Exact delay range for the stabilization of linear systems with input delays

This paper is concerned with the exact delay range making input-delay systems unstabilizable. The exact range means that the systems are unstabilizable if and only if the delay is within this range. Contributions of this paper are to characterize the exact range and to present a computation method to derive this range. It is shown that the above range is related to unstable eigenvalues of the system matrix. In the discrete-time case, if none of the eigenvalues of the system matrix is a unit root, then the above range is a finite set. If there exist some eigenvalues which are unit roots, this range may be a finite set or may be composed of several arithmetic progressions. When this range contains finite elements, the number of these elements is bounded by the geometric multiplicities of eigenvalues. When this range contains arithmetic progressions, the number of such progressions is bounded by the above multiplicities. On the other hand, our results can provide an upper bound for the well-known delay margin, which is the maximal delay value achievable by a robust controller to stabilize systems.