# Stability andl1-Gain Analysis for Positive 2D Systems with State Delays in the Roesser Model

Authors: Zhaoxia Duan; Hamid Reza Karimi; Zhengrong Xiang
Journal: Mathematical Problems in Engineering
 (2013)

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## Abstract

This paper considers the problem of delay-dependent stability andl1-gain analysis for positive 2D systems with state delays described by the Roesser model. Firstly, the copositive-type Lyapunov function method is used to establish the sufficient conditions for the addressed positive 2D system to be asymptotically stable. Then, l1-gain performance for the system is also analyzed. All the obtained results are formulated in the form of linear matrix inequalities (LMIs) which are computationally tractable. Finally, an illustrative example is given to verify the effectiveness of the proposed results.


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## Body

## 1. Introduction

2D systems exist in many practical applications, such as circuits analysis, digital image processing, signal filtering, and thermal power engineering [1–4]. Thus the analysis and synthesis of 2D systems are interesting and challenging problems, and they have received considerable attention; for example, 2D state-space realization theory was researched in [5], the stability and 2D optimal control theory was studied in [6, 7], and H∞ control and filtering problem for 2D systems were addressed in [8–11]. In addition, linear repetitive processes, a distinct class of 2D systems, have also been investigated. For example, the quasi-sliding mode control problem for linear repetitive processes with unknown input disturbance was solved in [12].The most popular models of two-dimensional (2D) linear systems were introduced by Roesser [13], Fornasini and Marchesini [5, 14], and Kurek [15]. These models have been extended to positive systems in [16–19]. A positive system means that its state and output are nonnegative whenever the initial condition and input are nonnegative [19–21]. Positive 2D systems are needed in many cases such as the wave equation in fluid dynamics and the heat equation which describes the temperature (using thermodynamic temperature scale) in a given region over time and the Poisson’s equation. These facts stimulate the research on 2D positive discrete systems. Reference [22] investigated the choice of the forms of Lyapunov functions for positive 2D Roesser model. The problem of stability analysis for 2D positive systems has been investigated in [17, 23–25]. It should be noted that although positive 2D systems have been discussed in control engineering and mathematics literature recently, there are still many questions which deserve further investigation.On the other hand, the reaction of real-world systems to exogenous signals is never instantaneous and, always infected by certain time delays. For general systems, even nominal stable systems when were affected by delays may inherit very complex behaviors such as oscillations, instability, and bad performance [26], and delayed systems have attracted many researchers’ attention [27–32]. The reachability, minimum energy control, and realization problem for positive 2D discrete-time systems with delays has been analyzed in [18, 33]. And the stability analysis for 2D positive delayed systems has been investigated in [34–36]. In addition, perturbations and uncertainties widely exist in the practical systems. In some cases, the perturbations and unmodeled errors can be merged into disturbances, which can be supposed to be bounded in the appropriate norms. It is important and necessary to establish a criterion evaluating the disturbance attenuation performance for the positive 2D discrete-time systems. However, to the best of our knowledge, there has been no literature considering the disturbance attenuation performance for positive 2D systems, which motivates the present study.In this paper, we will study the problem of delay-dependent stability andl1-gain analysis for positive 2D linear systems with delays. The main theoretical contributions of this paper are as follows (1) We use l1-gain to evaluate the disturbance attenuation performance of positive 2D linear systems. This important performance is firstly considered for positive 2D systems, and a delay-dependent stability criterion of these systems with state delays is developed. (2) Copositive-type Lyapunov function method is firstly used to analyze delay-dependent stability and l1-gain performance for positive 2D linear systems. (3) It is significant to characterize conditions under which the positive 2D delayed system is asymptotically stable. All the developed results are expressed in terms of feasibility testing of LMIs which is computationally tractable.The paper is organized as follows. In Section2, problem statement and some definitions concerning the positive 2D linear systems with delays are given. In Section 3, some theorems concerning the delay-dependent stability and l1-gain analysis of positive 2D linear systems are presented. In Section 4, a numerical example is given to illustrate the effectiveness of the proposed results. Finally, concluding remarks are provided in Section 5.Notations. In this paper, the superscript “T” denotes the transpose. The notation X>Y (X≥Y) means that matrix X-Y is positive definite (positive semidefinite, resp.). A≽0 (⪯0) means that all entries of matrix A are nonnegative (nonpositive). A≻0 (≺0) means that all entries of matrix A are positive (negative). Rn×m denotes the set of n×m real matrices. The set of real n×m matrices with nonnegative entries will be denoted by  R+n×m, R+n denotes the set of vectors with nonnegative entries, and the set of nonnegative integers will be denoted by Z+. The n×n identity matrix will be denoted by In. The l1 norm of a 2D signal w(i,j)=[w1(i,j),w2(i,j),…,wm(i,j)]T is given by
(1)∥w(i,j)∥1=∑k=1mwk(i,j).
And we say w(i,j)∈l1, if ∥w(i,j)∥1<∞.


## 2. Problem Formulation and Preliminaries

Consider the positive 2D Roesser model with state delays [25]:
(1a)[xh(i+1,j)xv(i,j+1)]=A[xh(i,j)xv(i,j)]+Ad[xh(i-dh(i),j)xv(i,j-dv(j))][xh(i+1,j)xv(i,j+1)]=+Bw(i,j),(1b)z(i,j)=Hx(i,j)+Lw(i,j),
where i and j are integers in Z+, xh(i,j) is the horizontal state inR+n1, xv(i,j) is the vertical state in  R+n2, x(i,j) is the whole state in R+n, w(i,j)∈R+m1 is the l1 norm bounded disturbance input, z(i,j)∈R+m2 is the controlled output, and A, Ad, B, H, L≽0 are system matrices with compatible dimensions. The matrices are
(2)A=[A11A12A21A22],Ad=[Ad11Ad12Ad21Ad22],B=[B1B2].dh(i) and dv(j) are delays along horizontal and vertical directions, respectively. We assume that dh(i) and dv(j) satisfy
(3)dhL≤dh(i)≤dhH,dvL≤dv(j)≤dvH,
where dhL, dhH and dvL, dvH denote the lower and upper delay bounds along horizontal and vertical directions, respectively. The boundary conditions are defined by
(4)xh(i,j)=hij,∀0≤j≤z1,-dhH≤i≤0,xh(i,j)=0,∀j>z1,-dhH≤i≤0,xv(i,j)=vij,∀0≤i≤z2,-dvH≤j≤0,xv(i,j)=0,∀i>z2,-dvH≤j≤0,h00=v00,
where z1<∞ and z2<∞ are positive integers, hij∈R+n1 and vij∈R+n2 are given vectors.Definition 1.
The 2D positive system(1a) and (1b) with w(i,j)=0 is said to be asymptotically stable if  liml→∞Xl=0 for all bounded boundary conditions (4), where
(5)Xl=sup{∥x(i,j)∥:i+j=l,i,j≥1}.Definition 2.
Forγ>0, the system (1a) and (1b)  is said to be asymptotically stable with the l1-gain index γ, if the following conditions hold.(1)
The system(1a) and (1b)  with w(i,j)=0 is asymptotically stable.(2)
Under zero boundary conditions, that is,hij=0, vij=0 in (4), it holds that
(6)∑i=0∞∑j=0∞∥z(i,j)∥1<γ∑i=0∞∑j=0∞∥w(i,j)∥1,∀0≠w(i,j)∈l1.Remark 3.
From (6), we see that γ can characterize the disturbance attenuation performance of the system (1a) and (1b). The smaller the γ is, the better the disturbance attenuation performance is.


