Passivity-based control for uncertain stochastic jumping systems with mode-dependent round-trip time...

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Journal of the Franklin Institute 349 (2012) 1665–1680

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Passivity-based control for uncertain stochasticjumping systems with mode-dependent

round-trip time delays

Hao Shena, Shengyuan Xua,n, Junwei Lub, Jianping Zhoua

aSchool of Automation, Nanjing University of Science and Technology, Nanjing 210094, PR ChinabSchool of Electrical and Automation Engineering, Nanjing Normal University, Nanjing 210042, PR China

Received 2 December 2009; received in revised form 31 July 2011; accepted 19 November 2011

Available online 31 December 2011

Abstract

This paper considers the passivity-based control problem for stochastic jumping systems with

mode-dependent round-trip time-varying delays and norm-bounded parametric uncertainties. By

utilizing a novel Markovian switching Lyapunov functional, a delay-dependent passivity condition is

obtained. Then, based on the derived passivity condition, a desired Markovian switching dynamic

output feedback controller is designed, which ensures the resulting closed-loop system is passive.

Finally, two numerical examples are provided to illustrate the effectiveness of the proposed results.

& 2011 The Franklin Institute. Published by Elsevier Ltd. All rights reserved.

1. Introduction

Time delays arise quite naturally in many practical systems [1,2,13,27]. It has beenshown that time delays may be the round-trip ones in some practical situations. A typicalexample for the existence of round-trip time delays is the bilateral teleoperation systems[17], where the forwarding and returning time delays of the communication channel areindependent essentially because of their different forming conditions. When dealing withsuch systems, researchers always combine the round-trip time delays together in order tomake the underlying problem tractable. Such an approach, however, generally leads to

2.00 & 2011 The Franklin Institute. Published by Elsevier Ltd. All rights reserved.

.jfranklin.2011.11.011

nding author.

dress: [email protected] (S. Xu).


dx.doi.org/10.1016/j.jfranklin.2011.11.011


dx.doi.org/10.1016/j.jfranklin.2011.11.011

mailto:[email protected]

H. Shen et al. / Journal of the Franklin Institute 349 (2012) 1665–16801666

conservative results, especially when the round-trip time delays are different from eachother [9]. On the other hand, a special type of time delays (i.e. interval time-varying delays),which vary in an interval and their lower bound may not be zero, are naturally taken intoaccount by many researchers.During the last decades, there has been extensive interest in the study of a class of physical

systems subject to random changes in their parameters. Such a class of systems can be modeledby the hybrid system dynamics known as Markovian jump systems [3,5,16,22,23,26]. ForMarkovian jump time-delay systems, there have been lots of results on the problems ofstability analysis and control synthesis [4,12,18–20,24,25]. Furthermore, considerable attentionhas also been devoted to the problems of passivity analysis and control synthesis due to thefact that the passivity theory can be used as an effective tool for the stability analysis andcontrol synthesis of systems. Different methodologies have been proposed for the solvability ofthis problem and many interesting results have been reported, see [6,8,10,11] and the referencestherein. Specifically, in the context of continuous-time Markovian jump systems, the passivityand passification problems were investigated in [7,14]. It should be noted that these papersonly considered the problem of state feedback passive control. Moreover, the round-trip timedelays were also not considered in [7,14].To the best of our knowledge, there have been few works undertaken on the passivity

analysis and passivity-based control problems for uncertain stochastic jumping systemswith mode-dependent round-trip time-varying delays. In this paper we will deal with theseproblems. By employing a novel Markovian switched Lyapunov functional, we present adelay-dependent passivity condition in terms of linear matrix inequalities (LMIs). In thelight of the derived passivity condition, a Markovian switched dynamic output feedbackcontroller is designed to guarantee the resulting closed-loop system is passive. Finally, inorder to demonstrate the effectiveness of our results, we provide two numerical examples.

Notation: Throughout this paper, for symmetric matrices X and Y, the notation XZY

(respectively, X4Y ) means that the matrix X�Y is positive semi-definite (respectively,positive definite); I is the identity matrix with appropriate dimension. The notation MT

represents the transpose of the matrix M; L2½0,1Þ is the space of square-integrable vectorfunctions over ½0,1Þ; where ðO,F ,PÞ is a probability space; O is the sample space, F is thes-algebra of subsets of the sample space and P is the probability measure on F . Ef�gdenotes the expectation operator with respect to P. Matrices, if not explicitly stated, areassumed to have compatible dimensions. The symbol n is used to denote a matrix whichcan be inferred by symmetry.

