Hindawi Publishing Corporation
Journal of Applied Mathematics
Volume 2012, Article ID 950590, 24 pages
doi:10.1155/2012/950590
Research Article
Exponential Passification of
Markovian Jump Nonlinear Systems with
Partially Known Transition Rates
Mengzhuo Luo1 and Shouming Zhong1, 2
1
School of Mathematical Sciences, University of Electronic Science and Technology of China, Chengdu,
Sichuan 611731, China
2
Key Laboratory for Neuroinformation of Ministry of Education, University of Electronic Science and
Technology of China, Chengdu, Sichuan 611731, China
Correspondence should be addressed to Mengzhuo Luo, zhuozhuohuahua@163.com
Received 24 August 2011; Accepted 28 November 2011
Academic Editor: Ying U. Hu
Copyright q 2012 M. Luo and S. Zhong. This is an open access article distributed under the
Creative Commons Attribution License, which permits unrestricted use, distribution, and
reproduction in any medium, provided the original work is properly cited.
The problems of delay-dependent exponential passivity analysis and exponential passification of
uncertain Markovian jump systems MJSs with partially known transition rates are investigated.
In the deterministic model, the time-varying delay is in a given range and the uncertainties are
assumed to be norm bounded. With constructing appropriate Lyapunov-Krasovskii functional
LKF combining with Jensen’s inequality and the free-weighting matrix method, delay-dependent
exponential passification conditions are obtained in terms of linear matrix inequalities LMI.
Based on the condition, desired state-feedback controllers are designed, which guarantee that the
closed-loop MJS is exponentially passive. Finally, a numerical example is given to illustrate the
effectiveness of the proposed approach.
1. Introduction
In recent years, more and more attention has been devoted to the Markovian jump systems
since they are introduced by Krasovskii and Lidskii 1. It is known that systems with
Markovian jump parameters are a set of systems with transition among the models governed
by a Markov chain taking values in a finite set. They have the character of stochastic hybrid
systems with two components in the state. The first one refers to the mode which is described
by a continuous-time finite-state Markov process, and the second one refers to the state
which is represented by a system of differential equations. Markovian jump systems have
got the virtue of modeling the abrupt phenomena such as random failures and repairs of the
components changes in the interconnections of subsystems, sudden environment changes,
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Journal of Applied Mathematics
and so forth, which often takes place in many dynamical systems 2–4. So due to extensive
applications of such systems in manufacturing systems, power systems, communication
systems, and network-based control systems, recently, many works have been reported
about MJSs, which including filtering problems 5–7, stability analysis problems 8–12, and
control problems 13–20, and so forth.
However, the aforementioned references almost considered that the transition
probabilities are known exactly. In some practical applications, the mode information is
transmitted through unreliable networks, it may be lost or observed simultaneously. That
means the systems mode is neither totally accessible or inaccessible. So the ideal assumption
on the transition probabilities inevitably limits the application of the traditional Markovian
jump systems theory. Therefore, whether in theory or in practice, it is necessary to further
consider more general systems with partially mode information 21–27.
Recently, the passivity problems for a variety of practical systems have been attracting
renewing attention 28–31. The passivity theory was first proposed in the circuit analysis
32 so it has played an efficient role in both electrical network and nonlinear control systems.
The main point of passivity theory is that the passive properties of system can keep the
system internal stability. Thus, the passivity theory provides a nice tool for analyzing the
stability of a nonlinear system, and the passivity analysis has received a lot of attention and
has found applications in diverse areas such as signal processing, complexity, chaos control
and synchronization, and fuzzy control 33–38. In 33 authors dealt with global robust
passivity analysis for stochastic interval neural networks with interval time varying delays
and Markovian jumping parameter; in 34 both delay-independent and delay-dependent
stochastic passivity conditions are presented for uncertain neural networks; in 35–37
authors discussed the robust passivity and passification of Markovian jump systems and
fuzzy time-delay systems; in 38, the exponential passivity of neural networks with timevarying are studied and the results are extended to two types of uncertainties.
In practice, input delays are often encountered in control systems because of the
transmission of measurement information. Especially, in networked control systems, sensors
controllers, and plants are often connected by a net medium hence it is quite meaningful to
study the effect of the input delay in the design of controllers. However, to the best of the
authors’ knowledge providing less conservative delay-dependent exponential passification
criteria for uncertain MJS with input delays and partially known transition rates to desired
performance are still open problems.
Motivated by this observation, in this paper, we study the exponential passification
problem of nonlinear Markovian jump systems with partially known transition rates,
including state and input delays, the aim of this problem is to design a controller such that
the resulting closed-loop systems satisfy a certain passivity performance index. Comparing
with the large amount of the literature on the analysis of stability of Markovian jump systems,
passivity analysis and passification for these systems have many obvious advantages. Thus,
research in this area should be of both theoretical and practical importance, which motivates
us to carry out the present work. Based on the LKF theory and the free-weighting matrix
method, some desired exponentially passification controllers are designed, which guarantee
that the closed-loop MJS is exponential passive. Finally, a numerical example is used to
illustrate the designed method.
Notations. The notations are quite standard. Throughout this letter Rn and Rn×m denote, resp.,
the n-dimensioned Euclidean space and the set of all n × m real matrices. The notation X ≥ Y
resp., X > Y means that X and Y are symmetric matrices, and that X − Y is positive
Journal of Applied Mathematics
3
semidefinitive resp., positive definite. · is the Euclidean norm in Rn . I is the identity
matrix with compatible dimension. If A is a matrix, λmax A respective λmin A means
the largest respective smallest eigervalue of A. Moreover, let Ω, F, Ft t≥0 , P be a complete
probability space with a filteration. Ft t≥0 satisfies the usual conditions i.e, the filtration
contains all P -null sets and is right continuous. E{·} stands for the mathematical expectation
operator with respect to the given probability measure. Denote by L2F0 −τ 2 , 0 : Rn the
family of all F0 measurable C−τ 2 , 0 : Rn -valued random variables ϕ {ϕs : −τ 2 ≤ s ≤ 0}
such that sup−τ 2 ≤s≤0 Eϕs2 < ∞. The asterisk ∗ in a matrix is used to denote term that is
induced by symmetry. Matrices, if not explicitly specified, are assumed to have appropriate
dimensions. Sometimes, the arguments of function will be omitted in the analysis when no
confusion can be arised.
2. Problem Formulation and Preliminaries
Consider the following uncertain MJS with time-varying delays
ẋt At, rt xt Ad t, rt xt − τt, rt B1 t, rt ut E1 t, rt ut − τt, rt D0 rt fxt, rt D1 rt ωt.
zt Ct, rt xt Cd t, rt xt − τt, rt B2 t, rt ut E2 t, rt ut − τt, rt D2 rt ωt,
2.1
2.2
Here xt ∈ Rn is the state vector, ut ∈ Rp is the control input, zt ∈ Rq is the control output,
and ωt ∈ Rl is the exogenous disturbance input which belongs to L2 0, ∞, {rt , t ≥ 0} is a
homogenous finite-state Markov process with right continuous trajectories, which takes value
in a finite-state space S {1, 2, . . . , N} with generator Π {πij }, i, j ∈ S and has the mode
transition probabilities
Pr rt
Δt j | rt i πij Δt oΔt
1 πii Δt oΔt
i
/ j,
i j,
2.3
where Δt > 0, limΔt → 0 oΔt/Δt 0, πij is the transition rete from i to j, and
πii −
πij ,
j/
i
πij ≥ 0, j / i.