## 3. Main Results

### 3.1. Stability Analysis

In this subsection, we focus on the problem of delay-dependent asymptotically stability analysis for the positive 2D discrete linear systems with state delays.Theorem 4.
For given positive constantsdhL, dhH, dvL, dvH, the positive 2D system (1a) and (1b)  with w(i,j)=0 is asymptotically stable if there exist vectors p, q, ς1, ς2, ζ∈R+n, such that
(7)Φ=diag{Φ1,Φ2,…,Φn,Φ1′,Φ2′,…,Φn′,Φ1′′,Φ2′′,…,Φn′′,Φ=diagΦ1′′′,Φ2′′′,…,Φn′′′}<0,
where
(8)Φk={(akT-Ek)p+(akT+(dhH-dhL)Ek)q+Ekζ+(dhH2(akT-Ek)-dhHEk)ς2+dhH2Ekς1,1≤k≤n1,(akT-Ek)p+(akT+(dvH-dvL)Ek)q+Ekζ+(dvH2(akT-Ek)-dvHEk)ς2+dvH2Ekς1,n1+1≤k≤n,Φk′={adkTp+(adkT-Ek)q+dhH2adkTς2,1≤k≤n1,adkTp+(adkT-Ek)q+dvH2adkTς2,n1+1≤k≤n,Φk′′={-Ekζ+dhHEk(ς2-ς1),1≤k≤n1,-Ekζ+dvHEk(ς2-ς1),n1+1≤k≤n,Φk′′′={-dhHEkς1,1≤k≤n1,-dvHEkς1,n1+1≤k≤n,
with k∈n_={1,2,…,n}, Ek=[0,…,0︷k-1,1,0,…,0︷n-k], and ak(adk) represents the kth column vector of matrix A(Ad).Proof.
Choose the following copositive Lyapunov-Krasovskii functional candidate:(9)V(i,j)=Vh(i,j)+Vv(i,j),
where
(10)Vh(i,j)=∑k=15Vkh(i,j),V1h(i,j)=xhT(i,j)ph,V2h(i,j)=∑r=i-dh(i)ixhT(r,j)qh,V3h(i,j)=∑r=i-dhHi-1xhT(r,j)ζh,V4h(i,j)=∑s=-dhH+1-dhL∑r=i+si-1xhT(r,j)qh,V5h(i,j)=dhH∑s=-dhH-1∑r=i+si-1ηhT(r,j)ςh,Vv(i,j)=∑k=15Vkv(i,j),V1v(i,j)=xvT(i,j)pv,V2v(i,j)=∑s=j-dv(j)jxvT(i,s)qv,V3v(i,j)=∑t=j-dvHj-1xvT(i,t)ζv,V4v(i,j)=∑s=-dvH+1-dvL∑t=j+sj-1xvT(i,t)qv,V5v(i,j)=dvH∑s=-dvH-1∑t=j+sj-1ηvT(i,t)ςv,ηh(r,j)=[xhT(r,j)δhT(r,j)]T,ηv(i,t)=[xvT(i,t)δvT(i,t)]T,δh(r,j)=xh(r+1,j)-xh(r,j),δv(i,t)=xv(i,t+1)-xv(i,t),
with ph, qh, ζh, ς1h, and ς2h∈R+n1,pv, qv, ζv, ς1v, and ς2v∈R+n2, ςh=[ς1hς2h]T∈R+2n1, and ςv=[ς1vς2v]T∈R+2n2.
Along the trajectory of the system(1a) and (1b), we have
(11)ΔV(i,j)=Vh(i+1,j)-Vh(i,j)+Vv(i,j+1)-Vv(i,j)ΔV(i,j)=∑k=15ΔVkh(i,j)+∑k=15ΔVkv(i,j),
where
(12)ΔV1h(i,j)=xhT(i+1,j)ph-xhT(i,j)ph,ΔV2h(i,j)=∑r=i+1-dh(i+1)i+1xhT(r,j)qh-∑r=i-dh(i)ixhT(r,j)qh=xhT(i+1,j)qh-xhT(i-dh(i),j)qh+∑r=i+1-dh(i+1)ixhT(r,j)qh-∑r=i+1-dh(i)ixhT(r,j)qh≤xhT(i+1,j)qh-xhT(i-dh(i),j)qh+∑r=i+1-dhHixhT(r,j)qh-∑r=i+1-dhLixhT(r,j)qh=xhT(i+1,j)qh-xhT(i-dh(i),j)qh+∑r=i+1-dhHr=i-dhLxhT(r,j)qh,ΔV3h(i,j)=∑r=i+1-dhHixhT(r,j)ζh-∑r=i-dhHi-1xhT(r,j)ζhΔV3h(i,j)=xhT(i,j)ζh-xhT(i-dhH,j)ζh,ΔV4h(i,j)=∑s=-dhH+1-dhL∑r=i+1+sixhT(r,j)qhΔV4h(i,j)=-∑s=-dhH+1-dhL∑r=i+si-1xhT(r,j)qhΔV4h(i,j)=∑s=-dhH+1-dhL[xhT(i,j)qh-xhT(i+s,j)qh]ΔV4h(i,j)=(dhH-dhL)xhT(i,j)qh-∑r=i-dhH+1i-dhLxhT(r,j)qh,ΔV5h(i,j)=dhH∑s=-dhH-1∑r=i+1+siηhT(r,j)ςh-dhH∑s=-dhH-1∑r=i+si-1ηhT(r,j)ςh=dhH∑s=-dhH-1(ηhT(i,j)ςh-ηhT(i+s,j)ςh)=dhH2ηhT(i,j)ςh-dhH∑r=i-dhHi-1ηhT(r,j)ςh=dhH2[xhT(i,j)xhT(i+1,j)-xhT(i,j)][ς1hς2h]-dhH[∑r=i-dhHi-1xhT(i,j)xhT(i,j)-xhT(i-dhH,j)]×[ς1hς2h],ΔV1v(i,j)=xvT(i,j+1)pv-xvT(i,j)pv,ΔV2v(i,j)=∑s=j+1-dv(j+1)j+1xvT(i,s)qv-∑s=j-dv(j)jxvT(i,s)qvΔV2v(i,j)=xvT(i,j+1)qv-xvT(i,j-dv(j))qvΔV2v(i,j)=+∑s=j+1-dv(j+1)jxvT(i,s)qvΔV2v(i,j)=-∑s=j+1-dv(j)jxvT(i,s)qvΔV2v(i,j)≤xvT(i,j+1)qv-xvT(i,j-dv(j))qvΔV2v(i,j)=+∑s=j+1-dvHjxvT(i,s)qv-∑s=j+1-dvLjxvT(i,s)qvΔV2v(i,j)=xvT(i,j+1)qv-xvT(i,j-dv(j))qvΔV2v(i,j)=+∑t=j+1-dvHj-dvLxvT(i,t)qv,ΔV3v(i,j)=∑s=j+1-dvHjxhT(i,s)ζvΔV3v(i,j)=-∑s=j-dvHj-1xhT(i,a)ζvΔV3v(i,j)=xvT(i,j)ζv-xvT(i,j-dvH)ζv,ΔV4v(i,j)=∑s=-dvH+1-dvL∑t=j+1+sjxvT(i,t)qvΔV4v(i,j)+=-∑s=-dvH+1-dvL∑t=j+sj-1xvT(i,t)qvΔV4v(i,j)=∑s=-dvH+1-dvL[xvT(i,j)qv-xvT(i,j+s)qv]ΔV4v(i,j)=(dvH-dvL)xhT(i,j)qvΔV4v(i,j)=-∑t=j-dvH+1j-dvLxvT(i,t)qv,ΔV5v(i,j)=dvH∑s=-dvH-1∑t=j+1+sjηvT(i,t)ςv-dvH∑s=-dvH-1∑t=j+sj-1ηvT(i,t)ςv=dvH∑s=-dvH-1(ηvT(i,j)ςv-ηvT(i,j+s)ςv)=dvH2ηvT(i,j)ςv-dvH∑t=j-dvHj-1ηvT(i,t)ςv=dvH2[xvT(i,j)xvT(i,j+1)-xvT(i,j)]×[ς1vς2v]-dvH[∑t=j-dvHj-1xvT(i,j)xvT(i,j)-xvT(i,j-dvH)]×[ς1vς2v].
Substitute the previously mentioned formulations into (11), and take
(13)p=[phpv],q=[qhqv],ζ=[ζhζv],ς1=[ς1hς1h],ς2=[ς2hς2h],DH=[dhHIn100dvHIn2],DL=[dhLIn100dvLIn2],x(i,j)=[xhT(i,j)xvT(i,j)]T,xd(i,j)=[xhT(i-dh(i),j)xvT(i,j-dv(j))]T,xH(i,j)=[xhT(i-dhH,j)xvT(i,j-dvH)]T,xs(i,j)=[∑r=i-dhH+1i-1xhT(r,j)∑t=j-dvH+1j-1xvT(i,t)]T.
Then we have
(14)ΔV(i,j)=xT(i,j){(AT-In)p+(AT+DH-DL)q+ζ+DH2((AT-In)ς2+ς1)-DHς2}ΔV(i,j)=+xdT(i,j){AdTp+(AdT-In)q+DH2AdTς2}ΔV(i,j)=+xHT(i,j){-ζ+DH(ς2-ς1)}ΔV(i,j)=+xsT(i,j){-DHς1}.
If condition (7) holds, one obtains
(15)(AT-In)p+(AT+DH-DL)q+ζ+DH2((AT-In)ς2+ς1)-DHς2≺0,AdTp+(AdT-In)q+DH2AdTς2≺0,-ζ+DH(ς2-ς1)≺0,-DHς1≺0.
It follows that ΔV(i,j)<0, which means that
(16)Vh(i+1,j)+Vv(i,j+1)<Vh(i,j)+Vv(i,j).
Summing up both sides of (16) from D to 0 with respect to i and from 0 to D with respect to j, for any nonnegative integer D≥max(z1,z2), one gets
(17)Vh(1,D)+Vv(0,D+1)+Vh(2,D-1)+Vv(1,D)+⋯+Vh(D+1,0)+Vv(D,1)=∑i+j=D+1Vh(i,j)+∑i+j=D+1Vv(i,j)=∑i+j=D+1V(i,j)≤Vh(0,D)+Vv(0,D)+Vh(1,D-1)+Vv(1,D-1)+⋯+Vh(D,0)+Vv(D,0)=∑i+j=DV(i,j).
Then from (9), we can conclude that
(18)limi+j→∞x(i,j)=0,
which implies that the system (1a) and (1b) with w(i,j)=0 is asymptotically stable.
This completes the proof.WhendhH=dhL=dh, and dvH=dvL=dv, the system (1a) and (1b)  with w(i,j)=0 is reduced to the following system:
(19)[xh(i+1,j)xv(i,j+1)]=A[xh(i,j)xv(i,j)]+Ad[xh(i-dh,j)xv(i,j-dv)],
where dh and dv are constant delays along horizontal and vertical directions, respectively, and the boundary conditions are defined in (4). Then we can get the following result.Corollary 5.
For given positive constantsdh and dv, the positive 2D system (19) is asymptotically stable if there exist vectors p and q∈R+n, such that
(20)Φ~=diag{Φ1,Φ2,…,Φn,Φ1′,Φ2′,…,Φn′}<0,
where
(21)Φ~k=(akT-Ek)p+akTq,Φ~k′=adkTp+(adkT-Ek)q,
with k∈K_={1,2,…,n}, Ek=[0,…,0︷k-1,1,0,…,0︷n-k], p, q∈R+n, and ak(adk) represents the kth column vector of matrix A(Ad).Proof.
Choose the following copositive Lyapunov-Krasovskii functional candidate for the system (19):
(22)V(i,j)=Vh(i,j)+Vv(i,j),
where
(23)Vh(i,j)=∑k=12Vkh(i,j),V1h(i,j)=xhT(i,j)ph,V2h(i,j)=∑r=i-dhixhT(r,j)qh,Vv(i,j)=∑k=12Vkv(i,j),V1v(i,j)=xvT(i,j)pv,V2v(i,j)=∑s=j-dvjxvT(i,s)qv,
with ph,qh∈R+n1, pv,qv∈R+n2. Then following the proof line of Theorem 4, the corollary can be obtained.