2. System descriptions and definitions

Consider the following class of uncertain stochastic jumping systems with mode-dependent round-trip time-varying delays ðSÞ:

dxðtÞ ¼ ½Aðrt,tÞxðtÞ þ A1ðrt,tÞxðt�hrtðtÞÞ þ B1ðrt,tÞuðtÞ þ B1vðrtÞvðtÞ� dt

þ½Eðrt,tÞxðtÞ þ E1ðrt,tÞxðt�hrtðtÞÞ� doðtÞ, ð1Þ

dyðtÞ ¼ ½Cðrt,tÞxðtÞ þ C1ðrt,tÞxðt�hrtðtÞÞ þ B2ðrt,tÞuðtÞ þ B2vðrtÞvðtÞ� dt

þ½Hðrt,tÞxðtÞ þH1ðrt,tÞxðt�hrtðtÞÞ� doðtÞ, ð2Þ

zðtÞ ¼DðrtÞxðtÞ þD1ðrtÞxðt�hrtðtÞÞ þ B3ðrtÞuðtÞ þ B3vðrtÞvðtÞ, ð3Þ

H. Shen et al. / Journal of the Franklin Institute 349 (2012) 1665–1680 1667

xðtÞ ¼ fðtÞ, 8t 2 ½�h2,0�, rð0Þ ¼ r0, ð4Þ

where xðtÞ 2 Rn is the state vector; uðtÞ 2 Rm is the control input; vðtÞ 2 Rp is thedisturbance input which is assumed to be an arbitrary signal in L2½0,1Þ; yðtÞ 2 Rq is themeasurement output; zðtÞ 2 Rp is the signal to be estimated; oðtÞ is a zero-mean real scalarWinner process on a probability space ðO,F ,PÞ relative to an increasing family ðF tÞt2½0,1Þof s-algebras F t � F satisfying

EfdoðtÞg ¼ 0, EfdoðtÞ2g ¼ dt: ð5Þ

frtg is a continuous-time Markovian process with right continuous trajectories andtaking values in a finite set S ¼ f1; 2, . . . ,N g with transition probability matrix C¼D fcijg

given by

PrfrtþD ¼ jjrt ¼ ig ¼cijDþ oðDÞ, iaj,

1þ ciiDþ oðDÞ, i¼ j,

(ð6Þ

where D40, limD-0ðoðDÞ=DÞ ¼ 0, and cijZ0, for jai, is the transition rate from mode i attime t to mode j at time tþ D and

cii ¼�XS

j ¼ 1,jai

cij : ð7Þ

To simplify the notation, we denote Ai ¼AðrtÞ and Ait ¼Aðrt,tÞ for each i¼ rt 2 S, andthe other symbols are similarly denoted. The mode-dependent time-varying delays hiðtÞ

satisfy

hiðtÞ ¼ hiðtÞ þ tiðtÞ, ð8Þ

0rh1rhiðtÞrh2,_

hiðtÞrm, ð9Þ

0rh1rhiðtÞrh2, _hiðtÞrm1, ð10Þ

0rt1rtiðtÞrt2, _tiðtÞrm2, 8i 2 S: ð11Þ

In Eqs. (1)–(4), fðtÞ is the initial condition. Ait, A1it, B1it, B1vi, Eit, E1it, Cit, C1it, B2it, B2vi,Hit, H1it, Di, D1i, B3i and B3vi are matrix functions of i for each i 2 S

Ait ¼Ai þ DAit, A1it ¼A1i þ DA1it, B1it ¼B1i þ DB1it,

Eit ¼Ei þ DEit, E1it ¼ E1i þ DE1it, Cit ¼Ci þ DCit,

C1it ¼C1i þ DC1it, B2it ¼ B2i þ DB2it, Hit ¼Hi þ DHit, H1it ¼H1i þ DH1it,

where Ai, A1i, B1i, Ei, E1i, Ci, C1i, B2i, Hi and H1i are known real constant matrices withappropriate dimensions for each i 2 S, while DAit, DA1it, DB1it, DEit, DE1it, DCit, DC1it,DB2it, DHit and DH1it are unknown matrices representing time-varying parameteruncertainties, which are assumed to have the following form:

DAit DA1it DB1it DEit DE1it

DCit DC1it DB2it DHit DH1it

" #¼

L1i

L2i

" #FiðtÞ½M1i M2i M3i M4i M5i�,8i 2 S,

ð12Þ


where L1i, L2i, M1i, M2i, M3i, M4i, and M5i are known real constant matrices, and FiðtÞ arethe uncertain time-varying matrices satisfying

F Ti ðtÞFiðtÞrI , 8i 2 S: ð13Þ

For convenience, we denote

f ðtÞ ¼AitxðtÞ þ A1itxðt�hiðtÞ�tiðtÞÞ þ B1vivðtÞ,

gðtÞ ¼EitxðtÞ þ E1itxðt�hiðtÞ�tiðtÞÞ:

Throughout the paper we shall use the following definition.

Definition 1 (Mahmoud [14]). The uncertain stochastic jumping system with mode-dependent round-trip time-varying delays ðSÞ

dxðtÞ ¼ f ðtÞ dtþ gðtÞ doðtÞ, ð14Þ

zðtÞ ¼DixðtÞ þD1ixðt�hiðtÞ�tiðtÞÞ þ B3vivðtÞ, ð15Þ

xðtÞ ¼ fðtÞ, 8t 2 ½�h2,0�, rð0Þ ¼ r0, hiðtÞ þ tiðtÞ ¼ hiðtÞ: ð16Þ

is said to be passive if there exists a scalar gZ0 such that

2EZ0

vT ðtÞzðtÞ dt

( )Z�g

Z0

vT ðtÞvðtÞ dt

( ),

under zero-initial conditions for any nonzero vðtÞ 2 L2½0,1Þ and all Z0.