2.4
For notational simplicity, which rt i, i ∈ S, the matrices At, rt , Ad t, rt ,
B1 t, rt , E1 t, rt , Ct, rt , Cd t, rt , B2 t, rt , E2 t, rt , D0 rt , D1 rt , and D2 rt will be
described by Ai t, Adi t, B1i t, E1i t, Ci t, Cdi t, B2i t, E2i t, D0i , D1i , and D2i . We
denote that
Ai t Ai ΔAi t,
E1i t E1i ΔE1i t,
Adi t Adi ΔAdi t,
Ci t Ci ΔCi t,
B2i t B2i ΔB2i t,
B1i t B1i ΔB1i t,
Cdi t Cdi ΔCdi t,
E2i t E2i ΔE2i t,
2.5
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Journal of Applied Mathematics
where Ai , Adi , B1i , E1i , Ci , Cdi , B2i , E2i , and D0i , D1i , D2i are known constant matrices with
appropriate dimensions. In this paper, the transition rates of Markov chain are partially
known, that is, some elements in matrix Π are unknown. We denote that
i
j : if πij is know
Ikn
i
Iuk
j : if πij is unknow
2.6
i
i
i
moreover, if Ikn
{k1i , k2i , . . . , km
}, 1 ≤ m ≤ N − 2.
/ ∅, it is further described as Ikn
Remark 2.1. kli ∈ N , l ∈ {1, 2, . . . , m} represents the index of the lth known element in the ith
row of transition rate matrix. The case m N − 1 is excluded, which means if we have only
one unknown element, one can naturally calculate it from the known elements in each row
and the transition rate matrix property.
Now the mode-dependent state-feedback controller is taken to be as follows:
ut Ki xt,
2.7
then, the closed-loop MJS can be represented as
ẋt Ai t B1i tKi xt Adi t E1i tKi xt − τi t D0i fxt, i D1i ωt,
zt Ci t B2i tKi xt Cdi t E2i tKi xt − τi t D2i ωt.
2.8
Before proceeding further, we will introduce the following assumptions, definition and
some lemmas which will be used in the next section.
Assumption 1. The uncertain parameters are assumed to be of the form:
ΔAi t ΔAdi t ΔB1i t ΔE1i t
ΔCi t ΔCdi t ΔB2i t ΔE2i t
T1i
F t N1i N2i N3i N4i ,
T2i i
2.9
where T1i , T2i , and Nki , k 1, 2, 3, 4, i ∈ S are known real constant matrices with appropriate
dimensions and Fi t, for all i ∈ S, are unknown time-varying matrix functions satisfying
FiT tFi t ≤ I.
2.10
Remark 2.2. It is assumed that all the elements Fi t, for all i ∈ S, are Lebesgue measurable.
The matrices ΔAi t, ΔAdi t, ΔB1i t, ΔE1i t, ΔCi t, ΔCdi t, ΔB2i t, and ΔE2i t are said to
be admissible if and only if both 2.9 and 2.10 hold. The parameter uncertainty structure as
in Assumption 1 is an extension of the so-called matching condition, which has been widely
used in the problems of control and robust filtering of uncertain linear systems.
Assumption 2. The time-varying delay τi t satisfies 0 ≤ τ1i ≤ τi t ≤ τ2i , τ̇i t ≤ μi , with τ1i , τ2i ,
and μi being real constant scalars for each for all i ∈ S.
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5
Assumption 3. For a fixed system mode rt i ∈ S, there exists a know real constant modedependent matrix Γi diagk1i , k2i , . . . , kni > 0 such that the nonlinear vector function f·, ·
satisfy the following conditions:
f T xt, i fxt, i − Γi xt ≤ 0.
2.11
Definition 2.3 see 39. The MJS 2.8 is said to be passive if there exists a constant δ such
that
T
2E
≥δ
z tωtdt
T
2.12
0
holds for all T ≥ 0.
Definition 2.4. The MJS 2.8 is said to be exponentially passive from input ωt to output zt,
if there exists an exponential Lyapunov function or called the exponential storage funtion
V defined on Rn , and positive scalars ρ, γ such that for all ωt, all initial conditions x0, all
t ≥ 0, the following inequality holds:
LV xt , rt ρV xt , rt − γωT tωt ≤ 2zT tωt.
2.13
Remark 2.5. From Definition 2.4, if ρ 0, then the MJS in the form 2.8 is passive, in other
words, exponential passivity implies passivity. It follows from 2.13 that
T
2E
z tωtdt
T
≥ −E{V x0 } − γE
T
0
ω tωtdt
T
δ.
2.14
0
Then from Definition 2.3, we can see that MJS 2.8 is passive. But the converse does not
necessarily hold, that is, we can not obtain the exponential passive if systems are passive.
Lemma 2.6 see 36. Let Qx QT x, Rx RT x, and Sx depend affinely on x. Then the
following linear matrix inequality:
Qx Sx
>0
ST x Rx
2.15
holds if and only if one of the following conditions holds:
1 Rx > 0, Qx − SxR−1 xST x > 0;
2 Qx > 0, Rx − ST xQ−1 xSx > 0.
Lemma 2.7 see 40. Let A, D, S, F, and P be real matrices of appropriate dimensions with P > 0
and F satisfy F T tFt ≤ I. Then the following statement holds.
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Journal of Applied Mathematics
1 For any scalar ε > 0
DFS DFST ≤ ε−1 DDT εST S.
2.16
2 For any vectors x and y with appropriate dimensions
2xT ADy ≤ xT AP AT x yT DT P −1 Dy.
2.17
Lemma 2.8 see 41. Let A, X be real matrices with appropriate dimensions. Then there exist a
matrix P P T > 0 such that P AT AP X < 0, if and only if, there exists a scalar ε > 0 and Z such
that
⎡
⎤
−Z − ZT ZT AT P ZT
⎣
∗
−ε−1 P X 0 ⎦ < 0.
∗
∗
−εP
2.18
3. Main Results
3.1. Exponential Passivity Analysis
In this section, we assumed the transition rates are partially known and given the statefeedback controller gain matrix Ki , i ∈ S, at first, we will present a sufficient condition, which
guarantees the MLS 2.8 is exponential passive.
Theorem 3.1. Given the state-feedback controller gain matrix Ki , the uncertain MJS 2.8
is exponentially passive in the sense of expectation if there exists positive definite matrices
Pi , Qi , Q1 , Q2 , Q3 , Q∗ , Z1 , Z2 , positive scalars γ, ε1i , ε2i , and for any matrices Gi , Mi , Ri , Ui , Vi , Hi
with appropriate dimensions such that the following matrices inequalities hold for all i 1, 2, . . . , N:
⎛ 1
⎞
Ωi,9 × 9 Λ1 Λ2 Λ3 Λ4k Λ5k
Λ6
Λ7
⎜ ∗
−Z2 0
0
0
0
2τ 2 Z2 T1i ε2i 0 ⎟
⎜
⎟
⎜ ∗
0
0
0
0 ⎟
∗ −Z2 0
⎜
⎟
⎜
⎟
0
0
0 ⎟
∗
∗ −Z2 0
⎜ ∗
⎜
⎟<0
⎜ ∗
0
0 ⎟
∗
∗
∗ −Z2 0
⎜
⎟
⎜ ∗
0
0 ⎟
∗
∗
∗
∗ −Z2
⎜
⎟
⎝ ∗
0 ⎠
∗
∗
∗
∗
∗
−ε2i
∗
∗
∗
∗
∗
∗
∗
−ε2i
k 1, 2.
