International Journal of Control, Automation, and Systems (2011) 9(3):581-587 DOI 10.1007/s12555-011-0319-8

http://www.springer.com/12555

Stabilization of Saturated Discrete-Time Fuzzy Systems

Abdellah Benzaouia, Said Gounane, Fernando Tadeo, and Ahmed El Hajjaji

Abstract: This paper presents sufficient conditions for the stabilization of discrete-time fuzzy systems,

subject to actuator saturations, by using state feedback control laws. Two different methods are pre-

sented and compared. The obtained results are formulated in terms of LMI’s. A real plant model illu-

strates the proposed techniques.

Keywords: Discrete-time systems, LMI, saturated control, stabilization, T-S fuzzy systems.

1. INTRODUCTION

It is known that the qualitative knowledge of a system

can be represented by a nonlinear model. This idea has

allowed the emergence of a new design approach in the

fuzzy control field. The nonlinear system can be

represented by a Takagi-Sugeno (T-S) fuzzy model [20-

22]. The control design is then carried out using known

or recently developed methods from control theory

[10,14,17,18,24].

T-S fuzzy models have proved to be very good

representations for a certain class of nonlinear dynamic

systems. The nonlinear system is represented by a set of

linear models interpolated by membership functions.

Then a model-based fuzzy controller can be developed to

stabilize the T-S fuzzy model, by solving a set of LMI’s

[11,13,16].

A main problem, which is always inherent to all

dynamical systems, is the presence of actuator

saturations. Even for linear systems this problem has

been an active area of research for many years. Two

main approaches have been developed in the literature:

The first is the so-called positive invariance approach,

which is based on the design of controllers that work

inside a region of linear behavior where saturations do

not occur (see [1,2,7] and the references therein). This

approach has been extended to systems modelled by T-S

systems [4,12]. The second approach, allows saturations

to take effect, while guaranteeing asymptotic stability

(see [5,15] and the references therein). This method has

been extended to T-S continuous-time systems in [9].

The main challenge in these two approaches is to obtain

a large enough domain of initial states that ensures

asymptotic stability of the system despite the presence of

saturations [3].

The objective of this paper is to extend the results of

[9] to discrete-time T-S systems subject to actuator

saturations. Thus, two directions are explored, based on

two different methods, one direct and one indirect,

leading to two different sets of LMIs. It is then shown,

by application to a real plant model, that the indirect

method, which uses the idea in [23] is less restrictive

than the direct one, that uses [9]. It is the first time that

this method is applied to T-S fuzzy systems.

This paper is organized as follows: Section 2 deals

with some preliminary results, while the third Section

presents the problem presentation. The main results of

this paper are given in Section 4 together with

application to a real plant model.

2. PRELIMINARIES

This section presents some preliminary results on

which our work is based. Define the following subsets of

:n

( ) 0n TP x x Pxρ ρ ρΩ , = ∈ | ≤ , > , (1)

( ) 1 [1 ]njF x f x j m= ∈ || |≤ , ∈ ,L (2)

with P a positive definite matrix and F∈m×n and fj

stands for the jth row of matrix F. Ω(P, ρ) is an ellipsoid

set, while L(F) is a polyhedral set. The set Ω(P, ρ),

which is an ellipsoid, will be used as a level set of the

Lyapunov function V(x(k))=xT(k)Px(k), while the

polyhedral set L(F) is the set where the saturations do

not occur.

Lemma 1 [15]: Let F, H∈m×n be given matrices, for

,

n

x∈ if ( )x H∈L then

( ) [1 2 ]m

i isat Fx co E Fx E Hx i

−

= + : ∈ ,

with i

E ∈V where

1 ,..., ,... m m

j mM M diag ζ ζ ζ×

= ∈ / =V

© ICROS, KIEE and Springer 2011

__________

Manuscript received January 25, 2008; revised July 30, 2010and January 18, 2011; accepted January 31, 2011. Recommendedby Editor Young-Hoon Joo. Abdellah Benzaouia and Said Gounane are with LAEPT-EACPI URAC 28, University Cadi Ayyad, Faculty of ScienceSemlalia, BP 2390, Marrakech, Morocco (e-mails: benzaouia@ucam.ac.ma, gounane.said@gmail.com). Fernando Tadeo is with Universidad de Valladolid, Depart. deIngenieria de Sistemas y Automatica, 47005 Valladolid, Spain (e-mail: fernando@autom.uva.es). Ahmed El Hajjaji is with University of Picardie Jules Vernes(UPJV), 7 Rue de Moulin Neuf 8000 Amiens, France (e-mail:

ahmed.hajjaji@u-picardie.fr).

