LMI characterization of fractional systems stability.

*(English)* Zbl 1125.93051
Sabatier, J. (ed.) et al., Advances in fractional calculus. Theoretical developments and applications in physics and engineering. Dordrecht: Springer (ISBN 978-1-4020-6041-0/hbk; 978-1-4020-6042-7/e-book). 419-434 (2007).

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:

93D05 | Lyapunov and other classical stabilities of control systems |

93D99 | Stability of control systems |

26A33 | Fractional derivatives and integrals (real functions) |

93B50 | Synthesis problems |