### 3.2.l1-Gain Analysis

The following theorem establishes sufficient condition of the asymptotical stability withl1-gain performance for the system (1a) and (1b).Theorem 6.
For given positive constantsdhL, dhH, dvL, dvH, and γ, the positive 2D system (1a) and (1b) is asymptotically stable with the l1-gain index γ if there exist vectors p∈R+n, q∈R+n, ς1∈R+n, ς2∈R+n, and ζ∈R+n, such that
(24)Φ-=diag{Φ-1,Φ-2,…,Φ-n,Φ1′,Φ2′,…,Φn′,Φ1′′,Φ2′′,…,Φn′′,Φ1′′′,Φ2′′′,…,Φn′′′,Τ1,Τ2,…,Τm}<0,
where Φk′, Φk′′, and Φk′′′ are denoted as in Theorem 4, and
(25)Φ-k={(akT-Ek)p+(akT+(dhH-dhL)Ek)q+Ekζ+(dhH2(akT-Ek)-dhHEk)ς21≤k≤n1,+dhH2Ekς1+∥hk∥1,(akT-Ek)p+(akT+(dvH-dvL)Ek)q+Ekζ+(dvH2(akT-Ek)-dvHEk)ς2n1+1≤k≤n,+dvH2Ekς1+∥hk∥1,Tε=bεTp+bεTq+bεTDhH2ς2+∥lε∥1-γ,1≤ε≤m,DH=diag{dhHIn1,dvHIn2},
with k∈n_={1,2,…,n}, ε∈m_={1,2,…,m}, Ek=[0,…,0︷k-1,1,0,…,0︷n-k], ak, adk, bk, and hk represent the kth column vector of matrices A, Ad, B, and H, respectively, and lε represents the εth column vector of matrix L.Proof.
It is an obvious fact that (24) implies the following inequality:
(26)Ψ=diag{Ψ1,Ψ2,…,Ψn,Ψ1′,Ψ2′,…,Ψn′,Ψ1′′,Ψ2′′,…,dΨ=diagΨn′′,Ψ1′′′,Ψ2′′′,…,Ψn′′′}<0,
where
(27)Ψk={(akT-Ek)p+(akT+(dhH-dhL)Ek)q+ζk+(dhH2(akT-Ek)-dhHEk)ς2+dhH2Ekς1,1≤k≤n1,(akT-Ek)p+(akT+(dvH-dvL)Ek)q+ζk+(dvH2(akT-Ek)-dvHEk)ς2+dvH2Ekς1,n1+1≤k≤n,Ψk′={adkTp+(adkT-Ek)q+dhH2adkTς2,1≤k≤n1,adkTp+(adkT-Ek)q+dvH2adkTς2,n1+1≤k≤n,Ψk′′={-ζk+dhHEk(ς2-ς1),1≤k≤n1,-ζk+dvHEk(ς2-ς1),n1+1≤k≤n,Ψk′′′={-dhHEkς1,1≤k≤n1,-dvHEkς1,n1+1≤k≤n,
with k∈n_={1,2,…,n}, Ek=[0,…,0︷k-1,1,0,…,0︷n-k], and ak(adk) represents the kth column vector of matrix A(Ad).
By Theorem4, we can obtain that the system (1a) and (1b) with w(i,j)=0 is asymptotically stable. Now we are in a position to prove that the system (1a) and (1b)  has a prescribed l1-gain index γ for any nonzero w(i,j)∈l1. To establish the l1-gain performance, we choose the same copositive Lyapunov-Krasovskii functional candidate as in (9) for the system (1a) and (1b). Following the proof line of Theorem 4, we can get that
(28)ΔV(i,j)+∥z(i,j)∥1-γ∥w(i,j)∥1=xT(i,j){(AT-In)p+(AT+DH-DL)qΔV(i,j)+∥z(i,j)∥1+ζ+DH2((AT-In)ς2+ς1)-DHς2}ΔV(i,j)=+xdT(i,j){AdTp+(AdT-In)q+DH2AdTς2}ΔV(i,j)=+xHT(i,j){-ζ+DH(ς2-ς1)}ΔV(i,j)=+xsT(i,j){-DHς1}ΔV(i,j)=+wT(i,j){BT(p+q+DH2ς2)}ΔV(i,j)=+∥Hx(i,j)∥1+∥Lw(i,j)∥1-γ∥w(i,j)∥1.
According to the definition of l1 norm, one obtains(29)Hx(i,j)=[h1,1h1,2⋯h1,n1h2,1h2,2⋯h2,n1⋮⋮⋯⋮hp,1hp,2⋯hp,n1h1,n1+1h1,n1+2⋯h1,nh2,n1+1h2,n1+2⋯h2,n⋮⋮⋯⋮hp,n1+1hp,n1+2⋯hp,n][x1h(i,j)⋮xn1h(i,j)x1v(i,j)⋮xn2v(i,j)]Hx(i,j)=[h1,1x1h(i,j)+⋯+h1,n1xn1h(i,j)+h1,n1+1x1v(i,j)+⋯+h1,nxn2v(i,j)h2,1x1h(i,j)+⋯+h2,n1xn1h(i,j)+h2,n1+1x1v(i,j)+⋯+h2,nxn2v(i,j)⋮hp,1x1h(i,j)+⋯+hp,n1xn1h(i,j)+hp,n1+1x1v(i,j)+⋯+hp,nxn2v(i,j)],∥Hx(i,j)∥1=∑k=1phk,1x1h(i,j)+⋯+hk,n1xn1h(i,j)+hk,n1+1x1v(i,j)+⋯+hk,nxn2v(i,j)=(∑k=1phk,1)x1h(i,j)+⋯+(∑k=1phk,n1)xn1h(i,j)+(∑k=1phk,n1+1)x1v(i,j)+⋯+(∑k=1phk,n)xn2v(i,j)=xT(i,j)[∥h1∥1∥h2∥1⋯∥hn∥1]T,where hk represents the kth column vector and hi,j represents the entry located at (i,j) of matrix H. Then, similarly
(30)∥Lw(i,j)∥1=wT(i,j)[∥l1∥1∥l2∥1⋯∥lm∥1]T,γ∥w(i,j)∥1=wT(i,j)[γγ⋯γ]T,
where lk represents the kth column vector of matrix L.
Substituting (29)-(30) into (28) leads to
(31)ΔV(i,j)+∥z(i,j)∥1-γ∥w(i,j)∥1=xT(i,j){[∥h1∥1∥h2∥1⋯∥hn∥1]T(AT-In)p+(AT+DH-DL)q+ζ+DH2((AT-In)ς2+ς1)-DHς2+[∥h1∥1∥h2∥1⋯∥hn∥1]T}+xdT(i,j){AdTp+(AdT-In)q+DH2AdTς2}+xHT(i,j){-ζ+DH(ς2-ς1)}+xsT(i,j){-DHς1}+wT(i,j){[γγ⋯γ]TBT(p+q+DH2ς2)+[∥l1∥1∥l2∥1⋯∥lm∥1]T-[γγ⋯γ]T}.
If condition (24) holds, we have
(32)Vh(i+1,j)-Vh(i,j)+Vv(i,j+1)-Vv(i,j)+∥z(i,j)∥1-γ∥w(i,j)∥1<0.
We know that
(33)ΔV(i,j)=Vh(i+1,j)-Vh(i,j)+Vv(i,j+1)-Vv(i,j).
For any positive scalars  kh, andkv∈Z+, it can be verified that
(34)∑i=0kh∑j=0kvΔV(i,j)=∑i=0kh∑j=0kv(Vh(i+1,j)-Vh(i,j))∑i=0kh∑j=0kvΔV(i,j)=+∑i=0kh∑j=0kv(Vv(i,j+1)-Vv(i,j))∑i=0kh∑j=0kvΔV(i,j)=∑j=0kv(VhT(kh+1,j)-VhT(0,j))∑i=0kh∑j=0kvΔV(i,j)=+∑i=0kh(VvT(i,kv+1)-VvT(i,0)).
When kh and kv=∞, we have
(35)∑i=0∞∑j=0∞(∥z(i,j)∥1-γ∥w(i,j)∥1)<∑i=0∞∑j=0∞ΔV(i,j).
The existence of solution for LMI (24) implies that the positive 2D system (1a) and (1b) is asymptotically stable. Together with the zero boundary conditions, one can get
(36)∑i=0∞∑j=0∞ΔV(i,j)=0.
Applying (36) to (35), one has
(37)∑i=0∞∑j=0∞∥z(i,j)∥1<γ∑i=0∞∑j=0∞∥w(i,j)∥1.
By Definition 2, the positive 2D system (1a) and (1b) is asymptotically stable and has the l1-gain index γ.
This completes the proof.Remark 7.
In Theorem6, the disturbance attenuation performance of positive 2D linear systems is analyzed, and sufficient conditions for the existence of l1-gain performance for positive 2D system to (1a) and (1b) are proposed in terms of LMIs which are computationally tractable. This is also the major contribution of our paper.