3.1
i
Case 1. If πii ∈ Ikn
πii Qi − Q∗ Qj
∗
−Qj
i
∀j∈Iuk
< 0,
3.2
Journal of Applied Mathematics
⎞
⎛⎛
⎜⎝1 πij ⎠πii Qi − Q∗ πik1i Qk1i
⎜
⎜
i
j∈I
kn
⎜
⎜
∗
−Qk1i
⎜
⎜
⎜
⎝
∗
∗
∗
∗
Pi Ai ATi Pi Pj < 0
7
···
0
..
.
∗
⎞
πikmi Qkmi ⎟
⎟
⎟
⎟
⎟ < 0,
0
⎟
⎟
⎟
⎠
0
i
−Qkm
3.3
3.4
i
∀j ∈ Iuk
.
i
Case 2. If πii ∈ Iuk
Qj − Q∗ > 0
3.5
i
∀j ∈ Iuk
, j i,
i
Qj − Q∗ < 0 ∀j ∈ Iuk
, j/
i,
⎞
⎛⎛
⎞
⎜⎝−1 −
πij ⎠Q∗
πik1i Qk1i · · ·
πikmi Qkmi ⎟
⎜
⎟
⎜
⎟
i
j∈I
kn
⎜
⎟
⎜
⎟ < 0,
∗
−Qk1i
0
0
⎜
⎟
⎜
⎟
..
⎜
⎟
⎝
⎠
.
∗
∗
0
∗
∗
∗
⎛⎛
Pi ⎝ ⎝ 1 3.7
−Qkmi
Pi Ai ATi Pi Pj > 0
i
∀j ∈ Iuk
, j i,
3.8
Pi Ai ATi Pi Pj < 0
i
∀j ∈ Iuk
, j/
i,
3.9
where
Ω1i,11
3.6
⎞
⎞
⎛⎛
πij ⎠Ai B1i Ki ⎠ ⎝⎝1 i
j∈Ikn
⎞
⎞T
πij ⎠Ai B1i Ki ⎠ Pi
i
j∈Ikn
πij Pj Qi τ2i Q∗ Q1 Q2 Q3 τ2i − τ1i Z1 GT1i G1i ,
i
j∈Ikn
Ω1i,12 −G1i GT2i M1i ,
Ω1i,13 R1i GT3i − M1i ,
Ω1i,14 −R1i GT4i U1i Pi Adi E1i Ki ,
Ω1i,16 −V1i GT6i H1i ,
Ω1i,17 GT7i − H1i
Ω1i,19 Pi D1i − Ci B2i Ki T ,
T
− M2i ,
Ω1i,23 −GT3i R2i M3i
T
− U2i ,
ptΩ1i,25 −GT5i V2i M5i
Ω1i,15 V1i GT5i − U1i ,
Ω1i,18 GT8i ε1i Γi Pi D0i ,
T
Ω1i,22 −GT2i − G2i M2i
M2i − Q1 ,
T
Ω1i,24 −GT4i − R2i M4i
U2i ,
T
Ω1i,26 −GT6i − V2i M6i
H2i ,
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Journal of Applied Mathematics
T
− H2i ,
Ω1i,27 −GT7i M7i
T
Ω1i,28 −GT8i M8i
,
T
− M3i ,
Ω1i,33 RT3i R3i − M3i
T
Ω1i,34 RT4i − R3i − M4i
U3i ,
T
Ω1i,35 RT5i V3i − M5i
− U3i ,
T
Ω1i,36 RT6i − V3i − M6i
H3i ,
T
− H3i ,
Ω1i,37 RT7i − M7i
T
Ω1i,38 RT8i − M8i
T
Ω1i,44 −RT4i − R4i U4i
U4i − 1 − μi Qi ,
T
H4i ,
Ω1i,46 −RT6i − V4i U6i
Ω1i,39 0 ,
T
Ω1i,45 −RT5i V4i U5i
− U4i ,
T
Ω1i,47 −RT7i U7i
− H4i ,
Ω1i,49 −Cdi E2i Ki T ,
T
,
Ω1i,48 −RT8i U8i
T
Ω1i,55 V5iT V5i − U5i
− U5i ,
T
Ω1i,57 V7iT − U7i
− H5i ,
T
Ω1i,56 −U6i
− V5i V6iT H5i ,
T
Ω1i,58 V8iT − U8i
Ω1i,66 −V6iT − V6i H6iT H6i − Q2 ,
Ω1i,68 −V8iT H8iT ,
Ω1i,78 −H8iT ,
Ω1i,29 0,
Ω1i,67 −V7iT H7iT − H6i ,
Ω1i,69 0,
Ω1i,79 0,
T
Ω1i,99 −D2i − D2i
− γI,
Ω1i,59 0,
Ω1i,77 −H7iT − H7i − Q3 ,
Ω1i,88 −2ε1i I,
τ 2 max{τ2i },
Ω1i,89 0,
τ 1 min{τ1i },
i∈S
i∈S
T
2τ 2 Z2 Ai B1i Ki , 0, 0, 2τ 2 Z2 Adi E1i Ki , 0, 0, 0, 2τ 2 Z2 D0i , 2τ 2 Z2 D1i ,
Λ1 Λ1 t 2τ 2 Z2 Ai t B1i tKi , 0, 0, 2τ 2 Z2 Adi t E1i tKi , 0, 0, 0, 2τ 2 Z2 D0i ,
T
2τ 2 Z2 D1i ,
Λ2 τ 2 − τ2i Hi ,
√
τ1i Gi ,
Λ41 τ2i − τ1i
Ri ,
2
T
Λ6 ε2i T1iT Pi , 0, 0, 0, 0, 0, 0, 0, −ε2i T2iT ,
Λ42 τ2i − τ1i
Ui ,
2
Λ3 Λ51 τ2i − τ1i
Mi ,
2
τ2i − τ1i
Vi ,
Λ52 2
Λ7 N1i N3i Ki , 0, 0, N2i N4i Ki , 0, 0, 0, 0, 0T ,
T
Gi GT1i GT2i GT3i GT4i GT5i GT6i GT7i GT8i 0
T
T
T
T
T
T
T
T
T
Mi M1i
M2i
M3i
M4i
M5i
M6i
M7i
M8i
0 ,
T
Ri RT1i RT2i RT3i RT4i RT5i RT6i RT7i RT8i 0 ,
T
T
T
T
T
T
T
T
T
Ui U1i
U2i
U3i
U4i
U5i
U6i
U7i
U8i
0 ,
Journal of Applied Mathematics
9
T
Vi V1iT V2iT V3iT V4iT V5iT V6iT V7iT V8iT 0 ,
T
Hi H1iT H2iT H3iT H4iT H5iT H6iT H7iT H8iT 0 ,
3.10
Proof. First, in order to cast our model involved in the framework of the Markov process,
we define a new process xt s xt s, s ∈ −τ 2 , 0, and let L be the weak infinitesimal
generator of the random process xt s, t ≥ 0 and
Lvxt , rt lim
Δ→0
1
{Evxt
Δ , rt
Δ | xt , rt i − vxt , rt }.
