Spherical µ with Application to Flight Control Analysis

Robust stability analysis problems for linear systems subject to real parametric uncertainties are treated. We assume that uncertainties are restricted to a hypersphere in a parameter space. This constraint is representedby an inequality with respect to the Euclidean norm of a parameter vector. From a statistical point of view, the use of a spherical constraint can be justified if uncertain parameters are Gaussian-distributed random variables. We define an extended version of the real structured singular value, which is referred to as spherical (real) μ, for a spherical uncertainty set. Geometrically, the reciprocal of spherical μ means the radius of a guaranteed stable spherical region. We newly develop an upper bound of spherical μ, which is similar to a well-known upper bound of standard μ for a cubical uncertainty set, that is, parametric uncertainties subject to interval constraints. As its counterpart, the spherical μ upper bound can be computed by solving a linear matrix inequality problem. We apply the standard and spherical μ tools to a flight control problem. Through the numerical study, it is shown that spherical μ is less conservative than standard μ. The main contributions are the following: 1) An upper bound is developed for spherical p. 2) It is shown that the use of spherical μ is rationalized from a statistical perspective. 3) The newly derived upper bound is applied to the robust stability analysis of a flight control system.