Abdellah Benzaouia, Said Gounane, Fernando Tadeo, and Ahmed El Hajjaji

582

with ζj =1 or 0, i

E− =I−Ei and co stands for the convex

hull function.

The main idea of [15] based on Lemma 1, is to build a

third set with matrix H as L(H). This polyhedral set will

be the set where saturations of the control are allowed

without destabilizing the system. It is generally shown

that the set L(H) is larger than the set L(F) [15].

Lemma 2 [9]: Suppose that matrices Di∈m×n i =1,

2,...,r and a positive semi-definite matrix m m

P×

∈ are

given:

if

1

1 0 1

r

i i

i

µ µ

=

= ≤ ≤ ,∑

then

1 1 1

r r r

T T

i i i i i i i

i i i

D P D D PDµ µ µ

= = =

≤ .∑ ∑ ∑

Lemma 3 [19]: Let n

x∈ ,m n

H×

∈ P=PT∈

n×n

such that rank(H) =r < n. The following statements are

equivalent:

) 0 0 0

) 0

T

n m T T

i x Px x Hx

ii X P XH H X×

< ,∀ ≠ , =

∃ ∈ : + + < .

3. PROBLEM STATEMENT

This section presents the problem to be solved.

Consider the fuzzy system described by the following r

rules:

Rule i:

IF θ1(k) is Mi1, …, θq(k) is Miq

THEN ( 1) ( ) ( ( )),i i

x k A x k B sat u k+ = +

where Mij is the fuzzy set and θ(k)=[θ1(k),...,θq(k)]T are

the premise variables.

By using the common used center-average defuzzifier,

product inference and singleton fuzzifier, the T-S fuzzy

systems can be inferred as

( 1) ( ( )) ( ) ( ( )) ( ( ))x k A k x k B k sat u kµ µ+ = + , (3)

where

1 1

11

( ( )) ( ) ( ( )) ( )

( ( ))( ) ( ( )) ( ( ))

( ( ))

r r

i i i i

i i

qi

i i ij jrjii

A k k A B k k B

kk k M k

k

µ µ µ µ

ω θµ ω θ θ

ω θ

= =

=

=

= , = ,

= , = ,

∑ ∑

∏∑

and Mij(θj (k)) is the grade of membership of Mij.

The saturation function is defined as follows:

1 if ( ) 1

( ( )) ( ) if 1 ( ) 1

1 if ( ) 1.

i

i i i

i

u k

sat u k u k u k

u k

, >

= , − ≤ ≤ − < −

(4)

Based on the Parallel Distribution Control (PDC)

structure [22], we consider the following fuzzy control

law for the fuzzy system (3):

Rule ii:

IF θ1(k) is Mi1 … θq(k) is Miq

THEN u(k)=Fix(k).

The overall state feedback fuzzy control law can then

be represented as:

1

( ) ( ) ( )r

i i

i

u k k F x kµ

=

= .∑ (5)

The objective of this work is to develop sufficient

conditions of asymptotic stability of the T-S fuzzy

system in closed-loop in presence of saturated control.

These conditions will enable one to obtain a large set of

initial values where the saturations of the control are

allowed.

4. MAIN RESULTS

This section presents the main results that consists of

two sufficient conditions of asymptotic stability of the T-

S system in closed-loop, under the form of two sets of

LMIs.

4.1. Direct method

In this subsection, a direct method is used to derive

sufficient conditions of asymptotic stability based on a

common quadratic Lyapunv function candidate.

Theorem 1: For a given fuzzy system (3), suppose

that the local state feedback control matrices Fj, j =1,...,r,

are given. The ellipsoid Ω(P, ρ) is a contractively

invariant set of the closed-loop system under the fuzzy

control law (5) if there exist matrices Hjm n×

∈ , j∈ [1,r]

such that

0 [1 ] [1 2 ]T mijs ijsA PA P i j r s− < ∀ , ∈ , , ∀ ∈ , (6)

1

( ) ( )r

j

j

P Hρ

=

Ω , ⊂ ,∩ L (7)

where

[ ]ijs i i s j s jA A B E F E H−

= + + .