Δ
3.11
Now consider the Lyapunov-Krasovskii functional as follows for rt i, i ∈ 1, 2, . . . , S:
vxt , i v1 xt , i v2 xt , i v3 xt , i v4 xt , i v5 xt , i,
3.12
where
v1 xt , i xT tP ixt,
v3 xt , i v4 xt , i v5 xt , i 0
−τ2i
t
t−τ1i
−τ1i
−τ2i
t
v2 xt , i t−τi t
xT sQixsds,
xT sQ∗ xsds dθ,
t
θ
x sQ1 xsds T
t
t
t
x sQ2 xsds T
t−τ2i
x sZ1 xsds dθ 2
T
t
θ
0 t
−τ 2
t
t−τ 2
3.13
x sQ3 xsd,
T
ẋT sZ2 ẋsds dθ,
t
θ
where
N
πij Qj ≤ Q∗ .
3.14
j1
In order to show the exponential passivity of the MJS 2.8 under the given controller gain
matrix Ki , we set
J ∗ Lvxt , i − γωT tωt − 2zT tωt.
3.15
10
Journal of Applied Mathematics
Notice that
N
Lv1 xt , i xT t Pi Ai t B1i tKi Ai t B1i tKi T Pi xt xT t πij Pj xt
j1
2x tPi Adi t E1i tKi xt − τi t 2x tPi D0i fxt, i 2xT tPi D1i ωt,
t
N
T
T
xT s πij Qj xsds,
Lv2 xt , i ≤ x tQi xt − 1 − μi x t − τi tQi xt − τi t T
T
t−τi t
t
j1
Lv3 xt , i ≤ τ2i xT tQ∗ xt − t−τi t xT sQ∗ xsds ,
Lv4 xt , i xT t Q1 Q2 Q3 xt − xT t − τ1i Q1 xt − τ1i − xT t − τ2i Q2 xt − τ2i − xT t − τ 2 Q3 xt − τ 2 ,
Lv5 xt , i τ2i − τ1i xT tZ1 xt 2τ 2 ẋT tZ2 ẋt −
−2
ẋT sZ2 ẋsds
τ2i − τ1i xT tZ1 xt 2τ 2 ẋT tZ2 ẋt −
t−τi t
τ2i /2
ẋT sZ2 ẋsds −
t−τ2i
−
−
xT sZ1 xsds
t−τ2i
t
t−τ 2
−
t−τ1i
t−τi t
τ1i /2
t−τi t
t
ẋ sZ2 ẋsds −
T
ẋT sZ2 ẋsds −
t−τ1i
t−τ1i
t−τ2i
ẋT sZ2 ẋsds
t−τ 2
t−τi t
t−τi t
τ2i /2
t−τ1i
t−τi t
τ1i /2
ẋT sZ2 ẋsds
ẋT sZ2 ẋsds
xT sZ1 xsds −
t
ẋT sZ2 ẋsds.
t−τ 2
t−τ2i
3.16
Then using Newton-Leibniz formula, for any matrices Hi , Gi , Mi , Ri , Ui , Vi we have
2ξ tGi xt − xt − τ1i −
T
t
0,
ẋsds
t−τ1i
2ξ tMi
T
τ1i τi t
xt − τ1i − x t −
2
−
t−τ1i
t−τ1i τi t/2
ẋsds
0,
t−τ1i τi t/2
τ1i τi t
− xt − τi t −
2ξ tRi x t −
ẋsds 0,
2
t−τi t
t−τi t
τ2i τi t
T
2ξ tUi xt − τi t − x t −
ẋsds 0,
−
2
t−τ2i τi t/2
T
Journal of Applied Mathematics
t−τ2i τi t/2
τ2i τi t
T
2ξ tVi x t −
ẋsds 0,
− xt − τ2i −
2
t−τ2i
t−τ
2i
2ξT tHi xt − τ2i − xt − τ 2 −
ẋsdt
11
0,
t−τ 2
3.17
where
ξT t τi t τ1i
τi t τ2i
, xT t − τi t, xT t −
,
xT t, xT t − τ1i , xT t −
2
2
xT t − τ2i xT t − τ 2 , f T xt, i, ωT t .
3.18
From the Lemma 2.7 2.2, it is easy to see that
−2ξ tGi
T
t
ẋsds ≤ τ1i ξ
t−τ1i
T
tGi Z2−1 GTi ξt
t
ẋT sZ2 ẋsds
t−τ1i
t−τ1i
τi t − τ1i T
ξ tMi Z2−1 MiT ξt ẋT sZ2 ẋsds
2
t−τ1i τi t/2
t−τ1i τi t/2
t−τ1i τi t/2
t−τ1i τi t/2
τi t − τ1i T
−2ξT tRi
ξ tRi Z2−1 RTi ξt ẋsds ≤
ẋT sZ2 ẋsds
2
t−τi t
t−τi t
t−τi t
t−τi t
−
τ
τ
t
2i
i
−2ξT tUi
ξT tUi Z2−1 UiT ξt ẋsds ≤
ẋT sZ2 ẋsds
2
t−τ2i τi t/2
t−τ2i τi t/2
t−τ2i τi t/2
t−τ2i τi t/2
τ2i − τi t T
T
−1 T
−2ξ tVi
ξ tVi Z2 Vi ξt ẋsds ≤
ẋT sZ2 ẋsds
2
t−τ2i
t−τ2i
t−τ2i
t−τ2i
ẋsds ≤ τ 2 − τ2i ξT tHi Z2−1 HiT ξt ẋT sZ2 ẋsds.
−2ξT tHi
−2ξT tMi
t−τ1i
ẋsds ≤
t−τ 2
t−τ 2
3.19
Now by Assumption 3, it can be deduced that for any positive scalar ε1i , i ∈ 1, 2, . . . , S,
2ε1i f T xt, i Γi xt − fxt, i ≥ 0.
Then from the above discussion, we can see that
t−τ1i
τi t − τ1i
τ2i − τi t
Φ1 t Φ2 t ξt −
xT sZ1 xsds
J ≤ ξ t
τ2i − τ1i
τ2i − τ1i
t−τ2i
t
ẋT sZ2 ẋsds,
−
∗
T
t−τ 2
3.20
12
Journal of Applied Mathematics
Φ1 t Ωi,9 × 9 t Λ1 tZ2−1 Λ1 tT τ 2 − τ2i Hi Z2−1 HiT τ1i Gi Z2−1 GTi
τ2i − τ1i Mi Z2−1 MiT Ri Z2−1 RTi ,
2
Φ2 t Ωi,9 × 9 t Λ1 tZ2−1 Λ1 tT τ 2 − τ2i Hi Z2−1 HiT τ1i Gi Z2−1 GTi
τ2i − τ1i Ui Z2−1 UiT Vi Z2−1 ViT ,
2
3.21
where
Ωi,11 t Pi Ai t B1i tKi Ai t B1i tKi T Pi
N
πij Pj Qi τ2i Q∗ Q1 Q2 Q3 τ2i − τ1i Z1 GT1i G1i ,
j1
Ωi,14 t Pi Adi t E1i tKi − R1i GT4i U1i
Ωi,19 t Pi D1i − Ci t B2i tKi T ,
Ωi,49 t −Cdi t E2i tKi T
3.22
other terms of Ωi,i×j t are similar to Ω1i,i×j . In order to get our results, we will describe that
the Φ1 t < 0 and Φ2 t < 0.
