Iterative Design of H∞ Loop-Shaping Weight for Single-Input Single-Output Lightly Damped Systems

Abstract

The $\nu$-gap metric has a clear frequency domain interpolation of system uncertainties. However, the $\nu$-gap metric between lightly damped systems may be very close to one, and causes difficulty for robust stability analysis as well as control design. In this article, the problem is addressed by adjusting the weight in the $H_{\infty }$ loop-shaping method to reduce the corresponding weighted $\nu$-gap metric at the resonance frequency (RF). To this end the selection of the weight is essential. The RFs of lightly damped systems with different perturbations often appear in a frequency band rather than at a single frequency. Thus, it is hard to investigate the weight's impact on the weighted $\nu$-gap metric based on any specific perturbation. Particularly, the weight $W$ may change the sequencing of $\nu$-gap metric of diverse perturbations to the nominal plant, and raises additional issue to the robust stability analysis. The expression of the maximum $\nu$-gap metric in the RF band is derived to tackle these problems. Then, the impact of the weight on the single-input single-output weighted $\nu$-gap metric is quantitatively analyzed. This provides a guidance to the $H_{\infty }$ loop-shaping of lightly damped systems. An iterative design of the weight optimization is applied by limiting the upper bound of the frequency response of the shaped plant. The feasibility and advantages of the iterative design is shown by an illustrative example of a single-phase pulse width modulation rectifier.