with Linear Cuts : Upper Bound

Three methods for computing the upper bound when linear cuts are performed on the uncertainty set are described. One of them is based on the computation of spherical , one involves the construction of an implicit system to implement the linear constraints, and the other reformulates the problem into a higher dimensional one such that the standard upper bound computation can be used. The quality and computational complexity of each method is discussed. Exact bounds are achieved for rank-one problems with all the methods. Comparison of the three approaches for general random matrices will be given through numerical examples. The computation of , the structured singular value, which was introduced by Doyle in 3](also Safonov in 8]), has been of many researchers' interests for years (see 9] and the references therein). By focusing on the worst case performance of the control systems with structured uncertainty, it gives guaranteed but sometimes conservative stability margins. The computation problem itself is in general NP hard in the number of parameters((1]), which means it cannot be computed in polynomial time in the worst case. However, polynomial time algorithms exist to compute the upper and lower bounds for , although the gap between both bounds are occasionally too big to make the bounds useful. One approach that has proven to be quite eeective in improving the bounds for is the Branch and Bound(B&B) method. As a general optimization tool, B&B divides the optimization domain into subdomains, performs the bounds computation on these smaller regions to get better global bounds. While B&B schemes have been used by a lot of researchers to compute stability margins with respect to real perturbations, Newlin and Young speciically addressed the problem of avoiding exponential growth of computational expense with problem size when they applied B&B to the mixed computation in 5], with numerical examples showing that B&B algorithms are typically eeective for medium sized problems with only moderate computation cost. In these examples, only axially aligned cuts were performed. Although more intelligent branching schemes can be explored , the numerical experience is it is not as critical as improving the original quality of the bounds. The need for performing non-axial cuts arose in the prob-abilistic robustness analysis, which aims at providing hard bounds on the probability of rare events, for which the traditional Monte-Carlo simulation would require an enormous number of samples for high conndence levels.((10],,7]) This is computationally more …