By the Schur complement, Φ1 t < 0 and Φ2 t < 0 under the restriction of 3.14 if and
only if
⎛
⎞
Ωi,9 × 9 Λ1 Λ2 Λ3 Λ4k Λ5k
⎜ ∗
−Z2 0
0
0
0 ⎟
⎜
⎟
⎜ ∗
0
0 ⎟
∗ −Z2 0
⎜
⎟
⎜
⎟
⎜ ∗
0 ⎟
∗
∗ −Z2 0
⎜
⎟
⎝ ∗
∗
∗
∗ −Z2 0 ⎠
∗
∗
∗
∗
∗ −Z2 k1,2
T
T1iT Pi , 0n×7n , −T2iT , 2τ 2 T1iT Z2 , 0n×4n Fi kN1i N3i Ki , 0n×2n , N2i N4i Ki , 0n×10n N1i N2i Ki , 0n×2n , N2i N4i Ki , 0n×10n T FiT k T1iT Pi , 0n×7n ,−T2iT , 2τ 2 T1iT Z2 , 0n×4n < 0,
3.23
where Ωi,9 × 9 is the nominal matrix of Ωi,9 × 9 t. Then from the Lemma 2.7 2.1, above matrix
inequality holds, which is equivalent to
⎞
Ωi,9×9 Λ1 Λ2 Λ3 Λ4k Λ5k Λ6
Λ7
⎜ ∗
−Z2 0
0
0
0
2τ 2 Z2 T1i ε2i 0 ⎟
⎟
⎜
⎜ ∗
0
0
0
0 ⎟
∗ −Z2 0
⎟
⎜
⎟
⎜
0
0
0 ⎟
∗
∗ −Z2 0
⎜ ∗
< 0.
⎟
⎜
⎜ ∗
0
0 ⎟
∗
∗
∗ −Z2 0
⎟
⎜
⎜ ∗
0
0 ⎟
∗
∗
∗
∗ −Z2
⎟
⎜
⎝ ∗
0 ⎠
∗
∗
∗
∗
∗
−ε2i
∗
∗
∗
∗
∗
∗
∗
−ε2i k1,2
⎛
3.24
Journal of Applied Mathematics
13
i
then 3.24 is equivalent to
Case 1. If πii ∈ Ikn
⎞
Ω1i,9 × 9 Λ1 Λ2 Λ3 Λ4k Λ5k
Λ6
Λ7
⎜ ∗
−Z2 0
0
0
0
2τ 2 Z2 T1i ε2i 0 ⎟
⎟
⎜
⎜ ∗
0
0
0
0 ⎟
∗ −Z2 0
⎟
⎜
⎟
⎜
0
0
0 ⎟
∗
∗ −Z2 0
⎜ ∗
⎟
⎜
⎜ ∗
0
0 ⎟
∗
∗
∗ −Z2 0
⎟
⎜
⎜ ∗
0
0 ⎟
∗
∗
∗
∗ −Z2
⎟
⎜
⎝ ∗
0 ⎠
∗
∗
∗
∗
∗
−ε2i
∗
∗
∗
∗
∗
∗
∗
−ε2i k1,2
⎛
⎛
⎞
3.25
⎛
⎞
15
15
⎜
⎜ ! "⎟
! "⎟
⎜
⎟
⎜
⎟
diag⎜
πij Pj , 0, . . . , 0⎟ diag⎜
πij Pi Ai ATi Pi , 0, . . . , 0⎟ < 0.
⎝ i
⎠
⎝ i
⎠
j∈Iuk
j/
i
j∈Iuk
j/
i
Obviously, we can see that if 3.1 and 3.4 hold, then Φ1 t < 0 and Φ2 t < 0 under the
restriction of 3.14. Next we will further consider the equivalent form of 3.14.
#N
∗
j1 πij Qj < Q is equivalent to
i
j∈Ikn
j/
i
πij Qj πij Qj πii Qi − Q∗ i
j∈Iuk
j/
i
πij πii Qi − Q∗ i
j∈Ikn
πij πii Qi − Q∗ < 0.
i
j∈Iuk
3.26
If we have the following matrix inequalities hold, we can have that 3.14 is satisfied
⎛
⎝1 ⎞
πij ⎠πii Qi − Q∗ i
j∈Ikn
πii Qi − Q∗ Qj < 0
πij Qj < 0,
i
j∈Ikn
j/
i
i
j ∈ Iuk
.
Obviously, 3.27 is equivalent to 3.2 and 3.3 by the Schur complement.
3.27
14
Journal of Applied Mathematics
i
then 3.24 is equivalent to
Case 2. If πii ∈ Iuk
⎛
⎞
Ω1i,9 × 9 Λ1 Λ2 Λ3 Λ4k Λ5k
Λ6
Λ7
⎜ ∗
−Z2 0
0
0
0
2τ 2 Z2 T1i ε2i 0 ⎟
⎜
⎟
⎛
⎞
⎜ ∗
0
0
0
0 ⎟
∗ −Z2 0
⎜
⎟
15
⎜
⎟
⎜
! "⎟
0
0
0 ⎟
∗
∗ −Z2 0
⎜ ∗
⎜
⎟
diag⎜
πij Pj , 0, . . . , 0⎟
⎜
⎟
⎜ ∗
⎟
⎝ i
⎠
0
0
0
∗
∗
∗
−Z
2
⎜
⎟
j∈Iuk
⎜ ∗
⎟
0
0
∗
∗
∗
∗
−Z
j
i
2
/
⎜
⎟
⎝ ∗
0 ⎠
∗
∗
∗
∗
∗
−ε2i
∗
∗
∗
∗
∗
∗
∗
−ε2i k1,2
⎛
⎞
⎛
⎞
15
15
⎜ ! "⎟
!
"
⎜
⎜
⎟
⎟
diag⎜
π P A ATi Pi , 0, . . . , 0⎟ πii diag⎝Pi Ai ATi Pi Pi , 0, . . . , 0⎠ < 0.
⎝ i ij i i
⎠
j∈Iuk
j/
i
3.28
Then if 3.1, 3.8, and 3.9 hold, then Φ1 t < 0 and Φ2 t < 0 under the restriction of 3.14,
furthermore, with the similar consideration, we can deduce that if 3.5–3.7 are established,
then 3.14 is founded. So there exists a positive scalar ρ1 , then
J ∗ ≤ −ρ1 xt2 − λmin Z1 t
xs2 ds − λmin Z2 t−τ 2
t
ẋs2 ds.
3.29
t−τ 2
On the other hand, it is easy to obtain that
$ $ $ $ $ $ $ $ $ $ $ $
vxt, i ≤ P xt2 Q Q∗ $Q1 $ $Q2 $ $Q3 $ τ 2 − τ 1 Z1 t
t
×
xs2 ds 2τ 2 Z2 ẋs2 ds,
t−τ 2
3.30
t−τ 2
where P maxi∈S {Pi }, Q maxi∈S {Qi }.
Let ρ > 0 be sufficiently small such that
ρP − ρ1 < 0,
$ $ $ $ $ $ $ $ $ $ $ $
ρ Q Q∗ $Q1 $ $Q2 $ $Q3 $ τ 2 − τ 1 Z1 − λmin Z1 < 0,
3.31
2τ 2 ρZ2 − λmin Z2 < 0.
So, by Definition 2.4, the MJS 2.8 is exponentially passive. This completes the proof.
Remark 3.2. It is easy to derive that the MJS 2.8 is exponential mean square stability with
ωt 0 if the MJS 2.8 is exponentially passive. Moreover, the result of Theorem 3.1 makes
use of the information of the subsystems upper bounds of the time varying delays, which
Journal of Applied Mathematics
15
may bring us less conservativeness, and from the free-weighting matrix and Newton-Leibnitz
formula, the upper bounds of μi are not restricted to be less than 1 in this paper. Therefore,
our result is more natural and reasonable to the Markovian jump systems.
