Construction of Periodic Counterexamples to the Discrete-Time Kalman Conjecture

This paper considers the Lurye system of a discrete-time, linear time-invariant plant in negative feedback with nonlinearity. Both monotone and slope-restricted nonlinearities are considered. The main result is procedure to construct destabilizing for system. If satisfies certain phase condition then nonlinearity can be constructed so that has non-trivial periodic cycle. Several examples provided demonstrate construction. represents contribution absolute stability analysis since provides less conservative upper bound than existing bounds literature.