Elsevier

Automatica

Volume 47, Issue 1, January 2011, Pages 235-238
Automatica

Technical communique
Reciprocally convex approach to stability of systems with time-varying delays

https://doi.org/10.1016/j.automatica.2010.10.014Get rights and content

Abstract

Whereas the upper bound lemma for matrix cross-product, introduced by Park (1999) and modified by Moon, Park, Kwon, and Lee (2001), plays a key role in guiding various delay-dependent criteria for delayed systems, the Jensen inequality has become an alternative as a way of reducing the number of decision variables. It directly relaxes the integral term of quadratic quantities into the quadratic term of the integral quantities, resulting in a linear combination of positive functions weighted by the inverses of convex parameters. This paper suggests the lower bound lemma for such a combination, which achieves performance behavior identical to approaches based on the integral inequality lemma but with much less decision variables, comparable to those based on the Jensen inequality lemma.

Introduction

In the field of stability analysis and control design, tighter upper bounds of various functions have been pursued: affine functions of polytopic-type uncertain systems, quadratic functions of T–S fuzzy control systems, and especially a special type of function combinations in delayed systems (Jiang and Han, 2005, Shao, 2009) which is the focus of our discussion.

This particular featured function originates from the relaxation based on the Jensen inequality lemma (Gu, Kharitonov, & Chen, 2003) in the delayed systems. Initially, the integral inequality lemma for matrix cross-products (Ko and Park, 2009, Moon et al., 2001, Park, 1999) has played a key role in guiding various delay-dependent criteria with the choice of the Lyapunov–Krasovskii functional introduced in Fridman and Shaked (2003). However, to fully relax the matrix cross-products, it has to introduce slightly excessive free weighting matrices. As a way of reducing the number of decision variables, at the sacrifice of conservatism, relaxations based on the Jensen inequality lemma (Gu et al., 2003) have been attracted, recently (Wu et al., 2009, Zhu et al., 2009). It directly relaxes an integral term of quadratic quantities into a quadratic term of integral quantities. Such relaxed quadratic terms appear as a linear combination of positive functions weighted by the inverses of convex parameters. As concerned about it, Shao (2009) has achieved an excellent work of reducing the conservativeness. The basic idea is to approximate the integral terms of quadratic quantities into a convex combination of quadratic terms of the integral quantities. However, owing to the inversely weighted nature of coefficients in the Jensen inequality approach, Shao (2009) has to introduce an approximation on the difference between delay bounds, th2th(t)(h2h1)f(α)dαth2th(t)(h2h(t))f(α)dα, h1h(t)h2, in the middle stage of the derivation.

This paper suggests a lower bound lemma for such a linear combination of positive functions with inverses of convex parameters as the coefficients. Based on the lemma, we develop a stability criterion that directly handles the inversely weighted convex combination of quadratic terms of integral quantities, which achieves performance behavior identical to approaches based on the integral inequality lemma but with much less decision variables, comparable to those based on the Jensen inequality lemma.

The paper is organized as follows. Section 2 provides a new lower bound lemma for a weighted linear combination of positive functions over the inverses of convex parameters. Based on this lemma, it considers a stability criterion to show how to handle the double integral terms of a Lyapunov–Krasovskii functional for delayed systems without introducing a convexizing anti-inverse process. Section 3 will show simple examples for the verification of the criterion.

Section snippets

Main results

In this paper, we concern a special type of function combinations, i.e. a linear combination of positive functions with inverses of convex parameters as the coefficients, which is defined below.

Examples

For simulation comparison, we derive a simple stability criterion, an interval delayed version of Park and Ko (2007), which is based on the integral inequality lemma.

Proposition 4 Criterion Based on Integral Inequality Lemma

The delayed system (9)(10) is asymptotically stable if there exist matrices P, Q1, Q2, R1, R2, X11, X12, X22, Y11, Y12, Y22, Z11, Z12 and Z22 such that the following conditions hold:[X11X12X12TX22]0,[Y11Y12Y12TY22]0,[Z11Z12Z12TZ22]0,X22h1R10,Y22h12R20,Z22h12R20,P>0,Q1>0,Q2>0,R1>0,R2>0,0>e5Pe1T+e1Pe5T+e1Q1e1Te3Q1e3T+e1Q2e1

Conclusion

This paper suggests a lower bound lemma for a linear combination of positive functions weighted by the inverses of convex parameters. Based on the lemma, we develop a stability criterion that directly handles such a combination of quadratic terms of integral quantities encountered in the Jensen inequality approach for guiding delay-dependent criteria of delayed systems, which not only achieves performance behavior identical to approaches based on the integral inequality lemma but also decreases

References (11)

There are more references available in the full text version of this article.

Cited by (2341)

View all citing articles on Scopus

This research was supported by the MKE (The Ministry of Knowledge Economy), Korea, under the ITRC (Information Technology Research Center) support program supervised by the NIPA (National IT Industry Promotion Agency) (NIPA-2010-(C1090-1021-0006) & NIPA-2010-(C1090-1011-0011)). This research was supported by WCU (World Class University) program through the Korea Science and Engineering Foundation funded by the Ministry of Education, Science and Technology (R31-2008-000-10100-0). The material in this paper was not presented at any conference. This paper was recommended for publication in revised form by Associate Editor Ulf T. Jonsson under the direction of Editor André L. Tits.

View full text