An optimal robust design method for fractional-order reset controller

In this paper, a novel design method for the reset controller structure (i.e., fractional-order proportional and integral plus Clegg integrator (PI alpha$$ {}<^>{\alpha } $$ + CI alpha$$ {}<^>{\alpha } $$)), is proposed for a second-order plus time delay plant. To this end, the designer can get an optimal fractional reset controller that gives the control system more phase margin over the base linear PI controller and robust to loop gain variation. The describing function method is used to investigate the capability of phase lead and the frequency domain properties of PI alpha$$ {}<^>{\alpha } $$ + CI alpha$$ {}<^>{\alpha } $$. The gain crossover frequency and phase margin specifications ensure the stability of the control system, and the flat phase constraint makes the control system robust to loop gain variations. Meanwhile, the integral of time and absolute error (ITAE) value is applied to achieve the optimal dynamic performance as the cost function. PI alpha$$ {}<^>{\alpha } $$ + CI alpha$$ {}<^>{\alpha } $$ is compared with its integer-order counterpart (i.e., proportional and integral plus Clegg integrator (PI + CI) controller) and their base controllers (i.e., integer-order PI and fractional-order PI controllers) in terms of the step response and robustness to loop gain variations. The simulation results illustrate that the PI alpha$$ {}<^>{\alpha } $$ + CI alpha$$ {}<^>{\alpha } $$ control system obtains lower overshoot and oscillation and better robustness to loop gain variations than others. The experiments are performed on the speed control of an air bearing stage. Experimental results show that the designed PI alpha$$ {}<^>{\alpha } $$ + CI alpha$$ {}<^>{\alpha } $$ control system behaves better than others. The proposed PI alpha$$ {}<^>{\alpha } $$ + CI alpha$$ {}<^>{\alpha } $$ design method can be applied to other general control plants easily.