1
Passivity-Based Asynchronous Control for Markov
Jump Systems
Zheng-Guang Wu, Peng Shi, Zhan Shu, Hongye Su, and Renquan Lu
Abstract—This paper studies the problem of passivity-based
asynchronous control for discrete-time Markov jump systems.
The asynchronization phenomenon appears between the system
modes and controller modes, which is described by a hidden
Markov model. Accordingly, the resultant closed-loop system is
named as a hidden Markov jump system. By utilizing the matrix
inequality technique, three equivalent sufficient conditions are
proposed to ensure the stochastic passivity of the hidden Markov
jump systems. Based on the established conditions, the design
of asynchronous controller, which covers synchronous controller
and mode-independent controller as special cases, is addressed.
A numerical example is given to demonstrate the effectiveness of
the derived results.
Index Terms—Markov jump systems, switching, hidden
Markov model, asynchronous control, passivity
I. INTRODUCTION
In the past few decades, significant progress has been
made on the theory of Markov jump systems, which is one
of the most active research areas in systems and control.
The reason is mainly that Markov jump systems are suitable
mathematical models to represent various practical systems
whose structure is subject to random abrupt variation mainly
due to, for example, changing in subsystem interconnections,
random component failures or repairs, sudden environmental
changes, and uncontrolled configuration changes. As a result,
fruitful results have been reported in this field. For example,
much progress has been made in the study of the stability and
stabilization problems for Markov jump systems; see, e.g.,
[1]–[6] and the references therein. When both the Markov
jump parameters and time delays appear in the systems, the
problems of stability and stabilization have been studied in
[7]–[9]. The problems of H
∞
control and sliding mode control
This work was partially supported by the Australian Research Council
(DP140102180, LP140100471), the National High Technology Research and
Development Program of China (863 Program 2012AA041703), the National
Natural Science Foundation of China (61573112, 61304072), the China
National Funds for Distinguished Young Scientists (61425009), and the
Zhejiang Provincial Natural Science Foundation of China (LR16F030001).
Z.-G. Wu and H. Su are with the National Laboratory of Industrial Control
Technology, Institute of Cyber-Systems and Control, Zhejiang University,
Yuquan Campus, Hangzhou Zhejiang, 310027, PR China (e-mail: nash-
wzhg@gmail.com;hysu@iipc.zju.edu.cn).
P. Shi is with the School of Electrical and Electronic Engineering, The
University of Adelaide, SA 5005, Australia. He is also with the College
of Engineering and Science, Victoria University, Melbourne, VIC 8001,
Australia; and College of Automation, Harbin Engineering University, Harbin,
150001, China (e-mail: peng.shi@vu.edu.au).
Z. Shu is with the School of Engineering Sciences, University of Southamp-
ton, Southampton SO17 1BJ, U.K. (e-mail: z.shu@soton.ac.uk).
R. Lu is with with School of Automation, Guangdong University of
Technology, and Guangdong Key Laboratory of IoT Information Processing,
Guangzhou 510006, China (e-mail: rqlu@hdu.edu.cn).
have been investigated for Markov jump systems in [10] and
[11], respectively. In [12], the robust H
∞
filtering problem
has been addressed for discrete-time Markov jump systems
subject to both randomly occurring nonlinearities and sensor
saturation. When time delays appear in the Markov jump
systems, the H
∞
and l
2
-l
∞
filtering problems have been
considered in [13] and [14], respectively.
It should be pointed out that most existing results on control
and filtering of Markov jump systems are based on an common
and important assumption that the mode information of plant
is fully available to the controller/filter at every instant of
time to ensure the switching of controller/filter synchronous
with that of plant. Accordingly, the designed controller/filter
is named as mode-dependent or synchronous controller/filter.
