Delay-dependent exponential stabilization of positive 2D switched state-delayed systems in the Roesser model

Delay-dependent

exponential stabilization of positive 2D

switched

state-delayed systems in the Roesser model

Zhaoxia

Duan

a

,

Zhengrong Xiang

a,⇑

,

Hamid Reza Karimi

b a_{School of Automation, Nanjing University of Science and Technology, Nanjing 210094, People’s Republic of China}

b_{Department of Engineering, Faculty of Engineering and Science, University of Agder, N-4898 Grimstad, Norway}

a b s t r a c t

This paper deals with the controller synthesis for a class of positive two-dimensional (2D) switched delay systems described by the Roesser model. This kind of systems has the property that the states take nonnegative values whenever the initial boundaries are nonnegative, some delay-dependent sufficient conditions for the exponential stability of positive 2D switched systems with state delays are given. Furthermore, the design of positive state feedback controller under which the resulting closed-loop system meets the requirements of positivity and exponential stability is presented in terms of linear matrix inequalities (LMIs). An example is included to illustrate the effectiveness of the proposed approach.

1. Introduction

Two-dimensional (2D) systems exist in many practical applications, such as circuit analysis, digital image processing, signal filtering and thermal power engineering [8,19,48]. Thus the analysis and synthesis of 2D systems are interesting and challenging problems, and have received considerable attention, for example, 2D state-space realization theory was researched in [14], the stability and 2D optimal control theory were studied in [7,12,17], and the H1filtering problem for 2D

Markovian jump systems was addressed in [42].

The most popular models of 2D linear systems were introduced by Roesser, Fornasini and Marchesini [14,15] and Kurek [27]. These models have been extended for positive systems in [20,21]. A positive system means that its states and outputs are nonnegative whenever the initial conditions and inputs are nonnegative [13,21]. Positive 2D systems are needed in many cases such as the wave equation in fluid dynamics, the Poisson’s equation, and the heat equation which describes the temperature (using thermodynamic temperature scale) in a given region over time. These facts stimulate the research on positive 2D discrete-time systems. The choice of the forms of Lyapunov functions for positive 2D Roesser model was inves-tigated in [23]. The problem of stability analysis for positive 2D fractional systems was invesinves-tigated in [24]. Furthermore, the reaction of real world systems to exogenous signals is never instantaneous and always infected by time delays [16,36–38]. The reachability, minimum energy control and realization problem for positive 2D discrete-time systems with delays was analyzed in [22]. And the stability analysis for positive 2D delayed systems was investigated in [3,25,26].

On the other hand, a considerable interest has been devoted to the research of switched systems during the recent decades. A switched system comprises a family of subsystems described by continuous or discrete-time dynamics, and a

switching law that specifies the active subsystem at each instant of time. Apart from the switching strategy to improve con-trol performance[10,35], switched systems also arise in many engineering applications[2,6]. Many techniques are effective tools dealing with switched systems, such as common quadratic Lyapunov function method, multiple Lyapunov function method, and average dwell time approach[1,9,28–33,39–41,49]. Recently,[34,45]studied the model reduction for linear switched systems, and[43,46]focused on the problems of stability and control synthesis by using sliding mode control method.

In addition, it is well known that the switching phenomenon may also occur in practical 2D systems, so 2D switched sys-tems have also attracted considerable research attention. There are a few reports on 2D discrete switched syssys-tems, Benza-ouia et al. firstly considered 2D switched systems with arbitrary switching sequences in [4] and investigated the stabilizability problem of 2D switched systems in[5], the generalized H2fault detection for 2D Markovian jump systems

was studied in[44]. Recently, the exponential stabilization of 2D switched Roesser model was firstly investigated in[47]. For positive 2D switched systems, a typical physical application is the thermal process with multiple modes. The positive value of temperature (using thermodynamic temperature scale) depends on position variable, time variable and the heating intensity (switching among multiple modes), so this system can be modeled as a positive 2D switched system. By using alge-braic techniques, sufficient and necessary conditions were first provided for the asymptotic stability of positive 2D switched systems described by the Roesser models in[11]. However, to the best of our knowledge, there has been little literature con-sidering the control problem of positive 2D switched systems with time delays, which motivates the present work.

In this paper we will investigate the problems of delay-dependent stability analysis and stabilization for positive 2D switched linear systems with delays. The main contributions of this paper lie in: (1) By constructing an appropriate co-positive Lyapunov function, we first analyze the delay-dependent exponential stability of positive 2D switched Roesser model with state delays. (2) Instead of using algebraic techniques[11]which have been employed for the analysis of positive 2D switched systems, the average dwell time approach is applied to our developments which are based on LMIs. (3) Based on the well established results of exponential stability analysis, equivalent conditions in terms of LMIs are obtained for the existence of stabilizing positive state feedback controllers. A remarkable advantage of these conditions lies in the easy verification by using some standard numerical software.

The remainder of the paper is organized as follows. In Section2, problem statement and some definitions concerning the positive 2D switched discrete linear systems with delays are given. In Section 3, some results concerning the delay-dependent exponential stability and stabilization of positive 2D switched linear systems are presented. In Section4, a physical example is given to illustrate the effectiveness of the proposed approach. Finally, concluding remarks are provided in Section5. The following notation will be used.

