Relative stability analysis of closed-loop SISO dead-time systems: non-imaginary axis case

We present a numerical method to analyse the relative stability of closed-loop single-input–single-output (SISO) dead-time systems on a given left complex half-plane for all positive delays. The well-known boundary crossing method for the imaginary axis is extended to a given vertical line stability boundary in the complex plane for these types of systems. The method allows us to compute the characteristic roots crossing the relative stability boundary and their corresponding delays up to a maximum predefined delay. Based on this method, we analyse the relative stability of the closed-loop system for all positive delays. Both numerical methods are effective for high-order SISO dead-time systems.