Robustness Issues Concerning Dipole and Doublet Root Loci

A dipole consists of a pole-zero pair for which the distance between the pole and the zero is relatively small in comparison to the distance between the pole-zero pair and other poles and zeros of the system. Likewise, when two zeros (or two poles) are relatively isolated from other system poles and zeros, they are called a doublet. Characterizations are given for root loci in the neighborhoods of dipoles and doublets, and robustness issues are addressed. The net phase contribution ¿net of other poles and zeros of the system plays a key role in the robustness of closed-loop poles in the neighborhoods of dipoles and doublets. Closed-loop uncertainty regions for dipoles are determined as a function of open-loop uncertainty regions and bounds on ¿net. Expressions are developed for the worst-case ratio of closed-loop pole-zero distance to open-loop pole-zero distance, and design considerations for selecting adjustable parameters of a dipole are explored.