Notations: In this paper, the superscript ‘‘T’’ denotes the transpose. The notation X > Y(X P Y) means that matrix X  Y is positive definite (positive semi-definite, respectively). A  0(0) means that all entries of matrix A are nonnegative (non-positive). A  0(0) means that all entries of matrix A are positive (negative). Rnmdenotes the set of n  m real matrices. The set of real n  m matrices with nonnegative entries will be denoted by Rnm

þ and the set of nonnegative integers

will be denoted byZ+. Rnþdenotes the set of vectors with nonnegative entries. The n  n identity matrix will be denoted by In.

2. Problem formulation and preliminaries

Consider the following 2D switched Roesser model with state delays:

xh_{ði þ 1; jÞ} xvði; j þ 1Þ " # ¼ Arði;jÞ x h_{ði; jÞ} xvði; jÞ " # þ Ardði;jÞ xh_{ði  d} hðiÞ; jÞ xvði; j  dvðjÞÞ " # þ Brði;jÞuði; jÞ; ð1Þ

where i and j are integers in Z+, xh(i, j) is the horizontal state in Rn1; xvði; jÞ is the vertical state in Rn2; xði; jÞ is the whole state

in Rn_.

_r

_{(): Z}

+ Z+?N = {1, 2, . . . , N} is the switching signal. N denotes the number of subsystems. Ak, Akdand B

k_{, k 2 N, are}

constant matrices with appropriate dimensions and can be represented as

Ak¼ A k 11 A k 12 Ak21 A k 22 " # ; Akd¼ Akd11 A k d12 Akd21 A k d22 " # ; Bk¼ B k 1 Bk2 " # ;

dh(i) and dv(j) are delays along horizontal and vertical directions, respectively. We assume that dh(i) and dv(j) satisfy dhL6dhðiÞ 6 dhH; dvL6dvðjÞ 6 dvH;

where dhL, dhHand dvL, dvHdenote the lower and upper delay bounds along horizontal and vertical directions, respectively.

The boundary conditions are given by

xh_{ði; jÞ ¼ h} ij;

8

0 6 j 6 z1; dhH6i 6 0; xh_{ði; jÞ ¼ 0;}

₈

_{j > z} 1; dhH6i 6 0; xv_{ði; jÞ ¼}

_v

ij;

8

0 6 i 6 z2; dvH6j 6 0; xv_{ði; jÞ ¼ 0;}

₈

_{i > z} 2; dvH6j 6 0; h00¼

v

00 ð2Þ

where z1< 1 and z2< 1 are positive integers, hij2 Rn1and

v

ij2 Rn2are given vectors.

In this paper, the switch can be assumed to occur only at each sampling points of i or j. The switching sequence can be described as

ðði0;j0Þ;

r

ði0;j0ÞÞ; ðði1;j1Þ;

r

ði1;j1ÞÞ; . . . ; ððip;jpÞ;

r

ðip;jpÞÞ; . . . ;

where (ip, jp) denotes the

p

-th switching instant. It should be noted that the value of

r

(i, j) only depends upon i + j (see the

references[5,47]).

Definition 1 [11]. System (1) is called a positive 2D switched model if xh_{(i, j)  0 and xv(i, j)  0 for any nonnegative}

boundary conditions hij2 Rnþ1and

v

ij2 Rnþ2.

The following lemma is a direct extension from positive 2D systems in[21]to 2D switched positive systems.

Lemma 1 [21]. System(1)is positive if and only if Ak 0; Akd 0 and B k

 0.

Remark 1. When N = 1, positive 2D switched system(1)will degenerate into the following positive 2D system[26].

xh_{ði þ 1; jÞ} xv_{ði; j þ 1Þ} " # ¼ A x h_{ði; jÞ} xv_{ði; jÞ} " # þ Ad xh_{ði  d} hðiÞ; jÞ xv_{ði; j  dvðjÞÞ} " # þ Buði; jÞ: ð3Þ

Definition 2 [18]. System(1)with u(i, j) = 0 is said to be exponentially stable under the switching signal

r

(), if for a given z P 0, there exist positive constants c and n such that

X

iþj¼D

kxði; jÞk 6 necðDzÞX iþj¼z

kxði; jÞkC; ð4Þ

holds for all D P z, where

X iþj¼z kxði; jÞkC, sup dhH 6hh60; d_vH6hv60 X iþj¼z

fkxði  hh;jÞk; kxði; j  hvÞk; kdhði  hh;jÞk; kdvði; j  hvÞkg;

dhði  hh;jÞ ¼ xhði  hhþ 1; jÞ  xhði  hh;jÞ;

dvði; j  hvÞ ¼ xvði; j  hvþ 1Þ  xvði; j  hvÞ:

Remark 2. From Definition 2, it is easy to see that, for a given z,Piþj¼zkxði; jÞkCwill be bounded, and

P

iþj¼Dkxði; jÞk will tend

to be zero exponentially as D goes to infinity, which also means kx(i, j)k will tend to be zero exponentially.

Definition 3. [47]. For any i + j = D P z = iz+ jz, let Nr(z, D) denote the switching number of

r

() on an interval [z, D). If

Nrðz; DÞ 6 N0þ

D  z

s

a

; ð5Þ

holds for given N0P0 and

s

aP0, then the constant

s

ais called the average dwell time and N0is the chatter bound. As

com-monly used in the literature, we choose N0= 0 in this paper.

