The Yakubovich quadratic criterion, F-stability and multi-agent consensus.∗

Generalizing numerous results on absolute stability of nonlinear Lurie systems, Vladimir A. Yakubovich established a fundamental abstract criterion of absolute stability under uncertain nonlinearities. His Quadratic Criterion offers elegant conditions for the uniform output L2-stability in a wide class of uncertain nonlinearities, obeying anytime ("local") or integral quadratic constraints; these conditions are based on either quadratic Lyapunov functions whose existence is proved via the KYP lemma, or the method of integral quadratic constraints dating back to the works by V.M. Popov. In the present paper, we extend the Yakubovich quadratic criterion, replacing the output L2-stability by boundedness of some quadratic integral performance index, defined by a quadratic form F and referred to as F -stability. We demonstrate that our F -stability criteria enables one to obtain easily many stability criteria for the “critical” case, where the frequency-domain inequality is non-strict and usual L2-stability of the solution cannot be proved, e.g., we derive the circle and Popov's stability criteria in this degenerate case. All of these criteria are especially effective for scalar nonlinearities, nevertheless, we demonstrate that our approach is also applicable to a class of large scale nonlinear multi-agent networks, and derive the “networked” circle criterion for such networks as a consequence of our results.