## 3.1. Stability Analysis

In this subsection, we focus on the problem of delay-dependent asymptotically stability analysis for the positive 2D discrete linear systems with state delays.Theorem 4.
For given positive constantsdhL, dhH, dvL, dvH, the positive 2D system (1a) and (1b)  with w(i,j)=0 is asymptotically stable if there exist vectors p, q, ς1, ς2, ζ∈R+n, such that
(7)Φ=diag{Φ1,Φ2,…,Φn,Φ1′,Φ2′,…,Φn′,Φ1′′,Φ2′′,…,Φn′′,Φ=diagΦ1′′′,Φ2′′′,…,Φn′′′}<0,
where
(8)Φk={(akT-Ek)p+(akT+(dhH-dhL)Ek)q+Ekζ+(dhH2(akT-Ek)-dhHEk)ς2+dhH2Ekς1,1≤k≤n1,(akT-Ek)p+(akT+(dvH-dvL)Ek)q+Ekζ+(dvH2(akT-Ek)-dvHEk)ς2+dvH2Ekς1,n1+1≤k≤n,Φk′={adkTp+(adkT-Ek)q+dhH2adkTς2,1≤k≤n1,adkTp+(adkT-Ek)q+dvH2adkTς2,n1+1≤k≤n,Φk′′={-Ekζ+dhHEk(ς2-ς1),1≤k≤n1,-Ekζ+dvHEk(ς2-ς1),n1+1≤k≤n,Φk′′′={-dhHEkς1,1≤k≤n1,-dvHEkς1,n1+1≤k≤n,
with k∈n_={1,2,…,n}, Ek=[0,…,0︷k-1,1,0,…,0︷n-k], and ak(adk) represents the kth column vector of matrix A(Ad).Proof.
Choose the following copositive Lyapunov-Krasovskii functional candidate:(9)V(i,j)=Vh(i,j)+Vv(i,j),
where
(10)Vh(i,j)=∑k=15Vkh(i,j),V1h(i,j)=xhT(i,j)ph,V2h(i,j)=∑r=i-dh(i)ixhT(r,j)qh,V3h(i,j)=∑r=i-dhHi-1xhT(r,j)ζh,V4h(i,j)=∑s=-dhH+1-dhL∑r=i+si-1xhT(r,j)qh,V5h(i,j)=dhH∑s=-dhH-1∑r=i+si-1ηhT(r,j)ςh,Vv(i,j)=∑k=15Vkv(i,j),V1v(i,j)=xvT(i,j)pv,V2v(i,j)=∑s=j-dv(j)jxvT(i,s)qv,V3v(i,j)=∑t=j-dvHj-1xvT(i,t)ζv,V4v(i,j)=∑s=-dvH+1-dvL∑t=j+sj-1xvT(i,t)qv,V5v(i,j)=dvH∑s=-dvH-1∑t=j+sj-1ηvT(i,t)ςv,ηh(r,j)=[xhT(r,j)δhT(r,j)]T,ηv(i,t)=[xvT(i,t)δvT(i,t)]T,δh(r,j)=xh(r+1,j)-xh(r,j),δv(i,t)=xv(i,t+1)-xv(i,t),
with ph, qh, ζh, ς1h, and ς2h∈R+n1,pv, qv, ζv, ς1v, and ς2v∈R+n2, ςh=[ς1hς2h]T∈R+2n1, and ςv=[ς1vς2v]T∈R+2n2.
Along the trajectory of the system(1a) and (1b), we have
(11)ΔV(i,j)=Vh(i+1,j)-Vh(i,j)+Vv(i,j+1)-Vv(i,j)ΔV(i,j)=∑k=15ΔVkh(i,j)+∑k=15ΔVkv(i,j),
where
(12)ΔV1h(i,j)=xhT(i+1,j)ph-xhT(i,j)ph,ΔV2h(i,j)=∑r=i+1-dh(i+1)i+1xhT(r,j)qh-∑r=i-dh(i)ixhT(r,j)qh=xhT(i+1,j)qh-xhT(i-dh(i),j)qh+∑r=i+1-dh(i+1)ixhT(r,j)qh-∑r=i+1-dh(i)ixhT(r,j)qh≤xhT(i+1,j)qh-xhT(i-dh(i),j)qh+∑r=i+1-dhHixhT(r,j)qh-∑r=i+1-dhLixhT(r,j)qh=xhT(i+1,j)qh-xhT(i-dh(i),j)qh+∑r=i+1-dhHr=i-dhLxhT(r,j)qh,ΔV3h(i,j)=∑r=i+1-dhHixhT(r,j)ζh-∑r=i-dhHi-1xhT(r,j)ζhΔV3h(i,j)=xhT(i,j)ζh-xhT(i-dhH,j)ζh,ΔV4h(i,j)=∑s=-dhH+1-dhL∑r=i+1+sixhT(r,j)qhΔV4h(i,j)=-∑s=-dhH+1-dhL∑r=i+si-1xhT(r,j)qhΔV4h(i,j)=∑s=-dhH+1-dhL[xhT(i,j)qh-xhT(i+s,j)qh]ΔV4h(i,j)=(dhH-dhL)xhT(i,j)qh-∑r=i-dhH+1i-dhLxhT(r,j)qh,ΔV5h(i,j)=dhH∑s=-dhH-1∑r=i+1+siηhT(r,j)ςh-dhH∑s=-dhH-1∑r=i+si-1ηhT(r,j)ςh=dhH∑s=-dhH-1(ηhT(i,j)ςh-ηhT(i+s,j)ςh)=dhH2ηhT(i,j)ςh-dhH∑r=i-dhHi-1ηhT(r,j)ςh=dhH2[xhT(i,j)xhT(i+1,j)-xhT(i,j)][ς1hς2h]-dhH[∑r=i-dhHi-1xhT(i,j)xhT(i,j)-xhT(i-dhH,j)]×[ς1hς2h],ΔV1v(i,j)=xvT(i,j+1)pv-xvT(i,j)pv,ΔV2v(i,j)=∑s=j+1-dv(j+1)j+1xvT(i,s)qv-∑s=j-dv(j)jxvT(i,s)qvΔV2v(i,j)=xvT(i,j+1)qv-xvT(i,j-dv(j))qvΔV2v(i,j)=+∑s=j+1-dv(j+1)jxvT(i,s)qvΔV2v(i,j)=-∑s=j+1-dv(j)jxvT(i,s)qvΔV2v(i,j)≤xvT(i,j+1)qv-xvT(i,j-dv(j))qvΔV2v(i,j)=+∑s=j+1-dvHjxvT(i,s)qv-∑s=j+1-dvLjxvT(i,s)qvΔV2v(i,j)=xvT(i,j+1)qv