Remark 3. Definition 3gives a definition of average dwell time for 2D switched discrete systems and the definition can be viewed as an extension of the proposed one in 1D (one-dimensional) switched systems. Similarly, if there exists a positive number

s

asuch that a switching signal has the average dwell time property, the average time interval between consecutive

switching is at least

s

a.

The aim of this paper is to design a state feedback controller for system(1)such that the resulting closed-loop system is positive and exponentially stable.

3. Main results 3.1. Stability analysis

In order to address the control problem, we first focus on the problem of delay-dependent exponential stability analysis for the following positive 2D switched discrete linear systems with state delays

xh_{ði þ 1; jÞ} xv_{ði; j þ 1Þ} " # ¼ Arði;jÞ x h_{ði; jÞ} xv_{ði; jÞ} " # þ Ardði;jÞ xh_{ði  d} hðiÞ; jÞ xv_{ði; j  dvðjÞÞ} " # : ð6Þ

Theorem 1. For given positive constants dhL, dhH, dvL, dvH, and a scalar 0 <

a

< 1, if there exist vectors pk2 Rnþ; qk2 R n þ;

1

k12 R n þ;

1

k 22 R n þ; f k 2 Rnþ; k 2 N, such that

U

k¼ diag

U

k1;

U

k 2; . . . ;

U

k n;

U

k0 1;

U

k0 2; . . . ;

U

k0 n;

U

k00 1;

U

k00 2; . . . ;

U

k00 n;

U

k000 1 ;

U

k000 2 ; . . . ;

U

k000 n n o <0;

8

k 2 N ð7aÞ where

U

k_l ¼ akT l 

a

El   pk_þ

_a

1_akT l þ ðdhH dhLÞEl   qk_{þ E} lfkþ d2hHEl

1

k1þ d 2 hHakTl  El dhH

a

dhHEl  

1

k 2; 1 6 l 6 n1; akT l 

a

El   pk_þ

_a

1_akT l þ ðdvH dvLÞEl   qk_{þ E} lfkþ d 2 vHEl

1

k1þ d 2 vH akTl  El    dvH

a

dvHEl  

1

k 2; n1þ 1 6 l 6 n; 8 > < > :

U

k_l0¼ a kT dlp k_þ

_a

1_akT dl 

a

dhH_E l   qk_{þ d}2 hHa kT dl

1

k 2; 1 6 l 6 n1; akT dlpkþ

a

1akTdl 

a

dvHElqkþ d2_vHakTdl

1

k2; n1þ 1 6 l 6 n; (

U

k00 l ¼ 

a

dhHE lfkþ

a

dhHdhHEl

1

k2

1

k1   ; 1 6 l 6 n1; 

a

dvH_E lfkþ

a

dvHdvHEl

1

k2

1

k1   ; n1þ 1 6 l 6 n; (

U

k000 l ¼ 

a

dhHd hHEl

1

k1; 1 6 l 6 n1; 

a

dvH_d vHEl

1

k1; n1þ 1 6 l 6 n; ( with l 2 n ¼ f1; 2; . . . ; ng, El¼ 0; . . . ; 0 zfflfflfflffl}|fflfflfflffl{l1 ;1; 0; . . . ; 0zfflfflfflffl}|fflfflfflffl{ nl 2 4 3 5, and ak

lðakdlÞ represents the l-th column vector of matrix A k

ðAk_dÞ, then system

(6)is exponentially stable for any switching signals with the average dwell time satisfying

s

a>

s

a¼ ln

l

 ln

a

; ð7bÞ where

l

P1 satisfies pk 

l

pf_{; q}k 

l

qf_; fk

l

ff;

1

k 1

l1

f 1;

1

k 2

l1

f 2;

8

k; f 2 N; ð7cÞ

Proof. Without loss of generality, we assume that the k-th subsystem is active. For the k-th subsystem, we choose the fol-lowing co-positive Lyapunov–Krasovskii functional candidate

Vk

ði; jÞ ¼ Vkhði; jÞ þ Vkvði; jÞ; ð8Þ

where Vkhði; jÞ ¼X 5 g¼1 Vkhg ði; jÞ; with Vkh1ði; jÞ ¼ xhTði; jÞpkh; Vkh 2ði; jÞ ¼ Xi r¼idhðiÞ

a

i1r_xhT ðr; jÞqkh; Vkh 3ði; jÞ ¼ Xi1 r¼idhH

a

i1r_xhT ðr; jÞfkh; Vkh4ði; jÞ ¼ X dhL s¼dhHþ1 Xi1 r¼iþs

a

i1r_xhT_{ðr; jÞq}kh_{; V}kh 5ði; jÞ ¼ dhH X1 s¼dhH Xi1 r¼iþs

a

i1r

_g

hT_{ðr; jÞ}

₁

kh_; and

Vkvði; jÞ ¼X 5 g¼1 Vkgvði; jÞ; with Vkv 1ði; jÞ ¼ xvTði; jÞpkv; Vk2vði; jÞ ¼ Xj t¼jdv ðjÞ