However, such an ideal assumption cannot always be satisfied
in real world applications. For example, in networked control
systems, network-induced delays and packet loss inevitably
make the mode information of plant not completely accessible,
which leads to the asynchronization phenomenon between
system modes and controller/filter modes. Thus, it is important
and necessary to study the asynchronous control/filtering for
Markov jump systems. The stabilization problem for discrete-
time Markov jump systems with time delay in the mode signal
has been investigated in [15] and a sufficient condition has
been proposed for the design of a controller with delayed mode
information such that the closed-loop system is stochastically
stable. The stabilization problem of discrete-time Markov
jump systems with a non-accessible jumping parameter has
been studied in [16], where the linear matrix inequality (LMI)
method has been proposed for designing a mode-independent
controller, which can be regarded as a special asynchronous
controller. In [17], [18], some approaches have been proposed
to design the mode-independent H
∞
filter for continuous-time
Markov jump systems. Very recently, the problem of asyn-
chronous l
2
-l
∞
filtering for discrete-time stochastic Markov
jump systems has been addressed in [19], where the existence
criterion of the desired asynchronous filter with piecewise
homogeneous Markov chain has been proposed. Although the
importance of asynchronous control/filtering for Markov jump
systems has been widely recognized, the related problems
have not been fully addressed, which is the motivation for
the current work.
In this paper, the problem of stochastically passive asyn-
chronous control for discrete-time Markov jump systems is
studied. The system mode is assumed to be asynchronous with
the controller mode, which is caused by various constraints
on signal processing and transmission in practice. A hidden
Markov model is introduced to describe the asynchroniza-
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2
tion phenomenon. As a result, the corresponding closed-loop
system is constructed as a hidden Markov jump system.
By adopting the matrix inequality approach, several criteria
are given to ensure that the hidden Markov jump system
is stochastically passive based on one of which the design
method of the desired asynchronous controller is proposed. A
numerical example is used to show the effectiveness of the
theoretic results obtained.
Notation: The notations used throughout this paper are fairly
standard. R
n
and R
m×n
denote the n-dimensional Euclidean
space and the set of all m ×n real matrices, respectively. The
notation X > Y (X > Y ), where X and Y are symmetric
matrices, means that X − Y is positive definite (positive
semidefinite). diag{. . .} stands for a block-diagonal matrix
and ||·|| denotes the Euclidean norm of a vector and its induced
norm of a matrix. (Ω , F, P) is a probability space, Ω is the
sample space, F is the σ-algebra of subsets of the sample
space and P is the probability measure on F. E {·} denotes
the expectation operator with respect to some probability
measure P. l
2
[0, +∞) is the space of square summable infinite
sequence.
II. PRELIMINARIES
Fix a probability space (Ω , F, P) and consider the follow-
ing discrete-time Markov jump system:
x(k + 1) = A(r(k ))x(k) + D(r(k))u(k) + B(r(k))υ(k),
z(k) = C(r(k))x(k) + E(r(k))u(k) + L(r(k))υ(k),
(1)
where x(k) ∈ R
n
is the system state, u(k) ∈ R
m
is
the control input, z(k) ∈ R
q
is the controlled output, and
υ(k) ∈ R
p
is the disturbance input that belongs to l
2
[0, ∞).
A(r(k)), D(r(k)), B(r(k)), C(r(k)), E(r(k)) and L(r(k))
are known real constant matrices with appropriate dimensions.
The parameter {r(k), k > 0} represents a Markov chain
taking values in a finite set N = {1, 2, ··· , N} with transition
probability matrix Π = {π
ij
} given by
Pr{r(k + 1) = j|r(k) = i} = π
ij
, (2)
where 0 6 π
ij
6 1 for all i, j ∈ N , and
N
j=1
π
ij
= 1 for
all i ∈ N .
In this paper, we consider the following controller:
u(k) = K(σ(k))x(k), (3)
where K(σ(k)) is a controller gain matrix to be determined,
and the parameter {σ(k), k > 0} takes values in a finite set
M = {1, 2, ··· , M} with conditional probability matrix Ω =
{µ
iϕ
} given by:
Pr{σ(k) = ϕ|r(k) = i} = µ
iϕ
, (4)
where 0 6 µ
iϕ
6 1 for all i ∈ N and ϕ ∈ M , and
M
ϕ=1
µ
iϕ
= 1 for all i ∈ N .