Proof: For any 1

( )r

jjx H

=

∈ ,∩ L since 1

1r

iiµ

=

=∑

and 0 1iµ≤ ≤ we have that:

1

.

r

j j

j

x Hµ

=

∈

∑L

Then by Lemma 1,

1

2

1 1 1

( ) ( )

( ) ( ) ( ) ( )

m

r

j j

j

r r

s s j j s j j

s j j

sat k F x k

k E k F E k H x k

µ

η µ µ

=

− = = =

= + ,

∑

∑ ∑ ∑

so, one can have

Stabilization of Saturated Discrete-Time Fuzzy Systems

583

2

1 1 1

( 1) ( ) ( )

mr r

ijs ijs

i j s

x k k A x kν

= = =

+ =∑∑∑

with [ ]ijs i i s j s jA A B E F E H−

= + + , and ( ) ( )ijs ik kν µ=

( ) ( )j sk kµ η .

Then, (3) becomes

( 1) ( ( )) ( )x k A k x kµ+ = ,

where

2

1 1 1

( ( )) ( )

mr r

ijs ijs

i j s

A k k Aµ ν

= = =

= .∑∑∑ (8)

Select the Lyapunov function candidate

( ( )) ( ) ( )TV x k x k Px k= .

Computing its rate of increase gives

2

1 1 1

2

1 1 1

2

1 1 1

( ( )) ( ) ( ( )) ( ( )) ( )

( ) ( )

( ) ( )

( ) ( ) ( ) ( )

m

m

m

T T

r rT T

ijs ijs

i j s

r r

ijs ijs

i j s

r rT T

i j s ijs

i j s

V x k x k A k PA k P x k

x k k A P

k A P x k

x k k k k A P

µ µ

ν

ν

µ µ η

= = =

= = =

= = =

∆ = −

=

× −

=

∑∑∑

∑∑∑

∑ ∑ ∑

2

1 1 1

( ) ( ) ( ) ( )

mr r

i j s ijs

i j s

k k k A P x kµ µ η

= = =

× −

∑ ∑ ∑

for all

1

( )r

j

j

x H

=

∈ .∩ L

By applying Lemma 2: ( ( ))V x k∆ ≤

2

1 1 1

( ) ( ) ( )

mr r

T Tijs ijs ijs

i j s

x k k A PA P x kν

= = =

− . ∑∑∑

This inequality is equivalent to ( ( ))V x k∆ ≤

( )2

1 1 1

( ) ( ) ( )

mr r

T Tijs ijs ijs

i j s

x k k A PA P x kν

= = =

− . ∑∑∑

It is easy to see that ( ( )) 0V x k∆ < if

0 [1 ] [1 2 ]T mijs ijsA PA P i j r s− < ,∀ , ∈ , , ∀ ∈ ,

and

1

( ) ( )r

j

j

P Hρ

=

Ω , ⊂ .∩ L

In order to synthesize the controller, we give the

following result:

Corollary 1: For a given fuzzy system (3), if there

exist a symmetric positive definite matrix Q∈n×n and

matrices Yj∈m×n, Zj∈

m×n, j∈ [1, r] and X∈n×n such

that

( )0

*

TT

i i s j s jX X Q A B E Y E Z

Q

− + − + + >

(9)

[1 ] [1 2 ]m

i j r s∀ , ∈ , , ∀ ∈ ,

and

1

0

*

jl

T

z

X X Q

ρ

≥ + −

(10)

[1 ] [1 ]j r l m∀ ∈ , , ∀ ∈ , ,

where * denotes the transpose of the off-diagonal

element, then the ellipsoid Ω(Q−1, ρ) is a contractively

invariant set of the closed-loop system (3), with

1 1 1and .

i i i iF Y X H Z X P Q

− − −

= , = =

Proof: Assume that conditions (9)-(10) hold. Then the

inequality (6) in Theorem 1 is equivalent to:

0T T T

ijs ijsX A PA X X PX− < ,

[1 ] [1 2 ],m

i j r s∀ , ∈ , , ∀ ∈ , and for all nonsingular

matrix X∈n×n.

