Colored Stability Crossing Sets for SISO Delay Systems

This paper considers the stability of a single-input single-output (SISO) system with an input/output (I/O) delay, called SISO delay system hereafter. We first derive an algebraic criterion to solve the complete stability w.r.t. the delay τ. Next, the stability analysis is extended to the case with additional free (plant and/or controller) parameters, denoted by a vector X. By adopting a mathematical tool for polynomial algebra, the discrimination system, the complete classification of the critical imaginary roots (CIRs) in the whole X-τ parameter space may be obtained. Furthermore, with the above results, we present the stability crossing sets (SCSs) in a new form, called the colored SCSs. For each point on the SCS the asymptotic behavior information of the corresponding CIR can be coded by a certain color (red, green, or yellow). Hence, the stability in the X-τ space can be analyzed easily and intuitively. Moreover, the geometry of the SCSs can be investigated from a new perspective, which will help not only to better understand the stability of the SISO delay system but also to considerably simplify the procedure for generating the colored SCSs.