a

j1t_xvT_{ði; tÞq}kv_{; V}kv 3ði; jÞ ¼ Xj1 t¼jdvH

a

j1t_xvT_{ði; tÞf}kv_; Vk4vði; jÞ ¼ X dvL s¼dvHþ1 Xj1 t¼jþs

a

j1t_xvT_{ði; tÞq}kv_{; V}kv 5 ði; jÞ ¼ dvH X1 s¼dvH Xj1 t¼jþs

a

j1t

_g

vT_{ði; tÞ}

₁

kv_; and

g

h_{ðr; jÞ ¼ x}hT_{ðr; jÞ d}hT ðr; jÞ  T ;

g

vði; tÞ ¼ xvT_{ði; tÞ d}vT ði; tÞ  T ; dhðr; jÞ ¼ xh_{ðr þ 1; jÞ  x}h_{ðr; jÞ; d}v_{ði; tÞ ¼ x}v_{ði; t þ 1Þ  x}v_{ði; tÞ;}

with pkh_{2 R}n1 þ; qkh2 R n1 þ; f kh 2 Rn1 þ;

1

kh1 2 R n1 þ;

1

kh2 2 R n1 þ;

1

kh¼

1

khT1

1

khT2  T 2 R2n1 þ ; f kv_{2 R}n2 þ,

1

v12 R n2 þ;

1

k2v2 R n2 þ, pkv2 R n2 þ; qkv_{2 R}n2 þ and

1

kv¼

1

k1vT

1

k2vT  T 2 R2n2

þ are real vectors to be determined.

Then we have

Vkhði þ 1; jÞ 

a

Vkhði; jÞ þ Vkvði; j þ 1Þ 

a

Vkvði; jÞ ¼X

5 g¼1 Vkhg ði þ 1; jÞ 

a

V kh g ði; jÞ h i þX 5 g¼1 Vkgvði; j þ 1Þ 

a

V kv g ði; jÞ h i : ð9Þ

Along the trajectory of system(6), one can obtain

Vkh1ði þ 1; jÞ 

a

V kh 1ði; jÞ ¼ x hT_{ði þ 1; jÞp}kh_

_a

_xhT_{ði; jÞp}kh_{; ð10Þ} Vkh 2ði þ 1; jÞ 

a

V kh 2ði; jÞ ¼ Xiþ1 r¼iþ1dhðiþ1Þ

a

ir_xhT ðr; jÞqkh

a

X i r¼idhðiÞ

a

i1r_xhT ðr; jÞqkh

¼

a

1_xhT_{ði þ 1; jÞq}kh_

_a

dhðiÞ_xhT_{ði  d}

hðiÞ; jÞqkhþ Xi r¼iþ1dhðiþ1Þ

a

ir_xhT_{ðr; jÞq}kh 

a

X i r¼iþ1dhðiÞ

a

i1r_xhT_{ðr; jÞq}kh

6

a

1_xhT_{ði þ 1; jÞq}kh_

a

dhðiÞ_xhT_{ði  d}

hðiÞ; jÞqkhþ Xi r¼iþ1dhH

a

ir_xhT_{ðr; jÞq}kh  X i r¼iþ1dhL

a

ir_xhT_{ðr; jÞq}kh 6

a

1_xhT_{ði þ 1; jÞq}kh_

a

dhH_xhT_{ði  d} hðiÞ; jÞqkhþ X idhL r¼iþ1dhH

a

ir_xhT_{ðr; jÞq}kh_{; ð11Þ} Vkh 3ði þ 1; jÞ 

a

V kh 3ði; jÞ ¼ Xi r¼iþ1dhH

a

ir_xhT ðr; jÞfkh

a

X i1 r¼idhH

a

i1r_xhT ðr; jÞfkh¼ xhTði; jÞfkh

a

dhH_xhT_{ði  d} hH;jÞfkh; ð12Þ Vkh4ði þ 1; jÞ 

a

V kh 4ði; jÞ ¼ X dhL s¼dhHþ1 Xi r¼iþ1þs

a

ir_xhT_{ðr; jÞq}kh_

_a

X dhL s¼dhHþ1 Xi1 r¼iþs

a

i1r_xhT_{ðr; jÞq}kh ¼ X dhL s¼dhHþ1 ½xhT_{ði; jÞq}kh_

_a

s_xhT_{ði þ s; jÞq}kh _{¼ ðd} hH dhLÞxhTði; jÞqkh X idhL r¼idhHþ1

a

ir_xhT_{ðr; jÞq}kh_{; ð13Þ} and

Vkh 5ði þ 1;jÞ 

a

V kh 5ði;jÞ ¼ dhH X1 s¼dhH Xi r¼iþ1þs

a

ir

_g

hT_ðr;jÞ

₁

kh_

_a

_d hH X1 s¼dhH Xi1 r¼iþs

a

i1r

_g

hT_ðr;jÞ

₁

kh ¼ dhH X1 s¼dhH ð

g

hT ði;jÞ

1

kh 

a

s

_g

hT ði þ s;jÞ

1

kh Þ 6 d2hH

g

hT ði;jÞ

1

kh  dhH

a

dhH Xi1 r¼idhH

g

hT ðr;jÞ

1

kh

¼ d2hH xhTði;jÞ xhTði þ 1;jÞ  xhTði;jÞ

 

1

kh 1

1

kh 2 " #  dhH

a

dhH Xi1 r¼idhH xhT_{ðr;jÞ x}hT_{ði;jÞ  x}hT_{ði  d} hH;jÞ " #

1

kh 1

1

kh 2 " # : ð14Þ Similarly, we can get the following formulations in vertical direction.

