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International Journal of Systems Science

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Stability and ℓ1-gain analysis for positive 2-DMarkov jump systems

Yong Chen, Chang Zhao, James Lam, Yukang Cui & Ka-Wai Kwok

To cite this article: Yong Chen, Chang Zhao, James Lam, Yukang Cui & Ka-Wai Kwok (2019)Stability and ℓ1-gain analysis for positive 2-D Markov jump systems, International Journal ofSystems Science, 50:11, 2077-2087, DOI: 10.1080/00207721.2019.1645229

To link to this article: https://doi.org/10.1080/00207721.2019.1645229

Published online: 02 Aug 2019.

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INTERNATIONAL JOURNAL OF SYSTEMS SCIENCE2019, VOL. 50, NO. 11, 2077–2087https://doi.org/10.1080/00207721.2019.1645229

Stability and �1-gain analysis for positive 2-D Markov jump systems

Yong Chena, Chang Zhao a, James Lam a, Yukang Cuia,b and Ka-Wai Kwoka

aDepartment of Mechanical Engineering, The University of Hong Kong, Hong Kong; bCollege of Mechatronics and Control Engineering,Shenzhen University, Shenzhen, People’s Republic of China

ABSTRACTThis paper investigates the problems of stability and �1-gain analysis for positive 2-dimensional (2-D)Markov jump systems. Themathematical model of 2-DMarkov jump systems is established basedon the Roesser model. Necessary and sufficient condition for stability and sufficient condition for�1-gain computation are derived. Furthermore, the stability and �1-gain conditions are extended toMarkov jump systemswith partially known transition probabilities. The effectiveness of the obtainedtheoretical findings is verified through two numerical examples.

ARTICLE HISTORYReceived 31 May 2018Accepted 14 July 2019

KEYWORDS2-dimensional systems;�1-gain; Markov jumpsystems; positive systems;stability analysis

1. Introduction

The 2-D model, in which the system state depends ontwo independent variables, has been used in differentfields to represent a wide range of practical systems. Thistype of systems can be found in image data process-ing and transmission, thermal processes, gas absorption,and water stream heating. Due to their theoretical andpractical importance, 2-D systems have attracted muchresearch attention in recent years. The stability problemof 2-D systems has been investigated inAnderson,Agath-oklis, Jury, and Mansour (1986), Lu and Lee (1985), andHinamoto (1997). Two popular models of 2-D systemsintroduced by Roesser (1975) and Fornasini and March-esini (1976, 1978) have been considered in those papers.TheH2 andH∞ control problems, which aim to study thesystem performance, have been considered in Du, Xie,and Zhang (2001) and Yang, Xie, and Zhang (2006) for2-D systems.

Besides 2-D systems, the variables of many dynamicsystems are always confined to the positive orthant inmany real processes, such as the drug therapy schedulingproblem in HIV infection (Hernandez-Vargas, Colaneri,& Middleton, 2014). Some research studies have beenfocusing on positive systems. For instance, in Zhang, Jia,Zhang, and Zuo (2018), a new model predictive controlframework has been proposed for positive systems withpolytopic uncertainty. The exponential stability, �1-gainanalysis problem and �1-induced controller for positiveTakagi–Sugeno (T-S) fuzzy systems have been investi-gated in Du, Qiao, Zhao, and Wang (2018) and Chen,Lam, andMeng (2017). In Zhu,Wang, and Zhang (2017),

CONTACT Chang Zhao zhaochang@connect.hku.hk

the stochastic finite-time �1-gain filtering problem fordiscrete-time positive Markov jump linear systems withtime-delay has been analysed. The dominant pole assign-ment problem, the dominant eigenstructure assignmentproblem and the robust dominant pole assignment prob-lem for linear time-invariant positive systems have beenconsidered in Li and Lam (2016). Some novel techniqueshave been developed in the literature to study positivesystems, for example, the linear co-positive Lyapunovfunction method. As the state of a positive system isnonnegative, a linear co-positive Lyapunov function hasbeen used in Lian, Liu, and Zhuang (2015) to analyse themean stability of the positive Markov jump systems withswitching transition probabilities. In addition, althoughthe �2-gain andH∞ performance index are important forcharacterising system performance of general systems,the 1-norm of the system state and the �1-gain of thesystem can provide more useful descriptions for positivesystems (Chen, Lam, Li, & Shu, 2013; Lian et al., 2015;Shen & Lam, 2016; Xiang, Lam, & Shen, 2017). The 1-norm of the system state has a good physical meaningfor positive systems because it is the sum of the val-ues of the components of the state, for example, the1-norm represents the sum of the number of materi-als if a positive system is used to model the numberof the materials in a chemical process. The monograph(Kaczorek, 2012) provides a brief introduction for posi-tive 2-D systems. In this monograph, somemathematicalmodels (including the general model, the Roesser model,and the Fornasini-Marchesinimodel) of positive 2-D sys-tems have been introduced, the controllability, minimum

