Stability and l2 gain analysis for 2D discrete switched systems with time varying delays in the second FM model

(1)

R E S E A R C H

Open Access

Stability and

l

_

-gain analysis for 2D discrete

switched systems with time-varying delays in

the second FM model

Shipei Huang and Zhengrong Xiang

*

*_{Correspondence:}

[email protected] School of Automation, Nanjing University of Science and Technology, Nanjing, 210094, People’s Republic of China

Abstract

This paper is concerned with the problems of stability andl2-gain analysis for 2D (two-dimensional) discrete switched systems with time-varying delays described by the second FM state-space model. Firstly, we introduce the definition of the average dwell time for a 2D discrete switched system, which is an extension of the ‘average dwell time’ concept of a 1D (one-dimensional) switched system. Secondly, based on the average dwell time approach, delay-dependent sufficient conditions for the existence of the exponential stability for the 2D discrete switched system are derived andl2-gain performance for the considered system is also analyzed. All the obtained results are formulated in a set of LMIs (linear matrix inequalities). Finally, a numerical example is given to illustrate the effectiveness of the proposed results.

Keywords: 2D systems; switched systems; time-varying delays;l2-gain; average dwell time; linear matrix inequality

1 Introduction

D (Two-dimensional) systems, which are a class of multi-dimensional systems, have re-ceived considerable attention over the past few decades due to their wide applications in many areas such as multi-dimensional digital filtering, linear image processing, signal pro-cessing, and process control [–]. The D system theory is frequently used as an analysis tool to solve some problems,e.g., iterative learning control [, ] and repetitive process control [, ]. The problems on realization, stability analysis, stabilization, filter design, and so on for D ornD systems have attracted a great deal of interest by many researchers. Xuet al.[, ] investigated the realization of D systems, and the problems of stability and stabilization for these systems were studied extensively in [–]. The observer and filter design problems have also been considered in [–].

It is known that modeling uncertainties and disturbances are unavoidable in practical systems, and it is important to investigate the problems ofH∞ control and robust

stabi-lization of D systems. Recently, many results onH∞ control for D systems have been

presented in [–]. Because time delays frequently occur in practical systems and are often the source of instability, theH∞control problem for a class of D discrete systems with state delays has also been investigated in [, ].

On the other hand, since switched systems have numerous applications in many ﬁelds, such as mechanical systems, automotive industry, switched power converters, this class of

(2)

systems has also attracted considerable attention during the past several decades [–]. Recently, some approaches have been applied widely to deal with these systems; see, for example, [–] and references cited therein. As stated in [, ], in many modeling problems of physical processes, a D switching representation is needed. One can cite a D physically based model for advanced power bipolar devices and heat flux switching and modulating in a thermal transistor. At present, there have been a few reports on D dis-crete switched systems. Benzaouiaet al.[] firstly considered D switched systems with arbitrary switching sequences, and the process of switch is considered as a Markovian jumping one. In addition, the stabilization problem of discrete D switched systems was also studied in []. In [], we extended the concept of average dwell time in D switched systems to D switched systems and designed a switching rule to guarantee the exponen-tial stability of D switched delay-free systems. However, to the best of our knowledge, no works have considered the disturbance attenuation property of D switched systems to date. Moreover, because of the complicated behavior caused by the interaction between the continuous dynamics and discrete switching, the problem of disturbance attenuation performance analysis for D switched systems is more difficult to study, and the meth-ods proposed in [–] cannot be directly applied to such systems. This motivates the present study.

In this paper, we are interested in investigating the issues of the exponential stability and l-gain analysis for D discrete switched systems with time-varying delays represented by

the second FM model. The main contributions of this paper can be summarized as fol-lows: (i) Based on the average dwell time approach, a delay-dependent exponential stabil-ity criterion for such systems is obtained and formulated in terms of LMIs (linear matrix inequalities); (ii) The Lyapunov-Krasovskii function with exponential term, which is dif-ferent from the previous ones, is constructed to investigate the stability of the considered systems; and (iii) In order to investigate the disturbance attenuation property of the con-sidered systems, we for the ﬁrst time introduce the concept ofl-gain for a D switched

system, which is an extension of thel-gain performance index in the D case. Thel-gain

performance index can characterize the disturbance attenuation property of the underly-ing systems, and then, based on the established stability results, delay-dependent suﬃcient conditions for the existence ofl-gain performance are derived in terms of LMIs, which

can be easily veriﬁed by using some standard numerical software. The proposed method can also be applied to non-switched D discrete linear systems.