Vk1vði; j þ 1Þ 

a

V kv 1ði; jÞ ¼ xv T_{ði; j þ 1Þp}kv_

_a

_xvT_{ði; jÞp}kv_{; ð15Þ} Vkv 2 ði; j þ 1Þ 

a

V kv 2ði; jÞ 6

a

1xv T ði; j þ 1Þqkv

a

dvH_xvT_{ði; j  d} vðjÞÞqkvþ X jdvL t¼jþ1dvH

a

jt_xvT ði; tÞqkv; ð16Þ Vk3vði; j þ 1Þ 

a

V kv 3ði; jÞ ¼ xv T_{ði; jÞf}kv_

_a

dvH_xvT_{ði; j  d} vHÞfkv; ð17Þ Vkv 4 ði; j þ 1Þ 

a

V kv 4ði; jÞ ¼ ðdvH dvLÞxvTði; jÞqkv X jdvL t¼jdvHþ1

a

jt_xvT ði; tÞqkv; ð18Þ Vkv 5 ði; j þ 1Þ 

a

V kv 5ði; jÞ 6 d 2

vH xvTði; jÞ xvTði; j þ 1Þ  xvTði; jÞ

 

1

kv 1

1

kv 2 " #  dvH

a

dvH Xj1 t¼jdvH

xvT_{ði; tÞ x}vT_{ði; jÞ  x}vT_{ði; j  d} vHÞ " #

1

kv 1

1

kv 2 " # : ð19Þ

Substitute the above formulations(10)–(19)into(8), and take

pk_¼ pkh pkv " # ; qk_¼ qkh qkv " # ; fk¼ f kh fkv " # ;

1

k 1¼

1

kh 1

1

kv 1 " # ;

1

k 2¼

1

kh 2

1

kv 2 " # ; DH¼ dhHIn1 0 0 dvHIn2 ; DL¼ dhLIn1 0 0 dvLIn2 ;

X

¼

a

dhHI n1 0 0

a

dvH_I n2 " # ;

xði; jÞ ¼ x hT_{ði; jÞ x}vT_{ði; jÞ}T_;

xdði; jÞ ¼ x hTði  dhðiÞ; jÞ xvTði; j  dvðjÞÞ T

; xHði; jÞ ¼ xhTði  dhH;jÞ xvTði; j  dvHÞ

 T ; xsði; jÞ ¼ Xi1 r¼idhHþ1 xhT_{ðr; jÞ} X j1 t¼jdvHþ1 xvT_{ði; tÞ} " #T : Then we have Vkh ði þ 1; jÞ 

a

Vkh

ði; jÞ þ Vkvði; j þ 1Þ 

a

Vkv_{ði;jÞ ¼ x}T

ði;jÞ ðAkT

a

InÞpkþ ð

a

1AkTþ ðDH DLÞÞqk n þ fkþ D2H

1

k 1þ ðD 2 HA kT  D2HXDHÞ

1

k2 o þ xT dði; jÞ A kT dp k þ ð

a

1_AkT d XÞq k þ D2HA kT d

1

k 2 n o þ xT Hði; jÞ Xf k þXDH

1

k2XDH

1

k1 n o þ xT sði; jÞ XDH

1

k1   : ð20Þ

If condition(7a)holds, one obtains

ðAkT

a

InÞpkþ ð

a

1AkTþ ðDH DLÞÞqkþ fkþ D2H

1

k 1þ D 2 HA kT  D2H

X

DH  

1

k 2 0; ð21Þ AkT d pkþ

a

1A kT d 

X

  qk_{þ D}2 HA kT d

1

k2 0; ð22Þ 

X

fkþ

X

DH

1

2k

X

DH

1

k1 0; ð23Þ 

X

DH

1

k1 0: ð24Þ

Inequalities(21)–(24)imply that

Vkhði þ 1; jÞ þ Vkvði; j þ 1Þ <

a

Vkhði; jÞ þ

a

Vkvði; jÞ: ð25Þ

Summing up both sides of(25)from D to 0 with respect to i and 0 to D with respect to j, for any nonnegative integer D > max (z1, z2), one gets Vkhð1; DÞ þ Vkvð0; D þ 1Þ þ Vkhð2; D  1Þ þ Vkvð1; DÞ þ    þ VkhðD þ 1; 0Þ þ VkvðD; 1Þ ¼ X iþj¼Dþ1 Vkhði; jÞ þ X iþj¼Dþ1 Vkvði; jÞ ¼ X iþj¼Dþ1 Vkði; jÞ <

a

fVkhð0; DÞ þ Vkvð0; DÞ þ Vkhð1; D  1Þ þ Vkvð1; D  1Þ þ    þ VkhðD; 0Þ þ VkvðD; 0Þg ¼