© 2019 Informa UK Limited, trading as Taylor & Francis Group

2078 Y. CHEN ET AL.

energy control, and realisation problems have been inves-tigated there. Moreover, the author of the monograph(Kaczorek, 2012) also considered the problem of stabilityfor positive 2-D systems with delays in Kaczorek (2009).In Kaczorek (2007), different forms of Lyapunov func-tions have been developed to study the positive 2-DRoesser systems. The stability problem for Roessermodelhas been treated in Kurek (2002). The authors in For-nasini and Valcher (2005) have focused on the controlla-bility and reachability problems for positive 2-D systemsby using a graph-theoretic approach.

On the other hand, many practical systems in thereal world have multi-mode features, such as the HIVviral mutation model and traffic congestion model inWang and Zhao (2017). Their multi-mode dynamics canbe captured by the so-called switched systems. Recently,many research results concerning 2-D switched systemshave been presented in the literature. The authors inXiang and Huang (2013) have investigated the problemsof stability and stabilisation of 2-D switched Roesser sys-temsunder average dwell time switching signal. The sameproblems for the Fornasini–Marchesini model have beenconsidered in Wu, Yang, Shi, and Su (2015). In addi-tion, the asynchronous control problem of 2-D switchedsystems under mode-dependent average dwell time hasbeen considered in Fei, Shi, Zhao, and Wu (2017). Asa special class of switched systems, Markov jump sys-tems are able tomodel practical systems subject to abruptchanges. The switching in a Markov jump system is gov-erned by a Markov process in the continuous-time caseand a Markov chain in the discrete-time case. The stabil-isation and H∞ control problems of 2-D Markov jumpRoesser systems have been studied in Gao, Lam, Xu,and Wang (2004). Wu, Shi, Gao, and Wang (2008) havestudied theH∞ filter design problem for the same class ofsystems. In Liang, Pang, andWang (2017), sufficient con-ditions are established for the filtering error system suchthat the system is stochastically asymptotically stablewith�1-gain. To the best of the authors’ knowledge, there arenot many research results on positive 2-D Markov jumpsystems in the literature. Thismotivates us for the presentstudy.

The main contributions of this paper are summarisedas follows:

(1) A necessary and sufficient condition for stabilityanalysis for positive 2-dimensional (2-D) Markovjump systems is proposed for the first time.

(2) Sufficient condition for �1-gain computation isderived.

The rest of this paper is organised as follows. The con-sidered dynamic systems and problems are formulated in

Section 2. The assumptions, definitions, and lemmas thatare used to derive the main results are also given in thatsection. Themain results on stability and �1-gain compu-tation are presented in Section 3. Numerical examples aregiven in Section 4 to verify the effectiveness of the the-oretical findings. Finally, the conclusion of this paper isgiven in Section 5.

Notations: The notations used throughout this paperare standard. Rn denotes the n-dimensional Euclideanspace. N0 denotes the set of nonnegative integers. E(∗)

denotes the expectation operator. ‘⊗’ denotes the Kro-necker product. The superscript ‘T’ represents matrixtranspose. x � 0 (x � 0) and A � 0 (A � 0) mean thatall elements of vector x and matrix A are positive (non-negative). 1 represents the vector [1, 1, . . . , 1]T. In standsfor the identity matrix. The 1-norm ‖xi,j‖1, where xi,j =[xi,j,1, xi,j,2, . . . , xi,j,n]T ∈ Rn, is defined as

∑nl=1 |xi,j,l|. A

vectorωi,j belongs to l1{N0,N0}means that ‖ωi,j‖1 < ∞.Vectors and matrices are assumed to have compatibledimensions for algebraic operations if their dimensionsare not explicitly stated.

2. Problem formulation and preliminaries

The considered positive 2-D discrete-time systems in theRoesser model with Markov jump parameters can bedescribed as follows:

S :

[xhi+1,j

xvi,j+1

]= A(ri,j)

[xhi,j

xvi,j

]+ B(ri,j)ωi,j,

yi,j = C(ri,j)

[xhi,j

xvi,j

]+ D(ri,j)ωi,j, (1)

where xhi,j ∈ Ru1 , xvi,j ∈ Ru2 are the horizontal and ver-tical state vectors, respectively; ωi,j ∈ Rv is the distur-bance vector which belongs to l1{N0,N0}; yi,j ∈ Rw is themeasured output vector.A(ri,j) � 0, B(ri,j) � 0,C(ri,j) �0, D(ri,j) � 0 are positive real-valued system matrices.They are determined by an homogenous Markov chainri,j, which takes values in a finite set L = {1, . . . , S} withtransition probabilities

Pr{ri,j = n, i + j = k + 1 | ri,j = m, i + j = k} = πmn,(2)

where πmn ≥ 0 and∑S

n=1 πmn = 1. Denote the transi-tion matrix by � = {πmn}.