-xvT(i,j-dv(j))qvΔV2v(i,j)=+∑t=j+1-dvHj-dvLxvT(i,t)qv,ΔV3v(i,j)=∑s=j+1-dvHjxhT(i,s)ζvΔV3v(i,j)=-∑s=j-dvHj-1xhT(i,a)ζvΔV3v(i,j)=xvT(i,j)ζv-xvT(i,j-dvH)ζv,ΔV4v(i,j)=∑s=-dvH+1-dvL∑t=j+1+sjxvT(i,t)qvΔV4v(i,j)+=-∑s=-dvH+1-dvL∑t=j+sj-1xvT(i,t)qvΔV4v(i,j)=∑s=-dvH+1-dvL[xvT(i,j)qv-xvT(i,j+s)qv]ΔV4v(i,j)=(dvH-dvL)xhT(i,j)qvΔV4v(i,j)=-∑t=j-dvH+1j-dvLxvT(i,t)qv,ΔV5v(i,j)=dvH∑s=-dvH-1∑t=j+1+sjηvT(i,t)ςv-dvH∑s=-dvH-1∑t=j+sj-1ηvT(i,t)ςv=dvH∑s=-dvH-1(ηvT(i,j)ςv-ηvT(i,j+s)ςv)=dvH2ηvT(i,j)ςv-dvH∑t=j-dvHj-1ηvT(i,t)ςv=dvH2[xvT(i,j)xvT(i,j+1)-xvT(i,j)]×[ς1vς2v]-dvH[∑t=j-dvHj-1xvT(i,j)xvT(i,j)-xvT(i,j-dvH)]×[ς1vς2v].
Substitute the previously mentioned formulations into (11), and take
(13)p=[phpv],q=[qhqv],ζ=[ζhζv],ς1=[ς1hς1h],ς2=[ς2hς2h],DH=[dhHIn100dvHIn2],DL=[dhLIn100dvLIn2],x(i,j)=[xhT(i,j)xvT(i,j)]T,xd(i,j)=[xhT(i-dh(i),j)xvT(i,j-dv(j))]T,xH(i,j)=[xhT(i-dhH,j)xvT(i,j-dvH)]T,xs(i,j)=[∑r=i-dhH+1i-1xhT(r,j)∑t=j-dvH+1j-1xvT(i,t)]T.
Then we have
(14)ΔV(i,j)=xT(i,j){(AT-In)p+(AT+DH-DL)q+ζ+DH2((AT-In)ς2+ς1)-DHς2}ΔV(i,j)=+xdT(i,j){AdTp+(AdT-In)q+DH2AdTς2}ΔV(i,j)=+xHT(i,j){-ζ+DH(ς2-ς1)}ΔV(i,j)=+xsT(i,j){-DHς1}.
If condition (7) holds, one obtains
(15)(AT-In)p+(AT+DH-DL)q+ζ+DH2((AT-In)ς2+ς1)-DHς2≺0,AdTp+(AdT-In)q+DH2AdTς2≺0,-ζ+DH(ς2-ς1)≺0,-DHς1≺0.
It follows that ΔV(i,j)<0, which means that
(16)Vh(i+1,j)+Vv(i,j+1)<Vh(i,j)+Vv(i,j).
Summing up both sides of (16) from D to 0 with respect to i and from 0 to D with respect to j, for any nonnegative integer D≥max(z1,z2), one gets
(17)Vh(1,D)+Vv(0,D+1)+Vh(2,D-1)+Vv(1,D)+⋯+Vh(D+1,0)+Vv(D,1)=∑i+j=D+1Vh(i,j)+∑i+j=D+1Vv(i,j)=∑i+j=D+1V(i,j)≤Vh(0,D)+Vv(0,D)+Vh(1,D-1)+Vv(1,D-1)+⋯+Vh(D,0)+Vv(D,0)=∑i+j=DV(i,j).
Then from (9), we can conclude that
(18)limi+j→∞x(i,j)=0,
which implies that the system (1a) and (1b) with w(i,j)=0 is asymptotically stable.
This completes the proof.WhendhH=dhL=dh, and dvH=dvL=dv, the system (1a) and (1b)  with w(i,j)=0 is reduced to the following system:
(19)[xh(i+1,j)xv(i,j+1)]=A[xh(i,j)xv(i,j)]+Ad[xh(i-dh,j)xv(i,j-dv)],
where dh and dv are constant delays along horizontal and vertical directions, respectively, and the boundary conditions are defined in (4). Then we can get the following result.Corollary 5.
For given positive constantsdh and dv, the positive 2D system (19) is asymptotically stable if there exist vectors p and q∈R+n, such that
(20)Φ~=diag{Φ1,Φ2,…,Φn,Φ1′,Φ2′,…,Φn′}<0,
where
(21)Φ~k=(akT-Ek)p+akTq,Φ~k′=adkTp+(adkT-Ek)q,
with k∈K_={1,2,…,n}, Ek=[0,…,0︷k-1,1,0,…,0︷n-k], p, q∈R+n, and ak(adk) represents the kth column vector of matrix A(Ad).Proof.
Choose the following copositive Lyapunov-Krasovskii functional candidate for the system (19):
(22)V(i,j)=Vh(i,j)+Vv(i,j),
where
(23)Vh(i,j)=∑k=12Vkh(i,j),V1h(i,j)=xhT(i,j)ph,V2h(i,j)=∑r=i-dhixhT(r,j)qh,Vv(i,j)=∑k=12Vkv(i,j),V1v(i,j)=xvT(i,j)pv,V2v(i,j)=∑s=j-dvjxvT(i,s)qv,
with ph,qh∈R+n1, pv,qv∈R+n2. Then following the proof line of Theorem 4, the corollary can be obtained.


