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\(\mathcal {S}\)-Procedure for Positive Switched Linear Systems and its Equivalence to Lyapunov–Metzler Inequalities

  • Junfeng Zhang
  • Tarek Raïssi
Chapter
Part of the Lecture Notes in Control and Information Sciences book series (LNCIS, volume 480)

Abstract

This paper presents the \(\mathcal {S}\)-procedure characterization for the stabilization of positive switched linear systems and establishes the relationship between the \(\mathcal {S}\)-procedure and its equivalent Lyapunov–Metzler inequalities. First, a piecewise linear co-positive Lyapunov function is constructed for positive switched linear systems. Under the Lyapunov function, the \(\mathcal {S}\)-procedure stabilization for positive switched linear systems in the continuous-time context is explored under a min state switching law. The \(\mathcal {S}\)-procedure conditions are formulated in the form of linear programming. Finally, an equivalence relationship between \(\mathcal {S}\)-procedure and Lyapunov–Metzler inequalities is presented.

Keywords

Positive switched linear systems \(\mathcal {S}\)-procedure Lyapunov–Metzler inequalities Linear programming 

Notes

Acknowledgements

The authors thank the anonymous reviewers and associate editor for their valuable suggestions and comments which have helped to improve the quality of the paper. This work was supported in part by the National Nature Science Foundation of China (61873314, 61503107, and 61703132), the Zhejiang Provincial Natural Science Foundation of China (S18F030001), and the Foundation of Key Laboratory of System Control and Information Processing, Ministry of Education, P.R. China.

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Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.School of AutomationHangzhou Dianzi UniversityHangzhouChina
  2. 2.Key Laboratory of System Control and Information ProcessingMinistry of Education of ChinaShanghaiChina
  3. 3.Conservatoire National des Arts et Metiers (CNAM)Paris Cedex 03France

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