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Stabilisation of discrete 2D time switching systems bystate feedback controlAbdellah Benzaouia a , Abdelaziz Hmamed b , Fernando Tadeo c & Ahmed EL Hajjaji da Faculty of Science Semlalia , LAEPT-EACPI,University Cadi Ayyad , P.B. 2390, Marrakech,Moroccob Faculty of Sciences Dhar EL Mhraz , LESSI, University Mohamed Ben Abdellah , BP 1796,Fes, Moroccoc Departamento de Ingenieria de Sistemas y Automatica , Universidad de Valladolid , 47005Valladolid, Spaind MIS – University of Picardie Jules-Vernes , 7 Rue du Moulin Neuf, 80000 Amiens, FrancePublished online: 26 Nov 2010.

To cite this article: Abdellah Benzaouia , Abdelaziz Hmamed , Fernando Tadeo & Ahmed EL Hajjaji (2011) Stabilisation ofdiscrete 2D time switching systems by state feedback control, International Journal of Systems Science, 42:3, 479-487, DOI:10.1080/00207720903576522

To link to this article: http://dx.doi.org/10.1080/00207720903576522

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International Journal of Systems ScienceVol. 42, No. 3, March 2011, 479–487

Stabilisation of discrete 2D time switching systems by state feedback control

Abdellah Benzaouiaa*, Abdelaziz Hmamedb, Fernando Tadeoc and Ahmed EL Hajjajid

aFaculty of Science Semlalia, LAEPT-EACPI,University Cadi Ayyad, P.B. 2390, Marrakech, Morocco; bFaculty of SciencesDhar EL Mhraz, LESSI, University Mohamed Ben Abdellah, BP 1796, Fes, Morocco; cDepartamento de Ingenieria deSistemas y Automatica, Universidad de Valladolid, 47005 Valladolid, Spain; dMIS – University of Picardie Jules-Vernes,

7 Rue du Moulin Neuf, 80000 Amiens, France

(Received 17 July 2009; final version received 6 December 2009)

This article deals with sufficient conditions of asymptotic stability for discrete two-dimensional (2D) timeswitching systems represented by a model of Roesser type with state feedback control. This class of systems cancorrespond to 2D state space or 2D time space switching systems. This work is based on common and multipleLyapunov functions. The results are presented in LMI form. Two examples are given to illustrate the results.

Keywords: switching systems; 2D time systems; Roesser model; stabilisability; multiple Lyapunov functions;LMIs

1. Introduction

In many modelling problems of physical processes,a two-dimensional (2D) switching representation isneeded. One can cite a 2D physically based model foradvanced power bipolar devices (Igic, Towers, andMawby 2004) and heat flux switching and modulatingin a thermal transistor (Lo, Wang, and Li 2008). Thisclass of systems can correspond to 2D state space or2D time space switching systems.

Switched systems are a class of hybrid systemsencountered in many practical situations that involve

different modes of operation (corresponding to differ-

ent subsystems). Thus, a switching system consists of a

family of subsystems and a set of rules that decides the

switching between them. This class of systems has

numerous applications in the control of mechanical

systems, the automotive industry, aircraft and air

traffic control, switching power converters and many

other fields. According to the classification given in

Blanchini and Savorgnan (2006), two main problems

have been studied in the literature: the first problem

is to obtain testable conditions that guarantee the

asymptotic stability under arbitrary switching rules

(Mignone, Ferrari-Trecate, and Morari 2000; Daafouz,

Riedinger, and Iung 2002; Bara and Boutayeb, 2006);

the second problem is to determine a switching

sequence that renders the switched system asymptoti-

cally stable (see Liberzon and Morse 1999; Lin and

Antsaklis, 2007 and references therein). The extension

to saturated switching linear systems is given in Boukas

and Benzaouia (2002), Benzaouia, Saydy, and Akhrif

(2004), Benzaouia, Akhrif, and Saydy (2006, 2010),

and Benzaouia, DeSantis, Caravani, and Daraoui

(2007), while the positive switching systems are pre-

sented in Benzaouia and Taedo (2010). The most

available methods for switching systems use the

quadratic common Lyapunov function. However, it

was recently proven for switching systems (Shorten

and Narendra, 1997, 1998; Branicky, 1998) that the use

of the multiple Lyapunov function leads to better

results in the sense that a common Lyapunov quadratic

function may not exist while a multiple one exists. This

class of systems has had much development to deal

with almost all the problems in control, as in Lian,

Zhao, and Dimirovski (2009), Liu, Liu, and Xie (2009),

Xiang and Xiang (2009), Wang, Zhao, and Dimirovski

(2009).In the last two decades, the 2D system theory has

been paid considerable attention by many researchers.The 2D linear models were introduced in the 1970s