Remark 2.1: In this paper, it is assumed that switchingoccurs only at each sampling point of i or j. In otherwords, the value of ri,j depends on i + j (Benzaouia,Hmamed, Tadeo, & Hajjaji, 2011; Duan & Xiang, 2014).

INTERNATIONAL JOURNAL OF SYSTEMS SCIENCE 2079

Denote system matrices A(ri,j), B(ri,j), C(ri,j), D(ri,j)as Am, Bm, Cm, Dm, respectively, if the considered sys-tem operates at themthmode, namely, ri,j = m. The stateof the system is denoted as

xi,j =[xhi,j

xvi,j

]∈ R

u. (3)

The boundary condition is (X0, R0) with

X0 = [xhT0,0 xhT0,1 xhT0,2 · · ·xvT0,0 xvT1,0 xvT2,0 · · ·]T , (4)

R0 = {r0,0, r0,1, r0,2, . . . , r0,0, r1,0, r2,0, . . .} . (5)

We make the following assumption on the boundarycondition.

Assumption 2.1: The boundary condition is assumed tosatisfy

limN→∞ E

{ N∑k=0

(‖xh0,k‖1 + ‖xvk,0‖1)}

< ∞. (6)

Next, we will introduce the definitions for positivity,asymptotic stability in the mean sense, and �1-gain in themean sense for system (1).

Definition 2.1: System (1) is positive if xi,j � 0, yi,j � 0for boundary condition X0 � 0 and disturbance ωi,j � 0.

Definition 2.2 (Kaczorek, 2011; Liang et al., 2017):System (1) withωi,j ≡ 0 is said to be asymptotically stablein the mean sense if

limi+j→∞ E

{‖xi,j‖1} = 0 (7)

for boundary condition satisfying Assumption 2.1.

Definition 2.3: Given a scalar γ > 0, system (1) is saidto be asymptotically stable with �1-gain γ in the meansense if it is asymptotically stable in the mean sense andunder zero boundary condition X0 = 0, ‖y‖E ≤ γ ‖ω‖1for all non-zero ω = {ωi,j} ∈ �1{N0,N0} where

‖y‖E = E(

∞∑i=0

∞∑j=0

‖yi,j‖1), ‖ω‖1 =∞∑i=0

∞∑j=0

‖ωi,j‖1.

(8)

Lemma 2.4 (Kaczorek, 2007): The positive 2-D deter-ministic system

[xhi+1,j

xvi,j+1

]= A

[xhi,j

xvi,j

](9)

is asymptotically stable if and only if there exists a vectorp � 0 such that

pT(A − Iu) ≺ 0. (10)

The stability of deterministic system (9) is defined asfollows.

Definition 2.5: System (9) is said to be asymptoticallystable if

limi+j→∞ ‖xi,j‖1 = 0 (11)

for boundary condition satisfying Assumption 2.1.

Lemma 2.6 (Zhu, Han, & Zhang, 2014): The positive 1-D Markov jump system

xi+1 = A(ri)xi, ri ∈ L (12)

is asymptotically stable in the mean sense if and only ifthere exist vectors pm � 0, m = 1, . . . , S, satisfying

pTmAm − pTm ≺ 0, (13)

where pm =∑Sn=1 πmnpn.

The aims of this paper are to establish

• the asymptotic stability condition for system (1);• the �1-gain computation condition for system (1).

3. Main results

3.1. Stability analysis

Theorem 3.1: Positive 2-D Markov jump system (1) isasymptotically stable in the mean sense if and only if thereexist vectors phm � 0, pvm � 0, m = 1, . . . , S, satisfying

pTmAm − pTm ≺ 0, (14)

where pm = [phTm pvTm ]T, pm =∑Sn=1 πmnpn.

2080 Y. CHEN ET AL.

Proof: Define the following indicator function (Costa,Fragoso, & Marques, 2005):

1{r(i,j)=l} ={1, if ri,j = l, l ∈ L,0, otherwise.

