Iterative Methods for Solving Feedback-Type Linear Equations under Small Phase Condition

This paper investigates iterative methods for solving feedback-type linear equations under small phase condition. We first develop an iterative solution scheme for linear equations with sectorial (semi-sectorial) matrices, i.e., matrices whose numerical ranges are contained in an open (closed) half complex plane. Exploiting the geometric properties of the numerical range, we introduce the notion of numerical-radius contraction as a convergence criterion. Such a scheme, with some extra effort, leads to an iterative method for the feedback-type equations under small phase condition, whose convergence condition is characterized in terms of phase information. This complements classical iterative methods for feedback-type equations based on the small gain condition. A case study on large-scale power networks is presented to illustrate the computational efficiency of the proposed method.