Remark 1: In practice, the information of system modes
accessible to controller is often inaccurate, that is, the actual
modes of system are hidden to the controller, which leads to
the fact that the controller modes that can be regarded as the
observation value of system modes do not synchronize with
systems modes. In this paper, we introduce stochastic variable
σ(k) to present the modes of controller, which is different
from the Markov chain r(k), but depends on it subject to
the conditional probability (4). It can be found that the set
(r(k), σ(k), Π, Ω) constructs a hidden Markov model [20].
To the best of authors’ knowledge, our work is the first
to make use of the hidden Markov model to describe the
non-synchronization phenomenon between system modes and
controller modes.
Remark 2: It is noted that when N = M and µ
iϕ
= 1 for
i = ϕ, the controller (3) becomes a synchronous controller
,
and when M = {1}, that is, the controller (3) only has
one mode, the asynchronous controller (3) reduces to the
mode-independent controller. Thus, the asynchronous con-
troller given in this paper covers synchronous controller and
mode-independent controller as special cases.
Applying the asynchronous controller (3) to system (1) leads
to the following closed-loop system,
x(k + 1) = (A(r(k )) + D(r(k))K(σ(k)))x(k)
+ B(r(k))υ(k),
z(k ) = (C(r(k)) + E(r(k))K(σ(k)))x(k)
+ L(r(k))υ(k).
(5)
Remark 3: It is worth mentioning that in this paper, the
closed-loop system (5) is named as hidden Markov jump
system due to the fact that the hidden Markov model
(r(k), σ(k), Π, Ω) is included in the system. Up to now, few
results have been proposed for such kind of system.
In this paper, the following definition will be used.
Definition 1: System (5) is said to be stochastically passive
if there exists a scalar γ > 0 such that for all k
0
> 0,
2
k
0
s=0
E {z(s)
T
υ(s)} > −γ
k
0
s=0
||υ(s)||
2
(6)
under the zero initial condition.
The objective of this paper is to design an asynchronous
controller (3) for system (1) such that system (5) is
stochastically passive.
For notational simplicity, in the sequel, for each possible
r(k) = i ∈ N and σ(k) = ϕ ∈ M , the matrices M(r(k))
and N(σ(k)) will be denoted by M
i
and N
ϕ
, respectively.
III. MAIN RESULTS
In this section, the deign method of asynchronous controller
will be given for system (1). Firstly, we will present some
conditions to ensure system (5) is stochastically passive.
Theorem 1: System (5) is stochastically passive, if there
exist matrices P
i
> 0 and a scalar γ > 0 such that for any
i ∈ N and ϕ ∈ M ,
Z
i
=
M
ϕ=1
µ
iϕ
(
ˆ
A
T
iϕ
X
i
ˆ
A
iϕ
− H
iϕ
S
−1
i
H
T
iϕ
) − P
i
< 0, (7)
S
i
= B
T
i
X
i
B
i
− L
T
i
− L
i
− γI < 0, (8)
where
X
i
=
N
j=1
π
ij
P
j
, H
iϕ
=
ˆ
A
T
iϕ
X
i
B
i
−
ˆ
C
T
iϕ
,
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3
ˆ
A
iϕ
= A
i
+ D
i
K
ϕ
,
ˆ
C
iϕ
= C
i
+ E
i
K
ϕ
.
Proof: Consider the following Lyapunov functional for
system (5):
V (k, x(k), r(k)) = x(k)
T
P (r(k))x(k). (9)
Letting E {∆V (k)} = E {V (k + 1, x(k + 1), r(k + 1) =
j)|x(k), r(k) = i}− V (k, x(k), i), we have
E {∆V (k)} = E {x(k+1)
T
X
i
x(k+1)}−x(k)
T
P
i
x(k). (10)
On the other hand, it can be calculated that along the solution
of system (5),
E {x(k + 1)
T
X
i
x(k + 1)}
=
M
ϕ=1
µ
iϕ
η(k)
T
Y
T
iϕ
X
i
Y
iϕ
η(k)
= η(k)
T
M
ϕ=1
µ
iϕ
Y
T
iϕ
X
i
Y
iϕ
η(k),
(11)
where
Y
iϕ
=
ˆ
A
T
iϕ
B
T
i
T
, η(k) =
x(k)
υ(k)
.
