Nonsmooth Methods for Control Design with Integral Quadratic Constraints

We develop an optimization technique to compute local solutions to synthesis problems subject to integral quadratic constraints (IQCs). We use the fact that IQCs may be transformed into semi-infinite maximum eigenvalue constraints over the frequency axis and approach them via nonsmooth optimization methods. We develop a suitable spectral bundle method and prove its convergence in the sense that, for an arbitrary starting point, the accumulation points of the sequence of serious iterates are critical. Our new way to look at IQCs is particularly suited for systems with large state dimension, because we avoid the use of Lyapunov variables. We present two numerical tests which validate our approach in the case of parametric uncertainties.