A multiloop generalization of the circle criterion for stability margin analysis

In order to provide a theoretical tool well suited for use in characterizing the stability margins (e.g., gain and phase margins) of multiloop feedback systems, multiloop input-output stability results generalizing the circle stability criterion are considered. Generalized conic sectors with "centers" and "radii" determined by linear dynamical operators are employed to enable an engineer to specify the stability margins which he desires as a frequency-dependent convex set of modeling errors-including nonlinearities, gain variations, and phase variations-which the system must be able to tolerate in each feedback loop without instability. The resulting stability criterion gives sufficient conditions for closed-loop stability in the presence of such frequency-dependent modeling errors, even when the modeling errors occur simultaneously in all loops; so, for example, stability is assured as loop gains and phases vary throughout a "set of nonzero measure" whose boundaries are frequency-dependent. The stability conditions yield an easily interpreted scalar measure of the amount by which a muitiloop system exceeds, or falls short of, its stability margin specifications.