Automatica 46 (2010) 1327–1333
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Automatica
journal homepage: www.elsevier.com/locate/automatica
Brief paper
Stability and stabilization of aperiodic sampled-data control systems using
robust linear matrix inequalities
✩
Yasuaki Oishi
a,∗
, Hisaya Fujioka
b
a
Department of Systems Design and Engineering, Nanzan University, Seireicho 27, Seto 489-0863, Japan
b
Graduate School of Informatics, Kyoto University, Kyoto 606-8501, Japan
a r t i c l e i n f o
Article history:
Received 9 May 2009
Received in revised form
18 January 2010
Accepted 16 April 2010
Available online 3 June 2010
Keywords:
Sampled-data control
Robust linear matrix inequality
Conservatism
Asymptotic exactness
Semidefinite programming
Adaptive division
a b s t r a c t
Stability analysis of an aperiodic sampled-data control system is considered for application to networked
and embedded control. The stability condition is described in a linear matrix inequality to be satisfied for
all possible sampling intervals. Although this condition is numerically intractable, a tractable sufficient
condition can be constructed with the mean value theorem. Special attention is paid to tightness of the
sufficient condition for less conservative stability analysis. A region-dividing technique for the reduction
of conservatism and generalization to stabilization are also discussed. An example demonstrates the
efficacy of the approach.
© 2010 Elsevier Ltd. All rights reserved.
1. Introduction
Sampled-data control is a mature research area and established
methodology is available both for analysis and design (Chen &
Francis, 1995). However, most of the existing results assume a
constant sampling interval and cannot be applied to networked
and embedded control systems, whose sampling intervals are
uncertain and vary with time.
For analysis and design of such aperiodic sampled-data control
systems, several approaches have been proposed. Some are based
on a continuous-time or hybrid framework (e.g., Fridman, Seuret,
& Richard, 2004; Mirkin, 2007a,b; Naghshtabrizi, Hespanha, & Teel,
2006). The stability conditions presented for these approaches
are rather conservative, although they are applicable to general
systems. Other approaches are based on a discrete-time framework
and provide less conservative stability conditions. In particular,
✩
This research is supported in part by the Grant-in-Aid for Scientific Research
of the Japan Society for the Promotion of Science and the Nanzan University Pache
Research Subsidy I-A-2 for the academic years 2009 and 2010. The material in this
paper was partially presented at the 48th IEEE Conference on Decision and Control,
Shanghai, China, December 2009. This paper was recommended for publication in
revised form by Associate Editor Mario Sznaier under the direction of Editor Roberto
Tempo.
∗
Corresponding author. Tel.: +81 561 89 2000; fax: +81 561 89 2082.
E-mail addresses: oishi@nanzan-u.ac.jp (Y. Oishi), fujioka@i.kyoto-u.ac.jp
(H. Fujioka).
Hetel, Daafouz, and Iung (2007) gave a stability condition using
a polynomial to evaluate the effect of aperiodic sampling. An
increase in the degree of the polynomial reduces the conservatism
of the result. On the other hand, Fujioka (2008, 2009) and Suh
(2008) gave stability conditions based on division of a region
where the uncertain sampling interval takes a value. Here, an
increase in the resolution of the division leads to a less conservative
result. Skaf and Boyd (2009) used such a division to evaluate the
degradation of the optimal quadratic performance of an aperiodic
sampled-data control system.
The approach in this paper inherits some ideas from Fujioka
(2008, 2009) and Suh (2008) but is improved through employing
the three following techniques. First, the stability condition is
based on the delta-operator representation (Middleton & Goodwin,
1990) and converges to the continuous-time stability condition
as the sampling interval goes to zero. Because of this property,
the stability can be analyzed without numerical difficulty even
when the sampling interval is small. Second, the effect of aperiodic
sampling is modeled as parametric uncertainty rather than
matrix uncertainty for less conservative stability analysis. Third,
a technique of adaptive division is introduced for reduction of
the computational cost. In this paper, the stability condition is
in the form of a linear matrix inequality (LMI) to be satisfied for
all possible sampling intervals. In general, a parameter-dependent
LMI to be satisfied for all possible parameter values is called
a robust LMI. Recently, there has been intensive investigation
of a robust LMI whose parameter dependence is polynomial or
0005-1098/$ – see front matter © 2010 Elsevier Ltd. All rights reserved.
doi:10.1016/j.automatica.2010.05.006