This paper is organized as follows. In Section , problem formulation and some nec-essary lemmas are given. In Section , based on the average dwell time approach, delay-dependent suﬃcient conditions for the existence of the exponential stability andl-gain

property are derived in terms of a set of matrix inequalities. A numerical example is pro-vided to illustrate the eﬀectiveness of the proposed approach in Section . Concluding remarks are given in Section .

Notations Throughout this paper, the superscript ‘T’ denotes the transpose, and the

no-tationX≥Y(X>Y) means that matrixX–Yis positive semi-deﬁnite (positive deﬁnite, respectively). · denotes the Euclidean norm.Irepresents an identity matrix with an appropriate dimension.diag{ai}denotes a diagonal matrix with the diagonal elementsai,

(3)

byZ+. Thelnorm of a D signalw(i,j) is given by

w=

∞

i=

∞

j=

w(i,j)

andw(i,j) belongs tol{[,∞), [,∞)}ifw<∞.

2 Problem formulation and preliminaries

Consider the following D discrete switched systems with time-varying delays described by the second FM model:

x(i+ ,j+ ) =A_σ(i,j+)x(i,j+ ) +A_σ(i+,j)x(i+ ,j) +Aσ_d_(i,j+)xi–d(i),j+ 

+Aσ_d_(i+,j)xi+ ,j–d(j)

+Bσ_(i,j+)w(i,j+ ) +B_σ(i+,j)w(i+ ,j), ()

z(i,j) =Hσ(i,j)x(i,j) +Lσ(i,j)w(i,j),

where x(i,j)∈Rn _{is the state vector,} _w₍_i_,_j₎_∈_Rq _{is the noise input which belongs to}

l{[,∞), [,∞)},z(i,j)∈Rp is the controlled output. iandjare integers inZ+. σ(i,j) : Z+×Z+→N ={, , . . . ,N} is the switching signal. N is the number of subsystems. σ(i,j) =k,k∈N, denotes that thekth subsystem is active.Ak

,Ak, Akd,Adk,Bk,Bk,Hk,

andLk_{are constant matrices with appropriate dimensions.}_d

(i) andd(j) are delays along

horizontal and vertical directions, respectively. We assume thatd(i) andd(j) satisfy

d_≤d(i)≤d, d≤d(j)≤d, ()

whered_,d,d, andddenote the lower and upper delay bounds along horizontal and

vertical directions, respectively.

In this paper, it is assumed that the switch occurs only at each sampling point ofiorj. The switching sequence can be described as

(i,j),σ(i,j)

,(i,j),σ(i,j)

, . . . ,(iπ,jπ),σ(iπ,jπ)

, . . . , ()

where (iπ,jπ) denotes theπth switching instant. It should be noted that the value ofσ(i,j)

only depends uponi+j(see the references [, ]).

Remark  If there is only one subsystem in system (), it will degenerate to the following

D system:

x(i+ ,j+ ) =Ax(i,j+ ) +Ax(i+ ,j) +Adx

i–d(i),j+ 

+Adx

i+ ,j–d(j)

+Bw(i,j+ ) +Bw(i+ ,j),

z(i,j) =Hx(i,j) +Lw(i,j).

(4)

For system (), we consider a ﬁnite set of initial conditions, that is, there exist positive integerszandzsuch that

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

x(i,j) =hij, ∀≤j≤z,i= –d, –d+ , . . . , ,

x(i,j) =vij, ∀≤i≤z,j= –d, –d+ , . . . , ,

h=v,

x(i,j) = , ∀j>z,i= –d, –d+ , . . . , ,

x(i,j) = , ∀i>z,j= –d, –d+ , . . . , ,

()

wherez<∞andz<∞are positive integers,hijandvijare given vectors.

Deﬁnition  System () withw(i,j) =  is said to be exponentially stable under the

switch-ing signalσ(i,j) if for a givenZ≥, there exist positive constantscandξ such that

i+j=D

x(i,j)≤ξe–c(D–Z)

i+j=Z

x(i,j)_C ()

holds for allD≥Z, where

i+j=Z

x(i,j)_C sup –d≤θh≤, –d≤θv≤

i+j=Z

_x₍_i_–θh,j)



,x(i,j–θv)



,

η(i–θh,j)



,δ(i,j–θv)



,

η(i–θh,j) =x(i–θh+ ,j) –x(i–θh,j),

δ(i,j–θv) =x(i,j–θv+ ) –x(i,j–θv).

Remark  From Deﬁnition , it is easy to see that whenZis given,_i₊_j₌_Zx(i,j)

C will

be bounded and_i₊_j₌_Dx(i,j)_{will tend to be zero exponentially as}_D_{goes to inﬁnity,}

which also meansx(i,j)will tend to be zero exponentially.