## 3.2.l1-Gain Analysis

The following theorem establishes sufficient condition of the asymptotical stability withl1-gain performance for the system (1a) and (1b).Theorem 6.
For given positive constantsdhL, dhH, dvL, dvH, and γ, the positive 2D system (1a) and (1b) is asymptotically stable with the l1-gain index γ if there exist vectors p∈R+n, q∈R+n, ς1∈R+n, ς2∈R+n, and ζ∈R+n, such that
(24)Φ-=diag{Φ-1,Φ-2,…,Φ-n,Φ1′,Φ2′,…,Φn′,Φ1′′,Φ2′′,…,Φn′′,Φ1′′′,Φ2′′′,…,Φn′′′,Τ1,Τ2,…,Τm}<0,
where Φk′, Φk′′, and Φk′′′ are denoted as in Theorem 4, and
(25)Φ-k={(akT-Ek)p+(akT+(dhH-dhL)Ek)q+Ekζ+(dhH2(akT-Ek)-dhHEk)ς21≤k≤n1,+dhH2Ekς1+∥hk∥1,(akT-Ek)p+(akT+(dvH-dvL)Ek)q+Ekζ+(dvH2(akT-Ek)-dvHEk)ς2n1+1≤k≤n,+dvH2Ekς1+∥hk∥1,Tε=bεTp+bεTq+bεTDhH2ς2+∥lε∥1-γ,1≤ε≤m,DH=diag{dhHIn1,dvHIn2},
with k∈n_={1,2,…,n}, ε∈m_={1,2,…,m}, Ek=[0,…,0︷k-1,1,0,…,0︷n-k], ak, adk, bk, and hk represent the kth column vector of matrices A, Ad, B, and H, respectively, and lε represents the εth column vector of matrix L.Proof.
It is an obvious fact that (24) implies the following inequality:
(26)Ψ=diag{Ψ1,Ψ2,…,Ψn,Ψ1′,Ψ2′,…,Ψn′,Ψ1′′,Ψ2′′,…,dΨ=diagΨn′′,Ψ1′′′,Ψ2′′′,…,Ψn′′′}<0,
where
(27)Ψk={(akT-Ek)p+(akT+(dhH-dhL)Ek)q+ζk+(dhH2(akT-Ek)-dhHEk)ς2+dhH2Ekς1,1≤k≤n1,(akT-Ek)p+(akT+(dvH-dvL)Ek)q+ζk+(dvH2(akT-Ek)-dvHEk)ς2+dvH2Ekς1,n1+1≤k≤n,Ψk′={adkTp+(adkT-Ek)q+dhH2adkTς2,1≤k≤n1,adkTp+(adkT-Ek)q+dvH2adkTς2,n1+1≤k≤n,Ψk′′={-ζk+dhHEk(ς2-ς1),1≤k≤n1,-ζk+dvHEk(ς2-ς1),n1+1≤k≤n,Ψk′′′={-dhHEkς1,1≤k≤n1,-dvHEkς1,n1+1≤k≤n,
with k∈n_={1,2,…,n}, Ek=[0,…,0︷k-1,1,0,…,0︷n-k], and ak(adk) represents the kth column vector of matrix A(Ad).
By Theorem4, we can obtain that the system (1a) and (1b) with w(i,j)=0 is asymptotically stable. Now we are in a position to prove that the system (1a) and (1b)  has a prescribed l1-gain index γ for any nonzero w(i,j)∈l1. To establish the l1-gain performance, we choose the same copositive Lyapunov-Krasovskii functional candidate as in (9) for the system (1a) and (1b). Following the proof line of Theorem 4, we can get that
(28)ΔV(i,j)+∥z(i,j)∥1-γ∥w(i,j)∥1=xT(i,j){(AT-In)p+(AT+DH-DL)qΔV(i,j)+∥z(i,j)∥1+ζ+DH2((AT-In)ς2+ς1)-DHς2}ΔV(i,j)=+xdT(i,j){AdTp+(AdT-In)q+DH2AdTς2}ΔV(i,j)=+xHT(i,j){-ζ+DH(ς2-ς1)}ΔV(i,j)=+xsT(i,j){-DHς1}ΔV(i,j)=+wT(i,j){BT(p+q+DH2ς2)}ΔV(i,j)=+∥Hx(i,j)∥1+∥Lw(i,j)∥1-γ∥w(i,j)∥1.
According to the definition of l1 norm, one obtains(29)Hx(i,j)=[h1,1h1,2⋯h1,n1h2,1h2,2⋯h2,n1⋮⋮⋯⋮hp,1hp,2⋯hp,n1h1,n1+1h1,n1+2⋯h1,nh2,n1+1h2,n1+2⋯h2,n⋮⋮⋯⋮hp,n1+1hp,n1+2⋯hp,n][x1h(i,j)⋮xn1h(i,j)x1v(i,j)⋮xn2v(i,j)]Hx(i,j)=[h1,1x1h(i,j)+⋯+h1,n1xn1h(i,j)+h1,n1+1x1v(i,j)+⋯+h1,nxn2v(i,j)h2,1x1h(i,j)+⋯+h2,n1xn1h(i,j)+h2,n1+1x1v(i,j)+⋯+h2,nxn2v(i,j)⋮hp,1x1h(i,j)+⋯+hp,n1xn1h(i,j)+hp,n1+1x1v(i,j)+⋯+hp,nxn2v(i,j)],∥Hx(i,j)∥1=∑k=1phk,1x1h(i,j)+⋯+hk,n1xn1h(i,j)+hk,n1+1x1v(i,j)+⋯+hk,nxn2v(i,j)=(∑k=1phk,1)x1h(i,j)+⋯+(∑k=1phk,n1)xn1h(i,j)+(∑k=1phk,n1+1)x1v(i,j)+⋯+(∑k=1phk,n)xn2v(i,j)=xT(i,j)[∥h1∥1∥h2∥1⋯∥hn∥1]T,where hk represents the kth column vector and hi,j represents the entry located at (i,j) of matrix H. Then, similarly
(30)∥Lw(i,j)∥1=wT(i,j)[∥l1∥1∥l2∥1⋯∥lm∥1]T,γ∥w(i,j)∥1=wT(i,j)[γγ⋯γ]T,
where lk represents the kth column vector of matrix L.
Substituting (29)-(30) into (28) leads to
(31)ΔV(i,j)+∥z(i,j)∥1-γ∥w(i,j)∥1=xT(i,j){[∥h1∥1∥h2∥1⋯∥hn∥1]T(AT-In)p+(AT+DH-DL)q+ζ+DH2((AT-In)ς2+ς1)-DHς2+[∥h1∥1∥h2∥1⋯∥hn∥1]T}+xdT(i,j){AdTp+(AdT-In)q+DH2AdTς2}+xHT(i,j){-ζ+DH(ς2-ς1)}+xsT(i,j){-DHς1}+wT(i,j){[γγ⋯γ]TBT(p+q+DH2ς2)+[∥l1∥1∥l2∥1⋯∥lm∥1]T-[γγ⋯γ]T}.
If condition (24) holds, we have
(32)Vh(i+1,j)-Vh(i,j)+Vv(i,j+1)-Vv(i,j)+∥z(i,j)∥1-γ∥w(i,j)∥1<0.
We know that
(33)ΔV(i,j)=Vh(i+1,j)-Vh(i,j)+Vv(i,j+1)-Vv(i,j).
For any positive scalars  kh, andkv∈Z+, it can be verified that
(34)∑i=0kh∑j=0kvΔV(i,j)=∑i=0kh∑j=0kv(Vh(i+1,j)-Vh(i,j))∑i=0kh∑j=0kvΔV(i,j)=+∑i=0kh∑j=0kv(Vv(i,j+1)-Vv(i,j))∑i=0kh∑j=0kvΔV(i,j)=∑j=0kv(VhT(kh+1,j)-VhT(0,j))∑i=0kh∑j=0kvΔV(i,j)=+∑i=0kh(VvT(i,kv+1)-VvT(i,0)).
When kh and kv=∞, we have
(35)∑i=0∞∑j=0∞(∥z(i,j)∥1-γ∥w(i,j)∥1)<∑i=0∞∑j=0∞ΔV(i,j).
The existence of solution for LMI (24) implies that the positive 2D system (1a) and (1b) is asymptotically stable. Together with the zero boundary conditions, one can get
(36)∑i=0∞∑j=0∞ΔV(i,j)=0.
Applying (36) to (35), one has
(37)∑i=0∞∑j=0∞∥z(i,j)∥1<γ∑i=0∞∑j=0∞∥w(i,j)∥1.
By Definition 2, the positive 2D system (1a) and (1b) is asymptotically stable and has the l1-gain index γ.
This completes the proof.Remark 7.
In Theorem6, the disturbance attenuation performance of positive 2D linear systems is analyzed, and sufficient conditions for the existence of l1-gain performance for positive 2D system to (1a) and (1b) are proposed in terms of LMIs which are computationally tractable. This is also the major contribution of our paper.