(15)

For (i, j) ∈ N0 × N0, l ∈ L, we introduce the followingnotations:

xhi,j(l) = E{xhi,j1{r(i,j)=l}}, xvi,j(l) = E{xvi,j1{r(i,j)=l}},

Xhi,j =

[xi,jhT(1) xi,jhT(2) · · · xhTi,j (S)

]T,

Xvi,j =

[xi,jvT(1) xi,jvT(2) · · · xvTi,j (S)

]T,

Xi,j =[XhTi,j X

vTi,j

]T,

X+i,j =

[XhTi+1,j X

vTi,j+1

]T,

Zi,j = [xi,jhT(1) xi,jvT(1) xi,jhT(2) xi,jvT(2)

· · · xhTi,j (S) xvTi,j (S)]T

,

Z+i,j = [xi+1,j

hT(1) xi,j+1vT(1) xi+1,j

hT(2)

xi,j+1vT(2) · · · xhTi+1,j(S) xvTi,j+1(S)

]T,

A = (�T ⊗ Iu)diag(A1,A2, . . . ,AS). (16)

Then there exists an orthogonal matrix

M =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

Iu1 0 0 0 0 · · · 0 00 0 Iu1 0 0 · · · 0 00 0 0 0 Iu1 · · · 0 0...

......

......

. . ....

...0 0 0 0 0 · · · Iu1 00 Iu2 0 0 0 · · · 0 00 0 0 Iu2 0 · · · 0 0...

......

......

. . ....

...0 0 0 0 0 · · · 0 Iu2

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

,

(17)such that Xi,j = MZi,j. For the orthogonal matrix M,it is easy to verify that the following statements areequivalent:

• qT ≺ 0;• qTM ≺ 0.

By using the above notations, for system (1) withoutdisturbances, we can get the following equation:

Z+i,j = AZi,j. (18)

Furthermore, we have the following positive auxiliarysystem:

X+i,j = MAMT

Xi,j, (19)

the boundary condition is obtained as

X0 = [XhT0,0 X

hT0,1 X

hT0,2 · · · X

vT0,0

XvT1,0 X

vT2,0 · · ·]T � 0. (20)

It should be pointed out that the obtained auxiliary sys-tem (19) is a positive 2-D deterministic system.

In addition, we have

‖Xi,j‖1 = 1TSuXi,j = 1Tu1

S∑l=1

xhi,j(l) + 1Tu2

S∑l=1

xvi,j(l)

= 1TuE{xi,j}= E{‖xi,j‖1}. (21)

Therefore, positive 2-D Markov jump system (1) isasymptotically stable in the mean sense if and onlyif auxiliary positive 2-D deterministic system (19)is asymptotically stable because limi+j→∞ ‖Xi,j‖1 =limi+j→∞ E{‖xi,j‖1}. By Lemma 2.4, auxiliary positive 2-Ddeterministic system (19) is asymptotically stable if andonly if there exists a vector q � 0 such that

qT(MAMT − ISn) ≺ 0. (22)

Let p = MTq = [pT1 pT2 · · · pTS ]T, then the above condi-

tion can be rewritten as

pT(A − ISu)MT ≺ 0, (23)

the above inequality is equivalent to

pT(A − ISu) ≺ 0. (24)

Substituting (16) into the above inequality yields⎡⎢⎢⎢⎣p1p2...pS

⎤⎥⎥⎥⎦T⎡⎢⎢⎢⎣

π11A1 − In π21A2 · · · πS1ASπ12A1 π22A2 − In · · · πS2AS

......

. . ....

π1SA1 π2SA2 · · · πSSAS − In

⎤⎥⎥⎥⎦

≺ 0, (25)

which is equivalent to pTmAm − pTm ≺ 0, ∀m ∈ L. Thuswe can conclude that positive 2-D Markov jump sys-tem (1) is asymptotically stable in the mean sense ifand only if condition (14) holds. This completes theproof. �

Based on Theorem 3.1 and Lemma 2.6, it is easy to obtainthe following corollary.

INTERNATIONAL JOURNAL OF SYSTEMS SCIENCE 2081

Corollary 3.2: Positive 2-D Markov jump system (1) isasymptotically stable in the mean sense if and only if posi-tive 1-D Markov jump system (12) is asymptotically stablein the mean sense.

3.2. �1-gain analysis

Theorem 3.3: Positive 2-D Markov jump system (1) isasymptotically stable with �1-gain no greater than γ inthemean sense if there exist vectors phm � 0, pvm � 0, m =1, . . . , S, satisfying

pTmAm − pTm + 1TCm ≺ 0, (26)

pTmBm + 1TDm − γ 1T ≺ 0, (27)

where pm = [phTm pvTm ]T, pm =∑Sn=1 πmnpn.