Deﬁnition [] For anyi+j=D>Z=iZ+jZ, letNσ(i,j)(Z,D) denote the switching

number of the switching signalσ(i,j) on an interval (Z,D). If

Nσ(i,j)(Z,D)≤N+ D–Z

τa

()

holds for givenN≥,τa≥, then the constantτa is called the average dwell time and

Nis the chatter bound.

Remark  Deﬁnition  is an extension of the ‘average dwell time’ concept in a D switched

system, which can be seen in []. In what follows, based on the extended average dwell time concept, we will investigate the problems of stability andl-gain analysis for a D

(5)

Remark  Similar to the D switched system case, Deﬁnition  means that if there ex-ists a positive numberτasuch that the switching signalσ(i,j) has the average dwell time

property, the average time interval between consecutive switching is at leastτa. The

av-erage dwell time method is used to restrict the switching number of the switching signal during a time interval such that the stability or other performances of the system can be guaranteed.

Deﬁnition  Consider D discrete switched system (). For a given scalarγ > , system

() is said to havel-gainγ under the switching signalσ(i,j) if it satisﬁes the following

conditions:

() Whenw(i,j) = , system () is asymptotically stable; () Under the zero boundary condition, it holds that

∞

i=

∞

j=

z_<γ

∞

i=

∞

j=

w_, ∀=w∈l

[,∞), [,∞),

where

w_=

∞

i=

∞

j=

w(i,j+ ) w(i+ ,j)



, z_=

∞

i=

∞

j=

z(i,j+ ) z(i+ ,j)



.

Remark  It is not diﬃcult to see that Deﬁnition  is an extension of thel-gain

perfor-mance index in the D case.γ characterizes the disturbance attenuation performance. The smallerγ is, the better performance is.

Deﬁnition  Consider D discrete switched system (). For a given scalarγ > , system

() is said to have weightedl-gainγ under the switching signalσ(i,j) if it satisﬁes the

following conditions:

() Whenw(i,j) = , system () is asymptotically stable; () Under the zero boundary condition, it holds that

∞

i=

∞

j=

αi+jz_<γ

∞

i=

∞

j=

w

, ∀=w∈l

[,∞), [,∞).

Remark  Similar to the D switched system case, Deﬁnition  means that system () can

also have disturbances attenuation properties when it satisﬁes conditions () and () in Deﬁnition .

Lemma  ConsiderD discrete switched system().Suppose that there exist a series of C

functions Vk:Rn→R(k∈N)and two positive scalarsλ andλfor which the following inequality holds:

λx(i,j)≤Vk

x(i,j)≤λx(i,j)_C, ∀i,j∈Z+,∀k∈N ()

if there exists a number <α< for which Vk(x(i,j))along with the solution of system()

satisﬁes the inequality

i+j=D

Vk

x(i,j)≤α

i+j=D– Vk

(6)

andμ≥such that

i+j=mπ Vσ(iπ,jπ)

x(i,j)≤μ

i+j=(mπ)–

Vσ(iπ–,jπ–)

x(i,j), π= , , . . . , ()

thenD discrete switched system()is exponentially stable for every switching signal with the average dwell time scheme

τa>τa∗=

lnμ

–lnα. ()

Proof Letχ=Nσ(i,j)(Z,D) denote the switch number of switching σ(i,j) on an interval

[Z,D), and let (iπ–χ+,jπ–χ+), (iπ–χ+,jπ–χ+), . . . , (iπ,jπ) denote the switching points of

σ(i,j) over the interval [Z,D). Denotingmi=ii+ji,i=π–χ+ , . . . ,π, it follows from

() and () that

i+j=D

Vσ(iπ,jπ)

x(i,j)< αD–mπ

i+j=mπ Vσ(iπ,jπ)

x(i,j)

≤μαD–mπ

i+j=(mπ)–

Vσ(iπ–,jπ–)

x(i,j)

< μαD–mπ_αmπ–mπ–

i+j=(mπ–)–

Vσ(iπ–,jπ–)

x(i,j)

=μαD–mπ–

i+j=(mπ–)–

Vσ(iπ–,jπ–)

x(i,j)<· · ·

< μχαD–Z

i+j=Z

Vσ(iπ–χ,jπ–χ)

x(i,j). ()

According to Deﬁnition , one obtains

χ=Nσ(i,j)(Z,D)≤N+ D–Z

τa

. ()

Then from (), we have

i+j=D

Vσ(iπ,jπ)

x(i,j)

<μχαD–Z

i+j=Z

x(i,j)

=μN_e–(–lnτaμ–lnα)(D–Z)

i+j=Z

x(i,j). ()

From (), we get

i+j=D

x(i,j)≤λ–_ λμNe–(– lnμ

τa–lnα)(D–Z)

i+j=Z

x(i,j)_C. ()