## 4. Numerical Example

Consider the positive 2D system with delays in the Roesser model(1a) and (1b), where
(38)A=[0.100.20⋮0.20.000.30⋮0.10⋯⋯⋯⋯0.000.10⋮0.40],Ad=[0.100.01⋮0.050.100.02⋮0.05⋯⋯⋯⋯0.030.12⋮0.03],B=[0.20.10.1],H=[0.100.2],L=0.1,dh(i)=4+2sin(πi2),dv(j)=5+2sin(πj2),dh(i)=4+2sinw(i,j)=e-(i+0.5j),
where state dimensions are nh=2 and nv=1. The boundary conditions are given by
(39)xh(i,j)=[0.10.1],∀0≤j≤52,-dhH≤i≤0,xv(i,j)=0.1,∀0≤i≤52,-dvH≤j≤0.
In this example, we can get dhL=2, dhH=6, dvL=3, and dvH=7. Given γ=4.5, then by using the LMI Control Toolbox [37] to solve the inequalities in Theorem 6, we can get the following solutions:
(40)p=[1.73101.68421.7364]T,q=[3.20772.87683.3026]T,ζ=[1.03341.10750.9957]T,ς1=[0.36750.40420.3552]T,ς2=[0.03850.04890.0330]T.Figures1, 2, and 3 show the state responses of the system; it can be seen that the corresponding positive 2D system is asymptotically stable. Furthermore, by computing, under zero boundary conditions, we have ∑i=0∞∑j=0∞∥z(i,j)∥1=4.0206∑i=0∞∑j=0∞∥w(i,j)∥1=1.0977. It is obvious that the prescribed l1-gain performance level γ=4.5 is satisfied.Figure 1
State response ofx1h(i,j).Figure 2
State response ofx2h(i,j).Figure 3
State response ofxv(i,j).


## 5. Conclusions

This paper has addressed the delay-dependent stability analysis withl1-gain performance for positive 2D systems with state delays in the Roesser model. A sufficient condition for the existence of the delay-dependent asymptotic stability of positive 2D linear systems with time delays has been established. Copositive-type Lyapunov function method has been used to get a computationally tractable LMI-based sufficient criterion which ensures that the system is asymptotically stable and has a prescribed l1-gain performance. A numerical example has been given to illustrate the efficiency of the results. Furthermore, our future work will be devoted to the l1-gain control problem for positive 2D systems with delays.



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*Source: 169713-2013-03-18.xml*