3. Passivity analysis

Theorem 1. Given a scalar g40, the uncertain stochastic jumping system in Eqs. (14)–(16) is

passive for any time-varying delays hiðtÞ and tiðtÞ satisfying Eqs. (10) and (11), respectively, if

there exist matrices Pi40, W40, Ql40, Nmi, Rmi, Smi, l ¼ 1; 2,3; 4,5; 6, m¼ 1; 2,3; 4,scalars e1i40 and e2i40 such that the following LMIs hold for each i 2 S

F11 F12 F13 F14 F15 F16 F17 F18

n F22 F23 F24 F25 0 F27 0

n n F33 F34 F35 0 F37 0

n n n F44 F45 F46 F47 0

n n n n F55 0 0 0

n n n n n F66 F67 0

n n n n n n F77 F78

n n n n n n n F88

2666666666666664

3777777777777775

o0,

ð17Þwhere

F11 ¼ PiAi þ ATi Pi þ

X3l ¼ 1

Ql þ aðh2�h1ÞQ1 þ aðt2�t1ÞQ2

þaðh2 þ t2�h1�t1ÞQ3 þ S1i þ ST1i þ

Xj2S

cijPj þ e1iMT1i M1i þ e2iM

T4i M4i,


a¼maxfjciijg, F12 ¼N1i þ ST2i, F13 ¼R1i þ ST

3i,

F14 ¼ PiA1i�R1i�N1i þ ST4i�S1i þ e1iM

T1i M2i þ e2iM

T4i M5i,

F15 ¼ ½�ðh2 þ t2�h1ÞN1i �ðh2 þ t2�t1ÞR1i �ðh2 þ t2ÞS1i�,

F17 ¼ ½ETi Pi AT

i J1 ETi W N1i R1i S1i�, F16 ¼ PiB1vi�DT

i ,

F22 ¼�ð1�m1ÞQ1 þN2i þNT2i , F23 ¼R2i þNT

3i , F24 ¼�R2i�N2i þNT4i�S2i,


F27 ¼ ½0 0 0 N2i R2i S2i�, F37 ¼ ½0 0 0 N3i R3i S3i�,

F33 ¼�ð1�m2ÞQ2 þ R3i þ RT3i, F34 ¼�N3i�R3i þ RT

4i�S3i,


F44 ¼�ð1�m1�m2ÞQ3�N4i�NT4i�R4i�RT

4i�S4i�ST4i þ e1iM

T2i M2i þ e2iM

T5i M5i,

F45 ¼ ½�ðh2 þ t2�h1ÞN4i �ðh2 þ t2�t1ÞR4i �ðh2 þ t2ÞS4i�, F46 ¼�DT1i

F47 ¼ ½ET1iPi AT

1iJ1 ET1iW N4i R4i S4i�, F67 ¼ ½0 BT

1viJ1 0 0 0 0�,

F66 ¼�B3vi�BT3vi�gI , F77 ¼ diagf�Pi,�J1,�W ,�W ,�W ,�W g,

W ¼ 2ðh2 þ t2ÞW ,

F55 ¼ diagf�ðh2 þ t2�h1ÞQ4,�ðh2 þ t2�t1ÞQ5,�ðh2 þ t2ÞQ6g,

J1 ¼ ðh2 þ t2�h1ÞQ4 þ ðh2 þ t2�t1ÞQ5 þ ðh2 þ t2ÞQ6 F88 ¼ diagf�e1iI ,�e2iIg,

F78 ¼0 LT

1iJ1 0 0 0 0

LT1iPi 0 LT

1iW 0 0 0

" #T

, F18 ¼ ½PiL1i 0�:

Proof. Let

U1i ¼ ½LT1iPi 0 0 0 0 0 0 0 LT

1iJ1 0 0 0 0�T ,

U2i ¼ ½M1i 0 0 M2i 0 0 0 0 0 0 0 0 0�T ,

U3i ¼ ½0 0 0 0 0 0 0 LT1iPi 0 LT

1iW 0 0 0�T ,

U4i ¼ ½M4i 0 0 M5i 0 0 0 0 0 0 0 0 0�T :

Then, noting Eqs. (12) and (13), and using Lemma 2.2 in [21], we have

U1iFiðtÞUT2i þ U2iF

Ti ðtÞU

T1i þ U3iFiðtÞUT

4i þ U4iFTi ðtÞU

T3i

re�11i U1iUT1i þ e1iU2iUT

2i þ e�12i U3iUT3i þ e2iU4iUT

4i: ð18Þ

It follows from Eqs. (17) and (18) that

Xito0, ð19Þ


where

Xit ¼ X1it þ ETit PiEit þ AT

it J1Ait þ ETit WEit þNiW

�1NTi þ RiW

�1RTi þ SiW

�1STi ,

X1it ¼

~F11 F12 F13~F14 F15 F16

n F22 F23 F24 F25 0

n n F33 F34 F35 0

n n n ~F44 F45 F46

n n n n F55 0

n n n n n F66

26666666664

37777777775,

~F11 ¼ PiAit þ ATit Pi þ

X3l ¼ 1

Ql þ aðh2�h1ÞQ1 þ aðt2�t1ÞQ2

þaðh2 þ t2�h1�t1ÞQ3 þ S1i þ ST1i þ

Xj2S

cijPj,

~F14 ¼ PiA1it�R1i�N1i þ ST4i�S1i, Si ¼ ½S

T1i ST

2i ST3i ST

4i 0 0 0 0�T ,

~F44 ¼�ð1�m1�m2ÞQ3�N4i�NT4i�R4i�RT

4i�S4i�ST4i,

Eit ¼ ½Eit 0 0 E1it 0 0 0 0�, Ait ¼ ½Ait 0 0 A1it 0 0 0 B1vi�,

Ni ¼ ½NT1i NT

2i NT3i NT

4i 0 0 0 0�T , Ri ¼ ½RT1i RT

2i RT3i RT

4i 0 0 0 0�T :