Therefore, according to Deﬁnition , system () is exponentially stable under the average

(7)

3 Main results

3.1 Stability analysis

Theorem  Consider system()with w(i,j) = .For a given positive constantα< ,if there

exist positive deﬁnite symmetric matrices Pk_h,Pk

v,Qkh,Qkv,Whk,Wvk,Rkh,Rkv,and matrices

with appropriate dimensions,k∈N,such that

⎡

hold,then under the average dwell time scheme

τa>τa∗=

lnμ

(8)

whereμ≥and satisﬁes

Pk_h≤μP_hl, P_hl ≤μPk_h, Qk_h≤μQl_h,

Ql_h≤μQk_h, W_hk≤μW_hl,

W_hl≤μW_hk, Rk_h≤μRl_h, Rl_h≤μRk_h,

Pk_v≤μP_vl, Pl_v≤μP_vk, Qk_v≤μQl_v,

()

Ql_v≤μQk_v, W_vk≤μW_vl, W_vl≤μW_vk,

Rk_v≤μRl_v, Rl_v≤μRk_v, ∀k,l∈N,

the system is exponentially stable.

Proof See the Appendix for the detailed proof, it is omitted here.

Remark  In Theorem , we propose a suﬃcient condition for the existence of

exponen-tial stability for the considered D discrete switched system (). It is worth noting that this condition is obtained by using the average dwell time approach.

3.2 l2-gain performance analysis

Theorem  Consider system().For given positive constantsγ andα< ,if there exist

positive deﬁnite symmetric matrices Pk

h,Pkv,Qhk,Qkv,Whk,Wvk,Rkh,Rkv,and matrices Nk= N_k

N_k

,Nk

= N_k

N_k

,Mk

 = Mk_

M_k

,Mk

= M_k

Mk_

,Xk₌XkXk

∗ X_k

> and Yk₌Yk Yk

∗ Y_k

> with

appropriate dimensions,k∈N,such that()and the following inequality

⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣

  T

(Pkh+Pkv)

dTRkh

dTRkv T ∗ –γI  BT

(Pkh+Pkv)

dBTRkh

dBTRkv LTk

∗ ∗ –γI BT

(Pkh+Pkv)

dBTRkh

dBTRkv LTk

∗ ∗ ∗ –(Pk

h+Pvk)   

∗ ∗ ∗ ∗ –Rk_h  

∗ ∗ ∗ ∗ ∗ –Rk

v 

∗ ∗ ∗ ∗ ∗ ∗ –I

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

< , ()

where

=

Hk _Hk _ _ _ __,

hold,then under the average dwell time scheme(),the system is exponentially stable and has weighted l-gainγ.

Proof It is easy to get that () can be deduced from (), and according to Theorem , we can obtain that system () is exponentially stable.

Now we are in a position to consider thel-gain performance of system () under the

(9)

(10)

where

˜

=

⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ ˜

 HkTHk     HkTLk HkTLk

∗ ˜     HkTLk HkTLk

∗ ∗ –αd_Qk

h     

∗ ∗ ∗ –αd_Qk

v    

∗ ∗ ∗ ∗ –αd_Wk

h   

∗ ∗ ∗ ∗ ∗ –αd_Wk

v  

∗ ∗ ∗ ∗ ∗ ∗ ˜ LkTLk

∗ ∗ ∗ ∗ ∗ ∗ ∗ ˜

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ ,

˜

= –αPkh+Whk+ (d–d+ )Qkh+HkTHk,

˜

= –αPkv+Wvk+ (d–d+ )Qkv+HkTHk,

˜

=˜=LkTLk–γI.

Then by the Schur complement lemma, we can obtain from () and () that

V_kh(i+ ,j+ ) +V_kv(i+ ,j+ )

–αV_kh(i,j+ ) –αV_kv(i+ ,j) +z_–γw_< . ()

Summing up both sides of () fromD–  to  with respect tojand  toD–  with respect toiand applying the zero boundary condition, one gets

i+j=D

Vσ(iπ,iπ)(i,j) <α

i+j=D–

Vσ(iπ,iπ)(i,j) –

i+j=D– (i,j)

<αD–mπ

i+j=mπ

Vσ(iπ,iπ)(i,j) –

D–

m=mπ–

i+j=m

αD––i–j(i,j)

≤μαD–mπ

i+j=(mπ)–

Vσ(iπ–,iπ–)(i,j) –

D–

m=mπ–

i+j=m

<

i+j=mπ–

μαD–(mπ–)_V

σ(iπ–,iπ–)(i,j) –μα

D–mπ

i+j=mπ– (i,j)

–

D–

m=mπ–

i+j=m

=

i+j=mπ–

μNσ(i,j)(i+j,D)_αD–(mπ–)_V

σ(iπ–,iπ–)(i,j)