Let Q = P−1 then we have

1 10

T T Tijs ijsX A Q A X X Q X

− −

− < ,

[1 ] [1 2 ]m

i j r s∀ , ∈ , , ∀ ∈ , .

By Schur complement, it is equivalent to:

1

0

[1 ] [1 2 ].

T T Tijs

ijs

m

X Q X X A

A X Q

i j r s

− > ,

∀ , ∈ , , ∀ ∈ ,

(11)

Since (X−Q)TQ−1(X−Q) > 0, it follows that X

TQ−1X ≥

XT + X − Q. Then ∆V(x(k)) < 0 if

( )0

*

[1 ] [1 2 ]

TT T

i i s j s j

m

X X Q X A B E F E H

Q

i j r s

− + − + + > ,

∀ , ∈ , , ∀ ∈ , .

(12)

To obtain an LMI, let Yj =Fj X and Zj =Hj X. Then

condition (12) will be equivalent to

( )0

*

[1 ] [1 2 ]

TT

i i s j s j

m

X X Q A X B E Y E Z

Q

i j r s

− + − + + > ,

∀ , ∈ , , ∀ ∈ , .

Consider now the condition (7) in Theorem 1, which is

Abdellah Benzaouia, Said Gounane, Fernando Tadeo, and Ahmed El Hajjaji

584

equivalent to [8]:

1 1 [1 ] [1 ],T

jl jlh P h j r l mρ

−

≤ , ∀ ∈ , , ∀ ∈ ,

where hjl is the lth row of Hj. This inequality is equivalent

to

1 1 1T T Tjl jlh XX P X X h

ρ

− − −

≤ ,

for any non singular matrix X∈n×n.

By Schur complement, one obtains equivalently

1

0

*

jl

T

h X

X PX

ρ ≥ .

As Q=P−1, then one can have

1

1

0

*

jl

T

h X

X Q X

ρ

−

≥ .

Thus, if

1

0,

*

jl

T

h X

X X Q

ρ ≥ + −

then, the inequality (7) of Theorem 1 is satisfied, hence

the result is obtained. Note that condition (12) implies

X

T + X > 0, that is, X is non singular.

4.2. Indirect method

In this subsection, an indirect method is used to derive

sufficient conditions of asymptotic stability by using a

common quadratic Lyapunov function candidate.

Theorem 2: For a given fuzzy system (3), suppose

that the local state feedback control matrices Fj, j =1,...,r,

are given. The ellipsoid Ω(P, ρ) is a contractively

invariant set of the closed-loop system under the fuzzy

control law (5) if there exist matrices Hj∈m×n, j∈ [1,r],

N1 and N2∈n×n such that

1 1 2 1

1 2

0

T T T Tijs ijs ijsN A A N P A N N

P N N

+ − −

<

− −

(13)

[1 ] [1 2 ]m

i j r s∀ , ∈ , , ∀ ∈ ,

and

1

( ) ( )r

j

j

P Hρ

=

Ω , ⊂ ,∩ L (14)

where [ ].ijs i i s j s jA A B E F E H−

= + +

Proof: Let ( ( )) ( ) ( ).TV x k x k Px k= Then

( ( )) ( 1) ( 1) ( ) ( )T TV x k x k Px k x k Px k∆ = + + −

( ( )) 0V x k∆ < is equivalent to

( 1) ( 1) ( ) ( ) 0

( 1) ( ( )) ( ),

T Tx k Px k x k Px k

x k A k x kµ

+ + − <

+ =

which is also equivalent to

[ ]

0 ( )( ) ( 1) 0

0 ( 1)( )

( )( ( )) 0.

( 1)

T TP x k

x k x kP x k

x kA k I

x kµ

− + < +

Σ − = +

By virtue of Lemma 3, ( )Σ is also equivalent to:

there exists a matrix X such that:

0[ ( ( )) ] [ ( ( )) ] 0.

0

T T TP

X A k I A k I XP

µ µ−

+ − + − <

Let X=1

2

N

N

then ( ( )) 0V x k∆ < if:

1

1 2

2

0 ( ( ))[ ( ( )) ] [ ] 0

0

T

T TNP A k

A k I N NNP I

µµ

− + − + < .

−

By using (8) one can get

2

1 1 1

1 1 1 2

2 1 2 2

( ) ( ) ( )

0

mr r

i j s

i j s

T T T Tijs ijs ijs

T Tijs

k k k

P N A A N N A N

N A N P N N

µ µ η

= = =

− + + − +

× < .

