LMI Stability Condition for Delta Fractional Order Systems With Region Approximation

Exploring the stability of delta fractional order systems is essential for them to be used properly in various applications. Since the existing researches usually focused on the system with
$\alpha \in (0,1)$
, it is natural to ponder a parallel case with extra order. Motivated by this need, this study addresses the stability of delta delay fractional order systems with
$\alpha \in (1,2)$
systematically. The main difficulties lie in the approximation of stable region and the derivation of the relevant linear matrix inequalities (LMI) conditions caused by the complicated stable region of the suggested system. Firstly, a novel clustering region is constructed and it is proved that such a region is the subset of the considered stable region. Besides, the scheme on how to construct alternative approximation regions is discussed tentatively. Secondly, the stability conditions are formulated in terms of LMIs, which are sufficient and necessary to evaluate all the eigenvalues of system matrix locating at the approximation region. Thirdly, the complex decision matrices are replaced by the real ones and the formulation is therefore more tractable. Finally, the validity and applicability of the proposed approaches are demonstrated by simulation study.