H2-norm of fractional transfer functions of implicit type

This paper studies the H2-norm (or impulse response energy) of fractional transfer functions of implicit type. Stability conditions are first shown to be identical as in rational systems with all poles located in the open left half complex plane. Then, analytical expressions of the H2-norm are derived for elementary fractional transfer functions of the first and the second kind cascaded with a pure fractional integrator. Next, general boundedness conditions are established in terms of transfer function relative degree. Three illustrative examples are finally proposed. The first one evaluates the quality of a rational approximation of a fractional model of implicit type on the basis of the H2-norm of the error signal. The second one evaluates the Integral Squared Error of a CRONE control loop and compares it to a classical proportional-derivative controller in a vehicle suspension. Finally, the third one allows to set up an implicit fractional preshaping filter for closed-loop control.