Research Article
L
1
/ℓ
1
-Gain analysis and synthesis of Markovian jump positive systems
with time delay
Junfeng Zhang
a
, Xudong Zhao
b,c,
n
, Fubo Zhu
d
, Zhengzhi Han
d
a
Institute of Information and Control, Hangzhou Dianzi University, Hangzhou 310018, China
b
College of Engineering, Bohai University, Jinzhou 121013, China
c
Chongqing SANY Highintelligent Robots Co. Ltd., Chongqing 401120, People's Republic of China
d
School of Electronic Information and Electrical Engineering, Shanghai Jiao Tong University, Shanghai 200240, China
article info
Article history:
Received 27 December 2014
Received in revised form
23 January 2016
Accepted 15 March 2016
Available online 7 April 2016
This paper was recommended for
publication by Dr. Dong Lili
Keywords:
Markovian jump positive systems
Stochastic copositive Lyapunov functional
Stabilization
abstract
This paper is concerned with stability analysis and control synthesis of Markovian jump positive systems
with time delay. The notions of stochastic stability with L
1
- and ℓ
1
-gain performances are introduced for
continuous- and discrete-time contexts, respectively. Using a stochastic copositive Lyapunov function,
sufficient conditions for the stability with L
1
=ℓ
1
-gain performance of the systems are established.
Furthermore, mode-dependent controllers are designed to achieve the stabilization with L
1
=ℓ
1
-gain of
the resulting closed-loop systems. All proposed conditions are formulated in terms of linear program-
ming. Numerical examples are provided to verify the effectiveness of the findings of theory.
& 2016 ISA. Published by Elsevier Ltd. All rights reserved.
1. Introduction
Positive systems are a class of systems whose states and out-
puts are nonnegative when the initial conditions and inputs are
nonnegative. Positive systems exist in many communities such as
economics, sociology, ecology, and engineering [1–3]. For more
details, one can refer to [4–10]. Some new research topics on
positive systems have also been investigated in recent years. In
particular, the hybrid systems consisting of positive systems
attract much attention in the past several years [11–15].
In engineering applications, control systems are always subject
to abrupt changes due to component failures, sudden environment
changes, and some other unexpected factors. The abrupt changes
often affect the performance of a system and even destroy the
stability. Thus how to deal with the abrupt changes is a key pro-
blem when investigating the systems with abrupt changes. Over
past decades, a quantity of results have verified that Markovian
jump systems are very effective for modeling the abrupt changes
[16–21]. To the best of our knowledge, a practical model of Mar-
kovian jump systems with positivity was firstly constructed in
[22], where the stochastic properties of a class of communication
networks were analyzed and the asymptotic behavior of the net-
works was characterized. A mathematical framework of
Markovian jump positive systems was introduced in [23], where
stability and stabilization were studied by using linear matrix
inequalities. Recently, stochastic stability and stabilization of
Markovian jump positive systems were further investigated in
[24], where the copositive Lyapunov function associated with
linear programming was employed as a new approach. The new
approach possesses two advantages. First, linear programming is
easier to be solved and can cope with a class of larger scale sys-
tems than linear matrix inequalities. Second, the new approach
can reduce the computational burden of present conditions.
Therefore, it has been regarded as a standard approach when
dealing with positive systems [11–15].
Time delay and disturbance input are two important char-
acteristics of a control system. Almost all practical systems contain
time delay and disturbance input, which will deteriorate the sys-
tem performances and even destroy the stability of the system. In
[25], necessary and sufficient conditions for exponential stability
of positive linear time-delay systems were established by using
the Perron–Frobenius theorem. Controller design for continuous-
time delayed linear systems was proposed in [26]. Linear supply-
rates were employed for robustness and performance analysis of
positive systems in [27] where L
1
- and L
1
-gain characterizations
were also proposed. The authors in [28] applied the L
1
-gain ana-
lysis of positive systems to stability of interconnected systems.
More results on positive systems with time delay and disturbance
input can be found in [29–33].
Contents lists available at ScienceDirect
journal homepage: www.elsevier.com/locate/isatrans
ISA Transactions
http://dx.doi.org/10.1016/j.isatra.2016.03.015
0019-0578/& 2016 ISA. Published by Elsevier Ltd. All rights reserved.
n
Corresponding author.
E-mail address: xdzhaohit@gmail.com (X. Zhao).
ISA Transactions 63 (2016) 93–102