Remark 3.3. In order to obtain the gain matrices Ki for convenience in the next section, 3.1 is
−1
and use the Lemma 2.7 2.1, we can obtain the equivalent
not LMI, if we substitute ε2i by ε2i
form of LMI.
3.2. Exponential Passification
In this section, we will determine the feedback controller gain matrices Ki , i ∈ S in 2.7,
which guarantee that the closed-loop MJS 2.8 is exponentially passive with partially known
transition rates.
Theorem 3.4. Given a positive constant ε, there exists a state-feedback controller in the form 2.7
such that the closed-loop MJS 2.8 is exponentially passive if there exist positive definite matrices
% 1, Q
% 2, Q
% 3 , Q∗ , Z 1 , Z 2 , positive scalar ε1i , ε2i , and for any matrices Gi , Mi , Ri , Ui , V i , H i , Zii
P i , Qi , Q
with appropriate dimensions satisfying the following LMIs under the two cases for all i 1, 2, . . . , N.
i
Case 1. If πii ∈ Ikn
πii Qi − Q
∗
⎛⎛
⎞
⎜⎝1 πij ⎠ πii Qi − Q
⎜
i
⎜
j∈Ikn
⎜
⎜
∗
⎜
⎜
⎜
⎝
∗
∗
⎛
1
Ω
Λ1
⎜ i,9 × 9
⎜ ∗
−Z
2
⎜
⎜ ∗
∗
⎜
⎜ ∗
∗
⎜
⎜ ∗
∗
⎜
⎜
∗
⎜ ∗
⎜
⎜ ∗
∗
⎜
⎝ ∗
∗
∗
∗
Λ2
0
Θ
∗
∗
∗
∗
∗
∗
∗
∗
Qj
−Qj
3.32
< 0,
i
∀j∈Iuk
πik1i Qk1i · · ·
−Qk1i
∗
∗
0
..
.
∗
Λ3 Λ4k Λ5k Λ6
Λ7
0 0
0
2τ 2 T1i ε2i 0
0 0
0
0
0
Θ 0
0
0
0
∗ Θ
0
0
0
∗ ∗
Θ
0
0
0
∗ ∗
∗
−ε2i
∗ ∗
∗
∗
−ε2i
∗ ∗
∗
∗
∗
⎞
πikmi Qkmi ⎟
⎟
⎟
⎟
⎟ < 0,
0
⎟
⎟
⎟
⎠
0
−Qkmi
⎞
Λ8
⎟
0⎟
⎟
0⎟
⎟
0⎟
⎟
0⎟
⎟<0
⎟
0⎟
⎟
0⎟
⎟
0⎠
Λ9
k 1, 2,
3.33
3.34
16
Journal of Applied Mathematics
⎛
⎞
−Zii − ZiiT ZiiT ATi P i ZiiT
0
⎜
0
Pi ⎟
∗
−ε−1 P i
⎜
⎟
< 0,
⎜
⎟
⎝
∗
∗
−εP i 0 ⎠
∗
∗
∗ −P j ∀j∈I i
3.35
uk
i
Case 2. If πii ∈ Iuk
∗
Qj − Q > 0
∗
Qj − Q < 0
⎛⎛
i
∀j ∈ Iuk
, j i,
3.36
i
∀j ∈ Iuk
, j
/ i,
3.37
⎞
∗ ⎜⎝−1 −
πij ⎠Q
πik1i Qk1i
⎜
i
⎜
j∈Ikn
⎜
⎜
∗
−Qk1i
⎜
⎜
⎜
⎝
∗
∗
∗
∗
⎛
2
Ω
Λ1
⎜ i,9 × 9
⎜ ∗
−Z
2
⎜
⎜ ∗
∗
⎜
⎜ ∗
∗
⎜
⎜ ∗
∗
⎜
⎜
∗
⎜ ∗
⎜
⎜ ∗
∗
⎜
⎝ ∗
∗
∗
∗
···
0
..
.
∗
⎞
πikmi Qkmi ⎟
⎟
⎟
⎟
⎟ < 0,
0
⎟
⎟
⎟
⎠
0
−Qkmi
⎞
Λ2 Λ3 Λ4k Λ5k Λ6
Λ7 Λ8
⎟
0 0 0
0
2τ 2 T1i ε2i 0
0⎟
⎟
Θ 0 0
0
0
0
0⎟
⎟
∗ Θ 0
0
0
0
0⎟
⎟
∗ ∗ Θ
0
0
0
0⎟
⎟<0
⎟
∗ ∗ ∗
Θ
0
0
0⎟
⎟
0
0⎟
∗ ∗ ∗
∗
−ε2i
⎟
∗ ∗ ∗
∗
∗
−ε2i 0 ⎠
∗ ∗ ∗
∗
∗
∗ Λ9
⎛
⎞
−Zii − ZiiT −ZiiT ATi P i ZiiT
⎜
⎟
< 0,
⎝
∗
−ε−1 P i − P j
0 ⎠
i
∗
∗
−εP i ∀j∈Iuk
k 1, 2,
3.38
3.39
3.40
ji
⎛
⎜
⎜
⎜
⎝
⎞
−Zii − ZiiT ZiiT ATi P i ZiiT
0
0
Pi ⎟
∗
−ε−1 P i
⎟
< 0,
⎟
∗
∗
−εP i 0 ⎠
∗
∗
∗ −P j ∀j∈Iuki
j
/i
3.41
Journal of Applied Mathematics
17
where
⎛⎛
1
Ωi,11
⎝⎝1 ⎞
⎛⎛
⎞
πij ⎠Ai P i B1i Yi ⎠ ⎝⎝1 i
j∈Ikn
⎞
⎞T
πij ⎠Ai P i B1i Yi ⎠ πii P i Qi τ2i Q
i
j∈Ikn
%1 Q
%2 Q
% 3 τ2i − τ1i Z 1 GT1i G1i ,
Q
⎛⎛
2
Ωi,11
⎝⎝1 ⎞
⎞
⎛⎛
πij ⎠Ai P i B1i Yi ⎠ ⎝⎝1 i
j∈Ikn
⎞
⎞T
∗
%1
πij ⎠Ai P i B1i Yi ⎠ Qi τ2i Q Q
i
j∈Ikn
T
%2 Q
% 3 τ2i − τ1i Z 1 G1i G1i ,
Q
1
2
1
2
1
2
1
2
1
2
1
2
T
Ωi,12 Ωi,12 −G1i G2i M1i ,
Ωi,14 Ωi,14
T
Ωi,13 Ωi,13 R1i G3i − M1i ,
1
2
T
T
−R1i G4i U1i Adi P i E1i Yi
Ωi,15 Ωi,15 V 1i G5i − U1i ,
T
Ωi,16 Ωi,16 −V 1i G6i H 1i ,
T
Ωi,17 Ωi,17 G7i − H 1i ,
T
Ωi,18 Ωi,18 G8i P i Γi D0i ε1i ,
T
1
2
Ωi,19 Ωi,19 D1i − Ci P i B2i Yi ,
1
2
T
T
% 1,
Ωi,22 Ωi,22 −G2i − G2i M2i M2i − Q
1
2
T
T
Ωi,24 Ωi,24 −G4i − R2i M4i U2i ,
1
2
T
T
Ωi,26 Ωi,26 −G6i − V 2i M6i H 2i ,
1
2
T
1
2
T
T
Ωi,34 Ωi,34 R4i − R3i − M4i U3i ,
1
2
T
T
Ωi,36 Ωi,36 R6i − V 3i − M6i H 3i ,
1
2
T
Ωi,23 Ωi,23 −G3i R2i M3i − M2i ,
Ωi,25 Ωi,25 −G5i V 2i M5i − U2i ,
1
2
T
T
1
2
T
T
1
T
Ωi,27 Ωi,27 −G7i M7i − H 2i ,
2
T
1
T
Ωi,28 Ωi,28 −G8i M8i ,
Ωi,33 Ωi,33 R3i R3i − M3i − M3i ,
Ωi,35 Ωi,35 R5i V 3i − M5i − U3i ,
1
2
T
T
1
2
T
T
1
T
Ωi,37 Ωi,37 R7i − M7i − H 3i ,
2
T
2
Ωi,29 Ωi,29 0,
1
T
Ωi,38 Ωi,38 R8i − M8i ,
2
Ωi,39 Ωi,39 0,