Proof: On the one hand, condition (26) implies con-dition (14) as Cm � 0. Then according to Theorem 1,system (1) with ωi,j = 0 is asymptotically stable in themean sense.

On the other hand, for �1-gain, consider the followingindex:

Ji,j = E

{[phT(ri+1,j)xhi+1,j + pvT(ri,j+1)xvi,j+1

−pT(ri,j)xi,j + 1Tyi,j]|(xi,j,ωi,j, ri,j = m)

}− γ 1Tωi,j, (28)

where p(ri,j) = [phT(ri,j) pvT(ri,j)]T � 0. Along the tra-jectory of system (1), we have

Ji,j = pTm(Amxi,j + Bmωi,j

)− pTmxi,j

+ 1T(Cmxi,j + Dmωi,j

)− γ 1Tωi,j

=(pTmAm − pTm + 1TCm

)xi,j

+(pTmBm + 1TDm − γ 1T

)ωi,j. (29)

Conditions (26) and (27) ensure Ji,j ≤ 0 for all xi,j �0, ωi,j � 0, and the ‘= ’ occurs at xi,j = 0, ωi,j = 0. Basedon the above relationship, we can easily obtain

E

{phT(r0,k+1)xh0,k+1

}= E

{phT(r0,k+1)xh0,k+1

},

E

{phT(r1,k)xh1,k + pvT(r0,k+1)xv0,k+1

}≤ E

{pT(r0,k)x0,k − 1Ty0,k

}+ γ 1Tω0,k,

E

{phT(r2,k−1)xh2,k−1 + pvT(r1,k)xv1,k

}≤ E

{pT(r1,k−1)x1,k−1 − 1Ty1,k−1

}+ γ 1Tω1,k−1,

...

E

{phT(rk+1,0)xhk+1,0 + pvT(rk,1)xvk,1

}≤ E

{pT(rk,0)xk,0 − 1Tyk,0

}+ γ 1Tωk,0,

E

{pvT(rk+1,0)xvk+1,0

}= E

{pvT(rk+1,0)xvk+1,0

}. (30)

Adding both sides of the above inequalities and con-sidering the zero boundary conditions xh0,i = 0, xvj,0 =0, i, j = 0, 1, 2, . . . , yield

E

⎧⎨⎩

k+1∑j=0

[pT(rk+1−j,j)xk+1−j,j

]⎫⎬⎭

≤ E

⎧⎨⎩

k∑j=0

[pT(rk−j,j)xk−j,j − 1Tyk−j,j

]⎫⎬⎭

+ γ

k∑j=0

1Tωk−j,j. (31)

Taking the sum of both sides of the above inequality fork from 0 to N, we have

E

⎧⎨⎩

N∑k=0

k∑j=0

1Tyk−j,j

⎫⎬⎭ ≤ γ

N∑k=0

k∑j=0

1Tωk−j,j

− E

⎧⎨⎩

N+1∑j=0

[pT(rN+1−j,j)xN+1−j,j

]⎫⎬⎭ . (32)

For N → ∞, we have

E

⎧⎨⎩

∞∑k=0

k∑j=0

1Tyk−j,j

⎫⎬⎭ ≤ γ

∞∑k=0

k∑j=0

1Tωk−j,j, (33)

for non-zeroω = {ωi,j} ∈ l1{N0,N0}. The above inequal-ity indicates ‖y‖E ≤ γ ‖ω‖1. Thus we can conclude thatthe �1-gain of positive 2-D Markov jump system (1) inthe mean sense is no greater than γ . This completes theproof. �

3.3. Further extension to partially known transitionprobabilities

In some circumstances, the mode transition probabilitiesare not always known (Sun, Zhang, & Wu, 2018; Zhang

2082 Y. CHEN ET AL.

& Lam, 2010). Next, we will extend the asymptotic stabil-ity definition for system (1) and the stability and �1-gainanalysis results to systemswith partially known transitionprobabilities. The transition probability matrix may takethe following form:

� =

⎡⎢⎢⎢⎢⎢⎣

π11 ? ? · · · π1S? π22 π23 · · · ?? ? π33 · · · π3S...

......

. . ....

πS1 ? πS3 · · · ?

⎤⎥⎥⎥⎥⎥⎦ , (34)

where ‘?’ represents an unknown transition probability.Form ∈ L, denote

Km = {n : if πmn is known} , UKm

= {n : if πmn is unknown} . (35)

In the following, when dealing with the partially knownprobability situations, it is assumed that UKm is a non-empty set.

Definition 3.4: System (1) with ωi,j ≡ 0 and partiallyknown transition probabilities is said to be robustlyasymptotically stable in the mean sense if

limi+j→∞ E

{‖xi,j‖1} = 0 (36)

for boundary condition satisfying Assumption 1.

