$$\fancyscript{H}_\infty $$ Control of Fractional Systems

Abstract

In this chapter, the solution for the standard \(\fancyscript{H}_\infty \) control problem for fractional linear time-invariant single-input-single-output systems is presented. The adopted approach consists of extending to the fractional case the procedure followed within the classical solution for the integer case. According to the classical route, the generalization to fractional systems of the standard Youla parametrization of all the stabilizing controllers is first considered. The \(\fancyscript{H}_\infty \) optimal controller is then found among the class of stabilizing controllers, recasting the control problem into a model-matching one. The obtained results naturally extend well-established results to a fractional setting, including both the commensurate and incommensurate cases, thus providing a framework where \(\fancyscript{H}_\infty \) design can be recast along the well-known and fruitful lines of the integer case.