Abstract
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 paper 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. We show a number of new and classical Lyapunov results can be obtained using our framework. In particular, we demonstrate that our results can be used to obtain compact time-domain versions of the SISO Kalman-Yacubovich-Popov lemma, the Circle Criterion, and stability multiplier criteria. Finally, we conclude by posing a number of open questions that arise as a result of our approach.
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References
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Shorten, R., Mason, O., Wulff, K. (2005). Convex Cones, Lyapunov Functions, and the Stability of Switched Linear Systems. In: Murray-Smith, R., Shorten, R. (eds) Switching and Learning in Feedback Systems. Lecture Notes in Computer Science, vol 3355. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-30560-6_2
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DOI: https://doi.org/10.1007/978-3-540-30560-6_2
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