| The research in this dissertation mainly focuses on stability analysis and robust control synthesis with generalized multipliers. For stability analysis with generalized multipliers, we first link the conic sectors of the topological separation framework to the integral quadratic constraints of the IQC framework. The forms of quadratic separating functionals used in both the topological separation framework and integral quadratic constraints (IQCs) framework for stability analysis are presented.;Next, we examine the quadratic functionals used for topological separation in the IQC framework for robust stability analysis. A canonical factorization result is used to establish the existence of a stable minimum-phase factorization of these IQCs. The factors are constructed via the solution to a Riccati equation, and the IQCs are proved to be always unimodularly congruent to a J-matrix. The Riccati equation result provides a method to construct the invertible causal and anticausal factors of the multipliers that arise in the classical multiplier stability theory.;For robust control synthesis with generalized multipliers, we formulate a new unstably-weighted robust control synthesis problem. In this problem formulation, we obtain a robust controller without having to factorize the generalized multipliers, i.e., the unstable-weighting matrices. Based on the positive real control approach, we present Riccati inequalities and Riccati equations solution for the unstably-weighted control synthesis problem. |