From (10) and (11), we get that
E {∆V (k)}
= η(k)
T
M
ϕ=1
µ
iϕ
Y
T
iϕ
X
i
Y
iϕ
η(k) − x(k)
T
P
i
x(k),
(12)
It is noted that
E {z(k)
T
υ(k)} =
M
ϕ=1
µ
iϕ
η(k)
T
U
T
iϕ
υ(k)
= η(k)
T
M
ϕ=1
µ
iϕ
U
T
iϕ
υ(k),
(13)
where
U
iϕ
=
ˆ
C
T
iϕ
L
T
i
T
.
Thus, we obtain from (12) and (13) that under the zero initial
condition
J
zυ
,
k
0
s=0
E
−2z(s)
T
υ(s) − γυ(s)
T
υ(s)
6
k
0
s=0
E
∆V (s) − 2z(s)
T
υ(s) − γυ(s)
T
υ(s)
=
k
0
s=0
η(s)
T
M
ϕ=1
µ
iϕ
Y
T
iϕ
X
i
Y
iϕ
η(s)
+
k
0
s=0
−2η(s)
T
M
ϕ=1
µ
iϕ
U
T
iϕ
υ(s)
+
k
0
s=0
−x(s)
T
P
i
x(k) − γυ(k)
T
υ(s)
=
k
0
s=0
υ(s)
T
M
ϕ=1
µ
iϕ
Ξ
iϕ
υ(s) − x(s)
T
P
i
x(s)
,
(14)
where
Ξ
iϕ
=
ˆ
A
T
iϕ
X
i
ˆ
A
iϕ
H
iϕ
∗ S
i
.
Furthermore, based on Schur complement, it can be found that
H
iϕ
S
−1
i
H
T
iϕ
H
iϕ
∗ S
i
6 0, (15)
which implies
Ξ
iϕ
6
ˆ
A
T
iϕ
X
i
ˆ
A
iϕ
− H
iϕ
S
−1
i
H
T
iϕ
0
∗ 0
. (16)
Thus,
M
ϕ=1
µ
iϕ
Ξ
iϕ
6
M
ϕ=1
µ
iϕ
ˆ
A
T
iϕ
X
i
ˆ
A
iϕ
− H
iϕ
S
−1
i
H
T
iϕ
0
∗ 0
.
(17)
Applying the above inequality to (14) yields that
J
zυ
6
k
0
s=0
x(s)
T
Z
i
x(s). (18)
Therefore, it follows (5) that
J
zυ
6 0, (19)
which implies that (6) holds for all k
0
> 0. This completes
the proof.
Note that in Theorem 1, a passivity condition is proposed
for system (5). However, the criterion cannot be directly
applied to design the desired asynchronous controller (3) due
to its complicated structure. To overcome this difficulty, some
equivalent conditions will be given in the following theorem.
Theorem 2: The following three assertions are equivalent:
(i) There exist matrices P
i
> 0 and a scalar γ > 0 such that
(6) holds;
(ii) There exist matrices P
i
> 0 and R
iϕ
> 0 and a scalar
γ > 0 such that
M
ϕ=1
µ
iϕ
R
iϕ
< P
i
, (20)
−R
iϕ
−(C
i
+ E
i
K
ϕ
)
T
Q
iϕ
∗ −L
T
i
− L
i
− γI D
i
∗ ∗ Y
i
< 0, (21)
where
Q
iϕ
=
√
π
i1
P
1
(A
i
+ D
i
K
ϕ
)
√
π
i2
P
2
(A
i
+ D
i
K
ϕ
)
.
