Matrix and frequency-domain inequalities for all guaranteed cost controllers of uncertain linear systems with unmatched uncertainties

A differential game approach is used in this paper to derive all guaranteed cost linear state feedback controllers for linear uncertain systems in which a norm bounded uncertainty enters both the state and the input matrices. Generalizing previous results, we do not assume the uncertainties to be ‘matched’, in other words they may enter the system dynamics through different matrices. The guaranteed cost control law for this general structure, as well as necessary and sufficient conditions for its existence, is shown to be expressed in terms of either Lur'e-Riccati inequality for some minimax problem or linear matrix inequality being dependent on two parameters. The frequency-domain inequality is derived in terms of a feedback gain immediately, under which a given state feedback corresponds to a guaranteed cost controller. A well-known result of Kalman on inverse optimality is generalized to the case of uncertain timeinvariant systems.