(Givone and Roesser 1972; Fornasini and Marchesini

1976, 1978) and have found many applications, such asin digital data filtering, image processing (Roesser

1975), modelling of partial differential equations

(Marszalek 1984), etc. It is well known that 2D systems

can be represented by different models such as theRoesser model, Fornasini–Marchesini model and

Attasi model. For a complete description of these

*Corresponding author. Email: benzaouia@ucam.ac.ma

ISSN 0020–7721 print/ISSN 1464–5319 online

� 2011 Taylor & Francis

DOI: 10.1080/00207720903576522

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models and methods to transform the system from onemodel type to another, one can refer the work byKaczorek (1985).

In connection with Roesser and Fornasini–Marchesini models, some important problems such asrealisation, controllability andminimum energy controlhave been extensively investigated (see e.g. Kaczorek,1997). On the other hand, the stabilisation problem isnot fully investigated and still not completely solved.

The stability of 2D discrete linear systems can bereduced to checking the stability of the 2D character-istic polynomial (Wu-Sheng and Lee 1985; Anderson,Agathoklis, Jury, and Mansour 1986). This appears tobe a difficult task for the control synthesis problem. Inthe literature, various types of easily checkable butonly sufficient conditions for asymptotic stability andstabilisation problems for 2D discrete linear systemshave been proposed (Yaz 1985; Lu 1994; Galkowski,Rogers, Xu, Lam, and Owens 2002; Hmamed, Alfidi,Benzaouia, and Tadeo 2008).

This article is interested in discrete 2D time switch-ing systems described with the Roesser model. To thebest of our knowledge, no works have directly consid-ered 2D switching systems with arbitrary switchingsequences before, except for the work of the sameauthors on the stability of 2D switching systems(Benzaouia, Hmamed, and Tadeo 2009). One can onlycite the work of Wu, Shi, Gao, and Wang (2008), wherethe process of switching is considered as a Markovianjumping one. First of all, the present work focuses onthe understanding of the switching occurring for the 2Dsystems. To this end, a new time basis definition isproposed. As a consequence of this, an adequateindicator function is derived to model the discrete 2Dtime switching systems. This background enables one tostudy the stabilisability problem of switching systemswhen the switching sequence is arbitrary. Hence,common Lyapunov quadratic and multiple Lyapunovfunctions are used for this class of systems for the firsttime in this work. In this context, sufficient conditionsof stabilisability are presented. Furthermore, theseconditions are presented in the form of a set of LMIsfor the state feedback control case.

This article is organised as follows. Section 2 dealswith the problem statement while some preliminaryresults are given in Section 3. The main results of thiswork with illustrative examples are presented inSection 4.

2. Problem statement

In this section, the system under study is presented.Particular attention is given to understanding theswitching of 2D systems. Hence, to model discrete-time

switching systems, a new time basis definition togetherwith the adequate indicator function are developed.

This article deals with switching systems describedby the following Roesser model:

xþðk, l Þ ¼ A�xðk, l Þ þ B�uðk, l Þ,

xhð0, l Þ ¼ f ðl Þ, xvðk, 0Þ ¼ gðkÞð1Þ

with

xðk, l Þ ¼xhðk, l Þ

xvðk, l Þ

" #, xþðk, l Þ ¼

xhðkþ 1, l Þ

xvðk, lþ 1Þ

" #, ð2Þ

A� ¼A�11 A�12

A�21 A�22

� �, B� ¼

B�1

B�2

� �, ð3Þ

where xh(k, l ) is the horizontal state in Rn1, xv(k, l ) is

the vertical state in Rn2, x(k, l ) is the whole state in R

n

with n¼ n1þ n2, u(k, l ) is the control vector in Rm, � is

a switching rule which takes its values in the finite setI :¼ {1, . . . ,N}, N is the number of subsystems (modes)and k and l are integers in Zþ. MatricesA�112R

n1�n1 , A�122Rn1�n2 , A�212R

n2�n1 , A�222Rn2�n2 , B�12

Rn1�m, B�22R

n2�m are constant.It is assumed that

. the switching system is stabilisable;

