LMI stability conditions for fractional order systems

By Jocelyn Sabatier, Mathieu Moze, Christophe Farges. Computers and Mathematics with Applications, 2010

436 citations recorded by the source corpus.

After an overview of the results dedicated to stability analysis of systems described by differential equations involving fractional derivatives, also denoted fractional order systems, this paper deals with Linear Matrix Inequality (LMI) stability conditions for fractional order systems. Under commensurate order hypothesis, it is shown that a direct extension of the second Lyapunov's method is a tedious task. If the fractional order @n is such that 0<@n<1, the stability domain is not a convex region of the complex plane. However, through a direct stability domain characterization, three LMI stability analysis conditions are proposed. The first one is based on the stability domain deformation and the second one on a characterization of the instability domain (which is convex). The third one is based on generalized LMI framework. These conditions are applied to the gain margin computation of a CRONE suspension.

Subfields: Control and Systems Engineering, Modeling and Simulation

View Paper: https://www.semanticscholar.org/paper/f3d102080791ebfc89e30441db5660234ebb3a3c

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Citations in the corpus

Fractional systems state space description: some wrong ideas and proposed solutions (2014, methodology): “It was demonstrated by Sabatier et al. (2008, 2010a) with a counter example, that Caputo’s definition does not permit one to obtain the exact system pseudo trajectories if usual initial conditions such as (3) are taken into account in description (2).”
Fixed points and asymptotic stability of nonlinear fractional difference equations (2011, background): “Some local asymptotical stability, Mittag-Leffler stability and linear matrix inequality (LMI) stability are discussed in [13, 15, 21, 22, 27], Chen and Zhou [13] considered the attractivity of fractional functional differential equations by Schauder fixed point theorem, Deng [15] discussed the attr”
Stability and stabilization of fractional-order linear systems with convex polytopic uncertainties (2012, background): “Stability is fundamental to all control systems, certainly including fractional order control systems [1],[2],[3],[5], [7],[8],[9],[15], [16],[26],[27],[29],[30].”
Stability analysis of Caputo fractional-order nonlinear systems revisited (2011, background): “In [14] the authors deal with Linear Matrix Inequality (LMI) stability conditions for fractional-order systems.”
Stability analysis of continuous-time linear systems consisting of n subsystems with different fractional orders (2012, neutral): “In [15] it was shown that if α1 = α2 = · · · = αn and the condition (16) holds then components of the vector x(t) decay to 0 not exponentially but following to the function t−μ, t > 0, μ > 0.”