A Geometric Study of the Phase Margin and its Implications in Design for Multivariable Control Systems

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9301196 Bar-on In robust control a collection of perturbations with a given structure can be viewed as a differentiable manifold with a nontrivial topology. This manifold can be divided into regions of stability and instability, where the identity matrix is a point in the region of stability and represents the unperturbed system. A stability margin for such a collection of structured perturbations can be defined as the smallest "distance" on the manifold from the identity matrix to the region of instability. Bar-on and Jonckheere have defined the multivariable phase margin by restricting the class of perturbations to the group of unitary matrices. It has been shown for the two-dimensional case that, when viewed geometrically, the phase margin can be represented as the minimum arc length on the three sphere from the north pole to the region of instability. The proposed research centers on generalizing these properties to the n-dimensional case with special focus on: (1) the geometric and topological properties of the phase margin; (2) the division of the manifold of unitary perturbations into stabilizing and destabilizing regions; and (3) the design of compensators to improve the phase margin. The main goal of this research is to develop a thorough understanding of the phase margin and to relate these abstract concepts back to engineering issues in robust control theory and design. ***