. at each time only one subsystem is active;

. the switching rule � is not known a priori but itsvalue is available at each sampling period.

As reported by the cited references, the third assump-tion corresponds to practical implementations wherethe switching system is supervised by a discrete-eventsystem, or operator, allowing for the value of � to beknown at each sampling period in real time.

The evolution of the 2D system (1) shows that (k, l )is varying according to one direction at each time: tocompute the horizontal state (resp. the vertical state) atcoordinate (kþ 1, l )(resp. (k, lþ 1)), one needs to knowboth the horizontal and vertical states at coordinate(k, l ). For example, to compute the states at kþ l¼ 2(coordinates of the points (2, 0), (1, 1) and (0, 2)situated on the same line, as shown in Figure 1), oneneeds to use the states for all the possible values of(k, l ) with kþ l¼ 1 (coordinates of the points (1, 0) and(0, 1) situated on the same line, as shown in Figure 1).That is, to compute the states xh(kþ 1, l ), xv(k, lþ 1),kþ l¼ 1, one has to compute the states xh(1, 1),xv(1, 1), xh(2, 0), xv(0, 2) by using xh(0, 1), xv(0, 1),xh(1, 0), xv(1, 0). This fact can be shown by Figure 1 fork, l¼ 0, . . . , 3. Thus, the 2D system causality imposesan increment depending only on �¼ kþ l. For this, theswitch can be assumed to occur only at each samplingof k or l, that is each �.

The following definition extends the one given for1D by Lygeros, Tomlin, and Sastry (1999) based on �.

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Definition 2.1: A hybrid time basis � is a finite or

infinite sequence of sets Ii ¼ f�2Zþ : �i � � � ��ig, with��i ¼ �iþ1 for i2L¼ {0, . . . ,L},

Si2L Ii � Zþ, where

system (1) evolves with �¼ i. If the sequence is finite,

one may take card(�)¼Lþ 151 and ��L ¼ 1, then

IL may be of the form IL¼ {�2Zþ : �L� �51}.

In this work, we are interested on the synthesis of a

stabilising controller for this class of hybrid systems.

The following state feedback control law is used:

uðk, l Þ ¼ F�xðk, l Þ ð4Þ

with

F� ¼ ½F1� F2

��, ð5Þ

where matrices F 1� 2R

m�n1 , F 2� 2R

m�n2 , which writes

the closed-loop system as

xþðk, l Þ ¼ ½A� þ B�F��xðk, l Þ: ð6Þ

Since the switching is assumed to occur only at each �,the following indicator function is proposed:

�ð�Þ ¼ ½�1ð�Þ, . . . , �Nð�Þ�T, ð7Þ

if the switching system is in mode i, then �i(�)¼ 1 and

�j(�)¼ 0 for j 6¼ i, so one can write the closed-loop

system (6) as follows:

xþðk, l Þ ¼XNi¼1

�ið�Þ½Ai þ BiFi�xðk, l Þ ¼ Acð�Þxðk, l Þ

ð8Þ

with

Acð�Þ ¼XNi¼1

�ið�ÞðAi þ BiFiÞ: ð9Þ

Note that �ið�Þ � 0;PN

i¼1 �ið�Þ ¼ 1. Hence, theproblem we are dealing with consists of designing thegains F(�) that stabilise the 2D closed-loop switchingsystem (8). Our goal is to propose an algorithm thatuses the LMI framework, to facilitate the computationof the feedback controller gains.

