Quadratic stability in the circle theorem or positivity theorem

In this technical note, we study the quadratic stability problem for linear time-invariant multi-input, multioutput systems which have time-varying sector-bounded or positive real uncertainty feedback. It is known that the circle theorem and positivity theorem are sufficient conditions for robust stability, for sector-bounded and positive real feedback, respectively. We show that the circle theorem is necessary and sufficient condition for quadratic stability when the feedback is sector-bounded, and that the positivity theorem is necessary and sufficient condition for quadratic stability when the feedback is positive-real.