluing domain of the fuzzy system
and an estimation of a symmetrical domain S with respect to 0 imposes:

1−c < h1 < c.
Now, consider the study of the impact of the t-norm T on the width of the neighborhood S of the equi-
librium point 0, and then the determination of the largest stability domain D included in S verifying the
previous condition.

For α = 0.345, we obtain the maximal value of c, c=0.82. Thus, the overvaluing matrix M(·) is
constant:

M(·) = M =
[

α 1
0 α

]

whitch is triangular and not irreductible. However, for c = 0.80, this irreductibility is skirted and for
α = 0.345, its comes the following overvaluing matrix

M =
[

0.345 1
0.01 0.343

]

whitch is irreductible, whose principal eigenvalue is λm = 0.446, the corresponding vector is:

um = [9.88 1]
T

and the largest estimated stability domain D is such as

D =
{

x ∈ S/pT (x)um = 9.88|x1|+|x2| ≤ xmax
}

where S is the square form domain and xmax is its width. S depends on the condition h1 < c with c = 0.8
and so on the t-norm T .



New Discrete Tanaka Sugeno Kang Fuzzy Systems Characterization and Stability Domain 17

Table 3: µmax and xmax corresponding to different t-norms

t-norm µmax xmax
Zadeh 0.80 0.60

Bandler 0.67 0.34

Hamacher (γ = 0) 0.75 0.50
Hamacher (γ = ∞) 0.58 0.16
Yager (ω = 2) 0.62 0.24
Dubois (α = 0.5) 0.70 0.40

For n = 2, table 3 gives µmax, corresponding to the membership values domain, and xmax, correspond-
ing to the square form domain S, with respect to different t-norms.

The results obtained in table 3 shows that for n = 2, the greatest value of µmax and so the largest
domain S is obtained specifically for the logic product min.

Figure 4 presents an estimation of the largest stability domain D respectively for Zadeh and Bandler
t-norms.

 

x
1

x
2

1−1

1

-1

S

0.6−0.6

0.6

−0.6

0.06−0.06
D

Ω

 

x
1

x
2

1−1

1

-1

0.34

S

0.34−0.34

−0.34

0.03−0.03
D

Ω

a) Zadeh’s t-norm (min) b) Bandler’s t-norm (prod)

Figure 4: Stability domains obtained for two different t-norms

7 Conclusion

In this paper, we are interested in the stability study of discrete TSK fuzzy systems and the corre-
spondent domain estimation relatively to the used conjunctive operator, materialized by a t-norm. The
stability approach is based on vector norms. When second order TSK models are considered, a mathe-
matical study permits to conclude that Zadeh’s t-norm, min operator, yields to the largest stability domain.
A DC motor with two discrete models is considered as an illustrative example and the estimated stability
domains for different t-norms confirm the former result.

However, it suits to remark that this study can be generalized for r-order TSK models, in the two



18 Mohamed Benrejeb, Dhaou Soudani, Anis Sakly, Pierre Borne

cases continuous and discrete, with r greater than 2 and then we must distinguish between the two cases,
r odd or even. For two cases, the stability study can be leaded in different symmetrical rectangular zones
surrounding the equilibrium point, from the smallest to the largest.

References

[1] Benrejeb M., Sakly A., Ben Othman K. and Borne P.: Choice of Conjunctive Operator of TSK Fuzzy
Systems and Stability Domain Study. To appear in MATCOM (MAThematics and COMputers and
simulation Journal).

[2] Benrejeb M., Borne P. and Laurent F. : Sur une application de la représentation en flèche à l’analyse
des processus. RAIRO Automatique, 16 (2). (1982) 133-146.

[3] Borne P., Richard J. P. and Radhy N. E. : Stabilité, stabilisation, régulation : approche par les normes
vectorielles, Systèmes non linéaires, tome 2 : stabilité-stabilisation. coordonné par Fossard A. J. et
Normand-Cyrot D., Responsable du tome : Borne P., 45-90. Editions Masson, Paris (1993).

[4] Butkiewicz B.S. : Control Error of Fuzzy System with Different Reasoning and Defuzzification
Methods. 7th International Fuzzy Systems Association World Congress (IFSA) , III. Praga, Czech
(1997).

[5] Cao S. G., Rees N. W. and Feng G. : Quadratic stability analysis and design of continuous-time
fuzzy control systems. Int. J. Syst. Sci., 27(2). (1996) 193-203.

[6] Cardenas E., Castillo J.C., Cordon O., Herrera F. and Peregrin A. : Applicability of T-norms in Fuzzy
Control. Busefal, 61. (1995) 28-37.

[7] Gupta M.M. and Qi J. : Design of Fuzzy Logic Controllers Based on Generalized T-operators. Fuzzy
Sets and Systems, 40. (1991) 473-489.

[8] Palm R., Driankov D. and Hellendoorn H. : Model Based Fuzzy Control. Springer-Verlag, Berlin
Heidelberg (1997).

[9] Sakly A., Ben Othman K. and Benrejeb M. : Sur le Choix de l’Opérateur de Conjonction en Com-
mande Floue Basé sur la Robustesse de Critères de Performances. 2ème Conférence Internationale
JTEA, Tome 1, Sousse Nord (2002).

[10] Sugeno M. : On stability of fuzzy systems expressed by fuzzy rules with singleton consequences.
IEEE Transactions on Fuzzy Systems, 7(2). (1999) 201-223.

[11] Tanaka K., Ikeda T. and Wang H.O. : Fuzzy Regulators and Fuzzy Observers : Relaxed Stability
Conditions and LMI-based Designs. IEEE Transactions on Fuzzy Systems, 6(2). (1998) 1-16.

[12] Tanaka K. and Sugeno M. : Stability analysis and design of fuzzy control systems. Fuzzy sets and
systems, 45. (1992) 135-156.

[13] Tanaka K. and Sano M. : A Robust Stabilization Problem of Fuzzy Control Systems and its Ap-
plication to Backing up Control of a Truck-Trailer. IEEE Transactions on Fuzzy Systems, II. (1994)
119-134.

[14] Yager R. R. and Filev D. P. : Essentials of Fuzzy Modeling and Control. John Wiley & Sons, Inc.,
USA (1994).



New Discrete Tanaka Sugeno Kang Fuzzy Systems Characterization and Stability Domain 19

Mohamed BENREJEB1, Dhaou SOUDANI1, Anis SAKLY1, Pierre BORNE2

1Ecole Nationale d’Ingénieurs de Tunis
Unité de Recherche LARA-Automatique

BP 37, Le Belvédère, 1002 Tunis, Tunisie

2Ecole Centrale de Lille, Cité scientifique
Laboratoire d’Automatique, Génie Informatique et Signal

BP 48, 59651 Villeneuve d’Ascq Cedex, France

E-mail: mohamed.benrejeb@enit.rnu.tn, dhaou.soudani@enit.rnu.tn,
sakly_anis@yahoo.fr, pierre.borne@ec-lille.fr

Received: November 10, 2006