Consider Eqs. (14)–(16), and define a new process fðxt,iÞ,tZ0g by xtðsÞ ¼ xðtþ sÞ,�h2�t2rsr0. Then, choose a Markovian switched Lyapunov functional candidate as

V ðxt,i,tÞ ¼X7l ¼ 1

Vlðxt,i,tÞ,

where

V1ðxt,i,tÞ ¼ xT ðtÞPixðtÞ,

V2ðxt,i,tÞ ¼

Z t

t�hiðtÞ

xT ðsÞQ1xðsÞ dsþ

Z t

t�tiðtÞ

xT ðsÞQ2xðsÞ ds,

V3ðxt,i,tÞ ¼ a

Z �h1

�h2

Z t

tþyxT ðsÞQ1xðsÞ ds dyþ a

Z �t1�t2

Z t

tþyxT ðsÞQ2xðsÞ ds dy,

V4ðxt,i,tÞ ¼

Z t

t�hiðtÞ�tiðtÞ

xT ðsÞQ3xðsÞ dsþ a

Z �h1�t1

�h2�t2

Z t

tþyxT ðsÞQ3xðsÞ ds dy,

V5ðxt,i,tÞ ¼

Z �h1

�h2�t2

Z t

tþyf T ðsÞQ4f ðsÞ ds dyþ

Z �t1�h2�t2

Z t

tþyf T ðsÞQ5f ðsÞ ds dy,

V6ðxt,i,tÞ ¼

Z 0

�h2�t2

Z t

tþyf T ðsÞQ6f ðsÞ ds dy,


V7ðxt,i,tÞ ¼X

k ¼ h2,t2,h2þt2

Z 0

�k

Z t

tþygT ðsÞWgðsÞ ds dy:

In the light of the generalized Ito’s formula cited by [15], we let L be the weak infinitesimalgenerator of the random process fxt,ig, and obtain the stochastic differential dV ðxt,i,tÞ as [15]

dV ðxt,i,tÞ ¼LV ðxt,i,tÞ dtþ 2xT ðtÞPigðtÞ doðtÞ, ð20Þ

where

LV ðxt,i,tÞ ¼X7l ¼ 1

LVlðxt,i,tÞ,

LV1ðxt,i,tÞ ¼ 2xT ðtÞPif ðtÞ þXj2S

cijxT ðtÞPjxðtÞ þ gT ðtÞPigðtÞ,

LV2ðxt,i,tÞ ¼ xT ðtÞQ1xðtÞ�ð1� _hiðtÞÞxT ðt�hiðtÞÞQ1xðt�hiðtÞÞ

þXj2S

cij

Z t

t�hj ðtÞ

xT ðsÞQ1xðsÞ dsþ xT ðtÞQ2xðtÞ

�ð1�_tiðtÞÞxT ðt�tiðtÞÞQ2xðt�tiðtÞÞ þ

Xj2S

cij

Z t

t�tjðtÞ

xT ðsÞQ2xðsÞ ds,

LV3ðxt,i,tÞ ¼ aðh2�h1ÞxT ðtÞQ1xðtÞ�a

Z t�h1

t�h2

xT ðsÞQ1xðsÞ ds

þaðt2�t1ÞxT ðtÞQ2xðtÞ�a

Z t�t1

t�t2xT ðsÞQ2xðsÞ ds,

LV4ðxt,i,tÞ ¼ xT ðtÞQ3xðtÞ þXj2S

cij

Z t

t�hjðtÞ�tjðtÞ

xT ðsÞQ3xðsÞ ds

�ð1� _hiðtÞ�_tiðsÞÞxT ðt�hiðtÞ�tiðtÞÞQ3xðt�hiðtÞ�tiðtÞÞ

þaðh2 þ t2�h1�t1ÞxT ðtÞQ3xðtÞ�a

Z t�h1�t1

t�h2�t2xT ðsÞQ3xðsÞ ds,

LV5ðxt,i,tÞ ¼ ðh2 þ t2�h1ÞfT ðtÞQ4f ðtÞ�

Z t�h1

t�h2�t2f T ðsÞQ4f ðsÞ ds

þðh2 þ t2�t1Þf T ðtÞQ5f ðtÞ�

Z t�t1

t�h2�t2f T ðsÞQ5f ðsÞ ds,

LV6ðxt,i,tÞ ¼ ðh2 þ t2Þf T ðtÞQ6f ðtÞ�

Z t

t�h2�t2f T ðsÞQ6f ðsÞ ds,

LV7ðxt,i,tÞ ¼ gT ðtÞW gðtÞ�X

k ¼ h2,t2,h2þt2

Z t

t�k

gT ðsÞWgðsÞ ds:

On the other hand, it can be seen that for each i 2 S

Xj2S

cij

Z t

t�hjðtÞ

xT ðsÞQ1xðsÞ dsra

Z t�h1

t�h2

xT ðsÞQ1xðsÞ ds, ð21Þ


Xj2S

cij

Z t

t�tjðtÞ


Z t�t1

t�t2xT ðsÞQ2xðsÞ ds, ð22Þ

Xj2S

cij

Z t

t�hjðtÞ�tjðtÞ


Z t�h1�t1

t�h2�t2xT ðsÞQ3xðsÞ ds: ð23Þ

Let

xðtÞ ¼ xT ðtÞ xT ðt�hiðtÞÞ xT ðt�tiðtÞÞ xT ðt�hiðtÞ�tiðtÞÞ xðtÞ vT ðtÞ� �T

,

xðtÞ ¼1

h2 þ t2�h1

Z t�hiðtÞ


f T ðsÞ ds1

h2 þ t2�t1

Z t�tiðtÞ


f T ðsÞ ds

�1

h2 þ t2

Z t


f T ðsÞ ds

�: ð24Þ

Then, we can obtain

2xTðtÞNi xðt�hiðtÞÞ�xðt�hiðtÞ�tiðtÞÞ�

Z t�hiðtÞ


dxðsÞ

� �¼ 0, ð25Þ

2xTðtÞRi xðt�tiðtÞÞ�xðt�hiðtÞ�tiðtÞÞ�

Z t�tiðtÞ


dxðsÞ

� �¼ 0, ð26Þ

2xTðtÞSi xðtÞ�xðt�hiðtÞ�tiðtÞÞ�

Z t


dxðsÞ

� �¼ 0: ð27Þ

Now, in view of isometry property, we derive from Eqs. (20)–(23) and (25)–(27) that

LV ðxt,i,tÞ�2vT ðtÞzðtÞ�gvT ðtÞvðtÞrxTðtÞ½X1it þ ET

it WEit þ ETit PiEit þ AT

it J1Ait�xðtÞ

�X

k ¼ h2,t2,h2þt2

Z t

t�k

gT ðsÞWgðsÞ ds�2xTðtÞNi

Z t�hiðtÞ


gðsÞ doðsÞ

�2xTðtÞRi

Z t�tiðtÞ


gðsÞdoðsÞ�2xTðtÞSi

Z t


gðsÞ doðsÞ

rxTðtÞXitxðtÞ:

Now, we introduce the following performance index:

J ¼ EZ0

½�gvT ðtÞvðtÞ�2vT ðtÞzðtÞ� dt

( ),

where 40. Under the zero-initial condition and for all non-zero disturbance variables vðtÞ,according to Eq. (5), we can deduce that for each i 2 S

J ¼ EZ0

½�gvT ðtÞvðtÞ�2vT ðtÞzðtÞ þ LV ðxt,i,tÞ� dt

( )�EfV ðx ,i, Þg

rEZ½�gvT ðtÞvðtÞ�2vT ðtÞzðtÞ þ LV ðxt,i,tÞ� dt

( )rE

ZxTðtÞXitxðtÞ ds

( ):

0 0


In the light of Eq. (19), we obtain Jo0 for any nonzero vðtÞ 2 L2½0,1Þ. This implies

EZ0

2vT ðtÞzðtÞ dt

( )Z�gE

Z0

vT ðtÞvðtÞ dt

( ):

Hence, system (14)–(16) is passive according to Definition 1. This completes the proof. &

4. Passivity-based control

In this section, we present a solution to the passivity-based control problem. For thispurpose, we consider the following dynamic output feedback Markovian switching controller:

dxðtÞ ¼AkixðtÞ dtþ Bki dyðtÞ, ð28Þ

uðtÞ ¼CkixðtÞ, ð29Þ

where xðtÞ 2 Rn is the controller state, Aki, Bki and Cki are matrices to be determined. Then,applying dynamic output controller in Eqs. (28) and (29) to the uncertain stochastic jumpingsystem (1)–(4), the closed-loop system can be obtained as

dZðtÞ ¼ ½ ~AitZðtÞ þ ~A1itGZðt�hiðtÞ�tiðtÞÞ þ ~Bv1ivðtÞ� dt

þ½ ~EitZðtÞ þ ~E1itGZðt�hiðtÞ�tiðtÞÞ� doðtÞ, ð30Þ

zðtÞ ¼ ~DiZðtÞ þD1iGZðt�hiðtÞ�tiðtÞÞ þ Bv3ivðtÞ, ð31Þ

where ZðtÞ ¼ ½xT ðtÞ xT ðtÞ�T , and

~Ait ¼Ait B1itCki

BkiCit Aki þ BkiB2itCki

" #, ~A1it ¼

A1it

BkiC1it

" #, ~Bv1i ¼

Bv1i

BkiBv2i

" #,

~Eit ¼Eit 0

BkiHit 0

" #, ~E1it ¼

E1it

BkiH1it

" #, ~Di ¼ ½Di B3iCki�, G¼ ½I 0�:

Theorem 2. Consider the uncertain stochastic jumping system with mode-dependent round-

trip time-varying delays ðSÞ and let g40 be a prescribed scalar. Then there exists an

admissible dynamic output feedback Markovian switched controller in the form of Eqs. (28)and (29) such that the resulting close-loop system is passive, if there exist Markovian switched

matrices Xi40, Yi40, W40, Ql40, Gi, Pi, Li, Nmi, Rmi, Smi, i 2 S, l ¼ 1; 2,3; 4,5; 6,m¼ 1; 2,3; 4, scalars e1i40 and e2i40 such that the following LMIs hold:

