Stabilization of Saturated Discrete-Time Fuzzy Systems

7/28/2019 Stabilization of Saturated Discrete-Time Fuzzy Systems

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International Journal of Control, Automation, and Systems (2011) 9(3):1-7DOI

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 LMIs. 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 hasallowed 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 knownor recently developed methods from control theory[10,14,17,18,24].

T-S fuzzy models have proved to be very goodrepresentations for a certain class of nonlinear dynamicsystems. 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 LMIs[11,13,16].

A main problem, which is always inherent to alldynamical systems, is the presence of actuator

saturations. Even for linear systems this problem hasbeen an active area of research for many years. Twomain 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 workinside a region of linear behavior where saturations do

not occur (see [1,2,7] and the references therein). Thisapproach has been extended to systems modelled by T-Ssystems [4,12]. The second approach, allows saturationsto take effect, while guaranteeing asymptotic stability(see [5,15] and the references therein). This method hasbeen 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 ensuresasymptotic stability of the system despite the presence ofsaturations [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 indirectmethod, which uses the idea in [23] is less restrictivethan the direct one, that uses [9]. It is the first time thatthis method is applied to T-S fuzzy systems.

This paper is organized as follows: Section 2 deals

with some preliminary results, while the third Sectionpresents the problem presentation. The main results of

this paper are given in Section 4 together withapplication to a real plant model.

2. PRELIMINARIES

This section presents some preliminary results onwhich our work is based. Define the following subsets of

:n

( ) { 0}n T

P x x Px , = | , > , (1)

( ) { 1 [1 ]}n jF x f x j m= || | , ,L (2)

with P a positive definite matrix and Fmn and fj

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

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

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

mn be given matrices, for

,

n

x if ( )x HL then

( ) { [1 2 ]}mi i

sat Fx co E Fx E Hx i

= + : ,

withi

E V where

1{ { ,..., ,... }}m m j mM M diag

= / =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: [email protected], [email protected]).

Fernando Tadeo is with Universidad de Valladolid, Depart. deIngenieria de Sistemas y Automatica, 47005 Valladolid, Spain (e-mail: [email protected]).

Ahmed El Hajjaji is with University of Picardie Jules Vernes(UPJV), 7 Rue de Moulin Neuf 8000 Amiens, France (e-mail:[email protected]).


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Abdellah Benzaouia, Said Gounane, Fernando Tadeo, and Ahmed El Hajjaji2

with j =1 or 0, iE =IEi and co stands for the convex

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

third set with matrixHas L(H). This polyhedral set will

be the set where saturations of the control are allowedwithout 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 Dimn i=1,

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

P

aregiven:if

1

1 0 1

r

i i

i

=

= ,

then

1 1 1

r r r

T T

i i i i i i i

i i i

D P D D PD

= = =

.

Lemma 3 [19]: Let nx ,m n

H

P=PT

nn

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

) 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 rrules:

Rule i:

IF 1(k) isMi1, , q(k) isMiq

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

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

where Mij is the fuzzy set and (k)=[1(k),...,q(k)]T are

the premise variables.By using the common used center-average defuzzifier,

product interference and singleton fuzzifier, the T-Sfuzzy systems can be inferred as

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

= =

=

=

= , = ,

= , = ,

andMij(j (k)) is the grade of membership ofMij.

The saturation function is defined as follows:

1 if ( ) 1

( ( )) ( ) if 1 ( ) 1

1 if ( ) 1.

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 controllaw for the fuzzy system (3):

Rule ii:

IF 1(k) isMi1 q(k) isMiq

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 k

=

= . (5)

The objective of this work is to develop sufficientconditions of asymptotic stability of the T-S fuzzysystem in closed-loop in presence of saturated control.These conditions will enable one to obtain a large set ofinitial values where the saturations of the control are

allowed.

4. MAIN RESULTS

This section presents the main results that consists oftwo sufficient conditions of asymptotic stability of the T-S system in closed-loop, under the form of two sets ofLMIs.

4.1. Direct method

In this subsection, a direct method is used to derivesufficient 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 matricesFj,j=1,...,r,are given. The ellipsoid (P,) is a contractively

invariant set of the closed-loop system under the fuzzycontrol law (5) if there exist matricesHj

m 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 H

=

, , L (7)

where

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

= + + .

Proof: For any1

( )r

jjx H

=

, L since 1 1r

ii

=

=

and 0 1i we have that:

1

.

r

j j

j

x H

=

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

=

= = =

= + ,

so, one can have


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Stabilization of Saturated Discrete-Time Fuzzy Systems 3

2

1 1 1

( 1) ( ) ( )

mr r

ijs ijs

i j s

x k k A x k

= = =

+ =

with [ ]ijs i i s j s jA A B E F E H

= + + , and ( ) ( )ijs ik k =

( ) ( )j sk k .

Then, (3) becomes

( 1) ( ( )) ( )x k A k x k+ = ,

where

2

1 1 1

( ( )) ( )

mr r

ijs ijs

i j s

A k k A

= = =

= . (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

= = =

= = =

= = =

=

=

=

2

1 1 1

( ) ( ) ( ) ( )

mr r

i j s ijs

i j s

k k k A P x k

= = =

for all

1

( )r

j

j

x H

=

. L

By applying Lemma 2: ( ( ) )V x k

2

1 1 1

( ) ( ) ( )

mr rT T

ijs ijs ijs

i j s

x k k A PA P x k

= = =

.

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

( )2

1 1 1

( ) ( ) ( )

mr rT T

ijs ijs ijs

i j s

x k k A PA P x k

= = =

.

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 H

=

, . L

In order to synthesize the controller, we give thefollowing result:

Corollary 1: For a given fuzzy system (3), if thereexist a symmetric positive definite matrix Q

nn andmatrices Yj

mn, Zjmn,j [1, r] andX

nn suchthat

( )0

*

TTi 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

+

(10)

[1 ] [1 ]j r l m , , , ,

where * denotes the transpose of the off-diagonalelement, then the ellipsoid (Q1,) is a contractivelyinvariant set of the closed-loop system (3), with

1 1 1and .

i i i i F Y X H Z X P Q

= , = =

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

matrixXnn.

Let Q =P1 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 (XQ)TQ1(XQ) > 0, it follows that XTQ1XXT+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 =FjX and Zj =HjX. 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


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Abdellah Benzaouia, Said Gounane, Fernando Tadeo, and Ahmed El Hajjaji4

equivalent to [8]:

1 1 [1 ] [1 ],T

jl jlh P h j r l m

, , , ,

where hjl is the lth

row ofHj. This inequality is equivalentto

1 1 1T T T

jl jlh XX P X X h

,

for any non singular matrixXnn.

By Schur complement, one obtains equivalently

1

0

*

jl

T

h X

X PX

.

As Q=P1, then one can have

1

1

0

*

jl

T

h X

X Q X

.

Thus, if

1

0,

*

jl

T

h X

X X Q

+

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

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

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

4.2. Indirect methodIn this subsection, an indirect method is used to derive

sufficient conditions of asymptotic stability by using acommon quadratic Lyapunov function candidate.

Theorem 2: For a given fuzzy system (3), supposethat the local state feedback control matricesFj,j=1,...,r,are given. The ellipsoid (P,) is a contractively

invariant set of the closed-loop system under the fuzzycontrol law (5) if there exist matricesHj

mn,j[1,r],N1 andN2

nn such that

1 1 2 1

1 2

0

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

P N N

+

Stabilization of Saturated Discrete-Time Fuzzy Systems

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