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Transactions of the Institute of
Measurement and Control
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DOI: 10.1177/0142331216636187
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Robustly resilient memory control for
time-delay switched systems under
asynchronous switching
Guangdeng Zong
1
, Qingzhi Wang
1
and Yi Yang
2
Abstract
The robustly resilient memory control problem is addressed for a class of switched systems with time delay under asynchronous switching. By
resorting to the piecewise Lyapunov–Krasovskii functional, an asynchronous memory controller is designed to ensure the exponential stability of the
closed-loop system. Here, the piecewise Lyapunov–Krasovskii functional means that for the activated subsystem, the associated Lyapunov–Krasovskii
functional on the unmatched interval is different from that on the matched interval. Then an asynchronously resilient memory controller is derived to
guarantee that the corresponding closed-loop system is robustly exponentially stable for all the admissible uncertainties. All the conditions are cast into
the form of linear matrix inequalities. Finally, a numerical example is provided to illustrate the validity of the proposed results.
Keywords
Asynchronous switching, piecewise Lyapunov–Krasovskii functional, resilient control, switched systems, time delay, exponential stability
Introduction
A switched system is a dynamical system that consists of a
finite number of subsystems and a logical rule that orches-
trates the switching between them. Due to its enormous theo-
retical and practical importance, increasing attention has
recently been given to research on the stability analysis and
controller synthesis of switched systems (Branicky, 1998; Hai
and Antsaklis, 2009; Hespanha and Morse, 1999; Liberzon
and Morse, 1999; Sun and Ge, 2005). However, time delays
actually arise in a wide variety of applications modelled by
switched systems, such as air traffic management, manufac-
turing systems, chemical processes, automotive engine con-
trol, and so on. We call this kind of system a switched system
with time delay. It is important to study such systems due to
the fact that time delay may deteriorate the performance of
switched systems and even make them unstable. The past
decades have seen considerable research activities into the
modelling, analysis and synthesis of switched systems with
time delay (Kim et al., 2006; Sun et al., 2006; Zong et al.,
2008; Zong, 2013).
Controller design in most of the existing literature assumes
that the switching between the controller and the system is
synchronous. However, it takes time to identify which con-
troller is activated in actual operation, which results in the
asynchronous switching issue. Here, ‘asynchronous switching’
means that the switching of the controller lags behind or
exceeds the switching of the system. Many related reports on
this issue are available (see, for example, Wang et al., 2013,
2014; Zhai and Yang, 2013; Zhang and Gao, 2010). Zhang
and Gao (2010) discuss asynchronously switched control for
switched linear systems with an average dwell time in the
continuous-time and discrete-time cases, respectively.
However, for each activated subsystem, the same Lyapunov
functions on the unmatched interval and matched interval are
chosen. Zhai and Yang (2013) extend this result to switched
time-delay systems, however, the Lyapunov functionals on
the unmatched and matched intervals for each activated sub-
system have the same form. Wang et al. (2013) further address
dynamic output feedback control problems for switched delay
systems under asynchronous switching, however, matrices in
the Lyapunov functional on the unmatched interval are the
same as those on the matched interval. This kind of candidate
Lyapunov function or functional for each activated subsys-
tem, to some extent, limits the application of these theorems,
which leads to more conservativeness. This inspired the
authors to adopt a piecewise Lyapunov functional for each
activated subsystem, in order to relax the restriction on the
Lyapunov functional. A piecewise Lyapunov functional
means that for each activated system, the corresponding
Lyapunov functional on the unmatched interval is different
from that on the matched interval.
Typically, most of the results on the controller design can
only deal with the situation in which the controller is precise,
and exactly implemented, such as Sun and Ge (2005) and
Zong et al. (2008). However, the controller should be able to
tolerate some perturbations due to the finite word length in
any digital system, A/D or D/A conversion, the imprecision
1
School of Engineering, Qufu Normal University, China
2
School of Reliability and Systems Engineering, Beihang University, China
Corresponding author:
Guangdeng Zong, School of Engineering, Qufu Normal University, Rizhao
276826, China.
Email: lovelyletian@gmail.com
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