O11 0 0 O14 0 O16 O17 O18 O19

n O22 O23 O24 O25 0 O27 0 O29

n n O33 O34 O35 0 O37 0 O39

n n n O44 O45 O46 O47 O48 O49

n n n n O55 0 0 0 O59

n n n n n O66 O67 0 0

n n n n n n O77 O78 O79

n n n n n n n O88 0

n n n n n n n n O99

266666666666666664

377777777777777775

o0, ð32Þ


where

O11 ¼AiXi þ B1iLi Ai

Gi YiAi þPiCi

" #þ

AiXi þ B1iLi Ai

Gi YiAi þPiCi

" #T

þ

ciiXi ciiI

ciiIPj2S

cijYj þP

j2S,jai

cijXj�P

j2S,jai

cij2I

24

35,

O14 ¼A1i

YiA1i þPiC1i

" #, O16 ¼

Bv1i

YiBv1i þPiBv2i

" #þ�XiD

Ti �L

Ti BT

3i

�DTi

" #,

O17 ¼0 XiA

Ti þ LT

i BT1i XiE

Ti 0 0 0

0 ATi ET

i 0 0 0

" #, O88 ¼ diagf�e1iI ,e1iI�2I ,�e2iI ,e2iI�2Ig,

O18 ¼L1i XiM

T1i þ LT

i MT3i XiM

T4i 0

YiL1i þPiL2i MT1i MT

4i 0

" #,

O19 ¼0 Xi

ffiffiffiffiffiffiffici1

pXi . . .

ffiffiffiffiffiffiffiffiffiffiffici,i�1

pXi

ffiffiffiffiffiffiffiffiffiffiffici,iþ1

pXi . . .

ffiffiffiffiffiffiffifficiN

pXi

0 Iffiffiffiffiffiffiffici1

pI . . .

ffiffiffiffiffiffiffiffiffiffiffici,i�1

pI

ffiffiffiffiffiffiffiffiffiffiffici,iþ1

pI . . .

ffiffiffiffiffiffiffifficiN

pI

24

35,

Oab ¼Fab, a¼ 2; 3,4; 5, b¼ 2; 3,4; 5, O27 ¼F27, O37 ¼F37, O66 ¼F66,

O29 ¼ ½NT1i þ S2i 0 0 . . . 0 0 . . . 0 �, O39 ¼ ½R

T1i þ S3i 0 0 . . . 0 0 . . . 0 �,

O46 ¼F46, O47 ¼ ½ET1i ET

1iYi þHT1iP

Ti AT

1i ET1i N4i R4i S4i �,

O48 ¼ ½ 0 MT2i MT

5i 0 �, O49 ¼ ½�NT1i�RT

1i þ S4i�ST1i 0 0 0 0 �,

O77 ¼ diag �Xi I

I Yi

" #,J1�2I ,W�2I ,�W ,�W ,�W

( ), O78 ¼ ½ J

T5 0 0 JT

6 �,

O59 ¼ ½ JT4 0 0 0 0 �, J2 ¼ ½E

Ti ET

i Y Ti þHT

i PTi 0 0 N1i R1i S1i �,

O79 ¼ ½ JT2 0 0 . . . 0 0 . . . 0 �, J6 ¼ ½L

T1i LT

1iYTi þ LT

2iPTi 0 LT

1i 0 0 0 �,

J3 ¼X3l ¼ 1

Ql þmðh2�h1ÞQ1 þmðt2�t1ÞQ2 þmðh2 þ t2�h1�t1ÞQ3 þ S1i þ ST1i�I ,

J4 ¼ ½�ðh2 þ t2�h1ÞN1i �ðh2 þ t2�t1ÞR1i �ðh2 þ t2ÞS1i�,

J ¼ ½ 0 0 LT 0 0 0 0 �, O ¼ diagf�I ,J ,�X . . .�X �X . . .�X g:
5 1i 99 3 1 i�1 iþ1 N


In this case, the desired Markovian switched dynamic output feedback controller in the form

of Eqs. (28) and (29) can be designed with parameters as follows:

Aki ¼ ðX�1i �YiÞ

�1ðGi�YiAiXi�PiCiXi�YiB1iLi�PiB2iLiÞX

�1i , ð33Þ

Bki ¼ ðX�1i �YiÞ

�1Pi, Cki ¼LiX�1i , i 2 S: ð34Þ

Proof. According to the closed-loop system (30) and (31), note that eliI þ e�1li I�2IZ0,l ¼ 1; 2, which implies �e�1li IreliI�2I . Similarly, we have �ðJ3 þ 2IÞ�1rJ3. Hence, byusing the Schur complements formula, it follows from Eq. (32) that for each i 2 S