–

D–

m=mπ–

i+j=m

μNσ(i,j)(i+j+,D)_αD––i–j₍_i_,_j₎

<

i+j=mπ–

μNσ(i,j)(i+j,D)_αD–mπ–_V

σ(iπ–,iπ–)(i,j)

–

D–

m=mπ––

i+j=m

(11)

≤

i+j=(mπ–)–

μNσ(i,j)(i+j–,D)_αD–mπ–_V

σ(iπ–,iπ–)(i,j)

–

D–

m=mπ––

i+j=m

μNσ(i,j)(i+j+,D)_αD––i–j₍_i_,_j₎

<· · ·

<

i+j=

μNσ(i,j)(i+j,D)_αD–_V

σ(,)(i,j)

–

D–

m=

i+j=m

μNσ(i,j)(i+j+,D)_αD––i–j₍_i_,_j_), _()

where

(i,j) =z_–γw_=

z(i+ ,j) z(i,j+ ) 



–γ

w(i+ ,j) w(i,j+ ) 



.

Under the zero initial condition, it holds that

i+j=

μNσ(i,j)(i+j,D)_αD–_V

σ(,)(i,j) = . ()

Thus we have

D–

m=

i+j=m

μNσ(i,j)(i+j+,D)_αD––i–j₍_i_,_j_{) < –}

i+j=D

Vσ(iπ,iπ)(i,j) < . ()

Multiplying the both sides of () byμ–Nσ(i,j)(,D)_{, we get the following inequality:}

D–

m=

i+j=m

μ–Nσ(i,j)(,i+j+)_αD––i–j₍_i_,_j_{) < .} _()

That is,

D–

m=

i+j=m

μ–Nσ(i,j)(,i+j+)_αD––i–j_z

<

D–

m=

i+j=m

μ–Nσ(i,j)(,i+j+)_αD––i–j_w

.

Note thatNσ(i,j)(,i+j+ )≤(i+j)/τa, then using (), we have

μ–Nσ(i,j)(,i+j+)₌_e–Nσ(i,j)(,i+j+)lnμ≥_e(i+j)lnα_. _()

It follows that

D–

m=

i+j=m

e(i+j)lnααD––i–jz_<

D–

m=

i+j=m

μ–Nσ(i,j)(,i+j+)_αD––i–jw

 ()

⇒

D–

m=

i+j=m

αD–z_<γ

D–

m=

i+j=m

(12)

⇒

According to Deﬁnition , we obtain that system () is exponentially stable and has

weightedl-gainγ. The proof is completed.

Remark  We would like to stress that thel-gain performance analysis problem of D

discrete switched systems is ﬁrstly considered in the paper. Although some results on l-gain performance analysis of D systems have been obtained in [–], the existing

methods proposed in these papers cannot be directly applied to D switched systems. In Theorem , suﬃcient conditions for the existence ofl-gain performance for system ()

are derived in terms of a set of LMIs.

Remark  As for the applicability of Theorem , it is easy to see that a largerα and a

largerγ will be favorable to the feasibility of matrix inequality (), while a smallerαis more expectable to decreaseτ_a∗, and a smaller γ means the better performance of the system. Thus for the ﬁrst time, we can chose a smallerαand a smallerγ, and then, by adjusting the values ofαandγ, we can ﬁnd a feasible solution.

Remark  It is noticed that whenμ=  inτa>τa∗=

common Lyapunov function exists for all subsystems. Then from () we get

(13)

Thus

Therefore,l-gain is achieved for switched system () under arbitrary switching. We state

the fact in the following corollary.

Corollary  Consider system().For given positive constantsγ andα< ,if there exist

positive deﬁnite symmetric matrices Pk

h=Ph,Pkv=Pv,Qk_h=Qh,Qkv=Qv,W_hk=Wh,Wvk=

> with appropriate dimensions such that()and()

hold for k∈N,then theD discrete switched system()achieves the l-gain under arbitrary switching signals.

4 Numerical example

In this section, we present an example to illustrate the eﬀectiveness of the proposed ap-proach. Consider system () with parameters as follows:

(14)

According to Remark , we can ﬁrstly takeα= . andγ= ., then solving () and () gives rise to the following solution:

P_h=

. –. –. .

, P_h=

. –. –. .

,

P_v=

. –. –. .

, P_v=

. –. –. .

,

Q_h=

. –. –. .

, Q_h=

. –. –. .

,

Q_v=

. –. –. .

, Q_v=

. –. –. .

,

W_h=

. –. –. .

, W_h=

. –. –. .

,

W_v=

. –. –. .

, W_v=

. –. –. .

,

R_h=

. . . .

