Commentary :
This paper considers quadratic surface Lyapunov functions in the study of global asymptotic stability of saturation systems (SAT), including those with unstable nonlinearity sectors. Quadratic surface Lyapunov functions were first introduced and successfully used to globally analyze asymptotic stability of limit cycles of relay feedback systems and later equilibrium points of on/off systems. Here, we show that quadratic surface Lyapunov functions can also be applied to analyze piecewise linear systems (PLS) 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 small gain theorem, Popov criterion, Zames-Falb criterion, IQCs, fail to analyze. In fact, existence of an example of a 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.
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