Stability and robustness conditions using frequency dependent half planes

This paper presents a sufficient condition that establishes closed loop stability for linear time invariant dynamical systems with transfer functions that are analytic in the open right half complex plane. The condition is suitable for analyzing a large class of highly complex, possibly inter-connected, systems. The result is based on bounding Nyquist curves by using frequency dependent half planes. It provides (usually non-trivial) robustness guarantees for the provably stable systems and generalizes to the multidimensional case using matrix field of values. Concrete examples illustrate the applications of the condition. From our condition, it is easy to derive a relaxed version of the classical result that the interconnection of a positive real and strictly positive real linear system under feedback is closed loop stable.