Efficient computation of a guaranteed lower bound on the robust stability margin for a class of uncertain systems

Sufficient conditions for the robust stability of a class of uncertain systems, with several different assumptions on the structure and nature of the uncertainties, can be derived in a unified manner in the framework of integral quadratic constraints. These sufficient conditions, in turn, can be used to derive lower bounds on the robust stability margin for such systems. The lower bound is typically computed with a bisection scheme, with each iteration requiring the solution of a linear matrix inequality feasibility problem. We show how this bisection can be avoided altogether by reformulating the lower bound computation problem as a single generalized eigenvalue minimization problem, which can be solved very efficiently using standard algorithms. We illustrate this with several important, commonly encountered special cases: diagonal, nonlinear uncertainties; diagonal, memoryless, time-invariant sector-bounded ("Popov") uncertainties; structured dynamic uncertainties; and structured parametric uncertainties. We also present a numerical example that demonstrates the computational savings that can be obtained with our approach.