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. 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 approach 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 transferred to admissible regions in the s-plane through preimage mappings. The approach is formulated entirely using complex transfer functions, 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. This interpretation establishes a direct link between frequency domain constraints and closed-loop pole locations, allowing a qualitative assessment of robustness and dynamic properties of control systems without introducing new stability criteria or design procedures.