.
.
√
π
iN
P
N
(A
i
+ D
i
K
ϕ
)
T
, D
i
=
√
π
i1
P
1
B
i
√
π
i2
P
2
B
i
.
.
.
√
π
iN
P
N
B
i
T
,
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4
Y
i
= diag{−P
1
, −P
2
, ··· , −P
N
}.
(iii) There exist matrices
¯
P
i
> 0 and
¯
R
iϕ
> 0 and a scalar
γ > 0 such that
−
¯
P
i
¯
R
i
∗
¯
X
i
< 0, (22)
−
¯
R
iϕ
−
¯
R
iϕ
(C
i
+ E
i
K
ϕ
)
T
¯
Q
iϕ
∗ −L
T
i
− L
i
− γI
¯
D
i
∗ ∗
¯
Y
i
< 0, (23)
where
¯
R
i
=
√
µ
i1
¯
P
i
√
µ
i2
¯
P
i
···
√
µ
iM
¯
P
i
,
¯
X
i
=
diag{−
¯
R
i1
, −
¯
R
i2
, ··· , −
¯
R
iM
},
¯
Q
iϕ
=
√
π
i1
(A
i
+ D
i
K
ϕ
)
¯
R
iϕ
√
π
i2
(A
i
+ D
i
K
ϕ
)
¯
R
iϕ
.
.
.
√
π
iN
(A
i
+ D
i
K
ϕ
)
¯
R
iϕ
T
,
¯
D
i
=
√
π
i1
B
i
√
π
i2
B
i
.
.
.
√
π
iN
B
i
T
,
¯
Y
i
= diag{−
¯
P
1
, −
¯
P
2
, ··· , −
¯
P
N
}.
Proof: (i)→(ii)
If there exist matrices P
i
> 0 and a scalar γ > 0 such that
(6) holds, then there must exist a scalar ε > 0 such that
M
ϕ=1
µ
iϕ
(
ˆ
A
T
iϕ
X
i
ˆ
A
iϕ
− H
iϕ
S
−1
i
H
T
iϕ
+ εI) − P
i
< 0. (24)
Choose
R
iϕ
=
ˆ
A
T
iϕ
X
i
ˆ
A
iϕ
− H
iϕ
S
−1
i
H
T
iϕ
+ εI, (25)
which implies (20) holds and
R
iϕ
−
ˆ
A
T
iϕ
X
i
ˆ
A
iϕ
+ H
iϕ
S
−1
i
H
T
iϕ
> 0. (26)
Applying Schur complement, we can get (21).
(ii)→(i)
If there exist matrices P
i
> 0 and a scalar γ > 0 such that
(20) and (21) holds, then based on Schur complement, we get
(29) hold. Thus
M
ϕ=1
µ
iϕ
(
ˆ
A
T
iϕ
X
i
ˆ
A
iϕ
− H
iϕ
S
−1
i
H
T
iϕ
) −
M
ϕ=1
µ
iϕ
R
iϕ
< 0.
(27)
Using (20) and (27), we can get (6).
(ii)↔(iii)
It can be found from Schur complement that (20) is equiv-
alent to
−P
i
J
i
∗
¯
X
i
< 0, (28)
where
¯
R
iϕ
= R
−1
iϕ
and J
i
=
√
µ
i1
I
√
µ
i2
I ···
√
µ
iM
I
. Denoting
¯
P
i
= P
−1
i
, and
pre- and post-multiplying (28) by diag{−
¯
P
i
, I, I, ··· , I
M
},
respectively, we can show (28) is equivalent to (22).
On the other hand, pre- and post-multiplying (21) by
diag{
¯
R
iϕ
, I,
¯
P
1
,
¯
P
2
, ··· ,
¯
P
N
}, respectively, we have (23) is
equivalent to (21). This completes the proof.