, R_h=

. . . .

,

R_v=

. –,  –,  .

, R_v=

. –. –. .

,

andμ= ., then we obtain from () thatτ_a∗= .. Choosingτa= , some simulation

results are shown in Figures -, where the boundary condition of the system is

x(i,j) = 

(j+ ), ∀≤j≤,i= , x(i,j) = 

(i+ ), ∀≤i≤,j= ,

(15)

Figure 2 Response of statex2(i,j).

Figure 3 Switching signal.

andw(i,j) = .exp(–.π(i+j)). It can be seen from Figures - that the system is asymptotically stable. Furthermore, when the boundary condition is zero, by computing, we get∞_i_=∞_j_=αi+jz

= . and

∞

i=

∞

j=w= ., and it satisﬁes

condi-tion () in Deﬁnicondi-tion . Therefore, it can be observed that the system has weightedl-gain γ = .. This demonstrates the eﬀectiveness of the proposed approach.

5 Conclusions

This paper has investigated the problems of stability andl-gain analysis for D discrete

(16)

Then some suﬃcient conditions for the existence of weightedl-gain for the considered

system are derived in terms of LMIs. Finally, an example is also given to illustrate the applicability of the proposed results. Our future work will focus on extending the pro-posed results to other kinds of D discrete switched systems such as D discrete switched stochastic systems or D discrete switched nonlinear systems.

Appendix

Proof of Theorem  It is assumed that the kth subsystem is active on the interval [mπ,mπ+), and thelth subsystem is active on the interval [mπ–,mπ). Now we consider

the Lyapunov function candidate for thekth subsystem

Vk

x(i,j)=V_khx(i,j)+V_kvx(i,j), ()

where

V_khx(i,j)=



g=

V_gkhx(i,j),

V_h_k(i,j) =x(i,j)TPk_hx(i,j),

V_h_kx(i,j)=

i–

r=i–d(i)

x(r,j)TQk_hx(r,j)αi–r–,

V_h_kx(i,j)=

i–

r=i–d

x(r,j)TW_hkx(r,j)αi–r–,

V_h_kx(i,j)=

–d

s=–d+

i–

r=i+s

x(r,j)TQk_hx(r,j)αi–r–,

V_h_kx(i,j)=



s=–d+

i–

r=i+s–

η(r,j)TRk_hη(r,j)αi–r–,

V_kvx(i,j)=



g=

V_gkvx(i,j),

V_v_kx(i,j)=x(i,j)TPk_vx(i,j),

V_v_kx(i,j)=

j–

t=j–d(j)

x(i,t)TQk_vx(i,t)αj–t–,

V_v_kx(i,j)=

j–

t=j–d

x(i,t)TW_vkx(i,t)αj–t–,

V_v_kx(i,j)=

–d

s=–d+

j–

t=j+s

x(i,t)TQk_vx(i,t)αj–t–,

V_v_kx(i,j)=



s=–d+

j–

t=j+s–

(17)

η(r,j) =x(r+ ,j) –x(r,j),

δ(i,t) =x(i,t+ ) –x(i,t).

Then we have

V_h_k(i+ ,j+ ) –αV_h_k(i,j+ )

=xT(i+ ,j+ )P_hkx(i+ ,j+ ) –αxT(i,j+ )Pk_hx(i,j+ ),

V_h_k(i+ ,j+ ) –αV_h_k(i,j+ )

≤x(i,j+ )T_Qk

hx(i,j+ ) –x

i–d(i),j+  T

Qk hx

i–d(i),j+ 

αd

+

i–d

r=i+–d

x(r,j+ )TQk_hx(r,j+ )αi–r,

V_h_k(i+ ,j+ ) –αV_h_k(i,j+ )

=x(i,j+ )TW_hkx(i,j+ ) –x(i–d,j+ )TWhkx(i–d,j+ )αd,

V_h_k(i+ ,j+ ) –αV_h_k(i,j+ )

= (d–d)x(i,j+ )TQkhx(i,j+ ) – i–d

r=i–d+

x(r,j+ )TQk_hx(r,j+ )αi–r,

V_h_k(i+ ,j+ ) –αV_h_k(i,j+ )

≤dη(i,j+ )TRkhη(i,j+ ) – i–

r=i–d

η(r,j+ )TRk_hη(r,j+ )αd_,

V_v_k(i+ ,j+ ) –αV_v_k(i+ ,j)

=xT(i+ ,j+ )P_vkx(i+ ,j+ ) –αxT(i+ ,j)Pk_vx(i+ ,j),

V_v_k(i+ ,j+ ) –αV_v_k(i+ ,j)

≤x(i+ ,j)TQk_vx(i+ ,j) –xi+ ,j–d(j) T

Qk_vxi+ ,j–d(j)

