This paper addresses the problem of determining the stabilizing proportional-integral-derivative controller regions for a general linear system with or without time delay in unity output feedback configuration. When such a problem is solved in the 3D space by a graphical method, it brings in the visualizing difficulties. This paper proposes a new method by projecting the stability boundary in the 3D to the parameterized stability boundary band in the 2D plane of while varies in an interval. The boundary band divides the plane to the regions for which simple and effective stability test is developed and the complete stabilizing regions in the 2D plane of are determined for any given interval on . The rules are presented to find conditionally stable band portions, within which the corresponding stabilizing sub-interval for can be obtained from an analytical formula for the case of no band intersections and from the root locus on for the case of the band intersections. Several examples are designed to cover different cases and illustrate the method.
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