Potential Implications of Mixing Perturbations on Robust Stability for Linear Uncertain Systems - Mao, Qi; Chen, Jianqi (Proceedings of the IEEE Conference on Decision & Control, 2024)
This paper is concerned with the stabilization problem of unstable systems with mixed gain and phase perturbations, thence elaborating on the exact computation of optimal robustness margins. We focus on non-minimum phase plants that are stabilized by proportional and proportional-integral (PI) controllers. Specifically, for such systems with mixed perturbations, we first show that the computation of optimal gain margin constitutes a constrained optimization problem. It is proved that the maximum gain margin is attained at zero integral gain, and the boundary value of proportional gain can be determined exactly. Via the Bilherz criterion, we next demonstrate that the maximal phase margin of non-minimum phase systems subject to mixed perturbations is also achieved at zero integral gain. It turns out that the calculation of optimal phase margin amounts to solving a concave optimization problem. Finally, we find that proportional-integral control and proportional control promise the same expressions of optimum robustness margins. Our explicit results clearly characterize the well-established dependence of the maximum robustness margins and/or the optimal controller coefficients on the system's unstable pole, nonminimum phase zero as well as uncertain perturbations.