3. Preliminary results

A useful stability result for 2D linear systems, which isbased on the Lyapunov function, is recalled in thissection. For this, consider the following 2Ddiscrete-time autonomous system:

xþðk, l Þ ¼ �Axðk, l Þ: ð10Þ

Theorem 3.1 (Anderson et al. 1986): The 2D system(10) is asymptotically stable if there exists a positivedefinite matrix of the form

P ¼P1 0

0 P2

� �, ð11Þ

with P12Rn1�n

1 and P22Rn2�n

2, such that,

�ATP �A� P5 0 ð12Þ

and the following is a Lyapunov function of the system:

Vðxðk, l ÞÞ ¼ xTðk, l ÞPxðk, l Þ: ð13Þ

Note that this result is obtained with the followingrate of increase:

4Vðxðk, l ÞÞ ¼ xþTðk, l ÞPxþðk, l Þ � xTðk, l ÞPxðk, l Þ:

4. Main result

This section presents sufficient conditions of asympto-tic stability using either a common quadratic ormultiple Lyapunov functions for 2D discrete-timeswitching systems given in the Roesser model. Thedesign using LMIs of the corresponding stabilisingcontrollers is also established.

4.1. Common quadratic Lyapunov function

The following result proposes a necessary and suffi-cient condition for the 2D closed-loop switchingsystem (8) to admit a common quadratic Lyapunovfunction.

Theorem 4.1: The following two statements areequivalent:

(i) There exists a Lyapunov function of the form(13) whose difference is negative definite,

0 1 2 30

1

2

3

k

l

Figure 1. How to compute the states at coordinates (kþ 1, l )and (k, lþ 1) situated on the same line with respect to theprevious ones at coordinates (k, l ) for k, l¼ 0, . . . , 3 situatedon the same previous line.

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proving global asymptotic stability of the 2Dswitching system (8) for any arbitrary sequenceof switching.

(ii) There exist positive definite matricesP12R

n1�n1 and P22Rn2�n2 such that

P1 0 ð�i11Þ

Tð�i

21ÞT

� P2 ð�i12Þ

Tð�i

22ÞT

� � P�11 0

� � � P�12

26664

377754 0 8i2I ð14Þ

with

�i11 ¼ Ai

11 þ Bi1F

i1, �i

12 ¼ Ai12 þ Bi

1Fi2,

�i21 ¼ Ai

21 þ Bi2F

i1, �i

22 ¼ Ai22 þ Bi

2Fi2

ð15Þ

and � denoting the transpose of the correspondingoff-diagonal term.

Proof 1: (ii)! (i): Consider the Lyapunov functioncandidate V(x(k, l)) given by (13) with the form (11)and compute its rate of increase:

4Vðxðk, l ÞÞ ¼ xþTðk, l ÞPxþðk, l Þ � xTðk, l ÞPxðk, l Þ

¼ xTðk, l Þ Acð�ÞTPAcð�Þ � P� �

xðk, l Þ:

This rate of increase is negative if

Acð�ÞTPAcð�Þ � P5 0: ð16Þ

Using the Schur complement, one obtains

P Acð�ÞTP

� P

" #4 0: ð17Þ

Substituting (9) in (17), leads to

XNi¼1

�ið�ÞP Ai þ BiFi½ �

TP

� P

" #4 0: ð18Þ

A sufficient condition to have (18) is given by

P Ai þ BiFi½ �TP

� P

" #4 0 8i2I : ð19Þ

Further, pre- and post-multiplying (19) by diag{I,P�1} while substituting matrices Ai, Bi, Fi and Paccording to (3), (5) and (11), respectively, inequalities(14) are directly obtained. That is, the 2D switchingsystem (8) is globally asymptotically stable for anysequence of switching.

(i)! (ii): Assume that function V(x(k, l )) given by(13) with the form (11) is a Lyapunov function forsystem (8). Then, its rate of increase 4V(x(k, l ))5 0.This implies that inequality (18) holds true. If theswitching system is in mode i, one has �j(�)¼ 1, for j¼ iand �j(�)¼ 0 for j 6¼ i. That is (19) is satisfied.

Following the same reasoning as before, condition

(14) is obtained. œ

This result can be used to synthesise the stabilising

controller under LMIs form.