The following two theorems give the robust asymp-totic stability and �1-gain computational conditions forsystems with partially known transition probabilities.

Theorem 3.5: Positive 2-D Markov jump system (1) withpartially known transition probabilities given by (35) isrobustlyasymptotically stable in the mean sense if and only if thereexist vectors phm � 0, pvm � 0, m = 1, . . . , S, satisfying

[pKm + (1 − πK

m)pn]T

Am − pTm ≺ 0, (37)

for all n ∈ UKm, where pm = [phTm pvTm ]T, pKm =∑n∈Km

πmnpn, πKm =∑n∈Km

πmn.

Proof: We state with the fact that 0 ≤ πKm ≤ 1. ForπK

m =1, the conditions in Theorem 3.5 reduce to the condi-tions in Theorem 3.1. For πK

m < 1, the left-hand side of

condition (14) in Theorem 3.1 can be rewritten as

�m = pTmAm − pTm ≺ 0

=⎛⎝pKm +

∑n∈UKm

πmnpn

⎞⎠

T

Am − pTm

=⎡⎣pKm + (1 − πK

m) ∑n∈UKm

πmn

1 − πKmpn

⎤⎦T

Am − pTm.

(38)

Since 0 ≤ πmn/(1 − πKm) ≤ 1, ∀n ∈ UKm, and∑

n∈UKmπmn/(1 − πK

m) = 1, we have

�m =∑

n∈UKm

πmn

1 − πKm

{[pKm + (1 − πK

m)pn]T

Am

−pmT}. (39)

Therefore, �m ≺ 0 is equivalent to condition (37). Then,according to Theorem 3.1, system (1) with partiallyknown transition peobabilities in the form of (35) isrobustly asymptotically stable in the mean sense ifand only if conditions (37) hold. This completes theproof. �

Theorem 3.6: Positive 2-D Markov jump system (1) withpartially known transition probabilities given by (35) isrobustly asymptotically stable with �1-gain no greater thanγ in the mean sense if there exist vectors phm � 0, pvm �0, m = 1, . . . , S,satisfying[pKm + (1 − πK

m)pn]T

Am − pTm + 1TCm ≺ 0, (40)[pKm + (1 − πK

m)pn]T

Bm + 1TDm − γ 1T ≺ 0, (41)

for all n ∈ UKm, where pm = [phTm pvTm ]T, pKm =∑n∈Km

πmnpn, πKm =∑n∈Km

πmn.

The proof of Theorem 3.5 can be easily obtained by asimilar method presented in the proof of Theorem 3.4.We omit it here for simplicity.

If the mode transition probabilities are completelyunknown, pKm and πK

m are no longer present in theanalysis. By removing them from inequality (36) inTheorem 3.4, we have the following two corollaries forpositive 2-D switched systems with unknown transitionprobabilities.

Corollary 3.7: Positive 2-D switched system (1) withunknown transition probabilities given by (35) is robustly

INTERNATIONAL JOURNAL OF SYSTEMS SCIENCE 2083

asymptotically stable if and only if there exist vectors phm �0, pvm � 0, m = 1, . . . , S, satisfying

pTnAm − pTm ≺ 0, (42)

for all (m, n) ∈ L × L, where pm = [phTm pvTm ]T.

Corollary 3.8: Positive 2-D switched system (1) withunknown transition probabilities given by (35) isrobustly asymptotically stable with �1-gain no greaterthan γ if there exist vectors phm � 0, pvm � 0, m =1, . . . , S, satisfying

pTnAm − pTm + 1TCm ≺ 0, (43)

pTnBm + 1TDm − γ 1T ≺ 0, (44)

for all (m, n) ∈ L × L, where pm = [phTm pvTm ]T.

4. Illustrative example

Example 4.1: Consider system (1) with parametersgiven in Liang et al. (2017). The parameters of system (1)are given as follows:

A1 =[0.2 0.50.2 0.3

], B1 =

[0.30.5

],

A2 =[0.2 0.60.5 0.2

], B2 =

[0.30.4

],

C1 = C2 =[0.1 0.00.1 0.6

],

D1 = D2 = [0.0 0.3]T .

Assume that the transition matrix is given by

� =[0.30 0.700.60 0.40

]. (45)

Using Theorem 3.3, the obtained minimum γ is γ ∗ =1.3165. The obtained value of P1 and P2 are

P1 =[0.68521.5243

], P2 =

[1.11011.4087

].

Assume zero boundary conditions, and let the distur-bance ωi,j be

ωi,j ={0.10, 0 ≤ i ≤ 3, 0 ≤ j ≤ 10,0, otherwise.

