Construction of Lyapunov functions in robustness analysis with multipliers

Robust control system analysis using a combination of multiplier theory and linear matrix inequality-based convex optimization has received considerable attention recently. The multipliers used in establishing stability are in general noncausal, and the underlying Lyapunov function that proves robust stability is not immediately apparent. Given an affine parametrization of a set of noncausal multipliers that prove robust stability in the presence of diagonal, linear time-invariant, passive uncertainties, we describe an explicit procedure for the construction of the corresponding set of quadratic Lyapunov functions that prove stability of the closed-loop system