αd

+

j–d

t=j+–d

x(i+ ,t)TQk_vx(i+ ,t)αj–t,

V_v_k(i+ ,j+ ) –αV_v_k(i+ ,j)

=x(i+ ,j)TW_vkx(i+ ,j) –x(i+ ,j–d)TWvkx(i+ ,j–d)αd,

V_v_k(i+ ,j+ ) –αV_v_k(i+ ,j)

= (d–d)x(i+ ,j)TQkvx(i+ ,j) – j–d

t=j+–d

x(i+ ,t)TQk_vx(i+ ,t)αj–t,

V_v_k(i+ ,j+ ) –αV_v_k(i+ ,j)

≤dδ(i+ ,j)TRkvδ(i+ ,j) – j–

t=j–d

(18)

η(i,j+ ) =x(i+ ,j+ ) –x(i,j+ )

For simplicity, we denote

V_kh(i,j) =V_khx(i,j), V_kv(i,j) =V_kvx(i,j), Vk(i,j) =Vk

Notice that the following equations hold for any matricesN_k=N k

with appropriate dimensions:

(19)

 = αd_xT₍_i_,_j_{+ )}_Nk

> , the following equations also hold:

 =dαd

(20)

+

Thus it follows from ()-() that

(21)

Thus () can be directly obtained. Moreover, by (), we can ﬁnd two positive scalarsλ

andλsuch that () holds, where

λ=min

k∈N

λmin

Pk_h+λmin

Pk_v,

λ=max

k∈N

λmax

Pk_h+λmax

Pk_v+dλmax

Qk_h+dλmax

Qk_v+dλmax

W_hk

+dλmax

W_vk+ (d–d)λmax

Qk_h

+ (d–d)λmax

Qk_v+d_λmax

Rk_h+d_λmax

Rk_v.

In addition, () can be deduced from (), thus by Lemma , we can conclude that D

discrete switched system () is exponentially stable.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

SH carried out the main results of this article and drafted the manuscript. ZX directed the study and helped with the inspection. All the authors read and approved the ﬁnal manuscript.

Acknowledgements

This research was supported by the National Natural Science Foundation of China under Grant Nos. 60974027 and 61273120.

Received: 3 November 2012 Accepted: 22 February 2013 Published: 14 March 2013 References

1. Du, CL, Xie, LH: Control and Filtering of Two-Dimensional Systems. Springer, Berlin (2002) 2. Kaczorek, T: Two-Dimensional Linear Systems. Springer, Berlin (1985)

3. Lu, WS: Two-Dimensional Digital Filters. Dekker, New York (1992)

4. Owens, DH, Amann, N, Rogers, E, French, M: Analysis of linear iterative learning control schemes - a 2D systems/repetitive processes approach. Multidimens. Syst. Signal Process.11(1-2), 125-177 (2000)

5. Li, XD, Ho, JKL, Chow, TWS: Iterative learning control for linear time-variant discrete systems based on 2-D system theory. IEE Proc., Control Theory Appl.152(1), 13-18 (2005)

6. Bochniak, J, Galkowski, K, Rogers, E, Mehdi, D, Kummert, O, Bachelier, A: Stabilization of discrete linear repetitive processes with switched dynamics. Multidimens. Syst. Signal Process.17(2-3), 271-295 (2006)

7. Galkowski, K, Rogers, E, Xu, S, Lam, J, Owens, DH: LMIs - a fundamental tool in analysis and controller design for discrete linear repetitive processes. IEEE Trans. Circuits Syst. I, Fundam. Theory Appl.49(6), 768-777 (2002) 8. Xu, L, Wu, LK, Wu, QH, Lin, ZP, Xiao, YG: On realization of 2D discrete systems by Fornasini-Marchesini model. Int.

J. Control. Autom. Syst.3(4), 631-639 (2005)

9. Xu, L, Wu, QH, Lin, ZP, Xiao, YG: A new constructive procedure for 2-D coprime realization in Fornasini-Marchesini model. IEEE Trans. Circuits Syst.54(9), 2061-2069 (2007)

10. Lin, ZP: Feedback stabilization of MIMOnD linear systems. IEEE Trans. Autom. Control45(12), 2419-2424 (2000) 11. Xu, L, Yamada, M, Lin, ZP, Saito, O, Anazawa, Y: Further improvements on Bose’s 2D stability test. Int. J. Control. Autom.

