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Zbl 1125.93051
Moze, Mathieu; Sabatier, Jocelyn; Oustaloup, Alain
LMI characterization of fractional systems stability. (English)
[A] Sabatier, J. (ed.) et al., Advances in fractional calculus. Theoretical developments and applications in physics and engineering. Dordrecht: Springer. 419-434 (2007). ISBN 978-1-4020-6041-0/hbk; ISBN 978-1-4020-6042-7/ebook

Summary: The notions of linear matrix inequalities (LMI) and convexity are strongly related. However, with state-space representation of fractional systems, the stability domain for a fractional order $\nu$, $0<\nu< 1$, is not convex. The classical LMI stability conditions thus cannot be extended to fractional systems. In this paper, three LMI-based methods are used to characterize stability. The first uses the second Lyapunov method and provides a sufficient but nonnecessary condition. The second and new method provides a sufficient and necessary condition, and is based on a geometric analysis of the stability domain. The third method is more conventional but involves nonstrict LMI with a rank constraint.

MSC 2000:: *93D05 Lyapunov and other classical stabilities of control systems
93D99 Stability of control systems
26A33 Fractional derivatives and integrals (real functions)
93B50 Synthesis problems

Keywords: linear matrix inequalities; LMI stability conditions

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