Combining loop-shaping and Laguerre development for rational approximation of a noninteger order, integral, analog controller

Fractional order systems are receiving more and more attention from the scientific literature as many real systems showed behavior which can be represented by fractional order differential equations. Besides many control design methods are being investigated since fractional order controllers guarantee robust stability in wide frequency ranges. This paper uses generalized Laguerre orthogonal functions to approximate a fractional integrator as a rational transfer function. A truncated Laguerre series is derived in the time domain to represent the approximated impulse response of the irrational transfer function. Then, a loop shaping is performed on the phase plot of the frequency response. Simulation results show how the compact and economic approximation gives good performance in a sufficiently wide frequency range