Hindawi Publishing Corporation
Journal of Applied Mathematics
Volume , Article ID , pages
http://dx.doi.org/.//
Research Article
Stabilization for Networked Control Systems with
Random Sampling Periods
Yuan Li,
1,2
Qingling Zhang,
1
Shuanghong Zhang,
1,3
and Min Cai
4
1
Institute of Systems Science, Northeastern University, Shenyang 110819, China
2
School of Science, Shenyang University of Technology, Shenyang 110870, China
3
School of Science, Jilin Normal University, Siping 136000, China
4
School of Science, Dalian Jiaotong University, Dalian 116028, China
Correspondence should be addressed to Qingling Zhang; qlzhang@mail.neu.edu.cn
Received November ; Accepted December
Academic Editor: Reinaldo Martinez Palhares
Copyright © Yuan Li et al. is 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.
is paper investigates the stabilization of networked control systems (NCSs) with random delays and random sampling periods.
Sampling periods can randomly switch between three cases according to the high, low, and medium types of network load. e
sensor-to-controller (S-C) random delays and random sampling periods are modeled as Markov chains. e transition probabilities
of Markov chains do not need to be completely known. A state feedback controller is designed via the iterative linear matrix
inequality (LMI) approach. It is shown that the designed controller is two-mode dependent and depends on not only the current
S-C delay but also the most recent available sampling period at the controller node. e resulting closed-loop systems are special
discrete-time jump linear systems with two modes. e sucient conditions for the stochastic stability are established. An example
of the cart and inverted pendulum is given to illustrate the eectiveness of the theoretical result.
1. Introduction
Networked control systems (NCSs) are feedback control
systems whose feedback paths are implemented by a real-
time network. Recently, much attention has been paid to the
study of NCSs, due to their low cost, reduced weight and
power requirements, simple installation and maintenance,
high reliability, and so on [, ].
A basic problem in NCSs is the stability of the systems. In
real-timecontrolsystems,threemainissuesarebandwidth
and packet size constraints, time delays, and packet losses,
which will degrade the performance of control systems and
even make systems unstable; so it is signicant to overcome
the adverse inuences of time delays, packet losses, and
dynamic bandwidth [–].
On the other hand, the Markov chain, a discrete-
time stochastic process with the Markov property, can be
eectively used to model NCSs with random time delays
and random sampling periods, which are modeled as the
Markovian jump linear systems (MJLSs). Recently, there have
been considerable research eorts into NCSs [–]. For
example, Xiao et al. present an V-K iteration algorithm to
design stabilizing controllers for specially structured discrete-
time jump linear systems with random but bounded delays in
the feedback loop []. Zhang et al. propose a promising two-
mode-dependent-state feedback scheme to stabilize section
with the current S-C delay (
𝑘
)andthepreviouscontroller-
to-actuator (C-A)delay(
𝑘−1
)modeledastwoMarkovchains
[]. Shi and Yu investigate the output feedback stabilization
and robust mixed
2
/
∞
control of NCSs, in which it is
assumed that at each sampling instant, the current S-C delay
(
𝑘
)andthemostrecentavailableC-A delay (
𝑘−1−𝜏
𝑘
)canbe
obtained [, ]. Huang and Nguang discuss the stabilization
problem for a class of linear uncertain continuous NCSs,
in which it is assumed that random communication S-C
delays (()) and random communication C-A delays (())
are modeled two continuous time-discrete-state Markov
processes [].
In the above references, the transition probabilities
are assumed completely accessible and considered as the
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