1
2
T
T
Ωi,44 Ωi,44 −R4i − R4i U4i U4i − 1 − μi Qi ,
1
2
T
T
1
2
T
T
1
2
T
1
2
T
1
2
T
Ωi,45 Ωi,45 −R5i V 4i U5i − U4i ,
1
Ωi,46 Ωi,46 −R6i − V 4i U6i H 4i ,
T
T
T
1
2
Ωi,49 Ωi,49 − Cdi P i E2i Yi ,
T
Ωi,48 Ωi,48 −R8i U8i ,
1
T
Ωi,55 Ωi,55 V 5i V 5i − U5i − U5i ,
T
2
Ωi,47 Ωi,47 −R7i U7i − H 4i ,
Ωi,57 Ωi,57 V 7i − U7i − H 5i ,
2
T
T
Ωi,56 Ωi,56 −U6i − V 5i V 6i H 5i ,
1
2
T
T
Ωi,58 Ωi,58 V 8i − U8i
1
2
Ωi,59 Ωi,59 0,
∗
18
Journal of Applied Mathematics
1
2
T
1
2
T
1
2
T
1
2
1
T
% 2,
Ωi,66 Ωi,66 −V 6i − V 6i H 6i H 6i − Q
1
T
Ωi,68 Ωi,68 −V 8i H 8i ,
2
Ωi,79 Ωi,79 0,
T
Ωi,99 Ωi,99 −D2i − D2i
− γ,
Λ1 Λ2 2
1
2
1
2
Ωi,88 Ωi,88 −2ε1i I,
T
T
1
2
Ωi,89 Ωi,89 0,
Θ J T Z2 J − J T P i − P i ,
T
2τ 2 Ai P i B1i Yi , 0, 0, 2τ 2 Adi P i E1i Yi , 0, 0, 0, 2τ 2 D0i ε1i , 2τ 2 D1i ,
τ 2 − τ2i H i ,
√
Λ3 τ1i Gi ,
τ2i − τ1i
Ui ,
2
Λ51 Λ41 τ2i − τ1i
Ri ,
2
T
Λ6 ε2i T1iT , 0, 0, 0, 0, 0, 0, 0, −ε2i T2iT ,
Λ42 T
% 3,
Ωi,77 Ωi,77 −H 7i − H 7i − Q
Ωi,69 Ωi,69 0,
1
Ωi,78 Ωi,78 −H 8i ,
2
Ωi,67 Ωi,67 −V 7i H 7i − H 6i ,
τ2i − τ1i
Mi ,
2
τ2i − τ1i
Λ52 V i,
2
T
Λ7 N1i P i N3i Yi , 0, 0, N2i P i N4i Yi , 0, 0, 0, 0, 0 ,
Λ8 πik1i P i
08n×n
πik2i P i · · ·
08n×n · · ·
πikmi P i
,
08n×n
Λ9 diag −P k1i , −P k2i , . . . , −P kmi ,
P i Pi−1 ,
Qi Pi−1 Qi Pi−1 ,
% 3 P −1 Q3 P −1 ,
Q
i
i
⎞
⎛
7
!
"
⎟
⎜
⎟
Σ diag⎜
⎝P i , . . . , P i ⎠,
Z 2 Z2−1 ,
Yi Ki P i ,
% 1 P −1 Q1 P −1 ,
Q
i
i
−1
ε1i ε1i
,
% 2 P −1 Q2 P −1 ,
Q
i
i
T
T T
T
Gi G1i , . . . , G8i , 0 P i GT1i , . . . , GT7i Σ, ε1i P i GT8i , 0 ,
T
T T
T
H i H 1i , . . . , H 8i , 0 P i H1iT , . . . , H7iT Σ, ε1i P i H8iT , 0 ,
T
T T
T
T
T
T
Σ, ε1i P i M8i
Mi M1i , . . . , M8i , 0 P i M1i
, . . . , M7i
,0 ,
T
T T
T
Ri R1i , . . . , R8i , 0 P i RT1i , . . . , RT7i Σ, ε1i P i RT8i , 0 ,
T
T T
T
T
T
T
Ui U1i , . . . , U8i , 0 P i U1i , . . . , U7i Σ, ε1i P i U8i , 0 ,
T
T T
T
V i V 1i , . . . , V 8i , 0 P i V1iT , . . . , V7iT Σ, ε1i P i V8iT , 0 ,
3.42
Journal of Applied Mathematics
19
when the LMIs are feasible, a desired state-feedback controller can be obtained in the form of 2.7 with
the controller gains given by Ki Yi Pi for all i ∈ S.
Proof. At first, we list the following fact:
T J T Z2 J − J T P i − P i J P i Z2 P i Z 2 J − P i Z2 Z 2 J − P i ≥ 0
3.43
which implies that
−P i Z2 P ≤ Θ J T Z2 J − J T P i − P i J.
3.44
Now perform a congruence transformation to 3.1 by
⎛
⎞
7
4
! "
! "
⎜
⎟
−1
diag⎝Pi−1 , . . . , Pi−1 , ε1i
, I, Z2−1 , Pi−1 , . . . , Pi−1 , I, I ⎠.
3.45
i
, then by the Schur complement and 3.44, we can infer that 3.34 is
If πii ∈ Ikn
i
, 3.39 is established.
established. In the same way, if πii ∈ Iuk
From Lemma 2.8, we can see that 3.35 is equivalent to Ai P i P i ATi P i Pj P i <
i
, so 3.4 can be established. Furthermore, using the same method that
0 for all j ∈ Iuk
proposed above, we can deduced that 3.32, 3.33, 3.36–3.38, and 3.40 are equivalent
to 3.2 3.3, 3.5–3.7, and 3.8, respectively. In conclusion, the gain matrix of desired
controller in the form of 2.7 is given by Ki Yi Pi . This completes the proof.
t
Remark 3.5. To reduce the conservatism, when estimating Lv5 xt , i, − t−τ 2 ẋT sZ2 ẋsds
t−τ1i τi t/2 T
t
ẋ sZ2 ẋsds,
is not simply enlarged as − t−τi t ẋT sZ2 ẋsds, but − t−τi t
t−τi t
T
− t−τ2i τi t/2 ẋ sZ2 ẋsds are considered as well, and different free-weighting matrices
are introduced. This method above may lead to obtain improved feasible region for
delay-dependent exponential passivity criteria.
