Stability Criteria for Time-Delay Systems From an Insightful Perspective on the Characteristic Equation

This article provides a delay-dependent and a necessary and sufficient delay-independent stability criterion for linear autonomous continuous-time systems with a discrete delay. We take a simple perspective on the two-variable formulation of the characteristic equation, which leads to the following advantageous but not widespread delay-independent criterion: The sum of the coefficient matrix of the delay-free term and the elementwise unitarily rotated matrix of the delay term must remain Hurwitz for all rotation angles. A graphical test for the latter is shown to require no more than three lines of code. Concerning delay-independent stability, our main contribution is to extend this sufficient criterion to a necessary and sufficient one. Concerning delay-dependent stability, the focus is on the critical delay that bounds the initial delay interval of stability. We formulate a constrained minimization problem that gives the exact value of this critical delay. The taken perspective is especially insightful in terms of how the coefficient matrices may look like, and, for scalar systems, the delay-dependent stability chart becomes obvious at a first glance. The presented criteria are complementary to the well-known frequency-sweeping test, which results from another of three possible perspectives on the two-variable criterion. Besides of a unified treatment of these different perspectives, the article also discusses corollaries of the one taken, which include Mori’s famous criterion.