$H_{\infty}$ Model Reduction for Fractional-Order Linear Systems

This paper focuses on the
$H_{\infty}$
model reduction problem of fractional-order Linear systems with commensurate fractional order
$0 < \alpha < 1$
. Firstly, resorting to the
$H_{\infty}$
bounded real lemma, a design method based on the linear matrix inequalities is proposed to construct an asymptotically stable reduced-order model for the given stable fractional-order linear system, such that the
$H_{\infty}$
norm of the error between the transfer function of the reduced-order model and the transfer function of the given stable fractional-order system is less than the prescribed
$H_{\infty}$
performance. Secondly, by introducing a free-weighting matrix and congruent transformation, the system parameters of the reduced-order model are decoupled with the complex matrix variable and parameterized by another free-weighting matrix variable. Finally, one numerical example are given to illustrate the effectiveness of the proposed theoretical results.