Time delays are present in control systems (sensing, actuation and decision-making) or they are inherent parts of dynamical systems (chatter in metal machining, heredity in biology). Delays are known to cause detrimental effects to system behavior, but they also favor system performance, assist controller's success and even recover stability. This counter-intuitive and intriguing phenomenon is a strong motivation for engineering, mathematics, biology and physics disciplines. This book addresses one of the timeliest and the most crucial problems studied since 1960s: the asymptotic stability of time delay systems. The main focus is on a novel paradigm, Cluster Treatment of Characteristic Roots methodology, which can treat the stability of linear time invariant (LTI) control systems with multiple delays. The author hopes that the methodology will not only help various disciplines to investigate similar problems, but it will also inspire new research directions in algebraic geometry, numerical techniques, stability of multinomials, mapping between finite and infinite dimensional systems, robustness and control design.

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