~O11~O12

~O13~O14

~O15 O16~O17 O18

n F22 F23 F24 F25 0 O27 0

n n F33 F34 F35 0 O37 0

n n n F44 F45 O46 O47 O48

n n n n F55 0 0 0

n n n n n O66 O67 0

n n n n n n ~O77 O78

n n n n n n n ~O88

2666666666666664

3777777777777775

o0, ð35Þ

where

~O11 ¼AiXi þ B1iLi Ai

Gi YiAi þPiCi

" #þ

AiXi þ B1iLi Ai

Gi YiAi þPiCi

" #T

þXj2S

cij

XiX�1j Xi XiX

�1j

X�1j Xi Yj

" #þ

Xi

I

� �J3 þ Ið Þ

Xi

I

� �T

,

~O12 ¼Xi

I

� �F12, ~O13 ¼

Xi

I

� �F13, ~O14 ¼O14 þ

Xi

I

� �ð�R1i�N1i þ ST

4i�S1iÞ,

~O15 ¼Xi

I

� �F15, ~O17 ¼O17 þ

Xi

I

� �J2, ~O88 ¼ diagf�e1iI ,�e�11i I ,�e2iI ,�e�12i Ig,

~O77 ¼ diag �Xi I

I Yi

" #,�J�11 ,�W

�1,�W ,�W ,�W

( ):

Now, from Eq. (32), it is easily seen that I�XiYi is invertible. Denote

U5i ¼Xi I

Xi 0

" #, U6i ¼

I Yi

0 X�1i �Yi

" #:

Then, we can set Pi ¼ U6iU�15i , i 2 S: Then, by Eq. (32), we have

Xi I

I Yi

" #40,

it can be verified that Pi40. Noting the parameters in Eqs. (33) and (34), pre- and post-multiplying Eq. (35) by diagfU�T

5i ,I ,I ,I ,I ,I ,I ,I ,U�T6i ,I ,I ,I ,I ,I ,I ,I ,I ,Ig and its transpose,


respectively, and according to the Theorem 1, we can conclude that the resulting close-loopsystems is passive. This completes the proof. &

Remark 1. Theorem 2 provides a delay-dependent condition for the solution to thepassivity-based control problem for a class of uncertain stochastic jumping systems. Notethat our results are related to the lower and upper bound of time delays and the mode, itis readily shown that information of mode switch and delays has been fully considered.Different from [7,14], we introduce some Markovian switched slack matrix variables inorder to reduce the conservatism and consider the dynamic output feedback case. On theother hand, if we suppose S ¼ f1g, namely, there is only one mode in operation, then thesystem ðSÞ degenerates into an uncertain interval round-trip time-varying delays systemwhere Markovian switching disappears. In this case, Theorem 2 can be also applied tosolve the passivity-based control problem via some simple manipulations.

5. Numerical examples

Example 1. Consider the uncertain stochastic jumping system in Eqs. (14)–(16) with twomodes with the following parameters:

A1 ¼�4:6810 0:2120

�0:1904 �1:0783

� �, A11 ¼

0:4318 0

0:1121 �0:0991

� �, Bv11 ¼

�0:2011

0:1121

� �,

E1 ¼1:3012 0

0:1998 1:1212

� �, E11 ¼

0:1008 0:1201

0:1121 �0:0993

� �,

Bv31 ¼ 1:2028, D1 ¼ ½�1:3012 0 �, D11 ¼ ½�0:1121 0:0993 �,

A2 ¼�2:4080 0:5026

0:4108 �1:4220

� �, A12 ¼

0:6885 �0:5125

0:0986 �0:1012

� �, Bv12 ¼

�0:1981

�0:1095

� �,

E2 ¼�0:1061 0

0:0986 0:0942

� �, E12 ¼

0:2686 0:2062

0:2896 �0:1082

� �,

Bv32 ¼ 1:1088, D2 ¼ ½�0:1998 �1:1212 �, D12 ¼ ½�0:1016 �0:1201 �,

L1i ¼0

�0:1000

� �, M1i ¼ ½ 0:2000 0 �, M2i ¼ ½ 0 0:1000 �,

M4i ¼ ½�0:1000 0 �, M5i ¼ ½ 0 �0:0500 �, C¼�0:6 0:6

0:4 �0:4

� �, i¼ 1; 2:

Theorem 1 is concerned with the passivity of the stochastic jumping system with twosuccessive interval time-varying delays. For example 1, the maximum allowable upperdelay bounds that ensure passivity of system in Eqs. (14)–(16) are tabulated in Table 1 fortwo different cases. For different levels h1, t1 and given t2 ¼ 1:0 and g¼ 0:8, the maximumallowable h2 max are shown in Case I, while in Case II, we assume g¼ 1:0 and t2 ¼ 1:0, andthen the maximum allowable h2 max are obtained for different levels h1, t1. From the CasesI and II, we can find that, the larger the value of the range of the time delay tiðtÞ is, the

Table 1

Theorem 1 m1 ¼ 0:2,m2 ¼ 0:6 m1 ¼ 0:4,m2 ¼ 0:3 m1 ¼ 0:4,m2 ¼ 0:5 m1 ¼ 0:8,m2 ¼ 0:6