− − −

∑ ∑ ∑

A sufficient condition to have ( ( )) 0V x k∆ < is

1 1 2 1

*2 2

0.

T T T Tijs ijs ijs

T

P N A A N A N N

P N N

− + + −

<

− −

In order to synthesize the controller, we give the

following result:

Corollary 2: For a given fuzzy system (3) and a given

,α ∈ if there exist a symmetric positive definite

matrix Q∈n×n and matrices Yj, Zj∈

m×n, j∈ [1,r] and

X∈n×n such that

2*

0

T Tijs ijs ijs

T

Q X

Q X X

α

α α α

Γ +Γ − − +Γ<

− −

(15)

[1 ] [1 2 ]m

i j r s∀ , ∈ , , ∀ ∈ ,

and

*

1( )

0 [1 ] [1 ]j lZ

j r l m

Q

ρ

≥ ∀ ∈ , , ∀ ∈ , ,

(16)

where [ ],ijs i i s j s jA X B E Y E Z−

Γ = + + then the ellipsoid

Ω(Q−1, ρ) is a contractively invariant set of the closed-

loop system (3), with

1 1 1and

T

i i i iF Y X H Z X P X QX

− − − −

= , = = .

Proof: In (13) let 1 2.T

i iN X i

−

= = ,

Pre and post -multiplying inequality (13) by

1 2( )T Tdiag X X, and

1 2( )diag X X, , respectively one

Stabilization of Saturated Discrete-Time Fuzzy Systems

585

can get equivalently:

1 2

*2 2 2 2

0

T Tijs ijs

T T

X A X

X PX X X

φ

−< ,

− −

[1 ] [1 2 ],m

i j r s∀ , ∈ , ∀ ∈ ,

where 1 1 1 1

T T Tijs ijs ijsA X X A X PXφ = + − .

Let 2 1

X Xα= and 1 1

.TQ X PX= Then,

1 1

21 1 1 1

0

T Tijs ijs

T Tijs

X X A

A X X Q X X

φ α

α α α α

− +

< ,− − −

[1 ] [1 2 ]m

i j r s∀ , ∈ , ∀ ∈ , .

Let X1=X then (15) is obtained. Note that (15) implies

that matrix X is non singular.

Further, one can show that the inequality (14) is

equivalent to [8]:

1 1Tjl jlh P h

ρ

−

≤ where jlh is the thl row of .jH

That is, 1 1 1T T Tjl jlh XX P X X h

ρ

− − −

≤ .

By Schur complement, one gets

1

0

*

jl

T

h X

X PX

ρ

≥ .

Since Q = XTPX and Zj=Hj X then

1

0

*

jlz

Q

ρ

≥ .

5. STUDY OF A PLANT MODEL

In order to illustrate the obtained results, consider the

balancing-up control of a simulated truck trailer

proposed in [21]:

1 1( 1) (1 ( )) ( ) ( ) ( ),x k v t L x k v t L u k+ = − . / + . /

2 2 1( 1) ( ) ( ) ( ),x k x k v t L x k+ = + . /

3 3 2 1( 1) ( ) [ ( ) ( 2 ) ( )].x k x k v t sin x k v t L x k+ = + . . + . /

The membership functions of this model are

represented as

1 2 1

( ( ))1

( )

sin kM M M

k

θ

θ= , = − .

The nonlinear model of the vehicle can be described by

the two following rules:

Rule 1:

if

1

2

( )( ) ( )

2

v t x kk x k

Lθ

. .= +

is about 0, then

1 1( 1) ( ) ( ).x k A x k B u k+ = +

Rule 2:

If

1

2

( )( ) ( )

2

v t x kk x k

Lθ

. .= +

is about π or –π, then

2 2( 1) ( ) ( ),x k A x k B u k+ = +

where

1 1

2 2

2 2

2 2

1 0 0

1 0 0 ,

02 1

1 0 0

1 0 0

02 1

v t L v t l

A v t L B

v t L v t

v t L v t l

A v t L B

d v t L d v t

− . / . / = . / , = . / .

− . / . / = . / , = . . / . .

with x=[x1 x2 x3]T, l =2.8m, L=5.5m, v =−1m/s, t =2s,

d = 0.01/π.

