Conditions for input-output stability of discrete-time Lur´e systems with time-varying delays

This paper derives a stability condition for a type of Lur´e systems with time-varying delays and a feedback nonlinearity. The case of discrete-time systems is considered, consisting of a LTI, fed back though a time-varying static nonlinearity. There is an additional delay before or after this nonlinearity, which delays the signals by a positive, time-varying number of steps. Either the time-delay needs to be bounded by a constant (if the LTI system contains a single integrator) or its rate need to be bounded (in case the LTI system is stable). It turns out that, if the LTI has a single integrator with non-decreasing impulse response, the derived stability criterion coincides exactly with the circle criterion for the corresponding constant delay system. The technical proofs rely on direct manipulation of the involved signals, and do not make use of traditional tools as the small gain theorem, Lyapunov functions or passivity results.