Geometric Interpretation of Frequency Domain Robustness Constraints and Closed-Loop Pole Locations

Requirements for robustness and performance in the frequency domain in control theory are usually formulated as constraints on the modulus of complex functions describing the open-loop system, the sensitivity function, and the complementary sensitivity function. These constraints generate circular sets that can be interpreted as admissible or forbidden regions in the complex plane, particularly in the Nyquist diagram. In engineering practice, they are often treated as method-specific constructions, without clarifying the general geometric mechanism by which they arise. This study develops a geometric interpretation in which a broad class of frequency-domain robustness constraints is represented as level sets of analytic and fractional-linear functions. The resulting circular sets in the Nyquist plane are characterized in a unified manner and mapped to admissible regions in the complex s-plane through preimage transformations. The approach is formulated entirely using complex transfer functions, remaining within the classical frequency-domain framework, without state-space representations, linear matrix inequalities, or optimization methods. Classical robustness measures, including gain margin, phase margin, and constraints on sensitivity and complementary sensitivity, are shown to be special cases of the same geometric structure. The main insight of this work is that these apparently different robustness constraints arise from the same underlying geometric mechanism. This interpretation establishes a geometric link between frequency domain robustness constraints and the location of closed-loop poles, allowing a qualitative assessment of robustness and dynamic properties of control systems without introducing new stability criteria or design procedures. The resulting admissible regions provide a geometric interpretation of frequency domain robustness specifications in terms of pole locations in the s-plane.