Novel Robust $H_{\infty}$ Stability and Stabilization Conditions for Fractional-order Systems with Convex Polytopic Uncertainties

This paper investigates the problems of robust stability and stabilization for fractional-order systems with polytopic uncertainties. It is assumed that the fractional-order
$a$
is a known constant and belongs to
$0 < \alpha < 1$
. Firstly, based on the
$H_{\infty}$
bounded real lemma for commensurate fractional-order control systems, a sufficient condition for the above stability problem is established in terms of linear matrix inequalities (LMIs). Secondly, on the foundation of this condition, sufficient LMI methods for the design of stabilizing controller are obtained for two cases where the polytopic coefficients are known and unknown. In the case of unknown polytopic coefficients, by introducing the additional matrices, the state matrix and the positive-definite Hermitian matrices are decoupled. In the case of known polytopic coefficients, by introducing parameter-dependent matrices, a less-conservative robust
$H_{\infty}$
stabilization condition is obtained. Finally, two different numerical examples are provided to compare the conservatism between the existing results and the results in this paper.