Delay-dependent methods: When does the first delay interval need a special analysis?

This paper deals with the solution bounds of time-delay systems via delay-dependent Lyapunov-Krasovskii methods. Solution bounds are widely used for systems with input saturation caused by the actuators saturation or by the quantizers with saturation. We show that an additional bound for solutions is needed for the first time-interval, where t < τ(t). This first time-interval does not influence on the stability and the exponential decay rate analysis. The analysis of the first time-interval is important for nonlinear systems e.g. for finding the domain of attraction. In the present paper, firstly regional stabilization of a linear plant with the input time-delay and saturation is revisited, where the saturation avoidance approach is used. Then the results are applied to the stabilization of Networked Control Systems (NCS) with actuators saturation under the Round-Robin (RR) scheduling protocol.