Syst.2(3), 319-332 (2004)

12. Lin, ZP, Lam, J, Galkowski, K, Xu, SY: A constructive approach to stabilizability and stabilization of a class ofnD systems. Multidimens. Syst. Signal Process.12(3-4), 329-343 (2001)

13. Fornasini, E, Marchesini, G: Stability analysis of 2-D systems. IEEE Trans. Circuits Syst.27(10), 1210-1217 (1980) 14. Ye, SX, Wang, WQ: Stability analysis and stabilisation for a class of 2-D nonlinear discrete systems. Int. J. Syst. Sci.42(5),

839-851 (2011)

15. Paszke, W, Lam, J, Galkowski, K, Xu, SY, Lin, ZP: Robust stability and stabilisation of 2D discrete state-delayed systems. Syst. Control Lett.51(3-4), 277-291 (2004)

16. Chen, SF: Delay-dependent stability for 2D systems with time-varying delay subject to state saturation in the Roesser model. Appl. Math. Comput.216(9), 2613-2622 (2010)

17. Feng, ZY, Xu, L, Wu, M, He, Y: Delay-dependent robust stability and stabilisation of uncertain two-dimensional discrete systems with time-varying delays. IET Control Theory Appl.4(10), 1959-1971 (2010)

18. Xu, HL, Lin, ZP, Makur, A: The existence and design of functional observers for two-dimensional systems. Syst. Control Lett.61(2), 362-368 (2012)

19. Xu, SY, Lam, J, Zhou, Y, Lin, ZP, Paszke, W: RobustH∞ﬁltering for uncertain 2-D continuous systems. IEEE Trans. Signal Process.35(5), 1731-1738 (2005)

(22)

21. Xie, LH, Du, CL, Soh, YC, Zhang, CS:H_∞and robust control of 2-D systems in FM second model. Multidimens. Syst. Signal Process.13(3), 265-287 (2002)

22. Du, CL, Xie, LH, Zhang, CS:H_∞control and robust stabilization of two-dimensional systems in Roesser models. Automatica37(2), 205-211 (2001)

23. Yang, R, Xie, LH, Zhang, CS:H2and mixedH2/H∞control of two-dimensional systems in Roesser model. Automatica

42(9), 1507-1514 (2006)

24. Xu, JM, Yu, L:H∞control of 2-D discrete state delay systems. Int. J. Control. Autom. Syst.4(4), 516-523 (2006) 25. Xu, JM, Yu, L: Delay-dependentH_∞control for 2-D discrete state delay systems in the second FM model.

Multidimens. Syst. Signal Process.20(4), 333-349 (2009)

26. Hespanha, JP, Morse, AS: Stability of switched systems with average dwell time. In: Proceedings of the 38th IEEE Conference on Decision and Control, pp. 2655-2660 (1999)

27. Zhai, GS, Hu, B, Yasuda, K, Michel, AN: Stability analysis of switched systems with stable and unstable subsystems: an average dwell time approach. In: Proceedings of the American Control Conference, pp. 200-204 (2000)

28. Xiang, ZR, Wang, RH: Robust stabilization of switched non-linear systems with time-varying delays under asynchronous switching. Proc. Inst. Mech. Eng., Part I, J. Syst. Control Eng.223(8), 1111-1128 (2009)

29. Xiang, ZR, Wang, RH: Robust control for uncertain switched non-linear systems with time delay under asynchronous switching. IET Control Theory Appl.3(8), 1041-1050 (2009)

30. Sun, XM, Zhao, J, David, JH: Stability andL2-gain analysis for switched delay systems: a delay-dependent method.

Automatica42(10), 1769-1774 (2006)

31. Wang, YJ, Yao, ZX, Zuo, ZQ, Zhao, HM: Delay-dependent robustH∞control for a class of switched systems with time delay. In: IEEE International Symposium on Intelligent Control, pp. 882-887 (2008)

32. Sun, YG, Wang, L, Xie, GM: Delay-dependent robust stability andH∞control for uncertain discrete-time switched systems with mode-dependent time delays. Appl. Math. Comput.187(2), 1228-1237 (2007)

33. Wang, R, Liu, M, Zhao, J: ReliableH∞control for a class of switched nonlinear systems with actuator failures. Nonlinear Anal. Hybrid Syst.1(3), 317-325 (2007)

34. Benzaouia, A, Hmamed, A, Tadeo, F: Stability conditions for discrete 2D switching systems, based on a multiple Lyapunov function. In: European Control Conference, pp. 23-26 (2009)

35. Benzaouia, A, Hmamed, A, Tadeo, F, Hajjaji, AE: Stabilisation of discrete 2D time switching systems by state feedback control. Int. J. Syst. Sci.42(3), 479-487 (2011)

36. Xiang, Z, Huang, S: Stability analysis and stabilization of discrete-time 2D switched systems. Circuits Syst. Signal Process.32(1), 401-414 (2013)

doi:10.1186/1687-1847-2013-56