Theorem 4.2: If there exist positive definite matrices

X1Rn1�n1 , X22R

n2�n2 and matrices Yi12R

m�n1 and

Yi22R

m�n2 such that

X1 0 ð�i11Þ

Tð�i

21ÞT

� X2 ð�i12Þ

Tð�i

22ÞT

� � X1 0� � � X2

2664

37754 0 8i2I ð20Þ

with

�i11 ¼ Ai

11X1 þ Bi1Y

i1, �i

12 ¼ Ai12X2 þ Bi

1Yi2,

�i21 ¼ Ai

21X1 þ Bi2Y

i1, �i

22 ¼ Ai22X2 þ Bi

2Yi2,

then the 2D switching system (8) is globally asympto-

tically stable for any arbitrary sequence of switching.

The corresponding controller gains are given by

Fi ¼ ½Yi1X�11 Yi

2X�12 � ð21Þ

with P1 ¼ X�11 and P2 ¼ X�12 .

Proof 2: The sufficient condition of stability of the

switching system is given by (16). Pre and post

multiplying by X¼P�1, leads to

X� XAcð�ÞTX�1Acð�ÞX4 0:

Applying the Schur complement gives

X ½Acð�ÞX�T

� X

" #4 0: ð22Þ

Substituting (9) in (22), leads to

XNi¼1

�ið�ÞX AiXþ BiFiX½ �

T

� X

" #4 0: ð23Þ

Let Yi¼FiX . A sufficient condition to have (23) is

X AiXþ BiYi½ �T

� X

" #4 0 8i2I : ð24Þ

By substituting matrices Ai, Bi according to (3),

X¼diag{X1,X2} with X1 ¼ P�11 ,X2 ¼ P�12 and

Yi ¼ ½Yi1 Yi

2� the LMIs (20) are directly obtained.

Finally, the stabilising controller gains of the 2D

switching system (8) are given by Fi¼YiX�1.

Expression (21) follows. œ

Example 4.1: Consider now the example of two long

transmission lines. We assume that one can switch

from a line to another arbitrarily. The system

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equations as presented in Kaczorek (1985) are given by

@uðx, tÞ

@x¼ Lð�Þ

@iðx, tÞ

@tþ vðx, tÞ,

@iðx, tÞ

@x¼ Cð�Þ

@uðx, tÞ

@t,

ð25Þ

where C(�), L(�) are the capacity and the inductance ofthe portion of the line of length 4x, v(x, t) is the controlassumed here to be the voltage per metre in the upsideof the line. Using the approximations:

@uðx, tÞ

@x’

uðk, l Þ � uðk� 1, l Þ

4x,

@uðx, tÞ

@t’

uðk, lþ 1Þ � uðk, l Þ

4t,

ð26Þ

one can use the following change of coordinates forthe current and the voltage: uh(k, l )¼ u(k� 1, l ) anduv(k, l )¼ u(k, l ), rð�Þ ¼ 4t

Lð�Þ4x and sð�Þ ¼ 4tCð�Þ4x. Define

xhðk, l Þ ¼uhðk, l Þihðk, l Þ

� �, xvðk, l Þ ¼

uvðk, l Þivðk, l Þ

� �:

The system (25) can then be transformed to a 2Dswitching Roesser model given by (1) with:

A1 ¼

0 0 1 0

0 0 0 1

0 �r1 1 r1

�s1 0 s1 1

26664

37775; BT

1 ¼ 0 0 0 �r1� �

;

A2 ¼

0 0 1 0

0 0 0 1

0 �r2 1 r2

�s2 0 s2 1

26664

37775; BT

2 ¼ 0 0 0 �r2� �

:

In this example, n1¼ 2, n2¼ 2 and m¼ 1. For r1¼ 0.5,s1¼ 0.8, r2¼ 0.1, s2¼ 0.4, the LMI solver of Matlabis used to solve LMIs (20) leading to feasible solutionsin X1, X2, Y1 and Y2. The matrix P and the gaincontrollers F1, F2 are then derived by using relations(21) as follows:

P1 ¼0:0030 0:0014

0:0014 0:0050

� �, P2 ¼

0:0242 0:0186

0:0186 0:0224

� �,

F1 ¼ �1:0052 �0:8278 3:3565 3:3766� �

,

F2 ¼ �0:9818 �0:8851 12:3782 13:4408� �

:

In order to save space, the trajectories of the states,which are asymptotically stable, are not plotted.