Case I Case II Case I Case II Case I Case II Case I Case II

h1 ¼ 0:1,t1 ¼ 0:1 1.4428 1.4495 2.0201 2.0215 0.6695 0.6711 0.4064 0.4076

h1 ¼ 0:1,t1 ¼ 0:2 1.4846 1.4906 2.0485 2.0508 0.6806 0.6812 0.4112 0.4121

h1 ¼ 0:3,t1 ¼ 0:3 1.6042 1.6105 2.1289 2.1415 0.7299 0.7311 0.4417 0.4425

h1 ¼ 0:3,t1 ¼ 0:4 1.6401 1.6517 2.1546 2.1644 0.7512 0.7520 0.5128 0.5146


larger the maximum value h2 max of the time delays hiðtÞ is, when h1, m1 and m2 are given.For the fixed h1, t1 and m1 ¼ 0:4, h2 max decreases as increasing m2, and h2 max decreases asincreasing m1 for the fixed h1, t1 and m2 ¼ 0:6. Furthermore, for the fixed h1, t1, m1 and m2,the value of the maximum allowable h2 max in the Case I is smaller than that in the Case II,this means the value of g affects the maximum allowable upper delay bounds of time delayshiðtÞ. Hence, according to aforementioned discussion, it is unreasonable to combine thesesuccessive mode-dependent time delays together in some cases due to the fact that eachdelay is constrained to another one and it results from variable conditions. Therefore, it issignificant to investigate the systems with successive time delay components.

Example 2. Consider the uncertain stochastic jumping system in Eqs. (1)–(4) with twomodes with the following parameters:

A1 ¼�1:0166 0:2120

�0:4904 �0:5618

� �, A11 ¼

0:3462 0:1000

0:1228 �0:0896

� �, B11 ¼

0:0516 0:2188

0 �0:1026

� �,

Bv11 ¼0:5242

0:0962

� �, E1 ¼

�0:3012 0

0:1998 0:1212

� �, E11 ¼

0:1008 0:1301

0:1027 �0:0993

� �,

C1 ¼�0:8214 0

0:1988 0:1681

� �, C11 ¼

�0:2811 0

0:1216 0:4241

� �, Bv21 ¼

0:1131

�0:0899

� �,

B21 ¼0:0516 0:2188

0 �0:1026

� �, H1 ¼

�0:8019 0:0866

�0:2862 0:2016

� �,

H11 ¼�0:5011 0

0:1986 0:7012

� �, Bv31 ¼ 2:2811, D1 ¼ ½ 0:0542 0 �,

D11 ¼ ½�0:1065 0:1612 �, B31 ¼ ½ 0:1000 0:1000 �,

A2 ¼�1:6680 0:5026

0:4108 �1:4720

� �, A12 ¼

0:6885 0

0:0986 �0:1012

� �, B12 ¼

�0:1984 0

0:1251 0

� �,

Bv12 ¼�0:1984

0:1251

� �, E2 ¼

�0:4286 0

0:5980 0:2281

� �, E12 ¼

0:3017 0:3982

0:3061 �0:1024

� �,

C2 ¼0:1231 0:3000

0 �0:1021

� �, C12 ¼

0 0:5168

�0:3671 0:1013

� �, B22 ¼

0:2398 0

0:1124 0

� �,


Bv22 ¼0:1116

0:0218

� �, H2 ¼

0:1124 �0:1340

0:2310 �0:0121

� �, H12 ¼

�0:2861 0

0:6220 0:1000

� �,

D2 ¼ ½ 0:2196 0 �, D12 ¼ ½ 0:1984 0 �, B32 ¼ ½�0:1251 0 �, Bv32 ¼ 5:1984,

L1i ¼ 0:3281I , L2i ¼�0:025I , M1i ¼ 0:01I , M3i ¼�0:01I , M2i ¼0 0:01

0 0

� �,

M4i ¼0 0:01

�0:01 0

� �, M5i ¼

�0:01 0

0 0:01

� �, C¼

�0:8 0:8

0:2 �0:2

� �, i¼ 1; 2:

In this example, we assume h1 ¼ 0:1, h2 ¼ 0:282, t1 ¼ 0:1, t2 ¼ 0:3, m1 ¼ m2 ¼ 0:1, c11 ¼�0:8,c22 ¼�0:2, g¼ 1:0 and use Matlab LMI control Toolbox to solve the LMIs in Eq. (32).The desired Markovian switched dynamic output feedback controller can be obtained as inEqs. (28) and (29) with

Ak1 ¼�4:7229 �0:1453

0:4340 �0:9115

� �, Bk1 ¼

0:0648 0:3184

�0:2637 0:0015

� �,

Ck1 ¼14:4911 �4:4483

�6:6723 0:7608

� �,

Ak2 ¼�10:2024 4:6870

5:4270 �4:5590

� �, Bk2 ¼

�3:2897 �1:2163

1:5490 1:6683

� �,

Ck2 ¼0:9946 2:9660

�0:0190 0:9944

� �:

6. Conclusions

In this paper, we have studied the problems of passivity analysis and passivity-basedcontrol for a class of uncertain stochastic jumping systems with mode-dependent round-trip time-varying delays. By using a novel Markovian switched Lyapunov functional andsome useful slack Markovian switched matrix variables, a delay-dependent condition forthe solvability of this problem has been presented. The desired Markovian switcheddynamic output feedback controller, which ensures that the resulting close loop system ispassive, has been constructed by solving a set of LMIs. Two numerical examples have beenprovided to illustrate the effectiveness of the proposed approach.

Acknowledgments

This work was supported by the Natural Science Foundation of PR China under Grants61074043 and 60904061, the Natural Science Foundation of Jiangsu Province under GrantBK2010493, and the Qing Lan Project.


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