Remark 4: It is noted that two novel passivity conditions
equivalent to the one in Theorem 1 are obtained in Theorem
2. The advantage of (20) and (21) lies in the separation
of µ
iϕ
and K
ϕ
. While in (22) and (23), not only µ
iϕ
and
K
ϕ
are separated, but also K
ϕ
and P
i
are separated. Thus,
from control design point view, the assertion (iii), which is
more flexible to parametrize K
ϕ
by introducing free parameter
matrices, is more desirable than the other two assertions.
Next, we shall make use of assertion (iii) to deal with
passivity-based asynchronous control for system (1), and a
sufficient condition is obtained for the existence of a asyn-
chronous controller.
Theorem 3: System (5) is stochastically passive, if there
exist matrices
¯
P
i
> 0,
¯
R
iϕ
> 0, matrices K
ϕ
, G
ϕ
, and a
scalar γ > 0 such that (22) and the following LMIs holds,
¯
R
iϕ
− G
T
ϕ
− G
ϕ
−(C
i
G
ϕ
+ E
i
¯
K
ϕ
)
T
ˆ
Q
iϕ
∗ −L
T
i
− L
i
− γI
¯
D
i
∗ ∗
¯
Y
i
< 0, (29)
where
ˆ
Q
iϕ
=
√
π
i1
(A
i
G
ϕ
+ D
i
¯
K
ϕ
)
√
π
i2
(A
i
G
ϕ
+ D
i
¯
K
ϕ
)
.
.
.
√
π
iN
(A
i
G
ϕ
+ D
i
¯
K
ϕ
)
T
.
Furthermore, if (22) and (29) are solvable, a desired asyn-
chronous controller (3) can be chosen with parameter as
K
ϕ
=
¯
K
ϕ
G
−1
ϕ
. (30)
Proof: Pre- and post-multiplying (23) by
diag{G
T
ϕ
¯
R
−1
iϕ
, I, I, ··· , I
M
}, respectively, yield that
−G
T
ϕ
¯
R
−1
iϕ
G
ϕ
−(C
i
G
ϕ
+ E
i
¯
K
ϕ
)
T
ˆ
Q
iϕ
∗ −L
T
i
− L
i
− γI
¯
D
i
∗ ∗
¯
Y
i
< 0. (31)
On the other hand, noting
¯
R
iϕ
> 0, we have
¯
R
iϕ
− G
ϕ
− G
T
ϕ
+ G
ϕ
¯
R
−1
iϕ
G
T
ϕ
= (
¯
R
iϕ
− G
ϕ
)
¯
R
−1
iϕ
(
¯
R
iϕ
− G
T
ϕ
) > 0, (32)
which implies −G
T
ϕ
¯
R
−1
iϕ
G
ϕ
6
¯
R
iϕ
− G
T
ϕ
− G
ϕ
. Combining
this with (29), we have that (31) holds. This completes the
proof.
Remark 5: It should be pointed out that the design method
proposed in Theorem 3 not only can be applied to design
the asynchronous controller, but also can be adopted to syn-
chronous controller
and mode-independent controller, that is, a
unified controller design method is provided for Markov jump
system (1).
Remark 6: It is noted that the LMIs in (22) and (29) are
not only over the matrix variables, but also over the scalar γ,
which implies that by minimizing γ subject to (22) and (29),
Limited circulation. For review only
IEEE-TAC Submission no.: 15-1570.2
Preprint submitted to IEEE Transactions on Automatic Control. Received: May 7, 2016 01:28:32 PST
5
we can obtain the optimal passivity performance γ
min
, and the
corresponding asynchronous controller gains as well.
Remark 7: It is worth mentioning that the results given in
this paper can be extended easily to dissipativity, which covers
passivity as a special case.
IV. NUMERICAL EXAMPLE
This section will present a numerical example to demon-
strate the effectiveness of the method given in the preceding
section.
Consider a DC motor device [21], which can be described
as system (1) with the following parameters:
A
i
=
a
i
11
a
i
12
0
a
i
21
a
i
22
0
a
i
31
0 a
i
33
, D
i
=
d
i
1
d
i
2
0
, B
i
= b
i
I, i = 1, 2, 3
The parameters can be found in Table I.