Consider now the following numerical example:

A1 ¼

3 0 1

1 �4 1

�1 5 1

24

35; B1 ¼

1 0

0 1

1 1

24

35,

A2 ¼

1 0 �1

0 2 1

0:4 1 �1

24

35, B2 ¼

1 1

0 1

1 0

24

35:

The LMIs (20) are not feasible.

4.2. Multiple Lyapunov function

Although the existence of a common quadratic

Lyapunov function for the various subsystems guar-

antees the asymptotic stability of the switching system,

such a function is not always possible and might lead

to conservative results. Thus, this subsection studies

the use of multiple Lyapunov functions. These multiple

Lyapunov functions are considered to be a strong tool

in the analysis of the stability of switching systems.

Consider the following multiple Lyapunov function

candidate.

#ðxðk, l ÞÞ ¼ xTðk, l ÞPð�Þxðk, l Þ

¼ xTðk, l Þ

�XNi¼1

�ið�ÞPi

�xðk, l Þ: ð27Þ

Define

Pi ¼Pi1 0

0 Pi2

" #: ð28Þ

The following result proposes a necessary and suffi-

cient condition for the 2D closed-loop switching

system (8) to admit a multiple Lyapunov function.

Theorem 4.3: The following two statements are

equivalent.

(i) There exists a Lyapunov function of the form

(27) whose difference is negative definite, prov-

ing global asymptotic stability of the 2D

switching system (8) for any arbitrary sequence

of switching.(ii) There exist positive definite matrices Pi

12Rn1�n1 ,

Pi22R

n2�n2 , i ¼ 1, . . . ,N such that

Pi1 0 ð�i

11ÞTð�i

21ÞT

� Pi2 ð�

i12Þ

Tð�i

22ÞT

� � ðP j1Þ�1 0

� � � ðP j2Þ�1

2664

37754 0 8ði, j Þ2I2, ð29Þ

where �ij are given by (15).

Proof 3: (ii)! (i): Consider the Lyapunov function

candidate #(x(k, l)) given by (27) with the form (28)

and compute its rate of increase:

4#ðxðk, l ÞÞ ¼ xþTðk, l ÞPþð�Þxþðk, l Þ�xTðk, l ÞPð�Þxðk, l Þ

¼ xTðk, l Þ Acð�ÞTPþð�ÞAcð�Þ�Pð�Þ

� �xðk, l Þ,

with

Pþð�Þ ¼XNj¼1

�j ð�þ 1ÞP j1 0

0 P j2

" #:

Note that

Pþð�Þ ¼XNj¼1

�j ð�þ 1ÞPj: ð30Þ

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This rate of increase is negative if

Acð�ÞTPþð�ÞAcð�Þ � Pð�Þ5 0: ð31Þ

Using the Schur complement, one obtains

Pð�Þ Acð�ÞTPþð�Þ

� Pþð�Þ

" #4 0: ð32Þ

Substituting (9) in (32), leads to

XNi¼1

XNj¼1

�ið�Þ�j ð�þ 1ÞPi Ai þ BiFi½ �

TPj

� Pj

" #4 0:

It follows that the set of inequalities

Pi Ai þ BiFi½ �TPj

� Pj

" #4 0 8ði, j Þ2I 2 ð33Þ

forms a sufficient condition to have 4#(x(k, l ))5 0.Further, pre- and post-multiplying (33) by diagfI,P�1j g,while substituting matrices Ai, Bi, Fi and Pj accordingto (3), (5) and (28), respectively, inequalities (29) aredirectly obtained. That is, the 2D switching system (8)is globally asymptotically stable for any sequence ofswitching.

