IET Control Theory & Applications
Research Article
Robust stochastic stability and
delayed-state-feedback stabilisation of
uncertain Markovian jump linear systems
with random time delays
ISSN 1751-8644
Received on 24th October 2014
Revised on 30th January 2015
Accepted on 23rd March 2015
doi: 10.1049/iet-cta.2014.1138
www.ietdl.org
Li Qiu
1
,Yang Shi
2
, Bugong Xu
3
, FengqiYao
4
1
Shenzhen Key Laboratory of Urban RailTransit, College of Mechatronics and Control Engineering, Shenzhen University,
Shenzhen 518060, People’s Republic of China
2
Department of Mechanical Engineering, University of Victoria, Victoria BC V8W 3P6, Canada
3
College of Automation, South China University ofTechnology, Guangzhou 510641, People’s Republic of China
4
School of Electrical Engineering and Information, Anhui University ofTechnology, Ma’anshan 243000, People’s Republic of China
E-mail: yshi@uvic.ca
Abstract: The problem of robust stochastic stability and delayed-state-feedback stabilisation of uncertain Markovian jump
linear systems with random Markov delays is investigated. Based on the Lyapunov stability theory and robust analysis
techniques, some robust stochastic stability criteria are derived in terms of linear matrix inequalities. Robust delayed-
state-feedback controllers that stochastically stabilise the uncertain Markovian jump linear systems is also proposed. The
state variable on the controller is assumed to be dependent on the Markov delay that has uncertain transition probabilities.
Finally, numerical examples are provided to illustrate the feasibility and effectiveness of the proposed methods.
1 Introduction
Markovian jump linear systems (MJLSs) represent an important
class of stochastic hybrid dynamic systems subject to abrupt
variations in their structure. This condition is partly caused by
phenomena such as changing subsystem interconnections, failures
or repairs, and abrupt environmental disturbances. Recently, MJLSs
have been an active research topic because of their extensive appli-
cations in networked control systems, electrical power systems,
communication systems, circuit systems, robot manipulator system
and economic systems. Many control topics of Markovian jump
systems or related systems with Markov process have been widely
investigated; notable developments have been achieved to ensure
stochastic Lyapunov stability, for example, robust control [1–4],
H
∞
control [5–8], fault-tolerant control [9–11], quantised control
[12], predictive control [13], H
2
/H
∞
control [14–16], synchroni-
sation [17–21], and stability and stabilisation [22–27]. The filtering
problem, such as filter design [28–30], Kalman filtering [31, 32],
H
∞
filtering [33–35] and exponential H
∞
filtering [36], have also
gained significant attention. Most of the reported research findings
on MJSs concern the Markovian transition probabilities that are
known a priori.
In practical processes, obtaining complete knowledge on the
Markovian transition probabilities is unlikely, or costly [37]. As
errors are usually generated through tests or experience, it is dif-
ficult to derive the exact transition probabilities of a real physical
system. Therefore investigating general Markovian jump processes
with uncertain transition probabilities further from robust control
perspectives is significant and challenging, especially when ran-
dom Markov time delays are included by communication network.
In the case of networked control systems with forward random time
delay, or random forward and feedback channel packet dropouts,
the random time delay in the forward and feedback channel or
the data packet dropouts in the forward and feedback channels are
modelled using a homogeneous Markov chain. However, finding
a probabilistic relation between the dropout phenomena that occur
in two channels is difficult. The same problems may arise in other
practical systems with Markovian jump processes. Thus, in-depth
study of more general jump systems with uncertain transition
probabilities from control perspectives, especially with parameter
uncertainties and random Markov time delays included, is signif-
icant. Recently, some well-established results have been obtained
for MJSs with unknown or partially known transition probabilities.
For example, a class of continuous-time MJSs with bounded or
unknown transition probabilities have been studied in [38–40]. The
fault detection problem and stabilisation problem for the discrete-
time MJLs with partly known transition probabilities have been
addressed in [41] and in [37], respectively. However, parameter
uncertainties are inherent features of many physical processes and
are often encountered in engineering systems. Also, the transi-
tion probabilities of random Markov time delays are difficult to
obtain but are bounded. The topic of delayed-state-feedback stabil-
isation problem for uncertain MJLSs with Markov time delays and
with uncertain transition probabilities has received limited atten-
tion. This observation motivates us to investigate the robust control
and delayed-state-feedback controller design problem.
This study focuses on the robust stochastic stability analysis
and delayed-state-feedback stabilisation synthesis problems for a
class of discrete-time uncertain MJLSs with uncertain transition
probabilities and random Markov time delays. The contribution of
this study is twofold. First, by using Jensen’s inequality approach,
a robust stochastic stability criteria are obtained for the MJLSs
with uncertain parameters and random time delays modelled as
a Markov process with uncertain transition probability. Second,
mode-dependent delayed-state-feedback control scheme is devel-
oped, and less conservative robust stability and stabilisation condi-
tions for the uncertain MJLSs are derived. The rest of this paper is
organised as follows. Section 2 provides the problem formulation
and Section 3 presents robust stochastic stability analysis. Section 4
establishes the robust delayed-state-feedback stabilisation condi-
tions for uncertain MJLSs with uncertain transition probabilities.
Illustrative examples are presented in Section 5 and the conclusion
is provided in Section 6 and two appendices containing formal
proofs of the results in the main body.
Notations: Throughout this paper, we let R
n
and R
n×m
denote the
n-dimensional Euclidean space and the set of all n × m real matri-
ces, respectively. The notation A > 0(A < 0) indicates that A is
IET Control Theory Appl., pp. 1–9
© The Institution of Engineering and Technology 2015 1
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