a

X iþj¼D Vk ði; jÞ: ð26Þ

Now let

t

= Nr(z, D) denote the switching number of

r

() on an interval [z, D), and let (ijt+1, jjt+1), (ijt+2, jjt+2),

. . .(ij1, jj1), (ij, jj) denote the switching points of

r

() over the interval [z, D). Denoting mp= ip+ jp, p =

j



t

+ 1,

j



t

+ 2, . . . ,

j

, thus, for D 2 [mj, mj+1), it holds from(26)that X

iþj¼D

Vrðij;jjÞ_{ði; jÞ <}

_a

Dmj X

iþj¼mj

Vrðij;jjÞ_{ði; jÞ: ð27Þ}

Using(7c) and (8), at switching instant mj= i + j, we have

X

iþj¼mj

Vrðij;jjÞ_{ði; jÞ 6}

_l

X

iþj¼mj

Vrðij1;jj1Þ_{ði; jÞ: ð28Þ}

In addition, according toDefinition 3, it follows that

t

¼ Nrðz; DÞ 6 N0þ

D  z

s

a

: ð29Þ

Therefore, the following inequality can be easily obtained by repeating the inequalities(27), (28)and using(29)

X iþj¼D Vrðij;jjÞ_{ði; jÞ <}

_a

Dmj X iþj¼mj Vrðij;jjÞ_{ði; jÞ 6}

_la

Dmj X iþj¼m j Vrðij1;jj1Þ_{ði; jÞ <}

la

Dmj X iþj¼mj1 Vrðij1;jj1Þ_{ði; jÞ}

_a

mjmj1 ¼

la

Dmj1 X iþj¼mj1 Vrðij1;jj1Þ_{ði; jÞ 6    <}

l

t1

a

Dmjtþ1 X iþj¼mjtþ1 Vrðijtþ1;jjtþ1Þ_{ði; jÞ} 6

l

t

a

Dmjtþ1 X iþj¼m ktþ1 Vrðijt;jjtÞ_{ði; jÞ ¼}

_l

t

_a

Dmktþ1X iþj¼z Vrðijt;jjtÞ_{ði; jÞ}

_a

mjtþ1z 6

l

t

a

DzX iþj¼z Vrðijt;jjtÞ_{ði; jÞ: ð30Þ}

Inequality(30)can be rewritten as follows:

X

iþj¼D

Vrðij;jjÞ_{ði; jÞ 6 e}ðlnsalþlnaÞðDzÞX

iþj¼z

Vrðijt;jjtÞ_{ði; jÞ: ð31Þ}

Moreover, considering the definition of Vr(i,j)_{(i, j) in(8), we can find two positive scalars}

_q

1and

q

2such that(32)holds.

q

1kxði; jÞk 6 V rði;jÞ ði; jÞ 6

q

2kxði; jÞkC; ð32Þ where

q

1¼ min ðl;kÞ2nNðp k lÞ;

q

2¼ max ðl;kÞ2nNp k l þ dH 2 ðdHþ dL 1ÞðdH dLÞ þ dH max ðl;kÞ2nNq k lþ dH max ðl;kÞ2nNf k l þ d 2 H max ðl;kÞ2nN

1

k 1lþ dHðdHþ 1Þ max ðl;kÞ2nN

1

k 2l with dH¼ maxðdhH;dvHÞ; dL¼ minðdhL;dvLÞ; pk¼ pk1;pk2; . . . ;pkn  T ; qk_{¼ q}k 1;qk2; . . . ;qkn  T ; fk¼ fk1;f k 2; . . . ;f k n h iT ;

1

k 1¼

1

k11;

1

k12; . . . ;

1

k1n  T ;

1

k 2¼

1

k21;

1

k22; . . . ;

1

k2n  T :

Combining(31) and (32), one can obtain

X iþj¼D kxði; jÞk 6

q

2

q

1 eðlnsalþlnaÞðDzÞX iþj¼z kxði; jÞkC: ð33Þ

By Definition 2, we know that the positive 2D switched discrete system is exponentially stable if

s

a>

s

a¼ lnl

 lna. This

com-pletes the proof. h

Remark 4. Note that when

l

= 1 in(7b), we have

s

a¼ 0, which means that the switching signal can be arbitrary.

Remark 5. It should be noted that a co-positive Lyapunov functional is constructed for the stability analysis in the derivation ofTheorem 1. The motivation for using this type of Lyapunov functional is that the state of system(1)is nonnegative and hence such a linear Lyapunov functional serves as a valid candidate. Compared with the existing stability result in[11], the one presented here is in the form of LMIs which can be conveniently verified. However, there exists the conservatism induced by Lyapunov functional(8)to some extent. The result can be improved by resorting to the delay-partition method for which a modified Lyapunov functional could be chosen.

3.2. Controller synthesis

This subsection studies the stabilization problem of positive 2D switched discrete Roesser model(1)for which the control law to be designed has the following state-feedback form

uði; jÞ ¼ Krði;jÞ xhði; jÞ

xvði; jÞ

" #

: ð34Þ

This control law will be designed to ensure the positivity and the exponential stability of the resulting closed-loop system:

xh_{ði þ 1; jÞ}

xvði; j þ 1Þ

" #

¼ ðArði;jÞþ Brði;jÞKrði;jÞ

Þ x h_{ði; jÞ} xvði; jÞ " # þ Ardði;jÞ xh_{ði  d} hðiÞ; jÞ xvði; j  dvðjÞÞ " # : ð35Þ

The following lemma will be useful in the subsequent development.

