A simplified proof of the multivariable Popov criterion and an upper bound for the structured singular value with real parameter uncertainty

The Popov absolute stability criterion is traditionally proved using a Lyapunov function and the positive real lemma. In this paper, a simplified proof of the multivariable Popov criterion based upon the multivariable Nyquist criterion is given for the case of one-sided, sector-bounded real parameter uncertainty. A loop-shifting transformation is then used to extend the Popov criterion to two-sided, sector-bounded uncertain matrices. Specialization of this result to norm-bounded uncertain matrices leads to an upper bound for the structured singular value for block-structured, real parameter uncertainty