(46)

The Markov chain, trajectories of the system and the�1-gain performance are shown in Figure 1. Figure 1(a)shows the switching signal generated by the transitionmatrix. Figures 1(b) and (c) show the state trajectories

of the system. The measured outputs are shown in Fig-ures 1(d) and (e). The obtained �1-gain under the switch-ing signal shown in Figure 1(a) is 0.8382. Define thefollowing function:

Li,j =∑i

p=0∑j

q=0 ‖yp,q‖1∑ip=0∑j

q=0 ‖ωp,q‖1.

The plot of Li,j is shown in Figure 1(f). It is obvious thatLi,j converges to 0.8382 as i and j increase. Running sys-tem (1) for 100 times, and the arithmetic mean of theobtained �1-gains from the 100 realisations is 0.8364,which is below the prescribed value γ ∗ = 1.3165.

Example 4.2: Consider system (1) with three operationmodes. The parameters are given as follows:

A1 =[0.2 0.50.2 0.3

], B1 =

[0.30.5

],

A2 =[0.2 0.60.5 0.2

], B2 =

[0.30.4

],

A3 =[0.1 0.30.4 0.2

], B3 =

[0.20.1

],

C1 = C2 =[0.1 0.00.1 0.6

],

D1 = D2 = [0.0 0.3]T .

C3 =[0.1 0.20.1 0.3

], D3 =

[0.20.1

].

The fully known transition matrix is given by

� =⎡⎣0.30 0.30 0.400.20 0.40 0.400.40 0.50 0.10

⎤⎦ . (47)

In the following, three cases are investigated to illustratethe efficiency of the proposed theorems and corollaries.

Case 1: considering the above positive 2-D Markovjump system, where the transition probabilities are fullyknown.Case 2: considering the above positive 2-DMarkov jumpsystem with partially known transition probabilities. Thepartially known transition matrix is given as

� =⎡⎣0.30 0.30 0.400.20 ? ?0.40 0.50 0.10

⎤⎦ ,

where the relation between known and unknown transi-tion probabilities is

π22 + π23 = 1 − π21.

Case 3: consider the above system with unknown transi-tion probabilities.

2084 Y. CHEN ET AL.

Table 1. Obtained minimum �1-gain γ ∗ for different cases.

Case 1 Case 2 Case 3

Stability Theorem 3.1 Theorem 3.3 Corollary 3.6analyse feasible feasible feasible

P1

[0.60161.3613

] [0.64531.4477

] [0.93121.8526

]

P2

[0.97181.3413

] [1.17881.5490

] [1.35301.7648

]

P3

[0.80701.0053

] [0.87791.0728

] [1.24871.4824

]

γ ∗ 1.1449 1.2336 1.5883

Using Theorems 3.3, 3.5, and Corollary 3.7, theobtained minimum �1-gain γ ∗ are shown in Table 1 fordifferent cases.

From Table 1, it can be found that the more infor-mation the transition matrix has, the better the system�1-gain performance is.

5. Conclusion

The problems of stability and �1-gain analysis for pos-itive 2-D Markov jump systems have been studied inthis paper. Sufficient and necessary asymptotic stabil-ity condition and sufficient �1-gain computation con-dition have been derived. The obtained results are fur-ther extended to positive 2-D Markov jump systemswith partially known transition probabilities. The use-fulness of the obtained stability and �1-gain condi-tions have been verified through simulation examples.

Figure 1. Switching signal and trajectories of the system. (a) Switching signal; (b) state trajectory of xhi,j ; (c) state trajectory of xvi,j ; (d) first

component of measured output yi,j ; (e) second component of measured output yi,j ; (f ) �1-gain.

INTERNATIONAL JOURNAL OF SYSTEMS SCIENCE 2085

Furthermore, our future work will extend the proposedresults to the 2-D positive Markov jump systems withstochastic nonlinearities, missing measurements, andstate-delays (Wang, Wang, Li, & Wang, 2016; Wei, Qiu,Karimi, & Wang, 2015).

Disclosure statement

No potential conflict of interest was reported by the authors.

Funding

This work was partially supported by GRF HKU (17205815,17206818), NTUT-SZU Joint Research Program (2019004) andTencent “Rhinoceros Birds”-Scientic Research Foundation forYoung Teachers of Shenzhen University.

