Exact delay range for the stability of discrete-time systems with applications in delay robust problems

This paper considers discrete-time scalar systems with delay. First, the stability of state-delay systems is investigated. The exact delay range, which means that the systems are stable if and only if the delay is within this range, is characterized analytically. A necessary and sufficient condition for the above range admitting a margin is established and characterizations of the margin are proposed. Secondly, the stabilization of input-delay systems under a state-feedback control are studied. The delay margin, which is the maximal $ d $ such that the systems can be stabilized by a control for all the delays less than $ d $, is presented. Finally, the consensus control of multi-agent systems with an input delay is considered. The consensus delay margin, which is the maximal $ d $ such that the multi-agent systems can achieve consensus under a control protocol for all the delays less than $ d $, is proposed.