Remark 3.6. In fact, Theorem 3.1 gives a exponential passivity criteria for MJS 2.8 with τ1i ≤
τi t ≤ τ2i , τ̇i t ≤ μi , where μi is a given constant. In many cases, μi is unknown. Considering
this case, a rate-independent criteria for a delay satisfying τ1i ≤ τi t ≤ τ2i is derived as follows
by setting Qi Q∗ 0, for all i ∈ S in the proof of Theorem 3.1.
20
Journal of Applied Mathematics
4. Examples
In this section, we will consider a interval time-varying delay MJS in the form of 2.8 with
three modes, and the parameters of the system are given as follows:
−0.05 −0.05
−0.05 −0.09
−0.03 −0.015
,
A2 ,
A3 ,
0.5 −0.5
1.5 −0.1
0.05 −0.01
0.11 0.24
−0.59 0.01
0.52 0.24
Ad1 ,
Ad2 ,
Ad3 ,
−0.53 −0.37
−0.07 −0.61
0.02 −0.45
0
0
−0.02
0
0 −1.2
D01 ,
D02 ,
D03 ,
−0.02 0
0
−0.02
0 0
2.0
1.0
1.0
0.5
B11 ,
B12 ,
B13 ,
E11 ,
1.0
0.5
2.0
1.0
0.8
1.0
1.0
1.0
E12 ,
E13 ,
D11 ,
D12 ,
2.0
0.5
0.2
1.0
0.5
C2 0.5 1.0 ,
D13 ,
C1 1.0 0.2 ,
0.5
Cd1 −1.0 0.2 ,
Cd2 0.1 −0.1 ,
C3 0.5 0.5 ,
Cd3 0.5 −0.5 ,
B21 1.0,
B22 −0.5,
B23 0.5,
0.02
D22 0.5,
D23 −0.5,
T11 T12 T13 ,
D21 1.0,
0.01
N21 N22 N23 0.01 0.02 ,
N11 N12 N13 0.02 0.01 ,
A1 N31 N32 N33 N41 N42 N43 0.01,
μ1 0.2,
μ2 0.3,
τ11 0.12,
τ12 0.11,
τ22 0.28,
τ23 0.25,
μ3 0.1,
τ13 0.13,
T21 T22 T23 0.1,
0.06 0
Γ1 Γ2 Γ3 ,
0 0.06
τ21 0.23,
4.1
The two cases of the transition rates matrices are described as follows:
⎛
Case 1 :
Case 2 :
−0.5
⎝
0.2
Π
0.5
⎛
−0.5
Π ⎝ 0.2
0.5
⎞
0.2 0.3
−0.6 0.4 ⎠,
0.3 −0.8
⎞
?
?
−0.6 0.4⎠,
?
?
4.2
Journal of Applied Mathematics
21
Table 1: Calculated the controller gains matrix for different cases.
Case 1
−0.2256, 0.0218
−0.4330, −0.1248
−0.2800, −0.1641
K1
K2
K3
Case 2
−2.4527, 0.2749
−0.4842, 0.1157
−1.0109, −0.1008
where ? means the unknown element. With the choice of ε 0.2 and J 0.3 0 , we can
0 0.3
obtain the feasibility solution of case 1 and case 2 as follows.
Case 1:
3.2525 −0.1821
,
−0.1821 4.2081
1.8478 −0.0727
,
−0.0727 2.0824
0.5074 −0.1035
,
−0.1035 0.7639
3.0280 −0.2085
,
−0.2085 4.1893
−0.8734 −0.5801 .
P1 Q1
%1
Q
Z2
Y3
2.6050
−0.2795
1.5779
Q2 −0.5056
% 2 0.3321
Q
−0.0760
P2 −0.2795
,
3.6119
−0.5056
,
1.7197
−0.0760
,
0.5493
Y1 −0.7376 0.1326 ,
3.2386 −0.2026
,
−0.2026 3.8813
1.4462 0.2109
Q3 ,
0.2109 1.8895
1.4315 −0.2159
Z1 ,
−0.2159 1.8214
P3 Y2 −1.0932 −0.3296 ,
4.3
case 2:
0.1304 0.3586
,
0.3586 2.6532
0.4962 0.7608
1.0e 004 ∗
,
0.7608 4.3848
0.1918 0.3925
1.0e 003 ∗
,
0.3925 3.7184
0.0658 0.3281
1.0e 003 ∗
,
0.3281 2.4649
0.2312 0.9273
1.0e 003 ∗
,
0.9273 7.3269
1.0e 003 ∗ −2.2126 −1.5022 ,
1.0e 004 ∗ −0.5783 −1.2112 .
P 1 1.0e 004 ∗
P3
Q2
%1
Q
Z1
Y1
Y3
0.5768 0.4664
,
0.4664 3.9812
0.0864 0.2330
Q1 1.0e 004 ∗
,
0.2330 1.7220
0.0537 0.2133
Q3 1.0e 004 ∗
,
0.2133 1.5653
0.0429 0.2206
%
Q2 1.0e 003 ∗
,
0.2206 1.6811
0.6754 1.4246
Z 2 1.0e 004 ∗
,
1.4246 5.4089
Y2 1.0e 003 ∗ −2.2530 2.3481 ,
P 2 1.0e 004 ∗
4.4
Under the two cases above, Table 1 lists the state-feedback controller gains matrix Ki ,
which can be determined by the method of Theorem 3.4. If the ρ is sufficiently small, we can
check that the MJS 2.8 is exponentially passive under the condition of Theorem 3.4. Given
0.6
0.4
0.2
0
−0.2
−0.4
−0.6
−0.8
−1
System mode
of case 1
Journal of Applied Mathematics
State response of case 1
22
0
5
10
15
20
25
Modes evolution r (k)
3
2
1
0
30
10
35
20
30
Time
40
40
50
45
50
Time (s)
x1
x2
0.6
0.4
0.2
0
−0.2
−0.4
−0.6
−0.8
−1
System mode
of case 2
State response of case 2
Figure 1: State response of case 1 and the switch signal.
0
5
10
15
20
Modes evolution r (k)
3
2
1
25
0 5 10 15 20 25 30 35 40 45 50
Time
30
35
40
45
50
Time (s)
x1
x2
Figure 2: State response of case 2 and the switch signal.
the initial condition as xt 2.0 − 2.0T and rt 2, from Figures 1 and 2, we can easily
see that the closed-loop system in 2.8 is mean square exponential stable with ωt 0.
Remark 4.1. In order to illustrate the effectiveness of the proposed approach, a numerical
example is given which included two cases, that is, case 1, the transition rate matrix is
completely known; case 2, some elements in the transition rate matrix are inaccessible. By
using Matlab Toolbox, we can obtain the gain matrix Ki , which guarantees that the Markovian
jump systems 2.8 is robust exponential passivity. If we choose the switch signal as
Figures 1 and 2, we can know that the closed-loop system 2.8 is exponentially stable in
the mean square under the state-feedback controllers obtained above, which have been listed
in Table 1.
5. Conclusions
In this paper, the problems of exponential passification of uncertain MJS have been
investigated. To reflect more realistic dynamical behaviors of the system, both the partially
known transition rates, state and input delays have been considered. With utilizing
the Lyapunov functional method and free-weighting matrix method, delay-dependent
exponential passivity conditions are established. Finally, an illustrative example has been
given to demonstrate the effectiveness of the proposed approach.
Journal of Applied Mathematics
23
Acknowledgments
This work was supported in part by the Opening Fund of Geomathematics Key Laboratory
of Sichuan Province scsxdz2011001 and the National Basic Research Program of China
2010CB732501.
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