Recent research on switched and hybrid systems has resulted in a renewed interest in determining conditions for the existence of a common quadratic Lyapunov function for a finite number of stable LTI systems. While efficient numerical solutions to this problem have existed for some time, compact analytical conditions for determining whether or not such a function exists for a finite number of systems have yet to be obtained. In this talk we present a geometric approach to this problem. By making a simplifying assumption we obtain a compact time-domain condition for the existence of such a function for a pair of LTI systems. Our conditions also relate the existence of such a function to the stability boundary of the underlying switched linear system (thereby indicating that requiring the existence of such a function does not, in a certain sense, lead to overly conservative stability conditions). We show that classical Lyapunov results can be obtained using our framework. In particular, we obtain simple time-domain versions of the SISO Kalman-Yacubovich-Popov lemma, the Circle Criterion, and stability multiplier criteria. Finally, we indicate how our approach can be used to analyse n-tuples of LTI systems and present preliminary results on the existence of common non-quadratic Lyapunov functions of a certain form.
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