Real structured singular value bounds using rational multipliers and scaling

Sufficient conditions for robust stability involving rational stability multipliers and scaling are presented for systems with sector- and norm-bounded, block-structured uncertainty. The frequency-dependent multipliers and scaling render the new robustness criterion less conservative than the multivariable scaled Popov criterion, which is a special case of the new criterion with a multiplier that is an affine function of frequency and scaling that is independent of frequency. Two rational parameterizations of the multiplier and scaling are considered and the robustness criteria for systems with block-structured, norm-bounded uncertainty are written as linear matrix inequalities. Upper bounds for the peak structured singular value over frequency are then derived. A numerical example provides a comparison of the peak upper bounds involving the two rational parameterizations of the multiplier and scaling. This example shows that the conservatism of the peak upper bounds is reduced by increasing the dynamic order of the multipliers and scaling.