The results X, Yi, Zi, Q of Corollary 1, obtained with the

LMI toolbox of Matlab, lead to:

[ ]1 21 2609 0 6759 0 0711 ,H H= = . − . .

[ ]1 22 4052 1 3751 0 1456 ,F F= = . − . .

4 5391 4 0483 0 4266

4 0483 6 2350 0 6541 .

0 4266 0 6541 0 1545

P

. − . . = − . . − . . − . .

The results X, Yi, Zi, Q of Corollary 2, obtained with the

LMI toolbox of Matlab, lead to:

[ ]1 20 0832 0 0900 0 0335 ,H H= = − . − . − .

[ ]1 24 5087 5 5010 0 4179 ,F F= = − . − . − .

0 0232 0 0107 0 0034

0 0107 0 0424 0 0029 .

0 0034 0 0029 0 0091

P

. . . = . . − . . − . .

The results of both corollaries are shown in Fig. 1.

Fig. 1. Inclusion of the ellipsoid inside the polyhedral set

given by direct (in black) and indirect method (in

grey).

Abdellah Benzaouia, Said Gounane, Fernando Tadeo, and Ahmed El Hajjaji

586

Comment: It is obvious that the use of Lemma 3

introduces an additional degree of freedom with the

parameter α leading to less conservative LMIs as

reported by [23] for linear systems. The obtained

ellipsoid sets of asymptotic stability obtained with these

two methods, for the studied example, are presented in

Fig, 1. It is clear that the one corresponding to the second

method is less conservative as expected, even the LMIs

are applied without any optimization program.

6. CONCLUSION

This paper presents stability analysis and design

methods for nonlinear systems with actuator saturation.

T-S fuzzy models with actuator saturation are used to

describe the nonlinear system. Two different methods,

one direct and one indirect are used to derive sufficient

conditions of asymptotic stability of T-S fuzzy systems

with saturated control. Finally, these design

methodologies are illustrated by their application to the

stabilization of a balancing-up truck trailer. It is shown

that the indirect method leads to less conservative LMIs

since it leads to more larger stability domains.

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Abdellah Benzaouia was born in At-

taouia (Marrakech) in 1954. He received

the degree of Electrical Engineering at

the Mohammedia School (Rabat) in 1979

and the Doctorat (Ph.D.) at the Universi-

ty Cadi Ayyad in 1988. He is actually

professor at the University of Cadi Ayyad

(Marrakech) where he is also the head of

a team of research on Robust and Con-

strained Control (EACPI). His research interests are mainly

constrained control, robust control, pole assignment, systems

with Markovian jumping parameters, hybrid systems and

greenhouses. He collaborates with many teams in France, Can-

ada, Spain and Italy

Said Gounane was born in Mesfioua

(Marrakech), Morocco, in 1979. He re-

ceived his master’s degrees with specia-

lisation in regulation of industrial system

from Faculty of Sciences Semlalia, Mar-

rakech, Morocco. He is preparing his

Ph.D. on fuzzy systems.

Fernando Tadeo was born in 1969. He

received his B.Sc. degree in physics in

1992, and in Electronic Engineering in

1994, both from the University of Valla-

dolid, Spain. After he received his M.Sc.

degree in control engineering from the

University of Bradford, U.K., he got his

Ph.D. degree from the University of Val-

ladolid, Spain, in 1996. Since 1998 he is

Lecturer (‘Professor Titular’) at the University of Valladolid,

Spain. His current research interests include robust control,

process control, control of systems with constraints and rein-

forcement learning, applied in several areas, from

neutralization processes to robotic manipulators.

Ahmed El Hajjaji received his Ph.D.

and “Habilitation à diriger la recherche”

degrees in automatic control from the

University of Picardie Jules Verne,

France, in 1993 and 2000, respectively.

He was an Associate Professor in the

same university from 1994 to 2003. He is

currently a Full Professor and Director of

the Professional Institute of Electrical

Engineering and Industrial Computing, University of Picardie

Jules Verne. Since 2001, he has also been also the head of the

research team of control and vehicle of Modeling, Information

and Systems (MIS) laboratory. His current research interests

include fuzzy control, vehicle dynamics, fault-tolerant control,

neural networks, mangles systems, and renewable energy sys-

tems.