Continuity properties of the real/complex structured singular value

The structured singular-value function (μ) is defined with respect to a given uncertainty set. This function is continuous if the uncertainties are allowed to be complex. However, if some uncertainties are required to be real, then it can be discontinuous. It is shown that μ is always upper semi-continuous, and conditions are derived under which it is also lower semi-continuous. With these results, the real-parameter robustness problem is reexamined. A related (although not equivalent) problem is formulated, which is always continuous, and the relationship between the new problem and the original real-μm problem is made explicit. A numerical example and results obtained via this related problem are presented