Lemma 2. Given the open-loop positive system(1)and the controller given by(34), the closed-loop system(35)is positive if and only if the following conditions hold for all k 2 N

Ak

þ BkKk

 0; Akd  0:

Proof. The result can be obtained by applying Theorem 8 in[26]to system(35). h

Theorem 2. For given positive constants dhL, dhH, dvL, dvH and a scalar 0 <

a

< 1, if there exist vectors

pk_{2 R}n þ; qk2 R n þ;

1

k12 R n þ;

1

k22 R n þ; f k_{2 R}n þ; w k_{2 R}n þ, such that e Uk¼ diag eUk1; eUk2; . . . ; eUkn; eUk 0 1; eUk 0 2; . . . ; eUk 0 n; eUk 00 1; eUk 00 2; . . . ; eUk 00 n; eUk 000 1 ; eUk 000 2 ; . . . ; eUk 000 n n o <0;

8

k 2 N ð36Þ where e Uk_l¼ a kT l 

a

El   pk_þ

_a

1_akT l þ ðdhH dhLÞEl   qk_{þ E} lfkþ d 2 hHEl

1

k1þ d 2 hH a kT l  El    dhH

a

dhHEl  

1

k 2þ Elwk; 1 6 l 6 n1; akT l 

a

El   pk_þ

_a

1_akT l þ ðdvH dvLÞEl   qk_{þ E} lfkþ d 2 vHEl

1

k1þ d 2 vH a kT l  El    dvH

a

dvHEl  

1

k 2þ Elwk; n1þ 1 6 l 6 n; 8 < : e Uk_l0¼ a kT dlpkþ

a

1akTdl 

a

dhHElqkþ d 2 hHakTdl

1

k2; 1 6 l 6 n1; akT dlpkþ

a

1akTdl 

a

dvHElqkþ d2_vHakTdl

1

k2; n1þ 1 6 l 6 n; ( e Uk00 l ¼ 

a

dhHE lfkþ

a

dhHdhHEl

1

k2

1

k1   ; 1 6 l 6 n1; 

a

dvH_E lfkþ

a

dvHdvHEl

1

k2

1

k1   ; n1þ 1 6 l 6 n; ( e Uk_l000 ¼ 

a

dhHd hHEl

1

k1; 1 6 l 6 n1; 

a

dvH_d vHEl

1

k1; n1þ 1 6 l 6 n; ( wk¼ KkTBkT _pk þ

a

1_qk þ D2H

1

k 2   ;

with DH¼ diagfdhHIn1;dvHIn2g; l 2 n ¼ f1; 2; . . . ; ng; El¼ ½0; . . . ; 0 zfflfflfflffl}|fflfflfflffl{l1 ;1; 0; . . . ; 0zfflfflfflffl}|fflfflfflffl{ nl , and ak l akdl  

represents the l-th column vector of matrix Ak _Ak

d

 

, then the closed-loop system(35)is positive and exponentially stable for any switching signals with the average dwell time satisfying (7b) and (7c). Under the above conditions, the desired controller gain matrices can be computed by wk¼ KkT_BkT _pk_þ

_a

1_qk_{þ D}2

H

1

k2

 

.

Proof. FromTheorem 1andLemma 2, the closed-loop system(35)is positive and exponentially stable if conditions(7b), (7c)and the following conditions hold.

ðAkT

a

InÞpkþ ð

a

1AkTþ ðDH DLÞÞqkþ fkþ D2H

1

k1þ A kT_D2 H D 2 H

X

DH  

1

k 2þ w k  0; ð37Þ AkTd p k_þ

_a

1_AkT d 

X

  qk_{þ A}kT d D 2 H

1

k 2 0; ð38Þ 

X

fkþ

X

DH

1

2k

X

DH

1

k1 0; ð39Þ 

X

DH

1

k1 0; ð40Þ where wk_{¼ K}kT_BkT _pk_þ

_a

1_qk_{þ D}2 H

1

k2    0, and Kk_{ 0, k 2 N.}

Then the conditions in(36)can be easily obtained from(37)–(40). This completes the proof. h The procedure for constructing the desired controller is given below.

Algorithm 1.

Step 1. Solve the LMIs in(36)to obtain pk_{, q}k_;

₁

k 1;

1

k2; f k_and

_w

k_{, k 2 N.} Step 2. By wk ¼ KkTBkT pk_þ

_a

1_qk_{þ D}2 H

1

k2   , compute Kk_{, k 2 N.}

Step 3. Compute

l

and

s

aby(7c) and (7b).

Step 4. The desired positive state feedback controller can be given as(34)with the obtained Kk_{, k 2 N.}

Remark 6. The condition(36)inTheorem 2is given in terms of LMIs which are computationally tractable by using the LMI toolbox and it is different from the results presented in[11], where the algebraic method is utilized for the case without state delays. Also, this paper takes time delay into consideration firstly, which is universal in the application.

Remark 7. We would like to point out that the algorithm we proposed may bring some computational complexities. It can be seen that we need to solve N matrix inequalities to obtain 6N variables in(36).

Remark 8. Many practical complicated systems can be modeled by the addressed system in this paper, among which a typ-ical example is the thermal process under the standard measurement of absolute temperature. The features of positivity for state, switching for the control law, and complexity for multi-dimensional systems constitute a challenging problem in the control field. The co-positive type Lyapunov functional method and typical average dwell time method are merged together to solve this significant problem in this paper. The proposed LMI approach is easy to be extended to more complex applica-tions. The feasibility of the proposed method will be illustrated by the example given in the next section.

