Feedback Stability of Generalized Positive Real and Negative Imaginary Systems

This article derives necessary and sufficient conditions for robust stability of a feedback interconnection of linear time-invariant (LTI) systems that exhibit positive realness on finite nonzero frequencies (excluding pole locations) through a weighting function, i.e., a multiplier. This class of LTI systems importantly includes the well-studied negative imaginary systems without poles at the origin as a subset. Under the assumption that the instantaneous gain of the loop transfer function is zero, it is shown that the aforementioned feedback interconnection is stable if and only if the static (a.k.a. DC) loop gain is less than unity. This condition is identical to the one that guarantees feedback stability of negative imaginary systems, but is applicable to a much wider class of systems beyond negative imaginariness. Our proof is based entirely on the multivariable Nyquist stability criterion—it does not make use of realizations of the transfer functions. Comparisons with integral quadratic constraint-based robust stability results are also made.