Design of a variable gain integrator with reset

This paper studies the properties of a variable gain integrator with reset, i.e. a nonlinear lag filter that is obtained by a) saturating the input, b) filtering the saturated input with a Clegg integrator, and c) add the filtered output to the unsaturated input before applying it to a PID-based controller. Depending on the amount of saturation, the corner frequency of the lag filter is reduced along with the associated phase lag. This follows from a describing function analysis in which at low frequencies a minus 20 dB/decade amplitude decay is realized with a phase lag of only 32.48 degrees. Conditions to assess global asymptotic stability of the closed-loop nonlinear control system are provided that are based on a circle criterion-like argument for the flow condition, which applies to the intervals without resets, combined with a jump condition at reset. The reset integrator design is demonstrated on a piezo-actuated motion system where its favorable phase and amplitude properties induce overshoot and settling times comparable to a single (linear) integrator, but with the disturbance rejection properties of a double integrator.