$H_{\infty}$ State Feedback Controller Based on Dynamic Observer Design for Singular Fractional-Order Systems

This paper focuses on the problem of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$H_{\infty}$</tex> state feedback controller design based on dynamic observer for singular fractional-order systems (FOS), where the fractional derivative order <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\alpha$</tex> lies between 0 and 1. First, a new form of dynamic observer with a non-singular structure is proposed, which is easier to implement physically. Secondly, the bounded real lemma corresponding to <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$H_{\infty}$</tex> norm of FOS is proposed via a set of linear matrix inequalities (LMIs). Compared to existing methods,the lemma employs real variable, which is easier to solve. Building upon the new lemma, the conditions for designing <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$H_{\infty}$</tex> state feedback controller based on dynamic observer of FOS are derived. Finally, numerical example is presented to validate the effectiveness of the proposed method.