Discrete-Time Periodic Monotonicity Preserving Systems

Abstract: Three nested classes of discrete-time linear time-invariant systems, which differ by the set of periodic signals that they leave invariant, are studied. The first class preserves the property of periodic monotonicity (period-wise unimodality). The second class is invariant to signals with at most two sign changes per period, and the third class results from the second by additionally requiring that periodic signals with zero sign-changes are mapped to the same kind. Tractable characterizations for each system class are derived by the use and extension of total positivity theory via geometric interpretations. Central to our results is the characterization of sequentially convex contours. Our characterizations also extend to the loop gain of Lur'e feedback systems as the considered signals sets are invariant under common static non-linearities, e.g., ideal relay, saturation, sigmoid function, quantizer, etc. In particular, our developments aim to form a base for signal-based fixed-point theorems towards the prediction of self-sustained oscillations. Our examples on relay feedback systems indicate how periodic monotonicity preservation give rise to useful insights towards this goal.