(i)! (ii): Assume that function #(x(k, l)) given by(27) with the form (28) is a Lyapunov function forsystem (8). Then, its rate of increase 4#(x(k, l ))5 0.This implies that inequality (33) holds true. If theswitching system is in mode i at � and mode j at �þ 1,one has �s(�)¼ 1, �r(�þ 1)¼ 1, for s¼ i, r¼ j and�s(�)¼ 0, �r(�)¼ 0 for s 6¼ i, r 6¼ j. That is, (33) issatisfied. Following the same reasoning as before,condition (29) is obtained. œ

Remark 4.1

. It is worth noting that inequalities (29) can begiven under the following compact form:

Pi ðAi þ BiFiÞT

� P�1j

� �4 0 8ði, j Þ2I 2: ð34Þ

. The results of Benzaouia et al. (2009a, b) can beobtained as a particular case of Theorem 4.3.

The synthesis of the controller can then be derived.

Theorem 4.4: If there exist positive definite matricesXi

12Rn1�n1 , Xi

22Rn2�n2 and matrices Yi

12Rm�n1 and

Yi22R

m�n2 such that

Xi1 0 ð�i

11ÞTð�i

21ÞT

� Xi2 ð�

i12Þ

Tð�i

22ÞT

� � Xj1 0

� � � Xj2

2664

37754 0 8 ði, j Þ2I2 ð35Þ

with

�i11 ¼ Ai

11Xi1 þ Bi

1Yi1, �i

12 ¼ Ai12X

i2 þ Bi

1Yi2

�i21 ¼ Ai

21Xi1 þ Bi

2Yi1, �i

22 ¼ Ai22X

i2 þ Bi

2Yi2,

then the switching system (8) is globally asymptotically

stable for any arbitrary sequence of switching. The

controller gains are given by

Fi ¼ ½Yi1ðX

i1Þ�1 Yi

2ðXi2Þ�1� ð36Þ

with Pi1 ¼ ðX

i1Þ�1 and Pi

2 ¼ ðXi2Þ�1.

Proof 4: By pre- and post-multiplying inequality (34)

by diagfP�1i , Ig while letting Xi ¼ P�1i with Pi1 ¼ ðX

i1Þ�1

and Pi2 ¼ ðX

i2Þ�1, Yi¼FiXi, it follows that

Xi ðArXi þ BiYiÞT

� Xj

� �4 0 8ði, j Þ2I2: ð37Þ

By substituting matrices Ai, Bi and Pj according to (3)

and (28), respectively, and using Xi¼ diagfXi1 X

i2g,

Yi ¼ ½Yi1 Yi

2�, the LMIs (35) are directly obtained.

Finally, the stabilising controller gains of the 2D

switching system (8) are given by Fi ¼ YiX�1i .

Expression (36) follows. œ

Example 4.2: Consider the same example as in

Example 4.1, the LMI solver of Matlab is used to

solve LMIs (35) leading to feasible solutions in

X11, X

21, X

12, X

22, Y1 and Y2. The matrices P1, P2 and

the gain controllers F1, F2 are then derived by using

relations (36) as follows:

P11 ¼

0:4009 0:0989

0:0989 0:7837

� �; P2

1 ¼3:9334 2:9613

2:9613 3:2969

� �;

P12 ¼

0:4394 0:0790

0:0790 0:4562

� �; P2

2 ¼2:0094 0:9370

0:9370 1:7379

� �;

F1 ¼ �0:9879 �0:5563 2:7369 2:9319� �

;

F2 ¼ �0:8221 �1:1374 13:0216 13:9148� �

:

Example 4.3: The second example of Example 4.1 is

studied. For the same data, the LMIs (35) are feasible.

The obtained solutions are given by

P11 ¼

0:0269 �0:0531

�0:0531 0:1086

� �, P1

2 ¼ 0:0011,

P21 ¼

0:0049 �0:0142

�0:0142 0:0476

� �, P2

2 ¼ 0:0016,

F1 ¼�0:6407 �4:2421 �0:5241

�0:0040 2:2068 �0:8029

� �,

F2 ¼�0:8375 1:1870 1:7299

0:0985 �2:4938 �1:1626

� �:

Figure 2 plots the evolution of the three states xh1ðk, l Þ,

xh2ðk, l Þ, xv(k, l ) and the corresponding arbitrary

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sequence of switching represented by the last figure.

One can notice that the 2D closed-loop switching

system is asymptotically stable.