4. Example

In this section, we present an example to illustrate the effectiveness of the proposed approach. Consider the thermal pro-cesses in chemical reactors with two modes, which can be expressed in the following partial differential equation with time delays. We assume that one can switch from a mode to another mode arbitrarily.

@Tðx; tÞ @x ¼  @Tðx; tÞ @t  a rðx;tÞ 0 Tðx; tÞ  a rðx;tÞ 1 Tðx; t 

s

Þ þ b rðx;tÞ uðx; tÞ; ð41Þ

where T(x, t) is the temperature at x 2 [0, xf] (space) and t 2 [0, 1) (time), u(x, t) is the input function,

s

is the time delay, and

arðx;tÞ 0 ; ar1, b

r_{are real coefficients with}

_r

_{(x, t) denoting the working subsystem at (x, t).}

Take

Tði; jÞ ¼ Tði

D

x; j

D

tÞ; uði; jÞ ¼ uði

D

x; j

D

tÞ;

r

ði; jÞ ¼

r

ði

D

x; j

D

tÞ; @Tðx; tÞ @x  Tði; jÞ  Tði  1; jÞ

D

x ; @Tðx; tÞ @t  Tði; j þ 1Þ  Tði; jÞ

D

t :

Denote xh_{(i, j) = T(i  1, j), xv(i, j) = T(i, j), where T(i, j) = T(iDx, jDt). It is easy to verify that Eq.(41)can be converted into a 2D}

Roesser model(1)with

Arði;jÞ ¼ Dt0 1 Dx 1  Dt Dx a rði;jÞ 0

D

t " # ; Ar_dði;jÞ¼ 0 0 0 arði;jÞ 1

D

t ; Brði;jÞ ¼ 0 brði;jÞ

D

t ; LetDt ¼ 0:1; Dt 6

s

6₂Dt; Dx ¼ 0:4; a1

0¼ 2:5; a20¼ 5; a11¼ 5; a21¼ 2:5, b1= 2.5 and b2= 5. The thermal process is

mod-eled in the form(1)with

A1¼ 0 1 0:25 0:5 ; A1d¼ 0 0 0 0:5 ; B1 ¼ 0 0:25 A2 ¼ 0 1 0:25 0:25 ; A2 d¼ 0 0 0 0:25 ; B2 ¼ 0 0:5

Then by using the LMI toolbox and following the steps ofAlgorithm 1, we make the following records. Step 1. Solve the LMIs in(36)to obtain pk_{, q}k_;

₁

k

1;

1

k2; f k_and

_w

k_{with k = 1, 2.} p1_¼ 0:7660 0:3613 ; p2_¼ 1:0881 0:5748 ; q1_¼ 0:1813 0:6627 ; q2_¼ 0:3274 0:4545 ; f1¼ 0:1654 3:5834 ; f2¼ 0:2728 2:4282 ;

1

11¼ 0:5383 0:1439 ;

1

21¼ 0:5935 0:3605 ;

1

1 2¼ 0:0405 1:8798 ;

1

2 2¼ 0:1065 1:4698 ; w1¼ 0:1004 0:1248 ; w2¼ 0:1879 0:2602 ; Step 2. By wk ¼ KkTBkT _pk_þ

_a

1_qk_{þ D}2 H

1

k2   , compute Kk_{with k = 1, 2.} K1¼ 0:0464 0:0576½ ; K2¼ 0:1075 0:1489½ :

Step 3. Compute

l

and

s

aby(7c) and (7b).

l

¼ 2:6290;

s

a¼ 5:9477:

Step 4. The desired positive state feedback controller can be given as(34)with the obtained K1_{and K}2_.

The boundary conditions are given by

xh_{ði; jÞ ¼ 0:1;}

₈

_{0 6 j 6 52; d}

hH6i 6 0;

xv_{ði; jÞ ¼ 0:1;}

₈

_{0 6 i 6 52; dvH}6_{j 6 0:}

Choosing

s

a= 6.12, the simulation results inFigs. 1 and 2show the state responses of the resulting closed-loop system under

the switching sequence depicted inFig. 3. It can be observed that the closed-loop system is positive and exponentially stable, which demonstrates the effectiveness of the proposed method.

0 10 20 30 40 50 0 10 20 30 40 50 0 0.05 0.1 0.15 j Xh i

Fig. 1. State response of xh

5. Conclusions

This paper has addressed the delay-dependent exponential stability analysis and stabilization for positive 2D switched delay systems described by the Roesser model. Sufficient conditions for the delay-dependent exponential stability of positive 2D switched linear systems with delays have been established. A co-positive type Lyapunov functional has been used to get a computationally tractable LMI-based sufficient criterion which ensures the exponential stability. A design methodology of positive feedback controller has been provided to ensure the exponential stability and positivity of the resulting closed-loop system. A numerical example has been given to illustrate the efficiency of the proposed approach. Furthermore, future work will be devoted to the robust control problem to achieve the delay-dependent stability and the disturbance attenuation per-formance of positive 2D switched systems.

Acknowledgments

This work was supported by the National Natural Science Foundation of China under Grant No. 61273120 and the Alex-ander von Humboldt Foundation in Germany.

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