Closed Stability-Margined Regions for the Second-Order System with Robust Stability

This paper generalizes the dented stability triangles for the second-order recursive digital system such that the borderlines of the dented stability triangles are included in the stability-margined (SM) region. This generalization enables the system designer to formulate the second-order system design more easily. More specifically, the existing inequalities for guaranteeing a specified stability margin are the inequalities that exclude the boundaries (borderlines) of the dented SM-regions, while this paper presents two sets of new inequalities that include the borderlines of the SM-regions and thus such inequalities produce a closed SM-region. In other words, the existing SM-regions are defined by using "less than" inequalities, while the new inequalities are "less than or equal to" inequalities. The added "equal to" constraints produces a closed region that includes borderlines in the SM-region. It is found that the resulting closed SM-regions are more convenient and more suitable for designing the second-order recursive system with robust stability.