The Structured Distance to the Nearest System Without Property P

For a system matrix M, this paper explores the smallest (Frobenius) norm additive structured perturbation δM for which a system property P (e.g., controllability, observability, stability, etc.) fails to hold, i.e., δM is the structured perturbation with smallest Frobenius norm such that there exists a property matrix R ∈ P for which M-δM-R drops rank. The Frobenius norm is used because of its direct dependence on the magnitude of each entry in the perturbation matrix. Necessary conditions on a locally minimum norm structured rank-reducing perturbation δM and associated property matrix R are set forth and proven. An iterative algorithm is also set forth that computes a locally minimum norm structured perturbation and associated property matrix satisfying the necessary conditions. Algorithm convergence is proven using a discrete Lyapunov function

On geodesics of the rotation group SO(3)

Geodesics on SO(3) are characterized by constant angular velocity motions and as great circles on a three-sphere. The former interpretation is widely used in optometry and the latter features in the interpolation of rotations in computer graphics. The simplicity of these two disparate interpretations belies the complexity of the corresponding rotations. Using a quaternion representation for a rotation, we present a simple proof of the equivalence of the aforementioned characterizations and a straightforward method to establish features of the corresponding rotations