Lyapunov functionals in complex μ analysis

Conditions for robust stability of linear time-invariant systems subject to structured linear time-invariant uncertainties can be derived in the complex it framework, or, equivalently, in the framework of integral quadratic constraints. These conditions can be checked numerically with linear matrix inequality (LMI)-based convex optimization using the Kalman-Yakubovich-Popov lemma. In this paper, we show how LMI tests also yield a convex parametrization of (a subset of) Lyapunov functionals that prove robust stability of such uncertain systems. We show that for uncertainties that are pure delays, the Lyapunov functionals reduce to the standard Lyapunov-Krasovksii functionals that are encountered in the stability analysis of delay systems. We demonstrate the practical utility of the Lyapunov functional parametrization by deriving bounds for a number of measures of robust performance (beyond the usual H. performance); these bounds can be efficiently computed using convex optimization over linear matrix inequalities.