Robust stability and performance analysis for skew-symmetric uncertainty using the shifted bounded real guaranteed cost bound

For unstructured time-varying or complex uncertainty, the small gain theorem provides a nonconservative test of robust stability. This stability condition, which is equivalent to quadratic stability, is captured by the bounded real bound. One example of structured uncertainty is linear system dynamics perturbed by skew-symmetric uncertainty. This uncertainty structure arises in representing nondestabilizing open-loop natural frequency perturbations in models of flexible structures. For this uncertainty structure the bounded real bound is known to be conservative. In this paper we present a modification of the bounded real bound and demonstrate its ability to reduce conservatism of skew-symmetric uncertainty. This modification, called the shifted bounded real bound was used in Tyan and Bernstein (1996) within the context of H₂ guaranteed cost controller synthesis. Our development is also carried out within the guaranteed H ₂ cost framework