Quadratic surface Lyapunov functions in global stability analysis of saturation systems

Abstract

This paper considers quadratic surface Lyapunov functions in the study of global asymptotic stability of saturation systems (SAT), including those with unstable nonlinearity sectors. We show that quadratic surface Lyapunov functions can be applied to analyze piecewise linear systems with more than one switching surface. For that, we consider SAT. We present conditions in the form of LMIs that, when satisfied, guarantee global asymptotic stability of equilibrium points. A large number of examples was successfully proven globally stable, including systems of high dimension and systems with unstable nonlinearity sectors, for which classical methods like the small gain theorem, Popov criterion, Zames-Falb criterion, IQCs, fail to analyze. In fact, an existing example of SAT with a globally stable equilibrium point that cannot be successfully analyzed with this new methodology is still an open problem. The results from this work suggests that other, more complex classes of PLS can be systematically, globally analyzed using quadratic surface Lyapunov functions.