A mathematical approach to Integral Resonant Control of second-order systems

Systems with colocated sensor-actuator pairs exhibit an interesting property of pole-zero interlacing. Integral Resonant Control (IRC) exploits this property to result in superior damping performance over multiple resonant modes by prescribing an adequate feed-through term to swap the pole-zero interlacing to a zero-pole one - thus enabling a simple integral feedback controller to add substantial damping to the system. Over the past few years, the IRC has proved extremely popular and versatile and has been applied to damp the resonance in a variety of systems. So far, a simulation-based manual search has been used to determine the three main parameters of the IRC scheme namely: (i) feed-through term, d, (ii) integral gain, k and (iii) resulting damping, ζ. In this paper, a full quantification of the effect of feed-through term on second-order colocated systems as well as a mathematical formulation for the relation between the feed-through term, integral gain and achievable damping are presented. These results add to the current understanding regarding the behaviour of colocated systems and facilitate the IRC design for a specified damping.