Delayed resonator with multiple distributed delays – Considering and optimizing the inherent loop delay

We generalize the distributed delayed resonator (DDR) concept by incorporating the previously unutilized but inevitable feedback loop delay, thus creating a so-called multiple-delay distributed delayed resonator (MD-DDR) for complete vibration suppression. The necessary parameter tuning and control stability analysis become more involved but remain analytically solvable. Particularly, we take the operability of control parameters imposed by hardware as an explicit condition for parameter tuning and show that the loop delay (even if it is small) can lead to incomplete vibration suppression and, counter-intuitively, even no suppression if the excitation frequency exceeds a threshold value. On the other hand, such negative effects are neutralized by correcting control parameters. The loop delay is then intentionally augmented to seek enhanced performance, yielding a considerably extended operable frequency band for the desired complete vibration suppression. Furthermore, it is optimized to expedite response speed by achieving the leftmost placement of the dominant (i.e., the rightmost) characteristic root. Instead of applying a brute-force sweeping procedure as in the previous works, we analyze the dominant root locus and seek if the jump phenomenon occurs to conduct exact optimization at a low computational cost. Finally, extensive comparisons using actual experimental parameters are performed to explore various effects of the loop delay on vibration suppression and the benefits of the proposed MD-DDR in handling such issues over the conventional DDR that treats a single delay. This study fully exploits the strength of the distributed delayed control logic, moving the DDR concept closer to real applications and aiming to establish a broader design and analysis framework for the delayed resonator from a multiple-delay perspective.