Notes on contributors

Yong Chen was born in Changsha, Hunanprovince, China, in 1988. He receivedhis B.Eng. degree in Automation (rank-ing No. 10 among the 126 studentsof the major) from Harbin Institute ofTechnology (HIT), Harbin, Heilongjiangprovince, China, in 2010 and the Ph.D.degree in Mechanical Engineering from

the University of Hong Kong in June, 2016. From 2016 to 2017,he was a Research Associate in the Department of Mechani-cal Engineering, the University of Hong Kong. Since 2018, heis an Assistant Professor in the School of Automation, Cen-tral South University, Changsha, China. His current researchinterests include switched systems, stochastic systems, periodicsystems, fuzzy systems, positive systems in control theory andmicro quadrotor in control engineering.

Chang Zhao received her B.E. degree inDetectionGuidance andControl Technol-ogy from Harbin Institute of Technology,Harbin, China in July, 2015 and theMasterdegree in Control Science and Engineer-ing from Harbin Institute of Technology,Harbin, China in July, 2017. She is cur-rently working toward her Ph.D. degree

in Mechanical Engineering at the University of Hong Kong.Her research interests include two dimensional systems, cyberphysical systems, state estimation and robust control.

Professor J. Lam received a B.Sc. (1stHons.) degree in Mechanical Engineer-ing from the University of Manchester,and was awarded the Ashbury Scholar-ship, the A.H. Gibson Prize and the H.Wright Baker Prize for his academic per-formance. He obtained the M.Phil. andPh.D. degrees from theUniversity of Cam-

bridge. He is a Croucher Scholar, Croucher Fellow, and Distin-guished Visiting Fellow of the Royal Academy of Engineering.Prior to joining the University of Hong Kong in 1993 where heis now Chair Professor of Control Engineering, he was a lec-turer at the City University of Hong Kong and the Universityof Melbourne. Professor Lam is a Chartered Mathematician,

Chartered Scientist, Chartered Engineer, Fellow of Institute ofElectrical and Electronic Engineers, Fellow of Institution ofEngineering and Technology, Fellow of Institute of Mathemat-ics and Its Applications, Fellow of Institution of MechanicalEngineers, and Fellow of Hong Kong Institution of Engineers.He is Editor-in-Chief of IET Control Theory and Applicationsand Journal of The Franklin Institute, Subject Editor of Journal ofSound and Vibration, Editor of Asian Journal of Control, SeniorEditor of Cogent Engineering, Associate Editor of Automatica,International Journal of Systems Science, Multidimensional Sys-tems and Signal Processing and Proc. IMechE Part I: Journal ofSystems and Control Engineering. He is a member of the Engi-neering Panel (Joint Research Scheme), Research Grant Coun-cil, HKSAR. His research interests include model reduction,robust synthesis, delay, singular systems, stochastic systems,multidimensional systems, positive systems, networked controlsystems and vibration control. He is a Highly Cited Researcherin Engineering (2014, 2015, 2016, 2017, 2018) and ComputerScience (2015).

Yukang Cui received his B.Eng. degreein Automation from the Harbin Insti-tute of Technology in July, 2012 and thePh.D. degree in Mechanical Engineer-ing from the University of Hong Kongin March, 2017. During 2017, he wasa Research Associate within the Depart-ment ofMechanical Engineering, the Uni-

versity of Hong Kong. Since 2018, he is an Assistant Profes-sor in the College of Mechatronics and Control Engineering,Shenzhen University, Shenzhen, China. His current researchinterests include positive systems, singular systems, time-delaysystems, robust control and multi-agent systems.

Ka-Wai Kwok has served as Assistant Pro-fessor in Department of Mechanical Engi-neering, The University of Hong Kong(HKU), since 2014. He obtained a Ph.D.at Hamlyn Centre for Robotic Surgery,Department of Computing, Imperial Col-lege London in 2012, where he continuedresearch on surgical robotics as a postdoc-

toral fellow. In 2013, he was awarded the Croucher FoundationFellowship, which supported his research jointly supervisedby advisors in The University of Georgia, and Brigham andWomen’s Hospital – Harvard Medical School. His researchinterests focus on intra-operative medical image processing,surgical robotics and their control techniques. His multidisci-plinary work has been recognised by various scientific com-munities through his international conference/journal paperawards, e.g. the Best Conference Paper Award of ICRA 2018,which is the largest conference ranked top in the field ofrobotics, as well as TPEL 2018, RCAR 2017, ICRA 2017, ICRA2014, IROS 2013 and FCCM 2011, Hamlyn 2012 and 2008, andSurgical Robot Challenge 2016. He also became the recipientof the Early Career Awards 2015/16 offered by Research GrantsCouncil (RGC) of Hong Kong.

ORCID

Chang Zhao http://orcid.org/0000-0003-4234-6547James Lam http://orcid.org/0000-0002-0294-0640

2086 Y. CHEN ET AL.

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