Stability Robustness Analysis of Multiple Time- Delayed Systems Using “Building Block” Concept

An intriguing perspective is presented in studying the stability robustness of systems with multiple independent and uncertain delays. It is based on a holographic mapping, which is implemented over the domain of the delays. This mapping considerably alleviates the problem, which is otherwise known to be notoriously complex. It creates a dramatic reduction in the dimension of the problem from infinity to manageably small number. Ultimately the process is reduced to studying the problem within a finite dimensional cube with edges of length 2pi in the new domain, what we call the building block. In essence, the mapping collapses the entire set of potential stability switching points onto a small (upperbounded) number of building hypersurfaces. We further demonstrate that these building hypersurfaces can be implicitly defined and they are completely isolated within the above mentioned cube. It is also shown that the exhaustive detection of these building hypersurfaces is necessary and sufficient in order to arrive at the complete stability robustness picture we seek. As a consequence, this concept yields a very practical and efficient procedure for the stability assessment of such systems. This novel perspective serves very well for the preparatory steps of the authors' earlier contribution in the area, cluster treatment of characteristic roots (CTCR). We elaborate on this combination, which forms the main contribution of the paper. Several example case studies are also provided