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 $\tau$. 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 $X$ - $\tau$ 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 $X$ - $\tau$ 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. [ABSTRACT FROM AUTHOR]