TABLE I
PARAMETERS OF SYSTEM (1) REPRESENTING A REAL DC MOTOR DEVICE
Parameters i = 1 i = 2 i = 3
a
i
11
-0.4799 -1.6026 0.6346
a
i
12
5.1546 9.1632 0.9178
a
i
21
-3.8162 -0.5918 -0.5056
a
i
22
14.4732 3.0317 2.4811
a
i
31
0.1399 0.0740 0.3865
a
i
33
-0.9255 -0.4338 0.0982
d
i
1
5.8705 10.2951 0.7874
d
i
2
15.5010 2.2282 1.5302
b
i
0.1 0.1 0.1
On the other hand, in this example, we choose
C
i
=
1 0 0
0 1.5 0
0 0 2.5
, E
i
=
0
0
0.5
, L
i
= 0
The transition probability matrix
Π =
0.9 0.05 0.05
0.36 0.6 0.04
0.1 0.1 0.8
and four cases of the conditional probability matrix Ω are
given in Table II.
TABLE II
DIFFERENT CONDITIONAL PROBABILITY MATRICES
Synchronous (Case 1) Partially synchronous (Case 2)
1 2 3
1 1.0 0 0
2 0 1.0 0
3 0 0 1.0
1 2 3
1 0.4 0.3 0.3
2 0 1.0 0
3 0 0 1.0
Partially synchronous (Case 3) Asynchronous (Case 4)
1 2 3
1 0.4 0.3 0.3
2 0.2 0.3 0.5
3 0 0 1.0
1 2 3
1 0.4 0.3 0.3
2 0.2 0.3 0.5
3 0.4 0.5 0.1
Based on Theorem 3, the optimal passivity performance
γ
min
is obtained in Table III for different conditional proba-
bility matrix Ω. It can be seen from Table III that for a higher
TABLE III
OPTIMAL PASSIVITY PERFORMANCE γ
min
DIFFERENT Ω
Case 1 Case 2 Case 3 Case 4
2.5688 3.0904 3.8323 3.9669
mode synchronization rate, the obtained optimal passivity
performance γ
min
is smaller.
As for the asynchronous case (Case 4), by solving the LMIs
in (22) and (29), one can get the following controller gain
matrices
K
1
=
0.2391 −0.9086 −0.0508
K
2
=
0.2350 −0.9046 −0.0512
K
3
=
0.2455 −0.8813 −0.0644
The possible time sequences of the system mode and
controller mode are given in Fig. 1, from which it can be found
that the system mode and controller mode are asynchronous.
Under the above controller gain matrices, the system state
x(k) and control input u(k) are presented in Figs. 2 and 3,
respectively, where x(0) =
−5 3 −2
T
and the exogenous
disturbance input υ(k) =
1/k 1/k 1/k
T
.
1
1.5
2
2.5
3
r(k)
0 20 40 60 80 100
1
1.5
2
2.5
3
Time (k)
σ(k)
Fig. 1. System and controller modes r(k) and σ(k )
V. CONCLUSIONS
In this paper, the problem of asynchronous control for
discrete-time Markov jump systems has been considered.
A hidden Markov model has been adopted to describe the
asynchronization phenomenon appearing between the system
modes and controller modes, which leads to the fact that the
closed-loop system is a hidden Markov jump system. Three
stochastic passivity conditions have been proposed for the
hidden Markov jump system via matrix inequality approach.
The existence criterion of the desired asynchronous controller
has been developed, which is formulated by LMIs and
thus
can be solved easily by available LMI toolbox. A numerical
example has been given to demonstrate the effectiveness of
the proposed design methods. It should be emphasized that
because the results of our paper are sufficient conditions only,
Limited circulation. For review only
IEEE-TAC Submission no.: 15-1570.2
Preprint submitted to IEEE Transactions on Automatic Control. Received: May 7, 2016 01:28:32 PST