Generalizing Hybrid Integrator-Gain Systems Using Fractional Calculus

The Hybrid Integrator-Gain System (HIGS) has recently gained a lot of attention in control of precision motion systems. HIGS is a nonlinear low pass filter/integrator with a 52° phase advantage over its linear counterpart. This property allows us to avoid the limitations typically associated with linear controllers, like the waterbed effect and Bode's gain-phase relation. In this paper, we generalize HIGS by replacing the involved integer-order integrator by a fractional-order one to adapt the phase lead from 0° (linear low pass filter) to 52° (HIGS). To analyze this filter in the frequency domain, the describing function of the proposed filter, i.e., the fractional-order HIGS, is obtained using the Fourier expansion of the output signal. In addition, this generalized HIGS is implemented in a PID structure controlling a double integrator system to validate the performance of the proposed filter in the time domain, in which by changing the fractional variable from zero to one, the output varies from the response of a linear control system to a nonlinear one.