Comments 4.1: Example 4.3 shows that the studied2D switching system is asymptotically stable for anysequence of switching without admitting a commonquadratic Lyapunov function. This confirms the claimproposed by many authors about this fact. Thus, theresults of Theorem 4.4 constitute a strong tool to studythe stabilisability of 2D discrete-time switching systemswith arbitrary sequence of switching.

5. Conclusion

In this work, 2D discrete-time switching systemsrepresented by Roesser’s model with state feedbackcontrol are studied. The analysis of the switchingphenomena associated to 2D discrete-time systems isdeveloped. A new indicator function is then adopted.These tools enable one to analyse the stabilisabilityof this class of systems. Hence, common quadraticand multiple Lyapunov functions are used to obtainsufficient conditions of asymptotic stabilisation. Theseconditions are then used to synthesise the requiredcontrollers under LMIs formulations. Three numericalexamples illustrate the results.

Acknowledgment

This work was funded by the ‘Convocatoria de Ayudas parala estancia de investigadores, resulto con 7-05-2008’ of Spain,AECI project A/7882/07 and CICYT project DPI2007-66718-C04-02.

Notes on contributors

Abdellah Benzaouia was born inAttaouia (Marrakech) in 1954. Hereceived his degree in ElectricalEngineering at the MohammediaSchool (Rabat) in 1979 and hisDoctorate (PhD) at the UniversityCadi Ayyad in 1988. He is aProfessor at the University of CadiAyyad (Marrakech), where he is also

head of a team of research on Robust and ConstrainedControl (EACPI). His research interests are mainly con-strained control, robust control, pole assignment, systemswith Markovian jumping parameters, hybrid systems andgreenhouses. He collaborates with many teams in France,Canada, Spain and Italy.

Fernando Tadeo was born in 1969. Hereceived his BSc degree in Physics in1992, and in Electronic Engineering in1994, both from the University ofValladolid, Spain. After he receivedhis MSc degree in ControlEngineering from the University ofBradford, UK, he received his PhDdegree from the University of

Valladolid, Spain, in 1996. Since 1998 he has been a Lecturer(‘Profesor Titular’) at the University of Valladolid, Spain.

05

1015

20

05

1015

20−30

−20

−10

0

10

20

kl

Sta

te v

ecto

r xh

1

05

1015

20

05

1015

20−15

−10

−5

0

5

10

kl

Sta

te v

ecto

r xh

2

05

1015

20

05

1015

20−30−20−10

010203040

kl

Sta

te v

ecto

r xv

0 5 10 15 20 25 30 35 401

2

i + j

Mod

e

Figure 2. The trajectory of the states xh1ðk, l Þ, xh2ðk, l Þ, x

v(k, l )and the corresponding sequence of switching obtained withTheorem 4.4.

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His current research interests include robust control, processcontrol, control of systems with constraints and reinforce-ment learning, applied in several areas, from neutralizationprocesses to robotic manipulators.

Abdelaziz Hmamed was born in SefrouMorocco, on July 6, 1951. He receivedhis Doctorate of Third Cycle andDoctorate of State degrees inElectrical Engineering from theFaculty of Sciences Rabat, Morocco,in 1980 and 1985, respectively. Since1986, he has been with theDepartment of Physics, Faculty of

Sciences Dhar ElMehraz, Fes Morocco, where he is currentlya Full Professor. His research interests are in delay systems,stability theory, systems with Constraints, 2D systems andapplications.

Ahmed El Hajjaji received his PhDand ‘Habilitation a diriger larecherche’ degrees in AutomaticControl from the University ofPicardie Jules Verne, France, in 1993and 2000, respectively. He was anAssociate Professor in the same uni-versity from 1994 to 2003. He iscurrently a Full Professor andDirector of the

Professional Institute of Electrical Engineering andIndustrial Computing, University of Picardie Jules Verne.Since 2001, he has also been also the head of the ResearchTeam of Control and Vehicles of the Modeling, Informationand Systems (MIS) laboratory. His current research interestsinclude fuzzy control, vehicle dynamics, fault-tolerant con-trol, neural networks, mangles systems and renewable energysystems.

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