Necessary and sufficient conditions of stability: a multiloop generalization of the circle criterion

A linear time-invariant system with a vector output and a vector input is described. This system is closed by an uncertain (nonlinear, time-varying) feedback. The only information on this feedback is given by several integral quadratic inequalities, i.e. the uncertainties under consideration generalize the so-called conic nonlinearities. Necessary and sufficient frequency-domain conditions of stability are obtained. An advanced version of the S-procedure losslessness theorem and some other tools of the absolute stability theory are used