Relays are common in automatic control systems. Even linear systems with re
lay feedback are, however, far from fully understood. New results are given
about the behavior of these systems via a state-space approach. It is prov
ed that there exist multiple fast switches if and only if the sign of the f
irst nonvanishing Markov parameter of the linear system is positive. Fast s
witches are shown to occur as part of stable limit cycles. An analysis is d
eveloped for these limit cycles that illustrates how they can be predicted.
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